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Undiluted Hocus-Pocus Reviewed by Andy Magid

Undiluted Hocus-Pocus: The Autobiography of coming one of two possible answers to Martin Gardner the trivia question, Press, 2013 who is the most im- US$24.95, 288 pages portant nonfiction ISBN-13: 978-0691159911 writer from Tulsa? The other possible At one time, algebraists used to entertain them- answer is the histo- selves with the pseudotrivia question: who is rian Daniel Boorstin, the most important ring theorist from Alabama? who was also born “Pseudo” because the point was not to test the in 1914 but arrived interrogee’s knowledge of Cotton State mathemat- in Tulsa as an in- ics, but to surprise them with the answer, namely, fant. Both men grad- Nathan Jacobson. That it should be a surprise uated from Tulsa carried more than a whiff of Yankee chauvinism; Central High School, anyone familiar with the Midwestern U.S. small although Boorstin, town merchants of the 1940s and 1950s who were something of a prodigy, seems to have graduated born in Poland and raised in the American South at age fifteen before Gardner entered. I do not would have instantly recognized from Professor know if they ever met. Incidentally, even though I Jacobson’s speech patterns that he shared that met these men only briefly and then near the end biography. And the algebra community being the of their lives, I spoke to them enough to know size it is, anyone who could have been surprised by that you couldn’t detect anything in their speech the answer was surprised long ago, although there patterns suggesting a Tulsa biography, not even in is a similar question still making the rounds—who­ the telltale second person plural. is the best group theorist from Arkansas?—that There’s also the academic historian John Hope reeks of the same chauvinism. Franklin, who graduated from Tulsa’s Booker T. Algebraists annoyed at being asked the Jacob- Washington High School. Franklin was known more son question too many times began to respond as a scholar than as a writer, although his 1947 with the question, who is the second most impor- book From Slavery to Freedom has sold millions tant ring theorist from Alabama? which is a cute of copies. He was born two days after the end of riposte but finesses a more fundamental point. 1914, which makes him almost a coeval of Gardner Is there any reason we should care to know that and Boorstin; again, I don’t know if Franklin ever Jacobson was from Alabama? Or, for that matter, met Gardner or Boorstin. know any biographical trivia about any mathemati- Gardner’s impact on mathematics in America cal figure? was profound, especially in the 1960s and 1970s, Martin Gardner, the author and mathematics when he wrote his widely popular column “Math- columnist who died in 2010, was educated in Tulsa, ematical Games” for . I was a Oklahoma, where he was born in 1914, thereby be- fan, beginning as a teenager, and loved reading the columns, especially those in narrative form Andy Magid is George Lynn Cross Professor of Mathemat- featuring the Dr. Matrix character. I also remember ics Emeritus at the . His email with pleasure reading in Gardner’s first collection address is [email protected]. of those columns, The Scientific American Book DOI: http://dx.doi.org/10.1090/noti1091 of Mathematical and Diversions, the

