The Marvelous Proof Still Lives in the Margin

Lex Fridman opens with an idea Luís Batalha had floated on Twitter: imagine if most papers carried a backstory section the way they carry an abstract. Knowing how the authors ended up on a problem, Luís argues, gives the work a context that the isolated PDF never will. People still hold a romantic picture of science as a string of eureka moments, two papers written in a row. Look closer and the papers are chapters of a much messier story.

Richard Feynman's quantum electrodynamics paper is the example Luís keeps coming back to. Before writing it, Feynman was in a dark stretch at Cornell, enthusiasm for physics gone. In the cafeteria he watched someone throw plates in the air and noticed two motions at once: the plate wobbled, and the Cornell seal on its face rotated. He wrote down the equations of those motions, which pulled him toward electron orbits in relativity, which became the QED paper that led to his Nobel Prize. A plate in a dining hall restarted a scientific life. Readers of the paper never see it.

Fermat's Library later ran an AMA on Ian Goodfellow's GAN paper, with Goodfellow answering questions live and retelling the bar-argument origin story. João Batalha adds Edsger Dijkstra, who invented the shortest-path algorithm in about twenty minutes on a café terrace in Amsterdam with no pen and no paper, the whole thing held in his head. That constraint, João says, is why the algorithm is so simple: a mind can only carry so much complexity. Cédric Villani's memoir of the road from graduate student to Fields Medal is, in João's reading, mostly a catalog of chance conversations over coffee. Backstories make the result memorable, and memorable knowledge is the kind that actually sticks.

Lex wonders how often the leap really is a night's work rather than an Andrew Wiles grind. Goodfellow, he thinks, carries a kind of sadness about that: in the early GAN days one person could go from idea to experiment on a single machine in a single night, and now many breakthroughs demand large-scale runs. João looked up arXiv papers tagged machine learning and found that about half had been posted in the previous twelve months. In a field still exploding, low-hanging fruit is everywhere. In a mature field such as mathematics, those bar-and-a-napkin moments get rarer.

The apple, they agree, is probably a later invention. What the record does show is a Newton who spent years on alchemy, took it seriously, wrote about it, and dug into biblical interpretation in search of prophecy. Those chapters get stripped out of the children's version, Luís says, but they were part of the man, and without them who knows what else he would have done. Lex offers alchemy a defense: it stands for dreaming of the impossible. Human flight once looked that way. Once you know atoms have protons, making gold is conceptually just shoving in another proton — and you can, at a ridiculous price. Standing in Newton's century, alchemy may not have been crazier than ideas that later came true.

The scientist who most marks Luís is Freeman Dyson. Fermat's Library asked him to annotate a paper. Dyson's first reply was that he had very limited knowledge, knew only a couple of things about certain fields, and was not sure he was qualified. This from someone Luís thinks should have won a Nobel, who had worked across quantum electrodynamics, reactor design, and the search for other civilizations. In later notes Dyson would say he was not there to answer questions, only to open more of them. That humility, Luís says, left a mark.

On Dyson's birthday — he was already ninety, maybe ninety-two, and deep into number theory — Luís turned a photograph into a giant prime. Darker regions became eights, lighter ones ones; cheap primality tests threw out the composites, then a full test finished the job. Dyson wrote back to ask how it was done, still curious. Lex has seen the same fire in Daniel Kahneman: the childlike refusal to believe the world is already figured out, still burning after the prizes. That stance, he says, is step one for a real conversation, a collaboration, or a scientific question.

Fermat's Library is, at heart, a platform for annotating papers. Academic PDFs are among the densest documents in circulation, João says, and often hard to follow. The interface is a browser PDF with notes down both sides; click one and a sidebar opens, with LaTeX and Markdown. An annotator can fill in a step the author treated as obvious, or add context a newcomer would never have. Over time the paper gets easier without rewriting the original. It started as a journal club among João, who studied computer science, Luís, who studied physics, and two other friends: they read papers and explained them to one another, translation built in. Then they tried to put that habit online.

