> Believing in standard mathematics basically means that you can not believe in absolute truth.

I agree that if one follows an axiomatic approach strictly and consider "truth" to be a shorthand for "provable from in some logic from some set of non-logical axioms" [1] then one is rejecting any notion absolute truth, since everything is relative to some set of axioms, but I don't agree with the charactedisation of this as "standard"; it seems to me to be a very Formalist stance.

I'd argue that most mathenaticians consider themselves Platonists, and believe that the mathematical objects they are describing are real enough to form some kind of metamathematical "standard model", and "absolute truth" can be defined in the model-theoretic sense relative to this standard model, even if this is somewhat unavoidably handwavy.

[1] : Even if you do think this, "truth" is generally used by logicians in the model-theoretic sense of "truth in some specific model/structure compatible with the language".

>but I don't agree with the charactedisation of this as "standard"

I used it as an objective term, defining mathematical objects on terms of ZFC and truth being relative only to ZFC is the standard mathematical foundation. If you ask a random mathematician what he thinks the foundations of mathematics are it will most likely be ZFC, even if he disagrees with it on any level, it is still what he would set his rival theory against.

Working, or at least claiming to work, in ZFC is fairly standard, but that doesn’t make it the definition of mathematical truth.

As a sibling comment mentioned, most mathematicians have a sense of truth that is not bound to any axiom system.

I don’t think it’s contradictory to work in ZFC whilst simultaneously having a non-axiomatic notion of mathematical truth.

I would hazard a guess that most (all?) working research mathematicians would accept the truth of the Gödel sentence for their preferred axiom system (and deductive calculus), be it ZF/ZFC or TG or something else entirely, so I cannot accept the claim that they see the “standard” notion of truth as being relative to all models of some axiom system.

You might think this is just nit-picking, but if it’s fine (in the sense that this is still “standard”) to add an arbitrarily large set of Pi_1 formulae to ZFC from repetitions of Gödel 1, then I don’t think we can say that ZFC is the standard basis of mathematical truth because this cannot be justified in ZFC; there must be some other (standard) notion of mathematical truth used to justify this.

>Working, or at least claiming to work, in ZFC is fairly standard

Which is why I called it "standard".

> I don’t think we can say that ZFC is the standard basis of mathematical truth

You literally just said that working in ZFC is "standard". A standard is a social agreement, standards can be completely false and absurds, while being standards.

>As a sibling comment mentioned, most mathematicians have a sense of truth that is not bound to any axiom system.

Which I agree with.

You said:

> truth being relative only to ZFC is the standard mathematical foundation

I think this is at odds with:

> most mathematicians have a sense of truth that is not bound to any axiom system

Re.:

> You literally just said that working in ZFC is "standard"

Nowhere did I say that considering ZFC to be the arbiter of mathematical truth is standard, in fact I’m claiming the opposite.

Deciding to adopt a particular set of axioms as standard just means that I’ll accept a proof from those axioms without question; it doesn’t mean that I believe mathematical truth is precisely that which is a syntactic consequence of ZFC/TG/whatever.

Perhaps I’m misinterpreting your claim? At the moment I’m reading it as: “The belief of the majority of working research mathematicians is that mathematical truth is defined relative to ZFC.”

>I think this is at odds with:

I don't think so. Mathenatical statements are almost always framed in the context of ZFC. This does not contradict that mathematicians think ZFC is not an absolute truth.

>Perhaps I’m misinterpreting your claim? At the moment I’m reading it as: “The belief of the majority of working research mathematicians is that mathematical truth is defined relative to ZFC.”

All I am saying is that mathematicians are framing their results in the context of ZFC and that this makes it the "standard" theory. I think that statement is absolutely not controversial, even alternative theories are framed in opposition to ZFC.

I absolutely do not think that mathematicians believe that ZFC is "true", as in it is the one and only perfect set of axioms.

My initial argument (and I am sorry if that was unclear) was that IF you believe that mathematical truth is about formal derivations from axioms (ZFC would be such a theory, same as ZF or any variation) then either you have to say that there is one perfect system and all truth is relative to it alone or that there are multiple equally true, but incompatible, theories.

The "IF" is of course important and I don't think many mathematicians actually agree with the IF clause. I actually completely agree with: "Deciding to adopt a particular set of axioms as standard just means that I’ll accept a proof from those axioms without question; it doesn’t mean that I believe mathematical truth is precisely that which is a syntactic consequence of ZFC/TG/whatever." and I am sorry if I wasn't clear. I actually do not think there is any disagreement here.

Yet still professional mathematicians have an underlying notion of truth outside of any axiom systems. I forgot who said it but if we were to find a contradiction using Peano’s axioms, we would say that the axioms were wrong, rather than arithmetic itself.

Even your comment references “perfect rules which totally correspond to reality” which seems to be another way to say “absolute truth”.

>Yet still professional mathematicians have an underlying notion of truth outside of any axiom systems.

I am certain about that, but it does not make my statement less true. I actually think that very few people believe in ZFC as either a formalist absolute or as an arbitrary set of rules. I think the most common view is that it enables other theories, that those mathematicians actually care about. The moment those theories rely directly on axioms things get difficult. I think the following quote describes quite well the state of ZFC: "The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"