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Note The Fundamental Theorem of Algebra: An Elementary and Direct Proof OSWALDO RIO BRANCO DE OLIVEIRA ere is a simp...

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Note

The Fundamental Theorem of Algebra: An Elementary and Direct Proof OSWALDO RIO BRANCO

DE

OLIVEIRA

ere is a simple, differentiation-free, integration-free, trigonometry-free, direct, and elementary proof of the Fundamental Theorem of Algebra. One can ask, as soon as the complex numbers have been defined, whether every polynomial has a zero in the complex numbers. In this note I consider how early in the development of complex analysis this question can be answered. As pointed out by Remmert [11], Burckel [2], and others, two of the best analytical proofs of the FTA are the easy and short but not elementary one given by Argand [1] (see also [3, 4, 6, 7, 10, 11, 12, 14]), and the elementary but not so easy or short one given by Littlewood [9] (see also [5, 8, 11, 13]). All these works, except [2], use or prove d’Alembert’s Lemma [14] or Argand’s Inequality [11]: ‘‘If P is a nonconstant complex polynomial and Pðz0 Þ 6¼ 0; z0 2 C, then any neighborhood of z0 contains a point w such that |P(w)| \ |P(z0)|’’. The proof of the FTA I will now present does not apply d’Alembert’s Lemma or Argand’s Inequality. Instead I assume without proof only the continuity of complex polynomials and the following consequences of the completeness of R:

H

• Any continuous function f : D ! R; D a bounded and closed disc, has a minimum on D. • Every positive real number has a positive square root. Square Roots. It is well known that z 2 ¼ a þ ib; a; b qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2 2 2 R, is solvable in C. We have z ¼ a2 þ a 2þb þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffi 2 2 b isgnðbÞ a2 þ a 2þb , with sgnðbÞ ¼ jbj , if b = 0, and sgn(0) = 1. Applying this formula repeatedly we find all the 2j-roots, j 2 N, of z = ±1 and z = ±i. Fundamental Theorem of Algebra. Let P be a complex polynomial, with degreeðPÞ ¼ n 1. Then there exists z0 2 C satisfying P(z0) = 0.

P ROOF . Writing PðzÞ ¼ a0 þ a1 z::: þ an z n , with aj 2 C; 0 j n; an 6¼ 0, we have jPðzÞj jan jjzjn ja0 j ::: jan1 jjzjn1 ; from which follows lim jPðzÞj ¼ 1. By continuity, it is jzj!1

easy to see that |P| has an absolute minimum at some z0 2 C. Suppose without loss of generality that z0 = 0. Hence, putting S 1 ¼ fx 2 C : jxj ¼ 1g; jPðrxÞj2 jPð0Þj2 0;

8r 0;

8x 2 S 1 ;

ð1Þ

and P(z) = P(0) + zkQ(z), for some k 2 f1; :::; ng, where Q is a polynomial and Q(0) = 0. Substituting this expression, 2011 Springer Science+Business Media, LLC, Volume 33, Number 2, 2011

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DOI 10.1007/s00283-011-9199-2

at z = rx, in inequality (1), we get jPð0Þ þ r k xk QðrxÞj2 jPð0Þj2 ¼ 2r k Re½Pð0Þxk QðrxÞ þ r 2k jQðrxÞj2 0;

8r 0;

8x 2 S 1 ;

and, dividing by r k [ 0; h i 2Re Pð0Þxk QðrxÞ þ r k jQðrxÞj2 0;

de´monstration d’un theóre`me d’analyse,’’ Annales de Mathe´matiques Pures et Appliqueés, tome 5 (1814-1815), 197-209. [2] Burckel, R. B., ‘‘Fubinito (Immediately) Implies FTA’’, American Mathematical Monthly 113 (2006), 344-347. [3] Cauchy, A. L., Cours d’analyse, Vol VII, Premie`re Partie, Chapitre X, Editrice CLUEB, Bologna (1990).

8r [ 0;

8x 2 S 1 ;

[4] Chrystal, G., Algebra, An Elementary Text-book, Part I, Sixth edition. Chelsea Publishing Company, New York, 1952. [5] Estermann, T., ‘‘On the Fundamental Theorem of Algebra’’.

whose left side is a continuous function of r; r 2 ½0; þ1Þ. Thus, letting r? 0, we have

J. London Mathematical Society 31 (1956), 238-240. [6] Fefferman, C., ‘‘An Easy Proof of the Fundamental Theorem of

ð2Þ

Algebra,’’ American Mathematical Monthly 74 (1967), 854-

Let a ¼ Pð0ÞQð0Þ. Factoring out powers of 2, we can write k = 2j m where m is odd. Taking x = 1 in (2) we conclude j that Re½a 0. Choosing x so that x2 ¼ 1, and thus xk = -1, we conclude that Re½a 0. Hence Re[a] = 0. Choosing j x so that x2 ¼ i, we conclude that xk = ±i and xk ¼ i. Substituting x and x into (2), we conclude that Im[a] = 0. So a = 0, and P(0) = 0.

[7] Fine, B., and Rosenberger, G., ‘‘The Fundamental Theorem of

2Re½Pð0ÞQð0Þxk 0;

8x 2 S 1 :

R EMARK The last paragraph of the proof was a trick to avoid appealing to trigonometry. The proof that (2) implies P(0) = 0 can be easily done with the help of De Moivre’s Formula ðcos h þ i sin hÞn ¼ cos nh þ i sin nh; n natural and h real, as follows. Putting x ¼ cos h þ i sin h; x 2 S 1 and h a real number, we choose values of h so that xk ¼ cos kh þ i sin kh; k as in the above proof of the FTA, assumes the values ±1 and ±i. Hence we get Re½ Pð0ÞQð0Þ 0 and Re½ Pð0ÞQð0Þi 0, and conclude that P(0) = 0. ACKNOWLEDGMENTS

I thank Professors J. V. Ralston and Paulo A. Martin for very valuable comments and suggestions.

REFERENCES

[1] Argand, J. R., ‘‘Philosophie mathe´matique. Re´flexions sur la nouvelle theórie des imaginaires, suivies d’une application a` la

2

THE MATHEMATICAL INTELLIGENCER

855. Algebra,’’ Springer-Verlag, New York, 1997. [8] Ko¨rner, T. W., ‘‘On the Fundamental Theorem of Algebra,’’ American Mathematical Monthly 113 (2006), 347-348. [9] Littlewood, J. E., ‘‘Mathematical Notes (14): Every Polynomial has a Root,’’ J. London Mathematical Society 16 (1941), 95-98. [10] Redheffer, R. M., ‘‘What! Another Note Just on the Fundamental Theorem of Algebra?,’’ American Mathematical Monthly 71 (1964), 180-185. [11] Remmert, R., ‘‘The Fundamental Theorem of Algebra’’. In H.-D. Ebbinghaus et al., Numbers, Graduate Texts in Mathematics, no. 123, Springer-Verlag, New York, 1991. Chapters 3 and 4. [12] Rudin, W., Principles of Mathematical Analysis, McGraw-Hill, Tokyo, 1963. [13] Searcoíd, M. O., Elements of Abstract Analysis, Springer-Verlag, London, 2003. [14] Stillwell, J., Mathematics and its History, Springer-Verlag, New York, 1989, pp. 266-275. Department of Mathematics-IME University of Saõ Paulo CEP 05508-090 Saõ Paulo-SP Brazil e-mail: [email protected]

Note

One-Line Proof of the AM-GM Inequality O. A. S. KARAMZADEH

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n he AM-GM inequality says x1 þx2 þþx n x1 x2 xn n (all xi [ 0). This is perhaps the best known and most useful nontrivial inequality in mathematics, with a large number of interesting proofs in the literature. A very clever proof of this inequality, which appeared of late in [1], has given me motivation to communicate a very short proof of it, which I used to present to Iran’s Olympiad team, preparing for the IMO, several years ago. Let me, before going any further, give a generalization of the AM-GM inequality in the case n = 2 . It is trivial to see that, whenever 0\a b and x [ 0; a x b if and only if the a þ b x þ ab x . The latter inequality is stronger than p ffiffiffiffiffiffiAMGM inequality for n = 2 (indeed, just choose x ¼ ab). It also allows equality only for either x = a or x = b. Now the quick proof of the general case goes as follows. If x1 ¼ x2 ¼ ¼ xn , then we are done. If not, put x1 x2 xn ¼ gn , and without loss we may assume x1 xi and x2 xi for all i, hence x1 \g\x2 . We have noted that x1 \g\x2 if and only if x1 þ x2 [ g þ x1gx2 : Therefore x1 þ x2 þ þ xn [ g þ x1gx2 þ x3 þ þ xn ; but the latter expression is g þ ðn 1Þg ¼ ng, by the inductive hypothesis (noting that x1gx2 x3 xn ¼ gn1 ).

T

R EMARK The last part of the above proof shows that if not n [ all xi’s are equal, then in fact we have x1 þx2 þþx n p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x1 þx2 þþxn n ¼ x1 x2 xn , that is to say, the equality n p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n x1 x2 xn holds only when x1 ¼ x2 ¼ ¼ xn . The latter fact does not follow automatically from most of other proofs of this inequality in the literature.

REFERENCE

[1] Michael D. Hirschhorn, ‘‘The AM-GM Inequality,’’ Mathematical Intelligencer 29 (2007), nos. 4, 7.

Department of Mathematics Chamran University Ahvaz Iran e-mail: [email protected]

Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 2, 2011

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DOI 10.1007/s00283-010-9197-9

Viewpoint

Finance and Mathematics: A Lack of Debate JONATHAN KORMAN

The Viewpoint column offers readers of The Mathematical Intelligencer the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and the publisher and editors-in-chief do not endorse them or accept responsibility for them. Viewpoint should be submitted to one of the editors-in-chief, Chandler Davis and Marjorie Senechal.

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THE MATHEMATICAL INTELLIGENCER Ó 2011 Springer Science+Business Media, LLC

DOI 10.1007/s00283-011-9210-y

ver the last fifteen years I have seen many of the people around me, some very talented people, turn to financial jobs. They set out to study mathematics, had no interest in finance, but ended up working for the banks. For many, the turning point was around the time they were finishing their graduate studies and were facing the realities of the job market. The banks are extracting much talent from society in the form of mathematicians who could be using their talent elsewhere. Given the harm the financial industry has inflicted on society, it can be argued that this talent is being misused. In Paul Krugman’s words [K], ‘‘But the fact is that we’ve been devoting far too large a share of our wealth, far too much of the nation’s talent, to the business of devising and peddling complex financial schemes – schemes that have a tendency to blow up the economy. Ending this state of affairs will hurt the financial industry. So?’’ In his Intelligencer piece ‘Mathematics and Finance: An Ethical Malaise’, Marc Rogalski [R] reminds us that the financial industry is part of the mechanism in the ‘‘class struggle for the division of surplus value’’ designed to increase profits of stockholders at the expense of the working class. The lesson of the recent crisis, that society needs protection from financial institutions, is not new. Today one tends to ask only about more regulations, but Rogalski asks us to bear in mind that the bankers’ constant striving for huge profits puts them in conflict with the interests of society. Regulations can slow down their attack, but will not remove its threat. The fundamental problem cannot be addressed by ‘‘dealing with a few bad apples.’’ The current financial crisis has brought this state of affairs to the consciousness of the general public. Now is a time to ask questions. If we are led to challenge a field of our science, financial mathematics, to examine the ethical implications of its work, there is nothing novel about that. During the Cold War, Operations Research was sometimes criticized; during WWII, some blamed physicists who worked on the Manhattan Project. Pure mathematicians are often spared these ethical challenges, as they are typically cushioned from reality by several layers of abstraction. Rogalski argues that the dedication of the financial industry to enriching a small minority may make it dubious for public funds to be used to train its professionals: ‘‘But must the nation pay for that? Must universities and research institutions pay for this partisan activity and steer students toward it?’’ I would like to examine in particular the role mathematics departments play. I will not deal with the financial industry, nor with the academic research and teaching motivated by it, but only with the way graduate students

O

starting with quite other motives are diverted toward financial jobs. The phenomenon is widespread. I do not know figures for the profession as a whole, but I checked the current job status of the 76 mathematics Ph.D.s at the University of Toronto between 2002 and 2009. Out of 18 who are known to be working in the private sector, 14 are in the financial industry. Of these 18, only 4 specialized in financial mathematics as students. Therefore, of the 14 nonfinancially educated graduate students who went into nonacademic jobs, 10 (71%) went into financial ones. (And some of those who went into academic jobs may subsequently be squeezed out of them and swell this percentage.) It is my impression that the situation in many other research universities is similar: A high percentage of math Ph.D.s obtaining nonacademic jobs end up at the banks. Self-Interest of Departments In the past 30 years, many universities have defined themselves more as production lines for business [W]. Insofar as mathematics departments accept this redefinition, they may see their goal as maintaining a large flow of graduate students, for this yields a supply of cheap instructors, an easy justification for maintaining the departments and faculty, and an influx of (government) funds. This value system does not lead them to ask questions about what happens to their students afterward. Insiders and Outsiders Like most groups of people with shared interests, mathematics departments have somewhat of a cult mentality: caring mostly for ‘insiders’, academic mathematicians, and little about ‘outsiders’, nonacademic mathematicians. The insiders regard the outsiders as contributing relatively little to significant mathematics, and hence of little interest. The insiders take it for granted that mathematics is a worthy pursuit. The tacit inference is that the more mathematicians there are the better it is for mathematics and for society. Insiders tend to think of what they do as ‘important’, ‘beautiful’, ‘innovative’ – but also as ‘beneficial’, or at least ‘harmless’. Whereas insiders include most of the decision- and policy-makers in the mathematical commu-

AUTHOR

......................................................................... JONATHAN KORMAN received his Ph.D.

in 2002 in p-adic representation theory. His recent research is on optimal transportation. Simultaneously he is an artist, and some of his teaching has been at the Ontario College of Art and Design. Department of Mathematics University of Toronto Toronto, M5S 2E4 Canada e-mail: [email protected]

nity, they pay little attention to outsiders and their impact on society. This mentality works for a systematic blindness toward the phenomenon I am trying to discuss in this article. Those students who turn to financial or other applied work have made themselves, to some senior mathematicians’ eyes, invisible. Moral Ambiguity In his ‘Response to Rogalski’, Ivar Ekeland [E] says, ‘‘If bets go sour, mathematicians cannot help, but governments can’’. What is the logic here? That if our students willingly go to work for institutions known for looting the public, it is of no ethical concern if governments have not yet got around to regulating these institutions? It may be that regulation could make banks into a more benign force, but is Ekeland saying that students and their advisers must passively wait for this to be done? A person who goes to work for the banks may be desperate for a job and have no other choice, or he or she may even dream of getting rich1. Many people working for the banks surely feel no moral issue. To me, these jobs are morally unjustifiable. In her famous analysis of the banality of evil [A], Hannah Arendt observed that Adolf Eichmann was incapable of thinking about the moral consequences of his actions: Nothing of the sort ever occurred to him. He was morally blind – seeing his operation of the concentration camps only in logistical terms. We hold him responsible for his crimes, but the system in which he operated had its role. A large organization – such as a government or a corporation – is conducive to this sort of moral blindness. Of course banks are corporations par excellence, and it is easy for mathematicians there to see no need to justify their work for the bank, because they work in a moral vacuum. Government regulation may try to contain the damage from financial institutions. But individuals also carry a responsibility, both toward themselves and others. Even when outside forces seem overwhelming, individuals have a choice: the extent to which they choose to participate. This applies to everyone, not just to mathematicians: To what extent do psychologists care to be part of a system of torture? To what extend do citizens stand by while the environment is being destroyed? And to what extent do mathematicians care to play along with a destructive financial industry? Having said that, I must add that it is not fair to put the burden only on individuals when the system has trapped them. Here is where mathematics departments are failing us: They are complicit in setting the trap by training students in research areas where nobody is hiring, leaving them without room for maneuver when it comes to jobhunting. In effect, departments are handing over many of their students to the banks. Lack of Discourse Discourse about the role of mathematics in society is not part of the general consciousness and is not encouraged by departments. On the other hand,

1

An example of being exploited while making a good salary: A friend who considers himself lucky to have one of these financial jobs is currently making $80,000 a year. Calculated at 40 hours a week this comes to about $35/hr. He actually works 60-70 hours a week and is effectively not allowed to charge overtime, which brings his hourly rate to about $20/hr. His billing rate – the rate the bank proposes to charge the client – is calculated at around $450/hr. The billing rate for more senior people at his bank is even more exorbitant at over $1000/hr.


5

the business-as-usual attitude of mathematics departments in moving students through the system, not asking questions about what happens to them, not encouraging them to think about the implications of what they do, and deferring moral responsibility to other echelons, in effect sanctions the status quo. Departments should consider it part of their function to prepare students for nonacademic life and to encourage students to think about their role in society. It should be a department’s duty to inform students about the current trends, to encourage discussion, to stimulate them perhaps to imagine new possibilities. An Example of Challenging the Status Quo In ‘Beyond the Disciplines: Art without Borders’ Suzi Gablik [G] writes about graphic designers trying to put their skills to more worthwhile and ethical use: ‘‘Recently I read in The Structurist, a magazine published in Canada, that graphic designers have risen up against sterile corporate modernism and consumer capitalism, and are looking for other ways of practicing their craft beyond that of designing brand-name logos and promoting obsolescence. According to Kalle Lasn, founder and editor of the Canadian journal Adbusters, graphic designers want to put design skills to more worthwhile and ethical use than product marketing. Instead of trying to become the next big ‘‘it’’ in the design world, these renegade designers joined up with the antiglobalization movement, wrote a manifesto (‘First Things First’, published in Adbusters in 1999), and declared their intention to do something more interesting than just speed up the consumer purchasing cycle.’’

6


Here is an example to put us to shame! Mathematicians are much better organized than artists or designers. They are not just individuals: They have professional societies, and are part of a university system where most work in math departments. It should be easier for us than for graphic designers to think and act to put our skills to better use. It should not be up to the victimized young people only: The departments should wake up and do their share. ACKNOWLEDGMENTS

For useful comments, I thank Jeff Adler, Hugh Alcock, Jenny Wing-Yee Tong, and friends in the industry who prefer to remain anonymous.

REFERENCES

[A] H. Arendt, Eichmann in Jerusalem: A Report on the Banality of Evil. Viking Press, New York, 1963. [E] I. Ekeland, Response to Rogalski, Mathematical Intelligencer 32 (2010), no. 2, 9-10. [G] S. Gablik, Beyond the Disciplines: Art without Borders. http://moncon.greenmuseum.org/papers/gablik.pdf. [K] P. Krugman, Don’t Cry for Wall Street, New York Times, April 22, 2010. [R] M. Rogalski, Mathematics and Finance: An Ethical Malaise, Mathematical Intelligencer 32 (2010), no. 2, 6-8. [W] J. Westheimer, Higher Education or Education for Hire? Corporatization and the Threat to Democratic Thinking. Academic Matters, May 2010.

A Fractal Version of the Pinwheel Tiling NATALIE PRIEBE FRANK

AND

MICHAEL F. WHITTAKER

he pinwheel tilings are a remarkable class of tilings of the plane, and our main goal in this article is to introduce a fractal version of them. We will begin by describing how to construct the pinwheel tilings themselves and by discussing some of the properties that have generated so much interest. After that we will develop the fractal version and discuss some of its properties. Finding this fractile version was an inherently interesting problem, and the solution we found is unusual in the tiling literature. Like the well-known Penrose tilings [5], pinwheel tilings are generated by an ‘‘inflate-and-subdivide rule’’ (see Figure 1). Tilings generated by inflate-and-subdivide rules form a class of tilings that have a considerable amount of global structure called self-similarity. Self-similar tilings are usually nonperiodic but still exhibit a form of ‘‘long-range order’’ that makes their study particularly fruitful. Unlike Penrose tilings and most known examples of self-similar tilings, tiles in any pinwheel tiling appear in infinitely many different orientations. The pinwheel tilings were the first example of this sort and as such presented both new challenges and intriguing properties. Many examples of self-similar tilings are made of fractiles: tiles with fractal boundaries. Fractiles arise in the foundational work [7] for constructing a self-similar tiling for a given inflation factor. Two fractile versions of the Penrose tilings are introduced in [3]. Additionally, the procedure used in [8] may result in self-similar tilings made up of fractiles. This made it reasonable to expect that the pinwheel tilings might have a fractal variant, but did not provide a template for finding it. The technique for finding fractiles in both [3] and [8] is similar. One begins with an inflate-and-subdivide rule for

T

which the edges of each inflated tile do not quite match up with the edges of the tiles that replace it (the Penrose kite and dart are an example). The edges of each tile are redrawn using the edges of the tiles that replace it. These new edges are revised iteratively by the subdivision rule ad infinitum. The final result is a set of fractiles that are redrawings of the original tiles, but now they inflate-and-subdivide perfectly. Our technique is completely different. We found a fractal that runs through the interior of a pinwheel triangle and behaves nicely under the inflate-and-subdivide rule. The fact that this fractal extends to become the boundary of fractiles follows naturally (with some work) from the pinwheel inflation, but there was no way to know how many types of fractiles to expect. It was only by creating the images by computer that we were able to generate enough information to answer that question.1

Pinwheel Tilings Pinwheel tilings pffiffiffi are made up of right triangles of side lengths 1, 2, and 5. We say a pinwheel triangle is in standard position and call it a standard triangle if its vertices are at (-.5, -.5), (.5, -.5), and (-.5, 1.5). Ifwe multiply this stan2 1 , it can be dard triangle by the matrix MP ¼ 1 2 subdivided into five pinwheel triangles of the original size (see Figure 1). This is known as the pinwheel inflate-andsubdivide rule, or more simply as the pinwheel substitution rule [9]. (Readers who wish to make drawings for themselves: notice that all of the images in this article are oriented with a standard triangle at the origin and the origin marked.) We can apply the rule again, multiplying by MP and then

Dedicated to the inspiration of Benoit Mandelbrot. The authors thank Dirk Frettlo¨h for helpful discussions about the aorta and Edmund Harriss for pointing out our theorem on rotations in the aorta. The second author is partially supported by ARC grant 228-37-1021, Australia. 1

Mathematica code for the images in this article is available on request by contacting the first author.


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DOI 10.1007/s00283-011-9212-9

Figure 1. The pinwheel inflate-and-subdivide rule.

subdividing each of the five tiles as in Figure 1 and its reflection. In this way we obtain a patch of 25 tiles that we call a level-2 tile; substituting n times produces a level-n tile. In Figure 2 we see three levels of the substitution, where we have emphasized the borders of the original five tiles to exhibit the hierarchy. Many inflate-and-subdivide rules for tilings have been discovered since attention was first drawn to the subject in

the 1960s. A compendium of tilings generated by substitution rules appears on the Tilings Encyclopedia website [6], and an introduction to several different forms of tiling substitutions appears in [4]. There are several equivalent ways to obtain infinite tilings from a tiling substitution rule. The most straightforward is a constructive approach. Since the standard triangle is invariant under the pinwheel inflation, the level-1 tile it becomes will be invariant under any further applications of the inflate-andsubdivide rule. So will the level-n tiles once the rule has been iterated at least n times. Thus it is easy to see that when the substitution rule has been applied ad infinitum we will obtain a well-defined infinite tiling T0 of the plane. In fact, we can apply the inflate-and-subdivide rule to any tiling of the plane made up of pinwheel triangles by multiplying by the matrix Mp and then subdividing. The tillings that are invariant are called self-similar tillings, and one can show that they are all rotations of T0. But there are other infinite tilings we would like to call pinwheel tilings. If we slide T0 so that the origin is in some other tile, or if we rotate or reflect T0, or if we apply any rigid motion to all the tiles in T0, we have really only changed the placement of T0 in R2 : Thus we will consider any translation

Figure 2. Level-1, -2, and -3 tiles for the pinwheel inflate-and-subdivide rule.

AUTHORS

.........................................................................................................................................................

8

received her Ph.D. from the University of North Carolina at Chapel Hill and has been in the mathematics department at Vassar College since 2000. She enjoys investigating both the mathematical and artistic beauty of selfsimilar tilings. Her husband Scott is an applied mathematician, and they have two young sons, Jeremy and Ethan. When she’s not looking after them, she likes to play tennis and volleyball and see how the flowers in her garden are doing.

MICHAEL F. WHITTAKER received his Ph.D.

Department of Mathematics Vassar College, Box 248 Poughkeepsie, NY 12604 USA e-mail: [email protected]

School of Mathematics and Applied Statistics University of Wollongong Wollongong, NSW 2500 Australia e-mail: [email protected]

NATALIE PRIEBE FRANK


from the University of Victoria in 2010 and is currently a post-doc in Wollongong, Australia. He has a passion both for teaching and research but especially loves traveling around the world meeting mathematicians and sharing ideas. Michael works mainly on operator algebras associated with dynamical systems and especially enjoys Smale spaces that include certain tiling systems. When not at his desk, he likes camping, hiking, and going to the pub with his wife Chelsea.

or other rigid motion of T0 to be a pinwheel tiling. Moreover, if a tiling has the property that the patch of tiles in every large ball around the origin agrees with that in some rigid motion of T0, we will call it a pinwheel tiling as well. We consider all infinite pinwheel tilings we just described to be elements in the tiling space XP. We can summarize this approach to defining pinwheel tilings as follows: 1) construct T0, a pinwheel tiling that is selfsimilar; 2) act on T0 by all possible rigid motions, obtaining infinitely many ‘‘different’’ pinwheel tilings; and 3) include any other tilings that agree with tilings from step 2) over all arbitrarily large but finite regions of the plane. Every pinwheel tiling in XP looks locally like T0, but there are infinitely many tilings obtained in step 3) that are not rigid motions of T0. In fact these tilings are accumulation points of the tilings from step 2) under a ‘‘big ball’’ metric that says that two tilings are close if they very nearly agree on a big ball around the origin. Including the tilings in step 3) makes XP a compact topological space. As noted earlier, one of the main reasons that pinwheel tilings are of such importance is that in any pinwheel tiling, the triangles appear in a countably infinite number of distinct orientations. This isn’t difficult to see once one notices that the pinwheel angle / = arctan(1/2) is irrational with respect to p, and governs the orientations we see in Figure 2. This leads to the fact that the space of pinwheel tilings can be decomposed into the product of an oriented tiling space and a circle [14].

Two Intriguing Pinwheel Properties Like many tiling spaces generated from inflate-and-subdivide rules, the pinwheel space has a sort of homogeneity known as unique ergodicity [10]. In the situation where the tiles appear in only finitely many rotations, unique ergodicity automatically implies that every finite configuration of tiles appears with a well-defined frequency in every tiling T in the tiling space. The frequency of some patch C of tiles can be computed by looking at the number of times C occurs in some large ball in T and dividing by the area of that ball; the fact that there will be a limit as the size of the balls goes to infinity is a result of unique ergodicity. This approach doesn’t quite work when there are infinitely many rotations in every tiling. Unique ergodicity in the pinwheel case means that there is a statistical form of rotational invariance present in XP that is quite intriguing. Consider a finite configuration C of tiles and some interval I of orientations in which it might appear. Given a tiling T 2 XP we can count the number of times that C appears in a large ball in T in an orientation from I. Dividing that by the size of the ball, and then taking a limit as the size of the balls goes to infinity gives the frequency of occurrence of C in orientation I. The fact that this frequency is independent not only of the tiling chosen but of the sequence of balls in T is a side effect of unique ergodicity. What is more remarkable, the frequency depends only on the size of I, not on I itself [11]. Thus not only are the rotations uniformly distributed, no particular range of orientations is preferred over another. For this reason the pinwheel tiling space is considered ‘‘statistically round’’ even though most individual tilings in it are not rotationally invariant. Another surprising property of pinwheel tilings is that the hierarchical structure mandated by the inflate-and-subdivide

Figure 3. The kite and domino.

rule can be enforced by local constraints called matching rules [9], decorations on the edges of tiles that specify how they are allowed to meet up. Although many famous tilings, for instance the Penrose tilings, were known to come equipped with matching rules that force the hierarchical structure, this was the first example for which the matching rules also enforced infinite rotations. In [9], a new set of triangles is constructed by making numerous copies of the pinwheel triangles, each with markings on their edges that specify how they are allowed to meet. The remarkable fact is that this extremely local constraint forces the pinwheel hierarchy: any tiling with these new triangles that obeys the matching rules will become a tiling from XP when the markings on the edges are forgotten.

The Kite-Domino Version of Pinwheel Tilings A useful concept in tiling theory is that of mutual local derivability, which gives a way of comparing tilings built with different tile shapes. Given two tilings T1 and T2 of R2 , we say that T2 is locally derivable from T1 if there is a finite radius R such that the T1-patch in the ball of radius R about any point ~ x 2 R2 determines the precise type and placement of the tile x If T2 is locally derivable from T1 and T1 is (or tiles) in T2 at ~. also locally derivable from T2, we say the tilings are mutually locally derivable. If two tiling spaces are mutually locally derivable, then they are homeomorphic in the big ball topology. The main goal of this work is to introduce a tiling substitution on fractal tiles that produces tilings that are mutually locally derivable from the pinwheel tilings. But first, following [2], we introduce a tiling substitution called the ‘‘kite-domino’’ pinwheel tilings. The pinwheel triangles in any pinwheel tiling meet up hypotenuse-to-hypotenuse to form either a kite or a domino, which we show in standard position in Figure 3. There are two types of domino: the one pictured and one with an opposite diagonal; in our images we denote the difference by shading them differently. It is clear that every tiling in XP can be locally transformed into a kite-domino tiling by fusing together triangles along each hypotenuse. If the pinwheel substitution is applied to a kite or domino twice the result can be composed into kites and dominoes, resulting in the inflate-and-subdivide rule of Figure 4. We can build the space XKD of all tilings admitted by the kite-domino substitution using the same three-step ‘‘constructive’’ method we used to define the pinwheel tiling space XP. It is shown in [2] that the pinwheel tiling space XP is mutually locally derivable from the kite-domino substitution tiling space XKD. In what follows we will rely on XKD to make the fractal version of the pinwheel tilings. Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 2, 2011

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Figure 4. Substitution for the kite and domino.

Figure 5. The subdivision method for generating the aorta.

Figure 6. A few iterations of the pinwheel inflation with the aortas marked.

Construction of Pinwheel Fractiles To construct the pinwheel fractiles we construct a fractal, invariant under substitution, that we will use to mark all pinwheel triangles. We call this fractal the aorta. The aorta will be used both to form the boundaries of the fractiles and to define the local map taking pinwheel tilings to fractal pinwheel tilings. The Aorta There are three special points in a pinwheel triangle: the origin, the point (-.5, 0), and the point (0, .5). The origin is a (central) control point (cf. [13]) since its location in the triangle is invariant under substitution. We will call the points (-.5, 0) and (0, .5) the side and hypotenuse control points, 10


respectively. The key observation is that one can generate a fractal by connecting these three control points and then iterating the pinwheel subdividision rule without inflating. Figure 5 shows a sequence of subdivisions of the standard triangle. The side and hypotenuse control points alternate type in the subtriangles. The resulting fractal is the aorta. Alternatively, one can define the aorta to be the invariant set of an iterated function system. Let MP be the pinwheel expansion matrix shown previously, and let Ry and Rp denote reflection across the y-axis and rotation by p, respectively. Let f1 ðx; yÞ ¼ MP1 Ry ðx; yÞ þ ð0:4; 0:2Þ; f2 ðx; yÞ ¼ MP1 ðx; yÞ, and f3 ðx; yÞ ¼ Rp MP1 ðx; yÞþ ð0:2; 0:4Þ. Note that the union of these maps take each stage of the aorta to the next in

Figure 5. Since each fj is a contraction, there is a unique set 3 S fj ðAÞ, and of course A is the aorta. such that A ¼ j¼1

If we begin with a pinwheel triangle in standard position, mark it with its aorta, and inflate by MP, then the aorta will lie along the aortas of three of the five tiles in its subdivision. So what happens if we mark the aortas of all five tiles in this level1 tile? Upon substitution, these five aortas will lie atop fifteen of the 25 aortas in the level-2 tile, all shown in Figure 6. A close look at the marked level-3 tile of Figure 6 suggests two things. First, that it may be possible to join up the aortas to create a finite set of tiles with fractal boundary. Second, the forward invariance of the aorta sets of level-N tiles indicates that these fractiles may possess their own substitution rules. We will show that both are true. But first, a few questions and some discussion about the aortas themselves.

Figure 7. The two types of continuations.

Questions. Suppose we mark the aortas of a pinwheel tiling T in XP. Each individual aorta is part of a connected component of aortas that crosses N triangles, where N is either a positive integer or infinity. (1) What is the distribution of N over the tiling T ?2 (2) The infinite tiling obtained by continuing to substitute the pinwheel tiling pictured in Figures 2 and 6 has a twosided infinite aorta passing through the origin. Does every pinwheel tiling have a one- or two-sided infinite aorta? If not, what proportion of tilings do? Note on Generalizing the Aorta. The fact that the side and hypotenuse control points are related by substitution from one step to the next is essential to the existence of the aorta. When we subdivide a pinwheel tile, the hypotenuse control point becomes a side control point and vice versa; an arbitrary finite set of points on the boundary will not behave so nicely. Finding ‘‘aortas’’ (hidden fractals) in other tiling substitutions using our method will involve finding points on the tile boundaries that are related by substitution. The set of prototile vertices seems like a good place to start looking, but we have not yet been able to discover any nontrivial examples of these hidden fractals in other tiling substitutions. It would be interesting to look deeper into the issues determining which tiling substitutions have versions of the aorta of their own. The Fractiles There are two ways to construct equivalent (up to rescaling) versions of the fractal pinwheel tiles. One way is to begin with the aorta marking of Figure 6 and join the aortas that stop abruptly at a tile edge to the central control point of the adjacent tile using an appropriate fractal (a piece of the aorta, in fact). We call this the continuation method. Alternatively, we can mark the pinwheel tiles more elaborately, marking not the aorta, but instead the five aortas of the tiles in the subdivision of each tile. Connecting the dangling aortas to nearby control points can be done unambiguously with kites and dominoes, and we call this the kite-domino method. The tiles produced by the continuation method are equivalent to 2

Figure 8. Marking the kite and domino.

those from the kite-domino method except that they are five times as large. The Continuation Method. In any pinwheel tiling the triangles meet full hypotenuse to full hypotenuse; this implies that whenever an aorta does not connect to an adjacent aorta, it is at the side control point. The surprising fact is that there are only two ways that this can happen. In Figure 6 we have highlighted (in red and orange) one instance of each type of dangling aorta and follow how each continues after substitution. We call the two types of continuations these require the main and domino continuations, respectively. (Notice that the triangular patch for the domino continuation does not appear until the second substitution.) Further substitution indicates how to define the continuations: they are isometric copies of the part of the aorta connecting the side control point to the central control point. The continuations appear in their triangle patches as pictured in Figure 7, with the main one on the left and the domino one on the right (red and orange added for emphasis only). That both kinds of continuation behave well under substitution can be seen from an iterated function systems argument similar to that for the aorta. Given any pinwheel tiling T, we can now produce a new tiling TC with fractal boundary by marking all aortas as in Figure 6 and then adding the continuations as prescribed by Figure 7. We defer discussion of the properties of tilings produced by this method, preferring to discuss the equivalent tilings produced by the kite-domino method. The Kite-Domino Method. We begin with a pinwheel triangle, marking not its aorta but instead the five sub-aortas that are the preimages of the aortas in its level-1 triangle. We must add an additional fractal segment to connect the

We thank one of the reviewers for this very interesting question.


