My own (personal) beef with the real numbers
Things are getting real here on the philosophy of mathematics forum. It's time to throw my own ball into the game (... or is two balls).
A few threads deal directly or indirectly with the construction of the reals.
When I was younger, and studying math, I spent a lot of time on foundational topics and had a keen interest in things like the construction of the reals.
It was only when I had to work on real problems, invariably using a computer that only deals with a small set integers, that I started to have my doubts about the real numbers.
I largely agree with fishry's assessment here:
Quoting fishfry
My beef is that the real numbers are introduced too early in education. Infinite processes and essentially 100% of numbers being infinitely complex are, though perhaps can be dealt with abstract rules, too difficult to conceptually grasp for most high school students. Moreover, most students studying high-school level calculus will not go on to study pure maths. Any applied math does not need real numbers to solve any real world problem. A finite sequence that gets one as close as required within the experimental error is sufficient for any real world engineering and, of course, is represented by a rational number.
Introducing the real numbers in high school, at best, imposes an unnecessary conceptual burden on students making learning harder (often contributing to "I don't get it" and choosing not to study math further), but worse, it breaks the chain of reasoning. High school mathematics simply posits the real numbers as the "kind of number calculus operates in" and does not go into how to construct them. The natural numbers and rational numbers are clear how to make them, and the chain of reasoning is smooth introducing natural numbers and then rational numbers and proofs built on these kinds of numbers.
Simply positing the real numbers creates a discontinuity in reasoning that is the anti-thesis of what mathematics is about. Filling in this reasoning gap is simply a waste of time for high-school level calculus, and, more importantly, the average teacher introducing calculus would not have the knowledge and skills to do so.
Why the conceptual burden is high, is that even when just dealing with "normal" real number behaviour such as mapping all the points between 0 and 1 to all the points between 0 and 10 000, there is simply no way to imagine what is happening (unlike most other high-school math that one can, after some effort, "see it" and "get it"). There is of course no real world problem where this feature of the real number system is needed.
However, it's also necessary to deal with questions like "why can't we just have rational numbers with infinite numerator and denominator; seems unfair to allow infinite decimal expansion but not infinite numerators?". Of course, we can have rational numbers with infinite numerator and denominator but it's then needed to explain how these aren't the "real" rational numbers we're talking about when we say the square root of two is irrational. Even if this explanation is successful (which I doubt the average high school teacher would be able to answer adequately) there's the followup problem of avoiding the claim that the rational numbers aren't finite, as there's clearly an infinite amount of them as they are countable by the infinite sequence of natural numbers. But ... if they aren't finite how do we avoid infinite numerators and denominators? If we're claiming the natural numbers aren't complete enough in some sense to simply go onto simply include every decimal expansion injected into every possible fraction (just take the decimals expansion of pi and place them over something else, like the decimal expansion of e to create a new rational number that comulates perfectly), then what do we make of the real number with a decimal expansion of the sequence of natural numbers, i.e. 0.12345678910111213...; is this not a correspondence between the a real number and natural numbers which we can then go onto make rational numbers with this natural number as numerator or denominator; if it is a real number with a decimal expansion "as big" as the natural numbers, why can't we make all the other real numbers through a process of permuting numbers in the natural number sequence along the way, and every time we permute we count 1? If we're "not allowed" to simply claim all the natural numbers is a new infinite natural number, why are we "allowed" to make infinite decimal expansion; they both seem very much the same process of assembling individual numbers on a line.
I'm aware these questions have answers. My point that I'm making is that these questions are perfectly natural for a student encountering real numbers for the first time to ask. In the rest of mathematics questions about proofs have answers; there's an answer to why theorems work (otherwise they're not theorems) and answering every critique possible of a theorem is what (in my view) mathematics is.
However, I do not believe it's possible for the average high school teacher, much less the average high school student, to be able to answer the above questions about the real numbers. The main problem, in my view, is that students simply can't imagine how big infinity is and do not have the prerequisite knowledge to represent what (maybe) is infinity with symbols.
So, my challenge is if someone can construct the real numbers in a concise and clear way that the average student starting calculus in high school would easily understand for then transcendental constants like pi to make perfect and clear sense and all the tricky questions above perfectly clear answers (just as clear as in geometry or proofs about discrete numbers).
If not, then my recommendation is to drop the real numbers and do calculus in the numerical regime of "as small as you want ... but not infinitesimal". I realize that pi is no longer pi in this regime, but some ornament could be added to all our precious transcendental constants to indicate that it is representing an approximate value (of "as approximate as needed", to be defined in the algorithm that will provide a numerical result); then, for students who go into pure maths, the process of how to remove those ornaments can be learned and students can then transcend to things like the purest e. Every other student that continues with math working essentially in the computer, those ornaments then serve as a constant reminder pi really isn't in the computer in any sense and thinking so is a mistake that will mess numerical recipes from time to time, and an additional step will be required of talking with the pure mathematics department if ever "real pi" is needed to solve a real world problem (they may lose count waiting, but that's the point).
This approach not only relieves the burden of the real numbers, but also teaches essential habits of numerical computation. It's also a critical educational error to solve equations with real numbers and then just "type into a calculator" to get a numerical answer; this isn't how numerical computation is done properly and makes learning both the pure math parts that are being attempted (however inadequately) as well as the numerical procedure to "get an answer" both simply wrong understanding of the maths involved.
TD;LR: we should teach ZF in high school and then add C later for pure maths students.
A few threads deal directly or indirectly with the construction of the reals.
When I was younger, and studying math, I spent a lot of time on foundational topics and had a keen interest in things like the construction of the reals.
It was only when I had to work on real problems, invariably using a computer that only deals with a small set integers, that I started to have my doubts about the real numbers.
I largely agree with fishry's assessment here:
Quoting fishfry
We can formalize the process of filling in the holes with various technical constructions of the reals. There are several, the two best known being Dedekind cuts and Cauchy sequences. The details aren't of interest. The point is that it can be done within set theory and it allows us to found calculus in a logically rigorous way, something that escaped Newton and Leibniz. We can also axiomatically define the reals as "the unique Cauchy-complete totally ordered infinite field." When you unpack the technical terms, you end up with an axiomatic system that's satisfied within set theory by the Dedekind cuts or Cauchy sequences. It's all very neat. One need not believe in it or care. It must be frustrating to you to both not believe in it, yet care so much!
My beef is that the real numbers are introduced too early in education. Infinite processes and essentially 100% of numbers being infinitely complex are, though perhaps can be dealt with abstract rules, too difficult to conceptually grasp for most high school students. Moreover, most students studying high-school level calculus will not go on to study pure maths. Any applied math does not need real numbers to solve any real world problem. A finite sequence that gets one as close as required within the experimental error is sufficient for any real world engineering and, of course, is represented by a rational number.
Introducing the real numbers in high school, at best, imposes an unnecessary conceptual burden on students making learning harder (often contributing to "I don't get it" and choosing not to study math further), but worse, it breaks the chain of reasoning. High school mathematics simply posits the real numbers as the "kind of number calculus operates in" and does not go into how to construct them. The natural numbers and rational numbers are clear how to make them, and the chain of reasoning is smooth introducing natural numbers and then rational numbers and proofs built on these kinds of numbers.
Simply positing the real numbers creates a discontinuity in reasoning that is the anti-thesis of what mathematics is about. Filling in this reasoning gap is simply a waste of time for high-school level calculus, and, more importantly, the average teacher introducing calculus would not have the knowledge and skills to do so.
Why the conceptual burden is high, is that even when just dealing with "normal" real number behaviour such as mapping all the points between 0 and 1 to all the points between 0 and 10 000, there is simply no way to imagine what is happening (unlike most other high-school math that one can, after some effort, "see it" and "get it"). There is of course no real world problem where this feature of the real number system is needed.
However, it's also necessary to deal with questions like "why can't we just have rational numbers with infinite numerator and denominator; seems unfair to allow infinite decimal expansion but not infinite numerators?". Of course, we can have rational numbers with infinite numerator and denominator but it's then needed to explain how these aren't the "real" rational numbers we're talking about when we say the square root of two is irrational. Even if this explanation is successful (which I doubt the average high school teacher would be able to answer adequately) there's the followup problem of avoiding the claim that the rational numbers aren't finite, as there's clearly an infinite amount of them as they are countable by the infinite sequence of natural numbers. But ... if they aren't finite how do we avoid infinite numerators and denominators? If we're claiming the natural numbers aren't complete enough in some sense to simply go onto simply include every decimal expansion injected into every possible fraction (just take the decimals expansion of pi and place them over something else, like the decimal expansion of e to create a new rational number that comulates perfectly), then what do we make of the real number with a decimal expansion of the sequence of natural numbers, i.e. 0.12345678910111213...; is this not a correspondence between the a real number and natural numbers which we can then go onto make rational numbers with this natural number as numerator or denominator; if it is a real number with a decimal expansion "as big" as the natural numbers, why can't we make all the other real numbers through a process of permuting numbers in the natural number sequence along the way, and every time we permute we count 1? If we're "not allowed" to simply claim all the natural numbers is a new infinite natural number, why are we "allowed" to make infinite decimal expansion; they both seem very much the same process of assembling individual numbers on a line.
I'm aware these questions have answers. My point that I'm making is that these questions are perfectly natural for a student encountering real numbers for the first time to ask. In the rest of mathematics questions about proofs have answers; there's an answer to why theorems work (otherwise they're not theorems) and answering every critique possible of a theorem is what (in my view) mathematics is.
However, I do not believe it's possible for the average high school teacher, much less the average high school student, to be able to answer the above questions about the real numbers. The main problem, in my view, is that students simply can't imagine how big infinity is and do not have the prerequisite knowledge to represent what (maybe) is infinity with symbols.
So, my challenge is if someone can construct the real numbers in a concise and clear way that the average student starting calculus in high school would easily understand for then transcendental constants like pi to make perfect and clear sense and all the tricky questions above perfectly clear answers (just as clear as in geometry or proofs about discrete numbers).
If not, then my recommendation is to drop the real numbers and do calculus in the numerical regime of "as small as you want ... but not infinitesimal". I realize that pi is no longer pi in this regime, but some ornament could be added to all our precious transcendental constants to indicate that it is representing an approximate value (of "as approximate as needed", to be defined in the algorithm that will provide a numerical result); then, for students who go into pure maths, the process of how to remove those ornaments can be learned and students can then transcend to things like the purest e. Every other student that continues with math working essentially in the computer, those ornaments then serve as a constant reminder pi really isn't in the computer in any sense and thinking so is a mistake that will mess numerical recipes from time to time, and an additional step will be required of talking with the pure mathematics department if ever "real pi" is needed to solve a real world problem (they may lose count waiting, but that's the point).
This approach not only relieves the burden of the real numbers, but also teaches essential habits of numerical computation. It's also a critical educational error to solve equations with real numbers and then just "type into a calculator" to get a numerical answer; this isn't how numerical computation is done properly and makes learning both the pure math parts that are being attempted (however inadequately) as well as the numerical procedure to "get an answer" both simply wrong understanding of the maths involved.
TD;LR: we should teach ZF in high school and then add C later for pure maths students.
Comments (533)
What??
First, the technical construction(s) of the reals are taught only to math majors in a class called Real Analysis. Nobody who's not either a math major or someone taking that class as an elective ever sees the technicalities. Perhaps they teach high school kids advanced real analysis in high school in Russia or China but truly I doubt that very much.
Second, if your complaint is with pedagogy it's not about math. And certainly there are many problems with the way math is taught. But that is not a beef with the real numbers. In fact I didn't see you present any beef with the real numbers. I only saw you beef with the teaching of the real numbers; a subject on which I'm in complete agreement with you in the large, if not necessarily every detail. When I'm emperor of the world the first thing I'm going to do is send all the math curriculum boards off to Gitmo. I've long held that idea.
And third, what does the axiom of choice have to do with anything? It's certainly not needed to define or construct the reals. Teach ZF to high school students? What, teach ordinals and cardinals to high school students? It would be fun to teach ZF to SOME high school students, the especially mathematically talented ones. The mainstream, no. I wonder what you are talking about here. Again, the axiom of choice is not needed to defined or construct the reals.
Quoting boethius
That IS the essence of the real numbers. There are no infinitesimals in the standard real numbers. I think perhaps you had an unhappy calculus class, as most students do. They don't teach the technical definition of the real numbers in calculus. Perhaps what you're unhappy about is that you didn't have a more rigorous class in calculus. But that's not done because calculus is a service class for a lot of physics, engineering, economics, and other majors. The math majors have to make the best of it till real analysis class.
In fact this is the very reason there's so much confusion about the real numbers. Nobody is ever taught what the real numbers are simply because it's not relevant to anyone's profession unless they're going to be a math major. So even technical professionals like physicists and engineers don't know what a real number is. And then when you get to online discussion forums, you get a lot of confusion on the subject.
I appreciate that you agreed with what I wrote, but what you wrote didn't have much if anything to do with it. I think what you are calling for is better teaching of the reals to high school students, which I'd also like to see. But to go into the full technical details is way beyond high school students.
[Disclaimer: Poster quotes me in AGREEMENT and I give him a hard time. I'm a terrible person. Forgive me].
It's worth noting that the pedagogy retraces the history.
Newton developed calculus to study the motions of the heavens. He was not able to drill down a logical explanation of his methods even to the standards of rigor of the day. He had no idea what his derivative (what he called a fluxion) was. If we contemplate the expression (in Leibniz's notation, which was better than Newton's) [math]\frac{dy}{dx}[/math], what exactly is it? If the numerator and denominator are not zero, that's NOT the derivative. If they are, then the expression is not mathematically defined.
The philosopher George Berkeley sarcastically called Newton's fluxions, "The ghost of departed quantities." What a great line. If these guys came back today they'd all be on the Internet flaming away at each other.
Newton struggled with the logical nature of infinitesimals and limits. In fact his publications clearly show that he didn't ignore the issue at all, but was rather keenly aware of the problem and tried hard over his lifetime, without success, to come up with a good explanation.
Dating from 1687, the publication of Newton's Principia, to the 1880's, after Cantor's set theory and the 19th century work of Cauchy and Weirstrass and the other great pioneers of real analysis; it took two centuries for the smartest people in the world to finally come up with the logically rigorous concept of the limit. For the first time we could write down some axioms and definitions and have a perfectly valid logical theory of calculus.
This was a very great achievement of humanity, one not very well appreciated. We don't teach the history of math. In addition to "Pull down the exponent and subtract 1" that we beat students over the head with, it would be great if we could impart the sense of mathematicians working on this problem of defining limits for 200 years before they worked it out.
It's no surprise that these are extremely subtle concepts to understand. So in high school and college we just tell people that real numbers are mysterious "infinite decimals," whatever that means, and no harm is done. And in calculus class we show people what limits are but we can't really be precise, so mostly we focus on techniques, which the physicists and engineers and economists will need for their work.
The development of the real numbers and the limit concept in the 19th century is one of our greatest intellectual achievements. I wish there were more appreciation of it.
Yes, did you even read my post, this is my complaint.
I have no issue with real numbers "existing" in whatever sense mathematicians using the real number system want to believe. I am not convinced that "the true infinity" or "the true continuum" is captured by these symbolic systems, but I agree with you when you say mathematicians need not care and usually don't care; you can use a different system if it suits your style or problem.
I even cite your own words on this subject and express my agreement.
Your not giving me a hard time, you just have poor reading skills of prose; but I don't mind that, you don't make any claims to be able to understand non-formal arguments and perhaps have formal reasons to believe this task is impossible.
The reason I presented my arguments in prose is because that's the sort of thinking a high school student will be equipped with starting to use the real numbers.
My challenge is that: is there any answers to these prose questions that doesn't involve an entire university course, which maybe not even enough. As someone who's taken these university courses and who works with math in my day job building numerical models, you seem to claim I don't understand these issues. Even if it was true, which I doubt, isn't this more evidence to my point?
Quoting fishfry
Again, terrible reading comprehension; mathematicians not learning any humanities really is a problem.
I do not claim the axiom of choice is needed to construct the reals.
My argument is above the tdlr which doesn't mention the axiom of choice. My tdlr is an over simplification of my argument in a recommendation that I believe most people who understand this subject and have good reading comprehension would get.
Which you seem to agree with, that ZF can be taught at a high school level, which is my recommendation. I think you would agree that most high school students would not be prepared to deal with C (which for me, is what then makes the real number system mathematically interesting; unless there's been some breakthrough since I last looked at this topic that C is no longer required).
Quoting fishfry
This is basically our difference.
I disagree that the pedagogy retraces the history. If it actually did, maybe I'd have less of an issue.
Newton did not have the real numbers to do calculus as you note, yet high school calculus students simply start with the real numbers.
Quoting fishfry
You realize you're just adding more weight to my contention in the OP here?
If you need to read Principia mathematica and two centuries of the smartest people to understand the real number system ... maybe this is too much of an ask to high school students?
Do you agree?
If not, my challenge is that you explain the answers to my questions in a way that a high school teacher and then students would understand. If you can't, just agree with my OP rather than try to prove your smarter than me, which I so far not seeing any evidence for: going off on random tangents, not addressing the point of the OP, cowardly hedging your own complaints etc.
For instance, I did not define "infinitesimal", it's just a word that I find perfectly suitable to use to refer to series converging to a point (i.e. the distance becomes infinitely small). My use of infinitesimal was to contrast using prose (using words most people here would understand) the definitions one would find in numerical calculus compared to what we usually just call calculus; not to conjure up 17th century philosophical debates.
To lift from wikipedia because I do basic "google the subject matter" research when engaging in internet debates.
From the wikipedia page on infinitismals:
Followed immediately by a section called "Infinitesimals in teaching":
So, not only is infinitesimal perfectly fine mathematical jargon to talk about things "infinitely small" in both a technical and a general sense (as wikipedia starts the article by saying: "In mathematics, infinitesimals are things so small that there is no way to measure them"), but it is a common notion (according to wikipedia) used to introduce students to calculus, as it's intuitive.
This, my contention is, is a pedagogical mistake unless there are answers to all the very normal questions students can have about the real number system (that are as easy to grasp as other associated concepts being introduced). I have yet to see them.
Why is there so much debate around these infinite related questions such as cardinals and continuums here on philosohy forum? And not about questions like solving the quadratic equation or any number of other theorems? Because, in my view, it takes very specialized knowledge to understand modern mathematics modelling of these questions, which as you say, need not bother anyone that specialists are building such systems, but it is bad mathematical pedagogy to introduce to students concepts that they are unable to fully grasp and have zero need for any of the tasks at hand; it serves only to mystify mathematics rather than build understanding.
An analogy would be introducing Euclidean geometry in the context of Reiman manifolds or rotation in quaternians because that's what the cool kids in university do, with neither having any basis to have any clue what a Reiman manifold or quaternion really is nor ever needing the extra things Reiman manifolds or quaternions provide to address the Euclidean problems being asked to solve; now, I understand why concepts got inverted historically (since we were computationally extremely limited until recently), in the development and subsequent teaching of calculus as opposed geometry (pending an answer to my questions), my point is it's now a completely fixable conceptual problem in our teaching methods: that finite computation is a much more basic concept than the real numbers, real analysis, metric spaces and so on (i.e. real numbers are not required for any high school level problem and there's no need to introduce them until they are actually needed).
Now, I'm not saying these issues should be kept secret or something, there could be extra material for students who want to get into it; but I see no high school level problem that is not perfectly addressed in the numerical regime which is far easier to understand; you can really "see" and "get" how a computer functions in principle and why algorithmic approximations that truncate at a suitable number of steps yields answers to real world problems that students can visualize even at a high school level; there is nothing remotely as difficult conceptually as an infinite decimal expansion. It's also critical to understand not just the algorithm that converges on the desired constant but under what conditions are correct to end such an algorithm for any given applied mathematical problem, which is what the vast majority of high school math students will be going into: engineering, computer science, programming, chemistry and even accounting requires intuition of the strengths and limitations of machine computation (i.e. what kinds of problems require special attention to the the finite nature of the computer, in terms of memory, floating point representation, iteration steps, economizing computer resources and so on; and what kinds of problems one can just paste code from stack overflow and let it ride).
So if you want to get back on track, answer my questions concerning the real numbers in a way that a high school teacher and student understands. I've claimed to understand the answers to these questions, but you seem to be arguing it's all too complicated for me and that you will explain to me why I don't in fact understand the issue and you're going to demonstrate that. Well, if this is true, I'll be the first to benefit from your addressing the point in the OP. I eagerly await.
The problem is that no math course has enough time to really take the time. Usually it's just "here's the proof, there, I showed it to you, now use this algorithm".
About pedagogy, when I started first grade in my country in the late 70's they had the wonderful idea to start math education with set theory. I found it a bit confusing then (I had a problem to learn the various Venn diagrams and their relations to addition and subtraction etc.) and I remember that the teachers weren't happy about the reform either. Well, that was the 70's and now they teach things to my children in the first grade basically the same way they taught things to my parents (and even grandparents).
Real numbers are one of those things that at first glance seem to be easy, but aren't at all. Just as, well, set theory.
Of course children first learn maths with the conflated maths; counting sheep etc. But perhaps around the time they enter secondary school ( around age 13) the distinction needs to be emphasised.
In pure maths one is dealing entirely with the manipulation of symbols following particular rules. Here the symbols used whether for integers, reals or imaginary numbers have no more connection to the 'real world' than any other; they are all abstract.
Then for the application to the 'real world' (applied maths) one takes a particular part of mathematics and applies a mapping between the abstract symbols and concepts that apply to the 'real world'.
With this clear distinction the complications of maths fade away.
Yes, this is basically what I am advocating, though with much heavier emphasis on the applied part in secondary school (and applied maths will still have plenty of symbol manipulation in it's own right and plenty of theorems that apply to both pure and applied maths).
Though more specifically I am singling out real numbers as the particular problem; though maybe there are others. Likewise, perhaps there is a pedagogical approach that accomplishes both, as you seem to be suggesting, but my feeling is that you can't really do pure maths without set theory, which as points out was a failed experiment to teach children.
However, if there's some simple explanation of the real numbers and all the questions that naturally arise from infinite decimal expansion, then I'd be proven wrong on this particular point.
Did this set theory experiment simply not work at all, or did it produce some small cadre of math geniuses?
It strikes me as both condesending and and an injustice to the excitment that math can elicit by treating kids as idiots just because they can't engage in construction. It seems a way to suck any exploratory spirit out of math, and kill any joy that might be gleaned from it.
Why set theory? Set theory is pretty uninteresting really, apart from Venn diagrams which are fun and useful. I presume that you are referring to the idea that set theory provides the 'foundation' to mathematics. But pure mathematics is abstract and doesn't need any foundations apart from its axioms which introduce the symbols and define the rules. (And admittedly these axioms are more implicit than explicit).
And as for the real numbers, they become necessary when one looks to divide (for example) 10 by 4. (10/4). although the task is in the domain of integers the answer is outside. It could be written as a fraction 2 1/2 and that is what ancient mathematicians did. They considered that all numbers could be written as integers or fractions or a composite of the two. It was quite a shock to them when they came to realise that the square root of two could not be expressed as a fraction! There was no alternative except to introduce real numbers.
More or less. There are alternative foundations, but set theory is the main one.
Quoting A Seagull
This is also true for applied mathematics.
Applied mathematics also has definitions and axioms, just focusing on those that generally have real world scientific application (applied maths is a subset of pure maths).
There are not infinite sets of anything in the real world (real world of scientific investigation at least); so how to deal with infinite sets can be excluded from applied mathematics. For me this is the main difference; there is no need to learn calculus in the real number system and then calculus in a finitist numerical system appropriate for a computer. It's, in my view, only historical accident that learning calculus with real numbers first seems to make sense. One can learn first a finitist numerical system for doing calculus.
Quoting A Seagull
This is the basic thing my pedagogical program would get rid of.
You never need an "exact" (i.e. infinite decimal expanded) value of 10/4 in an applied problem (first because it's 2.5, but I assume you intended an example like 1/3 or then an obtuse reference to infinite trailing zeros).
The numerical regime basically refers to replacing all calculations that can go on forever in a finitist setup tailored for the computing machine doing the calculations (values can be arbitrarily large or small, not infinite, and not more than can fit in the computer ... with a whole bunch of caveats) with algorithms that can be carried out to the required precision (the series sum or whatever the algorithm is, and a halting condition); in other words, those significant digits from physics class, can form the axiomatic basis of a completely adequate calculus for applied problems.
Learning the axiomatic setup of numerical computation rigorously is (until someone shows me how easy real numbers can be) a far better use of students time leading to, I believe, a better understanding of maths for both future applied maths and pure maths students. Understanding finitist maths well, I would argue is the correct basis to then going beyond finitism for students interested to do so; likewise, I would argue a more rigorous use of finitist maths wherever it is adequate in physics and other sciences is far better than a lazy use of more powerful models.
In other words, real numbers are not necessary when dealing with 1/3, or any calculation that can in principle go on for an arbitrary length; approximate solutions are fine for any real world problem.
The reason I'm posting here in logic and math and not politics, is not simply because of the theme but because my contention has a counter example of a very clear and simple presentation of the real numbers that high school teachers and students would find of appropriate effort to fully grasp. If there is such explanations that are graspable by the average student, then I'd capitulate.
We had just "numbers" (natural numbers), constructed by counting (the successor function). We could do addition and multiplication on them just fine and didn't need to worry about any other kinds of numbers.
Those then became "positive numbers" when contrasted with "negative numbers", which were introduced to fill out the set of numbers that could be constructed through subtraction, which were again just "all the numbers".
Those then became "whole numbers" (integers) when contrasted with "fractions", which were introduced to fill out the set of all numbers that could be constructed through division, which were again just "all the numbers".
Those then became "rational numbers" when contrasted with "irrational numbers", which could be made in a bunch of different ways; there wasn't just one kind of operation that resulted in irrational numbers sometimes. Between rational and irrational numbers, that was again just "all the numbers", and we never had to worry about having one method of constructing any given one of them, just that there was the stuff that could be constructed through division and the stuff that couldn't.
Those only became "real numbers" when contrasted with "imaginary numbers", which were introduced to fill out the set of numbers that could be constructed through taking roots, which were, by this point, finally not treated as "all the numbers" but as a set of their own, the "complex numbers", suggesting that the reals are still considered the normal set of all numbers, and the complexes are considered some kind of weird superset made of pairs of numbers, not just numbers simpliciter.
Perhaps you could state them succinctly. I prefer not to wade into this. You have an ax to grind and I've only succeeded in upsetting you. If you'll list some clear questions I'll do my best to respond. My sense is that you didn't actually read your own post. If you did, you'd realize that you have no beef with the real numbers, only with their teaching in high school. I share many of your concerns in that regard.
You did state that ZF should be taught in high school and ZFC in college. That does not make sense to me at all. ZF is a fairly sophisticated system. I wouldn't teach a full course in ZF to high school students except to the most mathematically motivated and talented students. ZFC actually adds very little in terms of complexity or teachability. If anything, the axiom of choice regularizes infinite sets and eliminates a lot of problems. For example absent choice, there's an infinite set that's not Dedekind-infinite. Surely you don't mean to claim we should be teaching this to high school students.
The whole point of my post is that high school students would have no way of stating their questions succinctly as you demand, but they are in my view meaningful questions to ask.
You could argue that they aren't meaningful questions and can just be dismissed not warranting an answer, or you could "not wade into it" as you suggest to yourself post-wading.
Quoting fishfry
I have no axe to grind. But you very much seem to have an axe to grind with projected axe grinders.
However, the continuous debate around this topic here on philosophy forum inspired me to post my own personal beef, which is that simply positing the real numbers without constructing them nor dealing with all the non-intuitive questions that can arise with completed infinities such as infinite decimal expansion, is poor pedagogy.
But, I'll play your game; perhaps it will satisfy the OP as all my questions will have simple and clear answers that an average high school student will have no problems understanding with a little effort.
Let's start with infinite numerators, denominators and exponents.
Instead of accepting the conclusion that root 2 is irrational (not a rational number), I'm going to solve root 2 using infinite denominators and numerators.
Where do I get these infinite natural numbers to make my rational solution to root 2? I simply take suitable real numbers and take out the decimal symbol and insert those infinite digit expansions into polynomials to represent values that solve my problem exactly, which I admit, I was unable to accomplish with any solution using finite natural numbers I could name.
Fairly simple procedure.
Please demonstrate how this infinite numerator and denominator either does not get counted in Cantor's diagonal proof, does not represent an irrational value, or there is something preventing finding and placing all the digits of suitable real numbers into a numerator and denominator to solve for root 2.
An infinite normal digit expansion (which I'll choose to use as suitable) is neither odd nor even, as is well known, and so there's no contradiction of division by 2 as is usually concluded in the run-of-the-mill finitist approach to proving the irrationality of root 2; I can just keep that 2 coefficient around no problem and divide by 2 to get rid of it. Therefore, root 2 is rational.
If I am given these infinite expansion of digits, seems I should be able manipulate and place them where I want if I have some procedure to do so (unless given suitable axiomatic conditions preventing me to doing what I want).
What axioms does a high school student possess to avoid the above issue of concluding root 2 is rational? If none really (if only because the reals aren't even constructed to begin with, just posited as a given) then I think we agree about the OP.
Followup question (as I believe this is what interests you to demonstrate) what axioms does a university student possess to avoid the above issue and how?
We're in deep and complete agreement on this. The mathematical definition of the real numbers is far beyond high school students; in analogy with the difficulties Newton and Leibniz had, which needed to wait 200 years for resolution. Quoting boethius
Quoting boethius
This fact has a proof, 2300 years old dating to Euclid's written account; but most likely at least several hundred years older than that.
The subject of the square root of 2 is part of my response to @Metaphysician Undercover in the bijection thread. I'm drafting a response that will expound at length on the mathematics and the mathematical philosophy of the square root of 2. I hope to corral my thoughts into a postable screed within a week or so. Meanwhile please forgive my lack of comment today on the mathematical existence of a positive real number whose square is 2. I believe in such a real number and I believe it's not the ratio of integers and I believe that it's computable, hence encodes only a finite amount of information despite its endless and patternless decimal representation. A real number is not its decimal representation. I hope you believe these things too.
ps -- Note well The irrationality of the square root of 2 does NOT introduce infinity into mathematics. All the irrationals familiar to us are computable, and have finite representations. The noncomputable reals do introduce infinity into math; but plenty of people who don't believe in noncomputable reals nevertheless DO believe in the square root of 2. Namely, the constructive mathematicians.
Quoting boethius
Euclid's proof of the irrationality of [math]\sqrt 2[/math] has nothing at all to do with Cantor's discovery of the uncountability of the reals. The rest of this paragraph, I confess, is not intelligible to me.
Quoting boethius
None whatever. In high school we mumble something about "infinite decimals" while frantically waving our hands; and the brighter students manage not to be permanently scarred for life.
The teaching of mathematics in the US public schools is execrable. How many times do I have to agree with you about this? I'd gladly join you in a protest down at the local school board, but it would do no good. The educrats have bought off and bamboozled the politicians. The teaching of math in the US is stupid by order of the government. I wish I could do something about it.
Quoting boethius
A university student in anything other than math: None.
A well-schooled undergrad math major? Someone who took courses in real and complex analysis, number theory, abstract algebra, set theory, and topology? They could construct the real numbers starting from the axioms of ZF. They could then define continuity and limits and I could rigorously found calculus. It's not taught in any one course, it's just something you pick up after awhile. The axiom of infinity gives you the natural numbers as a model of the Peano axioms. From those you can build up the integers; then the rationals; and then the reals. Every math major sees this process once in their life but not twice. Nobody actually uses the formal definitions. It's just good to know that we could write them down if we had to.
So, as I've noted already, people who study math in university get all their conceptual questions about the real numbers and the nature of calculus answered. The physicists, engineers, economists, pre-meds, and everyone else, do not. The formal constructions aren't important as long as you know the rules for manipulating real numbers. Even the mathematicians just use the real numbers according to the rules of addition and subtraction and so forth. The formal constructions are to demonstrate that the real numbers have logically valid mathematical existence. [As always please don't jump in just to point out that this is not necessarily actual existence. I quite agree].
Buildings and bridges are made of quarks. But architects and bridge builders don't need to know particle physics. Likewise nobody needs to care about the formal definition of the real numbers; except that if they ask, they can honestly be told that there is one. And of course it's all on Wikipedia.
https://en.wikipedia.org/wiki/Dedekind_cut
Construct in this context means build a thingie within set theory that behaves exactly like we want our thingie to do.
For this purpose, the construction of Dedekind cuts (linked earlier) is a construction of the real numbers. But how do I know that? It's because we first write down properties we want the real numbers to have; then everyone can use them, even though we don't know whether they exist mathematically. The construction shows that they do.
Let me expand this in full gory detail.
Here are the axioms for a totally ordered, Cauchy-complete, infinite field.
https://sites.math.washington.edu/~hart/m524/realprop.pdf
This PDF lists the properties of the real numbers. It's much better than the Wiki link on the same topic.
Briefly you can do arithmetic: a + b = b + a, and multiplication distributes over addition and ab = ba, and if b is nonzero then the quotient a/b exists. Then you have the order relations: for any two distinct reals either a < b or b < a.
Now the rationals satisfy these properties; so we need one more special property that fully characterizes the real numbers: The Least Upper Bound property, which says that every nonempty set of reals bounded above, has a least upper bound. This is also known as the completeness axiom. It says there are no "holes" in the real numbers.
Example. The rationals don't satisfy the LUB property. The nonempty set [math]\{x \in \mathbb R : x^2 < 2 \}[/math] does not have a least upper bound! This came up earlier. This is why the rationals aren't a mathematical continuum.
The real numbers -- whatever they are -- SHOULD have this property.
Now as far as we know, there is no such abstract object that satisfies these properties. Maybe the real numbers don't exist. But it doesn't matter. We can just use their properties. We can do physics, engineering, calculus, etc ... even if the real numbers don't exist. Just by using their properties!
But now a someone comes along and calls the entire enterprise null and void because for all we know, we're talking about something that doesn't exist. But if you believe in the rationals, you must believe in Dedekind cuts; and the set of Dedekind cuts satisfies all the real number properties. So we are justified in calling the set of Dedekind cuts the real numbers.
Having seen this construction once in our lives, we are confident that the real numbers are mathematically legitimate, because we can build an object using set theory that behaves exactly like the real numbers. We now forget all about it; till someone asks, "Oh yeah? How do we know there is any such thing as the real numbers?" Then we show them.
tl;dr: The real numbers are anything that satisfies the list of real number properties. But is there even any such thing at all? Yes. Within ZF set theory we can build up a set that has exactly these properties. We can in fact do this in several different ways. They're all isomorphic. We call any of these models the real numbers.
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I laughed out loud at this. It reminds me of my favorite line from the movie, The Incredibles 2, when Bob is trying to help Dash with his homework: "Why would they change math? Math is math!"
Yes, I think so too.
Furthermore, the mapping back to the real world must go through the regulatory framework of an empirical knowledge discipline, such as science. Direct application of mathematics to the real world should be discouraged, because mathematics does not seek to create such regulatory framework for empiricism, while such framework is clearly needed.
Therefore, real-world considerations are the domain of downstream users of mathematics, such as science, engineering, and so on. Mathematics itself should stay clear of those, in order to preserve its purity.
In my opinion, the most successful offshoot of set theory is relational algebra, for which the canonical language is SQL:
Quoting Wikipedia on relational algebra
Relational algebra is massively big. Very little modern software can do without.
Furthermore, separate from relational algebra (which is a niche application), there is a strong trend to moving to executable (general) set-theoretical expressions in modern programming. The flagship library in this regard is certainly underscore.js.
Set theory is an incredibly invasive species which, over the last two decades, has increasingly invaded the practices of contemporary software engineering. Set theory is slowly but surely turning into the primary foundations in programming. In fact, it is so intuitive that few people actually realize that all of that is almost pure ZFC set theory.
Yes, that was an attempt to teach set theory in grade school. Needless to say the teachers were confused and the students were confused. Big fail. Now we have Common Core. The teachers are confused and the students are confused. In the US, public school students learn math in spite of the curriculum, not because of it.
Quoting tim wood
Yes. And for example 1 is the least upper bound of the set .9, .99, .999, .9999, etc.
Quoting tim wood
Yes. Again consider [math]x \in \mathbb Q : x^2 < 2[/math] where [math]\mathbb Q[/math] is the rationals. No matter what upper bound you pick, there is no least upper bound in the rationals. So the rationals are not complete. (Or Cauchy-complete, or topologically complete, to distinguish this from other uses of the word complete).
But that set does have a least upper bound in the reals. In fact every nonempty set of reals bounded above has a least upper bound. That fact uniquely characterizes the real numbers among all totally ordered fields.
Quoting tim wood
Yes definitely. There are so many ways to develop the square root of 2. You don't even need least upper bounds. You can do it with pure algebra. If [math]\sqrt 2[/math] is a purely formal symbol that means nothing, but has the property that [math](\sqrt 2)^2 = 2[/math], then consider the set of all formal expressions
[math] S = \{a + b \sqrt 2 : a, b \in \mathbb Q \} [/math]
Define addition componentwise, and multiplication using the usual distributive law. Then it's easy to see that [math]S[/math] is closed under addition and multiplication, and that multiplication distributes over addition, etc.
What about inverses? It's not immediately obvious, but in fact if [math]a + b \sqrt 2 \neq 0[/math] then
[math] \frac{1}{a + b \sqrt 2} = \frac{a}{a^2 - 2 b^2} + \frac{-b}{a^2 - 2 b^2} [/math]
https://math.stackexchange.com/questions/821260/inverting-ab-sqrt2-in-the-field-bbb-q-sqrt2
In other words our set [math]S[/math] is a field (add, subtract, multiply, divide) that contains a square root of 2.
But, just as with the real number earlier, all we've done is define some formal symbols that have the right properties. Can we construct such a field using set theory? Yes. If you have seen some abstract algebra, here is the construction.
You start with [math]Z[x][/math], defined as the set of all polynomials in one variable having integer coefficients. We can add, subtract, and multiply any two such polynomials as we learned in high school. So [math]Z[x][/math] is a commutative ring.
Now the set of all multiples of the polynomial [math]x^2 - 2[/math] is an ideal in this ring. Ideals in rings are analogous to normal subgroups in group theory. You can "mod out" by an ideal to get another ring. In this case the ideal generated by [math]x^2 - 2[/math], denoted [math]\langle x^2 - 2 \rangle[/math], is a maximal ideal; and therefore (it's a theorem) that when you mod out [math]Z[x][/math] by [math]\langle x^2 - 2 \rangle[/math] you get a field. And what field do you get? You get a field isomorphic to our [math]a + b \sqrt 2[/math] field of formal made-up symbols.
This is a bit of abstract algebra that most people haven't seen unless they took that course. But the point is that we can whip up a field containing a square root of 2 in the usual two ways: We can invent it as a make-believe set of formal symbols that behave according to some rules; AND we can construct such a beast within set theory.
This gives [math]\sqrt 2[/math] all the mathematical existence it needs.
Apologies if this exposition was a bit too much abstract algebra. But perhaps someone reading this took a course in groups, rings, and fields but forgot this beautiful construction, which we can sum up in one equation:
[math]\{a + b \sqrt 2 : a, b \in \mathbb Q \} \simeq Z[x] / \langle x^2 - 2 \rangle [/math]
On the left we have a made-up collection of formal symbols that mean nothing; on the right, we have a concrete realization of that thing cooked up within set theory.
Again, note that this construction parallels how we define the real numbers. We make up a formal system using some rules; and then we show that such a thing can be built out of spare parts within set theory. That is mathematical existence.
Quoting tim wood
Seems that way to me too. Our friend @Metaphysician Undercover, who must be a neo-Pythagorean, is mightily vexed by the fact that the square root of 2 is (a) a commonplace geometric object, being the diagonal of a unit square; and (b) doesn't happen to be the ratio of any two integers.
What of it? Humans got over this about 2500 years ago.
Yup. New math, new new math, Common Core. One educrat failure after another. I have no idea what the answer is.
Yes, we're in agreement.
And, as I mentioned in the OP, I also agree with your position that there's no "problem" in the real number system, axiom of choice, well ordering, cardinals and the like; at least not some trivial contradiction I'm aware of.
My questions do have answers, and I'm only trying to demonstrate here that the answers are incredibly tricky and go far beyond high school maths.
I think continuing the debate is a good way to bring up how tricky these ideas are, and why simply positing the real numbers without the axiomatic framework to avoid these problems is bad pedagogy.
Quoting fishfry
Yes, it is quite clear to conclude root 2 is irrational in a finitist constructive approach.
My procedure only kicks in once I'm given the real numbers and have access to completed infinite decimal expansion. Given this, I now double back and ask "can I use these new values to prove root 2 is rational in my new system of rules that includes sets of completed infinite decimals".
I am now no longer satisfied by the proof by contradiction that originally brought me to believe root 2 was irrational, as I can solve the equation with values that are neither odd nor even. I can also now do the same thing to solve exactly for the roots of any polynomial that I was previously unable to do.
Quoting fishfry
Where this relates to Cantor, is that if I simply "have the real numbers" and can use my procedure to prove root 2 is rational (because I just have them and have no axiomatic system to prevent me from doing it), then I should be able to count it in Cantors diagonal proof. If I complete the count of the rationals I will "eventually get" to this rational number with infinite numerator and denominator; it's got to be there somewhere.
Quoting fishfry
Yes, we totally agree. The purpose of my questions is that none of these (what I view) as quite intelligible questions you can start to ask once you "have" the real numbers can possibly be answered in the context of high school maths in a reasonable amount of time. Therefore, it is a mistake to simply posit the real numbers in high school and only serves to mystify mathematics rather than build clear understanding of how the next idea relates to the previous ideas.
Quoting fishfry
This is what the OP is about, so from my point of view every time there's agreement on this point I am very satisfied.
The reason I'm not attacking as contradictory real numbers, Cantor's proofs, AC, in the other threads is because I know I won't succeed. I can only make a muddle of it here in the context of the lack of suitable axioms and understanding at the high school level, which as you've pointed out tool the smartest people hundreds of years to figure out how to prevent wild proliferation of contradictions.
Quoting fishfry
Yes, if anyone was having doubts about my recommendation that the real numbers in high school is bad pedagogy, take a long look at this statement.
Quoting fishfry
Yes, I agree, but my question is not how the reals are constructed. My exercise here starts with having the reals already.
My question is how exactly does one prevent the reals from breaking previous proofs by contradiction.
For, if we use proof by contradiction to establish the irrationals, then create the reals as existing between rationals, but then with the reals we have completed infinite decimal expansion and can go back and invalidate the proof by contradiction by just injecting suitable decimal expansions to solve the roots of the root 2 polynomial, then all the reals are now rational and there's no reals between rationals, and we now no longer have the reals, because they're all rational.
We then re-check Cantors proof that the rationals cannot count the reals and simply conclude that if "counted high enough" we would eventually go through all the rationals with infinite decimal expansions taken from the reals as numerator and denominator.
For me, constructing the reals isn't the tricky part, it's preventing the above things happening. Why it's way beyond high school math is that it's not at all intuitive what proofs by contradiction mean and mathematical induction means, when going from a finite to an infinite regime.
Solvable ... but extremely tricky.
Then once it's solved by preventing infinite decimal expansions from corresponding to natural numbers (that for every real number represented by infinite decimal expansion, there is not a natural number that simply lacks the decimal point, and that the reals are not "onto" the natural), then the next step is even more careful preventing the assertion that all the natural numbers placed after a decimal point do not correspond to a real number; we cannot simply take the completed set of natural numbers as a natural number, with even a single natural number corresponding to the decimals of a real number we can still carry out the scheme.
Asking high school students to understand that decimal expansion does not represent a natural numbers lacking a decimal point, is far beyond a reasonable task. For, both natural numbers and real numbers seem very much at first viewing just a list of numbers that you can continue as long as you like; there is no way to intuit some difference beyond "as far as you can practically go in any universe somewhat similar to ours given any amount of time".
Which we already agree on; I'm continuing the "high school devils advocate" simply to demonstrate, with your help, how far away from "simple, intuitive steps" resolving any of these problems are.
Now, if I had a simple clear answer to these kinds of contradictions that took hundreds of years to build up frameworks to prevent, I'd say so. My purpose here is to check that no one else on PF has a simple answers either.
Lol. Well, they took it back so I guess that the cadre was very small. And as Fishfry commented earlier, this experiment wasn't just limited to my country (Finland), but the US too. I'd suspect that we copied the 'new trends' during those progressive times from the US. From the viewpoint of teaching small children math, starting with counting sheep is the way to go. It is the natural way, I'd argue.
Quoting A Seagull
I think this is a bigger philosophical problem for mathematics. Basically mathematics has evolved from the necessity of counting, calculating and computation. Hence 'applied math' came first. Abstract mathematical thought has emerged only later. This makes us start mathematics from the natural numbers.
I think the main thing to understand here is that decimal numbers with infinite decimals can be considered as an extension of "regular" decimal numbers (finite list of digits), but infinite natural numbers (infinite list of digits) cannot be considered as an extension of "regular" natural numbers, since you cannot define on them arithmetic operations compatible with the ones defined on the "regular" natural numbers. Then, you can't build fractions with infinite integers because you cannot build infinite integers in the first place. In my opinion this is quite easy to understand. Did I miss something?
I think the infinities and infinitesimals of mathematics are the things that make it become more "magic" and interesting. The problem with teaching in my opinion is more related to the fact that the "magic" of the fact that infinities and infinitesimals really work is not explained, or worse, explained by pretending to have a simple logical explanation that, however, is not part of the school program.
It is a lot simpler just to start with the natural numbers as axioms. Introducing set theory just complicates things and achieves nothing.
I'm not building with infinite integers, I'm building with the infinite decimal expansion representation of real numbers and simply pruning off the decimal symbol. Sure, if we simply define integer as "not this" then it's not building an infinite integer, but it is building something that I can then do things with if I'm not prevented from doing so.
Now, clearly if the proof by contradiction of irrational numbers is constrained to using "regular" natural numbers or integers, I have no qualms. It checks out.
However, if we switch regimes to one where we now have access to the infinite digit expansion of real numbers, we can revisit every proof in the previous regime with our new objects; and now, revisiting the root 2 proof is irrational I am able to solve it with these new objects and not arrive at a contradiction as oddness / eveness is no longer defined upon which the classic proof by contradiction depends. This is what I am doing.
Am I prevented from doing this full stop? Am I unable to find a "suitable decimal expansion" to solve my problem? What exactly is preventing me from doing this, that is what I would consider a suitable answer in the context of learning maths. Given these infinite decimal expansion, I want to use them as what ways I see fit, unless I'm prevented by some axiom. Lot's of things may have been, and still are, true in the previous setup before I had these objects, but in the new setup where I can make use of these objects in equations, I want to take full advantage, and revisit every proof by contradiction as well as every mathematical induction proof.
Broad features and themes involved in rigorous proofs elsewhere I do not consider a good answer for learning math. For me, "learning math is" understanding the proof oneself, not understanding that others elsewhere have understood something.
Again, I am discussing high school students level of understanding and what's reasonable in terms of capabilities, time and relevance.
So your answer doesn't explain why I cannot do my method.
Moreover, your approach, would seem to me, to imply that the decimal expansion representation of a real number cannot be counted; is this your implication? or would you say the digits in a real number are countable?
Also, how do you maintain infinite sequences can be completed, there are no infinite integers, the sequence of integers is infinite, simultaneously within the system suitable for high school level maths. Do we simply elect not to use our "complete the infinity tool" on the integers, and add this axiomatically? What axioms do they have to work with? Do they know enough set theory do make a model that avoids all these problems, or do they have another suitable basis?
Yes, I agree with you here.
I'm not against touching on the infinity subject; there could be a whole class on it for students wanting to go into pure maths.
I think we agree that it's bad pedagogy to simply posit the reals with no explanation and no time or ability to answer very expected and natural questions. Instead of curiosity leading to better understanding, it leads to confusion and a sense maths is "because we say so", which is the exact opposite sense students should be getting. Students would be better served by a less ambitious (not actually having irrationals and transcendentals as objects) but more rigorous calculus in the numerical regime, which would make a much more solid foundation for students going on to use applied maths, who can simply stay in this regime (as they will likely be solving every problem with the computer), and better serving pure maths students as well (that mathematics is rigorous, and extensions are made to do new things in a rigorous way).
My very first course in abstract algebra (taken in my first semester in grad school) did something like this. Not being conversant with the various concepts, even groups, made it very challenging and also meaningless. Afterwards I took a course in group theory which was illuminating. Thereafter I avoided abstract algebra. :brow:
A mistake, looking at current complex variable theory!
Quoting jgill
Difficult to take abstract at grad level without undergrad. Abstract algebra shows up everywhere. Physics is a lot of group theory these days.
The opposite argument is that it's bad pedagogy to expect high school students to understand the sophisticated constructions of higher math. It's true in all disciplines that at each level of study we tell lies that we then correct with more sophisticated lies later. It's easy to say we should present set theory and a rigorous account of the reals to mathematically talented high school students. It's much less clear what we should do with the average ones. Probably just do things the way we do them now.
Human beings may have gotten over this, but they did not resolve the problem. Consider the problem this way. Take a supposed "point". Now measure a specific distance in one direction, and the same distance in a direction ninety degrees to the first. Despite the fact that you use the exact same scale of measurement, in both of these measurements, the two measurements are incommensurable. Why is that the case?
Doesn't this tell you something about the thing being measured (space)? What it tells me, is that this thing being measured (space), cannot actually be measured in this way. The irrational nature of pi tells us the very same thing. Two dimensional objects have a fundamental problem which demonstrates that space cannot actually be represented in this way.
We see a very similar problem in the relation between zero dimensional figures (points), and one dimensional figures (lines), as discussed in the other thread. So if we get done to the basics, remove dimensionality and focus solely on numbers, we can learn to understand first the properties of numbers, quantity, and order, without applying any relations to spatial features. Then we can see that it is only when we apply numbers to our dimensional concepts of space, that these problems occur. The problems result in establishing a variety of different number systems mentioned in this thread. None of these numbers systems has resolved the problem because the problem lies within the way that we model space, not within any number system. We do not have a representation of space which is compatible with numbers.
Quoting A Seagull
The problem though is that introducing real numbers does not actually solve the problem, it just offers a way of dealing with the problem. So the problem remains and mathematicians simply work around it with increasingly complex number systems.
Quoting boethius
So you ought to see, that there is no clear and concise way to construct the real numbers. "The real numbers" is an extremely complex way of working around a very simple problem. The problem, as explained above, is that we do not model space properly. This creates problems with applying numbers to spatial representations. So instead of addressing the real problem, which is our representation of space, mathematicians continue to create exceedingly complex number systems in an attempt to work around the problem. Of course the problem remains though, so new issues pop up, and mathematicians continue to layer on the complexity.
As philosophers, who have met the problems of mathematicians and have chosen philosophy instead, we might focus on the real problem.
Forgive me, but what is the problem, again? :worry:
The incommensurability which produces irrational numbers.
The context is that "this" is the discovery by the Pythagoreans that for any pair of integers, the ratio of their squares can not be 2.
@Meta as I mentioned I've been working on a response to something you wrote in the bijection thread, and I'm taking my time to sort out my thoughts. In fact many of my ideas have been leaking into my recent posts; and clearly I've spilled the beans that it's your paragraph about Pythagoras that really caught my eye.
I have a lot to say on the subject of the square root of 2 and I don't want to say it all here. I'll keep my remarks here brief, and please be aware that I am going to offer a more detailed response to your point about Pythagoras soon.
Quoting Metaphysician Undercover
Because not every number that pops up naturally is rational. This is just a fact of life. I could push a philosophical point and say it's a fact of nature; and that some mathematical facts, abstract though they may be, are nevertheless forced on us somehow; and that it's the job of the mathematical philosopher to figure out how that miracle occurs. 5 is prime even though there is no 5 and there aren't any primes. It's not fiction. We can make up our own math but some things are not negotiable. There are mathematical truths. "5 is prime " is one; "the square root of 2 is irrational" is another.
But your stance here is literally pre-Pythagorean. The Pythagoreans threw some guy overboard for making the discovery, but they accepted the fact of the irrationality of [math]\sqrt 2[/math]. You refuse even to do that. You're entitled to your own ideas, but to me that is philosophical nihilism. To reject literally everything about the modern world that stems from the Pythagorean theorem. You must either live in a cave; or else not live at all according to your beliefs. You must reject all of modern physics, all of modern science and technology. You can't use a computer or any digital media. You are back to the stone age without the use of simple algebraic numbers like [math] \sqrt 2[/math].
Quoting Metaphysician Undercover
No of course not. It tells me something interesting about abstract, idealized mathematical space. It tells me nothing about actual space in the world. Since all measurement is approximate, even in classical physics, we can certainly take all physical measurements to be rational numbers if you like. It's ok by me. There are no irrational distances in physical space for the simple reason that there are no exact distances at all that we can measure. So it's not meaningful to talk about them except in idealized terms.
Math [math]\neq[/math] Physics. When you get that you will be englightened.
Quoting Metaphysician Undercover
You're confusing physical measurement of the real world, with abstract idealized lengths in mathematics. It's an elementary error, easily corrected. Especially now that I've corrected it for you.
Quoting Metaphysician Undercover
That's a little word-salady for my taste. Couldn't parse it. But I gather from your Pythagoras paragraph in the bijection thread that you object to n-dimensional Euclidean space as well, for the same reason. It logically follows that if you don't like irrational distances you wouldn't like the n-dimensional Euclidean distance formula, so I don't think this is a separate issue.
As I say I am working on a more comprehensive response to your Pythagorean lament, which will be forthcoming soon. Meanwhile let me just note some key points.
* [math]\sqrt 2[/math] only encodes a finite amount of information. It's true that its decimal representation is infinite, but that's an artifact of decimal representation. There are many finite characterizations of [math]\sqrt 2[/math].
* It's computable, so its decimal digits can be completely described by a finite-length computer program. Therefore a constructive mathematician would accept its existence and properties.
* It's algebraic, so it can be realized in a finite extension field of the rational numbers, [math]\mathbb Q[x] = \{a + b \sqrt 2 : a, b \in \mathbb Q\}[/math]. I outlined this mathematical construction in the second half of this post ... https://thephilosophyforum.com/discussion/comment/368048
* Euclid's proof of the irrationality of [math]\sqrt 2[/math] requires only PA (Peano aritmetic) and not ZF. Therefore a mathematical finitist would accept its existence. The last time I tried to explain the Peano axioms to you, you were so triggered by a little symbology that you were unable to engage. I hope you'll grow past that. You can't be a philosopher of math without rolling up your sleeves and dealing with a little symbology now and then.
* [math]\sqrt 2[/math] has a continued fraction representation of [math][1; 2, 2, 2, 2, \dots][/math]. In other words it has a repeating pattern that can be described finitely: "one followed by all 2's."
For all these reasons, [math]\sqrt 2[/math] is essentially a finite mathematical object. You're simply wrong that it "introduces infinity," because you have only seen some bad high school teaching about the real numbers. Decimal representation is only one of many ways to characterize [math]\sqrt 2[/math], and all the other ways are perfectly finite.
Finally, if you said, "Math says there are noncomputable real numbers and that must be nonsense," I would still disagree with you but you would still have a much stronger case. All the constructivists and neo-intuitionists would agree with you. That argument would have the benefit of being a sophisticated attack on standard set theory.
But to simply say that you don't like [math]\sqrt 2[/math]; that's just a hopelessly naive viewpoint. [math]\sqrt 2[/math] shows up in many different finite constructions, from Turing machines to continued fractions to finite degree extension fields in abstract algebra.
[math]\sqrt 2[/math] has very solid mathematical existence. You're simply wrong that it doesn't.
If you said that you don't like noncomputables you'd have a better case. And if you said you don't like transcendentals, you'd still be wrong but it would be a slightly better case. But to reject the mathematical existence of a harmless algebraic number like [math]\sqrt 2[/math] is just a lack of sufficient mathematical understanding. You have no defensible philosophical point. It's too easy to construct [math]\sqrt 2[/math] by finitary means. Like ... um ... the diagonal of a unit square.
Well actually those were most of the points I'm making in my post-to-be. Most of it's just drilldown of these bullet points. Maybe I'm done now. I'll have to go back and look.
Let me ask a larger question. Is it mathematical abstraction that bothers you? Of course there's no physical length that can be measured as [math]\sqrt 2[/math]. But there's no physical length that can be measured as 1 either. Don't you know that?
Quoting Metaphysician Undercover
I just noticed this. It's the heart of your confusion. A representation is not the thing itself. We can and do "represent" space as the real line or pi or whatever. But we don't actually think space IS that mathematical representation. It's just a representation, imperfect from the start. It's an approximation at best, a convenient lie at worst.
Everybody knows this. Or at least I know this. And now that you know I know this, maybe you'll stop holding me responsible for opinions I don't hold. Your argument is with someone else.
ps -- Tegmark's wrong. Is this what you're on about? The world isn't literally math. A representation of reality is not reality. The map is not the territory. And not even Tegmark takes his own idea seriously, as witnessed by his rapid retreat from the mathematical universe hypothesis to the computable universe hypothesis, which is actually inconsistent with known physics. Don't worry about people who say the physical world "is" math. The physical world is only represented by math. Not the same thing.
It's feeding time in my vat now.
[math]\sqrt{x}=1+\cfrac{x-1}{2 + \cfrac{x-1}{2 + \cfrac{x-1}{2+{\ddots}}}}[/math] from which [math]\sqrt{2}[/math] can be calculated by iteration. This might be the algorithm used to obtain square roots on simple calculators. Or it may have been some time ago, replaced by better algorithms. CS people out there?
This expansion may be due to Omar Khayyam, the poet, rug maker, and mathematician from around 1100AD. :cool:
The decimal symbol is the thing that says which digit of the numerator matches which digit of the denominator. If you prune off the decimal symbol you have to specify in which sequence you add up digits up and down to "build" your infinite fraction. If you add one digit at a time up and down, you get a sequence of fractions that is (converges to) a well defined real number.
The difficult point to clarify here is that infinite expressions are not simply the set of symbols (or digits) that compose them, but that set of symbols PLUS the algorithm that says in what sequence you have to add them. And if you change the sequence you get a different result.
Quoting boethius
There is a much simpler proof for the irrationality of roots: take a fraction and write numerator and denominator as product of prime factors. Then square it. It's evident that every prime factor appears at least twice both in numerator and denominator. It means that every fraction squared has this characteristic of doubled prime factors. It follows (from "A ==> B" follows "not B ==> not A") that if a fraction is not done in this way, it cannot be the square of another fraction. Therefore, the square root of any fraction not containing doubled prime factors must be irrational. This proof has even the advantage to clarify how to check if a square root is irrational (not only root of two), and works even for cube roots, etc..
Now, the problem doing this with infinite integers is that you have to specify how to decompose them in prime factors. If you have an algorithm that decomposes the "partial" integer generated at every step (by the algorithm that specifies the sequence used to build the infinite number), the proof will work at the same way, and the result will be correct.
Quoting boethius
Perfectly right. Not sure if the previous explanation is enough... The problem with proofs by contradiction (the reason why it's easy to make mistakes using them) is that you should have only ONE assumption, and then if you reach a contradiction you know that THAT ONE assumption was false. Using them with infinite objects you very often introduce hidden assumptions about uniqueness of those objects that are not true.
Quoting boethius
I agree.
Quoting boethius
No, the infinite decimal expansion of a real number is a perfectly good real number: it's the limit of a convergent power series: 1.3762.... = 1 + 3/10 + 7/100 + 6/1000 + 2/10000 + ....
The digits in a real number should not be countable, but you have to say which algorithm you use to generate them, since they are infinite. Or, put in another way, you have to specify in same way a function "F: natural number ==> digit" that for each position (power of 10) says which digit to put in the numerator.
( or maybe I didn't understand your objection.. )
Quoting boethius
I would say:
- infinite sequences are the same thing as functions from integers to sequence elements.
- functions from integers to sequence elements are surely well defined if the rule to produce the Nth element is clear (is an algorithm)
( maybe explain that you can even assume the existence of non-algorithmic functions, with the axiom of choice, but you cannot use it freely without making use of formal logic )
- integers are defined as sums of powers of 10 (that is the DEFINITION of an integer in the standard notation, not some strange property. So, 2 is 1 + 1 BY DEFINITION: nothing to be proved). The problem with infinite integers is that you don't know which powers of 10 it's made of. If you have an infinite decimal expansion, you know the powers of 10 and everything works. If you are not convinced, try to write infinite integers in Peano notation: 1+1+1+1+.... (or SSSSS..0 - same thing): they are all the same number.
- the sequence of integers is infinite because is constructed by adding +1 at each step, and this is a non terminating algorithm that produces a well defined result at each step, so it's allowed as an algorithm.
