Grant Sanderson: There's never been a better time to go into math

The math world, Grant says, is on fire talking about AI. He has been thinking about it for a long time, and after he asked for questions last week, a lot of what came back was from students—PhDs and undergrads who had planned to go into math or nearby theory—asking what they are supposed to do next.

There is fear that math is dead, or solved, or that whatever it becomes will be unpleasant. Professors have been throwing up their hands too: what are they supposed to tell their students? One comment felt emblematic of the rest.

“A math PhD here, just tell me what to do tomorrow.” Grant wants to make the case—and he wants you to see if you agree—that there has never been a better time to go into math, or to fund it. The most radical change, he thinks, will be how we assign academic credit. The institution itself stays surprisingly intact, even in a hypothetical where machines can do anything mathematicians currently do, but better.

His framework is simple enough to reuse on any career. List what mathematicians actually do. List the ways math is valuable to the world. Then look at where the money comes from, and how those last two relate. Once AI is in the picture, you ask: suppose machines are better at each of these tasks, or all of them. What happens?

He will not claim to predict the future; the error bars on this video are wide. He also has no skin in the game for stuffing more people into academic math. If anything, he has long thought too many students pursue PhDs for the wrong reasons: school is comfortable if you have always been good at it, and a doctorate can feel like more of the same.

The better question is how you want to add value to other people, and how to use and develop your skills to do that. Academic research might be the answer. It might not. A lot of people leaving academia right now are doing it because they think the highest-leverage work is AI safety. If that is you, he says, great—run.

The people he wants to talk to are the ones considering leaving out of fear: fear that the career will not exist, or that they will not enjoy whatever is left. He believes it will continue, that it stands to be more useful than it has ever been, and that it can still be joyful.

What goes into math research? The obvious item is proving theorems. That is where a lot of people spend their time, and it is the source of the old joke that a mathematician is a machine for turning coffee into theorems. Everyone has known for a while that this is not everything, and that gap is now the center of the discussion.

A nice line on the subject: good mathematicians prove theorems, great mathematicians make conjectures, and the greatest write definitions. So alongside proving you have taste—deciding what is even worth asking, and sensing when an idea might be interesting enough to produce good math. Instead of “write definitions,” Grant adds a third category: theory building.

You can have a pile of proved theorems, or a sense of the right questions, and still want something broader: an understanding of what is really going on. When Descartes introduced Cartesian coordinates, it was not so much one question being answered as an entire framework for thinking.

When Galois planted the seeds of group theory, the significance had very little to do with the unsolvability of the quintic. It was a new way of conceptualizing the situation, and it produced a flowering of math that stayed useful for centuries. Grothendieck is another example.

One of Grothendieck's metaphors: if you want to get from the top of one mountain to another, you can solve every problem on the way down and up, or you can raise a sea between the peaks and sail across. That rising sea is theory—not one particular theorem, but a deeper overall understanding.

There is also applications: figuring out how the abstractions plug into science and engineering. And running through all of this is communication. It is not enough to vomit out a proof. You need a motivated exposition. In theory building it is not enough to list definitions; the great mathematicians convey why these are the structures worth engaging with, and they pass along some of their taste.

Where is the value? The obvious bucket is applications in science and engineering. In pure math, everyone writes in their grants that those applications can be decades or centuries downstream. A classic: Italians in the 1500s trying to solve cubics, with one weird trick that takes the square root of a negative seriously.

Propagate that theory a few centuries and you have imaginary numbers, which sit inside quantum mechanics, electrical engineering, waves, pretty much everything. Questions about Euclid's fifth postulate lead to non-Euclidean geometry, and then to general relativity. The pursuit of Fermat's last theorem builds algebraic geometry and elliptic curves, which then turn out to be useful in cryptography. People love to list these.

A second kind of value is individual, not societal. Training as a mathematician leaves you with problem-solving skills that look highly transferable. The economy is full of people in computer science, machine learning, AI safety, finance, cryptography, who had a lot of math—maybe a few classes, maybe a PhD they did not stay in. The way of finding the right abstractions seems to make people more valuable elsewhere.

