# AI, Mathematical Ability, and the Future Calculation, proof, conjecture, and the creation of mathematical worlds: as AI accelerates solutions and changes how people enter research, where will new questions—and their creators—come from? ## Metadata - HTML: https://glenzli.com/en/notes/ai-mathematical-ability-and-the-future/ - Markdown: https://glenzli.com/en/notes/ai-mathematical-ability-and-the-future.md - Collection: Notes - Language: en - Published: 2026-09-10 - Updated: 2026-09-10 - Tags: ai, mathematics, research, judgment, future ## Content Let me put on some armor first. I have no formal training in mathematics, no endorsement from anyone important, no great work to my name. I'm certainly no authority in the field. Just a nobody. If you disagree with what follows, feel free to laugh it off. There. Disclaimer done. Recently, a dispute involving OpenAI and mathematicians over a Millennium Prize Problem set the internet buzzing. Some suspected AI of plagiarism. Some announced the arrival of AGI. Others declared mathematics dead. Plenty worried about their livelihoods. It reminded me of a conversation I had with AI about six months ago. I said I divided mathematical ability into four kinds: Lv1 through Lv4. The numbers don't rank their worth. Their rarity, though, does differ. ## Lv1 — Calculation 1 + 1 = 2 was probably most people's first mathematics lesson. Word problems about chickens and rabbits sharing a cage, equations of every kind—the things that haunt a childhood. So what is calculation? Loosely speaking, working something out in a known world, under known rules. I've never thought of it as a lesser skill. A complicated limit, an abstract series, even an innocent-looking integral or equation can tie your brain in knots. People who can calculate rapidly in their heads or find an ingenious way through an equation are often formidable mathematicians. Still, calculation operates on known objects under known rules, a structure that lends itself naturally to programming. Plenty of deterministic computation had already come a long way before AI. Calculation also lends itself to training at scale: there are rules, there is a target, and often there is an answer you can check. Fairly friendly territory for AI. Those of us made of carbon still need to calculate. But when the objects, rules, and goals are given, and the job is to execute, compute, implement, and scale things up, competing on raw computational power puts humans at a disadvantage. ## Lv2 — Proof I suspect quite a few people see proof as mathematics' last fortress, a symbol of something uniquely human. Fair enough. At school, proof questions did far more damage than calculation exercises. Didn't every final question demand that you prove something bizarre, somehow also obvious, and impossible to find the right construction for? Even after accepting the existing definitions, axioms, language, and framing of a problem, proving or disproving something is hard. There are too many possible routes through a proof, and human bandwidth is too limited. With a difficult problem, we often need an extraordinarily clever point of entry before we can make substantial progress. AI may not have to work at that pace. Thousands of agents can try, judge, verify, and change direction around the clock. Nor must every attempt be brute force. Logical reasoning, construction, and search can work together. If these capabilities keep growing, AI may compress years of human attempts into a very short time, perhaps even blast through a problem that has tangled us up for centuries. “Generations of humans couldn't finish this, even in relay” might start to sound a little funny in front of AI. ## Lv3 — Conjecture What makes proof so compelling is often far removed from a textbook exercise with a predetermined answer. At the beginning, we may not even know whether a claim is true. There is only an intuition, an insight: perhaps something like this happens here. The great conjectures draw people in, one after another, consuming decades and sometimes producing wonderful things along the way. Saying “perhaps so” doesn't automatically produce a conjecture worth studying. What distinguishes Lv3 is the ability, still within an existing frame of reference, to pose fruitful new questions: questions we can make progress on, questions that may give rise to further structure. They may turn out to be wrong. Even then, they can show us where to go next. People often develop their Lv1 and Lv2 abilities by struggling with conjectures left by earlier practitioners of Lv3. That struggle can, in turn, kindle new sparks of Lv3. This isn't an ability you can easily train into people en masse. The tension now is that AI has already begun accelerating work on these conjectures. Some problems may have much shorter lives than we expected. Who, then, will pose the next questions? Humans? If we skip large parts of the process of solving problems, where will the human ability to ask new ones grow? ## Lv4 — Creating Worlds When a whale falls, a whole community of life grows around its remains. There is something strangely fitting about that image for Lv4. Someone working at Lv4 does more than walk through an existing world. They create a new one. This isn't random world generation. A world created at Lv4 will often make problems that previously lacked a common language visible together for the first time—open to calculation, proof, and conjecture. What is created is the mathematical world itself: new definitions, objects, languages, primitives, or structures. The most awkward thing about Lv4 is that you cannot set its exam in advance. The world doesn't exist yet. You don't even know what it will look like. What predetermined criteria could fully measure the ability to create