Quoting boethius
I would avoid set theory without speaking of logic before. Of course, you can do the set theory of FINITE sets without logic, but that is not useful to explain real numbers, or anything related to analysis.
Perfectly agree.
Quoting boethius
Well, I think a lot of interesting calculus at Euler's level could be done in a enough rigorous way, and just make the students aware of what are the really rigorous parts and which ones are the most "doubtful", when reasoning about infinities. But the most doubtful ones are even the most interesting! And if you explain that we don't know everything, that's the part that makes the subject of math worthy of studying. What for should I (as a student) loose time in a subject where everything hast just been discovered long time ago, and the only thing I can do is to learn by mind what others did? Math becomes interesting when you see that you can use it do discover new things that nobody said you. And there are still a lot of things to be discovered; only that you have to learn how to reason about them in the right way!
I agree this solves the problem, and this is for me the essence of what I've called conceptual inversion. Starting calculus with uncountable digit expansions as essentially prior knowledge isn't a good setup.
However, the other problem I've been alluding to is revisiting all previous theorems proven in a finitist regime; which is also essential part of understanding the infinite regime. Some theorems are abandoned. Choices must be made.
For instance, in Euclidean geometry we can have a theorem that sphere represented by an arbitrary amount of components, but not infinite, cannot be turned into 2 equal spheres of equal volume. We can also have a theorem that arbitrary amounts of lines never completely fill up area or volume. Going to the standard infinite regime we can revisit these theorems and prove them "false" in the sense that what we thought we couldn't do before we can do now in this new system. This, for me, these "side-affects" features that we didn't expect and didn't set out to make, is what makes these areas of mathematics difficult, even more than being able to construct objects we're intending to make like the real numbers, and high school students. So, even if there was time to explain infinite digit expansion is uncountable in some actual mathematical way involving definitions and proofs, and it's due to this uncountability that's we can assert they cannot be converted into integers ... while still having infinite integers but no "infinite integer" available to put in our set of rationals ... neither asserting that all integers in our set are finite in a sense of having a finite amount of them, which would be clearly false. Even if this was time to do this, it's still not a good understanding of this infinite regime with real numbers without reversing previous intuitions we'd have built up with finitist concepts.
Quoting Mephist
Yes, I'd pretty much agree with your program here.
By "numerical regime" I mean focus on these objects as algorithms and not "real numbers" that we simply have by writing down pi or e.
I think potential infinity is an intuitive concept. Though it may help some students to know that applied mathematics can also be done with only needing to imagine "what one could practically represent in our universe".
Quoting Mephist
I'd definitely be in favour of bringing everyone up to Euler's level.
I'm not advocating ultra-finitism in secondary education, mainly opposing the positing of real numbers as a "starting point" to doing calculus; I'd be willing to compromise on how rigorous the alternative can and should be.
I would take seriously ultra-finitist arguments that they have an even better educational setting to start, for I could be biased by my own familiarity with the subject matter and so think just potential infinity is easier than it is.
In either case, it makes perfectly good subject matter to discuss along with discussion of the kinds of problems one faces with infinities in your program. That there is diversity of perspective even among professional mathematicians I think is inspiring and engaging stuff to talk about.
But, when it comes to actually doing math to solve problems, building up the "intuition of what rigor is" in my view is paramount, and without it the average high school teacher is in a much worse position; in a rigorous system there really is answers to every question that can simply be looked up; which is a better position to be in than needing to say that one doesn't have the answer but "you'll understand when you do pure maths in university" or worse provide a wrong answer as you note is often the case.
I would also not be opposed to a pure maths course that build the real numbers, introduce uncountability, for students interested in pure maths. I'm not underestimating the capacities of high school students to engage with concepts from pure maths. However, it's a disservice to applied maths students to abandon reason for madness, simply because historically we went through lot's of mathematical ideas that turned out to go crazy (in the sense of proving A does not equal A).
Quoting Mephist
Although I agree with your sentiment here, I would argue such interesting questions are best approached from a rigorous foundation, which I don't think your contradicting.
For instance, the real numbers are best approached from a good understanding of natural numbers, integers, rational numbers and finite sets, and what limits these concepts have but also a good understanding of their intuitive strengths that may fail in different systems (what you see is what you get in finitist maths), as your program suggests to do.
So, infinite sets and real numbers could be something introduced at the very end of secondary maths when these foundational concepts are more familiar. But to start, understanding divergence and convergence and tangents and how series and sums and derivatives and integral functions relate to each other (and how to solve real problems with them), is challenging enough to learn in a finitist regime; my intuition is that doing this also with the conceptual challenge of infinity makes it much harder to "see" and to "get" what's going on, and students who start asking questions, even just pondering to themselves, that have no good answers available will much more easily get lost or believe their questions are seen as stupid by the mathematical community, simply because their teacher can't deal with them.
I wrote "The digits in a real number should not be countable". Well, the digits of (the decimal representation of) a real number are countable, since they are determined by a function of type "natural-number ==> digit".
It's the set (the totality) of all real numbers that is uncountable: meaning that there is no surjective function of type "natural-number ==> real-number".
But notice that even the set of all functions of type "natural-number ==> boolean" (for example) is uncountable. And Cantor's diagonal argument can be applied to whatever function of type "something ==> something else" to show that there are more functions than objects, even if the functions are simply well defined terminating algorithms: there's no need to use formal logic or set theory to prove this.
In my opinion, the thing that makes real numbers more difficult to grasp intuitively is that they don't have a normal form: there is no way to create an algorithm that decides if two arbitrarily defined real numbers are the same number or not, and that's because there are "too many ways" to build a real number (basically, you can use whatever algorithm you want, and in general there is no way to decide if two given algorithms produce the same output or not).
Quoting boethius
Well, I wouldn't start from the "pathological" cases to show that volume additivity doesn't work any more. On the contrary, I would start from the fact that you can calculate the volume of curved figures as if they were make of infinitesimal polyhedrons, and it always works! (Archimedes' volume of the sphere is very intuitive and beautiful).
OK, then there is this little "glitch" in the fabric of the universe named Banach Tarski theorem... :smile:
It doesn't work because in integral calculus you have to take "open sets" as infinitesimal pieces ( but I would prefer to not go into details about this issue, because surely @fishfry will read this and will not agree :wink: )
Anyway, the fact that infinite additivity works as if infinitesimal geometrical objects existed in reality is the really interesting and useful fact. The fact that it's so difficult to prove why it works and why in some cases it doesn't, maybe makes the problem even more interesting..
Quoting boethius
Sorry, but I don't understand when you say "due to this uncountability..." why is uncountability a problem?
Quoting boethius
I think more than the "finitist" regime they should be teached before in the 18th century "Eulerian" regime, where functions always work as if they were infinite polynomials and derivatives are made of infinitesimals. You have to see that all this staff with infinities really works before starting to wonder how is it possible that it works if infinities don't really exist. Then you have a motivation to study formal logics and set theory. More or less, following the historical development of mathematics. I think there is no sense in creating a theory of infinite sets if you don't see what for all this infinity staff is good for.
[math]a+b\sqrt{2}\ne 0[/math] then [math]\frac{1}{a+b\sqrt{2}}=\frac{a}{{{a}^{2}}-2{{b}^{2}}}+\frac{-b}{{{a}^{2}}-2{{b}^{2}}}[/math].
Banach-Tarski means nothing about actual space. It's a valid technical result that applies to mathematical Euclidean space. There's no reason to believe that the universe behaves exactly the way Euclidean space does. This seems to be a common theme here lately, but I think it's mostly a strawman argument. I don't think there's anyone seriously suggesting that the actual universe is exactly like the mathematical real numbers. I for one don't believe that in the slightest. I think mostly that the people who think that haven't given the matter much thought; and once you start thinking about it, it becomes perfectly clear that the real numbers are a mathematical model that works amazingly well, in spite of the fact that it's so unlikely to be anywhere near close to the "truth," if there even is any such thing as a truth of the matter. Most likely it's turtles all the way down.
There aren't any analogs of mathematical "points" in the real world, little zero-dimensional zero-sized thingies that somehow occupy a "location" in space. I don't believe that for a moment. I really don't think anyone else who's thought about the matter seriously does either. That's my opinion anyway. I like math but I never confuse it for reality. I think a lot of people in online forums are angry at math for making the claim of being a perfect representation of the universe; but math does not make that claim. Math asks to be taken on its own terms.
Banach-Tarski is a valid theorem. I heard that John von Neumann was the one who first noticed it in the 1920's. They were looking at how group theory interacts with geometry and measure theory, and this little paradox shows up. Mathematicians tend to delight in such results. They don't throw up their hands and go, "Oh woe is us, the physicists will make fun of us. Or even worse, the philosophers will!" They don't think that way. A cool result is a cool result. As Russell noted, math is about investigating the logical consequences of various sets of premises. It's not necessarily true or meaningful. Sometimes it is. Depends on what you use it for.
I'm with Hardy, who held that the the more useless a branch of math, the more beautiful it was. He applied this to his beloved number theory, which for over 2000 was regarded as a supremely beautiful and supremely useless part of math. How would Hardy feel if he came back and found out that in our very lifetimes, starting in the 1980's, number theory became the basis of Internet security, and is now the most applied branch of math you can imagine! I hope he'd have a sense of humor about it; and also a sense of wonder at how purely abstract math, considered useless for millennia, one day becomes the very heart of world commerce.
Hardy was the guy played by Jeremy Irons in The Man Who Knew Infinity, which if you haven't seen it, please do immediately. Besides being a mostly true account (for a Hollywood movie) of the miraculous and tragically short life of Ramanujan; it's also a meditation on the relation of intuition to formalism in math. Visions from the Goddess versus formal proof.
Oh gosh. Thanks for mentioning it.
I was idly skimming through the many posts in this thread that I hadn't read. And I swear this is how my brain works. I zeroed in on this particular comment like a laser. That's not an especially good quality because I often miss the larger points people are making. I'm an excellent proofreader too. Grammar and spelling errors literally jump right off the page as I read. Terrible affliction in a day and age when nobody gives a shit about spelling or grammar. Spelling and grammar are tools of patriarchical and colonial oppressors. Such is the zeitgeist.
You mentioned integrals. A lot of people think of an integral as the sum of the areas of infinitely many infinitely thin rectangles. I have no problem with that. That's how everyone thinks about them and that's perfectly fine. Professional physicists do in fact think exactly this way all the way up to the highest levels. It doesn't matter.
I have no beef with how anyone thinks about math or visualizes it or simplifies it in their minds.
But you did give a wrong and misleading definition of an open set. I do have to say that. Open sets are really important. An open set in the reals is just like an interval without its endpoints. What matters about it is that "all its points are interior points." It doesn't include any points of its boundary. That's what makes open sets have the interesting properties that they do.
They're not really infinitesimal. They can be arbitrarily small. But they aren't "infinitely" small. In fact that is the great "arithmetization of analysis," the great founding of the continuous world of calculus on the discrete world of set theory. Instead of saying things are infinitely small, from now on say they're arbitrarily small. For every epsilon you can go even smaller. But in any individual instance, still nonzero. That's the essence of open sets.
Ok I quibbled again. I had a great course in Real Analysis with a gifted professor. Open sets are very near and dear to my heart. But if you substitute "infinitely small" with "arbitrarily small," each time you do there will be that much more clarity and correctness in the universe. We can literally reverse entropy by fixing typos. Think about that.
I found what I wrote about six months ago: Quoting Mephist. It's still valid!
OK, I think I should give some explanation on this point:
I wrote you have to take "open sets" as infinitesimal pieces
What I meant is you should impose the restriction that the infinitesimal pieces are also "open sets"
The definition of open sets is of course what you wrote: "all its points are interior points", or "there are not isolated or border points in the set", or "each point of the set is surrounded by other points"
Now, if I wanted to explain under what assumptions additivity of volumes (or surfaces, or segments) works without using a formal logic system, I would say that it works even if you consider infinitesimals as really existing entities (with the appropriate rules of calculus: for example: integrating over a line, dx squared is zero), but you cannot take as dx isolated points: you have to take pieces that don't contain points that are isolated from each-other, because otherwise the topology of the object is not preserved (the functions are not continuous), and you can build a sphere using the points of a line, or two equal spheres using the points of one sphere. As I wrote in my explanation about Banach-Tarski mounts ago, the theorem works because it uses isometric transformations, but applied to set of points that are isolated from each other (not on open sets). If you impose the restriction that your isometric transformations should be even continuous (going from open sets to open sets), you can't do it any more.
Yes, I think this is the normal situation and what I was expecting. But now I believe the task is even more difficult as one now needs to explain to high school level math students that both the digits in integers "can be counted" and the digits expansion of real numbers can be "can be counted" (assuming they stick with you on what counting infinities mean), but you cannot count on making an infinite integer to make rationals.
The purpose of my series of questions is not to build ZFC or some analogue, but to demonstrate that without the context of ZFC there are no "simple answers" to questions about the real numbers. That there is no simple story to tell nor easy proofs to put on the board in the context of what high school students level.
I think this thread establishes pretty well that the average high school class room doing calculus for the first time would not be able to follow almost any of this conversation.
Quoting Mephist
My point is that these unexpected and non-intuitive theorems exist when going from the finite to the infinite regime.
Understanding infinite regimes means understanding these non-intuitive, arguably "undesirable" in some sense results, and doing that isn't achievable if students have not yet built up an intuition of the finite regime to be able to contrast with unintuitive results in the infinite regime.
Banach-Tarski is for me no less strange than being able to in some sense "stretch" the points in the interval 0,1 to cover 0, a billion; it only seems more strange if one has already gotten accustomed to the run-of-the-mill real number properties. But that's not understanding the real number system to just be given the real numbers and said "these numbers have these properties we want because we're doing calculus now".
Quoting fishfry
Yes, if we agree there's no simple enough answers to questions about the real numbers (defined as an appropriate amount of time for average teachers and students), your point here is a valid perspective that I'm not dismissing prima faci; certainly this has been the justification of doing calculus with real numbers.
My argument against this is that it breaks the chain of intuitions required to understand math. One step to the next should be clear, this is the "method" of mathematics; the rigor. With all the courses you mention needed to understand the real numbers well, this is the "method" employed, and the goal of these courses is to render what seems at first unintuitive (because they are not consequences of living in the real world) to something that is, step by step, intuitive consequences of the mathematical system.
In science classes, things are over simplified compared to advanced science, but the goal is to stick to the experimental and critical method (I would also argue this could be done a lot better). When this method is abandoned, I think we'd agree here on PF that it's not good science pedagogy. For instance, that creationism taught as a valid scientific theory is bad pedagogy because it is not verifiable by experiment; not that creationism should be "hidden" from students, but that it is doing philosophy and not science.
Also, in science there is no way to avoid starting with simpler "wrong" beliefs about the world that get fixed later. This isn't a requirement in math, there is no externally determined mathematical framework of truth determined by nature. Every step one takes in mathematics can be "true" in the sense of following from the previous steps. The infinite regime is, in my view, basically a restart with a new set of axioms; it is a different mathematical journey than the finite regime that students are naturally on due to living in a finite world and familiar only with finite objects.
My other argument would be purely practical, that focus on transcendentals and "exact" analytical solutions is a product of history due to 1. a lack of calculation ability, so analytical solutions were often the only practical way forward and 2. belief in a Newtonian world of a physical continuum (not to say that we can easily now do without a continuous mathematical framework in which to model discontinuous phenomena; just that we do not think that underlying framework is physical, as I believe you would agree).
However, with ubiquitous and incredibly powerful computing and no need for physicists to believe in a physical continuum, I would argue the average student is much better served by focusing on "what can the computer do for me", viewing constants algorithmically with arbitrary (to a physical limit of computation) precision potential determined in practice by one's problem, and building up intuitions around machine calculation (and analytical work including error bounds, computational complexity, along with analytical proofs of convergence when available, just in the "arbitrarily close to the limit" finitist framework); rather than, what we seem to all agree here, building up wrong intuitions about the real number system. Such a "numerical regime" can be made as rigorous as any part of pure maths, and so is also teaching (what is to me) the critical essence of pure mathematics, although of course additional material introducing infinity could be available for those interested in higher pure maths (whether starting from Euler or introducing ZFC; I don't have a strong opinion; my concern here is not serving those with mathematical aptitude heading for pure maths, but rather that everyone else has the best chance to be mathematically literate and also served by the mathematical community).
Yes, I see your point. Maybe you are right.
LOL Now you're trolling me seeing if I'll take the bait. There are no infinitesimals in the real numbers. If you're working in some other number system please say that. As I said earlier I have no problem with people informally thinking in terms of infinitesimals; but I do object to muddying the official formalism.
Quoting Mephist
I'm not sure what you mean. Can you please link your earlier post on B-T? The point sets in B-T are not isolated from each other, in fact the orbits are dense. They're disjoint from one another but not isolated. We should have a nice Banach-Tarski thread sometime, the subject keeps coming up.
I accept the fact that the square root of two is irrational. That's not the issue. And I actually use the Pythagorean theorem on a regular basis working in construction, laying out foundations. The issue is that I am inclined to ask why is it the case that the square root of two is irrational, and in doing this I need to consider what it means for a number to be irrational.
To simply say as you are saying, that some numbers are rational and some numbers are irrational, and that's a brute fact, does not express an understanding of what a "number" is. But then, to ask why is it that some numbers have the property of being rational and other numbers have the property of being irrational requires asking what it means to be an irrational ratio, and one might be faced with the prospect that what we call an "irrational number" ought not even be called a "number". Perhaps the Pythagoreans threw the baby out with the bathwater, saying we can't resolve this problem so let's just call them all "numbers" anyway, and get on with the project.
So, let me state clearly and concisely what the situation is. We have a very simple looking problem of division which cannot be solved because there is no "number" which can represent the solution. You say, the problem can be solved, the resolution is an "irrational number", so just forget about that problem, it's not a problem at all. And, you say it's just "a fact of life" that some numbers are irrational. I say it's a simple fact that a so-called "irrational number" is not a number at all, because it's quite obvious that there is not a definite number which expresses the resolution of the irrational ratio. See, the very simple looking problem of division has not been resolved, and it is a pretense to claim that it has been resolved to an "irrational number".
Quoting fishfry
If idealized mathematical space tells us nothing about space in the world, then physics has a big problem. But of course this is nonsense. That the square root of two is irrational, and that pi is irrational tells us something about idealized mathematical space, and that is that there is a problem with commensurability in idealized mathematical space. And, since idealized mathematical space is the tool by which we make measurements in real space, the problem of idealized space is simply ignored in application
Quoting fishfry
OK, let's talk about "irrational distances" in idealized terms then. Lets take a point A. Lets make a point B at a specific distance from point A, and a point C at the very same distance from point A, but in a direction at a right angle to the direction of point B. Do you agree that there is no definite distance between B and C? If you disagree then you are simply denying the fact. if you agree, then you might be inclined, like I am, to ask why this is the case. And so it appears to me, like there is a very real problem with "idealized mathematical space", making it less than ideal.
Quoting fishfry
You might say "?2" is a finite mathematical object, but until you define what a mathematical object is, it's you who's just typing word salad. In reality "?2" is an unresolved mathematical problem. That you call it a "mathematical object" doesn't mean that it is a "number", nor does it mean that it actually is an object. And, when one attempts to represent this so-called object as a number, infinity is introduced
However, I didn't say that it "introduces infinity" on this thread. If I mentioned that, it was another thread in another context. Perhaps I said that in a thread on infinity. What I am focused on here is simply the meaning of an irrational ratio, and whether it is appropriate to claim that the ratio has been resolved to a "number", called an "irrational number".
Quoting fishfry
It's not "?2" that I dislike, it's what it represents that is what I dislike. And it's not that I am simply saying this, I am giving you the reasons for my dislike. But you seem to be good at ignoring reasoning.
Quoting fishfry
Right! That's why we ought to seek a better one! That's exactly what I'm arguing. Don't you agree?
I linked it in the post just before this one. Here's the link:
https://thephilosophyforum.com/discussion/comment/302364
I don't know what this means. Matrix factorization? That's all there is to mathematics research these days? Surely you jest. :cry:
Ok. First, if you have been talking about mathematical objects and not physical space, my misunderstanding. But then all your mathematical objections will collapse.
Secondly, a terminological quibble. In math there is a thing called a measure. It's a generalization of the idea that the unit interval [math](0,1)[/math] has length 1, and a rectangle of sides 2 and 3 has area 6, and so forth. So it's a little better not to use that word.
If you're talking about distances, better to use the word metric. A metric is a distance function. For example in the Euclidean plane, the metric, or distance function, is given by the usual Pythagorean formula of the square root of the sum of the differences of the respective squares of the coordinates. That is, if [math]x_1, y_1[/math] and [math]x_2, y_2[/math] are points in the plane, their distance from one another is
[math]\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/math]. In n dimensions the formula is analogous. But there are also weirder metrics. A metric is the general name for any distance function.
Now back to your quote. Of course we measure mathemtical objects. A unit square has side 1, area 1, and its diagonal ... well you know what its diagonal is. In fact it falls out of the Euclidean distance formula as the distance between the origin [math](0,0)[/math]and [math](1,1)[/math].
Quoting Metaphysician Undercover
Cardinality is not a measure in the technical sense. It's a way of assigning a size to a set. I don't think there's a name for it.
Quoting Metaphysician Undercover
Of course we can assign a length or area or volume to mathematical objects. The unit interval has length 1, the unit square ... oh we've been over this already.
Quoting Metaphysician Undercover
Of course we can measure, or assign a length/volume/area to, or find the distance between pairs of, mathematical objects.
If I earlier thought you were objecting to physical measurement, my misunderstanding.
But of course we can measure or assign size to mathematical objects in many different ways, depending on context.
I'm fine with the latter principle so long as we maintain consistency.[/quote]
We do. The rules are laid out in the articles on measure theory and metric spaces that I linked.
Quoting Metaphysician Undercover
You are painfully misinformed about this. The square of the diagonal of a unit square is [math]\sqrt 2[/math]. It's a perfectly good real number; and metrics, or distance functions, are defined as functions between pairs of objects and the nonnegative real numbers (satisfying some distance-like properties).
Quoting Metaphysician Undercover
You keep saying an irrational number is "immeasurable" but that's simply false. You're just wrong about that. You're still humg up on decimal representations but I'll show you soon that you're wrong about this.
Quoting Metaphysician Undercover
No we're good, I thought you were saying physical distances can't be measured. But if you're talking about mathematical objects, my misunderstanding. And now that I understand what you're talking about, you're just wrong. We can use metric spaces or measure theory to measure distances and generalized volumes (length, area, volume, etc.)
Quoting Metaphysician Undercover
Understood.
Quoting Metaphysician Undercover
No, I'm pointing out that there only seems to be a difference depending on what age one lives in. If you live in the age of integers you don't believe in rationals. You're stuck in the age of rationals and don't believe in irrationals. Matter of history and psychology.
Quoting Metaphysician Undercover
We're in agreement then. But that's what a mathematical object is. A made-up gadget that, by virtue of repetition, gains mindshare.
Quoting Metaphysician Undercover
I've studied set theory and read a number of set theory texts. I've never read or heard of any such thing. Set theory in fact is the study of whatever obeys the axioms for sets. If you ask a set theorist what a set is, they'll say they have no idea; only that it's something that obeys whatever axiom system you're studying.
You're making stuff up to fill in gaps in your mathematical knowledge. Set theory doesn't assume anything at all. It doesn't assume it's "about" anything other than sets; which are things that obey some collections of set-theoretic axioms.
Quoting Metaphysician Undercover
But I already did. From the naturals to the integers to the rationals to the reals to the complex numbers to the quaternions and beyond. At each stage people didn't believe in the new kinds of number and though it was only a kind of "calculating device." Then over time the funny numbers became accepted. This is a very well known aspect of math history.
Quoting Metaphysician Undercover
Because I'm not making up wild stories about set theory as you are.
Quoting Metaphysician Undercover
Not at all. I know that in high school they tell you that a "set is a collection of objects." Nothing could be further than the actual truth. Sets as mathematicians understand them are very strange. They're simply abstract thingies that satisfy some axioms that we write down.
Quoting Metaphysician Undercover
It's something you made up. Or maybe someone told you that. They were wrong. Set theory doesn't assume any objects at all. In fact ZFC is a "pure" set theory, meaning that the only things that can be elements of sets are other sets. There are no other types of objects at all! Only sets, and we don't even know what those are!
For the record there are also set theories with urelements, meaning things that can be members of sets that are not themselves sets.
Quoting Metaphysician Undercover
What do you mean by real or actual things? In the physical world? Well, physicists use math to model electrons and gravity and quarks and stuff. Maybe you should ask a physicist.
But do you mean how can sets be used to model mathematical objects like numbers, functions, matrices, topological spaces, and the like? Easy. We can model the natural numbers in set theory via the axiom of infinity. Then we make the integers out of the natural, the rationals out of pairs of integers, the reals out of Cauchy sequences of reals, the complex numbers out of pairs of reals, and so forth. If you grant me the empty set and the rules of ZF I can build up the whole thing one step at a time.
Quoting Metaphysician Undercover
You haven't made any such case. On the contrary, your questions all have straightforward answers.
Quoting Metaphysician Undercover
No, I took pains to make a distinction. Driving laws are completely arbitrary. But many mathematical ideas are forced on us somehow, such as the fact that 5 is prime.
Quoting Metaphysician Undercover
Yes, I thought at that time that either yu were making a point of great subtlety, or else that you were insane. By reading your posts I have determined the latter. I don't mean to be pejorative here. But you said at one point that when we say "4 + 4 = 8", the two instances of the symbol '4' mean or refer to different things. That's ... Look man that's just bullshit. I can't be polite about this. That's a very bizarre idea.
Quoting Metaphysician Undercover
You're still hung up on decimal representations; which I've said (several times now) are NOT determinative of whether a given real number has infinitary nature.
Here's an example. Take 1/3 = .333333.... Would you say that 1/3 is not resolved or requires an infinite amount of information? But it doesn't. I could just as easily say, "A decimal point followed by all 3's." That completely characterizes the decimal representation of 1/3. I don't have to physically be able to carry out the entire computation. It's sufficient that I can produce, via an algorithm, as many decimal digits as you challenge me to.
Likewise there is a finite computer program that completely characterizes [math]\sqrt 2[/math]. I don't have to write out all the digits. I only have to write down a FINITE description of an algorithm that produces as many digits as you like. This is easily done.
Quoting Metaphysician Undercover
It has been completely resolved. You can't write down infinitely many digits any more than you can write down all the digits of .333... But in the case of 1/3, there's a finite-length description that tells you how to get as many digits as you want. And with square root of 2, there is ALSO such a finite-length description. Would you like me to post one?
Quoting Metaphysician Undercover
No.
Quoting Metaphysician Undercover
Bad analogy, nothing to do with the fact that computable numbers like 1/3, [math]\sqrt 2[/math], and [math]\pi[/math] only require a finite amount of information to completely determine their decimal expressions.
Quoting Metaphysician Undercover
I showed how to characterize pi a few days ago as the Leibniz formula. The square root of 2 has a very easy program to calculate its digits.
I really hope you'll consider the example of 1/3 and the fact that we can predict or determine every single one of its decimal digits with a FINITE description, even though there are infinitely many digits. Square root of 2 and pi are exactly the same. They are computable real numbers. There is a finite-length Turing machine that cranks out their digits.
Here's a Python program that prints as many digits of [math]\sqrt 2[/math] as you like. It uses a simple high/low approximation method. We know [math]1 < \sqrt 2 < 2[/math] because [math] 1^2 < 2 < 2^2[/math]. So we split the difference and guess 1.5. But 1.5 squared is 2.25, a little too big. So we split the difference between 1 and 1.25 and try that. The longer we run the algorithm the more digits we get. Just like the longer we write down 3's, the more decimal digits of 1/3 we get.
This simple little FINITE STRING OF SYMBOLS cranks out successively better and better approximations to [math]\sqrt 2[/math] the more iterations you do. Of course we can't physically write down all the digits because physical computations are resource-limited. But in principle we can; just as "keep writing threes" is a recipe for the decimal representation of 1/3.
I made some comments in the other thread so as not to pollute this one.
https://thephilosophyforum.com/discussion/comment/368991
In a sense, this is a form of compression of information: understanding something means compressing the information contained in something in a new simpler way (by using a different point of view, or definitions). That's mainly what mathematicians are doing today.
This is why I consider communication games to be the most important interpretation of logic. For it identifies the constructive content of logic with the permissible sequences of actions that can be taken by player A, and the 'non-constructive' content of logic with message-replies that A receives from a player B as a result of A messaging B. The existence of a message-reply from B is not-guaranteed a priori, and A's message to B is only said to be meaningful as and when A receives a reply from B.
A constructive real number refers to an algorithm constructed by A for generating a sequence of integers. In the case of a non-constructive real number, A invokes the "Axiom of Choice" , which is interpreted as A 'outsourcing' the creation of the real number, by sending B a message requesting B to send A an arbitrary sequence of integers. The sign for a non-constructive real number has no specific meaning or referent until A receives a stream of integers from B.
Quoting fishfry
OK, so we agree that if so-called "mathematical objects" are things which can be measured, Euclidian geometry creates distances which cannot be measured by that system. That agreement is a good starting point.
As a philosopher, doesn't the question, or wonderment, occur to you, of why we would create a geometrical system which does such a thing? That geometrical system is causing us problems, inability to measure things, by creating distances which it cannot measure.
Quoting fishfry
Maybe we can take this as another point of agreement. A "mathematical object" is nothing other than what you called a "funny gadget". Let's simplify this and call it a "mental tool". Do you agree that tools are not judged for truth or falsity, they are judged as "good" in relation to many different things like usefulness and efficiency, and they are judged as "bad" in relation to many different things, including the problems which they create. So a "good" tool might be very useful and efficient, but it might still be "bad" according to other concerns, accidental issues, or side effects. Bad is not necessarily the opposite of good, because these two may be determined according to different criteria.
Let's look at the Euclidian geometry now. In relation to the fact that this system produces distances which cannot be measured within the system, can we say that it is bad, despite the fact that it is good in many ways? How should we proceed to rid ourselves of this badness? Should we produce another system, designed to measure these distances, which would necessarily be incompatible with the first system? Having two incompatible systems is another form of badness. Why not just redesign the first system to get rid of that initial badness, instead of creating another form of badness, and layering it on top of the initial badness, in an attempt to compensate for that badness? Two bads do not produce a good.
Quoting fishfry
Come on, get real fishfry. Check Wikipedia on set theory, the first sentence states that it deals with collections of "objects". Then it goes on and on discussing how set theory deals with objects. Clearly set theory assumes the existence of objects, if it deals with collections of objects.
This is why it is so frustrating having a conversation with you. You are inclined to deny the obvious, common knowledge, because that is what is required to support your position. In the other thread, you consistently denied the difference between "equality" and "identity", day after day, week after week, despite me repeatedly explaining the difference to you.
Quoting fishfry
You have not explained how acceptance of a mathematical tool, through convention, converts it from a funny gadget, to an object. If you cannot demonstrate this conversion, then either the tool is always an object, or never an object. Then an extremely bad tool is just as much an object as an extremely good tool, and acceptance through convention is irrelevant to the question of whether the mental tool is an object.
Quoting fishfry
Until you recognize that an "element", or "member" of a set is an "object", you are simply in denial of the truth, denying fundamental brute facts because they are contrary to the position you are trying to justify.
Quoting fishfry
The case I made is very clear, so let me restate it concisely. You appear to agree with me that mathematical tools are not objects, they are "mind" gadgets, yet you defend set theory which treats them as objects.
Quoting fishfry
Mathematical ideas such as "5 is prime" are forced on us by the rules (laws) of the mathematical system, the definition of "prime" and "5" with deductive logic. So there is no difference. We create mathematical rules arbitrarily, as they are required for our purposes, just like we create driving laws arbitrarily as required for our purposes.
Quoting fishfry
This is nonsense. I can very easily say "the highest number". Just because I say it doesn't mean that what I've said "completely characterizes" it. We can say all sorts of things, including contradiction. Saying something doesn't completely characterize it.
Quoting fishfry
Unjustified, and false assertion.
Quoting fishfry
if you switch to a different number system, one which is incompatible with the first from which the irrational number is derived, like switching from rational numbers to real numbers, this does not qualify as a resolution, if the two systems remain incompatible.
For instance, if there is infinite rational numbers between any two rational numbers, and we take another number system which uses infinitesimals or some such thing to limit that infinity, we cannot claim to have resolved the problem. The problem remains as the inconsistency between "infinite" in the rational system, and "infinitesimal" in the proposed system.
Quoting fishfry
This has no relevant significance. To say "the square root of two", or "the ratio of the circumference of a circle to its diameter" is to give a 'finite description". We've already had the "finite description" for thousands of years. And, this finite description determines that the decimal digits will follow a specific order, just like your example of 1/3 determines .333.... The issue is that there is no number which corresponds to the finite description, as is implied by the infinite procedure required to determine that number.
So my analogy of "the highest number" is very relevant indeed. Highest number is a "finite description". And, the specific order by which the digits will be "computed" is predetermined. However, there is no number which matches that description, "highest number", just like there is no number which matches the description of "the square root of two", or "the ratio of the circumference of a circle to its diameter", or even "one third".
This demonstrates that there is a problem we have with dividing magnitudes, which has not yet been resolved.
Quoting fishfry
Let me return your attention to this remark. If you agree with me, that the representations are "imperfect" from the start, then why not agree that we ought to revisit those representations. Constructing layer after layer of complex systems, with the goal of covering over those imperfections, doing something bad to cover up an existing bad, is not a solution.
You're a funny guy.
You: The moon is made of green cheese.
Me: Actually scientists think it's made of dirt and rocks and stuff.
You: Ok good, now that we agree the moon's made of green cheese ...
Man why you do me like this?
Quoting Metaphysician Undercover
No. There are nonmeasurable sets. Some mathematical objects can be measured and some can't. I would never say that "a mathematical object is a thing that can be measured" since that's false, meaningless, and misleading. You just made it up and decided that I said it. You keep doing this. Why?
Quoting Metaphysician Undercover
I've told you half a dozen times already that:
* The length of the diagonal is the Euclidean distance between the points (0,0) and (1,1), which is [math]\sqrt 2[/math]. We can define this via a metric, which is what I just did. We can also define it in terms of measure theory. I gave the links to those subjects earlier.
You're hung up on the infinite decimal business but I've explained to you repeatedly that:
* The fact that the decimal representation of a number is infinite, tells us nothing about whether the number itself is essentially a finitary or infinitary object. For example 1/3 = .333... has an infinite decimal expression but it can be summarized as "a decimal point followed by all threes." That's a recipe to produce a arbitrary number of decimal digits of 1/3. Likewise there is a recipe to product the decimal digits of [math]\sqrt 2[/math]; as well as a recipe to produce the decimal digits of [math]\pi[/math].
* [math]\sqrt 2[/math] is computable; it has a finitely expressible continued fraction representation; and it lives in a finite extension of the rational numbers if one is an algebraist and doesn't like limits and infinite series. By these criteria, [math]\sqrt 2[/math] is a finitary object.
* You keep clinging to your mistaken belief, thinking that the rational numbers are good and the irrationals bad. This is a personal psychological condition that can be remedied with mathematical knowledge. If you so desire.
Quoting Metaphysician Undercover
LOL. My wonderment is that you consistently fail to engage with anything I say; repeatedly claim I said the opposite of what I actually said; and stubbornly cling to your misunderstood fractured math from high school.
You say "such a thing" as if [math]\sqrt 2[/math] is beyond the pale, whereas rational numbers are wonderful. You just made this up. Both classes of numbers are equally fake or equally real, depending on how you look at it. You don't want to engage with this point, go in peace then. I can't do any more for you.
Quoting Metaphysician Undercover
You keep repeating this without engaging with the fact that math says you're wrong. You keep repeating this over and over and over. I can't say anything beyond what I've already said many times.
Quoting Metaphysician Undercover
I would agree with that; except that utility is not the ONLY consideration. For pure mathematians, beauty and interestingness have higher virtue than utilility. Utility is for the physicists, and we know what THEY do with mathematics!! [They mangle the hell out if it for their own nefarious purposes].
Quoting Metaphysician Undercover
Yes, in general. But in this particular case, what you think is a defect is not. You're hung up on infinite decimals, but infinite decimals don't tell you anything about whether a number is rational or not. 1/2 = .5 = .49999.... It has TWO distinct decimal representions. That tells us nothing about the real number 1/2. It just tells us that decimal representation is a little buggy. Continued fractions are better. Turing machines are better. Infinite series representations are better.
Quoting Metaphysician Undercover
Yes all these generalities are wonderful but they're in service of a point that is wrong. Since [math]\sqrt 2[/math] is a finitary object, your general point doesn't apply here. There's nothing wrong with [math]\sqrt 2[/math] except your psychological block about it. Was a screechy math teacher mean to you? I can relate. I still can't do high school trig for shit because of my screechy trig teacher. She set my math development back years.
Quoting Metaphysician Undercover
For Christ's sake, knock it off with this point. You're absolutely wrong.
Quoting Metaphysician Undercover
No.
Quoting Metaphysician Undercover
Who will rid me of this meddlesome priest!!
Dude there is no badness. You had a bad high school math education -- not your fault, I'm sick at heart at the state of public math education -- but you refuse to move past it. You're just wrong on the facts.
Quoting Metaphysician Undercover
Please stop. You were wrong the first time, you're wrong the hundredth time.
Quoting Metaphysician Undercover
You are so full of yourself you won't stop to engage with the points I (and others) have made to correct your misunderstandings. [math]\sqrt 2[/math] is a finitary object. It only requires a finite amount of information to completely specify its decimal digits. Why won't you acknowlege this fact?
Quoting Metaphysician Undercover
I am not responsible for what people type into Wikipedia. Some math articles are very good, some are highly misleading.
In ZFC there is nothing called an object. There are only sets; and sets are an undefined term. ZFC consists of a collection of axioms about how an undefined operator called [math]\in[/math] behaves. You can verify this by checking any university or graduate text on set theory.
Once again you are giving the high school definition. It's confusing you.
Quoting Metaphysician Undercover
Nonsense. Set theory precedes objects. We use set theory to construct numbers, functions, matrices, topological spaces, and all the other "objects" of mathematics. An object literally is some gadget we construct out of sets. And sets are undefined. Nobody knows what a set is. We have private intuitions, but set theory itself supports no preferred interpretation.
I'm speaking sophisticated math to you and you just want to cling to what they told you in high school. That's your choice.
Quoting Metaphysician Undercover
I'm explaining to you what sets are, from the point of view of the mathematical discipline of set theory. You don't want to get it, fine by me. And you're right, we're pretty much at a point of completion here. I've made every point I have to make at least half a dozen times. I'm happy to abandon this thread.
When I started, because of your arrogance and certainty and wordiness, I thought perhaps that I was missing some subtle philosophical viewpoint.
Instead it turns out that you're just stuck on some psychological discomfort with what you learned (badly, and again not your fault) about the square root of 2 and its decimal representation.
Having satisfied myself that I'm not missing some subtle point of philosophy; I have turned my efforts to trying to educate you about mathematics. You don't seem to be receptive and now I'm just annoying you. So I'll happily withdraw from the conversation. I stand by everything I've written.
Quoting Metaphysician Undercover
Incredible. I went to great lengths to explain to you that mathematical equality is an expression of the law of identity. That's what my Peano axiom proof that 2 + 2 = 4 was all about. You didn't even engage.
Once again you have imputed to me a position that is the direct opposite of the one I expressed.
Quoting Metaphysician Undercover
I did. The passage of time. As the great physicist Max Planck said: science advances one funeral at a time. What he meant was that the old experts are not convinced by the new methods. Rather, the old guys die off and are replaced by a new generation that has grown up with the new ideas. That's how we came to accept rational numbers in the first place, and then irrationals.
Quoting Metaphysician Undercover
I already did, at least three times. Are you denying history? Read up on the history of negative numbers, zero, rational numbers, real numbers, complex numbers. I keep explaining this to you and you keep avoiding engaging with the point.
Quoting Metaphysician Undercover
You're so tied up in words you can't think straight.
Quoting Metaphysician Undercover
The truth is what you learned (badly) in high school or Wikipedia. Anything else is a lie. Whatever dude.
Quoting Metaphysician Undercover
There is no technical term called an object in set theory. It's something you made up. Sets don't contain "objects." They contain only other sets, if they contain anything at all.
Quoting Metaphysician Undercover
You're embarrassing yourself. Your mathematical philosophy is unsophisticated because your knowledge of math is nil. You are psychologically stuck to your wrong ideas and you're incapable of engaging substantively with any point that anyone makes.
Quoting Metaphysician Undercover
Whateva whateva. I hope I'm getting to the end of this soon. This is my last post to you. I'm a fool if I continue to engage.
Quoting Metaphysician Undercover
You are really good at writing stuff that sort of sounds intelligent, but doesn't hold up to scrutiny. That's why I was initially interested in your posts and why I took the trouble the read them. Now I've scrutinized them. You're ignorant about math and wrong on the philosphy.
Quoting Metaphysician Undercover
That doesn't even make any sense. It's a collection of words that seems to convey a coherent argument about something but simply doesn't.
So now you don't believe in 1/3? I think you just refuted yourself.
Quoting Metaphysician Undercover
I'm out of steam.
Quoting Metaphysician Undercover
Dividing magnitudes. So now you don't believe in rational numbers either. We are making progress. That's right! Rational numbers are just as fictional or just as real as irrational numbers.
Could understanding be dawning?
Quoting Metaphysician Undercover
There's nothing bad to cover up. The bad thing is some misinformation you got stuck with in high school or earlier. You have to let go of things you think you learned that don't happen to be true.
I learned a lot talking with you. Mostly I learned that I know a lot more about the philosophy of math than at least one person on this site who claims to know a lot about the philosophy of math. It's taken me years to get to this point. Thank you.
Peace, friend.
K-theory, Category theory, etc. might enforce this view. I remember years ago hearing a well-known algebraist joke that, "K-theorists will tell you, "All you have to do is believe me and I can prove it!'." However, moving up into more abstract or general levels with new definitions and relationships, while simplifying certain aspects of math below those levels, may or may not solve complicated problems at lower levels. For example, "Soft" analysis doesn't solve all the problems "Hard" analysis presents. I am well aware of this having done research in the latter. On the other hand, moving higher up, greater generality, in a subject can be wonderfully rewarding, and it certainly provides avenues of imaginative research for grad students. The lower level stuff has frequently been "mined out" and what remains can quite difficult.
However, this is a side track, unimportant in this thread. :nerd:
I mean: it's clear that finding the right definitions it's not all. And it's extremely conceited to say "I'll tell you what the whole mathematics is about!". But there's something true in what I wrote, and I wanted to see if somebody agrees without spending too much time to explain what I mean :smile:
I guess nobody replied to this one because everybody thought that it doesn't make sense at all :joke:
No, I most definitely would not want that, and I've already explained why. I don't think it's a good idea to do a second bad thing to cover up an original bad thing. So you'd have to demonstrate to me first that the original thing which I consider to be bad (irrational numbers), is not really bad, with reference to solid ontological principles, rather than referring to what I called the second bad, which is just a cover up of the first bad. I have no inclination to learn the cover up, call it a psychological condition if that makes you happy.
Quoting fishfry
I am arguing against accepted mathematical principles. How is "math says you're wrong" any sort of a counter argument? Of course math says I'm wrong, that's a given.
Quoting fishfry
OK, math says I'm wrong, but Wikipedia says you're wrong. Now we're even, both wrong.
Quoting fishfry
Fishfry! Get with the program, wake up and smell the coffee! We've engaged in these discussions for weeks now, you know it's pointless to speak sophisticated math at me. You're wasting your time, we're discussing philosophy on this forum, not sophistic math.
Quoting fishfry
Ok, so as time passes, a "funny gadget" is magically converted into a "mathematical object". Tell me another one.
Quoting fishfry
No, I got it from Wikipedia, someone else made it up. But how is that any different from your "funny gadgets", which someone makes up, and through the passage of time magically turn into mathematical objects?
Quoting fishfry
Actually, it's you who has not engaged in any of the substance of my post, and has regressed to ad hom, and repeated insistence of "your wrong".
Quoting fishfry
Huh, I don't see any evidence of that scrutiny, only repeated assertions, "you're wrong", "you're wrong", you're wrong".
Quoting fishfry
Right, we've been through this already they are fictions, like your "funny gadget". But in logic we look for consistency, along with the capacity to fulfil the purpose. Why would a geometrical system produce a distance which is impossible to measure? How is this consistent with the purpose of geometry?
Quoting fishfry
I thought we were talking about fictions. How is truth relevant?
Quoting fishfry
I never claimed to know a lot about philosophy of math. I didn't even know there is such a thing. I've been arguing ontology. No wonder we're on different pages.
I love coffee. My favorite part of the day is my morning coffee. I can't wait to wake up tomorrow morning and smell the coffee. Often I grind my fresh artisinal beans and then bring the container to my nose, inhaling the aroma. Ah, coffee. Nectar of the Gods.
I hope you will not mind too much if I refrain from commenting on other topics. If I said anything it could only be what I've said before. Little would be gained in further punishing the keys of my laptop.
Quoting Metaphysician Undercover
Ignorance as a debating point. "Your argument stands refuted because I'm incapable of understanding it." I can't top that.
Seems we have something in common.
Quoting fishfry
The problem is that you do not address the substance of the argument. You go off on some tangent using mathematical jargon, without addressing the issue.
I could respond but what would be the point? It is a logical truth that IF you believe in the rational numbers then you must necessarily believe in the rational numbers augmented by the square root of 2. It's a simple logical procedure to go from one to the other. You want to complain that this is a sophisticated mathematical argument. Actually it is. But I've just explained it in a most understandable way. You needn't follow the details of the procedure. What matters is that there is one. You can literally build a square root of 2 out of the rational numbers. I have in fact outlined the procedure a couple of times already.
I can't expect you to follow mathematical arguments. I am simply making you aware of the existence of these arguments.
Your preference not to engage with mathematical arguments does not give you the right to deny that such arguments exist. You don't have to follow the algebraic details. You do have to understand that from the standpoint of pure logic, the correctness of the rationals implies the correctness of a number system that includes the rational augmented with the square root of 2.
Else you really are trying to use ignorance as a weapon. "I don't understand it so don't waste your time explaining it to me," is acceptable if lame. But "I don't understand it therefore it's false," I hope you can see is not a sensible argument at all.
The construction procedure you described is never ending, just like the never ending digits. How is that any different? Without reaching the end, you have no definite solution, an approximation of something unresolved.
No, you're thinking of something else. I'm talking about the algebraic construction. Which I'll outline.
* Say you believe in the rationals. You believe in the rationals? Ok.
* Now arbitrarily make up a symbol, call it [math]\sqrt 2[/math], but bear in mind that this is an entirely arbitrary symbol that has no meaning. It's just some squiggles I type in.
* Now consider the set of all formal expressions [math]a + b \sqrt 2[/math]. Again, these are just marks on paper. They have no meaning.
* We can define "addition" on these expressions componentwise. So
[math](a + b \sqrt 2) + (c + d \sqrt 2) = (a + c) + (b + d) \sqrt 2[/math]
* Likewise we can define the "product" of two such expressions using the everyday distributive laws. FOIL if you learned that awful acronym designed to replace understanding with mindless drudgery. God I hate what passes for math education. My friend @Meta your mathematical ignorance is not your fault. I blame your teachers and the textbook committees and the educrats of your high school years.
* Now the set [math]\{a + b \sqrt 2 : a, b \in \mathbb Q \}[/math] happens to have a mathematical structure identical to that of the rationals; and in addition, it contains a square root of 2.
* If one then objects that these are "only formal symbols," well after all what are rational numbers but formal symbols that obey rules? And in fact we can go further and construct, out of bits and pieces of set theory, a mathematical structure that is the set-theoretic implementation of this set of symbols.
Fact: If you believe in the rationals, you must believe in the rationals augmented by the square root of 2.
You want to make some kind of distinction that the "rationals are actual" in some sense. But they're not. They're just as fictional.
ps -- You make a mathematical claim, "Sqrt 2 doesn't exist." Then you reject any mathematical counterargument. You can't lose that way, but you can't convince anyone else.
If you are going to make a mathematical claim you need to be able to accept a mathematical disagreement.
:gasp:
LOL
It appears like we need to go back over the law of identity, and the difference between identical and equal. Remember, I don't accept set theory on the basis that it violates the law of identity, so why give me a proof based in a set?
Quoting fishfry
Why do you believe this? "The square root of two" has no valid meaning in the rational number system. This means that taking a square root is not a valid operation. So your claim is like saying if you believe in the rationals then you believe in the rationals augmented by a tree. You can't augment a system by something which is inconsistent with the system, that creates a contradiction, or at best, meaninglessness.
Square roots are a problem in mathematics, as is demonstrated by "imaginary numbers". At first glance, it appears like a square root is simply the inversion of the power of two. But the power of two is a valid procedure whereas the square root is not. If we define "square root" as the inversion of the power of two, then we'll find many numbers which simply do not have a square root. Why not accept this as a natural fact of numbers, rather than trying to force a square root onto these numbers?
Quoting fishfry
So long as you spell out your premises, and they are acceptable, I'm good to go. But it's already been proven that the square root of two is not a rational number. Why flog a dead horse? Why not move on, and inquiry what this principle tells us about numbers and spatial relations, instead of trying to disprove it.
I am curious to know: do you have an answer to this question?
You want to make a mathematical claim (sqrt 2 doesn't exist) but you won't accept a mathematical response. Makes for pointless conversation.
Quoting Metaphysician Undercover
I have already explained to you at length that set theory is based on the law of identity; and that the mathematical equals sign expresses identity between two expressions.
Define "valid operation." You should have been around to make your current argument about 1700BC when the Sumerians were calculating the square root of two (and its reciprocal) on cuneiform tablets. They would have appreciated your perspective. :smirk:
I think there are two issues becoming evident. One is that we do not know how to properly represent space. The irrational nature of the "square", and the "circle", as well as the incompatibility between the "point" and the "line" indicate deficiencies in our spatial representations.
The other is that we do not know how to properly divide something. There is no satisfactory, overall "law of division", which can be consistently, and successfully used to divide a magnitude. We tend to look at division as the inversion of multiplication, "how many times" the divisor goes into the dividend. Because there is often a remainder, division really cannot be done in this way. The "square root of two" is amore complex example of this simple problem of division, the issue of the remainder.
Quoting fishfry
Your solution involves a violation of the fundamental laws of logic, the law of identity (as explained on the other thread), therefore I reject it. My argument is that the problem is fundamentally an ontological problem, and the objective ought to be to resolve the problem with principles which are ontologically sound.
Quoting fishfry
As I demonstrated in the other thread, the "identity" expressed here as "equals", is not consistent with "identity" as expressed by the law of identity. Therefore despite your claim that set theory is based in "identity", it uses a form of identity which is in violation of the law of identity.
To state the problem succinctly, set theory allows that two distinct things have the same identity, in the same way that we might say two distinct things are equal. The faulty premise is that things with the same value "2" for example, are the very same thing. In other words set theory premises that, "2" refers to an object, rather than a value assigned to an object. It is a category mistake to treat what "2" refers to, as a particular object, rather than as a universal principle.
Quoting jgill
A valid operation is one carried out with consistency according to consistent laws of a system. The example of imaginary numbers, as well as the various different attempts to prove the square root of two, demonstrate that there is a lack of consistency to the square root procedure.
At the time, I responded thoughtfully to your ideas. You never once engaged with the points I made. Now weeks later you're still repeating your claims without ever having responded to the points I made. It's not productive to engage with you.
Well, the "issue" of the irrationality of the diagonal of the square is the one that ancient greeks recognized: you cannot find any unit length that enters both in the side and in the diagonal of the square an integer number of times (no matter how little you take your unit length).
So there cannot exist any fundamental minimal length of physical space (kind of a microscopic indivisible stick) that can be oriented in any direction. If there is such a thing, every physical object at the microscopic scale should be made of tetrahedrons, or something similar. So circles and squares are really only approximations of the real "physical" shapes. Is your idea something of this kind? If not, in what other way can you make all the lengths be rational numbers?
If this is the idea, I think the problem with this kind of physical theory is that all laws of physics are expressed in terms of differential equations (even the ones that describe "quantized" entities), and if quantum mechanics is right, it doesn't even make much sense to speak about an exactly determined physical length: physical space appears to be much more weird than a simple 3-dimensional geometric structure.
Quoting Metaphysician Undercover
You mean that there is no defined physical procedure to divide a generic geometrical segment by another? If you take two generic segments of whatever length, you can always build a third segment that is proportional to their ratio (whatever it is, even irrational). That's in Euclides' elements. Can't be this counted as division? If not, what do you mean by "law of division"?
Sorry for the intrusion, but I am curious of this issue (only one premise: I didn't study philosophy :yikes:, so, for example, I don't really understand why this "law of identity" is so important...).
However, that's my question: how do you refer to an object instead of to it's value? I mean: if every symbol refers to a different object, even if the symbol is the same as the one that you used before, you can never refer to the same object twice, can you?
No, that does not follow. The irrationality of [math]\sqrt 2[/math] is a purely mathematical fact. It tells us nothing about the physical world.
1. Call Build_Side(N) the physical process of putting N sticks in line one after the other, along the side of a square. N is a natural number (abstract mathematical object), but the process of putting N sticks in line is a real, physical experiment.
2. Call Build_Diagonal(M) the physical process of putting M sticks in line one after the other, along the diagonal of the same square.
Try to find M and N such that the sticks arrive at the same point. Since M/N is irrational, you can't do it, and the physical process of trying to build the square withe the sticks cannot be realized. ( well, OK, you have to build two sides of the square and the diagonal at the same time, but you get the what's the point! )
All physical measurement is approximate. You can't have a physical stick of length 1. It's not only impossible, it's meaningless. There is no physical apparatus in the world, even in theory, that could do any better than to say that "The length of the stick is 1 +/- .00005 with 99.343% certainty. I'm making up the numbers but that is what the nature of physical measurement is: a number, an error tolerance, and a probability that the true value is within the tolerance.
In the real world you can't measure the diagonal and you can't measure the sides. You can't measure anything with absolute precision. In classical physics, you can't but God can. In quantum physics, even God can't. To clarify that: in classical physics, we can't possibly measure the exact length of a stick, but at least in theory the stick does have a specific length. In quantum physics, a stick has no length at all until we measure it; at which point, the classical problem of the inexactness of physical measurement kicks in.
You are right, we will never be able to check if this theory is correct with absolute precision, not even in principle, because all physical measurements must necessarily have a limited precision.
Nevertheless, in principle (if you have enough computing power and the model is complete and consistent - I know, that's a big if) you can use the mathematical model to make predictions about the result of experiments with arbitrary precision.
So, in a model of the physical world where all distances have to be multiple of a given fixed length (I don't know if such a model exists, but let's assume this as an hypothesis), there cannot be squares
made of unit lengths. I don't know what these unit lengths are made of: they are simply the building blocks of my model, the same as the "strings" of string theory or the "material points" of Newtonian mechanics!
By the way, to be clear, I don't believe in this theory! :smile:
I do not think that "there cannot exist any fundamental minimal length of physical space" is a reasonable starting point. If space has physical existence, then it has limitations just like any other physical things. So we ought to assume that space must have some fundamental "shapes" just like you suggest. Once it was believed that space is an aether, so the fundamental shapes were waves.
A wave is active, so it requires the passing of time, for its activity. So let's assume "space" is an active medium. Now suppose we try to make something static, like a circle or a square, within this medium which is active. The shape won't actually be the way it is supposed to be, because the medium is actively changing from one moment to the next. So if we want to make our shape, (circle or square), maintain its proper shape while it exists in an active medium, we need to determine the activity of the medium, so that we can adjust the shape accordingly. Understanding this activity would establish a true relationship between space and time, because defining this activity of space would provide us with a true measure of time.
Quoting Mephist
What I am talking about specifically, is dividing numbers. Divide ten by three, and you have a remainder of one. It is the remainder which is a problem. When we multiply numbers we never get remainders, yet we tend to treat division as the inversion of multiplication. It's actually quite different from multiplication because multiplication starts from premises of fundamental base units, whereas division presupposes no such base units.. So I think we need to pay close attention to this fact, that constructing a magnitude through multiplication is really a completely different process from destroying a magnitude through division.
Quoting Mephist
I don't understand how you would build an irrational length segment.
Quoting Mephist
What the law of identity says is that a thing is the same as itself. This puts the identity of the thing within the thing itself, not as what we say about the thing, or even the name we give it. This is a fundamental ontological statement about what it means to be a thing. First, to be a thing is to have an identity (but this is irrelevant to the identity we give the thing, it is the identity that the things has by virtue of being the thing that it is). Second, a thing is unique, and no two things are alike, and this is the principle Leibniz draws on. So the law of identity is not concerned with how we refer to objects, it is a statement concerning the real existence of objects, as the objects that they are, independent of what we say about them.
You never thought I ever had it, that's why you'd tell things like get back on your meds. Is the fact that you keep addressing my post with such nonsense evidence that you're losing it?
OK, but I don't understand how all this can be related to irrational numbers.
Quoting Metaphysician Undercover
Division between integers is repeated subtraction ( A/B you count how many times you have to subtract
B from A to reach 0 ); multiplication between integers is repeated addition ( A*B you add A B times starting from 0 ).
The definitions are quite symmetric between each-other. What do you mean by "division presupposes no such base units"? OK, A/B is not an integer ( there is a reminder ) if A is not a multiple of B. Again: what does this have to do with physical space-time?
Quoting Metaphysician Undercover
By using compass and straightedge (as described by Euclides) you can build all the lengths that can be obtained from the integers with a finite sequence of operations of addition, subtraction, multiplication, division, and taking square roots (https://en.wikipedia.org/wiki/Straightedge_and_compass_construction). Square roots are not so special from this point of view.
Quoting Metaphysician Undercover
Hmmm... :worry: maybe...
Quoting Metaphysician Undercover
OK
Quoting Metaphysician Undercover
OK, I translate this as: you can always distinguish a thing (meaning: physical entity) from all the other things. Not quite true in quantum mechanics, but let's assume it is.
Quoting Metaphysician Undercover
OK, but when you give a name to a concrete object, the name is a reference that identifies always the same concrete object, isn't it?
Anyway, my main objection to what you say is that you don't explain how to use the fact that square roots are irrational (some of them) to deduce something about physical space-time. A physical theory in my opinion (even if limited) should be falsifiable in some way (meaning: should be usable to predict that something should happen, or that something else can't happen). And if it's not physics but only mathematics, then there should be some kind of logical "proof". Don't you agree?
The problem of irrational numbers arose from the construction of spatial figures. That indicates a problem with our understanding of the nature of spatial extension. So I suggested a more "real" way of looking at spatial extension, one which incorporates activity, therefore time, into spatial representations. Consider that Einsteinian relativity is already inconsistent with Euclidian geometry. If parallel lines are not really "parallel lines", then a right angle is not really a "right angle", and the square root of two is simply a faulty concept.
Quoting Mephist
Ok, we can look at division as a matter of asking how many times we can subtracting B from A, as you say. The issue is that in many cases one does not reach 0, and this is what we call the remainder. So the problem is, how do we deal with the remainder. If we are dividing ten by three, we get a remainder of one. In this case, you might divide the unit into three. But in most practical circumstances, if you were dividing a group of objects, it would be unfeasible to split up one of the objects, rendering it useless. So the remainder is very often a problem in division.
Quoting Mephist
No, division and multiplication are not at all symmetrical, because you never have a remainder in multiplication. In multiplication, you take a designated number as the "base unit", a designated number of times, and you never end up with a remainder. You have no such "base unit" in division, you have a large unit which you are trying to divide down to determine the base unit, but you often end up with a remainder.
Evidence of this difference is the existence of prime numbers. These are numbers which we cannot produce through multiplication. We can still divide them, knowing there will be a remainder, but that doesn't matter, because there's often a remainder when we divide, even if the dividend is not prime.
Quoting Mephist
On paper you produce "a representation" of the Euclidean ideals. That representation is something completely different from the square root, which is part of the formula behind the representation which you draw on paper. When I want to lay out a square corner, a right angle, on the ground, I might use a 3,4,5, triangle. In this exercise I am not using a square root at all. I could make this square corner without even knowing the Pythagorean theorem, just knowing the lengths of 3,4,5. But if one side of the right angle is to be 5, and the other side 6, I'll need to know the Pythagorean theorem, and then figure the diagonal as the square root of 61 if I am going to make my right angle.
Quoting Mephist
That's not quite right. We, as human beings, cannot necessarily distinguish two distinct things, due to our limited capacities of perception and apprehension. So it's not quite right to say that you can always distinguish a thing from all other things. A thing is distinct from other things, but we cannot necessarily distinguish it as such. And that difference may be a factor in quantum mechanics.
