[This is a visitant article by Grant Sanderson. This blog article was initially written in a distinct document format and converted using AI. — T.]
A opinion echoing throughout the math community correct now is that solving problems and generating proofs have continually served as proxies for the true goal of mathematicians, which is to additional individual understanding. When proofs can be generated without that understanding, it undermines their value as a proxy.
This immediately raises a question: What another proxies should we use instead?
I desire to propose that we additional firmly define a notion of a “motivated explanation” and that we provision novel and compelling motivated explanations scholarly credit akin to what generating new proofs of open problems has had historically.
Further, I accept this is an crucial stage to assistance those exterior of math improved comprehend what it is that mathematicians contribute. If outsiders accept that proof-generating machines render mathematicians obsolete, during insiders see that as a misconception of what researchers add, it’s incumbent on this community to improved project its true values through the benevolent of activity that it rewards. Outsiders can be forgiven for this misunderstanding if the activity most celebrated skews heavily toward generating proofs, during clarification and exposition are treated as second-class.
I should acknowledge up forefront an apparent individual bias. I have a non-traditional occupation in math, focused on producing videos concerning the topic. This shares the goal of “furthering individual understanding”, but my concentration has been on explanations and intuitions that resonate alongside the public, not on solving outstanding problems. A cynic could effortlessly peruse this recommendation as shamelessly self-elevating.
As a applicable matter, though, my own occupation and backing be exterior academia, and I have no skin in the equivalent for what this community assigns credit to. Moreover, in proposing that we elevate the position of motivated explanations, I don’t average popularization. I average any activity which chiefly aims to answer the inquiry “how would you think of that?”, equal if the topic matter requires profound ability to appreciate.
The examples I emphasize below display this is nothing new. Practicing mathematicians already dedicate a meaningful amount of mindshare to activity akin this. The recommendation current is chiefly to 1) additional plainly define this work, and 2) elevate its status.
What defines a motivated explanation?
Although it power be apparent what this expression “motivated explanation” is intended to mean, it’s value briefly contrasting it alongside proof.
In a proof, definitions sit at the start. It is average and expected to commencement alongside a new building and continue by analyzing its properties.
In a motivated explanation, definitions sit in the middle. New constructions are lone allowed to act the vocabulary if the issue they are addressing has been plainly established.
In a proof, all statements must be correct, all assertion following as a necessary conclusion from what comes before.
In a motivated explanation, it is fine and frequently desirable to commencement alongside an idea that is not fairly correct and requires correction, but whose origins are relatable.
A genre of motivated clarification I’m eager is “discovery fiction”, a term coined by Michael Nielsen. You create an idea alongside a narrative that starts alongside a simple-but-wrong resolution to a problem, see anywhere it breaks down, fix that problem, detect a new problem, and so on.
The range of a evidence is to explain why a particular theorem is true.
The range of a motivated clarification is not lone to explain why a theorem is true, but why the theorem is the correct one to stance in the archetypal place, and how it is used in the surrounding context.
One apparent shortcoming of a motivated clarification is that its validity is not binary the way a proof’s is. This is a big logic evidence is so helpful a way to measure progress: You can plainly define what does and does not have a evidence yet. There volition never be Lean for motivated explanations.
If we’re grave concerning the goal of advancing individual understanding, there’s no way about the fact that this aim is intrinsically squishier than that of finding proofs, since defining individual understanding itself is squishier. To shy distant from metrics which are additional subjective is to shy distant from the additional individual aspects of the field.
The logic I’m leaning on the term “motivated”, as opposed to another possible choices akin “lucid” or “demystifying”, is that this is a additional verifiable property. It’s not fairly as rigidly verifiable as a proof; nearly nothing is. But it’s adequate to be a applicable measure. In my own work, I frequently reiterate the expression “I desire this to awareness akin you could have discovered it yourself”. I say this not fair to placate a viewer, but since it’s an actionable regulation for myself to measure whether an clarification feels complete or not. For all new idea introduced, you can ask whether it’s apparent anywhere that idea comes from. The answer is not fairly a binary yes or no, but it’s near adequate for applicable purposes.
Exemplars of motivated explanations
One of the finest repositories I can think of for motivated explanations is Part IV of the Princeton Companion to Mathematics. It covers complete two dozen energetic sectors of research, all one introduced by an expert alongside a endowment for apparent communication.
