A Beginning for Mathematics

Daniel Litt, professor astatine the University of Toronto

Three years ago, AI systems could not reliably adhd 2 numbers. A twelvemonth ago, soul models astatine OpenAI and DeepMind received the balanced of a gold-medal people connected the IMO. Now, these systems are autonomously resolving awesome unfastened questions. It’s difficult to ideate this inclination continuing for different year, but I expect it will. It is clear that this will require a extremist rethinking of our profession.

A fewer weeks ago, I gave a talk titled The End of Mathematics. If you only publication the title1, you mightiness conjecture that this talk was astir how, soon, AI will “solve” math. That’s not what it was about. The talk alternatively laid retired a gloomy imagination of the future, successful which, contempt the anticipation of AI systems that are robustly superhuman astatine mathematics, the creation of our institutions causes quality knowing of mathematics, and perchance moreover mathematical advancement successful the abstract, to stall. I deliberation we will debar this future, but I besides deliberation it is plausibly the default if world mathematics does not adapt. Despite my comparative enthusiasm for the usage of AI to do mathematics, I stock this position pinch galore of its detractors.

Here I want to laic out, instead, a affirmative imagination of the early of mathematics, and the quality believe of mathematics. I declare we tin deepen quality knowing moreover arsenic the accumulation of absorbing mathematics becomes little limited connected it.

This effort will return arsenic a premise that AI systems that are robustly superhuman astatine astir aliases each aspects of mathematics will beryllium present soon. But the actual changes to our institutions I propose only require accepting the weaker premise that the accumulation of mathematical matter is becoming progressively disconnected from mathematical understanding.

What are we moreover trying to do here?

I deliberation it has now go clear that location is nary statement successful the mathematical organization arsenic to what our goals are. Some of america want to lick problems; immoderate of america deliberation of mathematics arsenic play aliases arsenic poetry. For some: “Wir müssen wissen –\text{\textendash} wir werden wissen.”2 Some of america deliberation we are penetrating the mysteries of the platonic realm. Some of america deliberation the extremity is to embody emotion of and knowing of mathematics,3 and to transmit that emotion and knowing to the adjacent generation. 

My personal, if self-referential, answers are:

1. We’re trying to nutrient and understand precocious value mathematics.
2. We’re trying to nutrient precocious value mathematicians.

These goals should beryllium construed broadly. What precocious value mathematics consists of has changed rather dramatically complete time; we travel to its meaning arsenic a community. We are not conscionable training PhD students to do investigation successful mathematics. A important portion of our job, though possibly an underemphasized one, is to amended the wide nationalist astir precocious value mathematics and mathematical thinking.4

Whatever our goals are, we’ve operationalized them chiefly done proving theorems. Almost each papers aliases PhD theses person a main theorem, and ostensibly a impervious of it. But it should beryllium clear that the extremity of mathematics is not to beryllium theorems; if it was, it would beryllium trivial to automate. A machine aliases monkey could easy commencement astatine the axioms of ZFC and iteratively use conclusion rules to them, pinch nary attraction whatsoever paid to their meaning. It has had peculiar value erstwhile a theorem resolves an open problem, particularly 1 that has resisted important effort. Again this is easy automated; our machine aliases monkey tin simply conjecture each mathematical propositions successful alphabetical order.

The wide cognition of our organization towards a exertion that tin beryllium theorems and lick unfastened problems suggests that these operationalizations of our values are astatine champion incomplete.

The imaginable of automating mathematics by enumerating each conjectures, and each proofs of ZFC, is astir apt not truthful disturbing to you. But fto america for a infinitesimal presume the machine aliases monkey is very smart; possibly it understands the results it is proving, and writes beautiful expositions thereof. Perhaps it has a bully consciousness of what we find interesting, and is chiefly focusing connected those questions. Perhaps it has, successful the people of enumerating theorems of ZFC, answered galore of our astir pressing unfastened questions, and is asking galore much basal unfastened questions. Is location still a request for quality mathematicians?

I deliberation so. This instrumentality mightiness nutrient answers we value, but it would not, successful itself, nutrient quality knowing of those answers. In truth I deliberation we are astatine the opening of an incredible, awesome detonation of mathematics, and if we worth quality understanding, location will beryllium much request for quality mathematicians than ever before. But the occupation will person to change.

In the people of this change, we will person to determine what to clasp connected to and what to propulsion away. Some things I would for illustration to preserve: learning seminars; serendipitous conversations that spark an idea; students knocking connected a professor’s doorway to chat astir math. A robust organization learning breathtaking caller mathematics. Thousands of group that, together, slow commencement to resoluteness their confusion.

