A light study of the six problems, and the six members of the US team, at the 2001 International Mathematics Olympiad. Intertwined with Olson’s dissecA light study of the six problems, and the six members of the US team, at the 2001 International Mathematics Olympiad. Intertwined with Olson’s dissections of the math problems and an example solution for each by a team member are ruminations on what makes up what we think of as math “genius:” talent, creativity, competitiveness, dedication, and so on. Olson asks questions such as whether geniuses are born or made and why women do not excel at math as much as men, exploring various arguments fully without weighing heavily toward any one side. It’s a light read on an interesting subject. I must admit that from the title I was hoping for more of a suspenseful, drama-packed account of a grueling competition: brains as athletics, that sort of thing. So, not so much of the tough competition, and more on what the study of higher mathematics is all about. ...more

A biography of the famous mathematician Paul Erdos, the most prolific publisher of mathematical papers in the world. He spent his life travelling, livA biography of the famous mathematician Paul Erdos, the most prolific publisher of mathematical papers in the world. He spent his life travelling, living out of a single suitcase, staying at other mathematicians’ houses for a bit, collaborating on papers with them in blitzkrieg weeks of nineteen-hour days of work, then moving on, leaving his collaborators dazed and exhausted but proud. Famed for his work in probability, graph theory, and combinatorics, he is also remembered in math circles for his eccentric personality, most marked in his lack of practical and social skills and dependence on his various hosts. He is also remembered by the term “Erdos number,” which denotes how close a mathematician was to publishing a paper with the prolific Great One: an Erdos number of one means you have published with him as co-author; two means you have published with someone who was a co-author, and so on.

Full of amusing anecdotes and math jokes, the book is actually less a profile of Erdos than sort of “history of the eccentric or at least interesting in mathematics,” as Hoffman digresses at length into militant atheist and pure mathematician G.H. Hardy; his protégé, the prodigy Srinivasa Ramanujan; the conflict between Frege and Russell and the latter’s paradoxes in set theory; Fermat and the legion of mathematicians who tried to prove his Last Theorem; modern practitioner Ronald Graham and his work in finding the efficiency of worst-case scenarios in combinatorics for his employer AT&T; the strangeness of Euler’s identity; Georg Cantor and his hunt for the transinfinite (known today in Hilbert’s Grand Hotel problem); Marilyn vos Savant and her controversial solution to a problem on probability; and so on. Some of this has nothing whatever to do with Erdos; some is tangentially related (he read vos Savant’s answer and didn’t understand why it should be so). In any case, it’s a fascinating look into the world of pure mathematics and how work in the most remote corners of pure math, which can seem so rarefied, can have vast practical applications years or even centuries later (as in the study of prime numbers, which helps in military cryptography). The book gets back to Erdos’ personal life just in time to chronicle his death and the outpouring of tributes that came from the math world. It’s not a bad book at all, but it’s not a biography of Erdos, either. Rather, it’s a collection of interesting problems that popped up in the history of modern mathematics. Why Hoffman, who does an admirable job making some of the most baffling math concepts almost accessible to the layman, hung this (interesting, certainly) patchwork quilt on the bare-bones frame of a biographical sketch of Erdos, a mathematician he knew personally, is not very clear. Is it that there simply wasn’t much to say about the Great One beyond that he literally lived for mathematics? That Hoffman thought a book called “Eccentric Mathematicians of History and Their Puzzling Problems” wouldn’t sell? Well, maybe it wouldn’t. ...more

A history of zero and its counterpart the infinite, two ideas that have been regarded as dangerous through the ages but which unlock the secrets to caA history of zero and its counterpart the infinite, two ideas that have been regarded as dangerous through the ages but which unlock the secrets to calculus and the universe. Most interesting is Pythagoras’ and Aristotle’s vehement rejections of the idea. The Catholic Church's insistence on Aristotelian thought held Western science and mathematics back for centuries. ...more

It serves as both an engaging review of probability theory and an investigation of what mathematicians are still learning. Every aspect of the book is charming, from gambling odds to the probabilities of sharing a birthday with a group of other people. I was especially intrigued by Aczel’s scientific explanation of how people really can have nothing but good or bad luck. Informative and readable; I only wished some of the chapters were a bit more fully explained. ...more

A collection of articles from the 1960s on math, measurement, counting, or calendar. I was rather disappointed in this; I know a lot of it is out of dA collection of articles from the 1960s on math, measurement, counting, or calendar. I was rather disappointed in this; I know a lot of it is out of date, but I read it to be entertained by that crisp, clear Asimov style. But for the most part, these pieces are frothy squibs that aren’t particularly revealing, or are (for me, anyway) just too confusing to enjoy. Most egregious are the latter articles which are more or less lists of the longest or deepest things, which in this Internet age is not rare information. And again, I expected Asimov to say something a little more profound than he does here --- but with his prolificacy, they can’t all be riveting. ...more

The author describes the truly amazing abilities of dogs, bees, ants, birds and other creatures when it comes to eye-mouth coordination, navigation, lThe author describes the truly amazing abilities of dogs, bees, ants, birds and other creatures when it comes to eye-mouth coordination, navigation, locomotion, and so on. He uses this data, along with some studies on how infants pay attention to certain sets of things up to three, and some other studies on poorly-educated street vendors who can do complicated math procedures in their heads but not on paper, to argue that people have an innate instinct for mathematics.