The name comes from Fermat's Last Theorem. Pierre de Fermat was a lawyer who did mathematics on the side. In the margin of a Diophantus volume he wrote that he had a marvelous proof of a statement — no positive integers a, b, c satisfy a^n + b^n = c^n for n greater than two — but the margin was too small to contain it. The claim sat open for almost three hundred years, until Andrew Wiles published a proof in the 1990s that ran, Luís recalls, to nearly three hundred pages. The anecdote is the product thesis: a lot of knowledge is trapped in the margins, and there is potential energy to release if someone will sit with it. Journals then squeeze breakthroughs into page limits, so papers are not optimized for clarity. The margin is where the missing explanation can live.

Around the site they built three pieces. Journal Club is the public face: every week, an annotated paper from physics, computer science, or math. Margins is the same software, given away free, and universities use it to run their own clubs. Librarian is a browser extension laid over arXiv, so comments, reference extraction, and BibTeX export happen without leaving the preprint. Lex's reaction to Librarian is immediate: given how many people live on arXiv, why is this not everywhere? João's caution is that arXiv's growth has been almost a straight line for twenty years. Treat it like a startup, add growth hacks and paid features, and you might ruin the thing that made it last. Some scientific tools just take time. arXiv had just turned thirty.

arXiv began in mathematics and physics; over thirty years computer science passed both. For some math papers, João says, the wait from clicking upload on a journal site to seeing the article online can approach three years — possibly the longest upload on the internet — and during that time the work is locked. That is why preprints matter. Perelman's three papers on the Poincaré conjecture went up on arXiv, not in a journal. In machine learning the field moves so fast that people build on the preprint and never wait for the journal, which creates a second problem: everyone assumes the preprint is right. arXiv hosts PDFs well. João wants the layer on top — comments, annotations, code, some way to tell careful work from a fast dump.

Biology has a harsher history. In the 1960s the NIH backed an Information Exchange Group that let biologists share preprints by mail and through libraries. Six years later the project was cancelled under pressure from journals that feared the competition. Francis Crick, João notes, was among the famous scientists who opposed it. Decades on, only about two percent of biology papers appear first as preprints. A successful kill delayed a revolution by half a century. bioRxiv is trying again, but the field's habits were rewritten, and catching up takes time. The organization of science, on this telling, is not a natural evolution. It still carries the marks of an industry that fought to keep the gate.

Lex compares today's open-science tools to Craigslist: grateful they exist, frustrated they feel a decade behind. The pushback is Wikipedia, which has the same kind of simplicity and somehow works. Luís calls Wikipedia the cosmological constant of the internet: we are lucky to live in the parallel universe where it exists. Pitch a publicly edited encyclopedia twenty years ago and almost nobody would have bet it would become a reliable source — a Google Doc anyone can edit, as the authority on hundreds of thousands of topics. One user, he says, edited a third of the articles. Stack Overflow is the other miracle: a handful of obsessive contributors create enormous value and capture almost none of it. If you want arXiv or open review to work at that scale, Lex argues, design for the 10x person who will answer thousands of questions, not for everyone equally.

Tim Gowers, a Fields Medalist, ran the Polymath projects to test whether mathematics could be done as a mass collaboration on the internet. It can, João says, for problems that break into pieces and move step by step. Someone has to play maintainer: every so often reorganize the house, reframe the question, then let people throw in more ideas. That unglamorous role is the difference between a thread and a proof.

Zoom out, Luís says, and the publishing loop looks bizarre. Scientists on government grants give journals their work for free. Other scientists review it for free. Government-backed universities then buy the work back so researchers can read it. The government funds the research, the salaries, the reviews, and the product a second time. The equilibrium is hard to break because academic careers run through it. Hundreds of PhD students chase a faculty job, and the path is papers in high-impact-factor venues — in computer science, often prestigious conferences instead. Impact factor, the average citations a paper would get in that journal, is in Luís's phrase a nobility title: the paper is born with impact before anyone reads it. Researchers optimize for acceptance, not for whether the work will matter in ten years. Deans hire the people who publish in those journals because rankings and tuition follow. Mario Biagioli's *Gaming the Metrics* is the book Luís points to on how the numbers start to eat the science.

