If math is greater than proof, we have to higher have a good time the remainder of it

[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 all through the arithmetic group proper now could be that fixing issues and producing proofs have at all times served as proxies for the true purpose of mathematicians, which is to additional human understanding. When proofs might be generated with out that understanding, it undermines their worth as a proxy.

This instantly raises a query: What different proxies ought to we use as an alternative?

I wish to suggest that we extra firmly outline a notion of a “motivated clarification” and that we give novel and compelling motivated explanations educational credit score just like what producing new proofs of open issues has had traditionally.

Further, I imagine this is a vital step to assist these exterior of math higher perceive what it’s that mathematicians contribute. If outsiders imagine that proof-generating machines render mathematicians out of date, whereas insiders see that as a false impression of what researchers add, it’s incumbent on this group to higher challenge its true values by way of the form of work that it rewards. Outsiders might be forgiven for this misunderstanding if the work most celebrated skews closely towards producing proofs, whereas clarification and exposition are handled as second-class.

I ought to acknowledge up entrance an apparent private bias. I’ve a non-traditional profession in math, centered on producing movies concerning the matter. This shares the purpose of “furthering human understanding”, however my focus has been on explanations and intuitions that resonate with the general public, not on fixing excellent issues. A cynic may simply learn this proposal as shamelessly self-elevating.

As a sensible matter, although, my very own profession and funding exist exterior academia, and I’ve no pores and skin within the recreation for what this group assigns credit score to. Moreover, in proposing that we elevate the standing of motivated explanations, I don’t imply popularization. I imply any work which primarily goals to reply the query “how would you consider that?”, even when the subject material requires deep experience to understand.

The examples I spotlight beneath present that is nothing new. Practicing mathematicians already dedicate a significant quantity of mindshare to work like this. The proposal right here is principally to 1) extra clearly outline this work, and a pair of) elevate its standing.

What defines a motivated clarification?

Although it is likely to be clear what this phrase “motivated clarification” is meant to imply, it’s price briefly contrasting it with proof.

In a proof, definitions sit at the beginning. It is frequent and anticipated to start with a brand new building and proceed by analyzing its properties.

In a motivated clarification, definitions sit within the center. New constructions are solely allowed to enter the vocabulary if the issue they’re addressing has been clearly established.

In a proof, all statements have to be appropriate, every declare following as a vital implication from what comes earlier than.

In a motivated clarification, it’s okay and infrequently fascinating to begin with an concept that’s not fairly proper and requires correction, however whose origins are relatable.

A style of motivated clarification I’m keen on is “discovery fiction”, a time period coined by Michael Nielsen. You develop an concept with a story that begins with a simple-but-wrong resolution to an issue, see the place it breaks down, repair that downside, uncover a brand new downside, and so forth.

The scope of a proof is to clarify why a selected theorem is true.

The scope of a motivated clarification shouldn’t be solely to make clear why a theorem is true, however why the concept is the precise one to pose within the first place, and the way it’s used within the surrounding context.

One clear shortcoming of a motivated clarification is that its validity shouldn’t be binary the best way a proof’s is. This is an enormous cause proof is so helpful a method to measure progress: You can clearly outline what does and doesn’t have a proof but. There won’t ever be Lean for motivated explanations.

If we’re severe concerning the purpose of advancing human understanding, there’s no means round the truth that this goal is intrinsically squishier than that of discovering proofs, as a result of defining human understanding itself is squishier. To shrink back from metrics that are extra subjective is to shrink back from the extra human elements of the sphere.

The cause I’m leaning on the phrase “motivated”, versus different potential decisions like “lucid” or “demystifying”, is that it is a extra verifiable property. It’s not fairly as rigidly verifiable as a proof; nearly nothing is. But it’s sufficient to be a sensible measure. In my very own work, I typically repeat the phrase “I would like this to really feel like you possibly can have found it your self”. I say this not simply to placate a viewer, however as a result of it’s an actionable guideline for myself to evaluate whether or not an evidence feels full or not. For every new concept launched, you’ll be able to ask whether or not it’s clear the place that concept comes from. The reply shouldn’t be fairly a binary sure or no, but it surely’s shut sufficient for sensible functions.

Exemplars of motivated explanations

One of the most effective repositories I can consider for motivated explanations is Part IV of the Princeton Companion to Mathematics. It covers over two dozen lively fields of analysis, each launched by an professional with a expertise for clear communication.

