John Horton Conway (1937-2020)

Update (4/13): Check out the comments on this post for some wonderful firsthand Conway stories. Or for the finest tribute I’ve seen so far, see a MathOverflow thread entitled Conway’s lesser known results. Virtually everything there is a gem to be enjoyed by amateurs and experts alike. And if you actually click through to any of Conway’s papers … oh my god, what a rebuke to the way most of us write papers!

John Horton Conway, one of the great mathematicians and math communicators of the past half-century, has died at age 82.

Update: John’s widow, Diana Conway, left a nice note in the comments section of this post. I wish to express my condolences to her and to all of the Conway children and grandchildren.

Just a week ago, as part of her quarantine homeschooling, I introduced my seven-year-old daughter Lily to the famous Conway’s Game of Life. Compared to the other stuff we’ve been doing, like fractions and right triangles and the distributive property of multiplication, the Game of Life was a huge hit: Lily spent a full hour glued to the screen, watching the patterns evolve, trying to guess when they’d finally die out. So this first-grader knew who John Conway was, when I told her the sad news of his passing.

“Did he die from the coronavirus?” Lily immediately asked.

“I doubt it, but I’ll check,” I said.

Apparently it was the coronavirus. Yes, the self-replicating snippet of math that’s now terrorizing the whole human race, in part because those in power couldn’t or wouldn’t understand exponential growth. Conway is perhaps the nasty bugger’s most distinguished casualty so far.

I regrettably never knew Conway, although I did attend a few of his wildly popular and entertaining lectures. His The Book of Numbers (coauthored with Richard Guy, who himself recently passed away at age 103) made a huge impression on me as a teenager. I worked through every page, gasping at gems like eπ√163 (“no, you can’t be serious…”), embarrassed to be learning so much from a “fun, popular” book but grateful that my ignorance of such basic matters was finally being remedied.

A little like Pascal with his triangle or Möbius with his strip, Conway was fated to become best-known to the public not for his deepest ideas but for his most accessible—although for Conway, a principal puzzle-supplier to Martin Gardner for decades, the boundary between the serious and the recreational may have been more blurred than for any other contemporary mathematician. Conway invented the surreal number system, discovered three of the 26 sporadic simple groups, was instrumental in the discovery of monstrous moonshine, and did many other things that bloggers more qualified than I will explain in the coming days.

Closest to my wheelhouse, Conway together with Simon Kochen waded into the foundations of quantum mechanics in 2006, with their “Free Will Theorem”—a result Conway liked to summarize provocatively as “if human experimenters have free will, then so do the elementary particles they measure.” I confess that I wasn’t a fan at the time—partly because Conway and Kochen’s theorem was really about “freshly-generated randomness,” rather than free will in any sense related to agency, but also partly because I’d already known the conceptual point at issue, but had considered it folklore (see, e.g., my 2002 review of Stephen Wolfram’s A New Kind of Science). Over time, though, the “Free Will Theorem” packaging grew on me. Much like with the No-Cloning Theorem and other simple enormities, sometimes it’s worth making a bit of folklore so memorable and compelling that it will never be folklore again.

At a lecture of Conway’s that I attended, someone challenged him that his proposed classification of knots worked only in special cases. “Oh, of course, this only classifies 0% of knots—but 0% is a start!” he immediately replied, to roars from the audience. That’s just one line that I remember, but nearly everything out of his mouth was of a similar flavor. I noted that part of it was in the delivery.

As a mathematical jokester and puzzler who could delight and educate anyone from a Fields Medalist to a first-grader, Conway had no equal. For no one else who I can think of, even going back centuries and millennia, were entertainment and mathematical depth so closely marbled together. Here’s to a well-lived Life.

Feel free to share your own Conway memories in the comments.

90 Responses to “John Horton Conway (1937-2020)”

  1. Gil Kalai Says:

    It is sad to hear that John H Conway passed away. He was an amazing mathematician, among the few who invented a new simple group, among the few who invented a major knot invariant, among the few who invented a new system of numbers, and more than anybody he also invented many many many games. Conway was also a master of codes and sphere packings. And this is a very nice post, Scott.

    Here are few quick memories. (Last time when Scott asked for memories, about Boris Tsirelson, I was too slow and ended up writing them in a post. So I will try to be quick this time)

    1) The first (and most meaningful) time I met Conway in person was in 1979 in the common room at the (old) Cambridge (UK) mathematics building, when I came there, as part of an an after-army travel, to a combinatorics meeting in Cambridge. Conway showed me the draft of his monumental book with Berlekamp and Guy “winning ways” and offered me to play a board game called (as far as I can remember) “football”. We played for a while and simultaneously Conway also talked with others (some across the room), including with his wife. Conway set a special rule for me: Everytime I am convinced that I loose, we can switch sides. Needless to say that we switched sides several times; I was sure that my position is desperate beyond repair, we switched sides, and shortly afterward I was again sure that my position in the game is beyond repair.

    2) In the following decades I attended some highly entertainment talks by him (I remember that his 1994 plenary talk in Zurich was especially wild but I don’t remember the details).

    Once at Yale (in the early 2000s) we went after a lecture by Conway to dinner with Hillel Furstenberg. There, Conway showed us the ultimate riddle (I already knew it) and Hillel tried in vein to replicate it.

    3)* Once Conway thought that he can prove that every triangulated surface of genus g which is linearly embedded in space, must have at least a linear number of vertices in terms of the genus. I brought to his attention a 1983 paper by McMullen, Schultz and Wills with a construction of only g/log g vertices. This is still the world record, for topological embeddings, square root g vertices suffice and for linear embedding no better lower bound is known.

    4) Here is a famous problem asked by Conway in the late 60s (Conway offered 1,000$ for solving it.): A thrackle is a planar drawing of a graph of n vertices by edges which are smooth curves between vertices, such that nonincident edges cross exactly once, and no incident edges share an interior point. The conjecture is that a thrackle has at most n edges. (If the smooth curves are line intervals this is a famous result by Hopf and E. Pannwitz from 1934 .)

    In the mid 80s my academic brother Yaakov Kupitz workd with our supervisor Micha A. Perles on related questions (and their work initiated a rich area of geometric graph theory), and Kupitz thought that it would be a good idea for him to spend a year with John Conway. All the arrangements were made, including visa, and lodging, but then Yaakov discovered that he connected with a different (famous) mathematician named John Conway, and he cancelled the visit and went for a year to Aarhus, Denmark instead.

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  3. Robin Whitty Says:

    I heard him lecture once. He said “I worried for a long time about what the term ‘random variable’ means. In the end I concluded it means: ‘variable’.”

  4. Clive Says:

    John Horton Conway’s “Game of Life” was one of many amazing things Martin Gardner wrote about in his Mathematical Games column in Scientific American a few decades ago. I used to read those columns as a schoolboy in the 70s, and “Life” was something I found totally fascinating back then and have revisited periodically every since.

    I’m very sad to hear of his passing.

  5. asdf Says:

    I got to hang out with Conway a little as an undergraduate. He had a way to solve a Rubik’s cube that wasn’t the most efficient, but whose “subroutines” were so few and simple that he could teach it to a regular person (like me) in just a few minutes. So for a while, I knew how to solve a cube. Not any more, though, I’m sure.

    He asked me was I was doing and I showed him a differential equation someone had shown me, asking for a function whose derivative is the same as the function’s inverse, (hmm I thought there was a way to use TeX in here? I don’t see it). Anyway, you want f such that f'(x) is the same as f^{-1}(x). This is a cute and not too hard puzzle that I’ll leave as an exercise.

