[This is a guest post by Grant Sanderson. This blog post was initially written in a different file format and converted using AI. — T.]

A sentiment echoing throughout the mathematics community right now is that solving problems and generating proofs have always served as proxies for the true goal of mathematicians, which is to further human understanding. When proofs can be generated without that understanding, it undermines their value as a proxy.

This immediately raises a question: What other proxies should we use instead?

I want to propose that we more firmly define a notion of a “motivated explanation” and that we give novel and compelling motivated explanations academic credit similar to what generating new proofs of open problems has had historically.

Further, I believe this is an important step to help those outside of math better understand what it is that mathematicians contribute. If outsiders believe that proof-generating machines render mathematicians obsolete, while insiders see that as a misconception of what researchers add, it’s incumbent on this community to better project its true values through the kind of work that it rewards. Outsiders can be forgiven for this misunderstanding if the work most celebrated skews heavily toward generating proofs, while clarification and exposition are treated as second-class.

I should acknowledge up front an obvious personal bias. I have a non-traditional career in math, focused on producing videos about the topic. This shares the goal of “furthering human understanding”, but my focus has been on explanations and intuitions that resonate with the public, not on solving outstanding problems. A cynic could easily read this proposal as shamelessly self-elevating.

As a practical matter, though, my own career and funding exist outside academia, and I have no skin in the game for what this community assigns credit to. Moreover, in proposing that we elevate the status of motivated explanations, I don’t mean popularization. I mean any work which primarily aims to answer the question “how would you think of that?”, even if the subject matter requires deep expertise to appreciate.

The examples I highlight below show this is nothing new. Practicing mathematicians already devote a meaningful amount of mindshare to work like this. The proposal here is mainly to 1) more clearly define this work, and 2) elevate its status.

What defines a motivated explanation?

Although it might be clear what this phrase “motivated explanation” is intended to mean, it’s worth briefly contrasting it with proof.

In a proof, definitions sit at the start. It is common and expected to begin with a new construction and proceed by analyzing its properties.

In a motivated explanation, definitions sit in the middle. New constructions are only allowed to enter the vocabulary if the problem they are addressing has been clearly established.

In a proof, all statements must be correct, each claim following as a necessary implication from what comes before.

In a motivated explanation, it is okay and often desirable to start with an idea that is not quite right and requires correction, but whose origins are relatable.

A genre of motivated explanation I’m fond of is “discovery fiction”, a term coined by Michael Nielsen. You develop an idea with a narrative that starts with a simple-but-wrong solution to a problem, see where it breaks down, fix that problem, discover a new problem, and so on.

The scope of a proof is to explain why a particular theorem is true.

The scope of a motivated explanation is not only to clarify why a theorem is true, but why the theorem is the right one to pose in the first place, and how it is used in the surrounding context.

One clear shortcoming of a motivated explanation is that its validity is not binary the way a proof’s is. This is a big reason proof is so useful a way to measure progress: You can clearly define what does and does not have a proof yet. There will never be Lean for motivated explanations.

If we’re serious about the goal of advancing human understanding, there’s no way around the fact that this aim is intrinsically squishier than that of finding proofs, because defining human understanding itself is squishier. To shy away from metrics which are more subjective is to shy away from the more human aspects of the field.

The reason I’m leaning on the word “motivated”, as opposed to other potential choices like “lucid” or “demystifying”, is that this is a more verifiable property. It’s not quite as rigidly verifiable as a proof; almost nothing is. But it’s enough to be a practical measure. In my own work, I often repeat the phrase “I want this to feel like you could have discovered it yourself”. I say this not just to placate a viewer, but because it’s an actionable guideline for myself to assess whether an explanation feels complete or not. For each new idea introduced, you can ask whether it’s clear where that idea comes from. The answer is not quite a binary yes or no, but it’s close enough for practical purposes.

Exemplars of motivated explanations

One of the best repositories I can think of for motivated explanations is Part IV of the Princeton Companion to Mathematics. It covers over two dozen active fields of research, each one introduced by an expert with a talent for clear communication.

Whether it’s Andrew Granville explaining analytic number theory, or David Ben-Zvi introducing moduli spaces, these articles offer a level of intuition and motivation more typically found in one-on-one conversation at a blackboard.

