Genius at Play

A month or two ago I read the new biography of John Conway, Genius at Play, by Siobhan Roberts (whose book about Coxeter I reviewed here). Since then, writing about it has been on my to-do list, but I wasn’t at all sure what to say. In today’s Wall Street Journal Jordan Ellenberg has done a better job of this than I ever could, so I have a place to start: read Jordan’s review.

Probably the first thing to say about the book is that it’s an excellent portrayal of its subject, who is an unusual and well-known figure in the math community. Roberts spent a great deal of time with Conway, traveling with him and getting to know him rather well personally, then very ably turning that experience into a quite readable and enjoyable book. It’s hard to imagine that a better biography of Conway would be possible.

In his review, Jordan crystallized precisely for me why I was having trouble writing about Conway and the book:

Will you like this book? Here’s a simple test. What’s the rule that produces the sequence 1, 11, 21, 1211, 111221, 312211 . . . ?

This is Mr. Conway’s “look-and-say” sequence, so called because each number (after the first) is what you get when you look at the previous number and say it aloud: “one one; two ones; one two, one one; one one, one two, two ones . . .”

If that makes you laugh with surprise, as it did me, you’ll like Mr. Conway, and you’ll like “Genius at Play.” If not, you might want to quit here and go read something improving about the Greek debt crisis.

I’m afraid this didn’t make me laugh with surprise; it seems that I’m immune to the charm of this sort of thing. While there was a lot of Conway’s story I found interesting and which kept me avidly reading, his mathematical interests are very different than mine. Mathematical games make up a big part of his life and career, but the only aspect of this I’ve at any time found appealing was back in high school, when I remember writing a computer program to run the game of Life. I learned from the book that this is Conway’s most famous creation, a fact he’s not entirely happy with.

I also learned that my one personal experience with Conway is widely shared: at lunch with a group here at Columbia he mostly spent the time explaining how to calculate in one’s head what day of the week any date is. Unfortunately I just didn’t enjoy the idea of spending time on this then, and still don’t.

Conway is one of the main figures responsible for an important piece of mathematics, discovering and working out the properties of some of the sporadic finite groups. This isn’t something I’ve ever known much about, and I was quite interested to learn from the book some more about the subject and the history of how it came about.

I can’t think of any other biography that I’ve read that gives such a vivid impression of its subject. In Conway’s case this is somewhat of a mixed bag. He can be a very entertaining character, but his personal flaws are also apparent, with a suicide attempt and several failed marriages testifying to some real problems. Whenever books like this appear, I think one reaction of some mathematicians (not me…) is “is this good for the public portrayal of mathematics and mathematicians?”. Conway’s mixture of genius, highly accessible mathematical discoveries often related to games, and serious issues dealing with the outside world fit rather well with a certain caricature of what mathematicians are like. In my experience with great mathematicians, very few of them other than Conway fit the caricature. While any book about him would likely reinforce the caricature, at least this one gives a very well-written and comprehensive view of its subject.

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30 Responses to Genius at Play

  1. JSE says:

    It totally makes me laugh with surprise! I often teach that sequence to undergrads in discrete math; it really emphasizes the point that a sequence is an arbitrary list of numbers, generated however you please, not “a button on your calculator.” On the other hand, I find the idea of computing days of week totally boring.

    Anyway, thanks for kind words. One thing I’m sad about is that I addressed the point you make: “Conway in many ways fits a public stereotype about mathematicians, but many of the qualities that match the stereotype actually make him highly ATYPICAL among real mathematicians” more explicitly, but some of it got cut for length, so I hope the point is still somewhat clear.

  2. KJ says:

    I wouldn’t worry too much about this book propagating stereotypes. The set of people who 1) are not already interested in math 2) are the sort that is influenced by stereotypes and 3) decide to read this book has to be vanishingly small.

  3. Puffin says:

    What is your opinion of Conway & Kochen’s Free Will Theorem (which I believe he was alternately referring to as his “Free Whim Theorem” a while back.) This seemed to me to be saying something surprisingly deep about the relationship of observers with the principles of QM and relativity. It’s almost like having a theoretical bound on Bells inequalities if you believe observers can change their minds. I was surprised it didn’t receive more attention.

  4. Richard Séguin says:

    Damn, JSE’s review is behind a pay wall. It would have been nice to settle now with just that since I’m already off on a (ultra-)marathon read of something else and probably won’t get to the book for quite some time. I, for one, found the answer to the sequence problem amusing.

  5. Chris Lott says:

    Richard, google “wsj a fellow of infinite jest” and click on the first link.

  6. Ben R says:

    Puffin, I’m not PW but the “free will theorem” seems to be just a rebranded variant of Bell’s theorem. Bell’s setup was pretty awkward and I think Conway & Kochen’s is better, but I don’t think it says anything about free will that Bell didn’t understand in 1964. Scott Aaronson seems to feel similarly (

  7. Jeff M says:

    See and say is wonderful, even more so since Conway proved things about the sequence despite the fact that it has no “mathematical” definition. When I was in grad school Conway came and gave a talk about surreal numbers, which included his doing rope tricks to demonstrate the relation with knots. He’s much more than games. As for Free Will, it’s related to Bell’s, but they are different as best I understand. One can explain Bell in various ways, some of which have nothing to do with free will. The free will theorem, as best I remember, says that if you want to throw out free will, you have to throw out lots of other stuff too.

