> I suspect that most mathematicians are Platonists (this may be my bias creeping in) and they believe the objects they work with are real
> [...] 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.
I don’t think my position is contradictory: I believe that (non-technical, unqualified/alone) “true” is a property that exists independently of whether we are able to formally prove something in some kind of logic. This is because we are making assertions about “real” objects (e.g. the naturals), and I believe that these statements are either true or false.
I think this is a philosophical disagreement that we’re unlikely to resolve.
It’s not clear to me which definition of (non-technical, unqualified/alone) “true” you are using.
We’ve had a few:
1. Your original definition, where we say P is true iff it holds in all compatible models (which I claim is highly nonstandard);
2. The definition you started using later, where we say P is true iff it holds in all models of ZFC (which I claim is still nonstandard);
3. The definition I suggested, where we say P is true iff it holds in some “standard model”;
4. Something else.
Let’s consider the naturals; what is your opinion on the truth of the Gödel sentence for ZFC (let’s assume consistency, otherwise definition 2 cannot possibly be useful)? Under definition 3 is is true, but under definition 2 it isn’t.
If you think it isn’t true then you are saying that we don’t really understand the naturals intuitively and we can only understand them by axiomatisation.
I cannot counter that position mathematically, only philosophically, but I will say that we have seemingly used and understood the naturals for a very long time before they were first effectively axiomatised, so it seems to be a bold claim.
This is very far away from my original point, however, which was purely about your position that “true” means “true in all models” (i.e. definition 1), however it seems you are no longer adopting this position (in favour of definition 2)?
When used in a technical sense, as far as I am aware “true” is always qualified with respect to some specific model, which may be obvious from context and thus not explicitly stated, but there is always a formal model in mind; truth is generally seen as a model-theoretic concept, not as proof-theoretic one.
Re. ABC and IUTT, I’m fully behind Buzzard et al. pushing for machine verified proofs, as I say, I have been a big user of Coq. I just don’t think we can ever say that proof is what determines truth given we know from Gödel that proof is fundamentally limited.
Proof is a good way of convincing ourselves something is true, indeed anything we prove true is in fact true by soundness, but it’s not the arbiter of truth.
> 3. The definition I suggested, where we say P is true iff it holds in some “standard model”;
By the way, I wish you would answer my previous objection about that definition in the context of set theory. What is the standard model of ZFC? (or ZF?) As far as I know, you can't prove that a model for ZF exists (unless you assume some powerful axioms, in which case you won't be able to prove that a model for the extended theory exists).
Edit: Another situation where that definition is problematic is the case of an inconsistent theory. Obviously, an inconsistent theory cannot have a standard model since it does not have a model at all. Whereas with my definition, we get the usual "Ex falso" as expected.
The standard model that most set theorists have in mind is something like the Von Neumann Universe, V. Note that this is a proper class, so it's not a structure as usually considered in model theory.
We (hopefully) can't prove V is a model of ZF in ZF, because that would amount to proving consistency and fall foul of Gödel 2, but the axioms of ZF come from an attempt to axiomatise our understanding of set theory in the sense of it being the study of the objects that make up the Von Neumann Universe.
> Obviously, an inconsistent theory cannot have a standard model
Indeed. Paraconsistent logics are an attempt to deal with inconsistency from a proof-theoretic stance, but I'm far from an expert and I don't know what models of paraconistent theories look like.
Yes, the discussion has deviated, and I don't think we will resolve the disagreement, but I wanted to make my position clearer w.r.t to the claim that "most mathematicians are Platonists [...] and they believe the objects they work with are real".
> It’s not clear to me which definition of (non-technical, unqualified/alone) “true” you are using.
I may be elliptic and not very clear, but I have not changed my definition. We can't do mathematics in a vacuum. There is always a context, which consists of a language, i.e. a fixed set of constant, function and relation symbols, and a theory, which is a fixed set of statements of the language. Typical theories are ZF, ZFC, PA, etc. For me, "true" (alone) means satisfied in all models of the theory, and equivalently by completeness, provable from the theory. (And by the way, your notion of "true" (alone) as "satisfied in the standard model" is equivalent to requiring that the theory be complete.) That would be your definition 1, except for the "non-technical" part. Now, the discussion deviated towards set theory because to compare my idea of "true" (alone) with yours, I used your comment:
> “True in the standard model” is generally what most working mathematicians who are not logicians mean by “true”.
which lacked context and seemed to me to be especially problematic in the context of set theory. And also, "most working mathematicians who are not logicians" implies a context of set theory. So the "non-technical" definition would be your definition 2 although I think ZF+DC (the axiom of dependent choice) is closer to what most mathematicians won't have a problem with than ZFC (depends on the discipline I suppose). Probably a mistake to talk about "most mathematicians" though.
