Letters to the Editor
The Mathematical lntelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.
More on Vilnius
Zygmund at Mount Holyoke
I was outraged when I read the article
In the article "Vilnius Between the
titled "Vilnius between the Wars" in the
Wars" (Mathematical InteUigencer 21, 4, Fall 2000), the authors neglect to mention the five years, 1940-45, during
Fall 2000 issue of
The lnteUigencer.
Vilnius, or Wilno as the authors point out it was known in Polish, or Vilna, as
which Antoni Zygmund served on the
it was known by the 60,000 Jews who
Mount Holyoke College faculty. Indeed,
lived there-some
300-4> of the popula
tion-was a center of the flowering of
President
Ham
of
Mount
Holyoke
had been helpful in making it possi
Jewish life and culture in Poland. Jews
ble, through diplomatic channels, for
had made up a substantial proportion
Zygmund to leave Europe.
of the population for over 500 years; in
Moreover, emigration to the United
fact Napoleon had dubbed Vilna the
States for an academic at that time de
Jerusalem of Poland. All of this is doc
pended to a large degree on the as
From that Place
surance of a teaching position. Mount
and Time, a memoir by the noted his
Holyoke gave that assurance without
umented in the book
the advantage of a personal interview,
torian Lucy S. Dawidowicz. As for the University, it was re
having the courage to assume the risk,
nowned, not only as a center of learn
even though there was surely much un
ing, but as a hotbed of anti-Semitism.
certainty about Zygmund's suitability for
Its students repeatedly incited violence
an undergraduate college in which few
against
men had served as faculty members.
the Jewish
population
and
eventually pressured the administra
Norbert Wiener arranged for Zyg
tion into "ghetto bench" seating in the
mund to spend the Spring semester of
back of the classrooms for the few re maining Jewish students. The violence reached a peak in
1931 and led to the
1940 at MIT and in March drove him to him to
South Hadley to introduce
President Ham and the mathematics
death of a Polish student. Every year
chair, Marie Litzinger. In an amusing
thereafter, students from the University
notation by Marie Litzinger on a letter
would rampage in the streets on the an
from Zygmund, Wiener is identified for
niversary of his death, beating up Jews
President Ham's benefit as the son of
and breaking shop windows.
Leo Wiener, philologist and translator
None of this is mentioned in the ar ticle, which purports to be a capsule
of Tolstoy. In
1988 Mount Holyoke awarded
history not just of mathematics in
Zygmund an honorary doctorate and
Vilnius, but of the University itself, and
held a symposium in his honor at
of its role in the history of the city. This
which several of his former students
has the sad effect of appearing to make
participated, including Felix Browder,
the authors part of the ongoing attempt
Ronald
to expunge the role that Polish anti
Guido Weiss. Marshall Stone also came
Semitism has played in the unfortunate
to greet his old friend.
history of that country.
Coifman,
Peter Jones,
Upon Zygmund's death in
and
1992, the
larger part of his mathematical library,
Jacob E. Goodman
including many volumes in Polish, was
Department of Mathematics
left to the Mathematics Department at
City College, CUNY
Mount Holyoke, where it is displayed
New York, NY 1 0031
in the Department's seminar room.
USA e-mail:
[email protected] We also have a file of correspon dence dating from his last year in
© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 3, 2001
3
Poland (1939). A letter to J. D. Tamarkin of Brown University from Wilno (Vilnius), written on 5 November 1939, concludes with the paragraph:
I do not, of course, do any mathe matical work now. Even ifIforget my personal troubles, I cannot forget the horrors of the war and the immense suffering of people who had to leave Warsaw, sometimes with small chil dren, and were the whole time bombed by German aeroplanes. In particular, Wilno isfull of such people and I doubt if all of them will be able to survive the coming winter. Not to mention the thoughts of the barbarous treatment by the Germans of the Polish and Jewish population in the occupied territory. It is sometimes difficult to imagine the horrors. Finally, from our perusal of the College archives in preparation for the honorary degree, it became clear that Zygmund was extremely successful as a teacher at Mount Holyoke. One might have supposed that a research oriented mathematician speaking in a language not his own might be severely disadvantaged in teaching, but the record belies this. His departure at the end of WWII was deeply felt.
(the Madelung constant; see [1], [2]). Moreover, there are quite fast and sim ple formulas for r2 and r4: r2 is 4 times the number of divisors of n, unless there is a prime p congruent to 3 (mod 4) such that pm divides n, pm+l does not, and m is odd, and in that case it is 0; r4(n) is 8 times the sum of those di visors of n that are not divisible by 4. Ewell mentions that his algorithm 3 for r3(n) is O(n 12). One ought to point out that the same complexity is ob tained by simply examining all lattice points inside or on the sphere of radius Vn and counting those that are on it. But there is another approach that is much faster, having complexity O(n): count the number of representations of each possible n - i2 as a sum of two squares. Precisely,
r3(n) = r2(n) + 2
Vn
I
[r2 (n - i2)].
i=l
This function (with some further im provements as discussed in [3]) is in Mathematica as SumOfSquaresR, and it takes only about a second to deter mine the value of r3(n) for n < 107. REFERENCES
[1 ] D. Bressoud and S. Wagon , A Course in Computational Number Theory, Key College Press (Emeryville CA) and Springer-Verlag (NY), 2000.
Lester Senechal
[2] R. E. Crandall, New representations for the
Harriet Pollatsek
Madelung constant, Experimental Mathe
Mathematics Department
matics 8 (1 999), 367-379.
Mount Holyoke College
[3] E. Grosswald, Representations of Integers
So. Hadley, MA 0 1 075
as Sums of Squares, Springer-Verlag (NY),
e-mail:
[email protected] 1 985.
[email protected] Stan Wagon Macalester College St. Paul, MN 551 05 LaHice Points on Spheres,
USA
Quickly
e-mail:
[email protected] The article by John Ewell, "Counting lattice points on spheres," in the Fall 2000 issue raises the interesting ques tion of whether there is a good algo rithm to compute r3(n), the number of representations of an integer n as a sum of three squares. This is an im portant and intriguing problem in that there is an intimate connection be tween this function and the amount of energy holding a salt crystal together
4
THE MATHEMATICAL INTELLIGENCER
John Ewell comments: To Stan Wagon's informative letter I add one observation, pertinent when very large n are in question: my algorithms are purely additive, not requiring factoring of n or any other integers involved.
Granville Sewell's Opinion piece in our Fall 2000 issue provoked many reactions from readers-too many to handle. Here are two of them; others will follow in the next issue, with Sewell's rejoinder. -EDITOR's NOTE The Credibility of Evolution
Granville Sewell ("A Mathematician's View of Evolution," v. 22, no. 4) should stick to mathematics. Biologists who read this in The Intelligencer must feel the same way we mathematicians would were an article describing how to square a circle with straightedge and compass written by a biologist/amateur mathe matician to appear in the Journal of Population Biology. Sewell's article is riddled with errors; to refute all of them would require a response much longer than the original article. Nor are the er rors new: all the ideas have previously appeared in the creationist literature. I will limit myself to a few comments and provide pointers to sources where more detailed refutations appear. I do speak partly from the biologist's standpoint. Though a mathematician/ engineer by profession (thesis in ap proximation theory, employment at uni versities and Silicon Graphics), I have spent many months doing volunteer field work in biology, much of it with the Stanford Center for Conservation Biology. I ask you: how open-minded are you toward the fellow who sends your math department a 20-page angle-trisection technique? Oh, it could be that all pro fessional mathematicians have over looked a flaw in Galois theory for a hun dred years and that we'll have to toss out most of mathematics when the ama teur proves us all wrong. Theodosius Dobzhansky has written, "Nothing in bi ology makes sense except in the light of evolution." If Sewell is right, almost all of modern biology falls apart, and there is no theory to replace it There are ex actly zero articles published in the ref ereed biological literature about "intelli gent design." The creationists claim this is due to a giant conspiracy against them. It's pretty much the same as the conspiracy we mathematicians have against angle-trisectors. Look at Sewell's references-all are from the popular lit-
erature, and the publication dates are 1996, 1987, 1960, 1982, and 1956. Behe's "irreducible complexity" the ory, which Sewell expounds, basically amounts to statements of the form, "I can't figure out how this complex sys tem arose by natural selection, so it can not have done so." When someone goes to the trouble of providing an explana tion, that example is tossed out, and a new complex system is substituted. Responding to an infinite sequence of such challenges is pointless, especially since it's so easy to make up a challenge, and so difficult to respond. Suppose I were to claim that multiplication of in tegers is not commutative, and I'll show you by providing two 100-digit integers x andy such that xy -=!= yx. Suppose you then multiplied them both ways and showed me the result was the same, and I just gave you another pair to test. And another. And another. . . . The second law of thermodynamics does not say "natural forces do not cause ext;remely improbable things to happen"; it talks about entropy. With the sun's en tropy increasing at a fantastic rate, it's easy to spare a tiny portion of that to ac count for the decrease in biological en tropy on earth. lf I flip a coin 1000 times, and tell you the result ITHTHHT . . . , my result, whatever it may be, is ex tremely unlikely. It would only happen once in 21000 times. lf you flip coins for a billion years, it'll probably never hap pen again. The second law did nothing to prevent my unlikely result. One problem with Sewell's computer code example is that he assumes the program evolves toward a specific goal. In evolution, there is no specific target just something that works better than the competition. lf the universe were restarted, Sewell is right that it is in credibly unlikely that humans would evolve again. But something would. The National Center for Science Ed ucation (www.natcenscied.org) fights legal battles all over the country to keep creationism out of the public schools. The talk.origins archive (www.talkorigins.org) maintains a huge collection of articles that refute all the points made by Sewell and other creationists. Good popular books re futing Sewell's ideas include: Climbing
Mount Improbable by Richard Dawkins, Darwin's Dangerous Idea by Daniel C. Dennett, The Red Queen by Matt Ridley, and almost any collection of Stephen Jay Gould's essays from Natural History magazine. I deplore publication of this kind of junk, especially in a reputable journal. You can be certain that Sewell's article will be referenced for the next 20 years in the creationist literature as proof that evolution has been debunked by mathematicians. The fact that you have carefully labeled his article as an opin ion piece will have no effect. In addition, such articles make politicians think that there is merit to the creationist claims, and this leads to the exclusion of evolution, the big bang theory, and other religiously unaccept able ideas from high school science texts. There is a big enough problem with scientific illiteracy as it is, and having more people who believe that antibiotics and pesticides cannot be overused since it's impossible for bac teria and insects to evolve resistance to them will not help anyone. The Catholic church and many other major Christian churches have issued statements saying that there is no con flict between evolution and Christian faith. Only those who interpret the Bible as literally true have a problem with it. Of course the Bible also says the world is flat-see Revelation 7:1, for example. Tom Davis 24603 Olive Tree Lane Los Altos Hills, CA 94024 USA e-mail: tomrdavis@earthlink. net The Credibility of Evolution 2
Tom Davis's reply to Granville Sewell effectively responds to many of Sewell's dubious objections to evolu tion. Without attempting to be com prehensive, I would like to address sev eral additional points. 1. Any biologist can tell you how sys tems that appear to be "irreducibly complex" (i.e., all parts of the system are required for the system to function) could evolve: through scaffolding. As H. Allen Orr wrote in his perceptive re view of Behe's book:
An irreducibly complex system can be built graduaUy by adding parts that, while initiaUy just advantageous, be come-because of later changes--es sential. The logic is very simple. Some part (A) initiaUy does some job (and not very weU, perhaps). Another part (B) later gets added because it helps A. This new part isn't essential, it merely improves things. But later on, A (or something else) may change in such a way that B now becomes in dispensable. This process continues asfurther parts getfolded into the sys tem. And at the end of the day, many parts may aU be required [1]. (This and other reviews of Behe's book can be found online [2].) 2. Sewell draws an analogy between bi ological systems, coded by DNA, and computer programs. He writes "to anyone who has had minimal program ming experience this idea [that a signif icantly better program can be made by accumulating small improvements] is equally implausible." Sewell seems to be entirely ignorant of the field of genetic programming, where what he claims is implausible is routinely done [3]. The analogy between computer programs and biological systems is useful, as far as it goes. But like all analogies, it is imperfect. lf we draw conclusions from our analogy that con flict with very strong evidence- 2 there is a (2p + 1)-piece twist-hingeable dissection of a {2p} to a {p }. The dissection exhibits p-fold rotational sym metry. I discovered this family by taking the two tessella tions in Figure 16 and superimposing them as in Figure 25. Readers may note that in this superposition, points of twofold rotational symmetry coincide, although not in the same way as in Figure 16. This is less important for this dissection, because we do not use such points to position
Figure 26. Twist-hinged octagon to square. Figure 27. Twist-hinged pentagram to pentagon.