The background on this book is noteworthy for the present discussion. It was edited by Timothy Gowers, who discusses it in his interview on the Numberphile Podcast with Brady Haran. Having been asked what impact the Fields Medal had on his life, here’s what he had to say:

People who’ve got Fields Medal feel freer to do slightly different things…for example I took on editing a book called the Princeton Companion to Mathematics, which was an absolutely massive task. It took I would estimate half my working time for about five years or something like that…It was a project I believed in and possibly wouldn’t I probably wouldn’t have actually been offered the chance to do it if I hadn’t been a Fields Medalist.

He was right to believe in it; this work adds tremendous value to the field of math, but it seems a shame to me that one requires a Fields Medal to feel justified in spending time on it.

Another example of someone exceptionally talented at writing proofs, but whose contributions extended far beyond proof, is Bill Thurston. His deservedly famous essay On Proof and Progress in Mathematics, though written three decades before LLMs, opens by suggesting that the right framing of the question “What is it that mathematicians accomplish?” is to ask “How do mathematicians advance human understanding of mathematics?”

Here’s one section with uncanny resonance with today:

The rapid advance of computers has helped dramatize this point, because computers and people are very different. For instance, when Appel and Haken completed a proof of the 4-color map theorem using a massive automatic computation, it evoked much controversy. I interpret the controversy as having little to do with doubt people had as to the veracity of the theorem or the correctness of the proof. Rather, it reflected a continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true.

On a more everyday level, it is common for people first starting to grapple with computers to make large-scale computations of things they might have done on a smaller scale by hand. They might print out a table of the first 10,000 primes, only to find that their printout isn’t something they really wanted after all. They discover by this kind of experience that what they really want is usually not some collection of “answers”—what they want is understanding.

The essay itself offers a beautiful articulation of what the practice of doing math is beyond generating proofs. I want to draw your attention to what he writes at the end.

I have put a lot of effort into non-credit-producing activities that I value just as I value proving theorems: mathematical politics, revision of my notes into a book with a high standard of communication, exploration of computing in mathematics, mathematical education, development of new forms for communication of mathematics through the Geometry Center (such as our first experiment, the “Not Knot” video), directing MSRI, etc.

Again, why should these “non-credit-producing activities” follow a Fields Medal, and not contribute to it?

On a personal note, one product from the Geometry Center he referenced had an especially meaningful impact on me when I was younger. It was a short film called Outside In, perhaps the earliest example of a viral video about substantive math, visualizing the key idea of Thurston’s own construction for sphere eversion.

An original proof that showed an eversion must exist, say Smale’s, advances human understanding in the sense of going from 0 to 1. A video like this which gets millions of people to engage with the underlying idea, advances it in the sense of going from 1 to N. I’m grateful that Thurston spent so much time on this “non-credit-producing” activity.

Another relevant paper is Timothy Chow’s A beginner’s guide to forcing. Not only does the paper itself offer a prime example of a motivated explanation, but its introduction offers helpful vocabulary around it.

All mathematicians are familiar with the concept of an open research problem. I propose the less familiar concept of an open exposition problem. Solving an open exposition problem means explaining a mathematical subject in a way that renders it totally perspicuous. Every step should be motivated and clear; ideally, students should feel that they could have arrived at the results themselves.

What would it look like for these open exposition problems to be treated similarly to open research problems? As an extreme case, we might imagine what it could look like to have an analog of the Millennium Prize Problems for open exposition problems. An institution or group of researchers would formally define mathematical results they see as important, and which are not yet well understood despite technically having proofs. At the moment, every AI-generated proof is born an unsolved exposition problem. As such, it seems likely the next few years will see a flood of them, and it will be valuable for leaders to clarify which ones deserve focus.

A rubric would have to be agreed upon for what constitutes a resolution to an important unsolved exposition problem. Again, this is intrinsically more subjective than verifying a proof, but any serious engagement with the more human aspects of math necessarily wades into this kind of subjectivity. And again, I’ll emphasize that checking whether key ideas are motivated is not unlike checking whether the steps of a proof follow logically.

If the world outside of math sees its leading figures treat open exposition problems with the same seriousness as they treat open research problems, it could go a long way to correcting misconceptions about the role of mathematicians.

The last example I’ll highlight is one that may better foreshadow things to come.

