Topics Mathematics

Question I know that there are some serious problems concerning the idea that mathematics is grounded on logic. But computers can perform mathematical operations, and computers use logic, so I think that at least for practical purposes we can use logic to support mathematics. Am I right? My second question is this: can we infer that 2+2=4 from the principle of non-contradiction? Thank you!

Accepted:
March 5, 2009

Comments

You need to distinguish the claim that mathematics is grounded on logic, and the claim that mathematics uses logic.

The weaker second claim is evidently true, at least in this sense. Mathematical reasoning is a paradigm of good deductive reasoning. And standard systems of logic explicitly aim to codify, more or less directly, the kinds of good deductive reasonings that mathematicians use. (And computers might be used to echo some such reasonings too.)

But the fact that mathematics uses logical reasoning doesn't show that mathematics is grounded on logic if that is the much stronger thesis that at least arithmetic, maybe the whole of classical analysis, just follows from pure logic plus definitions of mathematical notions in logical terms. (I take it that it is this logicist thesis which you are thinking of, when you say that there are "serious problems" about the idea that mathematics is grounded on logic).

For example, you might think that in set theory we use logic to deduce what follows from the set-theoretic axioms. But you might suppose that is not a matter of logic that the axioms are true -- that depends on the nature of sets!