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The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.

T h e R e w a r d s of Honesty?.

This is a story of an ill-posed mathematical problem which seems to have led to serious injustice to an innocent young mathematician. We want to bring it to the attention of the readers of this journal, not only because it may carry a lesson, but also in the hope that international reaction could help the victim. Myung Ho Kim, a young U.S.-educated mathematician (Ph.D. from University of Michigan, 1988) returned to his homeland and assumed the position of Assistant Professor of Mathematics at Sungkyunkwan University in Seoul, South Korea, in 1991. In 1995, Kim participated in an entrance exam grading, an annual event usually taken very seriously in Korea, where the competition for entrance to college is fierce. During the grading, Kim found a serious mathematical error in the wording of one problem which counted 15 points out of a total of 100 points for the entire mathematical portion of the entrance exam. Here is the misstated problem: Three non-zero vectors A, B, and C in three-dimensional Euclidean space satisfy the following inequality:

IIxA+yB+zCII>I Ix AI t + I lY BII for all real numbers x, y, and z.

Show that the three vectors are perpendicular to each other. The difficulty pointed out by Kim is that the hypothesis is null: no three non-zero vectors exist satisfying the hypothesis. For distribution to students after the exam, the proposers had written out as the solution a (valid) proof of the conclusion A . B = B . C = A . C = O.

In response to Prof. Kim's observation, they called this part (i) of the solution,

4

THE MATHEMATICAL INTELLIGENCER 9 1997 SPRINGER-VERLAG NEW YORK

then appended as part (ii) the proof that also either A or B is O. But this seems still unsatisfactory, because it leaves the graders with no possible way to grade the problem fairly: the student who had given only the originally intended solution, part (i) of the posted solution, might claim full credit, yet the courageous student who wrote that the vectors can not be non-zero has given mathematically a much superior answer. Kim therefore persisted in recommending that no weight be given to this question in the grading. This seems to us to be a sensible position, mathematically and pedagogically. Unfortunately, the senior faculty members in the department who were responsible for the error chose instead to fight Kim. Since then, the department as well as the University began to take a number of disciplinary measures against Kim. He was first given a stern warning and was threatened to be barred from teaching for one academic quarter (three months) without salary. Later Kim was denied promotion to Associate Professor (necessary for continuing his appointment); therefore his employment at the University was effectively terminated. We are told that both the suspension and the refusal of promotion were unprecedented in his department. After Kim's firing, a number of younger mathematical faculty in Korean university circles rose to support him, and petitions were circulated protesting the University's unjustified action to the Ministry of Education as well as to the University. The petitions, however, did not help. Kim appealed to the courts, so far unsuccessfully. In his legal plea, Prof. Kim wanted to present to the court an independent authoritative statement that his objection to the contentious examination question was well founded. The Korean Mathematical Society (which naturally

has interlocking directorates with Sungksamkwan University) declined to give such a statement. Prof. Kim therefore turned abroad, and we willingly offered such a statement to the court. What are the lessons of this extraordinary case? As for making a minor blunder in setting problems for an exam, no reproach should be made. Mathematicians make mistakes. But when their mistake is noticed, they should be quick to apologize and retract. To the colleague who pointed out the mistake, the proper professional response is not punishment, but thanks. LAWRENCE A. SHEPP AT&T Laboratories Murray Hill, NJ 07974 USA e-mail: [email protected] SEYMOUR SCHUSTER Department of Mathematics Carleton College Northfield, MN 55057 USA e-mail: [email protected] CORA SADOSKY Department of Mathematics Howard University Washington, DC 20059 e-mail: [email protected] RONALD L. GRAHAM AT&T Laboratories Murray Hill, NJ 17974 USA e-mail: [email protected] CHANDLER DAVIS Department of Mathematics University of Toronto Toronto, M5S 1A1 Canada e-mail: [email protected] ZANG-HEE CHO Department of Radiological Sciences University of California Irvine, CA 92697, USA, and Department of Information and Communication Korean Institute of Advanced Science and Technology Seoul, Republic of Korea e-mail: [email protected]

The Burden of the Grant System Alfredo Octavio's article "The Indexed Theorem" (Mathematical Intelligencer, vol. 19, no. 3, 9-11) complains about the criteria used by the PPI, the grant program in Venezuela. We would like to point out that it is even worse than he says, in that the burden of systems like the PPI falls mostly on researchers who are starting their careers. While most mature researchers manage to get funded, even ff at inadequate levels, young researchers, arguably the ones who need support the most, easily fall through the cracks of the system. This is true in Venezuela in all the areas funded by the PPI. The effect is magnified by institutions using the level of PPI funding to make decisions about other funding sources, sometimes to the total exclusion of any other form of evaluation. Mr. Octavio fails to point out the most damaging implication of such a system to a scientific community: beginning researchers get discouraged by what they rightly perceive as an injustice, and in some cases they leave the country or abandon science altogether. MISCHA COTLAR Facultad de Ciencias Universidad Central de Venezuela Caracas, Venezuela L/~.AR O RECHT Departamento de Matem&ticas P. y A. Universidad SimSn Bolivar Caracas, Venezuela e-mail: [email protected]

W h a t the Citation Index is Good For Alfredo Octavio in "The 'Indexed' Theorem" (Mathematical Intelligencer vol. 18, no. 4, 9-11) complains of the use of the Science Citation Index in deciding on research grants. In his country, the granting agency judges an applicant's research merit by the number of papers listed in the SCI. But this is not the intended purpose of the SCI! It is a bibliographic database to be used in literature searching. By searching cited references, you may be able to fmd articles which you cannot find in other databases by searching by subject [8]. SciSearch, the

online edition of the SCI, has given us good results at the Freie Universit~t Berlin. It also seems to be more up-todate than other databases. Being multidisciplinary, it can also be used in fmding material on applications and cognate fields. Its drawback is that its coverage of mathematics is less complete than that of MathSci (online Mathematical Reviews) or MATH (on-

line Zentralblatt fi~r Mathematik). It can be helpful to do both citation searching (with SCI) and subject searching (with MathSci or MATH): you may avoid proving a theorem which is already published by somebody else. Unfortunately many mathematicians have not yet recognized this! The criteria for selecting journals for coverage in the SCI have been described by the founder and chairman emeritus of ISI, E. Garfield, in several articles: Citation analysis is only one criterion among others, for example journal standards and expert judgement[4]. Nowhere is it mentioned that you have to buy a $10,000 subscription to the SCI. Can that really be true? Secondly, Octavio's objections to using citation counts to evaluate merit get support from many articles of E. Garfield [2, 3, 5, 6] and others [1] in journals of information science. Sample: " . . . citations are only indicators of influence and impact, they are partial reflection of the interest of the academic community and the visibility of a person's work. They say nothing about intrinsic value. That is the role of human judgement."[5] Thus Octavio's ironic "Theorem" is not what is claimed by the proponents of SCI, only the following: THEOREM. A high number of citations indicates that a work is probably interesting or influential or useful to the scientific community. The converse is not true! There exist several reasons for a low number of citations, for example the "obliteration phenomenon'--taking a well-known paper for granted. A less cited article is not necessarily bad. "To be an un-cited scientist is no cause for shame.'[6] Therefore you cannot determine the quality of journals merely by a high impact factor. A recently published study

VOLUME 19, NUMBER 3, 1997

5

shows that top journals as qualified by 68 experts receive significantly higher citation rates than other journals [7]. "However, the application of these indicators is still l i m i t e d . . . They can never be used as a substitute for expert evaluations, nor are they a direct measure of quality." [7] Thirdly, ISI cannot be blamed for misuse and misinterpretation of the Science Citation Index as is done in the government grant program PPI. I hope Mr. Octavio will be successful in starting a serious discussion on alternatives. P.S.: On a personal note I would like to add that there are some interesting parallels between Mr. Octavio's career and my own. I too am married to a mathematician, and I am the mother of three children, two of whom are twins. REFERENCES 1. L. Baird, C. Oppenheim: Do citations matter? Journal of Information Science 20 (1994), 2-15. 2. E. Garfield: Citation Indexing: Its Theory and Applications in Science, Technology and Humanities. New York: Wiley, 1979. 3. E. Garfield: Uses and Misuses of Citation Frequency. In: Essays of an Information Scientist, Vol. 8. Philadelphia: ISI Press, 1986, 403-409. 4. E. Garfield: How ISI Selects Journals for Coverage: Quantitative and Qualitative Considerations. In: Essays of an Information Scientist, Vol. 13. Philadelphia: ISI Press, 1991, 185-193. 5. E. Garfield: Citation Data Is Subtle Stuff. A Primer on Evaluating a Scientist's Performance. In: Essays of an Information Scientist, Vol. 14. Philadelphia: ISI Press, 1992, 229-230. 6. E. Garfield: To Be an Un-cited Scientist Is No Cause for Shame. In: Essays of an Information Scientist, Vol. 14. Philadelphia: ISI Press, 1992, 390-391. 7. J.C. Korevaar, H.F. Moed: Validation of Bibliometric indicators in the Field of Mathematics. Scientometrics 37 (1996), 117-130. 8. M. L. Pao, D. B. Worthen: Retrieval Effectiveness by Semantic and Citation Searching. Joumal of the American Society for Information Science 40 (1989), 226-235.

D-14195 Berlin, Germany e-mail: [email protected]

EDITOR'S NOTE:

Dr. GSbel questions the allegation that ISI requires journals to subscribe to the Index in order to be listed. Dr. Octavio got this from the Scientific A m e r i c a n article he cites. Subsequently, in its October 1995 issue, p. 10, S c i e n t i f i c A m e r i c a n stated, "According to the ISI, it has never required that any journal, person, or institution purchase an ISI product to qualify for inclusion in its indexes." O p t i m a l Lacings

I am a student of Nara Women's University, Japan. I am submitting a master's thesis entitled "Variational Problems on Lacings," giving new proofs and extensions of Halton's theorem on the shoelace problem (The M a t h e m a t ical Intelligencer vol. 17 (1995), no. 4, 36-41). It was rather a shock when I recently found the article "Lacing Irregular Shoes" by M. Misiurewicz in The Intelligencer vol. 18 (1996), no. 4, 32-34, because essentially the same result with the same proof was obtained independently in my master's thesis. Perhaps your readers would like to see more of my results. The problem is in every case to choose a path going once to every A~ and every Bi (i = 0, 1. . . . , n), beginning at A0 and ending at B0, whose total length is shortest. The condition Every edge joins an Ai to a By (Alt) is imposed. Halton assumes (H) that the A i and the B j lie in two parallel equally spaced rows. I consider the weaker geometric hypothesis (C) that AkA1BmBn forms a convex quadrilateral whenever k < 1 and m > n, which is essentially the same hypothesis made by Misiurewicz. I also adopt other notation from Halton's article; in particular, (AM) is the usual lacing pattern, (EU) is the pattern Ao----> B1--> A1----~ B3---> A3----> . . . o B2 o A2 ~ Bo,

SILKE GOBEL Fachbereich Mathematik und Informatik Freie Universit&t Berlin Arnimallee 2-6

6

THE MATHEMATICALINTELLIGENCER

and (SS) is the pattern going directly Ao ~ Bn and then zigzagging back. Theorem A. Under the condition (C), the shortest lacing is (AM).

Theorem B. Assume Ai and B~ are connected for every i > 0. Then, under the condition (H), the shortest lacing is

(EU).

Theorem C. Assume A0 and Bn are connected. Then, under the condition (C), the shortest lacing is (SS). Theorem D. Under the condition (C), the longest lacing is Ao--) B n " ~ AI"-> Bn-1-"~ A2---> 9 . . ---->An-1--->BI"'>An'-->Bo.

The proofs of Theorems A, C, and D are similar. The proof of Theorem B needs a new idea, which also gives a simpler proof of Halton's original theorem. AKIK9 TETSUKA Department of Mathematics Nara Women's University Nara 630, Japan

M o r e on O p t i m a l Lacings 1

A method of lacing different from those considered by J.H. Halton can be shorter than the "American" method (AM) which he proves optimal. It is not a merely hypothetical method: I use it. I will call it the Jackson method (JM) after Sid Jackson who showed it to me. Let us assume the simplest geometry (H): there are n pairs of eyelets, the distance from Ai to Bi is w, and the distance from one eyelet to the next on the same side of the shoe is v. The method (JM) is A0--> A1---->B1--> B3--> A3---->... Bn_2 --->Bn --->/~n---->An_l ---->...--> A2 --->B2 --> B0. The condition (Alt) of alternating sides imposed by Haiton is not obeyed by (JM). (The expression given applies when n is odd. For n even, a diagonal path A2 ~ B0 is needed at the end.) The length is therefore (2v + w ) n for odd n > 1. If v = 1, w -- 2, we get a length of 4n for (JM) and 2~/-5n + 2 for (AM): the Jackson method always wins. If on the other hand v = w = l , we get lengths of 3n for (JM) and 2h/~n + 1 for (AM): the Jackson method wins only for n < 7. Similar results are found for other ratios and for even n. 1For references and notations, see the preceding letter.

Remark. It was pointed out by Ian Stewart ( S c i e n t i f i c A m e r i c a n 275 (1996), July, 94-97) that Halton's dodge of looking at "unfoldings" of paths reduces the shoelace problem to one of "Fermat optics": fmding the shortest path through a rectangular lattice. Shoes in which v < w - - a s is usual--make this a problem in an anisotropic medium: it is farther between nodes in one direction than in the orthogonal direction. D. SINGLETON Department of Physics Virginia Commonwealth University Richmond, VA 23284-2000 USA e-mail: [email protected]

Some readers of the S c i e n t i f i c A m e r i c a n presentation of Halton's problem also offered economical lacings violating condition (Alt); see their vol. 275 (1996), December, 1 1 8 . Editor's Note.

The Missing Past of the Not-so-missing Link The point of view heralded by Felipe Acker in The InteUigencer, vol. 18 (1996), no. 3, 4-9, is not so unprecedented as he asks us to believe. Clans Miiller's paper "Uber die Grundoperationen der Vektoranalysis," Math. A n n . 124 (1952), 427-449, defreed divergence and rotation of vector fields in 3-space via "rfiumliche Differentiation" and obtained the integral theorems in the same form as Acker, perhaps more directly. The step to n-space is straightforward. Since then, I have heard that Elie Cartan may have had the idea even earlier. On this basis, my own paper "Ein einfacher Beweis des Gaussschen Integralsatzes," Jahresber. D M V 66 (1964), 119-138, obtained the theorem for fairly general domains in n-space with fairly simple means. It is clear that the case n = 2 contains the full Cauchy-Goursat theorem. I have taught this to my students ever since, and it has been published in an appendix to Abstract A n a l y t i c F u n c t i o n Theory a n d H a r d y Algebras, Lecture Notes Math. 593 (1977), by Klaus Barbey and myself.

(These sources do not, however, involve the "extended mean value theorem" of Acker.) I must confess it is a relief to see that respected colleagues have overlooked something "well-known'--this happens to me again and again! HEINZ KONIG Fachbereich Mathematik Universit&t des Saarlandes D-66041 Saarbr0cken Germany

Felipe Acker Replies: I have nothing to add on questions of priority. The articles cited by Professor KOnig seem not to have made their way into current discourse on differential f o r m s - - a t least I m u s t confess being u n a w a r e o f them; we can conjecture that the same could be the destiny of my "The missing link." The real question is why these questions have been relegated to a marginal place and whether it is worth trying to reverse this. My answer is that Cauchy-Goursat alone is not enough motivation. I believe that the change in point of view should have consequences in the treatment of the equations of mathematical physics, where the search for solutions of partial differential equations seems to me a misguided approach. I conjecture that it will be more fruitful to dissociate exterior derivatives from partial derivatives. Departamento de Matem~,tica Aplicada Universidade Federal do Rio de Janeiro Caixa Postal 68530 21945-970 Rio de Janeiro, Brazil e-mail: [email protected]

A Shorter Derivation Jan van de Craats told me that my article on Escher in vol. 18, no. 4, pp. 42-46, could have been improved by observing that Equations (6) yield rlr2 = 2.

Since ( V 3 - 1)rl = (X/6 - 2)r2, it follows that r 2 = (X/6 - 2)(X/3 + 1), r 2 = (h/6 + 2)(X/3 - 1)

and

4x 2 = (r2 - rl) 2 = r 2 + r 2 - 2rtr2 = 2 V 2 ( V 2 - 1)2, leading immediately to (7). The "surprising" identities (1 u ~/2 -+ ~ / 3 ) 2 _2(V3 - 1)(V3 u h ~ ) , involving the units ~ ~ h/2, now come by comparing the expressions for r, and r 2. Moreover, AB could have been obtained without fn'st finding AC. In fact, the equation just above (5) could have been extended to d2 X/3-

r2 AB ~ - ( X / 3 - 1)/N/2 '

yielding

AB

=

(x/5

-

1)r2

= 2-3/4( - 1 + 2 V 2 + V~ - h/6). H.S.M. COXETER Mathematics Department University of Toronto Toronto, Ontario, Canada M5S 3G3

Dating the Cryptic Bone D. Huylebrouck's interesting article on the Ishango Bone [Math. Intelligencer 18(4) (1996) 56--60] does not mention the most recent scholarship on the date of this artifact. The following papers present a date of approximately 20,000 years ago (not 11,000 years ago as stated by Huylebrock) for the bone. A. S. Brooks and C. C. Smith, Ishango revisited: new age-determinations and cultural interpretations, A f r i c a n Archaeological R e v i e w 5 (1987), 65-78. A.S. Brooks, et. al., Dating and context of three middle stone-age sites with bone points in the Upper Semliki Valley, Zaire, Science 268 (1995), 548-553. The articles do not mention the Ishango bone specifically, but the dates nevertheless apply to it, as my telephone conversation with Professor Brooks revealed last year. JEFFREY SHALLIT Department of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1 Canada e-mail: [email protected]

VOLUME 19, NUMBER 3, 1997

7

Multivariate

Descartes Rule of Signs and

Sturmfels's Challenge Problem

D e s c a r t e s ' s rule of signs b o u n d s the n u m b e r of positive real zeros n + 0 0 o f a p o l y n o m i a l f ( x ) in one variable. If r

f(x)

= ~. gxm5 j=l

with 0 -< m l < m2 < 9 9 9 < m r and with all coefficients cj ee 0, t h e n the n u m b e r of positive real zeros o f f is u p p e r b o u n d e d by the n u m b e r of sign changes N+(f) b e t w e e n consecutive coefficients cj w h e n t a k e n in o r d e r o f increasing j ; s e e [2] a n d [5], C h a p t e r 6. There is a similar upp e r b o u n d N - 0 O for t h e n u m b e r of negative real zeros o f f, w h i c h follows b y applying this b o u n d to the p o l y n o m i a l f ( - x ) . Together, t h e s e give an u p p e r b o u n d N(f)

= N+(f)

+ N-(f)

for the total n u m b e r of zeros in R* = R\[0}. F o r a n y s e t {mi : 1 --< i --< r}, one can c o n s t r u c t p o l y n o m i a l s f ( x ) in w h i c h the u p p e r b o u n d s N + 0 0 and N(J) b o t h hold with equality. It i m m e d i a t e l y follows from D e s c a r t e s ' s rule of signs that the n u m b e r of n o n z e r o real zeros n 0 0 o f f is b o u n d e d in t e r m s o f the n u m b e r r of m o n o m i a l s a p p e a r i n g in f, namely n ( f ) -< 2 ( r - 1). This b o u n d d o e s n o t d e p e n d on the d e g r e e s of t h e s e m o n o mials. In contrast, the n u m b e r o f nonzero c o m p l e x zeros o f f is e x a c t l y m r - m l , b y the fundamental t h e o r e m o f algebra.

In the 1970s, A. G. Kushnirenko p o s e d the p r o b l e m o f b o u n d i n g the n u m b e r o f real z e r o s in (R*) n of a system o f multivariate p o l y n o m i a l s p u r e l y in t e r m s o f the m o n o m i a l s a p p e a r i n g in t h e various polynomials. Call a s y s t e m F = ( f l , 9 9 9 , f n ) of n p o l y n o m i a l s in n variables f i ( x l , x2, . . . , Xn),

l <- i <- n,

a r e g u l a r s y s t e m if its zeros in C n are i s o l a t e d a n d nonde-

generate, that is, if the s y s t e m F is a c o m p l e t e intersection. (Here n o n d e g e n e r a t e m e a n s t h a t the J a c o b i a n d e t e r m i n a n t of the s y s t e m is nonvanishing at e a c h zero.) Let M i d e n o t e the set o f m o n o m i a l s a p p e a r i n g in Ji(Xl, 9 9 9 , Xn). Kushnirenko [12, p. 123] c o n j e c t u r e d t h a t the n u m b e r o f r o o t s o f a c o m p l e t e intersection F ( x ) = 0 in ( R * ) n c a n b e b o u n d e d p u r e l y in t e r m s o f the cardinality Si :=~/i], b y n

YI ( s i - 1). i=l This c o n j e c t u r e r e m a i n s open. In 1980, A. G. Khovanskii [10] o b t a i n e d a w e a k e r u p p e r b o u n d d e p e n d i n g only on the Si, n a m e l y (n + 2)32 S(S + 1)/2,

in which S is the total number of distinct monomials appearing in all t h e f / ( x l , . . . , x n ) , so that S <- ~ in= 1 S i . Bounds on the n u m b e r o f real zeros have a p p l i c a t i o n s in computational c o m p l e x i t y theory; s e e [14]. In studying t h e s e questions it is n a t u r a l to l o o k for a

9 1997 SPRtNGER-VERLAG NEW YORK, VOLUME 19, NUMBER 3, 1997

9

multivariate a n a l o g u e of D e s c a r t e s ' s n i l e of signs. Consider, for a r e g u l a r s y s t e m with real coefficients, its sets o f m o n o m i a l s M : = {Mi : 1 -< i - n} a n d the p a t t e r n of signs e : = {ei : 1 - i -< n} a t t a c h e d to its coefficients; each ei : Mi---> {+1, -1}. Set ~. = (Mi, el), and let N+(/h, . . . , ~ ) a n d N(s 999, s d e n o t e the m a x i m a l possible n u m b e r o f zeros in the positive o r t h a n t (R+) n and in (R*) n, respectively. In view of Khovanskii's bound, these n u m b e r s a r e well defined and finite. A multivariate Descartes rule w o u l d be algebraic f o r m u l a s for the numb e r s N+(At, . . . , ~n) and N(At . . . . , An). Recent d e v e l o p m e n t s in real algebraic g e o m e t r y due to O. Viro [18] have s u g g e s t e d a p o s s i b l e answer. B. Sturmfels [17] d e v e l o p e d f o r m u l a s for Viro's m e t h o d a p p l i e d to regular systems. Using these, I. Itenberg a n d M.-F. Roy [9] prod u c e d c o n j e c t u r a l explicit c o m b i n a t o r i a l f o r m u l a s for the values of N+(z~t, . . . , ~n) and N ( s 9 9 9 , ~ n ) . These formulas are k n o w n to be c o r r e c t in a few special cases. To raise a w a r e n e s s o f the I t e n b e r g - R o y conjecture, Bernd Sturmfels p r o p o s e d a special case as a challenge p r o b l e m and offered a $500 r e w a r d for its solution; see Figure 4. In this article, w e give an overview o f the I t e n b e r g - R o y conjecture a n d t h e n p r e s e n t a solution to Sturmfels's challenge p r o b l e m . The c o m b i n a t o r i a l invariants in the I t e n b e r g - R o y c o n j e c t u r e involve triangulations o f the Newton polytopes a t t a c h e d to the polynomials3~(Xl,..., Xn). We first d e s c r i b e N e w t o n p o l y t o p e s and t h e i r relation to the c o m p l e x zeros in (C*) n of regular systems. In the following section, w e d e s c r i b e the I t e n b e r g - R o y c o n j e c t u r e in the special case n = 2 (to r e d u c e notation), a n d in the final section, w e solve Sturmfels's problem.

Fewnomials and Their Complex Zeros Polynomials d e s c r i b e d in t e r m s of their c o n s t i t u e n t monomials are called f e w n o m i a l s o r sparse p o l y n o m i a l s ; see [6] and [11]. [The t e r m f e w n o m i a l (malochlen) is due to Kushnirenko.] A m o n o m i a l x m = x~TM x~22 . . . Xn m~ is i n d e x e d b y t h e nonnegative integer v e c t o r m = (mr, m2, 9 9 9 m n ) E ~_~ o" The sparse r e p r e s e n t a t i o n of a p o l y n o m i a l f ( x ) is the d a t a [ c ( m ) : m E M}, w h e r e f(x)=

~

c ( m ) x m,

mEM

a n d all coefficients c ( m ) r 0 for m E M. The set M := n M [ f ] C E >-0 i n d e x e s the m o n o m i a l s p r e s e n t i n f ( x ) . A f u n d a m e n t a l invariant of a f e w n o m i a l is its N e w t o n polytope A = A(M), w h i c h is the c o n v e x hull o f the points of M. This p o l y t o p e is a lattice polytope in Rn; i.e., all its vertices are in 77". Figure 1 pictures the N e w t o n p o l y t o p e s At and A2 for t h e s y s t e m f l ( x t , x2) = 1 + XlX2 + 3x~lx~2, f 2 ( x l , x2) = 1 - Xl + 2xtx2. This e x a m p l e illustrates that N e w t o n p o l y t o p e s m a y be of l o w e r d i m e n s i o n t h a n n. In 1975, D. Bernstein [1] and A. G. K u s h n i r e n k o [12] obt a i n e d u p p e r b o u n d s for the n u m b e r o f zeros in (C*) n of regular s y s t e m s in t e r m s of the m i x e d v o l u m e s of their as-

10

THE MATHEMATICALINTELLIGENCER

A1

A2

A1 Jr A2

Figure 1. Newton polytopes.

s o c i a t e d N e w t o n polytopes. Mixed v o l u m e s can b e defined in t e r m s o f v o l u m e s of M i n k o w s k i s u m s of polytopes. The M i n k o w s k i s u m A t + A2 o f t w o p o l y t o p e s is A t + A2 : = {X 1 -I- X 2 : X 1 ~ At, X2 E A2}. Figure 1 pictures the M i n k o w s k i s u m of At and A2 for the e x a m p l e above. The M i n k o w s k i s u m of n p o l y t o p e s is def m e d similarly. (This definition m a k e s sense for a r b i t r a r y c o n v e x b o d i e s At and A2; s e e [16].) We define the combin a t o r i a l m i x e d v o l u m e V ( A t . . . . , A n ) b y the f o r m u l a V(A1, A2, . . . , An) := n "~ ~ ( - - 1 ) n + k ~ VOln(Ai,-b... k=l i 1 < i2 < . . . < ik

q- A i k ) ,

(1)

in w h i c h voln(.) d e n o t e s the n - d i m e n s i o n a l volume. In particular, V(A, A, . . . , h ) = n! voln (h). The c o m b i n a t o r i a l m i x e d volume differs b y a f a c t o r n! from the usual M i n k o w s k i definition o f m i x e d volume; see [16], L e m m a 5.1.3. THEOREM (Bernstein). The n u m b e r o f isolated zeros i n (C*) n o f a s y s t e m o f n p o l y n o m i a l s w i t h c o m p l e x coefficients fi(xl .....

Xn),

1 <- i ~ n,

(2)

is at m o s t the combinatorial m i x e d volume V(A1,A2,... , An), w h e r e A i is the N e w t o n polytope o f f i ( x l , 9 9 9 Xn). D. Bernstein [1] also o b s e r v e d t h a t a s y s t e m of "general position" polynomials (2) having Newton polyhedra A 1 , . . . An is a r e g u l a r s y s t e m a n d h a s e x a c t l y V ( A 1 , . . . , An) i s o l a t e d z e r o s in (C*) n. In particular, V ( A 1 , . . . , An) is an integer. I One i m p o r t a n t special c a s e is w h e r e e a c h J ~ ( x t , . . . , Xn) is a d e n s e p o l y n o m i a l o f t o t a l degree di, that is Mi = { m :Z~=I m j <- di}. In this case, a calculation r e v e a l s that V(A1, 9 9 9 , An) = dld2

...

dn,

w h i c h is the u p p e r b o u n d given b y Bezout's t h e o r e m for the n u m b e r of zeros of s y s t e m (2) in C n. In t h e t w o - d i m e n s i o n a l case, the c o m b i n a t o r i a l m i x e d v o l u m e f o r m u l a (1) b e c o m e s V(At, h2) = area(At + 52) - area(At) -- area(A2).

1Each Minkowski sum on the right side of (1) is a lattice polytope; hence, its ndimensional volume is of the form k/n! for some integer k. 13qis, however, only guarantees that nigh1. . . . . An) is an integer.

