what is mathematical truth?

[AntonioLao]

The discoveries of non-Euclidean geometries were the results of questions concerning the axiomatic bases of mathematics. Similar to religious faiths, the truthfulness of axioms and postulates are founded on mathematical faiths without logical proofs.

But by using these axioms as mathematical truths, all mathematical theorems can be provided with more than enough logical proofs, to which practical experiences and physical measurements can never be able to provide with absolute certainty.

Therefore, like religions, mathematics is also based on faith.

[Guille]

There is another thread I can't remember where, that you started, and I have replied, and it is about this.

Anyway, just to let you know, there is a part of math which is dedicated to make the opposite of what is ussually done, by this, I mean, that mathematicians get the theorems that they develop, unify them by finding axioms or statements that work for all. This is called axiomatization, and was tried by Hilbert's program, which failed due to godel's incompleteness theorems.
(I got knowledge of this information yesterday, whiles reading "Fermat's Enigma", great book). So, axioms aren't exactly believes. They are derived by logic.

[AntonioLao]

Godel asserted that logic cannot prove logic. Axiom are not proven by logic but accepted as truth by mathematical faith. Theorems are proven by using logic based on the axioms.

[Guille]

Godel asserted that logic cannot prove logic. Axiom are not proven by logic but accepted as truth by mathematical faith. Theorems are proven by using logic based on the axioms.

But there is an inverce method, from theorems to axioms, quote from Wikipedia: Mathematics; Notation, Language and Rigor:

"Even so, in the past it sometimes happened that something which had supposedly been proved turned out to be false. This was possible because mathematics was done using natural language. To prevent this from happening, mathematicians wanted their theorems to follow mechanically from a few simple incontrovertible truths and for this they invented axioms and axiomatic reasoning. An axiom is just a string of symbols which have an intrinsic meaning because of all derivable formulas. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory. But for most of mathematics this complete rigor is far too cumbersome and mathematical language and notation are supposed to suffice."

[AntonioLao]

the distinction is between the arguments among the formalists and intuitionists. In this respect, I am considering myself an intuitionist.

[Guille]

the distinction is between the arguments among the formalists and intuitionists. In this respect, I am considering myself an intuitionist.

Which (formalism or intuitionism) aproves the axiom to theorem or the theorem to axiom?

[AntonioLao]

maybe this site gives some ideas
http://mathworld.wolfram.com/IntuitionisticLogic.html
Russell might be an example of a formalist.

[Guille]

Thanks.

Now, what if:

There are logical proofs for the fundamental axioms of mathematics?

[AntonioLao]

GUILLE,

I thought Russell was a formalist. I was wrong. He was the founder of the logic school of mathematics. Your adhenrence to logic makes you a Russell's devotee.

[Guille]

GUILLE,

I thought Russell was a formalist. I was wrong. He was the founder of the logic school of mathematics. Your adhenrence to logic makes you a Russell's devotee.

Of course! I love the work of Bertrand Arthur William Rusell (1872-1970). I am reading: The Auto-Biography of Bertrand Russell (by himself (auto)). It's devided in 3 books, it's incredible. He is trully my favourite character in philosophy, a brilliant logician and a poore husband (in the book he has married so many times already, that I lost the count). But my philosophy is quite different to his, and I beleive that logic isn't good enough, not like him. Just for pleasure, I give here the logical form of Rusell's paradox, which I'm still, after 6 months, trying to solve without the theory of types being used or needed:

A/€A<-->A€R

R={A|A/€A}

R€R<-->R/€R

KEY: €: contians, /€: doens't contian, <-->: iff and only iff.