An Viable Alternative To Platonism In Math?

Science & Technology Physics and Mathematics
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To make vast generalizations about philosophers of mathematics, one should at least know of one.

As Hal mentioned, Russel, Kronecker, Hilbert were all great mathematicians but they were not such great philosophers, and many of them switched their stances on mathematics throughout their years. Also, there are many great philosophers, the greatest in my opinion being Wittgenstein, who did not know enough math to be able to talk about it in any non-trivial manner, but that did not stop them from doing so. My point in all of this is that for some reason or another non-mathematicians think that they know enough to provide a philosophical foundation for mathematics. That is why I say one should at least know enough algebra and topology to be able to understand non-trivial results in algebra and topology, once you know that much then go ahead and say math is this or that. Besides the thread was about an alternative to platonism in mathematics and my stance of mathematical ideas and concepts being some generalization of physical intuition, I think provides a great alternative, because I do not posit an existence of some realm that no one can touch or feel.

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mathematics is the science of measurement.

Then if some subject does not pertain to measuring things, in order not to confuse it with your concept, we’ll have to call this subject something other than mathematics, even though it’s in the mathematics department of virtually every university in the world, has the word ‘mathematics’ in the titles of the books and journals about it, and is classified by virtually every library in the world in a section called ‘mathematics’. I’m not arguing that your concept formation is necessarily incorrect, just that we should at least recognize that the word ‘mathematics’ is usually understood not to refer to your more specific concept.

It exists to work out the process by which we perform actual measurements.

Yes, this is one of the purposes people have in performing mathematics. But it is not the only purpose, and since humans first wondered about mathematical questions for just the sake of knowing, not necessarily measuring, it has hardly been the primary purpose of the people who are called ‘mathematicians’.

The way to keep it reality oriented, therefore, is to know what kind of things in reality it’s used to measure.

Do you think that the technological application of mathematical results are always known prior to their investigation?

In other words, you must keep in mind its applications.

You mean we should not wonder about mathematical questions from just our curiousity about them? Mathematicians should first check with engineers to see if the engineers can use the mathematics that the mathematicians are interested in investigating? And if there is no known application or even forseeable application, we should discourage the study of mathematics that, for all we know, is ahead of the technology and which we cannnot predict its applications? I don’t mean to put words in your mouth, but your statement does induce these questions.

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Russel, Kronecker, Hilbert were all great mathematicians but they were not such great philosophers

You don’t think they were great philosophers, but that addresses the CONVERSE of your assertion, not your assertion. You claimed that philosophers of mathematics were not mathematicians. When it was pointed out to you that that is incorrect, you’ve come around to say that some of the mentioned mathematicians aren’t much as philosophers. Anyway, the overwhelmingly vast number of philosophers of mathematics have been mathematicians. Wittgenstein is one example of a philospher who is usually not considered a mathematican. One of the very few. But what is the basis of your claim that he was not conversant in mathematics? In any case, the point stands that you are flat out incorrect in your original assertion, especially regarding the late nineteenth century and twentieth century. Even in earlier centuries, at least most of the famous philosophers who opined on mathematics weren’t exactly ignoramuses about the subject.

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The point is that a theoretician must never forget that the value of his work is in its application to human purposes – to man’s life.

To WHAT man’s life? To the lives of all men? To the lives of some collection of men? Or to his OWN life?

Most importantly, he must not oppose the application of his work to human life.

What mathematician ever opposed having his work applied to benefit human life? Okay, true, some scientists were and are disinclined to assist in the building of nuclear weapons and such. But I guess each of us would have to decide for himself whether that would qualify as an example here.

But this does raise an interesting question: how can a theoretical mathematician, who is working on math that does not yet have an application in reality, stay reality oriented?

First, we need a definition of ‘reality oriented mathematics’, which provides some way of distinguishing staying reality oriented in math from not staying reality oriented in math. Then, it would help just a little if there were some example of what one claims to be reality oriented math, as opposed to not reality oriented math.

He must keep in mind the ultimate purpose of his work (even if he doesn’t carry out this purpose) is the application of his work to reality.

