Mathematics- Discovered or Invented?

[Paul Potter]

I am interested in the ontological status of mathematical entities and statements, and the place for mathematics in a reductive, materialistic universe.

 

The mathematician Kronecker is supposed to have said 'God gave us the integers, all else is the work of man.' Translated into a secular form, this statement is the claim that the integers are somehow part of the 'ultimate furniture' of the universe. However, when we see two chairs, we will find on closer inspection that they are in fact not the same. I think that however natural it may seem to do so, and however ubiquitously it is done amongst many animal species, grouping objects with 'family resemblances' into categories for counting is something that requires some form of consciousness as a prerequisite. This argument may fail in the case that matter is composed of indistinguishable particles, but then again this is a claim that requires empirical science to justify.

 

Along another line of reasoning, it seems likely that an alien species that was entirely different to us might develop a system of pure mathematics that looked very different from our own. To some extent we proceed by making definitions that are interesting to study; constructing those objects that seem natural and that best fit with our intuitions. However, it also seems true that every statement of mathematics (taken to include a base set of axioms upon which it is predicated, rules of inference and so on) that is capable of consistent formulation is either true of false, and which one of these two is the case is in some sense necessary (so the the alien system would agree with ours where it intersects).

 

I would be interested to hear any informed opinions of the subject.

Edited June 10, 2009 by Phi for All
link removed, let's discuss this here

[ajb]

Already discussed here.