swing hinges. More importantly, the centers of the octagon and the large square coincide, as do the centers of the small squares. We can locate isosceles triangles by identifying line segments between vertices of the small square in the oc tagon tessellation and nearby vertices of the small square in the tessellation of squares. If we take the corresponding dissection, cut such appropriate isosceles triangles off cer tain pieces, and add them to other pieces, we can get the twist-hinged dissection shown in Figure 26. The reader has been left with the task of identifying these isosceles trian gl�s, but this is not so hard, since there is a twist hinge at the midpoint of the base of each one. Although we use tessellations to derive this dissection, we do not need to, and this is the key to dissecting any (2p 1 to a (pl. Just overlay the (2pl and the {p} so that their cen ters coincide and each side of the (pI intersects the mid point of a side of the (2pl. Then identify the bases of cor responding isosceles triangles, and infer the pieces. We can prove correctness with the use of fairly simple trigono metric identities. Because the method is related to the com pleting the tessellation method, but in general is not based on tessellations, I call it completing the pseudo-tesseUa
tion. Remarkably, there is another family of dissections of a similar nature. Consider any integers p > 4 and 2 ::;:; q ::;:; (p + 1)/3. Then there is a (2p + 1)-piece twist-hingeable dis-
Figure 28. Superposition of hexagrams, triangles, and hexagons.
section of the (p/q I to the {p 1. The dissection exhibits p-fold rotational symmetry. As an example, a twist-hingeable dis section of a pentagram to a pentagon is shown in Figure 27. The approach is the same as what we used for the previous family of dissections, if we treat the reflex angles of the (p/q I as vertices too. To ensure that each side of the (pi inter sects a side of the (plq I at a midpoint, p and q must satisfy the condition 4 cos(q1rlp) cos((q - 1)1r/p) ;::: cos(1r/p). For positive integers q > 1 and p ;::: 2q + 1, this condition is equivalent top ;::: 3q - 1. That the same approach works for both {plql and (2pl suggests that these can be unified. Indeed, this is the case, if we relax the constraint that q be a whole number and interpret (plql appropriately. The previous family of dissections includes a thirteen piece dissection of the hexagram to the hexagon. We can do better. In [ 1 7] , I gave a seven-piece unhingeable dissec tion of a hexagram to a hexagon, which has three pieces that we must turn over. However, the pieces that we turn over are the key to a nifty twist-hingeable dissection. I de rived my seven-piece unhingeable dissection by complet ing the tessellations, as shown in Figure 28. The solid lines in Figure 29 indicate the seven-piece un hingeable dissection. We add two isosceles triangles (indi cated by dotted lines) to each of the three small triangles, giving three triangles that we can twist-hinge. Producing the three new triangles does not yield a dissection that is completely twist-hingeable, because there is an equilateral triangle that we must transfer from the center of the hexa gram to the center of the hexagon. Following an approach
Figure 29. Derivation of a twist-hingeable hexagram to a hexagon.
VOLUME 23, NUMBER 3, 2001
17
Figure 30. Twist-hingeable dissection of a hexagram to a hexagon. Figure 31 . Intermediate configuration of a hexagram to a hexagon.
similar to that in Figure 17, we identify two irregular tri angles that can swap positions, as shown with dashed edges. To make these new pieces twist-hingeable, we in troduce more isosceles triangles (dotted edges). As luck would have it, we can glue all four of these isosceles triangles together, producing a trapezoid for our ten-piece twist-hingeable dissection (Fig. 30). An observant reader will see that I have cyclicly hinged eight of the pieces, and a skeptical reader may wonder if this actually works. I was not sure myself whether something so re markable was possible until I had constructed and tested a rough model out of a thin foamboard and toothpicks. Afterwards, I verified mathematically that it does indeed work Note that there are five pieces centered on the trape zoid that play somewhat the same role as each of the two large pieces. To convert the hexagram to hexagon, flip the two large pieces and the trapezoid-centered five, rotating them simultaneously about the axes shown with dotted lines, while partially turning the small triangles so as to ac commodate the differing levels of the twist hinges on the parallel edges. Figure 31 shows a perspective view of the configuration after rotating the pieces by 90° from their po sition in the hexagram. The three dotted lines identify an imaginary equilateral triangle that stays fixed as the pieces rotate. Each vertex of this triangle is the center of a smaller equilateral triangle (not shown) adjacent to the long edge of a small triangle. Furthermore, the axis of the twist hinge between a large piece and a small triangle pierces the cen-
Figure 32. Lindgren's unhingeable dodecagon to square.
18
THE MATHEMATICAL INTELLIGENCER
ter of that smaller equilateral triangle. As the large pieces complete their turning, the small pieces return to their orig inal side up. Again, Wayne Daniel crafted a wonderfully pre cise model of this dissection for me. Oftentimes, special methods produce twist-hinged dis sections that are not possible using the general techniques. A final treat, of a twist-hinged dodecagon to a square, il lustrates this point. Harry Lindgren [24] gave a six-piece un hingeable dissection of a dodecagon to a square (Fig. 32), on which I base a twist-hingeable dissection. Dotted lines in Figure 33 indicate isosceles triangles to switch from one piece to another. Add two isosceles triangles to the equi lateral triangle, and use two twist hinges to flip the result-
Figure 33. Add twists to a dodecagon to a square.
Figure 34. Twist-hinged dodecagon to square.
REFiiRiiNCES
A U T H O R
[1 ) Abu'I-Wafa' ai-Buzjanf. Kitab ffma yahtaju al-sani' min a' mal al handasa (On the Geometric Constructions Necessary for the Artisan). Mashhad, Iran: Imam Riza 37, copied in the late 1 0th or the early 1 1 th century. Persian manuscript. (2] Jin Akiyama and Gisaku Nakamura. Dudeney dissection of polygons. Res. Institute of Educational Development, Tokai Univ., Tokyo 1 998. [3) Jin Akiyama and Gisaku Nakamura. Transformable solids exhibi tion. 32-page color catalogue, 2000. [4) George Johnston Allman. Greek Geometry from Thales to Euclid. Hodges, Figgis & Co., Dublin, 1 889. [5] Anonymous. GREG N. FREDERICKSON
Department of Computer Science West Lafayette, IN 47907 USA
Ff tadakhul
al-ashkal
al-mutashabiha aw al
mutawafiqa (Interlocks of Similar or Complementary Figures). Paris: Bibliotheque Nationale, ancien fonds. Persan 1 69, ff. 1 80r-1 99v. [6) Donald
C.
Benson.
The Moment of Truth:
Mathematical
Epiphanies. Oxford University Press, 1 999.
e-mail:
[email protected] [7) Farkas Bolyai. Tentamen juventutem. Typis Collegii Reformatorum
Greg Frederickson was educated at Harvard (A.B. in eco nomics) and the University of Maryland (Ph.D. in computer sci
[8) Donald L. Bruyr. Geometrical Models and Demonstrations. J.
ence). He has been on the Computer Science faculty at Purdue University since 1 982. Most of his research is on the
[9] H. Martyn Cundy and A. P. Rollett. Mathematical Models. Oxford,
design and analysis of algorithms, especially approximation al
[1 0) Erik D. Demaine, Martin L. Demaine, David Eppstein, and Erich
gorithms for NP-hard problems, graph algorithms, and data structures. Formerly a tennis enthusiast and a bassoon player,
Proceedings of the 1 1 th Canadian Conf. on Computational
per Josephum et Simeonem Kali, Maros Vasarhelyini, 1 832.
he now plays squash and drives his children to piano lessons. He also creates harmonious motion in geometry.
Weston Walch, Portland, Maine, 1 963. 1 952. Friedman. Hinged dissection of polyominoes and polyiamonds. In Geometry, Vancouver, 1 999. [1 1 ] Henry E. Dudeney. Perplexities. Monthly puzzle column in The Strand Magazine (1 926) (a): vol. 7 1 , p . 4 1 6; (b): vol. 7 1 , p. 522; (c): vol. 72, p. 3 1 6 . [1 2] Henry Ernest Dudeney. The Canterbury Puzzles and Other Curious Problems. W. Heinemann, London, 1 907. [1 3] William L. Esser, Ill. Jewelry and the like adapted to define a plural
ing triangle around. Then use the conversion of two swing
ity of objects or shapes. U.S. Patent 4,542,631 , 1 985. Filed 1 983.
hinges to twist hinges, adding for each a piece that we turn
[14] Howard Eves. A Survey of Geometry, Allyn and Bacon, Boston,
over. Finally, use a twist hinge to bring the concave piece along, and slice and twist it to fit it in properly. The resulting nine-piece twist-hingeable dissection is shown in Figure
34.
1 963, vol. 1 . [1 5] Howard W. Eves. Mathematical Circles Squared. Prindle, Weber & Schmidt, Boston, 1 972. (1 6] Greg N. Frederickson. Hinged Dissections: Swinging and Twisting. Cambridge University Press, in production. [1 7] Greg N. Frederickson. Dissections Plane & Fancy. Cambridge
Conclusion With their visual and kinetic appeal, hinged dissections and their design techniques will continue to play a role in math
University Press, New York, 1 997. (1 8] Martin Gardner. The 2nd Scientific American Book of Mathematical
ematical recreation and education. They also invite sub
Puzzles & Diversions. Simon and Schuster, New York, 1 961 .
stantive research in mathematics and computer science.
[ 1 9] P. Gerwien. Zerschneidung jeder beliebigen Anzahl von gleichen
Hinges are the simplest of linkages, permitting only rela
geradlinigen Figuren in dieselben StOcke. Journal fOr die reine und
tive rotation between connected pieces; with hingeability
angewandte Mathematik (Grelle's Journal), 1 0:228-234 and Tat.
we address issues of transformation of objects which have wider relevance. In addition to the problem of generality discussed briefly in the introduction, there is the search for algorithms: procedures for determining whether a given dissection is hingeable, and for finding effectively a plan of motion that carries the hinged pieces from one of the fig ures to the other.
Ill, 1 833. [20] Branko GrOnbaum and G. C. Shephard. THings and Patterns. W. H. Freeman and Company, New York, 1 987. [21 ] H. Hadwiger and P. Glur. Zerlegungsgleichheit ebener Polygone. Elemente der Mathematik ( 1 951), 6:97-106. [22] Anton Hanegraaf. The Delian altar dissection. Elst, the Netherlands, 1 989. (23] Philip Kelland. On superposition. Part II. Transactions of the Royal Society of Edinburgh (1 864), 33:471 --473 and plate XX.
ACKNOWLEDGMENT This work was supported in part by the National Science Foundation under grant
CCR-9731758.
(24] H.
Lindgren. Geometric dissections. Australian Mathematc i s
Teacher ( 1 95 1 ) , 7:7-1 0.
VOLUME 23, NUMBER 3, 2001
19
[25] H. Lindgren. A quadrilateral dissection. Australian Mathematics
[32] lan Stewart. The Problems of Mathematics. Oxford University
Teacher (1 960), 1 6:64-65. [26] Harry
Lindgren.
