Mathematical Proof?

Question

Can we prove all mathematical statements?
What does Godel's incompleteness theorem have to do with mathematical proof?
Are there alternatives to rigorous proof?

I hope to have links

Answer 1

Godel's incompleteness theorem doesn't really have anything to do with the notion of rigourous proof per se.

In a nutshell, if you have a formal system of logic (e.g. mathematics), the system is considered complete if you can prove every true statement in it. However, Godel showed that every formal system will have true statements in it that cannot be proven by that particular formal system.

In other words, the statement of the Incompleteness theorem is that in any formal system (such as mathematics) there are TRUE statements that cannot be proven within that system which makes these formal systems incomplete.

For links you can start with this:

http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorem

There is also a good intuitive discussion of it in Roger Penrose's book The Emperor's New Mind where he provides examples of exactly what the theorem says.

Answer 2

The Incompleteness theorem deals only with "first-order logic" or logic involving basic arithmetic operations like addition, subtraction, multiplication and division.

The said theorem does not apply to advanced analysis like the one demonstrated by Andrew Wyles for his proof of FLT. Incompleteness Theorem does not render Mathematics per se as incomplete - it only says about arithmetic and its basic axioms.

Not all mathematical problems could be proved however since there are undecidable problems - problems that would either take infinite space for its solution or an infinite time.

Hope this explanation helps. Visit the following link for more info: http://www.jasoncrawford.org/goedel.html

Answer 3

Proof is a pretty relative term, although if we allow our logic to be completely defined by mathematics and nothing else, we can speak of proof in a more absolute sense. Godel surely realized that the logic of men is not whole, complete, or maybe even right. However, since ours is the only type of logic we can attempt to understand, we as human beings have no other choice. At least for now...