The Foundations of Mathematics

Dfpolis

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.

tim wood

Quibbles. Mostly concerning language. But quibbles are exhausting to chase down. An example:

and surely we cannot abstract universal truths that are not instantiated. — Dfpolis

Are you sure of that? Are not all instantiated truths particular truths, and are not all universal truths abstract? You have left out how to get from particular to "universal." And how do you instantiate the truths of transfinite arithmetic?

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. — Dfpolis

I doubt if one in twenty-thousand can say what a "Platonic" world is. Further, it may be that we can trace to a ground in nature, but is that all? And while the idea of concepts needing minds to have and hold them informally and intuitively seems right, is it altogether right? It seems to me there is here the fallacy of the half-argued argument, the appeal for both halves of something only half complete. Why the quibble? Mind-matter arguably requires a mind, but the content of thinking - the that that is thought about, is pretty much always already there prior to a mind thinking it. This means there is an accounting/definition problem.

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. — Dfpolis

I think maybe "unscientific" in this context is wrong usage. In any case, mathematics is not an experimental science. As to "games," again unfortunate usage. and against what you're saying, almost incoherent.

From wiki, "In 1938,[14] Kurt Gödel showed that the negation of the axiom of choice is not a theorem of ZF... thus showing that ZFC is consistent if ZF itself is consistent. In 1963, Paul Cohen... [showed] the axiom of choice itself is not a theorem of ZF.... [T]hus showing that ZF¬C is consistent[15].
Together these results establish that the axiom of choice is logically independent of ZF."

Thus indeed it would seem the axiom of choice is an axiom and not just a hypothesis. Therefore not "unscientific"; and the pursuit of the consequences of which, which may be "no different in principle," are in practice and in fact altogether and entirely different from a mere game.

But quibbles, I say, and until and unless you say something more substantial about all of these things, that can be more substantively addressed, then quibbles they stay. Next?

Wayfarer

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. — Dfpolis

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?

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 — Rebecca Goldstein

Fooloso4

In the Greek arithmos a number is always a number of something. A number tells us how many ones or units. This is why Aristotle says that two is the first number. In your example of two apples and two oranges, there is no problem of determining how many as long as we know what the unit of the count is. In this case pieces of fruit. In the same way you determine that there are two apples and two oranges, you determine that there are four pieces of fruit, that is, simply by counting them. Modern number theory and set theory axioms is anachronistic.

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.

RegularGuy

↪Dfpolis

I found your essay illuminating as I don’t have any experience in mathematical theory.

What is the axiom of choice?

ssu

Yet, Godel's work shows more: it shows that there are truths that cannot be deduced from any knowable set of axioms. — Dfpolis

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

I think maybe "unscientific" in this context is wrong usage. In any case, mathematics is not an experimental science. — tim wood

I agree here with tim wood, talking about scientific/unscientific here with foundations mathematics is totally out of whack.

Dfpolis

and surely we cannot abstract universal truths that are not instantiated. — Dfpolis

Are you sure of that? Are not all instantiated truths particular truths, and are not all universal truths abstract? You have left out how to get from particular to "universal." And how do you instantiate the truths of transfinite arithmetic? — tim wood

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.

Further, it may be that we can trace to a ground in nature, but is that all? — 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.

And while the idea of concepts needing minds to have and hold them informally and intuitively seems right, is it altogether right? — tim wood

It is, unless you have a different definition of "concept," but then, you're not talking about what I'm talking about.

the content of thinking - the that that is thought about, is pretty much always already there prior to a mind thinking it. — 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.

This means there is an accounting/definition problem. — tim wood

Specifically?

I think maybe "unscientific" in this context is wrong usage. In any case, mathematics is not an experimental science. — 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?

the pursuit of the consequences of which, which may be "no different in principle," are in practice and in fact altogether and entirely different from a mere game. — tim wood

What would the difference be? It's a game many mathematicians enjoy?

Dfpolis

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? — Wayfarer

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).

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'? — 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.

Surely that is something real, as real numbers are included in it, and irrational numbers are not. — 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.

And how does your account differ from run-of-the-mill evolutionary naturalism, in which there is nothing corresponding to what Aquinas 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. — 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.

Do you know that Godel considered himself a mathematical Platonist? — 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.

Dfpolis

↪Fooloso4

Thank you for your comments. I have no problem with the neoplatonic One Identified as God.

Dfpolis

What is the axiom of choice? — Noah Te Stroete

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.

Dfpolis

Yet, Godel's work shows more: it shows that there are truths that cannot be deduced from any knowable set of axioms. — Dfpolis

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 — ssu

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?

I think maybe "unscientific" in this context is wrong usage. In any case, mathematics is not an experimental science. — tim wood

I agree here with tim wood, talking about scientific/unscientific here with foundations mathematics is totally out of whack. — 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."

Wayfarer

Actual numbers exist only in minds actually thinking them. — Dfpolis

But, you also say

the principle of excluded middle reflects the nature of being and so is intelligible to anything capable of abstracting being as being. — 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.)

Naturalism has no problem describing neurophysically encoded contents, but it has no rational account of awareness. — 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.

ssu

So, you will have to explain why my criticism is "totally out of whack." — Dfpolis

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:

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. — 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.

ssu

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? — Dfpolis

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.

Fooloso4

Thank you for your comments. I have no problem with the neoplatonic One Identified as God. — Dfpolis

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.

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. — Wayfarer

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.

think it's reason that naturalism has no account of. — 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.

Dfpolis

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. — ssu

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.

Applicability of mathematics to the physical world isn't the logic that glues mathematics into a rigorous system, but logic itself. — 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).

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. — 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.

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. — 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.

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. — 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.

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. — 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.

Dfpolis

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. — Fooloso4

And that relates to my OP how?

