Immanuel Can wrote: ↑Sat Sep 26, 2020 10:47 pm
Well, I am content to mark our disagreement without impugning your character or intelligence.

I apologize for my tone. We exchanged a couple of civilities and I should continue to assume this is an honest misunderstanding. I commend you for taking the high road.

To keep this simple, please simply read this quote, which consistes of your own words; and either explain, clarify, retract, defend, or place into context your use of the word tautology in that sentence.

As you do so, bear in mind that Wiles's proof of FLT is a tautology.

These are your words.

Immanuel Can wrote: ↑Thu Sep 17, 2020 2:11 am
P=P is also a tautology. The fault is not that it is

*wrong*, or tells a

*lie*; it's that even if true, it's utterly uninformative of anything new. It adds no value to our thinking at all.

I am challenging your use of the word tautology and I'd like you to explain or retract or place in context etc.

Addendum: It occurs to me that some readers may not know what I mean by saying a proof is a tautology.

Wiles's proof of FLT is a written proof. In principle it could be drilled down to a formal step-by-step proof in set theory starting with the axioms and proceeding step-by-step till FLT was proven.

[In practice this would be very difficult and I don't think it's been done or that anyone's trying.]

As such, it is

*true in every model of set theory*. This is in fact the content of

Gödel's completeness theorem, not to be confused with his more famous incompleteness theorems.

Gödel's completeness theorem says that a statement has a proof from a given set of axioms if and only if it's true in every model of those axioms. This is the case with FLT and in fact with every formal proof in mathematics.

That is, the very fact that there is a proof of FLT from the axioms of set theory; is equivalent to the statement that it's true in every model of those axioms.

The Wiki definition of a tautology, as I mentioned earlier, is "In logic, a tautology (from Greek: ταυτολογία) is a formula or assertion that is true in every possible interpretation." By this definition. all proofs in mathematics are tautologies.

https://en.wikipedia.org/wiki/Tautology_(logic)
This sounds counterintuitive, till you realize that

*tautologies are not trivial*. They're not obvious, they're not simple, they're not intuitive. Just like a novel is made up of the same old 26 English letters and some punctuation; but the great writers are really good at arranging strings of those letters. In theory a computer (or infinitely many monkeys) could type out the works of Shakespeare; but in fact it took Shakespeare to do it.

It's the same in math. It's hard to find the proofs, even if they follow logically from axioms and could in theory have been cranked out by a computer. But computers don't know which proofs are important. That's what humans are for. And finding interesting tautologies is hard work.