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It has been said that it is remarkable that the world (at least, parts of it) can be described by mathematics, especially in physics. After reading another question I know that it was Wigner who spoke of "the unreasonable effectiveness of mathematics": Nature speaks the language of mathematics and can reply to the questions we ask in this language. It would be remarkable (unreasonable) if she couldn't reply in that language but that's my opinion.
There are people (like Tegmark) who think that because of the remarkable fact that Nature replies in a mathematical way (or because of something else), mathematics must reside somehow in Nature. It is even supposed that because of this there must exist other worlds in which different mathematics exists. We can think of different kinds of mathematics though, even if there is no part of Nature that replies to us in the thought math or math structures.
In what way is mathematics thought to exist independently of us human beings?

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There are at least two schools of thought on this topic, I invite the experts here to add more.

The first school holds that mathematics is a human construct and that connections between the natural world and mathematics are human inventions. In this view, without humans there would be no math.

The second school holds that mathematics is (frequently) embodied in phenomena occurring in the natural world, and that connections between the two which are uncovered by humans are not human inventions, they constitute discoveries about the (hidden) world of mathematics.

I reject the first view because to me it implies that without someone around to observe the "mathematicity" of the universe, it would not behave in a mathematically-describable manner: the tree does indeed fall even if there is no one around to hear it.

I accept the second view- that math somehow "exists" in the universe independently of the presence of humans to "hear" it. In my experience in the worlds of engineering and physics, I have seen how completely "abstract" fields in mathematics, which were invented by mathematicians, were decades later discovered to be just what was needed to furnish a logical structure for mathematical reasoning about a newly-discovered field or phenomenon in physics.

In my opinion, the two best examples of this are Niels-Henrik Abel's creation of Abelian and non-abelian group theory and its later application to making sense of quantum chromodynamics and the strong nuclear force, and Riemann's generalization of euclidean geometry to include the concept of a curved space which was exactly what Einstein needed to mathematically describe gravitation.

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    I agree with you on the existence of the physical world. I'm not sure if I agree with the math though. It would be strange if the physical world (and we belong as well to it; we eat it every day...) didn't answer in the language of math if we ask a question in this language. But the language doesn't say anything about existence. I know what you mean when you talk to me but to say the language resides in you, independently of you, is a different thing. How should the math exist in the physical objects it describes? Is language a true aspect of you?
    – user52804
    Jun 10 at 20:47
  • "didn't answer in the language of math if we ask a question in this language" Can you elaborate on what you mean by this? It seems easy to imagine a hypothical world where trying to predict/describe the world with math would just hit a wall at some point, like if we were in a Matrix style virtual reality and our "real" brains were outside the simulation and thus couldn't be investigated with measuring instruments within the simulation (which is not so far from what Cartesian dualists believe). Or an animist style world where every object was controlled by some intelligence outside it.
    – Hypnosifl
    Jun 11 at 1:57
  • @Hypnosifl Morning! (or evening!) I mean that if we look at the right places (determined by the math itself) we will see Nature (the stuff around us) correspond to it. This doesn't mean that it exists (the math) in the "real" world we can't investigate. The same holds for the gods. They live in a world we don't have access to (yet) but we (or the old Greek) could correspond to them. Do you think that math is like these gods? That it exists in a world of which this world is a shadow? Or does the math live in the stuff of Nature itself? If so, then how?
    – user52804
    Jun 11 at 4:35
  • When you say "look at the right places (determined by the math itself)", are you suggesting that--as in the virtual reality with external puppeteers--there might be other things we could look at that would resist all attempts to predict them using mathematical laws? Are you assuming the incorrectness of the reductionist view (see Einstein in the 5th paragraph here for ex.) that the laws of physics acting on basic physical states (arrangements of particles, say) could in princ. be used to derive all physical behavior?
    – Hypnosifl
    Jun 11 at 18:44
  • (cont.) "Or does the math live in the stuff of Nature itself?" Yes, I tend to favor the notion of structural realism--that the physical world is wholly defined by a structure of mathematical relations, w/ no other properties. I also favor the type of physical reductionism I talked about above. So for example if you could build a sufficiently powerful computer, a simulation which was given the initial physical state of all the particles in a human brain and body, along with the fundamental laws of physics, would reproduce human behavior.
    – Hypnosifl
    Jun 11 at 18:45

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