Quoting Mephist
Right, but to perceive a thing, name it "X", and then claim that it has the "identity" of X, is to use "identity" in a way inconsistent with the law of identity. You are saying that the thing's identity is X, when the law of identity says that a thing's identity is itself, not the name we give it. The law says a thing is the same as itself, not that it is the same as its name.
Consider that human beings are sometimes mistaken, so it is incorrect to say "the name is a reference that identifies always the same concrete object". The meaning of the name is dependent on the use, so when someone mistakenly identifies an object as "X", when it isn't the same object which was originally named "X", then the name doesn't always identify the same concrete object. And, there are numerous other types of mistakes and acts of deception which human beings do, which demonstrate that the name really doesn't always identify the same concrete object, even when we believe that it does.
Quoting Mephist
Do you recognize that Einstein's relativity is inconsistent with Euclidian geometry? Parallel lines, and right angles do not provide us with spatial representations that are consistent with what we now know about space, when understood as coexisting with time. My claim is that the fact that the square root of two is irrational is an indication that the way we apply numbers toward measuring space is fundamentally flawed. I think we need to start from the bottom and refigure the whole mathematical structure.
Consider that any number represents a discrete unit, value, or some such thing, and it's discrete because a different number represents a different value. On the other hand, we always wanted to represent space as continuous, so this presents us with infinite numbers between any two (rational) numbers. This is the same problem Aristotle demonstrated as the difference between being and becoming. If we represent "what is" as a described state, and later "what is" is something different, changed, then we need to account for the change (becoming), which happened between these two states. If we describe another, different state, between these original two, then we have to account for what happens between those states, and so on. If we try to describe change in this way we have an infinite regress, in the very same way that there is an infinite number of numbers between two numbers.
If modern (quantum) physics demonstrates to us that spatial existence consists of discrete units, then we ought to rid ourselves of the continuous spatial representations. This will allow compatibility between the number system and the spatial representation. Then we can proceed to analyze the further problem, the change, becoming, which happens between the discrete units of spatial existence; this is the continuity which appears to be incompatible with the numerical system.
.
If you mean electron microscope photos of a lattice of atoms, those are still subject to the quantum and classical measurement problems. To clarify what I said earlier:
* In quantum theory, nothing has an exact position at all. Before it's measured, it doesn't have a position. Sometimes that's expressed by saying that it's in a "superposition" of all possible positions. Then when you measure the particle, it (somehow -- nobody understands this part) acquires a position drawn randomly from a probability distribution.
This applies to all objects, large and small, though the effect is much more pronounced when an object is small.
For example you yourself are where you are in space right now because that's the most likely place for you to be. It is statistically possible that you might suddenly find yourself in a statistically improbable place. For example all the air molecules in your room could move to the corner of the room and you'd have no air. That is extremely unlikely, but it has a nonzero probability. It could happen.
So even if all instances of a given particle are the same, you still have no idea exactly where it is, or exactly how long a line made up of these particles is.
Atoms, by the way, are way too large and they're all different. I don't even know if two hydrogen atoms are exactly the same.
However it's interesting that every electron in the universe is (as far as we know) exactly the same. Why is that? It's another thing nobody understands.
There's a famous theory of Wheeler and Feynman, "not to be taken seriously" but evidently mathematically possible, that the reason all elecrons are the same is that there is only one electron in the universe. It moves rapidly backward and forward in time; and that's why whenever we see it, it appears to be in a different place. Like a point moving up and down from below a flat plane to above it. Every time it crosses the plane, you'd see an instance of the point. You'd think there are lots of points, when in fact it's only one point traveling perpendicular to your reality.
https://en.wikipedia.org/wiki/One-electron_universe
* And even in classical physics, a measurement is only an approximation.
So now I'd like to re-ask your question but pertaining to electrons, which are all exactly the same. But electons are very small and extremely subject to quantum effects. You simply can't say exactly where an electron is at any time. Only where it's statistically likely to be. One, because nothing is exactly anywhere at all in quantum physics; and even when it is, after a measurement, the measurement itself is subject to classical approximation error. You made the measurement in a particular lab with a particular apparatus, built and operated by humans. It's imperfect and approximate from the getgo.
The most accurate physical theory in the world, Feynman's quantum electrodynamics, has predicted some quantity or other to 12 decimal places. I read that somewhere. 12 decimal places is pretty good. But mathematically, it's not exact at all. If you had 12 decimal places of pi it wouldn't be pi.
Let me add that modern physics no longer thinks about "particles" like electrons and atoms. Rather, everything is interacting probability waves. An electron isn't a pointlike thingie. An electron is a probability wave, smeared all over the universe. When we observe it, we find that it appears to be in a particular location defined by a probability distribution.
There's even a current thread on this site on that very subject. "Fieldism versus materialism." We don't have particles or things or objects anymore. Just probability waves. Very strange, what the wise physicists are up to lately.
https://thephilosophyforum.com/discussion/7414/modern-realism-fieldism-not-materialism/p1
So the short answer to your question is, no. We can never know or measure an exact distance in the physical universe.
Quoting Mephist
Well there are no computers with arbitrary precision. That's the problem with the computational theory of the universe. There's too much it can't account for.
It's those pesky noncomputable numbers again, one of my favorite topics. If the universe is "continuous", in the sense that it's modeled by something like the real numbers; then it is most definitely not a computer or an algorithm. Because algorithms can't generate noncomputable numbers.
Quoting Mephist
Yes ok. I happen to have visited a world like that once. Manhattan. It's composed of a grid of mutually perpendicular streets and avenues. (Not entirely, but for purposes of discussion).
How far is it from 1st street and 1st Avenue to 2nd street and 2nd avenue? Well it's not [math]\sqrt 2[/math], because you can't drive or walk diagonally through the buildings. Rather, the distance is 2. You have to walk one block north and one block west.
There's a name for this: The taxicab metric. In the taxicab metric, the unit circle is a square. Next time some philosopher tells you there are no square circles, you can go, "Oh yeah? There are in the taxicab metric!" and thereby confound him.
But you know I still don't agree with you about squares. Of course there are square blocks in New York City. Actually they're rectangles because the streets are closer together than the avenues, but let's ignore that for sake of discussion. There are square blocks. You just can't walk along the diagonal! Your distance is the sum of your vertical and horizontal travel.
So in your hypothetical world there would be squares and if you want to go from (0,0) to (1,1) you simply have to move 2 units, one unit right and one unit up. You can't travel along the diagonal because at the finest level of the lattice, you can't move diagonally. I have no idea what that means physically but I think you are overthinking this or underthinking it. It's kind of tricky, which is a problem for the theory.
Quoting Mephist
Does the taxicab metric help your thinking?
Quoting Mephist
Some people do! There are some discrete or quantized theories of reality around, like loop quantum gravity. From the article: "The structure of space prefers an extremely fine fabric or network woven of finite loops."
The ultimate nature of our physical world is wide open to speculation, informed and otherwise. Even our wisest don't know.
But I don't speculate about the physical world. Math is so much simpler because it doesn't have to conform to experiment! In math if you want a square root of 2, you have your choice of mathematically rigorous ways of cooking up such a thing.
I see your point.
You are saying that there are 2 books and two fish and 2 schools of thought; but there is no 2 in the abstract.
Well, imagining or mentally conjuring up a "thing" that is 2, by itself, is one of the greatest intellectual leaps of humanity. As I've noted before, you appear to reject the very concept of abstraction.
The invention of the concept of number was a great leap forward for mathematics and also for civilization. That let us study 2 + 2 without having to say 2 fish plus 2 fish and then having to re-calculate 2 elephants plus 2 elephants, and then still not being sure about 2 birds plus 2 birds.
It's the power of abstraction that allows us to handle all these cases at once.
You reject abstraction. You're not wrong. It's just a nihilistic philosophy of math and of civilization. Everything about our lives is abstraction. We can't live without abstraction. How do you live without abstraction? How do you function, not believing in numbers?
Now if you want to say, "Yes but you admit numbers aren't real, they're only an abstraction!" I respond: Yes exactly. And traffic lights aren't real either, they're only an abstraction. Law is an abstraction. Government is an abstraction. Science is an abstraction. The Internet is an abstraction. Humans have the power of abstraction. It's how we crawled out of caves and built all this.
Let me restate your quote as a formal argument.
P: The square root of two" has no valid meaning in the rational number system.
C: This means that taking a square root is not a valid operation.
The conclusion doesn't follow from the premise. A valid conclusion would be, "This means that taking a square root is not a valid operation in the rationals. And of course that is correct.
One can, however, conceive of and build, with logical correctness. systems of numbers in which there IS a square root of 2.
I have an example on my mind, I'll toss it out there.
Do you know modular arithmetic, or the "integers mod 5" and so forth? Telling time is just the integers mod 12. If it's 11 now, what time will it be in 2 hours? The answer is 1. We add "mod 12," which means first do normal addition, then subtract out the largest multiple of 12 we can. In fact you've alluded to this phenomenon. When we divide two integers we get a quotient and a remainder. In modular arithmetic, we only care about the remainder.
We can do the same trick with any modulus, as it's called. Consider the integers mod 7. They consist of the symbols 0, 1, 2, 3, 4, 5, and 6, with addition and multiplication mod 7.
In the integers mod 7 we can add, subtract, and multiply. In general we can't divide. So the integers mod 7 are a perfectly valid system of numbers, not unlike the integers, but not like them either. [They're a quotient of the integers if you took abstract algebra].
Now, what is 3 x 3 in the integers mod 7? Well, 3 x 3 = 9 normally; and in the integers mod 7, the number 9 corresponds to the number 2.
So 3 x 3 = 2. That is, 3 is a number that, when squared, results in 2. So in the integers mod 7, 3 is the square root of 2. Just to startle people I'd go as far as to write
[math]3 = \sqrt 2[/math]
Like every statement in math, its truth value depends on the context. In the context of the integers, the statement is false. In the context of the integers mod 7, the statement is true.
[By the way what about -3? Well in the integers mod 7, we have -3 = 4. That's because 3 + 4 = 0. Now 4 x 4 = 16 = 2, after we've subtracted off 14. So even in the integers mod 7, it's true that if x is a square root of something then so is -x. That's a general rule that you can deduce just from the laws of basic arithmetic. If you took abstract algebra, we're talking about the ring axioms].
Now that's interesting, but it doesn't solve the problem of having a square root of 2 that also knows about the rationals. But there's a perfectly good number system called [math]\mathbb Q[\sqrt 2][/math] that is:
* A number system where we can add, subtract, multiply, and divide (except by 0); and
* It contains an exact copy of the rational numbers; and
* It contains a square root of 2.
There's no question that such an object exists in mathematics. It has mathematical existence by virtue of the fact that we can (1) characterize it axiomatically; and (2) construct it out of bits and pieces of set-theoretic operations. And even though you don't like set theory we can do the same thing in category theory or homotopy type theory or without any foundation at all simply by writing down the ring axioms and modding out the ring of polynomials having integer coefficients, by the ideal generated by the polynomial [math]x^2 - 2[/math]. I know you don't like technical stuff by I'm pointing out that I don't need set theory to build a square root of 2.
Now if we are talking about mathematics; and an object has mathematical existence, by what right do you require some other standard of existence?
By the way in the integers mod 5, we have
2 x 2 = 4 = -1.
So the integers mod 5 have a square root of -1.
Indeed, and one needs the abstraction of itself abstraction in order to complain about abstraction in the first place.
Quoting fishfry
Now those are strange entities, unlike the essentially finite square root of 2 (as you've already noted.) A dark ocean of infinitely informative numbers that can't be named is far more poetic and disturbing than little old [math] \sqrt 2.
Especially considering the algebraic approach that you presented. I find it intuitively satisfying without considering set theoretic foundations. And the Dedekind cut is satisfying if one can admit sets of rational numbers (intuitively self-supporting, IMO).
In Eastern Europe, software engineers studying for bachelor's degree have real analysis, abstract algebra, differential equations, etc, as mandatory subjects. That much is indeed true and many are dissatisfied with the curriculum for being too math heavy.
Right, my argument is that there is no such thing as an abstract object represented by "2". I replace this supposed object with something closer to the truth, "what 2 means". The symbol "2" has meaning. The key point I make, which some argue against, is that the meaning of "2" varies according to circumstances, context. This variance, or difference, indicates that it is impossible that "2" signifies an object. The Platonic realist argues that this is a difference which doesn't make a difference, but in making this argument, the realist has already invoked contradiction. This contradiction supports the realist's category mistake of failing to distinguish between a particular object, and a universal (meaning).
Quoting fishfry
I would say that you call this a "leap forward" as determined in relation to a pragmatist perspective. This move serves a purpose. But in relation to a true ontology, it is a blatant falsity put forward for a purpose. Therefore it is a move of deception. The pragmatist, from the perspective of the metaphysician, is a deceiver, a sophist.
Quoting fishfry
Such a study is the study of meaning, it is not a study of objects. When it is presented as a study of objects it is a deceptive presentation.
Quoting fishfry
I do not reject abstraction, I take it for what it is, and that is not a process of creating objects, it is a process of generalization. You don't seem to understand what "abstraction" means. Fundamentally, abstraction replaces particulars with generalizations, universals. If the universal (the product of abstraction) is presented as a particular (object), what is proposed is clearly a false proposition.
Quoting fishfry
This is not true. It's false to claim that 3 is the square root of 2, just because you've taken seven away, just like it would be false to claim that one o'clock in the afternoon is the same as one o'clock in the morning, just because you've taken twelve hours away. You've just presented a mathemagician's trick, pure sophistry.
Quoting fishfry
Yes, context is key, as I stated above, and the importance of context is reason why a numeral cannot refer to a number which is an object. The numeral has a meaning which is context dependent. The numeral "2" has a different meaning in mod 7 from the meaning it has in the rational numbers. In your example, you are conflating two distinct concepts. Your use of "the square root of two" is right out of context, because "2" has a different meaning in mod 7. Therefore your use of "2" is irrelevant to any common use of "the square root of two", because "2" has a different meaning in mod 7, like "one" has a different meaning as "one o'clock" from the meaning it has as a rational number. If you deny this, you arguing by equivocation.
Your objections apparently boil down to a demand that mathematicians revise their well-established technical terminology (existence, object, etc.) because some of the definitions conflict with those employed in your peculiar metaphysics. Good luck with that!
Do I strike you as a person who expects people to do what I suggest?
I'm not sure what this analysis. I have never heard a criticism of finitism that it is complex for applied applications. The whole point of finitism is that it aligns with practical application.
Engineering is not based on the real number continuum, it is largely based on differential equations that can be setup just as easily in a finitist framework.
Since I am concerned with high-school in this thread, can you give an example of an applied high-school level problem that cannot be addressed in a finitist framework of "arbitrary precision"?
As far as I know, all applied maths problem have precision constraints of their input data which results in precision limits of the output data of the algorithm (whether machine or human) solving the problem.
Even if there was a theorem that has no proof in the finite regime but does have a proof in the infinite regime (that we cannot prove the theorem for arbitrary precision, but we can prove the theorem for "all integers" or "all real numbers"), it is easy to borrow that theorem in a rigorous way by simply having the computer check the theorem up to some limit that we intend to use.
(Indeed, lot's of "theorems", i.e. conjectures, are used in applied math that have no pure math proof because they have been checked numerically to over the range in question; obviously, such a numerical check to some bound has nothing to do with the real number line.)
And this is how, in practice, applied maths work; people look up a theorem and use it, and if it works to solve the problem then that's the end of the thinking on that.
Of course, understanding what math actually is, is to understand proofs. But my whole point here is that simply positing the real numbers without a construction nor framework of rules that contains the paradoxes that otherwise occur with a naive approach, leads to a mystification of maths and not understanding of rigor. If the setup isn't rigorous, it is not a proof students are learning, but rather the applied method of "we're doing it because it works".
Quoting simeonz
My whole point is that students are not actually understanding the mathematical analysis if they do not actually understand real numbers. For me, simply positing the definition of real numbers in terms of some basic rules, does not lend any understanding of what real numbers are.
To repeat what I answered above to SophistiCat, the real numbers are not required to define convergence. Without the real numbers, convergence is to an arbitrary precision rather than a real number on the real number line. Arbitrary precision is perfectly adequate for any real world problem.
Because of this, the radical finitists desire to get rid of the real number system all together even at the university level. The ultra finitist dispute even "arbitrary precision" which in principle goes up to numbers with complexity that cannot be represented in the real world.
I am neither a finitist, much less ultra-finitist, for higher level pure maths. The real number system is, at the least, an interesting mathematical idea that lot's of effort has gone into developing and lot's of theorems are proven in the framework of ZFC that we have no reason to just throw away, and for me, pure maths is about the general question of "systems of symbols and rules" whatever they maybe, and so working with systems of rules that have unintuitive consequences is a good thing for the student of mathematics, as it opens the minds as to what can be done with this general "rigorous proof" based on rules and symbols.
However, for students encountering calculus for the first time, understanding the real number system is essentially impossible and a waste of time to attempt in anycase. In my experience (and it seems the experience of many posters here), teachers at the high-school level don't understand the real number line anyways and simply change from finitist "arbitrary precision" explanations while dropping in tidbits of the bizarre characteristics of normally distributed infinite decimal representations at best, and at worst provide wrong answers. I would wager most teachers and most students understand the real number line as just "numbers with decimals", such as the calculator provides, which is not the case; the calculator provides integers and fractions in decimal representation.
However, if we want to get into the discussion of the limits of this "numerical regime" approach in applied maths, it seems to be everywhere in practice.
For instance, I have always understood re-normalization in quantum physics to be exactly this "get rid of the infinities through the numerical regime" by simply measuring things and replacing divergent functions with constants. The justification of calculations outside the bounds of the reference experiments is basically a numerical regime game of "how far can the error be from this measured constant over here".
Likewise, the invention of "quanta" was due to abandoning the real number line and simply having a finite step (a numerical approach) which got rid of ultraviolet divergence and reproduced experimental result.
If physicists were not committed to a real continuum at this time, this would have simply been the obvious approach to define some "accuracy step" and then narrow in on the right value that matches experiment. And this is basically the argument of finitism in physics: there is no real continuum and so using the real numbers causes the confusions of the above kind, slowing theoretical advancements. I'm not sure if this causes confusions or not for physicists, but I have never heard a counter criticism of some example of a prediction that cannot be made without the real number line. As I mention, I'm not a finitist in higher maths, but if there's a criticism that some physics can't be done with finitist framework I have yet to hear it. The counter argument, is that keeping the real numbers around makes everything easier for both historical and abstraction reasons; just like in principle we could do physics without imaginary numbers, but no one advocates that because it would be clearly more confusing and harder to do physics without them.
As far as I know, physicists at the highest level do not need the real number line for making any prediction nor any theory, and it's simply historical accident that classical theories where developed with the motivation of a "no gaps" continuum, and it's important for physicists to learn these classical theories.
Ultimately, physicists do not justify the theorems they use with pure-maths proofs, but rather they borrow from pure maths "whatever works" and justify that in relation to experiment. Hence the "shutup and calculate" motto of modern physics (sometimes a pure maths proof extends the theory in a way that makes both intuitive sense and matches experiment ... and sometimes not, in which case no bother we'll just ignore that or say the theory breaks down at those energies). Which is why, as far as physicists I've talked to, this issue about real numbers they simply don't care about; it won't change how they calculate to get answers (unlike imaginary numbers, which would change a whole bunch of things and they would "care" about an argument to get rid of imaginary numbers, as it's clearly insane to do so).
However, regardless of whether a physicist thinks real numbers are a help or not, high school students, the subject matter of this thread, are neither learning rigorously about real numbers nor the numerical regime and, if what they are doing is not rigorous then it is not really understanding what mathematics is.
Therefore when a physicist makes the observation x = 0.14 +/- 0.0001, he could be equally described as stating an interval of rational numbers or as stating an interval of real-numbers. If this sounds wrong, "because the real numbers are uncountable, whereas the rational numbers are countable", recall Skolem's Paradox that the set of real-numbers actually possesses a model in which they are countable. The only important thing to know is whether the physicist fixed the result or whether he measured the result, for constructing a certain number is different to measuring an uncertain number - this difference isn't easy to express in either classical or constructive mathematics.
For real numbers, I consider two interpretations to be pedagogically correct. The first - algebra over totally ordered equivalence classes of Cauchy sequences/processes. This is the concrete/applied way to interpreting them. The fact that those equivalence classes have order and algebra defined over them does not imply that they stand on equal footing with the approximating elements of the sequences themselves. But nonetheless, they do act amorphously enough to allow us to think of them as "quantities". We call them numbers, but we also call complex numbers such, and they are not even totally ordered. So, to some extent, it is just a matter of nomenclature.
We deal with algebras on equivalence classes of Cauchy sequences, because we are interested in the convergence properties of the sequences, as well as how convergence interacts with transformations of various kinds - i.e. whether functions are continuous or not, whether operators are defined when acting on maps for those classes, etc. But ultimately, it still boils to our interest in the concept of convergence, not the reals for the sake of the reals. The real number in this sense is just a specification of the approximation process, whose behavior we need to analyze. Specifications can be, but need not be physically represented.
Alternatively, the notion of a real number from abstract algebra is one of a complete ordered field. Ultimately, it is the same concept. The properties are the same, except that the approach to the investigation is leaning more heavily towards non-constructivism. Which is fine, because this is what abstract algebra is all about. In fact, in some sense, the abstract definition is the proper definition, and the constructive one serves as an illustration. The latter is pedagogically necessary, but once understood, is not essential anymore.
As I said before, I cannot see how the study of complete ordered fields, being equivalent (up to isomorphism) to the ordered algebra of equivalence classes of Cauchy sequences can be replaced with something else, without reducing the scope of the theory. You either have analysis of your objects, or ad-hoc usage, but the latter is just a trial and error.
Yes indeed. I think of them as the "dark matter" of the real number line. We can't name them, we can't compute them, we can't use them for anything. But without them, there aren't enough reals to be Cauchy-complete. The reals lose their essential property. And without dark matter, the galaxies would fly apart. There are more things in heaven and Earth, Horatio, / Than are dreamt of in your philosophy -- Shakespeare.
Please explain this to @Metaphysician Undercover! I've had no luck.
I first read about them in Chaitin, before I had the training in math to really understand. It was clear even then that the real numbers had a certain magnificent unreality or ideality. When I studied some basic theoretical computer science (Sipser level), I saw the 'finitude' of now relatively innocent computable numbers like pi, which are of course countable. The measure of R is 0 without those unnameable numbers. Each is an oracle answering an infinite number of yes/no questions unpredictably. So does one believe in them? Within the mainstream game, of course! As pieces of the game that are one of its strangest features.
Quoting fishfry
Right! And there are more things in mathematics that are dreamed of by outsiders. I have taught math, and I occasionally hint at a world of strangeness awaiting those who wade in more deeply. It's a poetic enterprise, though one has to learn the grammar and spelling before one can understand the poetry.
Quoting fishfry
I know! I've followed your conversation. We human beings are sometimes stubborn as mules. We don't always want to know. Sometimes we'd prefer to 'win' an argument, however alone we are with that sense of victory. It's basically ridiculous to do philosophy of math without training in math: sex advice from virgins, marital advice from bachelors.
I always follow your posts. You know much more set theory than me, so I learn something. Though I agree that the rational numbers are synthetic, I find them intuitively satisfying enough to function as a foundation from the which the reals can be built as cuts or Cauchy sequences. I like cuts for not being equivalence classes. It's an aesthetic preference. Cuts are beautiful ('liquid crystal ladders').
I don't know, but I'm familiar with that excellent analogy.
Actually, yes. Dedekind cuts are another constructive approach. Not too different in spirit, I would say.
Quoting jgill
Rational numbers are actually quite nasty, if you want to work with them in computations. They are pleasant, if you are performing a finite number of arithmetic operations. But assuming a "fraction" representation, once you start evaluating some recursive formula, the numerator and denominator become unmanageable quickly. I am not sure how fast the periodic part in the repeating decimal representation grows, but I wouldn't want to work with that either. That is why software uses only fractions with denominator in exponent form (*) and represents the rest approximately. Correspondingly, algebraic numbers, and even computable transcendentals, are not that bad, when compared to arbitrary fractions.
I am only stating this, because rationals are looked upon so favorably, but I find that their simplicity is somewhat overstated.
* finite digits after the decimal point in appropriately chosen base;
One can construct the positive real numbers as a simpler version of the cuts. In the version I like we have a ray of rational numbers that starts from zero (a subset of Q that is closed downward with no maximum.) I like this for its intuitive connection to magnitude/length. The square root of 2 in this system of positive reals is just the rationals whose square is less than 2. The analogy is strong. We hold up some piece of Q^+ as a ruler, and we get all that Q itself leaves out. (I'm sure you already know this, so I'm just hyping the charms of this construction for the intuition. )
IMV, the rationals are quite difficult. We know how students hate fractions. But I like the idea of 1/n as a kind of flexible unit. Then m/n is just m of those units. We can adjust n to increase or decrease resolution. And we can do various conversions. So it's difficult but still (after much work and thinking) ultimately intuitive. At least for me.
This is exactly the geometric interpretation that got me into trouble. :) It assumes that rays have points corresponding to every non-negative real number (or lines have points corresponding to all real numbers.) To which, I remember my brain screamed, how do you know? Of course, if this is just analytic geometry, it would be true by construction, but then the argument becomes circular. So, I was asking essentially, how do we know that lines/rays, as they appear in real life, are complete. They could be, or they might not be, but how would a mathematical textbook use something like that, that we know very little about (i.e. space), and which is not axiomatic in nature, and use it to define a mathematical concept. At least for me, it didn't work, and caused me some difficulty.
I suppose this might be related to the objection that the OP has.
P.S. What I mean is - the definition (edit: interpretation) works if the person reading it has the right attitude. But I am rejecting its use on methodical grounds, of being informal. (And untested.)
P.S.(2): The reason why I got so heavily stuck on this interpretation was not so much because it prevented me from making sense of the mathematics presented, but because I started to question what was the goal - were numbers lexical entities or geometric properties? What was that we were trying to define - quantities, computations, geometrical facts? How could we validate them? At this point I didn't know much about algebraic structures and axiomatic systems. I went completely on a pseudo-philosophical tangent and refused to learn anything whose methodological grounds I did not understand completely.
I understand. After all, this is the rationals' whole gimmick - they are dense. Of course, the finite decimal [s]fractions[/s] rationals I talked about earlier are also dense, and easier to compute. But they are not a field. So starting with integers, you cannot scale yourself arbitrarily.
The even more interesting thing (that's why I talked about atoms) is that this is true not only for elementary particles as electrons, but even for atoms (of any element), and even for entire molecules, and this has been verified experimentally. Two atoms in the ground state (https://en.wikipedia.org/wiki/Ground_state) are EXACTLY IDENTICAL (as mathematical objects in the mathematical model of QM) if the ground state is not degenerate (https://en.wikipedia.org/wiki/Degenerate_energy_levels).
The tricky thing to realize experimentally is to obtain a non-degenerate ground state for a complex object as an atom: very low temperature, external magnetic field, confined position in a very little "box" (usually a laser-generated periodic electromagnetic field). But this is possible, and in this state the whole atom is COMPLETELY DESCRIBED from by one integer number: the energy level.
In this state you can put a bunch of atoms one over the other, if they are bosons (https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate) and the theory says that you can have N IDENTICAL objects all in the same IDENTICAL place.
The result is obtained by purely mathematical considerations on objects made of complex number functions (the states are the eigenvalues of the system's wave function), but the effects predicted using a purely mathematical abstract model generate real physical predictions in the form of measurable quantities. That seems very strange if mathematical objects are only symbols subject to arbitrary rules. In some way, the rules that we invented for the symbols correspond exactly to some of the "rules" of the physical (real) world.
Quoting fishfry
Yes, but the indeterminacy is only for the product position * momentum, and not the position alone (for example an electron emitted from the nucleus of an atom has an indeterminacy of initial position of the size of the nucleus from which it was emitted). And the curious thing is that the wave function, if you want the path-integral over the trajectories to be accurate enough, must be described with a much finer granularity of space than the size of the atom. The wave equation works the best if it's defined on the (mathematically imaginary) real numbers (at least for QED). The renormalization of electron's self-energy (https://en.wikipedia.org/wiki/Renormalization) is a mathematical theorem based on a mathematical model where space is the real euclidean space (real in the mathematical sense: vector space defined on real numbers) (I know the objection: it works even on a fine-enough lattice of space-time points, if you make statistics in the right way, but the lattice of positions have to be much smaller of the wavelength of the electron - that for "normal" energies is comparable with the size of an atom).
Quoting fishfry
Yes, however in same cases, the system is symmetric enough that you can use analysis to compute the results instead of making simulations, so you can get infinitely precise answers, (such as for example in the case of hydrogen atom's electronic
orbitals) that however you'll be able to verify experimentally only with finite precision.
Quoting fishfry
Well, that was a simple example that doesn't have much sense as a real theory of physics (and I absolutely don't believe that it can be a good model of physical space), but it's still a mathematical model suitable to be used to make predictions on the physical space (well, you should say how big are the sticks: surely there are a lot of missing details). However, as a model, you can decide to make it work as you want: in our case, the squares made with sides of one stick can't have a diagonal (so, let's say, nothing can travel along the diagonal trajectory, as in the Manhattan's metrics), and big "squares" can have diagonals but can't have right edges, or straight angles.
Quoting fishfry
Yes, but in loop quantum gravity loops are only "topological" loops: they are used to build the metric of space-time, not defined over a given metric space.
Quoting fishfry
I agree with you on the square root of 2, of course! But I am not so convinced that mathematical objects are only cooked-up fictions not related to physical reality.
I will allow myself to interject, although the physics involved in your discussion appears beyond my competence. In any case, just because something is not physical, doesn't make it purely fictitious.
For a lame example, if I define a geographical location called "not in London", which has the property that any statement exclusive to the London area is untrue for that abstract location, and all other statements remain undefined, this area would not properly represent any part of the universe. Why? Because many statements can be made about the Universe that are not specific to any location, and remain true for "not in London" in practice, but are not included in my structure/axiomatic system. However, I am constructing "not in London" not to express specific knowledge, but to express my lack of specific knowledge. I am creating an abstract entity which expresses my epistemic stance.
This is the way I look at mathematical objects in general, and real numbers in particular. They can be physically represented, if they happen to be. But generally, they are specifications more so then anything. As all specifications, they express our epistemic stance towards some object, not the properties of the object per se. Real numbers signify a process that we know how to continue indefinitely, and which we understand converges in the Cauchy sense. Does the limit exist (physically)? Maybe. But even if it doesn't, it still can be reasoned about conceptually.
Good point. The rays I mentioned have 'melting tips.' In some ways we are just sweeping the problem under the rug. What I like about the Cauchy sequence approach is that the real number is like a program that spits out rationals (closer and closer together in the long run.) There's another version where positive real numbers are increasing sequences of rationals that are bounded above. These have their advantages. If we hobble ourselves and just think in terms of computable functions from N to Q (whose outputs get closer together), we get maximum clarity but lose most of the line. (And we have no more reals than rationals cardinality-wise.)
But maybe there's no perfectly satisfying way to capture our intuition of the line. And sometimes I'm tempted to just enjoy the axioms of R as descriptive of intuitions of space.
Quoting simeonz
Ah, the curse of being a philosopher! I feel you. For me it was slightly different. I read a boatload of philosophy of math before studying math. I was vaguely anti-foundationalist by the time I was learning real analysis. With basic real analysis (pre-measure theory and Riemann integral), my intuition felt more or less satisfied. As I learned measure theory, all the sets of measure zero were a bit of a turnoff. More and more equivalence classes. Blah. I don't mind them in algebra, but in analysis there like film on a bathtub that needs cleaning. I'm a lapsed intuitionist. Math only has beauty to the degree that it corresponds (at some anchored point) to intuition. '[s]God[/s] intuition created the integers. The rest is the work of man.'
Quoting simeonz
It does seem to depend on an intuition of space, a space independent of physics' space (an ideal space.) We can prove (for the intuition) the commutative law for multiplication by turning a rectangle 90 degrees. At some point I'd like to see how many of the axioms of the real number system can be intuitively supported this way.
Quoting mask
Quoting simeonz
:up:
Great discussion. I don't really know if this contributes much to it, but I want to throw it among people I'm interested in reading talk about maths.
Something I find very interesting about these structures (and maybe this is part of what you were alluding to with "non-constructivism" @simeonz?) is that they need not be derived from more fundamental stuff (like set theory) in order to be understood in much the same way as if they were constructed from a more foundational object. Nevertheless, how you stipulate or construct the object lends a particular perspective on what it means; even when all the stipulations or constructions are formally equivalent.
I remember studying abstract algebra at university, and seeing the isomorphism theorems for groups, rings and rules for quotient spaces in linear algebra and thinking "this is much the same thing going on, but the structures involved differ quite a lot", one of my friends who had studied some universal algebra informed me that from a certain perspective, they were the same theorem; sub-cases of the isomorphism theorems between the objects in universal algebra. The proofs looked very similar too; and they all resembled the universal algebra version if the memory serves.
Regarding that "nevertheless", despite being "the same thing", the understandings consistent with each of them can be quite different. For example, if you "quotient off" the null space of the kernel of a linear transformation from a vector space, you end up with something isomorphic to the image of the linear transformation. It makes sense to visualise this as collapsing every vector in the kernel down to the 0 vector in the space and leaving every other vector (in the space) unchanged. But when you imagine cosets for groups, you don't have recourse to any 0s of another operation to collapse everything down to (the "0" in a group, the identity, can't zero off other elements); so the exercise of visualisation produces a good intuition for quotient vector spaces, the universal algebra theorem works for both cases, but the visualisation does not produce a good intuition for quotient groups.
If you want to restore the intuition, you need to move to the more general context of homomorphisms between algebraic structures; in which case the linear maps play the role in vector spaces, and the group homomorphisms play the role in group theory. "mapping to the identity" in the vector space becomes "collapsing to zero" in both contexts.
There's a peculiar transformation of intuition that occurs when analogising two structures, and it appears distinct from approaching it from a much more general setting that subsumes them both.
Perhaps the same can be said for thinking of real numbers in terms of Dedekind cuts (holes removed in the rationals by describing the holes) or as Cauchy sequences (holes removed in the rationals by describing the gap fillers), or as the unique complete ordered field up to isomorphism.
I very much agree. For someone who insists on math being beautiful, it has to sing for the intuition. For example, when learning group theory I really liked thinking of groups of permutations. Those were the anchor for my intuition. The operation on the group was 'really' functional composition, which is why groups weren't automatically commutative. The theory doesn't care. Epistemologically it's a non-issue. But it matters for motivation. Another approach is just to understand math as a chess of symbol manipulation. In some contexts this is satisfying enough. But one gets lost in real analysis without intuition guiding the construction of a semi-formal proof.
If you consider geometric spatial figures as real physical objects, there are a lot of "problems" with them: first of all, they are 2-dimensional (or 1-dimensional, if you don't consider the internal surface), and all real physical objects are 3-dimensional. The usual "trick" to make some sense of this kind of model is that they are so small that are not directly observable. Einstenian general relativity is the same as Euclidean geometry in this respect: world lines are just a mathematical abstraction to represent trajectories in space-time. They are not real objects, and there is no problem with the distinction between finite or infinitesimal distances: it works even if you consider space-time as discrete. In fact, in practice it's very common in GR simulations to approximate space-time as a 4-dimensional discrete grid of points.
The main point to keep in mind with physical models is that they don't have to be considered the real thing: they simply have to WORK as the real thing. So you have to choose which characteristics (or properties, or attributes) of the model correspond to characteristics of physical real objects and which ones are only mathematical approximations. For GR, the trajectories are only abstract 1-dimensional "lines": what's important (measurable) is only their length, and the angle between them, but only up to a certain approximation. Physical bodies can be represented as "points", or "spheres", just to make calculations simpler: the small-scale details are not important for the model, so they don't correspond to anything "real".
Now, if you think that the distinction between measures expressed with rational or with real numbers is essential in your theory (represents some important characteristics of the real physical space), I don't see any other way other than making lengths become discrete at the microscopical level.
If this is not your idea, in which way the use of rational numbers instead of real numbers could make a difference? I know that you think that real numbers do not exist, but what's the difference if they exist or not, if your model doesn't care of what happens at the smaller scales?
I agreed very hard on this in my heart. Tutorials and seminars in abstract algebra mixed between people who preferred algebra and people who preferred analysis; it's a shock to the intuition whenever an algebra lover presents the group operation "the other way around", and vice versa. Seeing groups as transformations was how I imagined them; but as for imagining [math]\{\mathbb{R},+\}[/math] as a group in the same sense it didn't work; those intuitions were tied to quantities and magnitudes, but they happened to correspond to translations along the "real line" of a given length, and that intuition could be passed up to vector spaces of low dimension (magnitude + direction, parallelogram rule).
Quoting mask
In general I have found that working over formalisms is one necessary part of developing understanding for a topic; don't just read it, fight it. Follow enough syllogisms allowed by the syntax and you end up with a decent intuition of how to prove things in a structure; what a structure can do and how to visualise it. Those syllogisms aren't the whole story, the visualisation matters.
What I want to pick a bone with, though perhaps this is a misreading on my part or a difference in emphasis, is whether such intuition development (associating a mental image or a shorthand for forming expectations regarding a structure) is merely aesthetic. We're quite well trained to think of mathematical objects as formal objects, symbol pushing, or as physically rooted (or obversely grounding reality in mathematical abstraction), but what of the required insight to, as you put it, anchor the intuitions of a structure?
Developing such anchors and being able to describe them seems a necessary part of learning mathematics in general; physical or Platonic grounding deflates this idea by replacing our ideas with actuality or actuality with our ideas respectively. In either case, this leaves the stipulated content of the actual to express the conceptual content of mathematics. This elides consideration of how the practice of mathematics is grounded in people who use mathematics; and whether that grounding has any conceptual structure; how is actual mathematics understood by actual people and does that have any necessary structure? Put another way; what is the structure of the conceptual content of mathematics?
Whitehead alludes to something similar regarding philosophical projects:
Why should this background of mathematics remain a secret? And is it merely aesthetic in nature (a consideration of mathematical beauty alone)?
This looks cool, the statistician in me likes including uncertainty into the operations of arithmetic, but dislikes characterising uncertainty as the range of a set.
This is a good point to bring up. I'm tempted to call it formal intuition. I used it for all the epsilon-delta stuff in basic real analysis. A continuous function was (in that context) something associated roughly with a set of logical moves. It's almost like playing a guitar. A certain know-how kicks in. A person can sit in class and start imagining how something might be proved. This is kind of what I meant by symbol manipulation. In this mode, the stuff of the proof itself is the medium of thought.
Quoting fdrake
I like this. On of the experienced mathematicians I know liked the math is language metaphor. I find this metaphor deeper than might be apparent. What is it to know English? Perhaps to know math is just as complicated, even if epistemologically the situation is simpler. A stupid computer can check a formal proof, but the world of mathematics is certainly not just a set of formal proofs. Squishy Heideggerian insights are valuable even here. Knowing math is like knowing English is like knowing how to ride a bike. That knowledge is largely tacit. When I've taught math, I mumbled to my students something about intuition coming with practice. 'Keep calculating. Faith will come!' I liked Hersh's mathematical humanism: https://www.amazon.com/What-Mathematics-Really-Reuben-Hersh/dp/0195130871 Maybe you've read that. He gets that math is a kind of intersubjective human practice.
OK, division and multiplication are not symmetrical for integers, because integers are "quantized": you can't give one candy to three children, because candies are "quantized". But physical space is not quantized, or is it? The mathematical description of continuous measures is not inconsistent: there are several ways to make them at least as consistent as natural numbers are.
So, if integers (quantized) objects exist in nature, why shouldn't continuous objects exist?
[quote= Sartre]
[The] impossible synthesis of assimilation and an assimilated which maintains its integrity has deep-rooted connections with basic sexual drives. The idea of "carnal possession" offers us the irritating but seductive figure of a body perpetually possessed and perpetually new, on which possession leaves no trace. This is deeply symbolized in the quality of "smooth" or "polished." What is smooth can be taken and felt but remains no less impenetrable, does not give way in the least beneath the appropriative caress -- it is like water. This is the reason why erotic depictions insist on the smooth whiteness of a woman's body. Smooth --it is what reforms itself under the caress, as water reforms itself in its passage over the stone which has pierced it....It is at this point that we encounter the similarity to scientific research: the known object, like the stone in the stomach of the ostrich, is entirely within me, assimilated, transformed into my self, and is entirely me; but at the same time it is impenetrable, untransformable, entirely smooth, with the indifferent nudity of a body that is beloved and caressed in vain.
[/quote]
I like to contrast the mathematician and the novelist. The novelist seeks a personal immortality. As a novelist, I want to crystallize my own precious experience of reality. As a mathematician, I lose myself in the inter-subjectively available object. I can't sign or claim this knowledge, or not in the same way. The ostrich swallows a stone that it cannot digest (as Sartre describes it in another passage of 'Existential Psychoanalysis.')
The philosopher is somewhere between the mathematician and the novelist. That Sartre quote tries to capture in words what he takes for a universal experience, a general structure of experience that is otherwise particular. This is basically the eternal 'behind' time, the invariant that is constantly present if we can grasp it.
So is the 3,4,5 triangle really straight or not? I don't understand...
Quoting Metaphysician Undercover
OK.
Quoting Metaphysician Undercover
OK, the identity cannot be identified with the name.
Quoting Metaphysician Undercover
OK, so what can I do with identities?
If I cannot refer to them with names, I would say that it's impossible to speak about identities. So, they surely cannot be used in logic arguments. Logic is basically manipulation (operations) of language, isn't it?
Quoting Metaphysician Undercover
But Einstein's relativity is based on differential calculus and real numbers. How can it be correct, if the whole system is wrong?
Quoting Metaphysician Undercover
OK, continuous change cannot be identified by a finite number of steps. But does this prove that continuous change cannot exist?
Quoting Metaphysician Undercover
Quantum physics is based on Hilbert vector spaces, that are infinite-dimensional continuous vector spaces (even "more" infinite than the infinite 3-dimensional euclidean space). I don't know if there is a way to express the same theory with similar results on first approximation making use only of mathematics based on integral fields. But even if there is a way, I suspect that it would become an extremely complex theory, impossible to use in practice. Would it then be more "real" then the current theory making use of real numbers? At the end, the only way to decide which theory is more "real" in physics is only agreement with experiments.
You know, that is a very interesting point of view. As I said I find it nihilistic because you must then reject all of the modern world that sprung from that basic act of abstraction. You would be back at the time of the Greeks, who expressed everything as ratios but did not actually have a concept of number as such.
Now that you mention it, that makes perfect sense relative to your neo-Pythagoreanism. By that I mean that you still profess to be "Shocked, shocked, I tell you!" at the fact that the square root of 2 is irrational. The rest of the world got over that a long time ago.
So ok, your mathematical viewpoint is that of an educated person in the time of ancient Greece, say between Pythagoras around 2700 years ago and Euclid 2400.
Question: Is your outlook on the rest of the world similarly situated in the past? Do you shake your fist in the air at the notion of heavier-than-air flight? How far do your convictions go? Have you heard about heliocentrism? It's the latest thing.
I don't want to comment on the specifics of your post at the moment. I dropped in to ask you a question that occurred to me:
Question: I get that you do not believe in the ontological existence, however you personally define that, of [math]\sqrt 2[/math]. My question is:
Do you believe in the mathematical existence of [math]\sqrt 2[/math]?
If you say yes, then our disagreement is over whether mathematical existence is sufficient for ontological existence.
If you say no, then our disagreement is whether [math]\sqrt 2[/math] has mathematical existence.
So, do you think [math]\sqrt 2[/math] has mathematical existence; even though you maintain that's not sufficient justification to grant it ontological existence as you define it?
Yes, great insight. The mathematical real numbers are very strange. I always noticed that the more I learned about the real numbers, the more unreal they got. It's very doubtful to me that such a structure has an analog in the physical world. And if it did, it would be quite a surprise.
Mathematicians have a tongue-in-cheek saying: The imaginary numbers are real; and the real numbers aren't!
Quoting mask
It's an extremely widely held false belief that pi encodes an infinite amount of information, when it of course does no such thing. Bad teaching of the real numbers in high school is the root cause of this problem. Whether there is a solution that would serve the mathematical kids without totally losing everyone else, I don't know.
Quoting mask
Another fine point to which I've endeavored to draw @Metaphysician Undercover's attention from time to time.
Quoting mask
Thanks. I loved all that stuff in school. I couldn't do calculus integrals for beans, but I took naturally to Zorn's lemma. Did you know that the proposition that every vector space has a basis, is fully equivalent to the axiom of choice? Isn't that wild?
What a great topic. If you Google "constructive physics," you find a small but nonzero number of paywalled articles on the subject . I believe there's a book about it, too. In fact here it is, the whole book.
https://arxiv.org/pdf/0805.2859.pdf
The table of contents is an awesome read. He has definitely done his homework.
I had heard that, but never studied the proof. It is indeed wild.
At my school we only had to learn the set theory that comes with analysis and algebra. I did look into ordinals on my own. I remain impressed by the usual Von Neumann constuction. I used it in visual art and I also think it has a philosophical relevance. It's a nice analogy for consciousness constantly taking a distance from its history. 'This' moment or configuration is all previous moments or configurations grasped as a unity. It works technically but also aesthetically.
Quoting fishfry
That would be surprising indeed. I think we agree on the gap between math and nature. As you mention, our measurement devices don't live up to our intuition and/or formalism. I have a soft spot for instrumentalism as an interpretation of physical science.
Quoting fishfry
I haven't heard that one. But I know a graph theory guy who thinks the continuum is a fiction and an analyst who believes reality is actually continuous. Another mathematician I know just dislikes philosophy altogether. I like philosophy more than math when I'm not occasionally on fire with mathematically inspired, though I have spent weeks at a time in math books, obsessed. (At one point I was working on different models of computation, alternatives to the Turing machine, etc. Fun stuff, especially with a computer at hand.)
Quoting fishfry
Right! Because of the infinite decimal expansion. One of my earliest math teachers had lots of digits of pi up on the wall, wrapping around the room. Some of the problem may be in the teaching, but I've wrestled with student apathy. Math tends to be viewed as boring but useful, the kind of thing that must be endured on the path to riches. Its beauty is admittedly cold, while young people tend to want romance, music, fashion, fame, etc.
The mathematical system being employed premises that a symbol represents an object, and that each time the symbol appears within an expression, like an equation, it represents the very same object. Any conclusions produced must uphold this premise.
Quoting Mephist
Right, clearly there are "problems" if we represent mathematical figures as real objects. Notice I removed your qualifier, "physical" objects. If we begin with a statement as to the nature of an "object", a definition, such as the law of identity, then we must uphold this definition. If the claim is that a "mathematical object" is fundamentally different from a "physical object", such that the same definition of "object" cannot apply to both, then we need to lay out the principles of this difference so that equivocation can be avoided.
Quoting Mephist
The problem is that any such "grid of points" is laid out on a spatial model. If a square is an invalid spatial model, then so is the Cartesian coordinate system Then "space-time" itself is improperly represented. If the claim is that "space-time" is not supposed to be an object, then we have nothing being represented except mathematical objects, and no grounding principle, defining what a mathematical object is.
Quoting Mephist
Pragmatism is not the answer, it is the road to deception. Human objectives often stray from the objective of truth. When we replace "the truth" with "they simply have to work", we allow the deception of sophism, because "what it works for" may be something other than leading us toward the truth..
Quoting Mephist
Yes, as I explained, I agree that this is the way to go. Once we recognize that this is what's needed, we can proceed towards a proper analysis of space and time, to establish some principles.
Quoting fdrake
The secret background is the intent of the author. So long as the intent remains a secret, pragmaticism remains unacceptable because we do not understand the end. "It works" has no meaning when the end remains a secret.
Quoting Mephist
This is good, a very good start. Suppose there actually is this distinction in "objects". Suppose there are both "quantized objects", and "continuous objects". We would need different principles to apply to each one. Then we would need to establish some principles of application, are we working with a continuous object, or a quantized object. Thirdly, we'd need some principles to relate the continuous to the quantized. For example, to me time appears to be continuous, and space appears to be quantized. If this is the case, then we need different principles for modeling time than we do for space, and some principles to relate these two systems to each other.
Quoting Mephist
The 3,4,5, triangle is just as faulty as the square, because it is validated by the faulty Pythagorean theorem. The point was that I can make a 3,4,5 triangle without any knowledge of the Pythagorean theorem. So let's say I am in the habit of doing this, and I know absolutely nothing about the theory. I am producing right angles at will, and I believe that the right angle is a perfectly natural thing. Then I learn about the square, and realize that there is a problem with the right angle, and therefore it is not a natural thing. Likewise, I could tie a string to a stake, and make a simple compass, and go around creating circles at will, thinking that the circle is a natural object, until I realize the irrationality of pi. This tells me that these are not natural objects, they cannot exist.
So now I want principles to explain why I can make a figure which is theoretically impossible to make. I make excuses, I rationalize that I am not making a "perfect" square, or a "perfect" circle, and the theory says that such "perfect" figures are impossible. But there's something fundamentally wrong with this rationalizing. The theory is supposed to give me the "ideal", the perfect geometrical figure, and my inability to construct it ought to be due to my imperfect procedure. But here, what is indicated by the mathematics is that this supposed "ideal" is actually less than perfect, so that the more perfect the procedure is, all it does is demonstrate how much less than ideal the ideal is. Therefore I can conclude that the "ideal" is not the ideal at all, and there is a fundamental contradiction here which tells me that we need is a better, more ideal "ideal".
Quoting Mephist
Aristotle established and used the law of identity as a fundamental tool against the deception of sophism. So let's assume as you say, that logic is the manipulation of symbols, and it doesn't say anything about any real things. To say something about a thing is an act of description, and this is distinct from logic which works with symbols. Therefore if a logician is claiming to say something about real things, we can charge that person with sophism, deception. This is what we do with "identity" then, use it to demonstrate that someone is falsely claiming identity. So how do we approach set theory, doesn't it look like sophism, a false claim of identity, to you?
Quoting Mephist
Good question, I think the jury is out still on that decision.
Quoting Mephist
No, it does not prove that continuous change does not exist. But it proves that numbers, which represent change in finite steps, are the wrong tool for representing continuous change. This is where mathematicians demonstrate their stubbornness. They want numbers to be capable of representing everything, so they twist and turn the systems, adding layer upon layer of sophistication, mixed with deceptive sophism, and voila, numbers represent the fundamental reality, or even more extreme, numbers are the fundamental reality. It's like physicalism, or scientism, but it's more properly called mathematicism. and it appears to be the root of the other two. People think that mathematics is all we need to describe reality, when in reality mathematics cannot describe anything.
So let's revisit this root problem. Numbers, which demonstrate finite increments of difference, cannot properly represent continuous change. Either we assume that continuity is not something real, therefore numbers can be applied to all of reality, or we assume that continuity is real, and numbers cannot be applied to all aspects of reality. I suggest the latter is what is really the case. But then mathematicism looms menacingly in front of me, and I feel it my duty to demonstrate the sophistry of the mathemagician.
Quoting fishfry
This is not the case. To reject that "the abstraction" exists as an object does not require that I reject abstraction. What I reject is any instance where an abstraction is presented as an object.
Quoting fishfry
I'm not "shocked" at the fact that the square root of two is irrational, what shocks me is that the rest of the world got over this.
Quoting fishfry
I'd answer this, but I really don't know what you would mean by "mathematical existence". Many things can be expressed mathematically, but what type of existence is that? I suppose the short answer is no. The symbol ?2 does not stand for anything with real "existence". I agree that is has a large amount of mathematical significance, and it is quite important mathematically, so the symbol definitely has meaning, but I don't think I'd agree that the symbol stands for anything which has "existence", in any proper sense of the word. All existence is "ontological existence" so it makes no sense to try and separate "mathematical existence" from "ontological existence".
Mathematicians and philosophers of mathematics, with the presumed exception of platonists, reject the premiss that all "existence" is ontological existence. Specifically, they acknowledge that mathematical existence does not entail ontological existence.
In my experience, formal intuition works more like an open neighbourhood around a proof than of proofs themselves. There are essential details and inessential details for the understanding of a structure. The essential details are what enable you to generate expectations of how the structure behaves; envisage theorems and ask questions about it.
EG: "We can label set elements however we like, so functions can be interpreted as ways of permuting the labels of the elements on a set... I wonder if every collection of functions on a set behaves just like a set of permutations on that set?"
It seems you can write something like mathematical pseudo-code to suggest an intuition and play about with it, an example for the above:
Let's take the function like f(x)=x+1 on the natural numbers, envisage it as a list of pairs:
(1,2)
(2,3)
(3,4)
and so on.
And you can read that as "relabel 1 with 2, relabel 2 with 3, relabel 3 with 4" and so on. If you have a familiarity with disjoint cycle notation for permutation groups, you might think both "those look a lot like permutations" and that whole thing looks like the permutation (123456...) in the cycle notation, which is the original function in another representation. The procedure to generate this intuition didn't look to depend on much besides the choice of set.
There are a lot of syntactic details that facilitate every step of this "pseudo-code", sometimes (often) I can get them wrong and that blocks the intuition from working. Furthermore, the "essential details" as interpreted by me might not (usually don't) generate all the behaviour of the structure- IE enable me to expect every provable theorem as provable. (They also often make me expect things are provable when they are not.)
So the essentials of a structure look like "necessary highlights", in a sense they cover the structure in question for the purpose with all relevant detail. These "highlights" I think, ideally, map onto your anchor points for formal intuition. In such a case it seems to me that someone would understand the conceptual content of a mathematical structure (insofar as it is relevant to the concerns) well. If you have mastery over the domain, I imagine the essential details allow you to generate expectations for many provable theorems from the structure, and allow you to easily see if something is inconsistent with it.
I mean: OK, you can reformulate all current physics theories in a constructivist logical framework, but is the result really different from the normal formulation?
I like constructivist logical frameworks based on dependent type theory because of the simplicity of formal proofs: that's the thing that makes the difference! But form the point of view of a physical theory, the equations and the computations, and of course the results, are exactly the same! ( I didn't read the whole book, maybe I missed something, but at first sight, that's the way it looks like ).
From my point of view, that's only another way of "encoding" physical formulas and procedures using a different logical framework. Of course encoding is important, if it "refactors" the same concepts in a simpler way: that is basically what I would call a better understanding of the theory. You can even "encode" these theories using only natural numbers using Godel's encoding if you want (https://en.wikipedia.org/wiki/G%C3%B6del_numbering) to obtain an absolutely incomprehensible theory that gives the same results: that's NOT a good formulation of the theory :smile:
But in this case, the two formulations are completely equivalent (in the sense of equivalence of categories, if you see theories as functors from formal systems to models), and from the point of view of physics the choice between two equivalent representations doesn't make any difference. And probably, after Voevodsky, it doesn't make much difference even from a mathematical point o view.
Yeah well, let's say so... The way QM is formulated is: there are "observables" that represent the objects (or better: the results of experiments), and then there are other mathematical "objects" (such as wave functions- https://en.wikipedia.org/wiki/Wave_function) that are not meant to represent something that normally we could call "objects" in physics.
Quoting Metaphysician Undercover
Yes, exactly!
Quoting Metaphysician Undercover
Well, the grid of points represents only the topology of space-time, not the metric.
Meaning: there are 4 integer indexes for each point (the grid is 4-dimensional), and then the ordering of the points, only defines which are the pieces of space-time that are adjacent (attached to each-other), not their size or the orientation of the edges.
Then you have to associate to each point two 4-dimensional tensors (a tensor, basically, is a 4x4 matrix of real numbers - approximated as floating-point numbers with limited precision) (for picky mathematicians reading this: these are the components of the tensor relative to an arbitrarily chosen base, not the tensor itself): one is the metric tensor (https://en.wikipedia.org/wiki/Metric_tensor_(general_relativity), defining basically the size of this piece of space-time in each dimension, and the other is the stress-energy tensor (https://en.wikipedia.org/wiki/Stress%E2%80%93energy_tensor), describing the matter that is contained in this piece of space-time.
But this is still a very over-simplified description: for example, to perform the calculations you have to calculate the equivalent of derivatives of these tensors (that are even larger sets of numbers - https://en.wikipedia.org/wiki/Levi-Civita_connection).
And there is the tacit assumption that the topology of space-time is the same as flat space-time (that is true for relatively "small" systems such as for example the solar system).
Quoting Metaphysician Undercover
Well, I wouldn't call this "pragmatism". That has more that only a practical meaning.
I mean: you can call "natural number" anything that "works" in a similar way as a given system of symbols and rules (there are a lot of equivalent systems: for example Peano arithmetics (https://en.wikipedia.org/wiki/Peano_axioms, or the usual decimal symbols with arithmetic rules for operations as they teach in primary school).
So, the fingers of my hand are a number (the number 5), if I specify how to perform the arithmetical operations with fingers. The electrical charges inside a computer are numbers too, because the computer has encoded the rules of arithmetic as logical circuits.
So, basically, every "object" that is composed of well-identifiable parts can be considered to be a natural number, if you specify how to perform the arithmetical operations with the parts.
Quoting Metaphysician Undercover
Well, the problem is that what you say doesn't seem to be in any way "compatible" with current physical theories. And current theories are VERY good at predicting the results of a lot of experiments.
To me, it seems VERY VERY unlikely that a simple physical theory based on a simple mathematical model can be compatible with current physics at least in a first approximation. The physical world seems to be much more complex than we are able to imagine...
[ TO BE CONTINUED ANOTHER DAY..]
I've never looked at the book and have no interest in constructive physics. You'll have to forgive me. Personally I think it's a fool's errand. I have disagreements with constructivists. I'm well aware that various neo-intuitionistic foundations are in vogue, homotopy type theory and so forth. What I'm saying is that I'm totally unequipped to respond in detail to your points about constructive physics; not only by knowledge, but also by interest.
I also wanted to mention that I'm falling a little behind in my mentions and a lot of interesting points are being made lately. In fact you wrote me a great reply that I wanted to get to. You know a lot more physics than I do and I didn't realize atoms can be identical. I hope to respond to that post at some point.
Quoting Mephist
Are you saying that classical and constructive physics are equivalent as categories? I'm afraid I don't know exactly how you are categorifying physics. I used to read John Baez back in his loop quantum gravity days, and I didn't understand how he was applying category theory to physics either.
Perhaps you can clarify exactly what you mean here. If you mean that you get the same physics, yes of course that would be the point. If I'm understanding you correctly. You want to be able to do standard physics but without depending on the classical real numbers. So if that's what you're saying, it makes sense.
But as I say, I am a little skeptical about the rejection of noncomputability. My own belief is that the next revolution in physics will involve going beyond our current notions of computability. So I don't think the constructivists are going to win. And from what little I've seen, every flavor of constructive math these days at some point has to sneak in at least a weak form of the axiom of choice; and I believe that would have to be true in physics as well. Even the constructivists will allow a certain amount of nonconstructability; because it turns out to be necessary to get a decent mathematical theory.
Quoting Mephist
I'm not sure exactly what you mean by that but I'm not sure I want to know. Voevodsky did a lot of things. But generally I don't like to jump down the constructive rabbit hole (having spent enough time learning about it to satisfy my own curiosity) so please don't feel obligated to write more than a few words here, if any.
That's what abstraction is! It's giving a name to something immaterial in order to manipulate it. We see 2 cows and 2 pigs and 2 chickens and 2 barnyard metaphors. So we abstract a thing, called 2. You may object and say that you only mean 2-ness. But I say that's no different than declaring a thing called 2. You either believe in abstraction or not. I don't buy the distinction you're making here.
Quoting Metaphysician Undercover
That's very funny. I do see your point of view. Like I say it's nihilistic because you must therefore reject the entirety of the modern world. You say you have found a loophole that allows you to accept the world yet reject ... something. The square root of 2. What on earth have you got against the square root of two? Even the Babylonians chiseled a few digits of the square root of 2 into a rock. It is in some sense inevitable that people will crawl out of caves and figure out how to use fire, and invent the wheel, and build cities and do commerce, and discover the square root of 2. It's not something some set theorist made up. It's out there in the world. It has an existence independent of us if anything does. Somehow. It's mysterious, I agree. But you seem to just reject it on grounds that you haven't explained to me yet.
Quoting Metaphysician Undercover
Since you don't know any math, perhaps you will take my word for it. [math]\sqrt 2[/math] has mathematical existence.
Now to describe to you what mathematical existence is, I would have to do some math. Which you don't like. But [math]\sqrt 2[/math] has mathematical existence because:
* If we believe in the rationals, we can build a totally ordered field containing the rationals in which there is a square root of 2.