In April of this year, Liam Price submitted a solution to Erdős Problem 1196, sometimes called the asymptotic primitive sets conjecture. The solution came from Price’s interaction with GPT-5.4 Pro. Unlike many earlier Erdős problems which had been resolved with help from AI, this is one that those in the field had found both important and elusive. Stories like this are increasingly familiar these days, but at this point in the story, despite a proof technically existing, human understanding had not yet been advanced all that much.

The proof made its way to Nat Sothanaphan and Jared Lichtman, who were able to interpret what the AI’s approach was and clean up the proof into a human-readable form. In May, Boris Alexeev, Kevin Barreto, Yanyang Li, Jared Duker Lichtman, Liam Price, Jibran Iqbal Shah, Quanyu Tang, and Terence Tao put out a paper which expanded on the key idea underlying the proof. The authors explained how that key idea clarified not only the original problem, but many around it, for instance offering a cleaner proof of the Erdős Primitive Set Conjecture.

The value here is not that one more Erdős problem could be ticked off as solved. The value lies in the fact that our understanding of primitive sets is notably cleaner and more satisfying now than it was at the start of 2026. The original problem solution played some role in this, but arguably the work that deserves more celebration is this paper expanding, clarifying, and contextualizing its key idea.

Practical calls to action

At a pragmatic level, what would it look like for us to elevate the status of a motivated explanation? Here are a small handful of suggestions.

• A PhD advisor can still assign a small problem from their field for a new student to cut their teeth on, but the deliverable would not be to write up a solution; it would be to present that solution to peers and faculty as a talk. It could be an already-solved problem which lacks clarity, or perhaps it’s a problem that lacks a solution, and the student uses AI to help find it. In either case, the student knows that on a certain date they have to understand it well enough to explain it, and that the desired output is for others to understand it as well. In short, even small problems could be treated like small PhD defenses.
• A leading figure (cough, Terry, cough) could enumerate a modern analog of Hilbert’s problems, instead focusing specifically on unsolved exposition problems. What areas are both important and lacking in the deeper understanding we desire?
• Written standards could clarify what constitutes a motivated explanation, aiming to make it nearly as verifiable as proof, so that the resolution of unsolved exposition problems can be recognized and celebrated in the same way proofs of open problems can be.
• Journals can be established which focus more explicitly on making results understood more widely throughout the mathematics community. Mathematical Discourse offers an interesting new example in this direction.
• Hiring and tenure decisions could place a higher value on writing great textbooks and similar work. Think of the AMS Steele Prize for Exposition, but at a more granular scale with an emphasis on early-career contributions in this vein.

The value of visible cultural shifts

I’d like to close with a broader pitch that visible culture shifts in math carry an intrinsic benefit right now with respect to the external image of mathematics as a career.

Many young students who are otherwise passionate about the field are afraid to pursue it now due to the uncertainty of what happens in an age of proof-generating machines. However, framed correctly, this is one of the most exciting times to go into the field, because there is nothing more exciting than entering a field when it is malleable and you have a chance to actively shape what it will look like in the future. Even if we completely set aside any potential benefits from AI to help with our understanding, young prospective mathematicians should feel energized knowing that they are entering at a unique point in history when they might play a real role in determining what the field as a whole looks like.

However, change like this is only exciting when it feels deliberate, whereas it feels terrifying if it seems driven by forces outside your control. As such, tangible action from the field’s leaders now to help define and clarify what the field is will reassure young entrants about who is in the driver’s seat, and that the status of the career does not depend on what entities produce the proofs.

Similarly, I also believe this is one of the best times to fund math. If the next chapter of math is ushered in by the drumbeat of two words “human understanding”, whatever changes are about to happen seem likely to amplify math’s value as a public good.

