[brazzy]

Math and reality are, in general completely distinct. Some math is originally developed to model reality, but nowadays (and for a long time) that's not the typical starting point, and mathematicians pushing boundaries in academia generally don't even think about how it relates to reality.

However, it is true (and an absolutely fascinating phenomenon) that we keep encountering phenomena in reality and then realize that an existing but previously purely academic branch of math is useful for modeling it.

To the best of our knowledge, such cases are basically coincidence.

[xanderlewis]

Opposing view (that I happen to hold, at least if I had to choose one side or the other): not only is mathematics 'reality'; it is arguably the only thing that has a reasonable claim to being 'reality' itself.

After all, facts (whatever that means) about the physical world can only be obtained by proxy (through measurement), whereas mathematical facts are just... evident. They're nakedly apparent. Nothing is being modelled. What you call the 'model' is the object of study itself.

A denial of the 'reality' of pure mathematics would imply the claim that an alien civilisation given enough time would not discover the same facts or would even discover different – perhaps contradictory – facts. This seems implausible, excluding very technical foundational issues. And even then it's hard to believe.

> To the best of our knowledge, such cases are basically coincidence.

This couldn't be further from the truth. It's not coincidence at all. The reason that mathematics inevitably ends up being 'useful' (whatever that means; it heavily depends on who you ask!) is because it's very much real. It might be somewhat 'theoretical', but that doesn't mean it's made up. It really shouldn't surprise anyone that an understanding of the most basic principles of reality turns out to be somewhat useful.

[brazzy]

I think you're not even disagreeing with me, we're just using different definitions of the word "reality". I meant it to use specifically "the physical world" - which you are treating as distinct from mathematics as well in your second paragraph.

[xanderlewis]

But then you must agree that it's not a coincidence.

[deterministic]

And yet another view:

Mathematics is an abstract game of symbols and rules invented by humans. It has nothing to do with reality. However it is quite useful for modelling our understanding of reality.

[lukan]

"that we keep encountering phenomena in reality and then realize that an existing but previously purely academic branch of math is useful for modeling it."

Would you have some examples?

(Only example that I know that might fit are quaternions, who were apparently not so useful when they were found/invented but nowdays are very useful for many 3D application/computergraphics)

[uzerfcwn]

Group theory entering quantum physics is a particularly funny example, because some established physicists at the time really hated the purely academic nature of group theory that made it difficult to learn.[1]

If you include practical applications inside computers and not just the physical reality, then Galois theory is the most often cited example. Galois himself was long dead when people figured out that his mathematical framework was useful for cryptography.

[1] https://hsm.stackexchange.com/questions/170/how-did-group-th...