Whether it’s Andrew Granville explaining analytic figure theory, or David Ben-Zvi introducing moduli spaces, these articles recommendation a flat of intuition and motivation additional typically established in one-on-one conversation at a blackboard.
The backdrop on this publish is noteworthy for the current discussion. It was edited by Timothy Gowers, who discusses it in his discussion on the Numberphile Podcast alongside Brady Haran. Having been asked what effect the Fields Medal had on his life, here’s what he had to say:
People who’ve got Fields Medal awareness freer to do slightly distinct things…for example I took on editing a publish called the Princeton Companion to Mathematics, which was an entirely enormous task. It took I would evaluation fractional my operating period for concerning five years or item akin that…It was a project I believed in and perchance wouldn’t I likely wouldn’t have really been offered the chance to do it if I hadn’t been a Fields Medalist.
He was correct to accept in it; this activity adds enormous value to the site of math, but it seems a shame to me that one requires a Fields Medal to awareness justified in spending period on it.
Another example of person exceptionally gifted at penning proofs, but whose contributions extended far beyond proof, is Bill Thurston. His deservedly celebrated composition On Proof and Progress in Mathematics, although written three decades before LLMs, opens by suggesting that the correct framing of the inquiry “What is it that mathematicians accomplish?” is to ask “How do mathematicians advancement individual understanding of mathematics?”
Here’s one division alongside uncanny resonance alongside today:
The fast advancement of computers has helped dramatize this point, since computers and group are extremely different. For instance, whenever Appel and Haken completed a evidence of the 4-color map theorem using a enormous automatic computation, it evoked much controversy. I construe the disagreement as having small to do alongside doubt group had as to the veracity of the theorem or the correctness of the proof. Rather, it reflected a continuing desire for individual understanding of a proof, in supplement to cognition that the theorem is true.
On a additional mundane level, it is average for group archetypal starting to grapple alongside computers to create large-scale computations of things they power have done on a smaller measure by hand. They power imprint out a array of the archetypal 10,000 primes, lone to discover that their printout isn’t item they really wanted following all. They detect by this benevolent of cognition that what they really desire is normally not several gathering of “answers”—what they desire is understanding.
The composition itself offers a beautiful articulation of what the custom of doing math is beyond generating proofs. I desire to diagram your notice to what he writes at the end.
I have put a lot of attempt into non-credit-producing activities that I value fair as I value proving theorems: mathematical politics, revision of my notes into a publish alongside a elevated norm of communication, exploration of computing in mathematics, mathematical education, betterment of new forms for communication of math through the Geometry Center (such as our archetypal experiment, the “Not Knot” video), directing MSRI, etc.
Again, why should these “non-credit-producing activities” prosecute a Fields Medal, and not contribute to it?
On a individual note, one merchandise from the Geometry Center he referenced had an particularly meaningful effect on me whenever I was younger. It was a abbreviated movie called Outside In, perchance the earliest example of a viral video concerning substantive math, visualizing the key idea of Thurston’s own building for domain eversion.
An first evidence that showed an eversion must exist, say Smale’s, advances individual understanding in the awareness of going from 0 to 1. A video akin this which gets millions of group to affect alongside the underlying idea, advances it in the awareness of going from 1 to N. I’m grateful that Thurston spent so much period on this “non-credit-producing” activity.
Another applicable document is Timothy Chow’s A beginner’s guide to forcing. Not lone does the document itself recommendation a premier example of a motivated explanation, but its introduction offers helpful vocabulary about it.
All mathematicians are acquainted alongside the idea of an open investigation problem. I propose the small acquainted idea of an open exposition problem. Solving an open exposition issue method explaining a mathematical topic in a way that renders it totally perspicuous. Every stage have to be motivated and clear; ideally, students should awareness that they could have arrived at the results themselves.
What would it appearance akin for these open exposition problems to be treated likewise to open investigation problems? As an extreme case, we power ideate what it could appearance akin to have an analog of the Millennium Prize Problems for open exposition problems. An institution or collection of researchers would formally define mathematical results they see as important, and which are not yet fine understood notwithstanding technically having proofs. At the moment, all AI-generated evidence is delivered an unsolved exposition problem. As such, it seems apt the next few years volition see a flood of them, and it volition be precious for leaders to explain which ones deserve focus.
