Is Mathematics invented or discovered?

How can we test this empirically?

Well, if it’s discovered, then every mathematical concept should have a physical analog. But that is not the case.

Take for example, infinity. We do not have tools to determine whether are an infinite number of objects exist in the universe. Similarly, there is no way for us to determine if time is truly infinitely long.

In addition, Cantor showed that the irrationals are uncountably infinite, and similarly the power set of any set is of strictly greater cardinality. In other words, there are infinities that are bigger than other infinities within mathematics, yet I don’t see a physical analog.

But why then is math so useful? For example, complex analysis — a calculus involving imaginary numbers — is often used in real world engineering problems.

Math is useful, precisely because it is an abstraction. I think the best analogy is that of a map. Maps model physical reality, but they are not physical reality. If they had all the details of physical reality, they would cease to be useful, but because they abstract away details, they elucidate spacial relationships that would otherwise be hard to grasp.

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