The third category is often underappreciated, and Grant thinks it will be worth appreciating more: something aesthetic. There is joy simply in doing math. He knows a specific emotion, a clicking in the head, when a good problem shows you how it will be solved, or when a new theory fits together and gives you a completely different way of thinking.

A lot of math has that feeling of reshaping thought. Anyone who has taken linear algebra starts speaking a little differently for the rest of their life; they call ideas orthogonal. People who go deep on category theory sometimes feel as if their brain has been turned upside down, in a satisfying way. Ask mathematicians why they went into the field, and for many of them this is actually the reason: an artistic pleasure.

That also helps explain why the reaction to AI in math so strongly echoes the reaction of artists to AI in art. It feels like an infringement on an art form. But if we are talking about a career, not a hobby, none of this matters without funding.

Where does the money come from? A big piece is teaching: mathematicians as professors, in departments that often teach far more people than just the majors, with an outsized claim on credit hours and tuition. Then public funding—in the U.S., NSF grants—private institutions such as the Simons Foundation, and sometimes industry.

Of those, teaching is the largest source, which is a little peculiar to math compared with other sciences: the research is comparatively cheap, and the flow of students through the department is broad. Before asking what happens if AI beats humans at pieces of the research stack, it helps to be clear how value and funding relate.

Problem-solving chops and aesthetic joy are individual goods. They are experienced by the person doing the mathematics. Public funding is not there to underwrite a prestigious hobby for an inner circle. A better picture is that those two goods flow through the money chain via teaching.

Students walking the halls of math departments are largely there to develop technical problem-solving they hope will be useful elsewhere. Some are also there for the aesthetic joy. Grant was. It was comforting to know the skill was generally useful, but an honest account of a lot of the value he was getting is on the aesthetic side.

The benefit to society at large comes from downstream applications. There is more shared benefit than that, though. Math popularization—this channel, Numberphile, talks—tries to share the aesthetic joy more broadly. And even though everyone writes grants around applications, Grant thinks an even larger positive externality is the institution training people with a certain kind of problem-solving ability.

It attracts them from around the world to prestigious universities. Most of them do not become mathematicians. They spill into other useful parts of the economy with those transferable skills. That component alone, he thinks, is a meaningful return on dollars pointed at the institution—at least insofar as it genuinely trains good problem-solving, and those skills are genuinely transferable.

Now the meat of it: what happens if AI gets good at various parts of the research stack? As he steps into the first block—machines better than mathematicians at proving theorems—he wants to summarize what just happened. A quiet project in the background of 3Blue1Brown, since about mid-2025, has been recording interviews with mathematicians about how they feel.

One broad conclusion: at any given snapshot, the position is overrated, but the velocity is not. In 2023 a friend texted him asking whether DeepMind had just crushed Olympiad-level geometry. There was a model that could answer IMO-level geometry questions. At the time the honest answer was: yes, sort of, but not in the spirit of what we mean by solving those problems. It felt bespoke and brute-force. The problems are supposed to require some creativity.

Fast-forward a couple of years and the natural-language models—not even the bespoke ones—could solve IMO-level problems by any reasonable definition of solve. Earlier this year, proofs started appearing for Erdős problems: questions posed by Paul Erdős, who had a good taste for what makes a conjecture, enumerated in a list that model companies naturally enjoyed as a benchmark.

In the early days a lot of people said look closely: these are low-hanging fruit, problems nobody had tried, or problems already solved and poorly documented, with the model repackaging the answer and skipping credit. By now, nobody is saying that. These are serious problems mathematicians had worked hard on, and the LLMs are outputting proofs.

That is true even if you set aside the headline results from OpenAI—the Navier–Stokes work, the hundreds of problems they dropped. Talk to mathematicians on the ground and, for some of them at least, you hear that they posed a serious research question to a reasoning model and got something they found insightful. It varies a lot by field and by person. The frontier is spiky at a fractal level, and not at all uniform.

There is also immense controversy around how some of these proofs are presented. Grant does not want to downplay that. The ten-thousand-foot view is still: four years ago these things could not even add numbers. They reached contest math faster than anyone anticipated, then jumped to anything relevant to research faster than a lot of people anticipated.

