The Foundations of Mathematics
In Questions V and VI of his Commentary on the De Trinitatate of Boethius, Thomas Aquinas distinguishes three degrees of abstraction as fundamental to the difference between physical science, mathematics and metaphysics. (See Armand Maurer, The Division and Methods of the Sciences for translation and comments.) Briefly: the first or physical degree of abstraction considers being as changeable, with the understanding that all material things are changeable; the second degree of abstraction considers being as quantifiable, prescinding the changeability of matter; The third degree of abstraction considers being as being, with no need for it to be material.
To get a sense of this with respect to the foundations of mathematics, we learn to count by counting concrete objects (pennies, apples, paper clips), but eventually we come to see that counting does not depend on what is counted. In coming to this insight, we fix on being as countable to the exclusion of the other notes of intelligibility found in what we are counting. In doing so, we see that certain (mathematical) relations inhere in countable beings (units, objects, elements) simply in virtue of being countable -- seeing the truth of number theory and set theory axioms. Similarly, in considering being as extended and measurable, we grasp the axioms of geometry, topology, analysis, and so on. In knowing these axioms, we see relations that obtain in nature, not in virtue of the kind of objects we are dealing with, but in virtue of the mathematical properties they instantiate.
Opposed to this view, we have (1) Platonism, which sees mathematical truths as existing in some abstract ideal realm; and (2) the movement characterized by Hilbert's program, which sees mathematical truths as reducible to logical truths.
The problems with Platonism have been known since Aristotle wrote his Metaphysics. I counted seventeen problems with Platonism when I read the Metaphysics. Chiefly: (1) Platonism leaves the process of learning universal truths and the epistemological role of examples inadequately explained. (2) It leaves unexplained how Platonic ideals are instantiated in nature. (The problem of "participation.") (3) It leaves unexplained how we can recognize a new instance of a universal we know. (If this example can evoke the concept now, why can't a similar example evoke the concept initially?) (4) It leaves unexplained how mathematical truths that exist only in the Platonic realm can apply to reality.
In this last point, how can the Platonic relationship 2 + 2 = 4 tell us that if we have two apples and two oranges, we have four pieces of fruit? Or, when physicists include mathematical premises in their arguments, how can these arguments be sound, if its mathematical premises say nothing of nature, and so are not "true" in a sense that makes them applicable to nature? (Not just relations that "exist" in the Ideal realm, but ones that can be trusted to tell us about the natural world?)
Hilbert's Program was effectively destroyed by Godel. Yet, Godel's work shows more: it shows that there are truths that cannot be deduced from any knowable set of axioms. Thus, there are intelligible mathematical truths that are not a logical consequence of the axioms that logicalists may take to be definitive of a mathematical theory. That means that there is more to math than is captured by the axioms. On the Thomistic view, what more there is, the unprovable truths, are the intelligible, but not yet actually understood, relations obtaining in nature.
Aquinas's abstraction theory solves another problem left open by Godel's work, viz. how we can know that an axiomatization is consistent. Since it is impossible to instantiate a contradiction, any set of axioms that are co-instantiated in nature are consistent -- and surely we cannot abstract universal truths that are not instantiated.
So, how does this apply to contemporary mathematics?
We may divide the axioms into three classes.
1. Most axioms are abstracted from our experience of nature as countable and measurable. To be concrete, children learn to count by counting particular kinds of things, but soon learn that the act of counting does not depend on the kind of thing being counted, only on its being countable. Thus, we abstract concepts such as unit and successor from the experience of counting real-world objects. This is the empirical basis of arithmetic and its axioms.
Since we are dealing with axioms abstracted from, not hypothesized about, reality, there is no need for empirical testing for them to be known experientially. Further, since the axioms are instantiated in reality, which cannot instantiate contradictions, we know that such axioms are self-consistent without having to deduce their self-consistency.
As we can trace our concepts to experiences of nature, and since there is no evidence concepts exist outside the minds of rational animals, there is no reason to posit a Platonic world. Doing so is unparsimonious and irrational.
2. Other axioms are hypothetical.
a. Some hypothetical axioms can be tested, e.g. the parallel postulate, which can be tested by measuring the interior angles of triangles.
b. The remaining hypothetical axioms can't be tested, e.g. the axiom of choice. These are unfalsifiable and unfalsifiable hypotheses are unscientific. As they are unscientific, pursuing their consequences is merely a game, no different in principle than any other game with well-defined rules, such as Dungeons and Dragons.
I am open to criticism and comments.
To get a sense of this with respect to the foundations of mathematics, we learn to count by counting concrete objects (pennies, apples, paper clips), but eventually we come to see that counting does not depend on what is counted. In coming to this insight, we fix on being as countable to the exclusion of the other notes of intelligibility found in what we are counting. In doing so, we see that certain (mathematical) relations inhere in countable beings (units, objects, elements) simply in virtue of being countable -- seeing the truth of number theory and set theory axioms. Similarly, in considering being as extended and measurable, we grasp the axioms of geometry, topology, analysis, and so on. In knowing these axioms, we see relations that obtain in nature, not in virtue of the kind of objects we are dealing with, but in virtue of the mathematical properties they instantiate.
Opposed to this view, we have (1) Platonism, which sees mathematical truths as existing in some abstract ideal realm; and (2) the movement characterized by Hilbert's program, which sees mathematical truths as reducible to logical truths.
The problems with Platonism have been known since Aristotle wrote his Metaphysics. I counted seventeen problems with Platonism when I read the Metaphysics. Chiefly: (1) Platonism leaves the process of learning universal truths and the epistemological role of examples inadequately explained. (2) It leaves unexplained how Platonic ideals are instantiated in nature. (The problem of "participation.") (3) It leaves unexplained how we can recognize a new instance of a universal we know. (If this example can evoke the concept now, why can't a similar example evoke the concept initially?) (4) It leaves unexplained how mathematical truths that exist only in the Platonic realm can apply to reality.
In this last point, how can the Platonic relationship 2 + 2 = 4 tell us that if we have two apples and two oranges, we have four pieces of fruit? Or, when physicists include mathematical premises in their arguments, how can these arguments be sound, if its mathematical premises say nothing of nature, and so are not "true" in a sense that makes them applicable to nature? (Not just relations that "exist" in the Ideal realm, but ones that can be trusted to tell us about the natural world?)
Hilbert's Program was effectively destroyed by Godel. Yet, Godel's work shows more: it shows that there are truths that cannot be deduced from any knowable set of axioms. Thus, there are intelligible mathematical truths that are not a logical consequence of the axioms that logicalists may take to be definitive of a mathematical theory. That means that there is more to math than is captured by the axioms. On the Thomistic view, what more there is, the unprovable truths, are the intelligible, but not yet actually understood, relations obtaining in nature.
Aquinas's abstraction theory solves another problem left open by Godel's work, viz. how we can know that an axiomatization is consistent. Since it is impossible to instantiate a contradiction, any set of axioms that are co-instantiated in nature are consistent -- and surely we cannot abstract universal truths that are not instantiated.
So, how does this apply to contemporary mathematics?
We may divide the axioms into three classes.
1. Most axioms are abstracted from our experience of nature as countable and measurable. To be concrete, children learn to count by counting particular kinds of things, but soon learn that the act of counting does not depend on the kind of thing being counted, only on its being countable. Thus, we abstract concepts such as unit and successor from the experience of counting real-world objects. This is the empirical basis of arithmetic and its axioms.
Since we are dealing with axioms abstracted from, not hypothesized about, reality, there is no need for empirical testing for them to be known experientially. Further, since the axioms are instantiated in reality, which cannot instantiate contradictions, we know that such axioms are self-consistent without having to deduce their self-consistency.
As we can trace our concepts to experiences of nature, and since there is no evidence concepts exist outside the minds of rational animals, there is no reason to posit a Platonic world. Doing so is unparsimonious and irrational.
2. Other axioms are hypothetical.
a. Some hypothetical axioms can be tested, e.g. the parallel postulate, which can be tested by measuring the interior angles of triangles.
b. The remaining hypothetical axioms can't be tested, e.g. the axiom of choice. These are unfalsifiable and unfalsifiable hypotheses are unscientific. As they are unscientific, pursuing their consequences is merely a game, no different in principle than any other game with well-defined rules, such as Dungeons and Dragons.
I am open to criticism and comments.
Comments (105)
So do you think the law of the excluded middle, or the Pythagorean theorem, only came into existence with h. sapiens; or that such principles are eternal, and are discovered by any intelligence sufficiently rational to discern them?
I think the error you're making with the 'Platonic world' is to try and conceive of it as a literal domain. But what of the 'domain of natural numbers'? Surely that is something real, as real numbers are included in it, and irrational numbers are not. So it is a 'domain', which one may or may not have knowledge of, but it's not a literal domain or 'place'; real, but not existing, is the way I think of it.
And how does your account differ from run-of-the-mill evolutionary naturalism, in which there is nothing corresponding to what Aquinas would deem the soul, which is 'capable of existence apart from the body at death'? Your account most resembles that of John Stuart Mill, whom I'm sure would not be the least inclined to agree with Aquinas.
Do you know that Godel considered himself a mathematical Platonist?
[quote=Rebecca Goldstein]Gödel was a mathematical realist, a Platonist. He believed that what makes mathematics true is that it's descriptive—not of empirical reality, of course, but of an abstract reality. Mathematical intuition is something analogous to a kind of sense perception. In his essay "What Is Cantor's Continuum Hypothesis?", Gödel wrote that we're not seeing things that just happen to be true, we're seeing things that must be true. The world of abstract entities is a necessary world—that's why we can deduce our descriptions of it through pure reason. 1 [/quote]
With regard to Platonic Forms, what the One itself is remains. But the questions of the One itself and the One and the many do not concern the mathematician.
The classic modern work on this is Jacob Klein's Greek Mathematics and the Origins of Algebra.
I found your essay illuminating as I don’t have any experience in mathematical theory.
What is the axiom of choice?
Does the Incompleteness Theorems say really this? Correct me if I'm wrong, but doesn't (the first Gödel Incompleteness Theorem) say that for any 'set of axioms' or consistent formal system there exists specific true but unprovable statements. That's a bit different
Quoting tim wood
I agree here with tim wood, talking about scientific/unscientific here with foundations mathematics is totally out of whack.
Yes, I am sure.
Yes, to be instantiated is to be particular. We move from the particular to the universal by removing particularizing notes of comprehension. For example, we ignore that we are counting apples, that apples are fruit, etc. and fix on the noted that matter to counting, i.e. that we are dealing with an instance of a unit and that unities are countable.
Contrast this with the Hume-Mill model of induction. In it you see 100 black crows, no white crows and posit "All crows are black." Whereas abstraction is a subtractive process, Hume-Mill induction is an additive process. We add the assumption that the cases we have not seen are like the cases we have seen. In abstraction, we add nothing. We merely remove notes of comprehension that don't interest us.
Re transfinite numbers: We come to the notion of Aleph-0 (countable) infinity by noting that the counting process has no intrinsic limit. We come to the notion of Aleph-1 (uncountable) infinity by proving that the numbers we assign to the points of continuous extents cannot be counted. This was done by Georg Cantor in his 1874 uncountability proof.
Quoting tim wood
I am not saying that nature is all that is, only that we have no reason to posit a Platonic realm of ideas.
Quoting tim wood
It is, unless you have a different definition of "concept," but then, you're not talking about what I'm talking about.
Quoting tim wood
Yes, the content may well be "there" as intelligible, as something capable of being known, but not yet known -- and so as not yet a concept. That's what it is for a concept to be grounded in reality.
Quoting tim wood
Specifically?
Quoting tim wood
Well, that is not how the game is played now, but that doesn't mean the game is played rationally, does it? The value of Popper's falsifiability criterion is that it restricts hypotheses to ones we can gain intellectual traction on. If you allow hypotheses that cannot be tested, then any guess, no matter how irrational, can be posited.
Of course, the test might not be experimental. While Godel's work means that we cannot prove the consistency of a set of axioms, it doesn't prevent us from proving their inconsistency. So, we could have mathematicians deducing hypothetical consequences in the hope of finding an inconsistency. If they do, they'd have proven an axiom was false. But, if they don't, they won't know that any of their hard-won conclusions are true.
As you note, we can't prove that the axiom of choice is true or false in the context of ZF. That leaves it unfalsifiable.
As for "games," what would you call playing by rules that are either false, or ungrounded in reality?
Quoting tim wood
What would the difference be? It's a game many mathematicians enjoy?
I am not positing that we're the only rational animals. I am saying that the principle of excluded middle reflects the nature of being and so is intelligible to anything capable of abstracting being as being. I am not saying that the Pythagorean theorem is true in general, only in flat spaces (where the parallel postulate holds).
Quoting Wayfarer
If someone wishes to define a "Platonic world" in non-literal way, I will be happy to comment on their effort. If it's undefined, it is irrational to appeal to it as an explanation.
The domain of rational numbers is countable objects. It is unlimited because we can partition unities into countable parts indefinitely.
Quoting Wayfarer
Irrational numbers are not based on countable objects. That does not mean that they are not based on other aspects of nature, i.e. measurable quantity. In neither case do actual numbers exist in nature. What exists in nature is the potential to be counted or measured. Actual numbers exist only in minds actually thinking them.