F o r the example in Figure 1, w e have a r e a ( A 1 ) = 0, area(A2) = 1, and area(A1 + A2) - ~ ,5 so that the N e w t o n p o l y t o p e b o u n d is V(At, A2) = 2. It is easy to c h e c k that this s y s t e m has no zeros with Xl = 0 or x2 -- 0; hence, all zeros in C 2 are in ((2*) 2. The N e w t o n p o l y t o p e b o u n d for this case strictly improves on the Bezout bound, which is 8. The b o u n d of B e r n s t e i n ' s T h e o r e m can b e e x t e n d e d to a b o u n d on isolated zeros in Cn; see [7] and [15]. Multivariate Descartes Rule F o r c o m p a r i s o n p u r p o s e s , w e reformulate D e s c a r t e s ' s rule o f signs for a univariate p o l y n o m i a l

w h o s e c o n v e x hull is A, t o g e t h e r w i t h an a s s i g n m e n t o f signs to the p o i n t s of M, i.e., a function e : M---) {+1, -1}. Itenberg and Roy obtained l o w e r b o u n d s n+(A1 . . . . , ~n, to) and n(s 9 9 9 , An, w) for the n u m b e r of zeros of (3) in (R+)n a n d (R*) n, respectively, which a p p l y w h e n t is positive a n d sufficiently close to 0. They t h e n defined n+(/X1,..., ~n):= max[n+(51,... to

n(51,...,

~):=

,An, to)],

m a x [ n ( 5 1 , . . . ,An, to)],

These quantities can be d e t e r m i n e d b y a finite calculation, d e s c r i b e d below. By definition, t h e s e quantities are l o w e r b o u n d s for N + ( s An) a n d N(A1, 9 9 9 An), i.e.,

if(X) : Z CjXmj, j=l in t e r m s of its signed N e w t o n p o l y t o p e A, w h i c h consists o f the N e w t o n p o l y t o p e A = [ml, m~], plus all its p o i n t s {mi : 1 <- i <-- r} C 77_<0, with a sign {e(m0 = sign(ci) : 1 -< i --< r}. The line s e g m e n t h is subdivided into r - 1 subintervals {Ji = [mi, m i + l ] : 1 -< i -< r - 1}, a n d e a c h subinterval is assigned a w e i g h t counting the sign change, namely

N+(s N(AI,

. . . , ~)

--> n+(s

. . . ,Z~n),

9 9 9 , A n ) --> n ( / ~ t ,

. .. ,~n).

The multivariate Descartes nile c o n j e c t u r e d b y Itenberg and Roy is t h a t equality always holds.

ITENBERG-ROY CONJECTURE.For a n y set o f signed N e w t o n

wt+(Ji) :=

1 f f e ( m i ) # e(mi+l) 0 if e ( m i ) = E(mi+l),

polytopes i n R n, N+(/~I, 9 9 9 , An) = n + ( A 1 , . . . ,An), N ( A 1 , . . . , ~n) = n ( 5 1 , . . . ,~n).

w h i c h is wt+(JO = 1 [1 § sign(cici+l)]. Then, r-1

N + ( f ) := Z wt+(Ji) 9 i=1

The n u m b e r of negative real zeros n - ( J ) is b o u n d e d a b o v e with a b o u n d N - ( f ) o b t a i n e d from the function f ( - x ) , w h i c h uses the weights w t - ( J 0 := 1[1 + ( - 1 ) mi § ml +i sign(cici+l)]. Now, set w t ( J i ) = wt+(Ji) + w t - ( J i ) . We o b t a i n the u p p e r b o u n d N(f) for the total n u m b e r of zeros in R* = R\{0} given b y r--1

goo:= X wt(y0. i=1

E a c h weight w t ( J 0 t a k e s on a value 0, 1, o r 2. Itenberg and Roy actually d e t e r m i n e d l o w e r b o u n d s for the n u m b e r of i s o l a t e d z e r o s of special s y s t e m s o f polyn o m i a l s in (R*) n. This t h e y did using the m e t h o d o f Viro [19], as e x t e n d e d to a p p l y to complete intersections b y Sturmfels [17]. They c o n s i d e r e d systems

This c o n j e c t u r e is b a s e d on the b e l i e f that the Viro cons t r u c t i o n includes e x t r e m a l systems. It is k n o w n that t h e r e are a r r a n g e m e n t s of zeros n o t p r o d u c e d by any Viro construction (3) in the limit as p a r a m e t e r values t---> 0 +, in s o m e a n a l o g o u s problems; s e e [8], Sect. 7. It r e m a i n s to d e s c r i b e f o r m u l a s for n+(A1 . . . . , An, to) and n ( A 1 , . . . , An, to). F o r simplicity, w e s u p p o s e that t h e d i m e n s i o n n = 2. The functions to = (Ol, to2) are required to be "sufficiently generic" in a s e n s e e x p l a i n e d below. First, the function tot induces a p o l y g o n a l subdivision ~1 o f h i as follows. L o o k at the c o n v e x hull F1 o f t h e graph o f tol on hi, i.e., o f the p o i n t s {(m, tol(m)) : m E M1} in R 3. This is a p o l y t o p e F1 in R 3, a n d its o r t h o g o n a l p r o j e c t i o n onto its fu'st t w o c o o r d i n a t e s h a s i m a g e AI. The orthogonal p r o j e c t i o n o f the l o w e r c o n v e x hull of F1 onto h i gives the p o l y g o n a l subdivision. Here the lower c o n v e x hull o f F1 c o n s i s t s o f the two-dimensional faces F of F1 such t h a t i f ( X l , X2, ts) E F, then (xl, X2, t) ~ F1 ff t < t3. The function to2 similarly induces a p o l y g o n a l subdivision T2 of A 2. Together, they induce a polygonal subdivision to of h i + A2 using the function tol,2 : M1 + M2--* R,

J~(Xl, 9 9

Xn, t) =

~" c ( m ) t ~ ra ~ Mi

x~,

1 - i -< n,

(3)

in w h i c h t is a p a r a m e t e r s u c h that t is positive a n d sufficiently close to 0, a n d toi : Mi---> R +,

l <_ i <_ n,

are essentially a r b i t r a r y functions. A signed N e w t o n polytope h is a set of p o i n t s M C 7/n 0

defmed by tol,2(m) : = min{tol(ml) + o~(m2) : m l + Ia2 = In, m l E M1, m2 e M2}.

E a c h t w o - d i m e n s i o n a l face F o f T~ h a s a unique representation F:=FI

+ F2,

VOLUME 19, NUMBER 3, 1997

11

with F1 a face of T1 and F2 a face of T2. We say that the function w = (wl,02) is sufficiently generic if the equality

As an example, consider the signed Newton polytopes given by

dim(F) = dim(Fx) + dim(F2)

M1 := {(0,5), (5,0), (3,5), (6,8)} and M2 := {(0,5), (5,0), (5,3), (8,6)}

holds for all two-dimensional faces F of r~. This condition holds for almost all functions ~0. In this polygonal subdivision an especially important role is played by the mixed cells. A mixed cell F of z~ is a cell F = F1 + F2 w h o s e representation 2 has dim(F1) = dim(F2) = 1. Note that the mixed cells and the polygonal subdivision T~ are completely determined by the monomial set M and the functions tot and 02. The signs 9 = (el, 999, 9 attached to M will enter in computing n+(A1, . . . , ~n,~O) and n(A1, 99 9 An, w) as a sum of contributions from the mixed cells.

2In the n-dimensional case, a mixed cell is a cell F = F~ + F2 + . . . A 1 4- A 2 4- . . . 4- An such that dim(F0 = dim(F2) = . . .

+ Fn of

= dim(Fn) = 1.

Figure 2. Signed Newton polytopes and subdivisions ~1 and

r2

with sign patterns 91 = { 1 , 1 , - 1 , 1 }

and

9

together with the functions Wl and 02 on M1 and M2 with values w1:={5,6,2,7}

and

02:={1,4,10,7}

corresponding to the vertices above. The resulting signed Newton polytopes A1 and A2 and the polyhedral subdivisions T1 and 1"2of 51 and h 2 , respectively, are indicated in Figure 2. The polyhedral subdivision z~ induced by tOl,2 on A1 + A2 is pictured in Figure 3, and the values of ~ol,2(') are also indicated. Note that for the points x = (5, 5) and (1 1, 1 1), we have ~ol,2(5, 5) = min(~oK0, 5) + 02(5, 0), oJ1(5, 0) + 02(0, 5)) = min(9, 7) = 7, ~ot,2(ll, 11) = min(wl(3, 5) + 02(8, 6), wt(6, 8) + 02(5, 3)) = min(17, 9) = 9. There are three mixed cells in T~, which are the three shaded parallelograms in Figure 3. The function ~o = (o~1,02) is sufficiently generic. The general formulas for n+(A1, A2, to) and n(A1, A2, w) consist of a sum of contributions from each mixed cell F of weights wt+(F) and wt(F), respectively. 3 Given a twodimensional mixed cell, its decomposition is into an edge F1 of T1 with vertices m(11) = ~rm(1)n,m~)) and m~1) = (m~), m(t)~22 ~ in M1 and an edge F2 of z2 with vertices m(12)and m(22) in M2. The edge F1 is alternating if the signs of its endpoints do not agree, i.e., if 9 1)) r 9 A mixed cell is alternating if all its edges are alternating. The weight wt+(F) for a mixed cell F is defmed by

Figure 3. Polyhedral subdivision of of ~1,2 are indicated.)

AI + h2.

(Function values

wt + (F)

:=

{10 if F is alternating otherwise.

The weight wt(F) for a mixed cell F in ~n is defined by wt(F) :=

Z

wt+(Fa)'

6 ~ {+1, --1}n

in which 3 = (31, 62, 999, an), and the system F s ( x ) is the system obtained from F ( x ) by replacing each variable xi by 6ixi. This has the effect of changing the sign pattern ei(ml, m2) of m = (ml, m2) to ~i(ml, m2) = ei(ml, m2)(60m'(&2)m2. Itenberg and Roy derived a combinatorial formula for wt(F) which shows that it can only take the values 0, 1, or a p o w e r of 2, up to 2 n. For the example in Figure 3, there is a mixed cell F -F1 + F2 in 1%,where F1 has vertices m~1) (3, 5) and m(21) = (6, 8), and F2 has vertices m(12) = (0, 5) and m(22) = (8, 6). =

3Here, w t + ( F ) and w t ( F ) d e p e n d on the subdivision ~-~,and the sign patterns ~1, ~2, . . 9, ~n.

12

THE MATHEMATICALINTELUGENCER

For this mixed cell, F1 is not alternating and F2 is alternating; hence, Wt+(F) = 0. The other two mixed cells in Figure 3 also have Wt+(F) = 0; hence,

Further computation yields wt(F) = 1 and n(51, 52, w) = 3. For example, if 6 = ( - 1 , - 1 ) , then the mixed cell F above has ~1(m(11)) = ( - 1 ) ( - 1)a( - 1) 5 = - 1, ~1(m(12)) = 1, ~2(I~1)) = 1(1)~ 5 = - 1 , and ~2(m(22)) = 1; hence, F is alternating. The calculations above depend on the signed Newton polytopes, and the dependence on w is completely specified by the polygonal subdivisions ~'1, 72, and T~. Consequently, only fmitely m a n y cases must be examined to determine n+(51, 52) and n(51, /~2)" The Itenberg-Roy conjecture is true for n = 1 by Descartes's rule of signs. Itenberg and Roy observed that it is true for N(A1, . . . , An) whenever n(51 . . . . , 5n) = V ( A 1 , . . . , An) by Bernstein's Theorem. The combinatorial mixed volume V(A1, . . . , An) is a l w a y s equal to the sum of the volumes of the mixed cells in any subdivision To of A1 + A2 + 9. . + An, i.e., V(AI' A 2 ' ' " " ' An) ---Z voln(F); F a mixed cell of ceo

see [6]. In the example T~ in Figure 3, the three mixed cells have areas 15, 21, and 27, and V(A1, A2) = 63. Itenberg and Roy observed that any mixed cell F of a subdivision which has vol(F) = 1 necessarily has Wt(F) = 1. It follows that if (51, 9 99, ~ ) has a sufficiently generic function ~o = ( o J 1 , . . . , wn) such that all the mixed cells of ~-~ have volume 1, then 5n) = V(A1, A2, 999,

Sturmfels offered a reward for resolving the Itenberg-Roy conjecture for the system x 5 = a l y 5 + a2xay 5 + a3x6y 8, y5 = b l x 5 + b2x5y3 + b3xSy6

n+(51, 52, w) = O.

n(/~l, / ~ 2 , . . . ,

Sturmfels's Challenge Problem

An),

and the Itenberg-Roy conjecture holds in this case.

(4)

with al, a2, a3, bl, b2, b3 > 0. Here, the Newton polytope bound of Bernstein's Theorem for the n u m b e r of roots in (C*) 2 is 63, and the combinatorial b o u n d for roots in (~.)2 is 3. Its signed Newton polytopes (51, 52) appear in the example in the preceding section. The system (4) was found by c o m p u t e r search to be one that exhibits a particularly strildng difference between the Newton polytope bound and the combinatorial bound. Sturmfels circulated his challenge problem at Oberwolfach and other places as an advertisement for the Itenberg-Roy conjecture; see Figure 4. The system (4) arises by removing a factor of x and a factor of y from his fLrst and second equations, respectively.] We solve Sturmfels's problem using ad hoc techniques involving Descartes's rule of signs in one variable. Perhaps this solution contains the seed of some idea relevant to the general case; however, this is not obvious. It does verify the Itenberg-Roy conjecture in a nontrivial case. We will actually prove the Itenberg-Roy conjecture for N(A1, A2) for all sign combinations of the coefficients for which the conjecture predicts that the system (4) has at most three solutions in (R*) 2. These consist of all positive coefficients; of a2 < 0, b2 < 0 with all other coefficients positive; and in the other cases resulting from these two by sign changes of the variables x ---) - x or y --~ - y . THEOREM. F o r al, a3, bl, b3 > 0, a2b2 > O, the s y s t e m (4) h a s at m o s t three roots i n (R*) 2. Proof. S t e p 1: Without loss of generality, we can assume a2 = b2. If a2 r b2, then we obtain an equivalent system in

Figure 4. B. Sturmfels at the Mathematisches ForschungsinsUtut O b e r w o l f a c h . ( P h o t o c o u r t e s y o f G. M. Z i e g l e r . )

VOLUME19, NUMBER3, 1997 13

which a2 = b2 as follows. Let T : = (b2/a2) 1/7. Substitute y --) yy into (4) and multiply through the second equation by T-5. S t e p 2: Let x -- y / m , and write the equations in the equivalent form (5) (6)

m = a i m 6 + a2m3y 3 + a3y 9, m 8 = b l m 3 + b2may 3 + bay 9.

This birational t r a n s f o r m a t i o n preserves roots in (•.)2. S t e p 3: Eliminate m a y 3 and y9, respectively, from these to obtain the following pair of equations:

follows ff we show that f has a single critical p o i n t (necessarily a m a x i m u m ) for m > 0; i.e., f ( m ) is unimodal. Differentiating m 7 u ( m ) v ( m ) -3 w i t h respect to m, we obtain f'(m)

= m 6 g(m) v ( m ) 4'

where g ( m ) := [7 u ( m ) + m u ' ( m ) ] v ( m ) - 3 u ( m ) m v ' ( m ) = - 7 r 3 m la - ( a l + 9 a l r3) m 12 - 3 a12m 1~ - 4rablm 9 +

(a3 - ba)y 9 = m - a i m 6 - ( m s - b l m 3) (1

-

a3)a2m3y3 = m - a i m 6 -

a3 (mS

b3

-

blm3)

(7) (8)

b3

If a3 ----b3, t h e n (7) reduces to m s + a~m 6 - bl m 3 - m = O.

By Descartes's rule of signs, this has at m o s t one root for m > 0 and n o roots for m < 0. It is clear that one positive root exists, a n d for this single positive root, there is a unique real value of y that solves (5) ( u n i q u e n e s s here follows from Descartes's n i l e of signs). Thus, there is one solution to the original system in this case. Henceforth, we a s s u m e a3 r b3, a n d set r3 := a3/b3 =/= 1.

In this case, (7) a n d (8) are equivalent to (5) a n d (6). Note that w h e n as r b3, t h e n for fixed m, there is a unique real value of y satisfying (7), a n d similarly for (8). S t e p 4: Eliminate y by cubing both sides of (8) and dividing them into the respective sides of (7). Then, multiply through by m 9 to obtain m 7 (1 - a i m 5 - m 7 + b l m 2) - b 3 (1 - r3) -2 a2 -3 = [ 1 - a i m 5 + r3 ( m 7 + blm2)] 3 " Note that the right-hand side does not d e p e n d o n a2. Thus, the left-hand side can be set to any n o n z e r o c o n s t a n t indep e n d e n t of the right-hand side. It is notationally more convenient to replace m with - m . Thus, we are left to consider O/=

m 7 (1 + a i m 5 - m 7 § b l m 2) [1 + a l m 5 - r3 ( m 7 - blm2)] 3 '

where ~ can b e a n y n o n z e r o constant. Let us write this equation as tim):=

m 7 u(m)

v ( m ) ~ - ~,

(9)

where u ( m ) : = 1 + a i m 5 + m 7 + b i m 2, v(m): = 1 + a i m 5 + r3(m 7 + blm2). By Descartes's rule of signs, u and v both have at most one zero in m < 0. It is clear that both have at least one negative zero; hence, both have exactly one negative zero. We will let m u and m y denote the negative zero of u a n d v, respectively. S t e p 5: Here, we consider the range m > 0. If a < 0, then there are n o solutions to (9) with m > 0 because f ( m ) > 0 in this case. We claim that if ~ > 0, t h e n there are at most two solutions to (9) with m > 0. Note that f(0) = 0 and f ( + ~ ) = 0. Because v ( m ) -> 1 for m - 0, the claim

14

THE MATHEMATICAL INTELLIGENCER

( - 6 bl a l + 14 - 14r3 + 6 a l r3 bl) m 7 + 4 a i m 5 + 3 bl 2 r3 m 4 + (r3 bl + 9 bl) m 2 + 7. The coefficients o f g for t e r m s of degree greater t h a n 7 are negative and the coefficients for terms of degree less t h a n 7 are positive. Thus, Descartes's rule of signs implies that g ( m ) has at most one positive zero; hence, f has at m o s t one positive critical point. S t e p 6: We consider the remaining range m < 0. We claim that for m < 0, eq. (9) has at most one solution if > 0 a n d at most three solutions if ~ < 0. This will complete the proof. We will first show t h a t f ' ( m ) = 0 can have at most one solution in the range m < 0. Note that here f ' ( m ) = 0 implies g ( m ) = 0. Now, m g ' ( m ) - 7g(m) = - r 3 m 4 (49 m 1~ + 8 bl m 5 + 9bl 2)

- (9 a l 2 m 10 + 8 al m 5 + 49) -- 5 (al + 9 alr3) m 12 - 5 (r3bl + 9 hi) m 2

<0. The inequality follows because 49m 1~ + 8 bl m 5 + 9 b2 is positive everywhere, being quadratic in m 5 without real zeros, and a similar statement applies to 9 a12m 10 + 8 a i m 5 + 49. We conclude t h a t i f m < 0 a n d g ( m ) = 0, t h e n g ' ( m ) > 0. Sinceg(0) > 0 and g(-or = - % we see that g ( m ) = 0 has exactly one real solution with m < 0. Let m g be this zero of g. Note that g ( m ) > 0 for m E (mg, 0), and g ( m ) < 0 for m E ( - % my). It is easy to see that m u = m y if and only if albl = 1. Ass u m e albl = 1; then, as u ( m ) = v ( m ) = 0 impliesg(m) = 0, we have mu = m y = m y = - b ~ . We then obtain the reduction mT(1 + b l m 2) f ( m ) = (1 + b i r a m 2 ) v ( m ) 2" We see t h a t f ( m ) is m o n o t o n i c a l l y decreasing o n ( - % m y ) a n d m o n o t o n i c a l l y increasing o n (my, 0), tending to 0 from b e l o w at both endpoints a n d t e n d i n g to - ~ from both sides at mg. We conclude in this case that, for m < 0, Eq. (9) has n o solutions for a > 0 a n d exactly two solutions for a < 0. Next, assume mu > mv. The argument here is s'Lmflarto the one above. In this case, we have m y ~ ( - % m u ) is t h e sole critical point o f f i n m < 0. On ( - % mu), we h a v e f < 0 and unimodal, tending to zero at the endpoints. On (mu, my), we have f > 0 and f ' > 0 with f ( m u ) = 0 and f ( m v - ) = +~. On (m~, 0), we have f < 0 a n d f ' > 0, with f(0) = 0 a n d f ( m v + ) = - ~ . We conclude that (9) has exactly one negative solution if > 0 and at most three negative solutions ff ~ < 0.

Finally, assume m u < mv. Because f ( m u ) = f(O) = 0 and f ( m ) < 0 for m E (mu, 0), w e observe that f ' ( m ) = 0 has a solution in (mu, 0). We conclude that my, the sole negative critical point o f f , lies in (mu, 0). Thus, on (mu, 0), w e see that f i s unimodal and negative and tends to zero at the endpoints. On (mv, mu), we have f > 0 and f ' < 0, whereas f ( m v +) = + ~ andf(mu) = 0. On ( - ~ , my), w e h a v e f < 0 and f ' < 0, w h e r e a s f ( m v - ) = -oo a n d f ( - ~ ) = 0. We conclude that (9) has exactly one negative solution if a > 0 and at most three negative solutions if a < 0. This completes the proof. Conclusion Sturmfels p a i d us t h e $500. Epilogue T.Y. Li a n d X. Wang [13] have j u s t a n n o u n c e d a count e r e x a m p l e to the Itenberg-Roy conjecture. REFERENCES 1. Bernstein, D., The number of roots of a system of equations, Funct. Anal. Appl. 9 (1975), 1-4. 2. Descartes, R., Geometrie, 1636, in A Source Book in Mathematics, Cambridge, MA: Harvard University Press, 1969, pp. 90-93. 3. Gelfand, I.M., Kapranov, M.M., and Zelevinsky, A.V., Discriminants of polynomials in several variables and triangulations of Newton polytopes, Leningrad Math. J. 2 (1990), 1-62. 4. Gelfand, I.M., Kapranov, M.M., and Zelevinsky, A.V., Discriminants, Resultants and Multidimensional Determinants, Boston: Birkh&user, 1994. 5. Henrici, P., Applied and Computational Complex Analysis, Vol. 1, New York: John Wiley & Sons, 1974.

6. Huber, B., and Sturmfels, B., A polyhedral method for solving sparse polynomial systems, Math. Comp. 64 (1995), 1541-1555. 7. Huber, B. and Sturmfels, B., Bernstein's Theorem in affine space, Disc. Comput. Geom. 17 (1997), 137-141. 8. Itenberg, I., Counterexamples to Ragsdale Conjecture and T-curves, in RealAIgebraic Geometry and Topology, Contemp. Math. Vol. 182, Providence, RI: American Mathematical Society, pp. 55-72, 1995. 9. Itenberg, I. and Roy, M.-F., Multivariate Descartes' rule, Beitr~ge Algebra Geom. 37 (1996), 337-346. 10. Khovanskii, A.G., On a class of systems of transcendental equations, Soviet Math. Dokl. 22 (1980), 762-765. 11. Khovanskii, A.G., Fewnomials, Providence, RI: American Mathematical Society, 1991. 12. Kushnirenko, A.G., The Newton polyhedron and the number of solutions of a system of k equations in k unknowns, Uspekhi Math. Nauk. 30 (1975), No. 2, 266-267. (in Russian) 13. Li, T.Y. and Wang, X., On Multivariate Descartes' Rule--A Counterexample, preprint, May 1997. 14. Risler, J.J., Additive complexity and zeros of real polynomials, SlAM J. Comput. 14(1985), No. 1, 178-183.. 15. Rojas, M., A convex geometric approach to counting the roots of a polynomial system, Theoretical Comp. Sci. 133(1994), 105-140. 16. Schneider, R., Convex Bodies: The Brunn-Minkowski Theory, Cambridge: Cambridge University Press, 1993. 17. Sturmfels, B., Viro's theorem for complete intersections, Ann. Scoula Normale Pisa, Sect. IV, 21 (1994), 377-386. 18. Viro, O., Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7, in Springer Lecture Notes in Math. No. 1060, Berlin: Springer-Verlag, 1984, pp. 187-200. 19. Viro, O., Plane real algebraic curves: Constructions with controlled topology, Algebra Anal 1 (1989), 1-73 (in Russian): English transl.: Leningrad Math. J. 1 (1990), No. 5, 1059-1134.

VOLUME 19, NUMBER 3, 1997

15

Proof: A Many-Splendored

Thing 0

n a recent article, A. Jaffe and F. Quinn, concerned that "today in certain areas there is again a trend toward basing mathematics on intuitive reasoning without proof" have suggested a framework for dealing with the issue which includes attaching labels to "speculative and intuitive" work. The article has engendered a fascinating

debate about the nature and function of proof in mathematics and, inevitably, about the nature of the mathematical enterprise, for it is sometimes (often?) difficult to isolate "proving" from the general fabric of doing mathematics [4,6,7,22,27,28,33,52,61]. A historical perspective on the subject would be useful: The current debate is a continuation of a 2000-year-old tradition! From antiquity onwards, mathematicians disagreed on how best to do mathematics, what methods to use for attacking problems and establishing results. Some employed formal, rigorous proofs, others intuitive, heuristic ones (and some did not see the difference between the two). Adherents of the synthetic method battled supporters of the analytic method. Idealists confronted empiricists and formalists opposed intuitionists. There were in general no established criteria of what constitutes an acceptable proof, and perceptions of the role of proof varied considerably. Observed Euler, Indeed, it cannot be denied that mathematics contains the sort of speculations which put eminent geometers in great disagreement. Not only applied mathematics but also pure abstract mathematics itself, strange as it may be, has supplied remarkable sources of d i s s e n t . . . [10, p. 143].