Citation indexes began as description. Eugene Garfield wanted a map of influence, links between papers, not a scoreboard. Larry Page and Sergey Brin have acknowledged that this work helped inspire PageRank. Once you count the links you get a market, and the SEO industry of science appears. The distribution of impact factors, Luís says, is two economies: *Nature* in the low forties, *Science* in the high thirties, a band of strong journals between ten and thirty, then a gigantic tail below two. In the tail, less famous universities — a lot of the Global South — push faculty to publish anywhere. At the luxury end, *Nature* cannot hold a factor in the forties on scientists' attention alone; articles have to be New York Times–grade, which pushes titles and abstracts past what the data will bear. Lex asks whether the disease is metrics or bad metrics. Amazon reviews work for computer parts because thousands of people contribute. Peer review is five, six, seven people, often fewer, anonymous, with no reward for doing it well. During COVID he watched people with basic data-analysis skills catch a weak step or a missing citation. João grants a Heisenberg problem: pick a metric and people start to game it. You can build more resilient ones. They still decay.

Overlay journals try to cut the middleman. They are, Luís says, collections of links to papers already on arXiv or bioRxiv, with a more transparent review process and no subscription bill for universities. Discrete Analysis does this in mathematics; Quantum does it in quantum information, and Fermat's Library partners with Quantum so those papers open for annotation. The hard part is critical mass. Young researchers still need impact factor. The workaround is to get tenured people — Gowers is named — to publish there first and lend the venue a reputation the next cohort can risk. Lex's blunter proposal is to treat paywalled journals as illegitimate: most of the work is publicly funded, the labor is unpaid, and most people cannot read the result. During 2020 some journals dropped the wall on coronavirus papers, which to him only proved the wall should never have been there.

Luís thinks the journal business model is already past the event horizon: a middle layer that adds less and less. Papers funded by the European Union are required to be open; the Gates Foundation makes a similar demand. Funders have noticed they are paying twice. Lex's comparison is sharp: open Spotify and you can hear almost any song ever recorded, even on the free tier, but a huge share of academic papers — he had seen an analysis putting more than seventy percent behind a paywall — remains closed, even though the public funded them. The music-rights problem was harder, and it got solved. Science did not.

If you want science to move, Lex has come to think, couple the paper with video. João holds the first issue of *Nature* next to a paper from today and sees the same object: a static PDF, code stuffed in as a typesetting element, data and process left outside. Papers 2.0, in his view, will have to hold code, datasets, and the moving pictures some ideas need. They even ran a small experiment on publishing a result as three or four tweets. Lex likes the constraint of a formal paper — it forces you to narrow the claim and put it somewhere others can cite — but he does not think the journal article should be the only legitimate unit of knowledge. A blog post, a thread, or a carefully recorded argument could be, if someone will give it a stable reference.

João's first advice is psychological. Papers are not written to be easy: size limits, format rules, and the dialect of the field all squeeze the text. If you are struggling, that may not mean the material is harder than you are. Fermat's Library often annotates work far outside the brothers' comfort zone, so they have to keep that fact in mind. Feynman had a hack: read the conclusion, then spend a few minutes trying to reach it himself before looking at the author's path. Fermi was famous for the same spirit, the back-of-the-envelope estimate.

The honest experience, João says, is worse than a slow climb. You open a paper, understand almost none of it, jump to a citation, then to that paper's citations, twenty tabs open, understanding stuck near zero percent. Then, at some point, it connects. He calls it a quantum jump: comprehension is not linear. You have to trust a process that looks like being lost. A lot of people quit before the jump.

Lex's method is breadth-first. Read the conclusion, then the related-work section, then the conclusions and related work of the papers it cites, down the tree until you hit dead ends or run out of coffee. Coming back up, you can place the original result where it belongs — and if the paper is truly revolutionary, that walk is how you see it. Related work, he says, usually carries the least agenda. The introduction is selling; methods are proving superiority. Related work is the map: where the field came from, where it is stuck, who refuses to cite whom. It is also, often, the most Wikipedia-like stretch of the paper, which makes it the easiest to read.

