But numbers, and other ‘objects of reason’, are real in a different way to sense objects. And that is a stumbling block for a culture in which things are said to either exist or not. There is no conceptual space for different modes of reality (leaving aside dry, academic modal metaphysics). Which is why we can only think of them as kinds of objects, which they’re actually not. They’re really closer to kinds of acts. — Wayfarer

Well said. Starting with the natural numbers, which are ways to distinguish objects and converse about quantities, mathematics has grown to virtually unimaginable proportions over the millennia. And it has changed character from a descriptive and predictive tool to an enormous game, unbounded in some aspects, with recently formulated foundational rules.

Some compare it to chess, where material pieces are moved around a board rather than the pen or pencil upon paper, or keys and screen of a computer. Where it might differ is in potential: mathematics awaiting discovery or creation versus possible strategies or moves on the chessboard. Chess players might comment on this.

Is a crossword puzzle real? Pondering how to fill in the spaces, then doing so with pencil. Sounds a little like math. Are emerging ideas real? Of course they are. Do mathematical objects exist in some exotic realm, awaiting discovery? I think of them as commonalities of minds, the way in which human brains have evolved.

Do infinitesimals exist (in the platonistic sense)? — Michael

I've always thought of these little critters as part of the metaphysics of mathematics. They now belong to a variation of the game called nonstandard analysis.

Do mathematical objects exist in some exotic realm, awaiting discovery? — jgill

As I said, I think ‘exist’ is problematical in the context. Not that they don’t exist, but the way in which they’re real is different to empirical objects. They are ‘objects of mind’ rather than ‘objects of sense’, but I don’t think the philosophical lexicon has an appropriate term. I tried this out on ChatGPT recently and it suggested ‘transcendentally objective’, although that is hardly an elegant expression.

And it has changed character from a descriptive and predictive tool to an enormous game, unbounded in some aspects, with recently formulated foundational rules. — jgill

Consider synthetic chemistry and genetic engineering. These too are grounded in traditional chemistry and biology but now have dimensions that would never be found in nature herself. It’s analogous in some ways.

I've always thought of these little critters as part of the metaphysics of mathematics — jgill

Maybe they are to natural numbers as viruses are to organisms ;-)

Hm. But how would one substantiate this idea that numbers exist in this different way?

And why would numbers be able to exist in this way, and not flying spaghetti monsters?

Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices.

I just want to point out that the bolded phrase is what's at stake. You can believe that numbers and other abstracta really and truly exist without being a mathematical platonist. You merely assert that they exist because we have created them, and they will cease to exist if we also cease. Whether you want to say this or not will depend on how you wish to use the word "exist." Clearly, if you are a friend of "existence = spatiotemporal objects or arrangements thereof", then you won't want to claim even a human-made existence for numbers.

I agree. I don't think mathematical platonism is supposed to be some big metaphysical statement. It's just reflecting our experience with math: that it's something we seem to discover, that it's not owned by particular people, in other words, it's not mental or physical.

That leaves the door open to trying to explain it anyway we want, kind of like gravity is a thing, but we're still working on how to explain it.