Press, Oxford, 1 987. Nostrand
[33] H. M. Taylor. On some geometrical dissections. Messenger of
[27] Ernst Lurker. Heart pill. 7-inch-tall model in nickel-plated alu
[34] Henry Martin Taylor. Mathematical Questions and Solutions from
Geometric
Dissections.
D.
Van
Mathematics, (1 905), 35:81-1 01 .
Company, Princeton, New Jersey, 1 964. minum, limited edition of 80 produced by Bayer, in Germany,
'The Educational Times, ' (1 909), 1 6:81-82, Second series. [35] William Wallace, editor. Elements of Geometry. Bell & Bradfute,
1 984. [28] W. H. Macaulay. The dissection of rectilineal figures (continued).
Edinburgh, eighth edition, 1 831 . First six books of Euclid, with a supplement by John Playfair.
Messenger of Mathematics, (1 922), 52:53-56. [29] Aydin Sayili. Thabit ibn Qurra's generalization of the Pythagorean
[36] Eric W. Weisstein. CRC Concise Encyclopedia of Mathematics. CRC Press, Boca Raton, FL, 1 998.
theorem. Isis, (1 960), 51 :35-37. [30] I. J. Schoenberg. Mathematical Time Exposures. Mathematical
[37] David Wells. The Penguin Dictionary of Curious and Interesting Geometry. Penguin Books, London, 1 991 .
Association of America, Washington, DC, 1 982. [31] Hugo Steinhaus. Mathematical Snapshots, 3rd edition. Oxford
[38] Robert C. Yates. Geometrical Tools, a Mathematical Sketch and Model Book. Educational Publishers, St. Louis, 1 949.
University Press, New York, 1 969.
S P R I N G E R F O R M AT H E M AT I C S Gregory J. Chaltln,
IBM Research Division,
PETER HILTON, State University of New York,
GEORGE M . P H I LLIPS,
Hawthorne, NY
Binghamton, NY; D ER EK HOLTON, University of
University
EXPLORING RANDOMNESS
Otago, Dunedin, New Z ealand; JEAN PEDERSEN,
Scotland
Santa Clara University, Santa Clara, CA
This essential companion volume to Chait in's highly
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An Introduction to Special
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the interest of bright people in mathematics. The
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THE MATHEMATICAL INTELUGENCER
•
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VISIT: your local technical
EMAIL:
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on textbook exam copies. YOUR 30-DAY RETURN PRIVILEGE IS A LWAY S GUARANTEED 4/01
PROMOTION #S2561
M a thematic a l l y Bent
C o l i n Ad ams , Editor
Hiring Season
by offering to sign us up for the Math
Dec.
Employment Registry.
1 : Big blow. Costa went away for
Thanksgiving and never came back.
ept.
S
The proof is in the pudding.
Halls have become a wasteland, lit
People saying hello to one another at
tered with crumpled letters of recom
the
faculty mailboxes.
Past malice
mendation
and
strategically placed
tacks. No hope of deciding this easily.
by warm summer winds. Ganser and I
Ganser and I have dug in for the long
Sept.
term. Ganser is determined, but I fear
are hopeful this year may be different. Mathematical
7: All-out war has commenced.
ment seems to be getting along.
seems to have vanished, blown away
Opening a copy of The
Dec.
Rumor has it he's now an actuary.
7: Everyone in the depart
12: Classes underway. Ganser
for his health. His hands have been
has finagled us both onto the hiring
shaking. He needs caffeine, and soon.
committee. Algebraists are upset two
Dec.
topologists are members. Bullman and
man's sister.
mathematical journal, or what?" Or
Klimkee
applied,
now despise each other. Ganser and I
you may ask, "Where am /?" Or even
Bullman because she is applied, and
take the opportunity to do a celebra
Intelligencer you
may ask yourself
uneasily, "What is this anyway-a
"Who am /?" This sense of disorienta tion is at its most acute when you open to Colin Adams's column. Rel03. Breathe regularly. It 's mathematical, it's a humor column, and it may even be harmless.
are
pushing
for
10: Klimkee is divorcing Bull Bullman and Klimkee
Klimkee because he is married to
tory dance in the corridor. Quite a
Oct.
show, but no one's there to see it.
Bullman's sister.
12: The hiring committee still
Dec.
15: The administration may have
can't come to agreement on whether to
to step in. Work has come to a stand
serve cookies or cheese and crackers
still.
at the meetings. Ganser just wants cof
Ganser and I are hunkered down in his
Even the students are afraid.
fee. I prefer the little frosted pink wafer
office. Departmental communication
cookies,
reduced to e-mail contact only, and
but no one
else
concurs.
Bullman and Klimkee are arguing for
most of it too coarse to repeat. Ganser
Oct.
continually paces, paces. This is al
wine.
23: The hiring committee chair
and the recording secretary are no longer on speaking terms, meaning
Dec.
most as bad as last year.
20: I tried to stop him, but Ganser
was desperate. Risked all for java run.
there are no minutes for the meetings.
Grabbed a jar of instant out of the
This allows committee members to say
lounge. Brought back my mail, includ
things they otherwise wouldn't dare.
ing the latest AMS Notices. Job listings
Nov. 2: Ganser and I have enlisted the
are meager. Both despondent.
support of Costa, who although not a
Dec.
topologist, has interests in Riemann
Nervously, Ganser and I attend the hol
surfaces. Perhaps we can broker a
iday party. Initially, everyone
deal.
Refreshments are a disappointment.
22: Cease-fire has been declared.
is civil.
Nov. 12: Algebraists are no longer co
No little frosted pink wafer cookies.
operating. Meetings are deteriorating.
Spirits low all around. We depart just
Bullman keeps kicking me under the
as yelling commences. Not much hope
table and then pretending it was acci
for the new year.
dental. It really hurts.
Jan. 2: Returnees look prepared for
Nov. 14: Ganser says faculty in his
the long haul. Several carrying cof
neighboring offices are becoming rude.
feepots. We are checking to see
He is uncomfortable entering and leav
fire marshal may prevent it.
if the
Column editor's address: Colin Adams,
ing the building. I fear for the direction
Jan. 15: Ganser and I are holed up in
Department of Mathematics, Williams
in which we are heading.
my office. Ganser is screaming for cof
College, Williamstown, MA 01 267 USA
Nov. 18: Ganser and I threaten to look
fee now. He's licked the instant jar
e-mail:
[email protected] for jobs elsewhere. Klimkee responds
clean. I'm feeding him chocolates and
© 2001 SPRINGER-VERLAG NEW YORK. VOLUME 23, NUMBER 3, 2001
21
cola, but he's begging for Guatemalan Mocha Supreme. Jan. 22: Algebraists have broken. Waving a white flag, they file out, headed for Starbucks most likely. Now it's down to Applied versus Topology. Ganser drops in and out of lucidity. Feb. 14: There is hope. Administration has promised funding for a fluid dy namics person, half in math, half in en gineering, making Topology the high est remaining priority. Ganser is elated. Chair calls to tell us the good
news. Doesn't seem too angry that we haven't met our classes in a month. Feb. 29: On pins and needles. Depart ment meeting slated for tomorrow. All will be in attendance. This could be it. March 11: It's official. Ganser and I have received permission to hire. We are jubilant. March 29: Best candidates are gone. We had three interview talks, and it's not clear any of them know the difference between a Mobius band and an annulus. April 17: We have hired. Although he
only speaks a Kurdish dialect, and he's actually in number theory, he does seem to be familiar with the torus, or so it appears from our communication via sign language. May 13: Due to visa problems, our can didate cannot come after all. We will have to repeat the process next year. It is disappointing, but we consider it a learning process. Ganser has installed a cappucino machine in his office. All told, it could have been worse. One can only hope for the future.
T H E M AT H B O O K O F T H E N E W M I L L E N N I U M ! B. Engquist, University of Cal i fornia, Lo Angele and Wilfried Schmid, Harvard Univer ity, Cambridge, MA (eds.)
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22
THE MATHEMATICAL INTELLIGENCER
JESUS CRESPO CUARESMA
Po i nt S p l itti n g and C o n d orcet C rite ri a "You can 't always get what you want" The Rolling Stones
group offriends have to decide on how to spend some money they won in the lottery. Three alternatives are put forward: going on holiday, buying a new car, and giving the money to charity. One of them (let us call him Alan) pro poses to vote upon the three alternatives by each one of the friends dividing fifty points among the three competing choices freely, be
when there is an alternative x which obtains a majority of
ing as accurate as they want in the division of points among
votes in pairwise contest against every other alternative (a
alternatives (one may vote, for instance,
40.99 points for
Condorcet winner), x is chosen as winner.
9.01 for the second, and 0 for the third alter
In an augmented version of Alan's points voting proce
native), and then choose the alternative that receives the
dure, the number of votes of the Condorcet winner alter
highest number of points. Charles (another member of the
native
the first one,
lucky group) replies that he has something against such a
(if there is one) is multiplied by a fixed number {3 (> 1 ), independent of the size of the voting population, that
way of deciding upon their money, because he once read
can be as high as we want. Intuitively, such a voting pro
that this "points voting scheme" may not give the same out
cedure would seem likely to satisfy the Condorcet winner
come as ordering the alternatives from most to least pre
criterion and have better features than the original points
ferred by each voter and then choosing the one that beats
voting procedure. Yet I will prove that, provided that indi
the most alternatives in pairwise comparison.
viduals have the possibility of being as precise as they want
Alan thinks about Charles's criticism and refines his
in distributing their points among alternatives (that is, the
original voting procedure: if there exists an alternative that
number of points available for distribution is infinitely di
beats all the others in pairwise comparison, its score will
visible), this class of voting procedure does not
be multiplied by a high number (say
20).
fulfill the
Condorcet winner criterion and/or the (symmetrically de
This paper proves that, surprisingly, Charles's reserva tions about the original procedure also apply to the modi
fmed) Condorcet loser criterion for any value of {3. I begin by formalizing the class of voting procedures,
and stating some criteria that are satisfied by these proce
fication. dure that Alan proposed, in which a certain number of
Pareto criterion and the monotonicity criterion. Then I prove that our class of voting procedures
points are available for distribution among the candidates,
satisfies neither the Condorcet winner nor the Condorcet
has the great advantage that voters' personal intensities of
loser criterion.
From the point of view of the voter, the "points" proce
dures, namely the
preference can be represented. Yet Charles is right: points voting has been proved not to satisfy several criteria that reasonable voting procedures might be expected to fulfill , such as the
Condorcet winner criterion.
A voting proce
dure is said to satisfy the Condorcet winner criterion
The Voting Procedure
Each of
N individuals has to distribute R votes among k
alternatives
if,
© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 3, 2001
23
All through
this article we will suppose that these individ
and for xn
uals are sincere, that is, that they will not misrepresent their preferences in order to get some other benefits from it.(For a study of the problem of preference misrepresentation in this framework, see
[3] or [8].) That means that if we de
fine for each voter an intensity-of-preference function
U(xi) > U(xj) if and only if Xi > Xj (alternative Xi is preferred to alternative Xj) , then voters would distribute Vij = R
[
Ui(xj) k =I Ui(Xh) Ih
]
(1)
is the number of votes assigned to alternative individual
Xj by each
i.
If there is a Condorcet winner (for the possibility of non
existence of a Condorcet winner, see, e.g., boost: the score for a given alternative
Xj is
[5]), it gets a
.
gJ with f3
=
{
this voting procedure.
DEFINITION 2. A voting rule satisfies the monotonicity criterion if, when x is a winner and one or more voters change their preferences in a way favorable to x (with out changing the order in which they prefer any other al ternatives), x is stiU a winner. PROPOSITION 2. The class of voting models defined by (1)-(3) fulfills the monotonicity criterion for every value of f3 > 1.
Proof If Xr is a winner under the voting procedure defined by (1)-(3), then
Vr > V8 (2)
where
gr as defmed in (3). It is straightforward to see that
under
where
their points in the following way:
with
Vr > V8, and therefore Xs would never be the social choice
Vs E { 1 , 2, ... , k - 1, k) \ {r},
and if one or more voters change their preferences in a way favourable to
Xn without changing the order in which they
prefer other alternatives, then
1
f3
if Xj is not a Condorcet winner if Xj is a Condorcet winner
(3)
where the asterisk denotes "after the change" values. On the other hand, as the preference on other variables re
> 1, fixed.
