Yes, that's exactly how pragmatists[1] answer the question. It depends entirely on what you mean by exists and what you mean by exists always depends on what you're trying to do, i.e. your goal. That's what the whole squirrel* anectdote that William James came up with was about.

1. https://plato.stanford.edu/entries/pragmatism

* The squirrel thing is in section 1 of the linked article.

Certainly looking at what mathematics is or could be in relation to the rest of the universe looks like a legitimate research area to then potentially guide the evolution of our mathematical tools.

I'm particularly interested in how one can reconcile naturalism/physicalism/materialism/monism with non-platonism/nominalism, i.e. if there is no such thing as Platonic ideals, what's the "physical" nature of math?

(Using lots of quotes here because I'm being lazy and not super careful with the terms I'm using — I'm aware.)

Even is we suppose what you're saying is correct — the goal isn't really to change how we use mathematics, but to understand what math is in a deeper explanatory sense. If that's not a project that interests you, that's totally fine.

This is an incredibly parochial point of view.

Kant's seminal "existence is not a predicate" argument has had a foundational impact after he put the rationalist vs. empiricist debate to rest. In fact, it's extremely deep, arguing that the P in ∃xPx can't be "exists." Modal logicians have tried to come up with more clever ways of circumventing this, e.g. E(t) := ∃x(x=t)—although this isn't entirely non-problematic, either.

For that matter, the question of anything existing is profoundly human, so asking "what is the point" kind of misses the point.

If one holds the view that mathematical objects can be understood as 'language games' rather than real ontological objects I think it creates a more pragmatic view on what is permissible, what constitutes a proof, whereas I think the more Platonic views might make someone more purist in how they approach maths.

One's philosophical / metaphysical worldview determines how one views reality.

As an example, some variations of Islam follow(ed) occasionalism:

> Occasionalism is a philosophical doctrine about causation which says that created substances cannot be efficient causes of events. Instead, all events are taken to be caused directly by God. […] The doctrine states that the illusion of efficient causation between mundane events arises out of God's causing of one event after another. However, there is no necessary connection between the two: it is not that the first event causes God to cause the second event: rather, God first causes one and then causes the other.

* https://en.wikipedia.org/wiki/Occasionalism

Whereas Christianity rejected it and went with secondary causation:

> Secondary causation[1][2][3] is the philosophical proposition that all material and corporeal objects, having been created by God with their own intrinsic potentialities, are subsequently empowered to evolve independently in accordance with natural law.

* https://en.wikipedia.org/wiki/Secondary_causation

So in in the first case asking "Why did X happen?" you answer "God willed it.", while the second case you would say "There is something in Object A that interacted with Object B." The latter then leads you down the path of examining objects and their relationships, as opposed to chalking events to spirits, gods, or God exclusively. Plants/crops growth because of something with-in themselves and not because of Ceres / Demeter willed it.

Without this worldview, you don't operate under (e.g.) the zeitgeist of being able to investigate Nature:

> That this objective reality is governed by natural laws;[35][36]

> That reality can be discovered by means of systematic observation and experimentation.[35][36]

> That Nature has uniformity of laws and most if not all things in nature must have at least a natural cause.[36]

* https://en.wikipedia.org/wiki/Naturalism_(philosophy)#Provid...

So on a day-to-day basis you may not deal with these axioms/beliefs, but they are there nonetheless.