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THE MATHEMATICALINTELLIGENCER9 1997 SPRINGER-VERLAGNEW YORK

Of course the tensions among "eminent geometers" have, in general, been healthy for mathematics, if sometimes less so for the protagonists. This paper consists of historical examples illustrating the pluralistic nature of mathematical practice, bearing in mind the debate initiated by the Jaffe and Quinn article. We shall return at the end to that debate. H e u r i s t i c s vs. Rigor

( a ) There is general agreement among historians of mathematics that the Babylonians of ca. 1600 B.C. knew the pythagorean theorem and knew how to solve quadratic equations, yet we have not the slightest evidence that they had any proofs of these results. Mathematics without proof?. Proof as deduction from explicitly stated postulates was conceived in ancient Greece, although when, how, and why this came about is open to conjecture. But there were apparently two distinct approaches to the subject in the crucial, formative period of ca. 450-350 B.C., what Struik calls Democritean materialism and Platonic idealism [51, p. 47]. Thus Democritus, a founder of atomism, is mentioned by Archimedes as having shown (ca. 430 B.C.) that the volume of a pyramid is one-third the base times the height, presumably using the nonrigorous method of indivisibles

employed subsequently by Archimedes in The M e t h o d and, centuries later, by Cavalieri and others. On the other hand Hippocrates, at about the same time, obtained rigorously the quadrature of lunes. This is the first result known to have been obtained deductively. Euclid was the paragon of rigor for over 2000 years, yet, as we now recognize, the proof of his very first proposition in Book I of the E l e m e n t s , which gives the construction of an equilateral triangle, is faulty: While Euclid implicitly assumes that two circles intersect, this requires an axiom of continuity, supplied two millennia later by Hilbert. Relativity of rigor! ( b ) We make a great leap forward and come to the "modern" period--the 17th century. It saw the introduction of (modern) number theory, probability, analytic and projective geometry, but above ail--the calculus. Among its early, illustrious practitioners was Fermat, whose methods of maxima and minima, tangents, and quadrature were conceptually the most advanced prior to the works of Newton and Leibniz. The following example illustrates Fermat's approach to tangents. Suppose we want to Find the tangent to the parabola y = x 2 at some point (x, x2)--a far from trivial problem for early17th-century mathematicians. Fermat lets x + e be another point on the x-axis and denotes by s the subtangent of the curve at the point (x,x 2) (see Figure 1). Similarity of triangles yields x2/s = k/(s + e). Fermat notes that k is "approximately equal" to (x + e)2; writing this as k ~ (x + e) 2, we get x2/s ~ ( x + e)2/(s + e). Solving for s, we have s e x 2 / [ ( x + e) 2 - x 2] = e x 2 / ( x 2 + 2 e x + e 2 - x 2) = ex2/e(2x + e) = x2/(2x + e), hence x2/s --~ 2 x + e. Note that x2/s is the slope of the tangent to the parabola at (x, x2). Fermat now "deletes" e and claims that the slope of the tangent is 2x. Fermat's method was severely criticized by some of his contemporaries, notably Descartes. They objected to his introduction and subsequent suppression of the mysterious e. Dividing by e meant regarding it as not zero9 Discarding e implied treating it as zero9 This is inadmissible, they rightly claimed. In a somewhat different context, but with equal justification, Bishop Berkeley in the 18th century would refer to such e's as "the ghosts of departed quantities," arguing that "by virtue of a twofold mistake FIGURE

1

x+e

x

8

e

[one] arrive[d], though not at a science yet at the truth" [30, p.428]. Of course Fermat's "mysterious e" embodied a crucial idea--the giving of a "small" increment to a variable. Newton's and Leibniz's methods were, logically, no better, and both were well aware of it. Newton affirmed of his fluxions that they were "rather briefly explained than narrowly demonstrated" [20, p.201], and Leibniz said of his differentiais that "it will be sufficient simply to make use of them as a tool that has advantages for the purpose of calculation, just as the algebraists retain imaginary roots with great profit" [20, p.265]. (In Leibniz's time, complex numbers had no greater logical legitimacy than fluxions or differentials.) Euler used infmitely small and infmitely large numbers, as well as divergent series, "with abandon" and with brilliance, achieving remarkable results, for example showing that the sum of the squares of the reciprocals of the positive integers is v2/6. P61ya wrote of the method Euler used to show this [44, p.21]: 9

In strict l o g i c . . . [it] was not justified. Yet it was justified by analogy, by analogy of the most successful achievements of a rising science that he c a l l e d . . , the 'Analysis of the Infinite.' Unsuccessful attempts w e r e made in the 18th century to provide rigorous foundations for the calculus, notably by Maclaurin via geometry and by Lagrange via algebra. D'Alembert, following a definition of limit (not sufficiently precise from our point of view), observed prophetically that "the theory of limits is the true metaphysics of the calculus," but it was left to Cauchy and Weierstrass a century later to work out the "details." In the Final analysis, 17thand 18th-century mathematicians were confident of the calculus because it yielded correct results---of this they had no doubt. ( c ) Rigorous mathematics as a going concern is a rather rare phenomenon in the 4000-year-history of mathematics (at least the way w e understand the term rigor). It was consciously practiced for around 200 years in ancient Greece and again for about the same time beginning in the 19th century. Both periods were preceded by two millennia (give or take several centuries) of often vigorous but rarely rigorous mathematical activity which produced "crises" (e.g., the pythagoreans' discovery of the incommensurability of the diagonal and side of a square, and Fourier's "proof" that a n y function is representable in a Fourier series). These created a need for examination of underlying foundations. Moreover, both rigorous periods sustained competing traditions in which intuition and heuristic reasoning were the animating forces of mathematical practice. Two shining exemplars of competing traditions in the second period of rigor are Riemann and Weierstrass. Both Riemann and Weierstrass (succeeding Cauchy) founded complex function theory, around the mid-19th century, but they had fundamentally different approaches to the subject. Riemann's global, geometric conception was

VOLUME 19, NUMBER 3, 1997

17

based on the notion of a Riemann surface and on the Dirichlet Principle, while Weierstrass's local, algebraic theory was grounded in power series and analytic continuation. Klein compares the nature of their approaches to mathematics [11, p.291]: Riemann is a man of brilliant intuition. When his interest is awakened he begins fresh, without letting himself be diverted by the tradition and without acknowledging the requirements of systematization. Weierstrass is first of all a logician; he proceeds slowly, systematically, step-by-step. When he works, he strives for the defmitive form. Indeed, Riemann had a penetrating geometric and physical intuition. But his two major "tools" the Riemann surface and the Dirichiet principle---were mathematically not well grounded. The former was defined rigorously only in 1913 (by Weyl), while the latter was severely criticized by Weierstrass, who produced a counterexample. Following this, Weierstrass's approach to complex analysis became dominant. "Only with the works of Klein and the rehabilitation of the Dirichlet Principle by Hilbert could the Riemannian theory again gradually recover from the blow delivered to it by Weierstrass" [43, p. 98]. As for Riemann's longer-term impact, Ahlfors wrote in 1953, "Riemann's writ~ ings are hill of almost cryptic messages to the future . . . . The spirit of Riemann will move future generations as it has moved us" [1, pp. 3,13].

Analysis vs. Synthesis The terms analysis and synthesis have changed meaning over the centuries, to the point where what they meant at one time was the opposite of what they had meant at an earlier time. Among the pairs of meanings of analysis-synthesis were, respectively, discovery-demonstration, algebra-geometry, manipulation-rigor, discrete-continuous, symbolic-visual, and rigorous-intuitive [13]. Several examples will illustrate some of these meanings. ( a ) In ancient Greece analysis was associated with discovery, synthesis with proof; they went hand in hand, two sides of the same coin. In the analysis stage one assumed what was required to prove or to construct in order to fend necessary conditions for the desired result or construction. In the synthesis stage one would try to reverse the steps, to show that the necessary conditions were also sufficient. The only extant Greek work on the subject is Pappus's On the Domain of Analysis (4th century A.D.), possibly because the Greeks were (apparently) reluctant to put their methods of discovery on public display (Archimedes's The Method is a counterexample). ( b ) Vi~te (1540-1603) called his groundbreaking work in algebra, in which he introduced symbols for parameters and unknowns, The Analytic Art, and Descartes believed that in his analytic geometry he was rediscovering the ancient method of analysis. But both Newton and Leibniz spoke out vigorously against Descartes's "analytic method" in geometry. Here is Leibniz:

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I am still not satisfied with algebra because it does not give the shortest method or the most beautiful construction in geometry. This is why I believe that, so far as geometry is concerned, we need still another analysis which is distinctly geometric or linear and which will express situation [simms] directly as algebra expresses magnitude directly. And I believe that I have found the way and that we can represent figures and even machines and movements by characters, as algebra represents numbers or magnitudes [47, p. 171]. It was Newton who was largely responsible for the extension of the term "analysis" to calculus. His frost exposition of the subject, based on infinite series, was entitled "On Analysis of Equations with Infinitely Many Terms." Although he applied this method to the solution of geometric problems (tangents, areas, volumes), he was disdainful of the algebra (coordinate geometry) of Descartes, calling it "the analysis of the bunglers" [13, p.456]. He was very emphatic about the desirability of solving (classical) geometric problems synthetically rather than analytically [57, p.380]: Men of recent times, eager to add to the discoveries of the ancients, have united the arithmetic of variables with geometry. Benefiting from that, progress has been broad and far-reaching if your eye is on the profuseness of output, but the advance is less of a blessing if you look at the complexity of its conclusions. For these computat i o n s . . , often express in an intolerably roundabout way quantities which in geometry are designated by the drawing of a single line. It is perhaps ironic that the two creators of the calculus--analysis par excellence--rejected the use of analysis

in geometry. ( c ) In the second half of the 18th century mathematicians began to debate the relative theoretical merits of the analytic versus the synthetic method. Some contended that the analytic method offered more power, generality, and economy of thought. Others countered that calculation (analysis) replaces thinking while pure geometry stimulates it. Analysis conceals the sources of discovery and is particularly unsuitable for solving problems in geometry. Indeed, geometry and intuition are inseparable, and the analytic method removes all intuition from geometry. Boyer [13, p.458] relates the vivid, although perhaps not unbiased, description of Malfatti, a foremost geometer of the period (ca. 1780), about the practitioners of the two methods: Malfatti compared the synthesist to a traveller who is in no hurry to reach his goal but stops frequently to enjoy everything beautiful which presents itself; the analyst, who shuts himself up in his carriage and is carried along mechanically, may reach his goal more quickly, but he will have seen less. The early 19th century witnessed a great resurgence of the synthetic method, largely dormant in much of the 18th

century, initially through the works in descriptive geometry of Monge, one of the founders of the Ecole Polytechnique. The battle of the synthesists versus the analysts was joined in full vigor in connection with the rise of projective geometry in the early decades of that century. "Perhaps no other controversy in mathematics," notes Boyer, "has exceeded in bitterness that between analysts and synthesists during the second and third decades of the 19th century" [13, p.459]. The fundamental Principle of Duality in projective geometry seems to have been a test case for the two schools of thought. The question was: On what grounds does one justify that principle? Why does it work? In this instance the analysts had the upper hand--the use of homogeneous coordinates made the Principle of Duality transparent. The Principle became synthetically transparent at the end of the 19th century when projective geometry was supplied with an axiomatic basis. P..re v s , A p p l i e d

Pure mathematics, in the sense of mathematics for its own sake, goes back (at least) to the Babylonians of ca. 1600 B.C., who solved quadratic equations and found pythagorean triples "for tim" (so it seems). Applied mathematics is, of course, older. In fact, since time immemorial there has been an intimate relationship, admittedly with discontinuities, between mathematics and the physical world. It has enriched our understanding of both. In ancient Greece that relationship took significant steps toward its flowering in the 17th century. It has since been sustained and strengthened. In the 18th century in particular, there was no clear distinction between analysis and several of its fields of application, such as mechanics, optics, hydrostatics, elasticity, and astronomy. Moreover, the very men who fashioned the infinitesimal concepts and m e t h o d s - - t h e Bernoullis, Euler, d'Alembert, Lagrange, et al.--also formulated and derived the laws governing the motions of fluids and of rigid bodies, the bending of beams, and the vibrations of elastic bodies. And they saw no distinction between the two types of activities. Here are some examples. The 18th century's vibrating-string problem and the 19th century's heat-conduction problem gave rise, among other important matters, to the study of partial differential equations. In fact, Truesdell asserts that the theory of partial differential equations is the great gift of continuum mechanics to analysis [53]. Klein claimed that much of Riemann's work was characterized by his striving to put "in mathematical form a unified formulation of the laws which lie at the basis of all natural phenomena" [12, p.4]. And Poincar~ pioneered the qualitative theory of differential equations to deal with the three-body problem.

In none of these cases would the mathematics stand up to logical scrutiny. We focus on two pivotal examples. ( a ) Vibrating strings and the musical tones they emit have been studied since antiquity. In the mid-18th century d'Alembert made a crucial advance by providing a model for the vibrating string in the form of the equation O2y/Ot2 = a2(O2y/Ox2) (called the wave equation), where y(x,t) describes the shape of the string with fkxed ends, released from some initial position. D'Alembert then proceeded to solve this equation--the first historical example of the solution of a partial differential equation. Euler wrote a paper on the same problem in 1748, in which he agreed with d'Alembert's solution but differed from him on its interpretation. Euler contended that d'Alembert's solution was not the "most general," as the latter had claimed, since it did not allow for the "initial position" of the string to have the shape of (say) an inverted V, nor of a curve drawn free-hand. Basing himself on physical and intuitive considerations, Euler asserted that such initial shapes were permissible. Although the Eulerod'Alembert debate centered on the allowable initial shapes of vibrating strings, it was driven by their differing approaches to mathematics as a whole. According to Langer [38, p.17], Euler's temperament was an imaginative one. He looked for guidance in large measure to practical considerations and physical intuition, and combined with a phenomenal ingenuity, an almost naive faith in the infallibility of mathematical formulas and the results of manipulations upon them. D'Alembert was a more critical mind, much less susceptible to conviction by formalisms. D'Alembert's "critical mind" dictated that the initial shape of the vibrating string must be a twice-differentiable function since it satisfied the wave equation O2y/Ot2 = a2(O2y/Ox2), hence he pronounced Euler's ideas to be "against all rules of analysis." Daniel Bernoulli entered the picture in 1753 by giving yet another solution of the vibratingstring problem. Bernoulli, who was essentially a physicist, based his argument on the physics of the problem and the known facts about musical vibrations (discovered earlier by Rameau, et al.). It was generally recognized at the time that musical sounds, and in particular vibrations of a musical "string," are composed of fundamental frequencies and their harmonic overtones. This physical evidence, and some "loose" mathematical reasoning, convinced Bernoulli that the solution to the vibrating-string problem must be given by

Pure mathematics goes back (at least) to ca. 1600 B.C. Applied mathematics is, of course, older.

y(x,t) = ~. bnsin(n ~rx/l)cos(n ~'at/1). 1

Although Bernouili's ideas were eventually vindicated, both d'Alembert and Euler declared them absurd. Bernoulli countered that d'Alembert's and Euler's solutions consti-

VOLUME 19, NUMBER 3, 1997

19

tute "beautiful mathematics, but what has it to do with vibrating strings?" [46, p.78]. Ravetz characterized the essence of the debate as one between d'Alembert's mathematical world, Bernoulli's physical world, and Euler's "no-man's land" between the two [46, p.81]. Beyond the specifics of the debate, however, its broader significance was to bring "the whole of eighteenth-century a n a l y s i s . . . under inspection: the theory of functions, the role of algebra, the real line continuum and the convergence of ser i e s . . . " [24, p.2]. ( b ) Fourier startled the mathematical community of the early 19th century with his work on what came to be known as Fourier series. His 1822 classic Analytic Theory of Heat had as its major mathematical result that a n y ftmctionf(x) over (-1,1) is representable over this interval by a series of sines and

cosines, f ( x ) = ao + ~. [anCos(nlrx/1) + 1

bnsin(n vrx/1)], where the coefficients an and an = X/lJ~lf(t) cos (n~rt/Odt and

bn =

1/l

b n are

given by

J~f(t)_sin(n.n't/1)dt. ,J

M

Fourier's proof of this theorem was loose even by the standards of the early 19th century. In fact, it was formalism in the spirit of the 18th century--"a play upon symbols in accordance with accepted rules but without much or any regard for content or significance," according to Langer [38, p.33]. Nevertheless, "it was, no doubt, partially because of his very disregard for rigor that he was able to take conceptual steps which were inherently impossible to men of more critical genius" [ibid.]. Fourier's result is, of course, false in the generality stated by him. Both Euler and Lagrange knew that some functions have Fourier-series representations, as above. But the "principle of continuity" of 18th- and early-19thcentury mathematics suggested that this could not be true for a// functions: Since the sine and cosine are continuous, the same had to be true of a sum of such terms (no distinctions were made between fmite and infmite sums). But to refute Fourier's claim one needed clear notions of continuity, convergence, and the integral. These were soon to be conceived by Cauchy, Abel, Dirichlet, and others. And of course Fourier's result, properly modified, remains one of the profound insights of analysis.

Legitimate vs. Illegitimate To outsiders, mathematicians appear rational and dispassionate, at least in dealing with their subject. But, of course, we know better. Such expressions about a colleague's work as "illegitimate," "against all rules," "outside of mathematics," "the work of the devil," are not unheard of. (a) Hamilton's invention (discovery) of the quaternions was conceptually a revolutionary development, for it liberated algebra from the canons of arithmetic. But like all revolutions, it was not received with universal approval. John Graves, Hamilton's mathematician friend, had this to say when Hamilton informed him of his discovery [26, p.300]: There is still something in the system [of quaternions] which gravels me. I have not yet any clear view as to

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the extent to which we are at liberty arbitrarily to create imaginaries and to endow them with supernatural properties. Although Graves and others soon came around to Hamilton's point of view by accepting operations which did not satisfy all the laws of arithmetic, this was not the end of the quaternion saga. A 50-year war ensued which pitted quaternionists against vector-analysts. Among the latter were Oliver Heaviside and Lord Kelvin, neither having kind words for the quaternions: "A positive evil of no inconsiderable magnitude," asserted Heaviside; "an unmixed evil to those who touched them in any way," echoed Kelvin [36, pp.9,10]. ( b ) The notion of function has a long and tortuous history. It was fu'st viewed as a formula--an "analytic expression" (although what that meant was not clear), then as a curve, a formula again, an arbitrary correspondence, a formula, a set of ordered p a i r s . . . [29]. Dirichlet's function D(x) = c for x rational and d otherwise, given in a paper on Fourier series in 1829, was a watershed in the evolution of the fimction concept. In an 1870 work on functions, Hankel laments the lack of clarity of this concept. He says that there are those who define it according to Euler, others according to Dirichlet, still others by saying that a function is a law, and some do not define it at all, "but everyone draws from this concept conclusions that are not contained in it" [11, p.198]. Hankel then proposes to set the record straight by distinguishing between "legitimate" and "illegitimate" functions, consigning Dirichlet's function D(x) to the illegitimate category [11,40]. Dirichlet's function was just the tip of the iceberg in a collection of "unacceptable" functions introduced in the second haft of the 19th century, designated (by some) as "pathological" [55]. ( c ) Continuity proved to be a very subtle concept in the faust half of the 19th century. For example, Cauchy "proved" that a convergent series of continuous functions is continuous. Everyone believed, and some "proved," that a continuous fimction is differentiable, except possibly at isolated points [54]. It was thus "shocking" when Riemann and Weierstrass gave examples of continuous, nowhere-differentiable functions. As expected, not everyone took to these examples: "I turn away with fright and horror from this lamentable evil of functions without derivatives," pronounced Hermite in 1893 [30, p.973]. Poincar~ was not as accommodating [30, p.973]: Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner.

In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that. If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratological museum. ( d ) In 1904 Zermelo introduced the Axiom of Choice and used it to prove that every set can be well ordered. In 1905, in a spirited exchange of letters, Baire, Borel, Hadamard, and Lebesgue discussed Zermelo's proof but also ranged more widely, dealing with such questions as: Does the definition of a mathematical object, say a number or function, ensure its existence? And does a (nonconstructive) proof of existence of a mathematical object legitimize its acceptance and use without further consideration? Baire, Borel, and Lebesgue lined up against Hadamard on the major issues (and against Zermelo's proof), although the first three differed on various matters of substance [41]. In particular, the use of uncountably many choices in definitions and existence proofs was classifted by Borel as "outside mathematics" [41, p.93]. Hadamard, who did not subscribe to this view, made a telling response [41, pp.96-97]: From the infinitesimal calculus to the present, it seems to me, the essential progress in mathematics has resulted from successively annexing notions which, for the Greeks or the Renaissance geometers or the predecessors of Riemann, were "outside mathematics" because it was impossibe to define them. ( e ) Closer to home, we have the now-famous 1964 review by Mordell of Lang's book, Diophantine Geometry, and a 1995 rejoinder by Lang, in the light of a 1964 letter from Siegel to Mordell which comments on Lang's book [37,42]. Siegel speaks briefly, and Lang extensively, about number theory and its relation to algebraic geometry. Siegel lashes out at what he sees as uncalled-for abstraction perpetrated in the subject by the likes of Lang. Here are several choice phrases: "I was disgusted with the way in which my own contributions to the subject had been disfigured and made unintelligible," and "I see a pig broken[sic] into a beautiful garden and rooting up all the flowers and trees," and again, "I am afraid that mathematics will perish before the end of this century if the present trend for senseless abstraction--as I call it: a theory of the empty set--cannot be blocked up" [37, p.339]. Mathematical passions! It is remarkable how differently three leading numbertheorists saw their subject in the 1960s, and how unaccepting some are of others' approaches. In fact, all five of the above episodes illustrate how resistant mathematicians can be to new points of view.

Idealists vs. Empiricists

These are terms introduced, in a mathematical context, by du Bois-Reymond in 1887 and again by Lebesgue in 1905 [41, p.100]. They signified roughly what we would now call formalists and intuitionists, respectively. This was prior to the formalist-intuitionist controversy of the first decades of the 20th century. The controversy had strong roots in the 19th century, some of which we spell out below. But we first want to observe that both formalist and intuitionist "tendencies" are already present in the mathematics of ancient Greece--in the forms, respectively, of the axiomatic method and of constructions. The latter, it has been argued, are in fact existence proofs [31]. ( a ) The foremost preintuitionist was undoubtedly Kronecker, and the foremost preformalist (probably) Dedekind. Their respective "philosophies" of mathematics--constructive versus conceptual--were reflected in their works [21]. For example, they developed distinct, but essentially equivalent, theories of algebraic numbers, Kronecker's based on divisors, Dedekind's on ideals. Dedekind defined an ideal as a collection of algebraic integers satisfying certain closure properties. The defmition was nonconstructive and used the completed infmite. Both ideas were anathema to Kronecker, who opted for the constructive divisors. But "for me", said Dedekind [21, p.10], the notion of a 'form with variables' [in terms of which divisors were defined] contains something far more abstract than that of an ideal, which seems to be thoroughly concrete as an assemblage of completely determined numbers having the two basic properties [of closure] . . . ( b ) We spoke earlier of the introduction in the 19th century of continuous, nowhere-differentiable functions. Among other "monstrous" functions which made their appearance around the same time were integrable functions which are discontinuous everywhere, continuous functions that are not piecewise monotonic, and nonintegrable functions that are limits of integrable functions. All were deemed artificial by Hermite [30, p.1035]: I believe that the numbers and functions of analysis are not the arbitrary product of our mind; I believe that they exist outside of us with the same character of necessity as the objects of objective reality; and we fmd or discover them and study them as do the physicists, chemists and zoologists.

(c) One of the prime examples of the computational versus the conceptual in the 19th century was invariant theory, an important and vigorous field of research in the century's second haft. First, specific invariants were found for certain forms (e.g., for binary quartic forms), but soon the problem shifted to fmding a (finite) complete system of invariants--a basis--for a given class of forms. Gordan found--that is, constructed---such a basis for binary forms, which was a major accomplishment. Hilbert astonished the mathematical community in 1890 by showing that forms of

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any degree, in any number of variable, have a basis. (This came to be known as the Hilbert B a s i s Theorem.) But Hilbert's proof was noncomputational--it did not exhibit a basis. Gordan protested: "This is not mathematics, it is theology" [30, p.930]. The theology of the 1890s, however, became the mathematical gospel of the 1920s. But not of the 1990s, according to Rota [48, p.95]: We witness today a return to the concrete mathematics of the nineteenth century after a long period of abstraction; algorithms and techniques which were once made fun of (as Hilbert did of Gordan's invariant theory) are now revalued after a century of interruption. Contemporary mathematics, with its lack of a unifying trend, its historical discontinuities and its lapses into the past, is a further step on the way to the end of the embarrassing Victorian heritage: the idea of progress and the myth of defmitiveness. ( d ) The formalist and intuitionist philosophies as expounded in the early 20th century by Hilbert and Brouwer, respectively, are well known [58]. Their effect on the thoughts and works of two of the foremost mathematicians of this century are perhaps less familiar: Outwardly it [the formalist-intuitionist controversy] does not seem to hamper our dally work, and yet I for one confess that it has had a considerable practical influence on my mathematical life. It directed my interests to fields I considered relatively "safe," and has been a constant drain on the enthusiasm and determination with which I pursued my research work (Weyl [59, p.13]). In my own e x p e r i e n c e . . , there were very serious substantive discussions as to what the ftmdamental pnnciples of mathematics are; as to whether a large chapter of mathematics is really logically binding or n o t . . . It was not at all clear exactly what one means by absolute rigor, and specifically, whether one should limit oneseff to use only those parts of mathematics which nobody questioned. Thus, remarkably enough, in a large part of mathematics there actually existed differences of opinion! (Von Neumann [56, p.480]). Speaking about "differences of opinion," here are three views of the Continuum Hypothesis: To the formalists it is independent (of the standard axioms of set theory), to the platonists it is true or false (in s o m e system of set theory yet to be determined), and to the intuitionists it is meaningless (no matter what set theory is under consideration). Democracy in mathematics! S h o r t vs. L o n g P r o o f s

The age of innocence is gone. Ideally, a proof should be simple, elegant, transparent, and relatively short--sufficiently short that a mathematician familiar with the subject should be able to convince herself or himself of its validity in reasonable time. But some proofs in the second

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half of this century have been anything but that. The star counterexample is the proof of the so-called Enormous Theorem, which gives the classification of finite simple groups [23]. The proof is about 15000 pages long, is scattered over some 500 articles, almost all published between the late 1940s and early 1980s, and was a combined effort of over 100 mathematicians from 6 countries [23]. Moreover, "gaps remain in the proof, even a decade after [the theorem] was 'established,' not to mention that some details depend on computer verification" [22, pp.672-673]. Yet the classification result is accepted as a theorem by those in the know, hence by almost everyone else, and important consequences have been drawn from it. For those reluctant to use it, and some are (see [15]), help is on the way: A second-generation short proof (3000-5000 pages) is expected "well into the next century" [15, p.794]. Speaking more generally of theorems with long proofs, Serre wonders: "What shall one do with such theorems if one has to use them? Accept them on faith? Probably. But this is not a very comfortable situation" [14, p.ll]. There are other examples of theorems with very long proofs, for example the Feit-Thompson Theorem, the FourColor Theorem, and Fermat's Last Theorem. Some believe that long proofs are becoming the norm rather than the exception. The reason is that there are, in their view, relatively few interesting results with short proofs compared to the total number of interesting mathematical results [32]. On the other hand, Spencer suggests that the mathematical counterpart of Einstein's credo that "God does not play dice with the universe" is that "short interesting theorems have short proofs" [50, p.366]. There are currently several counterexamples to this proposal. H u m a n s vs. M a c h i n e s

Another fundamental departure of the latter 20th century concerning proof is computer-assisted proofs. The unease which mathematicians feel about such proofs stems from the following: ( a ) Control over the subject must be shared with a foreign agent--a machine. In particular, how can a referee check the entire proof?. ( b ) Mathematics seems to resemble an experimental science. For example, to help increase one's confidence in a computer-assisted proof one "repeats the experiment" by letting another machine run the programme. ( e ) Computer-assisted proofs are not "surveyable": The computer programmes used in such proofs are (usually) not published and are thus not open to the traditional procedures of verification. ( d ) Both computer hardware and computer software are subject to error. "Computer programs always seem to have bugs," observes Thurston [52, p.169]. "To certify reliability of large systems is all but impossible," notes Babal [9, p.2]. ( e ) While the accepting of a traditional mathematical proof is a social process, that of a computer proof is not: "Being unreadable and--literally--unspeakable, verifications [of computer proofs] cannot be internalized, transformed, generalized, used, connected to other disciplines,

and eventually incorporated into a community consciousness. They cannot acquire credibility gradually, as a mathematical theorem does; one either believes them blindly, as a pure act of faith, or not at all" [17, p.34]. ( f ) The function of a proof is--or should be---to enlighten the reader, in addition to validating the result which it purports to prove. "A good proof is one which makes us wiser," observes Manin [39, p.18]. But surely a computer proof does not. It seems, then, that computer-assisted proofs are a major deviation from the traditional "hand-made" proofs. But on closer scrutiny, computer proofs resemble in several ways at least the long human proofs: One can exercise little control over both types of proof, both are unsurveyable, and both are subject to error. In fact, "mathematics is much less formally complete and precise than computer programs" [52, p.170], and "the probability of human error is considerably higher than that of machine error" [5, p.179]. In any case, computer-assisted proofs are clearly here to stay. D e t e r m i n i s t i c vs. P r o b a b i l i s t i c P r o o f s

An interesting recent idea concerning the concept as well as the practice of proof is that of probabilistic proof. It was (apparently) initiated by ErdSs ca. 1950 in connection with problems in combinatorics and graph theory [3]. Its use in number theory, mainly for primality testing, was pioneered by Rabin, Solovay, and others in the 1970s [8,45]. The idea of such proofs is to show rigorously that a given result is true with very high probability--in theory, as high as desired. In 1990 Babai and others introduced, in connection with work in complexity theory, the notion of "transparent" or "holographic" proof. The idea is to test the validity of extremely long formal proofs by transforming them into "transparent" proofs which are easy to "spot check." By making a number of such checks one can assure oneself with very high probability that the original proof is correct [8,16]. It seems, then, that when proofs are too long for humans we ask the aid of the computer, and when they are too long for the computer, we come back to humans for help. Some have argued that there is no essential difference between such probabilistic proofs and the deterministic proofs of standard mathematical practice [32]. Both are convincing arguments. Both are to be believed with a certain probability of error. In fact many deterministic proofs, it is claimed, have a higher probability of error than probabilistic ones. (E.g., can one be assured that the probability of error in the proof of the Enormous Theorem is less than 1/25~176176 as one can in proofs testing for primality?) Most mathematicians, however, would probably argue that qualitatively and conceptually the two types of proof are very different.

T h e o r e m s vs. P r o o f s

A better heading would perhaps be "Theorems, independent of proofs." It is remarkable that theorems have a permanence denied to proofs. Proofs have often been questioned, while the results obtained by them were not. If a proof is wrong, we usually modify the proof, and sometimes refine the concepts or conditions that went into the making of the theorem. Theorems are accepted not so much because of the rigor in their proofs as because of the reasonableness of the results, their importance, the light they shed on other work, the prestige of the author, and the endorsement of other mathematicians. The proofs of many of the theorems in Euclid's Elements are wrong from a m o d e m perspective: There are not enough axioms, hence the arguments are logically unjustified. But none of the results in the Elements has ever been disqualified. Most of the proofs (justifications) of results in the calculus of the 17th and 18th centuries, including those of Newton and Leibniz, are incorrect, yet the results were not rejected. Gauss's first proof of the Fundamental Theorem of Algebra--his Ph.D. thesis--had an "immense gap," filled only in 1920 [49, p.4]. Euler's theorem for polyhedra, v e + f - - 2, stands, but when he first published it in 1750 he stated that "he had no satisfactory proof but was convinced of its general validity by a wealth of examples" [25, p.122]. He later gave a proof, but it was inadequate. In fact, "no one was capable of giving a general defimtion of what should be understood by a ' p o l y h e d r o n ' . . . [until] Poincar~'s work in 1895" ([18, p.237; see also [34]). The examples could be multiplied. But the following two statements capture splendidly the essence of our remarks:

Proofs have often been questioned, while results obtained by them were not.