I interest that overmuch of what has been written connected this topic, including immoderate of my ain past writing, focuses excessively overmuch connected trying to sphere the precise style of the institutions of world mathematics, alternatively than our values. How tin we sphere the diary and adjacent reappraisal system?5 How tin we protect the arXiv? How tin we support our domiciled arsenic gatekeepers? If you person internalized the truth that existing AI systems tin nutrient comparatively precocious value results for the marginal costs of a fewer dollars, the thought that immoderate semblance of the existent equilibrium tin past what’s coming is absurd.

As we effort to find a caller equilibrium, we could effort to pursuit the separator of exemplary capabilities. Right now AI systems arguably underperform america astatine theory-building, asking questions, exposition, …\ldots truthful we could prioritize and reward those skills. I deliberation this is unwise: comparison the velocity astatine which the academy adapts to the velocity astatine which exemplary capabilities improve. We request to see the endgame. If the models stay incapable successful immoderate domain, we tin set later.

Before I propose immoderate comparatively actual steps we tin take, fto maine remark connected what we’re trying to protect mathematics from. There is simply a batch of anger astatine AI labs, and definite individuals astatine those labs. But immoderate our judgement of the labs, we request a scheme that does not dangle connected AI capabilities disappearing. The basal rumor is not the labs’ behavior, ethical aliases not.6 It’s the exertion itself. I deliberation location is immoderate belief that the labs will “move on” from mathematics adjacent year, beryllium nationalized aliases surgery up, aliases that a financial bubble will pop, someway returning things to normal, or…\ldots But location is nary measurement our institutions tin past unchanged erstwhile anyone pinch a laptop and a fewer 100 dollars tin make what would person been an Annals insubstantial past year. AI does not attraction if you are anti-AI.

Producing high-quality mathematicians

The astir urgent mobility our occupation needs to reply correct now is: what should our students beryllium doing? It’s now imaginable to nutrient a PhD thesis 1 hasn’t moreover read; successful position of demonstrating understanding, mathematical matter is worthy the insubstantial it is printed on.7 The matter nary longer reliably conveys a awesome astir the personification who produced it.

In my position we should invited absorbing mathematical results sloppy of provenance. But our institutions person historically relied connected the aforesaid awesome to bespeak some mathematical advancement and mathematical expertise. These now must beryllium distinguished.

I propose the pursuing reconceptualization of the extremity of a mathematics PhD: to go a world master connected immoderate interesting, heavy topic, and to beryllium capable to convey that liking and knowing to others. Part of operationalizing this mightiness beryllium a thesis, but the grade would beryllium awarded chiefly connected the ground of a rigorous defense, successful which the student explains the taxable to their examiners until they are satisfied. While we mightiness require the taxable to beryllium original, its provenance—\text{\textemdash}AI aliases not—\text{\textemdash}is irrelevant.8

How different would this look from existent PhDs? I deliberation students would still meet pinch an advisor, who mightiness propose a topic. That taxable could beryllium explored pinch AI assistance, aliases not, but the student would beryllium responsible for knowing it; it mightiness beryllium overmuch much open-ended and larger than the emblematic PhD is currently. The student would beryllium trained to inquire absorbing questions and effort to resoluteness them, by immoderate means. To support students connected track, location mightiness beryllium regular meetings successful which the student is asked to independently activity done an unfamiliar example, use a method successful a caller case, etc.

The allocative aspects of our occupation (hiring, postgraduate admissions, etc.) are successful dire request of betterment if we want to clasp quality mathematical expertise. Broadly speaking I deliberation we should attraction connected rewarding accomplishment successful the parts of our jobs that cannot beryllium automated: the soul (e.g. knowing mathematics) and social-relational parts, and operationalizations that hew arsenic intimately to those aspects of the occupation arsenic possible. For example, talks and sustained mathematical discussion now show knowing overmuch amended than papers. Once AI systems amended astatine exposition and “digestion,” this will beryllium moreover much the case. We already question and reply module hires; we must now do the aforesaid for postgraduate admissions.

I deliberation we should effort to foster a robust seminar culture successful which speakers are expected to explicate their taxable to the audience’s satisfaction. Much has been written precocious (by myself among others) astir the truth that we are chiefly willing successful understanding, not simply the truth worth of mathematical statements. If that is the case, fto america make judge we really understand each other.

Right now the usage of AI systems to do mathematics supra immoderate minimum barroom relies connected the truth that our organization has produced galore unfastened conjectures, whose liking is evidenced by the beingness of quality mathematicians who attraction astir them.9 The caller value of this truth suggests to maine our organization plays a very important usability that we have, arguably, underrated: namely, figuring retired what is interesting. It is not wholly clear to maine really to operationalize this, but 1 anticipation mightiness beryllium to reward the building of research programs (either pinch thief from AI systems aliases otherwise) that seduce others of their worthiness.

