Carl Hewitt (of Actor model fame) has this thing where he claims that contradiction alone allows you to defeat Godel Incompleteness, so he shops it around places and everyone's like, "man, wtf, this isn't how that works". I saw him shop it to D. Hofstadter and Hofstadter basically smiled and took the little piece of paper and threw it away when Carl wasn't looking.
This doesn't have anything to do with anything, just a thing that happened
I know that the above was written in jest anyway :) but just for completeness (har), a contradiction would help you defeat many things (i.e., it's not specific to incompleteness theorems, so to speak), because given it, (at least in traditional logic) you can prove anything. This is trivial and known, but to spell it out, if we have a contradiction formalized by p and not p (given), we can prove that q:
1. p and not p (given).
2. p (from 1).
3. not p (from 1).
4. p or q (from 2 (disjunction introduction)).
5. q (from 3 and 4).
So obviously this would upset quite a few things around :) that said, the interesting stuff is with Graham Priest's (et al.) paraconsistent logic systems wherein your system can tolerate a contradiction without exploding in whole. And (so the story goes) those systems may offer an actual insight into handling incompleteness (while still being usable). If anyone has looked into this more, would be interesting to hear about it!
That behaviour was desribed by the old masters as 'ex falso quodlibet': from falsity whatever (when else am i having the chance to be this pedantic). In the uni I was taught that hippie-era AI dealt with that with things called non-monotonic reasoning, abduction, truth maintenance systems.
“All difficult conjectures should be proved by reductio ad
absurdum arguments. For if the proof is long and complicated
enough you are bound to make a mistake somewhere and hence a
contradiction will inevitably appear, and so the truth of the
original conjecture is established QED.”
Hofstadter's examples 'this sentence is false' blah blah aren't inconsistent at all: it just semantic nonsense likes Gödel's made up logic and Cantor's ordinals before that? Mathematics exist without us no?
Wait, you are saying mathematics exists independent of humans but logic does not. But mathematics is build on top of logic, how can it gain independence from humans if its foundation does not have such independence? Or am I just misreading your statement?
To me this seems backwards. Don't we define some basic objects with axioms and then use logic to find out the consequences of the axioms? We make up some axioms for natural numbers, then we define operations like addition and multiplication on them, and finally we investigate the consequence of those definitions, we discover for example that some numbers have very special properties and can be used to uniquely decompose all numbers, we call them prime numbers.
I would therefore say that mathematics is a human invention and build on top of logic, it seems in no way universal. All the things we discover in mathematics are nothing but consequences of the axioms, the definitions, and the logic used to prove things. If there is something universal, at least so it seems to me, than it would have to be logic.
As a mathematician, I don't know if I agree completely with this statement. Logic in practice is essentially its own branch of mathematics with tools that are sometimes helpful in other branches, but it has its own internal motivations for research that are irrelevant to say an analyst or a number theorist. In my own opinion, it's as much a framework for the doing of mathematics as is the English language, and not too much more.
This doesn't have anything to do with anything, just a thing that happened