On notes, Luís's rule is not to underestimate how much you forget within six months. He will open a paper he annotated on Fermat's Library, read his own comments, and not recognize them. Right after you understand something, he says, you are the best possible teacher of the future self who will have forgotten it. You just walked from one to ten on that idea; you know which step hurt, which analogy helped, which symbol you misread. A textbook does not have that information. Only you do.

Lex finds written notes often fail him — he stares at a line and cannot recover the thought — while a short audio rant reloads the room, including the confusion. Video might be stronger still. The tools for a conversation between yesterday's expert self and tomorrow's forgotten self, he thinks, are still underbuilt. João adds that saying it out loud already helps: to make it understandable you have to restructure the thought, and the restructuring is the understanding.

They have not seen many people build videos on top of Journal Club annotations, though they have toyed with a podcast that distills a paper in half an hour, sometimes with the author. Lex points to Machine Learning Street Talk, multi-hour breakdowns, sometimes with the writer in the room, and to George Hotz of comma.ai live-coding a paper for ten hours. People watch the struggle, not the polished result — the same reason, Lex says, anyone listens to a podcast. A YouTube channel of a restorer quietly repairing old paintings pulled them in the same way. Domain is secondary. Excellence is contagious, and Twitch proved there is a market for watching someone good stay inside a hard problem.

Among the annotated papers João loves for their strangeness is one by the mathematician Tom Apostol on a tunnel on Samos, a Greek island off the coast of Turkey. In the sixth century BC the city sat on one side of a mountain and the springs on the other. An aqueduct was a bad idea: the terrain was wrong, and under siege an enemy could break it and dry the city out. They dug from both ends at once and planned to meet in the middle — a hard surveying problem with the mathematics of the time. For about two thousand years the accepted account, associated with Hero of Alexandria, was a right triangle whose hypotenuse joined the two mouths, walked out around the mountain on a grid, then scaled down at each entrance so the arrows pointed at each other. It looks perfect on paper.

Researchers who actually went to the mountain found the method fatal. Measurement error would accumulate; the crews would miss each other by meters. The plausible alternative João prefers is closer to an iron sight on a rifle: three sticks, lined up, carrying a level line along the mountainside until it could be cut through. A mathematician with no archaeology background reconstructed this. After the fall of Rome, João notes in an annotation, people did not build tunnels again for about a thousand years. On Fermat's Library the page grows instruments, later studies, and that historical gap as comments. The danger, they have heard from users, is that a beautiful set of notes can look finished, and later readers are afraid to add a stupid question. Lex's answer is to keep asking dumb questions in public. That, he says, is the only way to knowledge.

Some of the papers on the site are a page, or shorter — including what they call the shortest paper ever published in a math journal, a couple of words. One of Luís's favorites is Enrico Fermi's classified one-pager, *My Observations at Trinity*. A few miles from the first atomic explosion, Fermi tore a sheet into scraps, waited for the blast, and threw them in the air, estimating yield from how far they blew. He was perhaps twenty or twenty-five percent off; later work from the fireball's expansion did better. When a video of a Beirut bride went around, the shock wave lifting her dress, João used the same method, reading distance from a shop sign behind her and yield from the fabric's swing, and news outlets picked up the thread. Fermi problems, Luís says, are a style: one page, the whole argument in the open.

The first paper they put on Fermat's Library was the Bitcoin white paper, a text João had once spent weeks unpacking despite an undergraduate background in cryptography — language so spare he felt you could not pull a word without the rest falling down. The annotations walk an undergraduate through public-key cryptography, hash functions, and Merkle trees so the original can stay as Satoshi wrote it. Freeman Dyson's Dyson-sphere note is another one-pager: a civilization outgrows its star, wraps it in a collector, checks the shell's stability, and, by the second law of thermodynamics, predicts a sky full of waste infrared as a search strategy for intelligence. Concept, engineering, stability, and a detection method, Luís says, all on a single page. They also host the DNA double helix and the Higgs boson papers, not because readers will discover them there, but because those texts should be widely read with help in the margin.