    Anyway, Conway not surprisingly solved it in about a minute, so I asked if there might be other solutions (I had been trying to figure some out, getting nowhere). Within another minute or two he came up with a proof that any solution must have an infinite series of a certain form, and the puzzle solution was in fact unique (maybe modulo a constant or something). I was amazed. Being a dumb teenager at the time, I knew that Conway was a great combinitorialist, inventor of the Game of Life, did some famous thing with finite groups etc., but I thought that kind of mathematician was all about algebra and discrete math and would barely know what a derivative was. So it was cool to see him reach into his memory and immediately pull out the right trick from DE theory that I had never heard of. I looked it up today and I think it is called (TIL) the Method of Frobenius, so I’ll see if I can reconstruct the proof.

    I heard more recently that Conway apparently hated being best known for inventing Life. He did a lot of really terrific math that few people realize that. I guess it’s like von Neumann being best known for computers, or for cellular automata, when he had such huge impact in many other areas. Similarly with Shakespeare actor Alec Guinness being most famous as Obi-wan Kenobi, etc.

  6. Bernard Says:

    Conway was truly exceptional.

    In Spring 1967, during my first year reading mathematics at Cambridge, a fellow maths student told me I had to join him in a class on the foundations of mathematics being taught by a mad Liverpudlian professor with a huge ginger beard. Of course it was Conway who was only 3 years out of his PhD work. I was having great difficulty with the mathematics curriculum. Most other students were far better prepared that I had been and I was struggling to keep up, with limited success. But Conway’s enthusiasm was infectious and inspiring. He was also funny. All our other classes covered material from the early 19th century but John was teaching us about work that had been done in 1964 (Paul Cohen’s proof that the continuum hypothesis is independent of set theory).

    When I moved to Princeton in the 1990s I was astonished to find that John lived one street away. Despite the geographic proximity, it was years before I had a chance to speak to him. I told him I’d taken his class in 1967 and enjoyed it. His reply was Diracian in its brevity: “you survived”.

    Rest in peace.

  7. alpha Says:

    He is also to have said P=NP.

  8. Aaron G Says:

    Scott, I’ve been scouring the Internet for news about Conway’s death and haven’t found anything reported outside of social media. How reliable are the sources?

  9. Amin Says:

    This is very sad. I was also fascinated by Conway’s Game of Life when my maths teacher showed us some animations in middle school. Looking back it’s probably the first example of Turing completeness I saw. Although I must confess I had no idea what Turing machines were at the time :d

  10. Anon Says:

    Several years back, I met John Conway at a math summer camp for high school students. He gave several talks and also joined us for lunch sometimes. I remember thinking that this old man was a bit eccentric, but also really did see the fun in math (a phrase he used was “nerdish delights”). While I personally did not exchange many words with him, I was there while fellow high school students much more knowledgeable and passionate about math than I talked with him at length – this man truly could talk about math on any level to anyone. I really enjoyed him telling us about his Doomsday algorithm, a surprisingly simple way to figure out the day of a week for an arbitrary date (for the year or so after that, I would conscientiously always use the Doomsday algorithm to figure out the day of a week rather than actually check a calendar).

    Rest in peace, John Conway.

  11. mjgeddes Says:

    I had just been researching ‘Monstrous Moonshine’ – there’s actually a very large amount of background knowledge in algebra you need to build up to that – I had to read many wikipedia articles to lay the foundation. It’s thought to be important to String theory.

    Conway’s Game of Life, of course, very interesting, it’s a cellular automaton , gets you thinking about computation and complexity in a fun way ! My suspicion now is that it may be valid to consider the universe (as a whole) as a singular dynamical complex system – it’s not a closed system ! And Game of Life definitely nudged my thinking in the right direction, towards complex systems theory.

    Speaking of Stephen Wolfram, he’s apparently about to make a big announcement, he’s claiming that he’s made a huge physics breakthrough , and as far I can make out , it’s a new model of space and time that gives a new interpretation of quantum mechanics – looks quite intriguing. This could be quite big news, so look forward to hearing your thoughts on it Scott….

  12. Scott Says:

    Aaron G #8:

      Scott, I’ve been scouring the Internet for news about Conway’s death and haven’t found anything reported outside of social media. How reliable are the sources?

    In the end I made a judgment call that the many mathematicians and others reporting this, including on my Facebook, seemed trustworthy. I also reasoned that, in the sadly unlikely event that they were wrong, Conway—like Mark Twain—is the sort of guy who’d greatly enjoy reading his own obituaries.

  13. Vijay M. Patankar Says:

    What a sad news… I have been reading a lot about him and by him – his discoveries, his theorems, his perspectives. Remarkable, in so many ways. I was planning to teach basics from his book “The Sensual (quadratic) form” to my elementary Number Theory class.

    Rest in Peace…

  14. Barry Hayes Says:

    I wrote this a couple of years ago, after I had the pleasure of a few hours of John’s company at a G4G.

    I left a lot out, because he also talked openly about his personal life and his emotional arcs. There was a lot more I forget than I’ll remember.

    Oh, and we talked a bit about statements that were true but not provable — he firmly believed that the Collatz conjecture was in that class.

    https://barrystuff.wordpress.com/2014/03/22/johnconwayandicollaborateonaproof/

  15. Steve E Says:

    He helped me realize that lots of things which seem impressive at first glance are actually just sophisticated parlor tricks. This video gets no love but is a perfect example.

  16. Marek14 Says:

    He was my idol in my teenage years when I found Winning Ways. Despite the fact that I eventually ended in a different career, I still retained my love of math, especially of weird tessellations.

  17. A.G.McDowell Says:

    I did first year maths at Cambridge when Conway was lecturing there (I then switched to computer science). We had a term of lectures on vector spaces. At the last lecture it was announced that the lecturer was ill, and there was a stand-in. This turned out to be Conway, who brilliantly recapitulated the whole course in one lecture of amazing clarity. Much later I was told that the illness was diplomatic, and this had occurred more than once. I also heard Conway give a most entertaining talk at a Science Fiction convention. I think I remember that he covered Life, Monstrous Moonshine, and transfinite induction – this was nearly forty years ago, so if my normally appalling memory is correct, this is a tribute to Conway.

  18. Klaas Landsman Says:

    Dear Scott,
    Conway’s death has just been confirmed by his biographer, Siobhan Roberts. Thanks for drawing attention to this. Best wishes, Klaas Landsman

  19. Andrew Conway Says:

    I remember him confessing to making a mistake. “To borrow some language from Watergate, one of the proofs I gave in the previous lecture is now inoperative.”

    When he gave an optional evening talk on Life to one of the college mathematics clubs at Cambridge the room filled up, so we moved to a larger one. That filled up, so we waited till after dinner and moved to the college dining hall. Earlier versions of Life had two sexes, A and B, which stood for Actress and Bishop. “Then I realized that it was symmetrical in A and B, and that sex wasn’t necessary.” He went on from there to building a Turing machine out of Life components.

  20. Diana Conway Says:

    Thank you. Reading all this lovely tributes are making this day bearable for me and our son.

  21. Scott Says:

    Diana Conway #20: Thank you so much for leaving a note here. While nothing can make up for your loss, I think it’s safe to say that an entire planet’s worth of math enthusiasts are now mourning with you.

  22. Scott Says:

    alpha #7:

      He is also to have said P=NP.

    What I know is that, in Bill Gasarch’s P=?NP poll, Conway conjectured P≠NP, but with the following comment:

      In my opinion this shouldn’t really be a hard problem; it’s just that we came late to this theory, and haven’t yet developed any techniques for proving computations to be hard. Eventually, it will just be a footnote in the books.