Quoting Wayfarer
I distinguished abstraction from the Hume-Mill model of induction in my previous response (to Tim Wood). I wouldn't mind if my account were based on the aspects of reality fixed upon by naturalists, but, in point of fact, it does not. It requires the operation of an intellect in act (Aristotle's agent intellect) to make what is merely intelligible actually known.
Naturalism has no problem describing neurophysically encoded contents, but it has no rational account of awareness. Daniel Dennett showed in Consciousness Explained that naturalism cannot account for the experience of consciousness, and David Chalmers has pointed out the difference between the kind of progress made by neuroscience and the "hard problem of consciousness." In a previous thread I discussed the difference between intentional and material reality.
So, any account that hinges on the actualization of notes of intelligibility by awareness (as mind does) is beyond the scope of evolutionary naturalism.
Quoting Wayfarer
No, I did not, but as many mathematicians are, it's not surprising. Being a good mathematician doesn't make one a good philosopher.
The Wikipedia provides a good discussion. It says "Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if the collection is infinite."
The basic problem with it is the same as that with the parallel postulate -- we have no experience of the infinite per se, from which to abstract it.
Personally, it seems reasonable, but then so did the parallel postulate when I studied geometry. My criticism is based on purely its lack of epistemological justification.
It does not seem different enough to vitiate my point. In any lifetime, or finite number of lifetimes, we can only go through a finite number of axiom sets. So, there are true axioms we cannot deduce. Or, am I missing your point?Quoting ssu
Is it? If we cannot justify certain axioms, how can we rely on the conclusions? As a physicist, I want my mathematics to be not merely consistent, but applicable to the physical world. We know that, however reasonable, the parallel postulate is not so applicable if we define straight lines as geodesics. How are we to know that the consequences of ZFC fair any better?
So, you will have to explain why my criticism is "totally out of whack."
But, you also say
Quoting Dfpolis
What I'm questioning is the notion that an account can be given of intelligible objects (such as number) in purely mentalistic terms. I think that Platonic realism posits that numbers are real for anyone who can count. So they are only knowable to a mind, but they are not the product of an individual's mind. The same goes for all manner of logical and arithmetical principles; they are 'discerned by reason', but they're not only the product of the mind (although once we are able to understand such ideas we can think of many more.)
Quoting Dfpolis
I think it's reason that naturalism has no account of. What Chalmers and many other modern philosophers call 'experience', I think is actually the meaning of 'being' - as in 'human being'.
You (in particular) might appreciate an interesting passage that I have posted on forums many times over the years, Augustine on Intelligible Objects. I freely acknowledge my learning of these materials is sketchy at best, but this passage really resonates with me. I wonder what you make of it.
Fair enough.
First of all, with 'scientific' we describe that we are using the scientific method, an empirical way to make objective observations, experiments, tests or measurements, about reality, the physical world as you mention, to solve if our hypothesis are correct or not. Mathematics is logical system. Applicability of mathematics to the physical world isn't the logic that glues mathematics into a rigorous system, but logic itself. Above all, something that we have thought to be a mathematical axiom isn't shown to be false from physical reality, but with mathematical logic.
Especially if you use the term 'unscientific' it makes even a more confusing relation:
Quoting Dfpolis
Here you seem to have the idea that if the axiom of choice is independent of ZF, it is somehow 'unscientific' as if other axioms would be the 'scientifically' approved. Well, just look here on this site how utterly confused people are about infinity and try to then reason that axiom of infinity is scientific. But there there the axiom is, in ZF. On the other hand, the axiom of choice (AC) has a lot of equivalent findings in mathematics like Zorn's lemma, Tychonoffs theorem, Krull's theorem, Tukey's lemma and the list goes on and on. To say in this case that all of the math in all of those various fields of mathematics are unscientific is, should I say, out of whack.
In fact the independence of AC just shows how huge gaps we still have in our knowledge of the foundations of mathematics. If something is flimsy, it's more likely the whole notion of ZF, because the whole reason for ZF to have been made in the first place is to counter Russell's paradox. Yet when we have these independence results (and undecidability results), for me these seem to show that not all is there yet. It may be that ZF could likely be itself 'unscientific', but that will only be proven by logic, not with a physical test or measurement.
To make my argument short, scientific/unscientific is a poor definition in math, far better would be to speak of logical and illogical. We have had and can indeed still have illogical presumptions (or axioms) of the nature of math, just like some Greeks thought that all numbers had to be rational and were truly disappointed when finding out that there indeed were irrational numbers.
It's a subtle point only. Mainly that if for every consistent formal system there exists specific true but unprovable statements, that doesn't actually mean that there are true but unprovable statements in every formal system.
Again, in math we aren't confined to what is physically possible (physically countable, physically computable), as we use infinity so much everywhere in math.
The title of your topic is The Foundations of Mathematics. The neoplatonic One Identified as God has nothing to do with the foundations of mathematics or anything I said. You have completely ignored the foundation of Greek mathematics which makes your pseudo-problem of counting disappear. 2 + 2 = 4 is not a "Platonic relationship", at least not for Plato or the foundation of Greek mathematics.
Providing a purely mentalistic account is exactly what I am not doing. I am saying that our mathematical concepts have a foundation in reality. If there were no countable beings and no measurable beings, we could have no experience from which to abstract mathematical concepts.
While we may combine concepts in ways not found in reality, ultimately our concepts are traceable to the actualization of intelligibility found in nature. Objects act on our neural system is our senses, presenting intelligibility to awareness, Still, until we turn our awareness to these encoded contents, they are not actual concepts. It is our being aware of contents that makes what was merely intelligible actually known.
Further, engendered concepts are dynamically united to a corresponding intelligibility in nature, to the engendering object. First, the object's modification of my neural state is identically my neural representation of the object (encoding contents I can become aware of). Second, a single act of awareness simultaneously actualizes both the object's intelligibility (making it actually known) and the subject's capacity to learn (making it actually informed). So, there is no isolated mental construct here, but a (partial) ontological penetration of the intelligible object into the knowing subject.
This analysis does not, therefore, make numbers a mental product. It makes them the actualization of a prior intelligibility in nature -- eliminating the need for numbers to actually exist prior to being known -- in other words, the need for a Platonic realm.
Quoting Wayfarer
To me, "awareness" means agent intellect, without which there is no reason or conscious experience.
The link to Augustine is not available. I would like to see it if you have it (message it?). I am a great admirer of Augustine's insight, and think his account of the soul coming to know intelligibility is largely compatible with my understanding of Aristotle and Aquinas.
That is certainly the modern usage, but not the only one. Traditionally, scientia meant an organized body of knowledge -- organized in terms of explanations reducible to first principles. So, I would say that mathematics is a science in the sense of being an organized body of knowledge -- and that knowledge is an understanding of reality.
Second, "logical system" needs more explanation. All sciences proceed logically. I suspect that you mean a "closed system," i.e. one that simply elaborates an axiom set. Clearly, mathematics does more than that. I agree that mathematicians seek to put their science into a canonical form which is axiomatic, but they also have a history of examining and adding to their axiom sets. For example, we have the questioning of the parallel postulate and the development of non-euclidean geometries, the development of set theory, and discussion of the axiom of choice.
I'd suggest that the finished form of a science is a poor starting point for examining the nature of that science. Isn't it better to reflect on the process leading to the canonical form?
Also, "logical system" is inadequate if there are truths within the scope of the science that cannot be deduced axiomatically. The very existence of such truths implies that there are means of knowing truth more fundamental than the system's logical/axiomatic foundations.
Quoting ssu
No, math is not logic. That was Hilbert's view and Godel killed it. Math provides physicists with a set of assuredly true premises, which, combined with other, more empirical, premises will provide a sound conclusion if the empirical premises are true. What is special about mathematical premises is their reliability compared to less reliable Hume-Mill inductions and the hypothesis under consideration. Thus, when an experiment falsifies a conclusion, we can rule out the falsity of the mathematical premises as the cause, and fix our attention, first on our hypothesis, and second on our empirical inductions.
Typically a physicist uses a argument of the form:
Major (mathematical) premise: All systems instantiating mathematical property p are such that they instantiate mathematical property q.
Hypothesis: System A instantiates mathematical property p.
Conclusion: System A instantiates mathematical property q,
where we hope that the conclusion can be experimentally tested. The only "logic" here is a valid syllogistic form (Barbara), not mathematics. Our mathematical knowledge enters as a premise on the same footing as any other premise. For the argument to work, the mathematical premise must be adequate to reality (true).
Quoting ssu
This is inconsistent with your earlier claim that mathematics as a logical system. In such a system the truth of axioms is unquestioned. An axiom set may prove to be inconsistent, but if it is that only shows that some member of the set is false, not that a specific axiom is false -- unless one has metamathematical reasons for suspecting a particular axiom. But, if one does, then we are justified in examining the truth of axioms, and not merely accepting them as given a priori.
Quoting ssu
No, not at all. If you go back to the OP, you will see that I divide axioms into three groups: (1) justified by abstraction, (2) falsifiable, and (3) unjustified by abstraction and unfalsifiable, and so unscientific. The question is not whether we are doing math, but how we justify truths in general. It is special pleading to say that all scientific (in the older sense) truths need be justified, but mathematical premises get a pass.
Perhaps the problem here is that some mathematicians see themselves as enlightened by mystic insight into the realm of Platonic truth. I do not.
So, the problem is not that C is independent of ZF, it is that C is unjustified and seems to be unjustifiable. Perhaps I'm wrong on C being unfalsifiable. Perhaps some consequent of C can be falsified. Whether my example holds, however, is a contingent matter,and irrelevant to the need for justification.
Quoting ssu
I do not see this. After Laplace published his Celestial Mechanics and dispensed with Newton's ill-conceived "hypothesis of God," physics was seen as axiomatic and given canonical form by mathematical minds such as Hamilton and Lagrange. That did not make it true in any absolute sense. The classical mechanics they developed is still useful and taught today, but it is not true. Unjustified foundations necessarily give unjustified conclusions.
Quoting ssu
First, I have problems with the theory of types as a solution to Russell's paradox, but that is for another day.
What validly follows from axioms is necessarily "logical," but if the axioms are unjustified it has little claim on being true. If we understand science as organized knowledge, and what is unjustified does not count as known, then any consequent of an unjustified axiom is unscientific.
And that relates to my OP how?
Once again, the title of your topic is "The Foundations of Mathematics". Those foundations are not in modern mathematical theory or methodology. Greek mathematics is part of that foundation. Greek mathematics is not "Platonism".
To say:
Quoting Dfpolis
Is like saying a building has a foundation in the ground. It says nothing about that foundation.
Quoting Dfpolis
Instead, it would be more useful to direct the reader to Maurer's "Thomists and Thomas Aquinas On the Foundation of Mathematics", available free online: http://www.u.arizona.edu/~aversa/scholastic/Thomists%20and%20Thomas%20Aquinas%20on%20the%20Foundation%20of%20Mathematics.pdf
From that paper:
There are important consequences of Aquinas's placing the notions of mathematics in the second order of his quaestio disputata instead of the first. Unlike concepts on the first level, those on the second do not properly speaking exist outside the mind. Their proper subject of existence is the mind itself. They are not signs of anything in the external world. Hence mathematical terms cannot properly be predicated of anything real: there is no referent in the external world for a mathematical line, circle, or number. Finally, mathematical notions are not false; but neither are they said to be true, in that they conform to anything outside the mind. Aquinas does not suggest that they might be true in some other sense. (56)
First, sciences do not establish their own principles, so it would be very surprising if math did. So, we agree on the first part.
Second, I did not claim that Greek math was Platonism. So, I have no idea what you aim to show by the last sentence.
Quoting Fooloso4
If that were all I said, your criticism would be justified. it is not all I said. I said most of the foundations are the result of abstraction. In response to questions, I went on to explain how that worked. I also said that the rest were hypothetical -- and some of those were falsifiable and the rest not.
Quoting Fooloso4
Thank you for the reference, I did not know it.
I disagree with much of the quote you gave from Maurer. Every degree of abstraction is grounded in intelligible reality. It is true that there are no prefect triangles, etc. in empirical realty, but abstraction can leave behind the intelligibility of defects. I don't think anyone has ever come to the idea of a triangle without experiencing an imperfect instance in reality. The same is true of the other examples Maurer gives.
Where do you imagine these principles come from?
Quoting Dfpolis
After a full paragraph on Platonism you said:
Quoting Dfpolis
This is not a "Platonic relationship", it is simple arithmos, the counting of ones or units.
Quoting Dfpolis
To say what they are the result of is not to say what they are. The most basic concepts of of Western mathematics underwent a fundamental change with the origin of algebra, that is when numbers were replaced by symbols. Which leads to the question of whose mathematics?
Quoting Dfpolis
Are you disagreeing with his reading of Aquinas? If so, where do the mistakes lie? Or is it that you are disagreeing with Aquinas?
Yes, well, crocodiles and dragonflies have some degree of awareness, but zero intellect! I agree that science can't explain consciousness, but the problem boils down to the fact that consciousness is intrinsically first-person, something of which one is subjectively and immediately aware, or rather, 'that which is aware', and as such is never an object of experience (except for by abstraction). The precise reason why Daniel Dennett refuses to accept that it's real, is because it's not an object of experience.
