Objective Mathematics

Objective Mathematics is an ongoing project which aims to make mathematics objective, by connecting it to reality. This is being done by painstakingly going through each math concept, and thinking about the concrete, perceptual data to which it ultimately refers. If a math concept is not reducible to perceptual concretes, it is invalid; it is a floating abstraction. Objective Mathematics rejects Platonism: math concepts are not Platonic forms inhabiting an otherworldly realm, but rather they refer to physical, perceivable things. Objective Mathematics rejects Intuitionism: math is not a process of construction, but rather a process of identification. Objective Mathematics rejects Formalism and Logicism: math is not a meaningless game of symbol manipulation, but rather its statements have semantic content (just like propositions about dogs, tea, beeswax, or anything else).

Practical problems with standard mathematics

Standard mathematics is often useless.

In standard mathematics, what work is considered valuable is, to a large extent, decided by the sum of the subjective whims of mathematicians.

Standard mathematics is often boring.

In standard mathematics, there is a disconnect between proof and explanation. In practice, standard mathematicians must constantly translate back and forth between the conceptual way that they actually think, and the formalistic way that they write proofs. The geometer William Thurston wrote a nice article about this phenomenon.

In standard mathematics, it not always clear what constitutes a proof. Standard mathematicians will usually say it's a proof if it can be reduced to deductions from the ZFC axioms, but in practice, almost no one actually tries to do this. As a result, different communities in math have different standards of proof. For example, there is a disconnect between analysts who work on Fukaya stuff, and topologists who work on Fukaya stuff; there is stuff that the topologists consider proved, but the analysts consider unproved. For another example, consider the controversy over Mochizuki's supposed proof of the abc conjecture. [TODO be more specific about the disagreement]

Good things about standard mathematics

Standard mathematics has high standards of proof.

Standard mathematics does not use many undefined terms.

Standard mathematics is more conceptual than perhaps any other field.

Standard mathematics contains a lot of knowledge about the world.

Criticism of Objective Mathematics

In this section I will collect criticisms that I have heard, explain them, and refute them.

Objective Mathematics only works for basic math

Objective Mathematics is

The math concepts which Objective Mathematics has successfully connected to reality are, in the grand scheme of things, extremely basic. Most of them are math concepts that we learn about as children.

Objective

Standard mathematics is fictional, but that's okay

This position agrees with Objective Mathematics that standard mathematics is disconnected from reality, but thinks that it is not a problem. We can often learn many true things about reality from fictional novels, so why can't math be the same way? In standard math, we make up some characters, like strongly inaccessible cardinals, everywhere-continuous-but-nowhere-differentiable functions, and Fukaya categories, and then we proceed to tell a story with those characters. We can introduce new characters to the story if and when it is convenient to do so, or logic dictates that we must do so. Every once in a while, it may turn out that our story has something to do with real life, which is great! But in the meantime, we can have fun telling the story however we please, free from any constraints other than pure logic.

The problem with this point of view is that it is drawing a false analogy between novels and standard mathematics. It is of course true that novels are fictional. But the reason why novels are nonetheless able to tell us things about reality, is that they are written using real concepts. Unlike the concepts of standard mathematics, which are all ultimately defined in terms of set theory, most of the concepts of a novel can actually be reduced to perceptual concretes.

It is true that of the concepts of a novel---a slim minority of them---may be made up by the other.

(A novel could validly introduce some new concepts.

"A dragon is like a 10m long lizard that can fly and breathe fire."

vs

"A set $X$ is infinite if there exist functions $X\rightarrow X$ which are injective but not surjective."

you could really see a

with a morphism

and science fiction and fantasy novels often do

bu

---though many novels don't

Something like this view of math as fiction may be what people like Eugene Wigner have in mind, when they marvel at "the unreasonable effectiveness of mathematics in the natural sciences." Indeed, if mathematics is fictional, then the fact that it sometimes works so well in the real world, presents an unanswerable mystery.