Is mathematics something real like something so correlated in our universe which would be different in other theoretically universes or is it just an abstract universe-independent layer/framework we came up to with our minds for describing our universe?

Could the second Gödel's Incompleteness Theorem, the taking as axioms something we can not prove like Continuum hypothesis or the semidecidability of Hilbert system (K-Theory) inference be an hint to definitly say that it's just a working abstraction and not something real?

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    Possible duplicate of Was mathematics invented or discovered?
    – Eliran
    Nov 13 '18 at 18:24
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    Are traffic laws real? They're purely social conventions. But if you violate them you can die. So abstractions ARE real. Does a property owner own a piece of the earth? How can that be? There's no law of physics that says anything about it. Purely a convention. But you have to stay off other people's property unless you have permission, else you go to jail. Our entire civilization is about abstractions made real by social convention. A Martian physicist can use wavelengths to distinguish red from green. But she can't tell you which one's go and which one's stop. That's a social convention.
    – user4894
    Nov 13 '18 at 18:29
  • ps See Searle, the Construction of Social Reality. amazon.com/Construction-Social-Reality-John-Searle/dp/…
    – user4894
    Nov 13 '18 at 18:30
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    Hi, welcome to Philosophy SE. Please visit our Help Center to see what questions we answer and how to ask. Generic questions such as yours are not really suitable for our format. There is no answer, rather a centuries long philosophical controversy, which is best addressed by reading encyclopedias, see e.g. SEP's Philosophy of Mathematics. We take more pointed questions that can be more or less objectively answered within reasonable space.
    – Conifold
    Nov 13 '18 at 21:51
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    "Is mathematics something real ?" YES, it is. "is it an abstraction ?" YES, it is. Nov 14 '18 at 8:06

Before you begin to answer that you must be clear on what do you mean by real? Is anything you can think of isn't real and what you observe out there is? What is the domain where you define reality and separate it from fiction? If you consider fiction or thoughts and ideas to be equally real as the screen you are looking at then there is no problem to begin with. What I wanted to show is that this question has a history of confusion behind it and one must be careful before asking this question by having an expectation of what answer should look like.

One way of talking about math is that its just fiction (in our heads alone) that we use, like language, to label certain objects and that the way these objects interplay with each other, giving rise to new structures or behavior that wasn't there before, so you give it a new label, a product of an operation. Math also has the characteristic of being logical, which not surprisingly is deduced from our observations of the environment around us. We build our intuition using the world in which, we expect that if I have two apples and someone gives me a third I should say I have three apples. This can be readily transformed into a mathematical statement. And just like language is prone to lingual paradoxes, mathematics by its very nature, inherits those same kinds of paradoxes. If you believe that reality should be free of paradoxes, then you are right in saying that mathematics is an abstraction of the human mind.

To counter this idea you might propose mathematical realism or the kind of idea Max Tegmark likes to believe that reality IS math and not the other way around. But then reality(math) must have to be discovered and it seems the only way of doing so is using complex language and symbols and simulating the symbols using paper and pen. Then there is no surprise that nature seems very mathematical because guess what, it is math.

But what is common to either argument is that you cannot separate the similarities between language and math which implies math is not without paradoxes and this is the issue I have with mathematical realism. Paradoxes should exist as a feature of reality if math has to be reality. Math as fiction doesn't have this problem because it is free from carrying the burden of describing reality. I can conjure up a new set of rules for a set of objects and then I can describe totally bonkers of behavior, like a funny video game, that I should never expect to see in the world but it is still valid as an abstraction because I can play the video game! Again the answer boils down to what you will throw out or keep in your domain of reality. The confusion begins there.

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    This answer goes a long way to resolving this nagging question about the status of mathematics. Extremely well done! CMS
    – user37981
    Dec 14 '19 at 13:16

No, math, per se, isn't physically real. You can't measure "1", or hold it in your hand, etc. But you can measure 1apple, 1orange, etc. That is, you have to add units to mathematical abstractions in order to refer to reality. Those are typically meters, kilograms, seconds, etc, rather than apples and oranges, but it's fundamentally the same idea. Units added to math is typically called https://en.wikipedia.org/wiki/Dimensional_analysis

And then your "framework...for describing our universe" is exactly correct. Ordinary English is yet another such framework — English wouldn't be very useful if it couldn't describe the real world. But it turns out mathematical language is extraordinarily better at that than any natural language. Wigner famously calls this https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences Why is math so extraordinarily effective? Beyond the discussions in that link (or other googling on that phrase), let me know when you figure that out.

  • Great answer, with the exception of the first 2 sentences. Not being able to "hold something in one's hand" is not a precondition for realness. The orbit of the moon is real, as is the British Constitution, quarks, and World War 2.
    – Dcleve
    Mar 15 '20 at 17:15

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