Quoting Dfpolis
Thank you for very lucid explanation. We’re almost in agreement, but there’s a subtle point that I want to get at. I have the idea that numbers and other intelligible objects are not existent (as are sensory objects), but that they are real. Numbers do not come into or go out of existence, and when we know them, we know them purely intelligibly, i.e. they are only discernible to a rational intellect (which is the thrust of the passage in Augustine). So the statement that they 'exist in a Platonic realm' is misleading, because it is dealing with them on the same level as phenomena, i.e. as actual objects that exist in some domain. But they're not really objects, they're constituents of thought - so the word 'object of thought' is in some sense a metaphor. (I regard 'objects' as exactly that - things that you have a subject-object relationship with, i.e. everything around you.)
So I believe numbers and universals have a different mode of existence than do phenomena; but that the notion of 'modes of existence' has itself been lost to modern thought, as a consequence of the ascendancy of nominalism and then empiricism. The Thomists and neo-Thomists are about the only people who understand this, seems to me.
You should be able to find the Augustine quote by googling the exact phrase 'Augustine intelligible objects' which should find it in Google Books. There's a passage of numbered paragraphs in the Cambridge Companion to Augustine which lays it out. Marvellous.
Quoting Dfpolis
No. I think it is you who define "logical" or "logical system" in an extremely narrow way and as logical meaning as a "closed system". The way I see here math to be logical that simply every mathematical truth has to be logical. It doesn't state AT ALL that everything in math has to begin from a small finite set of axioms. What Hilbert was looking for was something else, especially with things like his Entscheidungsproblem.
Quoting Dfpolis
Ok, there's your problem right above. What you are making is a hugely reductionist argument that everything has to be deduced from the same axioms. If something doesn't fit to be the universal foundation, in your terms it has to be false and whole fields have to be false. Classical mechanics works just as classical geometry works. There being quantum mechanics or geometries of spheres etc. simply don't refute one another and make the other untrue or false. What is only wrong is the reductionist idea that everything can be deduced from one system or the other.
Math isn't like this. Mathematics has for example incommensurability, which is totally logical.
Each field of math assumes its principles (its postulates and axioms), but that does not mean that the principles can't be investigated and justified by nonmathematically. I have said how the principles of math derive -- mostly via abstraction, some hypothetically. Investigating and justifying these means is outside of the scope of mathematics and the axiomatic method.
Quoting Fooloso4
Please read sentences in context.
Quoting Fooloso4
What they are is not my present interest.
Quoting Fooloso4
The concepts that existed before the addition of unknowns, variables, functions and distributions continue in use today. Adding new concepts does not vitiate old concepts.
Quoting Fooloso4
Mathematics is not personal property. It is an intellible whole that becomes increasingly actualized (actually known) over time. At any time some people know more than others. That does not make them owners, but knowers.
Quoting Fooloso4
As I have thought about the topic, but not read Maurer's paper yet, I am not in a position to say where the errors originate. If they are in Aquinas, they would not be my first disagreement with the Angelic Doctor.
They have a degree of responsiveness that seems fully explainable neurophysiologically. We have no data implying such animals can actualize intelligibility as opposed to sensibility. I disambiguated my use of "awareness" by saying that it was the same as Aristotle's agent intellect -- whose function is the actualization of intelligibility.
Quoting Wayfarer
Yes, and no. I agree with most of what you say, but we would not know we were aware if we did not experience our own awareness. The problem, then, is not lack of experience, but lack of third person experience. Dennett rejects consciousness because he is unwilling to credit first person experience.
Logical positivists used to say consciousness/awareness was not intersubjectively available. It is. What is unavailable is multiple subjects observing the same token of consciousness. Science is not concerned with token availability, but with type availability. Natural science experiments are intersubjectively available even if only one person at a time observes a result -- as, for example, in the Rutherford scattering experiment. So, the number of observers of result token is irrelevant. What is relevant is type replicability -- and that is intersubjectively available.
Quoting Wayfarer
You're welcome.
Quoting Wayfarer
Yes, Augustine was inclined to Neoplatonism and that's a powerful intuition. I just don't see evidence to support it. It seems to me that we should rule it out on grounds of parsimony -- we don't need it to explain how we know mathematical truths.
Quoting Wayfarer
I agree with most of this, but "constituents of thought" bothers me. While we often reify ideas, it seems to me that the idea
I tried Googling "Augustine intelligible objects" but only found secondary sources so far.
https://books.google.com.au/books?id=3ISYAwAAQBAJ&lpg=PA24&ots=qR3TB2kKBD&dq=augustine%20intelligible%20objects&pg=PA24#v=onepage&q&f=false
We seem to be converging. I see good history as the result of rigorous method, but not as explaining events from first principles.
Quoting ssu
I have no problem with this. My point was that logic is necessary in all sciences. Of course, the amount of empirical data and the role of hypotheses varies widely. The point of my classical mechanics example was that it is a closed, axiomatic structure, within which one may deduce theorems in the same way that one deduces them in math. Still it is not math, and it is not true in any absolute sense.
Quoting ssu
Not at all. I believe in open philosophy -- the idea that we should be open to all sources of truth and not restrict our inquiries with a priori assumptions or conceptual spaces. I do, however, see each mathematical theory as defined by its axioms.
Quoting ssu
That is precisely the notion I reject.
I am saying that axioms are no different than any other claims. They are either justifiable, or not. Either adequate to reality (true) or not. Mathematics cannot be exempted from epistemological scrutiny just because it has a canonical, axiomatic form.
Quoting ssu
Of course the fact that classical mechanics fails to predict phenomena at quantum or relativistic scales, means that it is inadequate to these kinds of realities and so false in an absolute sense.
Geometry is a little different, as it lacks operational definitions of basic concepts such as
I am not a reductionist. For example, biology cannot be reduced to physics because some of the contextualizing data that is abstracted away in physics is the data on which biology is built. So, we must continually return to reality, to the experience of being, to correct our conceits.
Quoting ssu
I have no idea what you mean by "totally local." Are you claiming that the concept
It seems to me that you are looking at final canonical forms and forgetting the mental processes that got us there.
You are avoiding the question. Science does not simply "assume its principles". It determines them through observation, hypothesis, testing, theory, modeling, and so on.
Quoting Dfpolis
First, someone has to do the abstracting. Second, the properties of say a triangle are not determined by abstraction.
Quoting Dfpolis
You mean this context?
Quoting Dfpolis
2+2=4 does not exist only in the "Platonic realm", does not need to "apply to reality", and it is meaningless to call it a Platonic relationship. It does not apply to reality because it is counting something real.
Quoting Dfpolis
So, in a topic entitled The Foundations of Mathematics, the actual foundations of mathematics is not your present interest.
Quoting Dfpolis
It is not simply adding new concepts, it is a matter of different concepts. This does not vitiate old concepts in the sense that they are wrong, but that mathematics no longer operates according to the older concepts. But this is not simply an issue of mathematics but for philosophy.
If you are interested, the following will give you some sense of what is at issue: https://www.unical.it/portale/strutture/dipartimenti_240/dsu/Klein,%20Concept%20of%20Number%20Copy.pdf
Quoting Dfpolis
It is a question of assumptions and conceptual framework. As I pointed out, there is no number 0 or 1 in Greek mathematics. You might dismiss this as simply wrong, but in doing so what you miss is the ability to understand a way of looking at the world that is not our own.
This is just a verbal difference. Scientists certainly do, and that it my point: axioms need justification. The verbal difference is in how to define a science like math. Some would say that no science justifies its own assumptions, others look at what people who call themselves scientists or mathematicians actually do. I don't care how you define a science such as math. We seem to be agreeing on what is and needs to be done.
Quoting Fooloso4
Yes, we need knowing/observing subjects. And, yes, abstraction does not create content, it actualizes intelligibility already present in reality.
Quoting Fooloso4
The context was that of showing the consequence of the questionable claim that 2 + 2 = 4 exists in a Platonic realm. It was not me stating my own position.
Quoting Fooloso4
My interest is how the foundations are justified. The actual foundations are of interest only as examples of claims made and needing justification. I have used a number of actual postulates in that way. Enumerating all the postulates in all branches of mathematics would not help us understand the justification processes. It would only be a distraction.
Quoting Fooloso4
I am not sure how you distinguish different concepts that were not in prior use from new concepts. Perhaps examples would help.
Quoting Fooloso4
I think we are using "concept" in different senses. I am thinking of
Thank you for the link/reference.
Quoting Fooloso4
No, I don't dismiss different conceptual spaces as wrong -- they are just different ways of thinking about the same reality.
In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules. A central idea of formalism "is that mathematics is not a body of propositions representing an abstract sector of reality, but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess." According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other contensive subject matter — in fact, they aren't "about" anything at all. Rather, mathematical statements are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation (or semantics).
There may be an esoteric link between the abstract, Platonic world of mathematics and the real, physical world, but this hypothetical link cannot be used for any practical purpose.
[i]It has been claimed that formalists, such as David Hilbert (1862–1943), hold that mathematics is only a language and a series of games. Indeed, he used the words "formula game" in his 1927 response to L. E. J. Brouwer's criticisms:
And to what extent has the formula game thus made possible been successful? This formula game enables us to express the entire thought-content of the science of mathematics in a uniform manner and develop it in such a way that, at the same time, the interconnections between the individual propositions and facts become clear ... The formula game that Brouwer so deprecates has, besides its mathematical value, an important general philosophical significance. For this formula game is carried out according to certain definite rules, in which the technique of our thinking is expressed. These rules form a closed system that can be discovered and definitively stated.
We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise.[/i]
Hermann Weyl to David Hilbert:
What "truth" or objectivity can be ascribed to this theoretic construction of the world, which presses far beyond the given, is a profound philosophical problem. It is closely connected with the further question: what impels us to take as a basis precisely the particular axiom system developed by Hilbert? Consistency is indeed a necessary but not a sufficient condition. For the time being we probably cannot answer this question ...
My own opinion is that mathematics is a bureaucracy of formalisms that seeks to maintain consistency in its own symbol streams.
Mathematics is consistent by design while the real, physical world is consistent by assumption. Therefore, it is sometimes possible to construct consistency isomorphisms between both, that will be uncannily effective in mirroring some sector of reality inside an abstract, Platonic model. Physics is a heavy user of that principle.
I do not wish to defend mathematical Platonism, but I think you misrepresent the position. The problem stems, at least in part, from jumping from Aristotle's criticism of Plato's Forms to mathematical platonism.
From the IEP article on Mathematical Platonism:
Your example of counting fruit is a straw man.
Quoting Dfpolis
This strikes me as a form of Platonism, as if intelligibility is something somehow present in but other than the objects of inquiry.
Quoting Dfpolis
Do you mean different concepts that were in prior use? In the briefest terms, the arithmos is always a definite number of definite things,a collection of countable units, whereas in modern math a number, '4' for example, is itself an object. With the move to symbols, 'x' does not signify anything but itself.
This is a wide-ranging topic that goes far beyond the concept of number. The second part of this book review that addresses Klein will give a better sense of what is at issue as it relates to modern philosophy and science: https://ndpr.nd.edu/news/the-origin-of-the-logic-of-symbolic-mathematics-edmund-husserl-and-jacob-klein/
Quoting Dfpolis
No, I am speaking here specifically about the concept of number, that is, what a number is.
Quoting Dfpolis
What you said was:
Quoting Dfpolis
Either you think that each of these ways are retained in the development of the intelligibility of the whole or some are modified and rejected.
I agree, yet when modeling reality, it's apparent that there are approximations and generalizations etc. that simply don't make sciences as rigorously logical as mathematics. For starters, every measurement is an approximation. Logic is of course necessary. I studied myself economics and economic history and noticed that a lot of variables are rudimentary models of very complex phenomena, like 'inflation', 'GDP' or 'aggregate demand', and that one shouldn't forget it when calculating math formulas with them.
Quoting Dfpolis
Ok, then I think I've misunderstood your point.
Quoting Dfpolis
Perhaps now I understand your point. (I'm btw happy with pragmatism: usefulness is far more important than we typically think.) So if I understood you correctly, when you talk about 'unscientific' math that is "merely a game, no different in principle than any other game with well-defined rules" is that it's actually not applicable and/or the axioms simply aren't in line with reality. Like astrophysics using a helical model of the universe simply might not be useful...especially if the universe simply isn't optimally modeled using a helix.
The standard mathematicians answer would be "Well, it could be useful someday". Modelling the universe using a helix might have those not yet known nice 'mathematical properties' that future physicists make better models and can avoid today's problems. And some mathematicians are totally happy with the "math-is-just-various-kinds-of-rules" approach and declare every kind of math as worthy as long as it's logically correct.
Quoting Dfpolis
Logical (not local). No, I'm not saying that. What I'm saying that a field that has developed from the need to count and calculate to solve real world problems doesn't have it's axiomatic foundations solely on arithmetic as it has also incommensurability and uncomputability. So the foundations aren't so narrow that everything starts from simple arithmetic.
Some comments about your classification. You define in the first class to be math that has axioms rooted in our experience and reality.