I have had my results for a long time, but I do not yet know how to arrive at them (Gauss [34, p.9]). If I only had the theorems! Then I should find the proofs easily enough (Riemann [34, p.9]). The Current Debate

In a 1993 article in the Bulletin of the A m e r i c a n Mathematical Society, Jaffe and Quinn discuss what they see as the "start of the bifurcation of mathematics into theoretical [read: nonrigorous, speculative, intuitive, heuristic] and rigorous communities," claiming that "a new connection with physics is providing a good deal of the driving force toward speculation in mathematics" ([28]; all our subsequent quotations from Jaffe and Quinn refer to this article). They allow that "at times speculations have energized development in mathematics," but assert that "at other times they have inhibited it." The danger, in their view, is that "the failure to distinguish carefully between the two can cause damage both to the community of mathematics and to the mathematics literature." To meet the danger,

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they describe "practices and guidelines which, if carefully implemented, should give a positive context for speculation in mathematics." Their main objective is to ensure "truth in advertising," and to that end they propose, among other things, that "theoretical [nonrigorous] work should be explicitly acknowledged as theoretical and incomplete" (authors' italics). Jaffe and Quinn "turn to the history of mathematics for examples illustrating [both] the benefits and hazards of nonrigorous work." Among the hazards, the "cautionary t2]es," they mention 18th-century calculus and the "Italian school" of algebraic geometry (early 20th century). Concerning the former, they assert that "in the eighteenth century casual reasoning led to a plague of problems in analysis concerning issues like convergence of series and uniform convergence of functions." Here is our reading of this period in history: Newton and Leibniz invented the calculus in the late 17th century. Their 18th-century successors--Euler, Lagrange, and others-were well aware of its logical shortcomings (including issues of convergence), but nevertheless forged ahead, bravely and brilliantly, to exploit the marvellous subject bequeathed to them. They achieved extraordinary results, proved "speculatively." The absence of rigorous grounding in the calculus caused them hardly any problems. The watershed came with Fourier's work in the early 19th century. In this case the lack of clarity in fundamental concepts (fimction, convergence, integral) cried out for elucidation, which was brought off by Cauchy, Weierstrass, and others. Fourier's nonrigorous work was thus a godsend. As for the admittedly speculative Italian school of algebraic geometry, Jaffe and Quinn state that it "did not avoid major damage: it collapsed after a generation of brilliant speculation." But it was the attempts to tame this "brilliant [geometric] speculation" that inspired the splendid rigorous algebraic work of Zariski, van der Waerden, Weil, and others. Zariski speaks with gratitude of "the geometric theories which we were fortunate to inherit from the Italian school" [60, p.77], and Dieudonn~ points to "the new life in algebraic geometry brought by the arrival on the scene o f . . . Castelnuovo . . . [and] Enriques [members of the Italian school]" [19, p.288]. The Italian school of algebraic geometry was brealdng fundamentally new ground--the study of algebraic surfaces (rather than curves), and the major issues therein were what to prove rather than how to prove it. Hilbert observed that every mathematical discipline goes through three periods of development: the naive, the formal, and the critical. What we were witnessing in the case of 18th-century calculus was the formal period (formal manipulation with not much attention paid to rigorous justification), and in the case of the Italian school of algebraic geometry perhaps the naive and formal periods. Atiyah, responding to the Jaffe and Quinn piece, puts the issue thus [7, p.178]: If mathematics is to rejuvenate itself and break exciting new ground it will have to allow for the exploration of

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new ideas and techniques which, in their creative phase, are likely to be as dubious as in some of the great eras of the past. Perhaps we now have high standards of proof to aim at but, in the early stages of new developments, we must be prepared to act in more buccaneering style. Are the mathematical areas of concern to Jaffe and Quinn not now in their "creative phase," which will likely be followed by a "critical" (rigorous) stage, if history is any guide? But why not label speculative mathematics as such? The matter has been addressed in the various responses to Jaffe and Quinn, both as to the desirability and the feasibility of such a procedure [7], but perhaps (again) history can illuminate what we regard as a serious difficulty with labeling. Incidentally, sporadic attempts at "labeling" have occurred in the past: Recall the labels "illegitimate, .... outside mathematics," "pathological," "against all rules," etc. It is ironic that while Jaffe and Quinn suggest attaching labels to nonrigorous mathematics, most of the above labels referred to rigorous mathematics which was deemed too abstract. Of course, for much of its history mathematics thrived without any need for labels, often (we believe) in circumstances similar to those of the contemporary scene. Jaffe and Quinn are concerned that in the absence of labels, "failure to distinguish between the two types of activity [speculative and rigorous work] can lead students to emulate the more glamorous and less disciplined aspects and to end up unable to do more than manipulative jargon." There will, however, be high costs to be paid in the presence of labels. Heuristic work will inevitably be relegated to the status of a second-class activity, and this will discourage young (and perhaps not so young) mathematicians from pursuing it, to the great loss of mathematics. (Jaffe and Quinn grant that "speculation, if properly undertaken [i.e., if labeled], can be profoundly beneficial.") The following example from history has relevance. Mathematics in ancient Greece was not labeled, but in the time of its greatest masters---Eudoxus, Euclid, Archimedes, and Apollonius--there was an implicit if not explicit understanding that the proper way to do mathematics was to do it rigorously. This was the result, Struik conjectures, of "the victory of Platonic idealism [rigor] over Democritean materialism [heuristics] in the realm of mathematical philosophy" [51, p.47]. We speculate that it was because the Greeks could not (or would not) accommodate irrational numbers rigorously in the numerical realm, that their mathematics turned from a fruitful collaboration of number and geometry to an almost exclusive concern with the latter. This was surely to the detriment of further vigorous progress in mathematics. We can also surmise what turn mathematics might have taken had Archimedes's The Method (apparently an isolated singularity in classical Greek mathematics) not been lost, or had similar heuristic work been pursued by others. Wallis in the 17th century

was annoyed at the ancient Greek mathematicians for what he believed was the deliberate concealment of their methods of discovery. This forced Wallis and his contemporaries to reinvent them--2000 years after Archimedes's time! Another historical example of implicit labeling, this time rigorous work having been stifled, concerns Darboux. Much of his research was in differential geometry, but he wrote three papers "which are informed by the conviction that French analysis was not practiced with sufficient rigor" [2, p.179]. However, "the chilly reception afforded Darboux's three papers by the Paris mathematical community influenced Darboux's decision to publish no further articles on foundational matters" [2, p.180]. In the first haft of this century mathematics was rigorous, abstract, and "pure." Some (e.g., Courant, von Neumann, Weyl) warned of sterility setting in if connections with science were not made. In the last several decades important and beneficial connections have been made, but perhaps at a price of looser standards of rigor. The Jaffe and Quinn article has been useful in raising some of the resulting issues. To us the contemporary debate is not at all unlike those between (for example) Euler and d'Alembert, Riemann and Weierstrass, Kronecker and Dedekind, Gordan and Hilbert, or Borel and Hadamard. "We are a community", as Jaffe and Quinn note, but we do not, and need not, observe uniform "community norms and standards for behavior." Diversity is a sign of the health and vigor of mathematics. REFERENCES

1. L.V. Ahlfors, "Development of the theory of conformal mapping and Riemann surfaces through a century," Annals of Math. Studies 30(1953), 3-13. 2. D.S. Alexander, "Gaston Darboux and the history of complex dynamics," Hist. Math. 22(1995), 179-185. 3. N. AIon, J.H. Spencer, and P. Erd6s, The probabilistic method, Wiley, 1992. 4. G.E. Andrews, "The death of proof? Semi-rigorous mathematics? You've got to be kidding!" Math. Intelligencer 16:4(1994), 16-18. 5. K. Appel and W. Haken, "The four-color problem," in: Mathematics today, ed. by L. Steen, Springer-Verlag, 1978, pp. 151-180. 6. V.I. Arnold, "Will mathematics survive? Report on the Zurich Congress," Math. Intelligencer 17:3(1995), 6-10. 7. M. Atiyah, et al., "Responses to 'Theoretical mathematics: Toward a cultural synthesis of mathematics and theoretical physics', by A. Jaffe and F. Quinn," Bulletin AMS (2) 30(1994), 178-211. 8. L. Babai, "Probably true theorems, cry wolf?" Notices AMS 41 (1994), 453-454. 9. - - , "Transparent proofs", Focus: The Newsletter of the MAA 12:3(1992), 1-2. 10. E.J. Barbeau and P.J. Leah, "Euler's 1760 paper on divergent series," Hist. Math. 3(1976), 141-160. 11. U. Bottazzini, The higher calculus: A history of real and complex analysis from Euler to Weierstrass, Springer-Verlag, 1986. 12. U. Bottazzini and R. Tazzioli, "Naturalphilosophie and its role in Riemann's mathematics," Rev. d'Hist. Math. 1(1995), 3-38.

13. C.B. Boyer, "Analysis: Notes on the evolution of a subject and a name," Math. Teacher 47(1954), 450-462. 14. C.T. Chong and Y.K. Leong, "An interview with Jean-Pierre Serre," Math. Intelligencer 8:4(1986), 8-13. 15. B. Cipra, "At math meetings, Enormous Theorem eclipses Fermat," Science 267(10 Febr. 1995), 794-795. 16. - - , "New computer insights from 'transparent' proofs," in: What's happening in the mathematical sciences, vol.I, Amer. Math. Society, 1993, pp. 7-11. 17. R.A. De Mille, R.J. Lipton, and A.J. Perils, "Social processes and proofs of theorems and programs," Math. Intelligencer 3:1 (1980), 31-40. 18. J. Dieudonne, Mathematics-- the music of reason, Springer-Verlag, 1992. 19. - - - , "The beginnings of Italian algebraic geometry," in: Studies in the history of mathematics, ed. by E.R. Phillips, Math. Assoc. of America, 1987, pp. 278-299. 20. C.H. Edwards, The historical development of the calculus, Springer-Verlag, 1979. 21. H.M. Edwards, "Mathematical ideas, ideals, and ideology," Math. Intelligencer 14:2(1992), 6-19. 22. D. Epstein and S. Levy, "Experimentation and proof in mathematics," Notices AMS 42(1995), 670-674. 23. D. Gorenstein, "The Enormous Theorem," Sci. Amer. (Dec. 1985), 104-115. 24. I. Grattan-Guinness, The development of the foundations of mathematical analysis from Euler to Riemann, MIT Press, 1970. 25. B. Gr(Jnbaumand G.C. Shephard, "A new look at Euler's theorem for polyhedra," Amer. Math. Monthly 101 (1994), 109-128. 26. T.L. Hankins, Sir William Rowan Hamilton, The Johns Hopkins Univ. Press, 1980. 27. J. Horgan, "The death of proof," Sci. Amer. 269 (Oct. 1993), 92-103. 28. A. Jaffe and F. Quinn, "'Theoretical mathematics': Toward a cultural synthesis of mathematics and theoretical physics," Bulletin AMS (2) 29(1993), 1-13. 29. I. Kleiner, "Evolution of the function concept: A brief survey," Coll. Math. Jour. 20(1989), 282-300. 30. M Kline, Mathematical thought from ancient to modern times, Oxford Univ. Press, 1972. 31. W. Knorr, "Construction as existence proof in ancient geometry," Ancient Philosophy 3(1983), 125-148. 32. G. Kolata, "Mathematical proofs: The genesis of reasonable doubt," Science 192(June 1976), 989-990. 33. S.G. Krantz, "The immortality of proof," Notices AMS 41(1994), 10-13. 34. I. Lakatos, Proofs and refutations, Cambridge Univ. Press, 1976. 35. C.W.H. Lam, "How reliable is a computer-based proof?" Math. Intelligencer 12:1(1990). 36. J. Lambek, "If Hamilton had prevailed: Quaternions in physics", Math. Intelligencer 17:4(1995), 7-15. 37. S. Lang, "Mordell's review, Siegel's letter to Mordell, diophantine geometry, and 20th century mathematics," Notices AMS 42(1995), 339-350. 38. R.E. Langer, "Fourier series: The genesis and evolution of a theory," Amer. Math. Monthly 54 supplement (1947), 1-86. 39. Y.l. Manin, "How convincing is a proof?" Math. Intelligencer 2:1(1979), 17-18.

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40. A.F. Monna, "Hermann Hankel," Niew. Arch. voor Wisk. 21(1973), 64-87. 41. G.H. Moore, Zermelo's axiom of choice: Its origins, development, and influence, Springer-Verlag, 1982. 42. L.J. Mordell, "Book review: Diophantine geometry, by Serge Lang," Bulletin AMS 70(1964), 491-498. 43. E. Neuenschwander, "Studies in the history of complex function theory I1: Interactions among the French School, Riemann, and Weierstrass," Bulletin AMS (2) 5(1981), 87-105. 44. G. P61ya,Induction and analogy in mathematics, Princeton Univ. Press, 1953. 45. M.O. Rabin, "Probabilistic algorithms," in: Algorithms and complexity: New directions and recent results, ed. by J.F. Traub, Academic Press, 1976, pp.21-40. 46. J.R. Ravetz, "Vibrating strings and arbitrary functions," in: The logic of personal knowledge: Essays presented to M. Polanyi on his seventieth birthday, The Free Press, 1961, pp.71-88. 47. B.A. Rosenfeld, A history of non-Euclidean geometry, SpringerVerlag, 1988. (Translated from the Russian by A. Shenitzer.) 48. G.-C. Rota, "The concept of mathematical truth," in: Essays on humanistic mathematics, ed. by A.M. White, Math. Assoc. of America, 1993, pp.91-96. 49. S. Smale, "Algebra and complexity theory," Bulletin AMS 4(1981), 1-36. 50. J. Spencer, "Short theorems with long proofs," Amer. Math. Monthly 90(1983), 365-366.

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51. D.J. Struik, A concise history of mathematics, Dover, 1987 (4th edition). 52. W.P. Thurston, "On proof and progress in mathematics," Bulletin AMS (2) 30(1994), 161-177. 53. C. Truesdell, "The rational mechanics of flexible or elastic bodies, 1638-1788," in: L. Euler, Opera, ser. 2, vol. 11, 1960, section 2. 54. K. Volkert, "Zur Differenzierbarkeit stetiger Funktionen--Amp~re's Beweis und seine Folgen," Arch. Hist. Ex. Sc. 40(1989), 37112. 55.--, "Die Geschichte der pathologischen Funktionen--Ein Beitrag zur Entstehung der mathematischen Methodologie," Arch. Hist. Ex. Sc. 37(1987), 193-232. 56. J. von Neumann, "The role of mathematics in the sciences and society," Collected Works, vol. 4, ed. by A.H. Taub, Macmillan, 1963, pp. 477-490. 57. R. Westfall, Never at rest, Cambridge Univ. Press, 1980. 58. H. Weyl, "Axiomatic versus constructive procedures in mathematics," Math. Intelligencer 7:4(1985), 10-17, 38. 59. - - - , "Mathematics and logic," Amer. Math. Monthly 53(1946), 2-13. 60. O. Zariski, "The fundamental ideas of abstract algebraic geometry," in: Proceedings of the Intern. Congr. of Mathematicians, 1950, vol. II, Amer. Math. Society, 1952, pp.77-89. 61. D. Zeilberger, "Theorems for a price: Tomorrow's semi-rigorous mathematical culture," Notices AMS 40(1993), 978-981.

IlkV~lETii[;-inr-li[.-~-IllO'z.liiliit!l.ll|[:-l--ll Marjorie

Mathematics in Uganda Vincent Ssembatya Andrew Vince

This column is a foram for discussion of mathematical communities throughout the world, and through all time. Our definition of "mathematical community" is the broadest. We include "schools" of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from

Senechal,

Editor

lephant tusks mark the spot where Uganda's main road crosses the equator. A lush country, bordered by Kenya, Tanzania, Rwanda, Zaire, and Sudan, Uganda contains the ice-capped "Mountains of the Moon," some of the few mountain gorillas remaining in the world, and the elusive source of the Nile sought by such 19th-century western explorers as Speke, Grant, and Stanley. Makerere University, located in the capital Kampala, was founded in 1922. This article concerns how the Department of Mathematics at Makerere has coped with historical and financial constraints that make recent difficulties at western universities seem trivial. Visitors to Makerere University usually live in the campus Guest House, where the staff is friendly, there is an ample portion of potatoes at each meal, and it is easy to get accustomed to the stray dogs that bark late into the night. During our (Vince family) threeweek residence there, while waiting for a university flat, we met a number of academics who had visited Uganda during the 1960s and early 1970s. They had at least two recollections in common: game in Uganda as abundant as in the Serengeti in Tanzania, and the Eden that was Makerere University.

E

I

Winston Churchill called Uganda the pearl of Africa; Makerere was the Cambridge of Africa. The grounds were like a garden; the scholars were the most respected in East Africa, many renowned much further. Mulago was a prestigious teaching and research hospital. But two decades of political chaos and fmancial neglect have taken their toll. At the time of our visit in 1993-94, the library had almost no books published after the early 1970s; textbooks and computers were scarce; buildings were run down; overcrowded dormitories resembled tenements; very little research was being done. A defect in the water storage tank in our university flat resulted in a small Murcheson Falls flooding the place; our faculty neighbors kindly helped sweep the water into the street. We were embarrassed to learn later that our neighbors had no storage tanks at all, so water was available only two hours each day. The annual repair budget for the 1000 or so Makerere University housing units was about $7000. There was a two-week period during our stay when no one had water because the university had amassed too many arrears on its water bill.

mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.

Please send all submissionsto the Mathematical Communities Editor, Marjorie Senechal, Department of Mathematics, Smith College, Northampton, MA 01063, USA; e-mail: [email protected]

Main hall on the Makerere

campus.

9 1997SPRINGER-VERLAGNEWYORK,VOLUME19, NUMBER3, 1997

27

Erasers accidently left at a chalkboard would be stolen (as would toilet paper left in a restroom). Fire extinguisher cases were either empty or burglar-proofed (rendering the extinguishers inaccessible). Because surgical lighting is costly to replace, it was not unusual for a procedure in the operating theater to pause for a few moments while clouds passed overhead. Faculty salaries were so low (between $125 and $350/month) that most faculty had two or more jobs, and the faculty job was usually not the primary one. Second jobs among the mathematics department faculty included a computer consultancy in Kampala, teaching secondary school as far away as three hours by bus, and growing tomatoes in the village. Lecturers sometimes missed classes due to these other commitments and had little time for students. Students were demoralized. The faculty staged a strike for a "living wage"; it was not the first. When the students staged their own protest concerning an increase in fees, shots were fired in the air to disperse them. In a similar strike a few years earlier, two students were shot to death. The Colonial Years

The 1922 Annual Colonial Vincent Report on Uganda states that, brother "except for a Government technical college, which came into being during the year, educational work among the natives is in the hands of missionary societies." Because it was built on Makerere Hill, this technical school was called Makerere College. (There are numerous places in Uganda whose names are based on events in the lives of the Kabakas--kings of the Buganda kingdom. Concerning the origin of the name Makerere, the story goes like this. About 1680, Kabaka Jjuuko set out in secret one night on a journey to marry a girl from the village of Nabuititi. Although he intended to travel fiLrther, dawn found Jjuuko on the hill now known as Makerere.

28

THE MATHEMATICALINTELLIGENCER

for example, there were only five Makerere graduates on the academic staff (out of almost 100), none in mathematics. John Crabbe was chairman of mathematics during the 1940s and 1950s. Margaret Macpherson, in her chronicle of Makerere during these early years, recounts that, in addition to his departmental duties, Mr. Crabbe was in much demand at student dances. There were tutorials in ballroom dancing during the 1940s. In 1952 High Table in Hall was introduced and, at about the same time, students began performing in an annual Shakespeare play and wearing deep red gowns at academic processions. In 1950 Makerere officially became the University College of East Africa, with a total of 237 students, from Kenya, Tanganyika and Zanzibar, as well as Uganda. The teaching of university level courses at Makerere began in the 1950s, a period of emphasis on the upgrading of standards to levels of its partner school, the University of London. By 1961 there were 912 students at Makerere, 278 Ugandan (32 female). By 1966 the Makerere student population had grown to 1500 and by 1972 to 3427. In 1956 the new Zoology building was dedicated by L.S.B. Leakey, and in 1958 her Majesty Queen Ssembatya after his graduation, with sisters and Elizabeth the Queen Mother (at the equator). dedicated the new library. emphasis, however, remained on prac- During the 1950s and 1960s, life at tical aspects, mathematics as a tool Makerere was about the best that could and manipulative skill for persons des- be had in Uganda. Students paid no tutined to perform specific tasks in the ition, and in fact were given travel exBritish colonial economy. The college penses, a book allowance, and pocket provided vocational courses for med- money (popularly called "boom"). Each ical, veterinary, and enginee~_ng prac- student had his or her own room and titioners for the African Civil Service, considered it a right to have luxury items like honey and eggs served at and also for schoolmasters. Makerere College was run like a meals. Those who graduated could exBritish public school, students sport- pect, at minimum, what was referred to ing uniforms with tasselled caps and as the "4-3-2-1", a car (four wheels), a green socks with red stripes at the top. three-bedroom house, two children, The administrators and instructors and one wife. Many achieved more in during these early years were predom- several of these categories. inantly British, and this remained the Uganda gained independence in case until after independence. In 1954, October 1962, but aspects of the Jjuuko, annoyed to find daylight overcoming him so soon, exclaimed "Makerere," a Luganda word for "the place where dawn fmds someone.") In the 1920s, mathematics at Makerere was taught as an ancillary course at an elementary and practical level for students preparing for trades like surveying, agriculture, mechanics, and carpentry. In the 1930s and 1940s, teacher training and more advanced technical training became key objectives. The

age developments in the teachBritish academic system reing, studying, and usage of main throughout Uganda's edmathematics. We would like ucational system. Even now, to encourage communication invigilation (proctoring) of examong teachers at all levels on ams and oversight of grading topics like syllabus reform, by external examiners are the school text books and mathenorm. There are seven years of matics itself. This can be carprimary school, four years of ried out at conferences, semisecondary at "O" level (ordinars, or writing in the (Uganda nary), two years at "A" level Mathematical) Bulletin. An(advanced) for those passing other aspect of communicathe "O" level exams, and three tion we would like to foster is years at Makerere for those acone between users of mathecepted. Uganda also has numatics in industry and governmerous technical colleges, ment, and teachers of matheteacher training colleges and Patrick Mangheni, Chair of Mathematics (right), Daouda matics. This sort of crosscommercial colleges. About Sangare, Secretary General of the African Mathematical three million children (over Union from Ivory Coast (center), E. Opyene, lecturer at Mbale fertilization may lead teachers to teach the subject more 60%) attend primary school; University in Uganda (left). meaningfully and employers to less than 10% of those go on to gain an insight into the mathebrant mathematical society, a regular secondary school; and the enmatical training in this country." rollment at Makerere University is publication, and frequent meetings and lectures. As a young undergraduate about 8000. Dark Days By the end of the 1960s Makerere was student at Makerere in those days I rePolitical events at the time made it imoffering a wide range of university-level member attending several recreational possible for the Uganda Mathematical mathematics courses. The curriculum mathematics meetings devoted to Society to fulfill its objectives. Like and standard of instruction were quite games, puzzles, and mathematical most of the continent, Uganda has respectable by any criterion and the pro- talks, at which plans for a countrybeen jinxed by colonialism, corrupgrams were attracting some of the best wide mathematical society were distion, and political disasters. But even and brightest students. Visiting scholars cussed (albeit informally)." by African standards, Uganda's post-inThe Uganda Mathematical Society came from Europe, America. and the dependence history was brutal. MajorBritish Commonwealth. (UMS) was formally established on the General Idi Amin came to power in 25th of November, 1972. Since its in1971 through a military coup. By the The Uganda Mathematical ception, the UMS has enjoyed major end of 1972, the Asian population support from Makerere University and Society (about 50,000) had been expelled By the late 1960s there were groups, the Institute of Teacher Education in (Amin claimed that the final decision such as the Mathematics Ring (mostly Kyambogo for its infrastructure and to expel came to him as a directive primary and secondary school teach- leadership. from God in a dream). The only disers), and projects, such as the Entebbe Professor Paul Mugambi was elected senting voice was from the Makerere Mathematics Project for East Africa, first president of the UMS. He had gone University Student Guild. By the bewhich brought together active mathe- to the University of Southampton, ginning of 1973 Amin's administration maticians. Patrick Mangheni is the cur- England, for his masters degree and was, in the words of former Makerere rent chairman of the Department of later to the University of Rochester, historian Phares Mutibwa, "a combiMathematics at Makerere University U.S.A-, for his Ph.D. in non-linear dynation of guile, buffoonery, and utter and was chairman of the Uganda namics. He returned to join the faculty ruthlessness, killing anyone even reMathematical Society from 1993 to at Makerere in the early 1970s, where motely suspected by him or his subor1996. Being one of the most brilliant he was the only Ugandan among the dinates of being unfriendly." The army mathematicians in the country, he senior ranks for a number of years, was, in particular, suspicious of the quickly advanced through the school then became the first Ugandan to chair loyalty of educated people. Because and university system in Uganda and the Department of Mathematics. At the most of Amin's ministers were sollater attended Oxford University for time of his election to the chairmandiers, there was a general dislike by the his Ph.D. in functional analysis. ship of the UMS, he made the followmilitary regime of the intelligentsia of According to Professor Mangheni, "By ing statement: "Some of us felt a need the country. Victims of Amin's purges the early 1970s, there was a wide- to establish a national society which included Makerere administrators, and spread feeling, especially at Makerere would encompass more than a 'hobby' reportedly more than 100 university University, that Ugandan and Uganda- interest in mathematics. The society students were massacred in 1976. based mathematicians had attained a hopes not only to foster a lively interMany faculty left the country, often critical mass that could sustain a vi- est in the subject but also to encour-

VOLUME 19, NUMBER 3, 1997

29

tipped off about their imminent arrest just in time. The 1979 invasion by Ugandan guerrillas from Tanzania led eventually to the (second) regime of Milton Obote, a dictatorship many consider even more bloody than Amin's. During the years of Amin and Obote it has been estimated that one of every ten Ugandans was killed. The current crisis in higher education at Makerere---decay of the infrastructure, chronic shortage of staff, loss of faculty morale--is typical of universities in many African countries. What makes this particular situation poignant is that in 1971 Uganda was by far the most economically viable country in East Africa. The years of chaos squandered all the advantages the country possessed. By the time Yoweri Museveni, then leader of the National Resistance Army and now elected president, overthrew interim head General Tito Okello in 1986, the effects of political turmoil were extreme. Accompanying the economic collapse was a deterioration in professional ethics and a devaluation of education. Businesses previously owned by Asians were redistributed arbitrarily and essentially looted, leading to what was called "magendo," quick money without having to work for it. In the 1950s and 1960s a faculty member's status could be roughly measured by the altitude of his or her residence on Makerere Hill, with the most desirWandering the hills of Uganda.

able residences at the top. As salaries declined during the Amin years, the bottom became most desirable, where water still reached the taps and where gardens were large and fertile enough to grow food. According to Dr. Mutibwa, "Previously, students worked hard at school in order to make it to Makerere University. The Makerere graduate was assured a respected position within the community and a commensurate salary. During Amin's regime what came to be of value was money, and it did not matter how it was made. Positions of power were obtained through violence and influence. Makerere became a symbol of those who would never make it in life. A typical remark: our friend X will never be happy; she is going to marry a Makerere graduate." Standards for mathematics courses in the secondary schools collapsed, and there was a sharp decline in career opportunities in mathematics. The

Bulletin of the Uganda Mathematical Society made sporadic appearances until 1980 when it could no longer be published for lack of money and interest. According to Professor Mangheni, "The departure (and failure to return home) of most mathematicians crippled mathematical activity in schools, colleges and the university." Dr. L.S. Luboobi, a professor specializing in biomathematics, was elected Chairman of the UMS in July 1989. He had attended the University of Toronto, Canada, for his masters degree and then the University of Adelaide, Australia, for his Ph.D. But by the time Dr. Luboobi took over the chairmanship, the UMS had practically ceased to exist. N e w Life The motto of Makerere University is Pro futuro aedificamus, "we build for the future." In October 1995 the Uganda constitution was adopted, and in May 1996 a democratically held presidential