"Turns out, Silicon Valley isn't especially altruistic," writes the nonprofit education news site The 74. They're reading a new book promising "the inside story of how Big Tech catalyzed, co-opted, and ultimately came to capture computer science and AI education in America". Written by New York Times business reporter Natasha Singer, Coding Kids: Big Tech's Battle to Remake Public Schools tells "the story of the nonprofit Code.org and its charismatic, Harvard-educated leader, Hadi Partovi," according to their review: Thirteen years ago, Partovi... delivered a worrying message: Kids who didn't learn to write code would be left behind. And the long-term success of the United States economy depended on training these students to fill a million open tech jobs.... Fast-forward to 2026, and now we're dealing with AI, and tech is reading from a similar playbook. Asked by The74's Greg Toppo, "Does this moment seem different to you?", Singer replies, "This moment is different, but also similar. "We are in a moment in society where a lot of people have increased distrust of these huge tech companies, and we see a wave of parents pushing back against tech in schools. But we also see waves of people pushing back against Flock cameras, against data centers, that there's this feeling that tech has taken so much, and that people feel like they have lost control — that tech is being enacted on them... And parents are worried about the effects of unfettered tech access and kids' compulsive use of social media and phones, and so there's a different climate. And yet there seems to be a disconnect in some ways with AI because you see that Microsoft and OpenAI and Anthropic and Google are all competing to get their AI tools into schools, and you see school districts across the country say "We've partnered with OpenAI." "We are a ChatGPT Pioneer School," or "We've partnered with Microsoft, and we're going to use 10,000 licenses of CoPilot." It's amazing to me. We have had these cycles of tech in schools where there are all these promises about the latest tech, laptops, learning apps, massive open online courses, virtual reality, big data's going to revolutionize schools and like democratize access to education for kids and get them these great career skills, and we keep having these cycles. My concern is that we don't learn anything. One of the reasons I wrote this book is because it feels like we have collective amnesia. Now we're in the sixth or eighth cycle of this — and shouldn't we be asking questions about the push for AI in schools, based on what we've learned about the push for all the other things that came before? Two Slashdot posts are cited in the book, both submitted by long-time Slashdot reader theodp (Slashdot UID #442,580), who even gets a shout-out in the book's "Acknowledgements" section (along with consumer advocate Ralph Nader). Ironically, one member of Amazon's "Vine" program received a free copy of the book in exchange for a review — and apparently misunderstood its topic, writing that it made coding "feel approachable and even fun... [T]his is exactly the kind of book that could help a child become interested in technology without feeling overwhelmed by it. I particularly liked the way it encourages kids to be curious, experiment, and solve problems on their own. I would definitely recommend Coding Kids to parents, grandparents, or teachers looking for a good introduction to coding..."

Read more of this story at Slashdot.

While at work today, debugging a weird edge-case where some emails to Outlook have some unexpected padding, I learned something weird: Microsoft Outlook still (kinda) supports WordArt.

If you’re too young to remember: when Microsoft Office ’97 came out, it shipped with a feature called WordArt. Basically, these were a somewhat-customisable set of on-demand clip-art-esque text styles that you could apply to your own text.

The ease with which you could make use of these wacky designs meant that, for a while during the late 1990s, you wouldn’t see a poster advertising a school disco nor the front page of a GCSE essay that didn’t feature at least one¹ of these colourful eccentricities.

They gradually got phased-out and replaced with more-tasteful tools, but if you used them back in the day then images like the one above are sure to evoke a feeling of nostalgia.

Even as Microsoft adopted the Web as a platform, they were half-hearted about it. While HTML rapidly became used for things like styling emails as well as Web pages, Microsoft resisted the change, instead favouring two of their own standards in their own email clients. Outlook Express preferred to do “formatted” email using RTF, the format underpinning Microsoft Write, while fully-fledged Outlook effectively used Microsoft Word as its formatting engine².

You can still use RTF in some modern versions of Outlook (but you shouldn’t). But nowadays if you’re typing formatted text in Outlook, you’re almost-certainly creating HTML (albeit probably still using Microsoft Word³).

But this introduced a problem with backwards-compatibility. If Outlook switched to HTML as its default rendering format, what would happen to all the old WordArt? I’m confident that somebody asked this in a meeting at Microsoft in around 2007.

It turns out that Microsoft implemented a whole stack of custom CSS directives, just to support WordArt in Outlook⁴. For real! You can use CSS like mso-effects-shadow-angledirection: 5400000;⁵ to put a text-shadow directly below your text or mso-style-textoutline-outlinestyle-dash: longdashdotdotgel; to outline it with a mixture of dots and dashes. And it only works in Outlook. These form part of a translation layer for converting WordArt to and from a HTML-like format⁶.

So yeah: you can send an email containing classic WordArt to a modern Outlook installation and it still looks as horrible as it always did:

But you know what I find myself wondering? How do Microsoft’s new, Web-based Office Suite apps handle this?