Quoting Dfpolis
Yet we can have logical problems with those too: Zeno's paradoxes and the huge debate over infinitesimals have shown that we stumble to the problems of infinity from quite normal experiences. (And those who think limits have solved all the questions, well, how about the Continuum Hypothesis then?)
There's still a lot that we don't know.
The term "usefulness" is quite controversial in mathematics. I tend to agree with Hardy on the matter:
[i]I have never done anything "useful". No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.
We have concluded that the trivial mathematics is, on the whole, useful, and that the real mathematics, on the whole, is not.[/i]
As I see it, and in line with what Hardy said, while the low-hanging fruit is almost immediately useful, but only moderately so, the real game changers may take even centuries to find their way into applications. That is why it is necessary to abstract away "usefulness" when exploring the abstract, Platonic world of mathematics for new discoveries.
Hardy was a number theorist. At the time he wrote those words, number theory was regarded as beautiful but useless. Today it's the mathematical foundation of public key cryptography, underlying all Internet security and cryptocurrencies. I wonder what he would say if he came back and discovered that his belovedly useless number theory was intensely studied by the spies at the NSA.
For a discussion of the pragmatics of the axioms of set theory, see Penelope Maddy's Believing the Axioms parts I and II.
https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf
https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms2.pdf
All these bloody pure mathematicians trying to stop us from occupying their lawn. They forget the rest of us squatters were here first.
Which is why a consequent of formalism is that math, as a meaningless game, is of no intrinsic value. This view is incompatible both with our experience of learning math by reflecting on examples, and with the fact that mathematical propositions are treated as truths in scientific thought.
Quoting alcontali
The Aristotelian-Thomistic view also rejects actual numbers, sets, and triangles in extramental reality, but sees an alternative other that empty formalism. Reflecting on the role of examples in learning math and on its applicability in science, it sees that numbers, sets, and triangles are intelligible (potential, able to be understood) in reality. The act of abstraction, which is one function of awareness (the agent intellect), makes what was merely intelligible in nature actually understood. This provides a middle ground between Platonism and formalism.
Quoting alcontali
As there is no Platonic world, there is no possibility of a link to it, There is, however, a natural world with well-known links to mathematical thought.
Quoting alcontali
Your unnamed authority agrees that math is a science. Games are not sciences. Being a science (an organized body of knowledge) means that math is an understanding of reality.
Quoting alcontali
Clearly, this is nonsense, We know that there is no intrinsic necessity to the parallel postulate in the context of geometry, or to the axiom of choice in the context of Zermelo–Fraenkel set theory. If the rules are arbitrary, so are the results.
Quoting alcontali
No one can discover what does not pre-exist. Yet, the pre-existence of mathematical axioms is the exact premise formalists reject. So, again your authority is inconsistent.
Quoting alcontali
This is the problem of universals, solved by the moderate realist insight that ideogenesis involves the actualization of intelligibility in nature via abstraction by the agent intellect.
Quoting alcontali
The answer to this is, nothing. Abstraction fixes on certain notes of intelligibility and certain intelligible relations to the exclusion of others. Thinkers may have different conceptual spaces in light of their individual experiences and needs. So, the same reality can be understood in different, partial. ways -- depending on the perspective we take and the conceptual space into which we project our experience.
Quoting alcontali
No, the physical world is consistent in virtue of its existence. The nature of being is such that it cannot instantiate a contradiction. That does not mean that hypothetical theories, in physics or in math, need be consistent.
Quoting alcontali
This view makes the applicability of math to nature entirely accidental. If you think about it, you'll see that you can't construct such an isomorphism unless the relevant mathematical relations are already instantiated in nature -- and we can understand that they are. But, if they are already instantiated and intelligible, both Platonism and formalism are wrong. We can construct the relevant math on the basis of our understanding of those intelligible relations.
Quoting Fooloso4
My comment is directly on point, and does not attack a straw man, but premise ii. It misstates the conditions required for qa statement to be true, by taking the correspondence theory of truth too literally. If is not necessary that the predicates of true simple statements with singular terms as components to exist actually, but only potentially, That was Aristotle's insight in his definition of quantity in Metaphysics Delta. Quantity in nature is countable or measurable -- potential not actual numbers. "There are seven pieces of fruit in the bowl" is true, if on counting the pieces of fruit, we come to seven and no more.
This being adequate account of the numerical claim shows that we need make no appeal to an actual seven existing independently of a counting operation. In other words, "true only if the objects to which those singular terms refer exist" is false if we tale "exist" to mean "actually exist," but true it we take it to mean "potentially exist" or "exist as intelligible".
Quoting Fooloso4
It is a form of realism -- AristotelianQuoting Fooloso4
moderate realism, not Platonic extreme realism. Moderate realism sees content as deriving from objects (their intelligibility), and awareness of content as deriving from knowing subjects. So, I ask, does not data derive from what we are studying? And, is unexamined data thought?
I'm not saying "intelligibility is something somehow present in but other than the objects of inquiry." I'm saying that every note of intelligibility is an aspect of the object known. It is not the whole object, but an aspect (rubber is not all there is to being a rubber ball). I say "aspect" instead of "part" because parts can be separated in nature, but aspects may be separable only in the mind (by abstraction). E.g. we can separate rubber from the ball, but we can think of it in abstraction.
Quoting Fooloso4
No, I mean that concepts don't change. New concepts are necessarily different concepts. The may replace an old concept, but they are not the old concept transformed.
Quoting Fooloso4
This is an interpretive, not a mathematical, claim. If you're a Platonist, "4" is an object, if you're more reflective, you see that it's only an object of thought. No, "x" does not mean the letter "x." It has reference beyond itself. It may mean an unknown we seek to determine, a variable we can instantiate as we will, or possibly other things, but it never signifies itself, which is always a particular image -- because text images are not what math deals with.
I suppose you could mean that 'x' is just an object that can be formally manipulated according to a set of rules. That it is only that is also an interpretive claim, formalism. Nothing in the view I am proposing prevents rote, formal manipulation according to rules. My view just says "x" is usually more than that, but we can abstract away from its meaning in formal manipulation.
Clearly, mathematical symbols are not invariably free of meaning. Godel uses arithmetic forms to represent axiom sets, and his major theorems are restricted to systems representable in arithmetic.
Thank for the book review reference, It may take me a while to get to it.
Quoting Fooloso4
OK.
Quoting Fooloso4
Hypothetical understandings are modified and/or rejected over time. Abstractive understanding is partial and grows over time without need of replacement. Still, parts of it can be forgotten or fall out of vogue.
I do agree that physicists tend to think more eclectically and in a less structured way than mathematicians. Still, I think logic is logic and the validity of consequences depend only on the claims made in the premises, not on the accuracy of those claims.
Quoting ssu
I interpret Aquinas's veritas est adaequatio rei et intellectus in a way that spans from correspondence to pragmatism. Adaequatio means "approach to equality," not correspondence per se. The question is how close do we need to approach reality for our understanding to be true? My answer is that the approach has to be adequate to our needs in context. In metaphysics this is very close to correspondence. In science, it is very close to pragmatism.
Quoting ssu
Yes.
Quoting ssu
Agreed. I also agree that there is always more to learn.
2+2=4 is not a "Platonic relationship". That 2+2=4 is true, according to mathematical platonism is due to the nature of numbers. The relationship is made possible by their nature. The relationship itself is not another platonic object.
Quoting Dfpolis
The number of pieces of fruit in the bowl is undetermined until counted. This does not mean that the number of pieces is a potential number. It is an actual number that before we count we might say it could be six or seven or eight. There are actually seven pieces whether we count them or miscount them. They do not become seven by counting them. We are able to count seven because there are actually seven pieces of fruit in the bowl.
Quoting Dfpolis
So, an aspect of something known is that it is knowable. Aside from being tautological and trivially true it raises questions that go beyond the current topic and so I will leave it there.
Quoting Dfpolis
The question was about your wording. Whether the 'not' in "not in prior use" was a typo.
Quoting Dfpolis
Of course it is interpretative! What is at issue is the concept of number. That is an interpretive question.
Quoting Dfpolis
It does not have any reference until it is assigned one. That is the point. It is a variable that can stand for any unknown. In this sense it is different from both "4" as how many or "4" as an object.
[Added trivia note: I read somewhere that Descartes' publisher used x because he was low on letters an x was not frequently used in French. Whether that is true or not I did not verify.]
Quoting Dfpolis
Right.
Quoting Dfpolis
It is because it is indeterminate that it does not signify something other than itself, which is to say, unlike a number it has no signification until or unless assigned one. It could stand for any number or no number at all.
This unnamed authority was David Hilbert:
It has been claimed that formalists, such as David Hilbert (1862–1943), hold that mathematics is only a language and a series of games. Indeed, he used the words "formula game" in his 1927 response to L. E. J. Brouwer's criticisms: "And to what extent has the formula game thus made possible been successful? This formula game enables us to express the entire thought-content of the science of mathematics in a uniform manner and develop it..."
This letter predates Karl Popper's "Science as Falsification" by almost half a century. The dust hadn't settled yet on the impossibility of verificationism. Certainly the Circle of Vienna still happily amalgamated mathematics and science.
The other objections to David Hilbert's view came from Hermann Weyl: What "truth" or objectivity can be ascribed to this theoretic construction of the world ...
Quoting Dfpolis
Consistency is indeed assumed to be already instantiated in nature. The existence of consistency makes particular things impossible. These impossibilities give inescapable structure to nature. That is in my impression the core of the esoteric link between nature and mathematics. The structure visible in the Platonic world of math will therefore tend to be also visible in the real, physical world.
I personally refute neither Platonism nor formalism (Hilbert). They are a dual view on the abstract, Platonic objects versus the structures that constrain them in math.
Yes, the content of the Platonic realm is usually supposed to be prototypes of universal concepts, such as number and equality. Excuse my shorthand description. I don't think it impacts my point that the relation between the Platonic realm and empirical reality is fuzzy at best.
Quoting Fooloso4
Yes, the cardinality of the fruit in the bowl is seven whether we count or not. That does not mean that the concept
Quoting Fooloso4
It is not trivial that the intelligibility of an object does not constitute an actual concept. A state's potential for a seven count does not exclude is simultaneous potential for other counts when conceived in other ways. So, it is not trivial that states require further (mental) determination to be assigned actual numbers.
Quoting Fooloso4
Exactly, and so one that requires justification. It seems to me there is inadequate justification for both Platonism and pure formalism. Saying that mathematicians have such beliefs is not justification. One needs to look at how we learn and apply mathematics to have a theory that is coherent with the rest of our knowledge.
Quoting Fooloso4
It lacks determinant reference, but it has a reference type. That type may be a numerical value or something else that can be represented by the formalism. A variable might, for example, be assigned any real number, or perhaps, a complex tensor of rank 12, depending on its type. So it has a determinant (well-defined) potential reference -- just as does any universal term.
Again, we see the importance of distinguishing what is actual from what is merely potential.
Thank you. Recall that David Hilbert's "program" (concept of math) was destroyed by Kurt Gödel.
Quoting alcontali
The Vienna Circle hardly deserves to have its name attached to a movement started by Aristotle, and brought to fruition long before any of them were born.
Quoting alcontali
If so, we can certainly know that structure, and abstract it to form the axiomatic basis of mathematics -- making Platonism unnecessary and formalism inadequate.
My issue is with what you call "potential numbers". The number of pieces of fruit in the bowl or the number of seeds in the pieces of fruit in the bowl in never a potential but an actual number. We may have the potential to determine that number but that does not make it a "potential number".
Quoting Dfpolis
This is really convoluted and seems to be contradictory. The intelligibility of an object simply means that we are able to understand it in some way. That is not an aspect of the object. The way in which something is understood is not an aspect of the object but rather of our ability to see it or understand it in different ways. If a state requires mental determination then that determination is not an aspect of the object but rather something we say or know or understand or have determined about the object.
Quoting Dfpolis
No inquiry is free of assumptions. The ontology of mathematical objects is an open question. It is not that different theories of mathematical objects are without justification it is that there is no universal agreement regarding their justification.
Quoting Dfpolis
Which means that it differs fundamentally from a number, which is always has a determine value.
If numbers were objects in nature, you would be right, But they aren't objects in nature, they are the result of counting sets we chose to define. Why count only the fruit in this bowl instead of some other set we define? The objects in nature are fruit, bowls, and so on -- not integers. Integers are the counts of sets we arbitrarily define -- change your set definition, and the count changes. That makes the numbers partly dependent on us and partly dependent on the objects counted. So, numbers do not actually exist until we define what we're going to count and count it.
Universal ideas are not things. There is no "bigger than." There are pairs in which one is bigger than the other. In the same way there is no "seven." There are sets, some of which have seven elements, but that "seven=ness" ceases to be if we put those same elements in different sets.
Quoting Fooloso4
So, being rubber or spherical are not aspects of a rubber ball? Of course they are. Just because we can fix on the ball's matter or the form does not mean that the ball's intelligible properties depend on us (unless we're the ones defining the object). What depends on us is which notes of intelligibility we choose to fix upon.
Quoting Fooloso4
If it depends only on us, this is true, but knowing depends jointly on the properties of the object and what we choose to attend to. An object's properties do not force us to attend to them, nor does attending to an object typically create its properties.
