Sherman Stein, author of a calculus textbook, a monograph on the theory of tiling, a study of Archimedes, and Strength in Numbers (the latter two previously reviewed on MAA Online), here presents another installment of mathematics for the general public. How the Other Half Thinks: Adventures in Mathematical Reasoning consists of eight short chapters, each of which sets up and then solves a nontrivial mathematical problem.
Buffon's needle problem asks for the probability that a 1-inch needle thrown at random onto a grid of parallel lines, one inch apart, will cross one of the lines. The surprising answer is 2/π. Chapter 1 of Stein's book gives an elegant derivation of this result — a derivation so elegant, indeed, that it enjoys the honor of a chapter in Proofs from THE BOOK.
Chapters 2 and 4 deal with random strings of a's and b's. In Chapter 2, Stein asks how long such a string must be before the number of occurrences of one of the letters exceeds the number of occurrences of the other by 2. The expected value of this length is given by an infinite series. Stein evaluates the series by a clever rearrangement which goes back to the 14th century scholastic Nicole Oresme. The same series occurs in Chapter 4, where Stein computes the expected length of a run of a's or b's.
Another problem about probability is treated in Chapter 6: in an election involving two candidates, what is the probability that one candidate will lead during the entire count? The solution here is based on a geometric reflection argument.
Chapter 3 gives a proof of Sperner's lemma, and mentions some of its applications. Let T be a triangle, and label its vertices a, b, and c. Now triangulate T into smaller triangles. Label each vertex of the triangulation by a, b, or c, in such a way that vertices on edge xy of T must be labelled either x or y. Sperner's lemma (Emanuel Sperner, 1928) states that there is at least one small triangle in the triangulation whose vertices have all three labels. The proof — another Book proof — is by a counting argument.
Chapter 5 shows how to construct a circular string of a's and b's whose substrings of length 4 give all possible quadruplets of a's and b's. The solution involves finding an Eulerian walk on a graph.
Chapter 7 trots out those old war-horses, Cantor's diagonal argument and Russell's paradox — and none the worse for wear they are, after all.
Finally, Chapter 8 solves a problem posed by Axel Thue in 1912: can we construct arbitrarily long strings in a's, b's and c's which contain no pairs of identical consecutive substrings? (It is easy to see that this is impossible if we have available only two letters a and b.)
As Stein points out, the book differs from most books on mathematics written for the general public, in that it presents instances of actual mathematical reasoning, rather than, say, the lives of mathematicians, or descriptions of the applications of mathematics. It is, I suppose, praiseworthy to write about mathematics for a general audience, and even more so to use the occasion to discuss real mathematics. One might ask, however, how well this objective has been met.
Stein claims (p. viii) that "The thinking in each chapter uses at most only elementary arithmetic, and sometimes not even that." Of course, this cannot be true, and it is not. Thus, when he takes up Buffon's needle problem (p. 2), Stein repeats his promise that the solution will require no geometry at all. Yet the solution is based on the fact that the ratio of the circumference of a circle to its diameter is \(\pi\). Doesn't that count as geometry? Nowadays, to be sure, any schoolchild can recite the formula \(C=\pi D\), but for Archimedes (Measurement of a Circle, Prop. 1), even the fact that the ratio \(C/D\) was the same for all circles was a nontrivial result. And what about the evaluation of infinite series? Does this use "at most elementary arithmetic"?
What, after all, is the relation between arithmetic and algebra? We tend to associate algebra with the use of symbols and formulas, but it seems to me that, fundamentally, algebra is nothing but arithmetic carried out in generality. We can attain this generality by using formulas, or we can say everything in words. Is it the formulas that make the discussion difficult? Descartes, who developed the algebraic symbolism we use today, thought that it made mathematics easier. Comparing his newfangled method with the methods of Euclid and Apollonius, he comments, "This is one thing which I believe the ancient mathematicians did not observe, for otherwise they would not have put so much labor into writing so many books in which the very sequence of the propositions shows that they did not have a sure method of finding all, but rather gathered together those propositions on which they had happened by accident," (Smith and Latham, The Geometry of René Descartes, p. 17). More recently, in their classic paper "College Admissions and the Stability of Marriage", David Gale and Lloyd Shapley write (p. 15), "The argument [of their paper] is carried out not in mathematical symbols but in ordinary English; there are no obscure or technical terms. Knowledge of calculus is not presupposed. In fact, one hardly needs to know how to count. Yet any mathematician will immediately recognize the argument as mathematical, while people without mathematical training will probably find difficulty in following the argument, though not because of unfamiliarity with the subject matter."
