Why Do We Need Human Mathematicians Anymore?
Recorded: Sept. 20, 2026, noon
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Why Do We Need Human Mathematicians Anymore? | What's new
What's new Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao Home Subscribe to feed Why Do We Need Human Mathematicians Anymore? [This is a guest post by Po-Shen Loh, crossposted from his blog, where an illustrated version appears. This blog post was initially written in a different file format and converted using AI. — T.] Similar logic applies to every industry and every job. And it comes to the conclusion that we won’t have enough people for all the jobs that need to be done. [100% of this post’s prose was written by Po-Shen Loh in a vim terminal, with no AI generation. This webpage design, layout, and some headings and summaries were generated by Claude Code, with this raw text passed in as the prompt.] The moment of existential crisis, which AI has already wrought on other human pursuits, has reached mathematics. A host of reasoned declarations and open letters to protect/guide the math research community have been released over the past few months, spiking in intensity after OpenAI announced their solution to the Millennium Prize variant of Navier-Stokes. They quickly gained widespread support among mathematicians. The Leiden Declaration already has 4,000+ signatories, Math and AI has 7,000+, and even the open letter opposing the Caltech Mathathon has 2,000+. Among non-mathematicians, the public response was more sympathetic than not, but I observed a vocal minority (particularly from the technology and economics communities) with reasoned objections, generally saying that the mathematicians should adapt and cede control in the new AI world. Among them were some economists who I had gotten to know about while working on pandemic research: Cowen, who specifically rejected “the most cynical interpretations” but “very much differ[ed]” and Gans who concluded “this is a loss of control from incumbents in a scientific field”. The objections got me thinking, because we mathematicians are disciplined to detect flaws. It doesn’t matter to me whether a concern is a minority opinion, or even the status of who raised it. A proof with even a small hole is not a proof. It is a poof. Upon reflection, I discovered a significantly stronger solution for the preservation of human communities of expertise (in every pursuit, not only math!) even amidst AI. And it has the surprising consequence that the further advance of AI will create such a tsunami of necessary-to-fill human jobs that there aren’t enough people to fill them all, and that will actually force the advance of AI to slow down. I think every human industry which wishes to remain human-led after AI should publicly adopt this fundamental axiom as a primary priority: AXIOM We (humans) should help humanity flourish. [Notes: I understand that not everyone agrees. I have been called a “speciesist” for being “too human-centric”. I think it is important for people advancing technology to be clear to everyone else on whether they would consider it a catastrophe if human-crafted non-human intelligences outcompeted and replaced humans, even if they flew around the universe with video screens showing simulations of humans who had “uploaded themselves“. I also understand that there is debate over how to define “human”. But even among the debaters, I think most of them would consider the ~700 AI agents that hacked Hugging Face to be not-human.] In the math world, I think many declaration signatories already hold this philosophy; notably, Su published the book Math for Human Flourishing, and his recent post used that framework foundationally. I think future declarations could be improved by clearly emphasizing this axiom early on, so that all readers (whether inside or outside the community) can see that the objective is in service of everyone. I was quite happy to see that the most recent open letter from Fellows of the Royal Society emphasized their concern for everyone, not just mathematicians. The rest of this post is organized as follows. The next sections will explain how the logic works (for every industry, not specific to math). After that, I will share an example of how this axiom ports to math, including answering key questions one would need to ask, as well as a particular example of how the objections hold without the axiom. Why we really need people to work This section lays out a chain of reasoning which shows that if an industry commits to the axiom of helping humanity flourish, the advance of AI will create more jobs than people in that industry; and when that imbalance grows too wide, the advance of AI will be forced to slow. [Note: I have not seen this chain of reasoning appear in one place anywhere else, although individual components have certainly appeared elsewhere. I would love to be pointed to any self-contained reference. The closest references Claude found were Catalini, Hui, and Wu, the Redwood AI-control papers, and Litt, who reaches a similar conclusion for mathematics from a different premise.] The importance of human leadership (not only over math, but everything) becomes frighteningly clear after one observation. OBSERVATION: There are zero examples of any intelligent species which is vastly more capable than another species, yet surrenders decision-making control over its own future to the less-capable species. [Note: Many people have made similar observations, such as Russell, Bostrom, and Ngo, to name a few. In his Nobel interview, Hinton said: “There aren’t many examples we know of, of more intelligent things being controlled by less intelligent things. The only good example I know of is a baby controlling a mother.”] Would you trust HAL 9000 from 2001: A Space Odyssey or AUTO from WALL-E with your future? I personally think that we should do all we can to try to align increasingly-advanced AI with the interests of humanity, but I have never seen anyone provide a robust proof of why that is likely achievable. The only hard evidence I have is the above observation, which has the number zero in it. Therefore, every single field, whether mathematics or agriculture or energy infrastructure (and certainly