The IBM Manual Was So Bad He Became a Computer Scientist

Lex Fridman opens on Knuth's first large-scale program, written in the summer of 1957. Knuth corrects the premise at once: he wrote it in decimal machine language, not assembler, which almost nobody had heard of yet. You looked at the user manual. Sixty-nine meant load the distributor; then you gave the address. Each instruction was ten decimal digits: two for the operation, four for the address it acted on, four for where to fetch the next instruction. The whole machine had two thousand words of memory, addressed from 0000 to 1999.

The day before the taping, Dag Spicer at the Computer History Museum had sent Knuth a YouTube link: IBM's 1956 progress report. The film shows the assembly line around machine number 500, and classrooms at Stanford, MIT, and Knuth's own Case Tech after IBM donated 650s. A teacher writes 69 8000 on the blackboard and teaches students to program the decimal machine. Knuth says he started in a room like that. If the accompanying manual had been well written, he probably never would have gone into computer science. It was so bad that a freshman figured he could do better, took a job at the computing center, and began writing manuals.

The first program was not tic-tac-toe. It factored a number you dialed on the console switches and punched the factors onto cards. It started around twenty lines; debugging made it grow. At night, when nobody else was around, Knuth sat at the console, single-stepped, and set a stop address on four switches so he could watch the machine divide by two, then three, up to the square root. The first night he probably got 30 equals 2 times 3 times 5. Then the real bugs arrived: some numbers had more than eight factors, a card had only eighty columns, and ten columns per factor meant a second card he had never planned for. Feed it a large prime and the 650 might spin for ten minutes, too slow to tell a long calculation from an infinite loop.

His goal was not a record prime. He wanted to see the lights flash and watch a machine do what took forever by hand. The second program, binary to decimal, was much simpler. The third was tic-tac-toe. Those years, machine time meant night. Between freshman and sophomore year he barely needed sleep; in high school he had already stayed up every Monday putting out the student newspaper. Rodney Brooks later told Fridman he was intimidated when Knuth, very tall, stood waiting for a broken machine so he could get back to TeX. Knuth's own version is drier: the Stanford AI Lab's operating system was rewritten by talented programmers every day, so it got better and crashed more, and he wrote almost the entire TeX manual in the downtime.

Each cell of a three-by-three board can be empty, X, or O, so there are 3^9 positions. The 650 had twenty thousand decimal digits in total, and that budget had to hold the program, the prompts, and the output logic besides a memory of positions seen. Knuth used symmetry: rotate the board four ways and flip it, and eight layouts collapse into one, cutting the table by about eight. Each position stored a single digit, zero for bad and nine for good, starting around five or six. Win, and you raise the value for yourself and lower it for the opponent.

He split the program into three brains. Brain one moved at random into an empty square. Brain two used a canned strategy known to be optimal. Brain three was the learner. He could pit any pair against each other, or against a human. Two copies of brain one, knowing nothing, converged after about six hundred games to a safe draw. The result is easy to miss: random versus random ended in two players who no longer made mistakes. Knuth is explicit about why. The way he designed the learning, it learned how not to make mistakes. It did not try to win. It tried not to lose. A different reinforcement function, one that rewarded brilliant play, would have produced a different creature.

Charles Babbage, he later learned, had already thought about programming tic-tac-toe for the unfinished Analytical Engine more than a hundred years earlier. Knuth's own spark was closer to home: a Bell Labs relay exhibit at Chicago's Museum of Science and Industry that played the game, seen before he was a freshman. After the program, he felt he understood programming. He did not yet think about Turing machines or what computation *is*. That felt too philosophical for a freshman tinkerer. He took computability from Martin Davis as a senior because he needed to pass exams. The excitement was watching miracles happen on a machine he could control.

Fridman asks how Knuth can tell, reading a program, when the author changed. Knuth says the tells are not Hemingway-versus-Joyce. In a Carnegie Mellon compiler he once read, whole stretches used three instructions for a register move that one would have done; then came flashes where one instruction accomplished two goals because the writer knew the machine. He could identify authors by technical aptitude, not by rhythm. Asked what would mark his own code, he says it is literate programming now, English mixed with C. Looking only at the C, people would notice many global variables, a habit of expanding inline rather than calling, and a personal subset of the language.