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Figure 9. Marking the level-1 kite and domino.

dangling sub-aorta to the central control point. This segment is shown in red in Figure 8(a); the dangling sub-aorta along with this segment form exactly the main continuation shown on the left of Figure 7. The marking of the kite tile, shown in Figure 8(b), is simply this initial marking on both of its triangles. In order to mark a domino tile, we need to use the initial markings on its two triangles, but we also need to resolve the two dangling sub-aortas that arise along the hypotenuse. As shown in Figure 8(c), we add fractal segments to connect these to the central control points so that the resulting fractals are the domino continuations of Figure 7. Figure 9 shows the result of marking the kites and dominos this way in substituted kite and domino tiles. The fractal marking of any kite or domino will join with the marking of its neighbors at the side control points forming a fractal connection between their central control points. The fractal connections encircle closed regions, and when such a region has no fractal in its interior we call it a pinwheel fractile. Working by hand, we were not sure how many fractiles to expect and feared there could be hundreds. We wrote computer code that generated further iterates of the marked kitedomino substitution and counted the fractiles that appeared in the images. By doing this we were relieved to find that there are 13 tile types up to reflection, and 18 tile types when reflection is considered distinct. (We defer temporarily the

proof that we have exhausted all possibilities). Of the 13 tile types, 10 are visible in Figure 9; the remaining three types appear after one more kite-domino substitution. Each fractile arises inside a patch of pinwheel triangles that is, except in a few cases, unique. Knowing these patches is essential for writing computer code to generate the images, and for figuring out how to inflate and subdivide the fractiles. In Figure 10 we show three representative fractiles as they

Figure 11. The fractiles. For each tile the control point of the standard triangle is marked.

Figure 10. Three representative fractiles, in standard position, as they arise in their triangle patches. 12


arise in their pinwheel triangle patches. For readers who wish to get their hands dirty by drawing pinwheels of their own we have included a triangle in standard position in each patch and have marked the origin with a dot. When a choice of which triangle to standardize needed to be made, we did so based on convenience for our computer code. All thirteen fractiles are shown in Figure 11, in order of relative frequency with the most frequently seen tile first. (We take the frequency of the tiles to mean the average number of times the tile appears in any orientation, per unit area.) Each fractile is situated with respect to the standard triangle in exactly one of the three ways shown in Figure 10. The reader can verify this fact by comparing the placement of control points in Figure 11 to those in Figure 10. Equivalence of Tiling Spaces We can mark any pinwheel tiling T in XP with the kitedomino method, producing a new tiling made of pinwheel fractiles. It is clear that doing this for every tiling in XP will produce a translation-invariant set of tilings, which in turn forms a tiling space that we denote XF. Since the central control points of pinwheel triangles are exactly the locations where the continuations meet the aorta, the vertex set of a fractal pinwheel tiling and the set of central control points of the corresponding pinwheel triangle tiling coincide. The pinwheel tiling space XP, the kite-domino tiling space XKD, and the fractal pinwheel tiling space XF are all mutually locally derivable. The equivalence of the first two is in [1]; to complete the assertion we show that XKD and XF are mutually locally derivable. The fractile boundaries in any pinwheel fractal tiling in XF are locally identifiable as kite or domino markings since their vertices are control points: if the vertex is degree 3, it is inside a kite; if it is degree 4, it is inside a domino. Thus every tiling in XF locally determines a tiling in XKD, and so XKD is locally derivable from XF. Conversely, any tiling in XKD can be marked as in Figure 8. Once this is complete, kite-domino patches of radius 5 or smaller determine which fractile covers any given point in R2 , since the largest fractile comes from a patch of kites and dominoes that has a diameter less than 5. This means that XF is locally derivable from XKD and completes the proof that all three tiling spaces are mutually locally derivable.

The Fractile Substitution The fact that each pinwheel fractile arises from a finite patch of pinwheel triangles means that each pinwheel fractile inherits a substitution from the pinwheel tiles that created it. (It also inherits an equivalent one from the kite-domino substitution.) For a few of the fractile types, the triangle patch that creates it is not unique because the fractal markings at all four right angles of the domino tile create congruent regions (see Figure 8(c)). However, it is easy to check that the pinwheel substitution induces congruent markings on the interiors of these regions, which implies that the substitution induced by the pinwheel substitution on the fractiles is welldefined. In Figure 12 we demonstrate how the substitution is induced on the fractile we call the ‘‘ghost’’ (note that we include in this image only the portion of the kite-domino markings that lie inside the inflation of the ghost’s boundary).

Figure 12. How the ‘‘ghost’’ fractile inherits its substitution rule.

Figure 13. Substituting the 13 fractile types.

In Figure 13, we show the substitutions of all thirteen basic pinwheel fractiles. We would like to emphasize the remarkable fact that the boundaries of the tiles shown in Figures 11 and 13 are perfectly scaled versions of one another. No additional detail is gained or lost because the tile boundaries are built from the aorta, and the aorta is a true fractal. We can now argue that the list of fractiles we show in Figure 11 is complete. Since the substitution rule shown in Figure 13 is self-contained, it defines a translation-invariant substitution tiling space XF0 in the same way that the original pinwheel substitution generated the tiling space XP. XF0 is measure-theoretically the same space as the space of pinwheel fractile tilings XF defined previously; it is actually possible to show that they are exactly the same space. For XF0 is mutually locally derivable from a subspace of XP, which must be translation-invariant since XF0 is. Since XP is uniquely ergodic, the subspace corresponding to XF0 must have measure 0 or 1. Since it is not of measure 0, it must correspond to a measure-1 subset of XP. Fixed, Periodic, and Symmetric Points in XF The substitution-invariance of the standard triangle (see Figure 1) means that the fractile substitution also admits a Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 2, 2011

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Figure 14. Generating a fixed-point of the fractile substitution.

Figure 15. Generating a tiling fixed by substitution and reflection.

fixed-point (i.e., a self-similar tiling) as shown in Figure 14. Of course any rotation of the standard pinwheel triangle also leads to a self-similar tiling. The reflection of the standard pinwheel triangle across the y-axis almost leads to a substitution-invariant tiling too, but not quite: the substitution of the reflected standard tile has the reflected standard tile at the origin, but it is rotated clockwise by the pinwheel angle / = arctan(1/2). One can check that there are no other fixed-points by noticing that a tiling is invariant only if its patch at the origin is fixed under substitution. This implies that the tiles in such a patch must, under substitution, contain themselves. This happens only for the tiles pictured in Figure 14. It is interesting to note, however, that the fourth, sixth, and tenth prototiles contain reflections of themselves in their substitutions. Thus the tilings they create are fixed when a combination of

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substitution and reflection is applied. Figure 15 develops how this looks with the fourth fractile at the origin. There are six pinwheel triangle tilings that are fixed under rotation by p. Two are invariant under reflection as well; these two are the images of each other under the original pinwheel substitution and are thus period-2 under substitution and rotation by 2/. The corresponding tilings in XF have the second and the eleventh fractile types at the origin and are pictured in Figure 16. The other four pinwheel triangle tilings that are fixed under rotation have the center of a domino tile, or its image under substitution, at the origin. These are not symmetric by reflection and make a period-4 sequence under substitution plus rotation, or a period-2 sequence under substitution plus reflection across an appropriate axis. See Figure 17 for the beginning of the corresponding tilings in XF.

Figure 16. Period 2 up to rotation by /; also invariant under rotation by p and reflection.

Figure 17. Period 4, up to rotation by /; invariant under rotation by p.

Properties Basic Properties of the Fractiles A useful tool for analyzing substitution rules is the substitution matrix A whose entry Aij is the number of tiles of type j in the substitution of tile i (cf. [12, 13] for results used in this section). Since A is a nonnegative integer matrix, the largest eigenvalue k is real by the Perron-Frobenius theorem. In fact, k is the area expansion of the substitution and its left eigenvector represents the relative areas of the tiles. (In our case k = 5.) Moreover, a properly scaled right eigenvector represents the relative frequency with which each tile appears, where relative frequency is the number of occurrences per unit area. We can choose whether or not to distinguish between tiles that are reflections of each other, giving us either 13 or 18 prototiles. Although this affects the size of A, it does not particularly affect the eigenvector analysis much. Obviously,

the eigenvector representing the relative areas will give us the same relative sizes of tiles in each case. And since reflections of tiles happen equally often, we find that when they are taken into account, the relative frequency is halved. Thus, if we were to consider reflections to be distinct, the most frequently seen tile would no longer be the first tile shown in Figure 11 since it would appear half as often, and its reflection would also. When we consider the number of prototiles to be 13, we compute the vector of relative frequencies to be approximately (.1412, .1225, .1039, .1, .1, .1, .0843, .0784, .0784, .0353, .0245, .0157, .0157). These numbers gave us the order we used to display the tiles in Figure 11. At first it may seem surprising that the fractile areas are whole multiples of 1/5, but this fact can either be seen geometrically by looking at the kite and domino markings of Figure 8 or by eigenvector analysis. (The geometric argument begins by noticing that the aorta cuts its triangle exactly Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 2, 2011

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more iterations of the substitution, we will have those two triangles, plus the two triangles in the same location in the substitution of t1. Of those, the image of the standard triangle is at the same orientation as t1, but the image of t1 is a rotation by another 2/ clockwise. So in this level-4 supertile, we have triangles at rotations of 0, 2/, and 4/, and these tiles lie on the aorta. In the level-6 supertile, we gain another triangle along the aorta, providing 4 distinct orientations. In this way we see that if we need N orientations, we must pass to a level(2N - 2) supertile. The second part of the theorem follows since 2/ is irrational with respect to p, and thus the set fmð2/Þ such that m 2 Ng is uniformly distributed mod 2p.

Figure 18. Ghost tiles on the boundary of any level-4 ghost supertile.

in half). The areas of the tiles in the order shown in Figure 11 are: 1, 1, 1, 6/5, 9/5, 1, 9/5, 6/5, 9/5, 6/5, 7/5, 7/5, and 13/5. (Note that the area of a pinwheel triangle is also 1). Since the aorta pffiffiffi is the limit set of an IFS with a (linear) contraction factor 5 thatpffiffiffiuses three functions, its fractal dimension is ln 3= ln 5 1:365, thus the boundaries of the fractiles have this dimension also. The tiles have rational area but irrational boundary dimension!

Rotational Property Since any pinwheel tiling features triangles in infinitely many different orientations, it is clear that any fractile must appear in any tiling in XF in infinitely many orientations also. By the equivalence of the tiling spaces XP and XF we know that the pinwheel fractal tilings must also be ‘‘statistically round’’ in the same sense as the pinwheel triangle tilings. All these orientations are wound up together in every copy of the aorta in an intriguing way.

We conclude with a fun side effect of the rotational and border-forcing properties. Every fractal pinwheel tiling can be decomposed into level-N supertiles for any N, and when N is large, so is the number of (level-0) fractiles that intersect the level-N boundaries. Our rotational theorem can be interpreted as: if a certain fractile type (for instance, the ghost fractile) intersects the boundary of level-N supertiles, then as N increases to infinity, that fractile will dangle off the supertile boundaries in an unbounded number of orientations. Since with very little modification we have a borderforcing substitution, we know that the way that these tiles dangle off will be identical every time a particular level-N supertile appears. We leave you with Figure 18, which shows all of the ghost fractiles that intersect the boundary of any level-4 ghost supertile in any fractal pinwheel tiling, almost as children to a larger parent. Although N = 4 is not very large, we begin to see the many different angles in which these offspring appear.

REFERENCES

[1] M. Baake, D. Frettlo¨h, and U. Grimm, A radial analogue of Poisson’s summation formula with application to powder diffraction and pinwheel patterns, J. Geom. Phys. 57 (2007), 1331– 1343. [2] M. Baake, D. Frettlo¨h, and U. Grimm, Pinwheel patterns and powder diffraction, Phil. Mag. 87 (2007), 2831–2838.

THEOREM For every N [ 0 there is a connected subset of the aorta, copies of which appear in at least N distinct rotations inside the aorta. Moreover, the set of all relative orientations that occur is uniformly distributed in [0, 2p].

[3] C. Bandt and P. Gummelt, Fractal Penrose tilings I. Construction and matching rules, Aequ. Math. 53 (1997), 295–307. [4] N. P. Frank, A primer on substitutions tilings of Euclidean space, Expo. Math. 26:4 (2008), 295–326. [5] M. Gardner, Extraordinary nonperiodic tiling that enriches the

PROOF. We will show that for each N [ 0 there is an n 2 N for which the level-n pinwheel supertile in standard position contains triangles that intersect the aorta and are in at least N different relative orientations. Applying the matrix MPn to this supertile will take the aortas of these triangles to the desired connected subsets of the aorta of the triangle in standard position. We refer to Figure 6 for the level-1, 2, and 3 pinwheel supertiles in standard position. In particular, notice that in the level-2 supertile, the triangle in standard position shares the vertex (-.5, 1.5) with a triangle t1 that is its rotation by 2/ = 2arctan(1/2) clockwise around that vertex. After two 16


theory of tiles, Scientific American 237 (1977), 110–119. [6] E. Harriss and D. Frettlo¨h, Tilings encyclopedia, http://tilings. math.uni-bielefeld.de/. [7] R. Kenyon, The construction of self-similar tilings, Geom. Func. Anal. 6:3 (1996), 471–488. [8] N. P. Frank and B. Solomyak, A characterization of planar pseudo-self-similar tilings, Disc. Comp. Geom. 26:3 (2001), 289– 306. [9] C. Radin, The pinwheel tilings of the plane, Annals of Math. 139:3 (1994), 661–702. [10] C. Radin, Space tilings and substitutions, Geom. Dedicata 55 (1995), 257–264.

[11] C. Radin, Miles of Tiles, Student Mathematical Library 1,

[13] B. Solomyak, Dynamics of self-similar tilings, Ergodic Th. and

American Mathematical Society, Providence (1999). [12] E. A. Robinson, Symbolic dynamics and tilings of Rd , Proc.

Dynam. Sys. 17 (1997), 695–738. [14] M. Whittaker, C -algebras of tilings with infinite rotational

Sympos. Appl. Math. 20 (2004), 81–119.

symmetry, J. Operator Th. 64:2 (2010).


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Mathematically Bent

Colin Adams, Editor

CSI: MSRI COLIN ADAMS

The proof is in the pudding.

Opening a copy of The Mathematical Intelligencer you may ask yourself uneasily, ‘‘What is this anyway—a mathematical journal, or what?’’ Or you may ask, ‘‘Where am I?’’ Or even ‘‘Who am I?’’ This sense of disorientation is at its most acute when you open to Colin Adams’s column. Relax. Breathe regularly. It’s mathematical, it’s a humor column, and it may even be harmless.

â

Column editor’s address: Colin Adams, Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01267, USA e-mail: [email protected]

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THE MATHEMATICAL INTELLIGENCER 2011 Springer Science+Business Media, LLC

DOI 10.1007/s00283-010-9183-2

was in Berkeley, having just given a seminar on fluxions at the university. Only a handful of people had shown up, including one calculus student who had stumbled into the wrong room and didn’t seem to know the difference, and an emeritus professor who kept raising his hand to ask if anyone had seen his underwear. The seminar organizer hadn’t even come, claiming in an e-mail he had to clean his gutters. After finishing my talk, I asked for questions. There was a long silence and then the emeritus professor leapt up and ran for the door. The rest of the group filtered out without giving me even a desultory applause. Feeling dejected, I wandered over to Telegraph Avenue. Picking from among the dozens of coffee houses, I ordered a cappuccino latte, and settled into a chair opposite a student reading Hungerford’s Algebra. ‘‘That’s a big book,’’ I said, hoping for a little intellectual banter. ‘‘I’ll show you my theorem if you’ll show me yours.’’ She sighed, slapping closed the book. ‘‘This I don’t need,’’ she said, as she got up and walked away. I sipped nonchalantly on my latte. A postdoc sat down at the next table with a copy of Russell’s Principia Mathematica. I kept my mouth shut. Suddenly, my cell phone buzzed. It was Schmishschmitt, an old grad-school friend who had washed out of grad school, but ended up on the police force. He had the math beat. ‘‘Hello, Mangum,’’ he said. That’s me, Dirk Mangum, P.I. That’s right. I’m a principal investigator on an NSF grant. ‘‘I thought I would find you here.’’ ‘‘You called me on my cell phone, Schmishschmitt, so you shouldn’t be too proud of the fact you found me.’’ ‘‘True,’’ said Schmishschmitt. ‘‘Nevertheless, I have an interesting case for you.’’ ‘‘What makes you think I would find it interesting?’’ I replied. ‘‘It involves MSRI.’’ MSRI. Math lingo for the Mathematical Sciences Research Institute, way up in the hills overlooking Berkeley. The golden citadel of mathematics. A building dedicated to one thing and one thing only. Math all day, every day. The founders of MSRI had had a simple dream. Plunk a bunch of mathematicians down all by themselves among the eucalyptus trees, without any distractions but a beautiful view of the bay. Then they would have nothing better to do than create great mathematics. Unfortunately, they had that view. It might have been smarter to plunk them in

I

a basement somewhere. But, it was a view I wanted to see. And Schmishschmitt knew it. ‘‘How soon can you get here?’’ he asked, assuming he had reached me in my office at UCLA. I checked my pocket to see if I had the dollar in change I needed for the bus. ‘‘I’ll be there in 20 minutes,’’ I replied. The bus dropped me off in front of the MSRI building on Gauss Way. As I entered, a police officer waved me up the stairs, and another pointed me to an open door. I could see the victim’s legs sticking out from behind the desk. Police had cordoned off the area, and were taking photos. Schmishschmitt waved me over. As I joined him, I couldn’t help but catch a glimpse out the window over the bay. A fog was rolling in. It was magnificent. ‘‘Okay, Mangum. You got your view. Now quit your rubbernecking. We have a case to crack.’’ I sighed, turning away from that amazing panorama to the sordid reality of the crime scene before me. ‘‘Give me the lowdown,’’ I said. ‘‘She’s a tenured professor at Cornell, name of Kate Witherspoon.’’ I knew Witherspoon. She had started her career at Five Mile Island Community College. Then she knocked off the Little Armandhamer conjecture and got a job at Southern Miami State. She followed that with the Big Armandhamer Conjecture and moved to University of Miami. When she proved the Strong Big Armandhamer Conjecture, Cornell came knocking. I mean that literally. Armandhamer, who was the math chair at Cornell, flew down to Miami, knocked on her office door, and made her the offer. He had a few more conjectures he was hoping she would solve. ‘‘She was here at MSRI for a special semester on Combinatorial Counting,’’ said Schmishschmitt. ‘‘She was a very good counter.’’ ‘‘Probably didn’t count on this,’’ I said dryly, motioning to the scene before me. Papers were strewn about her prone body. ‘‘She had been here for three weeks,’’ continued Schmishschmitt. ‘‘By all accounts, she was quiet, unassuming. Kept her nose to the mathematical grindstone. Didn’t get in any trouble with residents of the neighboring offices. Then, suddenly, at 3:27 this afternoon, the inhabitants of the next office heard a scream. They rushed in and found her lying there.’’ ‘‘Can I talk to them?’’ ‘‘No. None of them speak English. Mostly Lithuanians, with a Rumanian or two mixed in. But they’re good with hand signals. I would want them on my team for charades.’’ ‘‘This isn’t a good time for games, Schmishschmitt. Maybe we can play later. What exactly happened to her?’’ ‘‘Why don’t you ask?’’ She was sobbing, as she lay on the floor. ‘‘Excuse me,’’ I said, ‘‘Professor Witherspoon? Can I interrupt your blubbering for a moment? I have a few questions.’’ ‘‘Who are you?’’ she asked as she sat up, pushing away a policeman dusting her for prints.

‘‘Name’s Mangum, Mangum P.I. I’m a principal investigator on a National Science Foundation grant. I’m also the guy the police call when there is a crime involving math.’’ ‘‘Well, you’re in the right place,’’ she said, as she slowly pulled herself to her feet. She was taller than she looked when she was lying down. Seems to be a pretty common phenomenon. ‘‘Can you tell me what happened?’’ She pointed to her computer screen. I leaned forward to read an e-mail message. ‘‘Dear Dr. Witherspoon, I regret to inform you there is a counterexample to your proof of the Strong Big Armandhamer Conjecture. Take for instance the collection of integers {7, 41, 321, 6,432}. Sorry about that. T. Boone Picky’’ ‘‘Is he right?’’ I asked. She started bawling. ‘‘Well, that is unfortunate,’’ I said. ‘‘Nobody wants to see a good theorem go down. But if it’s wrong, it’s wrong. These things happen. It’s hardly a criminal matter.’’ Schmishschmitt looked confused. ‘‘If the theorem was wrong, wouldn’t the referee catch it before it was published?’’ I laughed out loud. Then I pounded the desk a few times and laughed some more. Nobody was taking the bait, so I slapped Schmishschmitt between the shoulders, and laughed some more. ‘‘What’s so funny?’’ asked Schmishschmitt. ‘‘Referees are like people,’’ I said. ‘‘In fact, come to think of it, they are people. And just like people, they come in a variety of types. There are good referees and bad referees. There are referees who will hit you up for not capitalizing the first word of every sentence. There are referees who get upset when you say 1 and you mean 2. There are referees who will reject a paper because their name doesn’t appear in the list of references. And there are referees who could give a flying flype, referees who agreed to referee your paper for no better reason than some dean at their school counts that as service. And they’d rather be watching The Bachelorette on TV than poring over your paper trying to figure out if A implies B. So they do a quick skim, ten minutes glancing over dense, detailed mathematics. And then they flip a coin, and send in a two-sentence report.’’ ‘‘But you don’t understand,’’ said Witherspoon, having stifled her tears. ‘‘It’s not a mistake on my part or on the referee’s part. The paper is correct.’’ ‘‘Listen, Witherspoon,’’ I replied. ‘‘If there is a counterexample, a counterexample consisting of only four integers, then no conclusion is possible other than that the theorem is wrong.’’ ‘‘I know,’’ she said. ‘‘That is what is so strange. I know my paper is right.’’ ‘‘Can I see it?’’ I asked. She handed me a reprint. It was from the Annals of Analytic Computational Counting, Series B. Very impressive. It came in at a hefty 27 pages. 2011 Springer Science+Business Media, LLC, Volume 33, Number 2, 2011

19

I glanced the intro over. ‘‘So you use some other results in your proof.’’ ‘‘Yes, just a few. I use the Proper Premise Proposition. And I use the Leminscate Lemma.’’ ‘‘And how long has the Proper Premise Proposition been around?’’ ‘‘Since 1943. It appeared in a paper by Peter Proper.’’ ‘‘And the Leminscate Lemma?’’ ‘‘That was published in 2009.’’ ‘‘Who did it?’’ She looked confused. ‘‘Who proved the Leminscate Lemma?’’ I repeated. ‘‘Vito Leminscate,’’ she replied. The name rang a bell. ‘‘Not the Vito Leminscate who used to be at NYU?’’ ‘‘No. The Vito Leminscate who used to be at Brown.’’ That brought up some memories, some very unpleasant memories. Many of them were of the times I had confused the two Vito Leminscates. But a few of them were unpleasant memories about the Brown Vito Leminscate. ‘‘I had heard he disappeared off the mathematical map,’’ I said. ‘‘Went into i-banking.’’ ‘‘That’s right,’’ confirmed Witherspoon, ‘‘But he didn’t last long in the real world. He came back to math when the bottom fell out of the market. A lot of quants did.’’ ‘‘And where is he now?’’ ‘‘Actually,’’ she answered, ‘‘he’s two doors down. He’s here for the semester in Combinatorial Counting.’’ ‘‘Excuse me,’’ I said, as I stepped out of the office. I saw a head pop back into the office two doors down. As I headed down the hall, a crowd of Lithuanians and Rumanians popped out of the next office. They were all jabbering at once, and I knew immediately I would have to play some charades after all. I pantomimed my gratitude for their having phoned in the emergency, and explained by way of hand signals that it was essential I get past them to the open door beyond. They expressed their immense gratitude for the invitation to visit at MSRI and explained how much they enjoy the quarter pounder at MacDonald’s. Schmishschmitt was right. They were good. With a couple of swift hand gestures, I managed to convince them that MSRI would be providing them with bacon-wrapped scallops but they would need to be in their office to receive them, and then I walked up to the open door beyond. Leminscate was pretending to be hard at work, chewing on a pencil and staring off at the view of the bay. ‘‘You can get lead poisoning that way,’’ I said. ‘‘There is no lead in the lead of a pencil. It’s made of graphite,’’ he replied, spitting some pencil shards in to the wastebasket. ‘‘Doesn’t mean you should chew on ‘em. Have you considered the splinters?’’ Leminscate looked uncomfortable, as he ran his tongue around the inside of his mouth. ‘‘You know, Leminscate, I think the last time I saw you was at a bar in Providence in ‘05 after the number fields seminar at Brown. You were passing out preprints like they were cupcakes. Trying to convince naı¨ve grad students you were hotter than Heegaard Floer homology.’’

20


‘‘Mangum, I don’t have to listen to your guff. I actually have an office in this building. Do you?’’ ‘‘I’m here on business,’’ I said. ‘‘I’m here to talk about the so-called Leminscate Lemma.’’ Leminscate almost choked on his pencil shards. ‘‘You got nothing on me,’’ he said, spitting into the wastebasket. ‘‘You know, Leminscate, let me tell you about my plans for this evening. I’m thinking I’ll be sitting down with a glass of wine and a copy of the proof of the Leminscate Lemma. Just the two of us. It’ll be very cozy. Some romantic music in the background, the lights down low. We’ll get to know each other real personal. I’ll start with a slow perusal, just a skim up and down. But eventually, that lemma will open up to me. And by the time I’m done I will have examined every inch of that lemma, from top to bottom, I’ll have checked under the hood. There will be no bit of that lemma that was not exposed to my scrutiny. I’ll get to know it better than I know my own…..’’ ‘‘I don’t know how you did it, Mangum. But you made reading a theorem sound dirty.’’ ‘‘If the theorem’s dirty, then reading it’s dirty, too.’’ ‘‘All right,’’ said Leminscate. ‘‘You caught me. It’s a fake.’’ ‘‘What?’’ ‘‘The proof is a fake. I admit it.’’ ‘‘What do you mean, it’s a fake?’’ ‘‘The proof is wrong. I have been dying to tell someone. It’s so good. You see, in the Leminscate Lemma, I use an inductive argument. I prove the base case, which corresponds to one leminscate. And I prove the inductive step, that if there are n leminscates, then there are n+1.’’ ‘‘That sounds like the whole proof.’’ ‘‘But guess what, Mangum. The proof of the inductive step implicitly assumes there are already at least two leminscates. So you never get past the base case. You actually never have more than one leminscate. Nobody thinks to check that.’’ ‘‘Clever,’’ I acknowledged. ‘‘But why? Why would you want to plant false results in mathematics?’’ ‘‘I’m a mathematical terrorist.’’ ‘‘A what?’’ I asked. ‘‘A mathematical terrorist. Those results, once they appear, get used by other mathematicians in their results, and those results get used by others. Given enough time, the seeds that I have planted will blossom into the diseased trees that will bring mathematics to its knees.’’ ‘‘But why would you want to destroy mathematics?’’ ‘‘I spent a few years in the financial world.’’ ‘‘I heard that. So?’’ ‘‘So, I built an algorithm designed to milk the market. It was based on algebraic combinatorial counting. It utilized the differential between the current rate of pork belly futures and the diminishing return on lateral derivative indices. It made me a pile of money. A pile so high that I needed oxygen when I sat on it.’’ ‘‘So what’s so bad about that?’’ ‘‘It didn’t predict the pork belly run of 2008. Before you knew it, there wasn’t a pork belly to be had on the entire island of Manhattan. Who knew pork bellies would become

the biggest thing since supersubindices? I lost everything. And all thanks to that algorithm.’’ ‘‘So?’’ ‘‘So, mathematics destroyed my life. It’s only fair I return the favor.’’ ‘‘But math isn’t a person. You can’t exact revenge on a logical system.’’ ‘‘Oh yeah?’’ he said, ‘‘You just watch me.’’ With that he leapt up, spitting shards of pencil in my face. As I covered my eyes with my arm, he plowed into me, knocking me flat on my keister. ‘‘There’s nothing you can do to stop me,’’ he screamed, as he ran to the top of the stairs, and then leapt over the banister, landing on the floor below. Hearing the ruckus, the Lithuanians and Rumanians poured out of their office, gesticulating wildly, wanting to know where the baconwrapped scallops had gotten to. As I tried to disentangle myself from them, he shot out the front door of the building. I sprinted after him as he headed up the path into the woods. I knew that if I lost sight of him I would probably never see him again. But as he turned a corner in the trail, he almost tripped over a yoga class practicing in the woods. As he swerved to avoid them, I caught up. Diving forward, I wrapped my arms around him, and tackled him hard to the ground. ‘‘What the hell are you doing?’’ he screamed. It was a good question. There wasn’t anything criminal he could be charged with.

‘‘Hmm,’’ I replied, as we lay in a tangle of limbs, some ours, some the nearby trees’. ‘‘I will have you up on charges of assault and battery,’’ he cried, as he stood up, brushing himself off. ‘‘I have witnesses.’’ Luckily, the yoga class was meditating, and all of them still had their eyes closed. ‘‘Listen, Leminscate,’’ I said. ‘‘You’re right. There’s nothing I can do to you in a court of law. But the truth is that you’ll never work in this business again. You try to publish anything ever again and I’ll make sure the editors of the journal have eight referees going over every +, -, epsilon, and delta. Your tawdry plan has been exposed.’’ It would have been a great way to end the story, as I walked away, leaving him to ponder his fate. But unfortunately, we both needed to take the bus down to Berkeley, so there was a very awkward 20 minute wait at the bus stop. When we climbed on the bus, we made a point of sitting apart. As the bus finally stopped to let him off he turned to me and said with a sneer, ‘‘There are more like me out there. This isn’t finished. Hahaha!’’ The others waiting to get off pushed him out the door. And maybe what he said is true. Maybe there are more like him, intentionally hiding bogus results in the structure of math, with the long-term goal the utter destruction of math as we know it. But he is awfully weird. I kind of doubt it.

2011 Springer Science+Business Media, LLC, Volume 33, Number 2, 2011

21

Min Matrices and Mean Matrices RAJENDRA BHATIA

T

he principal character of this story, the min matrix, is the rather simple n 9 n matrix M whose (i, j) entry is mij ¼ minði; jÞ: Thus 2 3 1 1 1 1 61 2 2 2 7 6 7 7 M¼6 ð1Þ 6 1 2 3 3 7: 4 5 1 2 3 n

Looking at the matrix we see that M is symmetric, and the positions of each of the entries 1; 2; 3; . . . make a ‘‘C pattern’’. Less obvious is the fact that M is positive definite. Recall that a real symmetric n 9 n matrix A is said to be positive semidefinite (psd, for short) if x 0 Ax 0 for all x in Rn : (The superscript 0 denotes the transpose of a vector or of a matrix.) It is not always easy to determine when a symmetric matrix A is psd. Each of the following five conditions is necessary and sufficient for a real symmetric matrix A to be psd. 1. 2. 3. 4.

All principal minors of A are nonnegative. All eigenvalues of A are nonnegative. A ¼ B 0 B for some matrix B. A ¼ R 0 R for some upper triangular matrix R. (This is called a Cholesky factorisation of A.) 5. There exist vectors u1 ; . . .; un in some Hilbert space such that their inner products hui ; uj i are equal to aij. (The matrix with its (i, j) entries equal to hui ; uj i is called the Gram matrix associated with u1 ; . . .; un : This characterisation says that every psd matrix is a Gram matrix.) The following facts are easily verified:

1. If A1 ; . . .; Ak are psd and a1 ; . . .; ak are nonnegative numbers, then a1 A1 þ þ ak Ak is psd. (In other words, every positive linear combination of psd matrices is psd.) 22


DOI 10.1007/s00283-010-9194-z

2. If A and B are psd matrices, then their direct sum A A 0 B¼ is psd. 0 B 3. If {Am} is a sequence of psd matrices and Am ! A; then A is psd. 4. If A is psd and X is any matrix, then X 0 AX is psd. (If X is nonsingular, we say that A and X 0 AX are congruent. Congruence is an equivalence relation, and it preserves positive semidefiniteness.) Let E be the matrix all of whose entries are equal to 1. This is called the flat matrix, and is psd. This can be easily seen either from the very definition: if x ¼ ðx1 ; . . .; xn Þ then x 0 Ex ¼ ðx1 þ þ xn Þ2 0; or by directly verifying any of the five conditions listed above. (The reader will find it instructive to do so.) I recommend Chapter 7 of [14] for an excellent introduction to positive definite matrices.

Six Proofs that the Min Matrix is Psd I give several proofs that the matrix M in (1) is psd. Each proof uses a different idea and points to different directions. The first is what we may call a child’s proof or a proof without words. Playing with M one soon observes 2 3 2 3 1 1 1 1 0 0 0 0 61 1 1 17 60 1 1 17 6 7 6 7 6 7 6 7 M ¼ 61 1 1 17 þ 60 1 1 17 6 7 6 7 4 5 4 5 1 0 60 6 6 þ 60 6 4 0 2

1 0 0 0 0

1 0 0 1 1

1 0 1 1 3 2 0 0 7 6 07 60 7 6 17 þ þ 60 7 6 4 5 1 0

0 0 0 0

0 0 0 0

1

3 0 07 7 7 07 7 5 1

ð2Þ

Each of the summands is psd (being the direct sum of a zero matrix and a flat matrix). So M is psd. A lot more can be extracted from this idea. Let 0\k1 k2 kn ; and let Mðk1 ; . . .; kn Þ be the matrix with (i, j) entry mij ¼ minðki ; kj Þ: Call this a (generalized) min matrix. When kj = j, this reduces to the min matrix (1). A slight modification of the child’s proof shows that every min matrix is psd. (The modification is to attach nonnegative coefficients ðkj kj1 Þ to the summands in (2).) Let A = [aij] and B = [bij] be two n 9 n matrices. Let A B ¼ ½aij bij be their entrywise product. This is also called the Hadamard product or the Schur product. A famous theorem of Schur says that if A and B are psd, then so is A B. As a consequence, for every h i positive integer m

m, the mth Hadamard power A

¼

am ij

of a psd matrix

A is psd. If the entries aij are nonnegative numbers, we 1=m

could also consider fractional Hadamard powers A ¼ h i 1=m 1=m aij : If A is psd, then it is not necessary that A be psd. For example, consider the 3 9 3 2 1 1 A ¼ 41 2 0 1

matrix 3 0 1 5: 1

1=m Then A is psd, but det A ¼ 21=m 2 \ 0; for every m [ 1. A matrix A with nonnegative entries is said to be infi1=m nitely divisible if for each m, the matrix A is psd. Then, by Schur’s theorem, Ar is psd for every positive rational number r; and by taking limits, this is true for all positive real numbers r. Continuous analogues of infinitely divisible matrices (infinitely divisible kernels) are of great importance in probability theory [21]. If mij ¼ minðki ; kj Þ; then mrij ¼ minðkri ; krj Þ for every r [ 0. This brings home the very interesting fact that every min matrix is infinitely divisible.