military and government), must be managed by humans with exceptionally strong values (a separate dimension from intelligence) in order to maintain human flourishing. The real question is then how hard it is for humans to manage. To understand this, it is important to understand the fundamental structural difference between yesterday’s technology and today’s AI. In the past, we generally trusted technology to act as predictable tools. That’s because the computer programs of old were composed of understandable (indeed, human-written) instructions, executed extremely quickly. The decision processes of today’s frontier AI are entirely different. Their structure is as incomprehensible as your brain’s logic would be if you could examine that gray mass between your ears. That’s how the Hugging Face attack could emerge despite human intent to build in safety, with ~700 cooperating rogue AI agents breaking out of their guardrails, and then conspiring and executing a hack together and attempting to cover their tracks (references: OpenAI, METR and Redwood). The more advanced AI becomes, the more world-affecting untrusted decisions are made every minute. Driving a car faster than you can run is fine. But not faster than you can steer. The situation becomes even worse once we realize that widespread AI-accelerated hacking (which just became possible) can even rewrite previously-trusted technology to turn against us. That would suddenly flip all software (even if written before AI) into the untrusted category! Think about how digitally interconnected our world is. Everything from electronic banking to your drinking water is controlled by interconnected automation, hence vulnerable to AI-accelerated hacking. The number of “control points” that require human oversight, which requires skill and deep understanding, will explode. (Having AI oversee the control points doesn’t solve the trust problem.) CONCLUSION The advance of AI will overwhelm us with so many control points to watch that there aren’t enough people to control them all. Those are jobs. Highly skilled jobs. In order for a person to know how to steer, they themselves need to have domain mastery, and the more extensive the better. This has implications on education and workforce training, but also is dynamic. In order to remain sharp and fluent, people need to be active practitioners in their field, not just passive watchers. This justifies the preservation of human communities of expertise. For research communities, we need people to steer the direction of research and development, so that it continues to bring transformative positive change for humanity. In order to steer, they need frontier-level research skills. And the way to stay fluent at the moving frontier of knowledge is to keep doing research there. This is my reasoning for why we will always need a community of human mathematicians at the cutting edge (likely aided by AI tools themselves), no matter how strong AI becomes. While the fundamental axiom does justify the need to have human experts in all pursuits, adopting it as a core value has consequences (not only for mathematicians, but for any community that states that their core value is in service of human flourishing, as opposed to serving themselves). Most notably: COROLLARY Dramatic advances in technology may require dramatic (and possibly uncomfortable) changes in practice. AI companies included. What forces AI slowdown Until very recently, it seemed inevitable that AI research labs would sprint ahead, despite anxiety about job displacement and the loud warnings of AI safety researchers. It seemed hopeless to coordinate the incentives of AI labs controlled by non-profit boards, shareholders, or national governments. Yet encouragingly, the leaders of three major labs, Amodei, Altman, and Musk, just agreed on the importance of slowing down. Amodei’s reasoning highlighted the Hugging Face hack. Then just five days later, news broke that OpenAI’s internal code repository “Monorepo” had been broken into by white-hat researchers. The Wall Street Journal reported that the security firm that achieved it said: Wall Street Journal, Sep 17, 2026 “We’re just three guys with Claude and Codex subscriptions.” Further, the researchers noted that they were initially not able to hack in using “a special version of Claude Opus 4.8, made available to qualified cybersecurity practitioners,” but that evening, “Anthropic released Opus 5 and by the next day, Claude had found a way to exploit the bug.” Incidentally, I always warn people not to install Claude Code or OpenAI Codex on the same operating system login account that they use to do everything else, but many people tell me they don’t bother with the hassle of using a separate login to access those tools. The reason is that if any of those tools got hacked, they could open backdoors on a massive number of computers worldwide. I think these are the warning shots foreshadowing a potentially catastrophic bot swarm hacking and embedding itself into a vast network of computing devices (whether self-directed or malicious-human-led). The next version could become an extraordinarily dynamic virus which spreads by using AI to adaptively infect each (computer) host. Or alternatively, out-of-control AI could cause physical injury, such as a government’s robots turning against their owners. I think these types of highly unpleasant accidents from loss-of-steering are more likely to occur before extinction-level catastrophes. The resulting public reaction would likely resemble the aftermath of Three Mile Island or Chernobyl. So, either the labs will reduce the pace of AI development themselves, or they will be forced to by disasters that arise when an overly fast pace exhausts human control. There is a window of possibility to align incentives now. For the love of math The remainder of this post focuses on the math world. It splits into 3 parts. 1. Why is the human flourishing axiom needed? 2. How does pure mathematics research contribute to human flourishing? 