Our question is whether this augmented version of the
mains unchanged,
cumulative voting procedure has better features than sim ple cumulative voting. Some Positive Properties of the Voting Procedure
The study of the voting scheme defined above
will be done
It is straightforward to prove that
v; > v; and that there
exists no
Therefore,
Xg E X such that Vg > Vr·
Xr is still a
winner.
by analyzing which of the criteria that seem reasonable for a "good" voting procedure are satisfied. For a deeper in sight into the definitions displayed in this section and the following one see, e.g.,
[ 1 ] , [2],[4], [6],[7].
DEFINITION 1 . A voting rule satisfies the Pareto criterion if, when every voter prefers an alternative x to an alterna tive y, the voting rule does not produce y as a winner. PROPOSITION 1. The class of voting models defined by (1)-(3) fulfills the Pareto criterion for all values of f3 > 1.
Proof If every voter prefers an alternative native
Xs then
Ui(Xr) > Ui(Xs) I�=I Ui(xh) I�=I Ui(xh)
Xr to an alter
Some Negative Properties of the Voting Procedure
PROPOSITION 3. lf voters in the class of voting proce dures defined by (1)-(3) are not limited in their possi bility of discrimination among alternatives (that is, if the number of votes assigned to each voter is infinitely divisible), then for any f3 > 1 there exists a profile and a population such that this voting scheme does not satisfy the Condorcet winner criterion.
Proof An example constitutes the proof.Suppose the case in which M + 1 voters (M is an even integer) decide upon
Xi, and the preference profile is XI >j X2 >j X3 >j . . . >j Xk - I >j Xk
k alternatives
Vi ::S N,
a)
(j
which implies that
I Vir > I Vis. i i As every single voter prefers Xr to X8, Xs cannot be Condorcet winner.The number of votes for x8 is thus
24
Tl-IE MATl-IEMATlCAL INTELLIGENCER
b)
Xk >a (Xk- I
-a
Xk- 2 -a · · ·
1, 2,
=
2,
...'�+ )
-a XI)
( �+ �+ d
a
=
1 .
(4)
)
(5)
3, . . . , M + 1 .
where a >f b means "a is preferred to b by individual!," and a -f b means "individual f is indifferent between a and b." Suppose further that all individuals belonging to the
preference profile a) give a proportion to alternative
Kr of their R votes
an where Kr fulfills
Kr > 0 Kr+ l < Kr k Kr = 1 ,
Vr E { 1, 2, 3, . . . , k - 1 }, Vr E { 1, 2, 3, . . . , k - 1 },
L
(6) (7) (8)
r�l
Vr E { 1, 2, 3, . . . , k - 1 }.
Kr - Kr+ l = K (constant)
(9)
Voters in preference profile b), on the other hand, give
R, to alternative xk and zero to the rest. The relative number of votes for alternative x1 is
the totality of their votes, therefore
(!f + )
(M + 1)R '
.
tions concerning the axiomatic properties of the voting scheme studied. The four criteria studied until now are not the only ones that are usually taken into account when an alyzing voting procedures. Consider, for example, the fol lowing criterion:
DEFINITION 3. A voting rule is said to satisfy the ma jority criterion if, when most voters have an alternative x as their first choice, the voting rule chooses x as a winner.
a case such that the augmented cumulative voting proce
1 Kk
dure does not choose x1 as a winner. Thus, the voting pro
(M + 1)R
cedure defmed by (1}-(3) does not satisfy the majority cri
(4) and (5) that x1 is the Condorcet xk are
winner, therefore the scores of x1 and
{3
(!f ) +
terion. Summary and Conclusions
First impressions are not always right, especially in this
1 K1
field mostly inhabited by impossibility theorems. The ac
M+ 1
tual properties of the voting procedure constructed violate the intuition that motivated the rule. The Rolling Stones's
(!f + ) + !f 1 Kk
vk =
vk > VI. (vi > v2, . . , > vk - l is evident.) 4 have a number of further implica
Propositions 3 and
of a majority of voters, and for every value of f3 we can fmd
[(!f + ) + !f]R
VI =
sen so that
Proof See the proof of Proposition 3: x1 is the first choice
xk,
It can be seen from
C-"f + 1 against -"f) in pairwise contest. Neverthe
PROPOSITION 5. The class of voting models defined by (1)-(3) does not fulfiU the majority criterion.
1 K1R
and for alternative
alternative
less, as we saw, for any {3 the other parameters can be cho
sentence with which this paper started is quite apposite to Public Choice, where getting "what you want" is usually
M+ 1
more difficult than it seems.
I must now choose k, K, and M so that v1
, and the 8%. On 18 March 1919 Courant and Ack
war. Certainly you tried to weaken
terhalfen, quickly pushed them into
ermann spoke to a student meeting
the initial impact of your words by
the realm of actuality, where they
about their fact-finding trip to Berlin
explaining you meant war with spir
sadly faded as beautiful shadows.
for the Society of Freedom-minded
itual weapons. Now you can declare
Long bursts of applause thanked
Academics. Conservative student Georg
you never proclaimed a bloody war.
him, and disturbances broke out against him too. Speeches and
Schnath says these "known socialists"
But
disturbed the meeting by getting on the
reservation that if armed war was
left feeling these are quite ideal
spised the Republic got
ideas, indeed mere theories. The
Independents
sick at the core, to civil war. . . . On Friday you preached civil
you
immediately
made
the
stormier
agenda "explicitly against the will of the
"forced on you" the case would be
and some people took unseemly ad
majority." They reported that news of
different. . . . At the end of the meet
counter-speeches
grew
this, to charge that no
the situation in Berlin was wildly exag
ing, among many other comments,
one in the group had thought about
gerated in government lies, and warned
when you read out the latest unfor
social problems before, and again to
against using troops to fight workers.
tunate N oske edict, to provoke the
call for removing the "political suck
They said, in Schnath's words:
railway workers there to "action,"
vantage of
lings" ! Herr Miihlestein further pro
you seemed to me not quite clear on
voked the gathering by inflamma
The rule of class justice, and espe
tory outbursts. When he pointed out
cially the widely ridiculed way it is
war. You left me really expecting
the unfortunate author of the flyer,
administered, are decisive failures
you to give the signal within a few
the excitement rose so far that the
of Noske's gun-barrel politics, and
months.
great majority of academics began
greatly increase the membership of
the manner or timing of your civil
heavy protest against foreign med
Spartacus,
which, moreover, the
If you, Herr Miihlestein, really
dling and emphasized their own
speaker tried to distinguish strongly
mean well for the German people, if you are ready to commit yourself
Germanness in the strongest terms.
in
After this "uproar of the sucklings,"
Bolshevism.
every
respect
audience
entirely, then go carry the torch of
and insane pounding from Herr
showed they were not the ones to
world revolution in the countries of
Miihlestein's corner as he apparently
hear such claims, with the speakers
the Entente. Leave politics here to
took out his anger on the poor fire
taking
others.
screen, we had a rousing "Deutsch
Independents' interests. [5]
up
.
.
from
.
word
The
for
Russian
word
the
cheers for our Germanness, and the German Students, and Hindenburg, as we left the room.
A.R.
All we need add is that after wards
Herr
Professor
Willrich
GZ 3.7. 19
This is very like what anti-socialists
land, Deutschland tiber Alles" . . . and Courant
closed
his
part in the
said to
Social Democrats in
1918,
Revolution with two open letters to
and Miihlestein's reply was very like
newly Communist Hans Miihlestein.
Courant's defense in January
Courant wrote on the front page of the
did not urge civil war, but warned
Gdttinger Zeitung:
against it, and pressure from the gov
lauded the beginning of social legis
66
[or
1919: He
ernment and the military will make the
lation by the old Kaiser and Hinden
Your public debut as a Communist
workers fight. Courant's rejoinder does
burg as an act of the constitutional
agitator in Gottingen imposes a duty
not point to any difference between
THE MATHEMATICAL INTELLIGENCER
the cases nor to anything specific
teers. By July
Mtihlestein said. He treats these argu
very defenses he had
ments exactly as anti-socialists treated
for Revolution. A brief stint on the
Social Democratic arguments earlier.
Gottingen town council ended in spring
He ignores them. He and his friends will clarify what Mtihlestein said. The
1919 he was deaf to the
A U T H O R
earlier given
1920. After promoting the German vote 1921 plebiscite on whether Upper
in a
editors closed the debate with Cou
Silesia would be German or Polish, he
rant's rejoinder on 31 July, which began:
took only private interest in politics. [4]
First, I insist my representation
dropped all mention of "Marxism," as he
of Herr Mtihlestein's performance,
probably had very quickly after 1919. He
His
1933 report slanted a little. He
and especially the irresponsible pro
spoke of a "Bolshevist" menace, as the
vocative way he spoke of civil war,
right had in
corresponds entirely to the facts.
praised "the Social Democratic Party
Philosophy and Mathematics
This has been confirmed to me, both
under men like Noske." Noske was
Case Western Reserve University
before and after the letter was pub
strong in the party in
lished, by the most varied and in
criticized
1919 but he had not. And he
1919, but Courant
COLIN MCLARTY
Cleveland, OH
was unaware of what he said, and
him then. Like the reference to "Wehrmacht," a rare term before the Freikorps popularized it, and one Courant did not use in 1919, all this
Col in Mclarty is an Associate Pro
would otherwise have chosen dif
merely skewed the cast list towards Nazi
fessor of P h i losophy , and of Math
ferent words. But the content and
taste. The motives Courant claimed in
ematics, at Case Western Reserve
the effect of his speech were made
1933, he had shown in the Revolution.
disputable eye- and earwitnesses. I will happily believe Herr Mtihlestein
USA
e-mail:
[email protected] University. After a philosophy disser
tation on J. H. Lambert and Kant, he
so clear, not only in my letter but also and in the same way by others
REFERENCES
concentrated on mathematical and
who are much closer to him, that his
[1 ] Dahms, H-J. and F. Halfmann, "Die Univer
philosophic aspects of category the
attempt to deny it now seems rather
sit�i.t Gottingen in der Revolution 1 91 8/ 1 9 , "
ory and topos theory. He is currently
strange. It would seem more re
1 9 1 8: Die Revolution in SOdhannover.
working on the history of homology
spectable to me, if Herr Mtihlestein
(H-G. Schmeling, ed.) Stildtisches Museum
from Poincare through Grothendieck's
would quietly accept the fact that he
Gottingen (1 988).
algebraic geometry. This article grew
cannot defend his performance and his speech.
GZ 31.7.19
[2] Dick, A. Emmy Noether, 1882-1935. Basel: Birkhauser ( 1 98 1 ) .
from research into Ernmy Noether's role in that. His wife, Patricia Prince
[3] Peckhaus. V . Hilbertprogramm und Krit
house, is an historian of biology. To
ische Philosophie, Das G6ttinger Modell
gether they breed and show Pyrenean
acted throughout the Revolution to
interdisziplinarer Zusammenarbeit zwischen
Sheperds and Great Pyrenees dogs.
keep order and hold off radicalism and
Mathematik und Philosophie. Gottingen:
Courant said truly in
1933 that he
civil war. By late November
1918 he
agreed with Eduard Bernstein that the Revolution should end with elections,
Vandenhoeck & Ruprecht (1 990). [4] Reid, C. Hilbert-Courant. New York: Springer Verlag (1 986). dass ein weiblicher Kopf nur ganz aus
and socialist measures be postponed
[5] Schnath, G. "Gottinger Tagebucher Okto
until then. He never publicly mentioned
ber 1 9 1 8 bis Marz 1 91 9," G6ttinger Jahr
nahmsweise in der Mathematik schop
any specific socialist measures, unless
buch (1 976), 1 71 -203.
ferisch tatig sein kann," G6ttinger Jahr
we count the call for new Army volun-
[6] Tollmien, C. "Sind wir doch der Meinung,
buch 38 (1 990) 1 53-2 1 9 .