Whether it’s Andrew Granville explaining analytic quantity principle, or David Ben-Zvi introducing moduli areas, these articles supply a stage of instinct and motivation extra usually present in one-on-one dialog at a blackboard.

The background on this guide is noteworthy for the current dialogue. It was edited by Timothy Gowers, who discusses it in his interview on the Numberphile Podcast with Brady Haran. Having been requested what influence the Fields Medal had on his life, right here’s what he needed to say:

People who’ve acquired Fields Medal really feel freer to do barely various things…for instance I took on enhancing a guide referred to as the Princeton Companion to Mathematics, which was a completely huge process. It took I’d estimate half my working time for about 5 years or one thing like that…It was a challenge I believed in and presumably wouldn’t I most likely wouldn’t have truly been supplied the prospect to do it if I hadn’t been a Fields Medalist.

He was proper to imagine in it; this work provides large worth to the sphere of math, but it surely appears a disgrace to me that one requires a Fields Medal to really feel justified in spending time on it.

Another instance of somebody exceptionally gifted at writing proofs, however whose contributions prolonged far past proof, is Bill Thurston. His deservedly well-known essay On Proof and Progress in Mathematics, although written three a long time earlier than LLMs, opens by suggesting that the precise framing of the query “What is it that mathematicians accomplish?” is to ask “How do mathematicians advance human understanding of arithmetic?”

Here’s one part with uncanny resonance with as we speak:

The speedy advance of computer systems has helped dramatize this level, as a result of computer systems and individuals are very completely different. For occasion, when Appel and Haken accomplished a proof of the 4-color map theorem utilizing an enormous computerized computation, it evoked a lot controversy. I interpret the controversy as having little to do with doubt individuals had as to the veracity of the concept or the correctness of the proof. Rather, it mirrored a unbroken want for human understanding of a proof, along with information that the concept is true.

On a extra on a regular basis stage, it’s common for individuals first beginning to grapple with computer systems to make large-scale computations of issues they could have finished on a smaller scale by hand. They would possibly print out a desk of the primary 10,000 primes, solely to seek out that their printout isn’t one thing they actually needed in spite of everything. They uncover by this type of expertise that what they really need is normally not some assortment of “solutions”—what they need is knowing.

The essay itself provides an exquisite articulation of what the apply of doing math is past producing proofs. I wish to draw your consideration to what he writes on the finish.

I’ve put quite a lot of effort into non-credit-producing actions that I worth simply as I worth proving theorems: mathematical politics, revision of my notes right into a guide with a excessive commonplace of communication, exploration of computing in arithmetic, mathematical schooling, improvement of recent kinds for communication of arithmetic by way of the Geometry Center (resembling our first experiment, the “Not Knot” video), directing MSRI, and many others.

Again, why ought to these “non-credit-producing actions” observe a Fields Medal, and never contribute to it?

On a private notice, one product from the Geometry Center he referenced had an particularly significant influence on me once I was youthful. It was a brief movie referred to as Outside In, maybe the earliest instance of a viral video about substantive math, visualizing the important thing concept of Thurston’s personal building for sphere eversion.

An unique proof that confirmed an eversion should exist, say Smale’s, advances human understanding within the sense of going from 0 to 1. A video like this which will get tens of millions of individuals to interact with the underlying concept, advances it within the sense of going from 1 to N. I’m grateful that Thurston spent a lot time on this “non-credit-producing” exercise.

Another related paper is Timothy Chow’s A beginner’s guide to forcing. Not solely does the paper itself supply a chief instance of a motivated clarification, however its introduction provides useful vocabulary round it.

All mathematicians are aware of the idea of an open analysis downside. I suggest the much less acquainted idea of an open exposition downside. Solving an open exposition downside means explaining a mathematical topic in a means that renders it completely perspicuous. Every step must be motivated and clear; ideally, college students ought to really feel that they may have arrived on the outcomes themselves.

What would it not seem like for these open exposition issues to be handled equally to open analysis issues? As an excessive case, we would think about what it may seem like to have an analog of the Millennium Prize Problems for open exposition issues. An establishment or group of researchers would formally outline mathematical outcomes they see as essential, and which aren’t but effectively understood regardless of technically having proofs. At the second, each AI-generated proof is born an unsolved exposition downside. As such, it appears doubtless the following few years will see a flood of them, and it will likely be worthwhile for leaders to make clear which of them deserve focus.