THE MATHEMATICAL INTELLIGENCER

election took place. Even during my visit in 1993-94, there were conversations (some more serious than others) in the mathematics commons room concerning the possibility of restoring Makerere's former eminence. If it happens in mathematics, three senior faculty are largely to be credited: Paul Mugambi, Livingstone Luboobi and Patrick Mangheni. This trio has acted together as the invariant center of mathematics at Makerere University for 15 years. They are esteemed among their colleagues for perseverance that enabled them to get through the difficult years. Professor Mugambi is regarded as the grandfather of mathematics in the country. Over the past 25 years he has played a part in the training of most of the faculty now in the Department of Mathematics at Makerere University and has served as Dean of the Faculty of Science. Professor Luboobi was head of the Department of Mathematics before he was elected Dean of the Faculty of Science, a position he still holds. He is the exception at Makerere, the professor that can be found working in his office at all hours. He currently represents the UMS in the African Mathematical Union and is the Organizing Secretary for the PanAfrican Mathematical Olympiad. Professor Mangheni, in addition to his duties as chair of mathematics, is a partner in Rank Consults (a computer firm in Kampala), is on the board of directors of the Uganda Polytechnic Kyambogo, Uganda Airlines, and the Bank of Uganda, and is chairman of Uganda Computer Management. About 20 dynamic junior faculty are either under training or have just concluded their training at various universities, including the University of Bergen in Norway (with which Makerere University has a linkage), the International Center for Theoretical Physics in Italy, the University of Florida in the U.S.A., the University of Southampton in the U.K., and Makerere University in Uganda. Their fields of research include biomathematical modeling, combinatorial optimization, differential equations, numerical analysis, functional analysis, statistics, and computer science. The department hopes

Andrew Vince and companion near the summit of Mt. Margarita.

to have about ten faculty with Ph.D.'s and over thirty with masters degrees by the year 2000. Under Professor Luhoobi's tenure as chairman of the UMS, regular Executive Committee meetings were resumed, public lectures under the auspices of the Uganda Mathematical Society were inaugurated, and the Bulletin was revived. The society is now nmning a series of talks, the first given in 1996 by Geoffrey S. Watson, emeritus professor of statistics at Princeton University. Most of the secondary schools in Uganda participate in the mathematics contests sponsored by the UMS every year. Winners are given prizes. In 1995 two of the best students at advanced secondary school level represented Uganda in the African Mathematical Olympiad in Morocco, one finishing in third place. Thanks to NUFU (a Norwegianbased program for technological development) and the African Development Bank, most of the offices in the mathematics department have recently been equipped with computers. Plans are underway to link the department to the Internet. (Foreign aid, in general, has been pouring into Uganda over the past several years. A favorite donated 4-wheel drive vehicle on the streets of Kampala was the one with the "Rabbit Multiplier Project" logo.) The undergraduate curriculum now includes courses in calculus, probability and statistics, linear and abstract al-

gebra, classical mechanics, real and complex analysis, topology, differential equations, numerical analysis, and computer science. At present, the Department of Mathematics offers courses only to students majoring in mathematics, physics, chemistry, geology, economics, statistics or library science. However, the "Makerere University Five-Year Development Programme" addresses the problem of acute shortages of staff and lecture rooms, and there are plans to expand the mathematics curriculum to offer electives for all students. There is also a distance-learning project in progress and a proposal to establish a Center for Mathematical Sciences in Modeling and Biomathematics. The push for expansion of academic programs and improvements in infrastructure has been supported by the administration. Professor P.J.M. Sebuwufu, vice chancellor of Makerere University (Yoweri Museveni, president of Uganda, is the chancellor), has been a proponent of such programs. 1996 Uganda Mathematical Conference

The UMS has organized two major conferences in the last four years. The theme of the 1996 Uganda Mathematical Conference was "Issues in Mathematics and Mathematical Education." Over 300 participants came from countries such as Cameroon, Malawi, Nigeria, Zimbabwe, and Uganda. Funding for the conference came from the Ministry of Education in Uganda, The British Council, Uganda Airlines, and the NUFU project of Norway. The success of the conference was in large part due to the talents of the organizing secretary, Mrs. Janet Kaahwa, a lecturer at the Institute of Adult Education in Makerere University and current chairperson of the Uganda Mathematical Society. Many participants attended in order to listen to the main speaker, Professor John K. Backhouse of Oxford University, who has probably had as profound an influence as anyone on the direction of A-level mathematics in Uganda. The mathematics textbook co-authored by Professor Backhouse has been the standard for secondary school for over

30 years. Many secondary students held new copies of "Pure Mathematics," waiting for an opportunity to get Professor Backhouse's signature in them. Finding seats for all the participants was problematic. Primary and secondary school mathematics is a priority of the Uganda Mathematical Society. Most of the mathematics textbooks used in Uganda are imported and, despite their merit, lack a local flavor. Many in the UMS believe that a Ugandan student may better relate to a problem stated in terms of quantities of matoke (the main banana staple) sold at the Nakasero Market than to the same problem stated in terms of quantifies of Shetland sweaters sold at Harrods. The isolated attempts by Ugandans to write materials that are geared toward Ugandan students, however, have been frustrated by lack of funding, lack of support and motivation from the government, and failure to meet the standards set by publishing companies. The UMS intends to organize writers' workshops and invite resource people to facilitate them. At the conference, and in informal discussions, no uniform vision of the future of the university emerged among Makerere's faculty of mathematics. Certainly, for most the role of mathematics in a developing country is not simply a return to the path that was closed two and a half decades ago. Issues today arise from deep sources: ethnic rivalry for education opportunities (there are over 40 tribes in Uganda), elitism in a country with enormous material difficulties, the imposition of a student tuition, low faculty salaries, a vocational versus a liberal education. With respect to the latter, the secondary Alevel curriculum (primarily algebra and trigonometry, mechanics (particle kinematics and rigid body dynamics) and statistics) was designed with a view toward university entrance and ignores the more than 80% of the students who fail to make it to the university. More than 75% of the students who sit for the A-level mathematics exam fail on their first attempt, with resulting damage to morale. The issue of tuition, known as "costsharing," is a volatile one. A current crisis revolves about a 50,000 Uganda

VOLUME 19, NUMBER 3, 1997

31

M a t o k e on sale a t the Nakasero M a r k e t ,

shilling fee (about $50 US) imposed on the freshers of the 1996-97 class. Faculty and administration, in general, claim that a tuition is essential to the economic viability of the university and believe that the fee will not prove an unbearable burden on a student's family.* Students regard the move as a return of Makerere to its colonial past, to a university for the elite, and, more broadly, as "a retreat from the democratization of education in Africa." On October 23, 1996 a student general assembly called for a strike against costsharing, a decision that surely must have evoked memories of the two students killed by campus police on December 1, 1990 in the course of a demonstration against cuts in student allowances. The October 1996 strike was also in reaction to the recent closing of Northcote Hall, one of the student residences. Northcoters have, according to certain students, a "pride in their visibility" or, according to the administration, a proclivity toward "hooliganism." In any event, the closing of the hall was in reaction to alleged poisoning of food with pepper and glass by several Northcoters. As part of the strike, there was an attempt by demonstrators to reopen and occupy Northcote. By midnight of October 23, police with tear gas and batons had driven the students out of Northcote, and about 50 students

*In fact, secondary school fees are five to ten times as much. Students argue that the actual amount is not the issue, but that a precedent must not be set; once any tuition is introduced, the amount can subsequently be increased with a minimum of bureaucratic procedure.

32

THE MATHEMATICALINTELLIGENCER

were arrested and later released. In response to the demonstration, the Makerere Council has expelled 35 students, including Student Guild representatives. Students claim the expulsions were an attempt to silence protest against "costsharing." Kampala. Within the past year, faculty salaries at Makerere University have risen, and increased expectations for faculty participation in university life are sure to follow. Whatever occurs in the mathematics department will depend in large part on the teaching and example the junior faculty, themselves newly trained mathematicians, impart to their students. After a variety of talks at the UMS conference, the participants were entertained on the last day with a "Mathematical Cultural Show," presented by children in primary four and five from Kampala Parents School. The show was carefully crafted to challenge viewers to figure out the mathe-

matical patterns in the various dances. As is mathematically fitting, no solutions were revealed.

Acknowledgments We would like to thank all those at the Fulbright Program, at the Council for International Exchange of Scholars and at Makerere University for allowing one author the opportunity to lecture at Makerere University, and those at Makerere University and the University of Florida for supporting the graduate studies of the other author.

REFERENCES

1. H. Dinwiddy and M. Twaddle, The crisis at Makerere, Uganda Now, James Currey, London, 1988, pp. 195-204. 2. M. Macpherson, They Build for the Future, Cambridge University Press, London, 1964. 3. P. Mutibwa, Uganda Since Independence, Africa World Press, Inc., Trenton, New Jersey, 1992. 4. K. B. Richburg, Decay 101: the lesson of 'Africa's Harvard', Washington Post, August 11, 1994. 5. Departmental Handbook, Department of Mathematics, Makerere University, 1996. 6. Uganda, Report for 1992, His Majesty's Stationery Office, London, 1923.

The Geometry of

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n the short roster of m a t h e m a t i c i a n s o f the 1400's one n a m e stands out as a distinct surprise, that o f Piero della Francesca, the painter. Piero's reputation as a n artist has, i f anything, only increased w i t h time. The 500th a n n i v e r s a r y o f Piero's death i n 1 4 9 2 - - h e died on almost the very d a y Columbus reached the n e w world---

was marked with symposia and exhibitions devoted to his art, even including computer-generated demonstrations of his perspective illusions [1]. Paintings like the Flagellation of Christ (see cover), or the portrait of Duke Federigo da Montefeltro of Urbino, are icons of European cultural history. Piero's near cult status with many 20th-century painters and critics has itself become a subject of commentary. But mathematics? Vasari, writing the Lives of the Painters some 60 years after Piero's death, calls him "the greatest geometer of his time, or perhaps of any time" [2], and claims that he had written many books ("molti scritti"). Indeed, we do know of three books authored by Piero, rediscovered relatively recently. All were manuscript books, but all are now available in m o d e m printed editions [3], [4], [5] with introductions by their editors. They tend to confirm Vasari, at least in suggesting that Piero was the greatest geometer of his time. Of course one does not expect very much from Piero's time. Marshall Clagett, in Archimedes in the Middle Ages [6], documents in detail the very imperfect transmission of classical knowledge that was still the principal occupation of mathematicians as late as 1500. Piero's time was still the age of the manuscript. The fwst printed Euclid did not ap-

pear until 1482 [7], when Piero was past 60, and the first printed Archimedes was not until 1544 [8]. Before that the paucity of manuscript books makes it possible sometimes to know not only which edition was the source of a writer's knowledge, but exactly which volume. It is nearly certain, for example, that Piero studied Archimedes in the volume in the library at Urbino, now in the Vatican Library [9]. In addition to classical mathematics Piero knew another kind of mathematics, commercial arithmetic, taught in the so-called abacus schools [10]. By the indirect evidence of his own writings, Piero must have attended such a school, because his books resemble, in form, the abacus school texts: they are, like those texts, long series of worked examples, not really meant to be read, but to be worked through. (The texts for the abacus schools derive from the works of Leonardo Pisano, called Fibonacci, who introduced computation with Hindu-Arabic numerals into Europe around 1200 after himself learning the method in North Africa.) Piero was very proficient in arithmetic and very pedantic about it, another hint that he had attended an abacus school. However, the abacus school and the occasional look at a classical mathematical work in manuscript copy hardly form a promising basis for doing original mathematics, especially at a time when original

9 1997 SPRINGER-VERLAG NEW YORK, VOLUME 19, NUMBER 3, 1997

mathematics was hardly thought of. This is what makes Piero's books so surprising. Piero was a painter, and the great revolution in painting in the 1400's was the introduction of linear perspective. The basic idea was clear, as stated by Leon Battista Alberti [ 11]: light rays travel in straight lines from points in the observed scene to the eye, forming a kind of pyramid with the eye as vertex. The painting should represent a section of that pyramid by a plane. But this conception by itself does not immediately tell the painter how to paint. The first really practical treatise on perspective painting was Hero's De Prospectiva Pingendi [3]. The series of perspective problems posed and solved builds from the simple to the complex in a very methodical way. In Book I, after some elementary constructions to introduce the idea of the apparent size of an object being actually its angle subtended at the eye, and referring to Euclid's E/ements, Books I and VI, and Euclid's Optics, he turns, in Proposition 13, to the representation of a square lying flat on the ground in front of the viewer. What should the artist actually draw? After this, objects are constructed in the square (t'fiings, for example, to represent a tiled floor), and corresponding objects are constructed in perspective; in Book II prisms are erected over these planar objects, to represent houses, columns, etc.; but the basis of the method is the original square, from which everything else follows. Hero's exposition has been much criticized by his successors [12] and by art historians [13], [14]. What no one seems to say is that his construction is remarkably simple, efficient, and original. Piero's Basic Construction in Perspective A horizontal square with side BC is to be viewed from point A, which is above the ground plane and in front of the square, over the point D. The construction is shown in Figure 1. Although the square is supposed to be horizontal, it is shown, rather surprisingly, as if it had been raised up and were standing vertically, as the square BCGF. The construction lines AC and AG cut the vertical side BF in points E and H, respectively. These points give the crucial dimensions. BE, subtending the same angle at A as the horizontal side BC, represents the height occupied by the Figure 1. P e r s p e c t i v e construction of a horizontal square of side BC seen f r o m t h e elevation of the point A .

F

G

A

D

34

B

THE MATHEMATICALINTELLIGENCER

C

square in the drawing. EH, subtending the same angle at A as the far side of the square, CG, represents the length of that side of the square in the drawing. Now, Piero says, draw parallels to BC through A and E, and locate a point A' on the first of these to represent the viewer's position horizontally with respect to the edge of the square BC. Draw A'B and A'C, cutting the parallel through E at D' and E'. The construction is done! The square in perspective is BCE'D'. 1 It is clear that BCE'D' occupies the height BE in the drawing, but why is E'D' -- EH? Theorem:

E'D'

=

EH.

Piero gives the following proof, but leaves it partly to the reader to notice which similar triangles justify each equality: .

E'D' BC .

.

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.

.

.

AE AC

.

.

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EH CG

(1)

Thus, Piero says, E'D' and EH are either similar or equal: but they are equal, because BC = CG. This theorem is the first new European theorem in geometry after Fibonacci. It is a significant one, arguably the beginning of projective geometry. And it is beautiful, as the careful reader will surely agree. The result is not obvious, coming as it does through a series of similarity relations which move cleverly through the diagram. The way the perspective square suddenly appears, with a 90 degree twist in viewpoint, is unexpected and delightful. Even with the proof before us, are we sure it is correct? Can we really determine EH using a vertical side and then apply it to a horizontal side? Yes, but the reason is subtle: we see the painting foreshortened too. Because the result depends entirely on similarity, a powerful projective invariance is at work. Piero discusses none of this, but I have no doubt he thought of it. His laconic proof is aloof and cool: he could say more, but he stops. He has brought the reader to an intellectual result, but it seems more like the threshold of a deeper mystery. This is just the effect his paintings have on many viewers. The Mathematical Treatises Piero's next book, known now as the Trattato d'Abaco [4], claims to be an abacus text, or rather to present methods useful for merchants. In fact, though, it is extremely ambitious. Some of it is problematic, notably the algebraic part, which unsuccessfully tackles polynomial equations of 3rd, 4th, and 5th degree. Even this has interest, just as an attempt [15], [16], and it certainly belies the modest language of the introduction. The geometric section, on the other hand, is both ambitious and successful. The problems here were reworked by Piero in his old age, together with new ones, as his third book, dedicated to the young Duke Guidobaldo da Montefeltro at Urbino, 2 to be shelved together with De 1The choice of letters for points is Piero's own, except that he omitted the primes. Thus A, D, and E each label two distinct points in his diagram, one source of readers' irritation. 2It is this Guidobaldo's court which is so intimately portrayed in B. Castiglione's The Courtier.

x\ C Figure 2. Piero's basic approach to triangles is to find orthogonal projections of one side on another.

Prospectiva Pingendi in the great library, under the title LibeUus De Quinque Corporibus Regularibus, also called, for short, the LibeUus [15]. Piero says in his dedication that he intends the LibeUus to be his memorial, 3 a notion that will strike some as incomprehensible. The LibeUus survives in only this one manuscript copy. Its program is to complete the arithmetization of geometry which was begun in the work of Fibonacci, as Piero implicitly says in the dedication. 4 To understand his claim one must know that the geometry of the abacus schools Was very arithmetical. It dealt to a great extent with finding lengths, areas, and volumes. Geometrical figures invariably came with lengths given numerically, and the problems posed had numerical answers. The lack of a decent notation prevented these relationships from being expressed algebraically, but it was understood that the rules for doing the arithmetic of a given problem, which would be given as a numerical example, fell into patterns which amounted to algebraic formulae. In what follows I will sometimes represent Piero's statements algebraically, for this is their clear meaning, and lets us see that he is working toward an analytic geometry. For Piero, the natural thing to do with a triangle with sides a, b, and c, is to find an altitude h (see Fig. 2). He would do this in two steps, first fmding the length x which this altitude cuts from the side c (say), and then fmding the altitude itself from the Pythagoras theorem. The length x is given by c2 + b2 -

x -

2c

a2

'

(2)

recognizable as a version of the Law of Cosines stated entirely in terms of lengths, or, after some algebra, as an equivalent of Heron's formula. He got this from Fibonacci. Book I, Problem 1 of the LibeUus says you do this with any triangle. It amounts to saying you should routinely take orthogonal projections of line segments onto other lines (as one does in using Cartesian coordinates). From this starting point one can translate Euclidean 3"Eritque pignus et monumentum mei in te." 4"Etenim licet res apud Euclidem, et alios geometras nota sit, per ipsum tamen nuper ad arithmeticos translata est."

constructions into arithmetic, going, for example, from the construction of a regular pentagon to a formula for its area in terms of its side. The plane geometry of Euclid is transformed this way in Fibonacci, and the introductory B o o k I of LibeUus reviews this. What is not in Fibonacci is some of the three-dimensional geometry in the later books of Euclid, notably the regular polyhedra of Book XIII. This was the gap Piero set out to fill in Book II of LibeUus. He is never grandiose about it--indeed, in the exaggeratedly self-deprecating language of dedications he calls his w o r k "poor, empty fruit "5 and himself"a rude and clumsy farmer bringing rustic apples to an opulent and splendid table. "6 But he also says "in respect to its novelty, at any rate, it cannot displease. "7 This is the closest Piero ever comes to saying he has done something new. T h e Side of an I c o s a h e d r o n

An example of a typical problem is Problem 37 of Book II, which asks, given a regular icosahedron with volume 400 brachiae, what is the side x? Piero fmds x = (806400 - ~/597196800000) u6

(3)

Perhaps the first thing to say is that this is not something that any artist or merchant has the remotest need to know. Strangely, Piero adheres to a convention which requires all factors to be brought inside the radical, often leading to very large integers and other awkwardness. The computation seems to be at the limit of what quattrocento arithmetic is capable of handling. For this, as for all the other surface areas, volumes, containing spheres, etc., Piero describes three-dimensional constructions to guide the computation. He does this with great patience and precision; the very few errors appear to be copyist errors in the arithmetic--transposition of digits, for example. One is tempted again to make comparisons with his paintings, which are painstaldng in the extreme, even to the details in wall panels and ceilings which are too high and far away for any observer to be expected to see them. As Martin Kemp has said, for Piero " 'Getting it right' was an ethical imperative." [12] It seems like a good description of these computations as well, the first to treat the Platonic solids comprehensively. Doubts about Arithmetic?

In addition to results of this kind there are oddities. B o o k III is an arithmetization of Euclid's Book XV (no longer recognized as genuinely belonging to the Elements), which deals with the regular polyhedra in relation to each other. That one can neatly inscribe a regular tetrahedron in a cube, for example, by taking alternate vertices, is Proposition 1 of Book XV, and problem 2 of Piero's Book III. Problem 1 of Book III, however, is to fmd the side of the regular octahedron inscribed in a regular tetrahedron of

5"exiles et inanes fructus" 6-... in opulentissima et lautissima mensa, agrestia, e t a rudi et inepto colono poma suscepta" 7"Poterit namque, saltem sui novitate, non displicere."

VOLUME 19, NUMBER3, 1997

A Proof by Computation side 12, corresponding to In Book III Problem 9, conEuclid XV Proposition 2. The sidering an octahedron symsituation is illustrated in metrically inscribed in a doFigure 3: the midpoints of decahedron (Euclid XV, 9), the 6 sides of the tetrahedron Piero proves a new and are the vertices of the regupurely geometrical result lar octahedron. If the tetra(i.e., one that Euclid could hedron has side 12, the octahave discovered), but he hedron has side 6. We would proves it with a computation, not expect Piero to spend perhaps the first time a proof many words on this. Amazof this kind was ever done (in ingly, though, he gives it by Europe at least). The theofar the most complicated sorem is that a certain diamelution he presents to any ter of a regular dodecaheproblem. He drops altitudes dron is equal to the side plus in the faces and through the Figure 3. Libellus B o o k lU Problem 1 involves the octahedron the chord of the pentagonal interior, including one in a inscribed in a t e t r a h e d r o n of side 12. face. The diameter in quesvery asymmetric position. tion joins the midpoint of a He notes non-obvious simiside to the midpoint of a side, and is the diameter of the lar triangles, and at one point must recognize ~ containing sphere of the inscribed octahedron. His proof V ~ = V ~ (his cumbersome algorithm for knowing when is to compute both quantities in case the dodecahedron has such simplification is possible is given in the Trattato). In side 4, and show that they are the same number, namely the end he finds the side of the octahedron as V/-~, and he 6 + V ~ . He must have found this number as the diameleaves it that way. ter of the containing sphere of the inscribed octahedron Is this some kind of sophisticated joke? We cannot know and then, because of his thorough familiarity with the arithfor sure, but I believe Piero was concerned whether arithmetic of these situations, noticed the numerical coincimetic was really adequate to represent geometry. Such dence with the side plus the chord. The theorem is illusdoubts, going back to the discovery of the irrationality of trated in Figure 4. V2, were not really resolved until the construction of the real numbers. In Book III Problem 1 he seems deliberately The Volume of a General Tetrahedron in to use every single arithmetic method in his entire reperTerms of its Sides toire in a roundabout solution to a problem whose correct An unexpected problem of a different kind occurs in Book solution is obvious, perhaps as a test of consistency. If this II, Problem 11: fmd the altitude of a general tetrahedron is what he was doing, arithmetic passed the test. In his usual cool manner, he does not comment on what he has done. He never does it again. Figure 4. The result of the t h e o r e m in Libellus B o o k III P r o b l e m 9 can be visualized in this v e r y s y m m e t r i c a l orthogonal plane Another oddity may be related, however. In Book IV projection of the d o d e c a h e d r o n . The projected side x is exProblem 17 Piero suddenly departs from the methods he actly o n e half the side, or equivalently the projected chord y has been following to find volumes of increasingly comis e x a c t l y one half the chord, of the pentagonal face. Thus the plicated solids. For a very complicated solid, like a statue d i a m e t e r from top to b o t t o m is the side plus the chord. Piero's of a human figure, he suggests lowering it into a carefully proof m a k e s no r e f e r e n c e to a figure, but is a c o m p u t a t i o n . constructed rectangular vat of water and measuring the change in the water level in order to f'md its volume. This I suggestion jars. It seems out of place just because it is practical and approximate, hardly mathematical at all, unlike any of the foregoing problems. I suggest that this observation may serve a more mathematical purpose than at first appears. Piero might have wondered how far this process of finding volumes could go. If a shape is so complicated that the utmost ingenuity could still not decompose it into shapes with computable volumes, would it still have a volume? What seems a merely practical method may really be a "thought experiment" to prove that even in the most complicated case arithmetic can still produce a number which represents the geometric quantity "volume." It goes to the relationship between geometry and arithmetic. In more modern language it suggests the existence of the integral, with the water doing the integrating.

\

/

36

THE MATHEMATICALINTELLIGENCER

I

\\

A

a t h r e e - d i m e n s i o n a l analogue of H e r o n ' s formula, s 144 V2 = - a 2 b 2 c 2 - a2d2e 2 - b2d2f 2 - c2e2f 2 + a 2 c 2 d 2 + b2c2d 2 + a2b2e 2 + b2c2e 2 + b2d2e 2 + c2d2e 2 +a2b2f 2 + a2c2f 2 + a 2 d 2 f 2 + c 2 d 2 f 2

w

e

d

+ a2e2f 2 + b2e2f 2 - c 4 d 2 _ c2d 4 _ b4e 2 _ b2e 4 - aaf 2 - a2f 4 x-y

b

G

B

Figure 5. Piero's construction of the altitude o f a general tetrahedron in Libellus B o o k II P r o b l e m 1 1 . The point G o n the side Hi is shown closer t o H than it should be, in order to avoid clutter in the diagram. AG, perpendicular to Hi, is the altitude.

given its sides. Piero's result here w a s surely new. His sup e r b intuition for t h r e e d i m e n s i o n s guides the c o m p u t a tion, a n d is w o r t h seeing as an e x a m p l e o f his method, w h i c h u s e s only Eq. (2) a n d the P y t h a g o r a s t h e o r e m . It is illustrated in Figure 5. Construct altitudes of triangles, as usual, in this c a s e the t w o altitudes to BC, w h i c h are DE and AI. N o w t a k e HI parallel to DE a n d equal to it. Join DH. DHIE is a rectangle in the b a s e p l a n e of the tetrahedron, a n d AHI is in a p l a n e p e r p e n d i c u l a r to the b a s e plane, b e c a u s e it is perp e n d i c u l a r to the line BC. The altitude AG o f AHI is thus the altitude of the original tetrahedron. This implies the following algorithm for fending the altitude h: b2 § x

-

y --

c2 -

a 2

2c f2 §

C2 _

e2

2c

(4)

(5)

m 2 = b2 - x 2

(6)

n 2 = f 2 _ y2

(7)

w 2 = d 2 - (x - y)2

(8)

w 2 § m 2 -- n 2 z----

2m

h2 = w 2 - z2

(9)

(10)

This is Piero's result ( w h i c h he gives as a n u m e r i c a l example). He d o e s n o t give the volume of the t e t r a h e d r o n , although that is the u n s p o k e n motivation. It is truly u n f o r t u n a t e t h a t the a l g e b r a of Piero's time w a s so primitive. His f o r m u l a for the altitude t o g e t h e r with H e r o n ' s f o r m u l a for the a r e a o f the b a s e implies a fascinating f o r m u l a for the v o l u m e V of a general t e t r a h e d r o n ,

(11)

w h i c h manifestly has the s y m m e t r y o f the tetrahedron. P i e r o ' s result w a s never lost. It w a s printed, without the proof, b y Luca Pacioli [15] in his S u m m a A r i t h m e t i c a [17] in 1494, plagiarized from t h e T r a t t a t o d'Abaco i m m e d i a t e l y after Piero's death, and again in Pacioli's D e D i v i n a P r o p o r t i o n e [18] in 1509, plagiarized from the L i b e l l u s , w h e r e Piero h a d a d d e d the proof. It s e e m s n o t to have b e e n very well known, though. I believe t h e c o m p l e t e v o l u m e f o r m u l a d o e s n o t a p p e a r in print until the 19th century, w h e n it a s s u m e s d e t e r m i n a n t a l form. The first m o d e r n p r o o f is b y J.J. Sylvester in 1852 [19], w h o credits S t a u d t (1843) a n d Cayley (1841), a n d a d d s in a note at the b o t t o m of his first page, "Query, Is n o t this e x p r e s s i o n for the volu m e o f a p y r a m i d in t e r m s o f its sides to be found in s o m e p r e v i o u s writer? It can h a r d l y have e s c a p e d inquiry." The Cross Vault Piero's m o s t s o p h i s t i c a t e d g e o m e t r i c a l result is p r o b a b l y in L i b e U u s B o o k IV, P r o b l e m s 10 a n d 11: t h e volume a n d surface a r e a o f a cross vault, the c h a m b e r f o r m e d b y the i n t e r s e c t i o n of t w o equal right circular cylinders w h o s e a x e s i n t e r s e c t at right angles. This v o l u m e w a s also f o u n d b y A r c h i m e d e s , but Piero c o u l d n o t have k n o w n that, bec a u s e it only c a m e to light on a 10th-century p a l i m p s e s t d i s c o v e r e d b y Heiberg in C o n s t a n t i n o p l e in 1906 [20]. The A r c h i m e d e s m a n u s c r i p t c o n t a i n s no p r o o f o f the v o l u m e result. It is called On M e t h o d , and describes, in a letter to E r a t o s t h e n e s , s o m e t h i n g p e r h a p s m o r e interesting, A r c h i m e d e s ' s intuitive w a y o f thinking of p r o b l e m s o f this type. As T h o m a s Heath has said, "The m e t h o d will be s e e n to b e n o t i n t e g r a t i o n , as certain g e o m e t r i c a l p r o o f s in the great t r e a t i s e s actually are, b u t a clever device for a v o i d i n g . . , i n t e g r a t i o n . . , a n d m a k i n g the solution depend, instead, u p o n a n o t h e r integration, t h e result of which is alr e a d y known." [20] That is also a d e s c r i p t i o n o f w h a t Piero does. In brief, Piero relates the c r o s s vault a n d the sphere. 9 To visualize it, think o f the t w o c y l i n d e r s forming the c r o s s vault to be lying horizontally, as t h e y w o u l d b e in an architectural realization. Take the c e n t e r o f the r o o f o f t h e vault to c o r r e s p o n d to the n o r t h p o l e o f the sphere. Piero i n t r o d u c e s a right circular c o n e i n s c r i b e d in the hemis p h e r e with its b a s e on the equator a n d v e r t e x at the n o r t h pole, a n d the analogous p y r a m i d on the "equatorial" sec81 confess to using Mathematica here to reduce Piero's algorithm to a single expression. 9Marshall Clagett [9] has also tried to elucidate Piero's argument for this problem.