VOLUME 23, NUMBER 3 , 2001
67
MIHALY
T. BECK
Why I s Th e re N o M ath e m ati cal N o be l Prize ?
mong the many scientific awards, the Nobel Prize is of the highest rank. The question is frequently posed: why is there no mathematical Nobel Prize? The formal answer is simple: Alfred Nobel in his wiU determined that awards should be given scientists who made the most important discoveries in the fields of physics, chemistry, physiology/medicine; further, to au thors of excellent literary works and to persons who con tributed eminently to the cause of world peace. The real problem remains: why did Nobel not decide to reward the greatest contributions to mathematics? I have not found any authentic data in the available lit erature [ 1-3], but there are snippets of gossip and conjec ture in various autobiographies. In his memoir, Theodor Karman wrote [4] that although Oscar Prandtl would have deserved it, . . . he never received the prize apparently because the Nobel Committee didn't (and still doesn't) regard the sci ence of mechanics as sublime as other branches of physics for which they have provided many prizes. Einstein for example got the prize mainly for explaining the photoelectric effect, not for the brilliant mathemat-
68
THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK
ics underlying his theory of relativity. I have always per sonally suspected that this curious blind spot in the Nobel Committee arose because Nobel could not forgive his mistress for running off with a mathematician. The following explanation was given to Manfred von Ardenne [5] by Prof. B. Debiesse, then chief of the center for atomic energy in France: Nobel had a girl-friend younger by thirty years, whom he found tete-a-tete with a mathematician. Supposedly, this event prompted him to leave out mathematics in de scribing the statutes of the foundation.
It is not certain that the biographies give a complete ac count of the amorous affairs of Nobel, who remained a bachelor to the end of his life. However, in connection with
these assumptions it is worth mentioning that Nobel in fact
A U T H O R
had a long, most likely not merely platonic, connection with a Viennese woman, Sophie Hess, who was indeed thirty years younger than Nobel [6]. Sophie once told
him
that
she was expecting a baby and that the father was Kapivan Kapy, a Hungarian army officer. Nevertheless, Nobel was most generous with Sophie, and even provided for her in
his will. (It is surprising that there is no speculation in the literature that Sophie's seduction by an army officer con tributed to Nobel's antimilitaristic feelings.) Other possibilities, reported by Garding and Hormander [7], included a French-American and a Swedish explana tion for Nobel's neglect of mathematics: According to the
MIHALY T. BECK
former, Mittag-Leffler had an affair with Nobel's wife, while
Department of Physical Chemistry
according to the Swedish account, Nobel realized that cre
Kossuth Lajos
ation of a mathematics prize would mean that Mittag-Leffler
University
401 0 Debrecen
would be the first recipient-something Nobel did not fa
Hungary
vor. It seems to me that the two versions together afford
e-mail:
[email protected] an explanation, but neither works alone. Nobel was a lifelong bachelor, so version one is obvi
Mih8Jy T. Beck has dealt mostly with coordination chemistry and
ously wrong, although there could be suspicion of an affair
reaction kinetics. His other interests are ethical and method
between the wife of Mittag-Leffler and Nobel. It is much
ological problems of scientific research, and history of science.
more likely, however, if Gosta Mittag-Leffler had any role
1883 he and his wife will in which they left their villa in
in the decision of Nobel, it was that in had already drawn a
Djursholm to the Swedish Academy for the promotion of
1900, a Nobel
mathematical researches in general, but first and foremost
nowadays, or even just a few years after
in the Scandinavian countries.
would disregard the promotion of the development of math
I believe that Nobel's neglect of mathematics has a more
ematics.
prosaic foundation: his general scientific view. Nobel's pub lic schooling ended when he was
1 6, and he did not go on
to the university. He received some private instruction from Zinin, an excellent Russian organic chemist. In fact, it was Zinin who called Nobel's attention to nitroglycerol in
1855.
REFERENCES
1 . H. Schuck, R. Sohlman, A Osterling, G. Liljestrand, A Westergren, M. Siegbahn, A Schou and N.K. Stahle: Nobel, the Man and His Prizes, Elsevier, Amsterdam, 1 962.
Nobel was a typical ingenious inventor of the nineteenth
2. Erik Bergengren: Alfred Nobel, Nelson and Sons, London, 1 962.
century. His inventions needed profound knowledge of ma
3. Ragnar Sohlman: The Legacy of Alfred Nobel, the Bodley Head,
terials, resoluteness, and intuition, but not any knowledge of higher mathematics.
In
the second half of the century,
research in the field of chemistry in general did not demand higher mathematics. It is likely that Nobel's mathematical knowledge did not exceed the four arithmetic rules and the rule of three. The basic change in mathematical chemistry came about only after the death of Alfred Nobel. It is very unlikely that
London, 1 983. 4. Theodor von Karman with Lee Nelson: The Wind and Beyond, Little Brown and Co., Boston, 1 967. p. 32. 5. Manfred von Ardenne: Ein gluckliches Leben tar Technik and Forschung, Kinder Verlag, Zurich und Munchen, 1 972. p. 322. 6. Ref. 3, pp. 56-58. 7. Lars Garding and Lars H. Hormander: Mathematical lntelligercer 7 (1 985) pp. 73-74.
VOLUME 23. NUMBER 3, 2001
69
I\!/Mj,i§rr@ih$11@%§4611,J,i§,id
This column is devoted to mathematics forjun. What better purpose is there for mathematics? To appear here, a theorem or problem or remark does not need to be profound (but it is
Alexan d e r Shen, Editor
A T realise on
the Binomial Theorem by Prof
J.
Moriarty, M.A.
allowed to be); it may not be directed
REVIEWED BY SHALOM B. EKHAD
only at specialists; it must attract
REVISED AND ENLARGED EDITION, PRIVATELY PRINTED.
andfascinate.
CORK, 1 885
We welcome, encourage, and frequently publish contributions from readers-either new notes, or replies to past columns.
Please send all submissions to the Mathematical Entertainments Editor,
I
t has long been my hobby to find and peruse antique mathematics books and code books. The Internet makes it possible for immobile individuals such as myself to indulge this interest, by search of lists of book dealers or through electronic auction houses such as eBay. By this means I recently* acquired a copy of the rare book A Treatise on the Binomial Theorem (1885), by Prof. J. Moriarty. I had read of its existence, but did not think I would be so fortunate as ever to see a copy. The history of this work is quite cu rious. Its contents surely deserve to be better known, as they may prompt a modest rewriting of the history of mathematics. The edition was printed in a small run of only 50 copies, and the original copies of the book were ap parently distributed to a select set of British and continental mathematicians and natural philosophers, as a sort of manifesto. The copy I possess has the bookplate of L. J. Rogers. It is bound in half-calf, with some foxing of the inte rior pages, a damaged spine and rear cover, and with.all pages from p. 243 on being tom out. It could be that this is the sole surviving copy, due to singular circumstances that I shall relate. It might prove useful to trace the libraries of the other recipients above in the hope of locating a complete copy.
I
Mathematical Contents
The book is written in a difficult style, characteristic of amateur authors and mathematicians who do not wish their work to be read. The author appears to be an autodidact with an axe to grind. Prof. Moriarty's book very likely was given a careless examination by most of its recipients, who took it to be the work of a crank The treatise begins with a 120-page discussion of theories of the mind and a "Calculus of Possible Experience." The author argues that, humans being prone to error in logical reasoning, it is necessary and inevitable that all ratio cination be carried out by mechanical means. Such means, with proper exe cution, can reduce error to an unimag inably small level. The method should, moreover, be applied in all walks of life. In a prefatory "Advertisement" the author states that this book is being dis tributed to a select group of eminent natural philosophers, who are asked to join him in a secret society which shall represent the vanguard of those putting this vision of the future into practice. The author suggests that there may be considerable social disruption before they eventually prevail, and that all ob stacles should be ruthlessly removed. One perceives that this is no ordinary mathematics textbook To support his thesis, Prof. Moriarty proceeds to discuss various mathe matical and logical topics, in a some what obscure fashion. He asserts that all mathematical identities of a certain recondite form can be verified in a purely mechanical way. He asserts that his methods go far beyond what is available everi to the greatest mathe maticians of the day. He then states that he will illustrate this on two prob lems, one connected with identities for
Alexander Shen, Institute for Problems of
Information Transmission, Ermolovoi 1 9, K-51 Moscow GSP-4, 1 01 447 Russia;
*It was on April 1 , 2001 , a date of note, being an anniversary of my fabrication and the etching in of my ser
e-mail: shen@landau .ac.ru
ial number.
70
THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK
binomial coefficients and the other with certain properties of a function which can be shown to be the Riemann zeta function, which the author hints at in a rather secretive and elliptical way. The author uses his own notation for binomial coefficients and hyperge ometric series without explanation, ap parently treating it as self-evident. Once a dictionary is assembled be tween his notation and the standard one, this part of the book is surpris ingly comprehensible. On page 1 71 we discover the identities (my translation)
n2
oo
];0 (1 - q)(1 - q� . =
fl
n=O
(1
_
. . (1 - qn)
q5n + l) - l (1
_
q5n + 4) - l .
and
n2+n
oo
];O (1 =
-
fl
q)(1
n=O
(1
-
_
:2)
•
·
·
(1 - qn)
q5n + 2) - l c 1
_
q5n + 3r l.
These particular identities have since been discovered by a number of mathe maticians and are known by various names in the literature, cf. [6, pp. 90-91]. Prof. Moriarty provides short proofs of these which, although bizarre, can be checked line by line to be correct. The method, described in what can only be called crabbed prose, consists of the clever introduction of extra variables and telescoping identities, using a kind of symbolic calculus based on differen tial operators. The author supplies a long sequence of further identities and q-identities of various sorts, and for each he gives grotesque proofs whose origin is mysterious. At one point he describes these as "proof certificates, the bank notes of the future." With hindsight, it seems quite clear that Prof. Moriarty was in possession of the essence of the so-called WZ method for proving hypergeometric function identities, which is detailed in Petkovsek, Wilf, and Zeilberger [7]. A significant part of this method was cre ated by D. Zeilberger [8] in 1990, see
also [2] . The particular proof used by Prof. Moriarty for one of these identi ties is essentially identical with one published by my brother and his sili con companion [4] in 1990 using the later-to-be-named WZ-method. What methods Prof. Moriarty used to carry out his own computations are un known. In a less mathematical vein, I note that Doran Zeilberger [9] seems to be developing sociological opinions sliding in the direction of that of Prof. Moriarty. Perhaps such are the conse quences of thinking like a computer. I applaud this; but I digress. In the text that follows, beginning on page 220, Prof. Moriarty discusses the Riemann zeta function. He seems un aware of the work of Riemann, and in troduces it as a function of a real vari able, as "a curious binomial product function of the esteemed Russian sa vant Mr. L. Euler." He derives identities for the values of the function at even positive integers, and then gives various integral formulae, including one imply ing the functional equation, asserting that they were obtained by a mechani cal method for obtaining quadratures. He states that he has developed meth ods of "imaginary analysis," by which he seems to mean methods of a complex variable. Perhaps these also involve some ideas of renormalisation, because there are formulae resembling operator product expansions, but at this point the book (literally) breaks off. It seems fruitless to speculate on what the rest of the book may have contained, but I find some of the final formulae sugges tive and hope to communicate further on this subject in a year's time. The Author
My information about Prof. Moriarty's early career* is derived from Dr. J. Watson (vide [3]), who credits the fol lowing information to Mr. W. Sherlock Holmes:
[Prof Moriarty's] career has been an extraordinary one. He is a man ofgood birth and exceUent education, endowed
by Nature with a phenomonal mathe matical faculty. At the age of twenty one he wrote a treatise upon the Binomial Theorem, which has had a European vogue. On the strength of it, he won the Mathematical Chair at one of our smaller Universities, and had to aU appearance, a most briUiant ca reer before him. But the man had hereditary tendencies of a most dia bolical kind. A criminal strain ran in his blood, which, instead ofbeing mod ified, was increased and rendered infinitely more dangerous by his ex traordinary mental powers. Dark ru mours gathered round him in the University town, and eventuaUy he was compeUed to resign his Chair and to come down to London. ,
Certain features of the book under re view, however, have led me to a radi cal conclusion about its author. These features include some pencilled anno tations on page 173 and the impression of a square-toed boot of an unusual make on the inside of the back cover. My belief is that "Prof. J. Moriarty" is a pseudonym and that the true author is Mycroft Holmes. It is well known that Mr. Holmes, whose intellect was supe rior to that of his drug-addicted brother Mr. Sherlock Holmes, had presciently come to the conclusion that domination of the world by machines was inevitable. The author of the "Treatise" clearly be lieved that the creation of the new world order, however rational, would only be achieved by rivers of blood, and that a secret society to effect some guidance to the future world order was essential. Naturally enough, the book was pub lished under a pseudonym. Indeed I be lieve Mr. Holmes borrowed the title of Prof. Moriarty's earlier work to discour age uninitiated readers; the difference in tone from Prof. Moriarty's original which Prof. P. Gordan reviewed as "a capital example of mathematics, not theology"-would have been self-evident. The actual history of events, as I have reconstructed them through ex act logical deduction, closely parallels
•upon Prof. Moriarty's subsequent career, his successful investments in the stock exchange, his change of name, knighthood, and raising to the peerage, it is un necessary to dwell here.
