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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 a

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.

loicd 3y ago
> This is very far away from my original point 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.
- denotational 3y ago
  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...

loicd 3y ago

> This is very far away from my original point 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.

denotational 3y ago
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...