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I am a PhD students in physics, and my father is a Math researcher. One time, I asked him

"Doesn't the fact that we can use math to explain things that happen in front of us, tell us that math is not a human invention but rather a discovery?"

My father said that in his opinion, no - we were just able to find a very useful language to describe physical phenomena.

But then, I learned about Fraunhoffer diffraction, which tells us that if a light wave passes through a slit of a given shape, on a screen really far away from the slit we actually see the Fourier transform of the slit! I was mind blown.

I then went to my father and told him, "look! nature does Fourier transform, so Fourier transform cannot be man made!". He disagreed, and stated the same thing he told me before.

Does my statement make sense? Can we say, based on Fraunhoffer diffraction, that Fourier transform is an act of nature rather than a human invention?

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    But man is an "act of nature" Jan 4 at 17:12
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    Nature no more "does" Fourier transforms than it "solves" heat equations, even though heat propagation is described by their solutions. There is a difference between models and what they describe. And Fourier transforms and heat equations approximate many other processes too, that is how we design our modeling tools, flexible enough for multi-tasking. Our language is a masterpiece of that, it can describe anything we encounter. But it does not mean that nature "does" talking either.
    – Conifold
    Jan 4 at 17:32
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Mathematical objects are fictional. Nowhere in physical reality do you find a perfect circle or a polynomial equation.

But this does not prevent us from making true statements about what would be the case if these objects existed. Mathematical claims can be understood as truths about a counterfactual situation.

Truths - even truths about counterfactuals - are not a human invention. What is true, is true independently of anyone believing it. So while Fourier transforms are a human invention, theorems about Fourier transforms are discovered truths.

As a comparison, consider the rocket. The rocket is a human invention - it did not exist before humans assembled it. But the principles of a rocket's operation - the fact that the escaping gases produce a reaction force that can propel the rocket forward, the fact that spinning a rocket improves stability, and so on - were true before actual rockets existed, and will remain true after they are gone. Each principle is a human-independent truth about the properties of a certain type of object, even when that object is not physically instantiated at the moment. We can say that every invention is also a discovery of the principles behind the invention.

The same is true in mathematics. We define imaginary objects, but the principles concerning these objects are something we discover.

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This is an ancient controversy in philosophy. The Pythagorean and Platonic view holds that mathematical forms are realities of nature and thus "discovered" or, in Plato's case, "remembered" by rational beings.

In modern parlance, physicist Eugene Wigner famously dubbed this the "unreasonable effectiveness of mathematics." This idea is still by held by notable physicists like Roger Penrose and Max Tegmark, whose most recent book proposes that mathematics is the underlying reality of the universe.

In many varieties of idealism, foremost Kant's, mathematics is "synthetic a priori." It can function to derive or "synthesize" empirical propositions, yet it is not exactly "real" in some absolute sense, but part of the irreducible rational structure of human consciousness. Thus, we will "see" it wherever we look. This can easily be adapted to a naturalized view, like that of Chomsky's deep grammar.

The cases of "unreasonable effectiveness" are well known, including Maxwell's equations and Reimann geometry, in which purely theoretical math was later shown to correlate perfectly with physical phenomena. This is hardly conclusive. Were there no patterns and symmetries in nature, physics would not be possible.

Our brains recognize, create, and match intricate visual and tonal patterns, including those of mathematics. And there are many mathematical creations that do not have apparent counterparts in nature. So those that do undergo a kind of natural selection. While your father probably represents the majority view, you can oppose him and stand in very good scientific company.

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