Have we really proved the existence of irrational numbers?
All proofs of the existence irrational numbers (that I'm aware of) are proofs by contradiction. For example, we assume that ?2 can only be 1) a rational number or 2) an irrational number. Since we've proved that ?2 is not a rational number we conclude that it's an irrational number. Is it possible that this is a false dichotomy?
Comments (222)
What does it mean (to you) to prove that a number exists?
Hmm, you've revealed that my original post was quite vague. I believe that an object exists if it is being computed. I exist as an artifact of the laws of nature being evaluated and the number 3 exists within my mind because I'm thinking about it right now. With this view, the number ?2 (in its totality) cannot exist because it cannot exist in any finite computer. To be clear, I believe that a finite algorithm for computing ?2 can exist, but the output of that algorithm (which I'm calling the number ?2) cannot exist. The mainstream approach to giving the number ?2 existence requires us to assert the existence the Platonic Realm (which I equate with an infinite computer), but extraordinary claims should be backed by strong evidence, for which I have come across none. Thoughts?
But perhaps it is simplest if I take existence out of my original question: Have we really proved that ?2 is an irrational number?
Yes, Pythagoras gets credit for (but probably didn't personally have anything to do with) the proof that sqrt(2) is irrational. Is it proof by contradiction that you're concerned about?
It's interesting that you agree that sqrt(2) is computable, as are pi, e, and every other mathematical constant that anyone can name. Except for Chaitin's constant, which we can name but which isn't computable.
You seem to be applying a much stricter standard than even the mathematical constructivists. They would allow the existence of any computable number, since we can give an algorithm to approximate it to any desired degree.
But you want to strengthen that standard to say that a number only exists when we can not only give an algorithm for it, but that we can execute the algorithm to completion given the physical constraints of the universe. That's a terribly restrictive standard, for example 1/3 = .3333... doesn't exist according to you.
But why not? Why are you privileging decimal notation? The algorithm for pi is finite and expresses pi exactly. So what if the decimal representation's not finite? Why should that be the standard? If we have an algorithm, we have the number. That's the constructivist point of view.
For my own part, I take mathematical existence to mean anything that can be proven logically using the axioms of any given mathematical system. That includes all the computable numbers and all the noncomputable ones as well, which after all are necessary to ensure the completeness of the real numbers. And no mathematical objects exist at all in the physical world, since they're all abstractions.
Just checked. Here is a link to a discussion in Wikipedia about the proof that pi is irrational:
https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational#:~:text=In%20the%201760s%2C%20Johann%20Heinrich,is%20a%20non%2Dzero%20integer.&text=In%201882%2C%20Ferdinand%20von%20Lindemann,irrational%2C%20but%20transcendental%20as%20well.
Not claiming I understand it.
He only proved that ?2 is not a rational number. He did not prove that ?2 is an irrational number. Yes, I'm concerned with proof by contradiction. If I flip a coin and tell you that it is not heads that is not proof that it is tails because it may have landed on its side.
Quoting fishfry
Actually, I want to strengthen it even more so to say that a number only exists when it is being computed. If there is no computer currently thinking about 3 right now then the number 3 does not exist as an actualized number. It only has the potential to exist. I think this sort of view is required if we are to avoid actual infinity. Otherwise, how would a constructivist answer the question: how many numbers are there? My response to such a question is 'how many numbers are where? In what computer?'
Quoting fishfry
I believe that the decimal representation for 1/3 cannot exist but nevertheless the number certainly can. For example, it is LL on the Stern-Brocot tree. And we can do exact arithmetic using any rational number using the Stern-Brocot tree.
Quoting fishfry
But won't there always be undecidable statements? It seems like your definition is too restrictive as it would be missing some truths.
Quoting fishfry
Why is it necessary to have a number system which is complete?
Quoting T Clark
I have no doubt that pi is irrational (i.e. not a rational number). But a sandwich is also not a rational number. My argument is that just as a sandwich is not a number, perhaps neither is pi. Maybe it's an algorithm.
Constructivists deny the law of the excluded middle. You might be interested in this. For my own part I don't have any affinity for constructivism although it's enjoying a resurgence lately due to the influence of computer science and computerized mathematical proof systems. Brouwer's revenge, I like to call it.
https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_mathematics)
Quoting Ryan O'Connor
Different issue. Landing on side should be included in the outcome space. In high school statistics we were flipping coins once and a nickel started rolling on its side and slowing down and darn near landed on its edge, but it hit the wall and fell over. But I almost saw it happen.
Quoting Ryan O'Connor
I see your point, but then existence becomes contingent on what everyone's thinking about and/or computing. 3 might or might not exist depending on what 7 billion people and a few hundred million computers are thinking or computing at this exact instant. What kind of standard for existence is that? Not one that would have much support from ontologist, I'd wager.
Quoting Ryan O'Connor
If you deny the mathematical existence of the natural numbers you not only deny ZF, but also Peano arithmetic. That doesn't let you get any nontrivial math off the ground. Not only no theory of the real numbers, but not even elementary number theory. You may be making a philosophical point but not one with much merit, since it doesn't account for how mathematics is used or for actual mathematical practice.
Quoting Ryan O'Connor
Well then you're an ultrafinitist. You not only deny the existence of infinite sets; you deny the existence of sufficiently large finite sets.
I do actually have some sympathy for the ultrafinitist position, since it's the only mathematical ontology that is consistent with what's known about the physical world. But your particular flavor of ultrafinitism is untenable, granting existence to only those things that someone is thinking about or computing at any given instant. Under such a philosophy we can never say whether a given number or mathematical object exists.
Quoting Ryan O'Connor
Even the constructivists, with whom I've had many an interesting discussion in these very pages, believe in computable numbers. There is a countable infinity of them. Computable numbers, I mean, not constructivists.
The ultrafinitists don't put any particular upper limit on how large a number can be, only that there aren't infinitely many of them.
But you want to not only say that, but that whether a given number exists or not depends on whether someone's thinking of it. How can we ever determine that? It's an unverifiable standard. There is then no way to know whether any number exists and whether it still exists five minutes from now. It's impossible to hold such a view along with any kind of coherent ontology of numbers.
Quoting Ryan O'Connor
Why doesn't pi exist? It has a representation as a finite-length algorithm. By exist I mean mathematical existence of course, that's the only kind of existence I'm talking about.
Quoting Ryan O'Connor
Well in any sufficiently interesting mathematical system we are always missing some truths. That's just a fact. But at least it's not contingent. The Continuum hypothesis is always undecidable in ZFC. Now and five minutes from now and five million years from now. And in ZFC + CH, it's provable. Now and five minutes from now. We have logical certainty about what exists, unlike with your system in which we have to constantly poll 7 billion people and several hundred million computers.
Quoting Ryan O'Connor
Because otherwise the real number line has holes in it. The intermediate value theorem is false. There's a continuous function that's less than zero at one point and greater than zero at another, but that is never zero. Of course the constructivists patch this up by limiting their attention to only computable functions. The constructivists have answers for everything, which I never find satisfactory.
Quoting Ryan O'Connor
Of course LEM is always relative to a given universe of discourse. If pi is a real number that's not rational, then it's a real number that's irrational. Without the restriction to real numbers, you're right. It could be a sandwich. Time for dinner.
A = sandwich * r^2. Doesn't work for me. Arccos (-1) = sandwich. Nope.
The volume of a pizza of radius z and height a is pi z z a.
Are you proposing this as proof of the existence of God?
Most definitely. With extra cheese.
First answer:
There's a way you can play this where it is a false dichotomy. But you have to go out into the desert and live on locusts and honey. If you are doing philosophy of math, you can basically say whatever you want. Personally I don't believe that odd numbers are real. (That's a joke, and it would be harder to make a case against the legitimacy of odd numbers IMO.) [You can say whatever you want and call it philosophy. And if no one agrees and you are truly OK with that, then you win. I mention this because of experience with infinity-deniers, anti-Cantorians, etc. ]
Second answer:
If you are asking a technical question, then it seems you have too options. Believe one or more anonymous posters who sound knowledgeable or find out which books are considered 'classics' in the field and see what you can make of them. I know something about the subject (but why should you believe me?) and I can say that lots of ink has been spilled on the issue by clever people. They exist, but not in such a intuitively pleasing way as natural numbers do.
Quoting Ryan O'Connor
Not that you should believe me as a single anonymous voice, but no one has to embrace the Platonic creed to get degrees in math. People who know how to shuffle symbols correctly can still argue about what it all really means. Wittgenstein was heavy on phil. of math, probably because it's a toy model of philosophy in general. What people even mean by Platonic Realm is not exactly specifiable, is itself a kind of 'irrational number' in the everyday shuffling of words. Approximation and vagueness is the rule, and integers are glowing exceptions to just about everything else that's human.
We separate two questions:
(1) Is there a real number x such that x^2 = 2?
(2) Supposing there is a real number x such that x^2 = 2, is that real number rational or irrational? (Note that 'irrational' simply means, by definition, 'not rational').
Proof supplied to answer (1) depends on certain axioms. Usually, these are the set theoretic axioms used to prove the existence of the real numbers as a complete ordered field. The real numbers as a complete ordered field provide a foundation for the mathematics of the physical sciences. Axiomatization is a desirable approach to mathematics, as it provides an explicit objective algorithmic standard by which anyone may judge of a purported mathematical proof whether it is indeed a correct proof, as opposed to subjective standards such as what happens to be or not be in the mind at any given time of some human being or another. So, if you reject certain set theoretic axioms, then we may ask: What are your alternative axioms?
Perhaps (I don't opine for the purposes of this particular post) proof supplied to answer (2) may be called into question constructively by rejecting the dichotomy that a real number is either rational or it is not rational. However, as to proof by contradiction, the proof that sqrt(2) is irrational is of this form [let P be any proposition]: We wish to prove it is not the case that P (in particular, we wish to prove that it is not the case that the sqrt(2) is rational). We suppose it is the case that P. We then derive a contradiction. We conclude that it is not the case that P. That is a constructive proof form.
That is not to be confused with the non-constructive proof form: We wish to prove it is the case that P. We suppose it is not the case that P. We then derive a contradiction. We conclude that it is the case that P.
As to the overall contradiction form, the irrationality of sqrt(2) is of the former, constructive, form.
The proof that there is a real number x such that x^2 = 2 comes later in the history of mathematics. It is found in many a textbook in introductory real analysis. This is desirable, for example, so that we can easily refer to a particular real number that is the length of the diagonal of the unit square.
Quoting Ryan O'Connor
Do you think your version of non-axiomatic mathematics "decides" every mathematical statement? There are undecidable statements in the mathematics of the plain counting numbers themselves. Undecidability is endemic to the most basic mathematics even aside from the real numbers as a complete ordered field. So how does your version of non-axiomatic mathematics "decide" all mathematical questions? By polling among the billions of known existing people (and examining just completed computer computations) whether they have such and such mathematical answers or computations in their minds or in their computer output at some given time?
Quoting Ryan O'Connor
This was answered by another poster. I would add that a complete ordered field is the ordinary foundation for calculus, which is used for the ordinary mathematics for the physical sciences.
Quoting Ryan O'Connor
Arithmetic, sure. But you haven't shown how to do exact calculus without irrational numbers.
Quoting Ryan O'Connor
Mathematical platonism (roughly put) is the view that mathematical objects exist independent of consciousness of them and that mathematical propositions are true or false independent of conscious determination. Understanding the sense of mathematical existence statements - such as the existence of irrational numbers - does not require subscribing to mathematical platonism.
This I would disagree with. One can take the viewpoint that symbolic math is a human endeavor; and that a thing has mathematical existence whenever a preponderance of mathematicians agree that it does. No Platonism needed. We've seen this standard applied to irrational numbers, negative numbers, transcendental numbers, complex numbers, quaternions, transfinite numbers, and many other now-familiar mathematical objects.
In order to demonstrate that sqrt(2) has mathematical existence, I do not need to posit a mystical Platonic realm in which sqrt(2) lives. If I did, I might be challenged: What else lives there? The baby Jesus? The Flying Spaghetti Monster? Pegasus the flying horse? No, I don't need to sort all this out just to know that sqrt(2) exists.
I can show sqrt(2) exists as others mentioned, as real number such that [math]x^2 = 2[/math], once I've formalized the construction of the real numbers and shown their completeness.
We can simply make up an arbitrary symbol [math]\sqrt 2[/math] with the property that [math](\sqrt 2)^2 = 2[/math], and then consider the collection of all rational numbers [math]a + b \sqrt 2[/math] where [math]a[/math] and [math]b[/math] are rational. We will find that we've invented a set of numbers that obey all the rules of a field (I can add, subtract, multiply, and divide with all the usual properties) that contains a square root of 2. This works very nicely.
In fact this is exactly how we introduce the complex numbers to students, as the set of all [math]a + bi[/math] where [math]a[/math] and [math]b[/math] are real and [math]i^2 = 1[/math].
We can formalize the above idea by starting from the rational numbers, forming the ring of polynomials with rational coefficients, and mod out the ideal generated by the polynomial [math]x^2 - 2[/math]. The resulting object is a field in which 2 has a square root.
In other words the two previous paragraphs show that if you believe in the rational numbers, you can easily adjoin to them a square root of 2. That is: if the rationals exist then so does the square root of 2. One does not need any Platonic realm to perform these symbolic constructions.
If someone asks if sqrt(2) as I've defined it has mathematical existence, I just point to any textbook on real analysis or abstract algebra. I do not need a book on metaphysics! All the mathematicians in the world agree that these constructions are valid and that's what gives sqrt(2) mathematical existence.
Now let me give an objection to what I said, one raised by @Metaphysician Undercover when we had this convo a while back. And that is, that mathematical existence is now contingent on what people say. My idea isn't any better than @Ryan's idea of polling all the humans to see if they're thinking of sqrt(2) at this very instant. My standard is to poll all the mathematicians in the world to see if most of them are willing to agree that sqrt(2) exists. I'd be the first to agree that my criterion for mathematical existence has some problems.
But a Platonic world where non-physical things exist? That seems untenable. So we're left with the opinions of mathematicians.
I meant to write "[...] does not require [...]." I edited my post just now upon reading your post and realizing that I mistakenly left out the word 'not'.
Doh! That would have saved me a lot of typing! LOL.
I believe that the only measurable states of a proposition are true or false so in one sense I accept the Law of the Excluded Middle. Where my view deviates from the norm is that I believe there can also be an unmeasured 'potential' state where a proposition is neither true nor false (akin to Schrödinger's Cat). And returning back to numbers, numbers which are not a part of any computation are in this unmeasured potential state.
Quoting fishfry
Is it a different issue though? Isn't 'not a number' a reasonable option to be included in the outcome space of the proof ?2.
Quoting fishfry
Assume for the moment that our universe is like a computer simulation. Wouldn't your existence be contingent on the simulator 'thinking' about you? To me, it seems reasonable to think that, like us, numbers are contingent. Why must everything eternally and actually exist?
Quoting fishfry
Why do we need all mathematical objects to actually exist? And moreover, why do we need to know that all numbers actually exist. To me it seems sufficient to know that you have the potential to keep counting the natural numbers, we don't need to actually count to 'the end'...in fact we can't. Perhaps if we fully embrace potential infinity and potential existence, we will find that we don't need actual infinity or the platonic realm.
Quoting fishfry
Consider Euclid's Theorem. What of the following did Euclid actually prove?
Interpretation 1: Any finite list of primes is incomplete.
Interpretation 2: There exist infinitely many primes.
These two interpretations may seem equivalent but they're not because the second makes an unjustified leap to assert the existence of an infinite set. Nevertheless, the math is the same, it's the philosophy (the interpretation) that is different. I believe ZF and Peano arithmetic just need to be reinterpreted.
Quoting fishfry
While I don't believe in the existence of infinite sets, I wouldn't attach myself to the finitist label. Finitists usually have an uphill battle trying to establish mathematics as rich is infinite mathematics. I simply think we shouldn't interpret ZF in certain platonic ways.
Quoting fishfry
Pi the infinite digit number cannot exist. Pi the finite-length algorithm can certainly exist.
Quoting fishfry
With your view these truths are missing. With my view the 'missing' statements are not missing at all, they're fully accounted for in the unmeasured 'potential' state. It's just that some statements, like 'this statement is false' must permanently remain in the unmeasured 'potential' state.
Quoting fishfry
If we pluck the irrational numbers off the number line, we are not left with holes in between the rational numbers...we are left with lines in between the rational numbers. With this view, we don't have to believe that infinite 0-D points can somehow be assembled to form a 1-D object. Instead, lines (or more generally, continua) are fundamental, not points. Think of how you draw a graph: you start with a piece of paper (a continua) and you draw a grid on it. At each intersection (point) you label it with coordinates (numbers). Everything about this is finite. In between the points/numbers lies a continua. But somehow we think about it all backwards. We think that we start with infinite points and numbers and then they someone assemble to form a continua. It's because of this thinking why calculus seems so paradoxical. I don't believe that the intermediate value theorem is entirely false, it just needs to be reinterpreted to apply to continua instead of points.
Quoting fishfry
Ha! Love it!!
Quoting norm
I'm not okay with that, which is why I truly appreciate this discussion. I also don't consider myself an infinity-denier, after all I'm a huge proponent of potential infinity (I only reject actual infinity). And I also believe Cantor's work was extremely important, I just think it needs to be reinterpreted.
Quoting norm
Or could 'Approximation and vagueness' just be the norm because we haven't fully figured it out? Math has changed so much in the past ~100 years since Cantor, what reason do we have to think that all of the foundations have been set? There are still way too many paradoxes to think that.
Quoting GrandMinnow
Real numbers are never used in applied mathematics. Every number that we have ever used in a calculation is rational. Real 'algorithms' on the other hand are very useful. When I use pi in an equation I use it to refer to a potentially infinite series which I dare not try to calculate...so I keep it in algorithmic form.
Quoting GrandMinnow
I hope that my views are largely in agreement with ZF and that ZF just needs to be reinterpreted. This is largely an issue of philosophy, not mathematics.
Quoting GrandMinnow
Aren't you beginning your proof with an assumption, that irrationals are numbers?
Quoting GrandMinnow
I understand why we want it to be a number, but that doesn't mean it is. After all, when we write it out explicitly we always write it as some algorithm. Why can't it simply be an algorithm? Our classical intuitions have us wanting to be able to precisely measure any coordinate, for example, the coordinates of the points where y=0 and y=x^2-2 intersect. But with quantum mechanics, our intuitions have changed. With the uncertainty principle we know that there is a fundamental limit to the accuracy with which the values for certain pairs of quantities can be predicted. Is it possible that there is a similar limit to which certain pairs of coordinates can be measured in mathematics?
Quoting GrandMinnow
My view is that we don't need to 'decide' every mathematical statement because 'undecided' is a valid state. The undecided state is not a defect of my view, it is a feature. Returning to quantum analogies, it is like how the unmeasured superposition state of a particle is a valid state for the particle.
Quoting GrandMinnow
My view is in total agreement with the foundations of calculus. In fact, I believe that when calculus was reformulated based on limits (and potential infinity) to banish infinitesimals that we didn't go far enough because we failed to banish real numbers (which are inseparably tied to actual infinity). We should have replaced real numbers with real algorithms and interpreted calculus to be not a mathematics that outputs numbers, but a mathematics that outputs processes.
fishfry, norm and GrandMinnows, thanks for your detailed and educated feedback!
The context of modern logic and mathematics involves formal axiomatization. Anyone can come up with all kinds of philosophical perspectives on mathematics and even come up with all kinds of ersatz informal mathematics itself. But that does not in itself satisfy the challenge of providing a formal system in which there is objective algorithmic verification of the correctness of proof (given whatever formal definition of 'proof' is in play). I mentioned this before, but you ignored my point. This conversation is doomed to go nowhere in circles if you don't recognize that what you're talking about is purely philosophical and does not (at least yet) provide a formalization that would earn respect of actual mathematicians in the field of foundations.
There is actual work in formal constructivist, finitist, and computationalist, or multi-valued mathematics, but you don't reference or commit to any specific such formulation. So we don't have a basis for evaluating the merits of your notions compared to those of classical mathematics on the level playing field of "I'll show you my system, explicitly, without vernacular vagueness and you show me yours."
Quoting Ryan O'Connor
Again, please show your system of axioms and rules by which one may make an evaluation of such circumstances.
Quoting Ryan O'Connor
If it's an object, then it exists. At the very least, in formal terms, there is an existence theorem.
Quoting Ryan O'Connor
The justification is from axioms from which we prove that there exists an infinite set. You are free to reject, or be uninterested in, those axioms, but in the meanwhile you haven't said what your axioms are. And you are even free to reject the axiomatic method itself, or be uninterested in it. But then this conversation would be at a hard impasse between, on the one hand, mathematicians who understand the benefit of formal axiomatics and require that mathematical proposals be backed up not just by homespun philosophizing, and, on the other hand, you.
Quoting Ryan O'Connor
ZF and PA are formal systems. They are interpreted by the method of models. Whatever you have in mind by a reinterpretation of a formal system is not stated by you. However for you to state such a thing would require that you do know the basics of mathematical logic that is the context of such systems.
Quoting Ryan O'Connor
Fine. We don't need to. However, the mathematics itself stands whether the mathematician regards it platonistically or not.
Quoting Ryan O'Connor
That is patently false. Of course, usually physical measurements, as with a ruler, involve manipulation of physical objects themselves and as humans we can't witness infinite accuracy in such circumstances. But that doesn't meant that mathematical calculations are limited in that way.
Quoting Ryan O'Connor
ZF is a formal system. In ZF we prove that there exists a system, which we denote as 'the real number system' and we prove that it is a complete ordered field.
Quoting Ryan O'Connor
Yes, as I stated explicitly, "Supposing there is a real number x such that x^2 = 2."
I gave that as a supposition in the context of separating the two questions I mentioned, But, meanwhile "there is a real number x such that x^2 =2" is proven from the ordinary axioms.
Quoting Ryan O'Connor
If you show us your actual system for mathematics, then we could evaluate its heuristic advantages or disadvantages compared with classical mathematics. But just saying "it's an algorithm not a number" is a an informal thesis, not an argument.
Quoting Ryan O'Connor
Fine. Then your remark to the other poster about undecidability is not pertinent. By the way, in classical mathematics a statement that is undecided is not undecided simpliciter but rather it is undecided relative to a given system.
Quoting Ryan O'Connor
No it's not. Clearly.
This reminds me of the construction of the real numbers from Cauchy sequence of rational numbers. Equivalence classes are required though, because there are infinitely many approximations of the 'same' real number. For instance, consider f(n) = 1/n and g(n) = -1/n. Both converge to the rational number 0 and are representatives of the same real number 0. Errett Bishop went around the use of equivalence classes somehow. I can't remember how, and I'm not a specialist, but you'd probably really like the spirit of his constructive mathematics. Have you looked into Brouwer? Or Chaitin's Metamath: The Quest For Omega? In short, I think some experts have problems with the real numbers, but these problems are philosophical/intuitive rather than technical. A mathematician can always retreat to formalism, etc.. I distinctly recall conversations with one mathematician who disliked philosophy altogether. In math (IMO), you really can know that you are correct if you don't mind not knowing what it is you are correct about. [In general, we don't know exactly what we are talking about, but math tempts us to forget that.]
Why give special status to " a preponderance of mathematicians", granting them the capacity to determine the existence of things? Isn't there a preponderance of Catholic theologians who believe in Jesus, and a preponderance of Pastafarians who believe in the existence of the spaghetti monster? Why do you think that mathematicians, simply by believing in something have the capacity to grant that something "existence", while for other groups of people, simply believing in something is insufficient for the existence of what is believed in?
You ask a good question, and one I can't answer. This is the only thing you've ever said to me that has made me stop and think, and for which I have no good answer.
My standard response is that math is a formal game, like chess. A position is legal if and only if it follows from the rules, there's no right or wrong to it, nor any deeper reason.
But I must admit that math isn't really that simple. 5 is prime, and that seems to be true independently of the opinions of people. Mathematical truth has a necessity that's forced on us in some way nobody can understand.
Bowling balls fall down, and that's forced on us too, but bowling balls are physical. Mathematical objects are purely abstract entities, yet the facts about them seem absolutely true independent of their discovery.
I haven't got a good answer. A lot of smarter people than I don't have a good answer either. Have you?
Quoting fishfry
I hinted at this in my first reply to the OP. I'm amplify that answer here: We are social beings, profoundly interested in one another. We want to be respected, and for intellectual types that involves our words being respected.
Anyone can make up whatever philosophy or mathematics they like, but they are highly unlikely to be taken seriously, largely because they are highly unlikely to create anything impressive by starting from nothing (or rather by starting from inherited commonsense, a cage that the less uneducated won't even see as one.)