Literate programming is, he says, the most significant thing to come out of the TeX project: the realization that programs are to be read by people, not just by computers, and that typography can massively enhance that. He points listeners to *Physically Based Rendering* by Matt Pharr and colleagues, a book that won an Academy Award because movie graphics are algorithms, and the whole book is a literate program you can also run. Teaching programming, for him, is telling a story as part of the program. The acceptance test is personal. Does he still understand it easily a week later, or a year later?

What makes a program beautiful? He had dodged the question before, and he explains why: there are dozens of answers, and they change by the hour. It can be beautiful if it works at all, if it is easy to understand, if it is literate, if it makes you laugh. Fridman is with him on readability, elegance of thought, and wit. Knuth thinks humor belongs in comments, with a delicate standard he once had to defend to Addison-Wesley. Bob Floyd, the main reader of volume one of *The Art of Computer Programming*, flagged jokes; the publisher even ran focus groups. Knuth's rule: the ideal joke is one the reader knows is probably there if only they understood it, which is a reason to think harder.

He kept some of those jokes. Volume two hides a cryptogram that requires breaking an RSA key larger than people currently know how to break. If machines keep getting faster, someone in a hundred years may decipher the message and laugh. Humor, he says, is part of life, and a book about programming should be fun. Different centuries and cultures invent different kinds of it, so the tone has to be light rather than egregious. He even put little jokes into an organ composition based on the Bible. The point is not to clown. It is to leave a human being on the page.

Fridman describes GitHub Copilot: start a function with comments, and OpenAI Codex completes it, often well, though not provably. He hears an echo of literate programming, with the human moved into the seat of fixing rather than writing from scratch. Knuth's first question is the one that organizes the rest of the hour: how do you know whether it did a good job? Every year, he says, we lose more control over what machines are doing, and more people answer that it seemed to work. At Caltech in the 1960s he knew a colleague who gave inspiring lectures you could not reconstruct an hour later, and who really felt that it did not matter whether computers got the right answer, only whether they made you happy. Happiness, in that philosophy, outranked truth. The man did not believe in truth. He was a philosopher.

Knuth sees that attitude spreading. Competing interests make decisions nobody on either side understands. We still recognize that as bad; five or ten years later, each layer built on last year's opaque layer, the detachment is worse. Fridman offers the optimistic reading: automate the mess so humans keep the small part of programming that needs genius. Knuth will not write a book about that, and he will never recommend that students work on those systems. He is on the side of understanding. Medical diagnosis, a guided experiment, a cured disease: marvelous. Code that drifts, year by year, into autonomous weapons is a different story. Right now it looks like a little JavaScript for a website. He is trying to postpone the heat death of the universe a bit, and he would like some consciousness, preferably human, still around to appreciate it. Perhaps, he adds, that is why we pick up no signals from elsewhere. They invented this too.

Asked whether he is optimistic, he does not answer with a mood. Look at the room: pencils, books, tables, microphones, clothing, food, each invented by someone, millions and billions of artifacts inherited, each able to go wrong, and yet so much of it works. That amazement is the source of his optimism, including for artificial intelligence. We drive over bridges and use technology we do not understand; millions of people each own a small piece; it fails often enough to notice, and we improve it, often suboptimally. Then he qualifies the hope. The improvements he knows how to make require human beings to be rational, and he is losing confidence that they are. Fridman answers that people may not be rational but they are clever, and that most of them lean toward being good to each other if given the chance.

Fridman quotes the famous line from *The Art of Computer Programming*: programmers spend far too much time worrying about efficiency in the wrong places and at the wrong times; premature optimization is the root of all evil, or at least most of it in programming. Knuth first unpacks the word. He was a compiler writer, so optimization meant a better translation that ran faster on the machine, the flag you set on the compiler. An unabridged dictionary of the early 1960s defined *optimize* as to view with optimism. Whole fields later borrowed the word. When he used it, he meant tuning a program to a machine.

The mental bug is specific. A programmer tends to think the pages that were hardest to write will be hardest for the computer to execute. Ten pages of code, one of them a week's work, and the mind imagines the machine slowing down when it reaches that page. Of course that is silly, but while you write you do not know what the computer will actually spend time on. He once profiled a Fortran compiler and found it spending more than eighty percent of its time reading comment cards. The authors had worried about nested parentheses in complicated expressions, which was less than one percent. Ten minutes of work on the comments path could have more than doubled the speed. You only see that after the program exists, with measurements.

