Is Mathematics Real? (2020)

I'm quite surprised intuitionism is not brought into this conversation.

> Intuitionism is a philosophy of mathematics that was introduced by the Dutch mathematician L.E.J. Brouwer (1881–1966). Intuitionism is based on the idea that mathematics is a creation of the mind. The truth of a mathematical statement can only be conceived via a mental construction that proves it to be true, and the communication between mathematicians only serves as a means to create the same mental process in different minds. [^0]

The intuitionistic logic, through Curry-Howard-Lambek correspondence, serves an important role in proof theory and model theory.

[0]: https://plato.stanford.edu/entries/intuitionism/

This question can be applied to any concept and to me it translates roughly to "is language a good representation of reality?"

The answer I like: any model that is useful for practical applications I consider being real. You only have to remember that it is a model and thus has serious limitations. Take it out of context of the application is has been designed for and turns to nonsense immediately. So you must be careful to strictly limit model usage. But the same limitations also make it possible to attack any concept with solipsism / deconstructionism - which is a mistake in the opposite direction imo. Guess this is enlightenment vs postmodernism.

Also see: https://en.wikipedia.org/wiki/Map%E2%80%93territory_relation

It is if you round up.

One-ness is a physical property and that's why we can see it. The magic happens when we call them all one, give it a symbol, and make them all equal. Now "one" has a unique existence of its own on a piece of paper on a mathematicians desk, scribbled alongside other unique symbols, whose relationships come complete with proofs. And the only question then is whether what was scribbled says something true, as in, something that aligns with the reality it came from, because that would make it useful.

Math is a map. Maps are only real insofar as being a map, and only useful insofar as being true.

The drawing of the streets may be drawn with your pencil from memory, but the truth your friend relies on to get them where they need to go is real. The paper and pencil are real. And the physics of the informative truth that transcends from the streets to the paper is also real. Computers are the machines we've built based on the physics of logic, abstraction, and meaning. A computational value is something that means something to something else.

Constructivism is a less extreme position than finitism. It's hard to describe it, but it's got something to do with not believing that a mathematical object exists until it's been "constructed". I'm not sure if it can adequately be described without referring to formal logic. One motivation for considering this philosophy is that mathematics becomes more "computable" when you do, but there are other reasons as well.

The statement that there are infinitely many integers is true in constructive mathematics: Given any integer, you can construct some integer larger than it. Likewise, the statement that the real numbers are uncountable is also true: Given any sequence of real number, it's possible to construct a real number that's not in the sequence (using Cantor's method).

The claim that all real numbers are computable doesn't have any constructive content, so its truth depends on which version of constructivism you believe in. Personally, having looked into this, I prefer versions of constructivism in which the claim is false, because then you can claim things like every continuous function on R is locally uniformly continuous (which weirdly enough, implies that not all real numbers are computable, without asserting the stronger claim that an uncomputable one exists).