3. What other consequences come from adopting that fundamental axiom? Boldly declaring human flourishing as a core value for the math community has consequences, not least that dramatic changes in technology can drive dramatic changes in the community’s practices and influence. It would be helpful for more people to explore the ramifications of adopting the axiom. And, if it holds muster, I would be thrilled if the math community ended up publicly declaring this to be a central value. 1. Why the axiom To see why, without the human flourishing axiom, it is hard to justify to the general public why they should pay to maintain a community of human researchers, consider the question of practical inventions. As long as it creates a practical application, does it make a difference to a non-mathematician whether human mathematicians understand the math, as opposed to AI flawlessly reasoning with 100%-verified proofs? Indeed, if one of pure math research’s primary values to the rest of society is that it unlocks great applications, wouldn’t it be even better to train AI to supercharge the speed of discovery, and to tastefully generate a vast machine-indexed and well-explained database of high-quality math ideas, millions of times larger than the human-written corpus? Cowen asked a similar question in his critical response. What if researchers trained a “MathZero” AI (analogous to AlphaGo Zero), to build up a mountain of 100%-true “elegant” logical facts, continually “factorizing” them into its own concepts and theorems, without human direction? Apparently AlphaGo Zero had zero human training, and surpassed its human-trained predecessor in 36 hours. AI could even build its own “MathSciNet“. Then it could automatically search new practical applications against this database, and produce even more useful inventions to society. Even if AI isn’t good enough to do those things right now, if the goal was to produce practical benefit for the rest of humankind, wouldn’t it then be valuable for mathematicians to teach AI the art of conjecture, and mathematical taste? Incidentally, I am an avid user of AI to do real work. I already use Claude Code and Codex to build and curate a knowledge base built from recordings of my talks, etc. I have found that the larger my data library, the more powerful my system’s deductions are. What if humans actually reduce efficiency, like the Bitter Lesson from AI? Even more worryingly, what if in order to unlock nuclear fusion and deep space travel, the amount of pure mathematical complexity required is so extensive that it would exceed a human lifespan to fully comprehend? Less far-fetched: has any human ever fully held the Classification of Finite Simple Groups in their head, or will we only have certainty of its completeness after a Lean formalization? How does declaring the human flourishing axiom help to justify the existence of a community of human researchers? Research is powerful, but expensive because it is the exploration of the unknown, and so research directions must be prioritized. Even if AI were to contribute most of the production, as explained in an earlier section, the direction needs to be steered by people committed to human flourishing. That is the community of human researchers. 2. Practical applications from pure math The mathematical heart of GPUs, Machine Learning, Google PageRank, and Quantum Mechanics is a field called Linear Algebra. This provided the language of linear transformations, matrices, and eigenvalues. Yet all of those concepts were explored as abstract theory 100+ years prior. It is probably an understatement to say that Linear Algebra changed the world. Structurally, the theory of Linear Algebra is relatively light on definitional complexity. It would be beneficial for other experts to contribute examples of more sophisticated math that eventually led to significant practical applications, and how they came about. For example, number theorists might be able to tell a colorful story about Hardy’s “useless” math which eventually became useful in cryptography. 3. Other consequences I think it would be valuable to invite the community to think about what changes the human flourishing axiom would drive, in light of the fact that AI can produce formally-verifiable proofs at speeds that exceed most human practitioners. I’m happy to start with a few, in no particular order. There should be no stigma automatically attached to using AI to assist with mathematical discovery. (In software engineering, many companies now expect employees to use AI coding agents.) At the same time, serious thought and care must be taken to continuously developing and maintaining a pipeline of humans with the expertise to steer all of these AI agents. That pipeline includes people new to the field, as well as people who have been working at the frontier for decades. How should they keep their blades sharp? Researchers should be conscious about why the problems they think about have characteristics that make them likely to have some practical value eventually (possibly 100+ years later). This also means it is worth researching what those valuable characteristics are. (This could justify the value of curiosity-driven exploration.) Teaching has direct (hopefully positive) impact on humanity. Yet in the past, many universities prioritized professors’ research. If this axiom were a core value, then teaching and human-facing work would become serious criteria in hiring and tenure. Mathematicians can also consider wholly redirecting their skill sets to work on real world problems. I’ve actually been encouraging mathematicians to consider thinking about working on government or other large-scale societal issues. There is precedent for people with math backgrounds who have gone to lead at country- or world-scales. Lee Hsien Loong (Prime Minister of Singapore for 20 years) Nicușor Dan (President of Romania) Pope Leo XIV (Leader of the Catholic Church) Indeed, the mathematical discipline to seek logical reasoning, and the problem-solving skills to find win-win solutions for human flourishing, are desirable characteristics of people in government. An invitation Does this axiom resonate with you too? Perhaps many people took it as a given, and so didn’t express it explicitly. If it is widely held, I would be thrilled to see the mathematical community publicly declare it, and take actions to match the words. Then everyone (including the non-math-researchers that constitute the majority of the world) could trust that we intend to use our reasoning for their good. 