Where, then, lies the difficulty in mathematical reasoning? I believe that Gale and Shapley are correct to conclude that the difficulty lies in the logical precision of the argument itself, not in the auxiliary symbols we use to express it, which in fact we resort to just to make the argument clearer. Perhaps Stein recognizes this problem, for, at several points, when there are difficulties in his argument, he fudges them.
Thus, in the course of solving Buffon's needle problem, Stein considers a "needle" of an arbitrary polygonal shape, and asks for the total number of times the needle intersects the grid. He argues that the expected value of this number depends on the total length of the needle, but is independent of its shape. In the simplest case, take a bent "needle" consisting of two straight line segments. Let X be the number of times the first segment intersects the grid, and Y the number of times the second segment intersects it. Stein imagines a bug crawling down each of the segments and reporting the number of intersections (p. 9). "Neither bug knows about the other bug. Indeed, each knows only his own section and is not aware that there is another section. Each bug thinks that his section is being spun and thrown at random. The average number of crossings for the whole needle is the sum of the averages the two bugs report." Well, Stein's conclusion is correct. We know that E(X + Y) = E(X) + E(Y), regardless of whether or not X and Y are independent. But why doesn't the argument about what the little bugs "know" show equally well that, say, E(XY) = E(X)E(Y)? In view of the lack of public understanding of the concept of statistical independence — for example, as applied to DNA "fingerprinting" — I think it is too bad that Stein didn't present this argument with more clarity.
Or, what about infinite series? The concept of the sum of an infinite series depends on the concept of a limit. Here is how Stein explains that concept (p. 32): "We can imagine adding up more and more terms and watching what happens to the sums. If they look as though they are getting closer and closer to a fixed number, we will call that number the sum of the infinite number of terms." So "mathematical reasoning" is based on what "looks as though" it is happening?
Moreover, we know today (what Oresme did not) that the terms in an infinite series cannot always be rearranged without changing the sum of the series. In the present case, the terms are all positive, and consequently (as we know) the rearrangement is valid, but shouldn't the author at least caution the inexperienced reader that there is a subtlety here?
I am not suggesting that all these matters should be spelled out in detail in a book addressed to the "general reader"; what disturbs me is that Stein repeatedly assures that reader that all his arguments are based purely on "common sense".
In any case, the general reader will find this book tough going, and it is a little hard for me to see how a person who does not already have a particular interest in mathematics would be motivated to work through it. It is certainly no fault of Stein that the chapters are not easy reading; that is just the nature of mathematics. To follow the solutions requires close attention to an extended argument.
One way of motivating the reader to persevere through a difficult proof would be to start by considering interesting applications of the results. To be sure, Buffon's needle problem scarcely requires any motivation. Stein motivates the problems considered in Chapters 2 and 4 by asking, respectively, for the number of points scored in a volleyball game, and the length of a winning streak of a baseball team. On the other hand, though Sperner's lemma has a variety of applications — some of the more accessible of them have been written up recently in a beautiful Monthly article by Francis Su — Stein mentions these applications only in passing, at the end of Chapter 3; in effect, he takes up the problem in a purely detached way. In the final chapter, Stein quotes, evidently with approval, the remarks of Thue (pp. 137-138): "For the development of the logical sciences it is important to find large areas for speculation about difficult problems without any consideration of possible applications." I would soften the translation here to read, "without regard to possible applications". Thue is not disclaiming any consideration of applications, particularly since he goes on, in the next sentence (not quoted by Stein), to mention possible connections with number theory. It seems to me that Thue is in effect apologizing for the apparent lack of applications of his problem, not, as Stein makes it seem, claiming that we can do good mathematics by making up any old problem we like. I'm sure that Thue himself understood quite well why his problem was an interesting one, though of course the general reader will not have the benefit of Thue's experience in mathematical research.