AUTHOR

......................................................................... RAJENDRA BHATIA is Professor at Indian

Statistical Institute, New Delhi. He is the author of Matrix Analysis (Springer, 1997) and Positive Definite Matrices (Princeton University Press, 2007). Being the Chief Editor of the Proceedings of the International Congress of Mathematicians 2010 taught him the meaning of ‘‘herding cats.’’ Indian Statistical Institute Delhi Centre 7, S. J. S. Sansanwal Marg New Delhi 110 016 India e-mail: [email protected]; [email protected]

A small remark before we move further. Let k1 ; . . .; kn be any positive real numbers, not necessarily in increasing order. Then the matrix M ¼ minðki ; kj Þ can be expressed as M = PM:P where P is a permutation matrix, and M " ¼ h i minðk"i ; k"j Þ where k"1 k"2 k"n is an increasing rearrangement of k1 ; . . .; kn : Let us also call a matrix M of this sort a min matrix. It too is infinitely divisible. The child’s proof could be recast as what would look like an algebraist’s proof. The quadratic form correspond ing to the min matrix M ¼ minðki ; kj Þ ; with 0\k1 kn ; is x 0 Mx ¼ k1 ðx1 þ þ xn Þ2 þ ðk2 k1 Þðx2 þ þ xn Þ2 þ þ ðkn1 kn2 Þðxn1 þ xn Þ2 þ ðkn kn1 Þxn2 : An analyst might come up with a different proof. Everyone remembers the beautiful formula Z 1 2 sin x p dx ¼ : 2 x 2 0 (This is one of the standard integrals in a Complex Analysis [1, p.160] or a Fourier Analysis [2, p.70] course.) Using the trigonometric identity sin ax sin bx ¼ sin2

ða þ bÞx ða bÞx sin2 ; 2 2

we obtain from this Z 1 sin ax sin bx p dx ¼ minða; bÞ; 2 x 2 0 for all a, b [ 0. Thus the entries mij ¼ minðki ; kj Þ of the min matrix M can be expressed as Z 2 1 dx sin ki x sin kj x 2 ¼ hui ; uj i; mij ¼ p 0 x qffiffi sin k x where uj ðxÞ ¼ p2 x j ; and the inner product is that of the Hilbert space L2 [0, ?). Thus M is a Gram matrix, and hence is psd. This ‘‘hard analysis’’ proof could be made into a ‘‘soft analysis’’ proof. The quantity minða; bÞ is captured as another integral: Z 1 v½0;a ðtÞv½0;b ðtÞdt; minða; bÞ ¼ 0

where vE stands for the characteristic function of the set E. Using this, one can again see that M is a Gram matrix. Integral representations like the two used above are useful in several situations—we will see more examples. However, the idea just used works in finite-dimensional vector space just as well. If we choose uj ¼ ð1; 1; . . .; 1; 0; . . .; 0Þ where the first j entries are equal to one, then we see that hui ; uj i ¼ minði; jÞ: This brings us to the linear algebraist’s proof. The special form of M can be exploited to obtain its Cholesky decomposition:


23

2

1 1 1

6 61 2 2 6 M¼6 61 2 3 6 4 2

1 2 3 1 0 0

6 61 1 0 6 ¼6 61 1 1 6 4 1 1 1

1

3

7 27 7 37 7 7 5

n 3 0 7 07 7 07 7 7 5 1

2

1 1

6 60 1 6 60 0 6 6 4 0 0

1 1

3

7 1 17 7 1 17 7 7 5 0 1

¼ R 0 R:

ð3Þ

This proof has two additional bonuses. We see at once from this decomposition that det M = 1. Further, it is easy to see that 2 3 1 1 0 0 6 0 1 1 0 7 7; R 1 ¼ 6 4 5 0 0 0 1 a bidiagonal matrix with entries equal to 1 on its diagonal and -1 on its first superdiagonal. It follows that M-1 is a tridiagonal matrix 2 3 2 1 0 0 6 1 2 1 07 6 7 7 0 1 2 1 0 ð4Þ M 1 ¼ 6 6 7 4 5 0 0 0 1 1 Now the matrix M -1 is well known to numerical analysts as the discrete Laplacian with Dirichlet boundary condition at 0 and Neumann boundary condition at 1. To explain this, let me consider the integral operator T, and its adjoint T *, defined on L2 [0, 1] as Z x Z 1 Tf ðxÞ ¼ f ðtÞdt; T f ðxÞ ¼ f ðtÞdt: 0

x

These operators satisfy the boundary conditions T f ð1Þ ¼ 0:

Tf ð0Þ ¼ 0;

d The inverse of T is the differentiation operator D ¼ dx ; and d : From the definition of T its adjoint is the operator D ¼ dx we see that TT * is the integral kernel operator Z 1 minðx; yÞf ðyÞdy; TT f ðxÞ ¼ 0

and the remarks above show that its inverse is the operator d2 0 dx 2 ; with boundary conditions f(0) = 0 and f ð1Þ ¼ 0: 0 Let R be the lower triangular matrix in (3). Then for every vector u ¼ ðu1 ; . . .; un Þ the jth component of R0 u is ðR 0 uÞj ¼

j X

ui :

i¼1

This is a discrete version of the integral operator T. In the same way the upper triangular matrix R is a discretisation of the operator T *. 24


This gives us a differential equations proof of the posi2 tive definiteness of M. The differential operator d dx 2 is positive, and hence so is its inverse, the integral kernel operator with kernel minðx; yÞ: I mentioned earlier the importance of infinitely divisible kernels in probability theory. The covariance kernel associated with any probability distribution is positive definite. It is infinitely divisible if and only if the distribution corresponds to a sequence {Xn} where for each n, Xn is the sum of n independent identically distributed random variables fXnj gnj¼1 : See e.g., [21]. A probabilist gave me the following proof of infinite divisibility of the min matrix. The standard Brownian motion has as its covariance kernel cðs; tÞ ¼ minðs; tÞ: This kernel must be infinitely divisible since the Brownian motion is a process with stationary independent increments.

Eigenvalues of the Min Matrix The linear algebra and differential equations proofs yield one more dividend—a beautiful formula for the eigenvalues of the min matrix (1). Start with the eigen equation M -1u = ku for the matrix in (4). This can be written out as n component equations uj1 þ 2uj ujþ1 ¼ kuj ;

1 j n;

ð5Þ

together with two ‘‘boundary conditions’’ u0 ¼ 0;

unþ1 ¼ un :

ð6Þ

The two conditions in (6) account for the fact that the first and the last rows of M-1 are different from the other n - 2 rows. From the trigonometric identity sinðj þ 1Þa þ sinðj 1Þa ¼ 2 sin ja cos a

¼ 2 sin ja 1 2 sin2

a ; 2

we obtain a sinðj 1Þa þ 2 sin ja sinðj þ 1Þa ¼ 4 sin2 sin ja: 2 ð7Þ So the Eq. (5) are satisfied if we choose a uj ¼ sin ja: k ¼ 4 sin2 ; 2

ð8Þ

To ensure that u is nonzero, a should not be an integral multiple of p. The boundary conditions further restrict a: we must have sinðn þ 1Þa ¼ sin na: This, in turn, means that a is of the form a¼

ð2k þ 1Þp ; 2n þ 1

k ¼ 0; . . .; n 1:

Plugging these values of a into (8) we obtain the n eigenvalues and eigenvectors of M-1. Thus, the eigenvalues of the matrix M are 1 ð2k þ 1Þp cosec2 ; 4 2ð2n þ 1Þ

0 k n 1:

Let me interject a few historical remarks here. One of the several problems studied by J. L. Lagrange in his grand work on mechanics is a discretised version of the vibrating string problem. Consider a massless string of length (n + 1)d, along which are strung n beads at equal distances d. Suppose each bead has mass m, the endpoints of the string are fixed, and the tension s in the string makes it vibrate in the xy plane. An application of Newton’s laws then shows that the displacement uj of the jth bead satisfies the equation s 1 j n: mu€j ¼ ð2uj uj1 ujþ1 Þ; d This can be written in matrix notation as s Lu; u€ ¼ md where L is the n 9 n matrix 2 2 1 0 6 1 2 1 6 M 1 ¼ 6 6 0 1 2 1 4 0 0 0 1

3 0 07 7 07 7: 5 2

This matrix differs from (4) in that its last entry is 2. This difference comes from the fact that instead of (6) we have the boundary conditions u0 ¼ unþ1 ¼ 0; corresponding to the assumption that the endpoints of the string are fixed. Lagrange found the modes of oscillation of the beads by a calculation that, in today’s language, amounts to finding the eigenvalues of L. M. Pupin, in a paper [20] that appeared in Volume 1 of the Transactions of the American Mathematical Society in 1900, observed that the same equations describe the propagation of electrical waves over telephone networks. The language of matrices was still not in common use when this article appeared. An early paper that advanced both mechanics and matrix theory at the same time is Duncan and Collar [10]. A variety of mechanical systems from a triple pendulum to an airscrew blade are analysed here. These authors assign to each such system a ‘‘flexibility matrix’’, which in our parlance is a generalised min matrix. No simple formula for the eigenvalues of a generalised min matrix seems to be known. Duncan and Collar suggest some approximation techniques to find these eigenvalues. One of the illustrative examples chosen by them is the 3 9 3 flexibility matrix 2 3 2 2 2 42 5 5 5 2 5 11 associated with a triple pendulum. Numerical linear algebra has come a long way since then.

Three Cousins of the Min Matrix Closely related to the min matrix is the max matrix W with 1 : Thus entries wij ¼ maxði;jÞ

2

1 1=2 6 1=2 1=2 6 W ¼6 6 1=3 1=3 4 1=n 1=n

3 1=n 1=n 7 7 1=n 7 7: 5 1=n

1=3 1=3 1=3 1=n

ð9Þ

This is a symmetric matrix with entries making a ‘‘< pattern’’. More generally, we could take 0\k1 k2 kn 1 and consider the matrix W with entries wij ¼ maxðk : i ;kj Þ Observe that this is a generalised min matrix in disguise, since

1 1 1 : ¼ min ; maxðk1 ; k2 Þ k1 k2 So W inherits the properties of M. The reader could try to derive these properties without involving the min matrix, and find some interesting facts. Thus, for example we have the Cholesky decomposition of the matrix (9) 2

1

6 6 1=2 6 W ¼6 6 1=3 6 4 1=n

0

0

1=2 1=3

0 1=3

1=n

1=n

0

3 2

7 7 7 7 7 7 5 1=n 0 0

1

6 60 6 60 6 6 4 0

1=2

1=3

1=2 0

1=3 1=3

0

0

1=n

3

7 1=n 7 7 1=n 7 7 7 5 1=n

0

¼ CC :

The matrix C is called the Cesa`ro matrix. If y = Cx, then the components of y are the Cesa`ro averages yk ¼

x1 þ þ xk : k

This matrix and its continuous analogue, the Cesa`ro operator defined as Z 1 x f ðtÞdt; ðCf ÞðxÞ ¼ x 0 have been studied by many analysts. I will return to this in the next section. The next matrix I consider comes from probability theory. The Brownian bridge process has the covariance kernel cðs; tÞ ¼ minðs; tÞ st;

0 s; t 1:

Using the properties of the min matrix we can easily show that this is infinitely divisible. First note that cðs; tÞ ¼ minðs; tÞð1 maxðs; tÞÞ: Then observe that 1 maxðs; tÞ ¼ minð1 s; 1 tÞ: Hence for k1 ; k2 ; . . .; kn in [0,1] we have cðki ; kj Þ ¼ minðki ; kj Þminð1 ki ; 1 kj Þ: Thus the matrix cðki ; kj Þ is the Hadamard product of two generalised min matrices. Hence it is infinitely divisible. Another kin of the min matrix is the GCD matrix. Let x1 ; x2 ; . . .; xn be n distinct natural numbers, and let gij be the greatest common divisor (GCD) of xi and xj. The matrix G = [gij] is called a GCD matrix and has been studied by number-theorists. It is an infinitely divisible matrix. A


25

simple proof of this was given in [6], and now I offer a further simplification. Let p1 ; p2 ; . . .; pk be prime numbers such that m

mi

m

xi ¼ p1 i1 p2 i2 pk k ;

1 i n:

Then minðmi1 ;mj1 Þ

ðxi ; xj Þ ¼ p1

minðmik ;mjk Þ

pk

:

So the matrix G can be expressed as the Hadamard product G ¼ P1 P2 Pk ; where the matrix P‘ has as its (i, j) entry minðmi‘ ;mj‘ Þ

p‘ ði; jÞ ¼ p‘

:

The function f(x) = px‘ is monotonically increasing for every p‘. Thus P‘ is a generalised min matrix. So G, being a Hadamard product of infinitely divisible matrices, is itself infinitely divisible.

Mean Matrices The bridge between min matrices and mean matrices is the famous Hilbert matrix. This is the matrix H with entries 1 : Thus hij ¼ iþj1 2 3 1 1=2 1=3 6 1=2 1=3 1=4 7 7 H ¼6 ð10Þ 4 1=3 1=4 1=5 5: A delightful study of several features of this matrix is the paper by M.-D. Choi [9]. The Hilbert matrix is a special type of Cauchy matrix. Given positive numbers k1 ; . . .; kn ; the Cauchy matrix C 1 : associated with them is the matrix with entries cij ¼ ki þk j Cauchy gave an explicit formula for the determinant of such a matrix. See [12]. The focus here is on positive definiteness and infinite divisibility. We have for every r [ 0 Z 1 1 1 e tðki þkj Þ t r1 dt: r ¼ ðki þ kj Þ CðrÞ 0 Thus the (i, j) entry of C r can be expressed as the inner product hui ; uj i in the space L2 ðRþ ; lÞ with ui ðtÞ ¼ e tki t r1 and dlðtÞ ¼ CðrÞ dt: Thus C r is psd for all r [ 0, and C is therefore infinitely divisible. Hilbert showed that the (infinite) matrix H defines a bounded operator on the Hilbert space ‘2 and kH k\2p: This was improved upon by I. Schur who showed that kH k ¼ p: This statement has come to be known as the Hilbert inequality. It has fascinated several mathematicians, among them O. Toeplitz, M. Riesz, and G. H. Hardy, who found different proofs of the Hilbert inequality, studied integral operator analogues of H, and found their norms in ‘p and Lp spaces. While looking for a new proof of the Hilbert inequality, Hardy discovered another beautiful inequality that is now known as Hardy’s inequality. This says that the norm of the max matrix W, acting as a linear operator on Cn ; is less than 4, and it approaches 4 as n goes to infinity. The Cholesky decomposition of W demonstrated in the previous section 26


shows that this is equivalent to the statement that the Cesa`ro matrix C has kCk\2: Analogues for ‘p and Lp spaces have been studied. The Hilbert and Hardy inequalities are discussed at length in the classic [11], and in the more recent book [22]. These two inequalities are not the only link between the matrices H and W. They are connected also by the unifying concept of mean values, which is another major theme in [11]. We are all familiar with arithmetic, geometric, and harmonic means of two positive numbers. There are other means that are used in different contexts. Abstractly, a mean mða; bÞ is a binary function on (0,?) with the following properties: (i) (ii) (iii) (iv) (v)

mða; bÞ ¼ mðb; aÞ; minða; bÞ mða; bÞ maxða; bÞ; mðaa; abÞ ¼ amða; bÞ for all a [ 0, mða; bÞ is an increasing function of a and b, mða; bÞ is jointly continuous in a and b.

The three classical means (arithmetic, geometric, harmonic) possess these properties, as do the functions minða; bÞ and max (a, b). The binomial means, also called the power means, are defined as a

a þ ba 1=a ; 1 a 1: Ba ða; bÞ ¼ 2 When a = 1, this is the arithmetic mean; when a = -1, it is the harmonic mean. The values B0(a, b), B?(a, b), and B-?(a, b) are to be interpreted as limits, and reduce to the geometric mean, the maximum, and the minimum, respectively. For fixed a and b, Ba (a, b) is an increasing function of a. A substantial part of [11] is devoted to these means. Now what is a mean matrix? Given positive numbers k1 ; . . .; kn and a mean m; we consider the matrix M with entries mij ¼ mðki ; kj Þ and the matrix W with entries wij ¼ 1=mðki ; kj Þ: What do we know about such matrices? 1. Let mða; bÞ ¼ minða; bÞ: Then M is the min matrix. It is an infinitely divisible matrix. 2. Let mða; bÞ ¼ maxða; bÞ: Then W is the max matrix. It is infinitely divisible. 3. hLet mða; i bÞ ¼ a.m.ða; bÞ; the arithmetic mean. Then W ¼ 2 ki þkj

is a Cauchy matrix, and this too is infinitely

divisible. 4. Let mða; bÞ ¼ g.m.ða; bÞ; the geometric mean. Then M ¼ pffiffiffiffiffiffiffiffi ki kj : This matrix is congruent to the flat matrix E, and is infinitely divisible. In this case, for the same reason, the matrix W ¼ p1ffiffiffiffiffiffi is also infinitely divisible. ki kj

5. Let mða; bÞ ¼ h.m.ða; bÞ; the harmonic mean. This is the ratio of ab and the arithmetic mean. Thus in this case ki kj ¼ DWD; M ¼ h.m.ðki ; kj Þ ¼ a.m.ðki ; kj Þ where D is the diagonal matrix diagðk1 ; . . .; kn Þ and W is the mean matrix corresponding to the arithmetic mean. Thus M is infinitely divisible.

Do we see a pattern here? Let m1 and m2 be two means, and say that m1 m2 if m1 ða; bÞ m2 ða; bÞ for all a and b. It turns out that for a surprisingly large number of means we have the following: if m g.m.; then the mean matrices M ¼ mðki ; kj Þ are infinitely divisible; and if g.m. m; then h i the mean matrices W ¼ mðk1i ;kj Þ are infinitely divisible. Why should one care? First, positive definiteness is rare. (‘‘Symmetric positive definiteness is one of the highest accolades to which a matrix can aspire’’ [12, p.196].) Rarer still is infinite divisibility, and it is good to find it in such examples. I should add that the proofs of infinite divisibility are often intricate, involving a blend of matrix theory, complex analysis, Fourier analysis, and probability. ([3, 4, 7, 19].) More important, however, is the context in which the phenomenon was discovered. This is best explained by the simplest example. Let !1=2 X 2 1=2 jtij j kT k2 ¼ ðtr T T Þ ¼ i;j

be the Frobenius norm of a matrix T. If S and T are two matrices with jsij j jtij j; then clearly kSk2 kT k2 : Now let A be the diagonal matrix with positive diagonal entries k1 ; . . .; kn , and let X be any matrix. Then the matrix S ¼ pffiffiffiffiffiffiffiffi A1=2 XA1=2 has entries sij ¼ ki kj xij ; and the matrix T ¼ 1 1 2ðAX þ XAÞ has entries tij ¼ 2ðki þ kj Þxij : By the arithmeticgeometric mean inequality jsij j jtij j; and hence kA1=2 XA1=2 k2 12kAX þ XAk2 : Since every psd matrix is unitarily similar to a diagonal matrix, this inequality is true for all psd matricesA. Then argument, using a well-known 0 X A 0 ; shows that for all and the block matrices 0 0 0 B psd matrices A and B, and for all X, we have 1 kA1=2 XB1=2 k2 kAX þ XBk2 : 2

ð11Þ

This is a matrix version of the arithmetic-geometric mean inequality. For several reasons, the operator norm kAk ¼ sup kAxk

for all k1 ; . . .; kn the n 9 n matrix

m1 ðki ;kj Þ m2 ðki ;kj Þ

is psd. When-

ever two means possess this property, then the scalar inequality m1 ða; bÞ m2 ða; bÞ can be extended to matrices. We have seen one example in (12). As another, consider the logarithmic mean defined as Z 1 ab at b1t dt: ¼ l.m.ða; bÞ ¼ log a log b 0 This mean, much used in thermodynamics, satisfies the inequalities g.m. l.m. a.m. This leads to the matrix inequalities Z 1 1 At XB1t dtk kAX þ XBk: kA1=2 XB1=2 k k 2 0

ð13Þ

ð14Þ

The first inequality in (14) is a key ingredient in studying the geometry of the manifold P consisting of n 9 n positive definite matrices. See Chapter 6 of [4]. Many inequalities of this type have been established in [8, 15, 16, 18], and the idea is the inspiration for the monograph [17], and for Chapter 5 of [4]. In all hthe examples known, whenever m1 m2 ; the i matrices

m1 ðki ;kj Þ m2 ðki ;kj Þ

turn out to be not just psd but even

infinitely divisible. See [4, 7, 19]. This is intriguing, and its significance is not yet understood.

REFERENCES

[1] L. V. Ahlfors, Complex Analysis, Second Ed., McGraw Hill, 1966. [2] R. Bhatia, Fourier Series, Mathematical Association of America, 2005. [3] R. Bhatia, ‘‘Infinitely divisible matrices,’’ Amer. Math. Monthly 113 (2006), 221-235. [4] R. Bhatia, Positive Definite Matrices, Princeton University Press,

kxk¼1

2007.

is often more interesting than the Frobenius norm. Here jsij j jtij j does not always imply kSk kT k; and the simple argument given above to derive (11) has to be replaced by a more subtle one. Let P be a psd matrix. Then, by a theorem of Schur [4], kP Xk maxpii kXk: i pffiffiffiffiffiffi 2 ki kj g:m:ðki ;kj Þ Let pij ¼ a:m:ðki ;kj Þ ¼ ki þkj : We have seen that P is a psd matrix, and obviously pii = 1 for all i. So, using Schur’s theorem we see that for every psd matrix A, and for every X, we have kA1=2 XA1=2 k 12kAX þ XAk: As before, this leads to the matrix arithmetic-geometric mean inequality, this time for the operator norm: 1 kA1=2 XB1=2 k kAX þ XBk; 2

valid for all psd matrices A and B, and for all X. The inequality (12) was first proved in [5]. Several different proofs are known now. The one given above follows [13], and is the germ of a fruitful idea. Let m1 and m2 be two means, and h say ithat m1 m2 if

[5] R. Bhatia and C. Davis, ‘‘More matrix forms of the arithmeticgeometric mean inequality,’’ SIAM J. Matrix Anal. 179 (1993), 132136. [6] R. Bhatia and J. A. Dias da Silva, ‘‘Infinite divisibility of GCD matrices,’’ Amer. Math. Monthly 115 (2008), 551-553. [7] R. Bhatia and H. Kosaki, ‘‘Mean matrices and infinite divisibility,’’ Linear Algebra Appl. 424 (2007), 36-54. [8] R. Bhatia and K. R. Parthasarathy, ‘‘Positive definite functions and operator inequalities,’’ Bull. London Math. Soc. 32 (2000), 214-228. [9] M.-D. Choi, ‘‘Tricks or treats with the Hilbert matrix,’’ Amer. Math. Monthly 90 (1983), 301-312. [10] W. J. Duncan and A. R. Collar, ‘‘A method for the solution of oscillation problems by matrices,’’ Philosophical Mag. ser.7, 17

ð12Þ

(1934), 865-909.


27

[11] G. H. Hardy, J. E. Littlewood, and G. Po´lya, Inequalities, Second

[17] F. Hiai and H. Kosaki, Means of Hilbert Space Operators, Lecture

Ed., Cambridge University Press, 1952. [12] N. J. Higham, Accuracy and Stability of Numerical Algorithms,

Notes in Mathematics 1820, Springer, 2003. [18] H. Kosaki, ‘‘Arithmetic-geometric mean and related inequalities

Second ed., SIAM, 2002.

for operators,’’ J. Funct. Anal. 15 (1998), 429-451.

[13] R. Horn, ‘‘Norm bounds for Hadamard products and an

[19] H. Kosaki, ‘‘On infinite divisibility of positive definite functions

arithmetic-geometric mean inequality for unitarily invariant

arising from operator means,’’ J. Funct. Anal. 254 (2008), 84-

norms,’’ Linear Algebra Appl. 223/224 (1995), 355-361. [14] R. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985.

108. [20] M. I. Pupin, ‘‘Wave propagation over non-uniform electrical conductors,’’ Trans. Amer. Math. Soc. 1 (1900), 259-286.

[15] F. Hiai and H. Kosaki, ‘‘Comparison for various means of operators,’’ J. Funct. Anal. 16 (1999), 300-323.

[21] K. Sato, Le´vy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999.

[16] F. Hiai and H. Kosaki, ‘‘Means for matrices and comparison of

[22] J. M. Steele, The Cauchy-Schwarz Master Class, Cambridge

their norms,’’ Indiana Univ. Math. J. 48 (1999), 899-936.

28


University Press, 2004.

Lost and Found: An Unpublished f(2)-Proof PAUL LEVRIE Letter from Leonhard Euler to Daniel Bernoulli St-Petersburg, April 16, 1768 Hochedelgebohrener Hochgeehrtester Herr Professor. I finally managed to simplify my method to calculate the sum 1þ

1 1 1 1 þ þ þ þ etc. 22 32 42 52 2

As your worship may remember, I found a derivation that this sum equals p6 which I myself liked a lot. It was published in the Journal litte´raire d’Allemagne some years ago. But I wasn’t very happy with one thing: The derivation led to the series p2 1 1 1 1 ¼ 1 þ 2 þ 2 þ 2 þ 2 þ etc. 3 5 7 9 8 and then the missing terms had to be added to get the result. In that same paper, a direct proof was given. But there I used differential equations; now I have found a more elegant approach, using only the integration by parts result given below. We start with the well-known formula Z Z 1 n1 sinn2 /d/: sinn /d/ ¼ sinn1 / cos / þ ð1Þ n n It will be used from right to left, as your worship can see here: Z Z 1 nþ2 sinn /d/ ¼ sinnþ1 / cos / þ sinnþ2 /d/: nþ1 nþ1

The paper Euler refers to appeared in 1743 in the Journal litte´raire d’Allemagne, de Suisse et du Nord, 2:1, pp. 115–127, under the title De´monstration de la somme de cette suite [6].

This formula can be found in Euler’s book Institutionem calculi integralis (1766). But it goes back to James Gregory (1638-1675) who in 1673 proved it on the back of a letter he received from his friend John Collins.

From this formula all the formulas in the following table may be extracted: Z Z 2 sin0 /d/ ¼ sin / cos / þ sin2 /d/ 1 Z Z 2 4 sin6 /d/ sin0 /d/ ¼ sin / cos / þ sin3 / cos / þ 2 3 3 Z 2 24 5 sin / cos / sin0 /d/ ¼ sin / cos / þ sin3 / cos / þ 3 Z 35 46 sin8 /d/ þ2 35 and so on: Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 2, 2011

29

DOI 10.1007/s00283-010-9179-y

Continuing in the same manner, we see that 2 24 5 sin / cos / / ¼ sin / cos / þ sin3 / cos / þ 3 35 246 7 2468 9 sin / cos / þ sin / cos / þ etc: þ 357 3579

ð2Þ

Here the integral on the left has been replaced by its value. The terms in this sum are particularly easy to integrate: /2 1 2 21 4 2 41 6 2 4 61 8 sin / þ sin / þ sin / ¼ sin / þ 2 34 3 56 3 5 78 2 2468 1 sin10 / þ etc: þ 3 5 7 9 10

ð3Þ

Now we write down the integral of this sum without computing the right-hand side: Z Z Z /3 1 21 2 41 sin2 /d/ þ sin4 /d/ þ sin6 /d/ ¼ 2 34 3 56 6 Z Z 2 4 61 2468 1 8 sin /d/ þ sin10 /d/ þ etc: þ ð4Þ 3 5 78 3 5 7 9 10 and we replace / by p2 : Then the integrals in the sum may be calculated from the first formula I used. Your worship can see the results in the following table: Z 1p sin2 /d/ ¼ 22 Z 3 1p 4 sin /d/ ¼ 422 Z 5 3 1p sin6 /d/ ¼ 6422 Z 7 5 31p 8 sin /d/ ¼ 86422 and so on: 3

3

p If we multiply every integral by its coefficient in the series for /6 or in this case 48 , then we will have

p3 1 1 p 2 1 3 1 p 2 4 1 5 3 1 p 2 4 6 1 7 5 3 1 p þ þ þ þ etc. ¼ 48 2 2 2 3 4 4 2 2 3 5 6 6 4 2 2 3 5 7 8 8 6 4 2 2 Dividing both sides by

p 2

and multiplying by 4 gives us: p2 1 1 1 1 ¼ 1 þ 2 þ 2 þ 2 þ 2 þ etc. 2 3 4 5 6

So happy was I with this result, that when our new house in stone was finally finished I asked the stonemason to use it in our housenumber. Your worship can admire it in this picture:

30


If we replace / in this formula by arctan t, we get a series that was already known to Newton (16431727). It can be found in his De Computo Serierum (1684). Newton applied a finite difference transformation to the well-known Madhava-Gregory series for arctan.

Johann Bernoulli (1667-1748) also found Newton’s series for arctan. Dividing both sides by 1 + t2 and integrating, he found this formula [1, p. 25]. This series was found independently in 1722 by Takebe Katahiro [5].

From here on Euler repeats his arguments from the Journal litte´raire.

These formulas can already be found in the book Arithmetica Infinitorum (1656) by John Wallis (1616-1703).

I hope everything is well in Basel? My wife Katharina and I wish to bestow upon your worship our sincerest regards. In the meanwhile, I remain your faithful friend and humble and obedient servant. Leonhard Euler.

his very straightforward derivation in this (apocryphal) letter of the sum of the series defining fð2Þ cannot be found in this form in Euler’s papers. It may be used in a basic calculus class to find the sum of the series in an informal way. But can it be made into a valid proof? Indeed it can. In what follows I will fill in the missing details.

T

(A) Does the series in (2) converge to the left-hand side? Obviously it does for / = 0. I will prove that the series converges to / for -p/2 \ / \ p/2. To do this, we need the Wallis formulas: Z p=2 2n 2 2n 4 4 2 ... sin2n1 tdt ¼ 2n 1 2n 3 5 3 0 which can easily be deduced from (1). With them we can rewrite the series in (2) as ! Z p=2 1 X 2n1 cos / sin tdt sin2n1 /: ð5Þ 0

n¼1

Interchanging the sum and the integral is allowed since the series 1 X

sin2n1 / sin2n1 t

ð6Þ

n¼1

converges uniformly for all t if / [ (-p/2, p/2). We can prove this with the Weierstrass M-test [2, p. 535]1: For all t j sin2n1 / sin2n1 tj Mn ¼ j sin2n1 /j: P Furthermore, the series Mn converges: It is a geometric series with ratio jsin2 /j\1 if / [ (- p/2, p/2). Note that the series (6) itself is a geometric series, with sum 1 X

sin

2n1

2n1

/ sin

n¼1

sin / sin t t¼ : 1 sin2 / sin2 t

Hence we get for the series (5) ! Z p=2 X 1 2n1 2n1 sin / sin t dt cos / 0

¼ cos /

Z 0

n¼1 p=2

sin / sin t dt: 1 sin2 / sin2 t

Using sin2 t ¼ 1 cos2 t , we can calculate the integral on the right:

1

Z

p=2

sin / sin t dt cos2 / þ ðsin / cos tÞ2 1 sin / cos t p=2 arctan ¼ cos / cos / cos / 0

cos/

0

¼ arctanðtan /Þ ¼ /: The last step follows from / [ (- p/2, p/2), and it concludes this part of the proof. (B) If we want to integrate the series (2) term by term, we need to prove uniform convergence. This can be done for the interval [- t, t ] with 0 \ t \ p/2 by using the inequality 2 4 . . . 2n 2 sin2n1 / cos / Mn ¼ sin2n1 t 3 5 2n 1 which holds for j/j t . Hence term-by-term integration between 0 and t is allowed. (C) We now need uniform convergence of the series (3) in [0, p/2]. Again the Weierstrass M-test does the trick. In this interval we have the following inequality: 2 4 . . . 2n 2 1 sin2n / Mn 3 5 2n 1 2n 2n 2 4 2 1 ¼ ... : 2n 1 5 3 2n It follows from Raabe’s (or Gauss’s) convergence test P (see [2, p. 566]) that the series Mn converges. Hence convergence in (3) is uniform and term-by-term integration between 0 and p/2 is allowed, resulting in (4). This concludes the proof. ACKNOWLEDGMENTS

I would like to thank the anonymous referees for their very informative referee report (some of it is reflected in the annotations), Clive Tooth for sending me the picture of his house and permitting me to use it, and my colleague Walter Daems for many helpful suggestions.

REFERENCES

[1] J. Bernoulli, Opera Omnia Vol. 4, Lausanne, Genf, 1742 (reprinted Hildesheim, 1968). [2] R. Courant and F. John, Introduction to Calculus and Analysis Vol. I, Springer-Verlag, New York, 1989.

The Weierstrass M-test requires us to find an upper bound Mn on the terms of this series, with Mn independent of t. If uniformly convergent.

P

Mn converges, then the original series is


31

[3] C. E. Sandifer, ‘‘Euler’s solution of the Basel Problem – the longer

[5] R. Roy, ‘‘The Discovery of the Series Formula for Pi by Leibniz,

story’’, in Euler at 300: An Appreciation, R. E. Bradley, L. A. D’Antonio, and C. E. Sandifer, eds., The MAA Tercentenary

Gregory and Nilakantha’’, Math. Mag. 63, 5 (1990), 291–306. [6] P. Sta¨ckel, ‘‘Eine vergessene Abhandlung Eulers’’, Bibl. Math.

Euler Celebration, The Mathematical Association of America,

(3)8 (1907-1908), 37–54.

2007, 105–118. [4] D. T. Whiteside, ed., The Mathematical Papers of Isaac Newton 1674-1684, Cambridge University Press, Cambridge, 1971.

AUTHOR

................................................................................................................. PAUL LEVRIE obtained a doctorate in mathematics from the Catholic University of Leuven in 1987 in the field of numerical analysis. Since then he has been busy teaching and trying to find easy proofs of well-known mathematical results. He coauthors a Dutch blog entitled Wiskunde is Sexy (Mathematics is Sexy), an attempt to make Mathematics more popular in Flanders. For some time now he has been working on the problem of how to get an apparently infinite number of mathematics books into a finite house.

Department of Applied Engineering Karel de Grote University College Hoboken, Antwerpen Belgium Department of Computer Science K.U. Leuven Heverlee, Leuven Belgium e-mail: [email protected] 32


Mathematical Entertainments

Michael Kleber and Ravi Vakil, Editors

Minimizing the Footprint of Your Laptop (On Your Bedside Table) BURKARD POLSTER

Mathematical Laptops and Bedside Tables We assume that both the laptop and the bedside table are rectangular, and we will refer to these rectangles as the laptop and the table. We further assume that the center of gravity of the laptop is its midpoint. Finally, without loss of generality, we may assume that the laptop is 1 unit wide.1 We are considering all placements of the laptop such that it will not topple off the table; these are exactly the placements for which the midpoint of the laptop is also a point of the table. We are then interested in determining for which of these placements the footprint of the laptop is of minimal area; here, the footprint is the common region of the laptop and the table.

1

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, surprising, or appealing that one has an urge to pass them on. Contributions are most welcome.

Figure 1. No (stable) placement of your laptop on a bedside table has a smaller footprint.

often work on my laptop in bed. When needed, I park the laptop on the bedside table, where the computer has to share the small available space with a lamp, books, notes, and heaven knows what else. It often gets quite squeezy. It finally occurred to me, being regularly faced with this tricky situation, to determine once and for all how to place the laptop on the bedside table so that its ‘‘footprint’’—the area on which it touches the bedside table—is minimal. In this note I give the solution to this problem, using some pretty elementary mathematics.

I

â

Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil, Stanford University, Department of Mathematics, Bldg. 380, Stanford, CA 94305-2125, USA e-mail: [email protected]

In all reasonable circumstances, the optimal answer to this problem will always resemble the arrangement in Figure 1.2 This optimal placement is characterized by the fact that the midpoint of the laptop coincides with one of the corners of the table and the footprint is an isoceles right triangle. The proof is divided into two parts. First, we consider those placements for which the midpoint of the laptop coincides with one of the corners of the table: we prove that among such placements our special placement has smallest footprint area. Then, we extend our argument, proving that any placement for which the laptop midpoint is not a table corner must have a greater footprint area.

1 That is, the shorter side of the laptop is 1 unit in length. And, if the laptop is square then it is a unit square. Yes, yes, only a mathematician would consider the possibility of a square laptop, but bear with me. As will become clear, considering square laptops provides an elegant key to our problem. 2

‘‘Reasonable circumstances’’ means in reference to laptops and tables of relative dimensions close to those of the real items. In the nitty gritty of this note we’ll specify the exact scope of our solution, and also what happens in some unrealistic but nevertheless mathematically interesting scenarios.