Even if it is an exaggeration to say theorem proving is solved; even if most of the proofs we are seeing are short hops from recent human work; even if “solved” is not well defined—given that pace, for the sake of a first-year PhD trying to look ahead, let us just presume machines are categorically better at generating proofs. What happens?

In short: almost no influence on the value stack, and therefore on the funding stack, and a massive influence on the culture and credit-assignment system of math. Applications almost always stem from theory building. It is not the proof that the quintic is unsolvable that matters; it is the creation of group theory, which later has effects like people postulating that quarks might exist. It is not the yes-or-no of Fermat; it is the math that sprang from it.

Sometimes the statement of a theorem has utility on its own. If you are pursuing nuclear fusion and you want a magnetic field that is non-zero everywhere, the hairy ball theorem tells you that you cannot do it with spherical topology. That always feels like a stretch to Grant. By and large the applications come from theory. Downstream, public funding is still rational. If anything, mathematicians are now more leveraged to do their theory building. The same for private funding.

Building problem-solving muscles is essentially unaffected by whether machines are better. Plenty of humans are already better than you at generating proofs, and it is still helpful to create them yourself. The analogy is the gym: machines can lift heavier weights, and that does not negate the value of you lifting. As AI takes more of these abilities, this part of the value stack always requires that the human be engaged.

In the sci-fi futures implied by AI taking more and more of the stack, do we even need problem-solving abilities in humans? He postpones that. As long as there is value to it, it flows through funding via teaching. It is still valuable to have human experts who can generate proofs so they can train students, so you get those spillover benefits.

The aesthetic-joy side is more nuanced, because he has seen both reactions. Online especially, a common sentiment is that machine theorem-proving sucks the joy out of it. A mathematician pushing toward a particular result, only to see it among the hundreds OpenAI dumps, will say that takes the wind out of their sails. He gets that, and it is important to talk about.

In person he has also seen the other side: mathematicians who are exhilarated. They have questions they always wanted the answer to, and now an interaction with models is showing them those answers. When they care about the theory and not just the proof, it can feel like they have been unlocked. He shares a sample of a conversation with Constantin Kogler at the Institute for Advanced Study.

Kogler had been posing one of the most important problems in his field to whatever state-of-the-art model he could get. He describes two very serious successes, including, that Sunday, what he calls the major open problem in the field solved, and another with irreducible polynomials. It was marvelous, he says, and simple: a simple, powerful idea he had already seen on Thursday.

First he felt incredible excitement—about the math, about the power, about the truth he would learn, about the ideas. Then he thought about the math, slept less, and thought about the excitement, because he had the insights he had been craving. Grant notes the dichotomy: more negativity online, more joy in person. That is not unique to AI and math. It is the nature of discourse. He does not want to downplay the negative feelings.

It does have an implication for the PhD student asking what to do tomorrow. He promises a concrete answer at the end. The first piece of it is: be social. Close the laptop. Go on a long walk with your fellow students. As for the topics of that walk, he says, we will get there.

He is spending a lot of time on this one item because it is where most of the present discourse sits. What does the practice of math look like if machines are better at proof generation? Take the optimist's view, the one who is exhilarated, and it looks like leaning into theory building. You are no longer rate-limited by how quickly you can solve the problems along the way. You can ask all the questions you want and see the solutions much faster.

Then you parse them: how does this fit a bigger narrative, where is the simplicity among the proofs, how do you weave a new kind of theory, the way Galois introduced group theory or Grothendieck developed algebraic geometry. Maybe your PhD thesis is not a handful of proofs. It is that entire theory. An analogue: going through a textbook so much faster because there is an answer key at the back.

That is not a license to always look. You should still do the problems. But if you are stuck, a more coherent presentation of the answers helps you get unstuck, and you move faster than you would on total terra nova. You do have to be thoughtful. A lot of machine-generated proofs use ideas that originated in human papers, and attribution is not always automatic. That extra step—giving credit where it is due—is on you.

Acknowledging the real loss of joy, Grant suspects a lot of mathematicians would look at what he just described and say it sounds awful: sitting there parsing machine-generated slop, understanding questions somebody else has already answered. They wanted to find the solutions themselves. Addressing that feeling, he thinks, is one of the more important questions for the community right now.