Why must he do that? If his work is not applied mathematics, then why does he have to regard the ultimate purpose of his work as in application? Why does he have to even consider an ultimate purpose that is not his OWN purpose? That other men have certain or even general purposes should not mandate that he share that purpose, does it? If his purposes are not application, then I don’t see why he has to keep in mind the applicational purposes of OTHER men. Let me ask, if I want to know if the diagonal of a square with unit sides can be a rational number, then is it immoral for me not to first check whether answering this question will have some application for other men or even for myself?

If he does that, I don’t think he’s doomed to rationalism.

And he may not be obligated to rationalism even if he does NOT do as you recommend. If everyone read Shakespeare ten hours a week, then they’d not be doomed to illiteracy. It doesn’t follow that one must read Shakespeare ten hours a week to not be doomed to illiteracy. Further, a devotion to abstract mathematics is hardly even an INVITATION to rationalism, let alone a guarantor of it without the intervention of reminders to oneself about the purposes other men may or may not have.

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we can’t arrive at the concept 0 until we have 1 and a number of natural numbers

I’ve never heard of this.

In mathematics, we deduce our premises from a set of axioms, and then apply them by induction.

You must mean we deduce our conclusions from axioms. The axioms are the premises; the theorems are the conclusions. But what do you mean by saying that we use induction to apply these? Do you mean in the technological application or do you have some application within mathematics in mind?

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You don’t think they were great philosophers, but that addresses the CONVERSE of your assertion, not your assertion. You claimed that philosophers of mathematics were not mathematicians. When it was pointed out to you that that is incorrect, you’ve come around to say that some of the mentioned mathematicians aren’t much as philosophers. Anyway, the overwhelmingly vast number of philosophers of mathematics have been mathematicians. Wittgenstein is one example of a philospher who is usually not considered a mathematican. One of the very few. But what is the basis of your claim that he was not conversant in mathematics? In any case, the point stands that you are flat out incorrect in your original assertion, especially regarding the late nineteenth century and twentieth century. Even in earlier centuries, at least most of the famous philosophers who opined on mathematics weren’t exactly ignoramuses about the subject.

You don’t understand what my point is and keep pointing out that none of my assertions have any basis. In fact the basis is my personal experience and the books I have read on the matter. So here is my point. To talk about any subject, one should at least understand it and have some kind of intimate knowledge of the subject. Such knowledge comes after years and years of being immersed in the subject. Most philosophers do not have such a background in mathematics, and that is why when they “opine” on the matter, their “deep” philosophical inquiries seem quite childish. Plus, all the great mathematicians that were also philosophers, had very little to say on the subject of philosophical foundations of mathematics. If you read any of their works you will notice this. Most of their writings deal with logical foundations and there is very little actual philosophical inquiry. Most of their writing deals with trying to provide a foundation that is free of paradoxes. The little that does deal with philosophical foundations of mathematics is simple and nothing special.

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Of course I agree that people who don’t know anything about mathematics are up to their necks when they pontificate about the subject. But, the vast amount of published work (I don’t mean popularizations like the David Foster Wallace book and others) on the philosophy mathematics - journal articles, textbooks, et. al - is written by mathematicians or philsophers who have studied mathematics. Dedekind, Pierce, Frege, Hilbert, Russell, Brouwer, Heyting, Quine, Godel, Church, Carnap, Hintikka, Feferman, Putnam, Kripke, Dummet, Benacerraf and my favorite, Curry. One could list hundreds. And going back through the centruries. Hey, Plato was not exactly a mathematical dummy. There are entire shelves and sections of libraries and thousands of Internet pages full of philosophy of mathematics written by mathematicians. If you’ve not come across any of this stuff, then you need to get out a little more often.

You have an important point: too many people who don’t know about math are only too willing to spout nonsense about it. But you’ve overstated when you say that the philosophy of mathematics is bereft of mathematicians and the mathematically savvy. And especially of philosophers in the last fifty years or so. What example do you give of a philosopher whose writings on the philosophy of mathematics are insufficiently informed? (I mean philosophers, not merely popular writers and not post-modernists, or whatever they’re called.) Surely you’re aware that many (most? nearly all?) university philosophy departments have so much math swirling through them that they might as well be math departments.

the books I have read on the matter.

