What do you mean by "exist" here?
And I would dare to disagree right here. Math contains many structures that we don't know from our universe and that probably do not exist in our universe. If math is a model of universe, why is there a Mandelbrot set?
My map has a text written on it saying “Pacific Ocean”, yet I would not complain if I went to this place an couldn’t find a giant object in the ocean that would look like a letter P from the skies.
Except math can hypothetically model any consistent universe, not just our universe, which kind of undercuts the argument that it uses data that was mostly discovered, or that it's merely a model.
I think the most general view is that math is the study of structure, and some structures are real (in the sense that they exist in our universe), and some are not but we can still "discover" them by selective permutation or enumeration of axioms.
We can permutate and enumerate symbols for mountains, rivers and roads on a piece of paper. Maybe we would even get some “interesting” results like a map of the Lords of the Rings universe. How would that change anything?
Math is simply the logical conclusion of a set of conditions someone accepts as inherently true. If this, then that. Follow this logic far enough and you end up where we are today.
Certainly you can analyze such constructions using logic, but that's again conflating logic with mathematics. There's overlap, but they aren't strictly the same.
Math is nothing like a map -- maps are approximations of something real and they don't have any kind of internal consistency or complexity.
But there's a good argument that math is the fundamental nature of the universe, and mathematical discoveries lead to predictions of real-world behavior. While maps don't predict a thing.
The philosophical discussion isn't around whether math is useful for tracing the arc of a ball in the air, for which it always will be merely a useful approximation. It's more around math as the language of the universe, in things like quantum physics -- there's no "approximation" here, it's more the nature of reality itself.
I read my comment again and I was surprised, as I did not intend this tone. I’m sorry for being dismissive and for generalising too much about mathematicians.
Could you elaborate or point me to a formulation of the “language of the universe” argument you mentioned that avoids mentioning quantum physics? I don’t understand quantum physics and I’d like to avoid falling for the quantum physics fallacy [1]
[1]: https://www.logicallyfallacious.com/logicalfallacies/Quantum...
And a good place in general to start is always Wikipedia:
https://en.wikipedia.org/wiki/Philosophy_of_mathematics
And none of this has anything to do with the "quantum physics fallacy" at all. Philosophically, it's simply an argument about the most basic physical understanding of our universe, and right now that happens to be quantum physics.
Why is it that everyone thinks of mathematical models of quantum mechanics as much closer to the "nature of reality" than any other mathematical model? If anything the constant disagreements between quantum mechanics and physical models at other scales should make it clear that all the models we have are wrong by virtue of incompatibility.
That describes pre-1900s math we inherited from the greeks. With advent of non-euclidean geometry and abstract math, math is no longer bound to objective 'reality'.
Is a map real? Well, it is. I can see it on my desk. Is the earth real? It is too, but they are not the same. In that sense map is also not "real".
Is the map discovered? Well, it uses data that was mostly discovered, but some parts were "invented" or edited for simplification for the map to be useful.
The real question should be "is math useful" as a model. We all know most basic parts are, but some mathematicians forget that they are dealing with an imperfect model and keep finding paradoxes. It's like we would forget the imperfections caused by the mercator projection and be surprised the real world distances are not proportional to map distances.
That's the reason I always liked engineering more than maths. When programming you always "import" the libraries you need and find useful for the task. You only make sure that they are compatible with each other. Mathematicians "import" all axioms, call them maths, and are surprised they get paradoxes.