He is glad the sentence traveled, even though he said it in a limited context. It applies in a second sense: an optimization that helps running time can make the program impossible to change next week, because the data structure is no longer adaptable. A great principle of computer science, late binding, is to hold off decisions when you can. People have written theses on how valuable that is, from just-in-time manufacturing on out: there is often an optimality to laziness. The line resonated, he and Fridman agree, because programmers *enjoy* optimization. A well-tuned fragment has its own elegance, another kind of beauty, and the struggle is to keep that appetite from deciding too early.

Fridman brings up Roger Penrose: the claim that the way a human mind discovers mathematical ideas is more than a Turing machine, and that consciousness is part of that more. Knuth notes, mildly, that his daughter's children played with Penrose's children in Oxford. Asked whether he thinks there is such a limit, he says he can answer yes or no but he has no reason. He does not find it useful to carry an intuition here. In his view the question is unanswerable, or at least no better answered by him than by anyone else, and he does not want someone quoting Knuth as if expertise transferred. There are many things we can speculate about, including how many angels dance on a head. He would rather think about the Riemann hypothesis.

Fridman believes the workings of the human mind are within the reach of science eventually. Knuth does not deny it. He also jokes that maybe AI created the universe, intelligent design by an AI. What he refuses is to pronounce. He is a teacher who passes on knowledge, not someone with special standing on consciousness. John McCarthy, he thinks, was totally unimpressed by such statements. Turing, in the paper that introduces the imitation game, did not like the question "can machines think" either. Knuth does not consider it important: it would be nice, but he files it with things beyond knowledge, meaning not even well enough formalized to ask clearly. Last he checked, understanding consciousness still looked about two hundred years away.

Twenty years earlier, at a symposium of the American Academy, the opening line was that everything written about consciousness is hogwash. Knuth disagrees a little. He does not mind Fridman's engineering angle: once AI systems have personality and identity, the question stops being only philosophy. At those older lectures, neurologists already pointed out that signals go to the arm before a person is conscious of deciding. Leslie Valiant has an architecture for the brain. Christos Papadimitriou, with two colleagues in the *Proceedings of the National Academy of Sciences* a year before this conversation, offered a model that correlates with consciousness experiments and even has a machine language for writing and testing hypotheses. Knuth's own favorite picture is a continual survival-of-the-fittest competition inside the skull: while he speaks, possible next sentences vote, one wins, and it shapes the next. It feels like a clear story, like literate programming, while the messy computation underneath generates the narrative you get to believe.

Knuth thinks the Game of Life is wonderful for showing how things evolve without the creator understanding more than simple rules. For him the important lesson is free will. Life is totally deterministic, and he finds it hard to believe that anyone who has had children can fail to believe in free will; the game makes the tension crystal clear. John Conway wondered whether it was immoral to shut the computer off after a particularly interesting run. From simple initial conditions you know exactly how the system operates, yet if you allow yourself to forget some of the detail and watch the larger organisms, they seem living. If the universe is finite, Fridman says, we are all in a sped-up Game of Life. Knuth has a dozen exercises in Volume 4, Fascicle 6 that analyze it, and he points to Bill Gosper's algorithm that let the Golly simulator run thousands of times faster.

He had slept overnight in Conway's house more than once. They first met at Oxford in 1967, at a conference where Conway gave a talk on knots, work from high school he had never published, later the source of thousands of papers. Conway climaxed the lecture by assembling and then taking apart knots from little plastic pieces in front of the audience. In the spring of 1972 they happened to be in Calgary the same day, had lunch, and Conway covered a napkin with theorems about what he called numbers. Knuth thought it incredibly beautiful. That December, on sabbatical in Norway, he woke in the night with the thought that Conway's theory would be a way to teach students the joys of research. He had been reading a Socratic mathematical dialogue by Alfréd Rényi. In the morning he woke his wife Jill and said he wanted to write a book.

In January he rented a hotel room in downtown Oslo, run by a mission organization, and did nothing else for a week. He had lost the napkin, so he reconstructed the theory from memory, starting from two axioms, and every mistake he made, the characters in the book made too. The words flowed faster than he could think. After six days he finished; on the seventh he rested; then he tried to write his secretary instructions for typing and could not write anymore. The muse had left him completely, almost as if he had been channeling. Conway wrote back that one axiom was wrong: there is a difference, in a logical development, between "less than or equal" and "not greater than," and Knuth had chosen the harder one. That summer he took another week in the mountains of Norway and rewrote it. The book is *Surreal Numbers*.