It’s an interesting book, but not a cohesive one, and ultimately unsatisfying. The animal studies are fascinating, but nearly completely irrelevant to the matter at hand; the animals, as Devlin himself says, aren’t “doing mathematics” any more than the world is “doing physics” as it spins. The studies on humans, especially the differences between school math and real-world math abilities, are germane, and Devlin has a good case to make that people can do math, but are turned off it through poorly-done formal study. Finally, the book concludes with an exhortation for people to practice the basics more through memorization. Gosh, thanks. ...more

A look at Descartes’ life and work, written in a very breezy, popular style that supposes almost no math or philosophy experience. This material is peA look at Descartes’ life and work, written in a very breezy, popular style that supposes almost no math or philosophy experience. This material is peppered by a lot of winking references to Rosicrucians, the Inquisition, and some melodramatic insinuations about Descartes hiding his greatest discoveries (the “secret notebook” of the title, which Leibnitz decoded right after his death, so it’s not exactly secret, is it?).

This is the third Aczel book I have read (after the okay The Riddle of the Compass and the more interesting Chance), and while the others were both flawed, especially in matters of historical record, those flaws were outweighed by the stories Aczel had to tell. In the case of this book, though, the engine that drives the book, its raison d’etre, is not to tell Descartes’ story but to embellish it into something Dan Brown might write, a potboiler of secret societies and death threats. It is a sham of false drama, which the story of one of history’s greatest intellects, who was deeply religious, created Western philosophy, served as a gentleman soldier, and mingled with royalty, simply does not need. The book is riddled with errors, mostly minor (Aczel describes the Rosicrucians’ symbol right above an illustration of it that is quite unlike his description), and also contains many errors of logic (Aczel says Descartes had nothing to fear from the Inquisition regarding his cosmological model, since it was “not valid,” as if that has anything to do with a challenge to the Church’s authority); but Azcel’s main sin is his constant obfuscation – what he insinuates by not telling all the facts. For example, one of the main “reveals” of the book is that Descartes thought of, and hid, what would become Euler’s formula about three-dimensional solids, when it seems to be commonly accepted by scholars that Descartes did not make the final connection to the finished formula. He makes absurd, silly claims about November 10 being “eerily significant” in Descartes’ life, ironically adding mystical numerology to the life of a logical thinker. The false drama grates after only a few pages (the chapters and chapter breaks all ending with hyperbolic cliffhanger questions, all of which come to nothing in particular). And it is packed with uninformative filler (Descartes stopped to rest his horses! He went to an inn that was popular! This inn had people in it! He rested! He picked up his horses!). The book is deeply disappointing, because a sensible biography of Descartes for the layman, with Azcel using his expertise to make the math clear, could have been a treasure. ...more

The author, a mathematics and philosophy professor, writes about the basic concepts of simple arithmetic (addition, subtraction, multiplication, divisThe author, a mathematics and philosophy professor, writes about the basic concepts of simple arithmetic (addition, subtraction, multiplication, division), starting with the premise that numbers exist outside of human endeavor, then on to the definition of addition (which is just adding by one), lingering at the problem of zero, then through some rather convoluted proofs of various theorems, to stop at the abstract algebraic concepts of rings (structures which include sets of integers and provide the definition of addition and multiplication) and fields (which define division through multiplicative inverses).

If the summary above makes it seem as though this is a jaunt through the math you learned in elementary school, think again: “The recursion theorem justifies definitional descent by drawing a connection between the recipe or algorithm embodied in definitional descent and the existence of a unique function, the one that definitional descent has presumably defined.” Berlinski is often this recursive; I often found myself wondering what was being proved or defined, and what was being simply assumed. But aside from tortuous mathematical definitions, the book is written in an airy, conversational, sometimes jocular (sometimes smug) tone, with many sentences given their own paragraphs in order to give them Weight. Berlinski is even quite funny, as when he discusses Guiseppe Peano (whose axioms provide the groundwork for what Berlinksi attempts to show) and his bizarre simplified Latin that no one used or understood, or when he imagines early mathematicians’ dialogue when encountering the apparent absurdity that is negative numbers (“Can I do that?” “Why not?” “I’m just asking.” “What next? I mean besides giving up. That always works”). As a philosophical treatise on the concept of mathematics itself, the book makes some trenchant points (“across the vast range of arguments [in psychology, logic, physics, etc.]… it is only within mathematics that arguments achieve the power to compel allegiance because they are seen to command assent”). But as a tour of elementary abstract principles, it’s a bit abstruse for the layman. I enjoyed his insights on sets and some of the simpler chapters, but finished the book feeling as though Berlinski was a bit too clever for his own good, and yet not quite clever enough to make it all clear. ...more