VOLUME 23, NUMBER 3, 2001
71
this view. Mycroft and his brother, though congenial through much of their lives, had increasingly bitter ar guments during a continental tour they took in the spring of 1891. Mycroft as serted that freelance investigators of his brother's sort would soon be put out of business. He argued that "man aging your clients" was the really im portant thing for the success of any investigatory business; that logical thought was irrelevant, and was best left to machines; that actually solving cases was pernicious, though perhaps of some ephemeral intellectual amuse ment; and that drug use was interfer ing with Sherlock's ability to reason. Mycroft even claimed to have tired of Sherlock's statement "When the im possible was eliminated, what remains, however improbable, must be the truth" and that repeating it often was symptomatic of grandiosity and obses sion. Mycroft's untimely death at the Reichenbach Falls soon after brought his utopian hopes to an end. I believe that his vindictive brother tracked down and destroyed as many copies of Mycroft's book as possible. We are only fortunate that either L. J. Rogers or his manservant was present when Sherlock attempted to purloin his copy, and that in the ensuing struggle he was able to retain part of the book. Sherlock Holmes's death soon after brought an end to this whole sad story. It is true that events as I have de scribed them conflict with the narrative of Dr. Watson [3], but I am afraid his account must be dismissed as fiction. Taking the facts as he presented them, is it plausible that he would have sud denly left his young wife and a thriving medical practice that he was desirous of increasing, for a sudden trip to the Continent, of several weeks' length, at considerable expense, with an obvi ously paranoid individual? Would he not rather have diagnosed cocaine-in duced psychosis and immediately clapped his companion into a sanato-
72
THE MATHEMATICAL INTELLIGENCER
rium, where warm baths, hot milk, and cold turkey might restore him to san ity? Far more believable it is that Mr. Sherlock Holmes, having killed his brother, whether accidentally or wil fully, in a state of considerable agita tion telegraphed Dr. Watson for assis tance and money, and then disappeared for three years to Tibet. Dr. Watson's epistle, together with his fulsome obit uary to the Times of London, can only be viewed as an example of the sort of "disinformation" often produced during the Great Game. That Mycroft Holmes had a power ful intellect there is no doubt. However misguided he may have been in his pri vate life, this book proves he was a re markable mathematician. It is time to set the history of mathematics right, and pay him his due! In view of the con troversy currently raging over some of these issues [1], [5], [9], this is most timely. Sadly, my attempts to have his book reprinted, even in its mutilated form, have met with no success. REFERENCES
1 . G. E. Andrews, The death of proof? Semi rigorous mathematics? You've got to be kidding, Math. lntelligencer 16 (1 994), no. 4, 1 6-1 8. 2 . P. Cartier, Demonstration "automatique" d'i dentites et fonctions hypergeometriques (d'apres D. Zeilberger), Seminaire Bourbaki, Exp. No. 746, Asterisque No. 206 (1 992), 4 1 -91 . [3] A. C. Doyle, The Adventure of the Final Problem, The Strand Magazine, December 1 893. [4] S. B. Ekhad and S. Tre, A purely verifica tion proof of the first Rogers-Ramanujan identity, J. Comb. Theory, Series A, 54 (1 990), 309-31 1 . [5] S. B. Ekhad and D. Zeilberger, Curing the
A
=
B, With a foreword by D. E. Knuth,
A. K. Peters, Inc.: Wellesley, Mass., 1 996. [8] D. Zeilberger, A holonomic systems ap proach to special function identities. J. Comp. Appl. Math. 32 (1 990), 321-368. [9] D.
Zeilberger,
Theorems for a Price:
Tomorrow's Semi-Rigorous Mathematical Culture, Notices Am. Math. Soc. 40, No. 8 (1 993),
978-981 .
(Reprinted
Editorial Note. Shalom B. Ekhad has been incapacitated by a power surge. The review is presented just as it was re covered. I am obliged to point out that other evidence conclusively indicates that the reviewer's deduction about the identity of "Prof. J. Moriarty" is erro neous. It appears that during a visit to the University of Gottingen in 1879 Prof. Moriarty gained access to the Riemann Nachlass and may have absconded with some of its papers. The absence of any reference to Riemann's work in the "Treatise" would therefore appear to be a subterfuge. The editor's own surmise is that, enervated by onerous teaching duties and frustrated in his attempts to prove the Riemann hypothesis, Prof. Moriarty turned to a life of crime. Indeed the Riemann hypothesis is a dangerous problem to work on; let this example be a warning to the unwary. J. C . Lagarias
[6] G. H. Hardy, Ramanujan, Cambridge Univ. Press: London and New York,
1 940.
(Reprint: Chelsea). [7] M. Petkovsek, S. Wilf, and D. Zeilberger,
Math.
About the Reviewer. Shalom B. Ekhad is not to be confused with his identical twin brother, Shalosh B. Ekhad, the collaborator and sometime indentured servant of Doron Zeil berger. Being unable to travel (or even speak) without assistance, he takes great interest in the careers of other disadvantaged individuals such as the late Mycroft Holmes.
Andrews syndrome, J. Diff. Eqns. App/. 4 (1 998), 299-3 1 0 .
in:
lntel/igencer 1 6 (1 994), no. 4, 1 1 -1 4.)
A.T.&T Labs Florham Park, NJ 07932-0971 USA email:
[email protected] I a§!ii4'4i
J et Wi m p , Editor
I
What Is Random� Chance and Order in Mathematics and Life by Edward Beltrami $22.00
ISBN:
ing large in our everyday experiences" (p.
xi). It therefore behooves us to try to
understand it.
What mathematicians
over the centuries-from the ancients, through Pascal, Fermat, Bernoulli, and
NEW YORK: SPRINGER-VERLAG, INC., US
As we are told in the preface, "ran domness is the very stuff of life, loom
0-387·98737·1
1 999,
xx +
201 pp.
de Moivre, to Kolmogorov and Chaitin have discovered, is that it's a pro foundly rich concept. The more one
REVIEWED BY JERROLD W. GROSSMAN
delves into it, the more paradoxes
M
aybe I'm just too critical to be re
unknown and vice versa. Edward Bel
viewing mathematics books writ
trami, Professor of Applied Mathemat
arise, the more the known becomes the
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
ten for a lay audience. Here are my ex
(1) The
ics and Statistics at the State Univer
author is an expert
sity of New York at Stony Brook, has
in what he is writing about (or else a
mapped out a provocative and delight
pectations:
you would welcome being assigned
professional mathematics writer with
ful journey through this still-evolving
a book to review, please write us,
excellent contacts).
(2) The mathe
subject, taking the reader from the ba
telling us your expertise and your predilections.
matics is correct, both in substance
sic premises to speculations at the
and in detail, from the statements of
fringes of current research.
the theorems to the steps of the der
Chapter
1
contains a brief exposi
cares
tion of probability and how one can use
enough to get the typesetting right, us
the language and tools of mathematics
ivations.
(3)
The
publisher
ing appropriate fonts and layout for the
to explain uncertainty and chance
mathematical expressions. (4) The au
conveying what a random process
thor and publisher have taken the time
should be, the idea of the Law of Large
to proofread the final product so that
Numbers, and the use of statistical hy
there are very few mathematical mis
pothesis testing as a first approxima
prints
tion to detecting whether the output
(especially
confusing
ones).
(5) The exposition is understandable to
of a process is consistent with that
the target audience, both the general
process's being random. It ends with the
discussions and the technical points.
introduction of Borel's notion of normal
(6) Most of all, the book intrigues the
numbers-numbers whose decimal ex
intended reader, imparts a flavor of
pansions contain, in the limit, the ex
what we do for a living, and raises the
pected number of occurrences of every
mathematical
sequence of digits. Normality is a nec
awareness. Well, one-and-a-half out of
essary, but definitely not sufficient,
level
of the
public's
six isn't too bad-at least it's the last
What Is Random? gets bonus, this book should be
condition for randomness, as Champer
of these that
nowne's
right. As a
(Beltrami's version of it in binary is
interesting to those with the technical background to understand the finer points and to relate this material to
famous
constant illustrates
0.0 1 00 01 10 1 1 000 001 010 . . .). Chapter 2 starts with a nice explana tion of Shannon's 1948 theory of infor
their previous knowledge of mathe
mation and the definition of entropy (al
matics and statistics-mathematicians
though the presence of logarithms is
who know little of these topics, gradu
likely to challenge the less mathemati
ate students, undergraduate mathe
cally inclined readers). As usual through
Column Editor's address: Department
matics or computer science majors,
out this book, Beltrami provides good
of Mathematics, Drexel University,
maybe even a curious student in her
sound bites; here it is that "maximum
Philadelphia, PA 1 9 1 04 USA.
first calculus or statistics class.
entropy is identified with quintessen-
© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 3, 2001
73
tial randomness" (p. 43). Some good
unique
random
list has been carefully compiled, and
examples are provided to illustrate "ap
graph [2]). The level of philosophy and
countably
infmite
reading the recommended books and
proximate entropy," a rough-and-ready
mathematics is raised once again, and
articles
tool for testing binary strings for ran
the lay reader will probably have de
busy for months.
will keep the inspired reader
domness, and there is even MATLAB
spaired by now of ever understanding
Having said all this-and let me
code for computing it for your favorite
the finer points. Nevertheless, the fla
make it clear that I think it is good that
strings.
vor of what is going on here should
What Is Random? exists-let me get to
come through.
the gripes, following the list given in
Chapter 3 contains a rather confus ing, technical, and repetitive discus
The story could end there, but the
my opening paragraph. This book is
sion of generated and disappearing
author ventures on. In his words, "The
about probability (Section 60 of the
bits, algorithms running in reverse,
last chapter is more speculative. In
AMS 2000 Mathematics Subject Classi
pseudorandom
generators,
'The Edge of Randomness' [Chapter 5]
fication), a little statistics (Section 62),
gedanken experiments in physics
I review recent work by a number of
information theory (Section 94), Kol
(involving the Szilard engine and the
thinkers that suggests that naturally
mogorov complexity (Section 68), and
and
number
Second Law of Thermodynamics). The
occurring processes seem to be bal
computability (Section 03). According
author sums it up as follows: "Knowl
anced
to Mathematical Reviews, the only one
between
tight
organization,
edge of the past and uncertainty about
where redundancy is paramount, and
of Edward Beltrami's 24 publications
the future seem to be two faces of the
volatility, in which little order is possi
with any of these section numbers
same Janus-faced coin, but more care
ble" (pp.
