When it comes to mathematics, philosophers have divided conceptual space into two main categories, “Platonism” and “nominalism”, where Platonism is, roughly speaking, belief that there “really” are mathematical objects. The contemporary debate between Platonists and nominalists about mathematical objects is, of course, descended from the famous medieval debate about universals, which is in turn just a version of the old Plato-Aristotle debate. For example, besides my gray computer and the gray sky in Copenhagen, does “gray” itself exist?

In the twentieth century, the philosophical debate about mathematics had two main phases. In the first phase, practicing mathematicians tried to find their orientation to recent developments, with some of them adopting a formalist framework, others adopting an intuitionistic framework, and yet others adopting the newly developed set theory of Zermelo and Fraenkel.

In the second phase, philosophers — mostly under the influence of W.v.O. Quine — discussed “ontological commitments”, leaving many nominalists (most notably Quine himself) convinced that good-faith acceptance of the theories of natural science demands Platonism about mathematical objects.

Of course, not all philosophers liked being told that the only
rational view is Platonism.^{1} As a result, there have been great
efforts over the past fifty years to reformulate physical theories so
that they don’t “quantify over” mathematical objects (see especially
(Field 2016)).

Going off in a rather different direction, many contemporary philosophers take mathematics to provide a form of representation that is exempt from the pitfalls of natural language. See especially (Wallace 2022).

Niels Bohr’s view about (applied) mathematics has two main components:

Mathematics is a precisification [forfinelse] of natural language.

We are suspended in (natural) language.

Field, Hartry. 2016. *Science Without Numbers*. Oxford University Press.

Wallace, David. 2022. “Stating Structural Realism: Mathematics-First Approaches to Physics and Metaphysics.” *Philosophical Perspectives* 36 (1): 345–78. https://doi.org/10.1111/phpe.12172.

Contemporary philosophers seem to be deathly afraid of Platonism? But why? Perhaps they are worried that if they let in abstract objects, then God might be next. A more reasonable worry is that Platonism isn’t a good (scientific) theory about what exists. Or even more reasonably, even if abstract objects “exist”, the kind of existence they have would be very different from the kind of existence enjoyed by familiar physical objects. For example, even if I say that “there is a smallest even number”, I have no idea how to answer the question “how many number 2s are there?”↩︎