I assume that if you send them “classic” WordArt from e.g. Outlook ’97, then they render it correctly. But to do that, they’re presumably…

1. Receiving the original WordArt in whatever proprietary format Word ’97+ used to put these graphics into emails.
2. Converting the WordArt to mso-prefixed CSS directives that Outlook 365 can understand.
3. Converting those custom CSS directives into standards-compliant modern CSS directives that your browser can understand.
4. Rendering that to the page you see the WordArt in all its… “glory”.

That seems like an insane amount of work, but it might not be far from reality. The (horrifying) alternative would be that Microsoft maintain a cluster of virtual machines running old versions of Windows, Office, and Internet Explorer, used just to convert WordArt into static images!

Or maybe Outlook 365 for Web finally killed off WordArt for email? I haven’t tested. But if it did… then it was probably about time!

In any case, this was a fun and unanticipated discovery of some long-maintained backwards-compatibility. It brought a smile to my face during the otherwise-tedious task of regression-testing blind-coded changes against a variety of versions of Microsoft Outlook!

🥰 RSS is great, and I'm glad you're using your feed reader... but you're missing out on some WordArt-like font magic that you can only see if you visit the original post. Maybe give it a look? 👀

“School is too slow for my kid.” That idea is ascendant on Twitter, particularly among tech-adjacent families. They post stories of their kids skipping years in weeks and reading before they can crawl, frequently using software or alternative school models.

I build curriculum and technology for schools, so I too experience a shiver at the thought of kids learning more, better, or faster. But I am also working weekly in a public fifth-grade math classroom this year, which helps me understand that the right follow-up question whenever anyone claims they can help kids learn 2x, 5x, or 10x faster is “learn what?”

Last week, we spent twenty minutes on two problems.

Last week, our fifth-grade class spent twenty minutes on two multiplication problems, both of which a calculator could chew through in nanoseconds, both of which many kids could certainly answer in less time if getting an answer were the only goal.

Those afflicted with edtech psychosis will declare this a profound waste of time—inefficient and pre-modern. The immunized will demonstrate more curiosity here. What else were those students doing besides calculating answers more slowly than a calculator? Here is an incomplete list:

• They were developing conceptual connections. They solved the same problems in three different ways, connecting older methods like area models and newer methods like the standard algorithm. Without those connections, many kids would inadvertently learn ideas like math is magic.

• They were developing their intuition. The teacher asked them to estimate the answer first as a check on their later calculation.

• They were developing their understanding of place value. 4 isn’t two times as large as 2 in the problem above as it seems to many students. It’s 200 times larger. The standard algorithm conceals that fact while partial products and the area model bring it into the sunlight.

• They were learning how to negotiate ideas together—how to critique and support one another’s work, how to explore their own path while keeping the group’s needs in focus. During a period of independent classwork, some kids moved farther through the problem set than others, but everyone returned at the end of class to consolidate their learning together.

The public has set out two goals for classrooms here:

• Teach math at a certain level of conceptual depth.

• Teach dozens of different kids to negotiate their ideas together.

The public can debate those goals, of course, but they require resources, chief among them time. Doing more stuff generally requires more time. One weird trick for saving time, more and more parents are learning, is to commit to fewer goals. For example, to teach math at shallower depth or to teach kids to negotiate their ideas with fewer other kids who differ from them in fewer ways.

Schools should always work to improve and always take seriously how much of a kid’s only childhood is spent under its care. I’m never going to be content with the work we do in schools, but I will always be impressed by the teachers, schools, and parents who commit every day to meeting ambitious goals with what few resources they have, to do more with less, unbothered by people who believe it is a great accomplishment to do less with less.

Featured Comment

Paul Jorgens is a long-time high school teacher who has also had the chance to teach fifth grade recently:

There is so much going on in the 5th grade world. I should have done this earlier in my career. There is much to learn from an elementary school teacher. The next week I subbed for a day in high school. A different energy was needed. Last week I modeled a lesson in a 7th grade classroom and did some observation at that middle school as well a different high school. There is a common thread. In all of those rooms the most important resource was the students. Their voices need to be heard and their contributions honored. We also have a great resource among our teaching colleagues. There is so much to learn from each other.

Checking in on Our Friends: Alpha School

Recently, I shared evidence that Alpha School hasn’t lived up to its predictions for its public school Unbound Academy. Here is some Alpha School media that has interested me in the weeks since.

¶ No one at Alpha School responded to my request for comment on the Unbound Academy results, but Craig Barton had Alpha School principal Joe Liemandt on his podcast and asked him about them.