The broader question is: why is peer-review valued and important? Even a brilliant individual is just one little short-lived human being, likely too in love with themselves to be sufficiently self-critical. Envy and competition keeps people grudgingly honest, and thousands of minds together can cover far more intellectual terrain and see into one another's blindspots.
There is a special field of study which delves into the nature of being, existence itself, and this is called metaphysics. Metaphysicians, being trained in this field, are best able to say whether something exists or not.
Quoting GrandMinnow
My ideas are so many steps away from a formal mathematical theory and in any one of those steps my ideas can be revealed to be inconsistent or useless. With that said, it's unreasonable to expect a formal theory to be perfected in isolation. And while my ideas are far from publishable standards, I do think they're at a level where they can be discussed on a forum. So please allow me to pick your brain, and please demolish my ideas. All I can guarantee to you is that you are dealing with someone incredibly receptive to criticism so the probability of going in circles is likely far less than what you expect.
Quoting GrandMinnow
I don't have axioms for you and to be honest, I don't have the technical skills to do much of the mathematical heavy lifting. I'm an engineer not a mathematician nor a philosopher. But I still think I can contribute. When Descartes developed analytic geometry, I suspect that he didn't present axioms but instead presented an intuitive way of thinking that proved incredibly useful (I'm particularly referring to Cartesian coordinates graphs). I think I could do something similar.
I believe it was his work which catapulted us to our actual infinity point-based world-view. When we draw a graph, we think that it is completely filled with points. It is a 'whole-from-parts' view where a continuum is constructed from infinite points. But consider this alternate 'parts-from-whole' view. We start with a continuum, perhaps a 2D square whose dashed edges correspond to x=?, x=-?, y=?, and y=-?. This continuum has no points, only the 'pseudo-points' (?,?), (?,-?), (-?,?), and (-?,-?). Draw curve from (-?,?), passing through the interior, and ending at (?,?). Label this curve y=x^2-2. Next, draw a somewhat horizontal curve through the interior and crossing y=x^2-2 at 2 points. Label these points (?,0) and (?,0). Finally label the starting pseudo-point (-?,0) and the ending pseudo-point (?,0). This graph does not have the infinite points to give it a fixed geometry. Instead, it can be thought of a topological system, whereby the relationship between what was actually drawn is maintained through continuous deformations. This covers just an overview of where I'm coming from, but let me just say that so many paradoxes (especially Zeno's paradox, derivative paradox, dartboard paradox) are not paradoxical with this view because this view is void of actual infinity. Instead, this graph has limitless potential to be further refined by adding additional lines.
Quoting GrandMinnow
I argue that we cannot talk about existence without including a qualifier: actual or potential. An object can potentially exist. In my graph example, I can think of potentially infinite points that could be introduced should I decide to add more lines, but until I do so, those points only potentially exist. And so in the context of that graph the only number that actually exists is 0 (and perhaps 2 if you're including the function's definition).
Quoting GrandMinnow
Axioms are not my strength, but could we perhaps reinterpret the Axiom of Infinity to assert the existence of an algorithm for generating an infinite set, without requiring that the infinite set actually exists? For example, with a few lines of code I can create a program to list all natural numbers even though it is impossible for the program to be executed to completion. I hope but cannot demonstrate that we can replace the actual infinities in ZF with potential infinities. Cantor was a 'Whole-from-Parts' guy so he built up set theory from points. What if we instead took a 'Parts-from-Whole' view?
Quoting GrandMinnow
I believe that if we abandon Platonism by replacing the actual infinities with potential infinities that the mathematics will stand. This is what I mean when I say it's a philosophy issue, not a mathematical issue.
Quoting GrandMinnow
I think your comment is a result of me not communicating myself properly. All that I'm saying is that when I use pi precisely I do not evaluate it, I keep it as...pi. I consider this to be a real algorithm. But when I actually evaluate it to produce an actual number, the number that I produce is always a rational number.
Quoting GrandMinnow
Let's set this aside for now since I'm not in a good position to debate with you about what exactly is proven by the axioms.
Quoting GrandMinnow
With the 'Whole-from-Parts' view we can't put brackets around everything and call it math. There is no set of all sets. We cannot talk about division by zero. We can't avoid Gödel statements. With a 'Parts-from-Whole' view it's easy to talk about this stuff...they're just unmeasured potential objects. In a way, by embracing incompleteness nothing gets left out.
Quoting GrandMinnow
Limits have a precise meaning in a 'Parts-from-Whole' view of calculus - they describe potentially infinite processes. If you want the area under a curve, approximate it with a set of smaller and smaller rectangles, to no end. There's no need to rationalize how lines can be assembled to create an object of area. In this example, if we work with potential infinity, the objects of study already have area, because we are not studying points...we are studying continua.
Quoting norm
Have you ever seen the cauchy sequence of a non-computable real number? If I claim that that Cauchy sequence is for the number 42, how could you challenge that claim?
Quoting norm
Thanks for the recommendations. Yes, there are a few experts (e.g. Norman Wildberger) who have problems with real numbers, but they're few and far between. His criticism is quite technical but I agree with you that it's a philosophical issue.
Quoting norm
You may be right, but I'm of the view that we don't know exactly what we're talking about because there's more work to be done.
Quoting Metaphysician Undercover
My view is that we can only think objects into existence internally. For example, a computer program can simulate a reality where internal to that reality the flying spaghetti monster is real. But a computer cannot do any amount of computing to make the flying spaghetti monster real external to the simulation.
Quoting TheMadFool
At what point did we prove that it was a number?
Quoting fishfry
I like to think that there exist truths independent of consciousness, whether it's certain axioms of mathematics or the laws of nature. But I think '5 is prime' is a contingent truth...
Quoting norm
I like that idea that together we have no blindspots.
Quoting Metaphysician Undercover
Perhaps metaphysicians have an important voice, but I'm more inclined to say that philosophers of mathematics and philosophers of physics are best equipped on this matter.
Ahhhhh, so we shouldn't poll the general public as @Ryan suggests; nor the mathematicians, which I suggest; but rather the metaphysicians! Well that certainly makes a big difference. /s
So in the end you agree with the notion that existence is contingent on opinion, and you simply differ on which opinions count. You just lost the argument methinks.
And what if I find a metaphysician who, based on two years of dialog with me, clearly hasn't bothered to learn the most elementary facts of mathematics? Why should I trust that individual's judgment about anything?
Where do they live? And what else lives there? The baby Jesus? The Flying Spaghetti Monster? Pegasus the flying horse? Platonism is untenable. There is no magical nonphysical realm of stuff. And if there is, I'd like to see someone make a coherent case for such a thing.
Quoting Ryan O'Connor
Well this I don't understand. Contingent on what? If there is a Platonic realm after all, surely mathematical truths live there if nothing else.
Quoting Ryan O'Connor
Wildberger is a nut, his math doctorate notwithstanding. He does have some very nice historical videos and some interesting ideas. But his views on the real numbers are pure crankery. You should not use himin support of your ideas, since that can only weaken your argument.
Quoting Ryan O'Connor
No. The existence of an inductive set is specifically the content of the axiom of infinity. If all we wanted was a procedure for cranking out infinitely many numbers, Peano would suffice. 0 is a number and if n is a number then so is Sn, the successor of n. That gives us each of 0, 1, 2, 3, 4, ... without end.
The axiom of infinity lets us take all of the numbers given by the Peano axioms and put them in a set. That's the essential content of the axiom.
The Peano axioms gives us 0, 1, 2, 3, ...
The axiom of infinity gives us {0, 1, 2, 3, ...}
The former will not suffice as a substitute for the latter. For example we can form the powerset of {0, 1, 2, 3, ...} to get the theory of the real numbers off the ground. But we can't form the powerset of 0, 1, 2, 3, ... because there's no set.
Also I'm not sure what you intend by writing, "actually exists." We only mean that an infinite set has existence within our theory. There's nothing "actual" about it, of course. Personally I doubt that any sets at all have actual existence. I can see the apple on my desk but I confess I don't see the set containing the apple. Sets are strictly an abstract formal system. Existence is relative to whatever the axioms say. Perhaps you didn't intend for "actually exists" to be different than, "mathematically exists," in which case never mind this paragraph.
You're talking about how you'd like mathematics to be, but you do it entirely in a castles-in-the-air manner without regard to even a minimal understanding of the mathematical context. Philosophizing about mathematics is fine. But when the philosophizing concerns actual mathematical concepts, then, unless there is an understanding of the actual mathematics and the demands of deductive mathematics, that philosophizing is bound to end as heap of half-baked gibberish.
Ordinary calculus does use infinite sets. As I mentioned, one can provide finitistic or other alternative axiomatizations, but, as I said, to fairly evaluate the advantages and disadvantages of such alternatives, we would have to know really what those axiomatizations are. You should understand this point well: The set theoretic axiomatization of mathematics is very straightforward, easy to understand, and eventually yields precise formulations for the notions of the mathematics of the sciences. Meanwhile, as best I can find, many finitistic alternatives are either much more complicated, harder to grasp, and possibly fail rigor.
If you are sincerely interested in the subject, even from a philosophical point of view, you should learn the set theoretic foundations and then also you could learn about alternative foundations that bloom in the mathematical landscape.
I'm not up to untangling all of your comments, but here are some points:
Quoting Ryan O'Connor
I didn't say that initially in a discussion the formal aspect has to be perfect. It just has to be reasonably coherent and credible.
Quoting Ryan O'Connor
I haven't read the mathematical papers of Descartes, but I suspect that he presented some basic principles and reasoned deductively from them. Then, over centuries, the deductive principles and methods of mathematics became more and more sharpened, as we eventually articulated the notion of formality as recusiveness and algorithmic effectiveness.
Quoting Ryan O'Connor
And neither is there a Zeno's paradox with set theoretic infinity.
Quoting Ryan O'Connor
We already have that concept. It's called 'recursive enumerability'.
Quoting Ryan O'Connor
I wouldn't begrudge philosophical objections to the notion of infinity. My point though is that one does not have to be platonist to work with theorems that are "read off" in natural language as "there exists an infinite set". The axiom that is (nick)named 'the axiom of infinity' does not mention 'infinity' and, for formal purposes, use of the adjective 'is infinite' can just as well be dispensed in favor of a purely formally defined 1-place predicate symbol.
Quoting Ryan O'ConnorDivision by zero can be handled by the Fregean method of definition. And I addressed Godel previously; you don't know what you're talking about with regard to Godel. Morevover, as you mention what you consider to be flaws in classical mathematics, as I said before, you have not offered a specific alternative that we could examine for its own flaws. Again, set theory rises to the challenge of providing a formal system by which there is an algorithm for determining whether a sequence of formulas is indeed a proof in the system. So whatever you think its flaws are, that would have to be in context of comparison with the flaws of another system that itself rises to that challenge.
Mr. Public: One of the drawbacks of the current vaccines is that you have to stick a needle in the arms of people and many people don't like that.
Scientist: True. So what's your alternative?:
Mr. Public: I think we can do it with pills instead.
Scientist: What's your formulation? Where are your trials?
Mr. Public: I don't have those. I'm just approaching it from philosophy.
It's true, math is a social activity, but I bet a lot of it exists without a preponderance of mathematicians even being aware of it, much less agreeing it exists. A single practitioner can have an original idea, one that assumes mathematical existence at that moment.
Wiki: "Existence is the ability of an entity to interact with physical or mental reality."
On the other hand, it does take that kind of recognition to establish the importance of a mathematical idea in the mathematical community.
Once again, a topic of limited interest to practitioners. :smile:
In your professional research, did you have the feeling that you were investigating aspects of truths outside yourself that you were trying to find out about? Or that you were merely pushing around symbols in a formal game? I'd wager the former but I'd be interested to know.
I'm more or less continuously involved in light, unimportant research these days. As an example, I've defined a variation of linear fractional transformations in which the normally constant fixed points are themselves functions of the underlying variable. Then I search for criteria that produce convergence upon iteration. I definitely look for "truths" outside myself, and it's this exploration that's fascinating. Knowing that such truths depend ultimately on axiomatic structures has no bearing on my minor investigations, and the very thought of moving around symbols in a formal game is anathema.
I was once a rock climber and it was the delight of exploration that was intensely compelling. :cool:
We can embed the rational numbers in the real numbers this way. Let f(n) = 42 for all n in N (a constant sequence). Then the real number 42 is the equivalence class that contains f and all g such that |f(n) - g(n)| --> 0 as n --> inf. Another representative of 42 in R would be f(n) = 42 - 2/n. Clearly there are infinitely many representatives for each real number, and there are proofs that show that it doesn't matter which representative is used in computing sums and products, etc.
It's a pretty good system. The interesting stuff is (as you hint) the gulf between computable and noncomputable numbersQuoting Ryan O'Connor
The point I was getting at in this context is something like: we often think we are talking about numbers when we are really talking about talk about numbers.
I agree that there's more work to be done, but there's already a mountain of stuff out there. After years of formal study (proof writing), I still would argue that intuition is primary and that math is a language.
Quoting fishfry
I don't know much about Wildberger (I remember him being passionate and unorthodox), but you touch on 'the' issue here, social reality. Any philosophical conversation about numbers is perhaps bound to get back around to authority and who has it. For instance: obviously you have every right to call Wildberger a crank and indicate that you find him anti-persuasive in this context. But we are in the strange situation of talking anonymously. If Ryan can't trust a PhD who uses his own name and face, why should he trust you or me? There's just no substitute for looking around the space and seeing what those in power are actually saying. Being in power doesn't make anyone right, but typically what people actually want is recognition.
What's your background, Norm? Another math proof on the forum? If so, welcome. :smile:
Hi ! I'll private-message you about that.
Oh boy they're gonna gossip about the rest of us!
I'm not the only one, Google around. And FWIW, I'm a crankologist. I enjoy reading math cranks and am familiar with the work of most of the prominent ones.
Well, I hesitated when it came to posting that, but I thought it would be rude to not reply publicly to such a friendly hello. Seems silly now. At the same time, I really like anonymity. (Since I want to freedom to react quickly and maybe say something silly, that anonymity comes in handy.)
I believe you, and personally I think you know your math very well, significantly more than me, clearly. But still, it matters that all of this is anonymous. To profess really is to profess, to take the risk and burden of professing, subject to accusations of being a crank, for instance. That's his proper name, probably the one he was born with, and he's publicly called a crank. A little part of me cheers for the underdog, though I wouldn't want to be a crank myself.
Anyway, if information hygiene is really the issue, I don't see how we can unironically play doctor in our masks. I'm not at all suggesting that anyone unmask...quite the opposite. Academia already exists, so what's needed is a place where people can play with ideas, take some risks.
You're right, he's a professor of math and he puts his ideas out there under his own name, and the likes of me throws rocks from behind my anonymous handle. Can't deny it.
I decided to look him up.
[quote=Wildberger]
The pure mathematical community depends on these and other fancies to support a range of “theories” that appear pleasant but are not actually corresponding to reality, and “theorems” which are not logically correct. Measure theory is a good example –this is a subject in which the majority of “results” are without computational substantiation. And the Fundamental theorem of Algebra is a good example of a result which is in direct contradiction to direct experience: how do you factor x^7+x-2 into linear and quadratic factors? Answer: you can’t do this exactly — only approximately.
By removing ourselves from the seductive but false dreamings of modern pure mathematics, we open our eyes to a more computational, logical and attractive mathematics –where everything is above board, where computations actually finish in finite time, where examples can be laid out completely, and where we acknowledge the proper distinction between the exact and the only approximate. This is a pure mathematics which is closer to applied mathematics, and more likely to be able to support it. It also gives us many new insights, more precise definitions, and theorems which are actually …correct.
[/quote]
https://njwildberger.com/
My first reaction is that this guy is another Cantor crank! But if he taught at Stanford at one point and is about to retire at some other school, presumably as a full professor, then he must 'know better. ' He must know how crankish he sounds and how bold he is being to abandon traditional foundations in some kind of informal constructivism.
This isn't the time or place to discuss Wildberger's crankitude and I'll leave you to your research. FWIW he's one of two PhD-level math cranks I know, the other being Edgar Escultura. As I mentioned, Wildberger has some very nice historical expositions on Youtube and is a perfectly sane and smart guy, just cranky about the real numbers. Then of course there was the late Alexander Abian, a perfectly respectable mathematician who advocated blowing up the moon.
What argument have I lost? "Existence" is a word which is being used here as a predicate. So we need criteria to decide which referents have existence in order justify any proposed predication. Naturally we ought to turn to the field of study which considers the nature of existence, to derive this criteria, and this is metaphysics. Mathematics does not study the nature of existence, so mathematicians have no authority in this decision as to whether something exists or not, regardless of whether it is a common opinion in the society of mathematicians.
If you are arguing otherwise, then show me where mathematics provides criteria for "existence" rather than starting with an axiom which stipulates existence.
The eternal truths that I am referring to are different from your eternal truths because mine are finite. I don't need to assert the existence of an actually infinite entity beyond our comprehension. Nevertheless I believe that all truths are contingent on a 'computer', it's just that there exists a 'computer' that lives outside of time. For example, I believe that the laws of nature exist outside of time and that these 'eternal' laws reflect the 'personality' of the grand computer. If on the other hand the laws of nature somehow did exist within time, what laws allowed them to pop into existence? If such deeper laws exist, then it is those laws which I'm referring to as external truths. I believe that the only object which can live outside of time is the unmeasured wave function of the universe...or more generally, a continuum filled with infinite potential.
Quoting fishfry
I believe that everything in actual existence is finite, even 'the grand computer' in which eternal truths live. And so 'the grand computer' can not actualize the set of all natural numbers any better than us. The actual existence of 5 is contingent on a computer 'thinking' about it. When no computer is thinking about it, it does not actually exist and it is meaningless to say that it has definite properties (akin to Schrödinger's cat). With that being said, I am comfortable saying that when a computer is 'thinking' about 5 that it will certainly be prime.
Quoting fishfry
I believe that, like Zeno, Wildberger is able to take our 'whole-from-parts' view to its limit to suggest that there are fundamental problems with it. We should value people who identify paradoxes as paradox leads to progress. However, I don't agree with Wildberger's resolutions. They do not offer the richness that infinite math does. If there indeed are paradoxes, the resolution should not be to weaken mathematics.
Quoting fishfry
If there is no way to reinterpret the Axiom of Infinity to apply it to potential infinity, then I'm inclined to reject it on some level. However, that doesn't necessarily mean that I entirely reject ZF. When working with ZF, we are always dealing with finite statements. Is it possible that these statements are the mathematical objects, not the sets which they are talking about? By 'actually exist', I'm trying to identify the objects that computers are actually working with. A finite computer can never work with an actually infinite set, it can only work with finite objects and potentially infinite algorithms.
Quoting GrandMinnow
I agree that I can only half-bake a mathematical idea. But even when mathematicians 'bake' an original idea at some point it's half-baked. And why must I bake it all by myself? I understand that you may not want to invest your time in evaluating half-baked ideas, but isn't a forum like this a good place to discuss them?
Quoting GrandMinnow
Do limits require the existence of infinite sets?
Quoting GrandMinnow
Do you believe that there are any paradoxes related to the set theoretic axiomatization of mathematics, and if so, is it fair to conclude that it's straightforward? ZF Axioms are rarely if ever mentioned in applied math (science, engineering, etc.).
Quoting GrandMinnow
I have learned a lot of math independently, but I certainly have much to learn and realistically not enough time to learn what is needed to do everything by myself. May I ask, how much education should a person have before initiating a discussion on a philosophy forum like this?
Quoting GrandMinnow
What resolution of Zeno's paradox are you satisfied with? Limits can be used to describe a process of approaching a destination but they cannot describe arriving there. So how does one arrive at some new destination?
Quoting GrandMinnow
Are your and fishfry's posts in agreement? As I said to fishfry earlier in this post, I believe that 'there exists an infinite set' could be a valid mathematical object. But I think one does have to be a platonist if they think that such an infinite set exists.
Quoting GrandMinnow
I totally understand where you're coming from. I'm sure you've dealt with many infinity-cranks in the past and this probably feels like deja-vu. I get that. But I don't like that you are labelling my view incoherent. That is entirely my fault since I haven't communicated it well enough in my post. I've actually produced a collection of videos roughly explaining my views, would you consider reviewing a couple of them:
Derivative Paradox: https://youtu.be/PSzqoQ9J5yg
Dartboard Paradox: https://youtu.be/LQmwZZNUMNA
Zeno's Paradox: https://youtu.be/_96tczP_eaY
Please note that I'm not forwarding you these links in an attempt to generate views. I gave up on my channel last year and have no plans to do anything with it. I'm only linking you to it since it may better communicate my view. If you don't want to watch the videos I could try explaining my views further on this chat...
Also, even though you're providing much resistance, you (and others like fishfry) nevertheless have generously given me some of your time by reviewing my posts and writing responses so please note that I'm very appreciative of that.
Quoting norm
I understand your limit-based 'algorithm' but would there ever be an instant in time when you would be sure that it's not 42?
Quoting norm
Yes, and I think we do the same about actual infinity. We don't conceive of actual infinity, we conceive of conceiving of actual infinity (using potentially infinite algorithms).
I'd say you'd want to look into the details, but a couple points:
In the mainstream version, incomputable Cauchy-sequences of rationals are allowed.
There is a strict definition for < and > that includes something like that instant in time where 'not =' is established.
In (quite different ) computable analysis (which you'd probably like if you don't already), equality is not a computable function. It takes arbitrarily long to see whether two numbers are different. Just think of decimal expansions. I can't tell whether 0.999999...[?] is different than 1 until I finally find a non-9 in the expansion somewhere, so there's no bound on the check for equality. I'm far from an expert on computable analysis. It's just something I looked into and that's a piece I vaguely remember.
Also, have you looked into Zeilberger? He's a maverick too, a bit of a finitist.
Quoting Ryan O'Connor
Right! We are in some sense actually talking about talk. Wittgenstein's beetle in the box aphorism applies here, and it's significant that he spent so much time talking about math.
Not so, my friend. Norm is mathematically authentic, as are you and fdrake, and I will probably learn something from his posts, as I have from the two of you.
I'm of two minds about revealing anything about the expertise of math people on this forum. I realize the knowledge may intimidate some others and dissuade them from contributing their ideas. Or it might have the opposite effect of encouraging attacks on academia. Oh well, not a big deal.
Quoting norm
You can't tell by inspecting the digits, but at least 0.999... is computable so you can make some assessments by comparing the algorithms used to generate 0.999... and 1. The same cannot be said about non-computable numbers, which is what I was getting at.
Quoting norm
I've read about him and listened to him being interviewed on a podcast but I have only very briefly skimmed his website. I can't recall why but I left the podcast not being interested in pursuing his ideas further.
As I alluded previously, your "reinterpret an axiom" has no apparent meaning (surely not rigorous) other than as a vague personal notion. Axioms are formal syntactic objects; you have not provided any meaningful sense of what in general a "reinterpretation" of an axiom is.
Moreover, do you even know what the axiom of infinity is? Do you even know the basics of the first order language of set theory in which the axiom of infinity is formulated?
Quoting Ryan O'Connor
Formulas of ZF are finite sequences of symbols. But ZF itself is a certain infinite set of formulas. And an axiomatization of ZF is a certain infinite set of formulas. Yes, any given formula of ZF is finite, and any given proof in ZF is finite (indeed, not just finite, but algorithmically checkable). But the study of ZF goes on to considerations of infinite sets of formulas (and while the set of axioms itself is infinite, it is algorithmically checkable whether any given formula is or is not an axiom). .
Quoting Ryan O'Connor
The personal context you present is based in gross ignorance of the subject. I don't opine as to all discussions, but this discussion engenders an unfortunate trail of waste product - gross misinformation, misunderstanding, and confusion. Sometimes people who have an appreciation of the subject wish not to see such otherwise beautiful scenic trails left without being de-littered.
If you were sincere about this subject, rather than blindly swinging about flimsy claptrap, you would familiarize yourself with the basics of the subject. I know nothing about marine molecular biology, so I don't go into a thread saying "the notion of cell structures in mollusks needs to be reinterpreted." I don't go into a thread about neuroepistemology and say "the very notion of a neural network is not acceptable to me; I propose instead an inside-out interpretation instead of the classical outside-in framework."
Quoting Ryan O'Connor
The study of limits in ordinary calculus involves, among other things, functions on real intervals, which are infinite. Infinite sets, intervals, domains, ranges, functions, et. al are all over calculus.
Quoting Ryan O'Connor
Paradoxes are related to set theory and foundations, of course. But formal set theory does not have paradoxes; it has or does not have actual formal contradictions.