In the chapters which deal with random sequences, Stein frequently encourages the reader to experiment by tossing coins. He then provides data from his own experiments, and compares the results with the values given by the theory. For example (studying the number of points scored in a volleyball game, p. 37), "In short, the theoretical average number is 4. The experimental average, based on 64 tosses, was 4.66. The two averages are not far apart. Presumably, if we toss a penny thousands of times, the observed average would gradually move closer to 4 and away from 4.66. In any case, our common-sense method yields numbers that are in reasonable agreement with those from the tossed pennies." On the other hand, earlier in the same chapter (p. 28), "This equals about 4.66, lower than the 7.03 average for the volleyball games. That discrepancy is so large that it hardly could be due to chance." Well, I can see that a 51% discrepancy is a lot bigger than a 16% discrepancy, but the obvious question here is: how large must the discrepancy be, in order that it "hardly could be due to chance"? Stein does not address that question.
There is a serious mathematical error in the chapter on the diagonal argument. Stein states (p. 132) that Alan Turing used the diagonal argument to show that "there could never be a universal computing machine capable of duplicating the computations of every possible computer." On the contrary, part of Turing's accomplishment was precisely to prove the existence of a "universal Turing machine", which is indeed capable of doing any computation which can be done by any other machine. What Turing used the diagonal argument for was to prove that no Turing machine could solve (what is now called) the Halting Problem. Consider a Turing machine T having the following characterization: if T is given a description of any other Turing machine M, then T is to determine whether machine M ever halts. Turing proved that such a machine T cannot exist.
One might ask why the problem couldn't be solved by taking T to be a universal Turing machine: then it could simply duplicate the computations of M. The answer is that that is not enough to determine whether M will ever halt. If T, imitating M, does in fact halt, then of course that answers the question, but suppose T goes on for a long time without ever halting. We will have no way to tell whether M will halt eventually.
The title of Stein's book (perhaps chosen by the publisher?) seems to refer to the popular left brain/right brain dichotomy. As Stein writes (p. ix): "I hope this book will help bridge that notorious gap that separates the two cultures: the humanities and the sciences, or should I say the right brain (intuitive, holistic) and the left brain (analytical, numerical). As the chapters will illustrate, mathematics is not restricted to the analytical and numerical; intuition plays a significant role." Stein does well to avoid identifying mathematics with the activity of just one side of the brain. He would have done better, however, not to have endorsed the left brain/right brain ideology. While it does indeed appear to be the case that the two sides of our brain act in rather different ways, the idea that the right brain is "intuitive, holistic", while the left brain is "analytical, numerical", is a vast oversimplification, and goes far beyond the actual evidence.
Altogether, it seems unlikely to me that this book will succeed in reaching the audience to which it is directed. But it is not a bad book. Readers of MAA Online will probably enjoy it. I particularly liked the chapters on Sperner's lemma and on Thue's problem. And if you know any bright high-school students who can't decide between mathematics and, say, philology, give them a copy. You might change the course of history.
- Sherman K. Stein and Sándor Szabó, Algebra and Tiling: Homomorphisms in the Service of Geometry. MAA, 1994, ISBN 0-88385-028-1.
- Sherman Stein, Strength in Numbers: discovering the joy and power of mathematics in everyday life. Wiley, 1996, ISBN 0-471-15252-8.
- Sherman Stein, Archimedes: What Did He Do Besides Cry Eureka?. MAA, 1999, ISBN 0-88385-718-9.
