Utility/usefulness is never going to get you to truth — Leontiskos
I don't seek to reduce correct assertibility to utility. There are correctly assertible things which aren't useful in context - like using "Luke's father is Vader" in this discussion. And there are useful things which aren't correctly assertible in some contexts, like the idea that economic growth is always exponential.
It ultimately comes down to whether you see description relative to a frame as the same concept as description relative to an arbitrary frame. I don't. It seems you and
@Count Timothy von Icarus do. You both seem to want something "extra", in addition to norms of truth telling, knowledge and how people discover and find stuff out in the world, as a ground for knowledge.
I reject that in two ways, firstly that a ground in the sense you mean it is necessary to begin with - the concept is inherently "unrelative", so is a presupposition I don't have and don't need to adopt unless I'm trying to argue in your terms. Secondly that frames aren't arbitrary, and can serve as grounds for correct assertibility.
We've been going back on forth on me stating a presupposition, and you stating a presupposition, but neither of us are arguing which presupposition is "better" in manner which relates our perspectives {very well anyway}. Which, as fas I understand it, was
@Srap Tasmaner's point in the OP.
So let's go through why I like this perspective with a worked example or two.
I gave an example of something which is comprehensible in terms of a background which is nevertheless changing - a mountain path and the mountain. I also gave an example of correct assertibility with Ramanujan and Hardy. Which I'll go through in much more detail.
In which Ramanujan's statements prior to his collaboration with Hardy were correctly assertible but he did not assert them while following the norms of mathematical discourse at the time, and could not due to his lack of formal training - so they were not considered as true until that occurred. All there was to the truth of Ramanujan's claims was whether they were correctly assertible in accordance with the norms of mathematics taken in toto. They were, and were shown to be.
Note that "they were, and they were shown to" imputes a
somewhat fixed structure of justification to mathematical discourse, like proof. But it isn't a unitary phenomenon - like published mathematical proofs tend to be formally invalid, and corrections are submittable to published and believed proofs.
So there's enough normative character to do things with, but the nature of what can be done using those norms is not totally determined by their current state of expression - only what may be expressed with them fully determines their expression {given the current state of the assemblage}. Which is basically a tautology, but no one knows the scope of those rules without knowing all the theorems. An appeal to potential development, there, is an assemblage concept that
referenced, organisation in accordance with some abstract machine.
An assemblage like that which produces mathematical knowledge has pretty strict and durable rules, but many forms of them. And those rules are
identical with an idealised form of current practice. And current practice has both a potential and an actual component, as well as an idiosyncratic concreteness. People follow the rules in their own way, and improvise and revise the rules to more fully explore/extend the scope of mathematics.
The assemblage of mathematics production is also open ended in terms of its operation - if you deleted all universities and mathematical knowledge from the world, the norms would die too. That would impact the production of mathematical knowledge despite having nothing to do with the content of mathematical practice. It's also self organising, like the Langlands Program was generated within it and has ordered a lot of research and thought within it. It develops according to its own idiosyncrasies as well as its constraints, and is nevertheless sensitive to whatever milieu it finds itself in. Maths looks different if it's done on the tables of aristocrats or the laptops and whiteboards of universities and is submitted to different social forces.
Correct assertibility in that milieu has changed over time. Newton's proofs of his calculus principles were formally invalid, and it was known at the time - infinitesimals were 0 and not 0 at the same time in his proofs, at different stages. His principles were useful but not correctly assertible, but people believed them nevertheless, and just didn't give a crap about the self contradiction because the overall endeavour seemed cromulent and useful. The idea was
morally true {a term in maths scholarship}.
Nowadays we could interpret the proofs as a germinal form of something formally valid - his calculus with fluxions is much the same as our nonstandard analysis or formal calculus with limits. So we can "repair" his proofs and see his results as correctly assertible, as well as being morally true at their publication.
Hopefully the above worked example illustrates that although the background can shift, that doesn't render the justifications using it arbitrary.