Quoting Ryan O'Connor
I don't have a general answer to such a question. But I would suggest for you this undergraduate sequence:
Symbolic Logic (suggest: 'Logic: Techniques Of Formal Reasoning - Kalish, Montague & Mar; supplemented by chapter 8 ('Theory Of Definition') in 'Introduction To Logic' - Suppes)
Set Theory (suggest: 'Elements Of Set Theory' - Enderton, electively supplemented by 'Axiomatic Set Theory' - Suppes)
Mathematical Logic (suggest: 'A Mathematical Introduction To Logic - Enderton)
Quoting Ryan O'Connor
Zeno's paradox (at least as it is usually presented) is not a formal mathematical problem, but instead is a challenge to certain intuitive explanations of certain observable phenomena. 'arriving' and 'destination' are not, in this context, mathematical terms. Set theory is not responsible for disentangling every everyday common notion. Meanwhile, in mathematics we speak of the limit of a function at a point. It is quite clear and, as far as I know, it works for solving certain scientific problems.
Quoting Ryan O'Connor
You may reasonably ascribe to me only what I post myself. I imagine that holds for any poster.
Quoting Ryan O'Connor
I watched the one about Zeno's paradox up to the point you said, "mathematicians begin with an assumption [that] [an] infinite process can be completed". What exact particular quote by a mathematician are you referring to? Please cite a quote and its context so that one may evaluate your representation of it in context, let alone your generalization about what "mathematicians [in general] begin with as an assumption",
Set theory itself (at least at the level of this discussion), as formal mathematics, does not say "an infinite process can be completed". Set theory doesn't even have vocabulary that mentions "completion of infinite processes". And the assumptions of set theory are the axioms. There is no axiom of set theory "an infinite process can be completed".
FWIW, I agree with Chaitin that noncomputable numbers are suspicious. I can't even show you one. I can only talk about them indirectly. But if one does reject non-computable numbers, then R has measure 0, which completely breaks modern analysis.
For context, my overall view is that many alternatives are interesting (you might also like smooth infinitesimal analysis, which has a nice intersection with dual numbers for autodiff), but no foundation has ever seemed 'just right ' to me. There's always some ugly weeds. In the end I'm a relatively carefree antifoundationalist who enjoys math as an excellent if imperfect language. IMO, there's a 'know how' at the bottom of things that perhaps can never be formalized or made explicit. In some ways the quest for perfect mathematical foundations is a miniature version of the metaphysical quest. The impossible mission is to automate critical thinking, to capture that know-how in rules as clean as those for chess. Some of the philosophers I like have made strong cases against the possibility of this automation. They can't provide a decisive proof precisely because language is a soft machine.
In particular, the notion of a 'process' is imposed on axioms that don't mention 'process' at all. Granted, some mathematicians do mention stages of set building and levels attained (or, for example, see Potter's book in which he actually makes an equivalent set theory axiomatization as a theory of levels), for the purpose of providing an intuitive or philosophical framework work for thinking about the mathematics, while the actual mathematics is not itself liable for whatever difficulties may be found in such intuitive or philosophical frameworks.
Quoting GrandMinnow
That is a central point that I have made twice now. You have not responded.
Thanks! You'll probably learn more from the others, since I'm a philosopher/comedian at heart.
Quoting jgill
To explain my personal attachment to privacy: we live in polarized times and I'm still in the job market. Corporations and academia look very sensitive to me when it comes to exciting opinions. In other threads I talk about charged subjects like suicide, pessimism, war, etc. Even though I am a 'liberal,' I'm the Bill Maher type of liberal. He can get away with it, because he's a comedian. I don't want to limit my options. It's not only prudence though. There are other reasons for other threads that I value anonymity, which may be slipping away from us in general (and is only imperfect now, anyway.)
Perhaps we are both making this 'scenic trail' unpleasant for the other in different ways. Do you want to live in a country where the 'scenic trails' are exclusive to the 'privileged rich'? You are trying to find a way to reject my ideas without understanding them. Don't waste your time and simply disregard my posts.
Can you give me an example of what would break down without non-computable numbers?
Quoting norm
I like that you admit that there are ugly weeds. So you're just satisfied ignoring the weeds? But you must enjoy the philosophy to some extent, you're here after all? Actually, I'd love to hear what you think these weeds are...
Quoting norm
Can you explain what lies at the bottom that you don't think can be explained?
I appreciate that threads are open to posting by both well informed and less informed posters. That doesn't entail that misinformation, misconception, and confusion should not be called out for what it is.
Quoting Ryan O'Connor
I have explained exactly how certain of your ideas are ill-conceived and how you disservice the subject on which on which you opine while ignorant of its basics.
You asked me to look at your videos. Upon looking at one, I found that near the very start, you made a claim that "mathematicians begin with an assumption that an infinite process can be completed." I asked you to please say what specific statement by a mathematician you have in mind so that we can understand its context and to see how it fits your claim as to what mathematicians assume. Your critique of classical mathematics itself makes assumptions about classical mathematics - most of them quite ill-founded. You are the one who is critiquing ideas about mathematics without understanding them.
2. Keeps confusing technical points. But he keeps eliding the corrections presented by resorting to the cop-out "I'm only talking about it philosophically", even though the philosophizing is a critique of a technical subject. Or he just skips over the decisive corrections, as instead he replies by adding even more diverting tangents.
3. Projects onto others that they are not giving him a fair chance, that they don't try to understand him. Yet he keeps skipping over the actual explanations from others as to where he is confused, incorrect, and ill-informed.
4, Thinks he is presenting an innovative alternative in the subject. Yet he ignores the work already done in the subject over the recorded history of man - work by people who have dedicated truly incredible intellectual curiousness, creativity, rigor, and industriousness, while responsibly submitting their work to the most exacting standards of the peer-review method. This includes even ignoring serious work in alternatives to classical foundations - work that may be aligned with what he himself is ineptly stumbling to convey. The literature blooms with finitist, computationalist, constructivist and myriad other alternatives. Yet he won't inform himself about them.
5. Finally resorts to umbrage and the sophomoric instruction to ignore his posts, presumably then not to comment on them. Even though it was just explained that at least one motive in commenting on his posts is to not leave his falsehoods, misconceptions, and confusions uncorrected. Also, anyone should understand that it is the prerogative of posters arbitrarily to read and comment on whatever they want and that saying "then don't read my posts" is likely a doomed instruction anyway.
6. His misconceptions center on the usual crank bugbear: infinity.
So 1 through 6. But do we dare say 'crank'?
You may already know these things, but just in case:
The set of computable real numbers is countably infinite.
Countably infinite subsets of R have measure 0.
If we take out the noncomputable numbers in R, we are left with m(R') = 0.
This means that all subsets of R' would have measure 0, so that measure theory on the line would be dead.
The Lebesgue integral depends on measure theory.
It's 'the' mainstream integral (not the Riemann, whatever its old-fashioned charms.)
The mainstream real line is a vast darkness speckled by bright computable numbers, numbers we can actually talk about, numbers with names, while most of them are lost in the darkness and inferred to exist only indirectly.
For instance, if R has positive measure, then most real numbers are uncomputable, because the computable numbers have the cardinality of N (because we can enumerate Turing machines.)
Let me emphasize again that I don't specialize in foundations. Like every math student, I learned measure theory and the Lebesgue integral, so I can speak to the mainstream. It's plain to me that some of the other folks on this thread know much more about the nitty-gritty of logic and foundations.
Here's one example. If you follow the construction of the real numbers in set theory all the way from the construction of the natural numbers, you witness complexity stacked on complexity. You end up with something like equivalence classes of equivalence classes of equivalences classes. The process is spectacular really. I felt proud of myself when I could follow it all of the way. I even worked on a few of my own constructions of R starting from Q (nothing remarkable, but I was engrossed as if I were sculpting.)
But when I do math, I don't think of R in terms of that glorious set-theory mess at all (though I do think in terms of naive set theory and subsets of R), and of course these constructions of the real numbers came after many spectacular applications of the calculus. One of my favorite math books is Analysis By Its History. It's full of quotes from mathematicians on foundation issues in historical context along with early results. Altogether it's a living, breathing culture. Now we have more knowledge but at the cost of hyperspecialization.
In my POV, foundations is its own fascinating kind of math. It doesn't really hold up the edifice of applied calculus, IMO. It's a decorative foundation. Humans trust tools that work most of the time. Full stop. We could have taken a more empirical attitude toward math from the beginning. I'm not saying that we should have. It makes perfect sense that mathematicians want theorems and that results become more and more complex and presuppose more. It's a maddening mountain of knowledge, and it takes years of work to master a tiny piece of it, and only a few people understand what you are talking about (pretty lonely and dreary unless you fucking love the math,)
I don't want to derail the thread, but I'm talking about ideas in Wittgenstein, Heidegger,...others. Groundless Grounds is an excellent single book on the topic.
I'm not here to pick fights or spread misinformation. I'm here to learn. I've been quite open about my educational background and (I think) I've asked you many more questions than claimed that I have the answers. We don't need gatekeepers to 'scenic trails', we need people to help the litterers learn how not to litter. In joining this forum you did not sign up to teach others so feel free to ignore my messages, but if you're inclined to help then I welcome it. I certainly could benefit from someone with your knowledge.
FWIW, at the end of all 3 videos that I linked you to I include a message, like this one from my derivative paradox video:
"Now it’s worth repeating that the complete point-based construction is how we do math. The incomplete construction simply offers a different perspective on the paradox. We should not discard established ideas just because a different view might offer a more appealing resolution to a single paradox...Can we even do math with incomplete constructions? Or are there insurmountable problems with that approach? Let’s talk about it."
There are two issues being discussed here: (1) potential problems with the current philosophical foundations for math (2) potential problems with my proposed half-baked alternative to the philosophical foundation for math. I don't think you will enjoy us talking informally about (1) so I recommend that we set that aside.
Here are some replies to a subset of your comments which I think are most relevant:
Quoting GrandMinnow
I think the beauty of paradoxes, such as Zeno's, is that they capture the essentials of a profound problem in a way that anybody can discuss. If you don't want to talk informally, or if you want to disregard Zeno's paradox due to its informal presentation that is fine, we can go our separate ways.
Quoting GrandMinnow
You're right that I shouldn't have made that generalization. My apologies. Here's one quote by James Grime on Numberphiles video on Zeno's Paradox:
"I want to give you the mathematician's point of view for this, because, well, some say that the mathematicians have sorted this out........So something like this-- an infinite sum-- behaves well when, if you take the sum and then you keep adding one term at a time, so you've got lots of different sums getting closer and closer to your answer. If that's the case, if your partial sums--that's what they're called-- are getting closer and closer to a value, then we say that's a well-behaved sum, and at infinity, it is equal to it exactly. And it's not just getting closer and closer but not quite reaching. It is actually the whole thing properly."
To be fair, he follows that by saying that that's the paradox. Do you believe that infinite processes cannot be completed? If so, how can I move from A to B to C. I'll never get to C because I'll never complete the infinite steps required to get to B.
Anyway, I'm not sure where you want our conversation to go, but I'd be glad to hear your feedback on the rest of the videos that I linked if you care to give me a chance.
Quoting GrandMinnow
Agreed.
Quoting GrandMinnow
Should we move this discussion to a new thread?
Quoting norm
Very poetic, I like it!
Quoting norm
To me it is concerning that the foundations are so disconnected from the applications. Could this be an indication that further foundational work is required? I'm not sure if you're following this thread closely but I pointed GrandMinnow to a few links on my YouTube channel. Here's one that's somewhat related to our discussion on integrals (Dartboard Paradox). You might be interested in this perspective as it offers a different perspective (granted, probably wrong and certainly half-baked). Nevertheless, I'd love to hear your thoughts on this view, especially if you can find flaws in it...but no pressure at all!
I only have a moment just now, but I'll respond to the point above. It seems to me (consider Hume's problem of induction) that humans just are 'irrationally' inductive animals. Foundations that come later than the edifice are not really foundations at all, despite the metaphor.
I'm no expert, but my impression was the math moved toward being totally mechanized, totally formal, totally computer-checkable. The self-image of the mathematician changed, probably because math become its own art/science, not just part of physics, etc. But writing proofs 'feels' more like convincing the intuition of another mathematician and reassuring one's own.
It's unfortunate the word "foundations" is used in mathematics. Foundational set theory (that intersects with philosophy significantly) and all that revolves around it is a marvelous intellectual area, but a great deal of mathematics flows unimpeded by its pronouncements. Do most seasoned mathematicians worry about the existence of irrational or non-computable numbers? Or non-measurable sets? Not likely.
I've mentioned my own experience before: as beginning grad students sixty years ago we were required to take an introductory course in set theory (foundations). Afterwards the prof - a young energetic fellow - made the statement, "Unless you are really fascinated with the material in this course, I recommend you never take another foundations of mathematics class." I wasn't and I didn't. But I ultimately went into classical complex analysis. Had I gone into a more abstract realm of math I might have needed it.
Quoting Ryan O'Connor
It depends on which mathematician you ask. Let's hope not. It makes little sense to ask anyone outside the profession.
Quoting norm
And I'm sure Norm would agree, that movement would drive most mathematicians out of the profession. It can't be emphasized enough how much mathematics depends on intuition, imagination, inventiveness, and a spirit of exploration. Devising and proving theorems is an art form.
I liken it to physics. Many engineers get by with classical physics. They don't worry that it predicts singularities because it works great for them in their applications. But the singularities are the loose thread which suggest that classical physics is not fundamental. It doesn't lie at the foundation. We need to go quantum. If set theory properly lies at the foundation of mathematics then (I believe) it should have no loose threads (e.g. paradoxes). With that said, obviously set theory works to a significant extent so even if further refinements needed to be made to the foundation, I'm sure that the essence of set theory will play a significant role.
Quoting jgill
LoL. I'm sure the philosophy students in the room were aghast.
I don't think in a framework of "infinite processes being completed or not completed". The notion of "an infinite process being completed or not completed" is not a notion I find meaningful; I am not burdened with it.
Moreover I already addressed this with regard to set theory:
Quoting GrandMinnow
But you blow right past that. (You're too busy with things like explaining that I am free to ignore you, and extending my little metaphor of a 'trail' into an overblown conceit that is becoming ludicrous).
Quoting Ryan O'Connor
An infinite sum is the limit of a function. It is the unique number such that the terms of the sequence converge to that number.
The actual mathematics does not say "process" or "process completion". It doesn't need to be saddled with it. If you find problems with the intuitive view of an infinite sum as the completion of a process, then it's your intuitive framework that is problematic, not mine, since I don't have to resort to that framework and, as far as I know, mathematics may be understood without it.
So much more to unpack:
Quoting Ryan O'Connor
Imagine a discussion among intelligent and educated adults.
Quoting Ryan O'Connor
Among these educated and intelligent adults, if they post their opinions on certain matters that bear upon technical considerations, then one may point out where those technical points have been misconstrued and offer relatively concise corrections and explanations. If this is beyond the grasp of any of these educated and intelligent adults then they can consult any of a number of books that explain at a level any intelligent person can understand by simply beginning with chapter one and reading forward.
Quoting Ryan O'Connor
You have been occasionally exaggerating my points in order to knock them down. And you use other variations of the strawman argument, most saliently to me in the very first remarks in your video on Zeno's paradox, as I pointed out. (Though you have since retracted in this thread, I don't know whether you intend to correct the video itself).
To the point, I did not "pile on a dozen textbooks". You asked me about education. I gave you a list of three (plus some optional supplements) introductory undergraduate textbooks for a three step sequence: Symbolic Logic, Set Theory, Mathematical Logic. And I don't say that one has to have such a background merely to ask questions, speculate, ruminate, or convey ideas on the subject. Rather, my point is that when your philosophizing moves into matters that bear on technical points in mathematics, and you mangle those points or talk past right past them, then it's appropriate to point that out. And I recommended a few books in response to your question about education.
Quoting Ryan O'Connor
I never said that or anything equivalent to it.
Quoting Ryan O'Connor
Perhaps you are, but that's not all. You've also here for other people to be impressed with what you say.
Quoting Ryan O'Connor
I'm not a gatekeeper in the sense of saying that people may not post whatever they want to post. You can post as you like; I don't try to stop you. Meanwhile, I hope you are not a gatekeeper saying what I may post, including criticisms of your posts. And my purposes in posting are not determined by what you think a forum needs or doesn't need.
Quoting Ryan O'Connor
I have suggested ways you can abate your littering.
Quoting Ryan O'Connor
I already responded to your sophomoric protest that I may ignore you. Of course I feel free to ignore you, and I also feel free not to ignore you. Your personal preference in the matter is not relevant to me. When you post, others may reply or not reply arbitrarily at their own prerogative.
Quoting Ryan O'Connor
I've offered you help already. I've given you explanations at a pretty simple and straightforward level. But you blow right past most of the key points in those explanations. And I've given you a list of three books that constitute a truly splendid introduction to the subject on which you are posting. I am not even a mathematician, but at least I have made myself familiar with a number of books on the subject. If you are sincere about learning and benefiting from what I know, then the very best you could do is to accept my expertise in book collecting and get hold of the books I mentioned. Instead you take umbrage at the offer and whine as if you've been unduly sandbagged.
Quoting Ryan O'Connor
Quoting Ryan O'Connor
I am, as time permits, interested in philosophy of mathematics and informal discussion about mathematics and the philosophy of mathematics. And Zeno's paradox is of course important in the philosophy and history of mathematics. But this is the point you keep missing: When the informal discussion bears upon, or especially critiques, the actual mathematics and the actual foundational formalizations, then it is critical not to speak incorrectly, especially from ignorance, about the actual mathematics, formal theories, and the developments in set theory and mathematical logic. I surmise that you, like cranks, find poring through the actual technical development to be onerous but you prefer to opine about it in ignorance anyway. This is witnessed by the fact that no matter how many times one suggests to a crank that he consult the actual writings on the subject, he will never even look at chapter one.
I addressed that already. You blew right past it.
I very much agree (creative intuition is why it's beautiful and fun). I also agree with what you said about foundations. We never even covered constructions of R in the classroom. I actually have found 'foundations' (and mathematical logic) fascinating, but I never had the time or sufficient passion to really catch up with the present. I still love computability theory, but the most fun I've had mathematically is inventing things (like my own construction of R or various crypto systems or oddball never-before-seen (and not actually useful) neural networks. My blessing/curse is that I can't help approaching it like a sculpture. I don't care much about applications. I like beautiful machines made of pure thought.
I'll try to find time for it & definitely give some friendly feedback.
Quoting Ryan O'Connor
You don't know what set theory is.
You don't know about the symbolic logic in which set theory is formulated. You don't know what the language of set theory is. You don't know the axioms of set theory. You don't know the theorems of set theory and how they are derived from the axioms. You don't know the definitions in set theory. You don't know how set theory develops numbers and mathematics. You don't know how set theory axiomatizes calculus and other mathematics of the sciences. You don't know the purpose, motivation, and role of the axiomatic method. You don't know about constructive, intuitionist, predicativist, or finitist altermatives to classical mathematics.
You don't know anything about it.
Yet you have persistent critiques of it.
How do you do it?
That would be the empty set, Ryan. We were all math majors and the course was taught in the math department. :smile:
Quoting GrandMinnow
With a certain aplomb. I admire his spirit while avoiding his critiques. :cool:
That's a well made video. FWIW, I do like the continua-based approach. Have you looked into smooth infinitesimal analysis ? It seems similar. One issue worth noting is your description of a quasi-Riemann integral as an endless process. In an actual Riemann integral, for f which is continuous on [a,b], there exists a definite sum. In other words, we know that it's a particular real number, even if we only ever approximate it (like the areas under the standard normal curve.)
In SIA, certain issues are circumvented, because every function is smooth (infinitely differentiable). Some strange logic is involved.
Yes, and I can relate to theimpatience of someone who wants to talk about something now. Chances are that this thread will inspire some serious reading, and talking through things could help pick out just the right book. I mentioned a philosophy book and a math book, because for many people (whether they know it or not) it does turn out to be a philosophical issue, a perspective from which to interpret what math means. Or the issue here could be one of intuition and pedagogy and not really about nitty-gritty foundational work.
I don't mean to talk as you aren't 'here' with us. I'm curious about what kind of pure math that you have studied, if you feel like sharing. Have you wrestled with real analysis? I am nostalgic for basic real analysis on R, working through proofs of theorems about the beautiful Riemann integral. We didn't bother with constructions. We just used the axioms. I had the itch, so I learned the two classic constructions, and I was very passionate about grabbing those slippery real numbers in my intuition. It bothered me and it delighted me. (I was OK at writing proofs, but pretty good among my peers at reading them. Indeed, research can be a little dreary compared to enjoyed the condensed, finished product of generations --with only brief flashes of invention.)
Interesting! So, you think the square root of 2 could be something other than a number. Well, the square root operation is closed over real numbers i.e. a square root of a real number has to be a real number. Does that answer your question?
Mathies keep the fun stuff to themselves. :P
Quoting norm
Thanks for checking it out! I'm glad you like the approach. I have not looked into SIA. However, just a few points on the wiki page seem concerning to me, like I have no problems with discontinuous functions but I do have a problem with infinitesimals. Nevertheless, I will check it out. For Riemann integrals, how do we know that it corresponds to a real number if we are only ever able to approximate it?
Quoting TheMadFool
Yes, I think it's an algorithm for calculating a number (but the algorithm cannot be executed to completion). I'll need to look up a proof of your statement and hopefully I can understand it! Perhaps it will convince me.
Weird stuff, IMHO. Low priority in the world of mathematics.
I can understand your hesitation. As long as one demands a 1-to-1 map between individual mathematical inventions and metaphysical correlates, it's hard indeed to be satisfied. There's a thought (I think Hilbert's) that we should just mathematical systems as a whole for correspondence with reality. You mentioned something like points-at-infinity in your discussion with MU. I mentioned the convention that 0! = 1. I'm personally doubt l that there's a rigorous way to include the continuum without offending someone's intuition. Great mathematicians have wrestled with this. As I mentioned, you might like Errett Bishop more than anyone I'm aware of.
Quoting Ryan O'Connor
The 'magic' of least upper bound axiom does the heavy lifting in R. Every nonempty subset of R which is bounded above has a least upper bound, a 'supremum.' I can look at 'lower Riemann sums' (approximations with step functions <= to f(x)) and define the supremum of that set to be what the integral means, which is to say the number that is being approximated.
A related example is the greatest lower bound or infimum of a set. Consider the set {1/n : n >= 1}. All of its members are positive. It's infimum, however, is 0. It's the only lower boundary that makes sense. Any number great than 0 misses an element, and any number less than 0 is not as close and effective a bound as 0. This kind of thinking is IMO the essence of basic real analysis. (Note also that pi is defined geometrically but in terms of the zeros of a trigonometric function, which requires a detour through the convergence of power series first. I like Rosenlicht's little Dover book, Introduction to Analysis. It's cheap and is just good.
I'll just reiterate that if you aren't that concerned with proofs (deriving theorems from axioms), then mathematicians will hardly recognize that you are even interested in (pure) math. Lots of foundations can and even must 'spit out' the same engineering math, the stuff that keeps the planes in the skies. Being a mathematician, to anyone with mainstream training, means reading and writing proofs. I don't mean to be a snob about this. I'm personally just as interested in pedagogy for applied math as pure math (I'm an anti-foundationalist with empirical leanings. I like Wolfram, etc.)
That's been the case in my experience too. For applications, though, the dual numbers are actually important today. Some of the autodifferentiation powering machine learning uses the 'forward method', employing dual numbers to great effect (and even hyperdual numbers.) This allows one to compute high-dimensional f(x) and grad(f(x)) at the same time at low cost.
Infinitesimals are made rigorous with non-standard analysis derived with techniques of model theory or with internal set theory.
@Ryan claims that we should poll everyone in the world; I claim we should poll the professional mathematicians; and you claim we should poll the philosophers specializing in metaphysics. Doesn't seem like there's any qualitative difference between your position and the others.
Quoting Metaphysician Undercover
Don't start that! I'm sure you know the trouble one gets into using existence as a predicate. Existence is not a predicate.
Quoting Metaphysician Undercover
This is a mistake on your part. Doctors know more than the average person about doctoring; and baseball players know more about baseball. But metaphysicians don't know any more about existence than the rest of us. All they know is what other thinkers have said about the problem. Being a philosopher confers no special knowledge at all about what's true. That's one of the fundamental problems with philosophy. If I study math in school, I'll learn about math. If I study philosophy, I'll learn about what the great thinkers have said about philosophical problems; but I won't learn the truth about anything. So you have no credential whatsoever.
Quoting Metaphysician Undercover
Agreed. But mathematicians have total authority in terms of what has mathematical existence.
Quoting Metaphysician Undercover
That's the main criterion. And yes it's historically contingent, and yes it's somewhat unsatisfactory if one wants to believe in some kind of ultimate existence, but that's the position I'm taking.