  8. KenW says:

    That sequence reminds me of sudoku. Although I realize that you could play sudoku with mahjong tiles.

  9. Richard Gaylord says:

    i read the book, was not amused by the opening test and felt that while it might be an accurate portayal of Conway, it in no way, represents the way serious mathematicians work. I personally came away with the feeling that “it’s too bad that an obviously brilliant person would fritter away his professional life working on amusing but trivial problems”. i thought his own explanation of the history of the game of life which is referred to in the book ( was fascinating. i can understand his conflicted attitude to GoL since he would prefer to be known for his mathematical work rather than for GoL, the profound implications of which in regards to emergent phenomena, he apparently missed, but according to his biographer, Conway would prefer to be known for anything than for nothing at all so he’s come to terms with this notoriety . As for his free will theorem, it is vacuous and shows a basic lack of understanding of the philosophical nature of the issue (much as Wolfram’s free will argument does). physicists and mathematicians trying to do philosophy are always amusing (we see this in the work of Tegmark, Deutsh, Dawid and others you have mentioned in your blog postings).

  10. John Baez says:

    What made me “laugh with surprise” is not that Conway invented the look-and-say sequence, it’s that he showed the ratios of successive terms approach a constant 1.303577269034… which is the unique positive root of a polynomial of degree 71, and that he proved even deeper results about this idea.

    Anyone who thinks Conway has frittered away his time on trivial problems must not know the Atlas of Finite Groups, the book Sphere Packings, Lattices and Groups, and Conway’s profound analysis of the Leech lattice and its symmetry groups. I think anyone who did those things has done enough hard work – and if they want to goof off and have some fun, that’s fine with me.

  11. John Rennie says:

    If you’re interested in a mildly technical history of the finite group work then I recommend Mark Ronan’s book Symmetry and the Monster. This is a popular book not a maths book, but Ronan frequently slips and injects enough real maths to whet the appetite.

    Actually I have an idea I heard about the book on this site, though searching your site doesn’t find anything. My apologies if I’m telling you something you already know.

  12. Richard Séguin says:

    Chris Lott: It doesn’t depend on how you get to that page. If you’re not a subscriber, WSJ throws up a paywall. At this point, it might already be too late to find this issue on a newstand somewhere.

  13. JSE says:

    Richard, I promise, it depends how you get to the page. I’m not a subscriber and when I get there through a Google search, I can see the whole thing. Otherwise, just the intro.

  14. Montague says:

    Man, Ellenberg really missed the mark with his account of the Look-and-say sequence. The definition of the sequence is a trifle–I mean it can be amusing that a young child might be able to guess how it works where a Fields medalist would be stuck, but that is not the beauty of the thing. The beauty of the thing is how Conway takes this trivial-seeming property which does not promise any mathematics at all, and spins from it a whole theory of “audioactive decay” involving a startling analogy with the periodic table. The mind that is moved to construct such things from a child’s toything, alongside his deeper contributions to mathematics (see the Atlas of Finite Groups and the Sphere-packing book with Sloane) is a mind that I want to know more about!

  15. Anonyrat says:

    @John Baez, per Wiki, if L_n is the number of digits in the nth item in Conway’s sequence, then Limit n -> infinity of L_n+1 / L_n = 1.303577269034….

    Also, was his proof lost?
    Proof of Conway’s lost cosmological theorem
    Authors: Shalosh B. Ekhad and Doron Zeilberger
    Abstract: John Horton Conway’s Cosmological Theorem about sequences like 1, 11, 21, 1211, 111221, 312211,…, for which no extant proof existed, is given a new proof, this time hopefully for good. – See more at:

  16. Peter Lund says:

    Séguin: yes it does. Following the link gave me the paywall, going via google gave me the review with no paywall. This is my standard experience with WSJ.

  17. Richard Séguin says:

    Chris and Peter: Thanks! I originally did the search with DuckDuckGo, and I got the paywall. I switched to Google and got through. I’ll remember this trick in the future.

  18. Cosmonut says:

    Must agree with the comment about caricature mathematician.

    I’ve often found that while theoretical physicists are seen as “Brilliant minds unlocking the secrets of the Universe”, the perception of mathematicians is “Borderline nutcases obsessing over arcane stuff”.

  19. John Baez says:

    Anonyrat wrote: “if L_n is the number of digits in the nth item in Conway’s sequence, then Limit n -> infinity of L_n+1 / L_n = 1.303577269034….”

    Right, sorry: I said “ratios of successive terms”, but it’s “ratios of successive numbers of digits”.

    “Also, was his proof lost?”