Quoting Fooloso4
What we experience is not an assumption. It is data.
Quoting Fooloso4
Right. I never said that variables and determinate numbers were the same.
By definition none the less.
Quoting Dfpolis
Thinking of apples...
What are the requirements, the necessary pre-requisites, the sufficient pre-conditions...
What must also be the case in order for that to be?
What is the act of thinking existentially dependent upon?
That's a step in the direction of necessary elemental constituents.
We might say, for example, that the number of bacteria in a petri dish is potentially thousands or tens of thousands. Whether one is platonist or not, however, in such a case the number refers to the objects being counted. At any given moment that number is an actual number, even if we do not know what that number is. Here potential means we do not know what the actual number is.
What you said was:
Quoting Dfpolis
The number of bacteria in the petri dish or fruit in the bowl or whatever it is that we are counting cannot be counted if that number is not an actual number of items.
Quoting Dfpolis
What is dependent on us is what we choose to count. How many there are of whatever it is we choose to count is independent of us. Here we are not talking about the concept of number but how many of something.
Quoting Dfpolis
Rubber and spherical are properties of the object. Intelligibility is not a property.
Quoting Dfpolis
The intelligible properties are those properties we understand, rubber and spherical. Intelligibility is not another property that is intelligible.
Quoting Dfpolis
What depends on us is the ability to understand, to make the object intelligible to us.
Quoting Dfpolis
We are talking about what a number is, the concept or ontology of numbers. That is not an experience or data. We do not experience numbers, we experience objects of a certain if indeterminate amount.
Quoting Dfpolis
You were responding to the following:
Quoting Fooloso4
The point was the one you now acknowledge. Klein's insight is into the radical shift in mathematics from numbers to symbols. Although we treat them as interchangeable when we assign value to the variable, numbers and symbols are not interchangeable. We do not assign values to numbers, we must assign value to variables. Numbers are determinate. Symbols are indeterminate. 3+2=5 is true. a+b=5 may be true or false. 3+2=5 is not dependent on us. a+b=5 is dependent on the values we assign to a and b.
If "constituents" means preconditions, I have no objection to ideas having constituents.
There are two potentials here. One is our potential to be informed, which belongs to us. The other is the set's potential to have its cardinality known, which belongs to what is countable, and is the basis in realty for the proper number to assign to the set.
Quoting Fooloso4
I beg to differ. The items can be counted if and only if they are actual distinct items. The number that results is one, abstract, way we can think of the set.
Quoting Fooloso4
This is self-contradictory. If the number is "How many there are of whatever it is we choose to count," it is not independent of us.
Quoting Fooloso4
Necessarily, whatever is actually done can be done. If the ball is known, necessarily it can be known, and so is intelligible. As it can be known whether or not it is actually known, intelligibility inheres in objects. So, why do you say it is not a "property"?
Quoting Fooloso4
Don't we understand that balls are knowable?
Quoting Fooloso4
Rather, to make aspects of the object actually understood by us. Our understanding is not exhaustive and if we do choose not to look, we will not understand what we choose not to look at.
Quoting Fooloso4
And abstract arithmetic concepts from that experience. You let a child count four oranges, four pennies, etc., and she abstracts the concept
Both are dependent on us to determine, that is, to know or be informed of the number. In neither case is the number a potential number except with regard to our potential to know it.
Quoting Dfpolis
I am not going to get into methods of counting bacteria.
Quoting Dfpolis
What we choose to count is up to us, how many there are of what we count is not.
Quoting Dfpolis
You ignore a great number of questions. What does it mean to say the ball is known? When someone identifies an object as a ball is the ball known? If they cannot tell you whether the material is rubber or synthetic is the ball known? If they do not know the molecular or subatomic make-up is the ball known? If they know it is a baseball is being a baseball an intelligible property of the object? If some other ball is used to play baseball is being a baseball an intelligible property of the object? If the ball is used as a doorstop does someone who only knows it as it is used for this purpose know that it is a ball? A baseball? If they saw someone hitting it with a stick wouldn't they wonder why he was hitting the doorstop with a stick? Perhaps they might think that he does not know what a door stop is.
Quoting Dfpolis
She might be a platonist and assume that
Let's try this a different way. Surely the number does not inhere in the objects we count, for they can be grouped and counted in different ways to give different numbers. So, if it is already actual, and we agree that it does not pre-exist in our minds, where is it?
Quoting Fooloso4
I am not confining my claim to bacteria, nor discussing methods that apply to them in particular. So, do you agree that items can be counted if and only if they are actual and distinct?
Quoting Fooloso4
Think of it this way. Classical physics is deterministic. So, given the initial conditions and the laws of nature, the system state at a later time is fully determined. That does not mean the later state is now actual. It is only potential. So it is with counting. The number is predetermined, but not actual until the count is complete.
Quoting Fooloso4
It means that its intelligibility is actualized by someone's awareness.
Quoting Fooloso4
It has to be known as an object, as a tode ti (a this something) before it's classified.
Quoting Fooloso4
Yes, but not exhaustively. We never know anything exhaustively.
Quoting Fooloso4
Being a baseball is intelligible, but it is the ball as a whole, not a property of the whole.
Quoting Fooloso4
Not unless you change the definition of "baseball" to mean any ball you play baseball with. If you do, then the last response applies.
Quoting Fooloso4
It is not necessary to know everything about a this something to know it in some way.
Quoting Fooloso4
Perhaps.
Now that I've answered your questions, can you explain their relevance?
Quoting Fooloso4
That would not change how she came to the concept. It was by abstracting from her experience of counting real things -- not by mystic intuition.
Quoting Fooloso4
I am not saying that our conceptual space is independent of our cultural background. I am saying that whatever concepts we do have are abstracted from empirical experience.
The number is how many of whatever it is we are counting. If I count the number of fingers on one hand and I count correctly the number is 5. That is because I actually have 5 fingers on my hand. If one of my fingers was cut off I would count 4 and that is because I actually have 4 fingers on that hand.
Quoting Dfpolis
As I said from the beginning, the count depends on the unit. If we cannot determine the unit we cannot determine the count. It the items to be counted are actual then their total number is also actual.
Quoting Dfpolis
No wonder you are confused! Counting something has nothing to do with determinism.
Quoting Dfpolis
I would say that the number is not determined until we count, but what we are counting, the items, as you said, are actual. It is because there is actually this item and this item that we can determine how many there are. We can call this determination the count. It we count six and we count correctly that is because there are actually six of the items to be counted.
Quoting Dfpolis
This is evasive. Intelligible in what way? Which is to say, as I asked, what does it mean to say the ball is known?
Quoting Dfpolis
If you mean that it stands out (literally, exists) distinct from all else, that does not mean that intelligibility is a property of the object. To be is not a property of what is. To be is a necessary condition for having properties. "This" is not a property of this something. To be intelligible a thing must be distinguishable as separate from other things but to be intelligible there must also be some subject to which it is intelligible. Without subjects there is no intelligibility.
Quoting Dfpolis
What you said was:
Quoting Dfpolis
If someone from a tribe that knows nothing about baseball were to find a baseball what it it about it that would make it intelligible to the tribesman that it is a baseball? Its intelligibility as a baseball is not something that inhere in the ball. To be intelligible as a baseball one must know what baseball is.
Quoting Dfpolis
The relevance can be seen in the what I just said. If intelligibility inheres in the object then someone would know what a baseball is even if they did not know what the game of baseball is.
Quoting Dfpolis
No, it would not necessarily be by abstracting. I gave several different things she might assume, stories she might tell herself.
Quoting Dfpolis
I would say that since none of us are without experience we cannot say what if any concepts we would form, but that concepts are not always abstractions, the can be something we add to rather than something we take away from experience.
I am not denying that you have 5 fingers on your hand -- it is just that five fingers is not the abstract number 5 -- it is specific instance of five, not the universal five.
Quoting Fooloso4
If we cannot determine the unit, we can't count. The things we count are prior to our counting them.
Quoting Fooloso4
It does not have to do with physical determinism, but with the fact that things can be predetermined without being actual. The count of your fingers was predetermined to be five before anyone counted them, but there was no actual count of five fingers.
Quoting Fooloso4
I agree. There are six items -- a specific instance of 6 -- not the abstract number 6.
Quoting Fooloso4
I told you. The ball is intelligible as this kind of thing, with these specific properties, and someone has actualized part of its intelligibility by becoming aware of it. If it were not able to be known, no one could know it -- and if the knower were not able to be informed she could not be informed about the ball.
Quoting Fooloso4
This depends on how you define "property." What is intelligible is the whole, but we do not actually understand mall of it.
Quoting Fooloso4
The ball is a baseball because of its relation to the game. Knowing the ball in itself will not tell us its relation to the game.
Quoting Fooloso4
Quoting Fooloso4
The assumptions are all after learning. You have provided no alternate account of learning the concept.
What you seemed to be claiming is that the number, whether it is fingers or fruit, is not actual but potential until it is counted. One iteration of what you said is:
Quoting Dfpolis
What I said is that I actually have five fingers whether I count them or not. If I only get to three I still have five fingers.
As I said early on, I do not intend to defend platonic mathematics. For one, I am not well versed in the arguments. For another, I am agnostic on the matter.
Quoting Dfpolis
Agreed. I have said this from the beginning in my discussion of Greek mathematics.
Quoting Dfpolis
Here we go again. There is no actual count until they are counted, but there are actually five fingers, which is confirmed by the count.
Quoting Dfpolis
Knowledge is not passive reception of "intelligibility". Knowledge is conceptual.
Quoting Dfpolis
And it follows from this that the intelligibility of a baseball is not something that inheres it the object.
Quoting Dfpolis
The child has learned to count the objects. If she is not told, or as you would have it, learned what a number is, what she thinks a number is can vary. Is she is taught by a mathematical platonist what she learns the concept is is not what she learns if you tell her what you think it is.
Yes. No universal exists abstractly in nature. There is no actual humanity in nature. There are men and women with the intelligibility to engender the concept
Quoting Fooloso4
Yes, I was insufficiently clear earlier. I take responsiblity for the confusion. What is not actual is abstract fiveness, i.e. the pure number.
Quoting Fooloso4
I did n't say knowledge was passive. We have to actively attend to intelligiblity to make it understood. That is why Aristotle calls awarenss "the agent intellect." Our act of attending/awareness actualizes intelligiblity, converting it into concepts.
Quoting Fooloso4
We have to distinguish inherrent intelligiblity from relational intelligiblity. All objects have both.
Quoting Fooloso4
I have no problem with alternative conceptual spaces. There's nothing wrong with a concept of number that excludes 0 and 1. It just represents reality in a different way than a concept that includes them. Concepts aren't judgements and so they're neither true nor false.
The mathematical platonist does not claim that there is an actual five in nature.
Quoting Dfpolis
That is nothing more than an assertion. The platonist asserts that there is, but it is not in nature.
Quoting Dfpolis
I agree with those who say we construct concepts rather than actualize them.
Quoting Dfpolis
The intelligibility of an object is knowledge of its essence, that is, what it is to be the thing that it is. What it is to be a baseball is not something that inheres in the object. It is to that extent not intelligible unless we know it as a baseball. To know it in its role in the game is not relational, it is essential to what it is.
You said you were not a mathematical Platonist. I was explaining to you why the abstract five is not actual until abstracted.
Quoting Fooloso4
No, it is not a mere assertion, but an appeal to experience. Platonists have no basis in experience for their position.
Quoting Fooloso4
If we merely constructed concepts, there would be no reason to think they apply to or are instantiated in, reality. It is only because our concepts actual prior intelligibility that what we have in mind relates to reality.
Quoting Fooloso4
First, intelligibility is not knowledge. It is the potential to be known. Second, all human knowledge is partial, not exhaustive. We may, and usually do, know accidental traits rather than essences. Third, there is nothing intrinsic to a baseball that relates it to any particular game. The relation is a human convention, as games are human constructs.
I am not but your topic is an attack on mathematical platonism and if you are going to attack it you must accurately represent it.
Quoting Dfpolis
Your talk of potential and actual is misleading. If five is an abstraction from particular instances of five units or items then it is not actual except in that it is an actual abstraction.
Quoting Dfpolis
I think they might argue that the fact that mathematical truths are not dependent on experience is all the experience they need. Consider, for example, non-Euclidean geometries. They are not abstracted from experience. They were initially seen an useless, mere curiosities. But with the discovery of the curvature of space, they found their application. They work. They are not merely formally or internally consistent, they tell us something about the world without being dependent on it.
Quoting Dfpolis
First, see above regarding non-Euclidean geometries. Second, to some extent (Kant would say completely) experience is itself constructed. Third, concepts that are constructed are not all "merely" constructed, the construct may be based on experience but cannot be reduced to experience.
Quoting Dfpolis
Okay. Let me rephrase it: The intelligibility of an object is the potential to know its essence. This changes nothing about what I said that follows from this. To use your favored language, knowledge of an object's essence is the actualization of its intelligibility.
Quoting Dfpolis
The question is whether intelligibility inheres in the object. Whether or not our knowledge is partial is not at issue. The question is whether from the baseball alone it can be known that it is a baseball. An intelligence far greater than ours would not know this unless it also knows what the game is.