Which books? What are the books on the philosophy of mathematics, written by philosphers, that you’ve read that are not sufficiently informed?

Plus, all the great mathematicians that were also philosophers, had very little to say on the subject of philosophical foundations of mathematics. If you read any of their works you will notice this. Most of their writings deal with logical foundations and there is very little actual philosophical inquiry.

That’s a different complaint. You’re taking exception to the focus of these mathematicians’ philosophical writings. But that does not entail that they are not about philosophical foundations. Anyway, right off the bat, Quine and Putnam are two salient counterexamples. They’re very much concerned with the broad philosophical picture.

Most of their writing deals with trying to provide a foundation that is free of paradoxes. The little that does deal with philosophical foundations of mathematics is simple and nothing special.

The foundational issues are the Grand Central Station of the philosophy of mathematics, which makes perfect sense. But there is fervent debate on all kinds of issues. Yes, most of them touch back to the foundational questions, but in the same way that philosophy itself keeps touching back to certain central issues. Anyway, there is so much written on the philosophy of mathematics, and by mathematically astute people, that it just overwhelms me thinking about how much I’ll never even get a chance to look at, let alone read.

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That’s a different complaint. You’re taking exception to the focus of these mathematicians’ philosophical writings. But that does not entail that they are not about philosophical foundations.

But that is my point. Their writings, the ones I know of anyway, are so mathematically flavored and use so much mathematical intuition that they can hardly be considered philosophy. Any math student after spending some time with certain mathematical concepts would come to the same conclusions.

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Okay, I hear you. A lot of the stuff is technical. I mean, you know you’re in for a ride when a “philosophy” book has a three page index of mathematical symbols. But there’s plenty that has been written with hardly a formula in sight. How about Putnam’s terse little book? A classic platonist attack on nominalism. A great book. Or Quine? Six Dogmas Of Empiricism is a touchstone. How about Curry’s moderate formalism? I’d love to hear your thoughts on his brief book. How about the logical positivists? (But that might be too anti-non-analytic philosophy to have the metaphysical meat you’re looking for.) Kant? Can’t know the history of mathematical thought without at least a brief stopover in Konigsberg. Penrose? Descartes is a mathematician. Leibniz. Leibniz. I rest my case. The Greeks. I rest my case again. Pierce. How about Godel’s indispensibility argument? Maybe not, too many platonists in this list already. Three words: Alfred North Whitehead. And that’s just scratching the surface. There’s plenty of philsophy of mathematics to read. You just need to get that library card of yours out from the back of your wallet.

But that is my point. Their writings, the ones I know of anyway, are so mathematically flavored and use so much mathematical intuition that they can hardly be considered philosophy. Any math student after spending some time with certain mathematical concepts would come to the same conclusions.

How is that possible given that the philosophies are so divergent? How could a math student possibly conclude that Brouwer and Hilbert are both right?

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How is that possible given that the philosophies are so divergent? How could a math student possibly conclude that Brouwer and Hilbert are both right?

They both make valid points, one is a constructive mathematician and the other is in some sense a formalist, and both of their areas of research are now flourishing mathematical areas. And yes they are both right.

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They both make valid points, one is a constructive mathematician and the other is in some sense a formalist, and both of their areas of research are now flourishing mathematical areas. And yes they are both right.

I guess intuitionism is still vibrant, but it is still, as always, very much the minority view. A lot of work has been done since the foundational wars, by people like Gentzen, Kleene, and Godel, showing many of the important metamathematical relations between intuitionism and classical mathematics. It’s an exciting subject.

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Interesting discussion, I’m glad this thread has been revived. I am learning plenty since I am actually quite ignorant in math.

I agree with LauricAcid that many many of the great philosophers were also mathematicians. If they weren’t considered mathematicians primarily, their ideas did have an impact on mathematics. Kantian constructivism is a perfect example. Descartes, Leibniz, and Spinoza all have had a major impact on calculus if I’m not mistaken. Russell and Whitehead have had a huge impact in the 20th century (although I can’t say much for Russell’s philosophy).

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So we all agree then. I am right and the only philosophy of math that makes any sense is the one that says mathematical concepts are just generalizations of everyday physical intuition. Everyone who says otherwise just doesn’t know any better.

