Is logic the basis of all truth?

And if mathematics isn't logic (Gödel, etc) then what is it?

>>1

Mathematics uses logic. "all truth" is a meaningless concept.

"Mathematics" is the closure of a particular set of primitive logical sentences taken as axioms under the usual inference rules of propositional logic (modus ponens, etc.).  In the case of arithmetic, the relevant axioms could be the Peano Axioms, for example.

OP is all crazy on the dope!

Gödel showed that certain kinds of mathematics cannot be logical in a mathematical sense.  Mathematicians don't use mathematical logic to do their work.  Maybe to justify things later, but actually doing math does not rely on such things.

THEN WHO WAS MATH?!

>>5

Not at all.
Godel showed that any logical system that is "powerful" enough to include a model of the natural numbers is not both complete and consistent.

Now, if it were not consistent, that would be a worry. Given any system with the law of the excluded middle, if you can prove a statement is both true and false (Basically a system being inconsistent) you can prove that any statement is both true and false. That's bad for maths.

If it happens (and it does, as I'll explain later), that your model is not complete, that merely means there are statements in the language that cannot be proved within it.

Now anyone who studies maths can see this is actually rather plausible. I mean, a 10 year old can understand fermat's last theorem, the proof is horrendous. The fact that there might be statements that we cannot prove, never mind whether or not we appeal to a higher level system, is not unthinkable.

As it turns out, using second order logic you can prove, for example, peano's model of the natural numbers is consistent, which by godel's theorem implies it's incomplete. Which is much better than the other way round, or it being both.

Comprende?

The physical universe is the basis of all truth.  Wittgenstein had it backwards.