Later, giving seven lectures in Montreal on stable marriages, Knuth met Conway at a party between talks six and seven, sketched the topic, and watched Conway invent on the spot a theorem that the set of all stable matchings forms a distributive lattice, with a simple way to take greatest lower bounds and least upper bounds of two stable pairings. Knuth used it the next day. He and Jill had just celebrated sixty years of what he calls wedded bliss. They met because, in his sophomore year, he was dating her roommate and wanted Jill's advice on strategy; he enjoyed the advice more than the roommate. She is an artist; he is a geek. She had to take a computer science class and read a manual he had written. Every year they make a Christmas card and compromise their notions of beauty. Stable marriage, he notes, is a technical term, but the human version is the same: compromise, ups and downs, and no expectation of twenty-four hours of bliss. He adds, with a laugh, that he never really enjoyed kissing until he found how she did it.

At Caltech, Knuth and Richard Feynman went to each other's lectures. If Knuth saw Feynman in the front row, he would miss a few sentences. The story he returns to is Feynman in Brazil, arguing that physics students were learning to pass exams rather than learning physics. Feynman offered to open the textbook to any page and say what was wrong with it, and he did, in front of the host who had written the book, after first asking whether he was supposed to tell the truth. For Knuth it epitomizes how education goes wrong in every field and has to be pulled back, periodically, from giving credentials to giving knowledge. It is the same demand as believing a computer's answer. Bob Floyd loved a cartoon of two men in front of a large machine: this thing can do in one second what would take a million people a hundred years. The other man asks, "How do you know it's right?"

Knuth started as a physics major and switched to math. He could get an A-plus on a physics exam and have no idea how the teacher had invented the problems; in math he knew why the questions were set and could invent others. He is good at algebra, not geometry. Physicists think in waves; he can too, but it feels like a dog walking on hind legs. Computer science, he believes, is largely driven by people whose brains resonate with certain concepts, and quantum computing takes a different kind of brain, sitting at the intersection of physics and computer science. He could not understand quantum mechanics from *Scientific American*; eigenvalues in the language physicists use with each other made sense. Stephen Hawking said every formula loses half the readers, so Knuth could not understand that book at all.

Feynman hid deep calculation behind a show of intuition, and one exchange stuck with Knuth for decades. Knuth arrow notation names enormous numbers: addition repeated becomes multiplication, multiplication repeated becomes exponentiation (one arrow), and two arrows make a power tower. Even a small stack is already too large to comprehend. Feynman said, do not stop at integers; consider complex numbers. Interpolating a double arrow to 2.5 is not too hard, but what about a double arrow to *i*? Feynman claimed no analytic function would do the job. Knuth never saw why he could be so sure. Feynman's next question was whether the *number of arrows* could be complex. That is the kind of question Knuth remembered, calmly, for the rest of his life.

The Knuth-Morris-Pratt algorithm should, Knuth says, have been Morris-Pratt-Knuth; they used alphabetical order on the paper. The problem is the one every search engine user knows: find a word in a long string of text. The obvious method looks for the first letter, then the second, and on a mismatch backs up and starts over. Jim Morris noticed you had already read the letters after the first match, and if they cannot be a new start you need not reread them. With a pattern like *abracadabra*, where motifs repeat at the beginning, middle, and end, it gets subtle. Morris worked it out, put it into Berkeley Unix system software, and did not explain it. Months later someone looked at the code, decided it did not look right, and ripped it out. The algorithm existed. Because nobody understood it, it did not survive.

Vaughan Pratt found the same idea independently a year or two later, coming from palindromes rather than text search. Knuth's own path was stranger. Steve Cook, then a professor at Berkeley — Knuth calls denying him tenure the CS department's greatest mistake — had proved a peculiar theorem about stack automata. Such a machine can only see the top of a stack, push, or pop; it is far weaker than a Turing machine. Cook showed that if a stack automaton can recognize a language of strings of length *n* in any amount of time at all, a ordinary computer can recognize the same language in *n* log *n*. Knuth tried the construction on a language he knew a stack automaton could handle but could not see how to recognize fast. He thought he was a pretty good programmer. He filled a blackboard with Cook's construction, stared until the missing pattern appeared, and suddenly had an efficient program. He would never have written it without the purely abstract theorem.