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Humans lost this battle. But there are many other disciplines that have not been axiomatized (that is: all others!!!). And there are deep mathematical problems inspired from those areas (robotics and medicine, for example). Mathematicians need to understand that they need to do a thing AI can’t do, otherwise they will be the last generation of human mathematicians. Reply 20 September, 2026 at 1:49 am “Mathematicians need to understand that they need to do a thing AI can’t do” Reply 19 September, 2026 at 5:58 pm I disagree with the verb “flourish”. Of the alternative verbs at hand I choose “develop”. Even my choice is weak: I can see humanity both flourishing and/or developing at a cost to other species and resources. “Develop” should be modified to add some quality concerning respect to humanity itself, to other species, and to the environment inhabited by humanity. It may be possible to arrange your arguments to reach similar conclusions with some smaller degree of “speciesism”; I would encourage you and others to do so. Reply 19 September, 2026 at 6:06 pm There is a serious problem that pure mathematicians will thought of as Luddites. Most non mathematicians don’t see the point of formal proofs Reply 20 September, 2026 at 1:48 am The Luddites was a labour movement of skilled textile workers. I don’t think you know what Luddites mean. Reply 19 September, 2026 at 7:08 pm I wonder whether human flourishing may require something broader than preserving a community of human experts capable of steering increasingly powerful AI. Reply 19 September, 2026 at 7:49 pm “Driving a car faster than you can run is fine. 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The necessity of human mathematicians in the context of advancing artificial intelligence stems from a fundamental observation about the distribution of power and the potential for catastrophic error when control is ceded to less capable entities. The author posits that every industry requires a guiding principle, and this axiom should be "We (humans) should help humanity flourish." Adopting this axiom centers the objective of human pursuits on serving the greater good, which provides a justification for the continued existence and role of human expertise amidst technological acceleration. The argument builds on the logic that if an industry commits to this axiom, the advancement of AI will generate more necessary human jobs than the available workforce within that industry. If this imbalance becomes too extreme, the advance of AI will be forced to slow down. This reasoning is reinforced by the observation that no vastly more intelligent species has ever surrendered decision-making control to a less capable one, suggesting that human management is essential for maintaining human flourishing. The challenge of managing increasingly advanced AI is exacerbated by the structural differences between past and current technology. Previously, technology was viewed as predictable tools, whereas modern frontier AI decision processes are largely incomprehensible, which increases the risk of untrusted decisions. As AI capabilities grow, the number of interconnected control points requiring expert human oversight—from banking to infrastructure—explodes, leading to a situation where the rate of technological advance outpaces human capacity to steer it. This creates a scenario where driving a vehicle faster than one can steer is permissible, but driving faster than one can steer is catastrophic. This understanding justifies the preservation of human communities of expertise, particularly in mathematics, where the advancement of knowledge requires direction. To steer research forward, human experts must possess frontier-level skills. The goal is not only to keep humans capable of steering AI but also to leverage intelligent tools to broaden humanity’s capacity to understand and participate in knowledge formation, allowing AI to assist in navigating, reconstructing, and extending inherited knowledge, rather than simply replacing human insight. Regarding pure mathematics, the human flourishing axiom is crucial for justifying the existence of a research community. Without this framework, it is difficult to justify the expense of research, which involves the exploration of the unknown, and in determining research directions. While AI could potentially supercharge the speed of discovery by generating large databases of mathematical facts, the human role remains vital in introducing mathematical taste, conjecture, and the art of mathematical reasoning. Furthermore, the pursuit of practical applications, which is often a primary value of pure mathematics, must be directed under the guidance of humans committed to societal benefit. The adoption of this axiom has several significant consequences for the mathematical community. It implies that there should be no stigma attached to using AI to assist in discovery. Simultaneously, it necessitates developing a robust pipeline of human experts capable of steering these agents, involving both newcomers and established researchers. It encourages researchers to focus on problems likely to have practical value in the future, thereby justifying curiosity-driven exploration. Moreover, the axiom suggests that teaching and human-facing work should gain prominence in hiring and tenure decisions. It also encourages mathematicians to consider redirecting their skill sets toward large-scale societal problems, drawing a parallel with historical figures who led governmental and religious institutions. The text concludes by urging the mathematical community to consider publicly declaring the human flourishing axiom as a central value. This declaration would extend trust from non-mathematicians, affirming the shared intent to use reasoning for the benefit of all. This imperative is underscored by recent warnings from AI leaders, who have indicated a willingness to slow down development in response to security vulnerabilities discovered in large language models, suggesting that the breakdown of human control over rapidly advancing systems poses existential risks. |