Quoting Metaphysician Undercover
The entire history of mathematics is filled with examples, starting from the discovery of irrational numbers right through to the present day. Now you may well respond that if I'm admitting this was a discovery and not an invention, then irrational numbers were already "out there" waiting to be discovered. I have no good answer for this objection but neither does anyone else.
I love her videos and articles but felt that she entirely missed the meaning of complex numbers in that video. When physicists talk about math it's always a disaster. But she's always worth watching.
Just my opinion, I didn't want to give you a hard time for posting it, she's always worth watching. That particular video annoyed me but as you know I'm easily annoyed! It's good that you posted it and people should watch it. She does talk about the use of complex numbers in physics and that's definitely worth watching.
It seems like you don't know anything about metaphysics, which is the study of existence. Why would you think that someone who has not studied existence would know as much about existence as someone who has studied existence?
[quote=Wikipedia: Metaphysics]Metaphysics studies questions related to what it is for something to exist and what types of existence there are. Metaphysics seeks to answer, in an abstract and fully general manner, the questions:[3]
What is there?
What is it like?
Topics of metaphysical investigation include existence, objects and their properties, space and time, cause and effect, and possibility. Metaphysics is considered one of the four main branches of philosophy, along with epistemology, logic, and ethics.[4][/quote]
Quoting fishfry
So, where are your examples? Where is the criteria for existence found in mathematics?
See, I learned something from your post! Thanks, norm.
Quoting fishfry
What would you say the meaning is? Just curious. :cool:
I do agree, that we have not proved the existence of irrational numbers.
My argument is not only the problem of indirect proofs, but a deeper one:
We can not be sure that our logic is totally right.
It is astonishly easy to construct an alternative logic that gives in finite cases mostly the same results as classic logic, but has totally other results with infinite cases or indirect proofs or antinomies.
I myself constructed such a logic, the layer logic.
There we have an additional parameter, the layer,
and statements are not right or wrong but have a truth value in every layer.
And this values can be different in different layers without making a contradiction.
Therefore in indirect proofs (as for the irrationalioty of root 2) we need to get the contradiction in the same layer. But analysis shows, that different layers are used,
and therefore with layer logic we do have no contradiction and no proof anymore.
The same way we can show, that the diagonalization of Cantor does not work anymore.
So we need only one kind of infinity (that of the natural numbers) in layer logic or layer set theory.
But not all is better with layer logic:
The uniqueness of the prime decomposition (over all layers) can not be proved.
So we have a nice layer set theory where the set of all sets is a set,
but arithmetics may partly be time dependent:
My newest guess is, that there is a layer for all objects (quants) that can interact
(except interacting with gravity),
and if some interact than the layer for all objects is increased.
This way we get in the layers a kind of time arrow since the big bang,
and properties (even in math like prime decomposition)
can depend from it and change with time.
Most mathematicans will not like such a world,
but it seems possible to me.
The square root of 2 could be a rational quotient with different numinator and denominator
in different layers (or times).
Some details to layer logic you can find in:
https://thephilosophyforum.com/discussion/1446/layer-logic-an-interesting-alternative
Even more (in German) here:
https://www.ask1.org/threads/stufenlogik-trestone-reloaded-vortrag-apc.17951/
Yours
Trestone
This is similar to what I was telling Ryan in the other thread on Gabriel's horn. The classical way that mathematicians apply numbers to spatial representations (Euclidian geometry) assumes an eternally continuous, and static, space. But modern observations have produced a new concept called spatial expansion. Therefore we need to allow that space itself changes with time, and this means that the assumption of a static space is incorrect. So if we propose a number of points in space, and these points, if connected with lines, make a shape such as a triangle or square, and then we propose some passage of time, then these same points in space will no longer make the same shape.
In casual discussion, mathematicians may say things like "the square root of 2 exists". But in a more careful mathematical context, we don't say that. Instead we say, "There exists a unique x such that x^2 = 2." Then we may apply the square root operator to refer to sqrt(2).
So, indeed, in careful mathematics 'existence' is not a predicate. In careful mathematics It is not even grammatical to use 'existence' as a predicate. Instead, there is an existential quantifier that is applied to a variable and a formula (usually with that variable free in the formula); the formula specifies a "property".
Symbolically, existence is not a predicate in which we would write "the square root of x has the property of existing:
E(sqrt(2))
That is not even grammatical.
Instead, we write:
Ex x^2 = 2
Which reads, "There exists an x such that x^2 = 2".
'Ex' is the quantifier, and 'x^2 = 2' is the formula specifying the property.
Then we also derive a uniqueness quantifier and write:
E!x x^2 = 2
Which reads, "There exists a unique x such that x^2 = 2".
And that justifies using
sqrt(2)
as a term.
A quarter counterclockwise turn in the plane. That's the simple meaning. I was probably too harsh with my criticism of her video though, it's an excellent summary of the use of complex numbers in physical science. Just missed the mathematical essence IMO.
That's what happens when multiplying a+bi by i.
I play in the complex plane all the time, and I have always visualized figures and imagery and motion. Even created what might be considered art in the process. :nerd:
Right, that's the answer to "does the square root of -1 exist?" Just as the number 5 can be interpreted as stretching a line segment by five units; and multiplying by -1 preserves the length and reverse the direction of a line segment; multiplying by i rotates the segment a quarter turn counterclockwise. And if you do it twice in a row, you get the same effect as multiplying by -1. This in my opinion is what they should be explaining to every high school student. But they don't. And apparently they don't explain it to physicists either!
Quoting jgill
So what does the L mean in your equation earlier? Not familiar to me.
Give me a few moments. See the Gabriel's thread.
The problem of course, being that it is debatable whether there is such an x.
E!x x^2 = 2
is a theorem of ordinary mathematics.
Anyway, I made my point that existence is not a predicate..
Actually, I don't think you have.. You simply used "exists" as a verb, and verbs refer to actions which must be predicated of a subject to say anything truthful. So "there exists..." really doesn't say anything meaningful because you haven't properly identified the subject referred to when you point with "there".
Hello,
here the sight with layer logic instead of classical logic:
In layer logic you have to give a layer to the statement that should be possibly true:
In layer k the statement E! x^2 = 2 is true,
in another layer m the statement E!x x^2 = 2 may be also true, but with another x (say y).
So we can have x^2 = 2 is true in layer k and x^2 = 2 is false in layer m and y^2 = 2 in layer m.
Therefore the square root of 2 in layer logic is most probably not one irrational number
but many rational fractions in different layers (times).
Yours
Trestone
The "Sqrt 2" is at the very least, pragmatically useful as a moniker for the hypotenuse of a certain class of visually recognisable triangle, and it should be remembered that we have as much empirical justification for labelling the hypotenuse with sqrt 2 as we have for labelling its other sides with "1".
Zeno's reaction to his paradox is also similar to yours, in his conclusion that the existence of motion is impossible on the grounds that motion cannot be constructed from positional information. But the converse is also true: a position, in the logical sense, cannot be constructed by slowing down motion. Motion and position are irreconcilable concepts pertaining to mutually exclusive starting conditions of a system and mutually exclusive choices of disturbance of the system by an observer thereon after. Each concept can only informally represent the other as a "limit" that can only be approached but never arrived at.
The constructivist isn't forced into believing in the literal existence of hypertasks as the platonist might insists, rather the constructivist only needs to deny the existence of a universal constructive epistemological foundation.
I didn't write "E!x^". It doesn't make sense. I wrote "E!x x^2 = 2".
sorry, the citation at the beginning of my writing was an error and can be ignored.
Yours
Trestone
Didn't read back to find the source of the quote, but sqrt(2) is certainly computable. For example you can use a standard iterative procedure.
Because there are SOME subjects in which more study implies more knowledge; and others where it doesn't.
If you study more math you know more about math than someone who doesn't study math. Medicine, physics, and history are in this category.
But take, say, baseball. A non-athlete spends his life studying the game. Reads books on strategy and tactics, knows all the statistics, can name all the players on the 1928 Philadelphia Athletics and their batting averages. Another person knows nothing of the history of the game but has been playing all their life and has been toiling in the minor leagues for years (think Bull Durham). Who knows more about the game? I would say the practitioner and not the student.
Existence is the same. If someone's been existing for a few decades they know as much about existence than a philosopher. The philosopher knows the history of what great thinkers have written about the subject. But philosophy does not confer actual knowledge of its subject; only knowledge of what others have said.
Philosophers don't necessarily lead better lives than others, nor are they more moral, and they most definitely don't know any more about existence than the general public.
In particular, a philosopher who knows hardly anything about mathematics is in no position whatsoever to comment on mathematical existence. Many philosophers of mathematics are in this position. They simply don't know enough math to comment intelligently on the subject of mathematical existence.
yes, sorry, i should have clarified that i was referring to number in the extensional sense, i.e. as an obtainable state of a calculation. This is because i understood the OP as questioning the existence of sqrt 2 in this extensional sense.
Naturally, Sqrt 2 exists intensionally in the constructive sense of an algorithm that generates any finite cauchy sequence of computable reals, whose limit x fulfils the axiomatic definition
sqrt 2:= x : x^2 = 1.
Furthermore, it is constructively provable that the limit is irrational in the sense of being separated from any computable rational number. And the sqrt of 2 intensionally exists in this sense, irrespective of whether the limit of this process of calculation is axiomatically accepted as being a number in the extensional sense.
Of course, normally when we assign numbers to physical lengths we aren't resorting to logical construction, nor even necessarily to calculation, which means that our practical use of numbers is logically under-determined. We could for instance, declare the hypotenuse of unit length triangles to be "real" lengths that correspond to extensional numbers relative to some novel logical construction of numbers, in which the unit lengths of the other sides of the triangle only exist intensionally as limits in this number system.
In my opinion, constructive mathematics should permit a plurality of axiomatic systems to represent our use of numbers. That way we don't have to make arbitrary a priori decisions as to which numbers only exist as limits.
:up:
I didn't say it isn't perfectly fine English. I said you haven't properly identified the subject signified with "there", to which "exists an object" is predicated.
Quoting fishfry
All I can say is, wow! This is an unbelievable opinion coming from an educated person like yourself. Would you also say that a person who has been breathing for a few decades knows as much about breathing as a biologist?
So going to university and studying a subject of study only provides one with what other great thinkers have said about that subject, but it doesn't provide one with any knowledge of the subject? It only provides one with what those who've studied that subject, say about the subject? What about studying mathematics, wouldn't this be the same thing? Studying mathematics doesn't provide any real knowledge of mathematics, only what others who have studied the subject say about the subject. What do you think knowledge of a subject consists of, if not what those who study the subject say about the subject?
Quoting fishfry
The problem with this perspective is that "mathematical existence" means something completely different than "existence" in the philosophical sense. The op does not ask about "mathematical existence", it asks about "existence". If it asked about the mathematical existence of irrational numbers there would be nothing to discuss. Clearly irrational numbers are used by mathematicians therefore they have mathematical existence.
The op is asking a philosophical question about the existence of certain mathematical objects, not whether those mathematical objects have mathematical existence. That would be self-evident. So mathematicians who hardly know anything about existence, yet think they do because they know something about mathematical existence really seem to have very little to say about the philosophical question of whether certain mathematical objects which obviously have mathematical existence, have existence.
Whatever you have in mind linguistically is irrelevant since the sentence is linguistically perfectly correct.
Even more simply:
There is a unique object whose square is 2.
or
A unique object exists such that its square is 2.
or
An object exists such that its square is 2 and no other object is such that its square is 2.
or
A unique object exists such that it has the property that its square is 2.
Etc.
All perfectly grammatical and sensible English.
My rough impression is that professionals in the field of philosophy of mathematics usually do know about mathematics. Which philosophers in, say, the last 85 years do you have in mind?
1. They can be expected to exhibit a political bias towards inflating ontological claims in mathematics , in the same way that professional chess players can be expected to inflate the importance of chess and exhibit bias against other games. Traditionally this is bias is evident in the continued prevalence of classical logic in the justification of mathematical claims and the high percentage of mathematicians who are platonists, a position that is troubling to any engineer or computer scientist who unlike pure mathematicians have actual responsibilities regarding the physical actualization and interpretation of mathematical results.
2. The subject of ontological claims and commitments is the domain of logic rather than of mathematics, and the classical mathematician isn’t logic savvy, unlike today’s generation of mathematics undergraduates who are studying mathematics using theorem provers from the outset.
I agree.
Quoting Metaphysician Undercover
I believe I asked the OP what they meant by existence and don't believe I've gotten an answer.
Quoting Metaphysician Undercover
Ok then you are in agreement with my point. I only speak of mathematical existence. But why the square root of 2? How about the number 3? That has no more claim on existence than sqrt(2). That's why the OP is confused. They think 3 exists but not sqrt(2). I think they both have mathematical existence, and as far as "existence existence," I'd like to see a coherent argument one way or the other. I take no position at all. Clearly numbers don't have the same claim to existence as rocks or fish.
Quoting Metaphysician Undercover
This thread hasn't even begun to touch on the subtleties of that subject. I've seen no decent arguments one way or the other. And if that's what the OP really cared about, they'd have asked if 3 exists. Once you bring in sqrt(2) you are talking about mathematical existence.
Quoting Metaphysician Undercover
It's "above their pay grade" as Obama would have said. So make an argument. Do you think 3 exists? Of course the positive integers have a pretty good claim on existence because we can so easily instantiate the smaller ones. So how about [math]2^{\text{googolplex}}[/math]? That's a finite positive integer that could in theory be instantiated with rocks or atoms, but there aren't that many distinct physical objects in the multiverse. So make an argument, say something interesting about this. Forget sqrt(2). Do you think that extremely large finite positive integers exist?
https://en.wikipedia.org/wiki/Googolplex
Professionals, yes. Non-professionals (forum participants, for example) a lot weaker.
Quoting sime
What's a "theorem prover"? Computer program? A tutor?
Mostly the ones here. The famous philosophers of math know a lot of math, like Putnam and Quine and so forth. I may have overstepped a bit. Having just stuck my contrary toe in a political thread I think I've had enough for tonight. You know when you say something contrary to the dominant narrative, people don't just respond rationally. They come with insults and non sequiturs. It's like the 1950's all over again and I just came out as a Commie.
More generally, it is a dependently typed logical programming language, with clause resolution and other rules of logical inference, together with SAT solvers, methods of analytic tableaux and heuristics for automated or interactive theorem proving.
Lean with Mathlib, to my understanding is a state of the art approach to logic and mathematics programming that embodies the above principles , an approach which together with advances in deep learning theorem proving will undoubtedly revolutionise mathematics education , automation and research. Remarkably, the mathematics library of lean now codes the proofs of a lot of undergrad level mathematics.
https://leanprover.github.io/live/latest/
This is doubtful, and seems to contradict the rest of your statement. If we're talking mathematical existence, I do not think that natural numbers have more claim to existence than irrational numbers. In fact, I think the opposite is more likely the case. "The natural numbers" were in use prior to the Pythagoreans who are supposed to have demonstrated the "existence" of irrationals. So it was only by the work of the Pythagoreans that "existence" was assigned to numbers, and existence was stipulated in order to provide reality for the irrationals. Prior to this the natural numbers were in use without any assumptions that numbers exist, so the naturals are lacking in the claim of existence. There is no need for them to be stipulated as existing.
If we're talking "existence" in the philosophical sense, we'd have to first agree as to what existence means before we might judge whether one type of number has more of a claim to existence than another. If we do not find a definition of "existence" which allows that numbers exist in the first place, then the suggestion is meaningless.
Quoting fishfry
Why not? I don't see the reason for approaching the question with such a bias. It will only make a true answer more difficult to find. Plato demonstrated the pitfalls of this bias thousands of years ago. It is a mistake to assign a higher degree of being to something apprehended through the senses over something apprehended directly with the mind.
Quoting fishfry
This again shows some sort of bias toward natural numbers over irrational numbers. If "3" represents a number, and "sqrt(2)" represents a number, then why assume that the question is better asked of "3" than of "sqrt(2)"? That's just admitting that "3" is in some sense a better representation of a number than "sqrt(2)", and in doing this you undermine the concept of "mathematical existence". If some numbers have a better, or more valid "mathematical existence" than others, then there must be ambiguity within the concept which could allow for equivocation.
Quoting fishfry
As I said, there's really no point in making an argument as to the existence or nonexistence of something unless we have a workable definition of "existence". That's why the thread really won't get anywhere because all the members in this forum have wide ranging biases about what constitutes "existence", and a relatively small number of them have any formal training in this subject, so it will end up with people arguing to support their own prejudices.
I would be inclined to define "existing" as having either temporal or spatial extension.
Thanks for the information. From my perspective the project sounds dreadful, but for coming generations it may become standard. It's given me a moment to reflect on a theorem I am putting together and proving at present. The intuition and imagination I have used to both design the theorem, then prove it, presumably in the future could be generated in some computerized fashion. But of course I don't see how. :chin:
As for simply formatting the proof of a theorem in a computer language I might be able to do that in a version of BASIC. But why would I?
I can accept that the square root operation is closed over the 'reals', but that doesn't mean it's closed over the real numbers. I don't deny the value of irrationals or question the centuries of work on calculus, I'm just questioning whether we actually know that irrationals are numbers.
Quoting Metaphysician Undercover
A few have pointed to my original question so let me try to clarify. My view of existence encompasses mathematical existence. While I have doubts about the existence of irrational numbers, I don't doubt the value of our descriptions of irrational 'numbers' (e.g. Dedekind Cuts, Cauchy Sequences, infinite series, and so on). I know this seems like a contradictory statement since conventionally they're all equivalent. But I see an infinite series as an algorithm (described completely with finite characters) which if executed to completion would output an irrational number (described completely with infinitely many characters). Since that output could never be generated, the output (the number) cannot exist but the algorithm certainly can, for example on my laptop. This is why I think the number 3 can exist but not the 'number' sqrt(2). We never actually work with irrational 'numbers', we only work with their algorithms or rational number approximations. So why do we even need to assume that irrational 'numbers' exist? Why not assume that irrationals are the algorithms that we actually work with?
I think one issue is that some non-computable irrational 'numbers' don't have algorithms...
'The reals' means 'the real numbers'.
We construct the set of real numbers. It doesn't make sense to debate whether a real number is a number. Mathematics doesn't have a universal definition of 'number'. Mathematics doesn't really involve questions of what is or is not a number. Instead there are many different number systems. Each number system has its carrier set. And we may ask whether certain objects or are or not in the carrier set of a given number system. The carrier set of the real number system is the set of real numbers (sometimes just called 'the reals). Every real number is a number in the sense that it is in the carrier set of the real numbers system.
A version of this information was provided to you earlier in this thread.
Equivalence classes of Cauchy sequences. This has been mentioned to you previously in this thread
Quoting Ryan O'Connor
(1) We don't assume they exist. We prove they exist.
(2) I imagine that instead of constructing the real numbers, we could instead take the carrier set to be the set of algorithms. But limiting to only computable reals makes calculus a lot more complicated. Since calculus works with the irrationals included, and since it is more complicated for the calculus to regard the irrationals as algorithms and not as real numbers, we must weigh the advantages and disadvantages of both approaches. In particular, if we ask 'What is the length of the diagonal of a unit square?' we may answer 'sqrt(2)' rather than have to say 'Well, I can't tell you except there is an algorithm that computes successive approximations. Tell me what degree of accuracy would your like your approximation to be, and I'll tell you the answer to that degree of accuracy.' And I would say, 'Thanks, but I got a more succinct answer from the mathematician who said it is sqrt(2).'
And, by the way, if you show me an algorithm, then I may ask, 'What does it compute?' And how would you say what it computes without already having in mind that it computes approximations of ... wait for it ... sqrt(2). So, to get me sold on your algorithm, you would already have to presuppose that there is a thing that it approximates - and that thing is ... wait for it .. sqrt(2).
(3) There are proposed systems in constructivism, computationalism, and predicativism that may very well satisfy your desiderata. You only need to first inform yourself of the basics of the subject.
Most of this also has been mentioned to you previously in this thread.
Algorithms that execute to completion do so in a finite number of steps. As far as I know, what you may have in mind is not an algorithm but rather it is a supertask. I am not well informed about supertasks, so I don't know whether there is a rigorous mathematical definition of the notion or whether the notion is purely philosophical.
You've got to be kidding. Think about this statement for, say, five seconds. :roll:
Have I made a boo boo, a big one at that? Expand and explain.
I love Scott Aaronson's comments on this:
Quoting PHYS771 Lecture 9: Quantum - Scott Aaronson
Isn't this the difference between an object and a process? We'd say "3" represents a static object, a number, but "sqrt(2)" represents an operation. What would you say about "2+1"? Doesn't that represent an operation rather than an object? The difference between "2+1", and "sqrt(2)", is that the one process adequately resolves to an object. The question I see is what does it mean for a process to resolve to an object, and why does this make the process somehow more valid as a process? So, we say "2+1=3", and we are stipulating an equivalence between the process and the object. But we cannot produce the precise object which "sqrt(2)" is equivalent to.
What validates, or grounds numbers definitionally, is quantitative value "2+1" is equivalent to a definite quantitative value represented by "3". Having a definite quantitative value is what makes the number an object. If we do as you propose, and allow processes which do not have a definite quantitative value, to be "worked with", then we allow indefiniteness into our solutions. The solutions will contain indefinite quantitative values. This is counterproductive because the goal when using mathematics is to measure things, which is to assign to them definite quantitative values.
sqrt is an operation. sqrt(2) is the object that is the result of the operation applied to the object 2. sqrt is the operation, and 2 is the argument to which the operation is applied.
+ is an operation. 2+1 is the object that is the result of the operation applied to the objects 2 and 1. + is the operation and 2 and 1 are the arguments to which the operation is applied.
Quoting Metaphysician Undercover
Operations are functions. A function is a certain kind of ordered pair. The result of an operation is simply the unique second coordinate for the ordered pair whose first coordinate is the argument.
Quoting Metaphysician Undercover
We cannot finitely list the decimal expansion of sqrt(2). But sqrt(2) is a particular object. Also, what is important in this regard is not some object that the sqrt(2) is "equivalent to" (with whatever equivalence relation might be in mind) but rather with sqrt(2) itself.
Quoting Metaphysician Undercover
Your use-mention is inconsistent there. Yes, '3' represents a value. But so also does '2+1'.
3 and 2+1 are values. They are the same value. Exactly the same. 3 = 2+1. 3 equals 2+1. 'equals' is another word for 'identical with'.
'3' and '2+1' are names that represent values. '3' and '2+1' are not equal. They are different names. But they name the same object. They are two different names for the same object.
Look up the subject of 'use-mention'.
Quoting Metaphysician Undercover
A number is an object. If it's not an object, then what is it? If it is something that, according to you, might or might not be an object, then what is that something to begin with if not an object? How can we refer to something that is not an object?
Quoting Metaphysician Undercover
Mathematics may be used for purposes other than measuring things.
Quoting Metaphysician Undercover
We do mention a definite value when we mention sqrt(2). It doesn't have a finite decimal expansion, but it is a definite value.
No, if "sqrt" represents an operation, then "sqrt(2)" represents that operation with a qualifier "(2)".
Quoting GrandMinnow
This is incorrect, because "+" represents an operation. So there are two distinct values, "2", and "1" represented, in "2+1", along with the operation represented by "+".
Quoting GrandMinnow
No it isn't, that's a false assumption which I've discussed on many threads. You and I are equal, as human beings, but we are in no way identical with each other. "Equals" is clearly not another word for identical with.
Quoting GrandMinnow
Let's start with numerals, which are symbols. Do you agree that a symbol has a meaning, which is not necessarily an object? So there is no need to assume that "2" or "3" represent objects. We'd have to look at how the symbols were being used, the context, to determine whether they represent objects or not. When I say that there are 6 chairs in my dining room, "6" refers to a number, but this is the number of chairs; the chairs are the objects and the number 6 is a predication. The number is not an object, it is something I am saying about the chairs in my dining room, just like when I say "the sky is blue", blue is not an object.
I am telling you the terminology and framework of ordinary axiomatic mathematics. 'qualifier' is not the terminology used. Of course, you may set up your own terminology and framework, but the chance that it will make sense for ordinary axiomatic mathematics is slim since you don't know anything about ordinary axiomatic mathematics.
Quoting Metaphysician Undercover
(1) You are still making your use-mention mistake. Yes, '+' represents an operation and '2+1' is a representation of a value, but '2' and '1' are not values, they are representations of values.
Get it straight: The name of the object has quote marks and is not the same as the object. In cases where quote marks are used, we have a term that represents a value. When we mention the value itself, we don't use quote marks.