Grigori Perelman posted the Poincaré proof as three arXiv papers rather than a journal article, which Luís reads as a protest against paying to read science. For him Perelman is a Platonic ideal of a mathematician: he cares about the mathematics and about fair credit, and he disregards money. In his late twenties he proved the soul conjecture in four pages, lectured at Princeton, and when the chair, Peter Sarnak, tried to recruit him and asked for a CV, Perelman asked why, having just lectured on a hard theorem, he needed to send a résumé. Back in Russia, postdoc pay had fallen so far behind inflation that people did not bother collecting the checks. He had saved enough from the American postdoc to work for about seven years on Poincaré as Richard Hamilton had reframed it — a problem with clean boundaries, almost an olympiad statement, which suited him.

After 2002, two researchers, Luís believes from Harvard, sanded some edges of Perelman's argument and spoke as if they had settled the conjecture. By the 2006 International Congress of Mathematicians there was a real debate about credit. For Perelman, Luís says, the sadness was that anyone considered the question. That was enough to refuse the Fields Medal. He later refused the Clay Mathematics Institute's million-dollar Millennium Prize as well: the award required journal publication, and in any case Clay was a businessman who, in Perelman's view, had nothing to do with mathematics. Lex recognizes two musics in an MIT room: the sound of people talking about ideas, and the sound of funding, politics, and credit. Happiness, for him, is staying in the first. The moment drama about assignment enters, the exploration is already contaminated. Soviet-era math, done for little money, produced people like Perelman; capitalism, Lex admits, also works frighteningly well. The balance is not obvious. Poverty is not good for science. Neither is a culture that cannot talk without smelling money.

Terence Tao is the other pole: a public mathematician whose blog puts the research, the dead ends, and the failed attempts where anyone can see them. Working on the Erdős discrepancy problem, he posted a stall. A reader in Germany commented that the structure had a Sudoku flavor, and that the machinery used to solve Sudoku might apply. The comment was the key. Tao has also written about the downside of early gift. An IMO gold medal around age ten, classes he could ace without notes, and then Princeton, where the absence of study habits nearly ended the PhD; he almost failed the qualifying exam, and a conversation with his advisor — this is not working, you may have to leave — became a turning point. Lex pairs him with Scott Aaronson: different styles, same gift of a blog that lets outsiders watch a first-rate mind doubt itself. That material never makes the paper.

Fermat's Library's Twitter account is, Lex says, a trapdoor into rabbit holes. João's theory of it starts with mathematics' half-life: Euclid's *Elements* is as valid as it was two thousand years ago, which is not true of most scientific fields. Twitter is underexplored as a place to learn. A compact explanation delivers a dopamine hit, the same hit that makes a great lecture addictive, packed into 280 characters. Easy to consume, hard to make. Once the eureka lands, people come back. Dropout, compared with a course or a book, is roughly zero. Lex reads a recent post: 23456789 is the largest prime with consecutive increasing digits. It is a glimpse of some larger order, arbitrary and pleasant. Luís's line is that they are math drug dealers. They start you on primes and send you into number theory for the afternoon.

Andrew Wiles has said that most people can reach a good level of mathematics if they can handle being stuck. João applies the same sentence to research and to startups. The default state is stuck; you close doors until one opens. Social media, Lex thinks, makes that state intolerable, because the mind can always flee to something that feels better. Sometimes you have to sit in the stuckness and suffer a little. João cites Ed Sheeran on writing: turn on the faucet and dirty water comes first; you keep it open until the clear water arrives. The first twenty minutes are probably garbage, and you cannot skip them. Startups, João adds, do not have a linear reward function. School pays effort with grades. A company can take a mountain of work and return nothing, a Sisyphus boulder. Most people are not trained for that, and there is no clean rule for when to persist and when to pivot. His imperfect test: when you have no cards left, switch. Fermat's Library they treat as a twenty-year side project, not a three-to-five-year startup, because some scientific tools will not be rushed. Run Wikipedia as a startup, Lex says, and by now you would be paying $9.99 a month after twenty-four articles.

