Is it really Fermat's Last Theorem proven?

joinfree said:

What Does it Take to Prove Fermat's Last Theorem? Grothendieck and the Logic of Number Theory | Bulletin of Symbolic Logic | Cambridge Core

If it is necessary to use assumptions, or perhaps Unproven Axioms, why then we can not take the Fermat's Theorem as fact? Like was taken as fact the Darwin's Evolution? Just call Fermat's Theorem an Axiom. Like the 5-th postulate of Euclid.

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There is very little similarity between mathematical proofs and the weight assigned to scientific theories. You confuse your argument by introducing any reference to Darwin. I predict that if you do not edit your opening post to eliminate it this thread will be diverted into an interminable discussion about evolution.

joinfree said:

If it is necessary to use assumptions, or perhaps Unproven Axioms, why then we can not take the Fermat's Theorem as fact?

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Because there is not yet a proof that the theorem is true. Nor is it axiomatic.

joinfree said:

Like was taken as fact the Darwin's Evolution?

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Evolution has not been assumed true, it's been proven true.

chilehed said:

Evolution has not been assumed true, it's been proven true.

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If we're talking about the scientific theory, we don't prove theories, outside of math.

joinfree said:

But we can confirm a theory, if latter is not wrong.

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We can confirm that a theory satisfies all the known evidence, and the theory of common descent does.

As we've seen with Newtonian mechanics, it's always possible for new evidence to make a mess of things.

Yttrium said:

If we're talking about the scientific theory, we don't prove theories, outside of math.

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I'm talking about how the fact that evolution has been observed proves that it happens.

Ophiolite said:

There is very little similarity between mathematical proofs and the weight assigned to scientific theories. You confuse your argument by introducing any reference to Darwin. I predict that if you do not edit your opening post to eliminate it this thread will be diverted into an interminable discussion about evolution.

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I rest my case.

joinfree said:

What Does it Take to Prove Fermat's Last Theorem? Grothendieck and the Logic of Number Theory | Bulletin of Symbolic Logic | Cambridge Core

If it is necessary to use assumptions, or perhaps Unproven Axioms, why then we can not take the Fermat's Theorem as fact? Like was taken as fact the Darwin's Evolution? Just call Fermat's Theorem an Axiom. Like the 5-th postulate of Euclid.

Click to expand...

1.) Axioms are by definition unproven.

2.) Because, unless proven, it might be incompatible with what are usually taken to be the foundational axioms of mathematics.

3.) It would, to say the least, be undesirable to multiply axioms without number, every time there was a hard to prove theorem awaiting a proof. Especially as the possibility of incompatible axioms would increase exponentially as their number increased.

joinfree said:

What Does it Take to Prove Fermat's Last Theorem? Grothendieck and the Logic of Number Theory | Bulletin of Symbolic Logic | Cambridge Core

If it is necessary to use assumptions, or perhaps Unproven Axioms, why then we can not take the Fermat's Theorem as fact? Like was taken as fact the Darwin's Evolution? Just call Fermat's Theorem an Axiom.

Click to expand...

I think you're missing the point. Fermat's Last Theorem was proved, using the standard axioms of set theory.

The only problem is, it's a statement about numbers, not about sets. Can it be proved "with one hand tied behind your back," not using set theory? Nobody knows.

But that's more a question about proof elegance than about proof validity.