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The few Mathematics I have been studying so far is pure Mathematics. I happen to have some discussions with philosophers of Mathematics, but as they know I totally ignore their subject, we do not usually go through deep questions. However, I often find it very hard to even go beyond the very first steps of our conversations. I want to share this with you, so that you can help me to understand what I miss in their viewpoint (hopefully, this will happen before this question gets closed).

Let us look at just one particular instance. I was told Philosophy of Mathematics is concerned with questions like:

Q. When can we say that a mathematical object exists?

Well, I have an answer for the above question - maybe not a complete one, but a start. I would like someone to help me figure out what is wrong or unsatisfactory.

I would start by saying in Mathematics there is a crucial tool, that of a definition; if one finds an object fitting in the definition, that object exists as a mathematical object. "Mathematical objects" are always defined before one can say anything about them (this is another lovely peculiarity of Mathematics): to introduce a "mathematical object" is to give its definition! Then I say such an object exists if I can explicitly produce one fulfilling the definition.

Let me stress more about this with a stupid example: First, I tell you, say, what a prime number is. After that, any human being should in theory be able to pick any number $n$ and check through the definition whether $n$ is prime or not.

That is my idea of a definition. And it applies with no distinction to both

  1. mathematical objects one can try to visualize (spheres, triangles...) and
  2. to more abstract ones (all the remaining, e.g.: functors, Galois groups... whatever).

I believe that what explains the survival of question Q. (as an open question) is that many people believe that, just because they can handle an orange, spheres exist (in the perverse sense that one might, sooner or later, experience one).

They somehow prefer thinking about objects of type 1. Of course, spheres do exist, but not on earth! (The unique way one can get a sphere is, by definition, to write down an equation of the sort $x^2+y^2+z^2=1$.) I feel like there always is some sort of unconfessed attraction towards reality, or the secret hope that a resolution of singularities will eventually help us fixing our leaking sinks. This is what, in my opinion, makes question Q. survive, and my answer not an answer.

Let me end with a small but hopefully funny provocation: are we sure we still want Mathematics to be considered as a part of Science? After all, the main feature of Science is: it deals with the real world; no mathematical object properly exists in the real world. It would not be a shame at all not to be considered "scientific"...

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The real world is just one of the possible applications of math. Literally, Science deals not with reality,but with knowledge (from latin scire) –  Hagen von Eitzen Jan 28 '14 at 19:04
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What is your criterion for the existence of objects of type 2? I'm happy to say that no object satisfying a list of properties that leads to a contradiction exists. If we are confident this doesn't happen, does that suffice for existence? –  Hugh Denoncourt Jan 28 '14 at 19:18
    
Philosophy gets a bit more murkier than this. For example, try to explain exactly what you mean by the phrases "explicitly produce one" and "fulfilling the definition". My answer to the StackExchange question How do I convince someone that $1+1=2$ may not necessarily be true? should give you an idea of how deep the rabbit hole goes (as Morpheus in the 1999 movie "Matrix" would say). –  Dave L. Renfro Jan 28 '14 at 19:31
    
@HughDenoncourt: mathematics does not distinguish between type 1 and 2. (Humans do!) The criterion is the same. I agree that checking existence is often a delicate and very hard problem, though. –  Brenin Jan 28 '14 at 20:34
    
It seems to me that your suggestion, "if one finds an object fitting in the definition, that object exists," is dangerously close to circular, because "finding" involves the presupposition that what is found exists. For example, I find that Jewel (in "The Last Battle") satisfies every reasonable definition of "unicorn". Can I then conclude that unicorns exist? Shouldn't my "finding" be limited to exclude works of fiction? And how will you decide whether, say, arithmetic is a work of fiction until after you adopt some criteria for existence of numbers? –  Andreas Blass Jun 10 '14 at 18:22

3 Answers 3

up vote 1 down vote accepted

I recently read a summary of Carnap's position on existence and subsequently Carnap's Empiricism, Semantics, and Ontology as recommended by the summary:

Ontological questions (e.g. “are there propositions?”, “are there numbers?”, “are there properties?”, etc.) are, Carnap here argues, ambiguous. Under one disambiguation (internal), such questions have a yes/no answer. Under another disambiguation (external), they admit of no such binary determination.

The external disambiguation is mostly be dismissed for mathematical objects, so let's focus on the internal disambiguation. Carnap starts by introducing a prerequisite for internal questions:

If someone wishes to speak in his language about a new kind of entities, he has to introduce a system of new ways of speaking, subject to new rules; we shall call this procedure the construction of a linguistic framework for the new entities in question.

Here a mathematical object exists , if it's existence can be proved by the means provided by the system, as long as the system itself is consistent. However, even in this case there is still an important distinction to be made between cases where particular objects (solving a certain problem) can be presented explicitly (at least in theory) and cases where objects just (can be proved to) exist in general, but no particular object can ever be presented explicitly (not even in theory). For example, there exist prime numbers smaller than "3=S(S(S(0)))", namely "2 = S(S(0))". Contrast this with the existence of a Hamel basis of $\mathbb R$ over $\mathbb Q$, which can we prove to exist within ZFC. We know that no such basis can be presented explicitly, not even in theory (at least not by a predicate represented by a first-order formula in the language of ZFC).

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Thanks a lot, your answer was very informative! –  Brenin Jun 29 '14 at 13:17

One nice approach is to declare the question irrelevant: it doesn't matter if mathematical objects exist, what we care about is mathematical practice/structures involving those "objects". See my comments on Benacerraf here.

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I would start by saying in Mathematics there is a crucial tool, that of a definition; if one finds an object fitting in the definition, that object exists as a mathematical object. "Mathematical objects" are always defined before one can say anything about them (this is another lovely peculiarity of Mathematics): to introduce a "mathematical object" is to give its definition! Then I say such an object exists if I can explicitly produce one fulfilling the definition.

The problem is that the first order logic most mathematicians use is not necessarily explicit at producing objects. Demanding that you must explicitly produce an object actually takes you into a different sort of logic, which is "weaker" in its ability to prove things. The first order logic admits proof by contradiction to prove the existence of an object. A surprising number of people find this highly objectionable: the "finitists", the "constructivists"/computationalists, the "intuitionists". Historically, these people are not just philosophers hanging out on the side-lines. They were important mathematicians doing real work in analysis, topology, and all of that other good stuff.

Consider the axiom of choice. It says that any set admits a choice function. That is, for any set $A$, there is a mapping $c\colon \wp(A) -\{\emptyset\} \rightarrow A$. This is easy enough to check in the finite case. But it is impossible to prove in the infinite case. In fact, it is "independent" of ZF set theory -- you can construct models of set theory where it is true and others where it fails.

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You'll have a hard time choosing a choice function at $\emptyset\in\mathcal{P}(A)$. –  user72694 Aug 5 '14 at 15:07
    
Hmm, a choice function is supposed to choose an element from each set in the collection. If the set is empty then there is no element to choose. –  user72694 Aug 5 '14 at 15:59
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You should be subtracting off the empty set from P, bot from A. –  user72694 Aug 5 '14 at 16:09
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Strictly speaking you're subtracting off $\{ \emptyset \}$ from $\mathcal{P}(A)$. –  Ian Aug 5 '14 at 16:09
    
I prefer this "strict" objection over the vacuous one that there's nothing to choose from out of the empty set... at least this one isn't merely philosophical. –  nomen Aug 9 '14 at 22:16

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