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. – Mauro ALLEGRANZA Nov 14 '18 at 8:06

This is a false dichotomy. The two options you suggest are the central positions of Platonism and Formalism as approaches to math. But there are others. I would like to inject my favorite one of those at some length. (Pardon the repetitiveness and my occasionally running on. I feel the ideas are subtle and need to be explained multiple ways.)

From an Intuitionist/Constructivist point of view, mathematics is part of the structure of the human mind. As such it is evolved and shaped by the universe, not just made of nothing as the Formalist might suggest, but neither shaped perfectly to the universe, as the Platonist might expect.

From this perspective, mathematics studies the deepest aspects of human psychology, the primitive intuitions that allow us to make a combined whole out of our perceptual mechanisms and our linguistic structures. The sorts of things that all perception and all language must obey, whatever more situational adaptations enter into the mix at a higher level, or whatever compromises happen in the name of efficiency. Things like the nature of succession that allows us to represent time in language, or disjunction, conjunction and implication as natural ways of combining expectations without having to do any coping of the expectations themselves to make them fit together.

This encourages us to trust mathematics to a considerable degree, but not to be overwhelmed with surprise when we find that it contains paradoxes. We should expect honest paradoxes, places where two competing intuitions evolved for to different purposes compete and do not agree. Russel's paradox is not just a language trick. The paradoxes of measure like the Banach-Tarski theorem are not indicators that 'something is wrong' with either the axiom of choice or the notion of measurement. Natural needs do not have to fit together. Sometimes there are basic problems nature has not yet addressed.

It also explains the 'unexpected' success of math in our science. Something that is largely responsible for our survival as a species, perhaps so deeply embedded that we cannot really believe its most basic causes are false, has been challenged a great deal, and has successfully saved us over and over again. And we are still here. So none of its errors can be too near the surface, or can really matter all that much when they are applied to important things.

Then the ultimate gaps and paradoxes are problems that arise from intuitions that do not come together naturally and have not been simultaneously important often enough to have evolutionary forces apply to them and shape them into a common whole. We have never needed to know how an overarching logical system modeling human arguments in their most abstract forms works, and we should not be shocked if it just doesn't. That does not make the system itself any less real, or keep it from helping us negotiate contracts and decide which theories of causation are most likely when we encounter arguments about the relationships people build.


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.

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