    With the greatest of respect to a mathematical legend who recently departed us—if I’d had the chance to chat with Prof. Conway about this, I might’ve asked him: “and will the Riemann hypothesis also eventually just be a footnote in the books? Is the problem just that, like, we haven’t yet developed any techniques for proving all the zeroes to lie on the critical line?” 😀

  23. Sniffnoy Says:

    Much like with the No-Cloning Theorem and other simple enormities, sometimes it’s worth making a bit of folklore so memorable and compelling that it will never be folklore again.

    Man, it’s always worth writing down folklore theorems. Getting things into the literature is important.

    I’ve personally done a bit with surreal numbers and with Jacobsthal’s multiplication (which he didn’t discover but did rediscover). This is truly a loss for mathematics and th eworld.

  24. Andrei Says:

    Scott #22:

    In this vein, which solved long-open conjectures have become, if not footnotes, at least accessible enough to be incorporated in lectures? (I’m thinking e.g. of the recent sensitivity conjecture which, if I remember well, you did incorporate into your quantum information lecture.)

  25. Scott Says:

    Andrei #24: Generally, a solution to a longstanding open problem becomes a “footnote” when, and only when, the whole area around the problem dies out or the solution doesn’t lead to anything else.

    Huang’s proof of the Sensitivity Conjecture was the most spectacularly short and simple solution to a longstanding open problem that I’ve seen in my career—I haven’t taught it, but could easily do so in one lecture.

    The proof that all Robbins algebras are Boolean (a 60-year-old open problem at the time) was only half a page. And it was found by computer search!

    Anyone else have other examples of big open problems with surprisingly short answers?

  26. Asvin Says:

    Hi Scott,

    The way I interpret Conway’s comment is the following: We will eventually develop really good techniques for separating complexity classes and proving hardness and the specific question of P/NP will only be the first among a large class of such questions to be answered and perhaps the easiest.

    Comparing it to the Riemann hypothesis, we already have lots of techniques for showing that complex functions have zeroes in a prescribed domain. It’s just that none of them seem to apply to the Riemann zeta function and moreover, it seems like the question really belongs to a fundamentally different field (arithmetic geometry? Arakelov theory? Something new?) and the solution will come from there, not complex analysis.

    That is, the Riemann hypothesis is not emblematic of the field studying zeroes of complex analytic functions and a solution might not advance this particular field that much.

    On the other hand, there is a very special class of meromorphic functions (the L-functions) of which the Riemann zeta function is emblematic and one can hope that proving something about the Riemann zeta function will also extend to us proving things about this entire class of functions. In this sense, proving the Riemann hypothesis might well be a footnote to proving the Generalized Riemann hypothesis.

    Another example relating the zeta function might be Euler’s solution of the Bessel problem. It was the first determination of the special values of a zeta function but in the following 300 odd years of work, there has been amazing progress in this field (a lot of modern number theory comes under this heading – Iwasawa theory, Class number formulas, Birch-Swinnerton Dyer, Bloch-Kato and the list just goes on).

    While Euler’s result was very much celebrated in his time (and rightly so), it is definitely only a footnote today in the field.

  27. Scott Says:

    Asvin #26: If Conway’s point was just that P vs NP isn’t singularly important in itself, but is a “flagship” for a huge number of questions about proving computations to be hard (P vs PSPACE, BPP vs BQP, permanent vs determinant, complexity of matrix multiplication…), almost none of which we know how to answer—then not only would I emphatically agree, I’ve often made the same point! It’s just that the “footnote in the books” comment could be interpreted as Conway’s regarding the entire quest to prove computations hard as an eventual footnote. Thanks for helping me read his comment differently!

  28. Lior Silberman Says:

    Conway was a constant presence around the common room when I was a graduate student. There was a copy of the Altas hidden in one of the cabinets, which he would pull out if he was asked a question. When I tried to understand what the building of \PGL_3(\Qp) was he lent me his Zome set so I could build a model. After building a chunk of the apartment I realized I won’t get anywhere with a concrete picture and switched to geometry and algebra.

    Andrei #24 / Scott #25: The finite-field Kakeya problem was only open for 10 years, but it was a major open problem and Ze’ev Dvir’s half-page solution was truly amazing.

    Marcus–Spielman–Srivastava’s construction of Ramanujan graphs of every degree is also worth noting: the result is fairly elementary, certainly in comparison to the LPS method and its extensions. Unfortunately I don’t think MSS’s proof of the Kadison–Singer conjecture qualifies for our discussion.

  29. Wyrd Smythe Says:

    For me, it’s two: John Prine and now John Conway.

    I’ve written code for the Life game many times in many languages. It’s a good one to assign to students, since it’s got a couple “gotchas” in the algorithm — mostly the need for two buffers — plus it’s so much fun to watch in action.

    In his honor I feel like I should whip up another version…

  30. Nick Says:

    I interpret the “footnote” remark to mean that computational complexity theory will end up like syllogistic logic — fine as far as it goes, but ultimately a trivial fragement of a much larger theory, and one whose historical importance is way out of proportion to its actual value.

  31. Scott Says:

    Nick #30: If you want to get from syllogistic logic to Gödel and Cohen, would you agree that the obvious way to do that is to … think hard about logic? Likewise, if you want to get from complexity theory to some yet-unknown much larger theory of which complexity theory is a mere fragment, would you agree that the obvious way to do that is to … think hard about complexity theory? 🙂 (Or did you have a more specific suggestion?)

  32. Francis Fung Says:

    RIP John Horton Conway. My (brief) mathematical career is intimately linked to JHC. I believe I first met him at Kansas State University when he gave the Dressler Lecture on audiological decay (1988). I then saw him again at the Summer Joint Math Meetings, where he was ever-presently engaging students, in Boulder (1989), Columbus (1990), and then, fatefully, in Orono (1991), when he gave the Earle Raymond Hedricks lectures, on “The Sensual (Quadratic) Form”. I ended up sitting with him at dinner, along with Richard K. Guy (RIP just this year as well) and Don Albers from the MAA, who was in charge of MAA publications. As they discussed the upcoming lectures, they mused about how nice it would be if the subject matter could be written up and published, and they wished that there might be a “scribe” of some sort to write them up. I intrepidly volunteered to do so. I thus because his assistant in writing up a manuscript for the book (eventually published as a Carus monograph), which became a project for the next 6 years (and granted me an Erdős number of 2), and surely played a hand in getting me into Princeton, where I was his graduate student for a few years (ultimately I ended up pursuing different mathematical interests). Of the great many memories I have of time spent with him, one that stuck with me most strongly is that my father Daniel Fung (RIP), a food scientist who loved to dabble in many subjects, came up with a quick rule of thumb for converting powers of 2 to powers of 10 (roughly that 2^10 ~ 10^3, so you could take various powers of 2: 2, 5, 8, then 12, 15, 18, then 22, 25, 28….to roughly estimate powers of 10). He was very proud of this observation and eagerly told it to me, and in my teenage impudence, I rolled my eyes and pooh-poohed it as a triviality. When JHC came to visit Kansas State for the second time (and got to work on the book with him), my father was very eager to hear what a famous mathematician like JHC would have to say about his observation. I relayed this to JHC when we met, rather dismissively, assuring him that I did it because my father wanted me to ask him, but not because it was mathematically interesting. But he said “Oh, that’s quite interesting! Let’s figure out how good this approximation is!” And then enumerated it several tens out, to figure out how far this approximation held. I learned some humility that day, and also that one’s viewpoint and openness to exploring a problem is very important in searching for new knowledge. Farewell!

  33. Scott Says:

    Francis Fung #32: I’ve often used that same 210 ~ 103 approximation! (As, implicitly, do the very definitions of kilobyte, megabyte, etc.) When spitballing one could do a lot worse.