March 2014 Notices of the AMS 281 dedication to Gardner’s teacher at Tulsa Central platform is apparently a feature of express stops; High School, being an Oklahoma high school stu- hence the protagonist is said to live near such a dent myself at the time. I also have to say that, station) was unnecessary for most of his readers. with the possible exception of some word ladders, On the other hand, almost all the words in problem I can’t recall ever solving any of the problems in 2 are used in a specialized mathematical sense. Gardner’s column. This is no insult to Gardner—I Even if it were possible to present definition chains also never built any of the apparati described in of all the terminology involved, we would expect “The Amateur Scientist”, the wonderful Scientific years of study before a typical layperson would be American companion column by C. L. Strong—but comfortable tackling that problem. I hope those rather a testimony to his skills as a writer and the readers familiar with the terminology will agree pleasure his writing gave the reader. I suppose, with me, and I invite the rest to take my word for between the ends of the spectrum of readers like it, that the two problems are roughly the same sort me, who never solved any problems and read- of intellectual, or mathematical, challenge. ers, if any, for whom only the problems and not Month after month Martin Gardner’s column Gardner’s writing mattered fell the vast majority presented and discussed intellectual recreations of readers who made Gardner’s column the most that were on the level of what mathematicians popular feature of Scientific American in its day. thought about and yet were accessible to people Let’s agree that, whether he was a mathemati- without advanced mathematical training. He called cian or not, Martin Gardner wrote about math- these recreations—properly—mathematics. The ematical puzzles in an entertaining way that en- discipline has had no finer exponent. tranced a wide audience well beyond professional To further establish my Gardner fan credentials mathematicians and advanced students of math- before we turn to Gardner’s new book, I also re- ematics. Even when mathematics is admired by the call reading in high school with much satisfaction general public, as it was in post-Sputnik America, two other excellent Gardner books of the period, the gap between what the public thinks math- Machines and Diagrams and Fads and Falla- ematics is and what mathematicians understand cies in the Name of Science. Both of these, by the mathematics to be is vast. What Martin Gardner way, could well have been collections of articles, accomplished was to bridge that gap. although they were in fact unified projects; Gard- Here’s an illustration. The following two prob- ner just seems to write that way naturally. His lems are, in my judgment, about equal in intel- other great style is being the annotator, as he is lectual content. The first comes from Gardner’s in his most popular book, in first collection of columns mentioned above. The Wonderland. To illustrate its popularity and to second is an exercise in commutative algebra. keep Gardner’s mathematician fan base in perspec- 1. A young man lives in Manhattan near a sub- tive, it’s well to recall that since its initial release way express station. He has two girlfriends, one in 1960, the number of copies of The Annotated in Brooklyn, one in the Bronx. To visit the girl in Brooklyn, he takes a train on the downtown side of Alice sold is about twenty times the number of the platform; to visit the girl in the Bronx he takes mathematicians in or joining the profession from a train on the uptown side of the same platform. then until now. Since he likes both girls equally well, he simply So how did Tulsa schoolboy Martin Gardner takes the first train that comes along. In this way become the popular and admired writer on math- he lets chance determine whether he rides to the ematics and other topics? Undiluted Hocus-Pocus: Bronx or to Brooklyn. The young man reaches the The Autobiography of Martin Gardner tells the subway platform at a random moment each Satur- story. Like everything Gardner ever wrote, it is day afternoon. Brooklyn and Bronx trains arrive at entertaining, informative, witty, deft, and a joy to the station equally often—every ten minutes. Yet read, or so this fan assesses. Mathematicians need he finds himself spending most of his time with to be aware that the role that mathematics, or the the girl in Brooklyn; in fact, on the average he goes writing of the “Mathematical Games” columns, there nine times out of ten. Can you think of a good plays in Gardner’s story of his life is brief. Readers reason why the odds so heavily favor Brooklyn? also need to be aware that Gardner is concerned 2. The commutative ring S is an algebra over the with philosophical ideas, including theological commutative ring R. The ring R has no nontrivial ones. In his preface, quoting Lenny Bruce, Gardner idempotents, and S is finitely generated as an R identifies himself with people leaving churches module. Can S have infinitely many idempotents? and going back to . In his prologue, quoting One needs to understand some basic arithmetic himself from his 2007 Notices book review [“Do and have some notion of random and average to loops explain ?: Review of I Am a solve problem one, but beyond that no special Strange Loop”, August 2007], Gardner identifies knowledge is required. I suspect that even the himself as a mysterian, i.e., one who is convinced one specialized detail that Gardner puts in (the that “no philosopher or scientist living today has uptown/downtown choice from the same central the foggiest notion of how consciousness, and its