During the last season of *Game of Thrones*, over a company lunch, João wrote a serious little paper on the Battle of Winterfell. Each death in the living army adds a soldier to the dead; both sides start at about a hundred thousand. He wrote differential equations and found that each living fighter had to kill about 1.7 enemies to keep from being overrun. No dragons, no tactics — the show solved it with an assassin instead — but some of the cast retweeted it. What Luís loves is the intellectual democracy of the setup. The bounds are obvious, the initial conditions fit on a napkin, and a reader with no specialist training can follow the derivation. Fermat's Last Theorem has the same shape: a sentence a secondary-school student can understand. Problems like that, he says, put everyone at the same starting line. Someone had even taken Tolkien's map of Middle-earth, used distances between cities, and argued the map fits a sphere whose radius you can estimate. Lex jokes that Jocko Willink would reject the homogeneous-army assumption on tactical grounds. João never claimed the dragons were in the model.

Asked the most important question — both brothers are Portuguese — Luís takes Cristiano Ronaldo. Football, he says, will let pure talent reach the top with little discipline; Ronaldo had talent and then became famous for work, still elite near forty, having come from nothing. People resist crediting hard work, Luís thinks, because if success is grind, then failure is on them; if it is mysterious gift, it has nothing to do with their choices. Lex disagrees. He loves the Ronaldo story and will not pretend winning is nothing, but what he watches for is an aesthetic flash of genius that may not even show up as a goal. His ranking is Lionel Messi, then Diego Maradona, then Pelé. Maradona's drugs and temper do not cancel the greatness; the flaws are part of the figure. Winning, Lex says, was never the point. The moment was.

He has tried to quantify this. In college he worked on machine learning for sports, including camera tracking with the Boston Celtics. Baseball, he says, yields to statistics because it is discrete events in repeatable conditions; basketball is a continuous flow; soccer is harder still, almost a Poisson process, one or two goals a game, too few successes to hang a ranking on. Chess can score how deep in the tree a move sits — Bobby Fischer's "move of the century." A similar depth metric for a pass or a feint would need computer vision, and the nasty problem is identity across camera cuts: change the broadcast angle and the 3D reconstruction has to start again. Release tracking data for every Messi and Ronaldo match, Lex says, and the interesting papers would appear on their own.

João annotated an older basketball paper on the hot hand. The researcher surveyed about a hundred undergraduates, roughly sixty-eight percent of whom believed that making a free throw made the next one more likely, then looked at some five thousand free throws and could not find the effect. Players were simply a bit better on the second attempt, a practice effect, not momentum. João reran the calculation on a Kaggle set of about six hundred thousand free throws and saw the same pattern. Sports, he and Lex agree, are a constrained psychology lab: repeated actions, measurable outcomes, and a public that actually cares. That combination is rare in the wild.

Advice for the young, Luís borrows from Poincaré: among the things parents should teach is enthusiasm for the world's mysteries, and the hard part is dosage — enough wonder, not trauma. Start small and let curiosity grow. You might only care how numbers factor into primes; follow it and you fall into number theory, then cryptography, and one day you are buying bitcoin. Do it not for status or money. The iterative satisfying of curiosity is, for him, how you find happiness and something like meaning. João, who went to university in Europe, describes a rigid system: choose physics and you can barely take a computer-science class. Luís, who studied in the United States, had more room to wander. A good education, João thinks, should move with the student's interests, but only if the system first leaves space to explore. A physics training has a hidden dividend: after something that hard, finance did not feel intimidating. Lex's version is broader. Mathematics is probably the hardest, physics next, but the transferable asset is having done a difficult thing and proved to yourself that you can. He closes with Feynman: nobody ever figures out what life is all about, and it does not matter. Explore the world. Nearly everything is really interesting if you go into it deeply enough.