Quoting Dfpolis
Of course there is! Being a baseball is not incidental to it being a baseball. It is constructed according to specific rules for a specific purpose. 'Baseball' is not simply a name attached to it. But if there is nothing intrinsic to a baseball that relates it to any particular game then your argument fails. We could not tell from it that it is a ball designed, manufactured, and used for one specific purpose.
I wasn't representing it. I was telling you why abstract numbers do not occur in nature, which is what we were discussing.
Quoting Fooloso4
Exactly! At last we agree.
Quoting Fooloso4
People can argue whatever they like. There is no sound argument that "mathematical truths are not dependent on experience." How can we even know they are true unless they reflect our experience of reality?
Quoting Fooloso4
They can be. They are instantiated on spherical and saddle-shaped surfaces. If some axiom can't be, it's hypothetical.
Quoting Fooloso4
Nothing can tell us something of the world without being instantiated in it -- and if it's instantiated in it, it can be abstracted from it.
Quoting Fooloso4
Kant had no sound reason to claim that.
Quoting Fooloso4
My claim is that what we know is based on our experience of reality, not that everything we can or do construct is reflected in reality. That is why some hypotheses are falsified.
Quoting Fooloso4
Perhaps, but as counting never exhausts the potential numbers, so human knowing never exhausts anything's essence. There is always more to learn.
Quoting Fooloso4
Yes, that is the issue, but your argument is based on the fact that our knowledge is not exhaustive. That our knowledge is only partial does not show there is no potential to know more -- no greater intelligibility that that we have actualized.
Quoting Fooloso4
Its purpose is in the minds of humans, not in the ball. We can use the ball for other purposes, such as to be a display or even a paperweight.
I wasn't representing it. I was telling you why abstract numbers do not occur in nature, which is what we were discussing.
Quoting Fooloso4
Exactly! At last we agree.
Quoting Fooloso4
People can argue whatever they like. There is no sound argument that "mathematical truths are not dependent on experience." How can we even know they are true unless they reflect our experience of reality?
Quoting Fooloso4
They can be. They are instantiated on spherical and saddle-shaped surfaces. If some axiom can't be, it's hypothetical.
Quoting Fooloso4
Nothing can tell us something of the world without being instantiated in it -- and if it's instantiated in it, it can be abstracted from it.
Quoting Fooloso4
Kant had no sound reason to claim that.
Quoting Fooloso4
My claim is that what we know is based on our experience of reality, not that everything we can or do construct is reflected in reality. That is why some hypotheses are falsified.
Quoting Fooloso4
Perhaps, but as counting never exhausts the potential numbers, so human knowing never exhausts anything's essence. There is always more to learn.
Quoting Fooloso4
Yes, that is the issue, but your argument is based on the fact that our knowledge is not exhaustive. That our knowledge is only partial does not show there is no potential to know more -- no greater intelligibility that that we have actualized.
Quoting Fooloso4
Its purpose is in the minds of humans, not in the ball. We can use the ball for other purposes, such as to be a display or even a paperweight.
Of course abstract numbers do not occur in nature, nothing abstracted from nature exists in nature.
Quoting Dfpolis
At last? I have never said anything to the contrary. What was at issue was your denial five of something is actually rather than potentially five of something. You recently corrected yourself on that matter.
Quoting Dfpolis
It those truths precede in time our experience of reality then they cannot be dependent on experience. Such is the case with non-Euclidean geometries. As another example consider infinitesimal calculus. There is no experience of infinitesimals. Not only are they not found in experience, they confound experience, as Zeno's paradoxes show. They are not abstracted from nature, they are theoretical constructs applied to it. In addition, the experience of motion or change does not yield the mathematics that adequately describes it.
Quoting Dfpolis
Instantiation is not abstraction.
Quoting Dfpolis
The historical fact of the matter is that they weren't abstracted. Non-Euclidean geometries were first developed as purely formal systems.
Quoting Dfpolis
I won't bother getting into this. Do you imagine that neither Kant nor those who followed him were aware of this?
Quoting Dfpolis
What is at issue is your claim regarding the intelligibility of an object. Whether or not human knowing exhausts something's essence, if intelligibility inheres in the object then a sufficiently advanced intelligence should be able to know what a baseball is without knowing what the game is, or, perhaps, would know from the ball what the game is. But there is nothing in the ball that would provide this information.
Quoting Dfpolis
No, my argument has nothing to do with the limits of human intelligence. It has to do with what is knowable from the object itself. Not knowable within the limits of human intelligence but from an intelligence without our limits.
Quoting Dfpolis
That is right and that is why you cannot tell from the ball what its purpose is. To the extent the ball is intelligible its purpose is not part of that intelligibility. By your logic the intelligibility of a car does not include the potential to know that it is a means of transportation.
Quoting Dfpolis
Yes, we have been through this already.
Quoting Fooloso4
Truth is not a value, but a relation between mental judgements and reality. Since it depends on judgements, it can't be prior in time to them. Only being can be.
Quoting Fooloso4
There are no actual infinitesimals in calculus. There are limits as quantities tend to zero. That is the whole point of the epsilons and deltas in the formal definitions of calculus.
Quoting Fooloso4
Having read Kant's reasoning, he seems to have been unaware of the errors he was making.
Quoting Fooloso4
I did not say it was. I said that non-euclidean geometries could be abstracted from models instantiating them.
Quoting Fooloso4
If so, that would mean they had a hypothetical status until it was realized that they could be instantiated. The notion that one could be shown to be the true geometry of the universe was explicitly stated by János Bolyai. Quoting Fooloso4
According to the Wikipedia article: "Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean; this is a task for the physical sciences."
I have answered all this previously. Knowing an object's intrinsic nature need not entail knowing its relationships.
Quoting Fooloso4
One might figure it out, but only if one knew there were beings that could use it so.
There is no judgment of the truth of the deductions of non-Euclidean geometry that independent of reality, unless of course you maintain that there is a mathematical reality. They are formal logical truths. Whatever your theory of truth may be, non-Euclidean geometry works. They find their application in reality.
Quoting Dfpolis
The point is that they are theoretical constructs. They are not abstracted from nature.
Quoting Dfpolis
Him and several generations of Kant scholars. When are you going to publish your findings in a peer reviewed journal?
Quoting Dfpolis
But the fact that you are trying to dance around is that they didn't.
Quoting Dfpolis
They did not have a hypothetical status because they were not hypotheses. They were formal logical systems that were not intended to relate to anything else.
Quoting Dfpolis
This is besides the point. They were not constructed as models of the universe. The question is how is it that they do apply? Is it just coincidence?
Quoting Dfpolis
The problem is that a baseball being a baseball is not a relationship. It is intrinsic to what it is to be a baseball.
Quoting Dfpolis
You might claim that a car's being a mode of transportation is not intrinsic to it being a car, but that is only because you want to maintain your questionable claim about intelligibility. If not for that you would define it as everyone else does.
But there's a subtle recursion in this understanding, because it presumes we can attain a perspective where 'mental judgements' can be compared with reality.
'Truth, it is said, consists in the agreement of cognition with its object. In consequence of this mere nominal definition, my cognition, to count as true, is supposed to agree with its object. Now I can compare the object with my cognition, however, only by cognising it. Hence my cognition is supposed to confirm itself, which is far short of being sufficient for truth. For since the object is outside me, the cognition in me, all I can ever pass judgement on is whether my cognition of the object agrees with my cognition of the object”. (Kant, 1801. The Jasche Logic, in Lectures on Logic.)
That was not Hilbert's view. It seems you are confusing Hilbert with Russell.
Quoting Dfpolis
That is terribly incorrect. Godel's result is that, for any S that is a certain relevant kind of axiom system, there are true statements that cannot be deduced in S. However there are other systems, even of the relevant kind, in which the statement can be deduced. Then, in a followup:
Quoting Dfpolis
No, again, that is terribly incorrect. There is no axiom such that there is no system in which the axiom can be deduced. And it is not needed to refer to Godel to point out that we can only look at finitely many systems.
Quoting Dfpolis
It is reasonable to argue that certain central aspects of Hilbert's program were shown by Godel to not be achievable. But that doesn't destroy Hilbert's concept entirely.
Quoting Dfpolis
No, that is terribly incorrect. Godel's result is that for any S that is a certain kind of axiom system, the consistency of S cannot be deduced in S. But the consistency of S might be deducible by certain other systems.
Quoting Dfpolis
'aleph_1' is not synonymous with 'uncountable'. aleph_1 is the least uncountable cardinal. And showing that there are uncountable sets does not rely on proving the uncountability of the continuum, but comes even more simply from proving that the power set of any set has more members than the set, so if there is an infiinite set then there is an uncountable set. And, just to be clear, Cantor didn't prove that the cardinality of the continuum is aleph_1. The proposition that the cardinality of the continuum is alelph_1 is the continuum hypothesis, famously not proven by Cantor.
Quoting Dfpolis
If a consequence of C is falsified, then C is falsified.
Quoting alcontali
Hilbert didn't say that mathematics is only a language game. He regarded certain aspects of mathematics as a kind of language game. But he explicitly said that certain parts of mathematics are meaningful, and even that the ideal mathematics that he regarded as literally meaningless is still instrumental and crucial for the mathematics of the sciences.
Thank you. If you read the context, I was arguing against the position that math need only be logically self consistent, not Russell's more extreme position that math and logic were identical. In the SEP we read:
Quoting Richard Zach
Quoting GrandMinnow
One can always add a determinate and previously unprovable truth, or its equivalent (if one knows what it is and not merely that it is) to an axiom system and then "deduce" it. Still, the number of propositions we (all humans) can know is necessarily limited. So any knowable set of axioms is finite. No matter how large that finite set may be, there will be truths that cannot be deduced from it. Also, no computable procedure for generating new axioms will exhaust the possible axioms in a finite time. So, an exhaustive axiom set is unknowable. So there are truths we will never be able to deduce.
Quoting GrandMinnow
That was not my claim. I do not deny that any particular truth is deducible from suitable axioms. Rather, I am saying we cannot generate actual axiomatic sets sufficient to deduce all truths in a finite time -- for any finite set of axioms will leave some truths undeducable.
Quoting GrandMinnow
Nor did I claim that it was. I was merely trying to provide a clue as to what was being discussed to those unfamiliar with aleph-1.
Quoting GrandMinnow
Did I say it was? I pointed to Cantor's 1874 proof as one way of knowing that the cardinality of the reals is not countable. The question asked was how can we come to concepts of countable and uncountable infinity from experience, not what are the principal findings of transfinite number theory.
Quoting GrandMinnow
And do you think that an explanation based on the concept of power sets is more comprehensible to a general philosophic audience than what I said?
Quoting GrandMinnow
I did not say that he did, but that he proved that the cardinality of the continuum was uncountable. You seem to think that I need to provide excruciating detail when that detail is not relevant to the point I'm making, namely that the foundations of mathematics have an adequate moderate realist interpretation.
Quoting GrandMinnow
Again, I did not say that he did.
[quote="Wikipedia "Aleph Number";https://en.wikipedia.org/wiki/Aleph_number"]The cardinality of the set of real numbers (cardinality of the continuum) is 2^?o. It cannot be determined from ZFC (Zermelo–Fraenkel set theory with the axiom of choice) where this number fits exactly in the aleph number hierarchy, but it follows from ZFC that the continuum hypothesis, CH, is equivalent to the identity 2^?o = ?1. [/quote]
Quoting GrandMinnow
Isn't that exactly what I said?
Quoting GrandMinnow
If he was right, then the mathematical statements used by the natural science have to be instantiated in nature, and so are true in the sense of correspondence theory. That effectively vitiates formalism.
My question to you is, how do the details I have smoothed over serve to undermine my thesis? If they do not, then your criticisms are pedantic.
Boom! :up:
There is nothing wrong with the technical term "game".
In common parlance it is considered something unserious but that merely reflects the notorious ignorance of the unwashed masses who often tend to be inspired by their fake morality.
This is a very confused statement. If a mathematical theory applies to reality accurately, it is instantiated in reality and the adequacy of the theory to that instantiation shows the truth of the theory with respect to that instantiation. Further, since we presumably know the instantiation, we can abstract the theory from it. So, one need not "maintain that there is a mathematical reality." only that empirical reality has a mathematical intelligibility.
Quoting Fooloso4
Since they do not exist, they are not constructs. The theory uses small quantities tending to zero, while always remaining finite.
Quoting Fooloso4
Do you think that I'm the first to notice that Kant's arguments are inadequate?
Quoting Fooloso4
I have not read the original papers, so I don't know if they did or did not. I do know that the parallel postulate has been suspect since classical times precisely because it cannot be abstracted from experience -- which was my point.
Quoting Fooloso4
That is you view. I already noted that Bolyai discussed which geometry described reality, which means that he saw geometry as potentially reflecting reality, and the status of the parallel axiom as a hypothesis to be studied by physics. I am not denying that math can be treated formally once we posit our axioms. I am discussing how we come to posit its axioms, and their epistemological status.
Quoting Fooloso4
Yes, still, the name is not intrinsic to it, but assigned in light of its relation to the game.
The statement presumes that experience gives us access to reality -- which is an independent, not a recursive, assumption. Books have been written on this assumption, but that is a topic for another thread. I would simply say that one can't deny this without twisting the meaning of "reality" as what is revealed by experience.