* We can then construct such a field within set theory. If you prefer we can do it in category theory. In fact we can do set theory within category theory, so if you don't like set theory there's a more fundamental theory we can build it out of. There are plenty of alternative foundations about these days. They're very trendy in fact. Since you hate set theory you'll be glad to know that in some circles, nobody cares about it any more.
But see now you've got me ranting math again and it's a waste of time because you don't like it.
So forget everything I wrote that you don't want to read, and just know this: [math]\sqrt 2[/math] has mathematical existence because I say it does; and if you were willing to follow a symbolic argument I'd prove it to you, in fact I already have several different ways.
Quoting Metaphysician Undercover
Mathematical existence. Something you know nothing about because you don't know any math.
Quoting Metaphysician Undercover
Yes I know you already believe that. The question is whether you're willing to believe that it has mathematical existence. You ask me what that is but I've given you many demonstrations of the mathematical existence of [math]\sqrt 2[/math]; as the limit of a sequence, as an extension field of the rationals, as a formal symbol adjoined to the rational numbers. All these things are part of mathematical existence. You will either have to take my word for it, or work with me to work through a proof of the mathematical existence of [math]\sqrt 2[/math] .
Quoting Metaphysician Undercover
How can you tell, personally, whether that's your deep philosophical mind talking, or just your mathematical ignorance? From where I sit ... well let me tell you that you didn't score any points with me when you totally ignored my beautiful demonstration that 2 + 2 = 4 assuming only the Peano axioms. You showed me that you have a psychological block in dealing with symbology; leading to a massive area of ignorance of math; leading to making large errors in your philosophy. That's my diagnosis.
Quoting Metaphysician Undercover
Well see THAT is something we can agree to talk about. Whether there is any difference. I think a rock exists in a way that [math]\sqrt 2[/math] doesn't. The latter only has abstract mathematical existence.
Like Captain Ahab, who only has fictional existence in a novel. Nevertheless statements about him can can truth values, such as whether he's the captain of the Pequod or the cabin boy. So there's fictional existence.
Do you think Ahab has fictional existence? I've been meaning to ask you that, actually. If you say yes, then you accept at least one "lesser type" of existence, so why not accept mathematical existence? If you say no, then how can statements about him have definite truth values?
The context here is the fact that the statement "Every vector space has a basis," is fully equivalent to the axiom of choice. Each implies the other. The proof is a simple application of Zorn's lemma, an equivalent of choice. Given a vector space [math]V[/math], you consider the partially ordered set of all linearly independent subsets of [math]V[/math], ordered by set inclusion. You then convince yourself that if you have an upward chain of set inclusions, the union of the chain is in fact also linearly independent, and is an upper bound for the chain. Then you apply Zorn's lemma to conclude that there must be a maximal linearly independent set; which must therefore be a basis; because if it weren't, you could add an element to it so it wouldn't have been maximal. QED.
I started writing a much longer and more elementary explanation of the jargon to make this more accessible, but it quickly got way too long. It's all on Wiki or just ask :-)
That's one direction, that choice implies basis. The other direction, that the statement that every vector space has a basis implies the axiom of choice, was actually proven only as recently as 1980 I think. That surprised me when I looked that up a while back.
Quoting mask
Right, that makes sense. Everyone needs the basics of unions and intersections and so forth but even most math majors don't ever take a course in set theory. I did because I was interested. I always had some kind of affinity for that stuff. Like I say I can rattle off a Zorn proof like that but when I see those crazy freshman calc integrals my brain freezes.
Quoting mask
That's a cool idea. I get it. In von Neumann you go up one step at a time, by successors and by limits. You're suggesting a continuous analog of that. Every moment in time is the union of all that's come before. That's good
Quoting mask
Yes. It seems obvious to me. But then again there's that pesky "unreasonable effectiveness." And so often, the physicists discover the math before the mathematicians do. So the relation between nature and math is different than the relation between nature and, say, chess. Math and nature are intertwined, but they're not the same.
Quoting mask
I had to look that up. You mean science is valid insofar as it's useful. I'd disagree. I like math for the sake of math and science for the sake of science. In fact a lot of the uses of science are far more evil than the intent of the scientists. Atom bomb and all that, arising from beautiful theoretical work on the nature of the universe. The pacifist Einstein inventing the physics that led to the bomb. One of history's ironies I'd say.
Quoting mask
Exactly. Some working practitioners of math and physics have philosophical opinions and most don't care at all. And among those with opinions, they're all over the place. Like everyone else I suppose.
Quoting mask
I've heard a little about that. Continuous TMs and the like.
Quoting mask
I'm afraid that even if you totally reformed math education in such a way that everyone loved it, they'd still prefer music, fashion, fame, etc.
Basically, what I wanted to say is that there is a "trick" in his kind of "constructivist" theory. For example, from page 55:
"As in the classical logic, we can add to intuitionism the axioms of arithmetic or of the set theory, which gives the constructive versions of these logical theories"
All the results are exactly the same, and all theorems are equivalent, only reformulated in a different way (encoding the rules of logic in a different, but equivalent way)
For physics, if the formulas are the same and the method to calculate the results is the same, there's no difference: the difference is only in non-essential mathematical "details" (from a physicist point of view).
Quoting fishfry
Well, basically category theory can be used as a foundational theory for physics. It's rather
"fashionable" today, here's an example: https://arxiv.org/abs/0908.2469
One of the advantages is that equivalent formulations of a given theory can be seen as the same theory: pretty much the same of what Vladimir Voevodsky did with homotopy type theory and his univalence axiom ( https://ncatlab.org/nlab/show/univalence+axiom ).
That's interesting. I never looked at universal algebra but what you describe is a lot like category theory. There's a construction called a "product" which particularizes to the Cartesian product of sets, the direct product of groups or rings, etc. There are also some surprises. The coproduct, which is what you get when you take the definition of the product and simply reverse all the arrows, is the disjoint union in the category of sets; and the direct sum in the category of Abelian groups. Without the abstract point of view you wouldn't necessarily realize that disjoint unions and direct sums are essentially the same thing in different contexts.
Quoting fdrake
I've always felt that the symbology is how we communicate, but the intuitions are private. Some people gravitate to one visualization or another.
Quoting fdrake
I think you have to see the particular examples before you're ready to absorb the abstract viewpoint that integrates and clarifies things. There's no easy way through it.
Quoting fdrake
I guess the original example of all this is Descartes's great invention of analytic geometry. A problem in geometry can be turned into an equation, solved algebraically, and the result transferred back to the geometry. That was a great leap forward.
Quoting fdrake
With the reals, I think most people think of them via their axiomatic definition as a complete totally ordered field. The constructions are incidental, serving only as a proof that if we were called upon, we could cook up the reals within set theory. But in practice we don't care about the construction; only the properties. We add, subtract, multiply, divide, take limits, etc. It doesn't matter what set of Tinker Toys we use to build a model of them. It's their behavior that matters; and that's the start of modern categorial or universal thinking.
I do not see how this can be. The constructive real line is not Cauchy-complete. It's only countably infinite. It does not contain any of the noncomputable numbers. It can not possibly be an intuitively satisfying model of a continuum. I'm troubled by this and I'm troubled that the constructivists never seem to be troubled.
Quoting fdrake
Yes I used to read his articles on Usenet about how he uses n-categories in physics. I was quite amazed, having only seen category theory years before in math classes and not realizing it had escaped into the wild. Now it's a big deal in computer science too. It's taking over.
Quoting fdrake
I think Vovoedsky's name gets used way too much in vain in these types of discussions. It's a perfectly commonplace observation that isomorphism can be taken as identity in most contexts. The univalence axiom formalizes it but informally it's part of the folklore or unwritten understandings of math.
But really, that's not my point about constructive math. I don't care if it gives the right theory. The constructive real line is full of holes. The intermediate value theorem is false. It is not a continuum. Doesn't that bother anyone?
And since physics is supposed to be about the world, this is the kind of thing that should matter a lot! That's my thesis, based on a my admittedly limited understanding of these matters.
Quoting Mephist
Ok that's beyond my pay grade, but maybe I can tell you what I know about it. Say you have a hydrogen atom, one proton and one electron, is that right? The electron can be in any one of a finite number of states (is that right?) so if you take two hydrogen atoms with their electrons in the same shell (is that still the right term?) or energy level, they'd be exactly the same.
But you know I don't believe that. Because the quarks inside the proton are bouncing around differently in the other atom. Clearly I don't know enough physics. I'll take your word on this stuff.
Quoting Mephist
I don't believe you. I do believe that you know a lot more physics than I do. But I don't believe that there is an exact length that can be measured with infinite precision. I'm sorry. I can't follow your argument and it's clearly more sophisticated than my understanding of physics but I can't believe your conclusion.
Quoting Mephist
Wait, what? You just agreed with me. The "real physical predictions" are only good to a bunch of decimal places. There are no exact measurements. The theory gives an exact answer of course but you can never measure it. You agree, right?
Quoting Mephist
All of this is quite irrelevant to whether we can measure any exact length in the world. Since it's perfectly well known that we can't, it doesn't matter that you have this interesting exposition. There's some QED calculation that's good to 12 decimal digits and that's the best physics prediction that's ever been made, and it's NOT EXACT, it's only 12 decimal digits. Surely you appreciate this point.
Quoting Mephist
Well of course. I agree with that. It's like saying that if I have a circle with radius 1 its circumference will be 2pi but of course in the real world we can't measure pi.
If that's all you mean, you have gone a long way for a small point. Of course if we have a theory we can solve the equations and get some real number. But we can never measure it exactly; nor can we ever know whether our theory is true of the world or just a better approximation than the last theory we thought was true before we discovered this new one.
I think we're in violent agreement here but I'm not sure that you're fully appreciating the point. You can solve an equation using the real numbers. But you can't measure it to be correct to infinite precision; and you can't know that your model is true.
Quoting Mephist
I'm not following this.
Quoting Mephist
Ok.
Let's put it in this way: what you call "noncomputable" in boolean logic should be called "nonspeakable" in constructivist logic: they are not part of the language. You cannot say anything about them. There cannot be disagreement about sentences that do not exist in one of the two languages: you can only disagree about what you can say in both languages.
So, if you don't need to speak about the things that you cannot speak about, there's no problem.
If you need to speak about those things (for example the incomputable real numbers) you can add their existence to constructive logic as an axiom, and that axiom is independent from the other axioms. So it cannot cause inconsistencies, or alter the results that you just deduced using only the constructivist part.
[ sorry, I have to go now: I'll continue this another time ]
This paragraph seems to bear on my conversation with @Metaphysician Undercover.
I can indeed specify [math]\sqrt 2[/math]; but when I do so I am merely "expressing my epistemic stance" toward [math]\sqrt 2[/math]; yet not necessarily saying anything about [math]\sqrt 2[/math] itself, whatever that means. I think this is @Metaphysician Undercover's point perhaps.
And then "expressing my epistemic stance" towards a mathematical object is what I mean by endowing that object with mathematical existence. Perhaps this is the distinction being made.
If I have no internal strategy for constructing a basis from my formalism of a vector space, then I am reliant upon nature sending me a basis, which I have no control over. But suppose nature never sends me a basis?
Arguments between constructivists and classical logicians are caused by a fundamental disagreement about the nature of proof. The former equates proofs with fully-determined algorithms under the control of the mathematician, whereas the latter allows proofs to interact with nature in an empirically-contigent and indeterminate fashion.
Unfortunately, classical logicians are usually in denial about what they are actually doing. Instead of admitting that their notion of proof is empirically contigent and not internal, they insist their notion of proof is internally constructive in a transcendental platonic realm.
Now according to the SEP's article on the Axiom of Choice , AC implies LEM in the presence of two further axioms, namely Predicative Comprehension (PC) and Extensionality of Functions (EF). The former says that the image of every predicate applied to individuals is a set, whereas the latter says that every extensionally equivalent pair of sets has the same image under every Set Function.
https://plato.stanford.edu/entries/axiom-choice/#AxiChoLog
The author of the article proves that PC & EF => ( AC => LEM), but he then argues that whereas PC is constructively valid, EF isn't. See the argument, but I don't find his argument about EF persuasive, at least as I understand it.
In my current view, EF is also constructively admissable, implying that the precise difference between classical logic and intuitionistic logic is AC as much as it is LEM, which then cements the view that classical logic describes a game between two players, whereas intuitionisitic logic describes solitaire.
A further motive for my view (and indeed the most natural motive), is that classical logic involves sequents of the form (a AND b) => x OR y, where it isn't known which of x or y is true, in which the negation of one implies the existence of the other. On the other hand intuitionistic logic only involves sequents with a single conclusion, of the form (a AND b) => x. Thus there is indeterminism in the case of classical reasoning, but not in the case of intuitionistic reasoning.
For me, when we construct mathematical models, we are factoring in uncertainty, efficiency, capacity (for control, measurement, computation, etc), and we concoct a useful approach to solving a practical problem. This does not mean that our solution is completely disconnected from the underlying reality, but it is not literal representation of the natural phenomena either. It is mostly practical and has many interesting properties on its own. Those properties determine how it behaves when we use it and even if they are not necessarily properties of the original system (although they could be), they are worth investigating if we plan on applying the solution many times and want to have understanding of its behavior.
So, to me, algebraic structures, real numbers in particular, are "ways" of dealing with problems. They are inspired by nature, but are not necessarily literally representative of natural objects. The specifications of those algebraic structures, the solutions based on those specifications, etc, have interesting, in some cases paradoxical properties, that are worth investigating simply because of our usage of those structures. If they happen to coincide with nature's geometries on some fine-grained level, this is particularly interesting, and bodes further investigation, but it doesn't affect the originally intended utility.
Bit about the univalence axiom was in one of @Mephist's posts, no idea what happened there.
Yes, that's one of the most interesting subjects even for me :grin:
Quoting fishfry
Yes, everything right until now.
Quoting fishfry
No, because even the motion of the quarks inside the proton is quantized, at the same way as the motion of the electrons is. If the proton is in it's base state (and that's always the case, if you are not talking about high-energy nuclear collisions), ALL that happens inside of it is described ONLY by an eigenfunction of the Hamiltonian operator with the lowest eigenvalue: it's a well-defined mathematical object. And all protons in their base state are described by the same function. No other information is required to describe COMPLETELY it's state (even if quarks were made of "strings" and strings were made of "who knows what"). What would change in case quarks were made of strings is that the Hamiltonian operator would have a different form, probably EXTREMELY complex, but the wave-function would be the same for all protons anyway.
Quoting fishfry
No, there isn't an exact length that can be measured with infinite precision. But you don't need to be able to measure an atom with infinite precision to check if two atoms are exactly identical: identical particles in QM have a very special behavior: the wave-function of a system composed of two identical particles is symmetric (if they are bosons) or anti-symmetric (if they are fermions) (https://en.wikipedia.org/wiki/Identical_particles). Because of this fact, the experimental result conducted with two identical particles is usually dramatically different from their behavior even if they differ from an apparently irrelevant detail.
For example you can take a look at this: https://arxiv.org/abs/1706.04231
Quoting fishfry
Yes, I agree. There are no exact measurements, but there are exact predictions in QM. For example, the shapes of hydrogen atom's orbitals are regular mathematical functions that you can compute with arbitrary precision: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydwf.html
Of course, you cannot verify the theory with arbitrary precision, but the theory can produce results with arbitrary precision (at least in this case).
Quoting fishfry
Yes, that's because physical experiments are more and more difficult to realize if you want more and more precision, and in this case ( the exact measurement of electron's magnetic moment ) even the computational complexity of the theoretical computation grows exponentially with the precision of the result. But in theory the result can be calculated with arbitrary precision.
Quoting fishfry
Yes, exactly. That's what I mean.
Quoting fishfry
Well, let's abandon the discussion about this "theory" ... :smile:
P.S. In my opinion, that's one of the most interesting aspects of QM: the information required to describe exactly an atom is limited: it's a list of quantum numbers, each of them is an integer in a limited range. So, just a bunch of bits.
If the equations of QM were the same as in classical mechanics (orbits of planets depending on their initial conditions without any limit to the precision of measurement), chemistry would be a complete mess: every atom would be different from all the others (as every planetary system is different from all the others)
Ontology is the study of existence. Isn't it? How could there be a form of existence which isn't ontological existence? That sounds very contradictory to me.
Quoting Mephist
Actually, if you analyze this situation closely, "wave functions" are produced from observations, so they are still mathematical representations of the movements of objects. The wave function is a use of mathematics to represent observable objects. There is no such separation between the representation of a physical particle and the wave functions, the wave functions represent the particles. They are of the same category, and I think the physicist treats the particle as a feature of the wave functions. Wave functions are used because such "particles" are known to have imprecise locations which they can only represent as wave functions. With observed occurrences (interactions) the particles are given precise locations. Wave functions represent the existence of particles when they are not being observed.
Quoting Mephist
I don't quite get this. "Space-time" here is conceptual only, like "the square" we've been talking about, or, "the circle". Therefore, the positioning of the points is what creates the "object" called space-time, just like we could position points in a Euclidian system, to outline a line, circle, or square. What is at issue, is the nature of the medium which is supposed to be between the points, which accounts for continuity. The continuity might be "the real", what exists independently of our creations of points according to some geometrical principles.
So, we create a grid of points according to a geometry of space-time, like we might position points according to Euclidian geometry. There is an assumed figure produced by this positioning like a square etc., and we might assume that the figure could exist as a real physical object. If the geometry we use to position those points, is not consistent with what is allowed for by the real positioning of objects, because the medium is not continuous, or if the nature of the continuity is completely misunderstood, then I would say that this is a problem. This is what we see in Euclidian geometry. The geometry allows that we can place a point virtually anywhere. But when we create the figures which connect the points, like a square or a circle, we see that there is a problem with this assumption, that we can put the point anywhere we want. Notice that the problem is with the conception itself, it has nothing to do with "the real". The idea that we can conceive a point anywhere is false, as demonstrated by the square root problem. Conceiving of continuity in this way, such that it allows us to put a point anywhere is self-defeating. Therefore we need to change our concept of the continuity of "space".
Now. let's add time to the mix. We already have a faulty conception of space assumed as continuous in a strange way which allows us to create irrational figures. Special relativity allows us to break up time, and represent it as discontinuous, layering the discontinuous thing, time, on top of the continuous, space. Doesn't this seem backward to you? Time is what we experience as continuous, an object has temporal continuity, while space is discontinuous, broken up by the variety of different objects.
Quoting Mephist
Your analogy is faulty, because what you have presented is incidents of something representing what is meant by the symbol "5". So what you have done is replaced the numeral "5" with all sorts of other things which might have the same meaning as that symbol, but you do not really get to the meaning of that symbol, which is what we call "the number 5". The point being, that for simplicity sake, we say that the symbol "5" represents the number 5. But this is only supported by Platonic realism. If we accept that Platonic realism is an over simplification, and that the symbol "5" doesn't really represent a Platonic object called "five", we see instead, that the symbol "5" has meaning. Then we can look closely at all the different things, in all those different contexts, which you said could replace the symbol "5", and see that those different things have differences of meaning, dependent on the context. Furthermore, we can also learn that even the symbol "5" has differences of meaning dependent on the context, different systems for example. Then the whole concept of "a number" falls apart as a faulty concept, irrational and illogical. That's why you can easily say, anything can be a number, because there is no logical concept of what a number is.
Quoting Mephist
Prediction is not a good indicator of understanding. Remember, Thales predicted a solar eclipse without an understanding of the solar system. All that is required for prediction is an underlying continuity, and perhaps some basic math. I can predict that the sun will rise tomorrow morning without even any mathematics, so the math is not even prerequisite, it just adds complexity, and the "wow' factor to the mathemagician's prediction. So, continuity and induction is all that is required for prediction. Mathematics facilitates the induction, but it doesn't deal with the continuity. Real understanding is produced from analyzing the continuity. This is an activity based in description, and as I mentioned, is beyond the scope of mathematics.
Again, we encounter the problem of pragmatism. If prediction is all that is required, then we gear our epistemology toward giving us just that, predictability. If this is easiest done using false premises like Platonic realism, then so be it. But we do this at the expense of a real understanding.
Quoting fishfry
I believe you do not have a very thorough education in philosophy, or you would not characterize "abstraction" in this way. Abstraction is a process. That process is sometimes described as producing a thing which might be called "a concept", or "an abstraction". There might be a further process of manipulating that thing called "an abstraction", but notice the separation between the process which is abstraction, creating the immaterial thing called an abstraction, and the process which is fixing a name to the supposed "immaterial thing" (an abstraction) and manipulating it.
To begin with, we need to analyze that process of abstraction, and justify the claim that an immaterial object is produced from this process. If there is no immaterial object produced, then the name which is supposedly given to an immaterial object, simply has meaning, and there is nothing being manipulated except meaning. But if you are manipulating meaning you stand open to the charge of creating ambiguity and equivocation. This is why we separate logic, which is manipulating symbols, from the process of abstraction which is giving meaning to those symbols. So it is very good to uphold this principle. In logic we manipulate symbols, we do not manipulate "something immaterial" (meaning) which the symbols represent. What the symbols represent is determined by the premises. The "something immaterial" (meaning) precedes the logic as premises, and extensions to this, as new understanding, may be produced from the logical conclusions, but what is manipulated is the symbols, not the immaterial thing (meaning).
Quoting fishfry
I don't say that I've found a "loophole", I say that there is weakness. And, it's not me who found this weakness, which is a deficiency, it's been known about for ages. You look at this deficiency as if it is a loophole, and insist that the loophole has been satisfactorily covered up. But covering a loophole is not satisfactory to me, I think that the law which has that deficiency, that weakness, must be changed so that the loophole no longer exists.
Quoting fishfry
Until you provide me with a definition of "field" for this premise, your efforts are futile. If a field requires set theory, I'll reject it for the same reason I rejected your other demonstration. If you can construct a field with square root two, without set theory, then I'm ready for your demonstration. If you produce it I'll make the effort to try and understand, because I already believe that you would need to smuggle in some other invalid action, because that's what's occurred in all your other attempts.
Quoting fishfry
You never explained to me what you mean by "mathematical existence" that remains an undefined expression.
Quoting fishfry
It's not the case that I have a block in dealing with symbology, but what I need is to know what the symbol represents. Until it is explained to me what the symbol represents I will not follow the process which that symbol is involved in. I believe that whatever it is that is represented by the symbol, places restrictions on the logical processes which the symbol might be involved in. Supposedly, you could have a symbol which represents nothing (though I consider this contradiction, as a symbol must represent something to be a symbol), and that symbol might be involved in absolutely any logical process. However, once the symbol is given meaning, the logical processes which it might be involved in are limited. So if you start with the premise that a symbol might represent nothing, I'll reject your argument as contradictory.
Quoting fishfry
"Fictional existence" is contradiction plain and simple. To be fictional is to be imaginary, and to exist is to be a part of a reality independent of the imagination. If you are handing to "existence" a definition which allows that an imaginary, fictional thing, exists, then it's not the rigorous philosophical definition which I am used to. I think that if you cross this line, you have put yourself onto a very slippery slope, denying the principles whereby we distinguish truth from falsity.
Quoting fishfry
I interpret this as your "epistemic stance" requires Platonic realism as a support, a foundation. I deny Platonic realism, so I think your epistemic stance is ungrounded, unsound.
Yes, but the problem is that (for example) particles are always detected as little spots (such as a point on a photographic plate) and wave functions are spread all over the space, or on a space much larger than the observed spot. Nobody has never seen an elementary particle that looks like a wave function!
No, maybe I shouldn't have talked about "points": you simply split space and time in a lot of little "cubes" that are attached one to the other. Only that they are 4-dimensional "cubes": 3 dimensions of space and 1 dimension of time. These "cubes" are not all of the same size, they don't have straight angles and each edge in general can have a different length. The measures of these little "cubes" are described by the metric tensor. The information of which little cube is attached to which other on which side is described by the ordering of the "indexes" that I assigned to each cube:
for example: cube (1,1,1,1) is attached to cube (2,1,1,1) in this way: the right side of cube (1,1,1,1) is the left side of cube (2,1,1,1). Cube (1,1,1,1) is attached to cube (1,1,2,1) in this way: the up side of cube (1,1,1,1) is the down side of cube (1,1,2,1). I don't know if that gives the idea...
No, it is the study of being, which is not necessarily synonymous with existence. For example, one view is that ontological existence (i.e., actuality) is a subset of reality (which also encompasses some possibilities and some necessities), which is a subset of being (which also encompasses fictions).
Quoting Metaphysician Undercover
By defining "existence" in another context-specific way, obviously. There are plenty of other terms that mean something different in mathematics than in metaphysics or in other sciences.
Quoting Metaphysician Undercover
That is not just pragmatism, it is the scientific method. How else would you propose that we evaluate our hypotheses to ascertain whether they accurately represent reality?
Quoting Metaphysician Undercover
As I have explained to you several times now, no one except a platonist would claim that mathematical existence conforms to "the rigorous philosophical definition" of (ontological) existence. Everyone else understands this, so please stop belaboring your terminological objection.
Quoting Metaphysician Undercover
That is a misinterpretation, and you know it by now.
Yea, but I was speaking about a way to approximate space-time with discrete pieces to make computer simulations, not of the real equations. The real equations are partial differential equations defined on a continuous 4-dimensional space.
Quoting Metaphysician Undercover
Special relativity allows us to represent time as discontinuous?? Why? On the contrary, in special (and general) relativity space and time have to be "of the same kind", because you can transform the one into the other with a geometrical "rotation" ( https://en.wikipedia.org/wiki/Lorentz_transformation ) simply changing the point of view of the observer.
Quoting Metaphysician Undercover
Well, my idea was much simpler, I guess: just treat a number like an attribute of an object, like the color of the object, or it's volume. Do you agree that the volume is an attribute of an object? A lot of objects may have the same volume, or the same color. Well, an object can have even a number, if I consider the object as made of several distinct parts.
How would you teach a child what is 5? You show him a picture with 5 flowers and you say: this is 5. The child understands that there is some attribute of that picture that is called 5. Then you show him a picture with 5 trees and you say: that is 5 too! Then the child should understand what's the attribute (the characteristic) that the two pictures have in common.
You can do the same to show him what is a color: show two pictures of different objects with the same color.
I know, this explanation is not very "philosophical"... and, to say the truth, I don't really understand why is this such a philosophical problem :yikes: But what's the problem with this interpretation?
But I am afraid that's all what physics (at least contemporary physics) does: prediction. Nothing else!
Nobody knows how to make sense of the equations of quantum mechanics: physicists learned how to use them to predict the results of experiments. Maybe that is a problem, but it is a problem of physics since the beginning: Newton didn't know how to make sense of a "force" that acts from thousands of kilometers of distance.
I heard somebody say that now it's clear: everything is filled with a "field", and it's the exchange of particles of that field that transports the force. So, can you try to imagine how to generate an attractive force by exchanging an object?
The reality is that there are equations that work, and you can apply a mathematical theory made of imaginary things with imaginary rules that happen to give the right results. The real "ontological" reason why this system is able to "emulate" the experiments of the real world, nobody is able to explain. And it's not only about the use the square root of 2.
[continuation of the previous post]
So, you can see constructivist logic as an algebra of propositions built with computable functions (https://en.wikipedia.org/wiki/Heyting_algebra). You cannot build non-computable functions using only the operations of this algebra, but you can add elements that are not part of the algebra ("external" non-computable functions) to obtain a new algebra that uses all computable functions plus the function that you just added.
That's exactly the same thing that you do adding square root of 2 to the rationals: you obtain a new closed field that contains all the rational numbers plus all that can be obtained by combining the rational numbers with the new element by using the operations defined on rational numbers.
But I see that the main problem for you is not about the soundness of logic, but about the cardinality of the set of real numbers.
So, my question is: how do you know that the cardinality of the set of real numbers is uncountable?
- answer (let's speed up the interaction :smile: - you can add additional answers in the next post if you want): because of Cantor's diagonal argument (https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument).
Well, Cantor's diagonal argument is still valid in constructivist logic: it says that any function that takes as an argument an integer and returns a function from integers to integers cannot be surjective (cannot generate all the functions). The proof is exactly the same: take F(1,1) and change it into F(1,1) + 1, then take F(2,2) and change it into F(2,2) + 1, etc... This is a computable function if all the F(n,n) were computable, ( n -> F(n,n) - very simple algorithm to implement ) but it cannot be in the list: if it were in the list ( call it X(n,n) ), let "m" be the position of X ( X is the m-th function ). What's the value of X(m,m) ?
X(m,m) cannot be computable.
The problem is well known: you cannot enumerate all computable functions because there is no way to decide if a given generic algorithm stops.
So, computable functions are as uncountable as real numbers are. Where's the difference?
Yes in fact at the time I couldn't understand how your name got in there. Might have messed up the editing at my end.
I had a long convo about all this with @alcontali I believe, a while back. At that time I felt that I'd satisfied my curiosity about the subject of constructive math. I hope you'll forgive me but I prefer not to spend much time talking about this. It just doesn't hold my interest, trendy as it all is these days.
Quoting Mephist
You miss all the noncomputable numbers. You have holes in your real line.
Quoting Mephist
No, the cardinality argument is secondary. The primary argument is the lack of Cauchy-completeness of the constructive line. But it turns out that you can prove that Cauchy-completeness implies uncountability, so in a sense they're the same question.
Quoting Mephist
Cantor's theorem. [math]| X | < |\mathscr P(X)|[/math]. This is a theorem of ZF, so it applies even in a countable model of the reals. You mentioned Skolem the other day so maybe that's what you mean. Such a model is countable from the outside but uncountable from the inside.
[math][/math]
Quoting Mephist
It makes sense that a constructive version of Cantor's theorem is true. But as I said I'm not primarily concerned with cardinality arguments.
The constructive line is not Cauchy-complete. As an example consider the sequence made up of the successive finite truncations of the binary digits of Chaitin's Omega. This is a Cauchy sequence of computable numbers that fails to converge to a computable number.
Like I say, I am perfectly well aware thatl the constructivists have a million ways to wave their hands at this. I truly don't understand why they aren't bothered by a continuum with holes in it.
** EDIT ** I just realized what you'll say here. I cannot computably form the sequence of successive finite truncations of Omega because I can't computably determine the bits. The sequence I gave is noncomputable so you don't see it and it causes no problem for you. You can prove some version of "all computable Cauchy sequences converge," and that satisfies a constructivist. I'm learning to think like a constructivist! I don't know if that's good or bad.
On a different topic, let me ask you this question.
You flip countably many fair coins; or one fair coin countably many times. You note the results and let H stand for 1 and T for 0. To a constructivist, there is some mysterious law of nature that requires the resulting bitstring to be computable; the output of a TM. But that's absurd. What about all the bitstrings that aren't computable? In fact the measure, in the sense of measure theory, of the set of computable bitstrings is zero in the space of all possible bitstrings. How does a constructivist reject all of these possibilities? There is nothing to "guide" the coin flips to a computable pattern. In fact this reminds me a little of the idea of "free choice sequences," which is part of intuitionism. Brouwer's intuitionism as you know is a little woo-woo in places; and frankly I don't find modern constructivism much better insofar as it denies the possibility of random bitstrings.
Quoting Mephist
Well they're internally uncountable but actually countable. Analogous to the fact that Cantor's theorem still holds in a countable model of ZF. But the computable numbers are in fact countable. There is no computable enumeration of them but there is obviously an enumeration: by length and then by lexicographic order. So they are "countable but not computably countable." That's the very best you can do along these lines.
But again, I am not primarily making a cardinality argument. My two objections remain: One, the constructive line is not Cauchy-complete; and two, that constructivists must necessarily deny the possibility of random bitstrings.
Ok. It was only recently that I learned that protons have quarks inside them. Another thing I've learned is that gravitational mass is caused by the binding energy that keeps the quarks from flying away from each other. How that relates to Higgs I don't know. I've also seen some functional analysis so I know about Hilbert space. I have a general but not entirely inaccurate, idea of how QM works.
Quoting Mephist
If you agree that there's no way to measure an exact length to infinite precision, then you accept my point that the idea that measuring the square root of 2 is purely a mathematical exercise and not a physical one; as would be measuring a length of 1. But if you accept this point we're in agreement.
Quoting Mephist
So we're in agreement and there isn't actually any disagreement.
Quoting Mephist
And no way to exactly verify them. We're in complete agreement.
Quoting Mephist
Yes the mathematical theory gives an exact prediction. But that makes sense. And of course the relation of QM to reality has been the subject of physical and metaphysical argument for a century now.
Quoting Mephist
Yes that's the one, thanks.
Quoting Mephist
But this can't be, since calculating machines can't calculate ANYTHING with arbitrary precision. Where are you getting these mystical TMs? If the theory gives a result like pi, I'd accept that as a result having arbitrary precision. But if you are saying that even in theory there is a TM that can calculate anything with arbitrary precision, that's wrong. The best a TM can do is approximate a computable real number with arbitrary precision. That's much less than what you are claiming, if I'm understanding you correctly.
Quoting Mephist
We are in agreement. Though I'd like you to clarify your belief in magic Turing machines that can calculate "anything" with arbitrary precision. TMs can only approximate computable numbers. If you believe the universe is computable that's a proposition I disagree with.
Quoting Mephist
Second the motion.
Quoting Mephist
I've heard that without QM, atoms would collapse. So QM seems to be a good thing. But it can't be the ultimate answer. Something more is out there.
OK, I'll assume for the sake of argument that there is a type of existence, "mathematical existence", which is a different type of existence from "ontological existence". I'll assume two different types of existing substance, like substance dualism.
Quoting aletheist
So can you tell me what fishfry didn't seem to be able to tell me. How would I define "mathematical existence"? Do all fictional things (like fishfry's example) have mathematical existence, or is it only mathematical fictions which have mathematical existence?
Quoting aletheist
That should be obvious to you, we ought to evaluate through the criteria of truth and falsity. Do you not see a difference between "accurately represent reality", and "facilitate prediction"? A significant aspect of the "scientific method" involves "observation", and observation is meant to be objective. The goal of "prediction" introduces a bias into observation.
Quoting aletheist
I don't understand what you are saying. How do you propose that "mathematical objects" could have existence, except through some form of Platonic realism? It's very clear, that imaginary, or fictitious objects, do not have existence as objects. If you want to assign some sort of existence to imaginary, or fictitious scenarios, it would be rather strange to say that these exist as objects.
Quoting aletheist
I've never seen the existence of "mathematical objects" justified by any ontology other than Platonism. So there is no "misinterpretation". If you think that you can justify the existence of these so-called objects in some way other than Platonism, then I'd really appreciate the demonstration. I've actually been looking for this for years, to no avail. And, since I do not agree with the principles of Platonic realism, I've come to the conclusion that abstractions are not existent objects.
Quoting Mephist
But the issue is that they are not categorically distinct, as if one refers to a physical object and the other a mathematical object. It's different forms of the same thing.
Quoting Mephist
Special relativity is based in a discontinuous representation of time. Each frame of reference has a unique, and discrete "time" of its own. Therefore "time" in general consists of all these different unique and particular, "times", making it discontinuous.
Quoting Mephist
But that "rotation" is the faulty geometry we've been discussing. The circle is not real, pi is not real. Nor is the so called "continuity" assumed by this geometry real, because the idea that we can put a point anywhere in space is faulty. So general relativity has a double mistake. It starts from a discontinuous time (which ought to be represented as continuous), then it applies to this discontinuity the principles of a continuous space (which ought to be represented as discontinuous), to create the illusion of a continuous "space-time". In reality the concept of space-time consists of a faulty representation of time, with an attempt to fix it using a faulty representation of space.
Quoting Mephist
The "volume" is what we assign to the thing, a judgement we make based on our principles of measurement. So we cannot really say that it is something intrinsic to the object, it's a judgement, and that's why we have different standards to measure volume.
Quoting Mephist
Saying that we can assign the same volume to two distinct things, or that we can assign "5" to two distinct groups of objects, is really no different from saying that we can call two distinct things "a chair". This does not mean that there is an immaterial object which "chair" refers to, nor is there an object that "5" refers to, or an object which "volume" refers to.
But I think you are on the right track here, towards understanding what mathematics is incapable of helping us with, and that is description. Take your example of the two different pictures, where we use "5" to describe what's in the picture. From looking at one picture, and not knowing what five means, no one could ever know which aspect of the picture 5 is describing. So we look at two, and compare similarities, and perhaps deduce what "five" means. If "5" only gets meaning from referring to two or more different pictures, how is it describing anything in any of the pictures?
Quoting Mephist
Do you not see this as misguided, and not proper science? If the goal of science is to predict, and not knowledge, understanding, then observations will be made under that bias, therefore not truly objective. The observations will be made with the goal of being able to predict, rather than the goal of knowledge and understanding. As I explained in the other post, knowledge consists of a lot more than just being able to predict. Thales predicted the solar eclipse with very little knowledge of the solar system, and that capacity to predict did not give him a comprehensive understanding of the solar system.
Quoting Mephist
Perhaps this problem of pragmatism has always pervaded science to some exist, but I would argue that it has gotten much worse. There is a difference between establishing a principle, with the goal of applying that principle toward further understanding, and establishing a principle for the purpose of predicting.
Quoting Mephist
This is the same illusion propagated by the sophists of ancient Greece. The equations work, Thales predicted the eclipse. Nobody was able to explain why they work, the "real ontological reason". But that doesn't mean they all should have buried their heads in the sand, and not proceeded to investigate the real ontological reason. There's a matter of completion. Equations that work for prediction is only a part of a complete understanding.
This is surely no criticism of me since; for one thing, I have already admitted my ignorance of much of classical philosophy. And you are abysmally ignorant of mathematics, yet you insist on your right to make lofty pronouncements on mathematical topics. Why shouldn't I have the same right?
But in fact you are wrong that I don't know what abstraction is. You're right that I haven't read all the deep thinkers on how to define abstraction. But as someone who's had some small bit of exposure to higher math, I've seen a heck of a lot of absraction in the wild. I know what abstraction is. It's giving a name to something you can't kick with your foot. You don't like that definition but it seems quite serviceable to me. We can't see justice but we have a word for it and we can write down some of its properties and over time, we come to learn what justice is.
I suggest, based on our conversation, that you may be highly expert on what the great philosophers say about abstraction; but your actual experience and knowledge about how abstraction works is virtually nil. At least when it comes to mathematical abstractions. And what's more abstract than mathematical abstractions?
In high school algebra they use numbers, like 47. That's abstract. Then they ask you to find 'x'. A lot of students never get past that trauma. In college you look at derivatives and integrals, more abstractions. If you're a math major they tell you about sets, groups, rings, metric and topological spaces. Past that there's category theory, which is so abstract that mathematicians jokingly refer to it as, "abstract nonsense."
You may know what people have SAID about abstraction. But you know nothing of abstraction. You've proved that to me over and over.
Quoting Metaphysician Undercover
I see no reason to abandon my casual definition, paired with my experience of grappling with mathematical abstractions.
Quoting Metaphysician Undercover
This is all bullshit. It has nothing to do with the subject at hand. If you don't credit me with having a deep personal connection with the process of abstraction, that seems like more of a personal issue than anything else.
Aren't you conflating book learnin' with actual experience? How can you tell a math person they don't know abstraction? That's like telling a pizza chef he doesn't know marinara sauce.
Quoting Metaphysician Undercover
If you think the mathematical existence of the square root of two is a "weakness" or defect in mathematics, it is because you are so ignorant of mathematics, that you haven't got enough good data to reason soundly about mathematics. I would think someone in your position would be desirous of expanding their mathematical understanding. Think of it as "opposition research." Learn more so you can find more sophisticated ways to poke holes.
Quoting Metaphysician Undercover
Meta my friend sometimes you are a very funny guy. I mean that sincerely. I have in fact defined field once or twice, but the reason I haven't burdened you with the details is because you have expressly asked me not to burden you with mathematical details.
But I will be glad to walk you through this, a little later. Next post, after I've gotten your clear permission to walk you through a little math.
Quoting Metaphysician Undercover
Well, a field is typically defined as a type of set; but the definition really has nothing to do with set theory. It's about what algebraic operations are allowed. In fact I will be happy to build you a field, which briefly is a collection of numbers that you can add, subtract, multiply, and divide (except by zero) just like you can with the rational, that contains a square root of 2.
Quoting Metaphysician Undercover
I take this as an honest and brave statement on your part. If I'm understanding correctly, you are asking me to walk you through a mathematical argument, and that you will make a good faith effort to understand me. Is that correct? I have your permission to do this?
Because I can in fact, and without much difficulty at all, show you the square root of 2 without using any set theory; not even by a different name. In other words I won't just sneak in set theory without using the words.
If you will grant me the existence of the rational numbers; I'll build you a square root of 2.
Quoting Metaphysician Undercover
No actually I don't. All I need is for you to believe in the existence of the rational numbers.
Quoting Metaphysician Undercover
I thought I'd defined it several times; or at least given many examples of it. But perhaps you're right. Let me give some thought to what a formal definition of mathematical existence might look like.
Quoting Metaphysician Undercover
Symbols don't necessarily need to represent anything. If I have a symbol that behaves a certain way; that's just as good as a thing that behaves that way. At some level one can take the symbol for the thing.
That's abstraction.
Quoting Metaphysician Undercover
Why can't I write down a symbol.
[math]x[/math]
Why, pray tell, may I not type that symbol on the page? And say that it stands for a green thing? Why can't I do that? It's the foundation of civilization.
You know, I bet you failed the kinds of questions like:
If there are three fraggles in a snaggle; and four snaggles in a boodle; then how many fraggles are in a boodle?
Are you telling me that you would not be able to determine that it's 12 if I didn't tell you what a fraggle, a snaggle, and a boodle are?
If you assert that to me then there's no hope of communication here.
Quoting Metaphysician Undercover
You categorically reject abstraction. I can't work with you anymore. I'm not going to bother to show you the square root of 2 because you can't solve the fraggle problem.
Quoting Metaphysician Undercover
Ok at least you're consistent. You deny mathematical existence but you also deny fictional existence.
So I ask you? Is Ahab the captain of the Pequod? Or its cabin boy? Do you really claim to be unable to answer on the grounds that Ahab's a fictional character? Nihilism.
Ok to sum up:
1) You asked me to walk you through a construction of the square root of 2 that does not require set theory. I want to make sure I have your permission to do some math and that you'll engage with my exposition in good faith. Do I have that agreement from you?
2) In order to do (1) I require only one premise from you. You must grant me the existence, in whatever way you define it, of the rational numbers. If you'll do that I'll whip up sqrt 2 in no time flat, no set theory needed, and no cheating on that point. I will use no set theoretic principles.
3) We have a sticking point, which is that you don't accept a symbol unless it comes with a meaning attached. If you truly believe that then you can't solve the snaggle problem, which requires you to reason logically about symbols whose meaning is not defined.
If you are unwilling to do that, I can't give you a proof and frankly I can't continue our conversation. You reject rationality entirely with such a stance.
4) We have stumbled on an interesting point. You say that a symbol can never be conflated with the thing it's supposed to symbolize. But in math, we often do exactly that. We don't know "really" what the number 2 is. Instead, we write down the rules for syntactically manipulating a collection of symbols; we use those rules to artificially construct a symbol that acts like the number 2. Then we just use that as a proxy for the number 2.
There is even some philosophy behind this. When we talk about the number 2, we don't care about what the object is; we only care about how it behaves.
This is the modern viewpoint of math. We don't care what numbers are; as long as we have a symbol system that behaves exactly as numbers should.
I think we've just stumbled into category theory and structural thinking in math. A thing is not what it's made of; a thing is what it does. And even more, a thing is entirely characterized by its relationships to all other things. If you know the relationships you know the thing. That's category theory.
So anyway this got long but the last four numbered paragraphs enumerate the points that need reply. We needn't squabble about the definition of abstraction.
But really, reading over this, we're done. You won't allow a symbol that behaves like a thing, to be used as a proxy for that thing in a chain of reasoning, in order to get more understanding of the thing. It's like solipsism. I can't refute it but I don't waste my time arguing with solipsists.
In order to figure out things that we don't understand, we give them names. We write down the properties we want the names to have. We apply logical and mathematical reasoning to the names and the properties to learn more about the things. That's how science works. That's how everything works.
So we are ALWAYS writing down and using symbols without much if any understanding of the things we're representing. It's exactly through the process of reasoning about the symbols and properties that we LEARN about the things we're interested in. That's Newton writing that force = mass times acceleration. At the time nobody knew what force, mass, and acceleration were. Newton defined those things, which may or may not "really" exist; then he applied mathematical reasoning to his made-up symbols and terms; and he thereby learned how the universe works to a fine degree of approximation.
Of course no scientific theory is "true" in an absolute sense. But that's the point. That's how rationality works. We make up symbols for things we don't understand; and we come to understand the world we live in by means of reasoning about our symbols.
You reject all of this. If Newton had said F = ma to you, you'd have said, what's force? And Newton would describe it to you, and you'd say, well that's not real, it's only a symbol. You reject all science, all human progress, rationality itself. If I tell you all x's are y's and all y's are z's, and you REFUSE TO CONCLUDE that all x's are z's because I haven't told you what x, y, and z are, you are an absolute nihilist. You believe in nothing that can help you get out of the cave of your mind.
Wow, this keeps getting more and more ridiculous. No one is claiming that mathematical existence has anything to do with "existing substance." In mathematics--again, except for platonism--the term "existence" does not imply anything ontological whatsoever.
Quoting Metaphysician Undercover
Something exists mathematically if it is logically possible in accordance with an established set of definitions and axioms. The natural numbers, integers, rational numbers, real numbers (including the square root of two), and complex numbers all exist mathematically, in this context-specific sense.
Quoting Metaphysician Undercover
Nonsense, prediction is just as much a significant aspect of the scientific method as observation. Why do we have theories? How do we come up with them? Our observations prompt us to formulate hypotheses that would explain them; this is retroduction (sometimes called abduction). We make predictions of what else we would observe, if those hypotheses were correct; this is deduction. We then conduct experiments to determine whether our predictions are corroborated or falsified; this is induction.
Quoting Metaphysician Undercover
One more time: No one is claiming otherwise.
Using abstractions (concepts) is not the same things as the act of abstraction. To conflate these two is equivocation. To define abstraction as "giving a name to something you can't kick with your foot" is woefully inadequate, for someone accusing me of having no knowledge of how "abstraction works".
Abstraction is the act which creates that supposed thing which you cannot kick with your foot. And, you cannot create something by giving a name to nothing. Therefore the "something" which is created by abstraction, must exist prior to the act of giving it a name. The name is what you kick around (with logic), but "abstraction" (verb) refers to the creation of that thing which the name "abstraction" (noun) signifies.
But let's try it the other way around. Let's assume that we make a name which refers to nothing, and manipulating that name, with logical processes (kicking it around), creates a thing which that name represents. Let's say that this is the act of "abstraction", we take a name which refers to nothing, we manipulate that name using logical processes until the name refers to something, and this creates a concept, an abstraction. If this is the case, then all we have done is given meaning to the name. There never was anything which the name referred to, and there still isn't anything which the name refers to, but kicking the name around with logical proceedings has given the name meaning.
Either way you look at it, it's false to assume that we can create something which the name refers to, by starting with a name which refers to nothing. Either there is something immaterial there (Platonic object), which the name refers to from the start, or there is nothing which the name refers to, but the name is given meaning through use. These are two distinct ways of looking at this issue. To confuse these two, and say that we start with a name which refers to nothing, and then by using the name we create "something", and this something is an immaterial object, is to say that we create something from nothing. The point being that if the name starts out as referring to nothing, and logical processes are applied, it must end up as referring to nothing, or the logic would be invalid. therefore the symbol must have meaning from the time it is presented.
Quoting fishfry
See, you're using "abstraction" as a noun here. Please do not equivocate in your demonstrations. If a name refers to something (has a definition) that definition must be upheld or else the logic is invalid due to equivocation.
Quoting fishfry
Sure, your "casual definition" allows you to equivocate. Without equivocation you'd have no argument. Therefore you see no reason to abandon your casual definition.
Quoting fishfry
Are you familiar with Socrates? You have precisely described Socrates' MO. The person who knows how to do something, does not necessarily know what is being done. Such is my example of a 3,4,5 triangle. I know numerous people who can construct a right angle using the 3,4,5, formula, who have never even heard of the Pythagorean theorem. We can do the thing which the theory describes without knowing the theory. The pizza chef can very easily be making very good pizzas using marinara sauce, without knowing how to make marinara sauce, or even knowing the ingredients of the sauce. The mathematician, engineer, or physicist is most often using abstractions (concepts) without knowing what "an abstraction" is, because this is only studied in the field of philosophy. You should allow the truth of this matter. The philosopher who is trained in this, is much more likely to know what "an abstraction" is than the mathematician who uses abstractions, just like the person who is trained in making marinara sauce is much more likely to know what marinara sauce is than the person who makes pizza.
When a person (such as a mathematician) who doesn't know what abstraction is, not being trained in philosophy, starts to produce logical arguments based in unsound premises concerning the nature of "abstractions", this is called "sophistry". I call them mathemagicians, because their most famous trick is to make (mathematical) objects appear from nothing.
Quoting fishfry
Well, it's very clear, that it's a deficiency in spatial representation, just like the irrational nature of pi indicates a deficiency in that spatial representation. You are simply in denial. And as I said, covering up these deficiencies with complicated mathematics doesn't make them go away. That is where your denial leads you astray. Since you deny that these irrational numbers are the manifestation of a faulty spatial representation, you produce extremely complex numerical structures in a sophistic effort to cover up the truth of this fundamental flaw.
Quoting fishfry
You don't seem to understand the criticism. Algebra makes the same mistake as set theory, assuming that a symbol represents an object. Using algebra instead of set theory doesn't get you past my objection. From this premise, the square root of two is not a problem at all. We have a symbol, ?2 it represents an object, and the problem is solved. The real question though is whether the "object" supposedly represented by ?2 is a valid object. If you assume as a premise, that every symbol represents an object, then of course it is. But then that premise must be demonstrated as sound.
Quoting fishfry
Go ahead, but no algebra or other faulty premises.
Quoting fishfry
My OED: symbol, "a thing conventionally regarded as typifying, representing, or recalling something..."
If your symbol does not represent something, it simply "behaves" in a particular way, then the symbol simply has meaning, due to its behaviour. We can say that it is used as "recalling something". In this way, the symbol is the thing, as you say, and as I described above. But if we define "symbol" in this way, then we cannot use algebra or set theory, which require that a symbol represents something. We'd have equivocation.
Quoting fishfry
You can say that the symbol stands for whatever you want. It is "the symbol stands for nothing" which is problematic.
Quoting fishfry
Do you understand that predication is made of a subject, not an object. Whether or not there is a named object is irrelevant to the act of predication. There is no problem making predications of fictional characters, these characters are known as subjects. The problem is in claiming that the fictional character is an existent object.
Quoting fishfry
I will grant you the existence of symbols, and you define what the symbols mean.
Quoting fishfry
I have no idea what the "snaggle problem" is, but if you ask me to use symbols which represent nothing, then we must dismiss any mathematical premise which assumes that a symbol represents an object. Even if we allow that a symbol may or may not represent an object, we must dismiss such premises. If we allow that a symbol has meaning, rather than that it is representative of an object, then that meaning must be defined, or else the symbol will be dismissed as having no meaning and irrelevant.
Here's the issue I see. We can get past "the symbol must have meaning attached", by assuming that the symbol represents an object, and the object represented is unknown. However, if the object represented is unknown, then we also cannot know whether the symbol actually represents an object or not. Then we might say that a symbol may or may not represent an object, but that premise is useless as an epistemological principle.
Quoting fishfry
Are you confessing to the use of equivocation in mathematics?
Quoting fishfry
Symbols are passive entities, objects. The logician manipulates, moves the symbol. The "behaviour" of the symbol is a direct result of, a representation of, the logician's actions, so what is described here is the nature of the rules, and whether the logician follows the rules. Therefore "a symbol system that behaves exactly as numbers should", expresses nothing more than a judgement of the rules of the system. What is left unknown, is "as numbers should", and this might be an arbitrary criterion.
Quoting fishfry
Right, but there is something here, which the name has been given to. We do not assign the name to nothing, unless it is stated that this name signifies nothing. If it is stated that the name signifies nothing, it cannot change and evolve towards signifying something, that would break the stated rule. If, instead, we allow that the name has meaning, instead of representing something, then we can allow that logic enables the meaning to "grow". It cannot contradict the earlier meaning, just expand on that. But if this is the case, then the name must have some meaning from the beginning, and a symbol with no meaning is invalid, as providing nothing to grow.
Quoting fishfry
Clearly, you premise that symbols have meaning, and that the meaning may expand and grow. If this is truly what you believe, how can you make this consistent with the premises of algebra and set theory which dictate that a symbol represents a thing?
Quoting fishfry
What I reject is inconsistency and contradictory premises, not "all of this".
Quoting aletheist
Any claim of "existence" is validated (substantiated) with substance. If you think that there is a type of existence which is not substantial then please explain.
Quoting aletheist
Sorry, but "possible" does not necessitate existence. Do you not recognize that "possible" refers to what may or may not be, so it is contradictory to say that possible things are existing things.
Quoting aletheist
I didn't deny that. Both are essential parts of the scientific method. What I deny is that prediction is the goal of the scientific method. It is simply a part of it. Pragmatists allow that prediction is the goal of science.
Quoting aletheist
Then how do you explain set theory, which proceeds from the assumption that a mathematical symbol represents a mathematical object? It's easy to assert "no one is claiming that an abstraction is an existent object", yet everyone backs up set theory which clearly assumes that the abstraction which a symbol represents, is an object. If these objects are not supposed to be "existent", then why attempt to back up their existence with the idea of "mathematical existence"? Due to this behaviour of yours, I can find nothing else to say other than you are boldly lying.
Perhaps in metaphysics/ontology, but definitely not in mathematics.
Quoting Metaphysician Undercover
I have already done so, repeatedly.
Quoting Metaphysician Undercover
Perhaps in metaphysics/ontology, but definitely not in mathematics.
Quoting Metaphysician Undercover
I never claimed that it is. Prediction enables us to evaluate whether our hypotheses hold up to further experimental and observational scrutiny. The goal is knowledge, which consists of beliefs (i.e., habits) that would never be confounded by subsequent experience.
Quoting Metaphysician Undercover
You keep imposing your peculiar metaphysical terminology, as if everyone else is obliged to conform to it regardless of the context. In this case, you seem to be insisting that only an ontological existent can be the object of a symbol. In mathematics, and even in ordinary language, an abstraction can also be the object of a symbol, as long as the universe of discourse is established. The objects of the names "Pequod" and "Ahab" are a boat and its captain in the fictional world of Melville's novel. The object of the word "unicorn" is a horse-like animal with one horn; the fact that no such animal exists in the ontological sense does not preclude the word from having an object at all.
Quoting Metaphysician Undercover
Seriously? Due to this behavior of yours, I can find nothing else to say other than you are boldly ignorant (of mathematics and its terminology) and stubborn (about your rigid definitions).
Alright man. It's not set theory you object to, it's 10th grade algebra. It's not abstraction you object to, it's the very concept of using the symbol '2'.
I simply can't argue with such a nihilistic position. You claim that one must know and understand the referent of a symbol before being allowed to use that symbol. That flies in the face of the entire history of science. And nobody likes flies in their face. I bid you adieu.
Quoting Metaphysician Undercover
Let me try. Suppose that people have to compute the ratio between the lengths of the sides and the diagonal of an object that approximates a square, but lives in some unknown tessellation of space. You are not informed of the structure of the tessellation apriori and you know that the effort for its complete description before computation is prohibitive. You do however understand that the tessellation is vaguely uniform in size and has no preferred "orientation" or repeating patterns. It is random in some sense, except for the grain uniformity. The effective lengths for the purpose of the computation are the number of cells/regions that the respective segment divides. You also know that the length will be required within precision coarser then the grain of the tessellation/partitioning itself. With this information in mind, you want practical algorithm for the computation of the ratio between the sides and diagonal of a square, to an unknown precision, which is greater then the grain of the tessellation.
Now, honestly speaking, I cannot prove that the most effective generalized answer should be square root 2. But this is the kind of problem that an engineer would face. And if not theoretically, they could confirm empirically if square root 2 is a good general approximation. What would you advise? And when analyzing your algorithm, what object would you introduce to compare its convergence to other algorithms?
Yes, I agree, in mathematics some people make the unsubstantiated claim that the symbols represent existent objects. This is called Platonic realism
Quoting aletheist
That's right, in Platonic realism the abstraction is an object, and it is believed to exist as an object. In the "universe of discourse" called "Platonism", an abstraction is an object. You seem to believe that there is some other form of ontology, some other universe of discourse, which allows that abstractions have "mathematical existence", as objects, which is not Platonism. So I am waiting for you to produce the principles which distinguish this universe of discourse from Platonism. All you have done is stated Platonist principles and lied in asserting that no one is assuming Platonism.
Quoting fishfry
I don't object to using the symbol "2". But like any other language I might use, I want to know what the symbols are being used for. If you assert that the symbol "2" represents an object, I want a clear description of that object, so that I can recognize it when I apprehend it, and use the symbol correctly. If you are simply claiming that the symbol represents an object when you know full well that it doesn't, then you are engaged in deception.
Quoting tim wood
I went through this already, it assumes that the symbol represents an object. This is Platonic realism which is a faulty ontology.
Quoting tim wood
It's very clear that it works, I never disputed this fact. However, "works", and "it is designed to help us determine theh truth" are two distinct things. That is the problem with pragmatism, if the purpose is anything other than to bring us truth, then the premises employed will reflect that other purpose instead of the goal of truth. "It works" has no necessary relationship with truth, as deception clearly demonstrates.
Quoting tim wood
If it is being used as a system of logic employed toward determining the truth, it is faulty because it has a false premise. Platonic realism is false. Mathematical symbols do not represent objects. Aletheist seems to believe that the existence of mathematical objects can be supported by something other than Platonism, something called "mathematical existence". Perhaps you can assist and demonstrate how mathematical symbols represent some sort of objects which are other than the objects assumed by Platonic realism. As I told aletheist, I would be highly interested in this new ontology.
Quoting simeonz
Either the symbols represent objects or they do not. We might say that they may or may not represent objects, but then we would need to know whether or not they do, in order for the symbols to be useful. If we assume that there is an object represented by the symbol then the symbol is useful. But if there really is not an object represented by the symbol and we use it under the assumption that there is, then the use is deceptive.
Sorry simeonz, but your example seems to be lost on me. The question was whether whatever it is which is represented by ?2 can be properly called "an object". You seem to have turned this around to show how there can be an object which represents ?2, but that's not the question. The question is whether ?2 represents an object.
Indeed, that would be mathematical platonism, as I have acknowledged. However, I am not a mathematical platonist--I have quite explicitly denied that the symbols represent existent objects in the ontological sense.
Quoting Metaphysician Undercover
Platonism is by no means the only philosophy of mathematics that employs the well-established term "existence" when referring to abstract objects. As I have clearly and repeatedly stated, for those of us who are not mathematical platonists, ontology has nothing whatsoever to do with the "existence" of such objects.
Quoting Metaphysician Undercover
All you have done is obtusely stuck to your rigid terminology, refusing to pay any heed to the multiple explanations that I and others have offered to correct your evidently willful misunderstanding. I see no point in wasting my time any further.
You reject science. In science we DON'T know what something is. So we give it a symbolic name, write down the symbol's properties, and reason about it in order to learn about nature.
When Newton wrote [math]F = ma[/math] those were made up terms. Nobody knew (or knows!) exactly what force or mass is. Acceleration's not hard to define. But even then Newton had to invent calculus to define acceleration as the second derivative of the position function.
You reject all that.
Nihilism.
Yes, you can state that all you want, assert and insist until the cows come home, and then continue to assert some more. The challenge is yours, describe how abstractions, concepts exist as "objects", without invoking Platonism. To call the abstraction an "object" is already using a Platonic term. How are you going to show that an abstraction is an object, in any sense other than a Platonic object.
If we say that abstractions, and conceptions "exist", I have no problem with this. I very much agree that they exist. But "object" refers to a very specific type of thing, a unique individual, a particular, having an identity as described by the law of identity. Because an abstraction is not this type of thing, an object, mathematical axioms which assume that abstractions are objects, are simply wrong.
Quoting fishfry
I'm not rejecting science. When we name something which we don't know what it is, the name still represents a thing, it's just the case that we didn't know "what" that thing was at that time so we name it.. Once it is named, we know what it is, the thing called by that name. This is not the same as having a symbol which we do not know whether it refers to a thing or not.
So if you have a symbol "2", and you have apprehended a thing and assigned that symbol to this thing, then tell me something about this thing, so that I may use the symbol correctly, to refer to the thing named by that symbol.. It doesn't make sense that you would have assigned the symbol "2" to something and you know absolutely nothing about this thing which you have assigned the symbol to.
