Good talk. I agree with you (and the articles you linked which I read, BTW thank you for those there's a lot more reading for me to do) in explicit claims that the vast majority of real numbers cannot be rigorously constructed[0], and that there are many for which representation is not possible. To my knowledge this is a philosophical argument, and one for which we don't have any falsifiable theories which could be used to test one side versus another.
> You are arguing that an alternative number system should be 'infinitely dense', and I agree. But take e.g. the finite/constructive reals [1, 2], they are still 'infinitely dense'.
I'm only arguing that insofar as one can do useful things with holomorphic functions and the constructions we require to define them require the specific defining properties of the complex numbers (continuity, closure under important functions rather than clopensure, etc). If you want these properties then you're stuck with uncountable number systems and the baggage RE representation that comes along with them.
> That's exactly my point. Maybe approaching it from the point of view of 'what are the limitations of this model?' is helpful. Also see the discussion in [2].
> This is not an argument whether real numbers are useful, a good model, or interesting (there is not doubt they are all three).
Agreed, my argument is purely that the reals are "able" to "exist" in "reality" in the same way as the rationals. One of the more famous irrationals is Pi, which is of interest precisely because it is referenced by reality. The difference between the two is that the rationals are not continuous for useful definitions of continuity and so we fix that.
That "existence" is dependent on human interpretation of that word, and there's no particular reason to expect that we have the ability to see whatever the fundamental underlying reality[1] of our world even is. You could just as easily argue that negative integers do not exist because owing someone something generally requires a reference to the entity owed rather than just a indicating a lack of meaningful possession.
There are many intelligent species here on our planet which have internal models of reality, which we know are less than correct (e.g. good luck teaching anything beyond basic intuition of classical physics to a parrot), what makes us special? There are many humans who cannot handle the abstraction of charm and flavor and spin being very much real physical properties of the invisible objects underlying the reality we're able to observe.
You can argue forever over finitism and the holographic principle and what "really exists," but such discussions are fundamentally limited by the things having them. The thing we use for this analysis (logic) is itself a human construction and certainly not something which is even as real as the number 1. Our best understanding of the underlying structure of the universe right now is that it is probabilistic, which is almost antithetical to the idea of logic being the underlying set of rules by which it operates. I see the arguments people make regarding these, but what I do not see is a meaningful distinction between 1 in Z which maps to a human concept of possession of a singular instance of an object versus Pi in R which maps to a human concept of the ratio between specific properties of certain classes of objects. Both have their basis in reality grounded by human perception, both can be used to any precision you're able to, both are meaningless beyond the human constructions used to define them. The only reason we consider math to be a universal thing likely discovered by every sufficiently intelligent species is because it is of such high utility in constructing predictions which provide an evolutionary benefit.
[0] the constructability of some of the reals is likewise unimportant, they're in the set because we deemed their existence to be of future utility. there are infinitely many reals which can never and will never be specifically referenced, their purpose is no to be directly referenced but rather to be there in the background so that we can make useful assumptions concerning other numbers surrounding them. we do not have to directly reference, or even be able to directly reference for them to have value.
[1] which is under no obligation to even be the type of thing we consider to be an "underlying reality," that's just how it seems to present itself to us