VOLUME 19, NUMBER3, 1997 3 7

tion o f t h e cross vault ( w h i c h is square) having v e r t e x at the c e n t e r of the roof. There is no difficulty in fmding the v o l u m e o f this p y r a m i d o v e r the square. But then, Piero says, the u p p e r haft of the c r o s s vault m u s t have e x a c t l y twice t h a t volume, b e c a u s e the ratio of the v o l u m e o f the h e m i s p h e r e to that of the c o n e is 2:1. His p r o o f of this nonobvious assertion is to c o n s t r u c t a t r a n s f o r m a t i o n o f the s p h e r e into the c r o s s vault w h i c h he thinks of as a pers p e c t i v e t r a n s f o r m a t i o n on thin sections. He then refers to A r c h i m e d e s for the result t h a t the ratio of v o l u m e s in e a c h thin s e c t i o n is invariant u n d e r this transformation. (The s e c t i o n s a r e by p l a n e s w h i c h c o n t a i n the vertical axis o f the c r o s s vault: t h e y cut the cylinders in ellipses w h i c h are j u s t e l o n g a t e d versions o f t h e circles they w o u l d cut from a s p h e r e . ) The result is correct. Piero h a d already d e s c r i b e d the p e r s p e c t i v e r e n d e r i n g of a c r o s s vault in De Prospectiva Pingendi, and it is clear t h a t his g e o m e t r i c intuition for this situation is very g o o d - p e r h a p s t o o good. As in m a n y o t h e r places, one w i s h e s he h a d said more. His result for the surface a r e a of the inner c o n c a v e surface of the vault is even m o r e laconic. This a r e a has t h e s a m e relation to the v o l u m e of the c r o s s vault as the s u r f a c e a r e a o f a s p h e r e h a s to its volume: j u s t multiply the v o l u m e b y 3/R ( w h e r e R is the radius of the cylind e r in t h e case of the vault). This s e e m s sufficiently obvious to h i m that he d o e s n o t even give an argument. It is correct, o f course.

The Archimedean Solids It h a s long b e e n recognized t h a t Piero r e d i s c o v e r e d 6 of the 13 A r c h i m e d e a n solids [9], [21]. He did n o t think in t e r m s o f a c o m p l e t e classification, b u t simply o b t a i n e d n e w s e m i r e g u l a r p o l y h e d r a b y t r u n c a t i n g e a c h regular polyhed r o n at the vertices. The r e a s o n he found 6 A r c h i m e d e a n solids i n s t e a d of 5 is that t h e r e is m o r e than one w a y to truncate, and in going from the Trattato to the Libellus he s u b s t i t u t e d one n e w one. He d r a w s no attention to the cons t r u c t i o n itself, treating it a s obvious, but turns immediately to the c o m p u t a t i o n o f t h e sides, surface areas, a n d v o l u m e s o f these figures.

Conclusion Piero's b o o k s are a m a s s o f d e t a i l - - d e t a i l e d arithmetic, detailed instructions. In the c a s e of De Prospectiva Pingendi, though, w e have a n o t h e r medium, the paintings, to reveal w h a t it is really about. We s e e t h a t a simplistic reading w o u l d c o m p l e t e l y miss the point. With the m a t h e m a t i c a l t r e a t i s e s w e are not so f o r t u n a t e - - t h e r e is no o t h e r medium. If w e w a n t to k n o w the real meaning, w e have to c o n s t r u c t it from the t r e a t i s e s alone b y getting b e h i n d the superficial details and discovering the m a t h e m a t i c a l thought. B e n e a t h the surface, t h e thought is surprisingly deep. P i e r o w a s a real m a t h e m a t i c i a n - - - o n e can s a y it without apology.

Postscript Piero worked ambitiously in mathematics, proved new theorems, attempted difficult problems, and even formulated

38

THE MATHEMATICALINTELUGENCER

and completed a coherent research program, the arithmetization of the later books of Euclid. In the context of the fifteenth century this is impressive. In fact it is at odds with conventional histories of mathematics, which state that the fifteenth century produced no original mathematics at all (in spite of Vasari's well-known characterization of Piero). what happened? Was Piero forgotten so soon after Vasari? The truth is stranger than that: he was forgotten even before Vasari. In his Lives of the Painters Vasari is primarily concerned with art and artists, but the book's charm is enhanced by stories on many topics woven into the biographical sketches. In Piero's case, the story is of a lone mathematical researcher whose reputation has come to nothing because his work has been stolen by another and printed under his own name: Luca Pacioli. Vasari records not Piero's mathematical fame, as one might have thought, but rather his mathematical obscurity. This obscurity is borne out by contemporary documents. In 1559, for example, after Vasari's first edition and before his second, both of which tell the Piero story, Nicolo Tartaglia published an encyclopedic mathematical work called Trattato Generale di N u m e r i et Misure [22]. It cites Pacioli freely, and many other contemporary mathematicians, and is, on the whole, very gossipy, but it seems never to have heard of Piero. Tartaglia is very impressed by some of Piero's results, though. For example, he describes the construction of the altitude of a general tetrahedron twice, in two different contexts, giving the entire argument both times, which he calls "ingenious" [23]. The numbers and symbols he uses are the same as those of Piero, as printed by Pacioli, so there is little doubt where Tartaglia learned it, but he cites no one. A little further on he discusses what was then called the fifteenth book of Euclid, on inscriptions of one regular polyhedron in another. Although many believe the twelve inscriptions given there are the only ones possible, he says, he has found two more ("due altre ne ho ritrovate"), but does not say that he found the first one, the icosahedron inscribed in the cube, which is yet another nice construction due to Piero (Libellus Book III Problem 4), in Pacioli's De D i v i n a Proportione. He should have cited Pacioli, but the irascible Tartaglia never concealed his contempt for Pacioli, whose mediocrity was clear to him, and it appears he could not bring himself to give him credit. Vasari, with reference to Pacioli's plagiarisms from Piero, says Pacioli "covered his ass's hide with the glorious skin of a lion." Tartaglia undoubtedly knew that characterization: polemics on mathematical plagiarism were right in his line. If he saw that the lion's skin didn't fit Pacioli, though, he had no certain evidence that the lion was Piero either, whatever he might have suspected. After all, Pacioli plagiarized from others as well: Tartaglia specifically points out a possible plagiarism (although a very trivial one) from Piero Borgi da Venetia [24]. Tartaglia was far readier to cite others for their errors than for their accomplishments, and in these cases he cited no one. In effect, Piero had dropped out of the mathematical record.

Far from being the expression of a widely held opinion, Vasari's allegation that Piero had been defrauded of his righthfl mathematical reputation was completely unsubstantiated and widely doubted for the next 350 years. Naturally it was much debated, but the question was always whether an injustice had been done, not to Piero, but to Pacioli, and the usual verdict was that Pacioli had been maligned [25]. Pacioli's brother Franciscans were inclined to defend him. Vasari had said Piero's writings could still be found in Borgo San Sepolcro, or (a few pages later) in the ducal library at Urbino, which makes the story sound plausible, but as centuries passed, and no one produced these books, it seemed to suggest the opposite: that they had never existed. Vasari's inconsistency in saying where to look was derided, and his many other errors in matters of fact were called to witness how unreliable he was. What did an artist like Vasari know about mathematics anyway? Someone in Piero's family must have planted this exaggerated story, which the credulous Vasari accepted. Defenders of Pacioli pointed out that Pacioli, so far from being an enemy, lauds Piero highly, calling him Monarca della Pittura, an epithet which has stuck. (In retrospect Pacioli's praise looks carefully calculated to avoid mentioning Piero's mathematics.) As late as 1911 the Encyclopedia Britannica dismisses Vasari's allegation on grounds like these. By this time, however, the Libellus had already been found, and it confLrmed Vasari in every essential respect. It was found by G. Pitarelli in 1903 in the Vatican Library. Any doubt that it was the work of Piero vanished when the marginal corrections were found by G. Mancini to be in Piero's own hand. It is identical in content with a section of Pacioli's De Divina Proportione. Piero's Trattato d'Abaco came to light soon after, and the geometrical section was found to be incorporated into Pacioli's S u m m a Arithmetica. Despite the vague consensus that has always existed that Piero is "mathematical," it was only with the rediscovery of these books that it became possible to think of Piero as a mathematician who had actually done something. Thus it is no wonder that histories of mathematics written before 1920 do not mention him. The reassessment of Piero in light of these materials is still going on. Although he usually rates mention now, in standard histories of mathematics Piero is still a footnote to Pacioli. More space is given, as a nile, to explaining that Leonardo da Vinci did no mathematics than is given to anything that Piero did. It may seem that no reassessment is necessary of the period generally, since these works were never lost, only mislabeled, but this is to underestimate the baleful influence of Pacioli, who is considered emblematic of the fifteenth century. There is a quality to his work which makes it difficult to believe it could contain anything of merit. When Piero's work is segregated from it and correctly identified as the work of a different author, it takes on an integrity and intensity which appears altogether different. Most of the work on these materials has been done by art historians, but their opinions are also still evolving. The

VOLUME 19, NUMBER 3, 1997

39

only book-length study of Piero's mathematical treatises maintains that the geometry of the LibeUus derives from his art and intends to be useful to artists, a view which seriously misunderstands what is going on mathematically, and which is not representative of current thinking in the field. More recently there has been exploration of the subtler interplay of Piero's art and his mathematics [13]. This endeavor raises truly interesting questions of aesthetics and mathematics, in the context of a person and a period of great fascination. Acknowledgments This work was supported by NEH-NSF award EW-20327 for "Science and Humanities: Integrating Undergraduate Education," under the title "Mathematics Across the Curriculum." Thanks also to the Mount Holyoke College Archives and Special Collections. REFERENCES [1] M.A. Lavin, "The Piero Project," in Piero della Francesca and His Legacy, ed. by M.A. Lavin, University Press of New England, Hanover, NH (1995), 315-323. [2] G. Vasari, Le Opere, ed. G. Milanesi, vol. 2, Florence (1878), 490. [3] Piero della Francesca, De Prospectiva Pingendi, ed. G. Nicco Fasola, 2 vols., Florence (1942). [4] Piero della Francesca, Trattato d'Abaco, ed. G. Arrighi, Pisa (1970). [5] Piero della Francesca, L'opera "De corporibus regularibus" di Pietro Franceschi detto della Francesca usurpata da Fra Luca Pacioli, ed. G. Mancini, Rome, (1916).

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THE MATHEMATICALINTELLIGENCER

[6] M. Clagett, Archimedes in the Middle Ages, University of Wisconsin Press, 1971. [7] T.L. Heath, The Thirteen Books of Euclid's Elements, Cambridge University Press (1908), 97. [8] T.L. Heath, The Works of Archimedes, Cambridge University Press (1897), xxix. [9] M. Clagett, op. cit. vol 3, pp. 383-415. [10] P. Grendler, "What Piero Learned in School: Fifteenth-Century Vernacular Education," in Piero della Francesca and His Legacy, ed. M.A. Lavin, University Press of New England (1995), 161-176. [11] Leon Battista Alberti, On Painting, ed. Martin Kemp, trans. Cecil Grayson, London-New York (1991). [12] M. Kemp, "Piero and the Idiots: The Early Fortuna of His Theories of Perspective," in Piero della Francesca and His Legacy, ed. M.A. Lavin, University Press of New England (1995), 199-212. [13] J.V. Field, "A Mathematician's Art," in Piero della Francesca and His Legacy, ed. M.A. Lavin, University Press of New England (1995), 177-198. [14] J. Elkins, "Piero della Francesca and the Renaissance Proof of Linear Perspective," Art Bulletin 69 (1987), 220-230. [15] M.D. Davis, Piero della Francesca's Mathematical Treatises, Longo Editore, Ravenna (1971). [16] S.A. Jayawardene, 'The'Trattato d'abaco' of Piero della Francesca," in Cultural Aspects of the Italian Renaissance, Essays in Honour of Paul Oskar Kristeller, ed C. Clough, Manchester (1976). [17] L. Pacioli, Summa Arithmetica, Venice (1494) Book II, fol. 72r, Problem 36. [18] L. Pacioli, De Divina Proportione, Venice (1509). [19] J.J. Sylvester, "On Staudt's Theorems Concerning the Contents of Polygons and Polyhedrons, with a Note on a New and Resembling Class of Theorems,' Philosophical Magazine IV (1852), 335-345. [20] T.L. Heath, The Method of Archimedes, Recently Discovered by Heiberg, Cambridge University Press (1912). [21] P.R. Cromwell, "Kepler's Work on Polyhedra", The Mathematical Intelligencer Vol. 17, No. 3, New York (1995), 23-33. [22] Nicolo Tartaglia, Trattato Generale di Numeri e Misure, Venice (1559). [23] Nicolo Tartaglia, op. cit., Part IV Book 2, and Part V Book 2. [24] Nicolo Tartaglia, op. cit., Part I Book 13, fol. 107r. [25] In this paragraph I paraphrase arguments culled from 18th century sources by Gino Arrighi and quoted by him in the introduction to Ref. [4].

Note added i n proof: Proceedings of a 1992 conference held in Arezzo and Sansepolcro appeared after this work was done. Piero della Francesca tra arte e scienza, ed. by Marisa Dalai Emiliani and Valter Curzi, Venice, 1996. Cecil Grayson reports on a project to bring out new editions of all of Piero's treatises; J. V. Field's analysis of Piero's perspective square construction is virtually identical to mine. Another recent source is the World Wide Web: search on "de quinque corporibus regularibus" to find a page from Libellus showing Piero's construction of the icosahedron in the cube.

tion o f t h e cross vault ( w h i c h is square) having v e r t e x at the c e n t e r of the roof. There is no difficulty in fmding the v o l u m e o f this p y r a m i d o v e r the square. But then, Piero says, the u p p e r haft of the c r o s s vault m u s t have e x a c t l y twice t h a t volume, b e c a u s e the ratio of the v o l u m e o f the h e m i s p h e r e to that of the c o n e is 2:1. His p r o o f of this nonobvious assertion is to c o n s t r u c t a t r a n s f o r m a t i o n o f the s p h e r e into the c r o s s vault w h i c h he thinks of as a pers p e c t i v e t r a n s f o r m a t i o n on thin sections. He then refers to A r c h i m e d e s for the result t h a t the ratio of v o l u m e s in e a c h thin s e c t i o n is invariant u n d e r this transformation. (The s e c t i o n s a r e by p l a n e s w h i c h c o n t a i n the vertical axis o f the c r o s s vault: t h e y cut the cylinders in ellipses w h i c h are j u s t e l o n g a t e d versions o f t h e circles they w o u l d cut from a s p h e r e . ) The result is correct. Piero h a d already d e s c r i b e d the p e r s p e c t i v e r e n d e r i n g of a c r o s s vault in De Prospectiva Pingendi, and it is clear t h a t his g e o m e t r i c intuition for this situation is very g o o d - p e r h a p s t o o good. As in m a n y o t h e r places, one w i s h e s he h a d said more. His result for the surface a r e a of the inner c o n c a v e surface of the vault is even m o r e laconic. This a r e a has t h e s a m e relation to the v o l u m e of the c r o s s vault as the s u r f a c e a r e a o f a s p h e r e h a s to its volume: j u s t multiply the v o l u m e b y 3/R ( w h e r e R is the radius of the cylind e r in t h e case of the vault). This s e e m s sufficiently obvious to h i m that he d o e s n o t even give an argument. It is correct, o f course.

The Archimedean Solids It h a s long b e e n recognized t h a t Piero r e d i s c o v e r e d 6 of the 13 A r c h i m e d e a n solids [9], [21]. He did n o t think in t e r m s o f a c o m p l e t e classification, b u t simply o b t a i n e d n e w s e m i r e g u l a r p o l y h e d r a b y t r u n c a t i n g e a c h regular polyhed r o n at the vertices. The r e a s o n he found 6 A r c h i m e d e a n solids i n s t e a d of 5 is that t h e r e is m o r e than one w a y to truncate, and in going from the Trattato to the Libellus he s u b s t i t u t e d one n e w one. He d r a w s no attention to the cons t r u c t i o n itself, treating it a s obvious, but turns immediately to the c o m p u t a t i o n o f t h e sides, surface areas, a n d v o l u m e s o f these figures.

Conclusion Piero's b o o k s are a m a s s o f d e t a i l - - d e t a i l e d arithmetic, detailed instructions. In the c a s e of De Prospectiva Pingendi, though, w e have a n o t h e r medium, the paintings, to reveal w h a t it is really about. We s e e t h a t a simplistic reading w o u l d c o m p l e t e l y miss the point. With the m a t h e m a t i c a l t r e a t i s e s w e are not so f o r t u n a t e - - t h e r e is no o t h e r medium. If w e w a n t to k n o w the real meaning, w e have to c o n s t r u c t it from the t r e a t i s e s alone b y getting b e h i n d the superficial details and discovering the m a t h e m a t i c a l thought. B e n e a t h the surface, t h e thought is surprisingly deep. P i e r o w a s a real m a t h e m a t i c i a n - - - o n e can s a y it without apology.

Postscript Piero worked ambitiously in mathematics, proved new theorems, attempted difficult problems, and even formulated

38

THE MATHEMATICALINTELUGENCER

Ihl|[~ilLVA~--"i||[:linl~-||[*~-I|l.]|ldi--~J|

MoscowMGbius Istvan Hargittai and Lev V. Vilkov

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafd where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? I f so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may foUow in your tracks. Please send all submissions to Mathematical Tourist Editor, lan Stewart, Mathematics Institute, University of Warwick, Coventry CV4 7AL England e-mail: [email protected] Gy6rgy Doczi's MSbius strip [ t ]

lan

Stewart,

Editor

I

T

he m+biusband ~ and its preparation are illustrated here by a drawing of GyGrgy Doczi [1], a Seattle, WA, architect who used it to describe what he called "dinergic symmetry." It is about two people who lived on two sides of the band and could not climb over the edges of the band to meet. Then Dr. MSbius came along, according to Doczi, and connected the opposite ends of the band with a twist. Doczi's MSbius strip was symbolic, to show the way for the (then two) superpowers to overcome the barriers dividing them. This is also an example of the independent life the MSbius band has taken on, making this topological concept much more famous than other similar mathematical inventions are. Ironically, of the several important contributions by the Saxon mathematician August Ferdinand MSbius (b. 1790 in Schulpforta, d. 1868 in Leipzig), for this one he was probably preceded by a lesser known mathematician, Johann Benedict Listing (1808-1882) [2]. The purpose of the present paper is to introduce the reader to yet another appearance of the MSbius band, disThe artists of the M o s c o w M6bius: Vladimir VasiPtsov and Eleonora Zharenova

played as a facade decoration of the Central Institute of Economics and Mathematics of the Russian Academy of Sciences, located at the beginning of the huge avenue called Profsoyuznaya ulitsa in Moscow. The Institute was built in the late 1970s and started its operations in 1980. The original plan was to decorate the facade with a portrait of Archimedes, but then two artists came up with a suggestion of using a symbolic MSbius band for this purpose. The two artists were Eleonora Alexandrovna Zharenova and her husband Vladimir Konstantinovich Vasil'tsov.

9 1997 SPRINGER-VERLAG NEW YORK, VOLUME 19, NUMBER 3, 1997

41

signment. They have had a distinguished career. They received the State Prize of the USSR in 1984 and were also awarded the title of "honored artist" of Russia. They both now teach monumental painting, she in the Moscow State Academy of the P r e s s and he in the Highest ArtisticIndustrial School n a m e d after count Stroganov. They have a d a u g h t e r and a granddaughter.

Human Understanding. Pergamon Press, New York, 1986, pp. 39-62. 2. Biggs, "The development of topology." In J. Fauvel, R. Flood, and R. Wilson, eds., Mdbius and His Band. Oxford University Press, Oxford, 1993.

REFERENCES

Moscow State University 119899 Moscow, Russia e-mail: [email protected]

1. G. Doczi, "Seen and Unseen Symmetries." In I. Hargittai, ed., Symmetry: Unifying

Budapest Technical University H-1521 Budapest, Hungary e-mail: [email protected]

Close-up of the central part of the decoration (photo I. Hargittai, 1992)

Facade decoration of the Moscow Centrai Institute of Economics and Mathematics, 1980 (photo I. Hargittal, 1992)

The size o f the frame of the enorm o u s MSbius b a n d on the f a c a d e of the Institute is 13 x 13 meters, and the d e p t h o f the d e c o r a t i o n is five meters. The distance b e t w e e n the building a n d the frame is t w o meters. The metallic frame o f the d e c o r a t i o n is c o v e r e d b y c o n c r e t e and s m e l t e d mosaic. The central hole in t h e d e c o r a t i o n a c c o m o dates the w i n d o w of the d i r e c t o r ' s ofrice of t h e Institute. The a r c h i t e c t w a s the late Leonid Nikolaevich Pavlov. The m a t h e m a t i c a l s y m b o l s decorating the b a n d have no special meaning. They and the c o l o r s serve artistic purp o s e s only. Zharenova and Vasirtsov graduated from the Moscow V. I. Surikov State Artistic Institute in 1960. They had built another MObius decoration for an institute in Kaluga before their Moscow as-

42

THE MATHEMATICALINTELLIGENCER

ii'iFTl,[~,.~-i,[.-~-,i

: ; I~l k ( : --I l i

i':- I ~-,

l I I a -i [ ~-- I i i ( L |

Alexander

Shen,

Editor

n this issue we continue our collection of nice proofs. I present several examples where a simple but unexpected construction in a three-dimensional space provides a short solution of a plane problem which is rather difficult. 1. The first example is so famous that most of you surely k n o w it. However, it is too nice to omit. It is the Desargues theorem. Theorem. Consider two triangles AiBIC1 and A2B2C2. Assume that the straight lines AiA2, BIB2, and CiC2 go through a single point O. In this case the three intersection points of the corresponding sides ( o f A i B i and A2B2, of BiC1 and B2C2, and, finally, of AiCi and A2C2) lie on a straight line (Fig. 1).

ThreeI Dimensional Solutions for Two-Dimensional Problems

FIGURE

1

This column is devoted to mathematics for fun. 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 allowed to be); it may not be directed only at specialists; it must attract and fascinate. 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, Alexander Shen, Institute for Problems of Information Transmission, Ermolovoi 19, K-51 Moscow GSP-4, 101447 Russia; e-mail:[email protected]

44

THE MATHEMATICAL INTELLIGENCER 9 1997 SPRINGER-VERLAG NEW YORK

I

Let us see why this theorem is evident. Imagine a transparency on which the triangle A1B1Ci is drawn. This transparency is inclined in such a w a y that one side of it lies on a horizontal table. A lamp casts light on the transparency, and we see the s h a d o w of the triangle on the table. This s h a d o w is the triangle A2B262 . (This is without loss of generality: any two triangles can be related in this way.) The sides of this triangle are s h a d o w s of the sides of the original triangle and intersect them where the transparency touches the tables, i.e., on the line of intersection of two planes (transp a r e n c y and table). See Fig. 2. 2. The second example is the problem about three c o m m o n chords of three circles. Consider three intersecting circles in a plane. For each two of them we connect the two intersection points by a c o m m o n chord. We have to prove that these three chords go through one point (Fig. 3). To see w h y it is true, imagine a horizontal plane through the center of a sphere. This plane divides the sphere into two hemispheres separated by a circle. We need only the upper hemisphere. Looking at this hemisphere from above, we see a circle (Fig. 4). N o w consider two intersecting hemispheres of this type w h o s e diameter circles lie on the same horizontal

IGURE

FIGURE 3

IGURE

plane and intersect. Looking from above, we see two intersecting circles. A closer look reveals the common ,mamua_~ D

B

A C

chord of these two circles. Indeed, the spheres intersect each other along a circle that is orthogonal to the horizontal plane and therefore is visible from above as a straight line (Fig. 5) Now our problem becomes easy. Imagine three hemispheres based on a horizontal plane. Looking from above, we see three circles (which are diameter circles of those hemispheres). Consider the point where all three hemispheres intersect--in other words, the point where the circle that is the intersection of two spheres, intersects the third one. Looking from above, we see this point as the point of intersection of three common chords, so the problem is solved.

3. In our third example we start with a space construction and transform it into a plane problem. Consider a paper tetrahedron ABCD, with face ABC horizontal (Fig. 6). Let us cut the tetrahedron along the lines AD, BD and CD and turn the side faces about the horizontal edges until they are horizontal. We get a plane hexagon AD1BD2CD3 (Fig. 7) whose vertices include three copies D1, D2, and D3 of the vertex D. Let us follow the movement of these three copies while the side faces ABD, BCD, and ACD are turned around the horizontal sides. Each copy moves along a circle that is orthogonal to the horizontal plane and to one of the sides of the triangle ABC. Therefore, in the top view, the vertices D1, 02, and D 3 move along straight lines orthogonal to the sides of triangle ABC (Fig. 8). We arrive at the solution of the following problem: Consider a triangle ABC and three other triangles ABD1, BCD2, A CD3 that have common sides with it. Assume that the sides adjacent to any vertex of the given triangle are equal (AD1 = AD3, BD1 = BD2, C'D2 = CD3). Consider the altitudes of the three triangles orthogonal to the sides of ABC and going through D1, D2, Ds. Prove that these altitudes, continued, meet in a point. 4. This example is also about circles. Consider a black disc of diameter d drawn on a plane and a number of long white paper strips of different widths. Our goal is to cover the disc by these strips so that no black spot is ~sible.

D1 FIGURE 7

99 -

. . . . . ..

D

" ..... i: ............ """".. D1

B

A

".

D~

D3

D3

D2

VOLUME 19, NUMBER 3, 1997

45

If the total width of the strips is at least d, this is trivial--just put all the strips side-to-side. It turns out that if the total width is smaller than d, such covering is impossible. Why? To explain this, recall the following geometric fact. The area of the part of a sphere which lies between two parallel planes (intersecting the sphere) depends on the radius of the sphere and the distance between the planes, but not on the position of the planes. In other words, if we cut a spherical lemon into slices of equal thickness, the amount of skin will be the same for all slices (Fig. 9). What is the connection between this fact and our circle and strips problem? Imagine that the circle is a top view of a hemisphere. Then each strip that goes across the circle becomes part of the hemisphere lying between two parallel planes (visible as border lines of the strip); the area of this part is proportional to the width of the strip (Fig. 10). If the strips cover the circle, the corresponding parts of the hemisphere cover the hemisphere. Therefore, their total area is not less than the area of the hemisphere. And the whole hemisphere corresponds to a strip of width d. Therefore, the total width of all strips is at least d. Q.E.D. 5. Our last example is the classical problem of Apollonius: to construct a circle tangent to three given circles (Fig. 11). All the information below is taken from the paper by A.V. Khabelishvili in the Russian Journal

Istoriko-matematicheskie Issledovaniya ser. 2, vol. 1 (36), number 2 (1996), 6-81. I cannot vouch for the history, but the proof suggested in this paper is indeed nice. According to the paper, the problem of constructing a circle tangent to three given circles was stated by the Greek mathematician Apollonius (260170 B.C.) in his treatise "On tangents" in two volumes. However, this treatise as a whole and the solution given by Apollonius were lost. Many famous mathematicians worked on this problem later (including F. Vi~te, R. Descartes, I. Newton, L. Euler); however, all the solutions pro-

46

THE MATHEMATICAL INTELLIGENCER

posed involve some notions not available to Apollonius. (The best-known solution uses inversion. If two given circles intersect in the center of inversion, then after the inversion they become straight lines; this special case is easy.) A.V. Khabelishvili suggests a solution that he believes may be the originai solution found by Apollonius, who is famous as an expert in conic sections. Here it is. Assume the three given circles are drawn in a horizontal plane. Imagine three similar cones with vertical axes that intersect the plane along these circles (Fig. 12), like three conical volcanos of different height in the middle of the sea; the given circles are shorelines of these mountains. Consider one more cone. This cone is similar to the three given cones, and also has vertical axis, but its vertex is pointing down. Let us put this cone in between the three cones and then move it down until it touches them. At that point the intersection of this cone with the horizontal plane will be the required circle, and the vertex of this cone will coincide with the point of intersection of the given cones. So if we consider fmding the point of intersection of three cones as a legal operation, the Apollonius problem is easy. However, we are looking for a ruler and compass construction, so we continue to follow Khabelishvili's argument. Consider the plane that goes through the vertices of three given cones. In its final (desired) position, the fourth cone intersects this plane in an ellipse through the vertices of the given cones.