  34. Latham Boyle Says:

    Several years ago, I took my daughter to a memorable lecture by Conway in Toronto on the surreal numbers: https://www.youtube.com/watch?v=1eAmxgINXrE

    There was a touching moment in the middle (at roughly 37:26), when he brought himself to tears while explaining Cantor’s contributions to mathematics and his role in organizing the first international congress of mathematicians.

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  36. Gerry Myerson Says:

    Gil, I believe the correct spelling is “phutball”, a contraction for “Philosopher’s Football”, see https://en.wikipedia.org/wiki/Phutball

    I had the great good fortune to spend a year as a postgrad at Cambridge, 1974-5. Conway gave me his paper with Jones on trigonometric diophantine equations, aka vanishing sums of roots of unity, and set me to work on the rational tetrahedron problem (finding all tetrahedra whose dihedral angles are all rational). I didn’t get very far with that problem (and, so far as I know, it’s still unsolved), but what I learned along the way led me to the thesis I would later write under Don Lewis at Michigan, and also to a paper on rational products of sines of rational angles.

    I also attended lectures he gave on graph theory (I thought the contraction-deletion technique for finding chromatic polynomials was magical!) and on ordinal arithmetic, at the time he was working out the theory of surreal numbers. In those days the maintenance staff would wash the blackboards between lectures. Once, when the boards hadn’t completely dried, Conway sought out the dry patches to write on, scattered across the surface of the boards.

    I’m so lucky I had those opportunities 45 years ago.

  37. alpha Says:

    For all we know the Riemann Hypothesis may not be about primes at all. Perhaps for this problem we don’t have the right tools yet and current mathematicians are pushing wood with analysis. I do believe in what he says on NP. If we had a pen and a paper and enough time the only thing that is coherent that can be said about NP is that P is NP. For this the tools are already here and have been here. I disagree with him on this.

  38. mjgeddes Says:

    Nick #30 and Scott #31

    I think this great quote from Conway proves that he ‘gets it’ :

    “You know, people think mathematics is complicated. Mathematics is the simple bit. It’s the stuff we can understand. It’s cats that are complicated”

    It’s physics and (pure) math that’s simple compared to the everyday complex systems all around us… the weather, the economy, biological systems the brain and now just recently…. artificial intelligence 😉

    So the real question rationalists should have been asking all along, is the one that’s only just
    been posted to ‘Less Wrong’ now, decades too late really….

    “How should we rationally model complex systems?”

    Scott is close here….

    “if you want to get from complexity theory to some yet-unknown much larger theory of which complexity theory is a mere fragment, would you agree that the obvious way to do that is to … think hard about complexity theory?”

    Yes, indeed!

    Let me just elaborate a little. So imagine if you will , a “modelling machine’ (an oracle in other words) that generates ‘scientific paradigms’ – it comes with some knobs that you can turn to control the class of model that it will spit out.

    Each knob controls the ‘modeling scale’, in terms of a level of complexity that the model will have. A knob setting of 0, means the model will be perfectly ordered (minimal complexity), a knob setting of 100 means the model will be perfectly randomized (also minimal complexity), a maximum complexity model will occur at a knob setting of 50. And there’s more than one knob (say that there’s 3 knobs), for 3 different types of scale.

    So for instance, knob settings (20, 40, 75) might spit out a class of model we’d recognize, as ‘Chemistry’, settings (33, 45, 20) might spit out a class of model we’d recognize as ‘Physics’, setting (64, 10, 4) might spit out a class of model we’d recognize as ‘Algebra’ and so on.

    So here’s a puzzle for you all… for which knob sttings will the class of model that comes out be ‘Complex Systems Theory’ ?

    Hint: It’s exactly the same knob settings for which the class of model produced will be ‘Artificial General Intelligence’ 😀

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  41. Priscilla Roberts Says:

    I am a historian, not a mathematician, but I was quite a good friend of John Conway, at least towards the end of his life, and helped to organize one of his last international trips, to Hong Kong and Macau in April 2017. I’m not sure how much anyone else knows about that visit. So I’ve written down what I remember.
    John and I had met sometime one summer in the mid-1990s, when he helped me with my luggage on a train from Princeton to Washington, but we lost touch until late 2016, when I spent a semester at the Institute for Advanced Study. We ran into each other at the Panera coffee shop, a regular early morning port of call for John and several others, including myself, who used to chat together and comment on the newspapers before going off to do whatever each of us was doing.
    John and I were both English and had been students at Cambridge, albeit a good many years apart, which gave us something in common, and he enjoying talking about almost everything under the sun. John had already had at least one stroke and walked with some difficulty, using a stick, but was totally resolved to keep living an independent life, going in (usually on foot) to the Maths Department virtually every day that it was open. Very soon I came not just to enjoy his company but to admire his courage and vitality, plus his sheer bloody-minded determination to keep on going no matter what.
    I left Princeton in January 2017, to take up a position at City University of Macau. John had travelled to many places but never been to either Macau or Hong Kong, so I said that I’d try to discover if there was any chance of bringing him out to give some visiting lectures. Somewhat to my surprise, my new university approved the idea and I was able to put together a 2-week package that allowed him to spend a week in each city.
    John arrived in April 2017 on Easter Sunday, declining the wheelchair United Airlines had arranged for him and coming into the arrival hall at Hong Kong airport on his own steam. The first thing he said to me when he saw me was that when I left Princeton, he had never thought any such trip would really happen. To which I could only reply, You know, neither did I.
    John’s schedule in Macau was rather complicated, as he shuttled between three different universities, giving lectures in each. Arrangements were sometimes a bit chaotic. John was 79 and sometimes tired, but to my admiration and amazement he never complained, even when he had to change accommodation several times, or a hotel restaurant was closed for a special event and he had to rely on me to raid a local supermarket and produce a makeshift scratch meal. He was indeed remarkably undemanding. So long as he had a pad of paper and a pencil for mathematical calculations and an enjoyable book to read, he was quite content. Keeping a famously disorganized man united with his passport, wallet, watch, and mobile phone was a constant minor logistical challenge, but what had been mislaid always eventually reappeared.
    The great reward came in watching John give his lectures and seeing how his audience reacted. When he spoke at the University of Macau, the organizers asked me to orchestrate an interview format in which I would prompt him with questions to which he could respond at length. That meant I could watch the audience, which included classes of high-school students, as well as university students and academics. Their concentration and the rapt expression on many of their faces, as John explained his free will theorem and where he thought Einstein had got things wrong, was something I shall never forget. And one graduate student in particular was an ardent disciple, who spent hours talking maths with John when he had free time.
    After seven days John took the ferry to Hong Kong, to spend his second week at the Education University of Hong Kong. The mathematician who picked us up turned out to have been a graduate student of one of John’s closest friends. Much to my relief, throughout his last week John stayed in the very comfortable Marriott Hotel. Asked by his hosts in advance what they could do to entertain him, I see that I replied that, while he had never been to Hong Kong before so would probably like to see something of the place, “I suspect that what would keep him happiest, when he is not actually giving his public lectures, is to be set up in an office in your department, with a blackboard and some chalk, with people (academics, students, etc.) free to drop in on him ad lib and talk to him informally about Maths of every kind.” So that was what happened. Every day, someone picked him up and drove him to the Maths Department, where he conducted something of an impromptu seminar.
    Once I had seen John safely installed in his hotel, I had to return to Macau to teach for most of the week, while he spoke at the Education University and the HK Princeton Club. But I returned on his last full day, when he gave his final big lecture. I found John happily ensconced in an office, complete with blackboard, chatting to one or two other mathematicians. People from all over Hong Kong made their way to his lecture—I remember one French graduate student had spent several hours travelling there. John produced a bravura performance, answered questions, and when the event was over spent quite a bit of time explaining the concept of surreal numbers to a young female student. We went on to an excellent farewell banquet, which he enlivened by demonstrating magic tricks. The next day, he flew back to Princeton.
    Even at the time, I was rather awed by John’s willingness to fly halfway around the world (economy class at that) to places where he had never been before and really only knew one person, myself. Looking back, I feel honoured that he (and his wife Diana) trusted me enough to let me arrange his trip. It was a privilege that he came here.
    And soon afterwards, I would be exceptionally glad that I had made the effort when I did. John was indomitable but unfortunately not indestructible. In November 2017, less than seven months later, he had another stroke, which meant his travelling days were over. And on Easter Sunday three years later, I learned that he was gone. I wish John had remained in better health for longer. But I was very lucky to have the chance to know him.

