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CuriousCarrot
- 4
- 2
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?
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?