Fooloso4

↪Dfpolis

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:

... our mathematical concepts have a foundation in reality. — Dfpolis

Is like saying a building has a foundation in the ground. It says nothing about that foundation.

See Armand Maurer, The Division and Methods of the Sciences — 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)

Dfpolis

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". — Fooloso4

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.

To say:

... our mathematical concepts have a foundation in reality. — Dfpolis

Is like saying a building has a foundation in the ground. It says nothing about that foundation. — 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.

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 — 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.

Fooloso4

First, sciences do not establish their own principles — Dfpolis

Where do you imagine these principles come from?

I did not claim that Greek math was Platonism — Dfpolis

After a full paragraph on Platonism you said:

Platonic relationship 2 + 2 = 4 — Dfpolis

This is not a "Platonic relationship", it is simple arithmos, the counting of ones or units.

I said most of the foundations are the result of abstraction. — 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?

I disagree with much of the quote you gave from Maurer. — 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?

Wayfarer

To me, "awareness" means agent intellect, without which there is no reason or conscious experience. — Dfpolis

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.

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. — 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.

ssu

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. — Dfpolis

So in the way as history is a science? Some in the natural sciences would shudder at the idea, but I'm totally OK with it.

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 se — 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.

The classical mechanics they developed is still useful and taught today, but it is not true. Unjustified foundations necessarily give unjustified conclusions. — 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.

Dfpolis

First, sciences do not establish their own principles — Dfpolis

Where do you imagine these principles come from? — Fooloso4

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.

After a full paragraph on Platonism you said:

Platonic relationship 2 + 2 = 4 — Fooloso4

Please read sentences in context.

I said most of the foundations are the result of abstraction. — Dfpolis

To say what they are the result of is not to say what they are — Fooloso4

What they are is not my present interest.

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? — 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.

Which leads to the question of whose mathematics? — 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.

Are you disagreeing with his reading of Aquinas? If so, where do the mistakes lie? Or is it that you are disagreeing with Aquinas? — 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.

Dfpolis

Yes, well, crocodiles and dragonflies have some degree of awareness, but zero intellect! — Wayfarer

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.

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. — 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.

Thank you for very lucid explanation. — Wayfarer

You're welcome.

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). — 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.

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.) — 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 <apple> is simply the act of thinking of apples, not thing that can have constituent parts. Still, ideas can have multiple notes of intelligibility that can be separated in thought.

I tried Googling "Augustine intelligible objects" but only found secondary sources so far.

Wayfarer

↪Dfpolis

Here it is again

https://books.google.com.au/books?id=3ISYAwAAQBAJ&lpg=PA24&ots=qR3TB2kKBD&dq=augustine%20intelligible%20objects&pg=PA24#v=onepage&q&f=false

Dfpolis

So in the way as history is a science? Some in the natural sciences would shudder at the idea, but I'm totally OK with it. — ssu

We seem to be converging. I see good history as the result of rigorous method, but not as explaining events from first principles.

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. — 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.

What you are making is a hugely reductionist argument that everything has to be deduced from the same axioms. — 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.

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. — 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.

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. — 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 <straight line>. If we define "straight" to be the path taken by a light ray, then Euclidean geometry is inadequate on a cosmic scale and so false.

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.

Math isn't like this. Mathematics has for example incommensurability, which is totally logical. — ssu

I have no idea what you mean by "totally local." Are you claiming that the concept <incommensurability> came to be independently of any experience of reality? History would seem to rebut this.

It seems to me that you are looking at final canonical forms and forgetting the mental processes that got us there.

Dfpolis

↪Wayfarer

Thank you. I will read it when I have time.

Fooloso4

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 — Dfpolis

You are avoiding the question. Science does not simply "assume its principles". It determines them through observation, hypothesis, testing, theory, modeling, and so on.

mostly via abstraction — Dfpolis

First, someone has to do the abstracting. Second, the properties of say a triangle are not determined by abstraction.

Please read sentences in context. — Dfpolis

You mean this context?

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? — 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.

I said most of the foundations are the result of abstraction. — Dfpolis

To say what they are the result of is not to say what they are
— Fooloso4

What they are is not my present interest. — Dfpolis

So, in a topic entitled The Foundations of Mathematics, the actual foundations of mathematics is not your present interest.

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. — 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

Which leads to the question of whose mathematics?
— Fooloso4

Mathematics is not personal property. — 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.

Dfpolis

You are avoiding the question. Science does not simply "assume its principles". It determines them through observation, hypothesis, testing, theory, modeling, and so on. — Fooloso4

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.

First, someone has to do the abstracting. Second, the properties of say a triangle are not determined by abstraction. — Fooloso4

Yes, we need knowing/observing subjects. And, yes, abstraction does not create content, it actualizes intelligibility already present in reality.

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. — 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.

So, in a topic entitled The Foundations of Mathematics, the actual foundations of mathematics is not your present interest. — 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.

It is not simply adding new concepts, it is a matter of different concepts. — Fooloso4

I am not sure how you distinguish different concepts that were not in prior use from new concepts. Perhaps examples would help.

This does not vitiate old concepts in the sense that they are wrong, but that mathematics no longer operates according to the older concepts. — Fooloso4

I think we are using "concept" in different senses. I am thinking of <number>, <line>, <irrational number> and so on when I say "concept." You seem to be thinking rules of procedure. Yes, there are new procedures, but Euclidean geometry is still being pretty much as he wrote it in the Elements. So, the old procedures are not abandoned.

Thank you for the link/reference.

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. — Fooloso4

No, I don't dismiss different conceptual spaces as wrong -- they are just different ways of thinking about the same reality.

alcontali

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. — Dfpolis

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.

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.

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.

The Foundations of Mathematics

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