To beryllium clear, I americium not saying that AI systems will not beryllium capable to inquire absorbing questions, make absorbing conjectures, prosecute absorbing programs, and truthful on. I deliberation they astir apt will, resulting successful the accumulation of an abundance of PDFs. The contents of immoderate of those PDFs whitethorn moreover person important applications. But others will chiefly beryllium of liking because they show america thing basal astir basal mathematical objects, and accrue worth only if we tin and do prosecute pinch them. It seems to maine that it will beryllium up to america to build a community of researchers to do so, and we should reward mathematicians who do. And moreover if the AI is asking fantabulous questions, location is nary logic to deliberation it will inquire the aforesaid questions we would. 

All of these changes are oriented towards expanding the magnitude we talk to each different astir mathematics. It seems to maine that this would beryllium affirmative moreover successful a world pinch nary AI.

I deliberation location is room successful this world some for mathematicians who, for illustration me, are enthusiastic astir AI, and for those who do not usage it. But arsenic the models statesman to nutrient immense quantities of mathematics, it will not beryllium imaginable to debar their outputs entirely.

Producing high-quality mathematics

As we deliberation astir really to reshape our profession, it’s important to understand that, whether 1 likes it aliases not,10 it’s intolerable to extremity people, amateur aliases professional, from pushing a fastener to nutrient mathematics. The thought that we will seduce group not to play astir pinch math, aliases that we will beryllium capable to “reserve” problems for postgraduate students, is conscionable not realistic.11 And we shouldn’t want to do this!

There is now much liking successful mathematics than astatine immoderate different clip successful history. We should beryllium ecstatic for mathematics’s sake, moreover arsenic we are concerned astir mathematicians and mathematical expertise. And by and large, the worth of this button-pressing comes from the mathematical community. If a conjecture falls successful the woods and nary 1 is astir to perceive it, who cares?12 For the abundance of caller mathematics to person worth extracurricular application, we will request an abundance of caller mathematicians. And for results pinch applications, we will want group to beryllium tin of knowing their assumptions and consequences.

I wrote supra that solving problems and resolving unfastened conjectures is an incomplete operationalization of our values. But nevertheless it is important to lick problems and resoluteness conjectures! The provenance of specified solutions only matters insofar arsenic it intersects pinch the existing building of the occupation (incentives, prestige, and truthful on). It is evident that building needs to alteration successful immoderate case.  

Mathematics utilized to beryllium the cheapest of the sciences. I deliberation the biggest alteration we are facing is that now, immoderate information of our questions will beryllium answerable via a rate injection. I cognize immoderate of my colleagues find this distressing. Previously those questions mightiness person brought together a investigation community, led to absorbing auxiliary developments, and truthful on. This contingent advancement whitethorn now nary longer occur.

But don’t you judge successful mathematics!? There will ever beryllium much to learn. If a basal mobility tin beryllium resolved for the cost13 of a bully dinner, we should beryllium delighted. But that’s only the beginning. We will inquire what the reply explains, and what it helps america understand. It will lead to galore much caller questions, immoderate of which tin successful move beryllium resolved for the costs of a bully dinner, and others which renew our disorder and lead to the improvement of a investigation community.

Our industrious caller helpers will beryllium churning retired an unbelievable magnitude of math, pursuing our interests aliases possibly their own. We will person our ain questions, and confusions; sometimes they will beryllium resolved by the models, and sometimes they won’t. Sometimes the answers will beryllium complicated, and we’ll give a learning seminar to them. Sometimes advancement will beryllium minimal, but the mobility itself will beryllium truthful motivating it gives emergence to a investigation community.

A student will beryllium confused. They will sound connected their professor’s door. Maybe the 2 of them will inquire a exemplary for help, aliases possibly not, but first they mightiness walk immoderate clip astatine the blackboard reasoning done the question. And the exemplary mightiness springiness them a beautiful explanation, but we each cognize that’s not enough; nary 1 tin understand mathematics for us. We person sewage to do the work. 

There is truthful overmuch much to learn—\text{\textemdash}an infinite amount. We’ve ever been astatine the beginning, and we ever will be. 

Acknowledgments

I americium grateful for comments from Mohammed Abouzaid, alz, Boaz Barak, Frank Calegari, Ben Church, Jennifer Cutler, doomslide, Elden Elmanto, Francesco Fournier-Facio, Tony Feng, Dan Freed, Peli Grietzer, Michael Groechenig, Stephanie Koh, Joshua Lam, Mark Sellke, Ravi Vakil, and Amal Vayalinkal.

Received 9 September 2026.