(2) As I explained, and as you ignored, + is the operation; 2 and 1 are the arguments; and 2+1 is the value of the function for those arguments. Yes, the operation is represented by '+; and the arguments are represented by '2' and '1'. And the value of the operation is represented by '2+1'. And that value is 2+1. Again, you need to learn basic use-mention.
Quoting Metaphysician Undercover
Your personal, confused, incoherent and uninformed views in that thread were demolished and shown to be confused, incoherent and uninformed.
Quoting Metaphysician Undercover
You are conflating the meaning of the world 'equal' in various other topics, such equality of rights in the law, with the more exact and specific meaning in mathematics. Human equality means that your rights are identical with my rights, and yes, that does not entail that you and I are identical. But in mathematics, which is ordinarily extensional, the equal sign '=' stands for identity. Again, you can use terminology in your own way for your own notions, but the chance that it will make sense for ordinary axiomatic mathematics is slim since you don't know anything about ordinary axiomatic mathematics.
Quoting Metaphysician Undercover
Ordinary axiomatic mathematics is extensional. Each n-place operation symbol refers to a function on the domain of the interpretation, and functions are objects. The function might or might not be an object that is a member of the domain, but it is an object in the power set of the Cartesian product on the domain.
Quoting Metaphysician Undercover
Not in ordinary mathematics where the numerals represent natural numbers. Or, in greater generality, any constant symbols, such as '1' and '2' are either primitive or defined, and in either case they represent members of the domain of interpretation.
As for defined symbols ('1' and '2' are more often defined rather than primitive), it is true that we cannot define the symbol without first proving the existence/uniqueness theorem. That is, we prove that there exists a unique object having a given property, then we define the symbol as standing for that unique object.
So 1 is the unique object that it is equal to the successor of 0. And 2 is the unique object that is equal to the successor of 1.
Quoting Metaphysician Undercover
Ordinary mathematics does not view numbers as predictions.
Quoting Metaphysician Undercover
It is the mathematical object that is the number of chairs, and is the number musicians on the album 'Buhaina's Delight', and is the value of the addition function for the arguments 4 and 2 ...
Quoting Metaphysician Undercover
No what you are saying about the chairs is that the set of them has cardinality 6. You're not saying the chairs or the set of them has 6 or is 6. You are saying that the number of them is 6. 6 is not a property of chairs. Rather, a property of a (set of chairs) is that its cardinality is 6.
Quoting Metaphysician Undercover
'blue' is an adjective, which is a certain kind of word, which is a linguistic object. It does not alone stand for an object. When we say the sky is blue, we say that the sky has the property of blueness. The sky is the object and blueness is the property. When we say that 2 is even, we mean that 2 has the property of being even. 2 is the object, and evenness is the property. When we say that 2+1 equals 3, we mean that that 2+1 and 3 as an ordered pair <2+1 3> are in the reflexive relation of equality.
'2' is a constant (a kind of "noun") in mathematics.
So you are comparing apples and oranges when you compare the noun '2' with the adjective 'blue'.
The correct analogy is:
With 'the sky is blue' we have the noun 'the sky' that stands for the sky, and 'blue' is an adjective, and 'is blue' stands for the property that holds for the sky.
With 'My dining room has six chairs', we have the noun 'my dining room' that stands for your dining room, and 'has six chair's' stands for the property that holds for your dining room.
With '2+1 = 3', we have the nouns '2+1' and '3', and '=' stands for the 2-place property of equality and indicates in the equation that the property of equality holds for the pair <2+1 3>.
/
I suppose it is an absolute given that you will never look at even page one of a book on the subject of mathematical foundations.
Have we really proved the existence of irrational numbers? That's the name of this thread.
Is this a question people are debating for five pages here??? :snicker:
The issue I am looking at, is not how things are viewed by "ordinary mathematics", it is what is meant by the mathematical concepts. If we adhere to how things are viewed by mathematics, as if this is necessarily the correct view of things, as you seem inclined toward, then the discussion is pointless. You'll keeping insisting that I am not seeing things the way that mathematics sees things therefore I am necessarily wrong. I've been exposed to enough of this already, and see no point to it.
Quoting GrandMinnow
OK, I'll try to adhere to this formality. It is not the convention I am used to, but I'll try it.
Quoting GrandMinnow
No, "2" and "1" signify values. Or do they sometime signify values and other times signify arguments? If so how do we avoid equivocation? Anyway, I see no way that a function, which is a process, could have a value. That's like saying that + has a value.
Quoting GrandMinnow
Look, "2+1" means to put two together with one, and 2+1 equals "6-3", which means to take three away from six. You cannot try to tell me that to take three away from six is the exact same thing as putting two together with one, or I'll tell you to go back to elementary school and learn fundamental arithmetic properly.. Your claim is clearly false, equals does not mean identical to, or the same as, in mathematics.
Quoting GrandMinnow
I know that a function is a process. And I also know that the concept of process is incompatible with the concept of object. The two are distinct categories. Therefore it is fundamentally incorrect, by way of category mistake, to say that a function is an object.
Quoting GrandMinnow
I really don't know what you could possibly mean by this. The number of chairs is referred to by "6". There is a specific quantity and that quantity is what is referred to with "6". I don't see where you get the idea of an object from here. There are six objects which form a group. The group is not itself the object being referred to, because the six are the objects. Therefore the quantity must be something other than an object or else we'd have seven, the six chairs plus the number as an object, which would make seven.
Quoting GrandMinnow
More correctly, the quantity is six. You assume that this quantity is an object, a number, which is something other than the quantity. But that's not the case, the quantity is six, the number is six, and both "quantity" and "number" mean the same thing here. Why assume that there is something other than a quantity, an object called 6? That makes no sense, where and how are we going to find this object?.
Quoting GrandMinnow
You're just making an imaginary thing, like God, and handing a property, "even " to that thing, like someone might say God is omniscient. When we use the symbol "2", we use it to refer to a group of two things. like chairs or something. When we say that there is an even number of chairs, this means that the group of chairs can be divided into two groups. But the group clearly cannot be divided by three. If you say that "2" refers to some imaginary object, then you can assign to it whatever properties you like. You could make it infinitely divisible if you want. But if you're not adhering to any principles of reality, this is just useless nonsense. What good is assuming an imaginary object which you can attribute any properties to with total disregard for reality?
Quoting GrandMinnow
Now you're doing the same thing again, you're claiming two nouns, 2 and 1, are one noun signified as "2+1". Clearly this cannot be the case without equivocation. Either 2 is a noun and it refers to an object, or it is not. But you can't have it sometimes being a noun, and sometimes not without equivocation.
.
This is absurd, but then you do see things from an unusual vantage point, inaccessible to many.
How does "sqrt" change from signifying an operation, to signifying an object, simply by qualifying (or quantifying, if you prefer) it with a (2), without equivocation?
It's the same sort of problem which GM has with 2+1. Each of the symbols "2", and "1" refer to a distinct object. But GM claims that in the context of "2+1" there is only one object referred, and "2" and "1" do not each refer to a distinct object. How is this not equivocation?
What is meant by whom? What is meant by mathematicians is not what is meant by you.
Quoting Metaphysician Undercover
I haven't said here what is necessarily correct. There are formulations of mathematics - classical, constructivist, intuitionist, finitist - and then there are philosophical discussions about them. A formulation is not the same as the philosophical discussions about it. You are free to present a formulation (or at least an outline) of mathematics and then say philosophically what you mean by it. But lacking a formulation, I would take the context of a discussion of mathematics to be ordinary mathematics and not your unannounced alternative formulation.
Quoting Metaphysician Undercover
You're mixed up as to what I've said. Yes, I agree, and never disagreed, that '2' and '1' denote values.
Quoting Metaphysician Undercover
This is another case in which your nearly total ignorance of mathematics results in your confusions.
Those numbers are both values themselves and also arguments applied to a function that in turn has a value for those arguments.
Quoting Metaphysician Undercover
You don't see a lot of things, because you refuse to even look at an introductory book on the subject.
Do you know what a function is? (In mathematics for grownups, there is a more rigorous definition of a function than 'process'.) Do you know how mathematics develops the subject of functions?
Meanwhile, I didn't say the function has a value. I said the function applied to arguments has a value.
Quoting Metaphysician Undercover
It could not be more clear. 6 is the number of chairs in your dining room, and 6 is the number of musicians on the album 'Buhaina's Delight', and 6 is the number that is the value of the addition function for the arguments 4 and 2.
Quoting Metaphysician Undercover
We've gone over this multiple times already. 2+1 is the result of adding 2 and 1. 6-3 is the result of subtracting 3 from 6. The value (result) of adding 2 and 1 is the same exact value (result) as subtracting 3 from 6.
One more try to get through to you. What you get when add 2 and 1 is the same exact thing as what you get when you subtract 3 from 6.
2+1 is not a process. 2+1 is a number. It is the exact same number as 3, and the exact same number as 6-3. And '2+1' is a name of the number 2+1, and it is a name of the number 3, and it is a name of the number 6-3.
Meanwhile, a process in mathematics can be described as a certain kind of sequence of steps. Yes, the sequence of steps in adding 1 and 2 is different from the sequence of steps in subtracting 3 from 6. But the last entry in the sequence - the result - is the same. '2+1' and '6-3' are not names of processes; they are names of a number.
2+1 = 6-3. That is, 2+1 is 6-3.
'2+1' does not= '6-3'. But they are two different names of the same number.
Please do not misrepresent what I said. I said explicitly that '1' and '2' do each refer to a distinct object. My remarks should not be victim to misrepresentation by you.
The chairs are objects. And the mathematical object that is the number of chairs is the number 6. And the set of chairs also is an object, and it has cardinality 6.
Quoting Metaphysician Undercover
Mathematical objects and mathematical properties are abstractions. They are not theological claims like the saying that there are gods. Also, properties like 'blueness' and 'evenness' are abstractions. You are free to reject that there are abstractions, but I use abstractions as basic in thought and reasoning.
Quoting Metaphysician Undercover
We (not you though) understand the number 2 as not just the number of chairs or the number of any particular set of objects but also as a number onto itself.
Quoting Metaphysician Undercover
We prove from axioms that there is a unique object having a certain property, and we name it '6'. Why would we want to do that? Because it greatly facilities mathematics. I may refer to 6 itself rather than have to say "the number of chairs in Metaphysician Undercover's dining room".
In mathematics especially, names refer to things. The name '6' refers to something. It refers to a number. It refers to SSSSSS0. Mathematicians are undaunted by the fact that the thing named is an abstract object, not a concrete one.
Quoting Metaphysician Undercover
We don't find it by a physical search. We find it by a mathematical activity in abstract reasoning.
Quoting Metaphysician Undercover
You are thoroughly mixed up, not just about mathematics, but about plain and simple things I've just posted.
I did not claim that 2 and 1 are nouns. I did not claim that 2 and 1 are signified by '2+1'.
2 is a number. 1 is a number.
'2' is a noun. '1' is a noun.
2+1 is a number.
'2+1' is a noun.
'2+1' does not denote 1.
'2+1' does not denote 2.
'2+1' denotes 2+1, and '2+1' denotes 6-3, and '2+1' denotes 3.
Learn use-mention, no matter what your philosophy is.
I have no formulation, and no desire to present one. The op asks if something has been proved, therefore we are invited to be critical of formulations which claim to prove that. And there is no need to offer an alternative formulation to point out problems with an existing one.
Quoting GrandMinnow
As I said, you equivocate:
Quoting GrandMinnow
Quoting GrandMinnow
Which is the case, do "1" and "2' each signify distinct numbers, or does "2+1" signify a number? You can't have it both ways because that's contradiction. But I've been trying to go easy on you and settle for the lesser charge of equivocation. If "1" and "2" each signify distinct numbers, then there are two distinct numbers represented by "2+1", so it is contradictory to say that "2+1" represents one number, because there are two numbers represented here.
Quoting GrandMinnow
That the same quantitative value is predicated of the chairs in my dining room, and the musicians on that album, doesn't make that predicate into an object.
Quoting GrandMinnow
Sure, the resulting value of each is 3, but that's not the issue. Your claim is that "=" signifies identical to. 6-3 equals 2+1, but what is signified by "6-3" is not the same as what is signified by "2+1". You agree about this. Therefore it should be very clear to you that "=" does not signify identical to.
If you say that they have the exact same value, then we are using "equal" in the way I suggested. You and I have the exact same value in the legal system, therefore, as human beings we are equal, just like 6-3 has the same value as 2+1 in the mathematical system, but in neither case are the two equal things identical.
Quoting GrandMinnow
OK, let's go with this then. If you do something, and derive a result, this is necessarily a process. So you are very clearly talking about two distinct processes represented by "2+1", and "6-3". Two distinct and different processes can have the same end result, and so those processes can be said to be equal. Does this imply, that in mathematics you judge a process according to the end result? If so, then how do you propose to judge an infinite process, which is incapable of producing an end result, like those referred to in the op?
Quoting GrandMinnow
I don't see the difference. You are invoking an imaginary object represented by "2", just like a theologian might invoke an imaginary object represented by "God". Each of you will try to justify the claimed existence of your imaginary object. You are not showing the necessity required, which the theologians show, so you are not doing a very good job of it.
Quoting GrandMinnow
This is so contradictory to what you've been arguing. You've been arguing that 4+2 is 6, and 10-4 is 6, and that there is potentially an infinite number of different things which are 6. And it isn't just a matter of different names for the same thing, because "4" and "2" must each name a unique thing, so it's impossible that "4+2" is just a different name for "6". How can you now claim to be able to prove that there is a unique object named "6", when you've been arguing that all these different things are the same as 6, by virtue of equality. You are getting yourself so tangled up in a web of deceit, that's it's actually becoming ridiculous.
But you don't know anything about the formulation of classical mathematics.
Quoting Metaphysician Undercover
But your account of the meaning of mathematics is not compatible with the ordinary formulation of mathematics, so if your account were to have any consequence, then it would need to refer to some other formulation.
Quoting Metaphysician Undercover
You misrepresented me when you wrote that I said that '1' and '2' do not refer to distinct objects. I said that they do refer to distinct objects. I'll say again, please do not misrepresent what I've said. I don't expect you to have the intellectual honesty to retract your claim about what I said, but I ask that at least you don't do it again.
And I have not equivocated.
Quoting Metaphysician Undercover
A contradiction is a statement and its negation. You have not shown any contradiction in what I said. The fact that '1', '2' and '2+1' each denote distinct numbers is not a contradiction.
Quoting Metaphysician Undercover
'2+1' has '1' and '2' as parts in the term. The term '1' denotes the number 1, the term '2' denotes the number 2, and the term '2+1' denotes the number 2+1, which is 3, which itself is also denoted by the term '3'. That is not a contradiction.
Quoting Metaphysician Undercover
I didn't say 6 is an object on account of 6 being the number of chairs or musicians. That was not my argument at all. You're confused about the structure of this discussion itself.
Quoting Metaphysician Undercover
No I do not. You are completely confused. (1) I would put aside 'signify' since you might have some special sense for it. I have mentioned 'refer to' or 'denote'. (2) And I do say that '6-3' and '2+1' denote or refer to the same number. I've said that about a dozen times already and you still don't recognize that that is what I say.
Quoting Metaphysician Undercover
You are confused even about what you are saying yourself!
Saying that they have the exact same value is to say that they are identical.
Again '2+1' and '3' are not identical. They are different terms. But 2+1 and 6-3 are identical. They are the exact same number and they are the exact same referent of the terms '2+1', '6-3', and '3'.
Quoting Metaphysician Undercover
Again, 2+1 and 6-3 are not processes. They are numbers. They are the same number. They are the number 3. You skipped entirely my explanation that a process may be expressed as a certain kind of sequence of steps. The sequence of the steps is not the same as the last step itself, which is the result of the process. 2+1 and 6-3 denote the result. It is the same result.
Quoting Metaphysician Undercover
You have nearly everything backwards. A process is a sequence of steps. If one sequence of steps is different from another sequence of steps, then those are different processes even if their results are the same. Sequences are identical to each other if and only if every step in the sequences is the same. The sequence of steps in adding 1 to 2 is different from the sequence of steps in subtracting 3 from 6. But the results are the same for both. '1+2' and '6-3' do not denote processes; they denote numbers.
Quoting Metaphysician Undercover
No. Decidedly not. I just explained above, now for the second time.
Quoting Metaphysician Undercover
An infinite process that does not terminate is one that computes a recursively enumerable function. As to the notion of an infinite process that does terminate, I am not aware that this is a rigorous mathematical notion, though we are familiar with the philosophical notion of a supertask. I explained that also in a previous post to another poster.
Quoting Metaphysician Undercover
One difference is that mathematics works with algorithmically checkable axioms and rules of proof. Also, mathematics itself does not opine as to the ontological status of abstract objects but instead recognizes that we carry out mathematics with abstract reasoning regarding abstract objects - whatever we take such abstractions to be. Also, I tend to think that, strictly speaking, mathematicians could dispense even with the notion of objects, though it would make mathematical discussion clumsy.
Also, you have not answered how other abstractions could be acceptable, such as blueness or evenness or the state of happiness, etc.
Quoting Metaphysician Undercover
No they are not different things. '4+2' and '10-4' and '6' are different names for the same thing.
You just don't get the difference between names and the things named. That is a critical failure of understanding. You can't get past your confusions until you grasp the very simple distinction between a name and the thing that is named.
When we say that n is even we mean that n is a natural number such there exists a natural number k such that n = 2k. But, yes, that does imply that a set with even cardinality has a partition of 2 sets with equal cardinality.
But the property, in everyday discussion (not even necessarily mathematics), of evenness of numbers is itself an abstraction. Properties are not things that are physical objects. Yes, physical objects have certain properties. But the properties themselves are abstractions and are not physical objects. You can't point to the property of blueness as a physical object. You can only point to certain blue things as having that property, but then you are pointing to those particular objects and not to the property itself.
'2+1' and '6-3' are different terms, so, even though extensionally they name the same number, the terms themselves have different intensional meanings. But ordinary mathematics works extensionally.
There have been proposals for formulating intensional mathematics. But I don't know whether your remarks would be pertinent to such formulations. Meanwhile, your remarks are completely off-base when it comes to mathematics as it is ordinarily studied.
I don't understand intensional versus extensional with respect to math (or anything else for that matter) but I've been trying to explain to @Metaphysician Undercover for two years that 2 + 2 and 4 refer to the exact same mathematical object. Without success, of course, but never mind that.
Can you briefly explain to me what that means? How you can use the concepts of intensional and extensional to make the point that 2 + 2 and 4 are two names for the exact same thing?
Quoting GrandMinnow
Ah maybe this is a clue. @Meta keeps saying to me (at least before we stopped discussing the issue) that 2 + 2 refers to the process of adding two things together; and 4 refers to a single thing. Therefore they don't have the same meaning. So that must be "intensional" as you say.
I've heard these two terms for years without understanding. And also intentionality with a 't', that's Searle's point that the Chinese room doesn't have intentionality, or "aboutness." I gather this is an entirely different concept than intensionality with an s?
https://plato.stanford.edu/entries/logic-intensional/
And a classic brief introduction to the subject is:
Introduction To Mathematical Logic, pages 1-9, by Alonzo Church
/
Most simply, ordinary mathematics is extensional, so substitutability of terms holds. That is, the principle of "substitute equals for equals" holds. That is, roughly put, for any terms T and S, and formula F, from T=S we may infer F[x|S] from F[x|T]. For example:
from
4 is even
we may infer
2+2 is even
But consider intensional contexts, such as this:
4 is even
and
4 = (((182/2)-1)/2)-66
and
Bob knows that 4 is even
therefore
Bob knows that (((182/2)-1)/2)-66 is even
The premises are true, but the conclusion is false if Bob doesn't know that 4 = (((182/2)-1)/2)-66.
Putting in 'knows' throws us into an intensional context where substitutability may fail.
/
I am not well versed beyond such basics as that, so for more on the subject I recommend the Stanford article and the passages in the Church book.
As I said, if this point is of relevance then the discussion is pointless.
Quoting GrandMinnow
I can't believe that you do not understand the contradiction. Let' take the expression "2+1". Do the symbols "2" and "1" refer to distinct objects. If so, then there are two objects referred to by "2+1", and it is impossible, by way of contradiction, that "2+1" refers to only one object. Do you understand this?
Quoting GrandMinnow
This is false. A process may be described as a sequence of steps. The sequence of steps is not the process, it is the description of the process. That this is an important distinction is evident from the fact that the very same process may be described in different ways, different steps, depending on how the process is broken down into steps. That's why different people can use different methods to resolve the same mathematical equation.
Quoting GrandMinnow
I don't see any need to consider an abstraction as an object. Abstraction is simply how we interpret things, and there is no need to assume objects of meaning as a fundamental part of the interpretive process.
Quoting GrandMinnow
You agreed that they are different things which have the same result, or the same value. If they are different things, then having the same result, or the same value does not justify calling them the same thing.
Here's what you said:
Quoting GrandMinnow
Are you taking that back now? Why do you want to say that adding 2 to 1 is the exact same thing as taking 3 from 6, instead of what you already agreed, that they are distinct things with the same end result? You know the truth in this matter, why try to deny it?
Quoting GrandMinnow
Then why treat properties as if they are any sort of object? You treat numbers as if they are some sort of objects, when really they are a property of the thing which is numbered.
Quoting GrandMinnow
I've argued elsewhere that the axiom of extensionality is a falsity. It is the means by which you say that two equal things are the same thing, which is obviously false. So it's not necessarily that I do not understand extensionality, but I apprehend it as based in false premises.
Quoting GrandMinnow
In other words, equal things may be considered as the same thing. And that's clearly false.
You are repeating yourself without arguing specifically to the point I made. You argue by mere assertion. My point stands.
Quoting Metaphysician Undercover
Given your pattern of ignorance and confusion I can believe that you don't understand that you haven't shown a contradiction in my remarks. A contradiction implies both a statement and its negation. You have not shown how anything I've said implies both a statement and its negation.
Quoting Metaphysician Undercover
You skipped the answer I gave to that already.
Quoting Metaphysician Undercover
Since you have not given a mathematical definition of 'process', I am taking 'process in the sense of 'algorithm' or 'effective procedure'.
Mathematics addresses your point by recognizing that different processes may compute the same function.
Quoting Metaphysician Undercover
There are two different senses, e.g. (1) "I think by means of abstraction" and (2) "My thinking has resulted in arriving at the abstract concept of blueness."
Quoting Metaphysician Undercover
No, I definitely did not agree with that.
Indeed, I have explained for you that '4+2', '10-4' and '6' are not things that have results. They are NAMES, not numbers and not processes. Then, 4+2, 10-4, and 3 are the same number. What I do recognize is that, e.g., the process of adding 2 to 4 is different from the process of subtracting 4 from 10.
For about the fifth time:
4+2 is a number.
'4+2' is not a number; it is the name of a number.
If you simply refuse to understand the use/mention distinction, then you are doomed to continue in confusion.
Quoting Metaphysician Undercover
Please stop ignoring the distinctions I have said multiple times already.
I said the RESULT of adding 2 to 1 is the same as the RESULT of subtracting 3 from 6. I do not say that the processes are the same.
Quoting Metaphysician Undercover
The abstraction called 'blueness' is an abstract object. 'Blueness' can be the subject of a sentence in the manner of a subject that refers to an object. For example, the previous sentence itself is one in which 'blueness' is the subject. And 'blueness' refers to the abstraction blueness.
Quoting Metaphysician Undercover
The extensional nature of mathematics does not depend on the axiom of extensionality. They are different things. You just jump to the conclusion that because 'extension' is found in describing both things that one must depend on the other. You don't know what you're talking about.
Quoting Metaphysician Undercover
It's true by definition. You are welcome to define terminology in your own way, but I'm telling you in the meanwhile how the terminology is defined in mathematics.
Thank you for the references. Your post actually gave me a glimmer of understanding as to what @Metaphysician Undercover is talking about. And for that matter, what I'm talking about.
The denotation of 'the father of Jane Fonda and Peter Fonda' is Henry Fonda. The denotation is not Jane Fonda nor Peter Fonda nor the sibling relation nor the process of fatherhood.
'the father of Jane Fonda and Peter Fonda' refers to one specific person, and that person is Henry Fonda.
The denotation of '2+1' is 3. The denotation is not 2 nor 1 nor the process of adding 1 to 2.
Another way of saying '2+1' is 'the sum of 2 and 1'.
And the sum of 2 and 1 is 3. It is not 2 nor 1 nor the process of adding 1 to 2. Rather the sum is the RESULT of the process of adding 1 to 2. The sum is not a process; it is a number.
Henry Fonda = the father of Jane Fonda and Peter Fonda. Henry Fonda IS the father of Jane Fonda and Peter Fonda.