A rubric must be agreed upon for what constitutes a decision to an essential unsolved exposition downside. Again, that is intrinsically extra subjective than verifying a proof, however any severe engagement with the extra human elements of math essentially wades into this type of subjectivity. And once more, I’ll emphasize that checking whether or not key concepts are motivated shouldn’t be not like checking whether or not the steps of a proof observe logically.

If the globe exterior of math sees its main figures deal with open exposition issues with the identical seriousness as they deal with open analysis issues, it may go an extended method to correcting misconceptions concerning the position of mathematicians.

The final instance I’ll spotlight is one which will higher foreshadow issues to return.

In April of this 12 months, Liam Price submitted an answer to Erdős Problem 1196, generally referred to as the asymptotic primitive units conjecture. The resolution got here from Price’s interplay with GPT-5.4 Pro. Unlike many earlier Erdős issues which had been resolved with assist from AI, that is one which these within the discipline had discovered each essential and elusive. Stories like this are more and more acquainted nowadays, however at this level within the story, regardless of a proof technically current, human understanding had not but been superior all that a lot.

The proof made its method to Nat Sothanaphan and Jared Lichtman, who had been capable of interpret what the AI’s strategy was and clear up the proof right into a human-readable type. 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 important thing concept underlying the proof. The authors defined how that key concept clarified not solely the unique downside, however many round it, as an illustration providing a cleaner proof of the Erdős Primitive Set Conjecture.

The worth right here shouldn’t be that yet another Erdős downside may very well be ticked off as solved. The worth lies in the truth that our understanding of primitive units is notably cleaner and extra satisfying now than it was at the beginning of 2026. The unique downside resolution performed some position on this, however arguably the work that deserves extra celebration is that this paper increasing, clarifying, and contextualizing its key concept.

Practical calls to motion

At a realistic stage, what would it not seem like for us to raise the standing of a motivated clarification? Here are a small handful of options.

• A PhD advisor can nonetheless assign a small downside from their discipline for a brand new scholar to chop their enamel on, however the deliverable wouldn’t be to put in writing up an answer; it might be to current that resolution to friends and school as a chat. It may very well be an already-solved downside which lacks readability, or maybe it’s an issue that lacks an answer, and the coed makes use of AI to assist discover it. In both case, the coed is aware of that on a sure date they’ve to grasp it effectively sufficient to clarify it, and that the specified output is for others to grasp it as effectively. In brief, even small issues may very well be handled like small PhD defenses.
• A number one determine (cough, Terry, cough) may enumerate a contemporary analog of Hilbert’s issues, as an alternative focusing particularly on unsolved exposition issues. What areas are each essential and missing within the deeper understanding we want?
• Written requirements may make clear what constitutes a motivated clarification, aiming to make it almost as verifiable as proof, in order that the decision of unsolved exposition issues might be acknowledged and celebrated in the identical means proofs of open issues might be.
• Journals might be established which focus extra explicitly on making outcomes understood extra broadly all through the arithmetic group. Mathematical Discourse provides an fascinating new instance on this route.
• Hiring and tenure choices may place the next worth on writing nice textbooks and related work. Think of the AMS Steele Prize for Exposition, however at a extra granular scale with an emphasis on early-career contributions on this vein.

The worth of seen cultural shifts

I’d like to shut with a broader pitch that seen tradition shifts in math carry an intrinsic profit proper now with respect to the exterior picture of arithmetic as a profession.

Many younger college students who’re in any other case passionate concerning the discipline are afraid to pursue it now because of the uncertainty of what occurs in an age of proof-generating machines. However, framed accurately, this is among the most fun instances to enter the sphere, as a result of there’s nothing extra thrilling than coming into a discipline when it’s malleable and you’ve got an opportunity to actively form what it would seem like sooner or later. Even if we fully put aside any potential advantages from AI to assist with our understanding, younger potential mathematicians ought to really feel energized realizing that they’re coming into at a novel level in historical past after they would possibly play an actual position in figuring out what the sphere as an entire appears like.

However, change like that is solely thrilling when it feels deliberate, whereas it feels terrifying if it appears pushed by forces exterior your management. As such, tangible motion from the sphere’s leaders now to assist outline and make clear what the sphere is will reassure younger entrants about who’s within the driver’s seat, and that the standing of the profession doesn’t depend upon what entities produce the proofs.

Similarly, I additionally imagine this is among the greatest instances to fund math. If the following chapter of math is ushered in by the drumbeat of two phrases “human understanding”, no matter adjustments are about to occur appear prone to amplify math’s worth as a public good.