He suspects some of it refers to two different things. One is that the explanatory quality of the models often leaves a lot to be desired, and nobody wants to parse slop. He suspects that will get better—a temporary blip when you have proof generation that is almost incomprehensible. The model companies have a huge incentive for their models to explain well. Look at code: early generated code, even when it ran, was sloppy, and the trajectory has been toward cleanliness, even if people argue about the over-commenting.

The deeper issue is cultural inertia around what the community values and where credit has been assigned. It sucks if you were pursuing a problem and a model finds the answer—especially if it comes from the company, not from your own interaction—and then the problem feels solved and you are sweeping up after it. For so much of the history of math, we have used proof as the proxy for understanding. It is the north star. To prove something is to understand it. We treat the moment a proof is created as the moment a grand open problem is solved.

Of course that has never been everything, and everyone has known it. Submit a poorly written paper and the journal sends it back. Prove a very important theorem badly and they are frustrated, because they know they should care, but they still send it back. The onus is on you to explain it well.

At this point, Grant thinks, we should consider proof the same as numerical calculation: a form of raw data that is helpful for theory building, not an end in itself. That is not only his thought; other people have framed machine-generated proofs as something to be treated more like data. There is something liberating in taking theory building as the goal, and proof as raw material you plug into it.

He does not care about a forced blow-up in the Navier–Stokes problems. What he would care about is if somebody understands the ideas in that proof, and the mathematics that can flower from them into a coherent theory. If someone puts in that work, give them the Millennium Prize. That is why we cared in the first place. The proof is just data, like more digits of π.

This also changes how you do research. Sometimes you need one more proof, and interacting with the model is the best way. Often it is not. What is better is to close the laptop, scribble through your own understanding, and ask what is blocking a deeper theory: one more proof, or something about how you understand the existing ones? If machines generate proofs but do not communicate them well, you still need human mathematicians to make them understandable. That is probably not the most fun part. He assumes it is transitory. Let us say the machines also get better at communicating their ideas.

That is written communication. An equally valuable form is the social side, which often happens through oral presentation. Throughout this whole trajectory, the relevance of being able to present orally just gets higher. The machine cannot understand it for you. A big part of math is getting other people to understand and care, and we are social creatures. Even if the machine is better at explaining, a big part of what makes us care is seeing someone else who actually cares.

Next item: asking questions, deciding which theorems are worth proving. Right now that remains an incredibly vibrant, highly leveraged place for mathematicians to spend time. But imagine, again for the sake of argument, that machines get better at that too. You do not just ask a model to prove a theorem. It comes back with other questions worth asking, questions that might yield fruitful math, and by whatever definition we are using, it is better at asking them than the mathematicians are.

The line between this and theory building is blurry. Suppose, for a moment, that humans are still better at the theory side: making sense of the broad set of theorems and how they were proved. That is what is most relevant to downstream applications. From the perspective of funding, it not only still makes sense—you are much more leveraged, because each mathematician can push theory forward faster than before.

On problem-solving muscles, as always, it is weights in the gym. It does not matter if machines are better. You still want humans trained in how to do it, assuming human problem-solving is relevant at all. On aesthetic joy, the split continues. Some people will not find it fun to have a model ask its own questions and then answer them. Grant himself will probably spend his whole life trying to think of interesting questions in his own head and find his own ways to answer them, irrespective of what the models can do.

You will also have people who are exhilarated, less and less rate-limited in building theories and coming to understand them, maybe even starting to see applications on a cadence that does not look like waiting decades or centuries—seeking them out a little earlier than that.

But what if, the grad student asks, the theory-building side falls as well? Suppose the most effective way to push theoretical math is not just to have machines generate proofs or pick theorems, but to have them digest those proofs, find the simplicity, and understand what novel fabric you need in order to weave a coherent theory.

The analogue would be giving a model all of math up to the 1500s and having it not only solve cubics but invent the right notation, come up with complex numbers, and all of the downstream theory. Maybe you do not think that is likely. Maybe genuinely novel thinking is implausible from present LLMs, given their reliance on human training data. For the sake of the hypothetical: what does the field look like in that world?

