The Current State of Mathematical Rigour

[CuriousCarrot]

Sorry if that title doesn't match up well with my question. I think it captures roughly what I'm wondering about.

My uncertainty is to do with how much of mathematics is certainly true. Like, if I picked up any undergrad or grad textbook in mathematics, would everything in it be true based on the axioms for the areas the book explored?

Is Fermat's last theorem definitely true? Is it actually the case that a^n+b^n=c^n cannot be the case if a, b, c, n are natural numbers with n≥3? Did Andrew Wiles and Richard Taylor's work show that this is definitely true?

If I were bright enough and had unlimited time, Could I start with the most elementary theories in mathematics, state their axioms, and then start proving theorems from these axioms, then add other theories in mathematics, state their axioms and start proving their theorems until I got to all the current knowledge of mathematics?

Is it a valid method of proof to use other theories to prove theorems in one theory, e.g. Different theories to prove Fermat's last theorem in number theory. I guess it must be necessary if a proof is to exist in some cases by Godel's incompleteness theorems...is that right?

How would I go about studying mathematics from first principles?

 

[CuriousCarrot]

Another question, how can you be sure that a system of axioms does not lead to a contradiction when you develop the theory from them? E. G. Is it possible that the Peano axioms could lead to a contradiction? As you've probably guessed I'm a bit confused in general...

 

[mfb]

If I were bright enough and had unlimited time, Could I start with the most elementary theories in mathematics, state their axioms, and then start proving theorems from these axioms, then add other theories in mathematics, state their axioms and start proving their theorems until I got to all the current knowledge of mathematics?

In principle: Yes.

You would probably discover that some people made mistakes. I don't know the rate, but it is basically guaranteed that some published proofs are wrong. It is very unlikely for the famous ones as they have been checked over and over again, but there are also publications hardly anyone cares about.

How would I go about studying mathematics from first principles?

Every good introductory university lecture does this.

Another question, how can you be sure that a system of axioms does not lead to a contradiction when you develop the theory from them? E. G. Is it possible that the Peano axioms could lead to a contradiction? As you've probably guessed I'm a bit confused in general...

It is interesting that a system of axioms (with sufficient content to do something non-trivial) does not allow a proof that it is free of contradictions within itself. You might use a different set of axioms to prove the consistency.
Wikipedia article
Also relevant

 

[CuriousCarrot]

In principle: Yes.

You would probably discover that some people made mistakes. I don't know the rate, but it is basically guaranteed that some published proofs are wrong. It is very unlikely for the famous ones as they have been checked over and over again, but there are also publications hardly anyone cares about.Every good introductory university lecture does this.It is interesting that a system of axioms (with sufficient content to do something non-trivial) does not allow a proof that it is free of contradictions within itself. You might use a different set of axioms to prove the consistency.
Wikipedia article
Also relevant

Would the proof of consistency from the other set of axioms rely on the consistency of those axioms, which in turn relies on a proof of consistency from a further set of axioms, and so on? How do you get around this? Is it even an issue? Could you prove just that based on your axioms for the second system, the first system is consistent and there is no proof it is inconsistent? I'm a bit confused.

 

[mfb]

Is it even an issue?

The system of axioms used can be quite small, and no one expects a contradiction, but mathematically you cannot show it for all the systems you want to use.

 

[Svein]

Would the proof of consistency from the other set of axioms rely on the consistency of those axioms, which in turn relies on a proof of consistency from a further set of axioms, and so on? How do you get around this? Is it even an issue? Could you prove just that based on your axioms for the second system, the first system is consistent and there is no proof it is inconsistent? I'm a bit confused.

Alfred North Whitehead and Bertrand Russell tried to do exactly that in Principia Mathematica (https://en.wikipedia.org/wiki/Principia_Mathematica). Then Kurt Gödel came along and wrecked the whole project (https://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems).

 

[jbriggs444]

Alfred North Whitehead and Bertrand Russell tried to do exactly that in Principia Mathematica (https://en.wikipedia.org/wiki/Principia_Mathematica). Then Kurt Gödel came along and wrecked the whole project (https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems).

You might want to replace the link in the reference to "https://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems". It points to https://en.wikipedia.org/wiki/Principia_Mathematica instead.