  42. A1987dM Says:

    @Scott #33: pretty much all powers of 10^0.1 have easy-to-remember approximations (1.25, 1.6, 2, 2.5, 3.2, 4, 5 etc., see https://en.wikipedia.org/wiki/Renard_series)

  43. Infovore » Links for April 13th Says:

    […] Shtetl-Optimized » Blog Archive » John Horton Conway (1937-2020) Notes on the passing of John Conway, but also, a delightful set of comments sharing first-hand memories of Conway – particularly being taught by him – that bring him to life. (tags: math obituary johnconway mathematics science ) […]

  44. Scott Says:

    mjgeddes #11:

      Speaking of Stephen Wolfram, he’s apparently about to make a big announcement, he’s claiming that he’s made a huge physics breakthrough , and as far I can make out , it’s a new model of space and time that gives a new interpretation of quantum mechanics – looks quite intriguing. This could be quite big news, so look forward to hearing your thoughts on it Scott….

    God help us … I meant, uhh … can’t wait to hear more! 😀

  45. Nick Says:

    Scott #31

    > If you want to get from syllogistic logic to Gödel and Cohen, would you agree that the obvious way to do that is to … think hard about logic?

    No. Historically, getting from syllogistic logic to modern logic was not accomplished by thinking hard about syllogistic logic. That paradigm had to be scrapped and replaced with something better. Consider Quine’s shady dictum: “Logic is an old subject, and since 1879 it has been a great one.”

    Of course, I am only attempting to interpret Conway’s remark, not arguing for its truth. I doubt CCT is like syllogistic logic in this way, but then “footnote” is a strong word.

  46. Scott Says:

    Priscilla Roberts #41: Thank you so, so much for sharing that.

  47. David Corwin Says:

    Since others are sharing memories of him, I thought I should share mine, as I have quite a few. I knew him first at Mathcamp when I was in high school, then would hang out with him as an undergraduate at Princeton (especially around 2009-2011).

    1) In 2007, he gave a knot theory demonstration at Mathcamp. The most memorable part was when he ripped apart a plastic bag with his teeth to help teach us knot theory (see this photo https://www.facebook.com/dcorwin1/posts/4358648244354)

    2) One day at Mathcamp I came to him and told him that I had just learned about the Mordell-Weil Theorem. His immediate reaction was to recall a fun story of a time he spent at Mordell’s house in Cambridge, mentioning especially the wonderful cake(?) that Mordell’s wife made.

    3) For my first couple of years at Princeton, he would sit in the common room all day and talk math to anyone who would listen. I took advantage of that now and then, and it was always fun to see him there. I was a bit sad when, in later years, he moved out to one of the coves in the hallway with blackboards (maybe they told him not to spend all day in the common room?), but he was still there telling people about math.

    4) There was a funny old phone sitting in the common room (maybe even rotary dial?) that nobody knew what it was doing there. The one time I remember it ringing, it was for Conway.

    5) Conway’s office was what some would call a mess, but what I found quite delightful. I remember in particular a poster he kept in front that had “Conway” in big letters and some sort of price (I guess it was an ad). He always kept his office unlocked, so once or twice I went in just to look (though one time my friend discouraged me from it). I don’t think I ever saw him there.

    6) In the spring of my sophomore year, he taught a graduate seminar on finite simple groups. I have a few memories of the actual math (various stuff about card shuffling and the Mathieu groups), but my biggest memory was of how we figured out when it would be. It was originally scheduled for a certain time with the registrar, but that didn’t work for everybody. Conway told us he was happy to lecture whenever we wanted (‘even at 3am!’ he said). So I made a whenisgood (I still have the link in my email), and we initially decided on Wednesdays at 7:30-9 and on Fridays at 3:30-5. I told everyone in the class but not the registrar, who got upset that we had changed the time without telling them. For some reason, two weeks later the time changed to 9:30-11 on TuTh.

    In characteristic fashion, he would regularly forget to come. So he gave us his cell phone number (I guess that was after he stopped using the rotary phone in the common room), so that we could call him if he didn’t show up to class and couldn’t be found in his usual hangout spots.

    Here are a couple of memories I heard from others:

    1) One time at Mathcamp, he was lurking the hallways and heard someone say that every subset of the reals has a sup and inf. Conway shouted from the hallway “nonempty!”

    2) One time a friend of mine slept in the common room (as many of us undergraduates did on occasion). He had class with Conway in the morning. Five minutes before class, Conway came up to him and yelled something like “time to wake up!” right next to him.

    3) I was told that it was hard to follow his classes if you didn’t already know the subject matter, since he would talk about whatever he felt like talking about on a particular day (trying to keep it vaguely related to the title of the course). But if you already knew enough about the subject matter (or didn’t care), it was absolutely wonderful to take his class. I remember my friend saying that he talked about Bernoulli numbers a bunch in linear algebra class.

    That’s all for now. I spent time with him on a number of occasions, so there are probably other stories I forgot.

  48. David Corwin Says:

    Just realized there’s an important memory I forgot: in addition to his doomsday algorithm, he devised an algorithm for converting between Hebrew and Gregorian dates, which took only about 1.5 times as long as finding the day of the week (in his head). When he explained it to me, he displayed a surprisingly deep knowledge of the halakha (Jewish religious law) behind how the Hebrew calendar works (mentioning various religious constraints on the Hebrew calendar, including that Passover must be in the Spring, which days of the week certain holidays can’t fall on, etc). (Maybe he was even able to quote a specific rabbinic dispute or opinion? I don’t fully remember.) I don’t remember the algorithm, but I remember there were three important numerical constants one has to memorize to do the algorithm, and he called them “he”, “she”, and “it”.

  49. arch1 Says:

    My second exposure to Conway’s ideas (the first having been to Life via Martin Gardner’s column, as with many amateur enthusiasts) was unknowing: It was to the game of Sprouts, which figured prominently in Piers Anthony’s novel Macroscope, and which I later learned had been co-invented by Conway.

    Then for a long time it seemed that Conway would periodically lob incoming bursts of wonderment from many directions. Working hard with his wonderfully creative mind to create and share his and others’ beautiful ideas with many people in many ways, he really did add to his era.

    PS. Priscilla #41: Thanks for the great story.

  50. mjgeddes Says:

    Scott #44

    He did actually just put out quite an interesting YouTube video on ‘The History Of Physics’, as a lead up ….. and these quotes are the ‘teasers’ for his new theory…

    ‘There kind of is a luminiferous aether’

    ‘We now know why the path integral works’

    Let’s wait and see.

  51. Scott Says:

    Nick #45:

      Historically, getting from syllogistic logic to modern logic was not accomplished by thinking hard about syllogistic logic. That paradigm had to be scrapped and replaced with something better.