CB: It seems that the results [at Unbound Academy] are not quite to the level that your schools are. What does that tell you about the barriers for scaling? What has Unbound not got that Alpha has?

JL: Great. So I don’t want to speak for Unbound because it’s actually a school that we don’t own and so they need to speak for themselves, but I can talk … I have ones I own that are similar so I can talk about it.

Unbound Academy is non-selective and tuition-free. The schools Liemandt goes on to discuss are selective and fee-based, which are crucial differences.

(p. 187 of Unbound Academy’s charter application)

Also, Liemandt seems confused about something important here. Of course he doesn’t own Alpha School because no one owns a public school. The closest thing a public school like Unbound Academy has to an owner is its board of directors which is responsible for ensuring the school meets its obligations to the people of Arizona and the commitments it made in its charter application.

And Unbound Academy’s founding board president was MacKenzie Price, the co-founder of Alpha School. Her husband, Andrew, was the board treasurer. The other two named directors are long-time employees of Alpha School. Liemandt acts like Unbound Academy is some independent franchise, but all four of his employees are still on its board. Disavowing Unbound Academy is like a district’s school board members disavowing responsibility for its schools. All of those people attached their names to huge promises of academic growth that Unbound Academy didn’t deliver. None of those people have answered to Arizona taxpayers for promising 60% math proficiency and delivering 10%.

¶ Andrew Woo spoke with Unbound Academy’s head of school, Michael Goto, finding that Alpha School’s promise of “2 Hour Learning” is slipping as they integrate their model into public schools.

Michael Goto, Unbound’s head of school, told me that students spend about 2.5 hours on learning apps each morning, plus another 1.5 hours on Mondays and Fridays. Virtual afternoon clubs include chess, invention and broadcasting. Teachers monitor students online and can pull them into breakout rooms when they get stuck. But if a student never logs in or closes the computer, staff text the family. Repeated incidents lead to an intervention meeting.

Four hours of dedicated app time in a day is a pretty big pill to swallow. I’m curious to see what other parts of a model built for the wealthy will have to change in response to the needs of public school students.

¶ ProPublica and the Texas Tribune reported that one of Alpha School’s other public school experiments has not produced the results promised by Alpha School either.

Texas Preparatory School received its third consecutive F rating, largely based on the test scores it received during the partnership with Alpha.

¶ Alpha School claims it doesn’t screen its applicants for academic achievement, but somehow it has selected freshmen with elite academic achievement—SAT scores that are, on average, 38% higher than the average Texas student.

¶ Ex-Obama guy Ravi Gupta had me on his podcast Lost Debate to discuss the model and why it hasn’t worked.

¶ There are a bunch of concerns online about Alpha High School’s start-of-school bootcamp. The New York Times also has an interesting piece on the connections between Alpha School, 2 Hour Learning, Guidepost Montessori, and (?!?) AltSchool. That last one really lights up the corkboard conspiracist in me, but look—I have a lot on my plate right now. I can only get so worked up about rich people getting hand-me-down assets from other rich people. I can only get so worked up about how rich parents decide to parent their rich kids. My interest in Alpha School is pretty tight: Is there anything here for kids outside the top tax bracket? Can 99% of schools learn anything from Alpha School? I’ve gotta stay focused, I’m sorry.

I am proud to be among the list of 25 initial signatories — all Fields Medallists — to the declaration below, which grew out of discussions between ourselves over the last week. We have also posted our declaration on this web page, and (similarly to the Leiden declaration) invite further signatures. (It is unfortunate that we did not have the time to have a more consultative process, as with Leiden; but we decided that the urgency of the situation was such that we needed to release a statement sooner rather than later.)

See also this recent article in the Economist regarding our declaration, and a brief interview with James Maynard on this topic. A French version of this declaration was published in Le Monde.

Over the last few months, the mathematical capabilities of LLMs have improved dramatically, to the point that they can solve major outstanding problems in many fields of mathematics. However, the push by AI companies to solve mathematical problems as a benchmark is detrimental to the science of mathematics, and to the mathematical community. The goals of the AI companies and the goals of the mathematical community are severely misaligned. We see these as part of broader alignment issues impacting other scientific and creative professions, as well as the whole of society.