Quoting Wayfarer
This is why it is important to recognize that in both sensation and cognition we have an existential penetration of the subject by the object, Thus, Kant's claim that "the object is outside me" is only partly true. Every physical object is surrounded by a radiance of action, which is the indispensable means of our knowing it.
Kant's basic problem is that he wants knowing to be independent of knowers when it is actually a subject-object relation. Or, perhaps, he wants us to have divine omniscience of the noumena when we only have human knowledge -- knowledge, not of how reality is in se, but of how it relates to us. Yet, knowing how reality relates to us is exactly what humans need to know to be in reality.
How reality informs me, how I interact with its radiance of action, is immediately available to awareness -- not "outside me." So, Kant has misunderstood the issue.
Quoting Dfpolis
It is not a theory, it is a formal deductive system based on the negation of the parallel postulate.
Quoting Dfpolis
Once again,it is a purely formal, logical system that was developed prior to and independently of any instantiation.
Quoting Dfpolis
And in this case an intelligibility that was not empirically derived, suggesting that the physical world is structured mathematically, that the mathematics are fundamental, formative.
Quoting Dfpolis
This is nonsense. With regard to Zeno, it is the divisibility that is infinite. With regard to infinitesimals the quantity is smaller than can be measured. In neither case is it something derived from experience. They are theoretical constructs. Whether reality is continuous or discrete remains an open question.
Quoting Dfpolis
Your claim was that Kant had no reason to claim that experience is constructed. This was followed in another post by:
Quoting Dfpolis
What do you provide in support of that? That you read Kant's reasoning. My response was sarcastic - The title of your paper: Kant's Reasoning Regarding Experience Faulty. The text of the paper: I read Kant's reasoning.
Quoting Dfpolis
Was it? Your claim is that mathematics is an abstraction from experience. But now you say that the parallel postulate cannot be abstracted from experience. That would make it a theoretical construct, but you have denied that there can be such a thing. You also say that:
Quoting Fooloso4
So, now a central part of Euclidean geometry cannot be abstracted from experience but non-Euclidean geometry can.
The fact is, though, once again, that non-Euclidean geometry was not abstracted from experience. All of this leaves your claim about mathematics being an abstraction muddled. But I take it that was not your point.
Quoting Dfpolis
This is what Bolyai is quoted saying in that article:
The article also states:
Clearly they were not hypothesis about the physical world, or, as your prefer, reality. They were neither abstracted from or hypothesis about the physical world.
Quoting Dfpolis
And how do we come to posit the parallel postulate, if, according to you, it is not an abstraction from reality? Its negation is not an abstraction from reality either. Both, however, have their application in reality.
Quoting Dfpolis
We have been through this. It is not a name assigned to a ball that came to exist independent of the game. It is the name of a ball specifically designed and made to be used to play the game of baseball. If not for baseball the ball would not exist.
This is really a fundamental point. What you're arguing is British empiricism, per Locke and Hume. But does sensory apprehension qualify as 'revealed truth'? Certainly through scientific method, we can discover truth, but the assumption of the 'reality of the given' is precisely what is at issue in philosophy.
Quoting Dfpolis
I'm sorry but I think this is mistaken. Again I'm no Aquinas scholar, but I think I grasp some of the rudiments of his hylomorphism, which says that
Aquinas, Thomas; Truth, Vol. II, Qs. 10, Article 4
And this is because, in the view of Christian philosophy, material things have no intrinsic reality; creatures are, as Aquinas' Dominican peer Meister Eckhardt said, 'mere nothings'. The locus of reality, as it were, is not the empirical domain, the realm of sensible objects, but the intelligible order inhering in the divine intellect of which the sensable domain is a product or creation.
Quoting Dfpolis
I think it's more likely that you're misunderstanding Kant.
Well, since the scientific method cannot possibly discover any truths about itself, how would it be able to discover the complete truth? While mathematics does have the self-knowledge that it is necessarily incomplete, the scientific method is simply not capable of that kind of self-inquiry.
Scientism is the Dunning-Kruger effect on steroids:
In the field of psychology, the Dunning–Kruger effect is a cognitive bias in which people mistakenly assess their cognitive ability as greater than it is. It is related to the cognitive bias of illusory superiority and comes from the inability of people to recognize their lack of ability. Without the self-awareness of metacognition, people cannot objectively evaluate their competence or incompetence.
LOL We all can’t be John Nash or Alan Turing types. We need guidance from the intellectual elites because we don’t know what’s good for us.
I never would claim that science is omniscient and I myself am a critic of scientism. But it's implausible to deny the fact of scientific discoveries and principles. So we have to be able to grant science the considerable credit where it's due, without at the same time claiming that it is all-knowing, even in principle.
The Continuing Relevance of Immanuel Kant
I had a friend, Mike Zielinski, who received his PhD in mathematics from the University of Wisconsin-Madison. He tried describing to me his dissertation and it was completely over my head. I asked him out of curiosity if the subject of his dissertation reflected anything in the physical world. He simply said, “I don’t know.” Sounds like a game to me, but what do I know with my fake morality and all.
It may describe some as of yet unknown physical process or it may just be a mathematical unicorn. Could these mathematical discoveries still be used in, say, cryptography?
This is very profound. Who came up with this? Was it you? Also, could you flesh this out for me so I can understand it better: “Every physical object is surrounded by a radiance of action, which is the indispensable means of our knowing it.”
It’s totally up to your uncoerced will, of course.
Or more accurately, our experience of the physical world. I remain a Kantian until I hear better arguments.
Agreed.
Scientism points to the fact that science badly lacks humility. That is a fundamental problem. The scientific method does not allow them to know when they do not know. The method simply does not allow for self-awareness. That is why they are so delusional to believe that they know everything; which is obviously ridiculous.
Seriously, this impossibility of self-inquiry is an enormous flaw in the scientific method. As a result, the false belief in its own delusional omnipotence has been snowballing for centuries now, and has even gone mainstream as to infect the unwashed masses with this dangerous disease. For heaven's sake, who is going to save the delusional populace from their delusions? I do not think that it can be done.
In comparison, mathematics is incredibly humble.
While reasoning from its arbitrary starting points, mathematics admits that it is unable to answer many otherwise applicable questions about these arbitrary starting points. Gödel's discovery of this theorem turned him into one of the most admired grandees in the field. Furthermore, the formalist philosophy admits that on the whole a good mathematical theory is meaningless (has nothing to do with the real world) and useless (no direct application possible).
You might appreciate this thread and the essay it points to. Actually, there's a book from a couple of years back on a similar theme, The Blind Spot: Science and the Crisis of Uncertainty by William Byers - he's an emeritus professor of maths, I think with your background and interest in math you might find it interesting.
Quoting alcontali
'The beasts are driven to the pasture by blows', said Heraclitus.
It is had to say without even knowing the area of research.
Quoting Noah Te Stroete
I came up with it reflecting on Aristotle and Aquinas. Aristotle classes action as an accident as something inhering in a substance. If we reflect on any object that we encounter, we see that it is inseparable from its environmental effects -- its radiance of action. This includes its gravitational field, the light that it radiates and scatters and the odors it emits -- all the means making it sensible, observable. The quantum description of matter also shows no hard boundaries -- its material fields extend becoming ever more tenuous. This action on us modifies our our neural state, and that modification of our neural state is identically our neural representation of the object. So, so that part of us is also the object's action.
We can and usually do abstract the object from its radiance of action, leaving us with the impression that it is no more than a core with well-defined boundaries. Still, if we remove the radiance of action from an actual object, it no longer acts as it does and no longer is what it is. Instead of being an integral part of reality, it becomes an isolated monad.
Quoting Noah Te Stroete
Yes, I think this is the case, for example with the structures studied in abstract algebra. Ultimately, however, the foundations can be traced to abstractions from reality or to hypotheses.
Quoting Fooloso4
Intelligibility is a potential that exists prior to being actually known. So, it is not "derived." It is in nature.Quoting Fooloso4
I suggest you read a calculus book.
Quoting Fooloso4
I was challenging any Kantian to provide what they believed was an adequate argument. When an argument was provided I rebutted it.
Quoting Fooloso4
Reread the OP.
One can be right about some things, and wrong about others. While I am happy to allow Bolyai his joy, his assessment is clearly inaccurate. Human creativity consists in imposing new form on old matter, not creation ex nihilo. Most of the axioms in non-Euclidean geometry are from Euclid. Concepts derive meaning from experience. So, his achievement was to impose new form on prior, empirically derived, content.
Quoting Fooloso4
Yes, and no. I grant that most modern mathematicians are not thinking of the real world when they work. That does not mean that the content they work with is not derived from our experience of reality.
To be continued ...
This ignores the point. First, by derived I mean abstracted. Second, if the mathematical structure is in nature but that structure is knowable without being abstracted from nature then there is reason to think that structure might be independent of nature.
Quoting Dfpolis
I suggest you read why I said it was nonsense and respond to that. Here it is once again:
Quoting Fooloso4
First, Zeno's paradox is not something abstracted from nature. Second, both Newton and Leibniz used a concept of infinitesimals that was not abstracted from nature given that the infinitesimal is not measurable. Third, the question of whether reality is continuous or discrete is something that is dealt with in physics not mathematics.
Quoting Dfpolis
If you are referring to 2a, an axiom or postulate is not a hypothesis.
Quoting Dfpolis
Of course it is not creatio ex nihilo! He did not mean it literally. Nit picking does not address what is at issue. Once again, non-Euclidean geometries are not abstractions. The negation of the parallel postulate is not a hypothesis, it is an axiom. What is of interest is what follows from it, and what follows is completely independent of physical reality.
Quoting Dfpolis
The negation of the parallel postulate is not derived from our experience of reality, nor is what follows from it.
You seem to have forgotten the OP, where I used it as an example of a hypothetical postulate. It is derived by assuming that our small-scale experience with parallel lines can be extended to infinity.
Quoting Fooloso4
Whatever we know can be truly applied to reality can be abstracted from reality. We do not and cannot know that the parallel postulate is true because our experience is finite. We can only know if the space-time metric is approximately Euclidean.
We can abstract non-euclidean geometries from spherical and saddle shaped surfaces.
Quoting Fooloso4
I agree with all of this, The point is that none of it, including the name, is intrinsic to the ball.
---
Quoting Fooloso4
When intelligibility is abstracted it ceases being potential and commences being actually known. The whole point of intelligibility is that it is potential, not actual, knowledge.
Quoting Fooloso4
I do not understand this at all. If it is in nature, there is no reason to think that it is not intrinsic to nature. Green leaves are in nature and intelligible. Does that mean they also have a Platonic existence independent of nature?
Quoting Fooloso4
Yes, the potential to divide a continuous span is unlimited; however, any actual division is only finite. As we can only know what is actual, we cannot know anything infinitely divided. (Imagining an infinitesimal is not knowing it.) As I told you earlier, this is the reason for all of the epsilons and deltas in the definitions of calculus -- and it was to see those types of definitions that I referred you to a calculus book.
Quoting Fooloso4
The question is not measurability, which is one for physics, but of being finite or not. Any actual quantity quantity greater then zero is finite. If we use '0' to define the concepts of calculus, they will be indeterminate. So, we us the limits of finite quantities tending to zero.
Quoting Fooloso4
I did not claim it was.
Quoting Fooloso4
I have not read their derivations. I know that they were defective and have been replaced by those now found in most calculus texts.
Quoting Fooloso4
Physics might well find limits to what is actually measurable, given the laws of nature. That is not deciding whether reality is continuous or not. The concept of continuity abstracts from the question of actual measurability.
Quoting Fooloso4
Regardless of whether I am right or wrong, I did not claim that all mathematics is an abstraction.
Quoting Fooloso4
If he did not mean it literally, does not support your position. If he would agree that he was imposing new form on old matter, then he might agree that the matter of math was abstracted from experience.
You keep repeating your dogmas, but you do not support them with arguments. You have not said why my analysis does not work beyond saying it does not agree with your belief system. I agree, my analysis is incompatible with your beliefs.
No, I am not. I am arguing Aristotelian moderate realism.
Quoting Wayfarer
Experience is the data we have to work with. One can either work with experience, or one can simply cease thinking. The scientific method does not get one past this, as all it does is compare hypotheses to experience. Whatever you think reality is, experience is how we humans relate to it -- and we can only deal with it as we relate to it.
We do not and cannot have omniscience, so it is a trap to make omniscience the paradigm case of knowing. "Knowing" names a human activity. So as soon as you say "we do not know," you are abusing the foundations of language. "Reality" first means what we encounter in experience. So, if you say "we do not experience reality," you are again abusing language.
When you make "reality" mean more than, or something other than, what we encounter in experience, you are creating a mental construct. If you create that construct, and then claim that what you have constructed is inaccessible, you have said absolutely nothing about what we encounter in experience.
Doubt is an act of will. I can will to doubt anything, including my own consciousness, as eliminative materialists such as Dennett have chosen to do. What one cannot do is eliminate what we experience. We experience ourselves as subjects and everything else as objects. I know what I experience and no act of will, no doubt, can make me not know it.
Of course, I may misinterpret what I experience. I may think the elephant I experience is in nature rather than the result of intoxication. Still, if I did not have experiences I know to be veridical, I could not judge others to be errant.