String matching, he then realized, was another language a stack automaton could recognize, and the same translation told him exactly what to remember while scanning the text. He wrote a short note called "Automata Theory Can Be Useful." He had read plenty of automata papers that never improved his everyday programming; they were interesting journal material. This was a turning point: a problem he could not program, a theorem, then a program. He showed the note to Pratt, who said it was similar to what he was working on. Jim Morris was at Berkeley too. Each had discovered a different part of the elephant. Putting the parts together made one algorithm, and it was Knuth's job to write the paper.

Asked for the hardest problem he ever worked through, Knuth does not hesitate: the birth of the giant component. Start with *n* points, all equal, no geometry. Take the digits of π two at a time — 31, 41, 59, 26 — and draw an edge from 31 to 41, then from 59 to 26, or else just roll dice. π has no special status; nobody knows a pattern in it, so it is a random source. Edges accumulate. Paul Erdős and Alfréd Rényi called this the evolution of random graphs. When the number of edges is 0.49*n*, say a million points and 490,000 edges, the graph is almost all isolated trees, barely a loop. At 0.51*n* there is suddenly one component much larger than the others. It is a phase transition, and a double one: two things happen at once.

Dick Karp's students at Berkeley were simulating the process and taking snapshots. A rumor reached Stanford: almost every time they looked, only one component had loops. A million experiments, three or four exceptions. If that were true, Knuth's group thought, a whole theory was waiting. Looking closer, it was not quite true, but almost: there is a very short interval when two or three looped components exist, and they join quickly. If the shutter is slow, you miss it. The problem became how to slow down the big bang. Knuth defined the complexity of a component as edges minus vertices. A tree of ten vertices and nine edges has complexity minus one. A single cycle with trees feeding into it has complexity zero. One extra edge is a bicycle, two loops. Complexity plus one is the number of loops.

After the first cycle appears, the rumor said the next step should almost always be a bicycle. In fact the probability of splitting into two separate cycles is about five in twenty-four, substantial, though they soon merge. Knuth computed the first five transitions. About three in the morning the denominators came out, one of them 23023. He stared. From his first program he knew 1001 is 7 times 11 times 13; 23023 is 7 times 11 times 13 times 23. A random fraction does not factor like that. Small primes in the denominator meant there was a reason. He took logarithms, the formula simplified, and he went to bed feeling he could watch the explosion in slow motion. The next day Bill Gates visited Stanford, being courted to fund a new CS building. Knuth put the evolutionary diagram on the blackboard, the five-twenty-fourths and the rest. That evening Gates told the development office he had been really impressed by what Professor Knuth said about the giant component. Knuth loves the story because it is his exhibit that theoretical computer science is worthwhile.

With two coauthors he finished the analysis. The rumor's claim — that at every moment there is only one cyclic component — is not probability one. It is something like eighty-seven percent, and the exact value involves π, a form such as 12/π². It was the hardest theorem he ever proved, and it lit up a cluster of other questions about random graphs. The physics connection is Bose-Einstein statistics: how molecules bond together with no geometry and no distance. A day after a denominator factored, a billionaire on a fundraising tour heard a lecture about loops in a random graph. That, Knuth suggests, is what the subject is for.

When Knuth released TeX, the phrase *open source* did not exist. He did not want proprietary rights, because he had seen how they hold things back. In the late 1950s IBM developed Fortran and could have kept it for IBM machines; instead anyone who could translate Fortran onto their own hardware was allowed to write a compiler. Typography went the other way: each manufacturer had a page-description language, users were locked in, and printing equipment is amortized over twenty or thirty years while new machines appear in one. He also did not need the money. He had a job, the books sold, and his children's education was covered. Income is a threshold function: below it you starve; above it you start thinking about philanthropy, or about taking it with you. He was over the threshold, and he could see the advantage of making TeX open for everybody.

He does not think all software should be free. People should charge for non-trivial software and not for trivial software. Adobe Photoshop, in his example, has an undo that can walk back through a thousand complicated graphics steps, which means beautiful algorithms and, as he puts it, Nobel-prize-class intellectual property. Patents, at their best, publish so an idea cannot remain a trade secret. His own machine is a standalone Ubuntu laptop with no internet. He ferries flash drives to Macs for the network and for graphics. He trusts his family jewels only to Linux, because the version he runs is stable, even if it is old enough that he will have to upgrade one of these days.