282 Notices of the AMS Volume 61, Number 3 inseparable companion free will, emerge, as they staff, the house my late mother-in-law grew up in surely do, from a material brain.” a mile north on Owasso, in a less ritzy area, also Gardner has previously spoken to the math- had servants’ quarters in the rear. However, by ematical community about how he came to write the 1920s their domestic help were commuters, “Mathematical Games” in an interview with Allyn not live-in. (Although I never heard them mention Jackson in the Notices (June/July 2005) and in an him, my mother-in-law and her sisters were the interview with Donald Albers of the Mathematical right age to have passed Gardner in the halls of Association of America, which is included in the Horace Mann Junior High and Tulsa Central High.) MAA’s CD Martin Gardner’s Mathematical Games: The reader will also learn about the University of The entire collection of his Scientific American col- Chicago of the 1930s from the perspective of an umns (the disc also includes a biographical essay intelligent undergraduate with a deep interest in on Gardner by Peter Renz). As Gardner recounts and the philosophy of , about in Undiluted Hocus-Pocus, a friend of his showed Navy life in World War II aboard a destroyer escort, him a hexaflexagon, he decided to do an article on and about living the life of a writer in the Green- it that he sold to Scientific American, the response wich Village of the 1950s. to the article was such that the publisher asked What the reader will also hear about, but not Gardner to do a monthly column, and “Mathemati- enough, is the role that performing magic tricks cal Games” was born. This account, by the way, oc- played in Gardner’s life. Anyone who has ever trav- cupies the first two pages of Gardner’s chapter 15 eled with, say, mothers of small children or people (of twenty-one), which is entitled “Scientific Ameri- who keep kosher know how these folks can make can”. Gardner tells us that writing the column for contact with others of their kind almost instantly twenty-five years “was one of the greatest joys of in strange venues while the rest of us are still strug- [Gardner’s] life,” and that “one of the pleasures in gling to meet the local population. Gardner hints writing the column is that it introduced [Gardner] that the same type of radar connects magicians, to so many top mathematicians,” among whom and many of the pivotal events of his life seem to he mentions Solomon Golomb, John Conway, hinge on a connection made by magicians. (The , , and Donald friend who showed Gardner the hexaflexagon, for Knuth (in chapter 15), and Ron Graham and Persi example, was a fellow magic enthusiast. So are Diaconis in chapter 17, which is entitled “Math Diaconis and Randi.) Like followers of twelve-step and Magic Friends”. These chapter titles are not programs (another group who make instant con- rigid boundaries. Chapter 15 includes an account nections in new places), magicians apparently also of Gardner’s relationship with , and gather regularly to trade tricks and gossip. Gardner chapter 17 recounts his relationships with Salva- seems to have spent many hours in such gather- dor Dali and Vladimir Nabokov. As this last bit of ings, presumably with pleasure and involvement, name-dropping makes clear, Gardner’s long career but he doesn’t share much of this with his readers. as a Chicago- (where he attended the ) and New York-based writer brought him Gardner makes clear in this book, as he did in contact with a smorgasbord of top-tier intellec- in his MAA and Notices interviews mentioned tual celebrities. Of course he was one himself—or earlier, that he considers his book The Whys of a would have been had his proverbial public shyness Philosophical Scrivener his deepest work. It is an not kept him from certain spotlights. Here is an important book, which unfortunately seems not to example (not mentioned in Undiluted Hocus-Pocus): have reached the audience he sought for it. Perhaps according to Gardner’s son James, Gardner de- the format (each chapter is framed as a question clined an invitation from Stanley Kubrick to attend “Why I am not a …”, where the blank is replaced by the premiere of 2001: A Space Odyssey on the various philosophical systems) makes it look like grounds that he didn’t have a tuxedo. On the other a collection of discrete pieces in the style used for hand, Gardner must have been a wonderful small Fads and Fallacies or Logic Machines and Diagrams. group social companion, as the heartfelt Foreword It is not: it is a sustained account of Gardner’s by and Afterword by thinking leading up to his “mysterian” position. for this book make clear. Although various chapters of Undiluted Hocus- Except for the sections noted in the preceding Pocus could be read in isolation, the book is a paragraph, however, most of Gardner’s autobiog- sustained account of Martin Gardner’s eventful raphy, and life, was not about mathematics. The life. Despite the caveats noted above about the reader will learn about Tulsa in the 1920s from role magic played in that life, it is a remarkably the perspective of a bright and athletic schoolboy. open account for the publicly shy Gardner, perhaps And, it must be said, a privileged one: the fam- that being one of the reasons it is posthumous. ily home on South Owasso Street was in a pretty Gardner believed in God and an . He tells tony area (it’s still tony). Gardner tells that the us in his last pages that this is not grounded in third floor of the house was servants’ quarters. the “head”, which he is using as a metonym for Although Gardner doesn’t mention household rational inference, but in the “heart”, which he

