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loicd | 2 years ago
Yes I agree. There is always some context implied if we are being rigorous. But we do use the word "true" alone. Thus, the question is what is the implied context? I claim that this context consists of commonly agreed upon axioms. If I understand correctly, you claim it is a mental model.
Personally, I am not sure whether I qualify as a platonist. I do have a mental model that I use to evaluate mathematical statements, but that mental model is fluctuating. It is sometimes wrong (i.e. inconsistent) and therefore in needs of an update. Because of the mere possibility of errors, I (and this may be my personal bias) only consider statements "true" those that are proven (from some agreed upon axioms).
On the other hand, if you consider mathematicians as a community, I believe that mathematicians don't share the exact same mental model. So, a statement that mathematicians (as a community) will agree is "true", will be a statement that is satisfied in all their mental models. This is therefore a notion of validity rather than satisfiability. Of course, the mental models of mathematicians are unlikely to exhaust all possible models of a given theory. However, the ultimate arbiter of truth in the mathematical community is the satisfiability in all possible models of the theory, i.e. the proof.
denotational|2 years ago
As I mentioned in a sibling comment, I dispute your claim that most mathematicians work from foundational axioms such as ZFC. I think the number of mathematicians who can actually write down the axioms of ZFC is far smaller the number who claim that mathematics is founded on ZFC, and the number who actually do rigorous proofs in ZFC a discipline other than logic far smaller still.
If a mathematician has never seen a rigorous proof of a theorem from the axioms of ZFC, and either seen it machine checked, or hand verified it (probably intractible for most real maths), can they say it's true if they subscribe to your approach to truth? I don't think they can.
Given that these proofs don't even exist for most theorems (there are efforts to build libraries of machine checked proofs, but even ignoring the fact that the most recent of these don't start from ZFC, they are by no means complete) it would seem to me that most mathematicians must adopt a more Platonist approach.
To be clear, I like the idea of axiomatisation and I'm a fan of the introduction of proof assistants (indeed I've used Coq pretty heavily in the past), but I dispute that rigourous proof is what actually determines truth; the human understanding of the naturals predates it's formulation in PA, if PA cannot prove something that we know to be true from our understanding of the naturals then that is a limitation of the axiomatic process, we should not start changing our notion of the naturals to fit the axioms.
> This is therefore a notion of validity rather than satisfiability.
As I mentioned in another reply, I have not seen the term validity used in this context, only in cases where the formula is true in all models; this is regular semantic entailment.
Note that your original claim, however, was
> A statement is true by definition if and only if it is satisfied in every model.
Did you mean to say "satisfied in every model that satisfies ZFC"?
I am more comfortable with this as a valid mathematical position (although I still claim it is nonstandard terminology); I disagree, but my disagreement is purely philosophical, whereas I hold that the original claim is so nonstandard as to be incorrect.
loicd|2 years ago
> [...] I dispute that rigourous proof is what actually determines truth [...]
This is perhaps a bit out of topic, but to me these two statements are contradictory. I suppose that you should define what you mean by "real" (and Platonism). I certainly think that mathematical objects are real, but by that, I mean that they exist independently of my own mind. However, they can't exist independently of a mind if truth is determined by evaluation against a mental model. Even if that mental model is shared within a community, because that would turn mathematics into a belief system. Also, the human mind is fallible and prone to mistakes, so in my view, it is reasonable to doubt what comes out of it.
Sure, mathematicians agree on axioms for things like natural numbers, and deduction rules. However, I think that the reality of natural numbers and proofs (as mathematical objects) does not stem from a shared mental model, but from their finitary nature, which makes it possible to implement them on a computer. I am also skeptical that the human mind has any innate model for most advanced concepts in mathematics (I even doubt that it is true for real numbers). I think that the intuition we have of most mathematical objects is formed after exposure to simpler mathematical notions. That intuition is shaped by what is proved and disproved from prior mathematical knowledge. Yes, proofs written by mathematicians don't look very formal (and often, the more advanced are the maths, the less formal and detailed are the proofs), but I dispute that they are not rigorous and can't be translated into a formal framework. In my view, this is mostly a matter of efficiency and practicality.
To illustrate what I say, consider Mochizuki's claimed proof of the abc conjecture[1]. Here we have a claimed proof so difficult that most specialists fail to determine whether it is correct or not, although Scholze&Stix believe there is a gap. I say that most mathematicians don't have a mental model that allows them to determine whether the abc conjecture is true or not, and because of the fallibility of the human mind, it is reasonable to doubt those that claim they do. One can of course take sides, but in that case, we are no longer doing mathematics. The only thing that can resolve the issue will be a more readable and more rigorous proof. That's what determines truth.
[1]: https://en.wikipedia.org/wiki/Abc_conjecture#Claimed_proofs