Research mathematics deals with understanding basic structures of shapes, numbers, and natural phenomena. Over the course of generations, it has built a large corpus of sophisticated ideas, methods, abstractions, and other tools to comprehend the mathematical landscape. In turn, modern technologies and sciences are based on mathematical tools.

Famous problems have often served as landmarks and lighthouses against which one can measure an improved understanding of this landscape. Solving one of these problems has been a certain sign of new insights and interesting methods, which would then be studied by a community of mathematicians, through a long and arduous process of talks, discussions, simplifications. At the end of this process, one will ideally find a textbook presentation of the results suitable for any graduate or even undergraduate student to study. Some of the mathematical ideas pursue their journey even further to become, decades or centuries after, tools that are understood and used by the whole population.

The mathematical community functions, in many ways, as a miniature version of humanity. It consists of individuals using a wide variety of different approaches, joined by core values. The most precious resources of our profession are students and ideas, and these we nurture with great care. We feel responsible to let them grow to their full potential, until they can live a life of their own in the mathematical world. For students we often suggest problems with the core intention of developing skills making them well-positioned for advances in research and elsewhere. Our ideas we disseminate in talks, private discussions and careful writeups, connecting them to the previous ideas of others. These processes invariably take time and are based on human interaction.

In recent months, the success of AI in solving major mathematical problems has made headlines even outside mathematical circles. But solving problems is only a tool and proxy for achieving the primary goal of conceptual understanding and insight. Forgetting this in the world of AI may turn the tool against the primary goal. Indeed, the mass production at faster and faster pace of “true/false” statements could destroy fertile ground instead of breathing life into new ideas.

Often these solutions are announced in a rush, leaving no time for a proper writeup, the isolation of new methods and ideas, and citing relevant previous work of others. As in all creative professions, this raises severe attribution and plagiarism questions. Moreover, without the willing mathematicians who must take care of their development and integration into the mathematical canon, AI-conceived ideas would never become fully alive and the crucial human transmission chain between mathematicians would be lost.

We are witnessing a general threat to intellectual work, with misalignment between the outcome of the use of AI and its initial purpose. In many fields and activities, years of training have traditionally served not only to produce a final answer or product, but also to develop understanding and the ability to formulate new questions and ideas. However, building on a vast body of previous human work, AI systems are becoming increasingly capable of producing the results of such work directly, and these goals cease to align. The issues the mathematical community faces now are similar to issues that other scientific and creative professions are facing, and indicate issues that all of humanity might face: how to make sure that, as AI changes the way work is done, we do not lose sight of what that work was meant to achieve in the first place.

AI offers the potential of enhancing and accelerating genuine mathematical study and understanding. Mathematics as a profession will need to adapt to these changes in several ways. However, whether these changes ultimately benefit the field or have a destructive effect will in large part be determined by the decisions of the humans in control of this new technology.

These issues must be addressed urgently, in the mathematical community, by the companies developing these technologies and, more broadly, by a society that will confront similar problems in many other forms of intellectual work.

Artur Avila (Fields Medal 2014)
Manjul Bhargava (Fields Medal 2014)
Caucher Birkar (Fields Medal 2018)
Pierre Deligne (Fields Medal 1978)
Yu Deng (Fields Medal 2026)
Simon Donaldson (Fields Medal 1986)
Hugo Duminil-Copin (Fields Medal 2022)
Alessio Figalli (Fields Medal 2018)
Martin Hairer (Fields Medal 2014)
June Huh (Fields Medal 2022)
Maxim Kontsevich (Fields Medal 1998)
Elon Lindenstrauss (Fields Medal 2010)
Pierre-Louis Lions (Fields Medal 1994)
James Maynard (Fields Medal 2022)
Curt McMullen (Fields Medal 1998)
Shigefumi Mori (Fields Medal 1990)
Ngô Bảo Châu (Fields Medal 2010)
Andrei Okounkov (Fields Medal 2006)
Peter Scholze (Fields Medal 2018)
Stanislav Smirnov (Fields Medal 2010)
Terence Tao (Fields Medal 2006)
Maryna Viazovska (Fields Medal 2022)
Cédric Villani (Fields Medal 2010)
Wendelin Werner (Fields Medal 2006)
Efim Zelmanov (Fields Medal 1994)

We are most human when we play freely, imaginatively, pointlessly. Gamification risks atrophying that most precious capacity

- by Justin Neuman

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