xv-xvi). In other words, we
listed as either a primary or secondary
ful scrutiny establishes that what ap
have come full circle: Since total ran
classification is the book under review.
pears as predictable is actually ran-
domness is now as easy to understand
The book jacket states that his area of
as total order, meaningfulness lies be
expertise is applications of dynamical
tween these two extremes. As might
systems to things like blood clotting.
n encodes all
be expected in this free-wheeling
All other things being equal, I would
conclusion, we are asked to look at,
have preferred that Gregory Chaitin
information about
for example, biological processes, the
had written a book like this. Actually
music of Bach and Mozart, fractals in
he has (it's called
all com putations.
mathematics and nature, and "small
but I'll leave it for others to review that
world" networks [4]. The fmal sec
book [3], using their own criteria. On
tion, addressing the question "What
the other hand, Dr. Beltrami has also
good is randomness?", leaves the
written many erudite articles (and even
domness in disguise. I establish that
reader with an inspiring sermon, that
a book [ 1 ] ) on wine, and a sampling of
these two faces represent a tradeoff
this journey has been worth it after all.
one such article, and its recommended
between ignorance now and disorder
The book succeeds as a cross be
Chianti, left me impressed with his
later" (p. xiv).
tween a scholarly tome, a mathemati
The Unknowable),
breadth of knowledge. As the author claims, little in the
Things get back on track in Chapter
cal text, and a murder mystery. Many
4, where the big guns are rolled out
quotations from giants of the past and
way of formal mathematical
Kolmogorov's definition of complexity
present (philosophers, scientists, and
ground
(a given string is random if no shorter
popular writers, as well as mathemati
needed," p.
string encodes a computer program
cians) are sprinkled throughout. When
tends to skip the Technical Notes.
is
required
("no
back
calculus
xvii), especially if one in
that will produce the given string); Tur
the mathematical details get burden
What disturbs me as a mathematically
ing machines to capture the notion of
some, the reader is referred to (and can
sophisticated reader are the mistakes
computability, and their inherent in
choose to skip) Technical Notes and
in the mathematics: Big things, like
ability, demonstrated by Turing him
Appendices, which occupy about a
misstating
self, to analyze the computing process;
fifth of the volume. The lively non
theorem (we read that there is no proof
Chaitin's incompleteness theorem (the
technical discussions should for the
of the statement "for any positive inte
impossibility of verifying computation
most part keep the reader's interest
ger n there is a string whose complex
ally that a string is random); and the
through this short journey. An eight
ity exceeds
Law of Large Numbers ("with proba
Chaitin's
incompleteness
n, " p. 107) and the Strong
n (roughly, the prob
page postscript, "Sources and Further
ability that a randomly chosen com
Readings," points out some excellent
bility one . . . the sample averages
puter program will halt). As a conse
books and articles for filling in prere
Sr/n of a Bernoulli p-process will dif
quence of Turing's result, we cannot
quisites, reinforcing the message of the
fer from p by an arbitrarily small
n, and yet this one
present book, or going beyond it-clas
amount for all sufficiently large
sic
Scientific American articles, prob
30)-although in all fairness it must be
amazing number
know much about
number somehow encodes all infor
n," p.
mation about all computations. (I like
ability and statistics textbooks, and im
pointed out that the author para
to think of it as one of the mathemati
portant works by the leading writers in
phrases these results several times in
cian's answers to understanding the
the subjects being explored here and
different ways, and the other versions
meaning of life, the other being the
their offshoots. The 67-item reference
are correct. Medium-sized things, like
74
THE MATHEMATICAL INTELLIGENCER
sions (not to mention a handful of
usual misunderstandings. He will have
testing
omitted or misspelled words in the ex
been exposed to some of the profound
and criminal legal proceedings ("the
position), the reader has a right to be
mathematical questions that lie at the
prosecutor sets up a null hypothesis of
annoyed with the author and publisher.
heart of biology and physics. She may
getting tween
backwards statistical
the
analogy be
hypothesis
guilt, and the defending attorneys try
To cite just a few: there are three ty
even believe that she can begin to an
to reject that assumption on the basis
pos in the last four lines on page 189;
swer the question posed in the book's
of the available evidence," p. 27, which
m and m - 1 are confused on page
184;
title. Finally, the reader may well be
is also incorrect in the source Beltrami
the recurrence on page 1 70 defines Xn
left with the same pleasure upon fin
in terms of itself; on page 33 we are
ishing this book that one sometimes
requiring or
asked to believe that the three-digit bi
gets when experiencing the arts or pon
forbidding infinite (binary) decimal
nary numeral following 010 is 100; and
dering religious questions: I may not
expansions that end in all zeros (pp.
on page 26 we learn that
0.6.
understand all the details, but I can ap
185-187). Little things, like saying that
I'm hoping that if enough reviewers air
preciate that there is something very
says he took it from), and getting con fused as to whether he
is
t + -f;
=
probability one corresponds to mea
their frustrations in these matters, then
deep and beautiful and meaningful out
sure zero (p. 163) and that the word
publishers will feel the pressure and
there, and maybe I am now a little
outcome
institute better quality control.
closer to it.
in the context of a sample
space is interchangeable with the word
experiment (p.
10). The lay reader will
either be left with the wrong impres sion, or, more likely, be confused. The publisher is identified as "an im-
The perceptive reader wi l l real ize that com p uters can 't solve al l mathemat ical p roble m s .
Even when there aren't outright mis takes, the wording occasionally left me
REFERENCES
huh? (e.g., the integer n = Um -12m- l + Um -22m-2 + . . . + a12 + ao, where each ai is 0 or 1, is "never less
1 . Edward Beltrami and Philip F. Palmedo, The
asking
Wines of Long Island, Amereon Ltd. (2000). 2. Peter J. Cameron, The random graph, in The
than zero (simply choose all the coef
Mathematics of Paul Erd6s, Vol. /1, pp.
ficients to be zero)," p. 184). Some
333-351, Algorithms and Combinatorics 14,
times, especially in the last half of the
Springer, 1 997; MR 97h:05163.
book, the discussion becomes rather
3. Gregory J. Chaitin, The Unknowable, Springer
ethereal and mystical, and I fear that
Verlag, 1 999; MR 2000h:68071 ; see also
many readers
will
start to lose the
http://www.cs.auckland.ac.nz/CDMTCS/ch
drift. But for the most part, the writ ing
is fairly clear and compelling, with
aitin/unknowable/. 4. Duncan J. Watts, Small Worlds: The Dy
pithy statements that will appeal to the
namics of Networks between Order and
lay reader and the mathematician
Randomness, Princeton University Press,
alike, such as "strings that possess log
1 999; MR 2001 a:91 064; see also this writer's
ical depth must reside somewhere be
review in The American Mathematical Monthly
tween [the extremes of] order and dis
107 (2000), 664-668.
2
order" (p. 1 0), to pick one example print of Springer-Verlag" on the title
illustrating the author's main point in
Department of Mathematics and Statistics
page. Springer is certainly a respected
Chapter 5.
Oakland University
and important publisher of mathemat
And so we come to my sixth crite
ics (including, of course, this maga
rion, the bottom line for expository
USA e-mail: grossman@oakland .edu
zine). One would think, therefore, that
mathematics for the nonmathemati
Springer's expertise would be reflected
cian: Has this little book captivated the
in the typesetting of the book under re
reader, made her think more deeply
view. Alas, fonts are inconsistent, mi
about some of the ideas that drive our
nus signs often show up as hyphens,
professional lives, raised the level of
superscripts occasionally aren't raised,
his mathematical sophistication, and
spacing
is
sometimes inappropriate
conveyed at least the spirit of some of
in general, the layout makes the math
the
ematics ugly to look at and hard to
throughs? (For a good answer, we
exciting
new
research
break
read, for both mathematicians and lay
should probably tum to a review by a
readers. Is it too much to ask that
lay critic, but I was unable to find one
mathematics copy editors and typeset
anywhere, even though the book was
ters prepare the galleys?
published in late 1999.) I think that it
Everyone would agree that there is
has. The perceptive reader will realize
no way for a mathematics book to be
that computers can't solve all mathe
virtually error-free (well, maybe with
matical
problems.
She
will
Rochester, Ml 48309-4485
Squaring the Circle/ The War between Hobbes and Wallis by Douglas M. Jesseph CHICAGO AND LONDON, UNIVERSI1Y OF CHICAGO
1999, xiv + 419 pp. $80.00 (cloth), $28.00 (paper), ISBN 0-266-39899·4 (cloth), ISBN: 0-226-39900-1 (paper) PRESS,
US
REVIEWED BY GERALD L. ALEXANDERSON
have
Knuth's
brushed up against some of the trick
books). But when misprints repeatedly
ier parts of elementary probability and
T
creep into the mathematical expres-
perhaps be less likely to harbor the
author of
the
exception
of
Donald
homas Hobbes (1588--1679), the eminent English philosopher and
Leviathan,
is remembered
VOLUME 23, NUMBER 3, 2001
75
today for his influence in politics, law, and moral philosophy. His work in mathematics is much less well known and less studied-and, as we see in the work at hand and elsewhere, perhaps justifiably so. Douglas Jesseph is a philosopher and clearly is interested in exploring the implications of the Hobbes-Wallis dispute for philosophy in Hobbes's time and subsequently. At the same time we must recognize that Hobbes's concern for method and his at tempts to apply mathematical tech niques outside mathematics itself may be among his finest achievements. Hardy Grant in [G] said that, for Hobbes, "only the mathematicians' method, only strict deduction from sure premises, would serve. But this approach, so suc cessful in geometry and physics, had never (Hobbes urged) been applied outside those fields. Geometry is 'the onely Science that it hath pleased God hitherto to bestow on mankind.' Thus he did not blush to claim his own application of its method as his toric." And indeed it was. Unfortunately, as is often the case, a person's ability to appraise his or her own work is not always objective. Hobbes wrote of his mathematical work quite glowingly, citing, as we learn in the early pages of this book, his success in squaring the circle, di viding an angle in a given ratio, de scribing a regular polygon with any number of sides, and solving some other problems of well-known diffi culty. It is not entirely clear what he meant by these claims. Did he under standjust what constraints were put on the solution? Was he fully aware of what is involved in a euclidean con struction? Grant again: "Most notori ously, he claimed with complete confi dence the duplication of the cube, 'hitherto sought in vain,' and the squar ing of the circle; we may judge his grasp of this latter problem by his declaration that an 'ordinary' man might accom plish it better than any geometer, by simply 'winding a small thread about a given cylinder.' " So much for Hobbes's understanding of the problem! As every student of mathematics knows, there are three classical eu clidean construction problems dating from antiquity: (1) the squaring of a cir-
76
THE MATHEMATICAL INTELLIGENCER
cle (i.e., constructing the side of a square that has the same area as that of a given circle, or, to put it another way, constructing a root of x2 7T); (2) doubling the size of a cube, and (3) trisecting an arbitrary angle. An other such problem is: (4) constructing a regular polygon with n sides. In the early nineteenth century (2) and (3) were proved impossible for straightedge and compass, using properties of roots of cubic equations. Problem (4) was set tled by Gauss. But problem (1) re quired more: knowledge that 7T is a transcendental number. This was not proved until 1882 by F. Lindemann. Earlier work of J. Lambert (1761) had only shown that 7T is irrational, which says nothing of the constructability of =
"Thomas H obbes . came late in l ife to [mathematics] , and understood it poorly, but loved it m uch . " V";. Of course, none of this was avail able to Hobbes. The problem of the quadrature of the circle had plagued mathematicians since Anaxagoras (400-428 B.C.E.), who, we are told, worked on the problem in prison. Many mathematicians and lay men tried to "square the circle"-even Abraham Lincoln is reported to have worked on the problem-and people continue to try even to this day. Per haps the most risible attempt was pub lished in 1934 as a book with the half title, Belwld! The Grand Problem No Longer Unsolved: The Circle Squared beyond Refutation!, written by Carl Theodore Heisel, a 33rd degree Mason, in Cleveland, Ohio. The author seems to have substituted exclamation points for rigorous mathematical arguments. It is no wonder then that in the sev enteenth century this infamous prob lem should have attracted Hobbes,
who came to mathematics relatively late in life, though he had demon strated precosity in other disciplines. At the age of 14, for example, he had translated Euripides' Medea from Greek into Latin iambic. But his dis covery of Euclid was described thus by John Aubrey in [A]: He was . . . 40 yeares old before he looked upon geometry; which happened accidentally. Being in a gentleman's library in- Euclid's El ements lay open, and 'twas the 4 7 El. libri I. He read the proposition. 'By G-,' sayd he, 'this is impossi ble!' So he read the demonstration of it, which referred him back to such a proposition; which proposi tion he read. That referred him back to another, which he also read. Et sic deinceps, that at last he was con vinced of that trueth. This made him in love with geometry. Hobbes's conversion to mathemat ics in his 40th year did not result im mediately in a confrontation with the mathematical establishment. By 1646 his reputation in mathematics was such that he was named tutor in mathematics to the Prince of Wales. It wasn't until 1655, however, that he published his De Corpore, which he hoped would establish an honored place for him in mathematics. It con tained a collection of his mathematical achievements, including three admit tedly failed attempts at the quadrature of the circle. (Wallis eventually refuted over a dozen of Hobbes's attempts to solve this problem.) Hobbes worked in a period of great mathematical achievement: Barrow, Descartes, Mersenne, Cavalieri were all active at the time. These were ex citing times. Hobbes even met Galileo on a trip to Florence! Still, the De Cor pore was published eleven years before Newton's annus mirabilis, so greater things were yet to come. One can only conclude from this book that Hobbes was not an easy man to get along with. The focus of the book, his longtime dispute with Puri tan mathematician, John Wallis, was not the only contentious debate he en tered into. An ardent monarchist,
Hobbes "directed some of his most vit
lis did not always disagree on issues,
count of one of the great intellectual dis
riolic
no matter what the question, they
putes of history and of the social, cul
polemics
at the
universities,
largely because he saw them as en
sooner or later seemed to end up on
tural, religious, and intellectual life of
dangering civil peace by challenging
opposite sides. The dispute abated in
the period. Hobbes's mathematics may
the authority of the sovereign." Still,
1674 when Wallis may have grown
be largely discredited, but, still, Grant
the intensity of Wallis's animus toward
weary of Hobbes's verbal assaults, but
writes in [G], "Mathematicians should honor Thomas Hobbes, who came late
Hobbes is hard to understand without
Hobbes, when he died in
a larger context. Wallis went so far as
hind an unpublished manuscript writ
1679, left be
in life to their subject, and understood it
to obtain advance sheets of De Corpore
ten in his last year and still laying claim
poorly, but loved it much, and staked on
from the printer so that he could pre
to his having squared the circle. By this
the supposed sureness of its methods
pare his attack on it even before pub
time, though, Hobbes's reputation in
his hopes for the peace and good gov
lication. Wallis's intent was not only to
mathematics had been essentially de
ernment of mankind."