    OK, but Frege and Russell were thoroughly versed in the old syllogistic logic. At least in math and science, the creators of a new paradigm are usually pretty well versed in the old one.

  52. Dilip D'Souza Says:

    While the word was spreading but wasn’t yet confirmed, I kept hoping it would turn out to be false. Alas, that wasn’t to be. Of course every CS student likely remembers the wonders of Life, but John Conway was more than that too. He was an original, incessant thinker, a man who reminded us of the way we should consider and enjoy mathematics. And yet there was always a playful perspective too — I mean, did he really say this? “You know, people think mathematics is complicated. Mathematics is the simple bit. It’s the stuff we can understand. It’s cats that are complicated.” What a delightful mind to think like that!

    What a sad thing it is to lose Richard Guy of Life’s glider, and John Conway of Life itself, within a few weeks of each other!

  53. asdf Says:

    There is a joke going around that I guess he might have liked. He apparently died from being adjacent to either less than 2 or more than 3 other mathematicians.

  54. David Corwin Says:

    Conway once told me the following very illustrative piece of advice: “If everybody else is running north, then run south AS FAST AS YOU CAN.”

    He may have meant it primarily as advice about doing mathematics, but I wonder if he would extend it to life as a whole. I recognize there are many situations where that advice doesn’t make sense (literally, at least), but it gives an interesting point of view on life.

  55. John Figueroa Says:

    He was such an inspiration to me. Nobody wrote clearer and more enthusiastically about math than him. 🙁

  56. Robert Says:

    Some years ago, in connection with my development of an origami implementation of a surface Conway invented, I had an opportunity to spend a couple hours with him, ostensibly to learn the background about the surface and his exploration of it, but mostly to sit back and bask in the glow as he held forth on a wide variety of topics. John Conway had the remarkable ability to make one feel like a mathematical toddler every five minutes while enjoying every moment of it.

  57. william e emba Says:

    Looking through the list of Field’s medalists, I’d estimate roughly half of them solved their problems by working inside the paradigms their problems were set in and pushed them way past what anyone realized was possible, and the other half brought entirely new paradigms into existence (or at least borrowed from totally unexpected places) that proved incredibly powerful.

    For example, Smale’s proof of the higher-dimensional Poincare conjectures were brilliant extensions of existing manifold techniques, while Freedman’s proof in 4D combined Smale’s methods with “Bing” topology, consisting of various infinite constructions that had a workable limit, and Perelman’s proof in 3D used Hamilton’s Ricci flow.

    Score: Smale old-fashioned problem solving brilliance, Freedman half-and-half, Perelman off-the-charts new paradigm.

    To continue the story, 4D topology was also revolutionized by Donaldson using methods from Yang-Mills gauge theory (as started by Atiyah). Fake R4’s were a completely unexpected shock. And the extremely difficult progress begun by Donaldson received a completely surprising kick in the pants years later from the Seiberg-Witten equations.

  58. Ravi Boppana Says:

    The summer after my daughter Meena finished 7th grade, she went to a math camp (MathPath) in Colorado. John Conway gave a talk to the campers about Fibonacci numbers. Why did a world-famous mathematician travel all that way to speak to a bunch of middle schoolers? I don’t know why, but I am forever grateful. Thank you, Professor Conway.

  59. Lorenzo Sadun Says:

    I have a few clear memories of Conway. The oldest was of his giving a public lecture at a conference, involving exactly one overhead slide, with the pieces gradually revealed. This lecture also featured his jumping around the room, and eventually jumping onto a table, to demonstrate symmetry groups.

    Another involves visiting Princeton and his holding court in the common room. He taught me Phutball, and of course thrashed me. (But my son Jonathan eventually avenged that defeat at MathPath!)

    The last was when he visited UT Austin and did some work with Charles Radin and me. He seemed to enjoy talking with us about rotations by angles whose squared trig functions were rational, and we wrote a couple of papers on them, but what he REALLY cared about was teaching anybody who would listen (read: star-struck me) the Doomsday algorithm for figuring out days of the week.

    John Conway always believed that real math was real fun. Sometimes we forget that, but when I think of him it’s easy to remember.

  60. Daren Scot Wilson Says:

    Many scientists and engineers cite Star Trek when they were kids as inspiration for their career choices. I suppose many mathematicians and computer scientists would cite Conway, through Martin Gardner’s writings, as their source of inspiration. Or gateway drug, perhaps.

    I was fascinated with the Game of Life as presented in the issue of Scientific American from 1970. Worked it with pencil and paper, then my mom found a pad of cheap quadrille paper. At some point she made a nicely bordered cloth from an old sheet I could lay on the floor and use Lego blocks for “live” cells. Working the Game of Life gave me much to think about during my paper route in high school – determinism, causality, could reality be a cellular automaton at some level? How could relativity work in a cellular automaton?

    Later in high school, I turned my talent for electronics design toward building a “Conway Computer” from small scale TTL chips, a couple static memory chips, a calculator keyboard, and an oscilloscope for display. I built this, showed my electronics teacher, and entered it into the Detroit Metro Science Fair. Except I had taken it apart by the time of the Fair, needing the parts for other projects, but I did make a nice presentation board with block diagrams, sketches, and descriptions. This won me some sort of plaque.

    Conway’s simple scheme gave me years of deep thought and technical challenges. As I read of the other things he accomplished, especially in group theory and monster groups, I only became more impressed. He was a bright light in our world, and will be very much missed by many.

  61. mjgeddes Says:

    Another interesting idea related to complex systems theory from physicist Nicolas Gisin. The idea actually offers a different way to implement Scott’s ideas of Knightian uncertainty based on unknowable initial conditions, without needing quantum mechanics !

    When I first read QGTM, I got ‘half a super click’ sensation – I thought you were perhaps half right, half wrong. I liked the idea that free will should be related to Knightian uncertainty, and it was due to uncertainty about initial conditions, but I was never convinced that quantum mechanics was involved (free-bits seemed implausible to me).

    Now Gisin has proposed that the physical world runs on intuitionist math, and he relates the advance of time to the creation of information – he says that for precision measurements, the last digits (numerical values) don’t actually exist before they are physically relevant – so this allows a kind of uncertainty even in classical systems – specifically, for non-linear chaotic systems, the initial conditions can’t be known to unlimited precision even in principle.

    After making the connection between Gisen’s idea and Scott’s Kightian uncertainty, I get the full ‘super-click’ sensation- it intuitively seems like this might be how it’s done. I always vaguely thought it might be somehow related to complex systems theory rather than QM, but couldn’t see how until now.

    Quanta article:
    ‘Does Time Really Flow? New Clues Come From a Century-Old Approach to Math’

    https://www.quantamagazine.org/does-time-really-flow-new-clues-come-from-a-century-old-approach-to-math-20200407

  62. asdf Says:

    There is a lovely xkcd tribute: https://xkcd.com/2293/

  63. Mikko Kiviranta Says:

    There’s an ingenious tribute in https://xkcd.com/2293/ .

  64. Ehud Friedgut Says:

    I met Conway at a reception at the IAS where he entertained a bunch of us with a nice game for four people. You stand in a small circle, and then he ties a rope between a wrist of the person in the north and a wrist of the person in the south, and another rope to the wrists of the people in the east and west. There are only two allowed operations: the whole circle rotates clockwise, or the person in the west swaps with the person in the north, while bringing their tied hand over his/her head. The game begins by Conway giving us a series of instructions (swap! swap! swap! rotate! swap! rotate!….), and then we have to try to untangle using the two allowed operations.
    It turns out that the two operations correspond to the functions x -> x-1 and x-> 1/x, and that you untangle if and only if 1 is a fixed point of the composition of the functions you’ve applied.

