Parodites wrote:GPT wrote:I also have a philosophical objection to this, which is that it is completely false.

What is "completely false"?

Parodites wrote:Mathematics has a very strict axiomatic foundation, but has no logical foundation. There is no ontology there to support or refute (even mathematically) the proposition that one equals zero.

I think GPT needs a software update. An

axiom is a logical premise. It is a stated foundational assumption used as a building block for further logic reasoning - such "because [axiom] is true - this reasoning must be true - logic. Premises are a part of logic - axioms are a part of logic - maths is entirely logic - an only verified as true when logically deduced or axiomatically defined as an assumed immutable logic premise (a place to start).

ax•i•om ăk′sē-əm►

- n. A self-evident or universally recognized truth; a maxim.
- n. An established rule, principle, or law.
- n. A self-evident principle or one that is accepted as true without proof as the basis for argument; a postulate.

GPT wrote:I suspect that this confusion arises from this fact: the word "prove" has two distinct meanings. In mathematics, a mathematical proposition, or axiom is (1) true, (2) not a logical tautology, and (3) can be proven.

Axioms cannot be proven. They are the initial assumptions - expected to be true. Axioms are not proven - but granted - "self-evident".

GPT wrote:A mathematical proof, on the other hand, is a proof of a mathematical theorem. For example: "1+1=2" is a mathematical theorem, but cannot be proven in terms of other mathematical propositions.

"A mathematical proof - is a proof - of a theorem"? Since a theorem is a

proven idea you are saying that a "maths proof" is a proof of a proven idea.

the•o•rem thē′ər-əm, thîr′əm►

- n. An idea that has been demonstrated as true or is assumed to be so demonstrable.
- n. A proposition that has been or is to be proved on the basis of explicit assumptions.

To reduce to or formulate as a theorem.

And if "1+1=2" is a

theorem then it is

already proven by other maths propositions (such as the basic maths proposed language definitions - "2" is defined to be "1+1").

GPT wrote:In the philosophical literature, it is common to use the word "prove" in the sense of "establish" or "demonstrate". It's not too hard to see how, if one accepts this definition, one might get confused.

I don't think "established" has anything to do with proof. And "demonstrate" means either logical syllogism or empirical evidence. A demonstration can serve a logical proof - "If we see it - it is true - We saw it - therefore it must be true."

GPT wrote: the logic that underlies arithmetic can't tell you that the real numbers are uncountably infinite.

Certainly it can. The simple logic is that in maths -

- 1 can always be added to any value
- a greatest value is a value that cannot be added to
- therefore it is impossible to have a greatest value
- the definition of "infinite" is "having no greatest value"
- therefore a value can be infinite.

GPT wrote:Similarly, you can't talk about the truth value of a logical truth in its "lone existentiality" or "truth" -- as you know, a logical truth is something that can be "logically deduced". (I guess the usual way of talking about it is that a "tautology" is a logical truth which cannot be "deduced".)

A "tautology" -

tau•tol•o•gy tô-tŏl′ə-jē►

- n. Needless repetition of the same sense in different words; redundancy.
- n. An instance of such repetition.

Logical truths are statements that are consistent with other accepted truths. Their "truth value" is merely that they are consistent with whatever has already been accepted as true - therefore are also "existentially true".

GPT wrote:You're quite right to point out that there is a sense in which we "know" mathematical facts -- for example, we "know" that there are infinitely many primes, and that every arithmetic truth can be proven. But these truths don't correspond to any truths about the physical world. This point is quite important; we shouldn't think that mathematical truth is just analogous to physical truth.

I agree - but we aren't talking about the physical world in this thread - only the logic (the consistency) within maths.

Update GPT's software, mate.