- Martin Aigner and Günter M. Ziegler, Proofs from THE BOOK, Second Edition. Springer, 2001, ISBN 3-540-67865-4. The chapter on Buffon's needle problem (pp. 125-128) is new in the second edition. Aigner and Ziegler attribute the Book proof to Joseph-Emile Barbier, 1860. Sperner's lemma is on p. 140 (p. 132 in the first edition). There is a photograph of Sperner on p. 143 (135).
- David Eugene Smith and Marcia L. Latham, trans., The Geometry of René Descartes. Dover reprint, 1954, ISBN 0-486-60068-8.
- D. Gale and L. S. Shapley, "College Admissions and the Stability of Marriage", American Mathematical Monthly, 69, 1962, pp. 9-15.
- Francis Edward Su, "Rental Harmony: Sperner's Lemma in Fair Division", American Mathematical Monthly, 106, 1999, pp. 930-942.
- Oresme's summation of the series Σ n 2-n is translated in: Marshall Clagett, Nicole Oresme and the Medieval Geometry of Qualities and Motions. University of Wisconsin Press, 1968, pp. 412-417.
- For the ballot problem, see: William Feller, An Introduction to Probability Theory and Its Applications, vol. I, third edition, Wiley, 1968, ISBN 0-471-25708-7, p. 69. Feller attributes the problem to William Allen Whitworth, 1878, and Joseph Bertrand, 1887. He attributes the reflection argument to Désiré André, 1887.
- For Archimedes' evaluation of π, see: E. J. Dijksterhuis, Archimedes. Princeton University Press, 1987, ISBN 0-691-02400-6, pp. 222-240.
- For the question of statistical independence in DNA fingerprinting, see: Joel E. Cohen, "DNA Fingerprinting: What (Really) Are the Odds?", Chance, 3, 1990, pp. 26-32.
- Thue's article "Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen", is reprinted in Trygve Nagell, et al., eds., Selected Mathematical Papers of Axel Thue, Universitetsforlaget, Oslo, 1977, ISBN 82-00-01649-8, pp. 413-477. Here is Thue's own introduction to the paper: "Für die Entwicklung der logischen Wissenschaften wird es, ohne Rücksicht auf etwaige Anwendungen, von Bedeutung sein, ausgedehnte Felder für Spekulation über schwierige Probleme zu finden. Wir werden hier in dieser Abhandlung einige Untersuchungen aus einer Theorie über Zeichenreihen, die gewisse Berührungspunkte mit der Zahlentheorie darbietet, mitteilen."
- Turing's fundamental paper, "On Computable Numbers, with an Application to the Entscheidungsproblem", originally published in the Proceedings of the London Mathematical Society, series 2, 42, 1936-7, pp. 230-265, is reprinted in Martin Davis, ed., The Undecidable, Raven Press, New York, 1965, pp. 115-154. The construction of the universal Turing machine is given on pp. 127-132 of the reprint. The application of the diagonal argument to prove the unsolvability of the Halting Problem is on pp. 132-134.
- For a thorough discussion of the "left brain/right brain" dichotomy, see Joseph B. Hellige, Hemispheric Asymmetry: What's Right and What's Left, Harvard University Press, 1993, ISBN 0-674-38730-9, especially the section "The Quest for a Fundamental Dichotomy", pp. 54-61. Those interested in this question should also consult the recent paper by G. R. Fink, et al., "Hemispheric specialization for global and local processing: the effect of stimulus category", Proceedings of the Royal Society of London, Section B, Biological Sciences, 264, 1997, pp. 487-494, which describes the results of an experiment carried out to distinguish the activity of the two hemispheres, the results of which were just the opposite of what had been expected. Citing Hellige's book, Fink et al. write (p. 487), "Despite over 20 years of neuropsychological and neurophysiological investigations, little is known concerning which factors determine hemispheric specialization for global and local processing."
Stacy G. Langton (firstname.lastname@example.org) is Professor of Mathematics and Computer Science at the University of San Diego. He is particularly interested in the works of Leonhard Euler, a few of which he has translated into English.