discredit Hobbes's mathematics but by
stroyed by Wallis and others.
extension
to
attack
Hobbes's
Jesseph
con
tention that "his doctrines were suffi
conscientiously
covers
Hobbes's treatment of magnitude, ratio,
ciently well grounded that they would
and general quadrature, relating it to the
enable a solution of all problems." And
work of Mersenne, Robival, and others,
their feud, probably complicated by
and he devotes space to Hobbes's con
views on university reform, and per
temptuous views of analytic geometry.
haps by Hobbes's
But these writings of Hobbes are not
alleged atheism,
REFERENCES
[A] Aubrey, John, Brief Uves, ed. Oliver Law son Dick. New York, Penguin Books, 1 982. [G] Grant, Hardy, Geometry and Politics: Math ematics in the Thought of Thomas Hobbes, Mathematics Magazine 63 (1 990), 1 47-154.
moved well beyond circle squaring to
likely to be of much interest to modem
"fme points of Latin grammar (such as
day mathematicians, and, as the author
the proper use of the ablative case),
admits, much of the mathematical work
problems of Greek etymology, and
is largely viewed even by philosophers
Santa Clara University
as something of an embarrasm s ent.
Santa Clara, CA 95053-0290
questions
of
ecclesiastical
govern
It is
Department of Mathematics & Computer Science
ment"! So plenty of fuel was there for
not for the mathematics that one reads
USA
a good fight. Though Hobbes and Wal-
this book; instead one reads it for its ac-
e-mail:
[email protected] MOVING? We
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VOLUME 23, NUMBER 3, 2001
77
N EW B O O KS F R O M R O B I N W I L S O N ! ROBIN WILSON, The Open University, Milton Keynes, UK
Stamping Through Mathematics
An Illustrated History of Mathematics Through Stamps
Postage stamps are an attractive vehicle for pre enting mathematics and its development. For many years the author has pre ented illustrated lectures entitled Stamping through Mathematics to school and college groups and to mathematical clubs and societies, and has written a regular "Stamps Corner" for The Mathematical Intelligencer. The book contains almost four hundred postage stamps relating to mathematics, ranging from the earliest forms of counting to the modern computer age. The stamps appear enlarged and in ful l color with full historical commentary, a n d are listed a t the e n d of the book.
200 1 / 1 28 PP./HARDCOVER/$29.95/ISBN 0-387-98949-8
ROB I N WILSON and JEREMY GRAY, both, The Open Univer ity, Milton Keynes, UK (Eds.)
Mathematical Conversations
Selections from the Mathematical lntelligencer
"This fine book is a compilation of selected articles from The Mathematical lntelligencer, Springer's mathematical magazine about mathematics, about
mathematicians, and about the history and culture of mathematics. . . If you read mathematical books like I do, you will enjoy "the first reading, " i.e., browsing, of this book, because it covers so many interesting topics, has a lot of illustrations (including photographs), and displays formulas in a clear and readable format. ''The second reading, " i.e., reading the sections and articles that currently interest you, is even more enjoyable, because each gem in this collection was created by an expert in the respective field to be appreciated by -MAA Online
the general mathematical audience. "
Since its first issue, The Mathematical lntelligencer has been the main forum for exposition and debate between some of the world's most renowned mathematicians, covering not only hi tory and hi tory-making mathematics, but al o including many controver ies that urround all facets of the subject. This volume contains forty article that were published in the journal during its first eighteen years.
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�1flrri.CQ.h.i§M
R o b i n Wi lson
Mathematics and Nature
I
M
athematics
occurs
throughout
nature-from the arithmetic of
sunflowers
and
ammonites
to
the
from one end of a golden rectangle leaves another one; this process
is il
lustrated in the Swiss stamp, which
geometry of crystals and snowflakes. If
features the related logarithmic spiral
we count the leaves around a plant
found on snail shells and ammonites.
stem until we fmd one directly above
The delicate structure of a snow
the initial leaf, the number of interven
flake has sixfold rotational symmetry,
ing leaves is often a Fibonacci number
and no two snowflakes are the same.
(for example,
Their hexagonal form was recognised
5 for oak trees, 8 for poplar trees, 13 for willow trees). The
by the Chinese in the second century
arrangements of scales on a pine cone
BCE and was later investigated by
similarly
Johannes Kepler and Rene Descartes,
13
exhibit
left-hand
8 right-hand and
spirals,
while
larger
among others. Hexagons also appear in
(34, 55, . . . ) appear
honeycombs; the Pitcairn Islands re
in the spiral arrangements of seeds
cently issued a set of hexagonal stamps
in a sunflower head. The ratios of
featuring bees.
Fibonacci numbers
Fibonacci
As liquids crystallise they assume
sequence tend to the "golden ratio"
the form of polyhedra of various types:
1.618. . . . This ratio, the ratio of a di
fluorite crystals appear as regular oc
agonal to a side of a regular pentagon,
tahedra, while lead and zinc sulphide
successive terms of the
arises throughout mathematics and na
crystals appear as cuboctahedra and
ture. In particular, removing a square
truncated tetrahedra.
Bulgaria: snowflake
Switzerland: logarithmic
Pitcairn Islands: honeycomb
. . . . . .. .. 9 W
. • • • • • • •
Great Britain: sunflower
Switzerland: fluorite crystals
spiral
.. .
W" W W . W • • � I
..
Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics,
.
...... . �
The Open University, Milton Keynes, MK7 6AA, England e-mail:
[email protected] Israel: pine cone
.. . . ... �
... ......... . ......
Hungary: ammonite
Germany: sulphide crystals
© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 3, 2001
79
The YOK Bug Chandler Davis
B
y now everyone, even journalists, has come to terms with the problem of millennium change: the end of 2000 marked the lapsing of the second millennium, yet it was the beginning of 2000 that was marked by a change of digit in the thousands register. Journalists, especially non-mathematician journalists, sometimes sound re sentful of this, as if a perfectly healthy year had fallen victim to a crime. They may even confound it with that other outrage, the absence of a zero-th storey in many office buildings. We who know the relation between Z and R can dis abuse them of this. The North American custom of numbering storeys 1 , 2 , 3 , . . . ascending, and lB, 2B, 3B, . . . (or th e like) de scending, with no 0 in between, is admittedly absurd. Storeys in a building are discrete and in order, thus nat urally modelled by Z; it is natural to let the ground floor be G or the rez-de-chaussee* be RC, or simply 0, to let 1, 2, 3, . . . number the storeys above it, and to let IS, 2S, . . . (or the like) number those below. Time is naturally modelled by R. Humans choose a zero point and a unit of measurement of time, whereupon every instant acquires a labelling by a real number. Though the unit of measurement is determined by the observed ma neuvers of our planet, the choice of zero point is arbitrary. The most prevalent choice arose from the history of the Christian religion (as another much-used choice arose from Islamic religion), but it has no clear theological ra tionale even for Christians, and nowadays the labelling of time is usually divested of religious reference. I will call the present year simply 2001. +2001, that is. What then is a year? The year is the interval be tween the instants - 1 and so that the year 2001 is the interval between the instant 2000 and the instant 2001 . It is only slightly surprising that the custom is dif ferent for negative n, so that the year -34 (in which
n
'An
80
n;
n
some say Jesus was born) is the interval between the instant -34 and the instant -33. In a striking but su perfluous innovation, I write this explicitly: the year is the interval between the instant and the instant l) . And n =F 0. This convention divides R into disjoint intervals, and there is no room for a zeroth year. This for the same unmysterious reason that there is no zeroth cen tury, and no zet·oth millennium. Months, hours, and seconds are handled differently. They are counted forward, even in the territory of neg atives. Is that the hidden reason why this confuses peo ple? A century before 29 March 0053 is still 29 March, but we resist the impulse to call it 29 March - 0047. It is 29 March - 0046. In the same way, if my bank balance is $53. 15 and I am so careless as to write a check for $ 100, my new balance will be $53. 15 - $ 100.00 = - $46.85. But not - $46. 15: both the dollars and the cents are counted backward (like years) rather than forward (like months). Here my own reckonings part company with my cal culator's. I update my bank balance on an abacus, which would rather subtract $ 100 without touching any column smaller than the hundreds column. I ac commodate. I regard all dollar amounts as modulo $100000.00. If my balance is $53. 15 and then $100.00 goes out, my new balance on the abacus is $99953. 15. I know no way to adapt this idea to reckoning of time, where we will surely continue to count dates for ward, abacus-like, and names of years reversibly, cal culator-like. But contrast the zeroth storey of a build ing, which (like the 13th storey in some buildings) really was the victim of an idiotic tradition. The Mystery of the Missing Year Zero is not a case need ing investigation. It lacks a corpus delecti.
n(lnl - �nl
n
unmysterious word meaning street level-except for the mystery why rez instead of ras, as that word is spelled in every other context.
THE MATHEMATICAL INTELLIGENCER
n