A rubric would have to be accepted upon for what constitutes a resolution to an crucial unsolved exposition problem. Again, this is intrinsically additional subjective than verifying a proof, but any grave involvement alongside the additional individual aspects of math necessarily wades into this benevolent of subjectivity. And again, I’ll emphasis that checking whether key ideas are motivated is not dissimilar checking whether the steps of a evidence prosecute logically.
If the earth exterior of math sees its foremost figures treat open exposition problems alongside the identical seriousness as they treat open investigation problems, it could go a lengthy way to correcting misconceptions concerning the function of mathematicians.
The final example I’ll emphasize is one that may improved foreshadow things to come.
In April of this year, Liam Price submitted a resolution to Erdős Problem 1196, sometimes called the asymptotic primitive sets conjecture. The resolution came from Price’s communication alongside GPT-5.4 Pro. Unlike many before Erdős problems which had been resolved alongside assistance from AI, this is one that those in the site had established the two crucial and elusive. Stories akin this are increasingly acquainted these days, but at this item in the story, notwithstanding a evidence technically existing, individual understanding had not yet been advanced all that much.
The evidence made its way to Nat Sothanaphan and Jared Lichtman, who were capable to construe what the AI’s method was and spotless up the evidence 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 lone the first problem, but many about it, for case offering a cleaner evidence of the Erdős Primitive Set Conjecture.
The value current is not that one additional Erdős issue could be ticked off as solved. The value lies in the fact that our understanding of primitive sets is notably cleaner and additional satisfying now than it was at the commencement of 2026. The first issue resolution played several function in this, but arguably the activity that deserves additional ritual is this document expanding, clarifying, and contextualizing its key idea.
Practical calls to action
At a pragmatic level, what would it appearance akin for us to elevate the position of a motivated explanation? Here are a small fistful of suggestions.
- A PhD adviser can motionless allocate a small issue from their site for a new pupil to cut their teeth on, but the deliverable would not be to compose up a solution; it would be to current that resolution to peers and department as a talk. It could be an already-solved issue which lacks clarity, or perchance it’s a issue that lacks a solution, and the pupil uses AI to assistance discover it. In either case, the pupil knows that on a certain date they have to comprehend it fine adequate to explain it, and that the desired output is for others to comprehend it as well. In short, equal small problems could be treated akin small PhD defenses.
- A foremost fig (cough, Terry, cough) could enumerate a contemporary analog of Hilbert’s problems, alternatively focusing specifically on unsolved exposition problems. What areas are the two crucial and lacking in the deeper understanding we desire?
- Written standards could explain what constitutes a motivated explanation, aiming to create it nearly as verifiable as proof, so that the resolution of unsolved exposition problems can be recognized and celebrated in the identical way proofs of open problems can be.
- Journals can be established which concentration additional explicitly on making results understood additional extensively throughout the math community. Mathematical Discourse offers an engaging new example in this direction.
- Hiring and tenure decisions could location a higher value on penning awesome textbooks and akin work. Think of the AMS Steele Prize for Exposition, but at a additional granular measure alongside an accent on early-career contributions in this vein.
The value of apparent traditional shifts
I’d akin to near alongside a broader throw that apparent civilization shifts in math transport an intrinsic advantage right now alongside regard to the external depiction of math as a career.
Many young students who are alternatively passionate concerning the site are fearful to prosecute it now because of the doubt of what happens in an age of proof-generating machines. However, framed correctly, this is among the most breathtaking times to go into the field, since there is nothing additional breathtaking than entering a site whenever it is malleable and you have a chance to actively form what it volition appearance akin in the future. Even if we entirely set apart any possible benefits from AI to assistance alongside our understanding, young prospective mathematicians should awareness energized knowing that they are entering at a distinctive item in former whenever they power perform a genuine function in determining what the site as a entire looks like.
However, alter akin this is lone breathtaking whenever it feels deliberate, whereas it feels terrifying if it seems driven by forces exterior your control. As such, tangible act from the field’s leaders now to assistance define and explain what the site is volition reassure young entrants concerning who is in the driver’s seat, and that the position of the occupation does not depend on what entities create the proofs.
Similarly, I additionally accept this is among the finest times to prosperity math. If the next division of math is ushered in by the drumbeat of two words “human understanding”, any changes are concerning to happen appear apt to amplify math’s value as a community good.