'Henry Fonda' and 'the father of Jane Fonda and Peter Fonda' both refer to one identical person. Henry Fonda is identical with the father of Jane Fonda and Peter Fonda. There are not two different people - Henry Fonda vs. the father of Jane Fonda and Peter Fonda. There is one identical person with two ways of referring to him.
There are not two different numbers 2+1 vs. 3. There is one identical number with two ways of referring to it.
It seems that @Metaphysician Undercover is mistaking the symbols for the entity they refer to.
In general, "2" denotes a number, and "1" denotes a number, but in this particular circumstance, "2" does not denote a number, and "1" does not denote a number. Therefore you equivocate.
I did not say there is a circumstance in which '2' and '1' do not denote numbers.
'2+1' is a compound term made from the constants '2' and '1' and the operation symbol '+'.
'2' denotes a number. '1' denotes a number. '+' denotes an operation. '2+1' denotes the result of the operation + applied to the numbers 2 and 1. That result is a number. Therefore, '2+1' denotes a number.
Ok, we we have the numbers 2 and 1 denoted, and the operation + is denoted. Where is the result of the operation denoted? It seems to me like you're jumping the gun. Jumping to the conclusion, assuming that the some result of the operation, 3, is already denoted when clearly it is not denoted
That's the reason why we need to denote = 3, if we want to denote some result, because "2+1" on its own does not say 3. Otherwise there would be absolutely no purpose to the "=" because everything which 2+1 equals would already be said simply by saying "2+1". Therefore "2+1" would denote an infinite number of things, and that would make interpretation impossible. Furthermore, equations would be absolutely useless because the right side would just be saying the exact same thing as the left side, along with all the infinite other things that are equal. What would be the point to an equation in which the right side represented the exact same thing as the left? You'd never solve any problems that way, because the problem would be solved prior to making the equation. If you didn't know that the two sides signified the exact same thing already (meaning the problem is solved) you could not employ the equals sign.
I see that you are confused about the most basic aspects of mathematics, language and reasoning. On certain points, your understanding is not even at the level of a six year old child. I'm offering you help here, though I doubt you'll take it in.
Quoting Metaphysician Undercover
You just now quoted me with the answer to that question:
'2+1' denotes the result of the operation
Quoting Metaphysician Undercover
In a rigorous context, things are denoted by the method of interpretation of a language. In the usual interpretation, by the recursive method, the denotation of '2+1' is determined from the denotations of '2', '1' and '+'.
Quoting Metaphysician Undercover
'2+1' does not have '3' in it, but '2+1' and '3' name the same object. To express that '2+1' and '3' name the same object we write:
2+1 = 3
Or, put another way, to assert that 2+1 equals 3 ('equals' also said as 'is identical with', also said as 'is the same object as') we write:
2+1 = 3
Quoting Metaphysician Undercover
The purpose of '2+1 = 3' is to assert that 2+1 equals 3.
Quoting Metaphysician Undercover
You got it exactly backwards. Our method does not lead to '2+1' denoting infinitely many things. '2+1' denotes exactly one thing. On the other hand, 2+1 is denoted infinitely many ways:
2+1 is denoted by '2+1'
2+1 is denoted by '3'
2+1 is denoted 'sqrt(9)'
2+1 is denoted by '((100-40)/3)-17'
etc.
Quoting Metaphysician Undercover
The method of interpretation in mathematics does quite fine, thank you (nothwithstanding wrinkles such as Lowenheim-Skolem).
Quoting Metaphysician Undercover
The equation is the statement that the left side stands for the exact same thing as the right side. That is very useful.
If we want to know how much a company did in sales, the accountant starts by seeing that the company got 500 dollars from Acme Corp., and 894 dollars from Babco Corp, and 202 dollars from Champco Corp. Then the accountant reports:
500+894+202 = 1596
It's useful to know that '500+894+202' names the same number as named by '1596'.
Quoting Metaphysician Undercover
No, usually the problem is to find out or prove that the left side and right side are equal or not equal. Or to find another term to more easily represent, say, the left side.
From my sales reports I have 500+894+202. Then I follow a procedure to eliminate '+' and arrive at just one numeral: '1596;. And I conclude: 500+894+202 = 1596. So I then see that '500+894+202' and '1596' name the same number.
Quoting Metaphysician Undercover
One wouldn't honestly claim to know that the equation is true until one worked it out that it is true. Or to find a right side without '+' in it, then first one might have to perform the addition on the left side. This doesn't vitiate anything I've said.
Quite a soap opera. Can't wait for the next episode. :lol:
If it makes sense, I'll take it. But so far all you've offered is inconsistency. Let me see if I can follow you.
Quoting GrandMinnow
I see the number 2 denoted, and the operation + denoted, and the number 1 denoted. So there is clearly an operation denoted. What denotes that the operation has a result?
Would you agree that a finite operation is distinct from an infinite operation, the one having a result, the other not? If this is the case then there is a need to distinguish between an operation with a result, and one without a result.
Quoting GrandMinnow
It appears to me, like 2+1 demotes exactly nothing then. You can say the same thing in an infinite number of different ways, but none of these ways refer to anything. Each expression simply say I am the same as the infinity of others. If one of them refers to anything real, then they all must refer to something real, and you have an infinity of equivocation, with an infinity of different things referred to by on signification. Even the numeral "3" must refer to the result of an operation, exactly as the others, so there is nothing to validate any object
So what makes 2+1 different from 3+1 then? Each can be said in an infinity of different ways, and there is nothing which is actually being referred to be either one. How can they differ?
Quoting GrandMinnow
This is incorrect. The accountant writes out 500+ 894+202=?, or x, or some other placeholder for the unknown, because the sum is unknown. But if it were like you say, that "500+ 894+202" already says 1596, then the accountant would not have to sum up the numbers, because the result of the operation would already be stated.
That's why your way of looking at things, if it were true, would render the equation completely unnecessary and redundant. By the time the left side was stated, (500+ 894+202) the right side would already be known, because you claim that the left side states the result of an operation. Clearly this is false, because equations contain unknowns, and this is how we solve problems, by carrying out the operations required to determine the unknowns. Obviously you are spinning a web of deceit.
Quoting GrandMinnow
How do you apprehend a need to work things out? If "2+1" says sqrt(9), how is there any need to work out any equivalencies?
Quoting Metaphysician Undercover
Again, you argue by mere assertion while evading the replies given to you. This is a paraphrase:
You: You are inconsistent.
Me: To be inconsistent is to claim or imply a contradiction, which is a statement and its negation. You have not shown that I've implied both a statement and its negation.
You: You are inconsistent.
Quoting Metaphysician Undercover
That is just so daft!
The term itself doesn't denote that it has a result.
'+' is an operation symbol. An operation is a function. The usage "result of an operation" is an informal way of referring to the value of the function for the arguments. Every function has a value for arguments in its domain. That is, every function has a result when applied to arguments in its domain.
Quoting Metaphysician Undercover
For a while, in order not to split hairs, I went along with your term 'process', even though you have not defined it. That was okay for a while, but I was concerned that it would cause confusion, since there are actually two different notions: (1) a function. (2) a procedure for determining the value of a function applied to an argument. (I did touch on this earlier.)
So I'm not going to go along with your undefined terminology 'process'. Instead I'll use 'operation' (meaning a function) and 'procedure' (meaning an algorithm).
As I touched on before, there are:
(1) procedures that terminate, (2) procedures that do not terminate, and (3) supertasks that are not finite but terminate. (1) and (2) are mathematical, and (3) is philosophical (or I am not versed in whatever mathematics there might be about it).
Quoting Metaphysician Undercover
Use-mention! Please, I've pointed out a dozen times: use-mention!
2+1 is a number. '2+1' is a term that denotes.
Do you even know what the use-mention distinction is? I doubt you care to know even the most basic considerations in the subject of mathematical language.
There are an infinite number of ways to refer to the number 3. That doesn't mean they don't refer! Your argument is so daft!
Quoting Metaphysician Undercover
I pointed out in my last post that you have this exactly backwards. And you just repeat yourself again.
Quoting Metaphysician Undercover
Are you serious? Are you trolling?
Quoting Metaphysician Undercover
No he doesn't. If he does, he's wasting precious billable seconds. Instead, he just goes ahead to add the numbers.
Quoting Metaphysician Undercover
No, the term '500+ 894+202' already denotes 1596. It's just that the accountant doesn't know that until he performs the addition. The term doesn't start denoting only upon the knowledge of the account. The term doesn't spring into denotation every time some human being or computer somewhere in the world does a calculation.
Quoting Metaphysician Undercover
You have no idea how daft that is.
An equation might contain only constants such as:
2+1 = 3
or it might contain a combination of constants and variables such as:
2+1 = x
or it might contain only variables such as:
x+ y = z
Quoting Metaphysician Undercover
Because we might not know that 2+1 = sqrt(9). The fact that at some point we don't yet know that 2+1 = sqrt(9) doesn't mean that it wasn't true all along. Obviously!
OK, so what in the expression "2+1" denotes that there is a result. Grand Minnow was insisting that the expression denotes a result. I don't see it in the signification. Now you're pretending to be someone else, so that your inconsistency is not so glaring. Grand Minnow can argue that a result is signified and Deep Freeze can argue that an operation is signified. How's that?
Quoting TonesInDeepFreeze
A function is a process. Grand Minnow kept insisting that "2+1" does not signify a process. That's why I say there is inconsistency. But clearly an "operation" or "function" is a process, and that's what is signified with "+".
The "value" of the function is not signified, because it must be figured out by carrying out the operation which is signified. If I say add some sugar to water, and bring it to a boil, the value (result) of that operation is syrup. But I'm not telling you "syrup", I'm telling you the procedure to make it. To obtain that value, syrup, you must carry out the operation referred to first.
Quoting TonesInDeepFreeze
OK, I'm fine with "operation", so long as you recognize that what is signified is a a procedure, or operation, and as you say, this is "a procedure for determining the value of a function applied to an argument". The value is not signified, the "procedure for determining the value" is what is signified. Do you agree?
Quoting TonesInDeepFreeze
Of course I do. In philosophy we use a different convention. I use " " to signify a concept rather than a physical thing. I'm trying to conform to your convention but I'm a bit sloppy and missed one. Call it a typo.
Quoting TonesInDeepFreeze
Here's your inconsistency. You distinctly said "2+1" refers to "a procedure for determining the value of a function applied to an argument."
Now it's my turn to ask you, do you understand the difference between a procedure (function, or operation), and an object? Aristotle demonstrated a fundamental incompatibility between these two. A procedure cannot be an object, and an object cannot be a procedure because of this fundamental incompatibility. If "3" refers to the number three, and this is an object, then the procedure for determining a value, referred to with "2+1", cannot be the same thing as what is referred to with "3".
Quoting TonesInDeepFreeze
Have you ever seen a ledger? Every account must be stated and balanced. Call it redundancy if you want, but there must be no room for error.
Quoting TonesInDeepFreeze
Of course I'm serious. You just told me there is an infinite number of ways to say "2+1", and I assume an infinite number of ways to say "3+1", so I ask you what distinguishes one from the other? Why is "3+1" not just another one of the infinite ways of saying "2+1"? That you do not answer means that you do not know.
Quoting TonesInDeepFreeze
Here is your inconsistency. Above, you said things like "500+ 894+202" denote "a procedure for determining the value of a function applied to an argument", which I accept. Now you are claiming that it actually signifies the value. What you say now is clearly false, because the procedure must be carried out before that value is derived.
Quoting TonesInDeepFreeze
What the person knows, is that "500+ 894+202" signifies the operation required to determine the value. Your claim that "500+ 894+202" represents the value is nothing but a misrepresentation. And, if you proceed in a philosophical argument with that misrepresentation of what "500+ 894+202" is known to signify, you are guilty of equivocation.
(1)
Quoting Metaphysician Undercover
* I said at the very top of my last post that I am Grandminnow. I'm not pretending anything. And you lied by claiming that my motivation is to make anything less glaring.
* You have not shown any inconsistency.
(2)
Quoting Metaphysician Undercover
I distinctly did NOT say that. And you put that misrepresentation in quotes to fabricate something I did not say.
I said that one notion of a process is that of a procedure for determining the value of a function applied to an argument.
And I say '2+1' does NOT denote a function nor an operation nor a process nor a procedure for determining the value of a function applied to an argument.
So stop right here. Go back to what I actually posted and see that I did not say, as you pretended to quote me, "'2+1' [refers to] a procedure for determining the value of a function applied to an argument."
If you don't then recognize that you fabricated a quote, then I won't know that I not talking with someone plainly dishonest and/or with real cognitive problems.
I addressed that already. The term '2+'1' denotes the value of the function + applied to the argument pair 2 and 1. It denotes the result of any computation of the function applied to those arguments. But it does not say within itself "there is a result". A term denotes an object (in this case, the object is the value of the function for the arguments, or the result of a computation of the function); a term is not itself a statement that there is a value or result.
'The father of Peter Fonda' denotes the value of the function (call it 'the father of function') applied to the argument Peter Fonda. That value is Henry Fonda.
'The father of Peter Fonda' does not itself denote the claim that there is value for the function.
Quoting Metaphysician Undercover
As I mentioned, I'm not going to go along with your undefined terminology 'process'. Instead I'll use 'operation' (meaning a function) and 'procedure' (meaning an algorithm).
Quoting Metaphysician Undercover
An inconsistency would be:
"'2+1' does not denote a process" and '''2+1' does denote a process".
But I never claimed that '2+1' denotes a process or operation or function or procedure.
So there is no inconsistency.
For about the seventh time now: '2+1' denotes the value of the function + applied to the argument pair 2 and 1. '2+1' does not denote a procedure nor a process (whatever vague notion of 'process' you probably have in mind. 2+1 denotes the RESULT of the procedure, not the procedure itself.
Quoting Metaphysician Undercover
Yes, '+' denotes a function. But '2+1' does not denote the function. For about the eighth time now: '2+1' denotes the VALUE of the function for the arguments 2 and 1.
And rigorously a function is not a procedure. A function is a relation such that no member of the domain is related to more than one member of the range.
Quoting Metaphysician Undercover
I addressed that already. You skipped what I wrote about that and instead adduced an analogy that doesn't apply:
Quoting Metaphysician Undercover
Yes, that is a description of a process in the sense of a procedure (though, of course, only by analogy and not a mathematical procedure), and the result is syrup. But '2+1' is not a description of a procedure. A description of a procedure would be a statement of the recursive instructions for addition (and specifically for the inputs 2 and 1). '2+1' is not the name of a sequence of instructions. (Granted, in constructive mathematics, roughly put, there are notions of mathematical objects, such as numbers, being a construction. But we're not in that context, or to get to that context, you would need to understand a lot more about it.)
Quoting Metaphysician Undercover
No, and see above and my previous posts.
Quoting Metaphysician Undercover
Use-mention is a convention in philosophy. It's not a different convention from that used in mathematics. And "concept vs. physical thing" is not it at all.
Quoting Metaphysician Undercover
You've been doing it over and over again. Not just typos.
Quoting Metaphysician Undercover
I am the one who has been harping on that difference.
Quoting Metaphysician Undercover
You're babbling and again skipped my point. As I said, the accountant doesn't have to write '=?' or 'x' to add the numbers.
Quoting Metaphysician Undercover
Wrong. Use-mention again. I said there are infinitely many ways to denote 2+1, and '2+1' is one of those ways.
Quoting Metaphysician Undercover
Another use-mention error by you.
Quoting Metaphysician Undercover
You did it again. You fabricated what I said.
You have it completely backwards what I said.
Quoting Metaphysician Undercover
Yes, THAT is what I said. I said '500+ 894+202' denotes the value, not the procedure for determining the value.
This is probably around twenty times I've said it.
Now you got it right. So stop also fabricating that I said the opposite.
I apologize then, I misunderstood. I thought you meant that "2+1" could be interpreted as eithe of the following, (1) or (2).
Quoting TonesInDeepFreeze
Now I realize you are insisting that it is neither.
Quoting TonesInDeepFreeze
This is clearly incorrect. "The father of Peter Fonda" denotes that there is a person who has the position, the special relationship of being the father of the mentioned person, and this person who is the father of the mentioned person is your subject. It does not say that this person is Henry Fonda, so you cannot jump to that conclusion. If you knew someone named Henry Fonda, it would be a logical fallacy to jump to the conclusion that this man is the referred subject. You have not made the required logical connection, to determine that your subject is the same person as the one you know as Henry Fonda.
Quoting TonesInDeepFreeze
This is false as well, and you just don't seem to get it. Take a look at your example of "the father of Peter Fonda". The thing which you claim as "the value", is clearly not signified, because the premise required to produce the logical conclusion is not stated in the argument. We need a further "unstated" (that's the way I use quotations, to signify special significance) premise to make your assertion a valid conclusion. In your example, the required premise might be "the person you know as Henry Fonda is the father of Peter Fonda". Then you can validly conclude that when some one says "the father of Peter Fonda", this is the person you know as Henry Fonda.
The thing which you seem to have no respect for, is the fact that "the father of Peter Fonda" does not refer to "Henry Fonda". This is very clear from the fact that one stated premise in a logical argument cannot refer to a conclusion. "Socrates is a man" does not refer to the conclusion "Socrates is mortal". That is because the expression does not include everything required to make that reference. Nor does "2+1" refer to the value signified by "3", because it does not include everything required to make that reference.
When we jump to a logical conclusion without stating the required premises, error is possible. You know someone named "Henry Fonda"; you jump to the conclusion that this is the man referred to by "the father of Peter Fonda", and mistake is possible. Rigorous logic seeks to exclude the possibility of mistake, not to create the possibility of mistake. The principles you are arguing for create the possibility of mistake by removing the need for the statement of premises. If some premises can be taken for granted, and not stated, as you seem to believe, then those premises cannot be judged for truth of falsity, and error is possible.
Quoting TonesInDeepFreeze
Well, clearly "2+1" does not refer to a value. That is an invalid conclusion as I explained above. So, if it does not refer to a procedure, as I think it does, is it possible that we can find a compromise?
Rather than composing lessons for you on the subject, I recommend:
https://plato.stanford.edu/entries/frege/ [Section 3]
https://plato.stanford.edu/entries/logic-intensional/
'Introduction To Mathematical Logic' pgs 1-68 - Alonzo Church
Are you going to address the points I made or not, Tones?
Do you apprehend the flaw in your example, and the difference between what "the father of Peter Fonda" denotes , and what "Henry Fonda" denotes?
The terminology here is incorrect--these two signs denote the same object, even though what they signify about that object is different.
As I explained, they do not denote the same object. One denotes the father of a person called Peter Fonda. The other denotes a person named Henry Fonda. That they denote the same object requires a further premise, that the father of Peter Fonda is the person named Henry Fonda.
Without that premise, the conclusion that they denote the same thing is invalid. And adding that premise is to beg the question. So the argument that they denote the same object is fallacious.
Again, this confuses denotation with signification. In any and every proposition about "Henry Fonda," we could substitute "the father of Peter Fonda" without changing the truth value. That is what it means for two signs to have the same denotation.
Denotation is a form of signification.
Quoting aletheist
As I said, this is only the case if there is a premise which states that Henry Fonda is the father of Peter Fonda. But that is begging the question, which is respected as a fallacy.
Therefore the argument that "the father of Peter Fonda" denotes the same thing as "Henry Fonda" is a fallacious argument, by means of begging the question. The argument relies on assuming the conclusion.
Completely wrong, denotation and signification are two different aspects of a sign, corresponding respectively to its object and its interpretant. This is Semeiotic 101.
Quoting Metaphysician Undercover
I offered no argument at all, I simply stated a definition--if one sign can be substituted for another in any and every proposition without changing the truth value, then both signs denote the same object. This is also Semeiotic 101.
I have given you copious explanation. There's no point in me composing more explanation when it is better said anyway at the sources I offered you.
That's bullshit 101. In logic, there is no object, we have subjects. To denote is simply to be a sign of.
Quoting aletheist
Then you're not addressing the issue we've been discussing. We've been arguing the truth or falsity of of a very similar principle. The "Fonda" example was provided as an argument for the truth of it. As I've shown, it's a fallacious argument.
We were arguing the truth or falsity of the principle of substitution, which is the basis of extensionality. It is claimed that if two signifiers signify things of equal value, they are exchangeable, therefore they signify the very same object. It is very clear to me that this is a false principle because "equal" is assigned according to some system of judgement, so only the properties deemed significant within that system are accounted for, and this is insufficient for the conclusion of "the very same object". I find it utterly amazing, and rather distressing, the number of people in this forum who cannot apprehend this simple fact.
Now you are arguing a slightly different form of that principle. "Truth value" is something judged. Propositions are stated. Predications are of subjects. So within a logical system "truth value" concerns what we say about subjects, not objects. Unless absolutely every property of a given object is stated (a task humanly impossible), so that an infallible judgment can be made, your proposed principle: "if one sign can be substituted for another in any and every proposition without changing the truth value, then both signs denote the same object" Is clearly unacceptable as false. You have not provided the means for closing the subject/object gap.
2. I can create a number that lacks a repetend e.g. 1.01001000100001...
Ergo,
3. Irrational numbers "exist" (I just gave you an example)[1, 2 Modus Tollens]
Logic generalized is semeiotic, the science of all signs--not just arguments, but also propositions and terms; and not just symbols, but also indices and icons. Subjects are the terms within propositions that denote their objects.
Quoting Metaphysician Undercover
I never claimed otherwise, I was simply correcting a misuse of the technical term "denote." Again, "Henry Fonda" and "the father of Peter Fonda" denote the same object, even though what they signify about that object is different. If we were looking at a photo of the Fonda family, and someone asked me to point to Henry and you to point to the father of Peter, then we would both correctly point to the same person.
Quoting Metaphysician Undercover
This reflects more terminological confusion. What a sign signifies is not its object, but its interpretant.
Again, your argument that they denote the same object is fallacious. They may or may not denote the same object. They clearly signify something different, and we do not have the premises required to conclude that they denote the same object. Therefore your conclusion that they denote the same object is fallacious.
I agree that they signify different interpretants, but this does not preclude them from denoting the same object. It is a fact that Henry Fonda is the father of Peter Fonda, so by definition, it is also a fact that the signs "Henry Fonda" and "the father of Peter Fonda" both denote the same object. Someone who does not know the first fact would not know the second fact either, but that is irrelevant to their being facts.
I commend you for fighting the good fight against @Metaphysician Undercover. But here I find myself inclined to see his side of it. I might know who Henry Fonda is, but I might not know he's Peter Fonda's father. I can see @Meta's point that the "father of Peter" description conveys more information than merely saying "That's Henry Fonda."
Just as he claims that 2 + 2 conveys the idea of a process putting two numbers together in some way, which is different information than just referring to the number 4. He's wrong about that mathematically, but he may have a point about Henry and Peter.
A stronger example in natural language is Joe Biden and the president of the US. Joe is always Joe, but being president is contingent and others have held and will hold that office. Joe Biden has always been and will always be Joe Biden; but Joe Biden has not always been president and will not always be president. So these two statements are different somehow.
As I said, they may denote the same object, but we do not have the premises required to conclude that they do. In logic we cannot assume other premises which are not stated. We have a person denoted as "the father of Peter Fonda" and we have a person denoted as "Henry Fonda". We have no other information. So the conclusion that they both denote the same thing is extremely unsound, because it is not derived from valid logic. It is invalid.
Quoting aletheist
In logic, assertions do nothing for you. They are proposals, propositions which must be judged for truth or falsity. TonesinDeepFreeze has been asserting that "2+1" denotes the same object as "3" does, in a similar way. They very clearly each signify something different. The only attempt by Tones, to support this conclusion with a premise, was a vague reference to extensionality. But a premise which states that two equal things are the same thing is clearly false, making that argument unsound, by having such a falsity as a premise. To say that the person denoted as father of Peter Fonda, and the person denoted as Henry Fonda, are equal, as human beings, does not justify the claim that they are the same person.
If you just keep asserting as a proposition "it is a fact that...", and you expect me to take that proposition as a premise for an argument, then you're wrong. I will not. You need to demonstrate the truth of it. Adding the emphasis "it is a fact" does nothing for your case. I am very certain that two things with the same value are not necessarily the same thing, as I can give you endless examples, so that is a false premise.
Quoting fishfry
Fishfry! Never in a hundred years did I think I'd see this day. Let's go, I'll buy you a beer.
Quoting fishfry
The point is that "the father of Peter Fonda" gives different information from "Henry Fonda". The latter gives nothing, just the name of a person. The first expression also denotes a person, as well as the second expression denotes a person. But the information required to conclude that they are one and the same person is not provided. Even if we add the further premise, "Henry has a son Peter", the condition of reversibility, equality, is fulfilled, but we still cannot conclude that they denote the same person. There might be more than one Henry Fonda with a son Peter. Therefore there is still a possibility of error, which demonstrates why such conclusions are unsound.
