[denotational]

> Well, I suppose it depends on your definition of standard.

Of course :) I believe my distinction between validity and truth is the one generally used in the literature (I have listed four examples above), and the one that would be understood by most working mathematicians and analytic philosophers who care about mathematical logic.

We both agree that there is a clear distinction between formulae that are true in some model (specified, or inferred from context) and formulae that are true in all models; the latter are not particularly interesting to most mathematicians once one has agreed on the logic (e.g. classical, constructive, etc.) in which one operates, hence I think it’s reasonable to use “true” to refer to the former, as indeed many authors do.

> That's called satisfiability.

Many logicians say that a formula is true in a model (sometimes true in a structure) if it’s satisfied in that model under all assignments.

Can you find me a reference in the literature where “true” is used to mean “true in all models” consistently?

[loicd]

> We both agree that there is a clear distinction between formulae that are true in some model (specified, or inferred from context) and formulae that are true in all models; [...]

Sure. But I feel we are deviating from the subject. We have obviously been educated differently so it is pointless to argue about that, but there is a language issue. You insist on comparing what I mean by "true" (alone) with "true in a model". However, that's an apple to orange comparison. We should be comparing what I mean by "true" (alone) with what you mean by "true" (alone), and by that, you mean: "true in the standard model". (I don't think your references validate that use, although I don't have access to all of them at the moment.) The obvious problems with that are:

- I don't think there is such a thing as a standard model in set theory (actually you cannot prove that a model of set theory exists).

- When most mathematicians say something like "X is true", what they mean is "X can be proved from the axioms of set theory", which in logical terms means "X is valid". Are you really arguing against that?

- And of course (back to the original point), you get that confusing idea that "undecidable" means "true but unprovable" (I had never heard of the incompleteness theorem being presented that way before.). I argue "undecidable" is "neither provable nor disprovable".

EDIT: "X is valid" should read "X is valid in set theory".

[MrManatee]

> When most mathematicians say something like "X is true", what they mean is "X can be proved from the axioms of set theory".

I'm just one mathematician, but I certainly don't mean that.

Before we can prove anything about sets, we need to pick some axioms. Zermelo set theory (Z) would be enough for most of ordinary mathematics. If we need something stronger, there's Zermelo–Fraenkel set theory with the axiom of choice (ZFC). Or if I need something even stronger, there's, for example, Tarski–Grothendieck set theory (TG).

What I mean by "X is true" is technically difficult to define. The statements

(1) X is provable in Z.

(2) X is provable in ZFC.

(3) X is provable in TG.

are all increasingly accurate characterizations of "X is true", but none of them capture everything about it. And that's kind of the point. There is no proof system P such that "X is provable in P" would work as a faithful definition of "X is true". So the best we can get is this tower of increasingly sophisticated axioms that still always fail to capture the full meaning of "truth".

There is a convention among mathematicians: Anything up to ZFC you can assume without explicitly mentioning it, but if you go beyond it, it's good to state what axioms you have used. ZFC is not a bad choice for this role. It is quite high in the tower. In most cases ZFC is strong enough, or in fact, overkill. But still, it is not at the top of the tower (there is no top!), so sometimes you need stronger axioms. The fact that ZFC has been singled out like this is ultimately a bit arbitrary - a social convention. "X is provable in ZFC" may be the most common justification for "X is true", but that doesn't make it the definition of "X is true".

[denotational]

I think there are two points of disagreement here:

1. The definition of “truth”; and

2. My claim that most mathematicians who aren’t logicians use “true” in the sense of being true in some model, not in all of them.

Re. 1, I accept that this is just naming, but I will insist that your usage of “true” is nonstandard, and should carry a disclaimer as such :)

I think this is evidenced by the fact that “true but unprovable” is a very widely used characterisation of the Gödel sentence for a logic, and it is completely wrong given your definition of “true”, and further evidenced by the references I provided that use “valid” for true in all models.

I think point 2 is a lot more fuzzy because we’re trying to talk about what is happening inside other people’s minds, in particular their view of what a mathematical result actually is; I’m happy to concede that I may be guessing wrong regarding what most mathematicians are actually thinking.

As per 1, my position is that there is no such thing as “true alone”, at least not in mathematical logic as it is conventionally presented English speaking world (and apparently in the Spanish speaking world, if one source is enough to generalise); truth only exists in the context of a model.

Accordingly, when we talk about the truth of a formula, we have a model (or possibly a class of models) in mind; I claim that when a mathematician says the Gödel sentence is true, they mean it is true in the standard model of the naturals, which is why the “true but unprovable” characterisation is used. I don’t know if my references support this, since it’s harder to find than just looking for the definition of “validity” in the index, but if you Google “truth of the Gödel sentence” you’ll find a lot of people using “true” to mean true in the standard model (of the naturals).

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 certainly believe the naturals are real, and I believe that non-standard models of the naturals are not the (real) naturals, even though I’m perfectly happy to “play” with non-standard models as an intellectual exercise. I claim that most mathematicians have other (real) objects in mind when dealing with things other than the naturals, and those form the meta-mathematical model for the notion of truth.

> When most mathematicians say something like "X is true", what they mean is "X can be proved from the axioms of set theory", which in logical terms means "X is valid". Are you really arguing against that?

That doesn’t mean “X is valid”; if something follows from the axioms of set theory then it holds in all models of set theory (yes, assuming a sound deductive calculus, but that’s a given), but that’s not enough to consider it valid, it would need to hold in all models compatible with the language. So yes, I’m arguing against that, I think what you have written is factually wrong, unless it’s a typo?

> you get that confusing idea that "undecidable" means "true but unprovable"

I don’t think I ever said this, I agree that this is confusing, and in fact it’s just wrong as stated (under my usage of “truth”), since of course there are undefinable sentences that are false (but irrefutable), such as the negation of the Gödel sentence.

[loicd]

> As per 1, my position is that there is no such thing as “true alone”, at least not in mathematical logic

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]

> 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.

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]

> 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.

[1]: https://en.wikipedia.org/wiki/Abc_conjecture#Claimed_proofs

[denotational]

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]

> That doesn’t mean “X is valid”; if something follows from the axioms of set theory then it holds in all models of set theory

Yes, I was being elliptic. That should read "X is valid in set theory". The point being that it is a notion of validity (ie valid in all models of set theory) rather than a notion of satisfiability (ie valid in a particular model of set theory).

[denotational]

For some reason the "reply" buttons past a certain level of nesting were missing for me, but that appears no longer to be the case, so I'm moving a previous comment here

The notion of relative validity is just semantic entailment, no? I have never seen that referred to in terms of validity, which has been reserved strictly for formulae that are true in all models, not in some class of models.

I’m a Platonist, and I suspect most mathematicians fall towards that end of the spectrum, so I disagree that most Mathematicians see ZFC as the arbiter of truth. They certainly aren’t doing formal proofs in ZFC, and in fact I suspect that most non-logician mathematicians would have difficulty reciting the axioms of ZFC.

That’s not to say I don’t appreciate proof theory and the desire to work in an axiomatic framework, indeed in a past life I spent most of my time formalising various things in Coq, but I don’t think it’s relevant to fundamental mathematical truth, which I believe exists outside of axiomatisation (and I think most mathematicians would agree).

[Timon3]

> For some reason the "reply" buttons past a certain level of nesting were missing for me, but that appears no longer to be the case, so I'm moving a previous comment here

If I understand correctly, HN throttles reply speed by hiding the reply button for some time after a comment was posted. The deeper the thread the longer this timeout gets.