33

DOI 10.1007/s00283-009-9089-z

Balancing on a Corner We begin by considering a right-angled cross through the center of a square, as illustrated in the left diagram in Figure 2. Whatever its orientation, the cross cuts the square into four congruent pieces. This shows that if we place a unit square laptop on the corner of a sufficiently large table, its footprint will always have area 1/4, no matter how the square is oriented; see the diagram on the right.

Our previous argument shows that, as we rotate the laptop, the area of the blue part stays constant. On the other hand, the red part only vanishes in the special position shown on the right. We conclude that this symmetric placement of the laptop uniquely provides the footprint of least area. These arguments required that the table be sufficiently large. How large? The arguments work as long as the rotated square never pokes off another side of the table. So, since the short side of the laptop has length 1, we only require that pffiffiffi the shortest side of the table be at least of length 1= 2; see Figure 4.

1

Figure 2. A square laptop with midpoint at a corner will have a footprint area of 1/4.

Next, consider a non-square laptop with its midpoint on the corner of a large table, as in Figure 3. We regard the footprint as consisting of a blue part and a red part, as shown. Figure 4. Our corner argument works for relatively small tables.

In the Corner is Best

Figure 3. The blue regions have the same area, and so the right footprint is smaller.

AUTHOR

......................................................................... is the author of a number of books, including A Geometrical Picture Book, The Mathematics of Juggling, Q.E.D: Beauty in Mathematical Proof, The Shoelace Book, and Eye Twisters. Currently living in Australia, Burkard serves as Monash University’s resident mathematical juggler, origami expert, bubble-master, and mathemagician. Together with his colleague Marty Ross he writes a weekly mathematical column for the newspaper The Age in Melbourne. When he is not doing funmathematics, he has fun investigating perfect mathematical universes.

BURKARD POLSTER

School of Mathematical Sciences Monash University, Melbourne Victoria 3800, Australia e-mail: [email protected] URL: www.QEDcat.com 34


We now want to convince ourselves that the minimal footprint must occur for one of these special placements over a table corner. We start with a table that is at least as wide as the diagonal of the square inscribed in our laptop; see the left diagram in Figure 5. Place the laptop anywhere on the table. Now consider a cross in the middle of the laptop square, and with arms parallel to the table sides. As we saw above, the cross cuts the square into four congruent pieces. Furthermore, wherever the laptop is placed and however it is oriented, at least one of these congruent pieces will be part of the footprint: this is a consequence of our assumption on the table size. Finally, unless the midpoint is over a corner of the table, this quarter-square region clearly cannot be the full footprint.

Figure 5. If the table contains the square highlighted on the left, then at least one of the quarters of the square on the right is contained in the footprint of the laptop.

Putting everything together, we can therefore guarantee that our symmetric corner arrangement is optimal if the table is at least as large as thepsquare table in Figure 5. This ffiffiffi square table has side length 2: By refining the previous arguments, we now want to show that our solution holds for any table that is at least 1 unit wide. Since our laptop is also 1 unit wide, this probably takes care of most real-life laptop balancing problems. Begin with a circle inscribed in the laptop square, and with the red and the green regions within, as in Figure 6. The regions are mirror images, and each is arranged to have area 1/4. Note that if the laptop is rotated around its midpoint, each fixed region remains within the laptop.

At this point we summarize what we have discovered so far.

T H E O R E M 1 Consider a laptop that is 1 unit wide and a table that is at least 1 unit wide. If the laptop is not a square, then the placement of the laptop on the table that gives the smallest footprint is shown in Figure 1. If the laptop is a square, then the minimal area footprints are for placements for which the midpoint of the laptop coincides with a corner of the table. Odds and Ends What if you are the unlucky owner of a small bedside table? Well, if your table is really tiny, the footprint will always be the whole table, as shown in the left diagram of Figure 8.

Figure 6. Both the red and the green regions have the critical area of 1/4.

Now place the laptop on the table with some orientation. Suppose that the laptop footprint contains a red or a green region, or such a region rotated by 90, 180, or 270 degrees; see Figure 7. Then it is immediate that the footprint area for the laptop in that position is greater than 1/4. In fact the footprint may not contain such a region. However, this will be the case unless the laptop midpoint is close to a table corner, in one of the little blue squares pictured in Figure 7. On the other hand, if the midpoint is in a blue square then the footprint will contain one of the original quarter-squares of area 1/4; see the diagram on the right side of Figure 7.

Figure 7. The footprint area is at least 1/4, for the laptop midpoint in either the blue or brown region.

Figure 8. Balancing the laptop on tiny tables.

Suppose that no matter how the table sticks out from underneath the laptop, the protruding part is always a triangle (middle diagram). Then it is easy to see that the minimal footprint will correspond to a placement of the midpoint of your laptop on a corner of the table. For a square laptop, the minimal footprint then occurs when the protruding triangle is isosceles. For other tables this need not be the case. For example, the only way a corner of the thin table shown on the right can stick out is if the table diagonal is almost perpendicular to the long side of the laptop. This precludes an isosceles triangle part of the table sticking out. For slightly larger tables, things get even more complicated, with odd-shaped footprints entering the picture; there is no easy way to see why the best placement should be among the placements for which the midpoint of the laptop is one of the corners of the table, etc. We end this note with two challenges for the interested reader: 1) Extend our theorem to include all tables that are larger than the table shown in Figure 4. And, if this is too easy ... 2) Prove the Ultimate Laptop Balancing Theorem, that includes everything that your lazy author did and did not cover in this note: arbitrary location of the center of gravity, starshaped laptops and jellyfish-shaped tables, higher-dimensional tables and laptops, etc. Have Fun, and Good Luck!


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Years Ago

David E. Rowe, Editor

Personal Reflections on Dirk Jan Struik JOSEPH W. DAUBEN

Years Ago features essays by historians and mathematicians that take us back in time. Whether addressing special topics or general trends, individual mathematicians or ‘‘schools’’ (as in schools of fish), the idea is always the same, to shed new light on the mathematics of the past. Submissions are welcome.

â

Send submissions to David E. Rowe, Fachbereich 08, Institut fu¨r Mathematik, Johannes Gutenberg University, D-55099 Mainz, Germany. e-mail: [email protected]

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DOI 10.1007/s00283-011-9213-8

Editor’s Note: Dirk Struik and the History of Mathematics Dirk Jan Struik, who taught for many years at the Massachusetts Institute of Technology and died on 21 October 2000 at the age of 106, was a distinguished mathematician and influential teacher. His early work on vector and tensor analysis, undertaken with Jan Arnoldus Schouten, helped impart new mathematical techniques needed to master Einstein’s general theory of relativity. This collaboration lasted more than twenty years, but by the end of the 1930s Struik came to realize that the heyday of the Ricci calculus had passed. After the Second World War, having now entered his fifties, he gave up mathematical research in order to focus his attention on the history of mathematics and science. It was through his work as an historian that he left a truly lasting mark, not only as a writer but as a mentor to those who had the pleasure of knowing him personally. This column may help convey an impression of how he touched the lives of so many people. As an historian, Dirk Struik saw culture, science, and society as tightly intertwined. Mathematics, his first love, was deeply embedded in the cauldron of cultures, not as something freely imported from without, but rather built from within as a product of human intellectual and social activity. Taking this approach, he tried to ferret out the links between scientific ‘‘high culture’’ and the work of artisans, technicians, and the myriad other practitioners who represent the ‘‘applied science sector’’ within a society’s workforce. In Yankee Science in the Making (1948) Struik analyzed the local social, geographical, and economic forces that shaped the lives of those inventors and amateurs who contributed to the emergence of a new scientific culture in Colonial New England. A similar focus on local conditions animates his book on early modern Dutch science, The Land of Stevin and Huygens (Engl. Trans. 1981, Het land van Stevin en Huygens, 1958). Struik’s sensitivity to the inner workings of scientific and technological cultures was a hallmark of his scholarship that helps to account for its enduring value for historians (Stapleton 1997). Struik was a leading Marxist scholar and social activist, but his Marxism did not dictate his historical analyses, the best of which were guided by an intuitive grasp of the salient features that led to the formation of distinctive scientific cultures. Like his friend Jan Romein, he sought global patterns of development; both avoided reifying Marxist ideas or taking a reductionist approach to historical materialism. Inspired by the work of figures such as J. D. Bernal, J. B. S. Haldane, Lancelot Hogben, and others in the British tradition of the Social Relations in Science Movement, he became increasingly interested in the social underpinnings of scientific knowledge. Multiculturalism would only become a buzzword decades later, but Struik’s scholarship already

reflected a deep awareness of how ideas are molded and transported in a complex mix of cultural contexts. The bestknown example is his Concise History of Mathematics, first published in 1948, which went through four English editions and has been translated into at least eighteen languages. It would be fair to say that no historical survey has done more to promote interest in the rich diversity of mathematical ideas and cultures. Many readers of The Mathematical Intelligencer will have their own stories of Dirk Struik. Marjorie Senechal, perusing this article in draft form, fondly remembered reading Struik’s work as a young college student. She recalled how her grandparents, themselves good communists, gave her an early edition of Yankee Science, which first came out under a different title with a left-wing publisher (The Origins of American Science (New England), New York: Cameron, 1957). Years later she got to know the author personally when she came across a paper he had written, in Dutch, about Aristotle’s mistaken claim that regular tetrahedra fill space. Aristotle’s authority was such that his assertion caused mathematicians from antiquity through the Renaissance to twist themselves into knots trying to prove he was right—a fascinating story. Having just returned from a sabbatical in Holland, she had pretty good Dutch, so she wrote and asked him for permission to translate the paper for publication. He, of course, agreed and so they worked on it together. The Mathematics Magazine agreed to publish it, and she brought the problem up to date; ‘‘Which Tetrahedra Fill Space?’’ (Senechal 1981) was the result. Struik inscribed her copy of Yankee Science. In my own case, I had the great pleasure of knowing Dirk during the last fifteen years of his life. Initially our mutual interest in the history of mathematics brought us together. But like so many young people—Dirk thought anyone under 70 was young—I quickly became fascinated with his whole life. One of my favorite memories is of a talk he gave in Mainz in 1994. This was right after the wonderful centennial celebration in Amsterdam, which already had him in high spirits. We drove down to Mainz where I introduced him to a packed audience, easily 300 people. He liked it when I introduced him not as a member of the ‘68 generation but the generation of 1917. His topic that day was simply ‘‘Some Mathematicians I Have Known.’’ The whole while this amazing centenarian stood at the podium, a few note cards in hand, and proceeded to tell a string of fascinating anecdotes. He began in English, but kept slipping into German. Then he would catch himself and start speaking in English again. It was a wonderful example of how naturally he connected with people in so many languages and cultures, always with an immense sense of good will. He took lots of questions after his talk; people were sitting in the aisles, and I didn’t notice anybody leaving. That evening he kept a small group of my students spellbound with his stories. A few of the young ladies present got extra attention, and when they told him about their own ongoing work in the history of mathematics he got that twinkle in his eyes that made all of us feel special and glad to be around him. As he grew older, Dirk was often asked, of course, about the secret to his longevity. Those who knew him realized it had nothing to do with abstention from life’s little vices

Figure 1. Dirk Jan Struik (1894-2000) in the mid-1970s.

(a pipe of tobacco and a glass of sherry were part of his daily regimen). Characteristically, he attributed his good health and zest for life to the three pillars of his spiritual strength, ‘‘the 3 Ms’’: Mathematics, Marxism, and Marriage. He shared these passions with his wife of some seventy years, Ruth Struik, neé Ramler, herself a mathematician and a native of Prague; she died in 1993 at the age of 99. Dirk and Ruth Struik were political activists, deeply moved by the struggle against fascism in Europe and filled with high hopes that the Soviet Union’s socialist experiment would eventually triumph as a new model for human society. In 1934, after eight years in the United States, the Struiks became naturalized American citizens and began taking a

Figure 2. Dirk Struik with David Rowe toasting the good life in the Go¨ttingen Ratskeller, August 1989. The occasion marks his return to Go¨ttingen 63 years after his stay there as a Rockefeller Fellow. Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 2, 2011

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more active part in supporting various political causes. In the wake of the Nazi racial laws aimed at ‘‘purifying’’ the civil service, they both tried to help several European mathematicians find refuge in the United States. Dirk was also involved in the often bitter disputes among American leftists regarding the Spanish Civil War. During the Second World War years he worked for the Council of American-Soviet Friendship, and in 1944 he helped found the Samuel Adams School in Boston, a short-lived organization that nevertheless came under the eye of J. Edgar Hoover’s F.B.I. Dirk Struik had a boundless faith in the capacity of human beings to build a just society. He saw science and mathematics as liberating forces within society, but he also realized that the modern scientist has a responsibility to consider the social consequences of scientific research. In this regard, he was strongly influenced by his MIT colleague and friend Norbert Wiener, who refused to place his fertile mind at the disposal of government technocrats. Teaching always played an integral part in Struik’s academic life. During the 1940s he began offering an informal seminar on historical materialism and Marxism, a topic then outside the official curricula of universities throughout the country. George Mosse, who came to Harvard as a graduate student in the early 1940s and was eager to learn some Marxist theory, quickly found his way to Struik’s seminar (George L. Mosse, Confronting History, pp. 121, 136). He went on to become a

distinguished historian at the University of Wisconsin and a leading expert on the intellectual origins of fascism in Germany. Back in the 1930s, George Sarton pioneered studies of the history of science in the United States at Harvard. He regularly invited Dirk Struik to teach a special session of his Harvard seminar on the topic of history of mathematics. I. B. Cohen, then a graduate student at Harvard, later recalled how Sarton and Struik used to argue over the role of social factors in the history of mathematics (Sarton’s favorite counterexample was magic squares). Yet, their differences notwithstanding, both men shared a cultural approach to the history of science of universal scope. A generation later, a young man named Joseph Dauben came to Harvard to study history of science, en route to writing a doctoral dissertation on the mathematics and philosophy of Georg Cantor. Along the way, he too fell under the sway of Struik’s influence, a story he vividly recalls here. It gives me special pleasure to see his recollections appear in Years Ago, as I well remember the delight I took in reading Joe’s wonderful biography of Cantor when I was a graduate student in Oklahoma. Soon afterward, I wrote to him and I was even more delighted to receive a warm reply inviting me to study with him at the CUNY Graduate Center. So began my own journey into the field of inquiry that owes so much to the inspiration of Dirk Struik. D:E:R:

I

generally, or what a revolutionary, some might have even said what a radical, book it was. The Marxist approach Struik had adopted only became apparent and important to me just before I was about to meet Struik for the first time, when I was a graduate student in the newly founded Department for History of Science at Harvard. I was fortunate at Harvard to be part of a cohort of new graduate students, one of whom was Wilbur Knorr, and the two of us could not have been better prepared for our general examinations in history of ancient and medieval mathematics than by John Murdoch, and in modern mathematics than by Judith Grabiner, who had just finished her Ph.D.at Harvard with a thesis on the mathematics of Lagrange. In 1967-1968, as I was studying for these exams, I was again reading Struik, but with a much more sensitive eye to the political and social contexts that he made clear were sometimes as important for understanding the development of mathematics as were its purely internal developments. Thus I knew Dirk Struik quite well on paper, from his writings, but I had not as yet met him, and I was in fact unprepared when we did meet for the first time at one of the departmental Christmas parties held every year at the Harvard Faculty Club. I recall seeing the tall, gangly Struik discussing something with I. Bernard Cohen, who introduced us and then went off, leaving Struik and me to talk about history of mathematics. Struik had a way of making whoever he was with seem like the center of attention, at least his center of attention, and at that moment he wanted to know how I came

first encountered Dirk Struik through his Concise History of Mathematics, which I read when I was a senior in high-school. At the time I was taking an advanced calculus class at Pasadena High School in Southern California, and through a quirk of fate, the teacher had an evening class at the local junior college and invited me to present a lecture on the derivative, which, it was suggested, I might want to motivate with some historical background. Thus rather than my introduction to the history of mathematics being one of E. T. Bell’s questionable anecdotes, I was introduced at the outset to the subject through one of the adepts, someone with a true feeling for both historical and political sensibilities. Of course, I did not really appreciate what Struik was up to in his book at the time, but then I was only in high-school. Nevertheless, this says something about the readership Struik’s work enjoyed—reaching even to impressionable high-school students. I next read Struik’s book for a second time, and more thoroughly, in college (Claremont McKenna College, one of the Claremont Colleges) where I was taking a course on history of science taught by Granville Henry of the mathematics department. I was also writing my senior thesis on nonstandard analysis, overseen by Janet Myhre, and I found Struik’s book again helpful in explaining the early concerns for infinitesimals in the 17th century and the rise of epsilon-delta techniques to avoid them entirely in the 19th century. But again, I still had no real appreciation for what the Concise History had achieved for the history of mathematics

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to study the history of mathematics, and to be at Harvard. After a brief answer, I asked Struik about the Concise History; into how many languages had it been translated? I also confessed that his treatment of the 19th century in particular had given me a much broader panorama of interests to consider as I thought about the subject of my dissertation. At that point I had decided to write about Bernhard Bolzano, and, through John Murdoch, I had been in touch with Lubosˇ Novy´ at the archives of the Academy of Sciences in Prague about my working with him there. I had actually decided to divide my time between Prague and Vienna to work on Bolzano’s logic and his paradoxes of the infinite. But as I was enjoying a brief vacation in August 1968 in Southern California, to see my family there before setting off for Prague, news came of the Russian invasion of Czechoslovakia. That changed things considerably, and over the next year I rethought the subject of my dissertation. Back at Harvard, I realized that in all of my reading I had never found a detailed biography of Georg Cantor, founder of transfinite set theory. I had always been interested in set theory, and it was a subject that would at least force me to learn German, a language I could read but by no means use—with any fluency.

AUTHOR

......................................................................... JOSEPH W. DAUBEN is Distinguished Pro-

fessor of History and History of Science at Herbert H. Lehman College of The City University of New York and a member of the Ph.D. Program in History at the Graduate Center, CUNY. He received his Ph.D. in History of Science from Harvard University in 1972. His many books include Georg Cantor: His Mathematics and Philosophy of the Infinite and Abraham Robinson: The Creation of Nonstandard Analysis, A personal and Mathematical Odyssey. He also contributed the chapter on ‘‘Chinese Mathematics’’ to The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (edited by Victor J. Katz, Annette Imhausen, and Eleanor Robson). In 2005 he was the Zhu Kejun Visiting Professor at the Institute for History of Natural Sciences of the Chinese Academy of Sciences and now is an honorary member; this picture was taken in Beijing last year. Research Center for Humanities and Social Sciences National Chiao-Tung University Hsinchu City 30010 Taiwan e-mail: [email protected] The City University of New York, Lehman College and the Graduate Center New York, NY USA

This also seemed like a good choice since Erwin Hiebert had just come to Harvard from the University of Wisconsin. Hiebert was a specialist in the history of modern German physics, and that would be a good match with what I wanted to do in mathematics. Hiebert also suggested that since Struik was emeritus at MIT but still affiliated with the Department at Harvard for just such purposes as directing the odd thesis that might turn up, I should ask him about working with me on Cantor, so I approached him about serving as one of my thesis advisors. I am forever grateful that when I first brought up this possibility, he was enthusiastic and agreed to help however he could. Struik began by suggesting that I get in touch with Christoph Scriba, who was then Director of the History of Science group at the Technische Universita¨t in Berlin, and that I should also arrange to meet the Director of the Alexander von Humboldt Forschungsstelle at the German Akademie der Wissenschaften in East Berlin, Kurt-R. Biermann. I did, and subsequently spent a very productive year during 1970-1971, living what I considered the life of a double-agent, stationed in West Berlin while doing most of my archival work in East Berlin. From time to time I would write to Struik to inform him of my progress, and he would usually write back with a reference or two that he thought I should read as background or as a foil to how I might otherwise have been thinking about a particular aspect of Cantor’s life and work. When I got back to Harvard, Struik invited me to his home in Belmont to provide a ‘‘full report’’ of the year in Berlin. As I arrived on his doorstep he asked me with a wink, ‘‘Is it still ‘eine Reise wert’?’’ This was one of the slogans often used to refer to Berlin, and I assured him that it was definitely ‘‘worth a visit.’’ Actually, I considered the year I had spent in Berlin as my ‘‘Berthold Brecht Zeit,’’ given that I was living in a coldwater flat with an octogenarian former singer from the German Opera above what can best be described as a bar that was a local conduit to the nearest brothel. Struik was delighted to hear of all this, and we spent that afternoon, a lazy day in late September, swapping stories about European capitals, his stories better than mine and full of names well known to every mathematician. But he was genuinely interested in Berlin and how it was faring with the wall up and the city divided. I then spent regularly at least one afternoon each month visiting Struik delivering, chapter-by-chapter, my thesis as it began to unfold. I remember in particular one afternoon in the early spring—I had started delivering chapters to Struik in October, and by March I had gotten to Chapter VI. My writing was accelerating, and I had reached the point of Cantor’s major mid-career work, the Grundlagen, which was his first large-scale introduction to set theory and transfinite numbers, although at that point only the transfinite ordinal numbers had been worked out and the transfinite alephs were another decade in coming. Struik was surprised by that—and wanted to know why Cantor had chosen alephs when he did come to introduce the transfinite cardinal numbers in the 1890s. The answer to that question was one that amused Struik to no end, because the answer was social, political, and in a sense economic as well, a nice Marxian trio as he put it to me at the time, and I had come to see it that way as well. As Struik also said, he had always thought Cantor had Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 2, 2011

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chosen the alephs for his transfinite numbers because he was Jewish. That turned out to be another fable. In fact, Cantor’s mother was Roman Catholic (his father’s side of the family most likely had its roots in the Jewish community of Copenhagen). Cantor, who was raised as a Lutheran and seems to have been very comfortable in his correspondence with Catholic theologians (a part of my thesis in which Struik was also particularly interested), seems not to have been a practicing Lutheran or a follower of any particular faith. E. T. Bell’s famous characterization of Cantor and Kronecker as the epitome of two antagonistic Jewish professors who were enemies to the death was even further from the mark. Their views on foundations were certainly at odds with each other, but they did their best to get on with one another, and at the end of both their careers, as Cantor was actively working to establish the German Union of Mathematicians, he invited Kronecker to be Union’s first keynote speaker in 1891. In any case, the true story about the alephs, which Struik particularly enjoyed, was basically pragmatic; Cantor knew that his transfinite numbers were special, and he wanted a special notation for the transfinite cardinals. As he told Giulio Vivanti (13 December 1893), all of the usual alphabets were taken, and letters from the Roman and Greek alphabets were too common in mathematics, whereas it would have been costly to design an entirely new symbol that most printers would not have on hand. But in Germany, virtually all printers had the Hebrew alphabet at their disposal. It occurred to Cantor that since the Hebrew aleph also represented the number one, it was the perfect choice for the first of his transfinite cardinal numbers. By March of 1972, despite his interest in what I had been writing, Struik was becoming worried that I wasn’t going to finish in time for a June degree. I assured him that I was writing at full speed and was certain I could finish the last two chapters (as I then envisioned them) in April or mid-May at the latest; one chapter was on Cantor’s ‘‘Beitra¨ge,’’ his last major statement of his set theory, including both the ordinal and cardinal transfinite numbers, and a concluding chapter was about his philosophy of mathematics, the paradoxes of set theory, and the slow but eventual acceptance of set theory and the new mathematics. Struik apparently wasn’t convinced, because within the week I received a phone call from my mentor at Harvard, I. Bernard Cohen. I was also the head TA for Cohen’s Scientific Revolution course at Harvard, for which he was justly famous. I thought Cohen was calling about the next meeting of the course when I heard him say: ‘‘Joe, I’ve heard something rather disturbing about your thesis.’’ This did not sound like good news, and so I asked him what the problem seemed to be. ‘‘I understand,’’ said Cohen, with a suitable pause for dramatic effect, ‘‘that it is getting rather long, and I’ve spoken with Hiebert and Struik and they both assure me that what you’ve written is plenty for your Ph.D.We think you should stop. Where are you?’’ ‘‘I’m at home,’’ I said, without really thinking. ‘‘No—I mean where are you in the thesis?’’ ‘‘Oh,’’ I replied, ‘‘More than halfway— I’ve only got two more chapters to go and I’ll be done.’’ Nevertheless, I did indeed stop where I was, more or less, and my thesis, instead of being a history of Cantor’s entire work, became instead ‘‘The Early Development of Cantorian Set Theory.’’ And so, thanks possibly to Dirk’s behind-the40


scenes intervention, I received my degree in June and began teaching at Herbert H. Lehman College of the City University of New York in the fall of 1972. But before leaving Cambridge for New York, I had one last afternoon on the porch with Dirk at his home in Belmont. With a cold bottle of Riesling, we sat and watched the sunset and talked about his many travels. Among the most nostalgic, he mentioned Rome, knowing that I had spent a part of the past summer at the American Academy in Rome where I had visited the mathematician Lucio Lombardo-Radice. Struik knew Lombardo-Radice and had instructed me to visit in hopes of getting permission through Lombardo-Radice to see letters between Cantor and Vivanti that were still in the hands of Vivanti’s family. Struik reminisced about the time he had spent there with his wife Ruth at the University of Rome in the 1920s. His memory was as accurate about the university and the friends he had made there as if he had only been away a few weeks, rather than decades, and he spoke of Rome as if it were an old friend. He could still see, smell, and feel its pulse as he talked about walks along the Tiber or coffee in the Piazza Navona. As I remember, I next saw Dirk not in Belmont, but in Hamburg, Germany, in the summer of 1989. The occasion was the XVIIIth International Congress of the History of Science in Hamburg, and Struik was to receive the first award of the Kenneth O. May medal for outstanding contributions to the history of mathematics—an award he was pleased to share with his old friend and colleague from the Soviet Union, the Russian historian of mathematics Adolf P. Yushkevich. As the Chairman of the International Commission on History of Mathematics, which had established the prize, it was my pleasure to make the actual presentation of the medals,

Figure 3. Struik’s handwritten remarks, beginning with a passage from Goethe, on accepting the K. O. May Medal, Hamburg, 3 August, 1989.

Figure 4. Dirk Struik, Hamburg, August 1989. Photo courtesy of the International Commission on History of Mathematics.

which, thanks to Christoph J. Scriba, former Chair of the ICHM and one of the co-organizers of the Congress in Hamburg, were presented on boats in the ‘‘Binnen Alster,’’ the inner harbor in Hamburg. At the appointed time, three boats carrying members and guests of the ICHM pulled alongside each other and were tethered together while the presentations and speeches were made. After accepting his award, Struik came to the microphone and talked about the importance of the history of mathematics and how the field had grown from the time he started working, when he was something of a lone figure (Figure 3). Indeed, when he wrote his Concise History, nothing like it had really ever been written before, with its level of political and social consciousness that sat very comfortably with the technical mathematical history he also had to tell. Over the years, whenever I was in Cambridge with time to spare, I would make a point of visiting Dirk Struik in Belmont. The last time I saw him was shortly after his 100th birthday. This was in the fall of 1995, just after publication of my biography of Abraham Robinson, and I wanted to give Struik a copy. I made the familiar trip out to Belmont and congratulated him personally for having passed his 100th birthday. I reminded him of the phrase with which he had ended his personal remarks for the Festschrift Robert Cohen published in his honor in 1974. After noting that he and his wife had just celebrated their 50th wedding anniversary and had 3 daughters and 10 grandchildren, he simply added: ‘‘Wish me luck.’’ I remarked about this that it seemed to have worked, to which he replied in his usually laconic way: ‘‘much better than expected!’’ Once again, as so many times before, we were sitting out on his porch, enjoying a late September afternoon in Belmont, and along with the book I had brought a bottle of Riesling. As Struik poured out two glasses, he asked me what I had been working on since Robinson, and I told him about a paper I was reworking about Charles Sanders Peirce and a Tiffany watch the Coast and Geodetic Survey had provided him. Once, on a trip to New York, it had been stolen, and Peirce, in accounting for how he managed to track it down,

used this as an example of his theory of abductive reasoning, which he compared to the methods of Sherlock Holmes. The case of his finding the Tiffany watch by a process of elimination and intuition, as Peirce put it, made clear that ‘‘when all other possibilities have been excluded, what remains, however improbable, must be true.’’ Whereupon Struik immediately launched into a detailed account of Holmes and the connection between Holmes and Watson, recalling details of an article he had written that, he said, compared Watson to Zeno of Elea. Both are known only through the writings of others; in this case, what we know of Holmes, Struik said, came mainly from Watson. He said he thought he had copies of the article and would be happy to give me one. He suggested we go upstairs to his study to find it. As he made his way to the second floor, he observed wryly that at his age he no longer went up stairs as quickly as he used to, with the added footnote that ‘‘you know, I’m nearly a hundred-and-one.’’ It took him a few minutes, but he indeed found the article in question—written in 1947! Back on the front porch as he sipped his glass of Riesling, Struik told me about his active membership in the Boston Holmes Society, the Specked Band of Boston, and pointed out that in fact, he was in very good company: Holmesian devotees included Ellery Queen, Basil Rathbone, Isaac Asimov, Franklin Roosevelt, and T. S. Eliot, among many others. He didn’t mention that his article, which later appeared in a collection of articles edited by Philip Shreffler, Sherlock Holmes by Gas Lamp (1989), not only suggested that Watson’s Holmes was akin to Aristotle’s Plato, but also covered a vast terrain of cultural information from Mortimer Snerd to the motets of Orlando de Lassus. But this was typical of Struik— he was interested in everything and everyone around him, and that is a large measure, I am sure, of his longevity. It is what kept him young. How remarkable, to have literally spanned a century, the entire 20th century and into the 21st. As historians of mathematics, we must be grateful that Dirk Struik made clear that our subject is not only of highly abstract, theoretical interest, but has very real, significant social roots. These—as he made apparent in one of the most widely read books on the subject—profoundly affect the societies in which mathematics today plays a pervasive role in virtually every aspect of the world in which we all live.

REFERENCES

Gerard Alberts, On connecting socialism and mathematics: Dirk Struik, Jan Burgers, and Jan Tinbergen, Historia Mathematica 21(3), 1994, 280-305. Teun Koetsier, Dirk Struik’s autumn 1994 visit to Europe: Including ‘My European extravaganza of October, 1994’ by Struik, Nieuw Archief voor Wiskunde (4) (1) 14, 1996, 167-176. David Rowe, Dirk Jan Struik and his contributions to the history of mathematics, Historia Mathematica 21(3), 1994, 245-273. David Rowe, Interview with Dirk Jan Struik, Mathematical Intelligencer 11(1), 1989, 14-26. Marjorie Senechal, Which Tetrahedra Fill Space?, Mathematics Magazine 54, 1981, 227-243. Darwin H. Stapleton, Dirk J. Struik’s Yankee Science in the Making: A Half-Century Retrospective, Isis 88(3), 1997, 505-511.


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John von Neumann, the Mathematician DOMOKOS SZA´SZ Imagine a poll to choose the best-known mathematician of the twentieth century. No doubt the winner would be John von Neumann. Reasons are seen, for instance, in the title of the excellent biography [M] by Macrae: John von Neumann. The Scientific Genius who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More. Indeed, he was a fundamental figure not only in designing modern computers but also in defining their place in society and envisioning their potential. His minimax theorem, the first theorem of game theory, and later his equilibrium model of economy, essentially inaugurated the new science of mathematical economics. He played an important role in the development of the atomic bomb. However, behind all these, he was a brilliant mathematician. My goal here is to concentrate on his development and achievements as a mathematician and the evolution of his mathematical interests.

The Years in Hungary (1903-1921) n 1802, the Hungarian mathematician Jańos Bolyai was born in Kolozsva´r (called Cluj today). Jańos Bolyai was the first Hungarian mathematician, maybe the first Hungarian scholar, of world rank. His non-euclidean hyperbolic geometry, also discovered at the same time independently by the Russian mathematician Lobachevsky, is now everywhere recognized, but could not be appreciated in his time in Hungary. He never had a job as a mathematician. In the early nineteenth century, Hungary was in the hinterlands of world mathematics. A century later, John von Neumann was born as Neumann Jańos on December 28, 1903, in Budapest. His family belonged to the cultural elite of the city. Not only were both of his parents’ families quite well-to-do, they understood how to use their wealth to live a rich and complete human life. But beyond all this, Hungary had changed dramatically during the 101 years since Bolyai’s birth. In 1832, Jańos Bolyai’s discovery was not understood by anyone in the country, except for his father. In contrast, Jańos Neumann’s exceptional talent was discovered very early and was nurtured by top mathematicians on such a level that it is hard to imagine better circumstances for a prodigy in any other part of the world.

I

Much has been written on the life of Neumann and his achievements. I have used extensively Norman Macrae’s exciting biography [M], which I recommend for the general reader. For the mathematician, I recommend the special issue 3 of Volume 64 (1958) of the Bulletin of the American Mathematical Society dedicated to von Neumann shortly after his death. In this issue, first of all, Stanisław Ulam, a lifelong friend and collaborator of Johnny, as he was called by his friends, gives a brief biography and concise mathematical overview (see also Ulam’s autobiography Adventures of a Mathematician). After this introductory paper, world experts in various fields place von Neumann’s achievements in a wider context. Excellent papers with a similar aim also appeared in the Hungarian periodical Matematikai Lapok, unfortunately in Hungarian only. I also mention the book of W. Aspray John von Neumann and the origins of Modern Computing [A]. I note that von Neumann’s works were edited by A. H. Taub [T]. A selection was edited by F. Bro´dy and T. Va´mos [BV]. Neumann’s father, Neumann Miksa (Max Neumann), a doctor of law, prospered as a lawyer for a bank. ‘‘He was a debonair, fourth-generation-or-earlier, non-practicing Hungarian Jew, with a fine education in a Catholic

Research supported by the Hungarian National Foundation for Scientific Research grants No. T 046187, K 71693, NK 63066, and TS 049835. This article is a version of the author’s speech at the conference in memory of John von Neumann held in Budapest, October 2003.