Quoting Wayfarer
Only in post-Cartesian philosophy. The focus of pre-Cartesian philosophy was and continues to be being.
Quoting Wayfarer
What you quoted was a "difficulty" or objection Aquinas intends to resolve, not his position. His response is:
Quoting Aquinas De Veritate
Quoting Wayfarer
Eckhardt's is not Aquinas view. He sees material things as real and intrinsically good, as does Gen. 1, which sees God as judging each stage of creation as good.
Quoting Aquinas ST I Q 65 Art 6 ad 6
Quoting Wayfarer
If Kant is saying that we can know noumenal reality, but not exhaustively, I have indeed misunderstood him. I do not think he is saying that, do you?
I think the flaw is seeing the scientific method as the only acceptable means of inquiry. In its proper domain, the scientific method is fine.
Here is the problem in a nutshell. You refer to your "analysis" as if it is not based on your own dogmas and beliefs. The fact that you indefatigably argue them demonstrates nothing more than your willingness do so.
I have explained how ere abstract concepts such as that of number from the realization that counting does not depend on what we count. You have not shown that this is an inadequate explanation of our natural number concept.
Thank you for demonstrating my last point.
Why? This is not a flippant question.
I did not claim to address, let alone undermine, your thesis. I set straight certain of the mathematical subjects you mentioned. That is not pedantic. In some cases, your comments were incorrect, and in other cases, your comments were unclear, ambiguous, or confused so that they deserve response even if those responses are not onto themselves corrections. It was my fault not to distinguish between those cases. But in followup at this juncture, it would be unwieldy to always separate those cases, so not each of my responses should be taken necessarily as disputing you.
(1) In your first post you wrote, "the movement characterized by Hilbert's program, which sees mathematical truths as reducible to logical truths", and in a later post you stated that Hilbert viewed mathematics as logic.
But those are incorrect onto themselves, and they are incorrect even as you claim now that you meant only to refer to the position that mathematics only needs to be consistent. Moreover it cannot be discerned from what you originally posted that you did not mean, as you actually wrote, that Hilbert saw "mathematical truths as reducible to logical truths" but instead meant to refer to the very different position that "mathematics need only be logically self-consistent".
Moreover, I should add, while adequacy of consistency is, in a certain key aspect, an important part of Hilbert's view, he does not take mathematics to be merely a matter of consistency.
And your second comment about mathematics as logic (known as 'logicism'*) was in your context of the incompleteness theorem. Perhaps it can be argued that the incompleteness theorem refutes logicism, but that is not really related to Hilbert. The damage the incompleteness theorem does to Hilbert's concept is a different subject: The incompleteness theorem shows that the hope for a finitary consistency proof cannot be realized, but this does not, in itself, entirely refute the adequacy or role of consistency. (* Elsewhere you refer to 'logicalism'. I have never read that term before, so I take it that you mean 'logicism'.)
(2) You wrote:
"One can always add a determinate and previously unprovable truth, or its equivalent (if one knows what it is and not merely that it is) to an axiom system and then "deduce" it. Still, the number of propositions we (all humans) can know is necessarily limited. So any knowable set of axioms is finite. No matter how large that finite set may be, there will be truths that cannot be deduced from it. Also, no computable procedure for generating new axioms will exhaust the possible axioms in a finite time. So, an exhaustive axiom set is unknowable. So there are truths we will never be able to deduce."
There's a lot to sort through:
(2a) For any given system, we always know, by finitary construction, the specific Godel sentence. So I don't see the point of saying "if one knows what it is".
(2b) I don't know your point in putting the word 'deduce' in (scare) quotes. Any axiom, of course, is deducible from itself.
(2c) When writers in mathematical logic or in the theory of computability talk about such things as derivability, of course, this means derivability in principle, not limited to any given finite lifespan of human beings. And, of course, computable procedures are also not limited by finite "time" (such as a recursive enumeration that "runs" infinitely).
So to point out that
in finite time, in anthropomorphic terms or even in terms of physical computations running finitely long, there will always be unknown theorems
does not require invoking the incompleteness theorem.For the point you want to make, mentioning the incompleteness theorem is gratuitous. And the point you want to make is your defense of your original claim:
"There are truths that cannot be deduced from any knowable set of axioms".
I should have addressed the qualification 'knowable' last time. There is a difference between known and knowable.
Perhaps* at any given finite point, there are axiom sets that are not known, but that doesn't entail that they are never to be known (that they are unknowable). (In this context, by 'axioms' we mean recursive, consistent, arithmetically adequate axioms.) (* One might look into whether there is a finite description of the class of systems, as indexed by the ordinals, so that, in a sense, we do know the set (I'd have to brush up on that question).)
Quantifiers help:
There are truths that cannot be deduced from any knowable set of axioms. (False.)
From any particular set of axioms, there are truths that cannot be deduced. (True.)
You wrote:
"I am saying we cannot generate actual axiomatic sets sufficient to deduce all truths in a finite time -- for any finite set of axioms will leave some truths undeducable."
(2d) Axiom sets don't have to be finite to be recursive.
(2e) Again, we don't need the incompleteness theorem to tell us that the entire set of arithmetical truths (or any infinite set of statements) cannot be derived in finite time. Your point can be made even stronger: The incompleteness theorem yields that for any (recursive, consistent, arithmetically adequate) set of axioms, there are truths not provable from those axioms (provable period, not just in finite time). But your point doesn't change that your original statement "there are truths that cannot be deduced from any knowable set of axioms" is incorrect, or at best misleading pending explication of 'knowable' not just 'known', and anyway, that does not concern the incompleteness theorem.
(3) You wrote:
"Godel's work means that we cannot prove the consistency of a set of axioms"
As I mentioned, that is flat out incorrect. Thankfully, you have not disputed my correction.
(4) On Cantor and the continuum, you have conceded. Thank you. But I would like to answer your question:
"do you think that an explanation based on the concept of power sets is more comprehensible to a general philosophic audience"
I don't know what is more comprehensible to any given audience. I only mentioned that the power set proof is, in a sense, more basic and simpler. Proving by reference to the real numbers requires getting into the subject of representation by denumerable sequences (and in what base notation).
(5) I wrote, "If a consequence of C is falsified, then C is falsified" and you replied "Isn't that exactly what I said?" Maybe it was what you meant; I don't know because it was not clear to me what you meant.
(6) You wrote, "If [Hilbert] was right, then the mathematical statements used by the natural science have to be instantiated in nature, and so are true in the sense of correspondence theory. That effectively vitiates formalism."
Instrumentalism does not commit one to requiring that a mathematical statement onto itself must correspond to a corresponding statement about the natural world.
Yes, it is not a problem in itself to refer to 'games'. If is fair enough to say that Hilbert took mathematics, in a certain regard, as concerned with symbol games. But it is egregiously incorrect - blatantly against the clear evidence of Hilbert's writings - to claim that Hilbert took mathematics to be merely a matter of symbol games.
Right, standard analysis does not admit infinitesimals. However, calculus can be formulated in non-standard analysis or in internal set theory, as those approaches do formalize the notion of infinitesimals.
There are different variants of formalism. Only a quite extreme variant holds that mathematics is meaningless, let alone that it is useless.
In the section, formation rules, of the wiki page on first-order logic, you can see how they quickly gloss over a glaring issue in the current practice of mathematics:
The role of the parentheses in the definition is to ensure that any formula can only be obtained in one way by following the inductive definition (in other words, there is a unique parse tree for each formula). This property is known as unique readability of formulas.
The traditional Russell-Whitehead notation is not necessarily unambiguous.
That is probably normal, because the notation was originally introduced for human use and human consumption, in 1910-1913, during the publication of Principia Mathematica (PM).
PM obviously does not contain an EBNF specification for its notation, and is actually not even aware of the glaring problem of ambiguity in its notation. The BNF metalanguage was introduced only in 1959.
Just look at how the gloss over the following issue:
Each author's particular definition must be accompanied by a proof of unique readability.
In all practical terms, it means that the author is supposed to provide the EBNF grammar along with the generated parser tables in order to demonstrate the automaton associated with his grammar is conflict free. That is serious work.
Some people seem to be reinventing the wheel on this matter:
As a corollary, we see that the well-formed formulas of the classical propositional logic, written in Polish notation, are uniquely readable. The unique readability of wffs using parentheses and infix notation requires a different proof.
You can trivially prove that postfix Polish notation can be executed/verified by a stack machine, while infix first requires translation to postfix. Therefore, it does not require a different, but an additional proof, i.e. that all infix expressions can unambiguously be translated to postfix. Again, that can be achieved by demonstrating the the associated automaton is conflict free.
So, to come back to what you were saying about Hilbert, yes, there is a very important "language game" going on, indeed. Entire areas in mathematics do not pay enough attention to it, and still indulge in ambiguous notation that is not machine verifiable.
So, what about finally getting the "symbol games" right?
Meanwhile, mathematics is usually written in a combination of formal and informal notation along with natural language. This is not ordinarily problematic, since it is usually clear enough how one would formulate such semi-formal writings into pristine formalization (permitting proof of unique readability) if one wanted to do that.
Not only!
If your statement correctly parses into two different syntax trees, then it almost always has two different interpretations. Greek and Roman antiquity were already very well aware of this problem:
Ibis redibis nunquam per bella peribis
What exactly does that sentence mean? If your parsing strategy is greedy, it means exactly the opposite of what it means in a lazy parsing strategy:
[i]greedy: you will go, you will never return, in (the) war you will perish
lazy: you will go, you will return, never in war will you perish[/i]
It materializes as a shift-reduce conflict in the associated automaton. It is a problem with the grammar itself, and actually not just with this particular sentence. The conflict merely reveals itself in this particular sentence, but it is grammar wide.
This is also the material of which the more advanced network intrusion attacks are made. We can generally not ignore that kind of problems because the overall cost must now globally already run in the trillions of dollars.
Quoting GrandMinnow
Informal commenting versus more strictly regulated language is obviously normal practice. Still, it makes sense to machine-verify the regulated scope, at the very least, for well-formedness. It does not take much effort, and I wonder why anybody should be encouraged to publish formulas that are not even well-formed? E.g. MathJax could do more basic validation. That requires the use of more formal scope delimiters. Of course, we do not need to push it to the point that the editor only accepts expressions that are provable from the theory at hand, or so. That would obviously go too far.
Quoting GrandMinnow
That sounds very yagni ("You aren't gonna need it"). In principle, I agree with yagni, but sometimes it is not the right approach, while the really hard problem is to know when it matters and when it doesn't.
[i]Missing Comma Costs Millions
The essence: Delivery drivers who work for Oakhurst Dairy, a Portland, Maine-based company, will be entitled to an estimated $13 million in overtime pay after winning a three-year legal dispute with their employer. An appeals judge ruled that lack of a comma made interpretation of the phrasing of an agreement vague.[/i]
The post-mortem of a successful network intrusion almost always reads like that, and leaves the following impression:
What? Really? You must be kidding me!
Especially in cryptography, a lot of mathematical language ends up being implemented pretty much verbatim in software, without even checking the logic again; because that was supposed to have been done in the mathematical work anyway. An ambiguous syntax tree for a mathematical expression could then lead to billions of dollars in losses.
Unique readability affords definition by recursion, and definition by recursion affords the method of models, which provides that every statement has exactly one meaning per a given model.
You will find comments about unique readability, similar to mine at math.stackexchange.com. Unique readability plays a role in model theory, but it plays an even much bigger role outside of it.
Why is the unique readability of wff's important?
[i]The bottom line is that a statement cannot have unambiguous meaning unless you have a precise unambiguous way of interpreting (reading) it. Even in natural language almost all sentences have unambiguous grammatical structure and more or less clear semantic interpretation, which is not an accident, because otherwise it would fail to be a viable means of communication!
...
You will quickly realize that formal systems for mathematical logic are miniscule in complexity compared to programming languages, but you will also see clearly the reason for the exacting precision in defining a formal system.
...
If we didn't have unique readability, then the language would be useless - we wouldn't be able to say that a sentence is "true" or "false". We could have a sentence that was true under one reading, but not under the other.[/i]
Yes, and mentioning the issue "unique readability" forcibly drags the entire, seemingly endless field of formal language and therefore related automata theory into the fray.
That means that the "language game" now pushes you right into the middle of the mathematics that govern computer science and in front of the issues that arise in programming languages.
Automata theory is not something that was abstracted away from reality. There are no automata in nature. You have to painstakingly build them, and you have to know how they work, and why they work, before you can even build them. Furthermore, through the problem of unique readability, they strike at the core of mathematics.
If you cross over from the realm of computability into pure mathematics, the term "language game" does not sound controversial at all. Brouwer's accusation that Hilbert was merely "playing language games", gives the impression that Brouwer was seriously missing the point. Hilbert, on the other hand, was asking all the right questions, during the end of the 19th century and the first half of the 20th century, long before the first computer was even built.
I agree with you that Hilbert asked the great questions (or stated the great challenges). Hilbert was like the "master of ceremonies" of a great direction in mathematics and the philosophy of mathematics. But I am inspired by all the contributions, including the constructivists, intuitionists and predicativists toward the development of this deep and rich enquiry.