it? A better set of exam questions won't solve that problem. You can cultivate the soil; you cannot manufacture creators of worlds to specification. After a world is born, you can test whether it holds up and discuss what it makes possible. That doesn't necessarily give you a yardstick for recognizing the next Lv4. Historically, people who left new worlds behind were mostly monsters of ability. Their Lv1, Lv2, and Lv3 usually had to be strong enough too; otherwise, even a broken world would have been difficult to build. Creating a world feels wonderful. Without solid calculation and proof beneath it, though, you may have built a mirage that dissolves under the slightest scrutiny. And creation is so difficult that a single new world can sustain vast numbers of people working at Lv1 and Lv2, along with some at Lv3. Everyone works the land, making the world more complete. Its creator may never have intended to leave it unfinished. Life is simply too short; even a lifetime's work may not bring a world to completion. The fragments can feed generation after generation, perhaps even planting an idea for another Lv4. ## The AI Predicament What worries me about AI isn't how many mathematics problems remain. It's whether the path along which Lv3 and Lv4 used to grow might be cut off. Conjectures from Lv3 nourish new generations. The worlds left by Lv4 give those who follow somewhere to explore. A sufficiently powerful AI might rapidly harvest great swathes of that inheritance. But the inheritance is also the growth medium for future researchers. More answers do not automatically mean more opportunities to reach the frontier, endure uncertainty, and develop judgment of your own. If we routinely delegate the experience of battering ourselves against the old world's walls, where does the intuition for conjecture come from? Without dissatisfaction with the existing world, without things in it we cannot understand, what gives rise to the desire to create another? That is the question of how the inheritance gets divided. Turn instead to the living practitioners of Lv3 and Lv4, and another, not entirely pleasant possibility appears. Their cooperation with the mathematical community doesn't come solely from an innate desire to share. One very practical reason is that they haven't finished the proofs. The new world is too big. They cannot calculate everything, prove everything, or follow every implication to its limit. So other people enter the territory. Now give them an AI swarm with powerful Lv1 and Lv2 abilities. What do you think they will choose: more interpersonal wrangling and arguments at conferences, or quietly letting the swarm push onward until they can take it no further? Reality may stop short of the extreme. But some of the necessity that once sustained cooperation could disappear. A person and an AI might also form a kind of academic symbiosis. The person proposes a structure; AI develops it and uncovers contradictions; the results reshape the person's questions. Work that once required a group might be taken on by this combination. The entity doing the research has changed. How other people enter that research may change with it. Of course, AI could also produce fake Lv4s in bulk. Its scale is enormous: it can deliver real proofs while also churning out lavish, baseless flattery. People without enough ability to check the work can easily come to believe they have rewritten the world. If they also happen to have access to academic channels, this garbage can consume a great deal of the mathematical community's energy. Worse, while AI is still on its way to reliable, high-quality Lv1 and Lv2, humans will have to withstand a DDoS attack delivered at silicon bandwidth, in bodies made of carbon. ## The Future Vibe coding is here; even a dog can write software. AGI is here; even a dog can do research. That seems a fairly representative sample of this year's pronouncements. Learning is useless. Science students are finished. Mathematics is dead. Programmers are out of work. It's actually quite funny. Human bandwidth is limited, so we built an enormous cooperative society. Some people study problems. Others study the tools for solving them. Others study the process of solving them. Every step matters; together, they move the work forward. AI changes many of those intermediate processes and pushes past limits on bandwidth and ability that used to be beyond any one person. Replacing a process or a tool doesn't make the problem disappear. Cancer is still far from conquered. Energy remains a constraint. Interstellar travel is still out of reach. One pandemic is enough to throw human society into disarray. If an asteroid suddenly changed course and was going to hit Earth in ten years, what would we do? Wait to die? AI gives us new tools to take to the walls, to break through where we couldn't get through before. Livelihoods are a real problem, of course. Shouting “go explore” won't solve that. But old jobs being replaced and the world having no problems left are two different things. An ox has plowed the little patch outside your door. That doesn't mean there's no land left to till. Still, I have an uncomfortable answer of my own, much like the one facing Lv4. Even with AI beside you, a lifetime spent battering at those walls may yield no answer. You may not even know whether you're heading in the right direction. Every person and every AI might tell you you're right, while you need the clarity to kill the idea. Or the path you abandon might be the very last stretch before success. Everything is unknown. Humans aren't naturally all that fond of the unknown. A good evaluation within a familiar frame of reference can feel so peaceful, so satisfying. There should be more things to do in the age of AI.