Quoting fishfry
Force, mass, and acceleration are not things. They are not objects. They are concepts which describe properties. Properties are not things, and that's why you cannot point to the things which are referred to by these terms. You might provide a definition of the term, but having a definition does not make the term refer to an object. We might call it a logical subject then, the definition being what is predicated of the subject.
That I reject the notion that properties which are described by concepts like "force" "mass" and "acceleration" are themselves objects, doesn't make me nihilist. It just means that I understand the difference between an object and a logical subject.
Again, your peculiar metaphysical terminology is not binding on the rest of us.
Quoting Metaphysician Undercover
Apparently not--an object is whatever a logical subject denotes, which can be an abstraction or a concrete existent.
The challenge is open. All you do is assert without any justification. Where is your demonstration of an abstraction existing as an object, which is not a demonstration of Platonism?.
Quoting aletheist
Sorry if you misunderstood, but I was talking about what is represented by the symbol in logic, and that is a subject, not an object. Whether the subject denotes an object, a number of objects, or no object at all, is irrelevant to the point.
OK I'll stop arguing about intuitionism. But I think you didn't get my point here, so let me try one last time:
Cantor's theorem is valid in intuitionistic logic, but we know that intuitionistic real numbers are countable. In fact the theorem says: forall countable lists, there is an element that is not in the list, and we know that the set of elements missing from the list is countable because the list of all strings is countable.
Now you read the same theorem in ZFC and you interpret it as "there is an uncountable set of elements missing from the list". How do you know that the set of missing elements is uncountable? I mean: the symbolic expression of the theorem is the same, and the interpretation of the symbols is the same. How can you express the term "an uncountable set" in a language containing only the quantifiers "forall" and "there exists one" ?
And if there is no uncountable set of missing real numbers, there are no holes to fill..
Quoting fishfry
For the first part of the question, I guess your question is how do you say "a finite random sequence" in intuitionistic logic. You can't! (at the same way as you can't do it in ZFC: the axiom of choice does not say "random" function). If the sequence is finite it is always computable, so you can say "there exist a finite sequence of numbers" ( the same as in ZFC ).
There is a definition of randomness as "a sequence that is not generated by a program shorter than the sequence itself" (lots of details missing, but you can find it on the web), but this is about the information content and not about the process used to choose the elements of the sequence.
About the bitstrings that aren't computable: all finite bitstrings are computable of course. So probably you mean the bitstreams that contain an infinite amount of information (not obtainable as the output of a finite program). There is no way to prove that such strings exist using a formal logic system (even using ZFC): we can interpret the meaning of Cantor's theorem in that way, and maybe there is such a thing in nature, but you cannot prove it with a finite deterministic formal logic system.
Probably I don't understand the point of the conversation. But just to be clear. The space tessellation/partitioning was not to show how one mathematical construction can be derived from another. The tessellation corresponded to some unspecified physical roughness with uniformly spaced, but irregularly situated constituents. It aimed to illustrate that solving world-space problems imperfectly (due to efficiency constraints) results in the adoption of a modus operandi solution, whose own structure exists only in concept-space.
For analogy, similar utilitarian interpretations exist for probability and statistics. Most classical (pre-QM) applications of probability are not related to genuine physical indeterminacy, but to making decisions with imperfect knowledge of the conditions. The choices are made according to some sense of overall utility, independent of the true and objectively predictable, but unknown individual outcomes (Pignistic probability). In other words, we conceptualized indeterminacy, not because of its objective existence (aleatoric uncertainty), but due to our lack of specific knowledge in many circumstances and because introducing indeterminacy as a model was the most fitting solution to our problems.
I believe that all mathematics have utilitarian sense to them. Square root 2 is object of thought and human decision making, not of some independent world-state. Its existence may have physical grounds, but doesn't have to be perfectly accurate. It is put forward to deal with solving problems through computation. At least this is my way of thinking.
As I said, I may misunderstand the topic of the discussion altogether, which is fine. But just wanted to be sure that the intention of my example was clear. (That is - that the space tessellation was not intended as a mathematical structure, but as representation of some unknown coarseness of the physical structure, being ignored for efficiency reasons.)
Again, I do not hold than there is such a thing as "an abstraction existing as an object." I reject your peculiar terminological stipulation that an "object" can only be something that ontologically exists.
Quoting Metaphysician Undercover
No, a symbol in logic is itself either a subject or the predicate within a proposition. If it is a subject, then it denotes an object, which can be an abstraction or an existent. If it is the predicate, then it signifies the interpretant, which is a relation among the objects denoted by the subjects.
This is not a demonstration of algebra. So your example is irrelevant. Refutation complete.
What I dispute is that a symbol represents a number which is an object. In your example, X represents how many pennies in your left hand, and Y represents how many pennies in your right hand. Through abstraction we might reduce this to simple numbers, "6" and "8" for example. But those abstracted numbers is not what X and Y represent, as stated in your example. The numbers could only be produced through an abstraction from what is stated.as represented. You did not even state any numbers.
Quoting simeonz
The problem I've been discussing is that whatever it is which is expressed as "a square" does not actually exist in "concept-space" because the perpendicular sides are incommensurable. There is a deficiency in the concept which makes it impossible that there is a diagonal line between the two opposing corners, when there is supposed to be according to theory. The figure is impossible, just like the irrational nature of pi tells us that a circle is impossible.
This impossibility is not a case of us not being able to do in practise what can be done in theory, due to a lack of precision. It is the opposite of this, it is a defect inherent within the theory. The figures defined by the theory are impossible, according to the theory, just like a square circle is impossible.
Quoting simeonz
The problem in this situation is that the indeterminacy is created by the deficient theory. It is not some sort of indeterminacy which is inherent in the natural world, it is an indeterminacy created by the theory. Because this indeterminacy exists within the theory, it may appear in application of theory, creating the illusion of indeterminacy in the natural thing which the theory is being applied to, in modeling that natural thing, when in reality the indeterminacy is artificial, created by the deficient theory.
We can allow the indeterminacy to remain, if this form of "concept-space" is the only possible form. But if our goal truly is knowledge, then it cannot be "the most fitting solution" to our problems. When we proceed under an MO, which is essentially a habit, and we recognize that it is not the best, there is a certain laziness associated with "it works", which leaves us uninspired to seek a better operation. The usual way "works", in some instances because it is layered with a multitude of complexities, piled one on top of the other, exceptions to the rules etc.. And, regardless of these massive complexities we continue with the usual, extremely complex, way, because it works. In reality though, taking the time to analyze the fundamental problems at the base of the usual way might produce a much simpler, more efficient, and better way for revealing truth, by removing the indeterminacy from the theory.
Quoting simeonz
I'm not sure I actually understood your example. Maybe we can say that Euclidian geometry came into existence because it worked for the practises employed at the time. People were creating right angles, surveying plots of land with parallel lines derived from the right angles, and laying foundations for buildings, etc.. The right angle was created from practise, it was practical, just like the circle. Then theorists like Pythagoras demonstrated the problems of indeterminacy involved with that practise.
Since the figures maintained their practicality despite their theoretical instability, use of them continued. However, as the practise of applying the theory expanded, first toward the furthest reaches of the solar system, galaxy, and universe, and now toward the tiniest "grains" of space, the indeterminacy became a factor, and so methods for dealing with the indeterminacy also had to be expanded.
Now, to revisit your example, why do you assume "grain uniformity"? Spatial existence, as evident to us through our sense experience consists of objects of many different shapes and sizes. Wouldn't "grain uniformity" seriously limit the possibility for differing forms of objects, in a way inconsistent with what we observe?
Quoting aletheist
Then your beliefs are irrelevant to my concerns with algebra and set theory, which hold that the symbols represent objects.
Quoting aletheist
I'm afraid you have things backward. The symbol is itself an object. The symbol may signify a subject, and it may signify a predicate. It is impossible that the symbol "is" the subject, or "is" the predicate because then there would be no way to determine whether any given symbol is a subject or a predicate. It is only by the means of representing something (subject or predicate) that the distinction is made.
They do represent objects--abstractions, not existents.
Quoting Metaphysician Undercover
On the contrary, this is Semeiotic 101--in a proposition, the subjects denote objects, and the predicate signifies the interpretant.
Incommensurability does not preclude (mathematical) existence. Our inability to measure two different objects (abstractions) relative to the same arbitrary unit with infinite precision does not entail that one of them is (logically) impossible.
Quoting Metaphysician Undercover
Only according to your peculiar theory, not the well-known and well-established theory in question.
If you view mathematics as explanative device for natural phenomena, I can certainly understand your concern. However, I see mathematics first and foremost as an approximate number crunching and inference theory. I do not see it as a first-principles theory of the space-time continuum or the world in general. I see physics and natural sciences as taking on that burden and having to decide when and what part of mathematics to promote to that role. If necessary, physics can motivate new axiomatic systems. But whether Euclidean geometry remains in daily use will not depend on how accurately it integrates with a physical first-principles theory. Unless the accuracy of the improved model of space is necessary for our daily operations or has remarkable computational or measurement complexity tradeoff, it will impact only scientific computing and pedagogy. Which, as I said, isn't the primary function of mathematics in my opinion. Mathematics to me is the study of data processing applications, not the study of nature's internal dialogue. The latter is reserved for physics, through the use of appropriate parts of mathematics.
Quoting Metaphysician Undercover
As I said above, I don't think that mathematics should engage directly to enhance our knowledge of the physical world, but rather to improve our efficiency in dealing with computational tasks. It certainly is a very important cornerstone of natural philosophy and natural sciences, but it is ruled by applications, not natural fundamentalism. At least in my view.
Quoting Metaphysician UndercoverThis is what I mean.
Quoting Metaphysician Undercover
I do not see how our newfound knowledge about the universe will impact all of the old applications. How does it apply to the geometries employed in a toy factory, for example. The same computations can be applied in the same way. Unless there is benefit to switching to a new model, in which case both models will remain in active use. This is the same situation as using Newtonian physics instead of special or general relativity for daily applications. It is simpler, it works for relative velocities in most cases, and has been tested in many conventional applications. I am breaking my own rule and trespassing into natural sciences, but the point is that a computational construct can remain operational long after it has been proved fundamentally inaccurate. And therefore, its concepts remain viable object of mathematical study.
Quoting Metaphysician UndercoverOf course it would. I meant applications where the grain is indeed uniform, such as the atomic structure of certain materials. And even then, only certain materials would apply. The point being is - every construct which can be usefully applied as computational device in practice deserves to be studies by mathematics. As long as it offers the desired complexity-accuracy tradeoff.
I do agree that the use of mathematics in real applications is frequently naive. And that further analysis of its approximation power for specific use cases is necessary. In particular, we need more rigorous treatment that explains how accuracy of approximation is affected by discrepancies between the idealized assumptions of the theory and the underlying real world conditions. I have been interested in the existence of such theories myself, but it appears that this kind of analysis is mostly relegated to engineering instincts. Even if so - if mathematics already works in practice for some applications, and the mathematical ideals currently in use can be computed efficiently, this is sufficient argument to continue their investigation. Such is the case of square root of 2. Whether this is a physical phenomena or not, anything more accurate will probably require more accurate/more exhaustive measurements, or more processing. Thus its use will remain justified. And whether incommensurability can exist for physical objects at any scale, I consider topic for natural sciences.
I didn't answer to this yet, so I'll do it now.
In general, category theory can be used to represent formal logic systems and their interpretations, in the obvious way: an interpretation is a functor from a category representing the language to a category representing the model ( https://en.wikipedia.org/wiki/Categorical_logic ).
The formal logic system is represented as a category in this way:
- the objects of the category are the propositions of the language (all provable propositions)
- the arrows of the category are the derivations (all possible derivations A -> B from prop. A to prop.B)
Two categories A and B can represent different formal logic systems but be equivalent (https://en.wikipedia.org/wiki/Equivalence_of_categories). Basically, this means that there are two natural transformations X and Y (https://en.wikipedia.org/wiki/Natural_transformation) that map every derivation in A in a derivation in B and vice-versa.
X and Y are then adjoint functors (https://en.wikipedia.org/wiki/Adjoint_functors)
In this case, the correspondent propositions (objects) in A and B are different in general, but there is an 1-to-1 correspondence between derivations in A and derivations in B. The derivations on formal systems are (exactly) the computations needed to obtain the results of experiments.
In practice, it means that A and B use a different "encodings" (different languages) to describe the same experiment in equivalent ways. From the point of view of the physical predicting capacity of the model, it doesn't make any difference if you use A or B to perform the computations.
The binding energy due to the coupling between quarks and gluons is responsible for the most part of the mass, the rest of it (I don't remember now in which percentage) is due to the binding energy due to the coupling between quarks and the Higgs field.
Yes, to be precise, physical states are represented by rays in a Hilbert space (infinite-dimensional complex vector space). A ray is a set of normalized vectors (scalar vector X * X = 1 for every vector X), with X and Y belonging to the same ray if X = a * Y, where a is an arbitrary complex number with modulus(a) = 1.
The vectors of this Hilbert space are the wave functions (not observable).
Observables are represented by Hermitian operators on the Hilbert space.
And the results of experiments (the numbers corresponding to the measured quantities) are the eigenvalues o these Hermitian operators.
(P.S. it's impossible to understand how it works from this description, but that's the way it is, if you want to be mathematically accurate)
No, I didn't say you can calculate anything. You can calculate the magnetic moment of the electron in quantum electrodynamic with arbitrary precision, but only in theory (because the number of operations necessary grows exponentially with the number of calculated decimals), and only in QED (that is a part of the full standard model - in the full standard model (QCD + Higgs) I don't know. I never understood how QCD renormalization of path-integrals works).
But I wanted to point out that there are parts of QM that are in some sense "mathematically perfect". Meaning: there are a finite set of atoms corresponding to all the possible combinations of electrons' orbitals up to a certain number of electrons (82 stable elements? I don't remember). And that ones are "perfect shapes", in the sense that two of them of the same type are exactly the same shape, like two squares. Usually (before QM) physics was made of objects that only corresponded to mathematical objects in an approximate way (orbits of planets for example), but if you looked carefully enough, every object in the physical world was different, and different from the mathematical object that represented it.
Atoms, and particles in QM in general, are different: they are "digital" (quantized) and not "analogical" shapes. So, in some sense, they are "perfect" (mathematical?) objects.
For two people trying to end a conversation we're not doing a very good job.
I want to make a semantic point, which is that intuitionism is too vague. It's way more than constructivism. Intuition in Brouwer's formulation has a mystical component that I can never make sense of. I use the term neo-intuitionism to stand for all the contemporary attempts to revive the idea, minus the mysticism: constructive math, homotopy type theory, etc.
Quoting Mephist
But there are. It's a theorem that a Cauchy-complete totally ordered field must be uncountable. The constructivists pretend all the noncomputable numbers don't exist. But that's nonsense. Chaitin's constant exists (as a real number) and it's not computable. The Halting problem is not computable. Lots and lots of naturally occurring phenomena are noncomputable. Newtonian gravity is noncomputable. (The jury's still out on QM). You can't close your eyes to things then say they're not there. There's more to mathematical truth than proving theorems, as Gödel demonstrated. You can prove that the constructive real line is "computably complete," but it's still not complete, as in the example of truncations of Chaitin's constant shows.
Quoting Mephist
You seem a little off topic here. I asked you what principle of nature, or math for that matter, forces a sequence of coin flips to be computable. Of course I agree that any finite sequence of flips is computable, we can compress it just by writing down its base 10 equivalent. But if you flip infinitely many coins. you will get a computable sequence with probability zero. How can constructivists hope to get so lucky?
Nonsense. I can prove it easily. The measure of the unit interval is 1; the measure of the computable reals in the unit interval is zero. Therefore there must be a whole lot of of noncomputable reals in the unit interval.
You're claiming that if I flip infinitely many coins, they must land in a pattern that is computable. That's clearly nonsense. How would the coins know to do that? On the contrary, it's incredibly unlikely that an infinite bitstring is computable and "almost certain," as they say in measure theory, that it's not.
Here's a nice contemporary example of exactly that.
Do you happen to know what dark matter is? Don't worry if you don't, because nobody knows what dark matter is. It's a name given to something we can not understand but wish to study.
The story goes like this. Astronomers can estimate the amount of matter in a given galaxy. We can also measure the galaxy's rotational speed. It turns out that most galaxies are spinning so fast that they don't have enough matter to hold them together gravitationally. By all our known theories of physics, galaxies should have flown apart long ago. Why didn't they?
We have no idea. Being human, a creature with the power of abstraction (you must have not gotten your share) we give it a name even though we have no idea what it is or what it might be.
Dark matter is the name given to some hypothetical "stuff" that interacts with the gravitational field but no other fields. By contrast, a rock falls to earth so it interacts with the gravitational field. And you can see it, so it interacts with the electromagnetic field. That's normal for the stuff we call "stuff."
Dark matter must therefore be something that's matter, in the sense that it interacts gravitationally; but it's dark. It doesn't interact with electromagnetism. You can't see it. In fact "dark" is the wrong name, it should be transparent matter. But dark matter is the name that stuck.
So, is there dark matter? Maybe. We just don't know what it is yet.
Or maybe there is no dark matter at all. Perhaps the law of gravity needs to be tweaked so that at galactic distances, it has a little extra pulling power to hold the galaxies together. This fascinating idea is called Modified Newtonian Dynamics, or MOND.
"Dark matter" is therefore a symbolic phrase to stand for something that causes some effect, but we have no idea what it is, or even if it exists. Yet we reason about it and write scientific papers about it.
That's scientific abstraction. You know nothing of science. I'm impressed. The more I get to know you, the wider your sphere of ignorance seems to become.
Wrong theory. If something is computable or not, that's computability theory. If something is computable efficiently or inefficiently, that's complexity theory. You're conflating the two. Minor point but you've done it twice so I thought I'd clarify this point.
You can calculate the magnetic moment of the electron. Period. The efficiency of the calculation is a separate topic and has nothing to do with whether it's computable. I suspect you know that but forgot to make that distinction as you were typing.
But I don't know why you keep mentioning this. You can't measure the prediction with arbitrary accuracy in the real world. We're agreed on this point.
Quoting Mephist
Ok. Not exactly sure what you're saying here. I've already stipulated long ago that I know that all electrons are identical. That is in fact a highly strange phenomenon. You pointed out to me that atoms can be identical to other atoms. That's interesting. This last para I didn't quite follow.
As it happens, here is how I learned about what you're describing. I never had much physics background. A few years ago I had the opportunity to seriously study some functional analysis. Functional analysis is basically infinite-dimensional linear algebra combined with calculus, if you think of it that way. Normed vector spaces, Banach spaces, Hilbert spaces. For example you can recover the subject of Fourier series as a particular example of an orthonormal basis. One day I discovered that the mysterious bra-ket notation, which was something I thought I'd never be able to understand in this lifetime, turns out to be nothing more than a linear functional operating on a vector, written in inner-product notation. At that moment I realized I understood a lot of QM without having to study physics. So I actually understand all of what you said, from a mathematical point of view.
And now that you mention it ... that's one of my arguments against constructive physics! A Hilbert space is a complete inner product space. By complete we mean Cauchy-complete. So you can't even have such an object in constructive math, because the constructive real line is not Cauchy-complete.
Now if I'm understanding some of your comments correctly, you are saying this doesn't matter because even if we assume the constructive real line, we can still prove the same theorems. Constructive completeness is just as good as completeness, for purposes of calculations in QM. And even if there is ultimately a difference, we couldn't measure that difference anyway!
Perfectly sensible. We could do physics with the rational numbers and a handful of irrational constants if we needed to. No experiment could distinguish that theory from a theory based on real numbers.
This is a very interesting point I hadn't considered before. It makes the enterprise of constructive physics seem somewhat more reasonable to me. Am I understanding you correctly?
I think what I was getting at is that you made the claim that constructive and classical physics were equivalent categories; and I asked you to clarify how you were categorifying physics. I don't think you answered but it's not an important point. I would certainly take on faith that what you say is true.
Actually what you've convinced me of so far is that constructive math and standard math give the same theory of physics; since in QM we are only doing computable calculations anyway. Is that right?
I was just a little unclear about which category you're using. I know Baez and others use category theory in physics, but I don't know if there are official categories that describe gravity or QM or whatever. Doesn't really matter. Your main point is that constructive math is just as good as classical, since we only use computable calculations. And you are being agnostic about whether the actual universe is constructive or not. Is that a fair summary of your view?
Short answer: this is not a computable sequence.
- So how is this experiment described in a constructivist theory of physics? This is not an experiment, because it cannot be performed in reality: it never ends!
But there is even another problem: you cannot define the term "probability" as a mathematical function from a finite sequence of bits (results of partial experiments - the "total" experiment does not have a result, since it does not have an end) to real numbers (the probability) because of the limitation of the language - and this is true even for ZFC set theory: you simply define the probability of a sequence of N bits as the inverse of the number of possible sequences of N bits, such as if there were N results (many-worlds interpretation), but there is only one result. Probability is "a priori" in QM (not explained from other physical principles). Otherwise, if it's not "a priori", the result of the coin flip is derivable from the theory (such as in Newtonian mechanics), and then it is a computable function.
Short answer: for a finite experiment, "a priori" probabilities are simply functions that count the total number of possible results, "assuming" that each result has the same "probability" (yes, that's a circular definition: no formal definition of what "probability" is, even using ZFC set theory).
No, in QM the pattern is NOT computable: the pattern is NOT predictable from the theory, so you DON'T NEED any computable function to predict it!
:smile: Super! So we can speak about QM without equivocating the words!
Quoting fishfry
:cry: OK, let's just "pretend" that a Hilbert space is complete even in constructivist logic. Or maybe, let's stop arguing about constructivist theory: you said you are not interested, right?
Quoting fishfry
Yes, EXACTLY! :grin:
Quoting fishfry
:up: :smile:
Ok. So whether we use constructive or classical real numbers, we get the same physics. We get the same theorems and we can't measure any difference.
However, we do not necessarily have the same metaphysics. The world may be classical or constructive. It may consist only of computable things or it may contain noncomputable things. Our theories can't tell the difference and our experiments can't tell the difference. But ultimate reality may in fact be one or the other, computable or not. Which supports my belief that noncomputability is the next frontier in physics. If someone ever proves that a noncomputable real is necessary to explain some observable physical phenomenon, it's off to the races to find such a thing in the world. I'm talking a hundred years down the road, maybe longer.
Confused by this. Constructive physics wouldn't allow a random sequence.
You can certainly define a measure on the unit interval of reals and assign probabilities to sets of bitstrings. I didn't follow this post. You said you can't define probabilities for bitstrings but you can.
Yes, exactly! If there were an experiment that could tell the difference, then it's no more metaphysics!
But my guess about the future is that none of the two logics (constructive or ZFC set theory) will be the final answer (or maybe I should say the "next" answer), because both have the same common "defect": they assume that the wold is deterministic.
(OK, DO NOT ANSWER NOW please! I know the objection: mathematics is not the real world, it doesn't matter if the world is deterministic or not, I am contradicting myself!)
I don't have time now for a quick explanation, I'll get back to this when I came back from work... :pray:
Therefore the dualism of Platonic realism.
Quoting aletheist
And symbols represent subjects, so there's a double layer of representation, exactly what Plato warned us against, what he called "narrative", which allows falsity into logic, sophistry.
Quoting aletheist
I'm really tired of your unsupported assertions. As I explained It is not a case of imprecision in practise, it is a case of something being logically impossible within the theory. The theory dictates it as impossible, just like a square circle is impossible, by definition. It has nothing to do with our inability to measure with "infinite precision" (whatever that might mean), as it has been demonstrated that no degree of precision can give us that measurement. This is a defect of the theory, it gives us a so-called theoretical "object" which cannot be measured. Why would we produce a theory which presents us with an object that cannot be measured, when the theory is created for the purpose of measuring objects? It's self-defeating.
Quoting aletheist
It's not my "peculiar theory", it's the "Pythagorean theorem". This issue has been known for thousands of years, and I'm shocked by the level of denial in this thread. Accusing me of coming up with my own idiosyncratic theory, that's just ridiculous.
Pythagoras demonstrated how we can construct an abstract mathematical object, using accepted mathematical principles, which is impossible to measure. That is the diagonal of a square. Creating "impossible" abstract objects is nothing new, it is easily done through the use of contradiction. The problem here is that it is done through the use of accepted mathematical principles.
Quoting simeonz
To me, this is a new and refreshing perspective of mathematics, much more realistic than the Platonic realism defended by many mathematicians. From this perspective, we can see that mathematics is not the primary tool required for an understanding of "nature's internal dialogue". To continue this analogy we could say that nature speaks a different language, and if we want to translate nature's language into mathematics, we need to first recognize that the language is completely different from mathematics, then proceed to determine the differences and similarities to produce some principles for translation. I'm afraid that most physicists would not see things this way though. The prevalent theme here is Platonic realism, according to which, the physical, sensible universe, is a direct representation of the mathematical objects. Under the precept of Platonic realism therefore, learning mathematics directly enables one to understand the universe.
Quoting simeonz
I don't think that this is a realistic perspective. I don't know how often, if ever, there are individual atoms naturally existing, as they tend to come in molecules. And the "same" atom has a different structure in a different molecule. The first principle we understand about physical reality is the law of identity, and this recognizes that each particular thing is unique. To adopt a principle of grain uniformity would mean dismissing the law of identity, and I don't believe that would be consistent with our experience of the uniqueness of particular individuals.
The particular is unique, with a unique identity. In abstraction, we look beyond the uniqueness, and class unique things together as "the same" according to some principle of categorization. This allows that we might have "5", or "8", apples. Notice that "apples" is the qualifier, the principle of sameness, by which we class the things together as "the same", thus allowing for the abstraction to take place. If we allow that two distinct instances of particular objects are "the same" in an absolute way, like two distinct grains in "grain uniformity" or two distinct occurrences of the number 5, then we violate the law of identity. We would allow that we might have a group of those things classed together without any qualifier (principle of sameness), because we have already assumed that they are the same in an absolute way. There is nothing which makes them the same except the assumption that they are the same. Now we have entered into an extremely confused and contradictory conception within which distinct things are said to be distinct particulars, and they are treated by the application of the theory as distinct particulars, yet they are stipulated by the assumptions of that same theory to be the same in an absolute way. That's the kind of mess which "grain uniformity" might give us.
Quoting simeonz
I pretty much agree with what you say here. I think that there is no problem with making approximations in practise. This is common, and as an engineer one would know the acceptable limits of such approximations, established by convention. When I use pi for example I use 3.14, and this is my personal convention. When I use the Pythagorean theorem to lay out right angles, I might round off to about a quarter inch, more or less depending on the length of the perpendicular sides.
But approximation in practise is not the same as approximation in theory. Approximations within theory are employed when the theory cannot provide accuracy due to some deficiency of the theory. The approximations are used, and the theory proceeds from them. However, the approximations are covering over the original deficiencies, and as the theory extends and extrapolates, the effects of the deficiencies compound and magnify. If we refuse to recognize that the approximations are a manifestation of deficiencies in the theories, and address those deficiencies, we will never overcome the problems which inevitably result.
Quoting fishfry
That's not an example at all. We know a lot about dark matter, that's why we can name it. It's not at all like naming something which we have not apprehended. If we have apprehended it, it has an appearance to us, and we can describe it. I describe it as a manifestation of the deficiencies of the general theory of relativity. Maybe you recognize it as this as well, but there seems to be a convention amongst physicists and cosmologists making it taboo to mention deficiencies of general relativity.
Quoting fishfry
Yes! You do recognize it as a deficiency of the theory. Why hide this? Why not call it what we already know it is, rather than the mysterious "dark matter"?
No, I am not a platonist; I am not claiming that abstractions exist in the ontological sense. Why keep insisting otherwise?
Quoting Metaphysician Undercover
No, some symbols are subjects, while others are predicates, although the predicate of a proposition can also be embodied in the syntax rather than the symbols.
Quoting Metaphysician Undercover
I'm really tired of your willful obtuseness, insisting on your peculiar metaphysical terminology despite the fact that the same word often has different meanings in different contexts. "Existence" in mathematics is not the same as "existence" in ontology. An "object" in mathematics is different from an "object" in semeiotic, and both are different from an "object" in ontology. A "subject" in semeiotic can be an "object" (direct or indirect) in grammar. And so on.
Quoting Metaphysician Undercover
It is a feature, not a bug--it reveals a real limitation on our ability to measure things.
Quoting Metaphysician Undercover
What is the basis for the claim that mathematics is created for the purpose of measuring objects? On the contrary, the purpose of mathematics is to draw necessary inferences about hypothetical states of things. One such inference is that in accordance with the postulates of Euclidean geometry, the length of a square's diagonal is incommensurable with the length of its sides.
Quoting Metaphysician Undercover
"Impossible to measure" does not entail "impossible," full stop. There is nothing logically impossible about the diagonal of a unit square or the circumference of a unit circle. Again, given how those figures are defined, it is logically necessary for them to be incommensurable.
1. Enumerate the undecidable set of total functions within the entire set of enumerable Turing Machines of one argument {f1(x),f2(x),f3(x),..}, by running every Turing Machine in parallel on each input x=1,x=2,..., and shuffling their enumeration over time as necessary, so as to ensure that fn is defined when run on input x=n.
2) Define the Turing-Computable total function g(n) =fn(n)+1.
Congratulations, you've "proved" that the countable set of Turing machines is "larger" than the countable set of Turing machines.
[continuation of the previous post]
There is in fact a physical assumption that is necessary for both mathematics and logic to make sense: the fact that each time that you repeat the same computation (with the same input values) you obtain the same result.
At first, it would seem that this assumption is self-evident and has nothing to do with physics, but in reality there is no prove (nor even physical evidence) that this is true for very long and complex computations.
In fact the computations that are essential for mathematics and logic to exist are all based on the existence on non-linear physical processes. If the world were made only of continuous (non quantized) and linear (constant first derivatives) fields, there would be no way to perform any computation at all, and no intelligent human beings able to create - or discover - mathematical theorems.
Since we leave in a non-uniform universe, we can build objects that encode digital information, but it still seems to be likely that the quantity of information that can be contained in a portion of space-time and the speed at which this information can be digitally processed are both finite, due to some fundamental principles that seem to be unavoidable consequences of quantum mechanics, whatever the fundamental entities of the universe are (https://en.wikipedia.org/wiki/Quantum_indeterminacy, https://en.wikipedia.org/wiki/Quantum_information).
So, likely there exist quantities, or proves of theorems, whose computation (or prove) is physically impossible, even in principle. This is the some kind of question that we have in physics: does it make sense a model of the physical universe (or "multiverse") whose experimental detection is not possible, even in principle? Or, similarly, does it make sense to imagine an infinite non observable Euclidean space-time that contains the physical space (that is NOT incompatible, as a model, with general relativity), or is it better to imagine a finite non Euclidean space-time?
Now, both ZFC and intuitionistic logic (of whatever kind) assume that there exist some symbols that represent functions. And a function (in the model) is whatever "thing" with the following property: every time you give it the same input, it returns the same output.
But what if there is no such thing in nature, because all physical computational processes are in principle limited? There would be a process that is well defined as an experiment, but you cannot count on the fact that you always get the same output for the same input with absolute certainty: you get results that are statistically determined, but not deterministic.
So, you could have for example a theorem that is statistically true at 90%, but impossible to prove with 100% security even IN PRINCIPLE, due to some fundamental laws of physics (that we BELIEVE to be true, but of course have only an experimental - not mathematical - validity).
My question is: should this still be considered a mathematical theorem, or discarded because we do not have an absolutely certain proof of it? I believe that at the end it will be accepted as valid, but only in some kind of "quantum" logic, based on quantum-mechanical "experiments" instead of "real" proofs.
Constructive physics (constructivist logic) can ASSUME the existence of a function that you can call "random" (whatever it means: it's an axiomatic theory), representing a physical process. Only that you cannot DERIVE or COMPUTE this function. You have to assume it as an axiom of the theory. The point is that this is allowed by the logic because you cannot introduce inconsistencies in this way!
Yes, that's very confusing. And if you look at the mathematical literature, the terms "constructive" and "intuitionistic" seem to have changed meaning, and every author uses them with a different meaning even today.
What I mean by "intuitionistic constructive" logic (because this is the one that I know) is Martin-Lof dependent type theory ( https://ncatlab.org/nlab/show/Martin-L%C3%B6f+dependent+type+theory )
I can assign a number to an experiment (calculated with some well-defined algorithm) and call it "probability" (for example defined as the square of the amplitude of the wave function), but I cannot prove that this is random and has a continuous distribution: it could well be that the experiment gives always the same result!
You say that the probability will be zero for an infinite random experiment. Yes, but there are no infinite experiments!
You can assign a measure to sets of points, but you cannot prove that these sets are uncountabe! The measure does make sense in physics for open sets, but not for every set: not all sets of ZFC are measurable, so there is no corresponding experiment for non measurable sets! ( just to upset you the same old story :razz: - Banach-Tarsky theorem does not describe a physical experiment: all physical experiments are calculated with integrals over open sets). And in constructivist logic the set of all open sets IS COUNTABLE! You cannot test if it is countable or not with physical experiments limited in time (how do you know that you don't get the same results again after a thousand years?)
Huh? :roll: Really?? all?
P.S. I see there are several persons that studied physics visiting this site: maybe we could create a post especially on this point. I am pretty sure what I said is correct.
I cannot communicate with someone who doesn't speak my language. My apologies, as I am not inclined to learn yours. It strikes me that you have disregard for the fundamental rules of logic, and that's why I am simply not motivated toward wasting the effort.
You just make this shit up. How do you measure how much we need to know about something before we can name it? On the contrary, the day they discovered that the galaxies are spinning too fast to hold together, they named the cause "dark matter" while having no idea what it is or whether it exists at all.
In fact I already made that point to you. Read my post again.
Quoting Mephist
“No man ever steps in the same river twice, for it’s not the same river and he’s not the same man”, said Heraclitus in 544bc.
But you asked, "What if there exists no such thing in nature?" But that is not a problem for math, because math is not physics. In math there is certainty about the repeatability of functions.
I like the idea of a Boltzmann brain. The universe is completely random. Like static on a tv set. Once in a while, by perfectly random chance, a region of the universe suddenly coheres into a conscious mind. That's me. Or all of us. We'll be blinking out in an instant or two.
The idea that the universe is repeatable, or orderly, or understandable, etc., is a philosophical assumption. You can do science without making that assumption but most scientists prefer to believe their work is "about" something. I'm sure the bloodletters and proponents of the phlogiston theory of heat felt the same way in their time.
Quoting Mephist
My understanding is that the constructivists allow in nonconstructivity whenever they paint themselves into a corner. For example every type of constructivism includes some form of the axiom of choice, because you can't do math without it. Arguably you can't even do physics without it. There's a weak form of choice needed to prove the Hahn-Banach theorem, a key theorem in functional analysis; which, as I've mentioned, is the mathematical framework of QM. Whether you can do QM without any form of choice I do not know. But the constructivists aren't as pure as they claim to be. And in the end they'll be proven wrong about the world. That's my opinion, but it may take centuries for me to be right.
Why are you trying to convince me that math isn't physics? I'm talking about the measure of computable bitstrings in the space of all bitstrings. The measure of the computable bitstrings is zero.
Right. Constructive physics is physics based on constructive math. That's all it is. Instead of using the real numbers, you only use computable numbers. Or something.
You missed the point. You still do not seem to see the difference between describing something and measuring something. I said that if you're at the point of naming something, you ought to be able to say something about that thing.
Quoting fishfry
You said: "By all our known theories of physics, galaxies should have flown apart long ago. Why didn't they?"
I think the answer is very obvious, the theories are wrong. If the galaxies don't behave the way that the theories say they should behave, then the theories are wrong. We know what "dark matter" means, it means that the theories are wrong. But instead of facing this fact, that the theories are wrong, someone has dreamed up a name "dark matter", and they attribute the fact that the theories are wrong to this mysterious thing, "dark matter". Why not just call it like it is, "the theories are wrong", dump the theories, and the "dark matter" which the theories necessitate because they're wrong, and get on with producing a new theory which doesn't make this mistake?
That explains a lot. Why should I (or anyone else) accept the constraints of your peculiar language?
Quoting Metaphysician Undercover
How could you ever make such a determination, given your admission that you are unwilling even to try to understand my (or others') usage of the terms, simply because it is different from yours?
I hope you'll forgive me but I prefer not to engage with your scientific nihilism, which itself is driven by scientific ignorance. If we threw out everything we don't understand, we'd have never understood gravity or electromagnetism. You simply know and understand nothing about the things you're talking about.
The constraints of my language are the fundamental laws of logic, identity, non-contradiction, excluded middle. These are what facilitate discourse.
Quoting aletheist
You have demonstrated that you change the meanings of the words that you use, at will, as you use them, which results in the appearance of contradiction. You justify this with the claim that the same word has a different meaning in a different context. Yet there is only one context here, discussion between you and I. So all this does is lessen my charge against you from contradiction to equivocation, if it happens to be that the apparent contradictions are actually just ambiguous use, and not intentional deception.
Since you seem incapable of getting beyond this way of speaking, I see no point in continuing. I'm sorry, but I truly tried to understand your usage of terms, but I am incapable because your usage is extremely undisciplined. Let me make it clear though, it is just your usage, not others on this thread which I have this difficulty with. I disagree with others, but I understand their usage of language, not yours though.
You made the following statement:
If you cannot see that the truth of this statement indicates that the theories have been falsified, then I'm afraid your denial is beyond hope.
Name me a theory of science that hasn't been falsified or will not someday be falsified. You're a scientific nihilist. You deny the entire enterprise because it doesn't serve up absolute truth on a platter.
'Will someday be falsified', is not the same as 'has been falsified'. The falsification is what determines the faults, demonstrating the weaknesses of the theory, showing us where improvement is needed. There is no point in dismissing theories which have not yet been falsified, because we would not know what needs to be improved. That's the scientific method, observations which are inconsistent with what the theory predicts reveal the faults in the theory. But until those inconsistent observations come about, we don't know where the weaknesses of the theory lie.
Which has what to do with what we were talking about, the use of symbols to represent things we don't understand?
None of those dictate the peculiar metaphysical definitions that you insist on imposing for terms like "existence" and "object," even in the context of non-platonist mathematics where they entail nothing ontological whatsoever.
Quoting Metaphysician Undercover
On this we agree.
Yes but mathematics needs computations for proofs, and computations are physical processes. You may think that computations are not physics if you make them by mind, but if they are too long to be made by mind you need a computer, right? If a computation is too long to be performed by any computer even in principle, is it still valid? This sounds a little like the existence of parallel lines that never meet: they may not exist if the geometry of the universe is not Euclidean. OK, Euclidean geometry exists as a mathematical object, but can you use it to make deductions in general about any geometry? You can't, because there are sound geometries where that assumption is false.
At the same way, you can assume that unfeasible computations exist as a mathematical object, but can you use their existence to make deductions in general about any physical theory? You can't, because there are sound physical theories where that assumption is false.
How do you define this measure in pure mathematical terms? You cannot use probability, because probability is physics (unless you find a sound mathematical definition of probability)
https://en.wikipedia.org/wiki/Measure_(mathematics)
Measure theory is an abstraction of the concept of length, area, volume, etc; and also of probability. In fact probability theory is based on measure theory.
In fact if you know the Kolmogorov axioms of probability, that's basically measure theory.
A measure is a function from the collection of subsets of a given set, to the nonnegative real numbers satisfying some conditions. If in addition the measure of the entire set is 1, that's a probability measure.
The measure of the rationals in the reals is zero. And in fact the measure of any countable set must be zero. This follows from countable additivity. The measure of all computable bitstrings has measure zero for the same reason. That means if you pick a real number "at random," the probability is zero that you'll pick a computable real.
I have conjectured and proven lots of theorems - some quite challenging - that do not require computations beyond basic inequalities and a little arithmetic of complex numbers. However, creating examples and imagery in the complex plane usually requires programming skills and a computer. So, even if not for proofs, computations are necessary in many areas of mathematics. :cool:
A TM with a program too long to write down in the age of the universe is still a TM. Practical resource limitations do not apply to the theory of computability.
No computation is too long to be performed "in principle." In principle a TM is a finite sequence of instructions. No matter how long it is, as long as it's finite it's computable in principle, if not necessarily in practice.
And yet, the entire set of countable TMs can be diagonalised to prove that the set of countable TMs are "uncountable", by dynamically enumerating the halting TMs so as to ensure the termination of the diagonal TM for each of it's inputs; but in fact all that my proof of "uncountability" amounts to with respect to Turing Machines, is the construction of an enumeration of Turing Machines in such a way that the diagonal Turing machine cannot be part of the enumeration. This is analogous to enumerating the odd numbers and then diagonalising them to construct an even number.
To spell out the difference, in the case of Cantor's Theorem the constructed enumeration of the sets of natural numbers is considered to be prior to the construction of the diagonal set, but in the case of my method, the enumeration of TMs was constructed via the construction of the diagonal function. In other words, selectively constructing a non-exhaustive infinite enumeration via a diagonal procedure isn't a proof that a bijection with the natural numbers doesn't exist under a different enumeration. And in the case of Turing Machines we know that such a bijection does exist.
But this raises doubts about Cantor's original diagonal argument, for I might have been lucky enough with my original enumeration of TMs to produce a diagonal function without requiring any shuffling of the enumeration. Therefore Cantor's original argument isn't proof enough that the power-set of N is literally larger than N.
About the grain uniformity. let's say that we are talking about Voronoi partitioning of space, and not triangular tessellation, because it simplifies the definition. When I say uniform grain, what I mean is that the distances between the generator points of the partitioning follow some known distribution with finite variance. This is a simple example, and not a physical model. However, it does illustrate, I think, that the diagonal will usefully approximate some quantity over a rough spacial structure, despite being itself defined as a continuous object.
Quoting Metaphysician Undercover Which theory do you mean? For me, real numbers theorize some characteristics of computation. And do so imperfectly. The algebraic structure is defined over some converging computational sequences. It does allow for imaginary objects that do not correspond to actual computational processes, because the latter can not be specified procedurally. And thus it works with incomplete specification. But it is a best effort theory. I am not necessarily subscribed to the idea that real numbers correspond to the points of physical lines, whatever that may mean. It may turn out that this conceptualization works, but I am not convinced. I do think that it works for approximations however.
P.S. I want to clarify that I do actually think that our computational and logical ideals are naturally inspired. They are not literally representative of any particular physical structure, but they are "seeded" as concepts by nature, whether our sentience existed or not.
OK, so let's try to follow this definition of measure to calculate the measure of the two sets that you mentioned:
(1). - The set of computable bitstrings: there is an infinite number of computable bitstrings
(2). - The set of all possible bitstrings: there is an infinite (not countable) number of possible bitstrings
So, to calculate the probability of choosing a computable bitstring, I have to divide (1) by (2), and the result is zero. What kind of computation (limit, theorem, or whatever) should I use to divide (1) by (2)? How is this computation defined? That was my question.
Second question: supposing that the division of (1) by (2) is zero, what sense can I make of that measure?
- I cannot perform the experiments because they are infinitely long.
- I cannot even choose one bit at a time from each of the sequences, because both are computable at every step, until they are finite. But they will always remain finite forever: the experiment never ends.
The point is that a lot of interesting physical problems (for example computing the exact shape of biological proteins by following the lows of quantum mechanics) require an amount of computation that is beyond the computing power of any (classical) calculator, and probably even if you could use a computer big as the entire universe, if you want to be sure of the result at 100%. Maybe there are cases where the computation is easy because of particular symmetries of the system, but there are rather exceptional cases.
But what I am saying is that you can equally well assume as an axiom (that would be incompatible with ZFC) that Turing machines DO NOT exist!
So, what is the reason to assume the existence instead of non existence? It is possible (and even probable in my opinion) that the theory that assumes the non existence of Turing machines is even more interesting (both mathematically and for physical theories) than the theory assuming their existence.
And the alternative to TM's existence is not univocal: there are a lot of possible "intuitionistic" and "constructivist" theories that are not equivalent between each-other. But, at the same way, there are even a lot of non-Euclidean geometries. Constructivist theories correspond to elegant constructions in topology, represented as internal languages of certain categories. In comparison, ZFC axioms seem to be much more arbitrary, from my point of view.
- We know that we cannot enumerate all halting Turing machines, so for every supposedly complete list of halting Turing machines I can find another halting Turing machine that is not in the list.
Does that mean that the set of halting Turing machines in uncountable? No! It only means that there is no way to enumerate that list!
Now I have a second list: the list of all functions - even not computable - from N to N, and I have a theorem that says exactly the same thing: for every supposedly complete list of functions from N to N, I can find another function from N to N that is not in the list. But in this case I conclude that the set of functions from N to N is uncountable! Why? What guarantees that in this case I can build that list? The theorem only says that there have to be some more elements not present in the original list (the list cannot be complete), but not that there has to be an uncountable number of missing elements.
I've never heard of this idea, that TM's don't exist. I see no problem expressing TMs in set theory. An unbounded tape of cells is modeled as the integers. You have some rules that let you move right or left. It's pretty straightforward.
Can you say more about this? I have never heard this idea at all. It seems VERY restrictive. Perhaps it's like denying the axiom of infinity. Logically consistent but too restrictive to do math with.
I'm not even sure what that means, that TMs don't exist. The Euclidean algorithm to find the greatest common factor of two integers is 2400 years old. That's the world's first algorithm. It's a Turing machine. It exists. I think "TMs don't exist" introduces a contradiction.
Quoting Mephist
I'll stipulate that a lot of very clever people are on the neo-intuitionist bandwagon these days and that their viewpoint represent the future, while mine represents the past; at least for now. My unease with constructivism is psychological. I love the catechism of standard classical math. It's a matter of faith, not science. Or if faith is too strong a word, familiarity and preference.
Well, OK. Turing machines are a model of computation equivalent to Church's untyped lambda calculus.
Limiting Turing machines to have finite dimensions would in fact be too restrictive to do mathematics (including the infamous square root of two :wink: ). But you can limit Church's untiped lambda calculus by introducing a typed lambda calculus, and with dependent types this is a formal system powerful enough to do mathematics - and corresponding to some form of intuitionist theory (omitting details).
I don't know what would be the equivalent limitation to Turing machines that corresponds to dependently typed lambda calculus (if there is one). So, I should have said that we can assume that the "original" (non limited) Turing machine does not exist.
You're saying TMs don't exist but finite state machines do? Maybe so, but then you'll make your physics a lot harder if you can't even run an algorithm to approximate a constant of nature. I never studied lambda calculus so I can't comment on the rest. But since lambda calculus and TMs are equivalent, they either both exist or neither do. As far as I understand.
Yes, lambda calculus (the original one invented by Church - https://en.wikipedia.org/wiki/Lambda_calculus) and Turing machines are equivalent as computing power.
But there are various other less powerful (as computing power) forms of lambda calculus, that are called in general "typed" lambda calculi (https://en.wikipedia.org/wiki/Typed_lambda_calculus) that do not correspond to Turing machines as computing power, but are still more powerful than finite state machines.
One of the most powerful (as computing power and expressiveness of the resulting language) lambda calculi is called Martin-Lof dependent type theory (https://ncatlab.org/nlab/show/Martin-L%C3%B6f+dependent+type+theory). Yes, it's not called "dependently typed lambda calculus", but it is!
Meaning: it's still a version of lambda calculus using types with a very powerful type system. However, this is still less powerful than the original lambda calculus invented by Church, but powerful enough to do mathematics. The interesting thing is that it's not true that this limitation produces a logic that is difficult to use. On the contrary, the logic is easier to use than ZFC, and that's the reason why most of currently developed proof assistants (Lean, Coq, Agda) make use of this language (https://en.wikipedia.org/wiki/Category:Dependently_typed_languages).
Now, about Turing machines: a Turing machine equivalent to this kind of lambda calculus is basically one of these proof assistants (without hardware limitations of the real machines, of course), so probably it's not impossible to define. But I think that nobody described exactly these kind of logic in term of Turing machines. However, they are equivalent to a kind of Turing machine that is less powerful than the original Turing machine, but powerful enough to do all mathematics.
P.S. Here's a citation taken from wikipedia: (https://en.wikipedia.org/wiki/Typed_lambda_calculus)
All the systems mentioned so far, with the exception of the untyped lambda calculus, are strongly normalizing: all computations terminate. Therefore, they cannot describe all Turing-computable functions.[1] As another consequence they are consistent as a logic, i.e. there are uninhabited types. There exist, however, typed lambda calculi that are not strongly normalizing.
So the lambda formulation is more granular, able to support more nuanced theories? Something like that?
Anyway I know about Coq and the proof assistants and such, but my eyes glaze over every time I read the phrase, Martin-Löf type theory. So I know all about the existence of all this stuff, without necessarily knowing much about it. On the other hand, on something like homotopy type theory, I don't know anything about types. But I do happen to know what homotopies are, and that gives me a little insight into what they're doing.
But I think that now the picture is becoming quite clear (even thanks to Voevodsky's work): there is a very strict correspondence between topology and logic. But you have to "extend" the notion of topology to the one of topoi (a category with some additional properties). ZFC is the logic corresponding to the standard topology (where lines are made of uncountable sets of points). But ZFC and the "standard" topology are not at all the only logically sound possibility! (that in a VERY short summary)
I was reading through the thread and this caught my eye. Why does mathematics need computations? Because before there were computers, the humans did the computations. In math we still do. Computations with pencil and paper are no different in principle than the same computations done on a supercomputer, and vice versa.
So in fact mathematics itself, defined as everything mathematicians have written down or even thought about -- since thought is a physical process too -- is a "physical process." The work of mathematicians takes energy; and therefore it should be possible to study the nature of math as a physical process.
Or as the saying goes: Mathematicians are people who turn coffee into theorems! I think that pretty much captures the spirit of the idea.
(Eyes glaze).
Quoting Mephist
Name drops Voevodsky. Didn't we do this last week? Or was that someone else?
Quoting Mephist
Perfectly well known before Vovoedsky.
Quoting Mephist
Do you know topos theory? Can you "explain like I'm five" to someone who knows a little category theory?
But when did I ever say standard math is the only logically sound way of doing things? I don't think I hold the view you're trying to dispel.
Yes, exactly! (well, I don't understand what energy has to do with this, but more or less...) However, that's only MY idea (and it's more about philosophy than mathematics). So, even if there are lots of mathematicians that have similar ideas (as I believe), no serious mathematician is so courageous to say this loudly! :yikes: (just take a look at arXiv.org and try to find any word about philosophy!).
The correspondence between topology and logic instead, that's one of the most popular and ideas of today's mathematics! :cool:
It depends upon what you mean by "short." Or "normal." In the area I'm most familiar with theorem proofs vary from a few lines to a number of pages in length. And a short proof may be of an extension of a theorem which required many pages of reasoned articulation. I would rather use a pencil than a pen, however, so I can erase my errors or scribbling along non-productive paths of thought!
When you say computations can be considered proofs I'm not sure where you are going. Proofs of what? And "reduce an expression in a normal form" - what's that? Mathematical proofs are rigorously reasoned arguments in logic in which concepts and relationships play significant roles.
There are occasional exceptions in which computers are essential, like the Four Color Theorem in combinatorics. And then mathematicians attempting to verify the overall presentation of proof and conclusion are stuck with verifying computer algorithms and assuming the computers running them do not produce computational errors.
"The correspondence between topology and logic instead, that's one of the most popular and ideas of today's mathematics!"
I'll have to check this out. I've been out in the pasture too long I guess. :worry:
You mentioned topos theory in one of your posts. I read the Wiki page, or re-read it since I've looked at it before. It's abstract sheaf theory. What's sheaf theory? It's the idea of assigning an algebraic object to each point of a topological space or manifold. For example the set of continuous functions that vanish at a given real number is an ideal in the ring of continuous functions on the reals. So the ideals of the ring give you information about the points. This is a fairly sophisticated mathematical point of view.
And then topos theory is abstract sheaf theory. So mathematically, this is advanced grad student level. But apparently a lot of the terminology and concepts are trickling into computer science and other fields.
I was wondering in what context you'd seen topoi. I know there's a lot of category theory in computer science these days. But my sense is that topoi are fairly sophisticated mathematical objects, at least in their mathematical applications.
OK, I'll try to explain this point.
The fact that there is a relation between topology and logic (mediated by category theory) was well known even before, you are right. But Voevodsky's "homotopy type theory" (https://homotopytypetheory.org/) does not say simply that there is a relation between topology and logic: it says that "homotopy theory" (that is a branch of topology) ( https://en.wikipedia.org/wiki/Category:Homotopy_theory ) and Martin-Lof intuitionistic type theory with the addition of a particular axiom (the univalence axiom - https://ncatlab.org/nlab/show/univalence+axiom) ARE EXACTLY THE SAME THING (the same theory). Meaning: there is this axiomatic theory that speaks about homotopy between topological spaces, expressed in the language of category theory (and then in ZFC set theory - it is still valid in any topos, but I don't want to make it too complicated). So, the terms of the language are spaces, points, paths connecting points, equivalence classes between these paths etc...
Now, if you take whatever theorem from homotopy theory and RENAME all the terms of this theory, substituting the word "types" to the word "spaces", "proofs" to the word "points", "equalities" to the word "paths", etc... (lots of details omitted, of course), you obtain a theorem in type theory. And if you take any theorem in type theory you can reinterpret it as a theorem about topology.
At the end, something similar happens even with the usual set theory: a set can be interpreted as the set of all models that satisfy a given proposition, inclusion between sets can be interpreted as implication between propositions, intersections as "AND" operations, etc... But you know, the devil is in the details (expecially in "infinitary" details :wink:).
Well, in the case of topology and type theory all the details match exactly: it's "take two at the price of one". You prove something about topological spaces and you obtain a theorem about types, and vice-versa.
Thanks for saving me the effort of looking it up. That one sentence is enough for me. :brow:
OK, I'll try. But I think it will be a long post anyway. So, I'll do it after answering the other posts before.
But I can give you the reference to a very good book (in my opinion) on this subject that is easy to understand for somebody that has some basis of category theory:
- Title: "TOPOI THE CATEGORIAL ANALYSIS OF LOGIC"
- Autor: Robert Goldblatt
( I found it even on Amazon: https://www.amazon.com/Topoi-Categorial-Analysis-Logic-Mathematics/dp/0486450260 )
OK, let's start from examples of normal forms:
"245 * 10" is a number that is not in normal form. "2450" is the same number in normal form (one of the possible normal forms)
"5/10" is not in normal form. "1/2" is the same number in normal form.
If two strings of characters that represent numbers (in a given formal language) are in normal form, the numbers are equal if and only if they are the same string of characters.
I searched in wikipedia and found this: https://en.wikipedia.org/wiki/Canonical_form. But if you read a book about formal logic this is usually called "normal" form, and not "canonical" form.
The "proofs of what" question is a little harder to answer (but the concept is very simple).
Normally you think of a computation (for example a multiplication of two integers) as an algorithm that takes some arguments as input and returns an output.
But you can think of the same computation as list of transformations that starts from an initial expression ( "12 * 12" ) and transforms it into a second expression ( "144" ) by applying at every step one of the possible "rewriting rules" allowed by the logic. For example, take a look at the section "Arithmetic" here: https://en.wikipedia.org/wiki/Rewriting. In other words, a computation can be seen as a special kind of "prove": the prove that the term "144" is the normal form of the term "12 * 12".
Probably you think that I completely missed the "meaning" of what a mathematical proof. But that's the way computers (formal logic systems) see proofs. I agree that the formal part is not all there is in it. Just take a look at this discussion, for example: https://thephilosophyforum.com/discussion/6205/mathematics-is-the-part-of-physics-where-experiments-are-cheap
Probably you think that I completely missed the "meaning" of what a mathematical proof is. But that's the way computers (formal logic systems) see proofs. I agree that that's not all. Just take a look at this old discussion with Alcontali, for example: https://thephilosophyforum.com/discussion/6205/mathematics-is-the-part-of-physics-where-experiments-are-cheap
A sheaf S over a topological space X is a "fiber bundle", where the fibers over a point x in X are disjoint subspaces of S. Now, a section of the fiber bundle (https://en.wikipedia.org/wiki/Section_(fiber_bundle)) is what in type theory is called a "dependently typed function", that from the point of view of logic is interpreted as the proof of a proposition with a free variable x: the fiber bundle is the proposition (that depends on x) and a section of that fiber bundle is a proof of that proposition.
The fundamental idea is the same: you have a map in which each input value is defined on a different domain, and that domain DEPENDS on the input value in a "continuous" way. In your example the ideals are the fibers (dependent on the point x) and the ring of continuous functions is the total space S.
The point is: what continuity has to do with formal logic, where we speak only about symbols and rules?
Well, "continuity" in this context means "parametric" function. A "continuous" (or parametric) function is a function that doesn't jump from a formula to the other when you change the point x:
| x = 20 ==> "x is a multiple of 2"
| x = 21 ==> "x is a multiple of 3"
| x = 22 ==> "x is a multiple of 4"
this IS NOT a "continuous" function, because it has different formulations for different values of x
| x = 20 ==> "x is a multiple of 2"
| x = 21 ==> "x is a multiple of 2"
| x = 22 ==> "x is a multiple of 2"
this IS a "continuous' function, because it has the same formulation for every value of x.
The fibers of this bundle are the following ones:
| "20 is a multiple of 2"
| "21 is a multiple of 2"
| "22 is a multiple of 2"
The first and the third fibers are not empty (because they contain some points: the points are the proofs of the respective propositions)
The second fiber instead is empty (there are no proofs of this proposition)
OK, I'll stop here for the moment, because I am a little afraid of the answer "that's all bullshit, I'll not read the rest of it..." :worry:
However, the point is the following: this is an axiomatic theory; the same axioms can have completely different meanings that relate to complete different mathematical objects. Only they happen to behave in a similar way. That's the typical for category theory: every collection of objects that are related to each other in the same way are the same category. It is completely irrelevant what they are "made of"!
Re-reading your question I just realized that probably my answer about homotopy type theory could be misleading: this is really a weird and interesting theory, but topoi and sheaves are NOT related to computer science only because of the use of type theory as a logic to prove theorems! On the contrary, I think the use of formal logic in proof assistents is only a very limited field of informatics, with not many practical applications. On the contrary, sheaves are mostly used in data-analysis applications that have nothing to do with formal logic!
It is true that they can be very abstract objects in mathematics, but for a data-science person a sheave is mostly a data-correlation tool. A sheave can represent a cellular-phone network and relate each cell of the covered area with the set of users that are connected to that cell. Finite-shapes topology is still topology, and computer-science is first of all used for big and discrete sets of data! (here's an example of what I mean: http://www.drmichaelrobinson.net/sheaftutorial/)
OK, that's the "like I'm five" explanation: A topos is a category that "works" as the category of sets, but is not built using a set of rules that operate on symbols as a formal logic system: it's the formulation of set theory in terms of objects, arrows, and universal-mapping properties.
So, there is a main ingredient that is missing: points! Topos theory is a formulation of set theory where sets are not "built" starting from points. Sets (the objects of the category) and functions (the morphisms of the category) are considered as "primitive" concepts. The points are a "secondary" construction.
So, you can describe algebrically what is the intersection and the disjoint union of sets (as product and sum of objects), what is a subset, what is the powerset of a set (an exponential), etc.
The interesting part is that you can describe what are propositions and logical operators completely in terms of objects and arrows, only by assuming the existence of an object with some particular properties, called "subobject classifier". This is NOT the same thing as homotopy type theory, where you build a formal logic system in the usual way: by building strings of symbols with given rules. Here you describe logical operations and propositions in terms of universal mapping properties, as you do with operations between sets. The subobject classifier is the object that represents the "set of all the propositions", and the implications between these propositions are arrows that start and end in this object. Even the logical quantifiers forall and exists are only two particular arrows. Basically, everything is defined in terms of universal mapping properties.
I don't know which details should I add. What part of topos theory do you want me to explain?
What you have said about proving mathematical theorems may be the way computers see the process, but most theorems are not proven by computer programs. I don't see the connection to anything I have encountered in theorem-proving. But I am retired and way behind the times in really abstract math. Maybe the world has changed in my absence. Maybe not. I appreciate your efforts to explain, however.
"I agree that that's not all."
Which has to take a prize as an understatement. Is most of your experience in computer science?
OMG that sort of makes sense. Thank you for that example. I've been reading up on sheaf theory and every presentation that comes up on Google is heavily mathematical, so much so that I can't for the life of me understand how they're teaching this stuff to undergrads in non-mathematical fields. I will definitely read the link you supplied. Perfectly sensible ... to each individual cellphone you associate their call network. So it's an algebraic structure -- a network or graph -- assigned to each element of some set. Very nice example. And of course a great illustration of how wild mathematical abstractions so often end up being incredibly useful in the real world.
