A lot of the discussion regarding this topic sometimes revolves around "geometry". Since I don't have a reasonably informed opinion about that, I will skip that.
To me one of the biggest issues is "What kind subsets of ℕ are to be admitted as acceptable?". Quite honestly, this is one of the reasons why I have become more interested in mathematics in the first place (in last year or two). One would note that the question is equivalent to "What kind of reals are to admitted as acceptable?".
Basically a lot of opinions and varieties regarding math indirectly are related to this question. My current perspective (and obviously quite incomplete) perspective is that this takes one to the following three points:
(1) Strict Constructivism / Intuitionism
(1a) Strict Constructivism: As I see it, strict constrctivism is markedly different from classical math (to the point that it can't be made compatible). This is not to say that one can derive very many interesting theorems etc., but the understanding of basic idea of "reals" is radically different in constructivism.
Of course a classical mathematician might (and probably will) raise the objection that the set of objects admitted as "reals" are just too narrow (and hence don't represent the "actual continuum"). As I see it, if one doesn't pretend that one is dealing with some "actual continuum" (whatever that is supposed to mean) in constructivism, it is very close
"discovery".
There is still some space for variations though. For example, as I re-call, Markov assumed that every program either halts or doesn't halt (essentially LEM over halt set). But someone could be more conservative than this (and not assume that).
(1b) Intuitionism: I don't mean intuitionism in a some strict sense of variety here (because there would be many varieties), but in a very broad sense. For example, Brouwer made the distinction between "lawless" and "lawlike" seq., which I don't agree with (because my feeling is that such kind of distinctions can be made for "relative analysis" but not for "absolute analysis").
Nevertheless, the phrase "operations of human mind" most closely reflects my own views. And ofc this is why my personal hope/wish is that if we interpret intuitionism in a very broad sense, there might be hope for its reconciliation with classical math (to a fairly good degree). But indeed that's not how mathematics operates (not on hopes but what actually is the case). It is possible that such a reconciliation might not be possible in principle though.
(2) Objects below Ω/ψ
Now one might also bypass the previous discussions entirely (as I understand most mathematicians do) and focus on what makes "sense" (at least as long as one is always willing to assume a yes/no membership for reasonably well-defined subsets of ℕ).
Now it seems to me the bigger the set of ordinals below Ω one assumes one can go on to define more and more strong hierarchies of "subsets of ℕ". For example, in infinite program models the longer a program runs the bigger the collection of sets it can produce.
"Ultimately" I believe one would, at some point, have to face difficult questions regarding "bottom-up" constructions, "reasonably well-defined" constructions (and hence also the issue of
discovery/invention to an extent). And such questions I think might depend on what kind of view one has regarding sets.
It seems to me that some of the "multiverse" (as opposed to "platonism") view regarding maths comes from here. A fairly relevant
link (quite advanced, probably good as a reference for long term).
To be honest though, I have quite a lot of difficulty reconciling this with the possiblity of ℝ not having cardinality aleph_1 and how that's "really" possible (and maybe learning set theory properly might help alleviate this kind of difficulty to an extent ... though I really doubt it would change my own views completely ... perhaps partially though). There seems to be some aspect here where I just can't seem to relate to classical thinking.
(3) Objects above Ψ/Ω
Once again the issue what might count as bottom-up in a reasonably way or not would come up it seems. And this might interrelate to (2) in a complex way it seems.
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And lastly one has to face the difficult questions such as:
(i) How much and exactly what assumptions we have to make about the "reals" to derive such and such theorem (in analysis for example). Of course I don't know anything about this.
(ii) If truth/falsity of some number-theoretic statements is derived using some subtle assumptions regarding (2) and (3) above, what are to we make of it? Surely we could dispel the subtle assumptions and assume nothing about the truth/falsity of the corresponding number-theoretic questions (but the difficult question exactly is when we have to take a stance).P.S. To summarise the whole discussion in few sentences, it seems that one can keep a lot of the description of mathematical ideas very well-grounded if one is very clear about the assumption regarding what collection of sub-sets of ℕ (and hence exactly which reals) one is talking about (shouldn't matter if the collection is very large). But it seems that the generality of "real continuum" will be lost then.