I flip between agreeing and disagreeing with you! I'm not well qualified to comment so I expect an informed rebuttal to this will be instructive for me. So here goes:

I still find myself in the "number systems are a cultural artifact" camp. We choose which details to keep and which to throw away in our abstractions. There are anumeric cultures and it's not clear to me at all that the existence of integers is obvious (beyond what humans can subitize) unless you're immersed in a culture that has impressed them upon you since early childhood.

Do you have "2 cats" or actually just "this cat and that cat"? This cat is a bit bigger but that one looks meaner.

Deep down I think that an insistence on the "reality" of integers versus reals (say) is purely aesthetic. (Ignoring for the moment that our conventional constructions build one from the other in a certain way).

Anumeric cultures might not really have existed (not conclusive evidence, but some evidence: https://www.jstor.org/stable/10.1086/667452?seq=1 )

The integers represent perhaps the simplest case. Another example of their actual existence is the fact that properties of atoms and their nuclei critically depend on the number of the nucleons of particular types in them; note that this fact does not in any way depend on whether a culture is anumeric or not. With some effort you can extend this idea to other abstractions (this process is usually greatly hindered by one's leaning towards solipsism).

The existence proof you describe is in terms of a yet more complex abstraction; we can imagine aliens might describe a more useful atomic theory than our own which does not involve nucleons or their discrete counts at all (ok, unlikely as I acknowledge that may be).

I don't want to descend into solopsism and apologise if I veer too far in this direction. Also thanks for taking the time to rebut what is probably a sophomoric argument.

I suppose my fundamental objection is that if integers are "real" then it seems to me that quaternions must be similarly "real" (since I can describe useful things with them) and so on, I can't see a boundary which would let me say "ok integers are the real deal but infinitesimals are just a thing we made up".

I think we have a zone of possible agreement if we decide either that all these abstractions are "real" or none of them are.

This may seem a bit extreme, but, in my opinion, while in mathematics (and especially in physics) there is a fair amount of scaffolding, what they do in mathematics is, actually, they discover things that do exist in reality in one form or another. (It is important to understand that few, if any, things in mathematics are pure fantasy; the objects and relations studied there are pretty much forced upon us - which, incidentally, is yet another evidence in support of their having real significance, i.e. as something lying outside our consciousness and acting there independently from it.)

Yes, but the reals (or the power set of the reals) are really a bit dodgy. The Nana h tarski paradox shows a big disconnect with reality. There is other axiom models without AOC, which allow for every subset to be measurable, but they seem to have their own problems.

The reals exist in nature in the same sense as, say, 𝜋 does (which occurs as the natural limit in a particular Monte Carlo experiment). The paradox that you mentioned is, rather, disconnected from our intuition, from which reality itself is, too, disconnected (if you think of quantum mechanics, for example).

> Deep down I think that an insistence on the "reality" of integers versus reals (say) is purely aesthetic.

I disagree! In fact, I think the difference between countability and uncountability is pretty huge!

Only finite amount of integers like 10^91 (or so, number of protons in the universe) exist in this sense.

On the other hand, it is interesting to think about the fact that, say, the Avogadro number is usually represented as a real number; the problem with the number of protons or other particles is that their number is unbounded: given enough energy, another particle can always jump into existence...

That's still arbitrary (designed by humans) because you decided to count each proton as 1. We could also count every proton as 2 and it would be just as arbitrary.

Unfortunately, the number of dollar bills in your wallet cannot be thought of as arbitrary.

The notion of existence of ideas has been discussed in philosophy since Platon and has not settled. Consider that to express an idea even a simple one like the numbers one-two-three one needs to use a language. And language has infinite number of interpretations that may be entirely different. So how do one know that his ideas are not misinterpreted by others? Moreover, memory is based on the language, so how does one know that things that he remembers from a day ago mean the same things as it was then?