Again, what anyone knows or does not know is beside the point. Since it is fact that Henry Fonda is the father of Peter Fonda, by definition (in semeiotic) the two signs "Henry Fonda" and "the father of Peter Fonda" denote the same object.
Quoting fishfry
I have never denied this, but (in semeiotic) the information conveyed by a sign corresponds to its interpretant, not its object. If we were standing in a room with Henry Fonda--preferably back when he was alive--then we could point at him and truthfully say both "that is Henry Fonda" and "that is the father of Peter Fonda." Therefore, both signs denote the same object, despite signifying different interpretants.
Quoting Metaphysician Undercover
The only "premiss" required is the fact that Henry Fonda is the father of Peter Fonda. Someone previously unaware of this fact would learn that "Henry Fonda" and "the father of Peter Fonda" denote the same object upon being informed of it, but those two signs denoted the same object all along. Someone's ignorance does not affect the reality.
Quoting Metaphysician Undercover
Again, in that scenario, I agree that we do not know that the two signs denote the same object; but that was never the scenario that I was discussing. I was simply pointing out that since I do know that Henry Fonda is the father of Peter Fonda, I also know that "Henry Fonda" and "the father of Peter Fonda" denote the same object. These are facts, not opinions.
Quoting Metaphysician Undercover
I never said anything about persons or equality. I merely made the point--which is utterly uncontroversial (in semeiotic)--that since Henry Fonda is the father of Peter Fonda, the two signs "Henry Fonda" and "the father of Peter Fonda" denote the same object, regardless of whether someone else knows it.
Quoting Metaphysician Undercover
Again, I agree, the two signs signify different interpretants--i.e., convey different information--despite denoting the same object.
Did you mean a beer? Or did you mean an alcoholic drink made from yeast-fermented malt flavored with hops? According to you they're two entirely different things :-)
Quoting Metaphysician Undercover
I agree with you about this point of natural language. But I still disagree regarding 2 + 2 = 4, even thoug I do see the point you're trying to make.
Quoting Metaphysician Undercover
That latter is a bit disingenuous. If I say Socrates is a Greek philosopher, someone might object because they think I might have meant Socrates the cat philosopher. That's not really a good objection, if you fully qualified everything there would be no end to it.
Thank you. My granting of @Metaphysician Undercover's point is causing me to waver on 2 + 2 = 4. I need to read more of your posts so I can strengthen my backbone. Henry Fonda IS the father of Peter and that's that. There goes the beer @Meta was going to buy me, which according to @Meta is NOT the same as, "an alcoholic drink made from yeast-fermented malt flavored with hops."
.
Quoting aletheist
As you can see, @Meta has gotten into my head over the years. Must ... stay ... strong ...
Quoting aletheist
Thank you for this clarification. Must remember it. Must not weaken. 2 + 2 = 4. Please don't take me to room 101.
These are nonsensical assertions. You are asserting that it is a fact that these words refer to these objects regardless of how people use the words. The issue is whether or not "the father of Peter Fonda" and "Henry Fonda" necessarily represent the same object. That you can define them as representing the same object, and insisting that this is a fact, is irrelevant I don't see any point in discussing the soundness of a logical argument with someone like who, who simply insists that the conclusion is a fact, and that's all there is to it.
If you want to start with the premise, that Henry Fonda is the father of Peter, we can do that. But then the example is irrelevant to the question of whether "2+1" denotes the same thing as "3", because we are not starting with that premise. We are arguing whether or not this claim is true, or logically sound. So we cannot start with the premise that "2+1" denotes the same thing as "3" because that would be a fallacy of begging the question.
So, to make the example relevant, we must start with the two expressions, "father of Peter Fonda", and "Henry Fonda", and you need to demonstrate how they necessarily refer to the same object, without begging the question. Insisting that it is a fact is simply begging the question, and that is a logical fallacy. So your procedure up until now has been completely useless.
Quoting aletheist
Yes, yes, keep begging the question, it really doesn't bother me if you do. You're only fooling yourself.
Quoting fishfry
Let me give you a more relevant example. Let's consider an experiment in quantum physics. Consider that a photon is emitted by an emitter, and a photon is absorbed by a detecting machine. Each instance involves an equivalent amount of energy, so the assumption is that the two are the same photon. Then comes the difficult task of determining the continuous existence of that photon between point A and point B, which is produced by the idea that they are the same photon. But there is no need to assume that the two are the same photon, likewise there is no need to assume a continuous existence of the photon between point A and point B. It is only this (what I call odd) way of looking at things, that if there is a quantifiable value here, then an equal value over there, these must represent the same thing, which promotes the idea of the continuous existence of a photon between these two point.
Quoting fishfry
As explained above, to premise that "Henry Fonda IS the father of Peter" is no different from premising that "2+1 IS 3". But since what I am looking for is an indication that 2+1 really is the same thing as 3, some sort of logical argument, that's simply begging the question. So the issue is to demonstrate logically, how it is that the two distinct expressions "the father of Peter Fonda", and "Henry Fonda" both refer to the exact same thing, and how this is relevant to the case of "2+1" and "3".
You had me a little worried for a while there. :wink:
Quoting fishfry
You are welcome, I am glad that it was helpful. :cool:
Good grief, I never said that they necessarily denote the same object, I only said that they actually denote the same object. That is why I kept calling this a fact. If Henry Fonda were not the father of Peter Fonda, then obviously the signs "Henry Fonda" and "the father of Peter Fonda" would not denote the same object. What I mainly wanted to do was simply point out the difference between what a sign denotes (its object) and what a sign signifies (its interpretant).
Then the example is irrelevant to the issue we are discussing, that "2+1" denotes the same object as "3".
You joined the discussion a bit late, and seem to be missing the issue.
I have quite deliberately said nothing directly about that issue until now, because I was mainly interested in commenting on the other example that came up. Its relevance has to with the question whether "2+1" and "3" are likewise different signs that denote the same object despite signifying different interpretants. The problem, of course, is that we cannot even in principle point at something and say both "that is 2+1" and "that is 3." However, we can point at a collection of three apples and say both "that is 2+1 apples" and "that is 3 apples." Moreover, we can substitute "2+1" for "3" in any proposition without changing its truth value or in any equation without changing its result. What should we conclude from this?
That's correct in an extensional context, but not in an intensional context:
Suppose Alice doesn't know that Henry Fonda is the father of Peter Fonda. Then consider these two sentences:
(1) Alice knows that Henry Fonda is Henry Fonda.
(2) Alice knows that Henry Fonda is the father of Peter Fonda.
But (1) is true (since Alice knows that identity is reflexive), while (2) is false even though (2) just substitutes 'the father of Peter Fonda' for 'Henry Fonda'.
'knows that' creates an intensional context in which substitution may not preserve truth values.
That's not true, because the operation signified by "+" is not evident in the group of three apples, so it is not a true representation of "2+1". It is just a representation of "3". If you were teaching children you would not show them a group of three apples and tell them this is 2+1.
Quoting aletheist
We might say that expressions which signify equal value can be substituted, within that value system . We cannot conclude that because the expressions can be substituted, they signify the same thing. They are only "the same" in relation to that value assigned to them.
If I need assistance, and Tom, Dick, or Harry, will do, each having equal value for the task, I ask for Tom, Dick, or Harry, as they are interchangeable in relation to this value. They each make "the task will be done" true. This does not mean that each of them is the same thing as each other.
You're lying about me again:
Quoting Metaphysician Undercover
(1) I didn't make "vague references". Indeed, I posted an explanation of the notion of exentionsality vs. intensionality. And I gave references in the literature for you to read about it. Moreover, even if I had not done that, it is still the case that the notion of extensionality vs. intensionality is a well known basic notion in the philosophy of mathematics and philosophy of language. The fact that you're ignorant of such basics of the subject is not my fault and doesn't make my reference to them "vague", and especially not when I gave explanation and additional references in the literature anyway.
(2) I posted multiple times that proving that '2+1' and '3' denote the same object is the basis on which we justify claiming that they do. Or, for a better example (since the equation '3 = 2+1' has such a trivial proof), we say '6-3' and '2+1' denote the same object because we prove that they do.
Stop lying about me.
Again on this point:
Quoting Metaphysician Undercover
Again, this is conflating denotation is evidence, proof, demonstration and knowledge.
We conclude that two different names denote the same object by proving they do.
'2+1' and '6-3' have the same denotation. Of course, to legitimately assert that '2+1' and '6-3' requires first having grounds for the assertion, such as mathematical proof of the equation:
2+1 = 6-3.
That's what mathematicians do; they prove formulas, including equations. When the equation is proven, then we are justified in claiming that '2+1' and '6-3' denote the same object.
Fair enough, thanks. Indeed, an extensional context corresponds to denotation (object), which is the same for "Henry Fonda" and "the father of Peter Fonda"; while an intensional context corresponds to signification (interpretant), which is different for the two signs as I have acknowledged all along.
The example was given not so much as an argument but as an illustration for you to understand a basic idea.
Ordinary mathematics regards '2+1' and '3' as having the same denotation, because we prove
2+1 = 3
In general, for any terms T and S, we infer
T = S
when we prove it and then we may say that T and S have the same denotation.
As I mentioned before, this is the case both by ordinary mathematical practice and as made rigorous in mathematical logic. That's just a fact about certain conventions in ordinary mathematics. Whether ordinary mathematics should use that convention is a separate issue, of which I have not taken a position except to point out that alternatives are complicated.
A natural language example, such as the Fonda example, doesn't prove anything about mathematics, but, as I mentions, it illustrates the general principle.
I am treating "2+1" and "3" as signs here, and I already acknowledged that their signification is different. At issue is whether their denotation is different. What "+" represents in isolation is irrelevant, all that matters here is that I can point to the same group of items and truthfully say both "that is 2+1 apples" and "that is 3 apples."
Quoting Metaphysician Undercover
I actually might do exactly that, if I were teaching them basic addition such as 2+1=3.
Quoting Metaphysician Undercover
Of course not, but we can conclude (in an extensional context) that they denote the same thing.
Indeed, but as the Fonda example has brought to light, @Metaphysician Undercover apparently confuses denotation and signification. The result is wrongly denying that two different expressions signifying different interpretants can nevertheless denote the same object.
What he doesn't understand is that denotation is only one part of meaning. There is both denotation, which is extensional, and sense. 'Henry Fonda' and 'the father of Peter Fonda' denote the same thing. But indeed the they have different senses.
Ordinary mathematics deals with terms extensionally. For mathematics to deal with terms also with regard to sense requires a more complicated linguistic/logical system. As I mentioned there are proposals for such systems, but they are beyond ordinary mathematics.
You have it backwards again. Mathematics does not prove that objects are equal by showing they share all properties. Rather, we infer they share all properties from having first proved that they are equal. And whatever we prove, we do so from axioms.
Also, mathematical theories are in mathematical languages and regard models with objects and their relevant properties for particular areas of mathematical interest. In mathematics, one is free to state other languages, theories and models in which other properties are relevant besides those in some previous treatment.
That is ridiculously captious and sophomoric. It is deserves all three tropes: red herring, blowing smoke, and grasping at straws.
Of course in natural language and everyday discourse there may be vagaries that make definitive determinations difficult or impossible. So if you want to demand a context in which there are no vagaries, then of course all bets are off with natural language usage. So to proceed with understanding certain principles, of course we must assume, for sake of discussion, some context in which we are not thwarted by such vagaries as you mention.
Moreover, of course, in this context, we are assuming that there has been ample evidence that Henry Fonda is the father of Peter Fonda, and that we can agree on that for the purpose of the illustration regarding denotation of the names.
Again for about the fifth time: When I claim that 'Henry Fonda' and 'the father of Peter Fonda' denote the same person, I don't claim that I have shown that Henry Fonda is the father of Peter Fonda. That's not the point, and only someone pretty obtuse would miss this. To show that Henry Fonda is the father of Peter Fonda is a matter of empirical inquiry. Of course in this context I assume that we take it for granted that we know empirically (or by whatever means of such common knowledge) that Henry Fonda is the father of Peter Fonda, and on that basis, we make the linguistic observation that the names 'Henry Fonda' and 'the father of Peter Fonda' denote the same person.
Similarly, of course I take it for granted that we already understand that 2+1 = 3, either by proof or by common mathematical knowledge. It is from that understanding that we then observe that '2+1' and '3' denote the same number.
For about the dozenth time, get it straight:
First we find out that 2+1 = 3, and then we may justifiably claim that '2+1' and '3' denote the same number.
/
The philosophy of language does take on issues with problematic, indeterminate, equivocal, conflicting, temporally complicated, and paradoxical denotation. But denotation in ordinary mathematics is fixed, so it remains a simple fact that '2+1' and '3' denote the same number.
This is at the heart of it.
The claim that 'Henry Fonda' and 'the father of Peter Fonda' denote the same person is not an argument! It is a conclusion. It is a conclusion from the premise (however it has been established) that Henry Fonda is the father of Peter Fonda. No one every suggested otherwise!
You blame others for fallacies in arguments they never made! (I.e., variation on straw man.)
I told you already, extensionality provides a false premise. False premises produce unsound conclusions, which do not prove anything. When a human being judges two distinct things as having the same properties, and says therefore that they are equals, this does not make them into the same thing. The law of identity stipulates that the identity of a thing is within the thing itself, not a human judgement of the thing.
Quoting TonesInDeepFreeze
Clearly you, (and extensionality in general) have this backward. If we start with the law of identity, "a thing is the same as itself", as a fundamental premise, and we compare this with equality, which is a property that we assign to things, you ought to see this. When we judge two things as equal, we cannot assume that they are the same thing, because we need to allow for the fact that human judgements are deficient in judging sameness. We may not be able to account for all the potential differences between them, and thereby over look some, making a faulty judgement of "same".
So, we can truthfully say that a thing will be judged to be equal to itself, but we cannot truthfully say that things which are judged to be equal are the same thing. Therefore, when you say that T=S, you say that T has the same value as S within that system of judgement, and this means that the symbols have the same meaning within that system, but it does not mean that they denote the same object.
Quoting TonesInDeepFreeze
This precisely, is the false premise. Being equal is a human judgement, and being equal does not imply being the same. We can proceed the other way, and say that being the same implies being equal, but we cannot proceed from being equal to an implication of being the same. This is because two distinct things can be judged as equal, when they are not the same. Therefore proving that two things are equal does not imply that they are the same (share all properties). It only implies that they share the properties by which they are judged to be equal. And there is your false premise.
This is a basic fact of the way that we use signs and symbols. We use the symbol "2" here, and we use it later in some other application. These are two distinct instances of that symbol, they are not the same thing. However, they have an equality in what they signify. Each distinct instance of using that similar symbol signifies "the same value". This means that the two instances have the same meaning. It does not mean that they denote the same object.
So, proving that two distinct yet similar instances of a symbol "2", have an equal value, only proves that they have the same meaning. It does not prove that they denote the same object.
Quoting TonesInDeepFreeze
This is counterproductive. If in reality, language use is filled with vagaries, and we want to discuss the truth about language use, then we need to account for the reality of those vagaries. To assume a context without vagaries as your prerequisite premise for proceeding toward an understanding of certain principles of language use, is simply to assume a false premise. Therefore by adopting such a position we proceed toward a misunderstanding rather than an understanding.
Quoting TonesInDeepFreeze
In the case of Henry Fonda, we have observed with our senses, the very object being referred to. In the case of numbers we have not observed any such objects. You are requesting that I simply assume such an object, a number, so that we can talk about it as if it is there. Obviously, there are no such objects, the numerals have meaning dependent on the context of usage, just like any other symbols. They do not denote any objects, and your so-called understanding is actually a misunderstanding.
Quoting TonesInDeepFreeze
Claiming a denotation when there is only meaning, is a false premise.
Quoting TonesInDeepFreeze
Do you understand the fallacy of "begging the question", assuming the conclusion?
Quoting aletheist
To be clear, what is at issue is whether there is a denotation at all (when denote is defined as you do). Read the above.
The "+" is not irrelevant, it must be accounted for in your interpretation. You cannot simply leave words out of a phrase, in your interpretation, to make it say what you want it to say, or denote what you want it to. I really do not see any logic to your claim that two expressions can have distinct significations, yet denote the same object. I can see how "I did X", and "I did Y", both refer to the same object with "I", but each signify something different. Since each expression signifies something completely different, if we replace what is signified with "denoting an object" as you seem inclined to, then we do not come up with the same object. How do you come up with this idea that two phrases which signify something completely different actually denote the same object. I would call that contradiction.
As I stated clearly in the last post. A group of three apples does not truthfully represent "2+1". If that's not obvious to you, go back to grade school and find out how they represent "2+1".
Quoting aletheist
Actually it has become very clear now, that you and Tones are the ones confusing denotation and signification. Clearly, in our use of mathematics there is signification without denotation. You and Tones are seeing an object denoted by "2+1", when there is none. That is misinterpretation.
Whatever views you have about the distinction between extension and intension, and between denotation and sense, I gave you more than "vague reference" about them.
Quoting Metaphysician Undercover
You may think you've shown a false premise, but you haven't.
Quoting Metaphysician Undercover
I never said that we infer that distinct things are equal, let alone that they are equal due to having the same properties. You're resorting to strawman again.
Quoting Metaphysician Undercover
Identity is a reflexive relation. And I never said that things are identical due to human judgement. You're resorting to strawman again.
Quoting Metaphysician Undercover
We don't judge two things are equal. We judge that two terms refer to the same thing. And, of course, such judgements may be mistaken due to human error.
Quoting Metaphysician Undercover
No, the principle of the indiscernibility of identicals holds. It provides the method of "substitute equals for equals" that is fundamental in mathematics.
Quoting Metaphysician Undercover
Over and over you swing this "false premise" charge like a crudely made cudgel. It's mere assertion.
You haven't shown that it is false that mathematics does not have the kind of vagaries of natural language in everyday discussion.
Quoting Metaphysician Undercover
It was my point that the Fonda example involves first making an empirical determination. On the other hand, mathematics is deductive and axiomatic.
Quoting Metaphysician Undercover
Of course, if numbers are not at least abstract objects, then they cannot be referred to as objects. Then '2' has no denotation to a number. Go ahead and formulate your mathematics that way if you like, but you offer no formulation or even hint of one. And, I also mentioned that if we confine our attention to mathematics at the pure formula level, then object and denotation don't even need to be mentioned.
Quoting Metaphysician Undercover
Denotation is part of the meaning.
Quoting Metaphysician Undercover
I understand it better than you. And you've not shown I am question begging. Moreover, I showed exactly how I am not question begging, and you skipped that. This is like your claim that I was inconsistent - you never showed an inconsistency.
Quoting Metaphysician Undercover
I never used the term 'signification'.
Good, and they also both denote the same object with "did," which is the relation of doing. However, they presumably denote different objects with "X" and "Y," although since these are variables it is conceivable that they could also denote the same object--for example, the activity of exercise.
Quoting Metaphysician Undercover
We already went over this with "Henry Fonda" and "the father of Peter Fonda." These signs both denote the same object despite signifying different interpretants because it happens to be a fact that Henry Fonda is the father of Peter Fonda.
Quoting Metaphysician Undercover
As someone once said ...
Quoting Metaphysician Undercover
Do you know the law of identity? It states that a thing is the same as itself. It says nothing about equality or equivalence. That two things are equal is a human judgement.
And you said:
Quoting TonesInDeepFreeze
See, no strawman. You prove that they are equal (human judgement), then you infer from this, that they are the same. Let me put it simply, proving that they share one property, "are equal" does not prove that they share all properties. You need another premise, which states that equal things are the same thing. But we know that premise is false because we see all sorts of equal things (equal volume, equal weight, etc.) which do not make two things the same. Equality is not sufficient for a judgement of same.
Quoting TonesInDeepFreeze
You very clearly stated "having proved that they are equal". Therefore you do judge that they are equal, that's what proving is, providing the justification for judgement.
Quoting TonesInDeepFreeze
The indiscernibility of identicals does not provide the principle required for substituting equal things. Things are judged to be equal not on the basis that they are indiscernible. Clearly, as altheist agrees, and what ought to be obvious to you, what "2+1" signifies is not indiscernible from what "3" signifies. Since these two are judged to be equal, equal does not mean indiscernible. Therefore it is false to claim that the principle of the indiscernibility of identicals supports such a substitution. It does not.
Quoting TonesInDeepFreeze
Clearly mathematics has an extremely vague notion of identity, one not consistent with the law of identity, allowing that similar things which are judged to be equal may be substituted as if they are the same thing.
Quoting aletheist
I really can't see how a relation is an object. I think you are making things up as you go.
Quoting aletheist
This explains nothing. Words like "did" signify something. But you insist instead, that they denote an object. But you also allow that they signify things as well, and denote objects at the same time. On top of this you allow that two phrases might signify different things, yet denote the very same thing. This indicates very clearly that there are contradicting interpretations of the same phrases. One interpretation says that they are different, the other that they are the same. Yet you allow that the contradicting interpretations are both correct.
You mangle nearly everything.
(1) Claiming I've said things when I did not say them.
(2) Screwing up the direction of my explanation so that your representation of my explanation is not my explanation.
(3) Reversing the direction of conditionals.
(4) Ignore explanations and decisive points and instead keep repeating yourself past them.
(5) Ignore distinctions explicitly stated.
Quoting Metaphysician Undercover
I know about identity vastly more than you do. And your reply merely repeats your own thesis. And you did argue by strawman by trying to make me look as if I had said that identity holds based on human judgement.
Quoting Metaphysician Undercover
I said to stop claiming I've said things I did not say.
Quoting TonesInDeepFreeze
And now taking me out of context. Here is the context:
Quoting TonesInDeepFreeze
Of course, people make judgements of equality. But at this particular juncture in the discussion, I am pointing out that the activity is not that of judging equality itself but rather judging whether the terms refer to the same thing. Those activities are related but different.
Quoting Metaphysician Undercover
You're totally mixed up as to the order of the statements in my explanation. Again:
First we determine (by proof or whatever method) that 2+1 is 3. From that determination we are justified in claiming that '2+1' and '3' refer to the same number.
Don't screw up the sequence of statements in my explanation to thus mangle it.
Quoting Metaphysician Undercover
Sure it does. The indiscernibility of identicals is the general principle. Substitutivity is the formal application of the principle.
Quoting Metaphysician Undercover
Again, you got it backwards! For the third time I've told you, I did not claim to infer equality from indiscernibility. I claimed to infer indiscernibility from equality.
What I said:
equality -> indiscernibility.
But you keep saying that I say:
indiscernibility -> equality
even after I've told you that is not what I say.
Don't reverse the direction of my conditionals.
Quoting Metaphysician Undercover
There is both denotation and sense. Substitutivity holds as rule only for denotation. I was the one who gave the early example in this thread where substitutivity fails when the context is not extensional.
Meanwhile, if you don't recognize the use of substitutivity in even just basic math, then you can't do math.
Quoting Metaphysician Undercover
You can't see because you ignore mathematics. A relation is set of tuples. That set is an object.
You are ignorant of the view in which meaning has at least two components: denotation and sense.
Denotation is only part of the meaning of a term.
In ordinary mathematics, we concern ourselves only with denotation, which is the extensional aspect of meaning. And, as I've pointed out a few times already, if you wish to have mathematics that concerns also sense, or the intensional aspect, then you are welcome to formulate such mathematics or to look up formulations that have been given by other mathematicians and philosophers of mathematics.
That is because you evidently have an extremely narrow definition of "object" and refuse to accept how that word is defined as a technical term within the discipline of semeiotic. Anything whatsoever that is denoted by a sign is the object of that sign.
Quoting Metaphysician Undercover
Not at all, do some research into semeiotic (also called "semiotic" or "semiotics") and you will learn that what I have been discussing is a well-established field of study. I did not anticipate having to get into this level of detail when I offered the simple observation that "Henry Fonda" and "the father of Peter Fonda" denote the same object, which is utterly uncontroversial within that field.
Quoting Metaphysician Undercover
There is no contradiction because denotation and signification are not synonymous, they correspond to different functions of a sign. Again, what a sign denotes is its object and what a sign signifies its interpretant. The object of a term is whatever it stands for, while the object of a proposition is the collection of objects denoted by the terms that serve as its subjects. The interpretant of a sign is whatever it conveys about its object, and thus is usually what we have in mind when we talk about the meaning of that sign.
So get these straight already:
(1) My explanation runs in this order:
Determine equality, then it is justified to assert that the terms denote the same.
(2) Equality implies indiscernibility. I did not opine one way or the other whether indiscernibility implies equality.
(3) Substitutivity holds in extensional contexts, and it may fail in intensional contexts.