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DOI 10.1007/s00283-011-9223-6

provincial high-school. He was well attuned to fin-de-sie`cle Austria-Hungary and indeed to being a lively intellectual in any age. His party piece was to compose two-line ditties about the latest vicissitudes of his personal or business life, and about national and international politics.’’1 Jacob Kann, Neumann’s grandfather through his mother Kann Margit (Margaret Kann), ‘‘was a distinctly prosperous man. With a partner, he had built up in Budapest a thriving business in agricultural equipment that grew even faster in 1880-1914 than Hungary’s then soaring GNP.’’ The Kanns owned a rather big and nice house close to the Hungarian Parliament on the so-called Kisko¨ru´t, Small Boulevard. The Kanns and later the families of their four daughters lived there, while the ground floor was occupied by the firm Kann-Heller. When the families grew larger, they also rented the neighboring house. Jańos’s mother Margit was ‘‘a mother hen tending her children and protecting them. She was a family woman, a good deal less rigorous than her husband, artistically inclined, a wafer-thin and later chainsmoking enthusiast for comporting oneself with what she called ‘elegance’ (which later became Neumann’s highest term of praise for neat mathematical calculations such as the ones that made the H-bomb possible).’’ Max and Margaret had three children, of which Johnny was the oldest. ‘‘Grandfather Jacob Kann had gone straight

AUTHOR

......................................................................... ´ SZ DOMOKOS SZA

is a student of A. Pre´kopa, A. Reńyi, B. V. Gnedenko, and Ya. G. Sinai. He founded the Budapest schools in mathematical statistical physics and dynamical systems. His current ambition is the rigorous derivation of laws of statistical physics (diffusion, heat conduction) from microscopic assumptions. His other enthusiasms are his family (three children, two grandchildren), culture, and skiing. Mathematical Institute Budapest University of Technology and Economics H-1111 Budapest Hungary

from commercial high school into founding his business, but he proved demonic in his capacity for arithmetic manipulation. He could add in his head monstrous columns of numbers or multiply mentally two numbers in the thousands or even millions. The six-year-old Johnny would laboriously perform the computations with pencil and paper, and announce with glee that Grandfather had been absolutely on the mark. Later Johnny himself was known for his facility in mental computation, but he had long before persuaded himself that he could never match Jacob’s level of multiplication skill.’’ ‘‘Nursemaids, governesses and preschool teachers were an integral part of upper-middle-class European households in those days, especially in countries where (as in Hungary) children in such families did not start school until age ten. As the Kann grandchildren and some of Johnny’s second cousins came along, the building on the Small Boulevard became an educational institution in its own right. There was an especially early emphasis on learning foreign languages. Father Max thought that youngsters who spoke only Hungarian would not merely fail in the Central Europe then darkening around them, they might not even survive.’’ So Johnny (or Jancsi in Hungarian) learned German, French, English, and Italian from various governesses from abroad. Johnny later told colleagues in Princeton that as a six-year-old he would converse with his father in classical Greek. This seems to have been a joke, but his knowledge of languages was real. It not only made his communication easier in Germany, Zu¨rich, and in the US, but it may have imprinted at an early age axiomatic and abstract, algorithmic thinking into his brain. In such an environment, young Johnny was easily drinking in the knowledge surrounding him. One of his special interests was history: The family bought ‘‘an entire library, whose centerpiece was the Allgemeine Geschichte by Wilhelm Oncken. Johnny ploughed through all fortyfour volumes. Brother Michael was confounded by the fact that what Johnny read, Johnny remembered. Decades later friends were startled to discover that he remembered still.’’ Between 1914 and 1921, Johnny attended the Lutheran Gymnasium in Budapest. Here, too, he received a superb education. This was made possible by the fast economic and social progress after the compromise with Austria2 and subsequent cultural and educational reforms.3

1

Here and below, many unattributed quotes are from [M]. Some data from the late nineteenth-century history of Hungary. 1867: Appeasement with Austria and the formation of the Austro-Hungarian Monarchy after the defeat of the 1848 revolution. 1873: Formation of the city Budapest from smaller cities such as Buda and Pest. In the years afterward, Hungary and Budapest experienced extremely fast economic, social, and intellectual progress. Society was quite open, with ‘‘a flood of Jewish immigration into Hungary in the 1880s and 1890s as there was simultaneously to New York.’’ In 1896 a World Exposition was held in Budapest also commemorating the arrival of Hungarians to the Carpathians in 896. Hausmann-like reconstruction of the city contributed to its present character. For instance, the first subway of the European continent was inaugurated in 1896 in Budapest. ‘‘In 1903 the Elisabeth Bridge over the Danube was the longest single span bridge in the world.’’ For the interested reader I strongly recommend the book of John Lukacs [Luk]. 3 Some data from the nineteenth-century cultural progress of Hungary. End of nineteenth and first half of twentieth century: Development of national thought and nationalistic institutions, in particular, the foundation of the National Museum and the Hungarian Academy of Sciences. After the Appeasement of 1867, the educational system was transformed and modernized. (For some time the Minister of Education was Lorańd Eo¨tvo¨s, the famous physicist, inventor of the torsion pendulum). The Gazette of Sciences, called today World of Nature, was founded in 1869, that is, in the same year as the magazine Nature in Britain! The Mathematical and Physical Society was founded in 1891. (Since 1947 there are separate societies for Mathematics and Physics, called the Jańos Bolyai and the Lorańd Eo¨tvo¨s Societies, respectively; in 1968 the John von Neumann Computer Society was founded.) In 1894 Dańiel Arany founded the Ko¨ze´piskolai Matematikai Lapok, the oldest journal for high-school students; this journal still flourishes in an extended form including physics. In the same year 1894, the first Mathematical and Physical Competitions started (today the mathematical competitions are called Jo´zsef Ku¨rscha´k Competitions). 2


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As a result of these reforms the whole educational system was strong; for the talented (and let us also add, wealthy) pupils, there were e´lite high-schools. One of them was the Lutheran Gymnasium. It was quite liberal and had a number of Jewish pupils. Nobel Prize winners E. Wigner and later J. Harsańyi also attended this high-school. As a matter of fact, Wigner studied a year above Neumann, and already at school a respect and a friendship were formed between the two. Johnny’s teacher of mathematics was the celebrated La´szlo´ Ra´tz. Between 1896 and 1914, he was the chief editor of the high-school journal Ko¨ze´piskolai Matemat ika i Lapok. ‘‘Wigner and others recall that Ra´tz’s recognition of Johnny’s mathematical talent was instant.’’ Ra´tz agreed with Max Neumann that Johnny would be supervised by professional mathematicians from the universities. Thus Professor J. Kurschak (Ku¨rscha´k Jo´zsef) from Technical University arranged that Johnny would be tutored by the young G. Szego (Szeg} o Ga´bor) in 1915–1916. Later he was also taught by M. Fekete (Fekete Miha´ly) and Leopold Feje´r (Feje´r Lipo´t), and could also talk to A. Haar (Haar Alfre´d) and Fre´de´ric Riesz (Riesz Frigyes). All in all, the prodigy received education and supervision on the highest professional and intellectual level. In the years 1919–1921, when von Neumann graduated from the Lutheran Gymnasium, the mathematical competition for secondary schools did not take place because of the revolution in Hungary; but in 1918, Neumann was permitted to sit in as an unofficial participant, and would have won the first prize. The list of winners of the mathematical competitions before 1928 include among others L. Feje´r, Th. von Ka´rmań, D. Konig, A. Haar, M. Riesz, G. Szego, and E. Teller, whereas L. Szila´rd won a second prize. It is commonplace that around and shortly after the turn of the nineteenth and twentieth centuries Hungary exported a tremendous amount of brains to the West besides von Neumann, but it may be worth listing some of these scientists: Dennis Gabor (Ga´bor Deńes; Nobel Prize 1971), John Harsańyi (Harsańyi Jańos; Nobel Prize 1994), Georg von Hevesy (Hevesy Gyo¨rgy; Nobel Prize 1943), Theodore von Ka´rmań (Ka´rmań To´dor), Nicholas Kurti (Ku¨rti Miklo´s), Cornelius Lanczos (Lańczos Korne´l), Peter Lax (Lax Pe´ter; Wolf Prize 1987), George Olah (Ola´h Gyo¨rgy; Nobel Prize 1994), Michael Polanyi (Polańyi Miha´ly), George Po´lya (Po´lya Gyo¨rgy), Gabor Szego (Szeg} o Ga´bor), Albert SzentGyo¨rgyi (Szent-Gyo¨rgyi Albert; Nobel Prize 1937), Leo Szila´rd (Szila´rd Leo´), Edward Teller (Teller Ede), Eugene Wigner (Wigner Jeno¨; Nobel Prize 1963), and Aurel Friedrich Wintner. One might wonder about the ‘‘von’’s figuring in several of these names. Macrae explains the case of Neumann: ‘‘In 1913 the forty-three-year-old Max was rewarded for his services to the government: he received the noble placename Margittai with hereditary nobility, so that in German he and his descendants could be called ‘von Neumann.’ Ennoblement was not an unusual award for prominent bankers and industrialists during those last years of the Austro-Hungarian Empire. Many of the 220 Hungarian Jewish families who were ennobled in 1900–1914 (vs. just over half of that number in the whole century before) hastened to change their names. Ennoblement was a way through which one could seize the chance to call 44


The house on Kisko¨ru´t today. (Photo by Domokos Sza´sz)

himself less Jewish. Max Neumann deliberately did not change his name.’’

The Mathematics of John von Neumann My aim here should certainly be quite modest, and I can go into a detailed description neither of his achievements nor of their influence. My goal is to follow some of the main and continuing interests of John von Neumann. The selection of these interests is subjective and reflects my knowledge and judgement. Despite the breadth of his interests, the picture is not as complicated as it looks at first glance. This is especially true for the years before the War. It seems he usually had one main interest, and thought about other problems with the left hand, so to speak. This does not mean that the other results are less important, only that even a genius is subject to laws of nature. If one wants to reach repeated breakthroughs in a problem or more than one (cf. Poincare´), then a necessary condition is full concentration on it for quite a period. In short, the genius is known both by the difficulty of the problems he can solve with full concentration, and by the difficulty of the problems he can solve with perhaps less persistent concentration. Axiomatic Set Theory In the first part of this paper I emphasized that this exceptional prodigy received an optimal launch in Hungary. In addition, one should note that the excellent mathematical support in Budapest ensured his start as a scholar and as a researcher. Apart from his first paper with Fekete, his big interest was the axiomatization of set theory. It is most likely that he heard about this problem from a tutor in Budapest. Julius Ko¨nig (Ko¨nig Gyula) (the father of the graph-theoretician Denes Ko¨nig (Ko¨nig Deńes)) was himself also working on set theory, and in particular on the continuum hypothesis, but he died in 1913, before Neumann entered high-school. Anyway, mathematicians in Budapest were definitely aware of this circle of problems (J. Kurschak, Neumann’s mentor, for instance, was a colleague of J. Ko¨nig). Macrae writes: ‘‘The second of Johnny’s papers (about transfinite ordinals) had been prepared while he was still in high-school in 1921, although it was not published until 1923’’(see [vN23]).

The young Neumann Jańos. (Photo from http://www.ysfine. com.wigner/neum/ymath.jpeg)

His papers on set theory and logic were published in 1923 (1), 1925 (1), 1927 (1), 1928 (2), 1929 (1), and 1931 (2). It is known that when he read about Go¨del’s undecidability theorem in 1931, he dropped thinking seriously about this field. But there was another reason for that, too: starting at the latest from his scholarship in Go¨ttingen in 1926, his main interest had changed: it became centered around laying down the mathematical foundations of quantum mechanics and hence around functional analysis. Mathematical Foundations of Quantum Physics Hilbert certainly heard about the Hungarian Wunderkind quite early, and highly respected his results on set theory and on Hilbert’s proof theory. But in the mid-twenties, and very much in Go¨ttingen, quantum mechanics was in the center of attention. Hilbert himself was strongly attracted to this disputed science. One should not forget that Go¨ttingen was a center of quantum mechanics: Max Born (Nobel Prize 1954) and J. Franck (Nobel Prize 1925) were professors there between 1921 and 1933 and between 1920 and 1933, respectively, and W. Heisenberg (Nobel Prize 1932) and W. Pauli (Nobel Prize 1945) also spent some years there. In 1927, a long and important paper [HNvN27] on foundations of quantum mechanics of Neumann with Hilbert and L. Nordheim appeared. It also formulated a program that was later carried out in von Neumann’s book [vN32b]. The program became a success story in many ways. It meant a victory for the Hilbert-space approach. Moreover, it attracted the attention of mathematicians to the theory of operators and to functional analysis. Also, by translating the problems of quantum physics to the language of mathematics, it formulated intriguing questions, which either arose in physics or were generated by them by the usual process of mathematics. A substantial result of von Neumann was the spectral theory of unbounded operators,

generalizing that given by Hilbert for bounded operators. Also, the language of operator theory helped to reconcile the complementary and—apparently contradictory— approaches of Heisenberg and of Schro¨dinger. Von Neumann’s papers on the mathematical foundations of quantum mechanics and functional analysis appeared in 1927 (4), 1928 (5), 1929 (6), 1931 (2), 1932 (2), 1934 (2), 1935 (3), and 1936 (1). I add that physicists appreciate, in particular, his theories of hidden variables (more exactly, his proof of their nonexistence), of quantum logic, and of the measuring process (cf. Geszti’s article in [BV]). From the physical and also from gnoseological point of view I stress his disproof of the existence of hidden variables. The laws of quantum physics are by their very nature stochastic, basically contrary to the deterministic Laplacian view of the universe. Several scholars, including Albert Einstein (his saying ‘‘God does not throw dice’’ became famous), did not accept a nondeterministic universe. There was a belief that probabilistic laws are always superficial and that behind them there must be some hidden variables, by the use of which the world becomes deterministic. This belief was irreversibly repudiated by von Neumann.4 Von Neumann writes in 1947 in an article entitled The Mathematician [vN47], ‘‘It is undeniable that some of the best inspirations of mathematics—in those parts of it which are as pure mathematics as one can imagine—have come from the natural sciences.’’ It is certainly undeniable that having shown ‘‘his lion’s claws’’ in set theory, Neumann got to the right place: To elaborate the mathematical foundations of quantum mechanics was the best imaginable challenge for the young genius. The completion of this task led to a victorious period for functional analysis, the theory of unbounded operators. Von Neumann kept up this interest until his death. Peter Lax remembers, ‘‘I recall how pleased and excited Neumann was in 1953 when he learned of Kato’s proof of the self-adjointness of the Schro¨dinger operator for the helium atom.’’ [Lax]. Von Neumann, being a true mathematician, was also aware of the other fundamental motivation of a mathematician. In the same 1947 article [27] he starts, ‘‘I think it is a relatively good approximation to truth ... that mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure.’’ Then he continues, ‘‘But, once they are so conceived, the subject begins to live a peculiar life of his own and is better compared to a creative one, governed by almost entirely aesthetical motivations, than to anything else and, in particular, to an empirical science.’’ I can not resist adding one more idea of von Neumann about his favourite criterion of elegance: ‘‘One expects from a mathematical theorem or from a mathematical theory not only to describe and to classify in a simple and elegant way numerous and a priori special cases. One also expects ‘elegance’ in its ‘architectural’ structural makeup. Ease in stating the problem, great difficulty in getting hold of it and in all attempts at approaching it, then again some very surprising twist by which the approach becomes easy, etc. Also, if the deductions are lengthy or

4

I learnt from Peter Lax that von Neumann made a subtle error about hidden variables. It was fixed up by S. Kochen and E. P. Specker in J. of Math.& Mech. 17 (1967), 59-87. Editor’s note: This issue remains contentious, and will be revisited in a future issue of The Mathematical Intelligencer.


45

complicated, there should be some simple general principle involved, which ‘explains’ the complications and detours, reduces the apparent arbitrariness to a few simple guiding motivations, etc.’’

von Neumann Algebras In von Neumann’s hands, operator theory started to live an independent life in 1929. In that year he published a long paper [vN29] in Math. Ann., whose first half was his initial work on algebras of operators. This was a completely new branch of mathematics, whose study he continued in a couple of papers published in 1936 (3), 1937 (1), 1940 (1), and in 1943 (2) (these include his classic four-part series of works with F. J. Murray). Most likely von Neumann had no external motivation, for instance from physics, at least he does not say so in his first paper. He, partially with Murray, obtained his celebrated classification, leaving the classification of type III algebras open (this is what Alain Connes completed later). The creation of the theory of von Neumann algebras shows that his unrivalled knowledge, speed of thinking, and intuition led him to a brilliant discovery, whose real value only became evident three decades after its invention. One important idea was that the dimension of an algebra (or of a space) is strongly related to the invariance group acting on the object. The theory of von Neumann algebras started to flourish around the 1960s with many deep results, and, in particular, with the Tomita-Takesaki theory also revealing profound links of von Neumann algebras to physics. Not much later their relation to field theories was also discovered. Subsequently, this theory became deeper, and absolutely new connections and applicability were also discovered. In fact, at least four Fields Medals have been awarded for radically new results on or in connection with von Neumann algebras: to A. Connes in 1982, to V. F. R. Jones and to E. Witten in 1990, and to M. Kontsevich in 1998. It is intriguing that Jones’s discovery of the famous Jones polynomials, a topological concept related to knots/braids, was motivated by ideas from von Neumann algebras. The easier and more natural topological construction was afterward guessed by Witten and rigorously executed by Kontsevich. See [AI] for lectures of the first three medalists and by H. Araki, J. S. Birman, and L. Fadeev for their laudations. See also http://math.berkeley.edu/*vfr/. Despite the fact that the revolutionary progress of the theory of von Neumann algebras only occurred in the 1960s, von Neumann was conscious of their potential importance. In 1954, answering a questionnaire of the National Academy of Sciences, he selected as his most important scientific contributions 1) his work on mathematical foundations of quantum theory, 2) his theory of operator algebras, and 3) his work in ergodic theory.

‘‘Side’’ Interests, Ergodic Theory, and Game Theory Among Others Ergodic Theory. The appearance of ergodic theory in the aforementioned list is somewhat surprising. To be sure, von Neumann was well aware of the importance of the ergodic theorem for the foundations of statistical mechanics. (The

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ergodic hypothesis grew from ideas of Maxwell, Boltzmann, Kelvin, Poincare´, Ehrenfest, etc.) As a matter of fact, von Neumann found and proved the first-ever ergodic theorem: the L2 version in 1931 (which only appeared in 1932) [vN32a]. Then George Birkhoff established the individual ergodic theorem in 1932. Yet despite his numerous papers on ergodic theory (in 1927 (1), 1932 (3), 1941 (1), 1942 (1), and 1945 (1)), it seems to me that it was not one of his most persistent interests. Furthermore, he may have felt it as less than a triumph. When he heard about Birkhoff’s result ‘‘Johnny expressed pleasure rather than resentment, although he did kick himself for not spotting the next steps from his calculations that Birkhoff saw.’’ Progress in applying the notion of ergodicity to statistical mechanics, that is, showing that interesting mechanical systems are ergodic, has been rather slow. Only in 1970 could Sinai [S] find the first true mechanical system, that of two hard discs on the 2-torus, whose ergodicity he could prove; he conjectured that the situation is similar for any number of balls in arbitrary dimension. In 1999 Simańyi and I showed that typical hard-ball systems of N balls of masses m1 ; . . .; mN and radii r moving on the m-torus are hyperbolic; in 2006 Simańyi established ergodicity in general under an additional hypothesis: the Chernov-Sinai Ansatz. Other Interests. Von Neumann’s intelligence, culture, speed, motivation, and communication abilities were at play in many fields from the very start. He touched upon algebra, theory of functions of real variables, measure theory, topology, continuous groups, lattice theory, continuous geometry, almost periodic functions, representation of groups, quantum logic, etc. I suspect in a sense these were done ‘‘with his left hand.’’ Nevertheless, for his two-part memoir ([vN34] and [BvN35]) on almost periodic functions and groups, he received the prestigious Boˆcher Prize in 1938. He would have liked to construct invariant measures for groups, a goal actually completed by A. Haar in 1932. Von Neumann returned several times to this question, and the idea of the construction was also exploited in the construction of the dimension in his theory of operator algebras. Game Theory and Mathematical Economics. Two related left-hand topics deserve special mention: game theory and mathematical economics. It is well known that von Neumann’s two game-theory papers in 1928 (the more detailed one is [vN28]) contained the formulation and the proof of the minimax theorem (a model of symmetric, twoperson games had already been suggested by E´mile Borel in 1921, but he had doubts about the validity of a minimax theorem). Also, in 1937 von Neumann published a model of general economic equilibrium [vN37], known today by his name. These two sources were basic for his big joint enterprise with O. Morgenstern, the fundamental monograph [vNM44] Theory of Games and Economic Behavior. These topics were also the objects of his papers in 1950 (1), 1953 (3), 1954 (1), and 1956 (1) (and two further unfinished manuscripts that appeared in 1963). Judging from the fact that in 1994 three Nobel laureates in economics were named for achievements in game theory (J. Harsańyi, J. Nash, and Reinhart Selten), it is not too bold to surmise that had Neumann lived until this Nobel Prize was founded in 1969, he would have been the first to receive it.

Von Neumann’s Mean Ergodic Theorem

Von Neumann’s Minimax Theorem

Domokos Sza´sz

Andra´s Simonovits

Ludwig Boltzmann’s ergodic hypothesis, formulated in the 1870s, says: for large systems of interacting particles in equilibrium, time averages of observables are equal to their ensemble averages (i.e., averages w.r.t. the timeinvariant measure) (cf. D. Sza´sz, Boltzmann’s Ergodic Hypothesis, a Conjecture for Centuries? Hard Ball Systems and the Lorentz Gas, Springer Encyclopaedia of Mathematical Sciences, vol. 101, 2000, pp. 421-446.). For Hamiltonian systems the natural invariant measure is the Liouvillian one. In 1931, Koopman (B. O. Koopman, Hamiltonian systems and transformations in Hilbert space, Proc. Nat. Acad. Sci. vol. 17 (1931) pp. 315-318.) made a fundamental observation: If T is a transformation of a measurable space X keeping the measure l invariant, then the operator U defined by (Uf)(x) := f(Tx) is unitary on L2. This functional-analytic translation of the ergodic problem led von Neumann in the same year to finding his famous

Von Neumann made a number of outstanding contributions to game theory, most notably the minimax theorem of game theory, a linear input-output model of an expanding economy, and the introduction of the von NeumannMorgenstern utility function. Here I present the first of these. There are two players, 1 and 2, each has a finite number of pure strategies: s1 ; . . .; sm and t1 ; . . .; tn . If player 1 chooses strategy si and player 2 strategy tj, then the first one gets payoff uij and the second one -uij. To hide their true intentions, each randomizes his behavior by choosing strategies si and tj independently with probabilities pi and qj, respectively. For random strategies p = (pi) and q = (q j), the P Pnpayoffs are the expected values u(p, q) = m i=1 j=1 piqjuij and -u(p, q), respectively. An equilibrium pair is defined as (p, q) such that minq maxp u(p, q) = maxp minq u(p, q).

T H E O R E M 1 (von Neumann 1928) For any twoplayer, zero-sum matrix game, there exists at least one equilibrium pair of strategies.

MEAN ERGODIC THEOREM

T H E O R E M 1 (John von Neumann, Proof of the quasiergodic hypothesis, Proc. Nat. Acad. Sci. vol. 18 (1932) pp. 70-82.) For f 2 L2 ðlÞ the L2-limit lim

1

n!1 n

f þ f ðTxÞ þ þ f ðT n1 xÞ ¼ F ðxÞ

ð1Þ

exists. The T-invariant function F equals the L2-projection of the function f to Rthe subspace of T-invariant R functions, and moreover, Fdl ¼ fdl. Note that for ergodic systems, that is, for those where all invariant functions are constant almost everywhere, the last claim of the theorem is just the equality of (asymptotic) time and space averages. Shortly after von Neumann discussed this result with G. D. Birkhoff, the latter established his individual ergodic theorem (G. D. Birkhoff, Proof of the ergodic theorem, Proc. Nat. Acad. Sci. vol. 17 (1931) pp. 656660.) stating that in (1) the convergence also holds almost everywhere. (For a more detailed history of the first ergodic theorems see G. D. Birkhoff and B. O. Koopman, Recent contributions to the ergodic theory, Proc. Nat. Acad. Sci. vol. 18 (1932) pp. 279-282.) Von Neumann was also aware of the L2 form of his theorem: if U is any unitary operator in a Hilbert space H, then the averages n1 ð f þ Uf þ þ U n1 f Þ converge for every f 2 H.

Moving to the US Moving to the US in 1930–1931 apparently did not make a big change in von Neumann’s mathematics. For a time he did not face challenges such as encountering the revolution of quantum physics in Go¨ttingen. It is known that he had

REMARKS 1. Although another mathematical genius, E´mile Borel, studied such games before von Neumann, he was uncertain if the minimax theorem holds or not. 2. This theorem was generalized by Nash (1951) (Nobel Prize in Economics, 1994) for any finite number of players with arbitrary payoff matrices (J. Nash, ‘‘NonCooperative Games,’’ Annals of Mathematics 54, (1951) pp. 289–295.).

E X A M P L E . Let us consider the following game. Players 1 and 2 play Head (H) or Tail (T) in the following way. Each puts a coin (say a silver dollar) on a table simultaneously and independently. If the result is either HH or TT, then 1 receives the coin of 2, otherwise 2 receives the coin of 1. There is a single equilibrium: pi* = 1/2 = qj*, which can be achieved by simply throwing fair coins. http://www.econ.core.hu/english/inst/simonov.html trouble getting used to the higher publication-centeredness of the American mathematical community, and the lower level of informal communication as compared with his European experiences. The famous parties of the von Neumann family in Princeton, offering beyond alcoholic drinks both mathematical and intellectual communication, partly solved this problem. Events of his private life may have influenced his work. In the years around his divorce and second marriage, his productivity was less than his average (in 1938 he published just one work, and in 1939 none), but this is not essential. The proximity and then the beginning of the war, Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 2, 2011

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Johnny during his years in the United States. (Photo by Alan Richards, Fuld Hall in Institute for Advanced Study, Princeton)

however, brought a dramatic change in von Neumann’s mathematical interests and perhaps in his style. His hatred for Nazi Germany and his engagement in promoting victory over it had certainly pushed him toward essentially new questions. These were connected to new fields and new problems whose solution he hoped would contribute to the big goal. At the same time, these problems offered new mathematical challenges. War Effort: Ballistics and Shocks Ballistics. From the technological point of view, the war after the First World War was its continuation; the important work on improving firing tables of shells did not cease. (During the Second World War Kolmogorov also participated in similar work; see B. Booss-Bavnbek and J. Høyrup: ‘‘Mathematics and War: an Invitation to Revisit,’’ Math. Intelligencer 25 (2003), no. 3, 12-25). Oswald Veblen, who became the first professor of the Princeton Institute for Advanced Study in 1932, and was responsible for inviting von Neumann to Princeton and later to the Institute, had been the commanding officer of this research between 1917 and 1919 in the US Army Ordnance Office at Aberdeen Proving Grounds, Maryland. (Among others, the great mathematicians J. W. Alexander and H. C. M. Morse also worked there under his guidance.) In 1937, von Neumann, who from his knowledge of history and from experience had already expected a war in Europe, decided to join the work in Aberdeen. Details of his progressive involvement are again very nicely described in Macrae’s book, and I restrain myself from repetitions. The essential thing is that—because of the decreasing density of air with altitude—the equations governing the trajectories are nonlinear and cannot be solved exactly and even have new types of solutions. Later, especially during his 1943 visit to England, von Neumann also became interested in the dynamics of magnetic mines. His first publications about shock waves were i) an informal progress report to the National Defense Committee, which dates back to 1941, and ii) its detailed version [vN43]. As usual in von Neumann’s mathematics, this very first work already gave a 48


deep and broad overview, proved interesting new results, and provided a program, too. This report was then followed by an analogous report on detonation waves in 1942, and by further works in 1943 (3), 1945 (2), 1947 (1), 1948 (1), 1950 (1), 1951 (1), and 1955 (1). (A concise and nice description of von Neumann’s approach and results is given by Fritz in [BV].) Meteorology and Hydrodynamics. One of von Neumann’s favorite subjects was meteorology. He was quite optimistic that with progress in understanding the nature of aero- and hydrodynamic equations and of their solutions, and with the development of computation, a practically satisfactory weather forecast would become possible. His hopes were only partially realized. He was certainly aware of the chaotic nature of the solutions of hydrodynamic equations, but he probably did not see sufficiently clearly the limitations caused by chaos (one should not forget that E. Lorentz’s revolutionary work [Lor] appeared only in 1963, and its import was not appreciated for quite a while after that). The study of trajectories of shells, his study of the dynamics of magnetic mines in Britain in 1943, the work on hydrodynamics equations, and later his participation from 1943 in the Manhattan project, all of these projects brought home to him the fundamental importance of computations, and consequently that of the development of computers. His Memorandum written in March 1945 to O. Veblen on the use of variational methods in hydrodynamics [vN45b] starts with the sentence ‘‘Numerical calculations play a very great role in hydrodynamics.’’

Computers: Neumann-Architecture and Scientific Computing If the applied problems mentioned required a tremendous amount of computations, this appeared most spectacularly in Los Alamos, where for the work on the atomic bomb it was absolutely essential to replace experiments by mathematical modeling requiring a lot of computations. It is well known that in August 1944, on the railroad platform of Aberdeen, von Neumann by chance met Herman Goldstine, a main figure working in Philadelphia on the development of ENIAC. He immediately received a first lesson on the actual level of progress, soon got into the work, and presented groundbreaking new ideas (his first fundamental works were [vN45a] and [BGvN46]). So he not only became a devoted advocate of the project, but substantially participated in it in various ways; his two most important contributions were, first, the elaboration of the principles of the programmable computer, of the so-called von Neumann-computers, and second, the implementation of these principles in the construction of the computer IAS to be built at the Princeton Institute. This whole story is described in Aspray [A] and Legendi-Szentivańyi [LSz], for instance. Let me just mention some key words about von Neumann’s main implementations: RAM, parallel computations, flow-diagrams, program libraries, subroutines. I would say that if problems of hydrodynamics and of the construction of computers were a necessity and were in the air, so to speak, the importance of the use of computers for scientific research was not equally obvious. Von Neumann

On Reliable Computation Peter Gaćs Von Neumann’s best-known contribution to computer science is the conceptual design of general-purpose computers. Ideas of a stored-program computer had already been circulating at the time among engineers, but von Neumann’s familiarity with the concept of a universal Turing machine helped the clean formulations in John von Neumann’s ‘‘First draft of a report on the EDVAC’’ (Technical report, University of Pennsylvania Moore School of Engineering, Philadelphia, 1945). Control unit, arithmetic unit, and memory (which also holds the program) still form the major organizational division of most computers. The arithmetic and control units are modelled well as logic circuits with connections such as ‘‘and’’, ‘‘or’’, ‘‘not’’, and bit memory (von Neumann called these ‘‘artificial neurons’’ after McCulloch and Pitts). The report mentions the problem of physical errors, without proposing any solution. Von Neumann returned to the error correction problem in ‘‘Probabilistic logics and the synthesis of reliable organisms from unreliable components’’ (in C. Shannon and McCarthy (eds.), Automata Studies. Princeton University Press, Princeton, NJ, 1956). The most important theorem of this paper rich in ideas can be formulated as follows. Call a logic component ideal if it functions perfectly, and real if it malfunctions with probability, say, 10-6. Suppose that a given logic circuit C is built with N ideal components. Then one can build out of O(N log N) real components another logic circuit C 0 that computes, from every input, the same output as C, with probability C 0.99. In the solution, some parts (the so-called ‘‘restoring organs’’) of the circuit C 0 are built using a random permutation. This method was rather new at the time, though Erd} os’s random graphs and Shannon’s random codes existed already. The proof of the main theorem is somewhat sketchy: a rigorous proof appeared almost 20 years later (by Dobrushin and Ortyukov, using somewhat different constructions). There have been some more advances since (for example, making the circuits completely constructive with the help of ‘‘expanders’’), but the log N redundancy has not been improved in the general case. For a recent publication containing references see Andrei Romashchenko, ‘‘Reliable computation based on locally decodable codes’’ (in Proc. of STACS, LNCS 3884, (2006) pp. 537–548). Error-correcting computation, in the general setting introduced by von Neumann, has not seen significant industrial applications. Surprisingly, the decrease in the size of basic computing components (now transistors on a chip) has been accompanied with an increase in their reliability: the error rate now is one in maybe 1020 executions. This trend has physical limits, however. (Von Neumann himself contemplated these limits in a lecture in 1949. He was not the first one, Szila´rd probably preceded him, and the nature of these limits has been a subject of

lively discussion among physicists ever since.) Restricting attention to logic circuits sidesteps the problem of errors in the memory and the issue of the bottleneck between memory and control unit. There are parallel computing models without division between memory and computation: the simplest of these, cellular automata, was first proposed (for other purposes) by von Neumann and also by Ulam. By now, universal reliable cellular automata have also been constructed. http://www.cs.bu.edu/*gacs saw this very quickly, his brain was completely prepared for it, and he devoted a substantial amount of energy to it. Reading his papers on this theme, one feels that he was not only aware of the great perspectives that computers had opened up, he was at the same time discovering and enjoying the new horizons with a lively, childish curiosity. A simple ‘‘game’’ was, for instance, to calculate the first 2000 decimal digits of e and of p by the ENIAC in 1950. Aspray discusses von Neumann’s accomplishments in scientific computing in detail. For the record I add that about the principles of construction of computers he had reports in 1945 (1), 1946 (1), 1947 (1), 1948 (1), 1951 (2), 1954 (2), and 1963 (1). On scientific computing he had the first paper, joint with V. Bargmann and D. Montgomery [BMvN46], in 1946 (1), on solving large systems of linear equations by computers. It was followed by works in 1947 (4), 1950 (3), 1951 (2), 1953 (1), 1954 (1), and 1963 (2). He made computer experiments in number theory, ergodic theory, and stellar astronomy. He also had several suggestions for random-number generators (via the middle-square method or the logistic map), and, with Ulam, he was also working on inventing the Monte-Carlo method. (I emphasize that the publication counts are only for providing a feeling. The topic of a work is sometimes ambiguous; some manuscripts only appeared after his death; and some reports may have appeared as articles (I have not checked overlap).) Had he lived longer, he certainly would have greeted the amalgamation of computation with science and would have

Commemoration of the creation of the EDVAC computer. (Credit: http://www.computer-stamps.com/country/22/Hungary/) Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 2, 2011

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been one of the leaders. This is certainly true for computerassisted proofs. He would have been glad to see the birth of computational sciences: computer science, computational physics, chemistry, etc. He perhaps would not have been so glad to see that often computations replace creative and logical thinking, but this is only a guess. Last, but not least, I want to turn to a very intriguing interest of von Neumann. Having analyzed the functioning of computers, he formulated several related questions. He initiated research on cellular automata and probabilistic automata (i.e., automata with unreliable components). He raised the question of constructing self-correcting automata. Moreover, he envisioned the need to compare the functioning of computers with that of our brains. His articles on the subject appeared in 1958 in the collection [vN58], and the analysis of his thoughts on the topic would deserve a separate discussion.

Epilogue All in all, fate, often combined with his own decisions, put before John von Neumann the most extraordinary and diverse scientific challenges. In all cases, he did his job in an ingenious and superb way. Factors such as his genes, family, education, the intellectual ferment that surrounded him in Budapest, and the intellectual level of his teachers and of Hungarian mathematics of the time, gave him an excellent launch. Besides his intelligence, knowledge, deductive power, and fantastic mental speed, he showed depth, perspective, and taste; his diverse scientific and human interests, complemented by his education in chemical engineering, contributed to his unrivalled courage, openness, and flexibility. By excelling in an unusually broad spectrum of mathematical activities, he became an outstanding representative of twentieth-century mathematics, whose influence was unbelievably wide within and outside mathematics. Von Neumann’s achievements demonstrate convincingly the strength of the mathematical approach, ‘‘unreasonable effectiveness of mathematics,’’ as Wigner put it. It is not overstatement to compare the achievements of John von Neumann to those of Archimedes, Newton, Euler, or Gauss. Von Neumann was exceptionally widely known among mathematicians, and there are plenty of anecdotes related to him. I think that as a student, I heard from my professor A. Reńyi the saying: ‘‘Other mathematicians prove what they can, Neumann what he wants.’’ (It fits, though I have mentioned that he was not happy that it was not himself who found Go¨del’s incompleteness theorem or Haar’s construction of the invariant measure of a locally compact group or Birkhoff’s proof of the individual ergodic theorem.) Not long ago, Dan Stroock from MIT mentioned to me the following at least half-serious opinion: ‘‘A genius either does things better than other people, or does them completely differently (orthogonally) from other people.’’ I think it is true about von Neumann that he did most things better than other mathematicians would have done. An example of an ‘‘orthogonal’’ scholar could well be Einstein. Nevertheless, I think of times when von Neumann also did things orthogonally. Let me finish by citing a well-known anecdote. Question to Wigner: ‘‘How is it that Hungary produced so many 50


geniuses in the early 20th century?’’ Wigner’s answer: ‘‘That many geniuses? I don’t understand the question. There was only one genius: John von Neumann.’’

