political events of the second half of the did not tire of reading of his and his Random Curves: 20th century: The Cold War, but also the wife Ann Hibner’s happiness (she a quality of life in the former Soviet math historian with a profound Kova- Journeys of a Union; the American aggression in levskaia biography [7] and prolific Vietnam, but also the charisma of the anthropologist and gender researcher), solidarity and liberation movements; about their travels, their organiza- Mathematician racism and civil rights movements; tional work, their endowments, their by Neal Koblitz suppression in Africa and Latin America encounters with so many interesting, and counter forces; injustice, stupidity, impressing and sympathetic people BERLIN, HEIDELBERG: SPRINGER SCIENCE + brutality, arrogance of power—and the around the world. Koblitz is a good BUSINESS MEDIA, 2008, 392 PP., US $34.95, tactics and strength of the suppressed writer: People are described in a lively ISBN 978-3-540-74077-3 and how best to support them. Nothing way, with a lot of humor, but always REVIEWED BY BERNHELM BOOSS-BAVNBEK passes without getting Koblitz’s well- with high respect, whether he recalls argued ‘‘Another Look’’, be it a deserved the words of a schoolboy in rural Peru All men of any condition who have rap for widely accepted conditions or a or of the Vietnamese Prime Minister. done something of special worth or positive explanation of widely rejected This human touch hopefully reconciles something that may truly resemble circumstances. He is highly opinionated even a conservative reader who other- those things of special merit, should, and displays a total absence of the wise may feel repelled by the hard if they are truthful and good people, politeness associated to fashionable inexorable logic of Koblitz’s political write in their own hand the story of social constructivism and relativism. arguing. their lives, but they should not begin Throughout the book, Koblitz is There is not much about mathemat- such a fine undertaking until they opinionated with good reason: He ics in the book, just a few rather sketchy have passed the age of 40. writes only of things he seems to have comments about Koblitz’s personal investigated thoroughly, and he is, path into number theory at Harvard, more than most, aware of possible fatal Princeton and Moscow; then just 32 ith this verdict, the Renais- consequences of careless disregard for pages on his seminal work on elliptic sance goldsmith and sculptor logic and truth. There is no tolerance curve cryptography (ECC) and his con- WWBenvenuto Cellini (1500– for superficial political thinking. Thought- tinuing vendetta against claims by the 1571) opened his own autobiography lessness is perceived by Koblitz as proponents of the mainstream public- [3, p. 5], probably composed between almost worse than bad will or selfish- key cryptography algorithm RSA; and, 1558 and 1567. Neal Koblitz, the author ness, like the saying: ‘‘It is terrible to be finally, 21 pages on elementary math of the autobiographical memoirs Ran- knocked down by a car, but much education and math teacher education. dom Curves, is such a Renaissance more terrible to be trampled down by Contrary to model autobiographies like personality: A renowned top mathe- a hundred geese.’’ As in grading math- Norbert Wiener’s I Am a Mathemati- matician, a prolific author of widely ematical exercises, Koblitz vigorously cian [9] which enthusiastically discusses used text books in number theory and discloses the slightest weakness in com- control theory, prediction, Fourier cryptography, a harsh, polemic writer mon arguments; and as in mathemati- analysis and brain research, or Mark (also for 30 years for this magazine) cal proofs, and contrary to common Kac’s Enigma of Chance [6] which against ‘‘mathematics as propaganda,’’ political arguing and military tactics, explores the interface between different on elementary school mathematics Koblitz always attacks his adversaries’ fields of mathematics and statistical teaching, and on the mathematical strongest positions. That makes read- mechanics and explains, for example, and general cultural life in developing ing Random Curves sometimes offend- the combinatorial rules of phase transi- countries. ing, often demanding, but always tion, Koblitz seems convinced by Mark No doubt, Koblitz has both ‘‘done rewarding. Kac’s proclamation [6, p. xiii]: ‘‘The something of special worth’’ in his sci- Koblitz also displays a warm and autobiography of a mathematician must entific work and ‘‘something that may human enlightened self-interest asso- contain some mathematics. Yet a pre- truly resemble those things of special ciated with his extensive work for sentation in popular form of some of the merit,’’ though contrary to Cellini’s glo- solidarity and charity. He doesn’t problems and ideas with which I have rification of the bloody contemporary underplay the contradiction between been involved throughout my life is Florentine Medici dictatorship, Koblitz his almost hedonistic life as a highly unfortunately an impossible task.’’ praises the rights and the virtues of the gifted, tenured, respected and well- Nevertheless, Kac tried. That was good. suppressed, minorities, women, chil- paid university professor in mathe- Koblitz did not really try. That is a pity. dren. Another title for his book could matics and the miserable circum- Surely, Koblitz may have had good have been Another Look at Enlightened stances of the people he tries to help. reasons for that restraint. He describes Self-Interest: Because that is what the Random Curves is not so much about his life-long fight against overhyped book is about. his own sacrifices, renunciation, risks, ideas, often connected to improper For most of its pages, Random punishments (though they are there), numericity, when people dress up Curves delivers a long and fascinating as about the emotional reward for his poor understanding or even fraudulent array of very sharp, personal and un- political activism and the wide range of arguments by slick, pompous, false compromising comments on the great possibilities an individual really has: I and misleading manipulation of logic