~IGURE 1

FIGURE 9

FIGURE 10

IGURE

For any three points there are many ellipses going through them, so we need some additional information. Please note that all ellipses that are intersections of the moving fourth cone and the plane, are similar to each other and have the same ratio of long and short axes. Therefore, using a suitable projection (we will say now: affme transformation) we reduce the problem to the following one: construct a circle going through three given points. It remains to show that all required constructions may be performed using compass and ruler only (on a suitable plane). We will not go into details and mention only two basic facts needed: A. To find on the horizontal plane (that contains the three given circles) the line of intersection with the plane going through the cone vertices, we do the following: for each two circles we

1,

Indiscrete Thoughts G.-C. Rota, MIT F. Palombi, Univ. Bologna, Italy (Ed.) Indiscrete Thoughts gives a rare glimpse into a world that has seldom been described, the world of science and technology seen through the eyes of a mathematician. The period that runs roughly from 1950 to 1990 was one of the great ages of science, and it was the golden age of the American university. Rota takes pleasure in portraying, warts and all, some of the great scientists of that age. In portraying these men as they truly were, in revealing their weaknesses and insecurities, Rota deconstructs some of the cherished myths of our time. 1996 280 pp. Hardcover $29.50 ISBN0-8176-3866-0

Mathematical Encounters of the 2nd Kind P.J. Davis, Brown University "lf l ever were stranded in an airport, bored silly and searching for someone to talk to, I think l'd like Philip J. Davis to be stranded with me. Davis appears to have no artifice. His book is a compendium of several tales that caught his interest. It's also a book that, often subtly, reveals the sort of mathematical lore that keeps mathematicians talking as they gather for afternoon tea in universities throughout the country." ---SIAM Review 1997 304pp. Hardcover $24.95 ISBN0-8176-3939-X

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draw two common tangents and find their intersection; these three intersection points lie on the straight line in question (Fig. 13). B. Looking at the side view of a cone and a plane intersecting this cone, it is easy to construct the major and minor axes of the intersection ellipse and determine the ratio in which the cone's axis divides the long axis of the ellipse. Dear reader, what do you think? Was this the original solution of Apollonius? It would be interesting to hear from historians of mathematics. Acknowledgement. I want to express my gratitude to my friend Vadim Radionov (Moscow Center of Continuing Education Editorial House) who provided the PostScript drawings for this issue using METAPOST software.

Essays on the Future In Honor Of The 80th Birthday Of Nick Metropolis S. Hecker, Los Alamos National Laboratory &

G.-C. Rota, MIT, Cambridge, MA (Eds.) This volume represents a unique undertaking in scientific publishing. June 11, 1995 marked the 80th birthday of Nick Metropolis, one of the 3 survivors of the Manhattan Project at the Los Alamos National Laboratory. To celebrate this event, some of the leading scientists and humanists of our time have contributed an essay relating to their respective disciplines.

Contributors: H. Agnew; R. Ashenhurst; K. Baclawski; G. Baker; N. Balazs; J.A. Freed; R. Hamming; O. Judd; D. Kleitman; M. Krieger; N. Krikorian; P. Lax; D. McComas; T. Puck; M. Raju; R. Richtmyer; J. Schwartz; D. Sharp; R. Sokolowski; E. Teller; M. Waterman Summer1997 Approx.250 pp. Hardcover $44.50(tent.) ISBN0-8176-3856-3

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VOLUME 19, NUMBER 3, 1997

47

I'~[~.,.-tw_...[...

Jeremy

Gray

After 150 Years.. News from Jacobi about

Lagrancje's Analytical Mechanics Helmut Pulte

Column Editor's address: Faculty of Mathematics, The Open University, Milton Keynes, MK7 6AA, England

48

I

[Lagrange's] Analytical Mechanics is a book you have to be rather cautious about, as some of its content is more supernatural than based on strict demonstration. You therefore have to be prudent about it, ~f you don't want to be deceived or come to the delusive belief that something is proved which is actually not. There are only a few points which do not entail major difficulties; I had students who understood the M6canique analytique better than I did, but sometimes it is not a good sign if you understand something. arl Gustav Jacob Jacobi ([1996], 29) made this remark in his last lectures on analytical mechanics, which he delivered in Berlin in 1847/48, about three years before his death. The contrast with the earlier and much better known Lectures on Dynamics of 1842/43 makes his criticism of Lagrange seem quite astonishing. Indeed, Hamilton and Jacobi are always said to be the most successful mathematicians in the first half of the 19th century who developed mechanics along Lagrangian lines. When Hamilton called Lagrange's Mdcanique Analytique "a kind of scientific poem" he implied that he himself added some new stanzas to the same poem. More specifically, when Jacobi ([1884], 1) called Lagrange's textbook a successful attempt to "write down and transform" the differential equations of motion, he implied that his and Hamilton's contributions should be regarded as a necessary and sufficient complement, showing how to solve these equations. In this respect, Felix Klein was quite right when he said, "Jacobi's extension of mechanics is essential with respect to its analytical side," but it has to be criti-

C

THE MATHEMATICAL INTELLIGENCER 9 1997 SPRINGER-VERLAG NEW YORK

cised for its lack of physical relevance (Klein [1926], 203, 206-207). Nevertheless, if we take Jacobi's last lectures on analytical mechanics into account, this view is no longer tenable. These lectures have just been published, and I hope that they will lead to a change in Jacobi's place in the history of mechanics. Jacobi's criticism of Lagrange is the most explicit expression of what can be described as a shift from a physico-mathematician's view of mechanics to a "pure" mathematician's view. This shift has serious implications for the later philosophical understanding of what mechanical principles and the theory of mechanics are. Lagrange and the Tradition of "Mechanical Euclideanism" Like his predecessors Euler and d'Alembert, Lagrange attempted to give mechanics as an axiomatic science, starting from (seemingly) evident, general, and certain principles, and developing it in a deductive manner with a minimum of further assumptions. This abstract theory was presented as an expression of the intrinsic mathematical structure of nature itself. I will use Lakatos's term "Euclideanism" for Lagrange's concept of science, thus malting explicit that Euclidean geometry was the model for this ldnd of presentation of mechanics (Lakatos [1978], 28-29). On this view, mechanical knowledge of the world has the same status as any mathematical knowledge: it is infallible. It is well known that, in his Mdcanique Analytique, Lagrange eschewed geometry, though we know today that this applies more to his presentation and justification of mechanical propositions than to their invention or discovery. More importantly, it seems to me that, in restricting mechanics to the methods of analysis alone, Lagrange claimed not only to dispense with other

mathematical methods, but also to dispense with extra-mathematical methods. Indeed, Lagrange's Mdcanique Analytique is the first major textbook in the history of mechanics without any kind of explicit philosophical discussion. The metaphysical assumptions of his mechanics are not made explicit, nor is there any epistemological justification given for the presumed infallible character of the basic principles. This is in striking contrast to Lagrange's immediate predecessors Euler, Maupertuls, and d'Alembert (Pulte [1989], 232-240). This kind of "mechanical Euclideanism" contains a significant tension. Lagrange himself was partly aware of it, and his successors in the French tradition of mathematical physics were even more so. Lagrange not only adhered to the old Euclidean ideal of building up mechanics from evident, certain, and general first principles, he actually wanted to start with one (and only one) principle, the principle of virtual velocities. In order to achieve this aim, he formulated this principle in a very general and abstract manner, using his calculus of variations. In the first edition of his Mdcanique Analytique, he introduced this principle as "a

kind of axiom" (Lagrange [1788], 12). Jacobi's Changing Attitude But later he had to admit that this prin- toward Mathematical Physics ciple lacked one decisive traditional Jacobi was born in 1804 and started his characteristic of an axiom: It is "not university career around 1825. His sufficiently evident to be established early attitude toward mathematics de9~ra~ frnm the neo-humanism as a primordial p~,~nl,~- ct~_ minant in Germany, grange [1811], 23). made science and out of this dilemm; ~ntific education ends to clarify his princ themselves. Matheby referring to siml; uatics in particular mechanical processe should be regarded or machines. Later as an expression of critics, f~om Fourier pure intellectual creand de Prony to ativity, needing no Poinsot and Ostroother justification, gradsky, supported and the application him in this (Lindt of mathematics to [1904]). All these the natural sciences critics aimed at could even be seen better proofs, givas a degradation of ing the principle of mathematics (Knobvirtual velocities a loch/Pieper/Pulte more secure foun[1995]). dation and making In his early it more evident. C a r l G u s t a v Jacob Jacobi ( 1 8 0 4 1 8 5 1 ) career, Jacobi was They were not sus(from Meyers Enzyklop~idischea Lexikon) quite absorbed by picious about his this ideal of pure Euclideanism, they just tried to realise it better. As we mathematics. He was explicitly hostile shall see, Jacobi's deepest criticism to French mathematical physics as it was to doubt the validity of any such was successfully practised by Fourier, Laplace, Poisson, and others. On being attempt.

Where Jacobi gave his last lectures on Analytische Mechanik: K6nigliche Friedrich Wilhelms-UniversiUit, Berlin (about 1 8 4 0 )

VOLUME19, NUMBER3, 1997 49

for this proposition. In his first criticised by Fourier, who could see no tual velocities. In Lagrange's first atpractical use in Abel's and Jacobi's the- tempt, which referred only to statics, "demonstration" of its truth, Lagrange ory of elliptic functions, Jacobi gave he considered a system of connected introduced a set of pulleys to represent the famous reply: "A philosopher like masses. The single masses experience the forces (Lagrange [1798]; [1811]; him should have known that the central forces P, Q, and so on. A small 23-26). This set is to be understood as unique aim of science is the honour of impact to the system leads to virtual a mere thought-instrument, with massdisplacements of the mass points the human spirit" (Borchardt [1875], less and frictionless pulleys, an inex(these displacements are called virtual, tensible cord, and a unit weight. The 276)9 He kept to his ideal of pure mathe- i.e., possible, because they must be quantity Pdp + Qdq + . . . is then easmatics in his practical mathematical in- compatible with existing connections). ily expressed geometrically as the tovestigations. Even when he started ff the projection of the first displacetal change of length of the cord9 If this working on the theory of the differen- ment in the direction of the first force sum is zero, the weight obviously can't tial equations of motion go up or down when a displacement is applied to the around 1835, stimulated by W.R. Hamilton's investigasystem. tions, he was not interested Lagrange ([1811], 24) said: at all in the possible physical Now it is evident that as a implications of this theory. Mechanics at its best was for necessary condition to mainJacobi analytical mechanics tain the system--being subin the sense of Lagrange. jected to various pulling Par d. L. L.dGR~INC~, de Flnsthut des Sciences, Lem'es forces--in equilibrium, the There is not the slightest et .,4rts,.du Bureau des Longitudes; 3Iembre &t Sdnat trace of criticism of the founweight cannot descend as a Conservateur , Grand-Officier de la Ldgion d'Honneur , result of any infinitesimal disdations of Lagrange's meet Comte de l'Empire. chanics to be found in his placement of the system's work before 18459 points--whatever the n a c r e Later, in the last six or of this movement may be. As I~OU~TELLE EDITION, seven years of his life, Jacobi weight always has the tenREVUE ET AUGMENTEE PAB. L'.&UT~.UR. was more and more condency to descend, it will if fronted with problems of methere is a displacement of the chanics, astronomy, and system inducing it to deTOME PREMIER. physics in general, which s c e n d - a c t u a l l y and consedeal with the concrete bequently do so and produce haviour of physical objects. this displacement of the sysWhile he adhered to his ideal tem. of pure mathematics, he became more aware of the In the state of equilibrium, problem of how mathematics Lagrange argued, an infinias a product of our mind can tesimal displacement of the be applicable to natural realsystem can not result in a deity. He gave up the naive scent of the weight, and a deM,,, ys COUI~CIER,xm,Rx~t~va-Ltm~,~i~zrovr~ L~S}~,Tn~m~TtqUES. Platonism which he had scent of the weight implies ISII, propagated in his earlier cathat there is no equilibrium. reer, and came to a more This idea, he claims, is "exm o d e m and modest point of Where Lagrange's first "demonstration" was reprinted: pressed analytically in the view. His criticism of MGcanique Analytique, Vol. 1 (2nd ed., t 8 1 t ) principle of virtual velociLagrange's mechanics is the ties." most distinct expression of this P is dp, and so on, then Lagrange's "axIn his Berlin lecture from 1847/48, change, but it is totally ignored in the iom" of mechanics says that the sysJacobi ([1996], 29) quoted Lagrange's histories of mechanics---it is not just tem is in a state of equilibrium if the consideration. When he came to the Felix Klein who sticks to this picture s u m Pdp + Qdq § . . . vanishes: word "evident," he couldn't restrain of Jacobi! himself from commenting:

MECANIQUE ANALYTIQUE,

PARIS~

Pdp + Qdq + . . . J a c o b i ' s C r i t i c i s m of Lagrange's Mechanics

Jacobi devoted about one quarter of his lectures to Lagrange's two so-called demonstrations of the principle of vir-

THE MATHEMATICAL INTELLIGENCER

= 0

this is a bad word; wherever you f'md it, you can be sure that there are serious difficulties; [using] it is an evil habit of mathematicians, so old that I found it recently in the 9

In m o d e m terminology, if the system is in a state of equilibrium, then the virtual work must be zero. Evident truth can hardly be claimed

work of Diophantus, who applied it to a proposition which is very difficult to demonstrate even with moderu analysis. Where Lagrange asserts evidence and mathematical exactitude, Jacobi finds darkness and logical incorrectness. In outline, his criticism ran as follows. Lagrange's inference is based on two conclusions, or on a dichotomy of two cases: If an arbitrary infinitesimal movement is applied to the mechanical s y s t e m . . . ( A ) . . . the weight does not descend--state of equilibium: Pdp + Qdq + . . . = 0

(B) . . . the weight descends--no equilibrium: Pdp + Qdq + . . .

r O.

to probable truth for a restricted number of cases. This is the core of Jacobi's criticism, confronting mathematical physics with the strict standards he attributes to pure mathematics only. Lagrange himself was not very happy with his first attempt for various reasons. Shortly before he died, he gave a new proof, in the second edition of his Thdorie des f o n c t i o n s analytiques (1813).

THEORIE DES

law of gravity) on the other hand. These expressions are meant to bridge this gap. They should represent physical forces corresponding to certain geometrical constraints, and make them comparable with forces such as gravity. Technically, Lagrange's second construction is much more complex than the first, and so is Jacobi's second destruction. Now, the transition from statics to dynamics, which was hitherto out of focus, becomes important. Jacobi attacked the very substitution of a constraint F by a 'pulleyfunction'f. This substitution, he said, is by no means evident. Jacobi ([1996], 59) wrote:

FONCTIONS ANALYTIQUES, CONTENANT Los PrineiWs du Calcul diff@ent icl, d?gag/,s de t,*ute con~iddrati,,n d'iuliniment pctits, d'~Zvanouissans, de limites et d,, fluxi,m% et rl,duits '~ l'analyse alg6brique des quantitds linies.

The transition from statics to dynamics generally means a simplification of matters---and indeed reading the Mdcanique A n a l y t i q u e makes you believe that the equations of motion follow from those of equilibrium. This, however, is not possible if the laws are known only in respect to bodies at rest. It is a matter of certain probable principles, leading from the one to the other, and it is essential to know that these things have not been demonstrated in a mathematical sense but are merely assumed.

Jacobi's criticism can be summed up in these points: P.tn J. L. L.r E, de l'Institut des Sciences, Letlres et ..Irt;, (1) Conclusion (A) is probet du Bureau des Longitudes ; ~Iembre du &:nat-Conseruateur, ably correct for stable equiGrand-O./flcier de la I.~gton-d'Honneur, et Comte de I'Empirc. librium. (2) Conclusion (B) is deft~ O U V E L L E I~DITION, nitely wrong, because it doesrevue et au~'~ent~epar l'Auteur, n't take into account states of equilibrium which are not stable. (3) The argument cannot be restricted to stable equilibrium, because this restricPARIS, tion is not maintained when M "~ V ~ C O U R C I E R , Imprimeur-Libraire pour le$ M-Qh~matiques, the principle of virtual velocqual dt:s A.gustlus, n" 5 7. ities is extended from statics 1813. to dynamics. Jacobi started with the fop (4) (A) is merely based on lowing consideration: What experience and therefore not certain: " . . . you have to be Where Lagrange gave his second "demonstration": Th6orle happens if an instantaneous impulse of finite magnitude aware that these probable des Fonctions Analytiques (2nd ed., 1 8 1 3 ) is exerted on the system at considerations are not more This time Lagrange ([1813], 379- rest? Of course, the real movement of than probable, and must not be taken 385) used pulleys as a substitute for the the mass points must be modified acas a [mathematical] demonstration" inner connections or constraints be- cording to the constraints. To deter(Jacobi [1996], 32--33). tween the masses. Lagrange sought to mine the Lagrange multipliers and subexpress the forces of constraint by an sequently the real impulses, one has to Lagrange's "construction", as Jacobi equation of constraint (F = 0). This make use not only of the equations of repeatedly calls it, therefore can never was of the utmost importance for him, constraint, but also of their first total be accepted as a mathematical proof. because there is a gap between his derivatives in time. Jacobi showed in a His fourth point, in particular, makes clear that Lagrange mixed up mathe- purely mathematical representation of lengthy algebraic development that rigid, geometrical constraints on the this problem can be solved if the equamatical reasoning with empirical one hand and physical actions given by tions of constraint are independent. It knowledge, which cannot provide cerforce functions (for example, by the is obvious that the real momentary imtainty and generality, but can only lead

VOLUME19, NUMBER3, 1997 51

pulses of the mass points depend on the first partial derivatives of the constraints with respect to the coordinates (Jacobi [1996], 59-64, 78-82). Jacobi then discussed the initial state of the same system under a continuously acting force. Now, an analogous procedure has to be performed to determine the Lagrangian multipliers and to show that the real accelerations (forces) are compatible with the constraints, taking compatibility of the initial values of the velocities as given by the fLrst step. Without going into the details of these calculations, it is clear that the second total derivative of the constraints with respect to time has to be used. Consequently, in this case the Lagrange multiplier will depend not only on the given forces, but also on the velocities of the particles and on the first and second partial derivatives of the equations of constraint (Jacobi [1996], 83-86). But in Lagrange's second demonstration, the substitution of these constraints by pulleys depends only on first-order approximation of the corresponding surfaces. There are no conditions imposed on the second derivatives of the pulley function f whatsoever. To quote Jacobi ([1996], 86) again: From this results an objection to the transition from statics to dynamics. The principle of statics doesn't deal with points in motion, and a particular inquiry, a particular principle has to be premised, how the velocities are constituted and modified... According to Jacobi, Lagrange mixes up two kinds of mechanical conditions, which are in reality "quite heterogeneous," as he says: on the one hand, a mass can underlie certain physicai forces (as gravity, for example); on the other hand, a mass point is fixed on idealised, rigid curves or surfaces. Conditions of the second kind, that is, forces of constraint, can be replaced by Lagrange's pulley in the case of rest, but not in the case of motion. Therefore, Jacobi ([1996], 87) asks for a new principle, "according to which both conditions of movement can be compared and determined in their mutual

52

THE MATHEMATICALINTELLIGENCER

interactions." But such a principle certainly transcends Lagrange's very conception of analytical mechanics, as Jacobi ([1996], 193-194) sharply points out in a more general discussion of Lagrange's approach: Everything is reduced to mathematical operation . . . . This means the greatest possible simplification which can be achieved for a p r o b l e m . . . , and it is in fact the most important idea stated in Lagrange's analytical mechanics. This perfection, however, has also the disadvantage that you don't study the effects of the forces any longer . . . . Nature is totally ignored, and the constitution of bodi e s . . , is replaced merely by the defined equation of constraint. Analytical mechanics here clearly lacks any justification; it even abandons the idea of justification in order to remain a pure mathematical science. Mechanical Principles and Mathematics in Jacobi's View Why was it so important for Jacobi that he spent about 8 hours and more than 40 pages of his lectures on this demolition job? I believe that Jacobi systematically applied his analytical and algebraic tools in order to show that mathematical demonstrations of mechanical principles cannot be achieved. He does not say that all attempts of demonstration are in vain, or that one attempt of his forerunners is as bad as another (Jacobi [1996], 93-96). He accepts that such attempts can lead to new insights in the principles of mechanics. But Jacobi insists that Lagrange's conception of a mathematical mechanics stands and falls with the certainty of the principle of virtual velocities, and he wants it to fall. He wants to make clear beyond any doubt that Lagrange's "constructions" must not be regarded as mathematical demonstrations of the certainty of first principles, and that these principles are not to be taken as inevitable laws of nature. One might fmd this intention quite destructive, but Jacobi thought it unavoidable and positive.

This brings me to Jacobi's own views about mechanics, its principles, and the role of mathematics, which are quite different from Lagrange's. According to Jacobi, mechanics should not be regarded as a purely mathematical science, and its mathematically formulated principles should not be regarded as intrinsic laws of nature. Rather, mathematics offers a rich supply of possible principles, and neither empirical evidence nor mathematical or other reasoning can determine which of them is true. The search for proper mechanical principles always leaves room for a choice, which can be made according to considerations of simplicity and plausibility. It is thus Jacobi who calls these first principles of mechanics "conventions," exactly 50 years before Poincar~ did. I quote Jacobi ([1996], 3): From the point of view of pure mathematics, these laws cannot be demonstrated; [they are] mere conventions, yet they are assumed to correspond to nature . . . . Wherever mathematics is mixed up with anything outside its field, you will however fred attempts to demonstrate these merely conventional propositions a priori, and it will be your task to fmd the false inference in each case. Obviously, Jacobi here is still the pure mathematician, drawing a line between mathematics itself and "anything outside its field," as he says. Mathematical notions and propositions on the one hand and physical concepts and laws on the other hand are to be sharply separated. This is in striking contrast to Lagrange's physico-mathematician's point of view. According to Jacobi, we cannot expect generality, certainty, and evidence from statements about physical objects, but only from propositions of mathematics itself. It is to make this distinction unmistakably clear that he points out the shortcomings of Lagrange's so-called demonstrations. Jacobi's criticism is not restricted to the principle of virtual velocities, nor to the principles of analytical mechanics in general. He also applies it to

N e w t o n ' s principles. Let m e quote s o m e r e m a r k s a b o u t t h e law o f inertia (Jacobi [1996], 3-4): F r o m the point o f view o f p u r e m a t h e m a t i c s it is a circular argum e n t to s a y that r e c t i l i n e a r m o t i o n is the p r o p e r one, [and that] consequently all others require external action: b e c a u s e y o u c o u l d define as j u s t l y any o t h e r m o v e m e n t as the law of inertia of a body, if y o u only a d d that e x t e r n a l a c t i o n is responsible if it d o e s n ' t m o v e accordingly. If w e can physically d e m o n s t r a t e e x t e r n a l action in any c a s e w h e r e the b o d y deviates, w e are entitled to call the law of inertia, w h i c h is n o w at the basis [of o u r argument], a law of nature. As is well known, Poincar~ calls the principle o f inertia a "disguised defmition" to m a k e explicit t h a t it defines w h a t a force-free m o v e m e n t should be. AKain, w e see a similar view in Jacobi, b u t half a century earlier. Can the basic laws of m e c h a n i c s b e u n d e r s t o o d as empirical generalisations o r as synthetic principles a priori? J a c o b i and Poincar~ are not p r e p a r e d to a c c e p t this classical dichotomy, t h e i r comm o n a n s w e r is: n e i t h e r nor! Both hold the opinion that e x p e r i e n c e o r a p r i o r i reasoning c a n n o t l e a d us to first principles b u t that t h e s e p r i n c i p l e s are fLxed b y convention. I t h i n k it is justified to say that in J a c o b i ' s last lectures on analytical m e c h a n i c s w e c a n find at least s k e t c h e s of w h a t b e c a m e k n o w n as conventionalism after the turn o f the c e n t u r y (Pulte [1994]). The i m p o r t a n t difference, however, is that Poincar~ holds t h e opinion that w e can always stick to the c h o s e n conventions, that t h e y a l w a y s can b e maint a i n e d as a b s o l u t e l y valid. J a c o b i is not explicit on this point, b u t he obviously believes that empirical evidence is cap a b l e of falsifying principles. F r o m time to time he r e m a r k s t h a t t h e y are n o t certain, b u t only " p r o b a b l y valid." Lagrange's E u c l i d e a n i s m is no longer a m o d e l o f science for him. As far as I know, J a c o b i is the first in the analytical tradition o f m e c h a n i c s w h o says farewell to E u c l i d e a n i s m a n d a d o p t s s o m e form of fallibilism.

Concluding Remarks J u s t a s w e should t a k e the frequently d r a w n parallel b e t w e e n rational mec h a n i c s a n d g e o m e t r y seriously, w e s h o u l d p a y attention not only to the c h a n g e s in the foundations o f geometry b u t also to those in mechanics. T h e r e is a line o f m e c h a n i c a l none u c l i d e a n i s m from J a c o b i onwards, w h i c h later led to serious d o u b t s a b o u t the validity o f Newtonian mechanics. This tradition is quite i n d e p e n d e n t of E r n s t Mach's well-known criticism o f a b s o l u t e space, and p r e c e d e s it. Nevertheless, it is widely n e g l e c t e d in the h i s t o r y of m a t h e m a t i c s a n d physics. Let m e h e r e j u s t refer to B e r n h a r d R i e m a n n a n d Carl Neumann. R i e m a n n w a s one o f the students w h o a t t e n d e d J a c o b i ' s lectures, a n d he p i c k e d up J a c o b i ' s v i e w of the principles o f mechanics, b e f o r e he c a m e to geometry. (More precisely: Riemann's critical attitude t o w a r d s a x i o m a t i c f o u n d a t i o n s starts with mechanics, a n d n o t with geometry.) Carl N e u m a n n s t u d i e d J a c o b i ' s Analytical Mechanics in g r e a t detail s o m e m o n t h s before he gave his fam o u s inaugural lecture On the princi-

c o m m e n t s a n d his help in p r e p a r i n g the final v e r s i o n o f this article. REFERENCES Borchardt, W., Ed. [1875] Correspondance mathematique entre Legendre et Jacobi. J. Reine Angew. Math. 80, 205-279. Jacobi, C.G.J. [1884] Vorlesungen (~ber Dynamik. Kdnigsberg 1842/43. Ed. A. Clebsch. 2nd ed., Berlin, Gesammelte Werke, Supplementband. Jacobi, C.G.J. [1996] Vorlesungen (~ber analytische Mechanik. Berlin 1847/48. Ed. H. Pulte. Braunschweig/Wiesbaden 1996.

ples of the Galilei-Newtonian theory, w h i c h is r e m a r k a b l e in its logical analysis of the law o f inertia and the c o n c e p t o f a b s o l u t e space. This lecture m a r k s t h e starting p o i n t of a b r o a d a n d intensive discussion a b o u t the validity of N e w t o n i a n m e c h a n i c s t h a t lasted until Einstein. Therefore, Neumann's w o r d s ([1870], 22) n o t only reflect J a c o b i ' s p o i n t of view, t h e y are an a p p r o p r i a t e e n d to this p a p e r : 9 it is also n o t a b s o l u t e l y impossible that the Galilei-Newtonian theo r y will s o m e day be r e p l a c e d b y ano t h e r theory, by a n o t h e r picture.