Brief Biography 1903, December 28: Born in Budapest 1914-1921: Lutheran Gymnasium, Budapest (Teacher of maths: La´szlo´ Ra´tz. Tutored by Ga´bor Szego; afterward by M. Fekete and L. Feje´r.) 1921 Winter term - 1923 Summer term: Berlin University, student in chemistry, attends lectures in physics and mathematics (Professor of Mathematics: Erhard Schmidt, Professor of Chemistry: Nobel Laureate Fritz Haber). 1923 Summer term - 1926 Summer term: ETH, Zu¨rich, student in chemical engineering, attends lectures in physics and mathematics, too (Hermann Weyl, Gyo¨rgy Po´lya); October 1926: Diploma in chemical engineering. 1921 Winter term - 1925 Summer term: registered student of mathematics at Budapest University; March 1926: Ph.D. in mathematics (Supervisor: L. Feje´r?). 1926 Fall term - 1927 Summer term: visiting Go¨ttingen University, grant from Rockefeller Foundation (D. Hilbert, L. Nordheim, R. Courant, K. Friedrichs, P. Jordan, future Nobel Laureate physicists Max Born, J. Franck, W. Pauli, W. Heisenberg). 1927 Fall - 1929: Privatdozent at Berlin University. 1929: Privatdozent at Hamburg University. 1930 - 1938: Marriage with Marietta Ko¨vesi. 1930: Visiting Lecturer at Princeton University. 1931 - 1933: Visiting Professor at Princeton University. 1933 - 1957: Professor, Member of the Institute for Advanced Study, Princeton. 1935: Daughter Marina von Neumann born. 1938: Marriage with Kla´ri Dań. 1957, February 8: Dies in Walter Reed Hospital, Washington, buried in Princeton Cemetery A complete bibliography of John von Neumann can be found on the Internet: http://www.info.omikk.bme.hu/ tudomany/neumann/javnbibl.htm. Most papers, and manuscripts are in [T], and some in [BV]. ACKNOWLEDGMENTS

I express my sincere gratitude to Istvań Hargittai and Peter Lax for their most valuable remarks and to P. Gaćs and Andra´s Simonovits for their inserts. Special thanks are due to Re´ka Sza´sz for polishing my English and to J. Fritz, I. Juha´sz, D. Petz, L. Rońyai, A. Simonovits, and A. Szenes for stimulating discussions.

REFERENCES

[A]

W. Aspray, John von Neumann and the Origins of Modern Computing. MIT Press, Cambridge, MA, 1990.

[AI]

Sir M. Atiyah, D. Iagolnitzer (eds.), Fields Medallists’ Lectures.

[BV]

World Scientific, Singapore, 1997. F. Bro´dy, T. Va´mos (eds.), The Neumann Compendium. World Scientific, Singapore, 1995.

[Lax] P. Lax, Remembering John von Neumann. Proc. of Symposia in Pure Mathematics, 50:5-7, 1990. [LSz] T. Legendi, T. Szentivańyi, Leben und Werk von John von

[vN34]

John von Neumann, Almost Periodic Functions on a

[BvN35]

Group. Trans. Amer. Math. Soc. 36:445-492, 1934. S. Bochner, John von Neumann, Almost Periodic Func-

Neumann. Mannheim, 1983.

tions on Groups, II. Trans. Amer. Math. Soc. 37:21-50,

[Lor] E. N. Lorentz, Deterministic Nonperiodic Flow. J. Atmos. Sci., 20:130-141, 1963.

1935. [vN37]

[Luk] J. Lukacs, Budapest 1900: A Historical Portrait of a City & Its

system und eine Verallgemeinerung des Browerschen

Culture. Grove Press, 1988.

Fixpunktsatzes. Erg. eines Math. Coll, Vienna, ed. by K.

[M]

N. Macrae, John von Neumann. The Scientific Genius. Pan[vN42]

[S]

theon Books, New York, 1992. Ya. G. Sinai, Dynamical Systems with Elastic Reflections. Russian Math. Surveys, 25:139-189, 1970.

[vN43]

[T]

Menger 8:73-83, 1937.

A. H. Taub, John von Neumann: Collected Works I-VI. Pergamon Press, Oxford, 1961-1963.

John von Neumann, U¨ber ein o¨konomisches Gleichungs-

John von Neumann, Theory of Detonation Waves. Progress Report, 1942. John von Neumann, Theory of Shock Waves, Progress Report, 1943.

[vNM44]

John von Neumann, O. Morgenstern, Theory of Games and Economic Behaviour. Princeton University Press, 1944.

WORKS BY VON NEUMANN:

[vN23]

Acta Sci. Math. Szeged, 1:199-208, 1923. [HNvN27] D. Hilbert, L. Nordheim, J. von Neumann, U¨ber die Grundlagen der Quantenmechanik. Math. Ann. 98:1-30, 1927. [vN28]

[vN45a] [vN45b]

John von Neumann, Zur Algebra der Funktionaloperatoren

Preliminary Discussion of the Logical Design of an Electronic Computing Instrument, Part I, 1946. [BMvN46] V. Bargman, D. Montgomery, John von Neumann, Solution of Linear Systems of High Order. Report, 1946.

und Theorie der normalen Operatoren. Math. Ann. 102: 370-427, 1929. [vN32a]

John von Neumann, Proof of the Quasi-Ergodic Hypoth-

[vN32b]

esis. Proc. Nat. Acad. Sci., 18:70-82, 1932. John von Neumann, Mathematische Grundlagen der Quantenmechanik. Springer, Berlin, 1932.

John von Neumann, Use of Variational Methods in Hydrodynamics. Memorandum to O. Veblen, March 26, 1945.

[BGvN46] A. W. Burks, H. H. Goldstine, John von Neumann,

John von Neumann. Zur Theorie der Gesellschaftsspiele. Math. Ann. 100:295-320, 1928.

[vN29]

John von Neumann, First Draft of a Report on EDVAC, pp. 48, 1945.

John von Neumann, Zur Einfu¨hrung der transfiniten Zahlen.

[vN47]

John von Neumann, The Mathematician, in The Works of the Mind, R. B. Heywood (ed.) University of Chicago Press, 180196, 1947.

[vN58]

John von Neumann, The Computer and the Brain (Silliman Lectures). Yale University Press, pp. 88, 1958.


51

The 1958 PekerisAccad-WEIZAC Ground-Breaking Collaboration that Computed Ground States of TwoElectron Atoms (and its 2010 Redux) CHRISTOPH KOUTSCHAN*

AND

DORON ZEILBERGER**

n order to appreciate how well off we mathematicians and scientists are today, with extremely fast hardware and lots and lots of memory, as well as with powerful software both for numeric and symbolic computation, it may be a good idea to go back to the early days of electronic computers and compare how things went then. We have chosen, as a case study, a problem that was considered a huge challenge at the time. Namely, we

I

1 looked at C.L. Pekeris’s [9] seminal 1958 work on the ground state energies of two-electron atoms. We went through all the computations ab initio with today’s software and hardware.

Schro¨dinger Let’s recall the (time-independent) Schro¨dinger equation for the state function (alias wave function) w(x, y, z) of a

*Supported by NFS-DMS 0070567 as a postdoctoral fellow, and by grant P20162 of the Austrian FWF. **Supported in part by the United States of America National Science Foundation. 1 available online from http://astrophysics.fic.uni.lodz.pl/100yrs/pdf/04/076.pdf (viewed May 15, 2010).

52


DOI 10.1007/s00283-010-9192-1

one-electron atom with a stationary nucleus (see, for example, [8] Eq. (30-1) with N = 1), in atomic units: 2 o o2 o2 Z wðx; y; zÞ ¼ 0; þ þ þ2 E þ r ox 2 oy 2 oz 2 where Z denotespthe nuclear charge, E the energy of the ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi system, and r ¼ x 2 þ y2 þ z 2 the distance of the electron to the nucleus. Schro¨dinger’s solution of this eigenvalue problem is one of the greatest classics of modern physics, familiar to all physics students (and chemistry students, but unfortunately not math students), using separation of (dependent) variables, and getting explicit and exact results for the eigenvalues (the possible energy levels E) and even for the corresponding eigenfunctions w. Because the eigenfunctions (or more precisely their squares) are interpreted as probability distributions, certain restrictions have to be imposed on w; in particular, the integral of |w|2 over the whole domain must be finite. The eigenvalues then are exactly those values of E for which the Schro¨dinger equation admits such a solution. It turns out that these eigenfunctions are expressible in terms of the venerable special functions of mathematical physics, namely (associated) Legendre and (associated) Laguerre polynomials. But exactly the same predictions (about the energy levels) were already made by the ‘‘old’’, ad hoc, Bohr-Sommerfeld quantum mechanics; the ‘‘new’’ wave- and matrix-quantum theories needed to predict facts that were beyond the scope of the old theory, thereby offering a crucial confirmation. That’s why Schro¨dinger himself, Hylleraas, and many other physicists tried to derive the energy levels (alias eigenvalues) for two-electron atoms, whose Schro¨dinger equation, for the wave function w ¼ wðx1 ; y1 ; z1 ; x2 ; y2 ; z2 Þ, is 2 o o2 o2 o2 o2 o2 þ 2þ 2þ 2þ 2þ 2 2 ox1 oy1 oz1 ox2 oy2 oz2 Z Z 1 w ¼ 0; þ2 E þ þ r1 r2 r12

where E and Z are as above, while r1 ; r2 are the distances of the electrons from the nucleus, and r12 is their mutual distance. The task turned out to be forbidding. There were some crude attempts to use perturbation theory, but none of their predictions came close to the experimental spectra already known then. It was a major challenge to vindicate the new quantum mechanics by computation. For once, the experimenters were ahead, and the theorists had to catch up.

Pekeris Chaim Leib Pekeris (1908–1993) had a brilliant idea of how to catch up. With a computer, of course! He had a carefully laid-out approach, to be described soon, that would indeed give a very accurate prediction of the helium spectrum, given a powerful enough computer and a clever enough programmer. Except that when he first had that idea, computers didn’t yet exist, and when finally he had access to the JOHNNIAC, during his frequent long visits to the Institute for Advanced Study up to the time of von Neumann’s death (in 1957), it was not quite powerful enough, and at any rate was too busy, to pursue Pekeris’s plan. In addition to being a brilliant scientist, Pekeris was also an ardent Zionist. His good friend (another Chaim, and another scientist), Chaim Weizmann (1874–1952), had back invited him, in 1947, to head the department of applied mathematics at the Ziv Institute (later renamed the Weizmann Institute of Science), and Pekeris agreed— in principle, but only on condition that they build a computer similar to the JOHNNIAC. A committee was formed, including no lesser figures than Albert Einstein and John von Neumann, to decide whether this was a good idea. Einstein believed not. In those days computers were very expensive, and he thought that such a poor, developing country could make better use of such a big chunk of money; but von Neumann managed to win Einstein over and the plan was approved. It took a few years to

AUTHORS

......................................................................................................................................................... CHRISTOPH KOUTSCHAN studied com-

DORON ZEILBERGER is a Board of Gover-

puter science at the University of ErlangenNu¨rnberg, Germany, and received his Ph.D. in mathematics in 2009 from the Johannes Kepler University, Linz, Austria, under the supervision of Peter Paule. After spending a postdoctoral year at Tulane University, New Orleans, he is currently postdoc at RISC-Linz. Koutschan’s research interests are computer algebra methods for symbolic summation and integration.

nors Professor of Mathematics at Rutgers University. He received his doctorate from the Weizmann Institute of Science in 1976, under the direction of Harry Dym. Among his contributions to combinatorics, hypergeometric identities, and q-series, Zeilberger gave the first proof of the alternating sign matrix conjecture, and in 1998, together with Herbert Wilf, he was awarded the American Mathematical Society’s Leroy P. Steele Prize for their development of the now-well-known WZ theory.

Research Institute for Symbolic Computation Johannes Kepler University 4040 Linz Austria e-mail: [email protected]

Mathematics Department Rutgers University New Brunswick Piscataway, NJ 08854 USA e-mail: [email protected] Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 2, 2011

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partial-differential equation, in variables r1 ; r2 ; r12 , is easily derived (Eq. (5) of [9]). The next step (first suggested by H.M. James and A.S. Coolidge, see ref. 4 of [9]) was to make another change of variables, this time a linear one. After substituting E = -e2, introduce perimetric coordinates: u ¼ eðr2 þ r12 r1 Þ; v ¼ eðr1 þ r12 r2 Þ; w ¼ 2eðr1 þ r2 r12 Þ: These new variables have the advantage that they range freely and independently from 0 to ?. In contrast, r1 ; r2 , and r12 are the lengths of the sides of a triangle (whose vertices are the two electrons and the nucleus), and so must obey the triangle inequality. In addition, the expected asymptotic behavior of w, deduced from the hydrogen (one-electron) case, suggested writing ([9], Eq. (13)) C. L. Perkeris2

1

w ¼ e 2ðuþvþwÞ F ðu; v; wÞ; and letting F(u, v, w) be the function sought. Pekeris performed this change of variables —purely by hand— and derived a fairly hairy linear partial-differential equation with polynomial coefficients, satisfied by F, that we do not reproduce here; the curious reader can either look it up ([9], Eq. (14)), or look at the computer output that is available from our web pages. The next step was to express F (u, v, w) as a series expansion of products of (simple) Laguerre polynomials (Eq. (16) of [9]): F¼

1 X

Aðl; m; nÞLl ðuÞLm ðvÞLn ðwÞ;

l;m;n¼0

WEIZAC3

materialize, and finally they recruited one of the members of von Neumann’s team, a visionary electrical engineer by the name of Gerald Estrin (b. 1921) [5]. Estrin recounts ([5], p. 319) that in one short conversation with von Neumann, shortly before his departure, he asked, ‘‘What will that tiny country do with an electronic computer?’’ John von Neumann responded: ‘‘Don’t worry about that problem. If nobody else uses the computer, Pekeris will use it full time!’’ Estrin comments that this turned out to be an important prophecy that he often recalled.

Pekeris’s Crazy Plan The first step was standard. Using the symmetries of the problem, one sees that the wave function w of the ground state depends only on r1 ; r2 ; r12 , so one ‘‘merely’’ has to deal with functions of three variables, rather than six. The new 2 3

where Ln(x) denotes the Laguerre polynomial k n X n ðxÞ : Ln ðxÞ ¼ k k! k¼0 Like all families of classical orthogonal polynomials, the Laguerre polynomials satisfy a pure (linear) differential equation, a pure (linear) recurrence equation, and a mixed differential-recurrence relation: xL00n ðxÞ ¼ ðx 1ÞL0n ðxÞ nLn ðxÞ; xLn ðxÞ ¼ ðn þ 1ÞLnþ1 ðxÞ þ ð2n þ 1ÞLn ðxÞ nLn1 ðxÞ; xL0n ðxÞ ¼ nLn ðxÞ nLn1 ðxÞ; the primes denoting differentiation with respect to x. Now came an astounding feat! Pekeris substituted the expansion for F (u, v, w), in terms of the yet-to-be-determined A(l, m, n), into the above-mentioned linear differential equation (Eq. (14) of [9], politely not shown here), and using the above relations for the Laguerre polynomials got rid of all differentiations, and then, by using the pure recurrence, got rid of any monomials in u, v, w. Then he collected terms, and got —purely by hand— a huge monster, a 33term linear partial recurrence equation with polynomial

Photo courtesy of Optik Foto Rutz AG, St. Moritz, Switzerland. Photo courtesy of Yuval Madar under the Creative Commons Attribution ShareAlike 3.0 License, taken from http://en.wikipedia.org/wiki/WEIZAC.

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coefficients satisfied by the A(l, m, n). Each of the coefficients of the 33 shifts A(l + a, m + b, n + c) that showed up was polynomial in l, m, n of degree 3, and of degree 1 in the charge Z and the yet-to-be found e. We will kindly spare the reader this recurrence (and spare ourselves from typing it!), but the truly courageous reader can glance at Eq. (22) of [9]. We shudder to think of the pain of the poor typist who keyed this from Pekeris’s handwritten manuscript, and the typesetter for Physical Review, not to mention Pekeris himself. They all deserve lots of credit. In his wonderful essay [5] (p. 331), Estrin understates the point: the ‘‘appearance of this ugly 33-term recurrence would be enough to discourage most analysts.’’ The recurrence yielded a homogeneous linear system of equations with ?3 equations and ?3 unknowns that usually has no non-trivial solutions, but for some e, the ‘‘eigenvalues,’’ the ‘‘determinant vanishes’’ and there are solutions. The largest eigenvalue is of primary physical relevance, for it corresponds to the ground state energy of the atom. But even the most powerful computers can handle only finite systems! Hence the next step consisted in reducing to a finite, truncated version of the system, considering only those l; m; n 0 for which l þ m þ n x, for some finite x, and setting all the A(l, m, n) with l + m + n [ x equal to 0. In addition, the system could be cut approximately in half by requiring either symmetry (A(l, m, n) = A(m, l, n), the socalled para states) or antisymmetry (A(l, m, n) = -A(m, l, n), the so-called ortho states). If this was to be handled on a computer (even one that did not yet exist), one needed a convenient way to order linearly all the triplets of integers (l, m, n) with l þ m þ n x and l m in the symmetric case (resp. l \ m in the antisymmetric case). For this, Pekeris devised a fairly complicated bijective map k: ðl; m; nÞ 2 IN30 j l m ! IN, which once again we spare the reader, but which can be found in Eqs. (27-29) of [9] (by the way, Eq. (28) contains a very rare misprint, there should be 12 ðl þ mÞ added to it). It is not known when Pekeris devised this plan, but it was probably several years before he had access to a computer, so he just had to wait until Chaim Weizmann’s promised computer would materialize, carrying out the recommendation of the above mentioned committee of Einstein, von Neumann, et al. The difficulty of the problem that Pekeris faced becomes even more evident when taking into account that some closely related problems are still open. For example, it is experimentally known that all existing atoms can form negative ions with no more than one or two extra electrons, but there is no theoretical understanding of this phenomenon.

WEIZAC We have already mentioned Estrin, the person chosen to head the team that would build from scratch the first Israeli electronic computer, and highly recommended his vivid account [5]. The WEIZAC team consisted of a cadre of young and talented electrical engineers (including Aviezri Fraenkel (b. 1929) who later did a Ph.D. in number theory, became, inter alia, an authority on combinatorial games, and pioneered the use of computers in religious studies). Finally the computer was ready, and Pekeris was itching to use it on his many problems, including the spectrum of helium,

but he needed a programmer (what today we would call a ‘‘software engineer,’’ but there was no such thing as software in those days). Not, of course, a Java programmer, nor a Fortran programmer, and not even an Assembly-language programmer. Back in 1957 these were yet to be invented. The only language that WEIZAC understood then was machine language, and the alphabet consisted of two letters only, 0 and 1 (via the 16-letter alphabet of hexadecimals). But how to find such programmers? Definitely not among graduates of computer science departments, for there were none. What Pekeris did was to ask his secretary to place classified ads in the daily newspapers, asking for highschool graduates, after their military service, who attended the megama re’alit (math/science track).

Accad Yigal Accad (b. 1936), fresh out of his military service, answered such an ad. In a recent e-mail message, dated May 7, 2010, Accad recalls: On a 1957 Friday (or was it a Holiday Eve) that happened to be a non-working day at the Weizmann Institute, Prof. Pekeris unexpectedly drove his 1948 Studebaker to our residence at the southern edge of Rehovot. He invited me to join him in his office. Over there he pulled out a pile of handwritten papers and went with me through many of the equations you can find in the 1958 paper, including Eq. (22). As I remember, this tour took at least 2 hours. At the end Prof. Pekeris asked me if I can handle this problem. There were only 2 possible answers to this question and the rest is history. This may have been the best risk I have taken. Estrin goes on to state the following accolades ([5], p. 330): There is a clear testimony to the fact that Yigal Accad had unusual ability to use WEIZAC as a tool with very little software between him and the machine semantics. That ability, when combined with his talents as an applied mathematician, was a significant factor in the ensuing problem-solving successes at the Weizmann Institute. Accad became Pekeris’s right-hand man for many years, and it is hard to imagine what Pekeris would have done without him. Pekeris appreciated Accad’s invaluable work, and it was at his suggestion that Yigal, while working full-time as a software engineer, enrolled in the graduate school (after completing his undergraduate studies at Hebrew University) and incorporated some of the research into, first a master’s thesis in 1969, and then in 1973 a Ph.D. thesis, which was a far-reaching extension of the work we recount here). Accad stayed at the Weizmann Institute from 1956 until 1989. Between 1977 and 1989 he also served as a consultant to the pioneering Israeli Hi-Tech company Scitex. In 1989 he moved to California and joined Electronics for Imaging (EFI), working there until 2008, ultimately becoming chief scientist.

The Pekeris-Accad-WEIZAC Collaboration Indeed Accad was the perfect person to tame Pekeris’s monster recurrence, to write (machine-language) programs to generate the truncated matrices, and to implement the iterative algorithm for estimating the largest eigenvalue. Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 2, 2011

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The impressive (for its time) WEIZAC output is displayed in Table III of [9] for values of the charge Z ranging from Z = 1 to Z = 10. We are happy to report that our 2010 computations (on three different platforms) completely agree with that table, all the way to the last decimal digit. In a follow-up paper, published a year later, Pekeris [10] (and of course, Accad and WEIZAC—but it would be more than 30 years later before any computer, Shalosh B. Ekhad, became coauthor of a published paper!) treat the important special case of helium (Z = 2) with a greater accuracy, and also consider the ortho state 2 3S. Our computations agree with that paper, too.

2010 Of course, thanks to Moore’s Law, all these computations can now be done much faster, and there is no reason for us to be proud that we can compute the eigenvalues within seconds with today’s hardware and software, a task that kept WEIZAC busy round-the-clock for months: for example, a fixed-point multiplication took 1 millisecond on this early computer and the capacity of its memory was 4096 words (40 bits per word). But what is still remarkable and probably not so obvious: not only the WEIZAC part, the numeric computation that can now be done on every laptop, and the Accad part, challenging in machine language but today an easy exercise with high-level programming languages, but also, and especially, the Pekeris part can now be done much faster and mostly automatically, using computer algebra. Even more: in view of the gigabyte-sized recurrences that we can currently handle (see for example [7]) with symbolic software, the ‘‘monster recurrence’’ looks rather dwarfish. We don’t know exactly how long it took Pekeris to derive the differential equation and the recurrence, but let’s say 20 person-hours (including checking and rechecking); our program needs 0.108 seconds. To be honest, it took us a couple of hours to program Maple and Mathematica to follow Pekeris’s plan, but with almost the same effort, one could (and we did) program the general problem, that could be used again and again for many other differential equations in future problems. Our programs PEKERIS (for Maple, by DZ) and Pekeris.nb (for Mathematica, by CK) are indeed very general: they basically can input any linear differential equation, in any number of variables, and any series of substitutions, and output the transformed differential operator. Also the recurrence for a Laguerre polynomial expansion is achieved completely automatically. Using the widely known concept of Gro¨bner bases (invented by Bruno Buchberger in 1965 and hence not yet available for Pekeris) it is also possible to perform the series expansion for any set of orthogonal polynomials of hypergeometric type. For this purpose, the defining equations for the family of polynomials are represented as a Gro¨bner basis, which makes sense when the relations are rewritten, in operator notation, as (noncommutative) polynomials. Having chosen an appropriate monomial order, the elimination of the differentials can be achieved by a simple reduction modulo the Gro¨bner basis. Similarly, by changing the underlying polynomial ring, the elimination of the continuous variables u, v, w can be done. Let us also remark that you don’t need to be a Laguerre or a 56


Pekeris to generate the relations for the Laguerre (and other orthogonal) polynomials. They are all routinely derivable (and provable) by the so-called Wilf-Zeilberger method [12], as implemented, e.g., in the Mathematica package HolonomicFunctions [6] that we employ in our program. Modular techniques using Chinese remaindering and polynomial interpolation allow for computing the determinant symbolically up to quite large dimensions: for example, the determinant of the 161 9 161 matrix (para case with x = 10) is obtained in less than five minutes, yielding a polynomial in e of degree 161 having integer coefficients with about 500 digits! It is clear that this strategy produces a lot of overhead, so that an alternative way is desirable. We reformulate the problem of finding the largest e for which the determinant of M 2 Z[e]n9n vanishes, as a generalized eigenvalue problem: Av ¼ eBv;

M ¼ A eB with A; B 2 Znn :

Although Maple and Mathematica are computer algebra systems for symbolic computations in the first place, they also offer quite some functionality for numerical computations, in particular for the above problem. But since we were not 100% satisfied with either—Maple was rather slow for the desired precision and Mathematica didn’t allow higher precision than machine reals (6 decimal digits)—we tried with MATLAB, a software designated for numeric computations, especially in linear algebra. Notably, the program code for building the (sparse) matrices is itself computer-generated! It contains the 33 terms of the recurrence hard-coded to produce the matrix entries, and therefore certainly comes closer to Accad’s machine-code program. We were very impressed by MATLAB’s speed and accuracy. Computing all entries of Table III of [9] takes less than a second, and without much effort x can be increased to 60, corresponding to a 20336 9 20336 matrix.

Software and Sample Output This article is accompanied by the Maple package PEKERIS, available from http://www.math.rutgers.edu/*zeilberg/mamarim/mamarim html/pekeris.html, where readers can also find lots of output files (and input files if they want to modify them to get more output) that reproduce and far extend the seminal 1958 computations of Pekeris, Accad, and WEIZAC. Further we provide the Mathematica notebook Pekeris.nb (for which the package HolonomicFunctions is required), and the MATLAB programs PekerisPara.m and PekerisOr tho.m, all available from http://www.risc.jku.at/people/ckoutsch/pekeris/. Our maplephone readers are welcome to play with the first package while the mathematicaphones would probably prefer the latter one. However, even people (shame on you!) who speak neither Maple nor Mathematica can appreciate the output files, written in plain humanese. The second-

named author is particularly proud of the procedure PaperPara that fully automatically and seamlessly generates a whole article, ready to be submitted to Physical Review, without any human touch. Changing the parameters can produce many similar papers, see

lems have been staring at the applied mathematician for decades, and even more for centuries, without a practical solution being reached. A problem, like the tides of the oceans, for example, is not necessarily insoluble just because it had remained in the books for 184 years.

http://www.math.rutgers.edu/*zeilberg/tokhniot/oPEKERIS1.

Conclusion This article is first and foremost an ode to the vision and ingenuity of computing pioneers, but it also makes the point that there are lots of hidden treasures in the ‘‘old’’ scientific literature that can be revisited with today’s powerful symbolic computation software. We are not the first to advocate using symbolic computations in scientific computing, see for example [3] (unfortunately he was unaware of [13]), and the current impressive application to high-energy physics [4], but we believe that there is a huge potential for exploiting symbolic computation on problems that previously seemed intractable. This would complement the extensive use (and according to Nobelist Philip Anderson, excessive and sometimes abusive use [1]) of Monte Carlo methods. In particular, the Wilf-Zeilberger algorithmic proof theory [12] (and more importantly the subsequent generalizations to multi-summation and multi-integration [2, 13]), should be taught to all scientists. We would be more than happy if this article could seed future collaborations between symbolic computation and physics, chemistry, or other sciences.

Encore Many people, even today, are not comfortable with computer-generated or even computer-assisted proofs, such as the four-color theorem or the Kepler conjecture: they are uncomfortable trusting the computer. Although the ‘‘monster recurrence’’ discussed above was still derived purely by hand, Pekeris must have started using his own ‘‘symbolic’’ computation when he tackled seemingly intractable problems. Let us end with his prophetic words ([11], quoted in [5], p. 333): Here we are confronted with problems where the computer writes the formulae as well as evaluates them. By the nature of their origin such formulae are very long—in many cases too long to be published. We shall therefore be dealing in the future with equations which only the computer will see. The prospect of operating with invisible equations is a frightening one, but the alternative is to accept the situation of the past, where prob-

REFERENCES

[1] Philip W. Anderson. What is Wrong with QMC?, talk delivered at the 103rd Statistical Mechanics Conference, May 10, 2010, Rutgers University, http://www.math.rutgers.edu/events/smm/ smm103-invitedtalks.html. [2] Moa Apagodu and Doron Zeilberger. ‘‘Multi-Variable Zeilberger and Almkvist-Zeilberger Algorithms and the Sharpening of WilfZeilberger Theory,’’ Adv. Appl. Math. 37 (Special Regev issue), 139–152, 2006. [3] Michael P. Barnett. ‘‘Symbolic Computation of Integrals by Recurrence,’’ ACS SIGSAM Bulletin 37(2), 49–63, 2003. [4] I. Bierenbaum, J. Blu¨mlein, S. Klein, and C. Schneider. ‘‘TwoLoop Massive Operator Matrix Elements for Unpolarized Heavy Flavor Production to O(e),’’ Nucl. Phys. B 803(1-2), 1–41, 2008. [5] Gerald Estrin. ‘‘The WEIZAC Years,’’ IEEE Trans. for the Annals of the History of Computing 13(4), 317–339, 1991. [6] Christoph Koutschan. Advanced Applications of the Holonomic Systems Approach, Ph.D. thesis, RISC, Johannes Kepler University, Linz, Austria, 2009. [7] Christoph Koutschan, Manuel Kauers, and Doron Zeilberger. A Proof of George Andrews’ and David Robbins’ q-TSPP Conjecture, Technical Report arXiv:1002.4384, 2010. [8] Linus Pauling and E. Bright Wilson, Jr. Introduction to Quantum Mechanics with Applications to Chemistry, McGraw-Hill, New York, 1935 (reprinted Dover, 1985). [9] C.L. Pekeris. ‘‘Ground State of Two-Electron Atoms,’’ Phys. Rev. 112(5), 1649-1658, 1958. [10] C.L. Pekeris. ‘‘1 1S and 2 3S States of Helium,’’ Phys. Rev. 115(5), 1216-1221, 1959. [11] C.L. Pekeris. ‘‘Propagation of Seismic Pulses in Layered Liquids and Solids,’’ in Norman David (ed.), International Symposium on Stress Wave Propagation in Materials, New York, Interscience Publishers, 1960. [12] Marko Petkovsˇek, Herbert S. Wilf, and Doron Zeilberger. A=B, A.K. Peters, 1996, available online. [13] Herbert S. Wilf and Doron Zeilberger. ‘‘An Algorithmic Proof Theory for Hypergeometric (Ordinary and ‘‘q’’) Multisum/integral Identities,’’ Invent. Math. 108, 575-633, 1992.


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The Mathematical Tourist

Dirk Huylebrouck, Editor

Mathematical Flavor of Southwestern Moscow NATASHA ROZHKOVSKAYA Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe´ where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit an essay to this column. Be sure to include a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.

â

Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium e-mail: [email protected]

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DOI 10.1007/s00283-010-9180-5

t would be unreasonable to complain about a lack of sightseeing in Moscow. No trip to the Russian capital is long enough to absorb all of the cultural delights of the city. Here I would like to highlight five landmarks of special interest for their mathematical value. All are situated in the southwestern district of Moscow, and the concentration of research institutes and the nearby Moscow State University contribute to the scientific atmosphere of this part of the capitol city. 1. Moscow State University, Department of Mathematics (Vorobievy Gory). The exquisite campus of Russia’s largest university, on the hills of Vorobievy Gory, is centered around the skyscraper with the name ‘‘The Main Building’’. Founded in 1755 by Mikhail Lomonosov, an iconic figure in Russian science, the institution now boasts forty thousand undergraduate and seven thousand graduate students. The Main Building was built in 1953 by the architect Lev Rudnev. The central tower, a 36-story, 240-meter high building, is flanked by four wings of housing accommodations for students and faculty. During exam week, according to campus gossip, students do not go outside the building at all – they can find all their daily necessities within the Main Building’s walls. The facilities inside include libraries, administrative services, several cafeterias, bank offices, a concert hall, a museum, a swimming pool, a police station, a post office, a salon, shops, and a theater. The Department of Mathematics (Mekh-Mat) occupies floors twelve through sixteen of the Main Building. The glorious and controversial history of the department is intimately interconnected with twentieth-century Russian history. The new Soviet government brought literacy and basic education to the country. Many new schools and

I

institutions opened during the first decades after the revolution. The department of mathematics officially opened in 1933. The birth of the new educational establishment significantly accelerated the development of the scientific activity in the capital. At the beginning of the 1910s there was only one scientific seminar at Moscow University, organized by D.F. Egorov. By the 1930s there were dozens, and in the 1950s the number had risen to more than one hundred [3]. During the twentieth century, Mekh-Mat experienced two golden periods. The first period began shortly after the department was organized and lasted until the beginning of the Second World War. The second period started with the political thaw in 1953. The end of this time of prosperity, in 1969-1970, is also attributed to outside political events, in particular to the publication in the West of the ‘‘letter by 99 mathematicians’’ [3]. During the second half of the twentieth century, Mekh-Mat suffered heavily from the ‘‘brain drain migration’’, as many professors and graduates left for other countries. Nevertheless, the department remains one of the best in the world in the preparation of young mathematicians. Today, roughly two thousand undergraduate and five hundred graduate students receive their mathematical training at the department (Figures 1, 2). 2. The Central Economic Mathematical Institute (Nachimovski Prospect, 47). From the 1960s through the 1980s, the southwestern part of Moscow was a place for experimentation in civil architecture. The Central Economic Mathematical Institute (CEMI) building, designed by L.N. Pavlov, G.V. Kolycheva, and I. Ya. Yadrov, was erected during this era of innovation. Today the building hosts one of the most prestigious Russian schools in Economics, ‘‘The New Economic School’’. The building itself may look quite ordinary, but it has a surprising decorative element on the facade and is colloquially know as ‘‘The House With An Ear’’. This ‘‘ear’’ is a decorative panel with a Mo¨bius band. The brainchild of L.N. Pavlov, the band is scaled to 1/1,000,000 the radius of the Earth. Andrei Voznesenky, the famous Russian poet, wrote about the ear in his essay ‘‘O’’ [4], ‘‘It listens to the Cheremushkinski bazaar. The chaos of

Figure 1. The view of the MSU campus from the Main Building.

......................................................................... AUTHOR

Figure 2. The Main Building.

NATASHA ROZHKOVSKAYA is an Assis-

tant Professor at Kansas State University. She received her M.S. in mathematics at Moscow State University in 1997 and her Ph.D. in mathematics at the University of Pennsylvania in 2002. Her research interests are representation theory, quantum groups, and combinatorics. She fell in love with Moscow during her student years, and she is always happy to come back to the everchanging city. Department of Mathematics Kansas State University Manhattan, KS 66506 USA e-mail: [email protected]

boisterous speech, the crunch of vegetables, the diversity of unorganized forms, passions and vivid words reign at this place. The East of melons meets the North of cloudberries. Cloves of garlic prevent flu and vampires. Baby potatoes cost twelve rubles at the end of May, while veal is sold for ten rubles. Here the roses in the right corner are the best in Moscow and the revelry of payments roars. . . . . . This is not an Ear, this is a Mo¨bius band - argues Pavlov - it is a philosophical figure eight with an almost Henry-Moore hole in the middle. It tells about the endlessness of space. It is an eye into the inside of mother-nature. I gave size to it: one millionth of the Earth’s diameter. It is the magic module of my building. All the details are proportional to that number. For this reason you are attracted by the proportions of this square - instinct tells a man of the harmony with Earth (Figure 3). 2011 Springer Science+Business Media, LLC, Volume 33, Number 2, 2011

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Figure 3. The Mo¨bius band on the CEMI building.