And a topos is an abstract sheaf? And what I read was that topoi are inherently intuitionistic. I haven't followed that chain of reasoning yet.
I do not think this is so bad. I'm learning what a sheaf is and after that, topoi are the next step up. If I figure anything out I'll post it. What's interesting is that the highly super-abstract algebra has deep repercussions in logic. Even without any of the details, that's the takeaway.
Be advised that any paragraph containing the name Martin-Löf instantly glazes my eyes. I've had all these conversations too many times. I totally believe everything you say but can't actually figure out what point you are trying to make. I don't disagree with anything you say, and I'm aware in varying degrees with various aspect of the things you talk about. I just don't know why you're telling me this. I don't disagree with you on any of it and I can't relate this to whatever we are talking about.
Can you just remind me what is the point under discussion?
Thanks. I know Goldblatt as the author of Lectures on the Hyperreals. I'll add this book to my growing list of books to read someday. So far the list stretches into several of the next few lifetimes.
I am reading articles on sheaf theory (of the mathematical kind) and it's very straightforward, although most of the serious applications are beyond me. I will read the paper you linked about network topology, that example was very insightful. Steve Awody's book on category theory has a lot of applications to logic. Another book on my list.
Yes, I work as a programmer. And yes, a formal proof completely misses the essence of a proof: it's "meaning". I don't beprove lieve mathematics is changed at all in that respect! Computers do not PROVE theorems: they VERIFY the formal correctness of a proof, that is a completely different thing!
But I even believe that this distinction is not clear at all in today's mathematics, and it has never been understood: there is no definition of the "meaning" of a theorem, and many mathematicians (starting from Hilbert, I guess) think that there is no point in trying to identify the "meaning" as something different from a list of symbols.
I believe this fact is becoming evident, even thanks to artificial intelligence, and will be understood in a not very distant future. By the way, the effect of this misunderstanding of mathematical proofs is terrible for education: teaching only the formal part of mathematics completely misses the essence of mathematics!
That's what EVERYTHING is in category theory! So that didn't tell me anything about topoi!
You wrote me two detailed technical posts that I'll try to catch up to later. But I'm actually on my own path. I found a nice paper on sheaf theory that I can understand and I'm working through that.
I've also in the past seen a video with Steve Awodey where he worked out the definition of a Cartesian closed category then applied it to logic. So I don't even think you have to go all the way to topoi to get to applications in logic. I'm probably going to spend a few days reading papers and see if I can get a hold on a vertical slice of understanding.
Now that is interesting, because fiber bundles are a big thing in differential geometry. It's interesting that they lead directly to type theory. Thanks for pointing that out.
As with your other technical post, I'll defer comment for now but I'll read through them.
A section is a proof ... a section is a proof. I can't quite see that. I know that a section is a right inverse of a function. For example if [math]f(x) = x^2[/math] then a section is a function that, for each nonnegative real, picks out one of its plus or minus square roots. Is that about right? The section is the right inverse, so it's essentially a choice function on the collection of inverse images of all the points. Do I have that right? How does that become a proof?
A sheaf is a topos at the same way as a set is a topos: it's the "trick" of the Yoneda embedding! :smile: do you understand now? (sorry: bad example.. let's say that a sheaf can make everything - more or less - "become" a topos)
I'm going to spend the next week working through this material. I'm encouraged that I can understand what a sheaf is and know a few examples; and I know enough category theory to get by.
May I ask you a question? How does one come to know this material and not have heard of measure theory?
OK, never mind. Sorry for continuing to repeat the same things!
No problem, you've gotten me interested in sheaf theory and then on to topos theory. But I'm still probably more oriented to the mathematical applications than the logical ones.
Exactly! We are speaking about two different kinds of "representations" of logic.
No this is totally fascinating, very clear writeup, worthy of study. I know what fiber bundles and sections are. I can't quite grok the application to proofs but it may come to me.
Remind me of the defs. A section is a right inverse as I understand it. There can be a lot of right inverses to a function, you just keep choosing different elements in the preimages of points. Is that the bundle?
Ok now I know this idea as ETCS: The extended theory of the category of sets, which is an implementation of set theory on top of category theory. Is this the same thing as what you're talking about?
Yes it's wondrous that we can do set theory without talking about points! Deep philosophical implications since we no longer care at all what a thing is, only how it relates to everything else.
I know about measure theory since when I was at high school (I always liked that stuff), and I heard that sentence about the probability to find a rational among real numbers a thousand times :razz:
But then I realized that not everything that they teach you in high school has to be taken as the absolute truth :smile:
Yes! I should have added a formal definition, but I have an aversion to writing symbols on this site :confused: I added a link with a clear picture, I think.
This must be a significant difference between what you do and what a research mathematician does. Recently I've proven theorems related to compositions of functions in the complex plane, and with each I have a deep feeling, a strong sense of meaning, about the result and how the result comes about. A lot of geometrical mental imagery coupled with the essence to which the symbols point - much like reading literature and realizing all those symbols describe something that stirs the imagination.
However, my theorems are not profound - strictly what Wikipedia calls "Low" interest! :cool:
I think when I grok how fiber bundles can be likened to proofs, I'll be enlightened.
Did you learn all this from the CS viewpoint? Just wondering.
I have no idea what question is was a response to. But if you agree with me, you have excellent taste!
Where does measure theory (surely not taught in high school) intersect any of this? I've used it in various integration processes, the most interesting being functional integration. And Feynman constructed his sum of paths integral in more or less that concept.
No, I like "normal" mathematics: no computers involved. But having a theorem-prover as Coq to be able to verify if you can really write a proof of what you think is provable is very helpful.
I write math notes now as a hobby. Here's a couple :cool:
https://www.researchgate.net/publication/319057086_A_Primer_on_the_Elementary_Theory_of_Infinite_Compositions_of_Complex_Functions
https://www.researchgate.net/publication/324017489_A_Short_Note_Compositions_of_Linear_Fractional_Transformation_Forms
I'm reading the data science paper you linked. They teach sheaf cohomology to data scientists. That is so fascinating. To me, with my math background, sheaf cohomology is something that would take someone a long time to learn, at least a couple of years of grad school or more. But at the application level, the concepts have filtered down and you don't have to actually know any of the original mathematical context in which these ideas evolved. It's yet another illustration of the applicability of highly abstract math. The "unreasonable effectiveness" all over again. Category theory dates from the 1940's but it's only peeking its head into the real world in the past twenty years.
Oh, it doesn't. I mentioned that the computable numbers have measure zero in the space of bitstrings, and @Mephist asked me how that's defined mathematically, so explained it a little. Then he responded by saying he's known about this since high school. So @Mephist I apologize if I misunderstood you.
The subject came up in the context of my perennial hobbyhorse that there aren't enough computable numbers to make up a decent continuum; and why aren't the constructivists ashamed of themselves. I have never gotten an answer to this question that satisfies me. At some point they say "Martin-Löf type theory" and I know I've lost the argument. It's a cult. [mild humor intended].
Yes.
Ok a fibre bundle is the collection of all possible right inverses to a function. And a section is one of those right inverses. Yes?
I may be off the mark here but I wonder if it's relevant that we have introduced a little bit of nonconstructivism. How do you know you can always take a section of a surjective function? That claim is equivalent to the axiom of choice. It makes sense. Take the cosine or sine functions. Each real number between 0 and 1 has infinitely many inverses, separated by 2 pi. For each point in [0,1] we choose an element from its inverse image. So you can see how the axiom of choice comes up. Of course in this particular case we could take the smallest positive inverse, so we don't need the axiom of choice. But in the general case we do.
I can't see this. Can you give an example?
Quoting Mephist
I didn't understand which side you're agreeing with. Do you think that the meaning of a proof is to be found in its syntactic form? Or that mathematical meaning goes beyond formalization?
Ok back to the first point. Totally simplistic example. [math]f(x) = x^2[/math]. The fiber bundle is all the pairs like [math]\{5, -5 \}[/math] and so forth, all the sets that are the inverse images under [math]f[/math] of the various points on the real line.
A section is one particular choice of elements, like always choosing the positive square root, or flipping a coin to determine which of the two square roots to choose.
Now I am squeezing my brain but I don't quite see how the collection of all sections is a proposition, and an individual section a proof.
I hope this example is detailed enough so that you can straighten out any misunderstandings I might have about what's a fiber or section, and what's a fiber bundle. And how proofs relate.
This is plenty for now! I understand parts of all that from various perspectives. One question, homotopies are equivalence classes of paths. It's a topological notion. Wasn't sure what you mean by strings of symbols in that context.
I understand a subobject classifier as the set {0,1} that characterizes a subset of a set, say, as a map from the subset to 1 and everything else in the set to 0; that is, the characteristic function of the subset.
I'm still not totally seeing the part about propositions and proofs. But I must be getting close.
Wait -- the subobject classifier is the set of ALL the propositions? Not sure I'm following that.
"the implications between these propositions are arrows that start and end in this object." -- Tht seems to imply that the propositions form a category, and the arrows are implications. That much I know from Awodey. You lost me on the subobject classifier in this context; and how this relates to a section of a fiber bundle as a proof of a proposition. But I feel like I must very close to getting this.
I don't think I agree with your assumptions about the role which mathematics has. I believe that mathematical principles are always developed for purposes, goals, ends, and therefore utility in the physical world. Now, I discussed this to some extent in another thread, and I was shown that this is not necessarily the case. It was demonstrated that mathematical principles, as art, may be created purely for their beauty, without any intent to purposefulness, and then found to be purposeful posteriorly. Of course these arguments are still debatable, and we probably cannot get to the bottom of this. Even if a person creates a mathematical principle without intent or purpose, the demonstration of that principle to another person would be done with purpose. So the fact that we can argue that mathematical principles may be created without a purpose, is really just a demonstration of free will. When there is no necessary relation between the effect, and the cause, we have a demonstration of free will.
So I have a problem with your example, in the sense that it seems unrealistic. The purpose of the example is to illustrate something, to argue a point. But if you assume unrealistic "materials" to make that argument, how can the argument be acceptable? You have chosen a particular type of "material", one sufficient to make your argument, but one which is not real.
Quoting simeonz
So my argument was that "actual physical structures" do not approximate the mathematical ideals, in your example, because the "materials" are completely different. So, further to your example, I think that this problem is widespread in physics. The physicist applying mathematical principles believes that the "materials" of mathematical ideals are a good approximation of the "materials" involved in actual physical structures, when in reality they are not. This mistaken notion is a holdout from Platonism. Platonism assumes that the mathematical materials 'are' the materials of physical structures. When we let go of Platonism, we have a desire to maintain some semblance of truth within our conceptual structures, so we maintain that the mathematical materials provide an approximation of the actual physical materials, but there is no justification for this. "It works" justifies the means to the end, but it does not justify "an approximation of the actual physical materials".
Quoting simeonz
This is the point we need to iron out then. Let's assume that human beings came into existence through evolution. The kernel to the concepts which are "seeded", are what is instinctual to us, but these must have been produced through evolution, therefore they are purposeful. If there is always a seed prior to the instinct, and that seed is not necessarily for a purpose, then we'd go right back to the first life form and ask where did that first seed come from. So it really doesn't seem to make sense to say that the concepts are "seeded" by nature, and they are not representative of nature.
How could we proceed from here? If the seeds are part of nature, then we fall into Platonism. If they are something other than natural, wouldn't they necessarily have to represent something natural in order to be at all useful? How would a concept which has absolutely no representation of anything real become useful?
OK, I'll start from the example:
1. Let's call Unit our terminal object.
2. Let's call Prop the object that is part of our subobject classifier (usually called Omega, but I prefer Prop, from Coq's convention).
3. Let's call "true" the arrow from Unit to Prop that is part of our subobject classifier.
4. Assume that there is another object, called Nat.
[ this is the set of natural numbers ]
5. Assume that there is an arrow "s" from Nat to Nat.
[ this is the "successor" function, from Peano arithmetic ]
6. Assume that there is an arrow from Unit to Nat called "zero"
[ this is the number zero ]
7. Let Nat->Prod be the exponential of Nat and Prod
[ this represents all propositions with a free variable of type Nat ]
8. Let Nat->(Nat->Prod) be the exponential of Nat and (Nat->Prod)
[ this represents all propositions with two free variables of type Nat ]
9. Let "x_is_greater_than_y" be an arrow from Unit to Nat->(Nat->Prop)
[ this is a proposition with two free variables of type Nat ]
10. Let's call "x_is_greater_than_zero" the arrow "x_is_greater_than_y" * "zero"
[ * is the operation of arrow composition from category theory's axioms ]
[ "greater_then_zero" will be then an arrow from Unit to Nat->Prop ]
[ this is a proposition with one free variable of type Nat ]
11. Let's call "s(x)_is_greater_than_zero" the arrow "x_is_greater_than_zero" * "successor"
[ this is still a proposition with one free variably of type Nat ]
12. Let's call "one_is_greater_than_zero" the arrow "s(x)_is_greater_than_zero" * "zero".
.......
Sorry, I thought it was easier: I wanted to show you the complete structure of the category representing the proof of the proposition "2 > 0" derived from "forall x, s(x) > x" and "forall x, x > 0" but I didn't realize that it would take me ages to describe it in this way... and at the end it will be impossible to read. In Coq this is several lines of code, but is build automatically from a couple of commands. In reality, I never look at the real terms representing the proofs: they are built automatically. But I see that it would be a pain to build them by hand in this way!
I have to find a better way to give you a description of how it's made without writing all the details... :worry:
Certainly some sort of goals, but not necessarily physical ends. To a large extent it's curiosity about "what comes next?"
Let me just give you just some examples: "x >= 3" is a fibration from the object Nat to the subobject classifier Prop. The proposition "5 >= 3" is a fiber of this fibration. The elements of this fiber are the "proofs" that 5 >= 3. So, there will be a function F (built using the rules of logic) that allows you to build a proof of 3 >= 3, and a proof of 4 >= 3, and a proof of 5 >= 3, etc.. (meaning: to choose a proof for each proposition of the form "x >= 3"). This function is the section of the fibration from the space of proofs to the space of propositions of the form "x >= 3": it selects a proof (an element of the fiber) for each proposition of the form "x >= 3".
The function that generates all the proofs of the form "x >= 3", however, is not an algorithm that chooses a different list of rules for each value of x: it doesn't say "if x = 3, then apply the rule A to prove "3 >= 3";
if x = 4 then apply the rule B to prove 4 >= 3". Instead, it's a parametric program that says: if you have an x with such and such characteristics, then you can apply the rule A(x), where x is a variable of the rule. That means that the section of the fiber is "continuous".
Propositions containing variables (such as "x >= 3") are called "dependent" types because they are sets (of proofs) that depend on a parameter (x), and the function that is used to build the proof of "x >= 3" builds an element of a different type (a proof of a different proposition) for each value of x (builds the set and selects the point at the same time).
Fiber bundles of "normal" topological spaces (such as the famous Mobius strip - https://en.wikipedia.org/wiki/M%C3%B6bius_strip) are built in the same way: you choose a disjoint segment (set of points form a total space E) for each point of a circle (base space B) using a continuous function!
It's much more difficult to explain than I thought...
Some references:
https://ncatlab.org/nlab/show/subobject+classifier
https://ncatlab.org/nlab/show/type+of+propositions
P.S. notice that this form of the subobject classifier is typical only for type theories, where the propositions are represented by the collection of all their proofs:
( from https://ncatlab.org/nlab/show/type+of+propositions ):
"In type theory the type of propositions Prop corresponds roughly under categorical semantics to the subobject classifier. (To be precise, depending on the type theoretic rules and axioms this may not be quite true: one needs propositional resizing, propositional extensionality, and — in some type theories where “proposition” is not defined as an h-proposition, such as the calculus of constructions — the principle of unique choice?.)"
For classical (boolean) logic, instead, the subobject classifier is much simpler: it has only 2 incoming arrows (elements) from the terminal object: "true" and "false". That's because in classical logic, differently from type theory, the propositions are represented only by their truth value (true or false), and not by the list of derivations (or programs) used to prove them.
Yet saying that there is no way to enumerate the total computable functions is somewhat ambiguous, for as previously mentioned we can use brute force to simulate every TM on every input and enumerate on-the-fly the algorithms that have so far halted on all their inputs. Furthermore, the classical logician with realist intuitions will go further to argue that there is a definite matter-of-fact as to the set of computable functions and will therefore believe in the independent existence of a 'finished' enumeration, interpreting the limitations of finite constructive arguments to produce such an enumeration as being epistemic limitations rather than metaphysical limitations. Instead they will simply appeal to the Axiom of Choice to claim the independent existence of completed enumerations of the computable total functions.
As we said earlier, this full enumeration cannot be used if the diagonal function d(x) is to be both computable and total; otherwise if the realist diagonalizes the hypothetical full enumeration of computable total functions, then d(x) cannot be computable, for the enumeration ensures that d(x) is total. That d isn't computable is obvious, since it involves running nearly every Turing Machine for an infinite amount of time and then diagonalizing, meaning that it's godel number g is infinitely long and that it assumes an infinite value when evaluated at d(g)=d(g)+1
Maybe I am not on the same page with you here. Before I try to answer in any detail, I have to get clear on the substance of our conversation. Are you saying that you imagine a system of computations which does not involve algebraic numbers, and either has the same utility for our industrial and daily applications at a discount computational cost, or enhances our conventional applications at no additional computational cost? Or do you mean that the use of irrational numbers is conceptually inaccurate with respect to a first-principles analysis of physics? If the latter, as I said, I don't think that it matters for mathematics.
What I wanted to point out is that the sentence "there is an uncountable number of sequences", when expressed in the forall-exists language, does not contain the word "uncountable". And you can't say it in an unambiguous way. This is our interpretation in natural language of the sentence "forall lists of functions Nat->Nat there exists a function Nat->Nat that is not on the list". So, there exists ONE missing function. You are adding one function at a time! The translation of this sentence in natural language with the word "uncountable" is arbitrary, in my opinion.
As the induction principle "(P(0) and (forall N, P(N) implies P(N+1))) implies (forall N P(N))" can be seen as part of the axiomatic definition of the world "forall", at the same way there should be a second axiom that can be seen as the axiomatic definition of the world "uncountable": the same is done in modal logic with the word "necessarily". Doesn't it make sense?
Yah.
Quoting Mephist
Exactly what I was thinking as I read your exposition.
You know you still haven't told me exactly how you came by all this information and it's relevant to our problem. I spent last night reading through a couple of technical introductions to sheaf theory and (a) it is mathematically sophisticated, and (b) not that easy to get hold of. One viewpoint is that a sheaf is an abstraction of the charts and atlases of a manifold. Also it's an abstraction of differential forms and exterior algebra. That's the differential geometry point of view. It's connected with cohomology, which is a complicated subject in higher algebra and algebraic topology. A sheaf is a very sophisticated mathematical object.
You seem kind of glib about what you know, without actually convincing me that you know the mathematical aspect of what's going on. That's why I asked about your background. You sound like you know stuff but at some level you haven't convinced me. Not that you don't know stuff, but that you don't know the larger mathematical context sufficiently well to explain it.
Does that make sense? That the explanatory gap is because you know the logic but not the math. That is the sense I'm getting. I just trying to understand why your attempts to explain this stuff to me are so frustratingly opaque at my end.
As far as I know, Nat is a natural number category and not an object. Or maybe it's an object. Why don't you explain yourself more clearly. You jump in with jargon and frankly it makes me wonder how much you know. That's why I asked.
I really want to understand what you have to say, so I am not trying to give you a hard time. I'm asking you to write me some decent exposition. Start at the beginning.
* What's Nat?
* What is Prop? Is it the category of propositions? Is it an object in some other category? And in what way is it a subobject classifier, analogous to {T, F} in ordinary logic?
* What's a fibration? We've talked about fibers and sections and fiber bundles. I know about the Hopf fibration.
Can I ask your background again? Now I'm thinking that maybe you learned Coq but don't know the larger context of all these ideas. Or read a lot of Wiki pages. I don't mean to be provocative but in some vague way you are not convincing me that you know what you know.
I'm not saying that to give you a hard time. I'm saying that because I really want to understand what you're saying. If you tell me what you know and how you came to know this particular vertical slice of categorical logic, I can better understand where you're coming from.
Nat is the type of natural numbers. Types are represented by objects of the category, derivations are represented by the arrows. That's in the book from Awodey that you said you have read
Prop is the type of propositions. It's an object of the category. {Prop, true} is the subobject classifier of the topos
I know Coq. And I know type theory because it's the logic implemented in coq. And type theory is the internal logic of a topos. I read some books about category theory, because it's important for computer science. I am interested in mathematics and physics as an hobby, and I often read new publications from arxiv.org
I get that. So when you use jargon from categorical logic you don't know the greater mathematical context; and that's making it harder for you to explain your ideas.
I don't know a freakin' thing about type theory. I think it's time to drop this.
Thank you. That's what I suspected. It's a particular point of view. There is a much larger mathematical context in which these ideas developed, that you're not aware of. These ideas date from the 1930's and 40's. The CS people have discovered it recently and it's like a new toy. I have to admit I have spoken to other constructivists who learned math from Coq. Or was that you, a couple of months ago on this site.
So this is frustrating for me now, and probably for you.
If you know the categorical definition of something but you can't explain the bottom-up concept, you should fill in the blanks in your knowledge. The math you don't know, is why you can't explain this stuff to me. Like I said I did some heavy sheaf-theoretic lifting last night and still can't understand you. We're working different sides of the street.
No not at all. I'm vitally interested. I wonder if you'd be willing to meet me halfway; and realize that what you've learned in the abstract does not constitute knowing a lot about math. You know categorical logic. That's not the same thing.
The Wiki page gives some clues as to the difficulties ahead.
https://en.wikipedia.org/wiki/Sheaf_(mathematics)
This is helpful:
https://math.stackexchange.com/questions/2642231/what-is-an-intuitive-concise-explanation-of-a-sheaf
I've been looking at these two pdfs:
https://mast.queensu.ca/~andrew/teaching/math942/pdf/2chapter1.pdf
https://tlovering.files.wordpress.com/2011/04/sheaftheory.pdf
I'm not saying you need to know any of this. Just that ... I don't know. If you can go slower, define your terms, take things one step at a time, that would be helpful to me.
Quoting Mephist
I'd like to understand what you mean when you say Prop is a subobject classifier; given that all I know is that a subobject classifier is {T,F} in elementary logic or defining a subset of a set.
If you are talking about type theory, that won't be helpful to me.
Here's how to visualize it. You and I have almost disjoint backgrounds in math. From the point of view of math, categorical logic is off to the side somewhere. Mostly of interest to the computer scientists these days and not mathematicians. I think that if you could explain your concepts to me, you'd understand them better yourself. A sheaf, for example is defined only on the open sets of a manifold. So my example earlier of [math]f(x) = x^2[/math] is a little off. I was taking the fibers over the points; but sheaves are defined only over open sets. That's something I'm confused about at the moment.
It must be interesting to understand all of what you know, but without any of the mathematical context. Pretty abstract but perhaps this is just a matter of modern conventions versus classical. I know a little category theory but my brain's not hardwired for it as it is for people learning their math from a categorical perspective to start with.
So that's the explanatory gap. But it's not hopeless. You said that a proof is a section of a fiber bundle made up of propositions, or something like that. I believe it's possible for me to understand this in terms of things I know. Tell me what space we're working in. Define some terms clearly. This can be done. And tell me why Prop is a subobject classifier. In what was is a set or category or type of proposition, analogous to {T,F}?
A subobject classifier is a pair of an object and an arrow {Omega, "true": T->Omega} with the following property: every monomorphism m: A->B in the category (in the topos) is the pullback of the morphism "true" along a unique morphism x:B->Omega. It is isomorphic to a set with two elements in the category of sets (that is a topos), but not in all topoi.Quoting fishfry
Quoting fishfry
You don't need a manifold, you need only a topological space. A manifold is a topological space plus an atlas of continuous maps, right?
Quoting fishfry
Yes, that's the central point of the whole story: open sets are "more important" than points. In the category of sets an object is "made of" points, but in a generic topos this is not the case. In a topos, a point (of an object X) is an arrow from the terminal object to X. In the category of sets the terminal object is the singleton set and the set of arrows from the terminal object to X is isomorphic to the set of points contained in X, but this is not true in general for a topos.
P.S. I think I know what's the main reason of confusion: you are talking about a sheaf in the category of sets (no category theory needed: only topological spaces defined in set theory), an I am talking about a sheaf defined on a generic topos (category theory needed, set theory irrelevant)
Ok. Consider this. Earlier I gave the example of the ring of continuous functions on [math]\mathbb R[/math], and the fact that the zero-set of a point is an ideal. And you responded by saying that's a fiber. Which I think it is.
But it's not actually a good example of a sheaf, because sheaves are only defined on open sets. So now you are saying, "Oh yeah it's about open sets," but earlier you didn't remember that you know that, and you were seduced by my example and didn't realize it was inaccurate.
Likewise my example of the inverse images of [math]f(x) = x^2[/math]. I made the same error, taking inverse images of points. I think in differential geometry that's ok. But frankly I don't know much differential geometry either. My mathematical ignorance is vast.
That's what I mean by mathematical context. You are not being precise enough in your formulations, and that's making it harder for me to latch on to the ideas. So should I be thinking of a manifold with charts? Is the atlas the fiber bundle? I have no idea.
Re comma categories. I checked out a pdf book on categorical logic. In the table of contents they get to comma categories right away. In mathematically oriented category theory books, they get to them much later, and I have read the definition once but didn't understand enough to remember it.
In general, categorical logic seems to be taking a huge leap that bypasses several years of serious math study. So as I say, our knowledge is virtually disjoint. But a bridge can be built I'm sure.
By the way, I saw an Awodey video. I started to read his book but it was too oriented to logic for my taste so I spend more time looking at Leinster. So you see even by inclination I have remained ignorant of categorical logic. I'm trying to work my way up from the example of the ideals of continuous functions defined on open sets, not zero sets of points. That I believe is accurate. Now how do I shoehorn logic into that?
Quoting Mephist
That is a brilliant explanation. Totally lost on me.
I explained that what I know about subobject classifiers is this:
* {0,1} is a subobject classifier. That means if I have a set, say, I can define a subset by its characteristic function. That is, the function that maps the elements of the subset to 1 and everything in its complement to 0.
* In general you can use any interesting set as a subobject classifier, generalizing true and false.
* I can imagine that if we had a collection of propositions, and the {0,1} set, we could label propositions true or false.
Now that is what I know. Why don't you start from there.
By the way, when you quote chapter and verse of the technical definition of subobject classifier but don't say a word about what it means; is it because you don't relate to the meaning in some way? Or think the meaning is obvious from the jargon? Or know the words but not the meaning? Or think too much of my knowledge of category theory? I could unpack that definition if I wanted to. I know that much. But I'd have to work at it, and in the end I still wouldn't know what's a subobject classifier. I need to know how you are shoehorning logic into sections and fibers.
Please tell me if I'm being rude. Your communication style is very ... confusing to me. You lay down that paragraph as if you think you answered my question. You didn't. That's the mystery I want to clear up. Maybe it's just a style. You have all the textbook defs memorized but you are not telling me what they mean or what the ideas are.
It wasn't inaccurate, it was a particular case, as you usually do when you give an example..
Quoting fishfry
Yes, but you can easily find the precise definitions on Wikipedia. I usually don't think in terms of precise definitions.
Quoting fishfry
That's the advantage of category theory in comparison with set theory: it's more general, but even more simple: it speaks only about the essential features that are needed for proofs to work, and ignore the particular "implementation" (sorry, again a computer-science term). You don't think about real numbers in terms of set-theory, right?
Quoting fishfry
OK. I have to go now..
Yes but I haven't time right now to learn the category theory I'd need. I see a vertical thread of understanding from the idea of a fiber bundle over a manifold, to seeing how that idea generalizes to logic. You've pointed me in that direction several times. So it's not a matter of convincing me that your way is better. The only question is whether you want to explain this to me so I can understand it. I'm pretty close. Tell me the topological space, tell me the map from the open sets to some collection of algebraic structures, that represent propositions and proofs.
Right? A sheaf assigns to each open set of a topolgoical space, a data structure or algebraic object. Tell me the topological space, tell me the map, tell me which structures, represent propositions and proofs. I think that's a specific question we can meet halfway on.
I daresay one could put in a word for set theory as an antidote to too much abstract thinking! :-)
Quoting Mephist
Time for bed here too. I feel hopeful because I just articulated a very specific mathematical goal we can achieve. Top space, mapping, algebraic objects or data structures being attached to the open sets. That will clarify a lot of things for me.
Hmm... :chin: You want a topological space for classical logic. OK, a topological space is a set of all subsets of an "universal" set.
- The elements of the universal set are tuples of elements of our model (the things that we are speaking about: real numbers, for example).
- The subsets are our propositions: they represent the set of all tuples of elements for which the proposition is true (an example here: the proposition "3x + y = 6" is the set of (x,y) such that the equation is true).
- Inclusion between the subsets represents implication.
- Functions are represented in set theory as particular sets of pairs (surely I don't have to explain this to you).
- Relations are sets tuples of elements of our domain.
- There are some distinct points that correspond to the constants of the language.
- The set operations of Intersection, Union and Complement form a Boolean algebra on the subsets of the topology. ( no problem until here, I hope ).
Only in this case, what for is a topology needed? All subsets of this topology are both open and closed. This is a discrete topology (the most particular case of all).
And then the main problem: what about quantifiers? (forall and exists).
You don't want category theory, right? The quantifiers are naturally defined as adjoint functors in category theory, but you said you want only set theory. :roll: so I should reformulate the condition of adjunction of categories in terms of set theory... at first sight it will be a definition that will have to include in itself the algebraic structure of... a category! ( I don't know how to do this :gasp: )
P.S. of course you cannot allow infinite expressions (such as "forall" is an infinite intersection...), since our language is made of strings of symbols.
And then the subobject classifier is the usual set {"true", "false"} plus an evaluation function that for each proposition (subset) returns a value of the set {"true", "false"}
What about sheaves? In this case sheaves are unnecessary too, because we are in a discrete topological space.. I don't know if this program would bring some useful insight really, even if I am able to figure out how to find an algebraic structure on our "topological space" that includes quantifiers :sad:
I need at least the category of sets to be able to include logic, but sheaves are not really related to boolean logic, for what I know.
P.S. I forgot about proofs. Proofs in standard logic are not objects of the model (sets in our case), but it's only a partial order relation between our propositions determined by the rules of logic. Different situation in type theory, where they are represented as objects of our category - meaning: you cannot speak about proofs in standard logic; instead, you can speak about proofs in dependent type theory. And that's why the subobject classifier is not a simple set of values: you have to say not only if a proposition is true or false, but even what is it's proof.
Suppose we distinguish between "what comes next" in the physical world, meaning what happens in the next moment of time, and "what comes next" when you're counting 1,2,3,4,5..., meaning "6" is what comes next. The latter is really just a form of the former, you saying "six" is just an instance of what happens next in the physical world. One could argue that there are differences between natural, and artificial, in what comes next in the physical world due to intention and free will. But if such a difference is a real difference, we would have to validate the reality of the difference through some dualist principle like "the soul", which reaches in from some non-physical world to interfere with "what happens next" in the physical world. Otherwise "ends' are nothing other than physical ends.
So Platonism comes back on us, as 'the good", meaning what is intended, striven for. If we situate this "object", the end, the goal, in the non-physical, such that "what comes next" is a creation from the non-physical, rather than a natural process, then we cannot avoid dualism. However, we must maintain respect for the fact that the natural physical world imposes restrictions on what is possible, as a goal, object, or end. So for instance, the idea of "infinite possibility" is really an impossibility, due to those restrictions, and is therefore a contradictory idea. One might define "possibility" as a sort of logical possibility to avoid such a contradiction, but since what is possible and what is impossible is ultimately dictated by the physical circumstances, such a move would be an act of self-deception, telling yourself that "logical possibility" is not limited by "physical possibility'.
Quoting simeonz
I think that this is the more accurate representation of what I was saying. I don't agree with your conclusion though, because it requires that we make a complete separation between physics and mathematics. Suppose we assume such a separation. Mathematicians just dream up their axioms and principles for no apparent reasons, just because they are beautiful or something, so that the mathematical principles are somewhat arbitrary in this way, pure mind art. They leave these principles lying out there, and the physicists pick and choose which ones they want to use. It's like a smorgasbord of tools lying on the table, which the physicists can choose from. That's a fine start, but we must respect the fact that the process is reciprocal. So once the physicists choose their preferred tools, and start using them, then these are the principles that the mathematicians are going to concentrate on improving and fine tuning. Evidence of this is the fact that the real numbers came from the use of rational numbers, and the use of geometrical principles which created irrational numbers. Mathematicians did not have to allow irrationals into their numerical principles, but they were being used, so the mathematicians felt compelled to incorporate them . Therefore I think it is incorrect for you to say that a first principles analysis of physics is irrelevant to mathematics. .
Quoting Metaphysician Undercover I am not saying that it necessarily has to be pure mind art, but it wouldn't change the assessment we make of the final product. Even if our improvements in the scientific method (the philosophy of mathematics) remove obstacles to its continuous development, it should not come at the cost of the numerical properties of the prescribed computations. Just as long as the scientific method produces a framework of computations and analysis for our varied conventional applications that conserves the known accuracy-complexity tradeoff, the improvement can venture in any desired direction. But exploring concepts and algorithms of computations whose conventional (daily and industrial) utility is inferior to the ones currently in use is not an option, no matter what methodological grounds we have for that.
Quoting Metaphysician Undercover This is true. But mathematics is not focused only on fundamental physics. It serves economics, construction engineering, software design and electronics, government, etc. All those fields are served on equal footing by mathematics. Physics is undoubtedly the most fundamental of sciences. But that does not mean that mathematics is physics oriented. It serves any situation involving computing and conceptualizes theories that unify as many applications as feasible. If a given field requires specific axiomatic framework, it needs to engage the mathematical community deliberately and the other fields do not have to be impacted.
Quoting Metaphysician Undercover Yes, but this happened in times when people believed the earth was flat. How irrational numbers were originally conceived is irrelevant to their contemporary meaning.
Quoting Metaphysician Undercover The way I interpret irrational numbers today is as a property of the process you use to compute approximate quantity when a diagonal, perimeter, the intersections, etc, of various objects of various shapes are involved. The question is not what best explains the underlying physics, but how can we compute economically and socially interesting factors - the amount of material necessary to manufacture a part in the desired shape, the heat transfer occurring at the surface of a container, the sliding resistance a material with a given shape has, etc. We are not investigating the underlying physics, or trying to achieve precision in excess of what we need, but merely seek an efficient method for computing values within the desired accuracy. Irrational numbers are our "design" of this method, not our "interpretation" of the objects involved. We can still investigate the physical fundamentals, but this is not a concern for our procedures.
I have many questions. Let me first say exactly where I'm coming from, and what I'd eventually like to understand.
You said a while back, in a remark that started this convo, that you can do logic in a topos. I'm curious to understand that in straightforward terms. It doesn't seem that difficult once you know how the mathematical structures are set up.
So we start at Wiik:
Ok. So I need to find out what a category of sheaves of sets on a topological space is. But it can't be too complicated, because "Topoi behave much like the category of sets" and of course I happen to know a lot about the category of sets. So let's go read up on sheaves.
Wikipedia has a decent article but I find this Stackexchange quote very simple and clear:
It's ok if this isn't fully general or there are various other ways of looking at it. What we're trying to do is simply begin the process of developing some intuition about the subject.
So now you tell me you can use a topos to do logic. Of course I'm perfectly well aware this is true, but what I'm looking for is a straightforward exposition of how you tie your ideas to these definitions.
So, if a topos is like a category of sheaves, you must have some sheaves lying around. So what is the underlying set, what is its topology, what algebraic or other objects are being associated with open sets, and what is the mapping?
You say it's the discrete topology but you didn't convince me. Let me make some specific points.
Quoting Mephist
There is no universal set, so this needs clarification.
Quoting Mephist
Oh ok you are using "universal set" to mean the universe of discourse or the base set. Ok. So you have a set, which consists of tuples of elements. 1-typles, 2-tuples, 3-tuples? Need to be specific so I know what this set is.
But since we want to be able to do n-ary predicates in logic, maybe you want all possible finite n-tuples. So we have a set, let's call it [math]S = \mathbb R \cup \mathbb R^2 \cup \mathbb R^3 \cup \dots[/math]. That ok with you? It's all the 1-, 2-, 3-, etc. tuples of real numbers.
Quoting Mephist
Ok. Interesting. Question. How do you know what the propositions are? Suppose I have the subset {(1), (2), (3)} where those are three 1-tuples. What proposition is that the answer to? There are infinitely many for each tuple I'd think. For example "x = 1 or x = 2 or x = 3" is one such proposition: and "x is an integer strictly between 0 and 4" is another.
Also I think you have a "type error" in the sense that if you have a subset {(12), (3, 4, 5)} then you are conflating binary propositions with ternary ones. Does that cause trouble? Is this the model you are intending to communicate here?
- Inclusion between the subsets represents implication.
- Functions are represented in set theory as particular sets of pairs (surely I don't have to explain this to you).
- Relations are sets tuples of elements of our domain.
- There are some distinct points that correspond to the constants of the language.
- The set operations of Intersection, Union and Complement form a Boolean algebra on the subsets of the topology. ( no problem until here, I hope ).[/quote]
Ok. So you give the set of tuples the discrete topology, so that all sets are open. And to each set, you assign ... what was it you assign? Did you say?
Quoting Mephist
We need a topology because that's how we define a sheaf; and a topos is "category that behaves like the category of sheaves of sets on a topological space." In the end I need to understand this example in terms of the definitions of topos and sheaf. This is my mission.
Quoting Mephist
I'm not asking you to explain adjoint functors. I just want the broad outlines first of the sheaves involved and how topoi are "like" sheaves and what that means. You have the discrete topology on the set of all n-tuples of (say) real numbers; and a set of tuples represents (some, all, a random example of) the propositions that the subset satisfies. Modulo the confusion of 2-tuples and 3-tuples in the same subset.
Quoting Mephist
Ok.
Quoting Mephist
This is confusing. I believe that {T,F} is a subobject classifier, it's the only one I know. But then you say "for each proposition(subset)". I'm confused right there because subsets correspond to infinite collections of propositions. Are you perhaps equivalencing them? Or some other detail needs to be clarified?
Quoting Mephist
Well, you said you can do logic in a topos. A topos is "like a category of sheaves," and a sheaf assigns algebraic objects to open sets of a topological space. So in order to make your exposition legitimate, you have to say how you are using the topos/sheaf model to represent propositions.
So far we've got the set of n-tuples and some questions. Progress.
Quoting Mephist
Topoi are "like categories of sheaves" so if there aren't actually any sheaves around, maybe that should be clarified.
Quoting Mephist
Ok. I believe you are correct as far as it goes. I'm just trying to drill down these ideas to things I know. You said that proofs were fibers or sections or something like that; and that remark evokes manifold theory, and manifolds with charts seem like sheaves, to each open set you assign a copy of Euclidean space through a chart. So I believe there's an opportunity to flesh out this story from the bottom up as it were.
Discrete topology on the collection of n-tuples. So far that's what I've got, plus questions. But that's progress.
Lots of wiggle room in that "something." "Pure mind art" is good! :cool:
There are most definitely reasons. Penelope Maddy, the foremost authority on the philosophy of set theory, has a pair of papers, Believing the Axioms I and II, that describe the historical context and philosophical principles behind the adoption of the ZFC axioms. You might find these of interest.
https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf
https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms2.pdf
Sure, but the problem is that it does not actually exist as "pure mind art". The scientists, engineers, and others who use the mathematics force the existence of conventions which form the artist's medium. Then the mathematicians work with the existing conventions, and those conventions are a pollution to the notion of "pure mind art". Notice for example, that artists work with accepted (conventional) media, paint on canvas, music, etc.. The use of such conventions is a self-imposed restriction on the artistic expression, the artist chooses a medium. So if an artist chooses "mathematics" as a form of artistic expression, then by the very fact that it is mathematics and not some other art form which has been chosen, the artist has already restricted oneself to the use of certain principles, as the medium.
Quoting fishfry
That's what I am arguing, there are always reasons for such choices. If we insist that a true relation between the axioms and the reality of the physical world is not necessary in such choices, then I believe we restrict our capacity to properly understand the reality of the physical world.
Thanks for the references, I'm going to try to read some of that material. it looks interesting.
"Conventions" ? Can you be more specific?
Although the physical sciences have influenced quite a bit of mathematics, the intervention in the artistic process of mathematics as a medium that necessarily pollutes "pure mind art" is debatable. How does pure mind art make it to the public domain? Must it always involve sculpturing with one's bare hands? Or painting with colored oils that are extracted from plants? Or wait, for a novelist, does it entail writing out one's work with a pencil?
Here's the explanation in straightforward terms: a topos is an "extension" of the category of sets.
( probably it should have been called "setos" :grin: )
First of all, then, you have to know how you can do logic in the category of sets.
The category of sets is the category that has as objects ALL the sets and as arrows ALL the possible functions from any set to any set.
We just know how to do logic in ZFC set theory ( :confused: or should I start from the standard first order logic in ZFC? ), so we could do exactly the same thing here, but there is a problem: there are no "elements" inside an object ( the sets are represented by objects, but objects in category theory are a primitive notion: they are defined axiomatically, and not by describing how to build them starting from elements ).
However, there is a way to "represent" all the the logical operations of set theory ONLY in terms of the objects and arrows of a category. Here's the mapping:
- The empty set is represented by the initial object of the category
- The singleton set (all singleton sets are isomorphic, so there is only one singleton set, up to isomorphism) represented by the terminal object of the category
- An element of an object A is represented by an arrow from the terminal object to A.
- The the cartesian product of two objects A and B (the set of all ordered pairs of elements (a,b)) is represented by the categorical product of A and B
- The disjoint union of two objects A and B (the set that contains all the elements of A plus all the elements of B) is represented by the categorical coproduct of A and B
- The set of all functions from A to B is represented by the categorical exponential of A and B (notice that the set of all functions from A to {1,2} is isomorphic to the set of all subsets of A)
.....
There is a way to represent EVERY operation in set theory in terms of an universal property ( https://en.wikipedia.org/wiki/Universal_property ) in category theory, and the two representations work exactly in the same way... except for a detail: we cannot distinguish isomorphic sets between each-other. Meaning: we can distinguish them using the language of set theory (meaning: the set { {}, {{}} } is different from the set { {}. {{},{}} } in set theory, but in terms of universal properties, all sets that contain two elements are indistinguishable: you cannot even say how many two-element sets there are.
The arrows, instead, are assumed (by the axiomatic definition of a category) to be all distinguishable from each-other.
So, that's it! Category theory can be used to "represent" the operations used in ZFC set theory (except for this last limitation).
OK, so what for is the "topos"? The point is that WE DON'T WANT TO ALLOW THE USE OF EVERY POSSIBLE UNIVERSAL PROPERTY. We are looking for the MINIMAL SET OF UNIVERSAL PROPERTIES that are enough to be able to represent the operations used in set theory.
Well, it turns out that the minimal set is the following one:
(taken from Wikipedia: https://en.wikipedia.org/wiki/Topos )
"
A topos is a category that has the following two properties:
All limits taken over finite index categories exist.
Every object has a power object. This plays the role of the powerset in set theory.
"
There are a lot of equivalent definitions, but this is the simplest one that I found.
A topos is a category that contains the minimal set of universal properties necessary to encode all the operations required by the language of first order logic (including quantifiers).
Now, the most important point: the derivations that you can produce using this limited set of constructions are not all the derivations of classical logic. And you can build A LOT of different categories with the property of being a topos, by adding different requirements to the basic set of requirements called "topos".
At the same way, you can say that a given mathematical object is a "group" by giving the minimal set of operations and properties that a group must have (you have to be able to take inverses, to form products and to distinguish an element called "unit"), but then there are a lot of different additional requirements that you can add to restrict the set of mathematical objects that match your requirements.
Well, if you want to recover classical logic, you have to use a topos with the additional requirements:
- there are exactly two arrows from the terminal object to the subobject classifier
- there is an initial object (in general, by definition, there is no equivalence of empty set in a topos)
- ... I don't remember now .... Just look at the properties of the category of Sets here: https://ncatlab.org/nlab/show/Set
The category of Sets is the topos that has the additional properties required to make the logic work as the standard classical logic.
And that was only the first question :cry:
I don't think I have time for everything, but we'll see. Sheaves will be for the next time!
P.S. I re-read this and just realized that the correspondence between categories and logic theories that I described is not correct: here's the right correspondence: https://ncatlab.org/nlab/show/internal+logic (ZFC is not even in the list, but higher order logic can be used as an extension of ZFC where even iteration over subsets is allowed, and the corresponding category is called ELEMENTARY topos)
When numerous people use the same principle, it's a convention. So when we learn how to produce art, we are taught in the existing conventions. Do you not agree that to produce "pure art", "pure creativity", one would need to free oneself from the influence of such conventions?
Quoting jgill
Yes, art always requires a medium, otherwise it cannot be in the public domain, just like communication requires a medium. If not for the medium it would just be ideas in someone's mind. So words and symbols are just another medium. And the accepted symbols and words form the conventions of mathematics. But symbols are not the only medium for mathematics. For example, many mathematical ideas are expressed as music, rhythm, harmony, etc.. This is a completely different way of expressing mathematical ideas, distinct from putting symbols and geometrical figures on paper. But music is a very adept way of demonstrating some principles of division. The medium being divided is a temporal medium. What we can observe, is that the principles for dividing a temporal medium are completely different from the principles for dividing a spatial medium. So these two distinct forms of division require distinct mathematical systems
OK, now I see a real "use" for category theory. Nice presentation.
Quoting Metaphysician Undercover
You don't use symbols to express music? Well, one can play an instrument by ear I suppose.
Right, the symbols simply facilitate understanding of the expression which is the music itself, allowing others to join in the expression. The artist produces the mathematical expression as a composition of music, the symbols are not the expression, but represent the expression. In more conventional "mathematical expressions", the symbols represent the mathematical ideas. In music, the music represents the mathematical ideas (dealing with temporal extension), and the symbols represent the music
I have some specific comments, but first I want to say that I realize it may be an unreasonable expectation on my part to get the specific exposition I'm looking for.
On the other hand since last night I've been perusing a beautiful resource, Sheaves in Geometry and Logic by Saunders Mac Lane and leke Moerdijk, who isn't nearly as famous as Mac Lane. (Spelled with a space between Mac and Lane). Mac Lane invented category theory in the 1940's; and is very attuned to the philosophical implications of his work as well as being a brilliantly clear expositor.
Just reaing the first few pages of the Prologue, along with perusing the table of contents and some random reading, has given me a nice overview of categorical logic.
One thing that really got my attention was his presentation of a sheaf-theoretic version of Cohen's proof of the independence of the Continuum hypothesis. And his pointing out that Cohen's own proof, and his invention of the revolutionary method of forcing, is essentially sheaf-theoretic in nature. This was all a real revelation to me and gave me the high level view that I've been trying to get hold of. He pointed out that Cohen's approache relates to Kripke's work on intuitionist and model logic.
This all ties together a lot of amazing stuff. So Mac Lane made me a believer.
Unfortunately I still don't have the detailed technical example that I want to nail down; but perhaps that may have to wait.
Here is one really cool nugget that makes sense to me:
That is news I can use. It explains why if you consider the category of the open sets of a topological space, you get intuitionist set theory. That's the kind of insight I'm looking for. I never thought of it this way but now it's perfectly obvious. That's why I like reading Mac Lane. And that's only the prologue! The book is 629 pages. I imagine one could dive in and pretty much never come out, but be constantly enlightened on every page.
Now to your post.
Quoting Mephist
My latest understanding is that a topos is like a generalize universe of sets. You can have one kind of universe or a different kind of universe depending on the rules you adopt, but all set-theoretic universes fit into the topos concept.
In fact the category of sets IS a topos. That's very helpful. Sometimes the simplest examples are the ones to start with.
Quoting Mephist
Ok but I fear you're going off on a theoretical tangent again. What I was hoping for was specific clarification on things you've said. You've said proofs and propositions are like fiber bundles or sections. And at one point you used the word "fibration," which is a very specific thing in topology. I was hoping you would either explain those remarks with laserlike clarity, as if you were writing an exam; or else agree that for whatever reason we can't do that.
Likewise you claimed that the collection of n-tuples of real numbers with the discrete topology can be associated with propositions. You have my attention with that example, I just need to see the rest of it.
I will say for the record that it's ok if I never get clarity on these things. I'm more than happy discovering Mac Lane's Sheaves book, so I got my money's worth from the convo.
Quoting Mephist
I've read different points of view. For example the question arises, is the category of sets the same as the proper class of all sets? Well, not exactly. I've heard it expressed that the category of sets has "as many sets as you need" for any given application. But it's not exactly synonymous with the class of all sets. That's my understanding, anyway.
Quoting Mephist
I already noted a few posts ago that I'm familiar with ECTS, Lawvere's elementary theory of the category of sets. So most of this I'm aware of.
Quoting Mephist
To be fair, that's not it. I feel like you didn't address my specific question or expand on your remarks about fiber bundles, fibrations, and the n-tuples of real numbers taken as synonymous with propositions. Those are the kind of "bread and butter" things I'm trying to understand. But like I say I'm perfectly ok with that, because I got my money's worth from discovering Mac Lane.
Quoting Mephist
Minimal set of universal properties. That might be over my head. I know what universal properties are in terms of defining things like free groups, tensor products, and the like. I'm ignorant of what it would mean to select for certain universal properties. Or I'm not understanding you.
Quoting Mephist
Ok. Not getting your point but it's on me to understand what you mean.
Quoting Mephist
Ok.
Quoting Mephist
Ok. But just to save you some typing, I find that sometimes you tell me generalities about things I know, or that I consider tangential to the conversation, and we're missing each other that way.
And I think I'm winding down at my end because like I say, I think my expectation of clear answers to my questions might be unreasonable in this instance. I should just go read six hundred pages of Mac Lane, that would keep me out of trouble.
Yes!
Yes, but that correspondence is evident only in a dependent type theory, where you can make sense of the topology defined on your set of propositions (only open sets are propositions). In standard logic you cannot make sense of the topological structure of the space: no distinction between open and closed sets. All sets are both open and closed. That's the reason why taking the complement of the complement is an identity (boolean logic!). How can I show you the correspondence with dependent type theory without explaining dependent type theory?
That's pretty standard old-fashioned model theory and first order logic (the topology is irrelevant: forget about open sets and take simply the set of all subsets of a given set R). I noticed that other people on this site were starting some kind of "introduction to first order logic" thing. Maybe they can help to make clear this part.
Yes, well, the point is that you cannot "count" the objects of a category. You cannot distinguish between isomorphic objects. There is no "equality" relation defined on the set of all objects. How can you decide what's the cardinality of the set of all objects if you cannot associate them with another set? (no one-to-one correspondence possible between elements. Only equivalence makes sense, not equality!)
OK, as I said, I'll get to fiber bundles on the next episode.. :smile:
Part one: C is a category ( like A is an abelian group )
Part two: for each pair of objects of A, B there is a "product object" P ( like for each pair of elements (a,b) of A there is a product element: A is a ring ) ( omitting other needed properties, of course... )
Adding properties to the category I add structure! For example, each pair of objects has a product, the set of objects has to be infinite (pairs made of other pairs recursively). In general, without this requirement, a category may even be made of 3 objects and 4 arrows...
Well, I know what a fiber bundle is so if you claim something is a fiber bundle you could just explain what it is that's the fiber bundle. What is the underlying set, what are the fibers above each point, etc. But maybe there's too much of an explanatory gap and we're at a point of diminishing returns.
This is not about introduction to first order logic. This is about an explanatory gap. The topology is not irrelevant if you claim to have a sheaf. Perhaps we're done.
We're talking past each other. And this is not about cardinalities at all since neither proper classes nor categories (in general) have cardinalities. But I think between what you know and what you're able to explain, and what I know and what I'm able to understand of what you're saying, we have a gap that's not getting bridged.
ok
Quoting Mephist
And why would that be a problem? A group is a category with one element.
Been following along, maybe this helps.
I guess if we took [math]\Omega=\{0,1\}[/math], equipped it with the topology [math]T=\{\emptyset, \{0\},\{1\},\{0,1\} \}[/math], with the usual unions and intersections and complements, we could conceive that:
[math]\{T, \cup,\cap,{}^{c}\}[/math] is some sort of algebra (where the symbols have their usual meaning). Every open set is also closed.
If we have a proposition [math]P_i[/math], an interpretation [math]I[/math] is a mapping from [math]P_i[/math] to [math]\{0,1\}[/math]. If we imagine "starting with [math]\Omega=\{0,1\}[/math]", a proposition is something like pre-image of [math]I[/math]. If we have a family of such propositions, [math]Q = \{P_i, i \in J \}[/math], we could imagine each proposition being such a pre-image. If we consider propositional formulae from this alphabet containing at most [math]K[/math] symbols, [math]\times_K Q[/math], we could equip this with functions to [math]\{0,1\}[/math] (of up to arity [math]K[/math]) that evaluate to true or false. These are then truth functions. Like [math]f: Q \times Q \rightarrow \Omega: (x,y) \rightarrow 1[/math] iff [math]x=y=1[/math] else [math]0[/math] is the truth function for AND.
If we extended the topology to the product space [math]\times_{K} \Omega[/math], I thiiiink this ends up being the discrete topology? But then it's also the set of all truth table rows of propositional logic formulae containing at most [math]K[/math] proposition symbols. If we took the intersection of {1} with {1} we get 1, intersection {1} with {0} we get 0... and can construct AND as a truth function on the open sets of this topology.
If we set out the production rules of propositional logic on [math]Q[/math], an "open set" might be an interpretation of a syntactically valid formula. There looks to be a topology here: closed under finite conjunctions (intersections), finite disjunctions (unions), interpret the empty string as mapped to empty set.
The negation of a syntactically valid formula is a syntactically valid formula, so the closed sets are all open... Any subformula of a well formed formula is syntactically valid. Well formed formulae are closed under finite conjunction and finite disjunction (since it's a finite alphabet this gives the appropriate topology properties, or somehow corresponds to them). "Pulling back" a truth function along an interpretation might give the propositions which satisfy it. If we "pulled back" a tautology (the truth function which is 1 for all arguments), we would (probably) get a theorem - signified by all the truth table rows being in the pull back. Pullbacks and fibre products are intimately related (somehow, I'm not sure on the details, I've been working through Category Theory for Scientists and pullbacks/fibres were my last session).
This starts looking suspiciously like a correspondence is in play between the algebra of sets on the product topology of [math]\times_{K} \Omega[/math], and the production rules on the propositional symbols. The possible "propositional assignments" that satisfy an interpretation maybe float above an interpretation as an algebra (algebraic structure, anyway).
Maybe it doesn't help though, it's very scattered.
:smile: Thanks for trying to help! But it's not so simple...
I am afraid @fishfry has chosen the most complicated way to "build" a topos: the one that Gothendieck come up with at a time when (I believe) the notion of category didn't exist yet. And, what's worse, he wants to do it as a model of standard set theory. There is no model of standard set theory that can be seen as a sheaf! And this is for various reasons:
1. A sheaf is a continuous contravariant map from open sets to sets. The open sets should be thought of as the open sets of our model: (open sets of real numbers, for example, if we consider first order logic speaking about real numbers). But first order logic does not speak about sets of elements, but only (at most) about tuples of numbers! (forall x,y there exists z such that ...). To speak about sets you need higher order logic.
2. Even if you consider higher order logic, the algebraic structure corresponding to boolean logic (as you said) is a boolean algebra. Now, the topology that you get from a boolean algebra is a discrete topology: every set is both open and closed. This topology is the only particular case that does not contain any "information" about the "connectivity" of (the topological information about) the space: every set is simply a set of "isolated" points. Maybe I wasn't able to explain this: you can think of a topology as two different pieces of information: 1. The set of all subsets of a given "universe" set; 2. An (arbitrary) choice of which ones of these subsets are considered to be "open". In a discrete topology all subsets are open; so you don't really need a topology in case of a boolean algebra: it's enough to consider the set of all subsets of the universe, and that's what you do in boolean higher order logic.
Instead, if you consider a topos from a categorical point of view (a topos is simply a category with some additional structure), the set-theoretical operations (intersection, union, complement) are a Heyting algebra, quantifiers are an adjunction between categories, and open sets are naturally represented as types in type theory, that can be seen (in my opinion) as a nice generalization of high order boolean set theory, where you can make sense of the information related to the topology of the domain (not all subsets are open). Everything is much simpler to understand than Grothendieck's construction based on sheaves.
I am sorry if I am not able to explain this in a clearer way...
The underlying set is the set of all propositions. The fibers are sets of elements of our model.
Quoting fishfry
OK.
Quoting fishfry
Yes, unfortunately I am not able to follow your plan.
Quoting fishfry
It is not a problem. It's only an example: in general a category can have any number of objects, but a cartesian-closed category must have an infinite number of objects: the additional condition that all binary products exists implies a restriction on the possible number of objects.
If we continued with the propositional classical logic in my example, and we have an interpretation from [math]\times_K Q[/math] (the symbol set product-ed with itself [math]K[/math] times) to [math]\Omega=\{0,1\}[/math], and if we gave [math]\Omega[/math] the discrete topology as I said. If we then stipulated that the interpretation was "continuous" in the sense of topological spaces (the pre-image of every open set is open), would the pre-image of any open set of [math]\Omega[/math] with the discrete topology be a type, and thus a proposition? Or a collection of propositions which co-satisfy/are equivalent?
Or maybe did I miss something?
Quoting Mephist
I was hoping you'd be able to tell me, as it seems defining what a topology is in terms of the logic is precisely where the missing intuition is. I assumed there was some topology on the space of propositions, and tried to see if "pushing back" the open sets of the topology on [math]\times_K \Omega[/math] through the interpretation [math]I[/math] made sense to you.
Something like, if we have a logical connective's truth function [math]f[/math] from [math]\times_K Q \rightarrow \Omega[/math], and we glue together open sets on [math]\times_K \Omega[/math] 's discrete topology through the fibre [math] \{ I(a)=1=f(c) | a \in \times_K Q , 1 \in \Omega, c \in \times_K \Omega \}[/math]. I guess inducing a topology by pulling back a topology through a continuous function and a connective (dunno if that works at all).
It perhaps doesn't make much sense. Do you know the analogous construction for classical (or intuitionist) propositional logic? So we don't have to deal with the interpretation being a complicated set valued function. It should be in there somewhere and easier to talk about to exhibit a connection between a topology on the logic and a toplogy on the Omega product.
OK, I understand what you want to do. But in the case of fiber bundles you don't define the topology of the total space in terms of the topology of the base space. You assume a preexisting topological space E (the total space), and a preexisting topological space B (the base space), and a continuous function P from E to B, and then you define the fibers as inverse images of P.
Quoting fdrake
Yes, it's a Heyting algebra ( https://en.wikipedia.org/wiki/Heyting_algebra ) is the analogous of boolean algebra for intuitionistic propositional logic.
Quoting fdrake
But that's the whole point! You have to be able to talk about open sets to make sense of a logic that allows the existence of open sets not "built" as sets of points. The logic that we are talking about is (for example) the one that allows the existence of infinitesimal numbers that are not non-zero but whose square is zero.
Well, now that I think about it, I heard that you can see the modal connective of modal logic as an arrow of the form Omega to Omega (from propositions to propositions, the same as negation), and that this arrow is (represents) a topology on Omega (the set of truth values). So, the modal connective could be interpreted as "it is locally true that". But for me this is a little too "abstract" :smile:; and first of all, I don't know modal logic! Maybe there is a way to make a precise sense of your idea, and I just don't know about it.
I'm not sure I'm seeing that yet. I'm working on getting a bottom-up understanding of fiber bundles and this will take a couple of days or more for me to sort out a coherent argument. My general idea is to start with the Cartesian product of sets as an example of the fiber bundle idea; then work through the definition of a manifold, which (ignoring all the technical details) is an example of a Cartesian product. From there to sheaves is easy.
Now having laid out carefully a bottom-up understanding of fiber bundles, I'm going to want you to be very specific in putting your ideas in this context. I may or may not be successful in pulling this together because my knowledge of manifolds is weak but I think adequate to the task once I review some things.
Meanwhile I'll probably stay out of this for a while.
Quoting Mephist
Honestly there's much less here than meets my eye, at least, from the standpoint of knowing what a fiber bundle is in topology. You are not making any connection with the fundamental definitions and you're not supplying the details. My wild-assed guess is that your Coq teacher mumbled something about fiber bundles and you haven't thought the details through. Is that uncharitable? If not, and you do know how to drill your idea down to the definition of a fiber bundle, let's just say I wish you'd work harder on exposition. Hopefully I'll pull my idea together and that will give us something specific to work with.