    Not of this fact, but of a stronger fact of which this is a corollary: the Cosmological Theorem. The paper you mention is quite fun, and it adds to the reputation of mathematicians as borderline nutcases obsessing over arcane stuff. First, one of the authors, Shalosh B. Ekhad, is actually a computer. Second, the abstract says:

    Abstract: John Horton Conway’s Cosmological Theorem about sequences like 1, 11, 21, 1211, 111221, 312211,…, for which no extant proof existed, is given a new proof, this time hopefully for good.

    Third, it ends with a rant on the four-color theorem, titled “On a posteriori trivial theorems: The ultimate proof of the Four-Color Theorem should emulate our proof”.

    It explains the lost-and-found story:

    The Cosmological Theorem. There exists an integer N such that every string decays in at most N days to a compound of common and transuranic elements.

    Conway stated that two independent proofs used to exist, one by himself and
    Richard Parker (that only proved that N existed), and another one by Mike Guy
    (that actually proved that one may take N = 24, and that it was best possible).
    Unfortunately both proofs were lost. Here we announce a new proof (which establishes
    that one may take N = 29; with more computations one should be able to
    rederive (or else refute) Guy’s sharp N = 24).

  20. @John

    DZ goes on to write:

    “eventually one should be able to type Prove4CT();, and the truth of the theorem should be implied by the halting of the program. In order to check the validity, the checker would not need to see any specific configuration. Everything should be done internally and silently by the computer. All that the checker would have to do is check the program.”

    and this is now the case! Gonthier and Werner have of course completely formalised the proof. (for those who may not know:

  21. Peter Woit says:

    It does seem that the WSJ prefers Google to my blog: if you access the same URL via Google as the one in the link in the posting you get the full review.

    I confess to have pretty much paid no attention to the “free-will theorem”. It starts from assumptions that ignore anything interesting about the measurement problem, and claims to reach conclusions about something that to me is a question about consciousness and how brains work. As far as I can tell, Conway knows little about either physics or neurobiology. Maybe there’s something interesting going on in this “theorem”, but I’m happy to let others be the ones to spend their time on it.

  22. Martin S. says:

    Can anyone hint on the experimental setup of the spin-component-square measurements of the Free will theorem? I am into the basic physics they use, not the overall theorem herein.
    Regarding their spin axiom (on measuring squares of spin components), they cite several possibilities for such measurements at their article, endnote 1, page 27.
    I have looked at the Kochen/Specker work that is one of those cited possibilities. It seems that it is based on spectra measurements (of systems where Hamiltonian contains spin-based parts that alter the base Hamiltonian); see part 4 there, pages 13-16 (71-74 in journal page numbering).
    If I get it correctly, the Free will theorem takes entangled systems and looks at corresponding measurements. Does it mean that measurements of spectra on one of those entangled components can affect whether the other (entangled) component has (or does not have) its base Hamiltonian altered by the spin-based interactions?
    May be it is different for molecular beams that they cite as another possibility for the measurements. Yet when I looked at work of Erwin Wrede (that is cited by them), I have not seen anything on Hamiltonians or spin there; not being surprised, since that work of Wrede is from 1927.
    They mention interferometry with crafted coherent beams (see the endnote 1 at their article) too, but they do not go into details there.

  23. About “lost” proofs: This is the second “lost” proof story I’ve heard about Conway. Manjul Bhargava told a story at a talk I’d attended that Conway had also lost his original proof of the 15 Theorem. . Are “lost” proofs common among practicing mathematicians? Is Conway particularly disorganized?

  24. Peter O. says:

    Yes, common. If a proof reduces to checking six hundred trivial cases, one can sit down and check them. At the end one has proved to oneself that a certain statement is true, and can tell it everybody else. The proof is an unpublishable pile of waste paper, but others will find nicer proofs, or program a computer to check these cases.

  25. Shantanu says:

    Peter, OT to this, but some sad news, in case you were unaware.
    Jacob Bekenstein passed away yesterday.

  26. Chris W. says:

    Concerning Jacob Bekenstein, Jennifer Ouellette posted this scientific obituary on her SciAm blog yesterday, and Scott Aaronson posted a personal tribute today.

  27. Jeff M says:


    Yes, Conway is particularly disorganized. And I don’t believe his lost proofs were the “check 600 cases” types, though those proofs certainly exist. Mathematicians prove all sorts of things, and sometimes it isn’t even that the proofs are lost, it’s just that the result is not interesting. BTW, I know Conway is disorganized since when he was giving the talk on surreal numbers I mentioned before, someone had to drive to Princeton from NYC to make sure he would show up, and he had in fact completely forgotten he was supposed to give a talk 🙂

  28. Erraticon says:

    Interesting how people easily forgive a famous person his/her unpleasant excesses clearly caused by a personality disorder (most likely bordeline in this case).

  29. Neil says:

    Well, that did make me smile, if not laugh. I plugged it into the online encyclopedia of integer sequences ( and sure enough it came up, along with a few interesting tidbits (no integer larger than 3 appears, and 333 or longer does not appear, although 222 does). But the cosmological thing–I don’t know.

  30. One minor correction that may be implied by John Baez’s post (“…Conway invented the look-and-say sequence…”): Conway did not invent the sequence. The biography says it originated in a Math Olympiad in Belgrade.

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