    That was fun, but I’ve never tried bringing this game to a party of non-mathematicians…

  65. Jim Sturm Says:

    Some 5-10 years ago, over a period of a few years, my schedule and that of John were somewhat synchronized by chance. On my bike I would often pass John on my way the office around 7:30 AM, on a path which led through the back of campus to Nassau St in Princeton. He was walking slowly a with a boy perhaps 8 years old (his son?), I assumed to nearby St. Paul’s school on Nassau St. A few hours later I would often go to Small World Coffee (a block from the school) for my midmorning coffee, and usually John was there, on the bench by the door, usually doing number puzzles.

    Some years before that I went to one of the packed lecture series on free will that he held, alternating with Simon Kochen (my math prof about 45 years ago), I think in Fine Hall. Highly entertaining, both from a layman’s and intellectual perspective. As I recall, one moment was when John started writing right to left on the board, and no one dared correct him, until he noticed it himself about 10 minutes later.

  66. Judith Silverman Says:

    I was fortunate to cross paths with Conway both in Cambridge and in Princeton, where I was one of a crowd that gathered around him daily in the common room. My friend Aline joined us from time to time and her response to my message informing her of his death evokes memories of what it was like to be in his presence better than anything I could write. I am posting it here, with her permission.

    ************************************************************************************************

    Dear Judith,

    Thank you for letting me know about John Conway. It is so terribly sad that he has passed away.

    What a gift he was – is – to the world. Not only for his brilliance in his career, but also in sharing his passion for mathematics in such an accessible way. He brought people together around math.

    I know he has meant a lot to you.

    Even though I am not in mathematics, he is special to me, for his warm welcoming way of lighting up a person about math. One time when you and I were in the lounge in Fine Hall, he targeted me, probably because I was not in math, and did a pennies trick with me.

    Thank you for bringing me into his world. I will never forget, decades ago, the electric pickle experiment. It was “tea time” and people gathered around in great anticipation as to what would happen. Do I remember correctly that the juicy, salty pickle lit up? [Yes, I have a picture of the glowing pickle. –jhs] He had such an excitement for discovery and a joy in sharing it with other people.

    I will never forget your last time in Princeton — that wonderful chance to tag along with you on your visits with mathematician friends. I think our last stop was with John Conway in Fine Tower. Retired, he didn’t have his regular office, but he had a new office, an alcove in the hallway, which was furnished with two cushioned seats and a whiteboard. At one point he asked me to help him find something, saying that that was his filing cabinet, pointing to the seat, indicating that I should lift the cushion. Indeed under that cushion there were papers! I liked that he could see things in unexpected ways – even an office and a filing cabinet. Then he proceeded to explain his new discovery, doing calculations and writing numbers on the whiteboard. The numbers resulted in a visual configuration, a picture, almost like a building. Before he finished, though, he had to switch gears for a meeting with a magician, a part of which we got to hear.

    Decades ago, he held a restaurant math event which I had learned about and signed up for. In the restaurant, he had set up what looked like a maypole, with strings dangling from the top. Each of us was to hold the bottom end of one of the strings. He showed us how to weave in and out, with each move corresponding to a different mathematical operation, an algebra of sorts, where knots at the top of the pole could be done and undone.

    He made mathematics accessible and wondrous, he could reveal math around us in many different forms. He had an amazing mathematical charisma.

    He was an unforgettable character. As I recall, he used to store his thumbs under his armpits. Did I ever tell you that a friend, who used to work at The Institute for Advanced Studies’ nursery school, said that Conway’s little son who was a student there, used to store his thumbs under his armpits just like his father did.

    In his passing, I feel for you.

    With kindest regards,
    Aline Johnson

  67. aleph null Says:

    I highly recommend the book “Genius at play : the curious mind of John Horton Conway” by Siobhan Roberts. You don’t need to be a mathematician to experience the wonder and joy of Conway’s circuitous journey through life.

  68. Randall K McRee Says:

    FYI Wolfram’s big announcement…
    https://www.sciencenews.org/article/stephen-wolfram-hypergraph-project-fundamental-theory-physics

  69. aqsalose Says:

    Re: asdf #5:

    >Anyway, you want f such that f'(x) is the same as f^{-1}(x). This is a cute and not too hard puzzle that I’ll leave as an exercise.

    These kind of statements about puzzle difficulty are not kind if not quantified. Should I feel ashamed for not seeing how to do basic calculus-level manipulations with inverse function theorem or similar, or admit that I have not really studied diff eqs since undergrad and grudgingly give up, or admit I have never dealt with integral equations and give up with a sigh of relief?

  70. asdf Says:

    aqsalose #69, it’s basic calculus level. Basically guess at a reasonable form of the solution, then solve for the parameters. The resulting parameter values were (for me) surprising in a nice way.

  71. Michael Tsai - Blog - John Conway, RIP Says:

    […] Scott Aaronson […]

  72. John Horton Conway 1937–2020 | Gödel's Lost Letter and P=NP Says:

    […] group to modular forms, but those must be for another time. Also see Scott Aaronson’s tribute and its comments section for many more stories and […]

  73. Andrei Says:

    re: asdf

    You left out the fact that the domain of f is (0, \infty)

  74. Pete Says:

    Aqsalose – This is a puzzle to give someone who is using overly much machinery. If you try high-tech approaches you will suffer, because general things of this type are not going to have nice solutions. But if you try simple things, you have a chance. There are not many functions where you know both derivative and inverse off the top of your head, and most of those have derivative and inverse which look very different. The exception is $f(x)=ax^b$, for which both inverse and derivative are of the same form. So maybe you should see if there are some a and b which work…

  75. Freemon Sandlewould Says:

    Screw vote by mail. Dems already cheat enough will all the dead chicagoans who vote.

    And stop with the believing the corona hoax. The lock down will end when the people who give congress their orders have looted enough by way of bail out bills.

    Been tracking and running the numbers. This is going to be on the order of a standard flu season.

  76. Scott Says:

    “Freemon Sandlewould” #75: I’m leaving your comment up only so that my readers, especially those from outside the US, can better understand what we’re up against. Alongside respiratory viruses are memetic viruses that attack the brain, and I’m sorry to see that yours has been eaten. That you chose to spew your doofosity in an obituary post for a mathematician who just died of covid makes it only slightly more disgusting than it would’ve been otherwise. From now on, any further comments in the same vein will go straight to the garbage, along with Holocaust denial and all the rest.

  77. asdf Says:

    I’ve been looking at the Wolfram stuff a little. It’s kind of interesting. There’s a pretty bad smoke screen of marketing gubbish around it but, I don’t know how else to put this, if there’s anything to it then there might be something to it. It reminds me a little of causal dynamical triangulation, which is an approach to gravity along similar lines but with maybe less flexible machinery.

  78. Richard Says:

    Poem for Conway

    A whimsy that was wont to soar
    Will assault the heights no more
    While we below — bereft and chilly —
    Huddle in the memory of his supernal silly.

    Possessor of a mind apart
    (Nowhere found on normal’s chart) —
    He plied a sole and special sea
    With no regard for currency.

    Let us then bemoan a
    Victim of corona —
    And let fly the timeless cry:
    * Why do giants have to die? *

    ——————

    It bothers me greatly to think (And I may simply be wrong about this.) that Conway did not care very much for himself — and that he largely failed to appreciate the sheer * delight * he often occasioned in the lives of those around him:

    It galls me that his druthers
    Forbade a joy that others
    Colliered from the quarry of his ken —
    That the grandness of his vision
    Imposed such self-derision
    On one whose like we may not meet again.