Insofar as we want applications in the real world, by assumption the model is better at generating the theory. You would still need somebody who understands that theory well enough to see how it applies in surrounding fields. Is any of this relevant to curing cancer, making fusion a reality, or high-temperature superconductors? That is where the rubber meets the road. On the individual problem-solving side, same as throughout: you are at the gym. If human problem-solving muscles matter at all, you still want people going through the practice.

On aesthetics, Grant realizes he has been conflating two joys. One is the joy anyone doing math at any level can feel: seeing a clever solution click, or understanding a new topic in a way that changes how you think. That is everything happening on this channel. Separately, there is the joy of the researcher at the frontier, the feeling of pushing that frontier forward.

The first category feels to him more or less unaffected by whether AI is better at items on the list. It is analogous to the problem-solving slot: to get this value, you have to be the one doing the math in your own head. Models probably influence that for some people. He also thinks this joy is much more intrinsically communal, or at least it benefits from being so. It is easier to feel passionate when you are with others who are, or when you see someone else who is. Isolation works; the social component is always an amplifier.

For his part, he is always going to want to engage with more pieces of math, come to terms with them in his own head, and share them with you, regardless of any machine's capabilities. As to the researchers' joy, the split he has been describing just comes into sharper focus. You will probably always have people exhilarated by a machine that offers a new window onto math, and people who find that incredibly disheartening.

Because let's be real: something is lost. For a long time there was a tight coupling between advancing human understanding and all those downstream applications. In these hypotheticals those are getting more and more decoupled, and for some people that means advancing understanding for its own sake loses a little of its oomph. You may not want a textbook with an answer key at the back, even if you do not have to look. It defeats the pioneering spirit.

A lot of the mathematicians Grant talks to are not so interested in being first. They seem to find most of their joy in the community. So you will see more subcurrents inside math, arguing for pushing forward a human understanding of math for its own sake. Summing up on the diagram: insofar as there is always joy in learning math, and that can flow through teaching if people will pay to learn, that remains. Insofar as some of the joy was the hope of being a pioneer, of setting foot first in a new kind of theory—let's admit it, that would be a loss.

Play it all out. Suppose the machines are not just better at proofs, conjectures, and theory, but better than humans at seeing what applications those plug into, and at plugging them into science and engineering. Call this the anything-you-can-do-I-can-do-better future. Here things get more interesting, because for the first time on the value chain, all of the application benefits could in theory happen just in the machine.

Insofar as public and private funding was motivated by downstream applications, the most efficacious use of those funds would be to pour them toward a big blob of compute that just does math: creates the theories, plugs them into applications, pushes things forward. If Grant had to guess, even though that would be the rational use of application-driven money, he finds it pretty hard to imagine policymakers flipping a switch so that public funding no longer flows to humans but to a blob of compute.

What he can imagine is certain individuals, made highly wealthy by AI itself, starting private foundations that pour money toward that blob. That seems like a good chance. What seems much more likely on the public side is that people will talk more and more about the other positive externalities. An institution that trains problem-solvers who spill into the rest of the world is still incredibly valuable. That on its own gives positive return to public funding.

Or formal institutions might pop up around human math done for human flourishing: essentially embracing math as an art, which it always has been, with funding perceived the way the arts are funded, publicly and through private philanthropy. Keep perspective, though. Those are not even where most of the money for math comes from.

Most of it is teaching, and the value of teaching ultimately stems from the lower two elements in the value chain: training people to build problem-solving muscles, and instilling a very specific kind of joy by doing so. The future implied by anything-you-can-do-I-can-do-better is so wild it is hard to engage with seriously. As a side note, this whole conversation is contingent on a branch where things go well and there is not something catastrophic. So again, maybe some people think AI safety is just a more important place to work right now.

While we are engaged with this, and assuming it is a good branch of the future, it seems likely to him that you will still have something like existing university structures. Even if in theory the AIs are better at explaining, people will still want to learn from humans, because education is not rate-limited by the clarity of explanation. It is rate-limited by motivation. Do people actually want to learn the material, and what motivates them?

Sometimes it is the pure joy of learning. We are also social creatures, and a huge part of what motivates people is seeing somebody else deeply passionate about it, especially someone they respect or care about. Teaching in general feels to him like one of the safest positions against AI job displacement, because it is so relational, because it depends so much on human-to-human interaction. No amount of clarity generated through a screen is going to substitute for that.

