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 explain some criticisms of my project, and refute them.

Objective Mathematics is not practical

As of 2023, 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. Engineers, physicists, computer scientists, etc. routinely use advanced math concepts that are not yet on the Objective Mathematics wiki---or even concepts that Objective Mathematics says are invalid. They use these concepts to do practical things like build bridges, learn about the Milky Way, or create LLMs. Isn't Objective Mathematics undermining their work? And since we know that advanced mathematics works, aren't Objective Mathematics criticism of it just nitpicking?

This criticism misunderstands what Objective Mathematics is trying to do. The goal of Objective Mathematics is not to diminish practical things, or to say that they aren't actually practical. My goal is to understand, philosophically, why it is that the mathematical methods employed in science are practical.

An analogy can be drawn with work of Aristotle, the "father of logic." Before Aristotle, people still used logic. But there was no systematic understanding of which arguments were actually logical and which ones weren't. One would imagine that as a result, fallacious arguments ran amok through society, and that even people who knew that there was something wrong with them would have had trouble refuting them. Far from undermining the practical men who were making logical arguments, Aristotle's systematization of philosophy helped them. Aristotle helped practical men to avoid accidentally going down illogical rabbit-holes, and he helped them to defeat their intellectual opponents.

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 some of the concepts of a novel may have been made up by the author. Unlike in standard mathematics, only a slim minority of the concepts of a novel will be fictional, since the author must spend a lot of time describing to the reader each new concept that he creates, in terms of concepts that the reader already knows. To demonstrate this point, I will consider a typical example (1) of a fictional concept in a novel, and a typical example (2) of a fictional concept in a math paper:

1. "A dragon is like a 10m long lizard that can fly and breathe fire."
2. "A set $X$ is infinite if there exist functions $X\rightarrow X$ which are injective but not surjective."

You will never see an infinite set. It is totally unlike anything you ever have seen, ever will see, or ever could see. See the discussion of Hilbert's Hotel for one of the many possible demonstrations of that fact. By contrast, a dragon is similar to many real things. The physiology of a dragon is in some ways similar to that of a pterodactyl. The way dragons are treated by the people in the novel may be similar to how elephants were treated by Hannibal. The way that fire-breathing looks is in some ways similar to how a flamethrower looks. Etc. The imaginary concepts of a novel depend on some reference points in reality, some real things which weigh them down; the imaginary concepts of mathematics do not.

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.

Objective Mathematics is anti-scientific

This position notes that there are many things Objective Mathematics says, which seem to be incompatible with quantum mechanics, relativity, or some other well-established scientific field. Doesn't that mean that Objective Mathematics is denying The Science™?

Yes, in some respects Objective Mathematics is denying the scientific consensus, or is at least prepared to deny it if necessary. However, that does not mean that Objective Mathematics is anti-science; au contraire, Objective Mathematics is radically pro-science.

I do not doubt that the concrete experiments that led to e.g. quantum mechanics actually happened, and actually produced the concrete results claimed. However, to interpret any experiment, philosophical ideas are needed. Scientists, like everyone else, tend to absorb the philosophical ideas prevalent in their culture, and will agree with them unless they spend a great deal of time and energy independently thinking about philosophy. Objective Mathematics says that many of the philosophical ideas prevalent in the culture are wrong, so naturally, it is bound to disagree with most scientists about the interpretation of their experiments.

As of 2024, the extent to which Objective Mathematics actually contradicts e.g. quantum mechanics is unknown. It certainly contradicts some elements of the formalism, like Wiener measures and complementarity, but perhaps a large fraction of the concepts of quantum mechanics can be reformulated in a way that makes their connection to reality manifest.

Objective Mathematics ignores the field of applied mathematics

This criticism correctly notes that I seem to be referring to "pure" mathematics, as opposed to "applied" mathematics, in my many criticisms of "standard mathematics." Isn't it silly to criticize pure mathematics for not being practical, when there exists an entire field of practical mathematics, called "applied mathematics"? What's wrong with applied mathematics?

A small part of my answer to this criticism is that I do plan to talk about things that are currently categorized as "applied" mathematics---things like statistics, physics, computer science and more---but I haven't gotten around to much of that yet (writing in February 2024). The main part of my answer to this criticism, however, is that I think the problems with pure math are more fundamental than the problems with applied math. That is, I think that the latter are caused by the former. The reason for this is that in large part, applied mathematicians get their fundamental ideas---their concepts, methods, proofs, and calculations---from pure mathematicians.

Compared to pure mathematics, applied fields do not suffer as much from rationalism, but they suffer more from the opposite problem of empiricism. They have a tendency to not place much value on explanations, and instead like to focus on calculations ("shut up and calculate!"). They have a tendency to focus on arguments, without thinking much about precisely what statements are being proved by their arguments. They have a tendency towards collecting tons of disorganized details and trivia, without making an effort to organize these concretes into clear and distinct concepts.

It is understandable that applied mathematicians have these tendencies, because pure mathematicians have left them out to dry. Applied mathematicians know that many of the concepts of pure mathematics are impractical, but they know of no alternative; they think that's just how math needs to be. Furthermore, coming up better concepts is very difficult, and it will not immediately lead to the practical results that applied mathematicians value. For example, if the concepts of calculus were improved, I imagine that it would still take a very long time before they would improve the quality of weather simulations.

What I like about pure math, what I think drew me to it, is that some pure mathematicians take concepts very seriously. I have many problems with their view of concepts, but in some sense it is superior to the anti-conceptual haze of applied fields. Pure mathematicians are ultimately the ones who created that haze to begin with, and Objective Mathematics aims to be the spring wind that sweeps it away.