Equality is insufficient for a judgement of "same". That's very simple, clear, and obviously true, from all the instances where equal things are not the same thing. I'll expound on this below, but you ought to respect this principle instead of trying to deny it, and insist that equal things are necessarily the same thing.
Quoting TonesInDeepFreeze
If you can show that equality is something other than a human judgement, then you might have a case. Otherwise the charge holds.
Quoting TonesInDeepFreeze
You said that from a judgement of equality you can infer that they are the same. I'll quote for the third time:
"Rather, we infer they share all properties from having first proved that they are equal."
You are clearly arguing that if they are equal then they are the same.
Quoting TonesInDeepFreeze
Obviously this is false, 2+1 is equal to 3 but it is not the same as.
Quoting TonesInDeepFreeze
I explained why the indiscernibility of identicals does not support your assertion. Obviously 2+1 is not indiscernible from 3. Therefore you cannot use the indiscernibility of identicals to support your claim that they are identical.
Quoting TonesInDeepFreeze
This is false because what "2+1" signifies is very clearly discernible from what "3" signifies. There are two numbers denoted, 2 and 1, while "3" only denotes one number. We've been through this countless times already and you are in denial of the truth. Admit the fact, 2+1 is not indiscernible from 3.
Quoting TonesInDeepFreeze
This is the only way that the principle of indiscernibility could be used to support your claim that equality means the same as. So I assumed that this is what you meant. The other way, the way you claim to be using it, would work if it were true, but it is clearly a false premise. Equal is not sufficient for indiscernibility. That's obvious from all the cases of equal things which are discernible.
Quoting TonesInDeepFreeze
I assumed you were trying to make a sound argument. However, you've now corrected my to show that you are simply using a false premise. You admit now that your premise is that if things are equal they are the same. Therefore I'll take you back to what I asked days ago. Are you and I the same because we are equal? You have no special pleading now, for a special sense of "equal", which is supported by "indiscernible", because you've just admitted that you support "indiscernible" with "equal". By turning this around you have no special definition for "equal".
And to define that sense of "equality" with "value" doesn't help you because all senses of "equality" rely on a judgement of value. Quantitative value is no more special than moral value as an indicator as to whether or not two things are the same. The value which we assign to a thing is not a thing's identity.
Quoting TonesInDeepFreeze
See, you admit right here, that you only concern yourself with a part of what "2+1", and what "1" refer to. Therefore you ignore the other aspects, which are clearly different from each other, and you proceed to claim that what they represent is identical. What this really means is that they are the same in some aspect, but not in every aspect, so it is false to claim that they are the same.
Quoting aletheist
Sorry aletheist, but I must inform you that technical definitions narrow down a word's definition. That is because a broader definition allows for ambiguity. So your attempt to broaden the definition of "object", is not at all an attempt at a technical definition. It's an attempt to create ambiguity, which might be useful for the creation of vagueness and equivocation. So I see your definition as completely misguided because it's not conducive for understanding.
Quoting aletheist
I have, and I do not agree with the fundamental principles of that proposed field of study. It appears to be lost in ambiguity and category mistake. This opinion which I have, you might be able to detect. I am not one to dismiss things off hand, without some understanding of the fundamental principles.
Quoting aletheist
This appears to involve a fallacy of composition. And I think this is why your way of looking at "2+1" appears so incorrect to me. You say that "2+1" signifies something which is other than what "3" signifies, yet "2+1" denotes the same object as "3". You make this conclusion of denoting the same object through a fallacy of composition, concluding that the attributes of the parts within the statement "2+1" can be summed up into a collection, to make an object with the exact same attributes as 3.
It is not "my" definition, it is the well-established definition within the discipline of semeiotic.
Quoting Metaphysician Undercover
Then we can stop wasting each other's time.
Quoting Metaphysician Undercover
On the contrary, in my experience you routinely dismiss things out of hand, simply because they fail to conform to your peculiar, narrow, dogmatic definitions of terms.
Quoting Metaphysician Undercover
You also say things like this so that it sounds like you know what you are talking about when you really have no idea.
This is another instance of imposing your view as if it entails something I said that I did not say. You believe that equality holds based on human judgement. That doesn't entail that I said that equality or identity does. It's a strawman to represent me as saying something I did not say.
Quoting Metaphysician Undercover
You've mixed up two different issues, and got me wrong on both of them.
Issue 1: You claimed it is question begging to say that '2+1' and '3' denote the same object.
Of course if 2+1 and are equal 3 then they are the same. But what you claimed was that I was begging the question by saying that '2+1' and '3' have the same denotation. And I explained that it is not question begging since we infer that '2+1' and '3' have the same denotation from first determining that 2+1 equals 3.
Issue 2: You claimed that I said that indiscernibility implies identity.
But I did not. I said that identity implies indiscernibility.
There are three principles of identity/indiscernibility:
(1) If identical then indiscernible.
(2) If Indiscernible then identical.
And we may combine for:
(3) Identical if and only if indiscernible.
All I said is (1).
And again, not question begging.
Quoting Metaphysician Undercover
You have a cognitive problem that prevents you from discussing this without getting grievously mixed up about it. You keep reversing the direction of implication.
Again, I did not say that indiscernibility implies identity. I said the reverse direction of implication: identity implies indiscernibility.
Quoting Metaphysician Undercover
Still yet again, I am not using indiscernibility to support that 'equal' means 'identical' or that 'equal' means 'the same'. Stop mixing up what I've said and then representing your own mixed up version as if my own.
Quoting Metaphysician Undercover
I answered that many posts ago! Again you argue by just skipping past many key points in the replies to you.
Quoting Metaphysician Undercover
In ordinary mathematics, 'equal' is not defined but rather is a primitive. It is the sole primitive of first order identity theory. In that context 'equal' and 'identical' are two words for the same undefined primitive.
Quoting Metaphysician Undercover
And I don't define any sense of 'equality' with 'value'.
Quoting Metaphysician Undercover
Not so much what I concern myself with personally, but rather what ordinary mathematics concerns itself with.
I mentioned the extensional vs intensional distinction many many posts ago. And again the two different terms '2+1' and '3' refer to the same object. They have the same reference. However, of course, they do not have the same sense. Again, yes, ordinary mathematics is extensional and concerns only the denotation part and not the sense part. I referred you to the Stanford philosophy encyclopedia article that discusses this. I said many posts ago that you may also consider formulations in which intensionality is considered.
That is rich from someone who dismisses approaches in ordinary mathematics while insisting on remaining ignorant of understanding their fundamental principles or even reading a single page in a book or article about the subject.
I gave you a formal mathematical proof of this fact over two years ago, maybe three. You're telling a little fib here.
'S' stands for the successor operation.
def: 1 = S0
def: 2 = 1+1
def: 3 = 2+1
The proof in this case is utterly trivial, from the definition of '3'.
I asked you for an instance of equality which is not a human judgement. You didn't give me one. That's probably because you understand that such a thing is ridiculous.
Quoting TonesInDeepFreeze
In case you're having a hard time to understand, I see this as very clearly false. You and I are equal, as human beings, but we are not the same. Therefore we cannot conclude that if two things are equal they are the same. You seem to think that numbers are somehow special, so that if they are equal they are necessarily the same. I'm waiting for you to attempt to justify this belief, which to my understanding is demonstrably false.
Quoting TonesInDeepFreeze
Right, so the law of identity states that a thing is identical to itself. That's identity. Then we can proceed to say that a thing is indiscernible from itself, and this is consistent. Now, 2+1 is discernible from 3, so how do we conclude that they are the same?
Quoting TonesInDeepFreeze
Then how in hell are you supporting this obviously false assumption that "if 2+1 and are equal 3 then they are the same"?
Quoting TonesInDeepFreeze
The problem is that mathematicians do not use "=" in a way which is consistent with the law of identity. Therefore your "undefined" primitive is a violation of the law of identity. If the right and left side of the equation signified the exact same thing, as required by the law of identity and if equal signifies identical, then all equations would read like "X=X", or "Y=Y", or some other way of saying that the very same thing is represent on the right and the left. However, mathematicians use "=" to relate two distinct expressions with distinct meanings, which clearly do not signify the exact same thing.
Quoting TonesInDeepFreeze
This is the contradiction which altheist was trying to impose on me. If "2+1" signifies something different from "3", then it is impossible that what they denote is the same object by way of contradiction. If they are supposed to be signifying different predications, so contradiction is avoided, then no object is denoted, just two distinct predications without a subject, predicated of nothing.
Quoting TonesInDeepFreeze
That's nowhere near as bad as someone who routinely applies mathematics without recognizing the falsity of fundamental principles. I cannot understand the fundamentals because they are unsound. Contradiction or falsity make understanding impossible. But accepting contradiction, or falsity and proceeding to apply these principles is self-deception and misunderstanding.
Quoting fishfry
Talk about begging the question. That's what your so-called proof did.
Quoting TonesInDeepFreeze
All I see is "=" here. Where's the proof that "=" means the same as?
I don't recall you asking me such a question. If you did, then please link to the post where you asked it so that I can see the context. Anyway, there are infinitely many previously unwritten mathematical equalities that humans just happened not to have made judgements on yet. And I don't see any relevance to what I've said about equality.
Meanwhile, there are many decisive points I have raised that you have skipped.
Quoting Metaphysician Undercover
The case is that you can't read. I replied about the notion of human equality many many posts ago, and you skipped recognizing my reply, and I even mentioned a little while ago again that I had made that reply and you skipped that reminder too!
Quoting Metaphysician Undercover
I have not said that numbers are special regarding denotation.
Quoting Metaphysician Undercover
I have not use the term 'necessarily' in this context since 'necessarily' has a special technical meaning that requires modal logic.
Quoting Metaphysician Undercover
Yes we can, but that alone is not the principle of the indiscernibility of identicals.
Quoting Metaphysician Undercover
'2+1' and '3' have different senses but not different denotations. No matter how many times I point out the disctinction between sense and denotation, and even after I linked you to an Internet article about it, you keep ignoring it.
Quoting Metaphysician Undercover
Again, you skipped my reply much earlier in this thread. It is in the method of models that we have that equality is sameness.
Quoting Metaphysician Undercover
You've not shown any inconsistency.
Quoting Metaphysician Undercover
If I recall, I have not used the word 'signify'. Again, a term has both a denotation and a sense.
Quoting Metaphysician Undercover
Yes, since meaning includes both denotation and sense. I explained to you probably more than half a dozen times already that ordinary mathematics concerns itself only with denotation and that if you want to have sense handled also, then you need a more complicated framework.
Quoting Metaphysician Undercover
You've never even read page 1 in a book on the foundations of mathematics. So of course you can't understand anything about it.
Quoting Metaphysician Undercover
Again, I explained to you many posts ago that '=' maps to the identity relation per the method of models.
That itself is a judgement, that these unwritten equalities are equalities. Clearly equality remains a human judgement. See "equal" is a human concept. To say that there are equalities which humans haven't discovered, is to already judge them as equalities.
Quoting TonesInDeepFreeze
And I will continue to skip it because all you did was assert that equality in mathematics is more precise than equality in other subjects. The point being that in no subject does "equal to" mean "the same as", not even mathematics. As I explained the left side does not signify the same thing as the right. And your assertion of the precision of mathematics still doesn't get you to the point of being the same. Equal to, and the same as, are distinct conceptions.
Quoting TonesInDeepFreeze
This is exactly what you are saying. By insisting that "equal to" in the case of numbers means 'denotes the same object', you are saying that numbers have some special quality which can make two distinct but equal things into the same thing. You are claiming that numbers have a special status which makes equal things into the same thing.
.Quoting TonesInDeepFreeze
You don't have to use the word "necessarily", to mean it. When you say that being equal implies that they are the same, you refer to a logical necessity which dictates that if they are equal then they are necessarily the same. Otherwise it would be false to say 'if they are equal then they are the same'.
Which of course, is obviously false to say, because that necessity is based in a false premise.
Quoting TonesInDeepFreeze
I've already explained to you how you do not have the premise required to say that "2+1" denotes the same object as "3", when the two signify different things ("have different senses").
Remember your example? "The father of Peter Fonda" denotes a person in a particular relationship with Peter Fonda. That is the "sense". "Henry Fonda" also denotes a particular person. Again, that is the "sense". Now, you do not have the premise required to validly conclude that these two persons, indicated by those two senses, are the same person. The same thing is the case with "2+1" and "3". They signify different things (have different senses). Now, you do not have the required premise to conclude that they denote the same object. You can stipulate, as a premise, "Henry Fonda is the father of Peter Fonda", but that would be begging the question. Likewise, you can stipulate that "2+1" denotes the same object as "3", but that's simply begging the question. You are creating the premise required to support your desired conclusion, and that's a fallacy.
So no matter how many times you assert that despite the fact that "2+1" and "3" mean something different, they denote the same object, you have not produced a valid argument to prove this. All you've produced is the false premise that if they are equal then they are the same.
Quoting TonesInDeepFreeze
You mean that false premise?
Quoting TonesInDeepFreeze
Shouldn't we call this what it really is, the method of the false premise?
Over many posts, you keep telling me what I think or said, and you're wrong. You're a bane.
And you claimed that you asked me a question I didn't answer. I then asked you to link me to that, because I don't recall you asking me a question I did not answer recently. Please link me to it.
Quoting Metaphysician Undercover
What you just said, whatever its merit, doesn't vitiate anything I've said.
Quoting Metaphysician Undercover
Formal languages, including the language of identity theory, are more precise than natural languages. But the point I made was not so much about precision but that 'equality of human beings' in the sense of equal rights or whatever is a very different meaning of 'equality' in mathematics.
Quoting Metaphysician Undercover
Wrong. I'm not saying any of that.
Quoting Metaphysician Undercover
True, but I don't mean it.
Quoting Metaphysician Undercover
Nope. I am not bringing the notion of logical necessity into play.
Quoting Metaphysician Undercover
That's a non sequitur.
Quoting Metaphysician Undercover
You don't explain. You assert and then argue fallaciously.
Quoting Metaphysician Undercover
You said it both denotes and is its sense. Denotation and sense are different.
Quoting Metaphysician Undercover
One can stipulate premises and then infer conclusions. That is not question begging. Also, we don't have to stipulate that Henry Fonda is the father of Peter Fonda, since we can arrive at that claim by empirical or historical evidence.
Quoting Metaphysician Undercover
We prove that 2+1 = 3. Then we prove that '2+1' and '3' have the same denotation by the method of models. I've told you that about a half dozen times now.
Quoting Metaphysician Undercover
You've not shown any false premise in the method of models.
Now, please link me to the post in which you claim you asked me a question I did not answer.
If I did not ask it in the exact way that I repeated it, I apologize for the unclarity. But, here:
Quoting Metaphysician Undercover
Quoting TonesInDeepFreeze
These two senses utilize the same principle. They establish a value system and judge equality according to that value system. What differs is the value systems employed. In law they have legal values, rights, and in mathematics they have numerical or quantitative values. Since each refers to the specific aspect which it is designed for, neither provides what is sufficient for a judgement of "the same". Quantitative value is a single predication, therefore it does not suffice for a judgement of "same" which requires taking account of all attributes.
Quoting TonesInDeepFreeze
OK, so lets dismiss the notion of logical necessity. Let's assume that you say "if they are equal, then they are the same", and now you admit that you do not mean that they are "necessarily" the same, by any logical necessity. What good is such a principle? You apprehend things as equal, and you judge them as the same. But now you say that they are not necessarily the same.. So you are admitting now that your judgement of "the same" might in some cases be wrong.
Is this what you are arguing? You judge "2+1" as referring to the same thing as "3", because they are equal, but there is no logical necessity there, which proves that they are? If this is the case, then how do you know that they are the same? Don't you think that you might be mistaken just as often, or even more often then being correct in that judgement?
Quoting TonesInDeepFreeze
Sure, in that case we can refer to empirical judgement, but in the case of numbers we cannot, because we cannot sense numbers in any way. So your judgement that two equal things are the same thing is supported by no logical necessity, and no empirical evidence. Don't you think that this is a little flimsy?
Quoting TonesInDeepFreeze
I don't recall such a demonstration. Can you show me through your "method of models", how you prove that "2+1" and "3" have the same denotation? Then I can judge the soundness of that proof.
My points don't depend on whether equality is or is not independent of human judgement.
Quoting Metaphysician Undercover
To say that 2+1 and 3 are equal is saying that 2+1 is 3.
To say that John and Mary are equal (in the sense of equal rights) is not saying that John is Mary. Rather it is saying that the rights of John are the same as the rights of Mary.
These are very different uses of the word 'equal'.
Your argument is ridiculous.
Quoting Metaphysician Undercover
I don't "admit" in the sense of conceding or retracting some earlier point. I just never stated regarding necessity to begin with, and I don't state now because it would require a discussion about modality that is not needed to present the basic mathematical framework I've mentioned.
Quoting Metaphysician Undercover
It provides a clear and straightforward framework for doing mathematics.
Quoting Metaphysician Undercover
What do you intend the pronoun 'they' refer to there?
I don't say that '2+1' and '3' are equal. I say that 2+1 and 3 are equal.
But in a context where '2', '+', and '3' were symbols not standing for the number 2, the addition operation, and the number 3, then it may not be the case that '2+1' and '3' denote the same number in that context. And it is not necessary that symbols always denote the same. Denotation of symbols is by stipulation or convention not by necessity.
Quoting Metaphysician Undercover
(1) So in the empirical context, your objection was refuted.
(2) In the mathematical context, numbers are not physical objects. And over the course of this discussion I said that we arrive at mathematical conclusions by mathematical proof or by performing mathematical procedures. You are not caught up in the discussion because you ignore and skip.
Quoting Metaphysician Undercover
I can't cram it all into a post or even several posts. You need to read a book or other systematic presentation of mathematical logic in which the method of models is explained step by step, including the notions: concatenation functions, formal languages, signatures for formal languages, unique readability of terms and formulas, recursive definitions, mathematical induction, et. al. And prerequisite would be understanding basic mathematical notions, including: sets, tuples, relations, functions, et. al
As I explained, "equal" in both of these uses is based in a value system. If you truly believe that having an equal numerical or quantitative value justifies the assertion that the two things referred to, with the same value, are in fact the same thing, then you ought to be able to demonstrate to me your reasons for believing this. Suppose that I have two apples and you have two apples, are our apples the same, because they are equal quantity. Surely this is not the same as one of us having four apples despite the fact that there are four apples in that scenario. Are two objects which each weigh five kilograms the same object? I just don't understand where you get this idea that having an equal numerical value means being the same thing.
Quoting TonesInDeepFreeze
"They" refers to what "2+1", and "3", refer to it. I use "they" because it is plural, "2+1" refers to something, and "3" refers to something, hence there are two things referred to, and the plural "they".
Quoting TonesInDeepFreeze
This is incorrect, because there is no empirical object referred to by "2+1", or "3". So your act of introducing the empirical aspect of the Fonda example only makes the example irrelevant. To maintain relevance we must proceed, as I did, through logic only. Then to argue that the two phrases refer to the same object requires a question begging premise. This would make your argument invalid through that fallacy.
Quoting TonesInDeepFreeze
Right, there are proofs. Now I'm waiting for proof that "2+1" refers to the same object as "3". So far you've offered me only a false premise that if they refer to equal things, then they refer to the same thing. And you admit that you cannot back this up with any logical necessity, so it appears to me like you really recognize it as false. Of course you do, any rational human being of grade school education would recognize the falsity of that. Why argue so persistently that it's true?
The question at that point was about Henry Fonda and names for Henry Fonda, not numbers and names for numbers. You objected to my Herny Fonda example onto itself.
You're lost in the conversation.
Quoting Metaphysician Undercover
You skipped again. I told you that proof of same denotation is finalized in the method of models. You don't have to wait around to find out about it - you can find it many books. As I said, as you also skipped:
"I can't cram it all into a post or even several posts. You need to read a book or other systematic presentation of mathematical logic in which the method of models is explained step by step, including the notions: concatenation functions, formal languages, signatures for formal languages, unique readability of terms and formulas, recursive definitions, mathematical induction, et. al. And prerequisite would be understanding basic mathematical notions, including: sets, tuples, relations, functions, et. al."
I wouldn't expect someone to teach me, in the confines of a posting forum, the subject of molecular biology. You would be foolish to think I should do it for you about mathematical logic. You could educate yourself!
Come on TIDF, it must be a simple proof, if it exists, just like in the Fonda example, we look at the person denoted by "father of Peter", and also the person denoted by "Henry Fonda", and see that they are the same person. Why does this proof require concatenation functions, formal languages, signatures for formal languages, unique readability of terms and formulas, recursive definitions, mathematical induction, sets, tuples, relations, functions, et. al?
Those are concepts instrumental to a firm understanding of the method of models.
I find nothing about the "method of models" in my google search so I tend to think it is something you made up as a ruse, citing all these prerequisite subjects for understanding.
Under "Scientific Modelling" in Wikipedia I find this:
"Scientific modelling is a scientific activity, the aim of which is to make a particular part or feature of the world easier to understand, define, quantify, visualize, or simulate by referencing it to existing and usually commonly accepted knowledge."
Notice the explicit statement of "...to make a particular part or feature of the world easier to understand..". That's exactly what I said about the term "equal", it refers to a designated part, aspect, feature, or property of an object. Two distinct objects are said to be equal on the basis of modelling a part. The issue however, is how do you proceed from modelling a part, and concluding equality based on a model of that part, to making a conclusion about the whole?
You tend to think irrationally or not at all.
Below are some articles and a couple of online books. For a more systematic study, if I recall, earlier this thread I mentioned particular books I most recommend.
http://page.mi.fu-berlin.de/raut/logic3/announce.pdf
https://en.wikipedia.org/wiki/Structure_(mathematical_logic)
https://plato.stanford.edu/entries/model-theory/
https://plato.stanford.edu/entries/modeltheory-fo/
http://www.math.toronto.edu/weiss/model_theory.pdf
https://en.citizendium.org/wiki/Structure_(mathematical_logic)
https://faculty.math.illinois.edu/~vddries/main.pdf
I can't help it if your terminology is a little off the beaten path. You kept referring to a "method of models", and I couldn't even find that on google. Now I see you were really talking about model theory.
I took a look at your first reference. The book is directed at graduate students in mathematics, but it distinctly says in the preface that fundamental philosophical problems are not dealt with.
"Philosophical and foundational problems of mathematics are not systematically discussed within the constraints of this book, but are to some extent considered when appropriate."
I took at look at the second reference, and it does discuss "model theory", but I don't see how anything there can be used to prove that "2+1" denotes the same object as "3". The fact that mathematicians utilize that assumption does not prove that it is true.
I took a look at the third reference, and it tells me that in model theory the truth or falsity of a statement is understood to be dependent on the interpretation.
So, it appears to me, like you and I are both correct according to model theory. I interpret the statement "2+1" denotes the same thing as "3" as false, and you interpret it as true, and neither of us is wrong. We each interpret "2+1" differently and so, 'that "2+1" denotes the same thing as "3"' is false for me and true for you. Therefore we ought not even talk about whether it's true or false, because that's not something which could ever be determined. Is this conclusion correct? If so, then it clearly does not prove that "2+1" denotes the same object as "3".
I took a look at the fourth reference and it doesn't seem to be relevant.
I took a look at the fifth, and it just talks about structures as if they are objects, so it seems like this article simply assumes what you need to prove. By the way, most these articles you refer seem to have that problem. Your task is to prove that "2+1" refers to the same object which "3" refers to, not to show me instances where this is taken for granted. I already know, from your behaviour and the behaviour of others, that this is taken for granted. There is no need to prove that now.
And so I find the same problem with the sixth reference. It states right of the bat: "In this
course we develop mathematical logic using elementary set theory as given..."
What sort of proof is this, which takes what you are tasked with proving as a given? I think you are simply continuing with your fallacy of begging the question.
So you opted to suggest that I'm lying about the whole thing instead of just asking "Would you please provide some links?" Another example of your jejune approach to mathematics and discussion about it.
And what I wrote:
Quoting TonesInDeepFreeze [emphases added]
So you could have easily searched 'mathematical logic method of models' or just asked me. Instead you burden me with the suggestion that I'm lying about the subject because you can't be bothered to make a reasonable search. Then when you look at the links, you only skim a few parts of them, then pick a few items in them out of context and misconstrue them. And you're welcome, by the way, for the links I provided, even though your response to them is itself incorrect - including incorrectly summarizing their level, relevance, content and import.
The way to learn the subject is not by perusing articles and parts of books out of context, but rather by first starting with at an introductory level book on symbolic logic.
I still think you're lying. I don't believe there is any such thing as proof that "2+1" denotes the same object as "3" does. I think it's false, and I think you know it's false. But you're in denial, and you've come up with this proposition that the "method of models" provides a proof, as a ruse.