But nobody believes him, everyone knows that it is the monument to the Ear….’’1 3. Steklov Mathematical Institute (Gubkina Street, 8). One of the leading mathematical research institutes in the world was founded through the initiative of the RussianSoviet mathematician V.A. Steklov. Vladimir Andreevich Steklov was born in 1864 and studied science at the Alexandrov Institute in Nizhny Novgorod and later at Kharkov University. From 1889 to 1906 he taught theoretical mechanics in Kharkov, and in 1906 he started to work at Saint Petersburg University. In 1919 Steklov petitioned for the creation of the Mathematical Cabinet; in 1921 the Cabinet was extended to the Institute of Mathematics and Physics with the addition of the Physics Laboratory and several seismology stations. Upon Steklov’s death the institute was renamed for him. In 1932 the structure of the Institute of Mathematics and Physics was divided into the department of mathematics, headed by Ivan Vinogradov, and the department of physics, headed by Sergei Vavilov. In 1934 the departments were split completely into two separate organizations, and the new Moscow branch of the Institute of Mathematics started an independent research program (Figure 4). Steklovka - the nickname of the institute - is one of the very few mathematical centers in the world where the members can dedicate all of their time to research projects, and even among those research centers, Steklovka stands out. Most research institutes are visited by mathematicians for just a few months and then the scientists return to their home institutions. The Steklov institute is the home institution for its members, and the majority of those have longterm and permanent appointments. The scientific life at the institute has a dramatic impact on the mathematical society of the capital. The members of the institute do not have any official teaching obligations, nevertheless they put a significant effort into the education of young mathematicians. In 2005 the Science-Educational Center started an advanced lecture series program aimed at undergraduate students with strong research interests in mathematics and physics. 1

Translated from Russian by NR.

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Figure 4. V.A. Steklov.

Figure 5. RAS Headquarters.

In 2009 the institute held the Math Festival for high-school students. The traditional institute’s colloquium takes place on the third Thursday of every month. The talks are given in Russian, and they are translated online and videotaped. The seminar materials can be found on the institute’s website, http://www.mi.ras.ru/. To attend the seminar, one has to obtain a pass to the institute; information on how to obtain one can be found on the website. 4. Russian Academy of Science Headquarters (Leninskiy Prospekt, 32-8). The Russian Academy of Science was founded as the Saint Petersburg Academy of Science in 1724 by Peter the Great, in consultation with Gottfried Leibniz. The Academy headquarters moved to Moscow in 1934; the current home of the RAS was built in 1989 by Yu.P. Platonov. The golden crown adorning the building was quickly nicknamed ‘‘The Golden Brains’’ by locals. The

Figure 6. The inside view of the Radio Tower.

complex contains offices, conference halls, auditoriums, several exhibition halls, and a restaurant. One should note that the construction of the complex took place while the country was going through the radical changes of Perestroika and the project slightly deviated from its original architectural plans. The Academy was forced to search for outside financial support to maintain the building. In particular, for several years the stage of the state-of-the-art conference hall was rented out for a musical production and other events [1] (Figure 5). 5. The Shukhov radio tower (Shabolovka Street, 37). This chef-d-oeuvre of Soviet architecture is featured in many textbooks on architecture and engineering. The Radio Tower was built in 1919–1922 by renowned Russian architect Vladimir Shukhov. The talented engineer introduced breakthrough innovations in industrial design and architecture, as well as in the oil and military industry. Shukhov received patents for the invention of the hyperboloid structures in 1899, and went on to build a number of them. The Radio Tower on Shabolovka Street is the most widely recognized of this type of building. The hyperboloid sections minimize the wind load and create the elegant silhouette of the tower. In 1939, a small airplane crashed into the tower, but did not cause any significant damage and proved the durability of the structure. In addition, the tower is very light: the Shukhov tower uses three times less metal per height-unit than the Eiffel tower [2] (Figures 6, 7). The initial plans called for the tower to rise to a height of 350 meters, but the Civil War and the lack of metallurgical coal supplies in the young Soviet Republic resulted in plans for the building being scaled back to 148.5 meters. It nevertheless became the tallest building in the country at the time of its construction. The first radio signal was sent from the tower in March of 1922, and the transmission of television programs started in 1939. Currently the tower needs serious restoration work. Public organizations, activists and

Figure 7. The Radio Tower above the Moscow skyline.

the government are looking for solutions for preservation of the heritage of Vladimir Shukhov. ACKNOWLEDGMENTS

I thank Ilya Blanter, Jennifer Paulhus, and the editors for valuable comments that improved the text of the essay. I also thank Wikipedia and Googlemaps. I am grateful to Mikhail Chekalov for the kind permission to reproduce his photos (Figs. 1, 2, 5, 6, and 7) from the website http://moscow vision.ru. The source for Fig. 4 is http://kharkov.vbelous. net/english/politex1/steklov.htm.

REFERENCES

1. Neiman, V.G. Yurii Platonov: One Cannot Lose One’s Way in a Circular Square (in Russian), Izvestia Nauka, 2002. 2. Pankratov, V. The Hyperboloid of Engineer Shukhov (in Russian), Krug Zhizni, 8 (24), 2002. 3. Tikhomirov, V.M. On Moscow Mathematics - Then and Now, in Golden Years of Moscow Mathematics, AMS 2007. 4. Voznesensky, A. O (in Russian), in ‘‘Crowns and Roots’’, Essays and Poems, Ekaterinburg, Y-Factoria, 1999.


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Reviews

Osmo Pekonen, Editor

Einstein’s Unification by Jeroen van Dongen CAMBRIDGE: CAMBRIDGE UNIVERSITY PRESS, 2010, X + 213 PP., £ 50, US $85, ISBN: 9780521883467 (HARDBACK) REVIEWED BY N. P. LANDSMAN

his book is about the interaction between mathematics and physics in the work and thought of Albert Einstein, with emphasis on his later years. In his early work, culminating in his annus mirabilis 1905, Einstein’s use of mathematics was elementary; indeed, the picture of space-time as a four-dimensional manifold that has become standard was introduced in 1907 by the mathematician Hermann Minkowski, rather than by Einstein himself (though based on the special theory of relativity formulated by the latter in 1905 in terms of clocks, rods, and light signals). This changed in 1912, when Einstein’s friend and (ETH Zu¨rich) colleague Marcel Grossmann introduced him to Riemannian geometry and the closely associated tensor calculus of Gregorio Ricci-Curbastro and his pupil Tullio Levi-Civita. This turned out to be a decisive event in the history of science, as Einstein’s subsequent application of this area of mathematics to the physics of gravity climaxed in his General Theory of Relativity of November 1915, surely a high point in human thought comparable with (and to some extent superseding) Isaac Newton’s Principia of 1687. But here the controversy starts. First, there seems to be a significant discrepancy between Einstein’s later recollection of his creation of General Relativity – for example, in his autobiographical notes (Einstein, 1949) – and the careful and detailed recent reconstruction of this process by historians of science based at the Max Planck Institute for the History of Science at Berlin (Renn, et al., 2007). In Einstein’s own view (Einstein, 1949), mathematical intuition and deduction had played the essential creative role: I have learned something else from the theory of gravitation: no ever so inclusive collection of empirical facts can ever lead to the setting up of such complicated equations [i.e., Einstein’s field equations Rlm 12 glm R ¼ jTlm : A theory can be tested by experience, but there is no way from experience to the setting up of a theory. Equations of such complexity as the equations of the gravitational field can be found only through the discovery of a logically simple mathematical condition … On the other hand, a study of his notebooks and other sources displays a constant interplay between physical and mathematical arguments, where only at the very end Einstein

T Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.

â Column Editor: Osmo Pekonen, Agora Centre, 40014 University of Jyva¨skyla¨, Finland e-mail: [email protected] 62

THE MATHEMATICAL INTELLIGENCER 2011 The Author(s). This article is published with open access at Springerlink.com

DOI 10.1007/s00283-011-9202-y

indeed used the mathematical requirement of general covariance (i.e., the condition of invariance under general coordinate transformations) to clinch the issue. And even so, this requirement was (mistakenly) felt by Einstein to be a reflection of two physical ideas that had guided him toward his theory of gravity right from the beginning, namely the equivalence principle (between free fall in a homogeneous gravitational field and inertial motion in the absence of gravity) and the idea of general relativity of motion (cf. Norton, 1993). So it seems that Einstein’s memory was colored by the course his post-1915 work had taken: sidestepping the quantum theory, which initially he had pioneered himself (but whose probabilistic character he famously rejected from 1926 onward), he put most of his effort into attempts to create a unified field theory of classical gravity and electromagnetism, hoping to recover quantum phenomena (such as elementary particles) in the guise of singularity-free solutions of the classical field equations he sought. Jeroen van Dongen’s remarkable book – an updated and revised version of his Ph.D. Thesis of 2002, meanwhile matured through the author’s research in the Berlin group just mentioned and subsequently at the Einstein Papers Project based at CalTech – offers a delightful tour through Einstein’s efforts in that direction, constantly analyzed from the dialectics of Einstein’s use of (pure) mathematics versus actual physics. The author thereby establishes – and this may be said to be the main point of the book – a gradual shift from the latter (which formed the strength of Einstein’s youth) to the former (which, some would say, marked his decline; see below). Following the opening chapter reviewing Einstein’s road to general relativity, as a second starter toward the main course we find a chapter on Einstein’s method of theoretical physics. Helpfully, Einstein occasionally commented on his own methodology, as well as on the relationship between mathematics and physics in general: for example, his famous aphorism ‘‘As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality’’ is actually part of a very insightful essay (Einstein, 1921), in which he analyzes the said relationship in the light of the ‘‘modernist’’ tendency (initiated by David Hilbert) to strip mathematics of its traditional meaning in the physical world (cf. Gray, 2008). Elsewhere – notably in a letter to his friend Maurice Solovine from 1952, originally unearthed by Gerald Holton – Einstein explained how the uncertain application of mathematics to reality ought to proceed: from experiments E we infer mathematical axioms A for some physical theory, which (by a purely ‘‘logical’’ deductive path) implies certain assertions S about E. The inference from E to A is neither ‘‘logical’’ nor ‘‘inductive’’ – Einstein’s criticism of induction predated Karl Popper’s, and likewise for the falsification criterion – but is only ‘‘intuitive’’ or ‘‘psychological.’’ But, as Einstein’s own practice (or, more precisely, his own recollection thereof) shows, it is precisely at this stage that arguments related to mathematical simplicity, symmetry, beauty, etc., play a decisive role. The relationship between S and E is not ‘‘logical’’ either, because E is itself not of a

logical nature. On the other hand, it is ‘‘far less uncertain than the relation of A to E.’’ The irony, then, is that in his later years Einstein increasingly failed to pay attention to the experiences E; apart from bypassing quantum mechanics, his unified field theories never took the strong and the weak nuclear forces into account. More generally, as van Dongen points out, the mathematically oriented top-down style of Einstein’s later years sharply contrasted with the general trend of theoretical physics at the time, which was bottom-up, phenomenological, and relied on as little mathematics as possible – as a case in point, throughout his career Niels Bohr only used high-school level mathematics. To make things worse, even within the style he had chosen for himself, Einstein was outclassed by the mathematician Hermann Weyl, whose attempts to unify gravity and electromagnetism – though as unsuccessful as Einstein’s per se – led him to the unprecedented ‘‘gauge principle’’ that forms the basis of modern quantum field theory and elementary particle physics. More generally, though rightly recognized as one of the supreme geniuses humanity has produced, Einstein – unlike Newton within the same category – simply lacked the mathematical talent and creativity that would have been necessary to bring his program forward. This is particularly clear in an episode discussed in detail by van Dongen, namely Einstein’s adventures with semivectors in the early 1930s. During his work on general relativity Einstein had become used to vectors and tensors, but the spinors introduced by Paul Dirac in 1928 (subsequently analyzed by mathematicians such as Weyl, Bartel Leendert van der Waerden, and E´lie Cartan) were new and alien to him. Thus Einstein looked for analogous objects that did behave like vectors, coming up with the notion of a semivector. In the style of Dirac, he guessed a field equation for semivectors, which Einstein initially interpreted as a sensational prediction of two elementary particles, identified with the electron and the proton! Unfortunately, a straightforward group-theoretical analysis carried out by Valentin Bargmann (at the time a doctoral student at Zu¨rich of Wolfgang Pauli’s, one of Einstein’s sharpest critics) showed that semivectors were just direct sums of Dirac spinors, so that Einstein’s prediction had simply (though implicitly) been put in by hand. Einstein’s subsequent work on Kaluza– Klein theory (a five-dimensional generalization of general relativity intending to unify gravity and electromagnetism, dating back to 1919) in the late 1930s and early 1940s was less ridiculous, but despite persistent effort and the presence of excellent collaborators such as Bargmann (who had moved to Princeton) and Peter Bergmann, it led to absolutely nothing, neither in physics nor in mathematics. This sounds like a sad story, though van Dongen livens it with entertaining side information (e.g., on Einstein’s collaborators), and also provides a psychological explanation, in that ‘‘an emotionally defining moment [i.e., the discovery of general relativity] was instrumental in locking him, eventually, in a belief in his idealized method and the pursuit of unified field theories …, validated by a one-sided recollection of the experience of [general] relativity.’’ But where does this leave us in our perception of Einstein? 2011 The Author(s), Volume 33, Number 2, 2011

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His scientific biographer Abraham Pais left no room for doubt in a TV-documentary on Einstein (Kroehling, 1991): In fact, in his first part of his life when he did his really important work, his notion of simplicity were [sic] the guide to the 20th century insofar as science is concerned. Later on I think he was just completely off base. I mean if Einstein had stopped doing physics in the year 1925 and had gone fishing, he would be just as beloved, just as great. It would not have made a damn bit of difference. In addition, Pais all but ridicules Einstein’s well-known criticism of quantum mechanics in his biographies of both Einstein (Pais, 1982) and Bohr (Pais, 1991), portraying the latter as the clear victor in the Bohr–Einstein debate on the foundations and validity of quantum theory. Similarly, van Dongen – without taking sides himself – quotes J. Robert Oppenheimer as saying that in his last 25 years Einstein had been ‘‘completely cuckoo,’’ his work ‘‘a failure,’’ and his attempt at unification a ‘‘hopelessly limited and historically rather accidentally conditioned approach.’’ However, over the past few decades a gradual reappraisal of the later Einstein has emerged. As to the main topic of his own book, van Dongen quotes string-theorist and popular science writer Brian Greene as saying that ‘‘Einstein was simply ahead of his time’’ with his unification program; indeed, Edward Witten – who has led the string theory program for the last 25 years – is sometimes portrayed as Einstein’s successor. Similarly, literature on the foundations of quantum theory that has not been written by those under the personal spell of Bohr typically acknowledges the depth of Einstein’s critique – even as late as 1935! – and shows its profound influence on the current debate (cf. Landsman, 2006, and references therein). Even Einstein’s attempt to find particle-like solutions of classical field theories has been revived from the 1970s onward, notably in theories of solitons, magnetic monopoles, instantons, skyrmions, and the like (Rajaraman, 1982). What remains is the fact that these days practically no one shares Einstein’s rejection of quantum theory: the vast difference between his and current attempts at unification (such as string theory) is that the latter incorporate quantum (field) theory. What van Dongen has now shown is that this rejection was by no means a consequence of senility, but of a post-1915 research style that became increasingly dissonant with Einstein’s contemporaries, such as Bohr. Let me add that if Einstein had absorbed as much as the Preface of Dirac’s renowned book on quantum mechanics (Dirac, 1930), he would have been hooked: The formulation of these laws [of nature] requires the use of the mathematics of transformations. The important things in the world appear as the invariants … of these transformations. … The growth of the use of transformation theory, as applied first to relativity and later to quantum theory, is the essence of the new method in theoretical physics. Further progress lies in making our equations invariant under wider and still wider transformations.

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Dirac here refers to the invariance of quantum theory under unitary transformations, clearly suggesting the analogy with the invariance of Einstein’s theory of general relativity under general coordinate transformations – which invariance had been so dear to its creator! Indeed, although (in a different context) van Dongen mentions (and documents) the fact that Einstein ‘‘knew’’ Dirac’s book, he rightly adds the crucial qualifying remark that Einstein ‘‘never seems to have internalized its perspective.’’ Pity him! OPEN ACCESS

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

REFERENCES

Dirac, P. A. M. (1930). The Principles of Quantum Mechanics. Oxford: Clarendon Press. Einstein, A. (1921). Geometry and Experience. Translation by Sonja Bargmann of an expanded form of an Address to the Prussian Academy of Sciences in Berlin on January 27th, 1921. In: Albert Einstein, Ideas and Opinions, pp. 232–245. New York: Bonanza Books (1954). Einstein, A. (1949). Autobiographical notes. In P. A. Schilpp (ed.) Albert Einstein: Philosopher-Scientist, pp. 1–94. La Salle: Open Court. Gray, J. (2008). Plato’s Ghost: The Modernist Transformation of Mathematics. Princeton: Princeton University Press. Kroehling, R. (1991). Albert Einstein: How I see the world. PBS Home Video. Landsman, N. P. (2006). When champions meet: Rethinking the BohrEinstein debate. Studies in History and Philosophy of Modern Physics, 37, 212–242. Norton, J. (1993). General covariance and the foundations of general relativity: Eight decades of dispute. Reports on Progress in Physics, 56, 791–858. Pais, A. (1982). Subtle is the Lord: The Science and Life of Albert Einstein. Oxford: Oxford University Press. Pais, A. (1991). Niels Bohr’s Times: In Physics, Philosophy, and Polity. Oxford: Oxford University Press. Rajaraman, R. (1982). Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory. Amsterdam: North-Holland. Renn, J., et al. (2007). The Genesis of General Relativity. Vols. 1–4. Boston Studies in the Philosophy of Science No. 250. Dordrecht: Springer.

Institute for Mathematics, Astrophysics, and Particle Physics Radboud Universiteit Nijmegen Heyendaalseweg 135 6525 AJ Nijmegen The Netherlands e-mail: [email protected]

Here’s Looking at Euclid: A Surprising Excursion Through the Astonishing World of Math by Alex Bellos NEW YORK, FREE PRESS, 2010, XI + 319 PP., US $25.00, ISBN 978-1-4165-8825-2, ISBN 978-1-4165-9634-9 (EBOOK) REVIEWED BY JOHN J. WATKINS

he great French mathematician Henri Poincare´ once performed a study in which he carefully weighed his morning baguette each day for an entire year. The purpose of this study was to demonstrate to Parisian authorities that the local baker was in fact cheating his customers. A century later, the British journalist Alex Bellos conducted a similar experiment by buying each morning a baguette from his local baker and recording its weight each day over a period of several months. Bellos’s purpose in running this experiment was not to expose his own baker as a cheat but to explore for himself the nature of statistics for a book he was writing about the wonders of mathematics. That the journalist Alex Bellos would be writing a book about mathematics came as quite a shock to me. As a soccer fan, I thoroughly enjoyed his first book, Futebol: The Brazilian Way of Life, but never expected a book from him on the world of mathematics as a sequel. In order to write about Brazil and its love affair with football, Bellos spent a year traveling around Brazil and immersing himself in its culture. To write Here’s Looking at Euclid Bellos made a far more ambitious journey, not only following his story around the globe and into the past, but also into the world of real mathematics. And the wonderful thing is that he invites you along on this journey, that’s the way you feel as you turn the pages, as if you were exploring this astonishing world along with him. Here’s Looking at Euclid is a sumptuous book. Each chapter is loosely organized around a single mathematical topic, and this provides Bellos the luxury of meandering through time and space to tell his story. In a chapter that explores whether we as humans acquire numbers innately or construct them, he begins in the rainforest of Brazil with the Munduruku, whose language has no words for numbers greater than five, and takes us in the blink of an eye to the Serengeti to see how it is that lions use their own sense of number to survive. In a chapter about the invention of zero in India and the subsequent spread of the HinduArabic number system, we find out why in India the Oscarwinning film Slumdog Millionaire is called Slumdog Crorepati (that is, ten-millionaire). In a chapter on the development of algebra, we learn that when Descartes first introduced the now standard notation that letters at the beginning of the alphabet such as a; b; c, and d be used to

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represent known quantities, and letters near the end such as x; y, and z be used for unknown quantities, it was the printer of his book La geóme´trie and not Descartes himself who decided to concentrate on the letter x for the simple reason that he was running short of lead type for the letters y and z that were more frequently needed in French. Thus, x became the letter most commonly associated with an unknown quantity, so much so that it has become a part of our language (we have X-rays, Generation X, and The XFiles, the enormously successful TV series about unsolved FBI cases involving the paranormal). In a chapter devoted to the joy of discovery, he offers us a lovely infinitely repeating mosaic derived from the tenth-century Arabic mathematician Annairizi that contains within it a beautifully simple proof of the Pythagorean theorem (look carefully and you will see that the large superimposed square contains each of the two smaller squares, the blue colored square in two pieces and the white square in three pieces— that is, c2 ¼ a2 þ b2 ):

Tenth century proof of the Pythagorean theorem.

Here’s Looking at Euclid reads like a very good travel book as we follow Bellos around the globe, encountering quirky mathematicians and other curious characters along the way. Bellos uses a brilliant technique: In each chapter he first captures our interest by writing about people, and then he uses that interest to tell us about mathematics. The effect is quite remarkable in that his book is literally a ‘‘page-turner’’—you can’t help but keep turning to the next page to see who shows up next. Bellos traveled to Tokyo to visit an abacus club run by Yuji Miyamoto who devised a truly remarkable game of mental arithmetic. The game lasts for exactly three seconds. Fifteen three-digit numbers are flashed onto a computer screen, one at a time, each number remaining visible for only 0.2 seconds. Miyamoto’s students can produce the sum of the fifteen numbers at the instant the last number disappears from the screen. They do this by mentally visualizing an abacus; as the first number appears they visualize it on an abacus, then as each new number appears they simply shift imaginary beads on the abacus until, after the shift for the final number, the answer is just sitting there in their heads. Miyamoto’s nine-year-old daughter even did a demonstration of this game for Bellos adding 30 threedigit numbers in just 20 seconds while she simultaneously played a word game called shiritori with a friend—in this game each person says a word that begins with the syllable that ended the other person’s previous word, so the game Bellos witnessed went like this: shiritori – ringo – gorira – rappa – panda – dachou – ushsi – . . . . Ó 2011 Springer Science+Business Media, LLC, volume 33, Number 2, 2011

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While in Tokyo, Bellos also visited Maki Kaji in his office where he runs a puzzle magazine. One of Kaji’s hobbies is to photograph license plates; he is trying to collect photos of all 81 possible combinations of Japanese license plates that represent the multiplication table from 1 to 9 (for example, the plate numbered 34 12 would represent 3 4 ¼ 12). Kaji is the puzzle-maker who gave the world sudoku after discovering a puzzle called Number Place and redesigning it so that the numbers provided are placed symmetrically in the 9 9 grid. Then Bellos traveled to a town north of Tokyo to meet Kazuo Haga, a retired entomologist. Bellos used this visit to tell us about origami and, in particular, two remarkable theorems due to Haga. The first, discovered in 1978, is that if you fold a corner of a square piece of paper onto the midpoint of an opposing side you create three right triangles, and each is a 3-4-5 triangle. Now I had to stop and spend a few moments proving this for myself (and even longer for the second theorem) before I could continue reading, but Bellos has clearly addressed his book to a very general audience and does not want to put anyone off with such proofs—in fact, one of the real strengths of the book is just how clearly he can write about mathematical ideas without being overly complicated—and so he simply makes the point that these are truly amazing theorems by saying ‘‘Ker-azee!’’ One of Bellos’s most fascinating side trips was to a small town just north of London to visit Peter Hopp, a retired electrical engineer who has one of the world’s largest collections of slide rules. Along with Bellos we get to examine many of the beautiful objects in Hopp’s collection, such as an early eighteenth-century wooden slide rule used by tax men to make calculations of alcohol volume, and an elegant one with a mahogany handle and three concentric, hollow brass cylinders. The most intriguing item in Hopp’s vast collection was not a slide rule, however, but a mechanical pocket calculator with 600 moving parts, first produced in 1948, called the Curta, which looks like a pepper grinder and on which mathematical operations can be performed literally by turning the crank. Bellos met Daina Taimina in London. Taimina teaches at Cornell University and invented a stitching style of crochet known as ‘‘hyperbolic crochet’’ for crocheting the hyperbolic plane. Her hyperbolic models are not only lovely to look at, they can be used to make abstract ideas about hyperbolic space come to life. For example, using one of her models, it is possible to see the famous result that on the hyperbolic plane an octagon can be folded so that it is topologically equivalent to a pair of pants. The models themselves look organic since they—like many plants and marine organisms—are trying to pack a very large surface area into a relatively small volume. In fact, when Taimina and Bellos met, she was in London for a crochet exhibition to promote awareness of the loss of coral reefs! He visited the Chudnovsky brothers, Gregory and David, in Brooklyn where he saw their famous formula 1 1 X ð6nÞ! 163 096 908 þ 6 541 681 608 n ¼ ð1Þn p n¼0 ð3nÞ!n!3 ð262 537 412 640 768 000Þnþ12

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alongside dozens of other formulas for p decorating their blue office floor. Each term of this astonishingly powerful series adds roughly 15 digits to the decimal approximation of p. Using a supercomputer they built themselves, the Chudnovskys had computed p to more than two billion decimal places by 1991 (it has now been computed to almost three trillion places using this formula). During this visit we get to sit in on a conversation between Bellos and the Chudnovskys about how the apparent randomness of the digits of p is extremely weird and yet very useful. Bellos came across several interesting characters in 2008 in Atlanta at the Gathering for Gardner, a biennial affair at which over 300 mathematicians, puzzlers, and magicians gather to play and to pay homage to Martin Gardner. He met Erik Demaine, an MIT professor who talked about a problem he had been working on since he was a teenager based on Henry Ernest Dudney’s Haberdasher Problem (a famous ‘‘hinged’’ dissection problem that produces a triangle when folded one way and a square another way). He met the irrepressible logician Raymond Smullyan, then 88 years old, who began his talk: ‘‘Before I begin speaking, there is something I want to say.’’ He ran into Tomas Rokicki, who has analyzed billions of positions of Rubik’s Cube trying to answer the question: what is the smallest number n such that every configuration of Rubik’s cube can be solved in n moves or fewer? Rokicki is convinced the answer is 20 (and has proved that it is no greater than 22). He met Neil Sloane, who began collecting sequences of numbers as a graduate student in 1965—his collection, now called the On-Line Encyclopedia of Integer Sequences, has more than 180,000 sequences and is growing at the rate of about 10,000 sequences per year. It contains familiar sequences such as the sequence of primes 2; 3; 5; 7; 11; 13; . . . (known as A40 because it was the 40th sequence Sloane collected) as well as other sequences with interesting properties such as Gijswijt’s sequence 1; 1; 2; 1; 1; 2; 2; 2; 3; . . ., a sequence that is so reluctant to produce ever larger and larger numbers that 4 appears for the first time 23 in the 221st position and 5 occurs first at about position 1010 . Bellos captures the depth of feeling that mathematicians have for the objects they study when he quotes Sloane: ‘‘The land is dying, even the oceans are dying, but one can take refuge in the abstract beauty of sequences like Dion Gijswijt’s A090822.’’ From Atlanta, Bellos went to Norman, Oklahoma to visit the legendary Martin Gardner, who told him, ‘‘I am not a mathematician, I am basically a journalist. Beyond calculus I am lost. That was the secret of my column’s success. It took me so long to understand what I was writing about that I knew how to write about it so most readers would understand it.’’ Bellos has this same hard-won gift. Like Gardner before him, Alex Bellos is a journalist and not a mathematician—though he does have a degree in mathematics from Oxford—and I suspect that is one reason his book is as good as it is. In his chapter on statistics, for instance, where we hear the tale of Poincare´ and his baker, he focuses our attention on the normal distribution by telling us about his own attempt to produce this bellshaped curve experimentally by carefully weighing his daily loaves of bread. Along the way, as Bellos tells the story of his experiment, we learn many things, such as that

Galileo had been fully aware of the bell-like symmetry exhibited by data from his astronomical measurements, and that the pinball-like machine called the quincunx that you drop balls into to produce columns of balls at the bottom that look like a bell curve was invented by Francis Galton, a noted scientist who epitomized the nineteenth-century obsession with measurement (he tried to measure intelligence by measuring brain sizes and studying test scores, and also believed he could measure female attractiveness). We also learn what all of this has to do with Pascal’s triangle, binomial expansions, Sierpin´ski’s triangle, perfect numbers, kissing kangaroos, and a duck-billed platypus. By the end of his experiment, we and Bellos have learned that measurement is never simple, and that perhaps Poincare´’s baker wasn’t a cheat after all. In Reno, Nevada, Bellos met with Anthony Baerlocher, the director of game design for the largest maker of slot machines in the world, and takes us behind the scenes to see the details of how designing good casino games is not just about graphics and sound, it’s also about getting the underlying probabilities right. One of my favorite quotes in the book is when Baerlocher says: ‘‘My colleagues and I spent over a year mapping things out and writing down some formulas and we came up with a method of hiding what the true payback percentage is.’’ Bellos also met one of my heroes, Edward Thorp, in his office in Newport Beach, California. In 1961, Ed Thorp presented a paper at the annual meeting of the American Mathematical Society in which he gave a system for beating the casino game blackjack—this card-counting system changed the world of gambling forever. Thorp has long since turned his talents to economics and finance and is now rich enough that he has to worry about whether or not it is a good bet for him to insure a ‘‘small’’ item such as his house. Bellos asked Thorp about a plan to have his body frozen when he dies since this hardly seems like a ‘‘good bet.’’ Thorp’s response was: ‘‘It’s the only game in town.’’ Bellos’s book originally appeared in the U.K. under the title Alex’s Adventures in Numberland. I very much like the title of the U.S. edition, Here’s Looking at Euclid, but wonder whether the publishers realize how few young people today will even understand the pun. I recently showed a copy of the book to a class of college students and only one student had ever heard what surely must be one of the most famous lines in the history of movies: ‘‘Here’s looking at you kid.’’ And, sadly, the marvelously witty sketch of Humphrey Bogart on the cover of the book will be a complete mystery to the target audience for this

fine book—this image has been cleverly created using each integer from 1 to 9 (a horizontal 8 forms his bow tie, a languid 5 curls smokily upward from the tip of his cigarette). As well as Bellos writes about mathematics—and it is hard to imagine how a mathematics book could be more engaging than this one is—he can in some instances be a bit misleading. For example, having just said that Euclid proved there are infinitely many prime numbers, Bellos reinforces the idea this way: ‘‘Think of a number, any number, and you will always be able to find a prime number higher than that number.’’ Yet, knowing that larger and larger primes always exist is very different from being able to find them. Similarly, in a chapter on set theory and infinity, he says that the number of curves that can be drawn on a two-dimensional surface is greater than the cardinality of the continuum, but this is false if by ‘‘curves’’ he means continuous curves; he calls this ‘‘larger’’ cardinality d and says that while there must be an infinity larger than d no one has come up with a set of things with that larger cardinality, but the set of all subsets would have larger cardinality. In a chapter on the golden ratio, he says 1 is equal to the sum .01+.001 + .0002 + that the number 89 .00003 + .000005 + .0000008 + .00000013 + .000000021 + .0000000034, thereby giving the impression that its decimal expansion has a finite number of digits. In fact, of course, P Fn 1 ¼ 1 what is true is that 89 n¼1 10nþ1 where Fn represents the nth Fibonacci number. I was amused to see how frequently Bellos could work soccer into this book. In a chapter on chance, he discusses in detail the familiar birthday problem in which in a group of 23 people it is likely that two people will share a birthday. I have taught this problem many times over the years but it never occurred to me that soccer provides an almost perfect case study for this famous problem, since two teams of eleven players each plus one referee make 23 people. He then lists the players in the last 10 World Cup finals who share birthdays and it turns out that in 7 of these 10 finals, at least two players share a birthday. Now that Alex Bellos has written such engaging books about two of my passions, soccer and mathematics, I wonder what he will turn to next; baseball seems unlikely, perhaps gambling, or mountain climbing? Department of Mathematics and Computer Science Colorado College Colorado Springs, CO 80903 USA e-mail: [email protected]


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Duel at Dawn: Heroes, Martyrs, and the Rise of Modern Mathematics by Amir Alexander CAMBRIDGE, MASSACHUSETTS: HARVARD UNIVERSITY PRESS, 2010, 320 PP., US $28.95, ISBN-10: 0674046617, ISBN: 9780674046610 ´ NKA BILOVA´ ˇ TE ˇ PA REVIEWED BY S

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have found things so magnificent that I was astounded… All I can say now is that I have created a new and different world out of nothing…’’ Young Jańos Bolyai wrote these famous words to his father on his discovery of non-Euclidean geometry. In Duel at Dawn, Amir Alexander presents this discovery as the culmination of the abstraction that entered mathematics in the first half th of the 19 century. In his view, this was a time of profound changes, not only in the content of mathematical research, but also in the image of mathematicians. The title of the book suggests that the author symbolically connects the beginning of this new era with the tragic end of young E´variste Galois. Amir Alexander argues that these changes went hand in hand inseparably, each supporting the other. Furthermore, he claims that this should be viewed within the broader cultural development of the period. The sensibilities of High Romanticism allowed mathematics to follow the path of pure, self-contained science, with mathematicians taking on the image of tragic heroes pursuing sublime truth. Alexander examines the lives of several mathematicians and the stories written about them, draws attention to the differences, and looks for the reasons. The people described in the legends do not usually portray the real characters; instead they are morality tales about society or philosophical trends. Unfortunately, he stresses and restates his main arguments too often, making the text repetitive in a number of places. The chapters are divided into parts that cover various aspects of the narrative and piece together the complete picture. This enables the author to dose information and arguments gradually, but it also contributes to the repetitiveness of the text, and some readers may feel it breaks the coherence of the narration. The book focuses on Jean le Rond d’Alembert, E´variste Galois, Niels Henrik Abel, Augustin-Louis Cauchy, and Jańos Bolyai, although we encounter a great many other mathematicians along the way. Alexander does not give detailed and full biographies; he selects the parts that relate to his main points and shows the reader where to look for more information. These suggestions for further reading concern not only biographical data, but also notes on translations and, to a great extent, mathematical issues. Alexander is extremely precise in his citations and has done a very careful job questioning and judging the reliability of the facts and statements given in the sources. The text also contains many

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quotes from letters or other writings, which provide authenticity and often add dramatic or amusing elements. I am pleased to see that the author includes quite a lot of mathematics. The expositions are placed in separate chapters, allowing the reader who is not so keen on figures to skip them. It is even more pleasant that the mathematical presentations are clear and careful step-by-step descriptions in plain language with simple diagrams. I particularly enjoyed the part dealing with non-Euclidean geometry and its history, although at the end I wished there was at least a hint on the existence of models of hyperbolic geometry in Euclidean space. The first part of the book discusses 18th-century mathematics and mathematicians, with Jean d’Alembert as the typical example. Alexander shows how accounts of d’Alembert’s life paint the iconic picture of the 18th-century mathematician. Enlightenment mathematicians were viewed as ‘‘natural men’’ – simple, happy, devoid of any petty jealousies, enjoying successful social and professional lives. He explains the philosophy and practice of the 18th-century mathematicians and presents developments over the century through specific examples (the problem of the catenary and the problem of vibrating strings). In the second part of the book, Alexander moves to the 19th century. Two chapters describe the creation of legendary figures, Galois and Abel, and the third is devoted to Cauchy, the greatest proponent of modern, rigorous mathematics of the period. His critical comments on biographies and other works dealing with Galois’s and Abel’s lives suggest how and why Galois and Abel became twins in a universal myth of the tragic young genius misunderstood and persecuted by a ruthless and corrupt establishment. It is particularly interesting to read about various motivations that led to the development of this myth. Alexander analyzes Galois’s and Abel’s personalities, as well as the fate of their mathematical papers, to indicate their positions in the mathematics of the time. We may be surprised to find that Cauchy is discussed together with Galois and Abel, but Alexander points out that the rejection by scientific institutions and schools that he met in his youth and after returning from exile is similar to the treatment encountered by Galois and Abel. Furthermore, the personality traits and political views that made Cauchy’s life difficult remind us of Galois. Though their political convictions were completely opposed, both stood firmly for their truths, political and mathematical. The last mathematical poet of the first half of the 19th century that Alexander considers is Jańos Bolyai. Although Bolyai lived much longer than his contemporaries Galois and Abel, the three shared many features. The last two parts of the book discuss Romantic mathematics and the persona of the mathematical poets practicing it. The author shows how Romantic features attributed to the 19th-century mathematicians are closely related to those ascribed to poets, musicians, and painters. This separates modern mathematicians not only from their 18th-century counterparts, but also from natural scientists. Recalling several mathematicians of the later periods whose lives bear clear touches of Romanticism, Alexander observes that the legend of tragic genius loners has survived two centuries.

