Letters to the Editor

Hedgehogs and Foxes, Not particular field, trying to compensate theorem are unstable, i.e., a trajectory Birds and Frogs by deepness what they miss by lack close to such an unstable periodic ’s Einstein Lecture of extension and variety: Antoni Zyg- orbit diverges exponentially from it. (Notices, February 2009) is a beauti- mund. The spider type is Georg Can- Is this chaos? No, chaos occurs if the ful meditation on the distinction tor, proposing a personal construc- exponential divergence is present between two types of mathematical tion, with little reference to other for long-term behavior, i.e., on an thinkers. But in calling them “birds” authors. To the bee type belongs Paul attractor. Unstable periodic orbits and “frogs”, Dyson contravenes a Erdo˝s, moving permanently from in a repeller are physically invisible, metaphor that predates his by sev- flower to flower, changing always his and do not imply chaos. So, with the eral thousand years. The common problems. modern use of the word chaos, period terminology is that referenced by Dyson’s and Bacon’s typologies three does not imply chaos! can be combined: Bolyai: ant and frog; Interestingly, the theorem of Li the late philosopher Sir Isaiah Berlin von Neumann: frog and bee; Bour- and Yorke is a special case of a theo- in his essay “The Fox and the hedge- baki: bird and spider; Hilbert: bird rem by the Ukrainian mathematician hog”, which comes from the words and bee; Gödel: ant and bird; Poin- Sharkovsky (1964). In its glorious of the ancient Greek poet Archilocus: caré: bird and bee. Any researcher simplicity, the theorem of Oleksandr “The fox knows many things, but the combines in various proportions Mikolaiovich Sharkovsky states that, hedgehog knows one big thing”. different types, at different periods. if a continuous map of the real line —Kiran S. Kedlaya Open questions: Can we transfer to itself has a periodic point of (least) Massachusetts Institute of these metaphors from individuals period m​, then it also has a point of Technology to historical periods? For instance, period n​ whenever n​ is to the right [email protected] in the field of analysis, can we claim of m​ in the following unconventional (Received February 2, 2009) that the eighteenth century was pre- ordering of the natural numbers: 3, ponderently frog and ant, while the 5, 7, 9,…, 2.3, 2.5, 2.7,…, 4.3, 4.5,…, Zoological Metaphors for second half of the nineteenth century 16, 8, 4, 2, 1 (we start with the odd Mathematicians and the beginning of the twentieth numbers in increasing order, then century were predominantly a bee Freeman Dyson’s impressive “Birds have the odds multiplied by 2, 4, and a bird? Can we describe in such 8,…, and finally the powers of 2 in and frogs” (Notices, February 2009) terms the move, in algebraic geom- decreasing order). reminds me of another zoological ty- etry, from Castelnuovo and Severi to pology, proposed by Francis Bacon in —David Ruelle Zariski? etc. 1620, in his Novum Organum (Apho- Institut des Hautes Etudes rism I, 95): ants, spiders, and bees: —Solomon Marcus Scientifiques “Empiricists are like ants, who only Institute of Mathematics [email protected] Romanian Academy collect things and make use of them. (Received February 12, 2009) Rationalists are like spiders, who [email protected] weave webs out of their own bodies. (Received February 6, 2009) Separating Mathematicians But the bee has a middle policy; it In the February 2009 Notices, Free- extracts material from the flowers Some Comments on “Period man Dyson elaborates on the division of the gardens and meadows, and Three Implies Chaos” “birds” versus “frogs” among math- digests and transforms it by its own Freeman Dyson’s beautiful article ematicians. Usually, that division is powers. The genuine task of philoso- “Birds and frogs” (Notices, Febru- described as “bird’s eye view” versus phy is much the same. It does not ary 2009) refers to the well-known “worm’s eye view”, but then Dyson depend on or mainly on the powers paper “Period three implies chaos” avoids it, lest he may himself call a of the mind; nor does it deposit the by Li and Yorke (1975). This paper “worm”, since he is considering him- raw materials supplied by natural is at the origin of the current use self to be a “frog”. history and mechanical observations of the word chaos for differentiable Recently, a similar division in in the memory just as they are, but dynamical systems. Li and Yorke “seers” versus “craftspeople” was pro- as they have been worked over and proved that, for certain maps of the moted by Lee Smolin in his amusing transformed by the understanding. interval, the existence of a periodic book The Trouble with Physics. And Therefore there is much to be hoped orbit of period three implies the one of my good old and articulate for from a closer marriage (which has existence of periodic orbits of all colleagues likes to go even farther not yet taken place) between these periods. This is what Li and Yorke by dividing scientists into those who faculties, namely the experiential and called chaos. The use, however, has “think” versus those who “stink”. the rational.” changed and, as stated by Dyson, is No doubt, it is an irresistible urge of One can reconsider Bacon’s met- now that “neighboring trajectories many a human intellect to discrimi- aphors as follows: ants are those diverge exponentially”. Most of the nate, classify, and segregate; and scholars who remain involved in a periodic orbits arising in the Li-Yorke then of course, judge, sentence, and