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So we all agree then. I am right and the only philosophy of math that makes any sense is the one that says mathematical concepts are just generalizations of everyday physical intuition. Everyone who says otherwise just doesn’t know any better.

It depends what you mean by ‘generalizations of everyday physical intuition’ - thats a pretty vague term which could mean almost anything. Its not obvious that things like ideals, Mandelbrot sets, or transfinite arithmetic have anything to do with physical intution though. The Mandelbrot set is an especially interesting case, since it probably constitutes the single strongest argument for Platonism I’ve encountered (Roger Penrose, a platonist, tends to use it a lot for this reason).

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Of course I agree that people who don’t know anything about mathematics are up to their necks when they pontificate about the subject. But, the vast amount of published work (I don’t mean popularizations like the David Foster Wallace book and others) on the philosophy mathematics - journal articles, textbooks, et. al - is written by mathematicians or philsophers who have studied mathematics. Dedekind, Pierce, Frege, Hilbert, Russell, Brouwer, Heyting, Quine, Godel, Church, Carnap, Hintikka, Feferman, Putnam, Kripke, Dummet, Benacerraf and my favorite, Curry.

Well, I think there’s a good argument to be made that too much modern ‘philosophy of mathematics’ has been written primarilly by logicians rather than mathematicians. I think that people will be far more likely to lean towards some kind of formalist/axiomatic viewpoint if they approach mathematics from the point of view of formal logic or philosophy, instead of actually learning the profound and deep mathematics which tends to lead to the prevalence of Platonism amongst practicing mathematicians. When someone is thinking about mathematics as ‘all being reducible to arithmetic/set theory’ rather than actually looking at the beauty which lies in certain results of advanced mathematics, its a lot easier to take nonsense like ‘maths is just the study of formal systems’ seriously.

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Well, I think there’s a good argument to be made that too much modern ‘philosophy of mathematics’ has been written primarilly by logicians rather than mathematicians.

For example? Best would be an example of philosophy of mathematics written by someone actually remote from understanding even the importance of real analysis. Anyway, the fields of logic and mathematics intersect. Mathematical logic IS mathematics. One can’t even approach discussing, for example, incompleteness without knowing such basics as the fundamental theorem of arithmetic and the Chinese remainder theorem. Or for example, one cannot even approach model theory without understanding number systems and algebraic structures.

I think that people will be far more likely to lean towards some kind of formalist/axiomatic viewpoint if they approach mathematics from the point of view of formal logic or philosophy, instead of actually learning the profound and deep mathematics which tends to lead to the prevalence of Platonism amongst practicing mathematicians.

Your argument is based on what you think is likely. But what IS contradicts you. Mathematician/logicians such as Godel and Putnam are decidedly in the platonist camp. And whether they are the minority, I don’ know, but they are not vastly outnumbered. Platonism is quite common among logicians. On the other hand, the banner carriers for formalism are prominently mathematicians as well as logicians, as, of course, Hilbert, is the most salient example.

When someone is thinking about mathematics as ‘all being reducible to arithmetic/set theory’ rather than actually looking at the beauty which lies in certain results of advanced mathematics, its a lot easier to take nonsense like ‘maths is just the study of formal systems’ seriously.

But I don’t know of any writer on mathematics who doesn’t admire its beauty nor of any writer who contends that the study of formal systems precludes understanding mathematics in other ways. And the axiomatization of mathematics with set theory does not at all diminish one’s appreciation of the beauty of the various branches of mathematics, but rather, adds to it.

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Well, you have to ask where the axioms originally come from. Normally a branch of mathematics is only axiomised once it is fairly mature - when you have a large number of results, people will start to see if there is some minimum number of basic statements they can all be deduced from, and these will become the standard axioms. The way that mathematics is presentd in textbooks isnt the way which it actually evolves - mathematicians do not generally start with a set of axioms and try to formally deduce things from them. Euclid’s elements would be a classic example of this; most of the results contained in it were already known, although it was Euclid who first had the idea of reducing them to their basic premises. A more modern example would be something like group theory - the formalised notion of (eg) groups and ideals were postulated after many key results had already been found.