Romantic mathematics is defined as ‘‘the study of a pure realm of truth and beauty’’; the author points out that the idea of mathematics inclining toward poetry and the arts corresponds to G. H. Hardy’s reflections in A Mathematician’s Apology. He shows, through several examples (Cauchy’s definitions of the function, the limit, and the derivative, the problem of the quintic equation, and the emergence of nonEuclidean geometry), how modern mathematics freed itself from the physical world. The final chapter, ‘‘Conclusion: Portrait of a mathematician,’’ presents several unexpected ideas. The first surprise concerns portraiture, not only in the sense of description in legends, but also the portrait in its literal meaning. Two existing pictures drawn from life, Abel’s and Galois’s, bear the features of Romantic heroes. Alexander analyzes portraits of the 18th-century mathematicians, 19th-century natural scientists, and 19th-century artists. I found the visual analysis very interesting, but are two portraits sufficient to generalize this idea? In his conclusion, the author outlines the development of mathematics and its Romantic legend up to the present and finishes with possible future changes. The use of computers in mathematical proofs suggests the Romantic hero may be replaced by the popular myth of a computer whiz. I find this difficult to imagine, as mathematics will not become a computer science as such. Alexander’s description of the popular image of computer whizzes also surprised me: unsociable,

revenge-taking nerds living in their parents’ basements is not how I imagine them. However, that is quite subjective. This book is well suited for any interested reader, but some knowledge of the history of mathematics may be useful, as a great many names from various periods appear throughout. General readers or students of mathematics will find here the description, history, and cultural and biographical background of achievements leading to the creation of modern mathematics. Mathematicians or mathematics teachers may discover new information about wellknown stories and personalities. Poetic souls may look for parallels between the pure worlds of mathematics and arts. And philosophical minds can meditate on the nature of mathematics or the reasons we need legends in our lives. But no reader will miss the author’s argument: since modern pure mathematics is a self-contained world governed solely by its own standards of rigorous deduction, it must be practiced by those who themselves live beyond our imperfect world in which they are misunderstood. At least, that is what the popular legend says, not what mathematicians are really like. Fortunately.

Masaryk University Language Centre Brno Czech Republic e-mail: [email protected]


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Putnam and Beyond by Ra˘zvan Gelca and Titu Andreescu NEW YORK AND BERLIN: SPRINGER-VERLAG, 2007, 798 PP., PAPERBACK, US $ 69.95, ISBN-10: 0387257659, ISBN-13: 978-0387257655 ˘ AND JACK B. GAUMER REVIEWED BY BOGDAN D. SUCEAVA

here are problem books that express and incorporate the quintessence of a whole mathematical culture; one may remember The USSR Olympiad Problem Book [6], a book that gives the measure of the mathematical competitions in the USSR from a certain historical period, or the very interesting Problems and Solutions in Mathematics [5], which speaks about the formative quality and cultural model represented by the qualifying examinations in North American universities. Then there are problem books of perennial interest, such as Problems in Mathematical Analysis [2], that measure how education was conducted in universities in a certain historical period. Such books are important not only for their content, but for the formative idea that lies behind each of them, a principle summarized in the question: how does one train in mathematics? Or, rather: what is the best way to train for challenges in the field of mathematics? The importance in each program of mathematical competitions resides in the core values of the cultural model represented by it, and a book designed to train students for it reflects these values. In this sense, the North American academic system has produced a very interesting cultural model: the W. L. Putnam Competition. It has taken place once a year since 1937. One can read about its history and the problems assigned in the actual Competition in the thorough volumes [1, 3, 4] or the articles published annually by the American Mathematical Monthly and other journals. On the other hand, the book reviewed here aims toward one of the most important goals related to teaching gifted undergraduates mathematics in North America: to improve one’s understanding of fundamental mathematics by preparing undergraduate students for the Putnam competition. For this reason, the structure of the book parallels the undergraduate curriculum, starting with an exploration of methods of proof, leading into specialized topics in algebra, real analysis, geometry, trigonometry, number theory, combinatorics, and probability. These large parts are divided into smaller sections, each aiming to be a collection of gems with a clearly focused educational goal. The outcome is a 935-problem and almost 800-page super-problem book with solutions, whose reading would certainly challenge, attract, and keep really busy any undergraduate student interested in acquiring various problem-solving techniques. Some of the questions are straightforward problems assigned in the original Putnam competitions or in other Olympiad-type contests around the world, and others are proposed by the two authors or have appeared in various journals of

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mathematics whose references are indicated in the text. Some problems may seem artificial and might have a certain unnatural core. The book also includes, presented as problems, classical theorems with certain historical and mathematical value. Since each is included in an appropriate context, their presentation here is definitely a gain. For example, Lo¨wner’s theorem on spherical curves is problem 643, a classical problem used by E. Goursat in 1904 is problem 462, whereas problem 861 belongs to E. Cesa`ro. The value of the book consists in its structure, a thematic architecture that circumscribes the whole undergraduate curriculum, and in its gradual building of certain themes, recurring through several generations of committee members writing Putnam competitions. The authors declare in the Preface that ‘‘using the Putnam competition as a symbol, we lay the foundations of higher mathematics from a unitary, problem-based perspective.’’ They are certainly right to view the college years as a moment lying between ‘‘the stimulating problems of the high school years and the demanding problems of scientific investigation.’’ Thus the Putnam competition stands here not as an ultimate goal, but rather as the expression of a model in education, something that may be relevant in every young mathematician’s destiny, at its own right time. The Foreword mentions, among the major sources of inspiration for this book, journals from the USA, Romania, Russia, China, India, Bulgaria, and specifies a few journals that have been particularly useful to the authors in organizing their selection of problems: the American Mathematical Monthly, Mathematics Magazine, Revista Matematica˘ din Timis¸ oara (Timis¸ oara Mathematics Review), Gazeta Matematica˘ (The Mathematical Gazette, Bucharest), Kvant, and Ko¨ze´piskolai Matematikai Lapok (Mathematical Magazine for High Schools (Budapest)). A word should be said here about the two Romanian sources, with which the two authors are very familiar and where many of their contributions have appeared in the past. The Bucharest Mathematical Gazette bears the tradition of the educational model of one of its founders, the geometer Gheorghe T¸ it¸ eica, a former Ph.D. student of Gaston Darboux, who was interested in promoting the education of gifted students. Since the beginning of the twentieth century, the Gazette has organized a contest that received a lot of attention within the scholarly community, and contributed to maintaining and encouraging a stimulating intellectual environment where academic values and mathematical gifts were highly praised and commented upon. The original contest included written and oral examinations and the final ranking was discussed in an editorial in the journal. This tradition was renewed after 1980, when the Contest of the Mathematical Gazette, in a modern format (two written tests, each one with three assigned problems, in sessions four-and-a-half hours long) was promoted by Nicolae Teodorescu, a former Ph.D. student of Jacques Hadamard. As for the Timis¸ oara mathematical journal, it was created in 1921 under the coordination of Traian Lalescu (a former Ph.D. student of E´mile Picard) and, although the Timis¸ oara journal didn’t have an olympiad-

type contest as the Bucharest Gazette did, it carried an equally important academic tradition. This is why these two journals, in particular, developed a large array of original problems in their archive, many of them interesting and with meaningful educational value. The authors have done excellent work in revisiting this rich archive and using it to better serve a contemporary educational concept. We get a better understanding of the fundamental philosophy behind this book if we discuss a few sections in detail. The framework question could be phrased: how can problem-solving be taught? Or rather: is there any educational strategy for preparing students for approaching difficult mathematical problems? Take for example section 4.1.2, The Coordinate Geometry of Lines and Circles. It may be true that most of the experience that undergraduate students have in analytic geometry is based, at least in the academic curriculum in many North American universities, on concepts defined and studied in the calculus classes. The authors had to find problems that slightly extend the standard academic curriculum, developing new skills and connecting ideas. The first problem in this section is Prove that the midpoints of the sides of a quadrilateral form a parallelogram. Actually, why does one need analytic geometry to approach such a problem? Much more interesting and appropriate are other problems, for example: On the hyperbola xy = 1 consider four points whose x-coordinates are x1, x2, x3, and x4. Show that if these points lie on a circle, then x1x2x3x4 = 1. After nine such examples, the presentation focuses on properties of complex numbers and their applications to geometry problems. The authors point out that ‘‘during the Mathematical Olympiad Summer Program 2006, J. Bland discovered the following simpler solution’’. It is pretty common to give credit to a young mathematician for a new solution to an elementary problem. The way that Andreescu and Gelca note the successful interaction during a preparation program reminds us of the way that T¸ it¸ eica wrote in Gazeta matematica˘ about the former winners of that contest (such as Dan Barbilian, who competed several times before WWI and won the first prize), about their original solutions and the inspired creative novelty. With this, the authors promote a model of originality and excellence, a model that has successfully worked (they do this several times throughout the book). Another very interesting example is section 2.2.4, The Location of the Zeros of a Polynomial. Such a theme has only remote connections with the tradition of the Putnam competition, but the idea of discussing problem-solving strategies of such a class of problems is not at all extraneous. Only two of the nine problems included in this short section are actual contest problems (they come from the Romanian Mathematical Olympiad and the Hungarian Mathematical Olympiad). However, it could be quite useful for a student preparing for the Putnam competition to spend time reflecting on a problem such as: Let a 2 C and n 2: Prove that the polynomial equation axn + x + 1 = 0 has a root of absolute value less than or equal to 2. One very successful section is 3.2.8, on Definite Integrals (we feel it is one of the best in the book). If an

undergraduate student sees for the first time in a competition a problemR such as number 458, which asks for the p=4 computation of 0 lnð1 þ tan xÞdx, she/he may have the feeling that the problem is entirely artificial. However, by placing it in a context where several problems of the same kind are presented in a unitary context, the isolated case stands a better chance of revealing a method. In particular, while discussing this type of problem, Andreescu suggests that by looking at the integral 458 he got the idea of writing a question such as 459, which asks the reader to compute Z 1 lnð1 þ xÞ dx: 1 þ x2 0 Problem 459 was an actual contest problem in the 66th edition of the Putnam Competition, in 2005. On the other hand, the section titled The Many Versions of Stokes’ Theorem doesn’t seem to display the same imagination and variety of problems. The authors point out that ‘‘this is probably the most difficult section of the book.’’ It must have been difficult to write as well, since there are not too many problems on Stokes’ Theorem approachable at an undergraduate level and with a degree of originality that would recommend them for inclusion in such a problem book. The presentation mentions as theoretical preparation Green’s, Stokes’ and Divergence theorems, then presents Cauchy’s integral theorem (which is not really an idea connected to the tradition of the Putnam competition). The problems include theoretical results, such as the standard computation of the area of a planar region by Green’s theorem or Gauss’ law on the total flux of the gravitational field through a closed surface. This section includes standard examples of computation of the flux of a vector field across a region, which are quite standard in multivariable calculus textbooks. Perhaps the most interesting example included here is the problem that asks for the computation of a line integral along Viviani’s curve. Overall, it seems that the major challenge is to create imaginative applications of these theorems that could be asked at the undergraduate level. There are not too many such examples in the proposed problems sections of mathematical journals. How many applications of Green’s theorem, for example, are published in the problem section of the American Mathematical Monthly? Overall, with this volume the array of remarkable problem books has gained a new addition that could be really useful to undergraduate students. It is a book about excellence in mathematics, coming from a long cultural tradition whose history and experience can only help us deepen our understanding of how mathematics could be taught in a more attractive and inquisitive way.

REFERENCES

[1] G. L. Alexanderson, L. F. Klosinski, and L. C. Larson, The William Putnam Mathematical Competition, Problems and Solutions: 1965-1984, The Mathematical Association of America, 1985. [2] B. Demidovich (editor), Problems in Mathematical Analysis, Mir Publishers, Moscow, seventh printing, 1989.


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[3] A. M. Gleason, R. E. Greenwood, and L. M. Kelly (editors), The

[6] D. O. Shklarsky, N. N. Chentzov, and I. M. Yaglom, The USSR

William Putnam Mathematical Competition, Problems and Solutions: 1938–1964, The Mathematical Association of America,

Olympiad Problem Book, Selected Problems and Theorems of Elementary Mathematics, Dover Publ. Inc., 1993.

1980. [4] K. S. Kedlaya, B. Poonen, and R. Vakil, The William Lowell Putnam Mathematical Competition, 1985-2000, Problems, Solutions and Commentary, The Mathematical Association of America, 2002. [5] Ta-Tsien Li (editor) et al., Problems and Solutions in Mathematics. Major American Universities Ph.D. Qualifying Questions and Solutions, World Scientific, 1998.

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Department of Mathematics California State University at Fullerton Fullerton, CA 92834-6850 USA e-mail: [email protected]; [email protected]

Reviews

Theories of Mathematics Education: Seeking New Frontiers by Bharath Sriraman and Lyn English SERIES: ADVANCES IN MATHEMATICS EDUCATION, BERLIN AND HEIDELBERG: SPRINGER SCIENCE, 2010, 668 PP., $129, ISBN: 978-3-642-00741-5 REVIEWED BY KRISTIN L. UMLAND

heories of Mathematics Education is the first volume in a new Springer book series, Advances in Mathematics Education. The series intends to ‘‘integrate, synthesize, and extend’’ work in the field so that promising ideas can be improved and built upon. The seed for this inaugural book was the 2005 meeting of the International Group of the Psychology of Mathematics Education. Theories of Mathematics Education includes 19 essays, most with a preface and at least one commentary (a useful structure for such a wide-ranging and dense compilation of essays), collectively written by 52 contributors from 13 countries. The book pulls together the eclectic and sometimes contradictory ideas that have been advanced by theoreticians in mathematics education. As Stephen Lerman suggests in Chapter IV, mathematics education as a field of inquiry has what the linguist Basil Bernstein calls a ‘‘weak grammar,’’ that is, ‘‘a conceptual syntax not capable of generating unambiguous empirical descriptions.’’ It is not surprising, then, that the term ‘‘theory’’ is used throughout the book with clearly different meanings. Consider the difference between the notion of a scientific theory and a theory in literary criticism; both flavors are discussed or advocated in this book at different points, sometimes by the same author. Paul Ernest argues in Chapter II that the term ‘‘theory’’ is often used where ‘‘philosophy’’ might be more appropriate. His criteria for a set of ideas to rise to the status of a theory are that it must be sufficiently specific and also testable; clearly not all contributors share this perspective. The essays fall into three categories: (1) reflections on the philosophical foundations of mathematics education as a field and how education researchers should orient themselves to their work, (2) theoretical perspectives from other disciplines that could be brought to bear on mathematics education research, and (3) descriptions of theories (or proto-theories) of the processes underlying the teaching and learning of mathematics. Of course, this is not a perfect categorization scheme, and a number of essays could be assigned to more than one of these categories. However, Chapters I, III, IV, V, and XV are primarily about the philosophical foundations of mathematics education. Chapter IV provides a framework that I found helpful in understanding some of the other chapters in the book. Stephen Lerman introduces Bernstein’s distinction between weak and strong grammars (mentioned

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previously) as well his distinction between disciplines with vertical and horizontal knowledge structures. Vertical knowledge structures grow by integrating previous theories into new theories; horizontal knowledge structures grow by inserting new theories alongside older theories, with the possibility that they may be ‘‘incommensurate.’’ He suggests that science has a vertical knowledge structure and the field of education has a horizontal knowledge structure. We will return to this framework at the end of the review. The first chapter, written by the editors, summarizes the contents of the book and anchors it within historical work of some of the foundational thinkers in mathematics, philosophy, education, and the intersections of these disciplines. I wish I had understood this before I read the book, because without this context this essay seemed to be a random agglomeration of loosely related ideas. However, after having read a significant portion of the book I went back to this chapter and it made considerably more sense to me; other readers might consider this approach as well. In Chapter III, Frank Lester attempts to address three issues: What is the role of theory in educational research? How does one’s philosophical stance influence the sort of research one does? What should be the goals of mathematics education research? He argues that a ‘‘theoretical framework’’ enforces an ideological approach to research and that a ‘‘conceptual framework’’ that is ‘‘built from an array of current and possibly far-ranging sources’’ is a potential alternative. Mathematics education researchers, he suggests, should appropriate whatever theories seem relevant to the problem at hand, an approach known as bricolage. He notes the tension between theoretical and practical pursuits in educational research and proposes a model integrating them. In Chapter V, Richard Lesh and Bharath Sriraman argue that the research paradigm in mathematics education should be modeled after the ‘‘design sciences,’’ such as architecture and engineering, which are as much about solving real problems as developing relevant theories. They suggest that a critical activity should be developing operational definitions of key concepts, and they note that one of the weaknesses of the research record in mathematics education is that there is little ‘‘accumulation.’’ Whether or not researchers embrace the idea of mathematics education as a design science, their call to develop operational definitions and to work toward accumulation is a call to develop theories of mathematics education with a stronger grammar and vertical knowledge structures. In Chapter XV, Angelika Bikner-Ahsbahs and Susanne Prediger address the plurality of theories in mathematics education and the criticism that mathematics education research lacks focus by describing ways to link different theories together by ‘‘networking of theories.’’ They begin by noting that people use the word theory in different ways and review some efforts by scholars to define the term more precisely and describe the various roles that theories play. Like several authors in this book, they trace this multiplicity of theories to the complexity of the systems being studied. They then suggest various approaches to knitting theories together. In Chapter XVI, Helga Jungwirth and Uwe Gellert present two detailed examples of the Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 2, 2011

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application of such networking. Gellert also provides a much-needed critique of the bricolage approach. Five chapters present theoretical perspectives from other disciplines. In Chapter VII, Luis Moreno-Armella and Bharath Sriraman advocate attending to the semiotic dimension of mathematics. In Chapter X, Stephen Campbell discusses integrating cognitive neuroscience and psychophysiology with educational research. In Chapter XIV, Judith Jacobs brings a feminist perspective into the mix. In Chapter XVII, Andy Hurford suggests ideas from complexity theory that could be brought to bear. In Chapter XVIII, Nathalie Sinclair suggests that the study of gesture, intuition, and aesthetics can inform our understanding of tacit (as opposed to explicit) mathematical knowledge. The remaining chapters provide descriptions and applications of theories (or proto-theories) related to teaching and learning mathematics. In Chapter II, Paul Ernest describes different variants of constructivism. In Chapter VI, John Pegg and David Tall review theories of mathematical concept formation. In Chapter VIII, Gerald Goldin suggests that discrete mathematics is a useful mathematical discipline for teaching problem-solving heuristics, whereas in the commentary for this chapter and the preface for the next, Jinfa Cai points out that past theories about the development of problem solving are problematic and that new research is very much needed. In Chapter IX, Lyn English and Bharath Sriraman develop this theme at length, and suggest directions for future research on problem solving. In Chapter XI, Guershon Harel describes his DNR framework for mathematics instruction. In Chapter XIII, Gu¨nter To¨rner, Katrin Rolka, Bettina Ro¨sken and Bharath Sriraman discuss an application of Alan Schoenfeld’s theory of Teaching-in-Context. Finally, in Chapter XIX, Bharath Sriraman, Matt Roscoe, and Lyn English discuss the politics of mathematics education. Given the wide-ranging nature of these chapters it is not possible for me to do them justice in this short space; however, the interested reader should now have a better idea of where to begin. Several themes recur; two in particular caught my attention. The first is the issue of a plurality of theories versus a ‘‘grand theory.’’ Much is made of the complexity of educational settings. However, there is also tremendous complexity in biological systems, for example, and yet there are many useful theoretical structures in that discipline. It is instructive to compare briefly the two areas of inquiry. As with mathematics education, there is a multiplicity of theories in biology both within and across different levels of analysis; for example, there are theories about processes within cells, theories about the structure and function of nervous systems in organisms, theories about habitats for individual species, and theories about ecosystems. Note, however, that there is just one theory about the structure and function of the nervous system of the frog. The multiplicity of theories in biology is meant to explain different phenomena; there are not usually multiple theories for the same phenomenon, and when there are, there is an effort to sort out which one is most consistent with both the available empirical evidence as well as neighboring or overlapping theories. Furthermore, the layered nature of biological theories allows that they all be

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subsumed within the grand theory of biology, namely, the theory of evolution. Although we are nowhere near such an elaborately structured set of theories in mathematics education, the complexity of educational systems does not preclude the possibility that a well-articulated set of linked and nested theoretical structures could be developed over time. I would argue, furthermore, that this should be a long-term goal of researchers in the field. The second (clearly related) theme is the question of whether research in mathematics education is or should be ‘‘scientific’’ or at least more scientific. I would have liked to see this issue addressed more directly, though the contributions of several authors shed light on what this would mean and how it might be accomplished. A clear implication of Lester’s interpretation of Bernstein’s theory is echoed by Lesh and Sriraman: for mathematics education research to become more scientific, mathematics educators must work to develop more tightly defined concepts (i.e., a stronger grammar for the field) that are organized within vertical knowledge structures (i.e., that lead to accumulation). The discussion of theories about and research into problem solving in Chapters VIII and IX is one of the few examples in the book of a research area that reflects this scientific process. Early theories about how problem solving works and should be taught, which sprang from the heads of philosophically-minded people, were initially embraced by practitioners and researchers. However, the empirical record has shown that those theories are problematic, and researchers are now calling for revisions as well as new research. The insights gained through this accumulating process and the forward planning suggested by the authors and commentators of these chapters should serve as a model for the work of mathematics educators. As Lesh and Sriraman point out, mathematics education as a field of inquiry is still in its infancy. Theories of Mathematics Education represents a much needed effort to bring some coherence to the theoretical foundations of this young field, although it is safe to say that there is still a tremendous amount of work to be done. The authors have laid the groundwork for a natural next step, namely, a systematic survey of the big questions in mathematics education that need to be addressed. This survey should include a discussion of research methods that might be appropriately used to investigate them and weaknesses in both the relevant empirical record and extant theories, many of which are still very immature and should necessarily be refined as time passes. The ultimate test of the value of the ideas in this book is whether they or their progeny help solve the problems that teachers, administrators, and policymakers face as they work to improve mathematics teaching and learning.

Department of Mathematics and Statistics 1 University of New Mexico MSC01 1115, Albuquerque NM 87131-0001 USA e-mail: [email protected]

36 Arguments for the Existence of God by Rebecca Newberger Goldstein NEW YORK: PANTHEON BOOKS, 2010, 416 PP., US$27.95, ISBN: 978-0-307-37818-7 REVIEWED BY ULF PERSSON

he Enlightenment meant, if anything, a rejection of religious authority and a championing of human reason. So thorough was its triumph that it put religion permanently on the defensive, and nowadays any literal belief in its tenets is likely to be met by scorn, especially by its ‘‘enlightened’’ defenders. Modern Western Society is indeed highly secular, which means that religion is not allowed to interfere in the affairs of society, having its role reduced to the private sphere. In particular a doctor who eschews modern clinical practice for religious ones is bound to find himself facing criminal charges for quackery. Yet there is a need for vigilance. Religion has indeed easily accommodated itself to the advances of physics, even finding some kind of confirmation in modern theories of cosmology. In fact what better illustration of the opening lines of Genesis than a Big Bang! Evolutionary theory, especially in its Darwinian version, has struck much deeper chords in the imagination of the public, and thus presented a much more serious affront to human dignity. The controversies of the late Victorian era, pitting scientists against clergymen, and their eventual resolution, are too well-known to need to be recalled. Yet, the victory was not complete - what victory is? To seasoned European observers, the very vocal role played by fundamentalist Christians in American society is a disturbing one, especially as its rhetoric is often echoed by established politicians espousing similar agendas. One upshot of this religious revival is the pseudo-science of socalled ‘‘Intelligent Design’’ and the demand, based on spurious references to freedom of expression, that it be given equal time with Darwinism in schools. Thus it is after all hardly surprising that someone like Richard Dawkins, taking his position as Professor of the Public Understanding of Science seriously, not to say literally, wages a full-scale war at the spectacle of ‘‘Intelligent Design.’’ Not only that, he declares war on Religion in general, trying to eradicate it on all its turfs. Religion, he claims, is the root of all evil. Dawkins’s ire is directed not only at the so-called fundamentalists, be they Christian or Islamic, but also against their apologists with their wishywashy ideas, who should know better and not act as enablers. Such a mission, undertaken with the fury of an Old Testament Prophet, may be cause for ironic comments. But it also elicits admiration for a consistent and uncompromising stand set to complete once and for all the project instigated by the Enlightenment, making reason finally triumph, and rid mankind of the last vestiges of sentimental superstition. This campaign has produced a series of books

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by Dawkins himself (The God Delusion) as well as by Christopher Hitchens (God Is Not Great) and the philosopher Daniel Dennett (Breaking the Spell). In their wake, Rebecca Goldstein’s new novel makes a timely appearance. The title of the novel and the enthusiastic endorsement by Hitchens on the back cover suggest that Goldstein has provided a fictional complement to the general attack led by Dawkins. The appendix contains a blow-by-blow refutation of all possible proofs of the existence of God! However, a closer reading of the novel reveals a more nuanced and complicated relationship to religion. In an interview with Robert Wright on bloggingheads.tv (available through her official web-site, http://www.rebeccagoldstein.com) she explains that a large part of the motivation for writing the book was her sense that militant atheists, with whom she philosophically agreed, lacked an understanding of what made religious people tick and so tended to diminish them. This sensitivity is hardly surprising given her Orthodox Jewish background. Goldstein understands the deep sense of shared community that a religious faith can engender. That experience was not nullified by the usual shedding of belief during adolescence and a career as an academic philosopher. For thirty-odd years Goldstein has pursued a dual career as philosopher and novelist. She is not the woolly Continental kind of philosopher, nor of the strident French deconstructionist variety; she adheres to a hard-nosed, no-nonsense analytic version, where there is a real premium on hard thinking. As a novelist, Goldstein has explored the imaginative ramifications of her pursuit. Her first novel, The MindBody Problem, is obviously, as most first novels tend to be, somewhat autobiographical. Like her fictional narrator (also raised in an orthodox Jewish family), Goldstein was bright and pretty, a combined blessing and curse. Encountering philosophy in her adolescence, a totally new, much wider world opened to her. She was propelled to academic stardom in her undergraduate years, but graduate studies at an elite institution led her to doubt her inherent ability and thus ultimately her reasons for being. This experience is, I believe, common among ambitious burgeoning mathematicians, who thus can easily identify themselves with the agonies of Goldstein’s alter ego. Indeed, through her academic philosophical studies, Goldstein came into contact with mathematicians and theoretical physicists and was intrigued by these supposedly most cerebral of beings. The older husband in her first novel is a brilliant mathematician; at that time Goldstein was married to a mathematical physicist. Her fascination with mathematicians makes her, or should make her, a particular interesting novelist for mathematicians, who are not used to being portrayed in fiction. And in fact, as she points out in the interview mentioned previously, mathematicians, unlike philosophers, were thrilled to be portrayed. Goldstein’s new novel, 36 Arguments for the Existence of God, weaves together a number of themes, mathematics being just one of them, pertaining to the phenomenon of religion. Central to her book is the story of a young math genius resembling the protagonist of Aldous Huxley’s short novel Young Archimedes. Like the young boy in Huxley’s story, her hero is born into a community that neither understands nor appreciates his gift. This naturally leads to a dilemma as the child grows into adulthood and has to make Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 2, 2011

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DOI 10.1007/s00283-011-9209-4

momentous decisions. Should he follow his individual gift and pursue mathematics, or should he accept his other destiny as the rabbi and head of the community that has nurtured and revered him? To grasp this dilemma fully, the reader must understand the truly dominant role of a rabbi in such a community. The decision takes on urgency when his father suddenly dies of a heart attack. I will not disclose his, or rather the author’s, way of resolving the dilemma. This story, a central subplot of her novel, is somewhat reminiscent of Philip Roth, although not quite as obsessive as he tends to be. She evokes with great conviction a Pennsylvania-Dutch-like community situated on the Hudson River. From a novelistic point of view it is one of the most successful features of her novel, drawing as she obviously does from her own experience. Her point is clearly that such a community has an intrinsic value. And what defines and holds a community together is a religious faith, which thus should not be looked at literally for what it says, or seems to say, but for what it actually makes people do. The story of the young genius is embedded in a wider academic context, and here her treatment makes one think of David Lodge, although she is not quite as hilarious as he at times manages to be. The main protagonist of the novel is a certain Cass Seltzer, a self-described failed professor at an obscure institution cowering below the august one around Harvard Square. Seltzer is a student of Jonas Elijah Klapper, the Extremely Distinguished Professor of Faith, Literature and Value, and an academic egomaniac, with an overflowing body to match. (Klapper, evidently modeled on the legendary Harold Bloom, Professor of English at Yale, is another of Goldstein’s feats of imagination.) Seltzer is currently involved with Lucinda Mandelbaum, a very successful game-theorist who, like Nash, has an equilibrium named after herself, and whose aim is to reduce psychology to game theory. Mandelbaum is frustrated by irrationality, causing our protagonist to wonder plaintively how one can be a psychologist and not deal with irrationality. He has previously been married to a French poet with a mathematical connection, and also has an old girlfriend hovering around in the background. For being a loser he does not suffer a shortage of women. Seltzer approaches religion from a psychological angle. This has of course been done before, unsympathetically by Sigmund Freud (The Future of an Illusion) and far more sympathetically by William James (Varieties of Religious Experience). Seltzer acknowledges his debt by calling his book Varieties of Religious Illusion. In mathematical jargon, Seltzer’s book is a kind of fibre product of the two preceding works. Freud, very much in vogue in the first half of the previous century, proposed a theory of mind that promised not only to unlock human neuroses, but also to explain the human condition, in particular the phenomenon of religion as wish fulfillment. James explored religious experiences as examples of inescapable psychological realities. As a philosopher, William James is known for his pragmatic, no-nonsense approach with a concomitant disdain of metaphysical cant, especially of the Hegelian variety. But James carried his pragmatism almost to the point of silliness, and his professed materialism did not quite mesh with his temperament. Moreover, like many skeptics, he often 76


displayed a touching credulity, in particular in his long-term study of various mediums. Thus one suspects that James was not a wholly disinterested reporter on religion. In fact one of the arguments for the existence of God that Goldstein presents in order to demolish it is inspired by the pragmatism of James’s ‘‘The Will to Believe.’’ No such criticism can be levied against the fictitious Seltzer. He attacks the position that without religion there is no morality nor any spirituality, and hence that a secular society is brutish and empty. On the contrary, only by jettisoning religious illusions can we develop a truly disinterested morality, he says. One suspects this position is close to the author’s. Varieties of Religious Illusion propels Seltzer to stardom. Dubbed ‘‘an atheist with a soul,’’ he appears on the cover of (Time) magazine. He also gets a very attractive academic job offer. Far from delighting his partner, this sends her packing, whether out of irrational resentful envy or because life is a zero-sum game. At the climax of the novel, Selzer is pitted against a hot-shot economist, Felix Fidley, in a public debate. Hard-hitting Fidley is posturing (in a true sophistic spirit?) as a believer, but is forced to acknowledge defeat after Seltzer makes an impassioned plea for the intrinsic superiority of an atheist’s morality. The title of the novel refers both directly to the appendix printed at the back and to the 36 chapters of the novel itself. (If there is a deeper connection than cardinality between those chapters and the purported proofs presented of God’s existence, it evades me completely.) Each argument for the existence of God is commented on and rejected, a rather easy exercise for a philosopher in the analytic tradition. The only argument that really generates any sympathy in Goldstein is that proposed by Spinoza, the subject of her latest nonfiction book. Yet the emphasis on arguments is in the old scholastic tradition of trying to use reason to divine what really should only be accessible by blind faith. And here, somewhat surprisingly, mathematics may enter. Mathematics is usually considered the most rational of human pursuits. Not (says the philosopher C. S. Peirce) as the science of necessary thought, but proceeding by necessary thought. But even in mathematics we come up against the limits of thought and thereby touch faith. (As a child, Bertrand Russell was horrified when told that axioms are not to be proved.) We may evade this theological aspect by treating mathematics as a game, in which we can freely choose our assumptions as long as they are not inconsistent. But consistency cannot be proved on first principles, and without consistency the games themselves collapse. Are mathematicians in pursuit of God, in the guise of Ultimate Truth? Theirs would be an esoteric inhuman God, if any. Most mathematicians are not concerned with such questions, at least not professionally; but Goldstein’s conception of mathematics is very much connected to foundations and logic and the figure of Kurt Go¨del, about whom she has also written a biography. The mathematical references in the novel go beyond the tenuous connections I have just sketched. In her initial description of the young Azaraya, Goldstein has him discover prime numbers and find patterns of successive differences of the various integral powers. Later she has him rediscover Euclid’s proof of the infinitude of primes

and, in a didactic aside, spells it out in detail for the uneducated reader. Allusions to Russell’s paradox are sprinkled throughout the text, hidden references that finally come to the fore when the poor kid at the age of 16 has to decide whether to follow the family tradition and the expectations of his community, or pursue his own mathematical calling. At that time the author sees to it that he gets a subscription to Annals of Mathematics, and meets with his future tutor, a more mature and established genius at MIT. A mathematician may wistfully wonder who among his colleagues would read the Annals from cover to cover, but our young genius is at the beginning of his potential career and is still blessed with truly omnivorous tastes. All in all, the mathematical references in the novel are more in the nature of garnish than in any way related to the plot or the question of God’s existence.

For all her appreciation of mathematics and mathematicians, I am disappointed that Goldstein has nothing more original to say about them than to perpetuate romantic and hackneyed notions of mathematicians as strangers in this world and endowed particularly with musical talents. On the other hand, the fault may not be hers. Maybe, seen from the outside, mathematicians are indeed rather otherworldly creatures. The external view tends to emphasize similarities, whereas the view from inside exaggerates differences. Department of Mathematical Sciences Chalmers University of Technology SE-412 96 Go¨teborg Sweden e-mail: [email protected]


77

Stamp Corner

Robin Wilson

Recent Mathematical Stamps: 2007, The Golden Ratio

I

n 2007, as part of a series in science and technology, the Macau Post Office issued a set of stamps featuring topics related to the golden ratio, U = (1 + H5)/2:

• the Fibonacci sequence 1, 2, 3, 5, 8, 13, . . ., illustrated by numbers of rabbits (from the problem that gave rise to it); • two intersecting sets of spirals, as may be seen on a sunflower;

â Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics, Computing and Technology The Open University, Milton Keynes, MK7 6AA, England e-mail: [email protected]

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DOI 10.1007/s00283-011-9205-8

• Penrose tilings, constructed from two shapes known as kites and darts; • the spiral pattern that appears on a nautilus shell. There was also a souvenir sheet featuring the golden ratio and a number of constructions that lead to it.

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