2009 Springer Science+Business Media, LLC, Volume 31, Number 4, 2009 65 and numbers. Understandably, but considered highly rewarding. It seems troops and civilians. Moreover, on unfortunately, he restrains himself then that Koblitz doesn’t know about the April 9, 2003, one month into the Iraq from giving more detailed explana- desperate situation of the editors of war, the then president of Koblitz’s tions of his own work. He must have formerly highly-respected journals professional organization, the Ameri- been afraid that such explanations (names can be provided) who are can Mathematical Society, took notice, could give the impression of adequate deserted by their referees and, conse- not so much of the military role of description while in reality and by quently, must stick to short superficial mathematics in general, as of mathe- necessity remaining superficial. checks of submitted papers by the matics as a key component in the Perhaps it is not a bad choice opinion leaders in a field. (Often, it preservation of US military superiority, Koblitz has made for this book, rather seems, the opinion leaders do not have implicitly promising the loyal assis- carefully describing the math environ- the time to check more than whether tance of his members (and asking for ment instead of the mathematics itself. and how they are quoted.) adequate payment): ‘‘…for a military There, in the mathematician’s everyday II. Even more mysterious is the commander to have secure communi- life, he has observed much and has author’s reluctance to address the role cations in the field depends on funda- much to say: On the nice and generous of mathematics in the various wars mental advances in number theory…. personality of many mathematicians, fought by his country, in particular in Future progress in this seemingly how easy it was and is from the rather Vietnam three decades ago, then in abstract area, by us or by hostile forces, remote and protected position in a Kosovo (Yugoslavia) and now in Iraq could threaten the security of all these math department to mix with politics and Afghanistan. Koblitz brushes aside communications.’’ (David Eisenbud, even on the outspoken left, and how G. H. Hardy’s concerns regarding mil- statement to a US congressional sub- to make use of ties with sympathetic itary applications of mathematics (long committee overseeing the funding of professionals. Strangely enough, when before the cryptography potential of the US National Science Foundation [5]). writing about the mathematical life, he number theory became evident), by III. Another strange aspect of Ran- sees everything through rose-colored one single claim regarding cryptogra- dom Curves is what may appear as a spectacles and seems ready to make phy: ‘‘Earlier systems for scrambling naı¨ve perception of corporative busi- any compromise, even to close his messages worked well in military or ness. Apparently, he sees only the eyes and shut his ears, like so many diplomatic applications, where there blessings of setting the math supported otherwise very critical mathematicians, was a fixed hierarchy of people who turbo on modern capitalism. Once such as Laurent Schwartz [8]. For a were authorized to know the secret again, for Koblitz, clearly, capitalism has possible explanation, I quote Elias keys. By the 1970s, with major sections its bleak sides. He advocates socialist Canetti [2]: ‘‘Don’t tell me who you are. of the economy rapidly becoming ideas and, to a surprisingly large extent, I want to worship you.’’ computerized, the limitations of clas- but rather convincingly, also socialist Three Examples: sical cryptography were coming to the practice as he witnessed it. But he closes I. With reference to C. P. Snow, fore’’ (p. 297f). Koblitz elaborates that his eyes once again to the role of Koblitz notices the gap between claim of the purely commercial rele- mathematics in the modern economy, different cultures, and he praises, vance of mathematical work for public- for example, in the dawn of the present unreservedly, a math community with key cryptography in a speculation financial crisis. To be fair, the book was values solely associated to ‘‘intellectual about what, in Koblitz’s perspective, published early in 2008, a few months achievements.’’ Snow was not so one- would really upset Hardy today, namely before the financial crisis became visi- sided, and even less so was his source the war in Iraq—‘‘much more than the ble to a large population, and with it the Benjamin Disraeli [4], who coined the use of number theory in cryptography’’ fatal role of the math-supported belief ‘‘Two Nations’’ concept when he was a (p. 320). in the security of hedge funds and social critical writer, before becoming I cannot judge. I’m not an expert investing in real estate. How come he the conservative British politician and on the high-speed cryptography now closes his eyes to the role of mathe- Prime Minister. indispensable for real-time control of maticians? Here we have a man, feeling Where has Koblitz been the last military operations, and I doubt that responsible for his students, his prod- 30 years? On what Mount Olympus? Koblitz is. More generally, it seems to uct, and for the society, his customers. He doesn’t seem to be aware of the me that the mathematics historian Jens What customers? In recent years, breakdown of the peer referee system Høyrup and I have put sufficiently rich financial business has most probably in mathematical journals, now that all material together [1] to expect a polit- employed more than half of each year’s mathematicians are pressed by their ically attentive mathematician to make ‘‘products,’’ also from Koblitz’s depart- deans to publish more and shorter the connection between mathematics ment. A few mathematicians were con- articles; to make a small epsilon varia- and modern war when speaking of cerned about the emerging contradic- tion into a separate note at once rather and against war. There is ample evi- tion between the math-based triumph than wait for full solution; to establish dence that only the superstitious belief of rational pricing of options and other friendship circles of mutual citation in the math-supported pin-point accu- derivatives and the evolving impene- for higher impact factors; not to racy of modern weaponry could create trability of the financial markets. Appar- waste time as a referee for uncredited the necessary public and political ently, Koblitz didn’t belong to them. and time-consuming reading, learning support for an aggression promising to Perhaps Koblitz is right: Perhaps and checking, an activity previously be clean and gentle with one’s own we need not pay attention to the

66 THE MATHEMATICAL INTELLIGENCER mathematical aspects of military aggres- Koblitz’s recollections and will be able [4] Benjamin Disraeli. Sybil: Or the Two sion and the capitalist economy. Per- to compare them with their own expe- Nations. Oxford World’s Classics, Oxford haps it suffices to protest aggression riences. The book might also be University Press, Oxford, 1998. and exploitation. It seems that Koblitz attractive to young readers (possibly at [5] David Eisenbud. Statement to a US Con- says, ‘‘Don’t mix!’’ I see his point, but I the advanced high-school level but gress subcommittee overseeing the can’t agree. more probably college age) who like to funding of the US National Science Foun- Who will be interested in this book? read the intelligent and sensitive eye- dation, Notices of the AMS, June-July Any mathematician or historian with a witness and reflections ‘‘of a student 2003: 704f; http://www.ams.org/notices/ desire to immerse herself/himself in and later a scientist caught up in the 200306/inside.pdf. the vanished world of the American tumultuous events of his generation,’’ [6] Mark Kac. Enigmas of Chance: An Auto- civil-rights movement, in solidarity move- as the back cover reads. This is the kind biography. Sloan Foundation Series, ments, in national liberation move- of autobiography that I read avidly Harper and Row, New York, 1985. Pub- ments, and in the differences and when I was a teenager, and although lished posthumously with a memoriam parallels between intellectual and cul- prior knowledge of mathematics and note by Gian-Carlo Rota. tural life in different nations and cryptography might be helpful, it is [7] Ann Hibner Koblitz. A Convergence of different segments of society will find certainly not essential for the enjoyment Lives: Sophia Kovalevskaia, Scientist, Wri- Random Curves absorbing. This is not of Koblitz’s moving stories. a meticulous documentation of politi- ter, Revolutionary. Birkha¨ user, Boston, cal moves and reactions like Noam 1983. [Reviewed in The Mathematical Intelligencer, 7/4:69–73, 1985.]. Chomsky’s writing. Koblitz delivers REFERENCES [8] Laurent Schwartz. A Mathematician Grap- mostly oral history, with its charm and [1] Bernhelm Booß-Bavnbek, Jens Høyrup, pling with His Century. Birkha¨ user, Boston, its limitations. Happily, some stories editors. Mathematics and War. Birkha¨ user, sound really old and passe´. This is 2001. A translation in English of Laurent Basel, 2003. All chapters can be down- particularly true for Koblitz’s reported Schwartz’s autobiography, Un mathe´ mat- loaded for free at http://www.springer. hardships in the US Civil-Rights Move- icien aux prises avec le sie` cle, originally com/birkhauser/historyofscience/book/978-3- ment. Regarding other events, one is published by editions Odile Jacob, Paris, 7643-1634-1. tempted to recall Zhou Enlai’s alleged 1997. [2] Elias Canetti. Nachtra¨ ge aus Hampstead; quip to Henry Kissinger (not reported in [9] Norbert Wiener. I Am a Mathematician. Aus den Aufzeichnungen 1954–1971. Carl Kissinger’s autobiography), ‘‘It is too Victor Gollancz Ltd., London, 1956. early to say,’’ when asked for his assess- Hanser, Munich, 1994); Notes from Hamp- ment of the 1789 French Revolution. stead: The Writer’s Notes: 1954–1971. This may be partly true for Koblitz’s Translated from German by John Har- comments on the rise and fall of socialist graves. Farrar Straus Giroux, NY, 1998. Department of Science, Systems ideas—in spite of the fact that he has [3] Benvenuto Cellini. La vita—My Life. Trans- and Models / IMFUFA passed the ominous age of 40 required lated from Italian by Julia Conaway Bonda- Roskilde University by Cellini for a balanced view. nella and Peter Bondanella, Oxford DK-4000 Roskilde Older liberal and left-wing mathe- World’s Classics, Oxford University Press, Denmark maticians will recognize many of Oxford, 2002. e-mail: [email protected]

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