Asked for a favorite letter, number, or symbol, he first flips to the last page of an index and points at a mark from Dr. Seuss's *On Beyond Zebra*, a book he prefers to *The Cat in the Hat*; as a child he knew *Bartholomew Cubbins* by heart. Then he names π, for a practical reason: when he needs a random example, π has no known pattern. He wanted Masyu, a Go-like loop puzzle more addictive than Sudoku, as an example in *The Art of Computer Programming*, and set himself the task of placing stones on the pixels of the Greek letter π, then coloring them black or white so the puzzle had a unique solution. There was exactly one such coloring. π, he says, is how he proves he did not start from a canned instance.

More recently he was writing about graceful graphs: label vertices so that the absolute differences on the edges are exactly 1 through *m*, each used once. He took a thoroughly unsymmetric graph, the contiguous United States plus the District of Columbia, forty-nine regions you can drive between, Alaska and Hawaii left off. He had labels whose edge differences sat between 1 and 117 with gaps, short of the perfect 1-to-about-105. He thought of a contest. His friend Tom Rokicki found a perfect graceful labeling in an afternoon, so the contest was over. Two days later Rokicki came back: Washington was 31, Idaho 41, and along the northern border the labels were the digits of π, still graceful. Knuth was delighted. He likes to use π, and π has a way of turning up in the middle of the hardest problems.

Knuth's scheduling principle is to do the thing he hates most on the to-do list. By week's end he is happy, because the unpleasant tasks are finished. During the pandemic his productivity fell by about half: communication by writing is slow, he will not send a bad sentence, and libraries were closed. His mother taught a second rule, different from robotic planning: see something that needs doing and do it, instead of ranking a list. The two coexist as a heuristic. At any moment there is a set of possible tasks; prefer the one you have no reason to think you will do better later. Some work improves if you do something else first; some gets harder because you forget the groundwork. He had just finished a painful stretch of the book and was on the fun parts, but the reader must never see the pain. The tone has to stay upbeat, with humor and the occasional *this is cool*. The road to that excitement is laden with pain, and he has to disguise it.

Advice to the young is the same as he has always given, offered with the preface that other people do many of these things better than he does. Do not do a thing because it is trendy. Do what you personally feel called to do, not what someone else expects. How do you know you are called? You try it, and it works or it does not; you learn about yourself. Life is a binary search. One experiment shows you have background that helps, or that more effort might get you there; another shows you have no intuition and the problem does not have your name on it. He probably overreacts the other way when he sees everyone going in one direction and thinks the competition looks fierce. Mostly he played with what interested him, and later realized the most important skill he learned was how to be interested in almost anything.

It makes him sad to hear children say something was boring. A person should feel upset, he says, at having to admit he could not find something interesting: it means he has not learned to enjoy life and needs someone else to entertain him. Fridman quotes David Foster Wallace on being unborable. Knuth agrees it is a skill, harder some days than others. During COVID he went many days without seeing another human being and still had a pretty fun time. He arrived a few minutes early to the recording, walked around Foster City, and found beautiful flowers at a Home Depot nursery a few blocks away. Life, he says, is full of amazing things. Sometimes he just sits and stares at a tree.

Fridman will not let him skip the meaning of life. Knuth bats away 42, then answers for himself. He believes God exists, though he has no idea what that means: something beyond human capabilities, perhaps some AI, and he can try to learn how to resonate with whatever that being would like him to do. He does not choose strategically. He tries to imagine he is following someone's wishes even though he is not smart enough to know them, trusting that the clues, and the journey from clue to clue, can be made fun. Reading Ken Follett's *The Man From St Petersburg*, with Churchill needing a Russian alliance and an anarchist trying to stop it to save ordinary lives, he realized that in a job responsible for other people's lives he would have no confidence in any of his decisions. Fridman says the coupling of uncertainty with the duty to decide is actually a good thing. Knuth has faith that great leaders will arise; that is the hope of democracy. Fridman hopes algorithms enhance them. Knuth leaves the audience with a line of his own: science is what we understand well enough to explain to a computer. Art is everything else we do.