I agree; I’m not sure if this was intended as a disagreement with me or not, but I don’t really see that this is inconsistent with what I said. The axioms are induced, but the laws, theorems, what-have-you that contribute to a complete mathematical theory are deduced from those axioms, which are then applied by induction.

You can also have disagreements about which axiom sets should be used - different axioms will have different consequences, so people can disagree about the best abstraction in a pariticular case.

That’s true, too. People often disagree on applications of the same principles.

This is true psychologically, but not logically. Yes, its a psychological fact about humans that we learn the concept of the natural numbers before that of 0.

Well… that was kind of the point. That is how we, as humans, form the concepts. That is the fundamental starting point. That is the connection to reality, which is the subject that Don raised.

But this doesnt imply that the natural numbers come first from a mathematical point of view; the positive integers are normally defined in terms of the empty set with zero coming first: 0 = {}, 1 = {0} = {{}}, 2 = {0,1} = {{}, {{}}} and so on.

Set theory is valid and useful in a great many contexts (and brilliant!), but it isn’t a metaphysical base. It is an epistemological method.

Rational numbers do come after natural numbers in the standard constuction, since they are defined as equivalence classes of ordered pairs of integers, but there are different constructions where this is not true. For instance, when you construct your number system using something called surreal numbers, you start by creating a particular class of rational numbers (dyadic fractions) which you can then use to define the integers (and later, the real numbers).

Surreal numbers are far more abstract than either natural numbers or rational numbers, and are epistemologically much more complex; they are actually concepts of method. Using them in relation to real numbers is primarily a process of integration, not one of truly deriving the natural numbers from them. One cannot form the concept of “surreal number” without a long chain of abstraction leading back to natural numbers, which may be abstracted directly from reality, i.e. from the things we are using mathematics to measure.

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I disagree; I often use ‘imagistic intuition’ when solving math problems, and many mathematicians report the same. Sometimes the feeling of what the answer should look like occurs well before youve worked out how to actually reach it. There’s a flash where you somehow ‘see’ the answer, even though you cant quite formulate the route which brings you there.

It is hard to describe, and in a sense, it’s entirely proper to use the term intution. I’ve used the term myself regarding mathematics. I generally use “intution,” in this context, to mean a subconscious suggestion deriving from automatized thought processes.

I do think that some strict Platonists regard it as “intuition” in the sense of direct access to knowledge by means of non-thought, but what they are actually experiencing is what I described.

Well thats certainly encouraging; I might order one of the backissues. The problem I’ve always found in the past with attempts to ‘construct mathematics rationally’ (or whatever) is that they tend to be exercises in psychologism, largely focusing on how humans acquire knowledge of mathematical concepts. And while that can be interesting from the point of view of cognitive/developmental psychology, it has very little to do with either mathematics or logic.

This view of mathematics has a great deal to do with mathematics and logic. It doesn’t discount other studies, it strengthens them, by keeping them grounded in reality. The importance isn’t necessarily one of theoretical mathematics, but one of applied mathematics. The lack of such a grounding is how we ended up with applications to non-reality such as all this hocus-pocus we have going on in physics right now (like in string theory).

It’s the bridge between philosophy of science and foundations of mathematics (which, in my view, is a branch of the former).

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\You must mean we deduce our conclusions from axioms. The axioms are the premises; the theorems are the conclusions. But what do you mean by saying that we use induction to apply these? Do you mean in the technological application or do you have some application within mathematics in mind?

Axioms are axioms. They are the most fundamental premises. We arrive at the theorems by deducing from axioms, and then use those theorems as premises which we apply inductively. A proper theorem is universally valid; it is applied inductively, by acting as a single unit integrating all applications within a given context. While theorems may be applied inductively, we may also deduce further from them and arrive at various rules, laws, and formulas, all of which have an inductive purpose.

The point is that epistemologically, their purpose is primarily to integrate an innumerable quantity of individual applications into a single, manageable idea.

Specific applications… take your pick. L’Hospital’s rule, the pythagorean theorem, the law of cosines… the list goes on and on.

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A proper theorem is universally valid; it is applied inductively, by acting as a single unit integrating all applications within a given context.

I think that you are including both generalization and instantiation in your concept of “induction”. Most of us consider instantiation to be deduction, not induction.