EXACTLY a question I put to @Mephist the other night. That's identical to my understanding of what's being proposed, but I don't entirely believe it till I work through fiber bundles from the bottom up, which I'm working on. I'm gratified to find agreement on our interpretation of this idea.
Oh you DO know this material. It must just be your exposition that I can't understand. I'm happy to see this paragraph. I understand it, I agree with it, and I'm hopeful you'll frame the logic argument in terms like this. Maybe you already have. I'll work at this some more.
I haven't worked through this yet but it looks very promising.
Bleh, constructing the connection from the set theoretic side is extremely indirect. I think there're three components.
(1) Showing that the open sets of a topology are a Heyting algebra (or associate with one).
(1a) This should obey some restriction property, so that if we restrict the topology to a subspace, it will also be a Heyting algebra. This should follow from (1) directly, as if all open sets in any topology form a Heyting algebra, then so too will the subspace topology induced through a restriction. This probably turns out to let us take subsets of a language, like individual well formed formulae, and give them models in exactly the same way. Such a restriction would probably be continuous, as if we endow the subspace with the subspace topology and treat the mapping between them as an indicator function, the pre-image of every open set would be open just by definition. So the "continuity" of the interpretation looks to be a by product of (1) and the continuity of the inclusion map from a space with a topology to a subspace which is the image of that map with the subspace topology. The restriction map is probably quite similar to a subobject classifier, since it is a characteristic function for the set being used in the restriction. The overall intuition I have is that if we have a model, we can go to a larger model by "pushing back" along the restriction map.
(2) Showing that a model of intuitionist logic is a Heyting algebra (or associates with one)
(2a) The ordering relation in the Heyting algebra respects syntactic entailment, since we're using a Heyting algebra as a model, the logic is complete, it also respects semantic entailment. If it "respects semantic entailment", it should "respect the mapping from the logic to the Heyting algbra", because that's how we provide the formulae with models.
(3) (1) and (2) together give us that the open sets of a topology are a model of intuitionist logic. The interpretation function would then be taking the well formed formulae of the logic and associating them with their corresponding open sets, which are isomorphic to set "domains" that satisfy the formula, and have their own Heyting algebra.
The category theoretic construction lets you come at this some other way. If we re-parse everything in category theoretic terms the same things would hold. But it is probably true that it holds more generally for topoi that aren't representable as sets.
Edit, trying to translate between @Mephist's intuitions and our stodgy old analysis ones: if we drew out a category-theoretic diagram that had intuitionistic logic models on it as a category, then topological spaces as a category, then the syntax of intuitionist logics as some other object, we could construct the arrows:
Intuitionistic logic syntax ---interpretation>> intuitionistic logic models
Topological spaces ---replacement of union and intersection by join and meet and other stuff>> intuitionistic logic models
We could construct a model of an intuitionistic logic through a pullback of the interpretation and the other map? So that the "fibres" (pullback elements) are the open sets which model our specified syntactical objects. The topological space would model the whole logic when and only when the whole space is in the fibre product, maybe anyway.
There's a comment that goes through some of the construction here.
Still unclear, but I'm starting to see how it could be the case.
The BASE SPACE is constituted of 3 propositions (propositios are types):
P1 := {A;B;C}
P2 := {C;D;E}
P3 := {E;F;A}
Capital letters are the "constructors" of the types, that you can see as the simplest possible kind of "rules" of our logic.
- to prove P1 you can use A, B or C (a proof of P1 would be written as "A: P1")
- to prove P2 you can use C, D or E
- to prove P3 you can use E, F or A (the order is irrelevant)
From the topological point of view, of the space of the propositions is made the following open sets:
P1, P2, P3, the empty set, the set {A;B;C;D;E;F}, and all possible unions and intersections of P1, P2 and P3. (the letters are the points of the base space)
The total space is made of the hours of the day, from 1 to 12 (no distinction between morning and afternoon to make it simpler).
Our model is an object made as a clock, with 12 hours painted in circle, and an arrow indicating a "set of hours". Let's say that the arrow in general does not indicate a precise hour, but a set of them.
The possible sets of hours that can be indicated by the arrow are the following ones:
{1;2}, {3;4}, {5;6}, {7;8}, {9;10}, {11;12},
{1;3}, {2;4}, {3;5}, {4;6}, {5;7}, {6;8}, {7;9}, {8;10}, {9;11}; {10;12}; {11;2}; {12;1} (the last two sets are not a mistake)
We define the topology of the total space in the following way:
- all the sets of the previous list are open sets;
- the empty set and the set of all hours are open sets;
- all possible intersections and unions between sets of the previous list are open sets.
No other subset of hours is an open set.
Now, we define our projection function X: a map from the total space of the hours of the clock to the base space of the propositions describing the result of our experiment:
1 => A; 2 => A; 3 => B; 4 => B; 5 => C; 6 => C; 7 => D; 8 => D; 9 => E; 10 => E; 11 => F; 12 => F.
( notice that this is a continuous function )
The inverse image of X is the following one:
A => {1;2}; B => {3;4}; C => {5;6}; D => {7;8}; E => {9:10}; F => {11; 12}
Let's check the homotopy type of our fibration.
We start from point A - 1 and follow the continuous path A-B-C-D-E-F-A in the base space.
- when you are on B, you can move only to 3 on the total space because there is no open set containing both 1 and 4 (the rule, of course, is that you have to follow a continuous path in the total space)
- then, from B - 3, you are forced to move to C - 5 (because {3,6} is not an open set)
and then, continuing in this way, we see that the only continuous path (the only possible section of the fiber bundle) is the following one:
A-1; B-3; C-5; D-7; E-9; F-11; A-2; B-4; C-6; D-8; E-10; F-12; A-1
We see that the fiber bundle is not trivial (double covering of the base space).
Now, let's come back to the logic interpretation.
The meaning of our propositions is given by the inverse image of X (let's call it Y).
So, P1 means that our arrow points to the set of hours {1;2} or {3;4} or {5;6} (then, it means "we are in the first third of the day")
P2 means "we are in the second third of the day"
P3 means "we are in the third third of the day"
But now, let's check which propositions we can form starting from P1, P2 and P3.
"P1 and P2", for example, means that the arrow is on {5;6}, corresponding to the open set {C} of the base space.
But there is no way to say that the arrow is on {3;4}, because the {B} is not an open set in propositions space. B is a point of the base space, but the propositions correspond to open sets, not to single points. And not all subsets of {A;B;C;D;E;F} are open sets!
You see, the open sets have meaning but not the points of the space
Or, better, the points of the base space correspond to proofs of our propositions, but what meaning can you give to the points of our model? (the hours of the clock). Our clock's arrow is too "fat" to distinguish between single hours, so only sets of them are meaningful hour indications. So, for example, you'll never be able to say the difference between 1 o'clock and 2 o'clock using our logic, even if in the topological space of the clock they are different points, and the space is "made" of points in our case: this is standard point-set topology.
[OK, it's just become way too long...] But at least NOW WE HAVE A CONCRETE EXAMPLE.
P.S. The types the proofs in this example are the simplest possible example of inductive types.
But of course this is not all: for example, it would be impossible to represent the set natural numbers in this way. But I cannot write a book on type theory on this site....
If a simple explanation is not possible, say so. That's OK :chin:
(The logic however, is not the standard logic of set theory)
P.S. we lost the topic of the thread a long time ago... :grin:
Well, not only is useful, but if you find a relation between apparently completely different areas of mathematics, maybe those concepts have in some way a deeper meaning.
Yes, I'm familiar with the notion, although I have no experience in an algebraic venue. For instance, many years ago I showed that convergence of complex limit periodic continued fractions useful as functional expansions could be accelerated by employing a feature of dynamical systems: attracting fixed points (Proceedings of the AMS). And could be analytically continued by using repelling fixed points (Mathematica Scandinavica and Proc. Royal Norwegian Soc. of Sci. & Letters). There are deeper meanings here by locating these concepts in theory of infinite compositions of complex functions.
I also showed that the traditional Tannery's theorem makes far more sense when embodied in more general infinite compositions rather than merely series and products, or integrals. Not quite what you are stating, but close. You guys are on a roll! :cool:
WOW!!! I understand only partially what the terms mean: analytical continuation of a complex function has something to do with chaotic systems? Did I understand correctly? Why didn't I even here anything about this? And YOU proved it? Now I am impressed! Really! ( or maybe I didn't understand a thing of what you just said... )
Dynamical systems! Actually, I was more general, showing certain sequences of linear fractional transformations can have their regions of convergence expanded by the use of fixed points. Continued fractions are a special case.
Here is a simple example illustrating the use of a fixed point:
Given [math]F(z)=\frac{1}{1-z}[/math]
Then [math]F(z)=1+z+{{z}^{2}}+\cdots ,\text{ }\left| z \right|<1[/math]
For [math]\beta=0[/math] the T-fraction expansion of the power series is
[math]{{T}_{n}}(\beta )=1+\frac{z}{1-2z+\frac{z}{1-z+\frac{z}{1-z+\ldots +\frac{z}{1-z+\beta }}}}[/math]
The continued fraction value is by definition
[math]{{T}_{n}}(0)\to CF[/math]
Here it is found that
[math]{{T}_{n}}(0)\to F(z),\text{ }\left| z \right|<1[/math] and
[math]{{T}_{n}}(0)\to \tfrac{1}{2},\text{ }\left| z \right|>1[/math]
Whereas
[math]{{T}_{n}}(\beta )\to F(z),\text{ }\left| z \right|>1,\text{ }\beta =z[/math]
Here
[math]\beta =z[/math] is the repelling fixed point of the function [math]t(\zeta)=\frac{z}{1-z+\zeta }[/math]
Exactly not what you said the other day, when you started out by saying that the underlying set consisted of all n-tuples of real numbers then chose not to respond to any of my questions. That's why I"m trying to get you to nail down your definition. If you say you have a fiber bundle I'm entitled to ask what is the base set, what is the total space, what is the map? It's a perfectly sensible question.
Which model? You haven't said anything about models.
Is your previous claim now retracted and you are now making this different claim? The one with the clock?
If yes, can you please try to explain this in a better way? I don't even have much time for this, sorry. I have even to go to the hospital for a couple of days next week.
I'm sorry to hear of your personal health issues.
Have you ever heard it said that if you can't explain something clearly, you don't actually understand it? That's the sense I get from your posts. I could be wrong. Hope your health issues turn out well.
The Heyting algebras are for intuitionist logic (of some sort). I wrote that previously.
Hope you feel better soon.
What do logic and topology have to say about each other?
Specifically; if a logic has a model is there a correspondence between a topological space on the set which models it and how proof works in the logic?
I can summarize. Short answer is that these days you can do logic via category theory; and when you do that, you get intuitionist logic (denial of the law of the excluded middle (LEM) and all that) in a natural way.
This relates to topology via the idea of fiber bundles from differential geometry if you know what those are. If not just ask). The examples presented so far aren't clear to me and I haven't worked through @fdrake's promising-looking examples. @Mephist may or may not have presented coherent examples but there's a gap between his expositions and my understanding that only gets worse over time. The fault may all be mine.
There's a very nice illustration of how this works if you consider any topological space and restrict your attention to the open sets. The "complement" of an open set in a topology, if you only care about open sets, is the interior of the complement of the set. (In other words in general the complement is not an open set, but if we only look at the open sets, it makes sense to define funny complements this way).
So it is in general NOT true that the complement of the (true) complement of an open set is going to give you back the original open set. This corresponds to a failure of the law of the excluded middle.
For a very nice overview of the history and meaning of all this I recommend the prologue of Mac Lane's Sheaves in Geometry and Logic. One need not understand the details to get the big picture from this very clearly written book.
Intuitionism was developed in the 1930's but didn't get any mindshare in mainstream math. Now with the advent of computers (where the complement of a noncomputable set of natural numbers may also be noncomputable), denial of LEM is back in fashion, especially in computer science and categorical logic. I call this Brouwer's revenge. Brouwer invented intuitionism in the old days but it didn't catch on. His form of intuitionism had a touch of mysticism to it, but the modern versions are mathematically solid. Fifty years from now (or sooner) they'll be teaching this to undergrads and set theory will be a relic of the past like Euclidean geometry. Set theory of course won't become wrong, just out of fashion.
Another thread of development is that mathematicians want to use computers to check their proofs for accuracy. It turns out that intuitionist type theory (which I know nothing about) is the key. In homotopy type theory one uses the idea of homotopy from topology (continuously deforming one path into another) to do intuitionist logic in such a way that you can build working computerized proof assistance for professional mathematicians. Also see intuitionistic type theory.
These are the broad outlines I've picked up, but I haven't spent much if any time on the details. One name you'll hear a lot is Vladimir Voevodsky, a Fields medal winning mathematician who became frustrated at longstanding errors in published proofs and devoted himself to the project of computerized proof assistants. He died tragically young just recently, in 2017.
Voevodsky's contribution is the Univalent foundations of mathematics. The idea here is that mathematicians routinely conflate equality and equivalence whenever it's convenient. For example there's only one cyclic group of order four, even though there are lots of isomorphic copies of it that are not equal as sets. For example the integers mod 4 and the powers of the imaginary unit [math]i[/math] are the "same" group.
I should mention that this kind of thing bothers @Metaphysician Undercover greatly, and he's right that mathematical equality can sometimes be stretched past what he would consider true equality. The answer to this is that if it quacks like a duck it's a duck, and if it's isomorphic to the cyclic group of order four, it doesn't matter what representation we choose. They're all the "same" in the appropriate technical sense.
In category theory this conflating of equality and equivalence becomes semi-formalized via universal properties. Any two objects that satisfy the same universal property are isomorphic and are regarded as the same thing.
Univalent foundations takes this one step farther by formalizing an axiom that says that equivalent things are equal. My high-level understanding is that the univalence axiom makes mathematically precise the informal practice mathematicians have been accustomed to for decades.
https://en.wikipedia.org/wiki/Univalent_foundations
All of what I've written is of course hopelessly vague and should not be relied on as gospel. But I've hit most of the buzzwords and major concepts in their broad outlines.
ps -- The immediate subject of the thread recently is to see how we can view intuitionist logic as an example of a fiber bundle. Maybe that was the short answer to the question.
I see that there is a misunderstanding between us on what it means "a logic has a model".
A logic is a bunch of rules that describe how you can build sentences that speak about "something".
What I call model is that "something". For example, the real numbers can be the model. The model is the thing that we are speaking about. The rules of logic have nothing to do with it! If we speak about the waves of the sea, then "the set of all waves of the sea" is the model. It is "the real thing" that we are speaking about.
Now, the essential change in the point of view that allows you to see the correspondence between topology and logic it this one: consider sets to be more "fundamental" than their elements.
So, if our model are the real numbers, the sets of real numbers are more "fundamental" than the single real numbers. If you think about it, that's what boolean algebra does: boolean algebra speaks about sets and operations between sets (union, intersection, complement): you build sets starting from other sets, without mentioning their elements.
From this point of view, we have an "universe" set (in our example the set of all real numbers), and the set of all sets of real numbers (the powerset of the "universe"), and a boolean algebra defined on the powerset of the "universe".
Now, we can generalize logic by substituting the powerset of the "universe" with a topological space. A topological space is in general defined as the powerset of the "universe", plus a choice of which elements of the powerset are "open" (this choice of the open sets is what's usually called the "topology").
A topological space, then is a generalization of the powerset of the "universe": instead of considering as the fundamental elements of your algebra the full powerset of the "universe", you consider as your fundamental elements the open sets of the universe (in our example, the open sets of real numbers). The algebra built on the open sets is a Heyting algebra. The algebra built on the full power set of the universe is a Boolean algebra. Of course, you can consider the Boolean algebra as a particular case of Heyting algebra by choosing as open sets all the subsets of the "universe" (the full powerset of the universe). This is what is called a "discrete" topology.
So, logic built as an algebra based on a topological space is a generalization of the logic built as an algebra on the powerset of the universe, at the same way as a topological space is a generalization of the powerset of the "universe".
( I'll describe the part related to proofs the next time )
This is the mistaken procedure called Platonism. A set is a human creation, produced to categorize. If we produce an empty set, which may or may not be filled, and this is implied if a set is more fundamental than its elements, then the type, or universal Form, is prior to the particular, or individual. However, as Plato demonstrated, then the type, or universal, what you call "the set", must itself have some existence, and this would be as a particular, individual object. So the empty set has been created with the purpose of being a universal Form, a type, but upon creation, it actual exists as a particular object. To uphold the premise, that a set, or universal Form is more fundamental than an element, this created object, the empty set must already exist as an element of an existing larger set. Aristotle demonstrated this premise, that the Form, as a universal type, (what you call "the set") is more fundamental than its elements, leads to an infinite regress and is actually impossible, therefore false. This is because the set itself can only be represented and understood as a particular object, and understanding a particular requires relating it to a more universal, categorizing it.
The Neo-Platonists get beyond this problem by producing an Ideal particular as the most fundamental. The Ideal particular is the most fundamental, as a type, a universal Form, or a set, which is also an individual, or particular. It is both. However as the most fundamental, it cannot be an element of a further set, or part of a more fundamental or universal Form. As a particular, and also the most fundamental universal, it is identified as the "One".
Making the "One" the most fundamental resolves the inherent contradiction of having the empty set as fundamental. The empty set is inherently contradictory because it is something, an object, which at the same time must be nothing.
What I wrote is only an idea, that (in my opinion) is important to understand the "meaning" of a theory, but from the point of view of mathematics all explanations that you can give by words are worth nothing: at the end, the only thing that counts in a mathematical argument are proofs. If what you say cannot be proved, it's not mathematics. I know, neither of us presented any proof of what we said here, but we are on a philosophy forum here, right? :wink:
What I want to say is that your argument "... leads to an infinite regress and is actually impossible, therefore false" (Aristotle's argument) would not be accepted as a valid prove in today's mathematics.
In mathematics you are free to "invent new worlds" (I believe this is what Grothendieck was saying about his work), but you have to do it using proofs that are rigorous enough to be accepted by peer reviewers.
Then, you can discuss if what you invented is "important", and what's it's "meaning".
Quoting Metaphysician Undercover
The empty set is only an abstract construction defined by a set of axioms that has nothing to do with the "One" of ancient Greek philosophers.
I really appreciate your explanation. Thank you. I looked briefly at your first link to get an idea of the univalent approach. I was completely unaware of this, being happily non-constructive at times! :smile:
Combinations, not permutations; i.e., the different proper subsets, and the order of the members does not matter. For a set with n members, its power set has 2^n members.
Quoting tim wood
This was Cantor's view, which is fairly standard among mathematicians today. However, there is a power set for the real numbers, and a power set for that power set, and so on ad infinitum. That being the case, some argue that the real numbers are not truly continuous, despite comprising what is conventionally called the analytical continuum.
Quoting tim wood
There are no points in a truly continuous line, period. As a one-dimensional continuum, its parts are all likewise one-dimensional, rather than dimensionless points. We could hypothetically mark points on a line of any multitude--including that of the real numbers and that of their power set--or even beyond all multitude.
It's not true that words are worth nothing in mathematics, because the axioms are written in words. My demonstration was a proof, a logical proof that a set cannot be more fundamental than its elements, because that creates an infinite regress. If you are satisfied with an infinite regress you have an epistemological problem. Such mathematics is not supported by sound epistemology.
(continuation: correspondence between a topological space and how proof works in the logic)
The rules of logic should be valid for ANY topology and ANY subset of points. So, for example, they should work even for a discrete topology.
Now, for intersection and union everything works fine, since the intersection of two open sets is again an open set, and the union of two open sets is again an open set. But the complement operation does not preserve the openness, so it cannot be a primitive operation in an algebra of open sets.
Well, the "trick" to make it work is very simple: do NOT considered it as a primitive operation but define "complement of A" as "the largest OPEN set X such that the intersection of X and A is empty" (the largest open set with no points in common with A, but we don't need to speak about points)
With this definition the two logics are exactly the same in the limit case when all subsets are open, but the second one (intuitionist logic) is weaker: every intuitionist derivation is even a standard derivation, but there are boolean logic derivations that are not intuitionistic derivations. The difference is due to the fact that intuitionistic logic does not "see" the points that are not part of open sets (the points on the border of an open set, or even isolated points). If you take a look at the example with fiber bundles that I posted a couple of days ago, you can see what I mean: even if you think of your model as a set of points, you can't speak about single points using this language: every eventual single point not included in an open set is simply "ignored".
The effect on the logic rules, as @fishfry pointed out, is that double complement (double negation) does not give you back the original set: if you had a set A that was not open nor closed (let's say an open set plus some part of the border), the first complement operation gives you the complementary OPEN set; and then the second negation gives you back A WITHOUT BORDER (that now has become open. Taking again the complement a third time, now you obtain the same result as taking it once, and so on.
The effect on the rules of logic (one of the effects) is that the excluded middle is not valid rule. But if you add the rule of excluded middle to intuitionistic logic as an additional axiom you don't obtain an unsound system: you simply get back boolean logic. This is equivalent to choosing as topology of the space the discrete topology. It's an additional assumption: intuitionistic logic works for any topology; boolean logic, instead, works only in the particular case of the discrete topology.
I meant words in plain english language (or in another natural language): you have to use a formal language to express mathematical theorems.
I see mathematical axioms expressed in plain English.
There is a way to translate any mathematical proposition (or axiom) into plain English, but there is no way to translate any English proposition into a mathematical proposition: formal languages are more limited than natural languages.
Sure, but how's that relevant? What is at issue is the postulate that a set is more fundamental than its elements. That's plain English.
P.S. That's a very important point to understand: the words used in mathematical sentences are not chosen at random: they are carefully chosen to give some "intuition" of the things that we are speaking about. However, you cannot use that intuition in proofs. Proofs have to be completely "formal": they have to be valid even if you substitute the words with random strings of characters.
The powerset of a set is the set of all subsets of the set. So for example [math]\mathcal P(\{1,2,3\}) = \{\emptyset , \{1\}, \{2\}, \{3\}, \{1,2 \},\{1,3\}, \{2,3\},\{1,2,3\}\} [/math].
It's easy to show that [math]| \mathbb R | = | \mathcal P(\mathbb N)| [/math]. For any subset of [math]\mathbb N[/math], create a bitstring that has 1 in the n-th position if n is in the subset, 0 otherwise. Put a binary point in front of the string and you have the binary expression of a real number in the unit interval, and vice versa.
For example the real number whose binary representation is .10101010101... corresponds to the set {1, 3, 5, 7, ...}. So we have a bijection between the real numbers (in the unit interval) and the subsets of the natural numbers. (For convenience I'm numbering positions to the right of the binary point starting at 1, and excluding 0 from the natural numbers).
[We can ignore dual representations (.5 = .4999...) because there are only countably many of those and countable sets don't make any difference to the cardinality of an infinite set].
Quoting tim wood
That doesn't work because for example the set of permutations of {1.2.3} is 123, 132, 321, 312, 213, and 231. It's not the same thing.
Quoting tim wood
No not really. We just proved above that the cardinality of the reals is [math]2^{\mathbb N}[/math], the set of functions from [math]\mathbb N[/math] to the set {0,1} (which we can think of as the set of bitstrings). Now if the transfinite cardinals are [math]\aleph_0, \aleph_1, \aleph_2, \dots[/math], the question is which Aleph is [math]2^{\mathbb N}[/math]? The claim that it's [math]\aleph_1 [/math] is the Continuum hypothesis. For all we know it's some other Aleph, perhaps a very large one. The answer is independent of the usual axioms of set theory.
Quoting tim wood
I'm not sure I follow your idea of a particular point on the line.
Quoting tim wood
I don't follow your idea but that's not what CH is. CH is just the question of which Aleph is the cardinality of the reals.
Quoting tim wood
The points on the real line are just the real numbers and vice versa.
Quoting tim wood
I'm afraid I don't follow this idea. The "real line" is just a synonym for the set of real numbers.
I found a paper that indicated the the fibers are "L-structures." Not too sure what those are, or what the base set is. I'm not sure I entirely believe it's a discrete topological space. I'm thinking you've probably explained this point to me several times over but I still don't get it. My apologies for giving you a hard time out of frustration at my inability to understand how fiber bundles can be used to model logical structures.
I'm afraid I share @Metaphysician Undercover's misgivings about this remark. I understand the categorical viewpoint of sets, but I would not characterize that viewpoint via this particular way of phrasing it.
That's a Peircean view and not a standard mathematical view; and I think it's important to make that distinction when explaining things. The standard mathematical view is that "the continuum," "the real line," and "the set of real numbers" are synonymous. Philosophical considerations do not alter the conventional mathematical meanings.
No, your idea only lists all the bitstrings of FINITE length, of which there are only countably many. For example 1010101010101010... never appears on your list.
I don't know what are "L-structures", but I think I know what's the source of misunderstanding: the words "discrete" and "continuous" used to refer to finite structures. In my example the "space" of the model is made of 12 points, but it's NOT a discrete space: not all subsets of the set {1...12} are open sets BY DEFINITION. The definition of which sets are open is arbitrary: the only required conditions is that it has to include the empty set, the full set, and all possible unions and intersections.
You should see the topology as a kind of "blurring glass" that is put over the set {1...12} and does not allow you to distinguish the individual points: you can see groups of points, but not individual points. Think of the set of real numbers when they are interpreted as results of physical experiments: you explained this to me very clearly: you can never get a real number as the result of an experiment (and you can't split a physical sphere in distinct points as in the Banach-Tarsky theorem).
The same thing can be true for the set of 12 points in my example: you cannot distinguish the point 1 from the point 2, because there are no open set {1} and open set {2} in the topology.
Of course, the most interesting cases of open sets are infinite sets (as real numbers), not finite ones as in my example. But I especially chose a finite set to make it crystal clear: topology is not about the cardinality of the "universe" set.
There's one decimal place for each natural number. A decimal expression .abcdef... means a/10 + b/100 + c/1000 + ... There's one place for each negative power of 10.
Quoting tim wood
Your list is countable. You've listed all the FINITE bitstrings. Where is .10101010101010... on your list? It's not there.
Not clear to me. I literally and honestly did not understand what you said in this post. Perhaps it's a lost cause.
A fiber bundle is like the collection of tangent planes to a sphere. Somehow, one can replace the tangent planes with logical structures of some sort, and the points of the sphere with .... something, and intuitionist logic drops out. Perhaps it's not explainable in elementary terms. But I couldn't relate what you wrote with any attempt to clarify this point.
All right. Fair point.
Well, OK, never mind. However, the book that I gave you the link is very clear and contains proofs and exact definitions. Surely that's easier to understand than my explanations...
Ok I will have a look. Many links have been posted recently. Can you repost the one you want me to look at please?
The usual intuition is more like an "airbrush" ( https://en.wikipedia.org/wiki/Fiber_bundle ). The fibers are seen as stick wires coming out from a common surface; they are separated from each other.
I stand by my remark. The tangent bundle of a sphere is most definitely a fiber bundle.
https://en.wikipedia.org/wiki/Tangent_bundle
Can you give me the link you want me to look at? There's been so much back and forth and so many links.
https://www.amazon.com/Topoi-Categorial-Analysis-Logic-Mathematics/dp/0486450260
Unfortunately, it's not downloadable for free
Yes, of course it is!
That's the intuition I'm working with at the moment, special case that it may be.
Quoting Mephist
Yes I just checked that out. I'll keep searching around for an insight.
In my opinion, the misleading part of that example is that the tangent planes seem to have some points in common, since they are immersed in an ambient 3-dimensional space. That's not true! The tangent vector spaces are completely separated from each-other (no points in common).
(even the origins of the tangent vector spaces are not in common: they are not seen as subspaces of a common ambient vector space in a higher dimension)
Oh I see. Good point. Funny but it never occurred to me to be confused by that. The tangent planes are conceptual thingies attached to each point but they don't "intersect in 3-space" at all. The technical condition is that the total space is the disjoint union of the fibers. I suppose I like this example because it's nice and concrete. For example a vector field is a choice of a single vector from each fiber. So if we have a vector at each point of a sphere that gives the wind direction and velocity at that point, that's a section of a fiber bundle. In set-theoretic terms a section is a right inverse of the projection map. That's how I think about all this.
In set theory class many moons ago I proved that "every surjection has a right inverse" is equivalent to the axiom of choice. That makes sense because it says we can always make a simultaneous choice of a tangent vector from each tangent plane. When I found out that a section is what differential geometers call a right inverse, I was enlightened.
Yes! :up:
Quoting fishfry
Yes! (even if this is not related to the topology of your sets)
Moreover, in this case the topology of the total space (the space made of vector spaces) is "inherited" from the one of the base space: in this sense this is a rather "artificial" example. My example is the most "clean" that I can think of: base space and total space have pre-existing and independent topologies. And it's much simpler than vector spaces: only sets of sets, and functions between sets!
P.S. If you don't like my example because it's made of finite sets, you can "fill the squares" of the total space (it will become a Mobius strip), and connect the points of the base space to make it become a loop! :smile:
The problem is that your demonstration, through this technique, produces a misunderstanding of the theory, rather than an understanding. So the criticism is of your technique. You describe topology through reference to set theory, but to understand set theory requires an understanding of extensionality. You demonstrate a misunderstanding of extensionality. The fundamental assumption that a set has extension negates the possibility of an empty set. Therefore your demonstration, which places the set as more fundamental than its elements, implying an empty set, is a demonstration of misunderstanding.
The axiom of extension dictates that a set's identity is established by its elements. Therefore a set without elements can have no identity as "a set", and is therefore not a set. Some set-theorists are wont to obscure this fact by saying that the empty set is unique, when in reality it is distinct from all other sets because it is not a set at all; it has no extension. As I explained, this problem was overcome thousands of years ago by making "One" the fundamental "unique set", as the term "unique" implies.
Quoting fishfry
When you and I agree on something, that's really something to be afraid of; better move the hands on the doomsday clock. But I think the appearance of agreement is based in different principles, so there's really nothing to worry about.
Yes, and I acknowledged as much.
Quoting aletheist
OK, so I have a question: does the number zero exist? Where's the difference between the number zero and the empty set?
Quoting Metaphysician Undercover
In category theory sets are described without the making use of the axiom of extension.
Quoting Metaphysician Undercover
Then I think you should like topos theory: in a topos the object that represents the empty set (the initial object) is not in general required to exist. You can assume it's existence, but it's not required by the definition of a topos.
:lol: :rofl: :lol:
Quoting tim wood
That's not what 'countable' means. Here are the definitions:
S is countable iff (S is 1-1 with a natural number or S is 1-1 with N)
S is denumerable iff S is 1-1 with N
S is countably infinite iff (S is countable and S is infinite)
So it's easy to prove that S is countably infinite iff S is denumerable.
So 'countable' does not mean "denumerably infinite", and "denumerably infinite" is redundant, and 'countably infinite' is equivalent to 'denumerable'.
Quoting tim wood
In this context, for convenience, by 'string' we mean 'denumerable binary sequence'. That said, here we go:
The length of any string = card(N) = aleph_0.
Quoting tim wood
Each string in the list has denumerable length. And there are denumerable lists of such strings. But there is no denumerable list of strings such that the list has every possible string in the list.
Quoting tim wood
IF the list includes every string, then the diagonal argument doesn't work. But that's not saying much, because we have not shown that there is a list that includes every string. Indeed the diagonal argument proves that there does not exist a list that includes every string. There is no force to an argument that says "If there is a complete list then there is a complete list".
Instead, you start with the question "Is there a complete list?" Then you prove that there is no complete list.
REVIEW of all this:
Each string has denumerable (i.e., countably infinite) length.
The question is "What is the cardinality of the set of all strings?"
If there is a denumerable list that has every string as an entry, then the cardinality of the set of all strings is card(N).
So is there a denumerable list that has every string as an entry? The diagonal argument proves that the answer is No.
And the notation 2^N stands for 'the set of functions from N into {0 1}", which is exactly the set of all strings.
We also prove that the card(2^N) = card(PN).
So the cardinality of the set of all strings = card(2^N) = card(PN).
And the cardinality of the set of all strings does NOT equal card(N).
The axiom of extensionality is:
If for all z we have z is in x iff z is in y, then x = y.
That does not contradict the theorem:
There exists a unique x such that for all z we have that z is not in x
and then we have the definition:
the empty set = the x such that for all z we have that z is not in x
Actually, this vector field is a good example of a dependently-typed function. The domain of the function is the surface of the sphere, but what is it's codomain? For each point of the sphere the codomain is a different vector space. But all these vector spaces are identical, except for the fact that they are associated to a different base point. This in type theory is called a parametric type: a type that depends on a parameter in an "uniform" way. And the value of the function is the vector representing wind's direction and velocity, that of course vary with the point on the sphere.
These kind of programming languages can be used as logic languages for mathematics. And a mathematical proof can be expressed as a program in a dependently-typed programming language.
This is a symbol, "0", or "zero". As you seem to be fairly well educated in mathematics, you'll know that it means different things in different contexts. Despite your claim that mathematical languages are very "formal", there is significant ambiguity concerning the definition of "zero". Do you agree that when we refer to "zero" as an existing thing, a number, like in "the number 0", it means a point of division between positive and negative integers? How is this even remotely similar to what "the empty set" means?
Quoting Mephist
It's not "the object which represents the empty set" which I am concerned about, it is "the empty set" itself which bothers me. It is a self-contradicting concept. If a set is to be something, an object, then, as an object, it cannot be empty because then it would be nothing. You would have an object, a set, which is at the same time not an object because it's composed of nothing.
So there is a distinction to be made between the definition of the set, "the set of...", and the actual set, or group of those things. If there is none of those defined things, then there is no group, or set of those things, such a defined set is non-existent. There is none of the describe things and therefore no set of those things. There is a defined set, "the set of..." which refers to nothing, no things. It is not an empty set, it is a non-existent set. Only through the category mistake of making the defined set ("the set of..."), into the actual set, can you say that there is this set which is empty. So if we allow that there is this actual set, the set of nothing, then the set becomes something other than the collection of things which forms that set. And we'd have no way to identify any set because the set would not be identified by the things which make it up.
I was referring to the natural number zero. Natural numbers in set theory are defined as sets: the natural number N is a set that contains N elements. If there is no empty set, there is no zero, right?
So, you say that zero is not like the other natural numbers (that are sets), but is only a symbol not well defined. I understand this, but then you say - in "the number 0", it means a point of division between positive and negative integers - but what are negative integers then? Aren't they just symbols? Following your reasoning, I would say that only positive natural numbers are real and all other kinds of numbers are just not well-defined symbols. OK, then how should they be defined correctly? I mean: it seems to be a little "restrictive" to throw away all mathematics except from positive natural numbers...
Quoting Metaphysician Undercover
OK, I understand! NO EMPTY SET OF REAL THINGS EXISTS IN REALITY. I agree. But the problem remains: how can you define the other mathematical entities except from positive natural numbers? I think you have to allow the use of symbols that are NOT REAL THINGS if you want to do mathematics, don't you agree?
I see that I didn't answer on the main topic here, that was about extensionality.
The fact that "sets are more fundamental than their elements" is true for topos theory, not for topology based on set theory, of course.
In set theory a set is identified by it's elements, and extensionality is an axiom.
In topos theory instead, the "category of sets" (that you can interpret as "the class of all sets") is defined axiomatically as an algebraic structure (a category with some special properties).
An analogous thing to "the class of all sets" is for example "the class of all groups" (in the sense of group theory). You don't describe groups by saying what a group is "made of", but only saying what are the properties of groups: how they relate to each other, and not what they are "made of".
The same is true for sets in topos theory: the theory describes how sets relate to each-other, and not what a set is "made of".
P.S. To summarize:
- axioms of group theory ==> axiomatic description of groups
- axioms of category theory + axioms of topos theory ==> axiomatic description of sets and functions (sets are represented objects and functions are represented by arrows)
This is exactly why it is contradictory. If there is a set without any elements it is not a set at all. With zero elements the supposed set is non-existent. But if you propose that the number zero is itself an element, such that there can be a set with "zero" as an element, you are saying that there is a set which has an element "zero", but also has zero elements. That is contradictory.
Therefore if we adhere strictly to the method of definition provided, then within set theory, there ought to be no natural number "zero". The natural numbers are defined by the sets which have those elements. There is no such thing as a set which has no elements, this is contradictory, as a collection of things without any things. So zero is excluded as a natural number. by the precepts of set theory. As I explained already, this very same problem was exposed by Aristotle, in slightly different terms, so the Neo-Platonists established "One" as the fundamental Form. To place "zero" as the fundamental form, or "set", is to base the system in contradiction.
Quoting Mephist
The symbol has a different meaning depending on how it is used or defined. The natural numbers are used for counting objects. We name a type, apples or oranges etc., and count the number of that type. Since we can name a type and also have no object of that type, we can have zero of that type. "Zero" allows the named type to have meaning, when there is none of them, by allowing that we have the potential for a quantity of that named type, without actually having any of them now. This concept of zero, as the "potential" for objects of a specified type allows us also to count negatives of that type.
Do you see that if we make the number which is named by the symbol, an object itself, then we lose the capacity to use "zero" as the potential for a number of the named type of objects? Zero is itself the object which is named, so there is no such thing as none of those objects. The set of "0" already has an object so it cannot be an empty set.
Try looking at it this way. The natural numbers are used for counting objects. The "count", the number or quantity, is distinct from the objects themselves. "Five", as the quantity of apples on the table, is distinct from the actual objects on the table, it is not a property of the apples. It is only by apprehending the quantity as distinct from the objects, that we can use "zero" as a quantity (natural number). If the quantity of objects was not distinct from the objects, if it were a property of the objects, then there'd be no such thing as zero objects, because there'd be no objects to have that property, "zero". It is only by postulating that the quantity is something distinct from the specified objects that we have the capacity to say that there is zero of the specified type of object. Do you recognize that in set theory, the quantity itself, (as the natural number), is the specified object, therefore we have no capacity to talk about none of those objects? Mentioning that object, as the object talked about, necessitates that there is not zero of that object.
Quoting Mephist
It's not a matter of what exists in reality, it's a matter of what is contradictory in principle. To say " I am going to talk about this object, but this object is not really an object, because there is zero of them", is blatant contradiction. To bring this expression out of contradiction we must amend it. I might say for instance, "I am going to talk about a type of object, of which there are none", or I might say "I am going to talk about a quantity, and this quantity is zero". But if I make the category mistake of conflating these two options to say "I am going to talk about this quantity, zero, as an object itself, and assert that there is none of these objects", then I contradict myself.
Quoting Mephist
Do you see that this proposition denies the possibility of an empty set? The empty set has no identity as a set, and therefore cannot be a set.
Quoting Mephist
So consider that we have a defined property. A "group" is all the members which have that property. We can establish relation between individual members of groups, based on the different groups that they are in. However, the properties are properties of the members, they are not "properties of groups", that would be a composition fallacy. So we cannot proceed toward establishing relations between groups this way, that would be a relation based in a fallacy. Suppose one property is "red", and the other is "hot", and we find that many red things are also hot things, it would be invalid to establish a relation between the property "red", and the property "hot", in this way.
See, "the set", or "group", is based in the defined property. To deal with "the group" as if it were a whole, an object, means that we are dealing with types, the defined property, a universal, rather than the individuals of the group. As an object, the universal, or type is a Platonic Form. A Form, as a universal, is completely different from a particular, an individual. The rules for relating universals to each other are completely different from the rules for relating individuals, because we relate individuals by determining their properties, but a universal is nothing other than a property already.
So here's an example of how we relate Forms or universals. In the Aristotelean way, the more general is "within" the less general, as an essential property, by definition. For example, "animal" is within "human being", as an essential property, like "polygon" is within "triangle", by definition. Further, "human being" is within "Socrates", Socrates being a specific human being. The particular, being the specific thing referred to, the individual human being who bears the name Socrates, is not within anything, and so is called primary substance. Do you see that it is possible for something to be within nothing (within no set), as the more general is always within the less general, so the most specific is not within anything, as primary substance? But that which is within nothing is still something. Now, at the other extreme is the most general, that which is within everything. Never is there the possibility of a set, or defined property which has nothing within it. So we might ask what it means for something to not be within a set, but it makes no sense to talk about a set which has nothing within it.
Ah. Thank you. That was very interesting and helpful.
Who argues that, exactly, besides the Peirceans on this forum? I've actually never run across this point of view except for here. And how does that square with the intuitionist continuum, which has even fewer points than the standard reals? They can't all be right.
LOL. @Mephist was making the point that one can do "set theory without elements" as in Lawvere's elementary theory of the category of sets, which unfortunately doesn't have a Wiki page. But basicaly you can do most of set theory in a purely categorical way. As I understand it you get a slightly weaker version of set theory than the standard theory.
A closet is an enclosed space in which I hang my clothing.
One day I remove all the clothing from my closet.
Do I still have a closet?
Do I not in fact have a perfectly empty closet?
How is that relevant? As Mephist said, a set is identified by its elements. That's the reason why an empty set makes no sense. Clearly a closet is not identified by its elements..
Ok you're right. Closets and empty grocery bags aren't really on point, even though they can be helpful visualizations, such as a grocery bag containing an empty grocery bag to visualize [math]\{\emptyset \}[/math].
So how about an axiomatic approach? The axiom schema of specification says that if [math]P[/math] s a unary predicate, and [math]X[/math] is a set, then [math]\{x \in X : P(x) \}[/math] is a set.
Consider the unary predicate [math]x \neq x[/math]. Let [math]X[/math] be any set whatsoever, say the natural numbers or the real numbers or whatever set you might happen to believe in. Then we can define
[math]\emptyset = \{x \in X : x \neq x \}[/math].
So if you believe in the existence of any set at all, and you accept the axiom schema of specification, then you must accept the mathematical existence of the empty set.
What say you?
That's the problem, I don't believe in the existence of any set. That any set has real existence has not yet been demonstrated to me. And axioms which allow for the demonstrably contradictory "empty set" lead me away from believing that sets could be anything real.
I've previously called your philosophy mathematical nihilism, and once again you confirm it. You start by saying you don't believe in the empty set; but it doesn't take long to get you to agree that you don't believe in the existence of any sets at all.
If you don't believe in sets, why go to the trouble of explaining why you don't believe in the empty set? I wonder if that shows that you haven't thought your idea through. Why bother to argue about the lack of elements, when you don't even believe in sets that are chock-full of elements?
Quoting Metaphysician Undercover
But this is a strawman argument, "... giving the impression of refuting an opponent's argument, while actually refuting an argument that was not presented by that opponent."
Nobody has claimed sets have "real" existence, whatever that is. Sets have mathematical existence, and that's the only claim I'm making.
I could easily take you down the rabbit hole of your own words. Is an electron "real?" How about a quark? How about a string? How about a loop? And for that matter, how about a brick? Are there bricks? When we closely examine a brick we see a chemical compound made of molecules, which are made of atoms, which contain protons, neutrons, and electrons, which themselves are nothing more than probability waves smeared across the universe.
Do you believe in the existence of bricks? Physics tells us that even bricks are nothing more than probability waves smeared across the universe. We see a brick in its location simply because that's the most likely location for it to be found. Once in a long while, a brick appears someplace else where it has a low probability of being found. I hope you know that this is standard doctrine of modern physics. Do you deny science along with math?
You are painting yourself into an ontological box. Not for the first time, I might add.
Aha! So, finally, a resolution to the question I posed about PWs on another thread. The medium through which they travel are brick roads. And they culminate on the shores of the Emerald City!
Thanks! :nerd:
The problem with the "empty set", which I have described, is a demonstration of the reason why I don't believe in the existence of sets. So I take the trouble of explaining the problem with the empty set to justify why I do not believe in sets. A set is a Platonic Form, and it's nonsense to speak of the Form of nothing. Yet we use "zero", and "nothing" commonly. Therefore I do not believe in this type of Platonic realism because it doesn't give us an appropriate way to represent what "zero" means.
Quoting fishfry
Sorry fishfry, but you stated quite clearly "if you believe in the existence of any set at all, and you accept the axiom schema of specification, then you must accept the mathematical existence of the empty set."
You're making up nonsense if you are asserting that there is some sort of existence which is not real existence. When I say "real existence", I mean existing, categorically, as distinguished from not existing. What is it, if it's not real existence, an illusion? Real existence is opposed to the illusion of existence. I'd call it deception, claiming that something exists when you know it's not real existence but an illusion. And it's completely nonsensical to claim that there is a type of existence "mathematical existence", which is not real existence. How would that work in set theory? You have a set of existing things, and you have a subset, "mathematical existence". Then you say that "mathematical existence" is somehow outside of the set of existence, because it's not a real existence.
I went through this with aletheist already. Altheist was trying to distinguish different types of existence corresponding to different subjects (fields of study), but refused to recognize that all of these are a member of a single, more general category of "existence" itself. So aletheist refused to recognize ontology as the study of existence in general, what all different sorts of existents have in common, and insisted on placing "mathematical existence" outside of, separate from 'ontological existence". But that is nothing other than an assertion that there is no such thing as ontology. Now you attempt the same move, by saying that mathematical existence is not real existence, you place it outside the field of ontology, which studies "being", "existence", rendering ontology useless by saying that there is a type of existence which cannot be studied by the field of study, ontology, which studies existence in general.
Quoting fishfry
We can look at these concepts, "electron", "quark", "string", "brick", and see if there are inconsistencies, contradictions, or other forms of fallacious logic, and if not we can say that the thing referenced most likely has some form of existence. So I'm not nihilist, I just believe that contradiction negates the possibility of existence, such that it is impossible that a contradictory thing exists. So things referred to have a probability of existing, until they are proven impossible, then that probability is removed.
Quoting fishfry
If you define "brick" in this way, as "possibility waves smeared across the universe" it probably doesn't have existence, because physics uses a lot of contradictory mathematics and inconsistent principles, as I am arguing here. But there are other ways that we can define "brick", and use the term, which do not involve logical inconsistencies, and so a "brick" in this sense would have a reasonable probability of existence.
Quoting fishfry
Science which uses faulty math is obviously faulty, don't you think? The soundness of the conclusions is dependent on the soundness of the premises.
Quoting fishfry
The Peirceans who are not on this forum, for starters; but it goes back at least as far as Aristotle, who recognized that numbers of any kind are intrinsically discrete, rather than continuous. The key word here is "truly"; I have acknowledged that the real numbers are an adequate model of continuity for most mathematical and practical purposes. Nevertheless, conceptually a line is not composed of points, a surface is not composed of lines or points, and a solid is not composed of surfaces or lines or points. Instead, the parts of a line are one-dimensional lines, the parts of a surface are two-dimensional surfaces, and the parts of a solid are three-dimensional solids. Anything of lesser dimensionality is not itself a part (or portion) of that which is truly continuous, but rather a connection (or limit) between its parts.
I think this would end up giving precisely the same mathematical theorems, no? You just restate things in terms of connections and parts.
Like: the metric topology on the interval (0,1) consists of open sets defined by: the empty set (a limit) is in the topology, the whole (0,1) is in the topology, unions of parts and connections/limits are in the topology, countable intersections of parts (be they connections/limits or parts) are in the topology.
The vocabulary of sets lets you phrase all these concepts already. "Parts of a line? They're finite intersections of its interval subsets which have cardinality greater than than 1".
A set is a bottom-up conception, assembling a whole from discrete parts. True continuity is a top-down conception, such that the whole is more fundamental than the parts.
Let's say I have some "True continuity" X. Like a line X=(0,1). Let's say I can take "parts" of it in the above manner; arbitrary subintervals. (0,a), (a-1/n,1) are subintervals for any a less than 1 and greater than 1/n. Since they're parts of a true continuity, the true continuity ensures the existence of their intersection; limit the process of intersection over n and get the limit {a}. If I take the union of all such limits, since a was an arbitrary member of (0,1), it produces (0,1). So starting from a true continuity, like a line segment, you can get discrete numbers, then build up the true continuity out of the individual numbers through a union. It's top down and bottom up at the same time.
Do you agree this process is legitimate?
If not, do you reject sets as a concept?
Introducing numbers already imposes discreteness. Numbers are for measuring, they cannot constitute a truly continuous line.
Quoting fdrake
No, again, a line is not composed of points corresponding to numbers. We can only mark them on (not in) a truly continuous line. They then serve as arbitrary and artificial limits/connections between distinct portions/parts.
Quoting fdrake
Not at all; again, set theory can be quite useful as the basis for an approximate model of continuity. However, it cannot serve as the basis for true continuity, because it requires discreteness at the outset.
Mephist draws on a top-down description of a set; the set as more fundamental than its elements, to explain topology. This is the only way that "empty set" makes sense, if the set is a top-down construction. So a set may be a bottom-up construction, but set theory employs sets as if they are actually top-down, by utilizing the empty set.
Makes sense. I've never met any besides here. Must not hang out among the right philosophers.
I can certainly see Peirce's objection that a true continuum could never be made up of individual points. Did Aristotle reject the notion of an instant of time? Or did Peirce? You could't accept instants if you reject points, I'd imagine.
Not sure about Aristotle, but Peirce indeed explicitly rejected the notion that continuous time is somehow composed of durationless instants. They are artificial creations of thought for marking and measuring time, just like discrete points on a line.
Do you know of any current mathematical objects that behave more like true continuity in your view?
Line figures, surfaces, and solids can be understood in geometry as truly continuous. We use points to model and analyze them, but they are not composed of points.
More fundamentally, my understanding is that category theory is broader than set theory and can serve as a basis for alternative approaches to mathematics that recognize true continuity. The one that currently seems to come closest to Peirce's views is synthetic differential geometry, also known as smooth infinitesimal analysis.
Thanks!
You are reading more into what @fdrake proposed than there is. He didn't say anything about numbers constituting a line; on the contrary, he was going with your paradigm of continuous line figures - nothing else. And he was trying to show how, with assumptions that seem reasonable even in that paradigm, you still end up with a system that is isomorphic to the set construction. (Surely, we can still talk about the lengths of those line figures? Those are all the numbers that we need to get going.) I am not sure whether it can actually work out that way, but that was the idea, if I understand him correctly.
Regardless, I think the synthetic differential geometry pdf suggested an intuition closer to it. Roughly, augment the real numbers with some set of symbols [math]D[/math].
You then stipulate that any function [math]f[/math] defined on [math]D\rightarrow \mathbb{R}\cup{D}[/math] has some unique constant [math]b[/math] such that for all [math]d \in D[/math] [math]f(d)=f(0)+db[/math]. There's also a property that [math]d^2 = 0[/math] for all [math]d \in D[/math]. This looks like placing a family of infinitely small line segments around every point in [math]\mathbb{R} \cup D[/math]. For functions, it gives a "tangent space" of a function at a point confined to infinitesimal neighbourhoods around it. It "smears out" the real line (and function values) into something that can't be element-wise disassembled in the same way (if you fix a point of [math]\mathbb{R}[/math], this corresponds to an infinitesimal neighbourhood around that point in [math]\mathbb{R}\cup D[/math])
But you can't start from ANY real number "a". If you define real numbers as limits of rational numbers, "a" should be rational, or should be itself a limit of a sequence of rationals. In a constructivist logic you have to define how "a" is "built".
Quoting fdrake
Exactly. This is a sheaf of linear tangent spaces built on the base space of Cauchy sequences of rational numbers. Similar to the sheaf of all vector spaces tangent to a sphere.
Quoting tim wood
?o. That's the cardinality of infinite DISCRETE sets.
?1 is the cardinality of powersets of sets whose cardinality is ?o.
Quoting Metaphysician Undercover
If you want to prove that ZFC is inconsistent you have to derive "false" using the rules of ZFC's logic. You can't do it using english language, as you are trying to do.
You can't be an art critic without looking at the paintings!
Thanks. That would make sense. Physics would get more difficult I imagine.
Here is one especially succinct argumentation from Peirce.
It then follows from the first sentence that since the present is not an instant, there is no such thing as an instant at all.
Here is another passage that I found very enlightening when I first came across it in an unpublished manuscript.
Physical reality is a dynamical process of continuous motion, while psychical reality is an inferential process of continuous thought; more generally, continuous semeiosis. Positions and propositions are artificial creations for describing hypothetical instantaneous states of motion and thought/semeiosis, respectively.
You do recognize that "false" and "true" are assigned to the premises, not by what is determined by the logical system, (which is validity), don't you? Inconsistent, or contradictory premises, is not determined by the logic of the system.
I criticize the axioms according to how they are expressed in English. If the fundamental axioms could not be expressed in English, or other natural languages, they would be meaningless. Terms need to be defined.
If a piece of art has any meaning at all, it would be recognizable by someone other then the artist. So, you don't need to be an artist to be an art critic. The critic assesses the meaning.
You might argue that the axioms have a different meaning in mathematics from what is represented in the English expressions of them, just like the artist might argue that the critic doesn't understand the piece of art. But what type of meaning could this be, if when it is represented in English it is contradictory? Sure, an artist might represent a person as being happy and sad at the very same time, but such illogical nonsense ought to be excluded from a logical discipline such as mathematics.
NO. "false" and "true" in first order logic (the logic used in ZFC) are purely SYNTACTICAL expressions. They are determined ONLY by the logic of the system. That's the way it works!
Look at the definition from wikipedia ( https://en.wikipedia.org/wiki/Consistency ).
"""
although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states a theory T is consistent if there is no formula "phi" such that both "phi" and its negation "not phi" are elements of the set of consequences of T.
The set of axioms A is consistent when "phi" and "not phi" belong to "sentences derivable from A" for no formula "phi".
"""
Then, there is a theorem ( https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem )
"""
that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It makes a close link between model theory that deals with what is true in different models, and proof theory that studies what can be formally proven in particular formal systems.
"""
Quoting Metaphysician Undercover
In a formal logic system TERMS DON'T NEED TO BE DEFINED. That's why it is called "formal" logic.
Quoting Metaphysician Undercover
It can be a meaning that has nothing to do with the English meaning of the words. The proof of the theorem shows that a model always exists (if no contradiction is derivable) because it can be built using the strings of symbols of the formal language itself!
Probably that's the part that you strongly disagree with. But if you want to criticize the proof of Godel's completeness theorem, you should at least read it! That's what I meant by "looking at the paintings" before.
Ok. But isn't he conflating human experience with reality? He's right that for humans, the present is an experience of flow. But we have no idea what the underlying reality is. Why should human experience be privileged above nature?
Perfectly sensible. Our physics is an approximation or conceptual model to help us describe reality. It's not to be confused with reality. A point I've made many times.
Quoting aletheist
Yes of course. Perfectly well agreed. But if all Peirce is saying is that the map is not the territory, that our mathematical and conceptual models are useful fictions to help us manage or conceptualize reality. then of course he's right; but is that all there is? I thought this point was fairly well agreed, even in science. Einstein supersedes Newton supersedes Aristotle. You never get to the end of the process, you just get increasingly better models.
The only reality that we can know is what we learn from experience. We formulate hypotheses to explain our experience (retroduction), work out their necessary consequences and make predictions accordingly (deduction), then test whether those predictions are corroborated or falsified by subsequent experience (induction).
The problem though, is that I am talking about judging those premises or axioms which establish those definitions of "true" and "false". If I am to judge them, I must judge them in relation to something else, something outside the system If you are asking me to accept the precepts of the system without judging them, then you are being unreasonable. That's the way it works! We're grown adults, we have free choice to judge these things. It's completely unreasonable for you to say that I must accept the system's axioms in order to judge the system's axioms, when acceptance is dependent on judgement, and acceptance precludes the possibility of fair judgement. A conclusion cannot be incompatible with the premise, so if I accept the axioms, it is literally impossible for me to produce a judgement against them. Therefore you are being completely unreasonable.
Quoting Mephist
Logical systems use symbols. A symbol which represents nothing is contradictory nonsense, just like the empty set. So you're just spewing more contradictory nonsense in an effort to justify your earlier contradictory nonsense.
Quoting Mephist
Yes, that's what I strongly disagree with. Strings of symbols without definitions is nonsense. A symbol which represents nothing is not a symbol. If you are merely talking about a set of rules by which symbols are related to each other, then there is no reason why we can't discuss these rules in plain English. I think that your refusal to discuss this in plain English is evidence that you know that there is deception within the system. So, either you discuss these rules in plain English or I level the accusation that you're attempting to hide deception behind your language.
I don't see how your art analogy works for you. An individual can glance at a painting, and find it ugly without analyzing it, just by apprehending a few prominent features of it. Likewise, we can hear a piece of music, and right away form a dislike for it based on some fundamental aspect of it. Why would you insist that the person must make a thorough analysis, attempting to empathize with the artist's intent, wasting one's time, and even torturing oneself, to justify one's dislike for the piece? When the person can point to a few fundamental, and prominent features, and explain why these features make such an effort unappealing and unwarranted, why not simply accept that, rather than insisting that the person cannot make such a judgement. If fundamental and obvious aspects of the art are unappealing, why insist that the critic must analyze all the finer aspects before making a judgement of dislike?
Didn't Plato point out that what we experience is but shadow on a cave? And that the true reality lies outside, unseen and unseeable by us?
But is what we're talking about simply the question of whether what we experience is the same as reality? Did Peirce argue that it is? But how can that be? Science is historically contingent; and the better equipment we have, the better experiments we can do. Our theories of the universe keep changing. The universe, presumably, stays the same.
Were you like this when you learned to play chess? "This is the knight." "But no it's not REALLY a knight. Real knights don't make moves like that, they slay dragons and rescue damsels. I refuse to accept the rules of your game till you tell me what they mean outside of the game."
And sometimes a conclusion is indistinguishable from the premise.
:roll:
Yes, I confess that I am trying to hide a deception behind MY language :rofl:
The things that I wrote can be found in any introductory book to mathematical logic, and they are very clear for most of the people that write on this forum. If you really wanted to understand it, just buy a book and read it!
Quoting Metaphysician Undercover
Yes, but you are not even trying to take a glance at the painting! You don't have to be an expert to understand how mathematical logic works. And unlike other parts of math, you don't even need to learn some other more fundamental concepts before.
Quoting Metaphysician Undercover
You don't have to accept the axioms. You have to prove that assuming those axioms leads to a contradiction. You have to use the rules of logic to produce a sentence of the form "A and not A" (I am not sure if "true" and "false" are terms of first order logic, maybe I made a mistake before saying that you
have to derive "false"). As @fishfry has explained to you many times, this is like in the game chess: you have to show that, starting from a given position, white can checkmate. If it's not possible to produce a sentence of the form "A and not A", it means that the axioms are consistent (not contradictory).
The interpretation of the terms as sets (and then the meaning of the sentences) is a different issue.
You can argue that the terms that ZFC calls "sets" are not exactly correspondent to what we "intuitively" think to be sets, and a lot of people (even mathematicians) have this kind of objections to ZFC. But this is not about the consistency of the theory; this is about it's "meaning".
That's not quite right. I learned how the game was played, then decided I didn't want to play it. The fact that it was a game, and the rules referred to nothing "real" probably made me think of it as a waste of time.
Quoting Mephist
It's not a matter of whether or not the axioms lead to contradiction, it's a matter of whether or not the axioms themselves are contradictory. I demonstrated precisely this, that the axioms are contradictory, with the "empty set". If you've forgotten already, go back and take another look at those posts. Sure, there was some mention that the contradiction might potentially be avoided by introducing exceptions to the rules. But exceptions to the rule just serve to disguise and hide the contradiction in the rules, behind sophisticated complexities. They do not resolve it.
Quoting Mephist
This is where you are wrong Mephist, and I can't seem to get this through your head. You cannot use the rules of the logic without first accepting the axioms. The axioms state the rules. You must agree to play by the rules in order to use the logic. If the axioms themselves are contradictory, then you accept those contradictions, when you proceed to use the logic. Therefore you cannot prove that there is something unacceptable about the axioms, i.e. that they are contradictory, through the use of the logical system, because by accepting the axioms you consent that there is nothing unacceptable about them, i.e. they are not contradictory.
Use of contradictory axioms may lead to absurdities like paradoxes, but a paradox does not prove that any axioms are contradictory, the appearance of paradox could be caused by something else. So when a paradox appears one might go back to the axioms, and determine whether or not there is a contradiction, using principles outside the logical system, the principles in which the system's axioms are based. The axioms are the rules of the logical system and they may just lead to an unsolvable paradox within the system. We do not necessarily know the cause of the paradox though, and sometimes analysis of the paradox cannot lead us to its cause. That's why it's a "paradox". if the paradox is caused by a problem with the axioms, then to solve the problem requires going outside the logical system to determine how the axioms are founded, the relations between them, interpretations of them, etc., and removing inconsistencies.
