2026-06-16

Abstract: This article defines math and math education, and argues for a lower bound of human effort required to learn mathematics (or any other abstraction-heavy subject) regardless of LLM capability.

LLMs are evolving rapidly; within a year AIs are able to tackle math problems - we had thought of them as the hardest for AI to automate - but they are getting there. Let it be OpenAI's marketing bluff or not (of course humans helped, how much? we can never know...), we can at least say that frontier models are useful for assisting with math research. The natural question to ask here is that, is AI helpful at math education? To what extent?

Mathematics is a truly unique subject from all other sciences, in that it is not natural at all, despite the fact that universities like to put math in a physical science building of some sort. While it originates in counting and measuring the physical world, it has evolved out of its physics context into a discipline purely focused on a-priori reasoning and abstraction during the formalism movement. My preferred way to define math is "the study of a-priori constructions". I'd also like to think of all math knowledge as an infinitely large graph of theorems where one node points to another if one can be deduced from the other by the axioms chosen. In this interesting perspective, math is much like art, poetry, or music, where every theorem already exists somewhere and we are just discovering them. An implication of this view is that, humans have to occupy a position in math research, since we are the ultimate judge to say whether an abstraction or theorem is interesting and worth developing or not. Math is tightly connected to personal and collective taste and intellect: "The product of mathematics is clarity and understanding. Not theorems, by themselves." ^[1]

As a result, calculation or theorem proving is only a small part of doing math, and the goal is rather to cultivate good instinct ^[2] - the ability to fluently navigate and manipulate some levels of abstraction, and thus "sense" how to get from one node to another or which nodes are worth exploring. People used to develop abstraction out of physical properties, such as the invention of calculus which was used to describe continuous physical phenomena. But now, the abstraction is so far removed from the physical world that it is often the case that mathematical abstractions are invented before any application is found: Riemannian geometry, a 19th-century invention, is now the language of general relativity (1915).

Math education gets hard here, since the properties of good math education, in contrast to math itself, the most rigorous of subjects, are interestingly ill-defined, heavily depending on human creativity and interpretation. I can only name properties of good math education: I learn math best when I am in the middle of the material, and suddenly I "click" and can predict what comes next. The "moment of insight" reminds me of Grant Sanderson's repeatedly emphasized "want you to feel like you could have reinvented <math topic> yourself" in his channel. It also aligns with the "generation effect" ^[3] in cognitive psychology, which states that people remember better if they generate the answer themselves instead of just reading it. In my experience, it is non-trivial to write a prompt as it is non-trivial to write a textbook which is good enough to guide students to have the "moment of insight".

Regardless of LLM capability, it still requires a non-trivial minimum human effort to learn math; since math is all about building intuition about abstractions, the old, usual, and perhaps the only way is to see and practice a lot of concrete examples, after which the motivation for building some abstraction can be understood, and after which the abstraction itself can be fully grasped. For example, the "group" abstraction requires one to see a lot of integers, reals, polynomials, modular arithmetic, matrices, and so on before knowing why we want such a thing. It's unskippable.

I was motivated to write this after reading the Daily Californian's report on UCB that soaring failing grades correlates with increasing AI usage. It is consistent with my above point that one always needs to grind through to build math skills, and also reveals the problematic side, not on LLM itself but on the problem of laziness in human. It does not imply students have gotten more lazy because of AI though, but rather that AI removes a lot of friction for laziness: people used to copy each other's homework, google an answer key, and now they can simply ask AI to solve arbitrary math problems. Since there do exist people who are genuinely willing to throw their entire lives into math, laziness may not be a human nature but rather a product of a flawed education system. The solution is beyond the scope of this essay, but it certainly won't be found by simply trying to "ban AI".