> If you think it isn’t true then you are saying that we don’t really understand the naturals intuitively and we can only understand them by axiomatisation.
I mean something more subtle. I think we understand the naturals intuitively but only to some extent. Enough to write some axioms, but not enough to reliably answer many seemingly simple questions about them. I also think that our intuitive understanding is not static but grows as we study mathematics.
> we can ever say that proof is what determines truth given we know from Gödel that proof is fundamentally limited.
This is perhaps where the disagreement is? I don't have a problem with the fact that proofs are fundamentally limited.
I think we're understanding the phrase "true in all models" in different ways.
I understand it to mean: "true in all structures compatible with the language".
On the other hand, I think you understand it to mean: "true in all models of some latent theory left implicit", where the theory may be ZF(C) or something else depending on context?
I'm basing that on your comment:
> For me, "true" (alone) means satisfied in all models of the theory, and equivalently by completeness, provable from the theory. [...] That would be your definition 1 [...]
That isn't my definition 1, because I'm refering to all structures compatible with the language, not all models of some theory. This is probably an abuse of terminology on my part because usually we reserve the term model for structures that model a theory [1], sorry for the confusion!
> And by the way, your notion of "true" (alone) as "satisfied in the standard model" is equivalent to requiring that the theory be complete.
Please can you explain this? I don't think that's what I mean. We know that PA isn't complete, but when I say the Gödel sentence is true I mean that it's true in the standard model of the naturals.
> There is always a context, which consists of a language, i.e. a fixed set of constant, function and relation symbols, and a theory, which is a fixed set of statements of the language
I completely disagree with the idea that this context always existed, it's too Formalist. There is a rich history of mathematics before the concept of a formal language and a formal theory existed; if you were to ask Gauss if he worked in ZF or ZFC or TG I don't think he would have an answer, but clearly he had some concept of mathematical truth.
[1] : Although all structures are vacuously models of the empty theory, so technically they are models, but that's not very convincing or useful...
> On the other hand, I think you understand it to mean: "true in all models of some latent theory left implicit", where the theory may be ZF(C) or something else depending on context?
Yes, that's what I mean. (For me, "structure" is preferred to "model" when nothing is implied.)
> The standard model that most set theorists have in mind is something like the Von Neumann Universe, V.
Now I am getting confused. Isn't that equivalent to requiring the axiom of regularity? I have a book on set theory by JL Krivine with the theorem: "V is the whole universe iff the axiom of regularity holds". This book also proves that if "U is a universe (i.e. a model of ZF) then the collection V inside U satisfes ZF+axiom of regularity" (which proves the relative consistence of the axiom of regularity).
To talk about the Von Neumann Universe, you must assume some "surrounding" universe which is a fixed but arbitrary model of ZF. Thus, X is true in the Von Neumann Universe if and only if X is satisfied in all models of ZF+axiom of regularity. That certainly matches my idea of "true", albeit with a weaker set of axioms... (I proposed ZF+DC as a least common denominator because a large part of analysis can't be done without some form of axiom of choice.)
> Please can you explain this?
Let us call S your standard model of PA. I understood your idea of "X is true" as "S satisfies X". Now, let T be the set of all statements satisfied by S. Then T is a complete, consistent theory that extends PA and "X is true" if and only if "T proves X". (Of course, T is much larger than PA, and in fact, by incompleteness, there are no recursively enumerable theories equivalent to T.) This correspondence between complete consistent theories and models is not one-to-one though, a complete consistent theory may have infinitely many models.
> if you were to ask Gauss if he worked in ZF or ZFC or TG [...]
Fair enough, but I think he was familiar with Euclid's elements, and would have agreed on the fact that there are things that are assumed to be true because they are intuitive and things that are proved to be true. In my view, ZF is the culmination of an effort to minimize that intuitive part. By constrast, the notion of model (and Tarski's notion of truth) are more modern.
> [...] 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