Aha, it’s an ontological question. Most mathematicians believe that numbers, sets, etc don’t exist in the world we live in, but exist in a special world called “Platonic world”. That’s what Platonism in philosophy of mathematics is about.

I agree that numbers don't exist in the real world. However, we humans have an ability to perceive the world through numbers. The question then is about our perception of the world if you wish.

There is a subtle issue lurking behind. And that can be framed as a question: can one access things(say, numbers) that do not exist? If the answer is "Yes", then we don't need the Platonic world. Otherwise, we need to postulate the existence of numbers and sets in another world (Platonic world). You can also see people who are looking for Platonic love.

In other words, it is an issue between access and existence. Does access need existence? For instance, when X says 'John is charismatic', is 'charisma' like 'neurosis'? Definitely not, and X sees John's charisma. And this charisma doesn't play the causal role.

I strongly doubt that "most" mathematicians are Platonists.

Most modern cryptography relies on number theory.

Math is the formal study of patterns. The universe is built on patterns. The universe is beautiful. Math is beautiful.

Maybe we don't know today if there's an application, but notice that proof mentioned using Fourier Transforms to study a set, the same way you might use it to study a radio signal. There was probably a time when people asked what use is FT?

Good point, but bad example. Fourier developed his transform for the sole purpose of attacking the heat equation, which is definitely quite a physically important problem!