688 Notices of the AMS Volume 56, Number 6 Letters to the Editor when possible, why not, also execute. Replies and Correction have taken pains to motivate the And the most primitive and brutal I am grateful to the authors of the long-term goals of their project as way, needless to say, is to divide in above letters for their criticisms and well as to present the state of the merely two categories. Dyson appears corrections. I am especially grateful art in its accomplishments as well to enjoy himself quite a bit with this to David Ruelle for telling us about as limitations; they have managed “apartheid” venture, and on top of it, the Sharkovsky theorem and explain- the difficult feat of writing clearly seems to be convinced to be doing ing its meaning. I am sad to learn that about this highly technical subject something useful, if not in fact, even the clarion statement of Yorke and for non-specialist readers while pro- important. Among others, he judges Lee, “Period three implies chaos”, is viding enough substance to make von Neumann to be a “frog” and not no longer true. formalization credible. Whether or a “bird”, just like his old professor Surprisingly, none of these four not Wiedijk’s prediction is realistic at . Be- letters calls attention to the worst that, in “a few decades suddenly all yond all such “apartheid” excursions, errors in my lecture, which were mathematicians will start using for- however, Dyson misses quite a few pointed out by two other authors, malization for their proofs,” I have crucial facts related to the work of Adrian Bondy and Manjit Bhatia, in no doubt that a project capable of some of the mathematicians he sets personal letters to me. I am grateful attracting so many talented people out to segregate. With von Neumann, to these two gentlemen for identify- around such clearly defined objec- for instance, he misses two of his ing my mistakes, which occur in the tives for one of our central activities extraordinary insights. One of them discussion of the P ​ N​P​ ​problem on will ultimately change the practice of = is the basis of present and future com- page 217. To set the record straight, our profession in ways that are both profound and unpredictable. putation, namely, that a computer pro- here is a description of the mistakes. Writing for an audience of math- gram is allowed to act not only upon I was wrong to say that the traveling- ematicians, the authors may be for- the data, but also upon itself, and do salesman problem, as usually formu- getting that among their readers so in ways dependent on the data. This lated, the problem being to find the will be those who do not necessarily manifestly self-referential nature of shortest route visiting a given set share or sympathize with a com- computer programs has only come re- of cities, is N​P​. To find the shortest mon assumption they see no need cently to a more systematic attention route is probably harder than N​P​. To to make explicit, namely that human within a wider mathematical context, obtain an N​P​ problem, one should understanding of proofs is of inter- namely, with the emerging theory of ask a more modest question, for ex- est for its own sake. I am not mainly the so-called non-well-founded sets, ample, whether there exists a route visiting the cities and not exceeding thinking of future mechanical proof presented in the 1996 book Vicious a given length. In addition to this assistants themselves, whose coming Circles of Jon Barwise and Lawrence mistake, I made a second mistake role in determining our priorities is Moss. Indeed, ever since the ancient when I said that the traveling sales- scarcely addressed. Of more immedi- Greek Paradox of the Liar, not to men- man problem is conjectured to be ate concern are decision-makers who tion its modern set theoretic version an example of a problem that is P may well be convinced by the special in Russell’s Paradox, there has been a ​ but not N​P​. Here I should have said, issue to take the attainment of a given considerable reluctance among math- “N​P​ but not P​”. These mistakes oc- benchmark in the mechanization of ematicians to deal with any form of curred because I fell into the trap of mathematics as a signal to begin self-reference. After all, it indeed can- talking about a subject of which I am phasing out human mathematical not be treated lightly, being nothing ignorant, quoting some remarks that research as a superfluous luxury. else but the name of God in Exodus I heard from a friend who is equally Harrison writes that a formalized 3:14 of The Old Testament. Well, von ignorant. Thanks to Adrian Bondy proof “can be presented to others in Neumann not only introduced self- and Manjit Bhatia, I am now a little a high-level conceptual way,” but a referential programs into effective less ignorant. pure cost-benefit analysis might see computation, but managed to do even this as an unnecessary expense. one better when he proved the exis- —Freeman Dyson Harrison sees “the traditional so- tence of self-reproducing automata, Institute for Advanced Study, cial process” for verifying correct- and showed that such automata can in Princeton ness of proofs as “an anachronism fact be rather simple, having less than [email protected] to be swept away by formalization.” a few hundred elements. So much for (Received March 7, 2009) I would argue that this “woolly com- applying hard and fast segregation munity process” is precisely what methods of “apartheid” to truly re- gives meaning to the peculiar practice markable scientists. Human Understanding and of proving theorems. On the alternate Formal Proof view, human intervention in proof- —Emeritus Professor Elemer E. By devoting a special issue to an making can easily be construed as a Rosinger extended discussion of prospects temporary inconvenience. The risk is University of Pretoria, South Africa for formalization of mathematics, real that we will lose one of the few [email protected] the Notices has done its readers a fragile means we have evolved to (Received February 12, 2009) great service. The articles’ authors come to terms with our experience