So where does this leave us? For the foreseeable future, research needs to place more and more emphasis on the theory-building side of math, which it always has—that always has been the point. The main difference is that proof was a very valuable proxy for that, and it is losing its value as a proxy. At this point it is a form of raw data, a useful form, and yes, we have to change how we measure human academic contribution.

That is probably a good thing. Proof in and of itself has probably been overvalued for decades. Similarly, insofar as the institution is valuable for sharpening problem-solving skills, it throws into question how exactly to do that, because previous methods no longer apply. Assigning and grading problem sets, on its own, does not really make sense right now.

Even if you are worried about a sci-fi future where all of math can be mechanized, or that the training for problem-solving somehow will not be useful—seems like a stretch, but let's engage—math will always be valuable to humans insofar as there is joy in doing it. One reason he is bullish on that last point standing as a pillar, even in wild futures, is this channel.

Think about what is going on here. There are millions of you willing to engage, and what we are doing is wallowing in the beauty of math, finding elegant problems whose solutions are satisfying, trying to instill new ways of thinking. Admittedly it is not as if millions of you are willing to pay for that—that would be great—but thousands of you are, and that makes all the difference.

Looking beyond this channel, one of his favorite examples of public appetite is Brady Haran's channels: among all of them, the most popular by far is the one about math. Stepping back, look at Martin Gardner, the godfather of modern math popularization. His column started at the end of Scientific American, and they had to move it to the beginning because so many people were getting their subscriptions specifically for it.

Once people get a taste for what math really is, there is an immense appetite from society at large. The mathematics community would be wise to turn itself outward and lean into satiating that hunger. For most of this video Grant has been laying out why the institution is valuable and will continue to exist. He opened with a stronger claim.

He said this is one of the best times to go into math. He guesses many viewers anticipated that he meant AI will unlock new opportunities. That is not what he is referring to. The reason it is one of the best times to go in is that the community as a whole knows it needs to change, and so many people are leaning back, looking for guidance and examples of what that change should be.

You, as a new entrant, stand to actually influence what the field becomes, at one of the most pivotal times in its history. He thinks that is incredibly exciting. So to that comment from the PhD student—tell me what to do tomorrow—he can tell you what he would do if he were a PhD student in 2026.

He would do 3Blue1Brown. He would do exactly this. If ten years ago he had walked into a math department with a concept for explaining math in a highly visual-first way, and asked them to make that an academically recognized PhD thesis, not a hobby, he would have been laughed out of almost every department. He is not positive his younger self could pull that off today. But if ever there was a time people would be open-minded to it, now would be it.

He is not saying that is what you should do. You should lean into whatever makes you love math. He loves explaining it. He loves visualizing it. That is his passion, and that is what he wants to run with. But what is yours? If you love the theory-building side, if you want to be the next Galois or the next Grothendieck, great, run with that. You have a proof-generating machine that is going to get better over the next four years. Maybe you can use it to create a thesis the likes of which the world has never seen, forging territory you never could have in a previous world.

If you do not like the models, if they leave a bad taste, great—lean into that. There are undoubtedly going to be subcurrents that want to establish what math by humans for humans would be. You can plant a flag in the sand. You can decide what a PhD thesis emblematic of that goal would even look like. The other thing he would do is recognize that if machines are better at any subset of the research stack, the most effective way to measure a human's understanding, and their potential to help push the field, is going to shift more and more toward oral presentation.

So give talks. Give lots of talks. Get really, really good at giving talks, and make that a regular part of how you practice your math. His most concrete suggestion for what to literally do tomorrow is to go on a long walk. Find a friend, maybe someone else in your program, and reflect with them: what is it you like about math? Why did you go into it? Ask them what they love, and why they went into it.

Together, brainstorm how you want the next few years of your life to contribute to various parts of this value stack. For his part, he still loves math. He is always going to love math. He is always going to love finding ways to visualize it and share them with you on this channel. His biggest hope is that you can find a way to lean into your own passion. The community has never been more open-minded, and they will be listening to what you have to say.