March 2014 Notices of the AMS 283 uses as a metonym for direct emotional percep- Jewish Publication Society’s of 1982, renders the tion. Gardner says: verse “The benighted man thinks….” Could Mar- tin Gardner, the master annotator and occasional As for God and an afterlife, our head pseudonymous prankster, not know this? This tells us both are illusions. An Old Testa- reviewer, an afterlife skeptic, is willing to suspend ment psalm (14:1), Unamuno reminds disbelief long enough to wonder if in some form or us, does not say “The fool hath said other Gardner is watching to see how many readers in his head there is not God.” God is a catch this “glitch”. Plus of course I wonder if there hope only of the heart. are other glitches Gardner has planted that I’ve In the King James translation of Psalm 14 that missed. For example, about that express subway Unamuno is referencing, the italicized word is station: are there any stops in New York where rendered heart, which is indeed the literal transla- trains go in two directions from a central platform? tion of the Hebrew word in question. On the other In keeping with the prankster tradition, I con- hand, the biblical writers understood that organ to fess that the Arkansas group theorists’ query be the location of the intellect, not the emotions, above is also a trick question. There’s a three-way which is why a modern translation, such as the tie for first (some say two-way).

Donoho Awarded 2013 Shaw Prize

The Shaw Foundation “A central problem in statistics is to devise has awarded the 2013 optimal and efficient methods for estimating (pos- Shaw Prize in Math- sibly nonsmooth) functions based on observed ematical Sciences to data which has been polluted by (often unknown) David L. Donoho of noise. Optimality here means that, as the sample size increases, the error in the estimation should “for his profound con- decrease as fast as that for an optimal interpola- tributions to modern tion of the underlying function. The widely used mathematical statistics least square regression method is known to be and in particular the nonoptimal for many classes of functions and development of opti- noise that are encountered in important appli- mal algorithms for sta- cations, for example, nonsmooth functions and tistical estimation in non-Gaussian noise. Together with Iain Johnstone, the presence of noise David L. Donoho Donoho developed provably almost optimal (that and of efficient tech- is, up to a factor of a power of the logarithm of the niques for sparse rep- sample size) algorithms for function estimation in resentation and recovery in large data sets.” The wavelet bases. Their ‘soft thresholding’ algorithm prize carries a cash award of US$1 million. is now one of the most widely used algorithms in The Shaw Prize in Mathematical Sciences Selec- statistical applications. tion Committee released the following statement “A key theme in Donoho’s research is the about Donoho’s work. recognition and exploitation of the fundamental “For more than two decades David Donoho has role of sparsity in function estimation from high- been a leading figure in mathematical statistics. dimensional noisy data. Sparsity here refers to a His introduction of novel mathematical tools and special property of functions that can be repre- ideas has helped shape both the theoretical and sented by only a small number of appropriately applied sides of modern statistics. His work is chosen basis vectors. One way to characterize such characterized by the development of fast compu- sparsity is to minimize the L0-norm of the coeffi- tational algorithms together with rigorous math- cients in such representations. Unfortunately, the ematical analysis for a wide range of statistical and 0 engineering problems. L -norm is not convex and is highly nonsmooth, DOI: http://dx.doi.org/10.1090/noti1097 making it difficult to develop fast algorithms for

284 Notices of the AMS Volume 61, Number 3