Intro. I have kind of felt an urge to write about this theme for a while, ever since I read The Big Reset in Mathematics, a blog post by Daniel Spielman (whom I have not met in person, but whose name I had known through a research network close to my own). I also guess that any mathematician who takes responsibility for their standing as a mathematician might want to have their own version of a “Reset in Mathematics” in the AI era; at the very least, I want to be able to respond when asked.
Indeed, this theme is very important to me, and presumably to any mathematician—as is evident from the frequent national and international conferences on AI use and AI regulation over the past year, not to mention that social media has been full of such arguments.
I’m writing in English despite my nationality (Japanese), since English is, to me, the default mode for scientific writing. I also declare that I have not used AI to write this article, except for the most basic linguistic corrections, without which clarity would be lost; only by doing so can I do justice to what this particular theme is all about.
Before continuing, let me clarify my identity for the record. If one must be affiliated with an institution for pure mathematical research, or hold a doctoral degree in mathematics, in order to be called a mathematician, then the title does not apply to me: I am not a mathematician in that sense. Here, however, I dare to call myself a “mathematician”—not for the sake of a self-gratifying illusion of fame, nor because of the amount of effort I have ever put into the subject, but to clarify the philosophical mindset on which I acknowledge my thinking to be based: given the choice, it is clearly more comfortable and natural for me to speak about this as a mathematician.
This article is not meant to be complete; it only expresses my thoughts at this moment (which will change in the near future).
What is understanding? Out there, we can find hundreds of thousands of threads where people discuss why we need human mathematicians and, ultimately, why we humans need to understand math at all. We can also find a large body of literature on cognitive, social, philosophical, psychological, and even anatomical studies of understanding.
To be clear, I think we need experts who understand math, both now and in the future. This is one of the points where people who have not been mathematically trained—or, more precisely, who have never worked through a single mathematical theory (set theory, for instance, would suffice, I think)—can easily get sloppy in their arguments, and where careful treatment of primary sources (e.g., the voices of mathematicians themselves) is likely to be dismissed.
Those who express delight in understanding math, myself of course included, almost exclusively have a background in math and speak of the understanding of math, not the understanding of concepts in general. In this respect, the phrase “mathematical understanding” is something of a red herring: “mathematical understanding” and “mathematical” “understanding” are different things. Failing to make that distinction would cause confusion between defensible institutional criticism and the stronger, considerably less substantiated allegation that mathematicians are receiving a deserved punishment for their past complacency.
Mathematical understanding may be defined, in one respect, as a visualization of a concept, made observable through an appropriately gap-filling construction of that concept in the language of math, for which other mathematician(s) would cast a vote of “understandable”.
This (tentative) definition of mathematical understanding may sound circular, far narrower than one would expect, and perhaps inside. It must unarguably be demonstrated in mathematical language; harsher still, it is characterized by a blockchain-like consensus mechanism internal to the community, which turns out to be rather opaque (quite a few mathematicians may not even be aware of it) and may seem inconsistent enough to give non-mathematicians the impression that “it is insular”.
This impression, and the criticism it invites, is legitimate. The math community now faces the need to adequately explain the distinctive value of mathematical research as a human, institutional, and intellectual activity; nevertheless, I believe this is really good for us—for math and for humanity!
As for the other side of the criticism—the unjustified one—it seems to be partly rooted in a dilemma of perception inherent to one’s standpoint: one cannot know, or even imagine, what mathematical understanding is like (for humans) unless one has worked hard enough at math to reach the point of genuinely knowing what it is like for oneself.
The fact that AI (or machines in general) has survived a series of mathematical probes, in an exercise analogous to the classical Turing test, does not verify that AI understands math, simply because the definition of mathematical understanding above applies only to humans under a prescribed assumption, not to others. For AI (or machines), it would have to be defined on their own terms to make sense. Imagine a student taking a math exam with computer use permitted: this obviously changes the meaning—or at least the qualitative meaning—of the understanding demonstrated by their answers. In this sense, the property of mathematical understanding seems surprisingly local, domain-specific, relative, and diverse.
Math is a form of art through understanding. Why humanity needs mathematicians to understand math is an unsolved question—perhaps the next step of the argument—on which I attempt to explain my personal view here. I am not, however, claiming any formal protocol that attests to the value of the profession: there is none, and I would never want such a thing to be implemented in general, for the sake of human dignity. I believe it is important for both mathematics as an institution and individual mathematicians to be able to explain their contributions to others.
The latest objective analyses by a number of mathematicians who have spoken out on social media have shown, at least partially, how OpenAI’s recent promotion of its resolutions of long-standing math problems (the 722 manuscripts released in October 2026) was full of slop, and how some of its arguments fall short of mathematical standards. By comparison, what I will discuss here is very limited, but in the end, it’s all about humanity.
I trust in humanity: I believe the majority of people are careful, and not so stupid as to end up ignorantly and optimistically pursuing short-term profit, driven by a kind of tech-hype fanaticism, in a way that effectively means giving up on fostering those who want to do math, or giving up on understanding math entirely.
To casually present how I exactly see math in a meta-perspective, I may want to ask myself what art is. Without consulting an encyclopedia, I would say that it is a type of cultural practice that evokes in a person a sense of pride in being human.
When I read a book, listen to music, or think about a beautiful formulation of a tiny bit of mathematical theory, I find myself deeply appreciating the moment: I feel thankful to a life, and I feel respect for the creators and for all those predecessors who have put enormous effort into preserving these cultural activities in their present form.
What actually evokes such a sense in you will differ from person to person, but the point is that it is a concrete form of cultural inheritance that we now cherish and that enriches our lives. In this light, disrespect for mathematicians who have devoted decades of effort to training, teaching, and writing appears to be a self-destructive act, which would make no one happy.