# Is it possible to apply coherentism to the philosophy of mathematics?

I asked this question on the math site and was told it'd be better suited to the philosophy forum:

If I'm not mistaken, one of the most basic issues in the philosophy of mathematics is that of the truth criterion, i.e., what, precisely, it is that enables us to discern a "true"/"valid" mathematical statement from one that is "false"/"invalid". I'm aware that a similar question is explored in epistemology more generally, and that one possible solution is to adopt coherentism. So, my question(s) center around the marriage of coherentism and the philosophy of mathematics.

First question: Given that a coherentist epistemology could lead to the idea of truth being relative, couldn't a coherentist mathematical epistemology likewise lead one to something like "mathematical relativism"?

Second question: Does anyone know of any thinkers who may have explored this idea? (Just to be clear, I've been searching for days and can't seem to find anything on the subject.)

I'll appreciate any help anyone might be able to give.

I think coherentism is an important way at looking at how mathematics is actually done in practise, although I don't know of any philosopers that have looked at this in this way.

Truth is obviously important, but one has to add a qualifier that the truth is actually significant. There are lots of true theorems that mathematicians aren't interested in.

Take an example: Algebraic geometry.

a. Its generally taken that it was the Italian School in the 19C that originated this subject. They had a series of results that were 'intuitive' that is there were gaps in there reasoning or places where the reasoning was obscure. For example the idea og a 'generic element'. But the whole seemed to hang together - that is it formed a coherent system of results. And they were important as they pointed to a new way of looking at algebra & geometry together.

b. It was then Zariski that decided these results had to be put on a serious & formal basis. This impelled his work on commutative ring theory.

c. Grothendieck then applied the newly established Category Theory to place algebraic on what seemed the correct framework - the theory of schemes.

What this shows is that as a set of results which points towards a new theory and conjectures which one hopes & expects to be true to build this theory new horizons are breaking out, some of which transform how one looks at old results and some of which consolidates.

As one builds a theory one can build a series of conjectures that establish the framework within which one expects the theory to be built.

For example: representation theory of groups. This is a huge area with important links to physics. For finite groups the theory is easily worked out, for compact ones although using different methods they have exactly corresponding results. This bear out the mathematical intuition or rule-of-thumb that in the continuous case compactness is similar to finiteness in the discrete case. In the construction of the theory conjectures would be established bearing out this rule-of-thumb.

I think the whole issue of mathematical relativism is a deep and difficult issue. I prefer the idea of plurality & process. Mathematics is a tradition, and there are many different perspectives within this body of thinking which have their own standards of truth, justification & methods of proof.

• Thank you for your answer. Correct me if I'm wrong (and I'm open to the idea that I may very well be), but assuming epistemic coherentism and "mathematical relativism," then wouldn't it be as theoretically possible (even if not metaphysically), given the right set of axioms, to prove the truth of the statement "2+2=5" as, given a different set of axioms, it would be to prove the truth of the statement "2+2=4"? I know, I'm quite possibly making myself look like an idiot, but it seems a question worth asking, at least in terms of its value as a thought experiment. – Chris Dec 3 '13 at 20:45
• If you think the answer is useful you should vote it up :). Yes, sure. I wouldn't think of it as mathematical relativism but plurality. In the trivial ring for example 0=1; and in Z/5Z 5=0. There isn't a set of axioms where 2+2=5, but no doubt it cold be thought out & formalised. I expect it wouldn't be useful which is why no-one has gone to the trouble. All this is due to decoupling mathematics from its metaphysical basis - which is called formalism. That is all that is required of an axiomatic system is that it is consistent - that is one can't deduce a contradiction. – Mozibur Ullah Dec 3 '13 at 21:00
• To return to metaphysics one could demand an interpretation in the real world. When one does this with 2+2=5, that is 2 bottles & 2 bottles is 5 bottles. This is obviously wrong. And one is asked for is a physical contradiction. One could suppose given we have a reasonably complete description of physics that one could hypothesise a possible world whose physics is different - for example some people look at p-adic physics. That is physics developed on the real line but topologised in a different way so that it is in effect a different number line. Its a hypothetical situation, mind. – Mozibur Ullah Dec 3 '13 at 21:05
• But to return to this real world, one could ask how is p-adic systems useful. They are to mathematicians who investigate them; and that surely is real enough. The problem about 2+2=5 is that no-one has discovered a context in which this is a useful result in a natural way. One could invent one - but that would be arbitrary & un-natural. – Mozibur Ullah Dec 3 '13 at 21:07