To take fishfry's example of the chess game, imagine contradictory rules in the game. This could lead to unsolvable problems within the game. From within the game, the problems cannot be solved because the rules leading to the problem are set. However, because the premise of my example is that their are contradictory rules, it appears obvious that the problems are caused by contradictory premises, as contradictory premises can cause problems. So you might think that if there are contradictory rules in any game, the problems they cause would demonstrate clearly, the very contradiction which exists in the rules. But this is not the case, because an equivalent problem might arise as a matter of a difference in interpretation. The "paradox" within the game, by the very nature of a "paradox" doesn't necessarily reveal the source of the problem. And although the rules of the game might include rules of interpretations, those rules of interpretation cannot have rules of interpretation, ad infinitum. Therefore contradiction, and other problems in the rules, or axioms, can only be determined and resolved by reference to principles outside the system composed of those rules, because they might equally be problems of interpretation.
Quoting Mephist
Now perhaps you are starting to grasp what I am arguing. I am arguing the "meaning". Contradiction within the meaning of a theory's axioms is clearly a matter of inconsistency. We might have two distinct senses of "inconsistency" here though, whether the proceedings in a logical system are consistent with the axioms, and whether the axioms themselves are consistent. I am arguing the latter.
So you might hand me the rules of a game, and ask me to play. If I look over the rules and see that there is blatant contradiction in the rules, as in set theory, or even that the rules are open to contrary interpretations, or there are holes, possible situations not covered, I might tell you that I have no desire to play your game. And, I might proceed to show you what I see as blatant contradiction, the empty set. You might accept this and say ok, don't play then. You might also claim that this is only a matter of interpretation, and attempt to work out a satisfactory interpretation with me. But your insistence, that I join the game, even when I see a glaring problem in the rules, and then we try to work out this problem with the rules only after we develop a problem within the play of the game, is completely unreasonable.
You reject all formal systems not based strictly on physical reality? Do you drive on the correct side of the road appropriate to your jurisdiction? Traffic laws are a made-up game too.
Again, nihilism. You reject games, you reject abstraction, you reject science. Fun to argue all day long on an Internet forum but I truly doubt you actually live this way. I bet you obey traffic laws even though they are not laws of nature.
What you call "contradiction", the impossibility to identify the terms of the language with physical objects, is not considered as a problem in mathematics: it's simply ignored.
Here's a famous quote from Bertrand Russell about mathematics:
(https://www.brainyquote.com/quotes/bertrand_russell_402437)
This is at the same time both a limitation and an advantage: it gives you the freedom to invent new concepts, but you lose the relation between mathematical concepts and the physical world!
This is a choice that mathematicians have done at the beginning of 20th century, mainly (I believe) to get rid of the paradoxes arising from the use of infinity and infinitesimals.
However, in my opinion this is the natural development of Aristotle's logic: the formalization of the rules of deduction. The rules of deduction (used in proofs) should not depend in any way on the meaning (or correspondence to real physical objects) of the words.
So, you say that this is all wrong, because you are allowed to create axioms that don't have any correspondence to reality. That's true. But what is the alternative? After all, these mathematical constructions based on "nothing real" happen to be very useful to build models that agree with experiments.
As far as I know, I think it would be possible to reformulate all mathematics without making use of empty sets at all. But would this make any difference?
Uh, no, laws are not games. What kind of game do you get penalized for not playing?
Quoting Mephist
But I haven't yet accepted your mathematical rules. I am judging them as to whether or not they ought to be accepted. So I can only judge "contradiction" according to what it means in English.
Quoting Mephist
"Contradiction" has nothing to do with identifying things with physical objects, it relates to how words are defined. So for example, if "set" is defined as something having extension, and "empty set" is defined as a set having no extension, then there is contradiction here. "Empty set" breaks the rules expressed in the definition of "set", and therefore cannot be a set.
Quoting Mephist
"Meaning" is not necessarily dependent on correspondence with physical objects, it might be derived from relations within a conceptual structure. That's why I outlined two distinct types of "consistency", consistency within a particular structure, and "consistency" in how that structure relates to outside principles. Notice there is no necessity for correspondence with physical objects. But when correspondence with physical objects (what some call "truth") is one of those outside principles, then the conceptual structure might be judged in relation to this principle.
Quoting Mephist
That's not what I'm saying is wrong. What's bad is if there is contradicting axioms, like in my example above. Suppose people are creating axioms, and the axioms are not necessarily corresponding with reality. There's no inherent problem with that. Now suppose a problem in application of the axioms appears, possibly because the axioms don't correspond with reality, perhaps some sort of paradox appears or something when people try to apply the axioms. So the people creating axioms decide that if they change this axiom, or create another axiom, the problem can be avoided. But maybe they don't realize that the new axiom contradicts another axiom, or if they do, they might still be inclined to accept it because it makes that particular problem go away. However, I think the contradictory axioms are bound to create other problems further down the road.
OK, let's follow you definition of "set" (that is not the definition used in ZFC set theory, but we are considering an alternative definition because we do not accept MY mathematical rules).
Definition 1: "a set" is something having extension
Definition 2: "an empty set" is "a set" having no extension
Substitute the word "a set" from D1 in D2 and you get P1:
Proposition 1: "an empty set" is something having extension having no extension
P1 could be rewritten as: "an empty set" is something "having extension" AND NOT "having extension"
So we get a contradiction "H and not H" where H is "having extension".
Then, the two definitions D1 and D2 cannot be used at the same time.
Let's follow your reasoning and keep only D1: there is no empty set with this definition.
But now we are not finished yet: we have still to define what is "extension".
I think you have two possibilities:
1. define "extension" in terms of another property (something like "occupation of space"? I don't know..)
2. consider "extension" as an undefined "primitive" notion
- In case of 1. you end up in an infinite chain of definitions (of course you cannot define "extension" in terms of "a set", right?)
- In case of 2. you just did what today's mathematics do, just replacing the primitive notion of "set" with the primitive notion of "extension" and changing the definitions accordingly.
But now what prevents me to consider a "null extension"?
"extension" at this point is an undefined notion, so "null extension" does not generate any contradiction now.
And if you allow "null extension", why not allow "empty set" and consider "set" to be a primitive notion instead?
Of course you can say that the term "null extension" is not allowed (meaning: you are not allowed to use the attribute "null" with the word "extension"). But this is now an arbitrary limitation of the terms (a choice that you made in defining the concept of "extension"), and not a necessary condition to avoid a contradiction.
Following this argument that "null extension" is not allowed, you could say for example that a segment with "null length" is not allowed, so a point is not a segment. That is OK, but it's not due to a contradiction: it's only a choice of your definitions. Defining a point as a segment with no length does not create any contradiction, if you consider a segment as a primitive notion and a point as a derived notion.
Do you agree?
If you don't agree, then try to derive a contradiction due to the introduction of the concept of "an empty set" without making use of other undefined terms, such as "extension".
P.S. Try to take a look at Euclid's elements (https://en.wikipedia.org/wiki/Euclid%27s_Elements)
Here are the definitions, from Book 1 (taken from the book 'The elements of Euclid" by Oliver Byrne)
1. A point is that which has no parts
2. A line is length without breadth
5. A surface is that which has length and breadth only
Are these definitions contradictory?
Quoting Mephist
I recognize that is not the proper definition, I wrote something simple as an example.
Quoting Mephist
I agree that the definition of "extension" is in principle irrelevant. But no matter how "extension" is defined, it doesn't resolve the contradiction which is involved with something that, at the same time, both has and has not extension. To define "extension" as a property which something can both have and have not, at the same time, is just a trick of sophistry, designed to dodge application of the law of non-contradiction. If this is the case, then the definition becomes relevant.
Quoting Mephist
Do you not apprehend the trick of sophistry here? The law of non-contradiction says that the thing cannot be categorized as both "having extension" and "not having extension". Now, you introduce "null extension" as if it allows that the thing to be categorized as "having extension", as if "null extension" is a sort of extension, when "null extension" really means "not having extension".
Quoting Mephist
I don't see this. I cannot see how you made the contradiction go away. All I see is a trick of sophistry, which hides the contradiction behind the illusion that zero extension is some sort of extension. But it cannot be, because if zero extension was some extension it would not be zero. I do understand that it is a matter of definition, but I do not see how defining a property, whatever that property is, extension, length, or whatever, in such a way so that a thing can be said to both have that property and not have that property, at the same time, is anything more than a trick of sophistry designed to circumvent the law of non-contradiction.
So let's look at this example of the segment and the point. You define "point" using "segment". A point is a segment without any length. So the property here is "length". The definition of "length" as you described with "extension" is irrelevant. But from my perspective we need to ensure that "length" is not a sort of property which a thing can both have and not have, at the same time, or else the definition of length would become relevant, as a sophistic trick. The subject, or category is "segment", a point is defined as a type of segment. So we now need a definition of "segment" to make sense of what a point is. Remember that we have just allowed for a segment with no length, so "length" cannot be a defining feature of "segment". How would we proceed to define "segment" now?
I submit that this is a similar situation to what we have with "set". If we define "empty set", such that it is a real set which has no extension, then "extension" cannot be a defining feature of "set" without allowing that "extension" is a sort of property which defies the law of non-contradiction..
No I don't see any contradiction here. There is nothing to imply that a point is a line without length. That would be contradictory when #2 says that a line is length
Still sets tho. [math]x+db \in R \cup D[/math].
Let's look at it this way. The "line" introduces a new property which the preceding "point" has not, "length". The "surface" introduces a property which the preceding "line" has not, "breadth". But the "surface" also maintains the property of the "line", which is "length". Following this pattern, the "line" ought to maintain the property of the "point". But "no parts" is a sort of negation of a property, instead of a proper property. We can say that this negation is the property which the "point" has. So if we understand "no parts" as a negation of all properties, we'd have to understand the property of the point as "no properties", and this would be contradictory. It would be contradictory, to say that the property of a thing is that it has no properties. However, we do not understand "no parts" as "no properties", so the point is defined by what it does not have, and what it does have is left empty or undefined.
In the case of the "empty set", the property which it does not have is extension. However, being designated as a "set", it also has whatever property is proper to a "set". If extension is a defining property of a set, we have contradiction because we talking about a thing which is said to have extension (by the type of thing that it is said to be), yet it is also said to have no extension by the value given to that property.
In Anders Kock's book ( https://users-math.au.dk/~kock/sdg99.pdf ) D is an infinitesimal interval centered on x = 0 and defined algebrically by x^2 = 0 (see the definition at page 2).
R instead is the base space, defined agebrically simply as a commutative ring (built starting from two fixed points 0 and 1). The "real" real line is made of pairs of elements (a,b) of R (see definition 1.1 at page 3), where "a" is the point from the base space (the finite part of the number) and "d * b" is the fiber over "a" (the infinitesimal part of the number). "d" is an element of D.
I'm only pointing out that if sets can't provide a model for @aletheist 's intuition of continuity, since they consist of distinct entities (you can distinguish infinitesimals from each other, and make claims like [math]x+db[/math] lays in some set), then neither can the synthetic axioms presented in that link, as they concern sets.
(Though, they can be presented in a more general context. What I'm pointing out is that while sets aren't necessary to talk about continuity in that sense, they don't preclude it by themselves either.)
OK, now I read it, but I don't quite agree on all that he writes
For example example this part:
Quoting aletheist
"true continuity" can be defined even using standard set theory. Actually, even category theory can (and usually is) be based on standard set theory.
The fact that you have to use intuitionistic logic with it's weird rules about double negation, to make sense of "d^2 = 0" even if "d =/= 0" in my opinion is just a "wrong" definition of the negation operator: it should be called "complement of" instead of "not" (for example we should say "d belongs to the complement of 0" instead of "d =/= 0"). The complement of 0 is an open set. The complement of "the complement of 0" is another open set, disjoint from the first one, that includes not only zero, but zero and the infinitesimal neighborhood of zero.
It is true that the whole theory is usually formulated in terms of topos theory, and it seems way too abstract to be used as the "normal" theory of real numbers, but in my opinion it doesn't have to be presented in this way.
Quoting Mephist
It's not really about the reals and analysis as usually thought of, the construction doesn't satisfy the field axioms. Moreover, every function from the constructed number line with infinitesimals to itself turns out to be smooth (so, continuous and differentiable). If I've read right anyway.
Why not? Which of the field axioms are not satisfied?
Quoting fdrake
Yes, that's true. All subsets of the real line are open, so all functions are continuous (and differentiable).
Of course it's not equivalent to the "normal" real line, but calculus works at the same way. Why is this a problem?
Multiplicative inverses, you have zero divisors from the infinitesimals and that blocks it.
Quoting Mephist
It's not, it's just a point of difference. Having a number system that's built to embed differential geometry intuitions and make differential equation diagrams rigorous is very cool.
Ok. Assume d had a multiplicative inverse. Also d^2=0. Then d.d^-1=0.d^-1. Which gives d=0. But d is nonzero. So d does not have a multiplicative inverse. All nonzero elements need a multiplicative inverse for it to be a field.
d is the base vector of a vector space attached to each number (ring element) on the base space.
The real numbers [math]\mathbb{R}[/math] are a field (without 0). The real numbers augmented with the set of numbers defined in the previous post's link are not.
All you need to do to turn any non-zero dividing element into a zero dividing element is to multiply it by d. So all elements zero divide. If you quotient out the infinitesimals by mapping to standard parts it's a field again, as it's just the reals. Zero divisors can't have multiplicative inverses.
If you have two nonzero ring elements a and b, and a.b = 0, assume wlog that b has an inverse, then a.b.b^-1 = 0.b^-1, then a=0, so b has no inverse.
But you cannot multiply by d. You can multiply by (0, d*1), for example, not by d. All non-zero elements are all the elements of the form (x, d*y) where x is not zero.
Page 4. Properties.
"However if xd = 8d for all d then x=8". In that link, they are multiplying by d.
This implies that if one reads for example xd1 = 8d1 this not necessarily means x = 8. However if
xd = 8d ?d ? x = 8.
I don't really understand this. For what I understand, d should be treated as a differential operator and 8d is another differential operator. The elements of the real number line are pairs made of an element of a ring and a differential operator: to each element of a ring is attached a linear vector space. Maybe I missed something...
It seems to let you rigorously think of dx as an infinitely small translation/length with the expected connections to calculus and differential geometry.
:smile: good to know. Of course you don't have to believe me as a matter of principle. Usually I make a lot of mistakes when I write.
Quoting Metaphysician Undercover
Yes, of course. A property such as "X has extension" is a boolean value (true or false) associated to X. It cannot have both values at the same time.
Quoting Metaphysician Undercover
"null" is an attribute of "extension": an extension can be "null" or "not null". A set can be defined as something having extension (following your definition). null extension => empty set; not null extension => not empty set. Maybe this is a trick of sophistry, but it avoids the contradiction.
Quoting Metaphysician Undercover
A segment is defined by giving two points. if the two points are coincident (the same point), then it's the same thing as only one point. If the two points are distinct, the length is the measure of their distance from each-other. I don't see any problem with this definition.
Quoting Metaphysician Undercover
Yes, that's the same thing: you don't define what a set is, but just give some properties that any set should have, and one of the attributes of a set is the fact to be empty or not: this is just an attribute of any set: no need to define the word "extention".
Quoting Metaphysician Undercover
In my opinion to say "A line is length without breadth" is like saying "A line is a rectangle with zero width". He means that real objects have 3 dimensions, but a line is like a real object that has only 1 dimension. The other two dimensions are missing.
Quoting Metaphysician Undercover
Well, we could do the same with sets: adding properties instead of subtracting
- an "point-set" is a set with no parts
- a "line-set" is an extension of the "point-set" that introduces a new property: the number it's parts.
For me, that's the same logical construction. Why this should not be allowed?
At the end, they are all similar ways to do the same thing: attach some properties to an object to describe it without giving an explicit definition in terms of other objects!
No, the whole point of talking about "true continuity" is to distinguish it from (analytical) "continuity" as defined in accordance with standard set theory. The real numbers do not possess true continuity, because numbers of any kind are intrinsically discrete. However, they serve as a useful model of continuity, adequate for most mathematical and practical purposes.
Quoting Mephist
I am not a mathematician, but my understanding is that this is exactly backwards. Set theory can be established within category theory, but category theory cannot be established within set theory. "Set" is one of the categories, but there are others that need not and do not conform to standard set theory.
I'm not personally in accord with this point of view.
The relationships among category theory, set theory, various flavors of type theory, and other candidate foundations is not a simple matter to be summed up in a phrase.
I found an informative and insightful thread here.
Are category-theory and set-theory on the equal foundational footing?
The entire thread is well worth your time for anyone interested in contemporary foundations. In particular Derek Elkins's response is comprehensive and mind-expanding.
A few points. First, CT does not model ZFC. Rather, Elkins notes that "ETCS [Elementary theory of the category of sets] is equivalent to Bounded Zermelo set theory (BZ) which is weaker than ZFC."
Secondly, CT doesn't properly account for mathematical existence. This quote is a comment by Michael Greinecker:
"Set theory is full of axioms that guarantee that some things exist, which can be used to show that other things exist and finally that all the mathematical objects we want to exist do exist. Category theory doesn't really do that. You can formulate existence statements in categorical terms, but it is much less clear what kind of foundations category theory is meant to supply."
So it's not fair to say that you can "get set theory from category theory" or "do set theory within category theory." Those are facile statements. Facile means, "appearing neat and comprehensive only by ignoring the true complexities of an issue; superficial." That's apt.
On the other hand, can we do category within set theory? The conventional wisdom would be that classes are too big to be sets. The category of sets, for example is surely not a set. The category of Abelian groups is not a set. One way around this is to consider only "small" categories in which the objects and morphisms form sets. Another way is to say that we are agnostic as to whether a category contains "all" possible instances of an object type; but rather contains "enough" for any argument you need to make. I have seen this point of view expressed but don't have a reference at the moment.
Or we could just say that categories in general are proper classes, in the sense of "Predicate satisfiers that are too big to be sets." And then the argument is that since ZFC doesn't have proper classes, you can't do category theory within ZFC.
That is the argument. But what about a set theory like Morse-Kelly or Von Neuman-Bernays-Gödel set theory, two set theories that DO incorporate proper classes? Can you do category theory in those set theories? Good question.
This brings up a larger question: Which set theory, and which version of category theory? There are various flavors of each. The Stackexchange thread brings out this point in more detail. I truly hope people will read that page to get a sense of the many interrelated and nontrivial issues.
Set theory and category theory are not in a cage match to the death, as some seem to think. They're complementary ideas in a toolkit. It's like the programmers arguing over functional versus object-oriented. They're tools, not religions.
Thanks for the insights and the link.
Yes, I would say it is a trick of sophistry. To say of an attribute that it is "null", is to say that the attribute is non-existence. So to say that the attribute of extension is null, is to say that the attribute of extension is not there. So all you are saying is that the thing has the attribute of extension, but the attribute of extension is not there because it's null. Of course that's contradictory.
Quoting Mephist
Do you not see, that if the supposed "two points" are really one point, then they are not two points at all, they are one point? That's directly from Leibniz' identity of indiscernibles. So if they are really one point, then there is no segment.
Quoting Mephist
OK, we could take this route, but I think I've followed it before. Perhaps you could lead me to something new. Let's say that a set does not necessarily have extension, extension is not an essential feature. We'll say that a set may have extension or it may not have extension. Let's define "set" then. Isn't a set a collection of objects? Doesn't it appear contradictory to you, to speak of a collection of objects with no objects? If we define "set" as a possible collection of objects, such that the set is the defining terms rather than the actual objects, then no sets would actually have any objects and they would all be empty sets.
Quoting Mephist
Wow, I find that a very strange way of looking at this. Instead of imagining a line exactly how it is defined, length with absolute purity, no width, you imagine a wide long thing, then subtract the width off it.
Quoting Mephist
As I explained above, a set with no parts seems contradictory, and this is due to what a set is. Do you see the difference? A point is defined as having no parts. But a set has parts, according to what a set is, so the set with no parts doesn't make sense unless we changed what a set is. If a set is something other than a thing with parts, what is it?
Maybe you are right: sets cannot be empty. So you have to define another thing, named "set_or_nothing", that is a set or it's nothing. Just substitute the word "set" with the word "set_or_nothing" everywhere, and everything will be fine!
Sorry, but I have no more ideas to explain this... I prefer using programs that manipulate symbols without wondering what those symbols really mean :razz:
What this means is that zero is a very strange type of number, unique and different from the other numbers. So we ought to be careful in the way that we use it.
So what do you make of the set [math]\{x \in \mathbb N : x \neq x \}[/math]?
You reject the axiom schema of specification? You don't think [math]\mathbb N[/math] is a set? I really want to hear this.
This is a "set_or_nothing", not a "set"... :smile:
If it's not a set, which do you disagree with: The axiom schema of specification? Or that the natural numbers are a set?
Does the smiley mean that you don't actually believe what you wrote but that talking to @Metaphysician Undercover has caused you to lose your grip? What does the smiley mean? Why did you claim there is no empty set? If you so claim, what do you do with the brief existence proof I just gave?
When I put the same question to @Metaphysician Undercover, he admitted that it's not the empty set he objects to, but rather the entirety of set theory. That's a nihilistic position but at least it's a position. You have none that I can see.
Yea, this discussion is going in circles without any hope of a conclusion. I would like to finish discussing about empty sets!
You retract or stand by your claim that there is no empty set?
Mephist seems to have no rebuttal to the arguments which demonstrate that the "empty set" is a contradictory concept, and unlike you, seems about ready to face the reality of this.
Quoting fishfry
What I object to is the claim of "existence" for objects which have a contradictory description. This is not nihilistic, but a healthy skepticism. The attitude demonstrated by you, that we might assign "existence" arbitrarily is best described as delusional.
I didn't change idea: there is no contradiction in the axiomatic definition of sets given by ZFC, at least for what has been discovered until now. It has not even been proved that ZFC is not contradictory, however; but since nobody has found any contradiction in ZFC after 100 years of using it, I would guess that it is consistent.
By the way, dependent type theory - at least a subset of the version used in coq - has been proved to be consistent (but of course it is not complete - no way to avoid Godel's incompleteness theorem).
However, I don't see any problem in the definition of an empty set, and the fact that the name "set" could suggest that it has to be composed of at least some elements is not a problem for me, since this is just a name, and names have no role in a formal logic system. If it was called "asodifj" nothing would change, except that it would be more difficult to remember this absurd name.
Quoting Metaphysician Undercover
I understand that your objection is more about the philosophical interpretation of the idea of "set". This in my opinion is not about mathematics, and to say the truth I don't really see the point of this kind of issues. In my opinion, there are only two points of view:
- mathematics, that don't care about the "real" meaning at all
- physics, that cares only about the correspondence between symbols and results of physical experiments.
Anything that has no direct correspondence with the results with physical objects or results of experiments (such as the sets of ZFC, that can be infinite) is an useful mathematical entity, but it doesn't need to have any meaning at all: it's just an useful abstraction.
By the way, if you consider only finite sets, I don't see any problem at all with the obvious interpretation of sets as physical containers of something (that can be even empty).
Quoting Metaphysician Undercover
I think the word contradictory is not the right term in mathematics. The right term should be "inconsistent", and it has a very precise meaning in of formal logic system. Your "proof" of inconsistency, as I just said before, is not something that contemporary mathematics would accept as valid. Maybe it's valid from a philosophical point of view, but I don't fill qualified enough as a philosopher to discuss about it.
However, I really have no other new ideas how to object to @Metaphysician Undercover arguments, and I don't see the point in repeating continuously the same things...
Was that a yes or a no? Stop dancing. You're wrong on the facts, wrong on the math. Why are you trying to placate @Metaphysician Undercover's nutty ideas?
What do you think a set is, if not anything that obeys the rules of set theory?
When I challenged you on this point, you admitted that it's not only the empty set, but set theory in its entirety that you object to. Then you added that you reject modern physics as well. Nihilism. You are speaking nonsense. For @Mephist's part, he read a book on category theory but knows very little actual math. The fact that he's confused about the empty set, even when shown its existence proof from the axioms of set theory, supports that conclusion.
I don't understand what I am wrong about. I said there is no proof that ZFC is inconsistent (meaning: nobody has never derived a contradiction from ZFC's axioms), but there is even no proof that ZFC is consistent. That's why I prefer type theory to ZFC. Type theory is weaker but is provably consistent.
Can you show me what I said wrong?
Quoting fishfry
I think the sets that are defined in ZFC are a hierarchical tree-like structure that can be used to model the relation "belongs to" at the same way as the leaves of a tree "belong to" it's root. It lacks symmetry and is too complex. I think in the future it will be substituted by a more elegant and simpler definition. I think it does not correspond to anything in the physical world, so basically yes: it's just an imaginary gadget that obeys the rules of set theory, ad it could be substituted by other similar gadgets that logically equivalent to it.
Quoting fishfry
Can you show me a proof of consistency of ZFC set theory that doesn't make use of another even more complex and convoluted set theory?
I would like to hear the opinion of other real mathematicians about what I wrote. For example @jgill or anybody else that can be surely qualified as a mathematician. Could you please point out what I said that is not correct? (this, or even one of my previous posts).
It's perfectly possible (and probable) that I wrote something wrong, but I would like to know what's the mistake that I made.
I'd like to chime in with words of wisdom, my friend, but category theory is terra incognita to me. Read about it years ago and decided to give it a pass. Don't worry about the empty set. It can take care of itself. If it feels lonely, it can con an element from a comrade. Life goes on.
Thanks for the answer, anyway!
I went through this already. You cannot use the logic derived from the axioms to judge the axioms of the system, because valid logic will not allow one to produce a conclusion which is inconsistent with the premises. Therefore we need to refer to some other principles, and philosophy provides us with those judgements and proofs. If you're not interested, that's fine, but the assertion that such proofs are not valid in mathematics is not a sound rebuttal. I'm demonstrating to you, that the premises of your logic are false, and you reply, that doesn't matter because for me, and for everyone who uses my system the premises are true, and unless you can prove that they are false, by starting from the premise that they are true, you have no argument.
Quoting fishfry
Of course, the nature of the empty set is essential to understanding what a "set" is, and if a theory has contradictory premises, then I object to the theory in its entirety, it needs to be reformulated
You get something really similar to that with any mapping [math]t(k):D\rightarrow M[/math], where [math]M[/math] is some manifold in which [math]x[/math] is a point. With the constraint that [math]t(0)=x[/math], the collection of all such maps forms a module ("vector space with elements from a ring"). It's an infinitesimal tangent space attached to the point. It might mutilate intuitions of the real number line, but that doesn't matter, as it seems designed to simplify language and proofs about smooth functions. Whether it's "wrong" or not is just a question of taste and application.
Quoting Mephist
As I'm sure you know, if a theory's consistent and has arithmetic, it can't prove its own consistency. You always have to go outside a theory to prove that theory's consistency; so consistency of system X is always just relative consistency to some other system Y. If you want to find a model of some axiom system, you need to construct the model through other rules (even if they're incredibly similar).
A weaker criterion might be that you want some theory which is equivalent to ZFC in the sense of having statements which are inter-derivable with all of its axioms. But then all consistency proofs here would establish are that "equivalent axiomatisations of ZFC are consistent at the same time"; like being reassured that ZFC is consistent because you can replace the C with Zorn's Lemma and that ZF+(Zorn's Lemma) is consistent when ZFC is.
I'd've thought you'd be quite happy with small categories? :brow: Aren't the models of intuitionistic logic Heyting algebras (from earlier) anyway? They're sets, or categories which are represent-able as sets. So I'm reading this like: "I don't like sets because I don't like the structures that establish ZFC has models. But I like intuitionist dependent type theory because it has a model! (Which is a collection of sets.)"
If you want a theory of sets, you want to be able to compare pairs of sets. You want to know if they have any elements in common, and aggregate all elements they have in common into some set at the same time; taking an intersection of sets.
Let's compare the interval of real numbers (1,2) and the interval of real numbers (3,4). Assume that the intersection of those sets must be non-empty, then there is is a number which is both strictly less than two and strictly greater than three. There is no such number.
The empty set comes quite naturally from two principles:
(1) The ability to state what elements sets have in common.
(1a) The elements that sets have in common must always be equal to a set.
(2) That two sets might be disjoint.
(as @fishfry pointed out, you can get contradictions if you disallow an empty set like object)
One possibility, seemingly proposed by @Mephist (if I understood your responses to MU anyway) is that you put in the empty set as a proper class primitive into the theory, so that (1a) is denied but (1) and (2) are still true. It's pretty much a distinction without a difference though; "intersections of sets contain at least 1 element or are the empty class", I'd imagine you'd have transfer results from this "empty set is declared a proper class for no reason + ZFC" theory to the usual ZFC theory. My guess is that it would be a distinction without a difference.
Yes, exactly. The intuition of the real number line, in my opinion, is not mutilated but simply different: you have a base space of points that can be built in type theory as limits of convergent sequences of rational numbers. Each of these points is "covered" by an infinitesimal open set.
The weird thing at first sight is that there are no closed intervals such as [0,1], and all functions have to be continuous. But if you think of a function as a model of a physical process, in my opinion that's the ideal mathematical object. In physics, there aren't really discontinuous processes, even in principle, since infinite precision is not measurable. A discontinuous function of time, for example, would mean an infinite velocity of some kind of process.
Quoting fdrake
Yes, that's true. But the proof of consistency of a dependent type theory (such as for example Calculus of Constructions) is simply a proof that a given class of programs always terminates and is "strongly normalizing" (meaning: any two programs can always be compared with each-other by reducing them to a normal form, and then checking if they are syntactically equal). See for example https://prosecco.gforge.inria.fr/personal/hritcu/temp/snforcc.pdf.
In reality, in CC the induction principle P(0) and ( forall n:Nat, P(n) ==> P(n +1) ) ==> ( forall n:Nat, P(n) ), and then the definition of the type of natural numbers, is not an internal part of the logic, but is assumed as an axiom. So, strictly speaking, the logic of CC does not include natural numbers. But you only have to "trust" the principle of induction, not the existence of abstract infinite sets.
Quoting fdrake
Heyting algebras represent only the propositional part of intuitionistic logic, not including variables and quantifiers.
Quoting fdrake
Yea, this seems not to make much sense.
OK, I'll try to explain.
1. To speak about the elements of an arbitrary set, you need only first order logic. You don't need a set theory. Meaning: first order logic (the rules and axiom of the logic) define the meaning of "forall" and "exists" quantifiers. For example, you can axiomatize a generic group, or a generic ring, using only first order logic. No need of ZFC. ZFC is needed to speak about the relations between sets (inclusion, intersection, subsets, etc...), or the relations between groups, or rings.
2. In principle, the definition of a category requires the existence of two sets: Objects and Arrows. So, two sets, not one.. However, the definition can be reformulated to use only one set (the set of arrows), and the objects can be identified as the identity arrows. So, in principle a generic category can be defined making use only of first order logic, without any set theory.
The problem is that the most important theorems and constructions of category theory (such as functors, natural transformations, Yoneda lemma, etc..) are related to the category of sets. And the category of sets is "the category that has as objects (all possible) sets and as arrows (all possible) functions between these sets". In other words, the category of sets is not defined in terms of category theory itself (meaning: in terms of objects and arrows, as in the definition of a topos, for examle), but is defined in terms of an underlying set theory (that is what it's called a "concrete" category - https://en.wikipedia.org/wiki/Concrete_category).
If you suppose to have ZFC as an underlying theory, I believe (not completely sure) that there would be a problem with this formal definition: you can't say that the objects of the category of sets are the set of all sets, because there is no such set in ZFC. So, if you want to avoid this kind of problems you should use an underlying set theory that assumes the existence of a class of all sets.
Or, for example, you can build category theory based on type theory. Then, the category of sets in this case will have as objects all the types of a given "universe" of types, and as arrows all computable functions. Most of the results of category theory are independent from the preexistent "set theory" that you use to build the category of all sets, but not all of the results.
I don't know the details of these differences, but from a foundational point of view, category theory is not an univocally defined theory. There are even several ways of defining a category for a fixed intuitionistic type theory, because you can "build" the types of arrows and objects in more than one way.
Fortunately, most of the results (practically all of that I know) are independent of the underlying set (or type) theory that you use to define the category of sets. So, if you have a "real numbers object" in category theory you don't have to worry about the underlying logic that you use: you can choose the easiest one to reason about; one where all functions are computable, for example.
Yes, that's correct. You can consider a "topos" as a generalized class of all sets. So, the sets are the objects of the category. The final object of the category corresponds to the set that contains only one element (singleton), and the initial object of the category corresponds to the empty set.
In the general definition of a "topos", the existence of an initial object is not required (because in the definition you want to have largest possible generalization of the category of sets), but the "normal" examples of topoi (such as the category of sheaves, or the category of sets - that are all particular examples of a topos) all have an initial object (corresponding to the empty set).
I can't take that as much of a criticism, since by your own criterion you regard the entire community of working professional mathematicians as delusional, and perhaps the physicists too. You have put me into some great company, that, frankly, I hardly deserve.
I do agree that I have not provided a definition of mathematical existence that you would find satisfactory. I'm thinking the issue over but it's tricky. However you are someone who regards the simple adjunction to the rational numbers of a formal square root of two as completely beyond the pale. I confess that I'm at a loss to respond to such philosophical nihilism. The principles by which we accept the rational numbers are no different and no logically simpler than those by which we adjoin a formal square root of two.
Take it up with Frege, Russell, Zermelo, von Neumann, and all the other brilliant 20th century set theorists including those working at the forefront of knowledge today such as Hamkins, Steele, Woodin, Shelah, and others. Your childish objection to modern math and science is noted. You don't have to repeat it. I heard you the first 20 times. Your nihilistic point does not gain sanity by repetition.
I'm sorely behind in responding to my mentions but I am getting to this point soon. I will explain in detail why you are wrong in your response to my demonstration of the existence of the empty set.
This point must be profoundly wrong or disingenuous on your part. If anything at all -- dependent type theory, Coq, a tuna sandwich on rye -- can prove its own consistency, then it must necessarily be entirely useless for representing modern mathematics.
If on the other hand you mean that it's been proven consistent using assumptions outside of itself, then the same is true of set theory. In 1936 Gerhard Gentzen proved the consistency of Peano arithmetic by assuming the consistency of transfinite induction up to the ordinal [math]\varepsilon_0[/math].
And in algebraic geometry, the branch of math that led to the original discovery of category theory by Mac Lane in the 1940's, the existence of an inaccessible cardinal is assumed. This in effect amounts to the assumption that ZFC is consistent, since an inaccessible cardinal is a model of set theory; that is, a set that satisfies the axioms of set theory.
In category theory and algebraic geometry the inaccessible cardinal shows up in the definition of a Grothendieck universe. The reason universes are the natural setting for categorical algebraic geometry is that they ensure that the categories in question contain enough sets to make the theory sensible.
Per the Wiki article: "The concept of a Grothendieck universe can also be defined in a topos." So if you're as as big a believer in topos theory as you say, you have a ready-made proof within topos theory of the consistency of ZFC.
Professional mathematicians, even category theorists -- especially category theorists -- understand that if you can't ground your categorical theory in set theory one way or another, you don't have a good theory.
First let me put this in context. You said the empty set doesn't exist. I gave a short existence proof from the axiom schema of specification. That's a valid proof in ZF of the existence of the empty set. You then objected to my proof by saying ZFC can't prove itself consistent. Which would result in your rejecting the entirety of modern mathematics.
So I'll explain why you're wrong First, your response is a total deflection, changing the subject. Second, your response has the same slippery slope problem as @Metaphysician Undercover's response to the same question. Namely, that it's not only the empty set that's not deserving of being called existent. Rather it's the entire enterprise of modern mathematics. Surely you must realize that such reasoning is untenable because it's so broad. You both want to reject the empty set on narrow terms -- "it makes no sense to have a collection that doesn't collect anything," etc. -- but you each end up saying that math itself is flawed therefore there's no empty set. There must be a name for such an argument. You want to argue a very narrow technical point and your only argument is to blow up the entire enterprise.
Third, your overall understanding of what math is about is inverted, in exactly the sense @jgill notes. Many people who come to math through foundations believe math is about the foundations. It's the other way 'round. Mathematics comes first and foundations are just our halting and historically contingent attempts to formalize accepted mathematical practice. Archimedes, Newton, Euler, and Gauss never heard of set theory. Were they not doing math? You see the absurdity of trying to put foundations logically prior to mathematics.
First we discover the math; then we make up the axioms that let us formalize it.
That's how math works. My sense is that professional philosophers of math (Maddy et. al.) perfectly well understand this; and that it's only the amateur enthusiasts on the message boards who believe otherwise.
And fourth, you're wrong on the math and logic of the situation.
So let me lay out some talking points in support of my four reasons you are wrong.
Quoting Mephist
The horrors. I suppose when Andrew Wiles solved Fermat's last theorem you said, "Harrumph, poppycock, we don't even know if ZFC is consistent." I hope you see the absurdity of your own position. For that matter it might interest you to know that Wiles's proof is done in the framework of Grothendieck's approach to modern algebraic geometry; which as I mentioned to you in another thread is done within a Grothendieck universe, a model of set theory that (a) assumes ZFC is consistent; and (2) posits the existence of an inaccessible cardinal, a transfinite cardinal whose existence is independent of ZFC. There's a lengthy and famous Mathoverflow thread about whether or not an inaccessible cardinal is necessary to Wiles's proof. Consensus is that it's not.
Likewise when Maryam Mirzakhani became the first woman and the first Iranian to win the Fields medal for "the dynamics and geometry of Riemann surfaces and their moduli spaces," you of course shouted, "Doesn't she know ZFC hasn't been proven consistent? She shouldn't have bothered."
If you are arguing anything different than this please let me know. Else retract your nonsensical point that since ZFC can't prove itself consistent, it must be fatally flawed. And that you can use this as a trump card to win any mathematical argument "The empty set exists." "No it doesn't, ZFC can't be proven consistent."
Man is this what you are arguing to me?
I want to add that when @Metaphysician Undercover makes the same argument, I have less of a problem with it; because he at least openly admits he does not engage with symbolic arguments. Please correct me if I have mischaracterized that in any way.
@Mephist, on the other hand, you seem perfectly willing to claim mathematical and symbolic knowledge. So your argument here is just awful. The empty set doesn't exist because ZFC can't prove itself consistent. Said by someone claiming math sophistication.
Am I missing your point here? Please tell me if I'm going off on the wrong thing. Because if that's your argument then you are a nihilistic as @Metaphysician Undercover, but with less of an excuse. You both want to throw out the entirety of modern mathematics just to defend your point that the empty set is not deserving of existence. You must not have much of an argument, either of you.
* Note that even if ZFC is inconsistent, then the empty set exists! The derivation from the axiom schema of specification is valid. So your own logic is screwed up. If ZFC is consistent the empty set exists, and if ZFC is inconsistent the empty set exists. Or rather in either case, the proof of its existence is valid. And what more do you ask for in terms of mathematical existence? You both want to reify the empty set. What nonsense. That's sophistry, to pretend to reject mathematical abstraction.
Quoting Mephist
You thanked me for posting the Stackexchange thread the other day but I'm not sure you got its message. The example of synthetic differential geometry was given to show that the point of alternative foundations is to shed light on problems, not to brag about which foundation is more fundamental.
Likewise he gave the example of someone saying that set theory's more fundamental than topology so they don't need to study topology. That's silly, right?
So when you say, "I prefer type theory" because of a spurious understanding of ZFC's inability to prove its own consistency, you sound like you're clinging to what you know because you can't understand what you don't know. So far your logic is "the empty set doesn't exist because ZFC can't prove its own consistency and that's why type theory is better."
You're making a poor argument and only showing the limitations of your own understanding.
Quoting Mephist
I responded to this in more detail in another post. This claim cannot possibly mean what you say it does. If type theory or any other theory can prove itself consistent, then à la Gödel it's useless for doing modern math.
On the other hand if you mean it can be proven consistent using means outside of itself, so can ZFC, as is commonly and standardly done in the modern categorical approach to algebraic geometry as pioneered by Mac Lane and perfected by Grothendieck.
Didn't they mention any of this in your category theory book? This is what I mean by your having a lack of overall understanding of math. It's part of the wrongness of your reply. Category theory and type theory don't invalidate 20th century math. They view it from another perspective. The math itself is the thing represented by the representations. You're trying to privilege one particular representation over another simply because you know one and not the other and don't get that the representation is not the thing itself. Not a good argument, not making points with me.
Again: Math precedes foundations. Not the other way 'round.
Quoting Mephist
I've said my piece, and if it was too long, it's because "I didn't have the time to make it shorter," as some clever person said once.
Quoting Mephist
Maybe, but not what I was looking for. A set is anything that obeys the axioms of set theory; in exactly the same way that point, line, and plane are undefined terms in Euclidean geometry. As Hilbert noted: "One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs." That is how we regard sets.
You want to somehow reify sets. You think a set should refer to the real world. I for one don't believe that. There's no set containing the empty set and the set containing the empty set in the real world. In set theory we call that set '2'.
Quoting Mephist
I'd argue the contrary. Sets as an abstraction of collections are very natural. You can teach sets to school kids in terms of unions and intersections of small finite sets. Type theory and category theory are more sophisticated concepts that require some mathematical training to appreciate.
But so what? Are you honestly rejecting the entirety of contemporary math because you have some kind of personal issue with set theory? That makes no sense. Set theory, type theory, and category theory are various tools in the toolkit for exploring the world of mathematical entities.
Math precedes foundations. Not the other way 'round.
Quoting Mephist
But of course. Foundations are always historically contingent. Set theory in its current form is less than a century old dating from Zermelo's 1922 axiomitization. By the way Cantor always gets the credit, but it's Zermelo who did the heavy lifting in the development of modern set theory. Before Cantor there was no set theory. A few decades from now category theory and type theory will be much better known and perhaps set theory will fade into history. It won't be wrong, just out of fashion. That's inevitable.
Quoting Mephist
Well of course. Was someone thinking set theory refers to the physical world? It's a formal game. It's the chess analogy I constantly use (to no effect) with @Metaphysician Undercover. You are standing on a soapbox fervently preaching something so obvious it barely needs to be said. Set theory is an attempt at a formalization of math. What of it?
Quoting Mephist
Not complex or convoluted? Sure. Grothendieck universes are very plausible and straightforward, and are the standard everyday mathematical framework in much of modern math. Wiles's proof of Fermat's last theorem is presented in the framework of universes, even though that's probably not strictly necessary. The proof of the consistency of ZFC via assuming an inaccessible cardinal is part and parcel of modern math. Of course we DO have to assume an axiom in addition to ZFC; but that axiom is by no means unintuitive or unbelievable. It's rather natural.
And what of it? You are making a TERRIBLE argument. That because you have some technical objection to the empty set (which you have not articulated) therefore the entirety of modern math is rejected because, "Nyah nyah type theory is better." And this to a simple technical question, does the empty set exist. And you go, "Well no, because the entirety of contemporary mathematics is bullshit."
That's your argument?
If I may make an analogy, it's like a beginning programmer arguing that their favorite language is better, just because it's the only one they know. And you, an experienced developer with a dozen languages under your professional belt, can only shake your head and remember when you were that young and dumb.
Oh and Columbo would say, One More Thing.
The empty set is the unique initial object in the category Set. You do believe in the category Set, don't you?
https://en.wikipedia.org/wiki/Initial_and_terminal_objects#Examples
I'll not discuss about the empty set any more. Yes, you are right. The empty set exists. You win!
I don't care. If anything whatsoever can prove its own consistency, it's useless as a foundation for math. If it requires something external to itself to prove its consistency, it's no better off than ZFC, which can easily (not complex or convoluted, but naturally) do the same.
But I am curious. Which of these is the case? Is the Calculus of Constructions useless as a foundation for most of modern math? Or relies on something outside itself for a proof of consistency?
If it's not one of these, then I stand to learn something. So please explain how this thing, whatever it is, defies Gödelian incompleteness.
Quoting Mephist
I was just surprised the other day when you agreed with @Metaphysician Undercover that it doesn't. I'm glad I was able to nip that in the bud, if in fact you mean it and are not just placating me. I should also point out that the reason denying the empty set entails denying so much more of modern math, is because the formalization of pretty much all of modern math depends on the existence of the empty set! This is a fact. Not of math itself, which is agnostic as to foundations. But to our formalizations of math. You can't do without the empty set. It exists on pragmatic grounds.
But now @Metaphysician Undercover has a good question. What does it mean that the empty set exists? Is it ONLY that its existence formally follows from somewhat arbitrary axioms? If so, he has a point. Can we do better in terms of defining mathematical existence?
(1a) would be better stated as, "if" two sets have elements in common this must be a set. There is no reason to interpret a situation where two sets have nothing in common, as meaning that this nothing ought to be a set. That is arbitrary, and actually illogical. If two sets have nothing in common, then why must this nothing be a set? That's nonsense, by the described situation, they have "nothing" in common. Why try to make nothing into something (a set)?
Quoting tim wood
A set is not analogous with a warehouse where things are stored, or any such container, because there is no separation between the set and its elements. Unlike a warehouse which has an identity as a warehouse, with the potential to store things regardless of what is actually stored within it, a set is identified solely by its elements, and is therefore inseparable from its elements. That's why an empty set is contradictory nonsense.
Quoting fishfry
Don't cast this in the wrong light. I don't say "math itself is flawed therefore there's no empty set". I say "there's no empty set therefore math is flawed". I point out the specific flaws to justify the more general claim, that math is flawed. So it's not the case that I am arguing that math by its very nature is inherently flawed (what you call nihilism), I think the very opposite, that math by its very nature is perfect, "ideal". And, because it has this status of being ideal, we must hold it to the highest standards of perfection. Therefore we are obliged to reject any imperfections.
Quoting fishfry
How could you conceive of this?
I think I'll have to complete my work in a more formal way and present it to a math forum.
Sorry for bringing trouble. You can continue without me.
Ah ... was it something I said? I have no sense of having suppressed any thoughts you may have. I had no idea you were presenting anything original. If I somehow crushed your creativity I apologize. Is that what I did?
Just for my understanding, can you point to a post where you presented your original ideas?
Surely you may have noticed that you can write the most simple, commonly understood things here and have many people not understand you. Happens a lot to me.
Quoting Mephist
Just so I have some idea what this is about ... can you just summarize your idea? I can't for the life of me remember any interaction where you presented an original idea and I crushed your spirit. Truly sorry if anything came across that way.
ps -- I've reviewed your earliest posts from 9 months ago. I see no evidence that you ever put forth an original idea and got shot down or discouraged by anyone. What the heck is this about? I simply can find no evidence of your assertion.
Quoting fishfry
I am just pulling these out to highlight them as excellent observations. After all, Peirce--the founder of pragmatism--was decidedly non-foundationalist in his own philosophical system. SDG/SIA captures certain aspects of true continuity that modern analysis using the standard real numbers does not.
Tell me something of pragmatism. Let me say first where I'm coming from.
I've read Maddy's great papers Believing the Axioms part I and part II. These papers provide a historical overview of how and why the various axioms of set theory came to be adopted; along with a number of pragmatic (in the everyday sense of the word) criteria for deciding whether to adopt an axiom. For example one principle is Maximize, which says that we choose the axioms that give us the richest theory.
I'll give an example. The axiom of choice (AC) may be taken or denied with equal logical consistency. Both AC and its negation are consistent with ZF.
Why do mathematicians simply accept AC? I'm not talking about specialists who investigate the consequences of the negation of AC, but rather everyday working mathematicians who never give a thought to foundations. They're taught in grad school to freely apply Zorn's lemma, the Hahn-Banach theorem, and other applications of AC in their work, and they do so as a matter of course.
Philosophically, we accept AC because it gives us more and better theorems. That is the reason. This is a very pragmatic (in the everyday sense, not necessarily the technical sense) way to view the axioms. We adopt the axioms that give us a good theory.
Is that Peircean pragmatism? Or what exactly is it, given what I already know about the practical or pragmatic (everyday sense) reasons to adopt or reject axioms?
Knowledge of the history of math.
But it's the same in any discipline. There's science, and then there's the philosophy of science. One can and does do science without regard for its philosophy. That's true in every field. X precedes the philosophy of X.
Quoting Metaphysician Undercover
Yes I understand this. I get that you reject math and, when pressed, physical science as well. I'm happy for you. I can't take such a point of view seriously. The task of the philosopher is to explain how it comes to be that math and science are abstract yet useful. You can't solve the problem by rejecting math and science; not unless you live in a cave. Without Internet access. Even then you'd draw a square in the sand and eventually discover the square root of two; and from that, abstract algebra and group theory and modern physics. You haven't got a serious philosophy, just sophistry.
As applied to mathematics, yes. Charles Peirce adopted his father Benjamin's definition of it as the science that draws necessary conclusions about hypothetical states of things. There are no restrictions on conceiving such hypothetical states, other than that they be suited to the purposes for which we wish to analyze them. Applied mathematics seeks to formulate hypothetical states that resemble reality in certain significant ways, but pure mathematics does not have that particular limitation.
Ok. Not too far from Bertie's quip that math is the subject where we never know what we're talking about or if what we say is true.
Quoting aletheist
Ok I perfectly well agree with that. Completely. Have explained it to many people on many forums over the years. May be coming to understand it myself!
But now Maddy's pragmatism (in the everyday sense) goes further. According to Peirce, we can accept or reject AC at will, which of course we can. But that does not explain why mainstream math accepts it and never gives it a second thought. For that you need Maddy's Maximize principle, which says that assuming AC gives a better or richer theory. I'm paraphrasing my understanding of Maddy and didn't double-check her article so I might be mangling her ideas a bit but I think I'm in the ballpark.
Point being that Maddy says that given two equally logically consistent but mutually inconsistent axioms; we adopt the one that's more fun, more interesting, gives more good theorems. That's a pragmatic justification for choosing AC over not-AC.
Is that an example of Peircean pragmatism or is that an expression of something else?
Also ... given that I'm perfectly in agreement with Peirce's point of view here, and Maddy's as well; can you help me to understand @Metaphysician Undercover's point of view? He rejects the idea of taking math on its own terms; insists that it must refer to something outside of itself. Nobody believes that, not about pure math. Why does @Metaphysician Undercover believe that? What is the basis for his ideas? Do they have a name? Is there a reference? I asked him this once and did not get a satisfactory answer. I can't tell if he is asserting the ideas of a particular school of thought, or just venting over a bad experience with his screechy third-grade math teacher.
It is consistent with Peircean pragmatism in the sense that one's purpose dictates how one formulates the hypothetical state of things to be explicated--in this case, the particular set of axioms to adopt.
Quoting fishfry
I can try. As I noted several times a few weeks ago, MU employs a rigid metaphysical terminology and seeks to impose it on everyone else. "Existence" has only one meaning, and that is its ontological definition; so asserting mathematical existence is, according to MU, asserting some kind of ontological existence--no matter how many times and in how many ways we explain that this is not what anyone actually means by mathematical existence. More broadly, my impression is that MU--as the name suggests--is a philosophical foundationalist who begins with metaphysics, deriving everything else from that. Accordingly, MU has been quite dismissive of pragmatism on various occasions.
By contrast, as I said before, Peirce was a philosophical non-foundationalist, and he had harsh words for the dogmatic metaphysicians of his day. He classified the sciences in accordance with their nature and purpose, such that the more basic ones furnish principles to those above them. Mathematics comes first, because its subject matter consists entirely of hypotheses that may or may not have any basis in reality. Every other science thus relies on mathematics to some extent, not as a foundation but as a necessary tool. The first positive science is phenomenology, which deals only with what appears to the mind and identifies three irreducible elements--quality, reaction, and mediation. Then come the normative sciences of esthetics (feeling), ethics (action), and logic (thought), although Peirce generalized the last of these to semeiotic--the theory of all kinds of signs, not just symbols. Only then--after we have established the proper method for discerning truth from falsehood--do we reach metaphysics, the science of reality.
The claim was that math precedes foundations. But math consists of concepts, and concepts require foundations, so it is impossible that math is prior to foundations. It should be obvious to you, a structure is built on foundations, not vise versa. The fact that the philosophy of math follows math is irrelevant, because it is not the philosophy of math which produces the foundations. The philosophy of math might study the foundations, but it does not produce the foundations.
Quoting fishfry
You don't seem to grasp the problem. "Usefulness" is judged in relation to a goal or end, and that goal or end may be contrary to truth. When principles are used in a way which is contrary to revealing the truth (i.e. hiding the truth), this is deception. The philosopher seeks truth, so the task of the philosopher is to prevent such deception. Principles of math, axioms, might be useful for obtaining truth, or they might be useful for deception, if they are produced without any goals whatsoever. Since the philosopher seeks truth, the mathematical principles which are not useful for revealing truth, and are only useful for hiding the truth,(i.e. deception), ought to be rejected.
You seem to be under the illusion that if mathematical principles are useful they ought to be accepted by the philosopher, simply because they are useful. But you misunderstand philosophy, which seeks to distinguish truth from mere usefulness. The philosopher seeks to know, without regard for the usefulness of that knowledge, knowing for the sake of knowing.
Quoting fishfry
Look, just above, you refer to the "usefulness" of math. "Usefulness" of something is determined by relating that thing to something outside itself. That's what usefulness is, it's putting the thing, as a tool, toward a further end, something outside itself. So, if philosophy to you, is explaining how something like mathematical abstractions might be "useful", you need to apprehend how mathematical abstractions relate to things outside of mathematics. This is what it means to be useful. If you assert that no one believes mathematics ought to relate to anything outside of mathematics, then all you are doing is denying that mathematics ought to be useful. Then your philosophical dilemma, of how it is that math is useful, is self-imposed by your faulty belief, that math does not relate to anything outside itself. By this belief mathematical abstractions cannot be useful because usefulness requires relations to other things..
However, true philosophers take for granted that math is useful, because the evidence is everywhere. Therefore we take for granted that mathematical abstractions relate to things outside themselves. Then we can proceed to analyze these relations, instead of denying that there is such relations and wondering how it is that math can be useful when it doesn't relate to anything outside itself.
I happen to be a practicing structural engineer. What might not be obvious to you and others is that a structure is always designed first, followed by its foundations. We have to establish what needs to be supported before we can go about determining how it will be supported. Often there are multiple options--footings on native soil, footings on special fill materials, footings on improved ground, driven steel piles, driven precast concrete piles, auger-cast grout piles, drilled shafts filled with cast-in-place concrete, etc. Typically what we end up selecting is either the least expensive solution or the one that in our judgment properly balances cost with risk. In other words, the "best" foundation can only be determined relative to a specific purpose. Why would philosophy be any different?
Quoting Metaphysician Undercover
A philosophy of mathematics identifies the foundations that have already been established by the practice of mathematics. Most people who use mathematics, including most professional mathematicians, have little or no knowledge of its foundations--and little or no need to have such knowledge. Likewise, most people in general have little or no knowledge of philosophy in general--and little or no need to have such knowledge.
Quoting Metaphysician Undercover
Some philosophers define truth as mere usefulness. I am not one of them, and neither was Peirce, although many of them call themselves "pragmatists"--one reason why he eventually started calling himself a pragmaticist instead.
Quoting Metaphysician Undercover
In other words, philosophy is useful in relation to the goal or end of knowing for the sake of knowing. Well, so is pure mathematics in relation to the goal or end of knowing what follows necessarily from certain axioms, purely for the sake of knowing it.
"Risk" here means the possibility of a law suit against the engineering firm. Real risk contains unknown factors which cannot be assessed. What we really end up with "typically", is an extremely expensive building designed to protect the engineer from lawsuits. The exorbitant expense is not balanced by the building's owner being protected from risk, because the real risk, the unknown, has not been assessed.
Quoting aletheist
As I said already, the purpose of philosophy is truth. I don't try to disguise the fact that this is a "specific purpose". But if we justify the use of false principles on the basis that false principles may still be useful for some other purpose, this particular usefulness would be inconsistent with the specific purpose of philosophy, being truth. Therefore they would be rejected by good philosophers..
Quoting aletheist
That people have little knowledge of the foundations of mathematics, is not an argument supportive of fishfry's claim that mathematics precedes its foundations. As you explain, the design is prior to the structure. If people move into houses, having no knowledge of the design or blueprints, or even that there were blueprints, this does not mean that the house is prior to the design.
The practise of mathematics itself, requires that a foundation already be laid in order for that practise to occur. These are the theoretical principles employed in the practise. So it is a mistake to say that the practise produces the foundation. In reality, the theory which provides the principles for practise, is the foundation.
Quoting aletheist
Do you see what happens when you qualify "knowing" with "what follows necessarily from certain axioms"? The goal of philosophy is an unqualified sense of knowing, and truth is implied by "knowing". If the axioms involved are false, then what follows from them cannot be sound conclusions. Therefore we cannot even call this a form of "knowing". This problem becomes very evident in the use of counterfactuals in logic. Even when we recognize the counterfactual as false, it is doubtful whether the use of counterfactuals provides us with any real knowledge.
Thanks for confirming that you have no idea how building design and construction actually work.
Quoting Metaphysician Undercover
History demonstrates otherwise, as @fishfry has pointed out.
Quoting Metaphysician Undercover
The axioms of pure mathematics are neither true nor false, and claiming otherwise is a category mistake. If a certain theorem follows necessarily from a particular system of axioms, then it is true within that system.
Actually history has not demonstrated that. You and Fishfry have simply made the assertion that history has demonstrated this. I demonstrated how that assertion is illogical..
Quoting aletheist
Another assertion without justification.
One assertion after the other, without explanation or justification, together with an ignorance of logic; you just reminded me of how difficult it is to hold a discussion with you.
LOL! Right back at you.
Assuming that by "foundations" you mean the acceptable accumulated knowledge and practices up to any particular moment of mathematical history. Weierstrass and Cauchy laid the critical foundations for my interests, above and beyond what came before.
Formal foundations, such as PAs and ZFCs are another matter. Apart from certain mathematical specialties, they are dispensable. Others might differ. It's a philosophy thread. :cool:
I think "foundation" refers to principles which "the acceptable accumulated knowledge and practises" are based in, built upon. So Weiersfass and Cauchy produced some new principles and also built upon some existing principles, and this would be the foundation. Notice that no conceptual structure is completely new, in an absolute sense, it's always built with some already existing principles. Because of this, there is an appearance of an infinite regress in the creation of such principles, y, as a principle, was created using x, but x was created using w, and so on.
The infinite regress of dependency was avoided by the Pythagoreans by assuming that the principles are eternal, have always existed (platonic realism), and are simply discovered rather than created. But since these ideas or Forms are necessarily dependent on a mind for their existence, this conceptual structure is closely tied to the idea of an eternal soul. So Plato describes the discovery of the principles as remembering what is already known (Platonic theory of recollection, "Meno").
From the principles of modern science, we reject the eternal soul, along with the eternal Forms, and Plato's theory of recollection. This puts us back toward an ontology of mathematical principles which describes them as actively evolving. The problem though, as described above, is the appearance of infinite regress, because there must always be something which supports the changing structure, the underlying kernel (what I call the foundation). So when a new type of structure is created, such as in your example of what interests you, it is not a completely new creation, it is still supported by underlying principles which have already been tested by practise.
If we follow this type of evolutionary theory, rather than platonic realism, we might consider that there was a first mathematical idea created. We see that other animals don't use mathematics, and human beings evolved from other animals, so there must have been a time when mathematical ideas were first thought up. It might be a first group of ideas which we could call the first mathematical ideas. So at some time, mathematical ideas were thought up for the first time. Would you agree that these first mathematical principles would also have some underlying principles, being not mathematical in nature, but already tested by practise, and these non-mathematical principles would support the first mathematical ideas? We could say that these principles are the foundation for mathematics.
Thanks for your detailed response. It helped put a lot of things into context. So MU is an anti-Peircean. That actually helps me understand Peirce. Sounds like I'm a Peircean and never knew it.
My late advisor would say that there is nothing really new in mathematics. I would disagree, but to some extent much is preordained by the past. There has long been an ongoing discussion as to whether mathematics is discovered or created. I think it is both.
" we might consider that there was a first mathematical idea created"
Counting fingers and toes by some means, perhaps. Think how much of the modern world flows from that. :chin:
That's not a logical conclusion.
Furthermore, I actually agree with a lot that Peirce has said. in particular his very coherent method of laying out particular epistemological problems. What I disagree with is the metaphysics he proposes to resolve those problems. He provides no real solution, only the illusion of a solution. Aletheist resorts to contradiction in an effort to support Peirce's illusory solutions, because Peirce draws on dialectical materialism, or dialetheist principles which decline ruling out contradiction. Aletheist refuses to acknowledge this.
How would you know? You have not demonstrated any familiarity with his voluminous writings.
Quoting Metaphysician Undercover
Nonsense. As usual, bare assertion with no basis in fact.
Quoting Metaphysician Undercover
I refuse to acknowledge this because it is blatantly false.
I'm surely unqualified to discuss Peirce. But his view on math as outlined by @aletheist appears to track mine. Mathematical existence is pragmatic. Useful/necessary for our theory. Agreed on by a preponderance of professionals.
I want to highlight the references in the linked post as being excellent. They give you a real sense for what concepts people were wrestling with at the dawn of set theory.