June/July 2009 Notices of the AMS 689 Letters to the Editor

AmericAn mAthemAticAl Society of the given world, and even more so Then I should find the proofs easily of the virtual reality we all inhabit as enough.” I would like the mathemati- Mathematical participants in the society of human cians of the future (particularly those beings. who are not of Riemann’s caliber) to be able to say: “If only I had the broad —Michael Harris outline of a proof! Then I should have Moments Université de Paris my proof checker verify the details [email protected] A series of posters that easily enough.” (Received February 18, 2009) promote appreciation —John Harrison and understanding of the [email protected] role mathematics plays in Reply to Harris (Received March 4, 2009) science, nature, technology Michael Harris reminds us that it is and human culture sometimes beneficial to say things that go without saying. For the re- www.ams.org/mathmoments cord, I did not intend to disparage the understanding of proof for its own sake, nor the creative activity of human mathematicians generally. Neither, I’m sure, did any of the other authors of papers in the special issue. I would draw a sharp distinction between (i) verification of a proof, and either (ii) its conceptual under- standing, or (iii) the creative process that led to it in the first place. My critique of the “social process” re- lates solely to its role in verifying the correctness of proofs, as the text following the “anachronism” remark tried to make clear. As a vehicle for conveying understanding, I cannot seriously contemplate an alternative to communication between people. My goal with mechanical proof- checking isn’t to put mathematicians out of work or eliminate the need for human creativity. On the contrary, the goal is to free the creative spirit from worrying about whether great Correction imaginative constructs are invali- The September 2008 issue of the dated by small errors in detail. Notices carried a brief article I Riemann is supposed to have wrote about Grothendieck and the said “If only I had the theorems! 75th anniversary of the Institut des Hautes Etudes Scientifiques (IHES). The article called the oc- Submitting Letters to the casion the “sesquicentennial” of Editor the IHES. Thanks to Jordan Bell, The Notices invites readers to sub- a mathematics graduate student mit letters and opinion pieces on at the University of Toronto, for topics related to mathematics. pointing out that the word “ses- Electronic submissions are pre- quicentennial” refers to a 150th ferred (notices-letters@ams. anniversary, not a 75th. Bell knows org); see the masthead for postal his Latin: He has translated forty of mail addresses. Opinion pieces are Euler’s papers from the Latin and usually one printed page in length posted them on the arXiv.org with (about 800 words). Letters are nor- the author name “Euler”. mally less than one page long, and shorter letters are preferred. —Allyn Jackson

690 Notices of the AMS Volume 56, Number 6