  79. Todd Trimble Says:

    Just a few reminiscences: I met Conway just as he was starting his series of lectures for the Nemmers Prize at Northwestern. I was seated close to the very front, in fact I just behind him before he was about to be introduced, and for some reason we started a conversation. I was immediately struck by his friendliness in engaging with a complete stranger.

    After the talk I found him in the hall and buttonholed him with a mathematics question, which in those days I would pose to anyone that I thought might be able to answer, but never got one. (It wasn’t such an impressive question looking back, but still I could never get an answer.) To my amazement he was answering my question before I even finished asking it! I pressed him if he had a proof, and no, he didn’t have a proof right away, but if I’d be willing to give him a lift to his hotel room, he’d be happy to think about it. Of course we were hardly out of the building when it was all clear to him and he was outlining to me how it would go…

    So there we were in my car, he with a complete stranger, thinking of things to say. He glanced at my car clock which read 5:41 and said, “541, the one hundredth prime!” I asked him how he happened to know that, and he explained that at Princeton he used to compete in speed contests with others in Doomsday calculations, and was getting a little annoyed at how good some of the graduate students were getting, and decided he needed something new to be best at. Only, he decided it would also have to be something that could also be useful to him mathematically, so he settled on factorizing numbers. Obviously as part of that, he would need to know his primes pretty cold, so there you go. He also mentioned that to memorize the primes up to 1000, you’re actually better off memorizing which numbers are non-obviously composite (i.e., not obviously composite based on the usual tricks).

    Maybe a year later or so he came back to Northwestern and was giving a talk on rational tangles, the type of thing Ehud #64 was describing, where four volunteers are holding pieces of rope and creating tangles according to certain rules called “twist” and “turn” (instead of swap and rotate). It’s based on continued fractions – if you keep in your head the rational number as you go along, you can shout out the correct sequence of instructions to get the knot untangled again. At some point I called out “twist”, then called out “turn”… oops, mistake! took it back, “no, I mean twist again!” Immediately he shot back, “like we did last summer?”

  80. Tim Keys Says:

    Cambridge, UK, 1982 — I still remember vividly having supervisions from John Conway. He had an immediate clarity for what maths was really about. For example, I hadn’t really understood why normal subgroups were important until he fished out his Rubic cube and said that the moves that fix the corners formed a normal subgroup and it was natural because you could define it without saying which way round you were holding the cube. And then there was his advice about tackling a difficult problem (“tickle it”), the Klein bottle in his office he kept peanuts in (“it’s rather hard to clean”), the game of talking using only words of one syllable (started just before entering a coffee shop), Conway’s Mornington Crescent (you win if you say “Mornington Crescent” just before your opponent was going to) and the time he gave up trying to find matching socks and went out and bought 28 identical pairs (“they thought I was running an orphanage”).

    A few morsels, but enough to last a lifetime.

  81. To cheer you up in difficult times II: Mysterious matching news by Gal Beniamini, Naom Nisan, Vijay Vazirani and Thorben Tröbst! | Combinatorics and more Says:

    […] mathematical hero,  passed away a few days ago. There are very nice posts on Conway’s work by Scott Aaronson (with many nice memories in the comment section), by Terry Tao, and by Dick Lipton and Ken Regan. […]

  82. John Conway (1937-2020) | Delightful & Distinctive COLRS Says:

    […] Great memories of Conway available here:  https://www.scottaaronson.com/blog/?p=4732 […]

  83. Richard Says:

    Re: Comment #78:

    The second stanza of “Poem for Conway” should be changed to:

    Pilot of a mind apart —
    Nowhere marked on normal’s chart —
    He plied a sweet and secret sea:
    Short shrift’s gift to infamy.

    ——————-

    Diana and Gareth,

    It’s a shame death did not spareth
    The Jedi djinn we call “John Horton Conway” —
    But this you surely know:
    Wherever he did go,
    His soul’s not now at rest — but at * play * .

  84. Richard Says:

    The second stanza of “Poem for Conway” should be changed to:

    Pilot of a mind apart —
    Nowhere marked on normal’s chart —
    He plied a sweet and secret sea:
    Short shrift’s gift to infamy.

    ——————-

    Diana and Gareth,

    It’s a shame death did not spareth
    The Jedi djinn we call “John Horton Conway” —
    But this you surely know:
    Wherever he did go,
    His soul’s not now at rest — but at * play * .

  85. fred Says:

    Not sure if this was posted yet

    https://www.quantamagazine.org/john-conway-solved-mathematical-problems-with-his-bare-hands-20200420/

  86. John Horton Conway passes - British APL Association Says:

    […] There is a blog post about his passing with some good comments: scottaaronson.com […]

  87. Randy Says:

    Regarding the j-function featuring in the monstrous moonshine work by Professor Conway and others, my impression of the plot in the complex plane ( https://en.wikipedia.org/wiki/J-invariant ) is the upper part (above unity on the vertical axis) features some verticals extending ever up whereas the lower part features coarse “marbles” of ever-decreasing size. A wanderer on the plot appears to have a choice: travel northward and experience unbounded length, or stay south and enjoy bounded lengths, but accept an unbounded whole number of regions.

    Is the j-function suggesting a particular recipe for compactifying a dimension, but with a warning that a fractal penalty will appear in some way?

    Some years ago I heard Professor Kaplunovsky of UT Austin point out that based on Noether’s Theorem, the Monster symmetry group means something is conserved, but at least in physics we don’t know exactly what might be conserved yet. This still bothers me. Could it be about compactifications? I’m aware of the Conformal Field Theory work but I’m not understanding if any of it answers the conserved quantities question.

    This is my probably-childlike offering this time.

    Professor Conway’s remarks on the Monster made an impression on me when I first watched them years ago:

  88. John H. Conway (1937-2020)​​​ un prolífico matemático británico conocido por el «juego de la vida» - La Ciencia de la Mula Francis Says:

    […] Te recomiendo leer también a Michael Aschbacher, «The Status of the Classification of the Finite Simple Groups,» Notices AMS 51: 736-740 (2004) [PDF]; Ron Solomon, «On Finite Simple Groups and Their Classification,» Notices AMS 42: 231-239 (1995) [PDF]. Y, por supuesto, sobre Conway a Siobhan Roberts, «Genius at Play: The Curious Mind of John Horton Conway,» IAS, 2015; Siobhan Roberts, «A Life in Games,» Quanta Magazine, 28 Aug 2015; Terence Tao, «John Conway,» What’s new, 12 Apr 2020; The Blog of Scott Aaronson, «John Horton Conway (1937-2020),» Shtetl-Optimized, 12 Apr 2020; […]

  89. Deepa Says:

    We were tired of watching sitcoms and movies as a family and recently started watching interesting talks. This weekend we watched John Conway’s Google talk, which led to a good family debate on random versus deterministic versus free. And then immediately after that we watched his talk on surreal numbers at the University of Toronto. He said he was tired of being associated with the game of life, so he would have been happy we listened to this other big idea of his :).

    Thanks for introducing me to him (sadly, via this post). p.s: My son went to a presentation at Texas Advanced Computing Center as a 7th grader where he was shown the game of life, so I heard about it then, but not the name John Conway (until now). He is such an engaging speaker. He has a great talent for telling stories. My husband said he loves how passionate, deep and thorough he is, and also that he is not afraid to express his emotions (in the Toronto talk, he teared up at one point).

  90. Shtetl-Optimized » Blog Archive » The Busy Beaver Frontier Says:

    […] explain to a child lost two of its greatest giants: John Conway (who died of covid, and who I eulogized here) and Ron Graham. One thing I found poignant, and that I didn’t know before I started writing, […]

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