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Not trying to be inflammatory at all, this is a genuine (maybe dumb) question.

Especially in regards to the genesis of the surreals, which was Conway thinking about Go endgames. They seem among the least platonic numbers/mathematical objects. They have the benefits of down to earth construction but also the capabilities of the reals and beyond.

I also can’t find what Conway thought about platonism if anyone knows.

I imagine some philosopher must have said something about them in regards to ontology.

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    I haven't gone over the details at all, but I think Hamkins uses surreal numbers in one of his essays. He's a Platonic realist. What do you mean by "down to earth construction"? Having nontrivial division, root extraction, logarithms, and other things besides, applying to transfinite ordinals, amazes me, and I wouldn't say it's more or less "Platonic" than if we didn't have these operations on those numbers, but I'm wondering what the criteria are for comparing the degrees of Platonism respecting various mathematical categories. May 11, 2022 at 16:45
  • @KristianBerry In the same way Hartry Field said something like in Science Without Numbers, “when the chips are down…we can do physics without numbers”, because we can use non-abstract, concrete, physical and linguistic objects. Aren’t “go endgames” “down to earth constructions” and non-abstract objects? That is, when using surreals one may always have in mind they are using a linguistic or physical tools based on games, which are physical and concrete.
    – J Kusin
    May 11, 2022 at 17:08
  • Maybe less a matter of ante rem and more an in re realism (or construably so), but an interesting paper could be this one on ultratasks. Another paper relates No as the proper class of surreals to NBG theory and talks about an "absolute" kind of continuum (that word "absolute" indicating, though not implying, a transcendentally absolute ante rem reality). May 11, 2022 at 17:08
  • The epigraph for the second essay also suggests a more peculiar idea, something like modal neo-logicism, though addressing all that would mean clearer and more useful distinctions between abstract and concrete information, the status of 'mere possibilia', etc. May 11, 2022 at 17:11
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    Kadvany makes such a point in his review of Badiou's Numbers and numbers:"Badiou would like to think that Conway's surreal numbers, being "constitutive" of ordinals, real numbers, and their wild arithmetic, avoids that constructivist stain... But surreal numbers don't arise from dust, they rely on combinatorial rules via Conway's game-theoretic constructs, and these are axioms manqué. Computation and symbolic manipulation cannot be ignored because every formalism relies on them."
    – Conifold
    May 11, 2022 at 20:03

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A lot of mathematicians consciously hold on to Mathematical Platonism. For example, Penrose in The Emperor's New Mind said there appears:

to be some profound reality about these mathematical concepts, going quite beyond the mental deliberations of any particular mathematician. It is as though human thought is, instead, being guided towards some external truth - a trith which has a reality of its own.

Others hold it unconsciously. Fir example, Eugenia Cheng has said in a paper that when she thinks about it, then she doesn't think numbers or triangles live in some Platonic realm but when she goes about her daily work as a mathematician, this is exactly how she treats it: they're there, they exist.

Now the surreal thing about mathematical platonism is that all mathematical concepts are at the same ontological level. Plato himself placed it between the matter of the world and his higher id3as/forms.

Thus for mathematical platonism, there is no difference between surreal numbers or the ordinary integers: like 1, 2, 3 ...

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