Acknowledgments An e x t e n d e d a n d m a t h e m a t i c a l l y m o r e d e t a i l e d v e r s i o n of this p a p e r will s o o n appear. It w a s w o r k e d out during m y stay at the D e p a r t m e n t o f History a n d P h i l o s o p h y of Science, University of Cambridge, as a fellow of the Alexand e r v o n H u m b o l d t Foundation. I a m also grateful to J e r e m y Gray for his

VOLUME 19, NUMBER 3, 1997

Klein, F. [1926] Vorlesungen Oberdie Entwicklung der Mathematik im 19. Jahrhundert, Vol. I. Berlin. Knobloch, E./Pieper, H./Pulte, H. [1995] " . . . das Wesen der reinen Mathematik verherrlichen". Reine Mathematik und mathematische Naturphilosophie bei C.G.J. Jacobi. Math. Semesterber. 42, 99-132. Lagrange, J.L. [1788] Mechanique Analytique. Paris. Lagrange, J.L. [1798] Sur le principe des vitesses virtuelles. J. Ecole Pol. (1)2, Cah. 5, 115-118. Oeuvres III, 317-321. Lagrange,J.L. [1811] Mecanique Analytique, Vol. 1. Nouvelle edition. Paris. Oeuvres XI. Lagrange, J.L. [1813] Th6orie des fonctions analytiques. Nouvelle edition. Paris. Oeuvres IX. Lakatos, I. [1978] Mathematics, science and epistemology (Philosophical Papers, VoL 2). Ed. J. Worrall/G. Currie. Cambridge. Lindt, R. [1904] Das Prinzip der virtuellen Verr0ckungen. Abh. zur Gesch. Math. Wiss. 18, 145-196. Neumann, C. [1870] Ueber die Principien der Galilei-Newton'schen Theorie. Leipzig. Pulte, H. [1989] Das Prinzip der kleinsten Wirkung und die Kraftkonzeptionen der rationalen Mechanik. Stuttgart. Pulte, H. [1994] C.G.J. Jacobis Verm&chtnis einer 'konventionalen' analytischen Mechanik. Ann. Sci. 51,498-519. Ruhr-Universit&t Bochum Fakult&t for Philosophie, Pb.dagogik und Pubrizistik D-44780 Bochum Germany

LEN BERGGREN, JONATHAN BORWEIN and PETER BORWEIN, all of Simon Fraser University, Canada

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54

THE MATHEMATICAL

INTELLIGENCER

ReferenceNumber11201

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Mathematica 3.0 software issued by Wolfram Research, Inc. 100 TRADE CENTER DRIVE, CHAMPAIGN, IL 61820

The Mathematica Book, 3rd Edition Wolfram Media CAMBRIDGE UNIVERSITY PRESS, 1996, xxtv+ 1403 PP, ISBN 0-521-588889-8 (HDBK), ISBN 0-521-58888-X (PBK)

REVIEWED BY STAN WAGON

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o o d c o m p u t e r h a r d w a r e h a s b e e n with us for several decades. But there w a s a s e r i o u s lag time b e f o r e available s o f t w a r e m a d e g o o d use o f t h e s p e e d and convenience fast b e c o m i n g available to e v e r y o n e with a d e s k t o p computer. Mathematica is a v e r y p o w e r f u l program, y e t I p u r c h a s e d it in 1989 b e c a u s e no o t h e r Macintosh p r o d u c t could even d r a w a sine curve to a r e s o l u t i o n a p p r o p r i a t e for a laser printer. Little did I k n o w t h e t r e m e n d o u s i m p a c t that this one p i e c e of software w o u l d have on m y future m a t h e m a t i c a l life. R e a d e r s no d o u b t k n o w that Mathematica is a mathem a t i c a l s o f t w a r e p a c k a g e of t r e m e n d o u s b r e a d t h (see [HI). And while v e r s i o n 2 was r e a s o n a b l y e a s y to use, the comp a n y w a s sensitive to u s e r requests for the addition o f m a t h e m a t i c a l typesetting to the environment. The j o b w a s n o t an e a s y one; the d e v e l o p m e n t o f v e r s i o n 3.0 has t a k e n five years, and the addition o f typesetting is only a small p a r t o f w h a t the n e w front e n d c a n accomplish. To b e frank, m y own interest r e s i d e s a l m o s t entirely with the kernel, the p a r t o f the p r o g r a m that carries out the m a t h e m a t i c a l computations. I w a n t the kernel to w o r k faster, to have s o m e useful n e w functions, and to have certain functions w o r k in a w i d e r v a r i e t y o f cases. Thus this review will focus m o r e on the kernel, a n d in p a r t i c u l a r on the n e w capabilities of v e r s i o n 3.0, on the a s s u m p t i o n t h a t the r e a d e r is a l r e a d y a w a r e t h a t p r e v i o u s versions c o u l d handle the f u n d a m e n t a l s of integration, plotting, animation, a n d s y m b o l i c algebra. That said, w h a t h a s b e e n acc o m p l i s h e d on the typesetting front is i n d e e d impressive, and I a m sure I will quickly b e c o m e a d d i c t e d to the sop h i s t i c a t e d typesetting features, the ability to include hyp e r l i n k s in n o t e b o o k s , a n d the like. An i m p o r t a n t feature of Mathematica is portability. To s e n d a c o m p l i c a t e d image via e-marl, j u s t s e n d the Mathe-

G

9 1997 SPRINGER-VERLAG NEW YORK, VOLUME 19, NUMBER 3, 1997

59

matica c o d e a n d the recipient can run it a n d generate the image. One c a n n o t easily do the s a m e with, say, a Microsoft Word document. So one virtue of the n e w typesetting features is that one can s e n d an entire n o t e b o o k (they are p o r t a b l e a c r o s s p l a t f o r m s a n d consist o f ASCII c h a r a c t e r s only) Via e-mail to a recipient who has only to o p e n the file in Mathematica to s e e p r o p e r l y f o r m a t t e d text. Since I a m TeX-illiterate, I have s p e n t m a n y hours o v e r the y e a r s convetting m y nicely f o r m a t t e d Word d o c u m e n t s to r a w text so I can send t h e m via e-mail. That s h o u l d n o longer be necessary. Moreover, b i t m a p images are n o w platform-independent. This is e x t r e m e l y important, for it m e a n s that one can c r e a t e c o m p l i c a t e d images on a Macintosh, say, and convert t h e m to b i t m a p s before saving the file, thus removing the p o s t s c r i p t that t a k e s up so m u c h space. The IBM-PC u s e r can t h e n o p e n the file a n d s e e y o u r images without having to r e g e n e r a t e them. A n o t h e r n e w feature that s h o w s the p r i o r i t y the developers have given to the u s e r interface is the r e d o n e Help facility. Essentially all of the Mathematica b o o k (1300 pages) is e n c o d e d a n d available online, a n d t h e r e is a very s o p h i s t i c a t e d s e a r c h engine with live links to guide the u s e r through the m a s s o f data. Of course, t h e s e fancy features m e a n that the p r o g r a m uses a large c h u n k o f disk s p a c e and internal m e m o r y . But c o m p u t e r s do i n d e e d get larger and f a s t e r at a fairly predictable rate a n d all t h e special features run s e a m l e s s l y on m y P o w e r Macintosh. The redesign m e a n s t h a t even exp e r i e n c e d u s e r s will n e e d s o m e time to learn the n e w interrace. On the o t h e r hand, the m a t h e m a t i c a l k e r n e l has no serious c h a n g e s a n d I believe that well over 99% of version 2 code will run u n c h a n g e d in version 3.0. The I n t e r p o latlngFunctlon o b j e c t s have b e e n redesigned, however, and since I w o r k with those quite a bit, I will have to r e w o r k s o m e of m y code. But a 2500-line p a c k a g e I w r o t e for visualizing differential equations r e q u i r e d only a b o u t 10 changed lines, so it s e e m s that a s e r i o u s effort w a s m a d e regarding d o w n w a r d compatibility. e
There are s o m e important changes in the numerical underpinnings of the system. For example, 2. + E n o w returns 4.71828; in version 2.2 it returned 2. + E. In other words, it used to be the case that the perfect symbolic nature of E w a s kept, so that if 2. w a s added and then subtracted, the result would stillbe the symbol E. N o w the result of such cancellation will be an approximate real. Another amusing quirk in the old version w a s the following: E<

Pi

E<

Pi

In short, Mathematica w a s saying it did n o t k n o w that e is less than ~-. One could get the right result b y using N[E] < N [Pi], but this behavior invariably caused problems for programmers who might use things like If [x < Pi, a, b], which w o u l d n o t w o r k w h e n x was symbolic. There was a

60

THE MATHEMATICALINTELLIGENCER

r e a s o n for this o d d b e h a v i o r but, happily, t h e y have figured out a w a y a r o u n d it, and n o w Mathematica k n o w s that e < T'.

The r e a s o n for the old b e h a v i o r w a s that it is nontrivial to p r e d i c t h o w m a n y digits the real a p p r o x i m a t i o n s n e e d to be before the inequality can be decided. Think of the c l a s s i c e x a m p l e e " ~ - ~ , w h i c h is d e s p e r a t e l y c l o s e to an integer. Mathematica's internals n o w allow it to d e c i d e that it is n o t really an e x a c t integer.

N[~ "~'~, 291 2. 6 2 5 3 7 4 1 2 6 4 0 7 6 8 7 4 4 0 0 0 0 0 0 0 0 0 0 0

x 1017

E x 1~q'~ < 2 6 2 5 3 7 4 1 2 6 4 0 7 6 8 7 4 4

True A more important example concerns a request such as N [ S i n [ 1 0 ~~ , 30]. Since successful evaluation of this requires knowing where 1060 is relative to the nearest multiple o f ~r, using a 30-digit a p p r o x i m a t i o n to 1060 (as w a s d o n e in p r i o r versions) is clearly insufficient. But 3.0 k n o w s to a d d e x t r a p r e c i s i o n as n e c e s s a r y and return 30 digits' w o r t h o f the c o r r e c t answer. N[Sln[106~

30]

0.83038976521934266466406178542

The p r e c e d i n g points a r e a n e x a m p l e of the sorts o f comp r o m i s e s n e c e s s a r y in a s y s t e m with the b r e a d t h a n d app e a l o f Mathematica. H e r e ' s a n o t h e r interesting e x a m p l e . Would a n y o n e w a n t a s y m b o l i c p r o g r a m to not simplify to 10V~ w h e n e v e r it s h o w s up as the result o f a comp u t a t i o n ? Should the u s e r b e e x p e c t e d to a s k for s u c h simplifications explicitly? Well, t h e r e are t w o s c h o o l s of t h o u g h t on this. On the one h a n d it is not p o s s i b l e to c a r r y out s u c h simplifications for arbitrarily large integers, since that requires factoring the integer. So, simplification being in g e n e r a l impossible, c o n s i s t e n c y requires that p e r f e c t s q u a r e s n e v e r be pulled out o f a square r o o t (note: p u r e p e r f e c t squares are always simplified). And Mathematica d o e s try to be c o n s i s t e n t in h o w its functions work. Nevertheless, here is a p l a c e w h e r e a c o m p r o m i s e w a s instituted: the simplification t a k e s p l a c e only if it c a n b e d o n e b y discovering divisors less t h a n or equal 2 TM+ 1.

Number Theory Determining primality o f large n u m b e r s is a beautiful int r o d u c t i o n to m a n y issues o f c o m p u t a t i o n a l c o m p l e x i t y a n d d e e p n u m b e r t h e o r y (such as the Riemann hypothesis). Here is h o w Mathematica 3.0 does it. P r i m e Q u s e s t w o s t r o n g - p s e u d o p r i m e t e s t s (base 2 and 3 [BS]) a n d a Lucas p s e u d o p r i m e test. It is h a r d to k n o w w h e t h e r t h e s e t h r e e t e s t s have a realistic c h a n c e of providing a c o m p l e t e c h a r a c t e r i z a t i o n of p r i m a l i t y o v e r the entire range o f integers. But w h a t can be said is t h a t the c h a r a c t e r i z a t i o n h o l d s up to 1016. A n d - - t h e m a i n p o i n t - - i t is v e r y fast. F o r numb e r s b e y o n d 1016 one could u s e P r i m e Q (or p e r h a p s a simple2-pseudoprimetest:PowerMod[2, n-1, n ] = = 1)

to get an almost-certain primality criterion; t h e n one could a p p e a l to the P r o v a b l e P r i m e Q function (in the N u m b e r T h e o r y P r l m e Q p a c k a g e ) that a t t e m p t s to p r o v i d e a p r o o f (usually called a certificate) of p r i m a l i t y using elliptic curve methods. The n u m b e r theory p a c k a g e s also contain the n e w functions SumOfSquaresR and SumOfSquaresRepresentations. The firstcomputes the number of representations of n as a sum of d squares (denoted r4(n));the second generates allthe representations. First one loads the package. Needs ["NumberTheory" NumberTheoryFunctions" "] ; Then one can call on the functions in the package. SumOfSquaresR [2, i0 I~ ] 44

SumofsquaresRepresentatlons [2, 10 I~ 440, i00000}, 47584, 99712}, 428000, 96000}, 435200, 93600}, 453760, 84320}, 460000, 80000}} SumOfSquaresR[4,

1000]

3744

SumOfSquaresRepresentations[4, 440 , 42, 44, 48,

I000]

0, i0, 30], 40 , 0, 18, 26}, 40, 6, 8, 30}, 40, i0, 18, 24}, 42, 4, 14, 28}, 8, 16, 26}, 42 , 14, 20, 20}, 42, 16, 16, 22}, 44, 4, 22, 22}, 44 , I0, i0, 28}, i0, 20, 22}, 46, 6, 12, 28}, 46, 8, 18, 24}, 46 , 12, 12, 26}, 48, 8, 14, 26}, 14, 16, 22}, {i0, i0, 20, 20}}

These functions work only for modest sizes of d and n (but if d = 2, then n can be quite large). Here is an interestingapplication that is related to the Madelung constant for salt (see [BW]). That c o n s t a n t has fascinating n u m b e r - t h e o r e t i c connections, since it is closely related to conditionally c o n v e r g e n t series related to the sum-of-squares functions. We focus on a variant here: ~ . ( - 1) n n=l function can help us a p p r o x i m a t e this series. 10000

(-l)a SumofsquaresR[2, n]

n.*

N[n]

Z

r2(n)/n.

Of course, the sum-of-squares

-2.17828 This d o e s n o t s e e m p a r t i c u l a r l y interesting (the digits of e are a coincidence!) until one divides by ~, a l w a y s a g o o d i d e a w h e n f a c e d with an unfamiliar constant. %171" -0.693368 Could this really be - l o g 2? In fact, it is: the series does indeed converge to - ~r log 2. But this is quite shocking! F o r it is n o t h a r d to see t h a t the a s y m p t o t i c average value o f r2(n) is ~-; this u s e s little m o r e than A = ~-r 2 for a circle. If,~m the original series, w e r e p l a c e r2(n) by this average value, the series b e c o m e s ~- Z ( - 1)n/n, o r ~r log 2. n=l So this blatantly illegal r e p l a c e m e n t turns out to yield the c o r r e c t answer! F o r m o r e information on t h e s e t o p i c s see [BW, BBP1]. As often happens, interesting c o m p u t a tions can l e a d the c u r i o u s h a c k e r to s o m e fascinating a r e a s o f m a t h e m a t i c s . F o r the r e a d e r w o n d e r i n g h o w these sum-of-squares c o m p u t a t i o n s a r e carried out, I will say here only t h a t t h e y are related to a v e r y fast s c h e m e using the E u c l i d e a n algorithm f o r m u l a t e d b y G. C o r n a c c h i a in 1908. This allows one to f a c t o r over the

VOLUME 19, NUMBER 3, 1997

61

Gaussian integers, w h i c h yields s u m s of s q u a r e s as in the following example. Because 153 - 1 ( m o d 4), it h a s the form a 2 + b2; since that is (a + bi)(a - bi), the a and b can be found a m o n g t h e divisors o f 153 in the G a u s s i a n integers (see [Wl]). Divlsors[153,

GaussianIntegers

{i, I + 4 I, 3, 3 + 1 2 I , A Profoundly Depressing

4+I,

~ True]

9, 9 + 3 6

I, 1 2 + 3 I, 17, 3 6 + 9

I, 51, 153}

Proof

Symbolic c o m p u t a t i o n can have a big effect on certain sorts of m a t h e m a t i c s . In late 1995 David Bailey, P e t e r Borwein, a n d Simon Plouffe [BBP] d i s c o v e r e d the following truly shocking formula:

z =

( 8k+I

8 k+4

8 k+5

8

6 )'

k=O

The r e a s o n this f o r m u l a is shocking is t h a t it allows one to c o m p u t e the dth base16 digit of 7r w i t h o u t computing any p r i o r digits! And if d is, say, a billion, t h e n machine p r e c i s i o n is sufficient. All p r e v i o u s m e t h o d s of getting s u c h a digit required a c o m p u t a t i o n using d or m o r e digits o f p r e c i s i o n throughout. So this is a big improvement! F o r an e x p l a n a t i o n of the simple i d e a s that turn the f o r m u l a into an algorithm see [BBP] o r [BBBP] o r [AW]. Unfortunately, it s e e m s as if no s u c h f o r m u l a s exist for base-10 digits. Now, BBP u s e d n u m e r i c a l a p p r o x i m a t i o n s to discover this formula. Once discovered, the f o r m u l a w a s e a s y to prove b y e l e m e n t a r y calculus. Of course, the difficult thing is to fmd the f o r m u l a in the first place. I tried to s e e if Mathematica w o u l d spit out ~r w h e n fed the BBP formula. Here is an a p p r o a c h t h a t succeeds. First w e c h a n g e the series to four integrals by the following e l e m e n t a r y transformations. The first equality is a simple integration and the second an interchange. 1

~1

= 21/2 ~

x.o 16k ( S k + i )

Z 8k§

dZ

=

Z8 k~i'l

2 i/2

k.o ~o

~Z

~o

Now w e define g [ i ] to be the integral c o r r e s p o n d i n g to one t e r m o f the series.

g[t_]

:=2~"fox/~(~.oZ'k'*X)4z

When w e f e e d the BBP e x p r e s s i o n to Mathematica, out p o p s 7r. simplify[4g[1]

- 2o[4]

- g[S]

- g[6]]

71"

There is s o m e t h i n g p r o f o u n d l y d e p r e s s i n g a b o u t this calculation. It gives us absolutely no i d e a why this almost-geometric series should equal 7r. This is a c o n s t a n t danger in s y m b o l i c computation. One gets c a r r i e d a w a y b y the p l e a s u r e in seeing the software p r o v e a result, not realizing t h a t true u n d e r s t a n d i n g has d i s a p p e a r e d . Yet, further c o m p u t a t i o n n o t only e n h a n c e s o u r understanding, but also l e a d s to n e w results. F o r we can c o n s i d e r arbitrary s u m s o f the form:

-: 1~._ (

al

k=O |6k ~ ' ~

a2

a3

a4

+ 8k +"''"~+ 8k +'----~+ 8k +"""""~+

as a~ 8k +'--------+'~ 8k +-----~+

a7

a s ,

8k +'~ +

Mathematica is c a p a b l e of evaluating this in t e r m s of the u n s p e c i f i e d coefficients. The result, after t r a n s f o r m i n g to integrals a n d carrying out s o m e simplifications, is

62

THE MATHEMATICAL INTELUGENCER

1

-~- rr (%/~ a [I] + 2 a[2] - 2 ~/~ a[3] + 4 ~/~ a[5] - 8 a [6] - 8 ~/~ a[7] ) + 1 ~- (a[l] - 2 a[2] + 2 a[3] - 4 a[5] + 8 a[6] - 8 a[7]) ArcTan[2] + (-a[l] + 2 a[3] - 4 a[5] + 8 a[7]) ArcTan[%/~] 24Y + 1 8 a[8] Log[2] + ~- (a[2] - 2 a[4] + 4 a[6] - 8 a[8]) Log [3] + 1 ~- ( a [ l ] - 2 a[3] + 4 a[4] - 4 a[5] + 8 a[7] - 16 a[8]) Log[5] + (a[l] + 2 a[3] + 4 a[5] + 8 a[7]) L o g [ l + ~ / ~ ]

247 This l o o k s like a mess, b u t really is quite beautiful. One can a s k collect the coefficients, w h i c h leads to:

Mathematica

to

"[41] + a~3l + a [ 5 ] + 2 a [ 7 ] .[81] _ a[2 ] + .[241 - a%51 +at7]-2a[8]

.~2_-~+] +a[61-2a[s] 8a[8] -

a~l] + a[23] - a [ 5 ] "[41|

-

+2a[7]

a/2] + a123] - a [ 5 ]

+2a[6]

-2a[7]

al~+ I 4 -a[6] .11!_ a~,, + .~5,-a[71

w h e r e the rows are the coefficients of log(1 + X/2)/ 8, log 5, log 3, log 2, a r c t a n [ V 2 ] / 8 , arctan 2, ~r, and ~-~f2, respectively. Now we j u s t solve the linear system obtained b y setring the coefficients equal to {0, 0, 0, 0, 0, 0, 1, 0}. If w e find a solution w e will, by force of cancellation, have discovered a formula for 7r. In fact this works, a n d w e get the foll o ~ n g one-parameter family of identities for ~r ( k n o w n to BBP).

7r= ( 2 - r ) g [ l ] +

(2

+ r ) g[2] + ( i + 2 ) g[31 + r g [ 4 ] +

1

1

1

(--+-- +

+ (-.+- + §

+( 1,

r .+) +[.7]

Setting r = - 2 o r 2 yields the BBP f o r m u l a s ~- = 4 g[1] - 2 g[4] - g[5] - g[6] and ~r = 4 g[2] + 2 g[3] + g[4] - 89 More exciting, this m e t h o d of u n d e t e r m i n e d coefficients leads to the following n e w 3-term f o r m u l a for qr of the BBP type: oo

21" "-

E

,-,,k 4k

( ~

2

2

"1" 4k+2 -i-

),

k=O as well as n u m e r o u s o t h e r formulas. See [AWl for details. The p o i n t h e r e is that powerflfl s y m b o l i c s allow one to e x p l o r e and m a k e d i s c o v e r i e s for w h i c h the p r o o f s c o m e along for free!

Huygens Verified The symbolic p o w e r o f 3.0 s e e m s m u c h improved. Indeed, n o w a d a y s s o m e calculators can do s o m e symbolic integration, so the onus w a s on Mathematica to go farther in this direction, and it s e e m s that it has. One of m y favorite e x a m p l e s is the integral that p r o v e s that the time it t a k e s a b e a d to slide d o w n a cycloidal wire is i n d e p e n d e n t of the starting position (Huygens, 1673). Older versions o f Mathematica could not do this easily. Here is h o w to do it in 3.0. The cycloid is given b y f(t) = (t - sin(t), cos(t) - 1), and an e l e m e n t a r y argument s h o w s that the d e s c e n t time for a b e a d that starts at f(t0) is:

]f/ /x-'(t)2 +-B'(t)2 at V 2 g(Yo - y[t]) VOLUME19, NUMBER3, 1997 63

First we g e n e r a t e the integrand.

f[t_] -. { t - s i n [ t ] , norm[u_]

co.[t] -I)

-= norm[ f" It] ]

speed = simplify[

~/2 g Vi

(~"[tO]

[2]

-f

[t]

[2])

- cos [t]

qg ( - c o s [ t ] + c o s [ t 0 ] ) Now w e evaluate a n d plug in the e n d p o i n t s to and 7r. There is a singularity at to (because the initial s p e e d is 0) a n d so w e m u s t u s e a limit as t --~ to. The i n t e r m e d i a t e outputs are s u p p r e s s e d , b u t t h e y are quite c o m p l i c a t e d . timeRaw :

fspeeddt

((timeRaw/. x

/ / Simplify;

t ~ ~) - Limit[timeRaw, ,.,

t ~ tO]

/.

Sign[_]

:~ 1 // PowerExpand)

/.

//simplify

2T

% The to has totally d i s a p p e a r e d , as Huygens p r o v e d it must! While it is satisfying to w o r k out the s e q u e n c e of simplifications that l e a d s to the simple answer, this manipulation gives no insight into why the c y c l o i d is a tautochrone. F o r t h a t o n e w o u l d have to study it geometrically, or p e r h a p s via differential equations. But n o t e that the use of s o m e clever s u b s t i t u t i o n s is e x a c t l y h o w m a n y calculus b o o k s verify the t a u t o c h r o n e property.

Solving Equations Of course, Mathematica can solve equations in the usual way: symbolic m e t h o d s for lowdegree polynomials and s o m e simple nonpolynomial equations, numerical m e t h o d s for general equations. There is no built-in function to find a / / t h e a p p r o x i m a t e solutions to a system of equations. In general, this could b e quite difficult, but there is a m e t h o d that w o r k s very well for arbitrary systelns of t w o equations. This a p p r o a c h illustrates the w a y one can use the high-level graphics capabilities of Mathematica t o g e t h e r with its openness in allowing the u s e r to access the inner d a t a structure of graphics objects. I have w o n d e r e d if Maple's graphics s c h e m e w o u l d allow a similar approach, and I believe it would not. This particular p r o b l e m w a s v e r y useful for a differential equations p a c k a g e I was working on, as it allowed us very quickly to determine the c o m p l e t e set of equilibrium points for an a u t o n o m o u s system of two ordinary differential equations. Note that this also allows one to solve for all zeros of a c o m p l e x f u n c t i o n f ( z ) : just use the real and imaginary p a r t s (but beware, since the m e t h o d is not designed to fred solutions corresponding to tangencies of the curves). So s u p p o s e one w a n t s the s i m u l t a n e o u s solutions o f f ( x , y) = 0, g(x, y ) = 0 in the rectangle [a, b] • [c, d]. First use C o n t o u r P l o t to generate the curve(s) c o r r e s p o n d ing to f = 0. Then one can go inside the g r a p h i c s object and e x t r a c t the d a t a that defme the curves. Simply evaluate g along t h e s e curves and use the p l a c e s w h e r e g changes sign as s e e d s to N e w t o n ' s m e t h o d via F i n d R o o t . Here is a quite c o m p l i c a t e d example. F i r s t w e plot f = 0 and g -- 0. f = - C o s [ y ] + 2 y C o s [ y 2] C O S [ 2 X ] ; g = -Sin[x] + 2 Sin[y 2] S i n [ 2 x ] ; S e t O p t i o n s [ C o n t o u r P l o t , Contours ~ {0}, P l o t P o i n t s ~ 60, C o n t o u r S h a d i n g ~ F a l s e ] ;

fZero = C o n t o u r P l o t [ f ,

{x, -3.5, 4}, {y, -1.8, 4.2}];

gZero = ContourPlot[g,

{x, -3.5, 4}, {y, -1.8, 4.2}, C o n t o u r S t y l e ~ A b s o l u t e T h i c k n e s s [ 2 ] ] ;

Show[fZero,

gZero];

THE MATHEMATICAL INTELLIGENCER

4

2

J

0

-2

0

2

4

-2

0

2

4

Figure 1. The graphs of - c o s y + 2 u oos[y ) cos(2 x) = 0 (thin) and - s i n x + 2 s i n ~ sin(2 x ) = 0 (thick). T h e r e are lots o f crossings.

Figure 2. This image shows black disks at all the points found by Equilibria.

3~here are lots of c o m m o n zeros. The m e t h o d outlined a b o v e is i m p l e m e n t e d as ~ . q u i I i b r i a in the V i s u a l D S o l v e p a c k a g e by Dan S c h w a l b e and m e [SW]; it finds all 67 o f t h e m in 8 seconds. The result is s h o w n in Figure 2. A very i m p o r t a n t n e w feature is the ability to h a n d l e algebraic numbers. Mathematica r e p r e s e n t s s u c h as R o o t objects. I will give h e r e only t w o e x a m p l e s o f h o w the rel a t e d function, R o o t R e d u c e works. If one wishes the m i n i m a l p o l y n o m i a l satisfied b y an algebraic number, p r o c e e d as follows.

Sophisticated Mathematics So far I have m e n t i o n e d c o m p u t a t i o n s that can be done in j u s t a few lines. Figures 3-7 s h o w h o w the pinpoint control over graphics that Mathematica allows can be u s e d to create illuminating images in s o m e very s o p h i s t i c a t e d mathematical settings. The explanations are in the captions.

Roo,,,~..,,,,.-. [ (~ 9 ~ ) / , ' , n ~ ' ~ I ] [ U [x] 128

+ 1200

x 2 + 4077

x 4 + 4090

x 6 - 6318

x e - 10140

x I~ + 8 7 8 8

x I~

A n d here is h o w one would attempt to simplify a radical. Working out this simplification was inherent in problem B4 of the 1995 Putnam competition. (x/e}

m

RootReduce

1

2

(3

[

2207

+ "~ 2 2 0 7 -2

s - 4

]

+'~)

Figure 3. A c a r t w i t h s q u a r e w h e e l s can roll on a road m a d e up of c a t e n a r i e s so t h a t its axle has no u p / d o w n m o v e m e n t ! Indeed, one can build nice w o r k i n g models! (Joint w o r k w i t h Leon Hail [HW~

Conclusion Of course, a r e v i e w should p o i n t out s o m e a r e a s w h e r e the software c o u l d b e improved. There w e r e s o m e front-end bugs in t h e b e t a v e r s i o n that I p l a n n e d o n mentioning, b u t they have b e e n fLxed. Still, the official r e l e a s e v e r s i o n d o e s c r a s h u n e x p e c t e d l y from time to time (these are front-end p r o b l e m s , n o t k e r n e l problems; the k e r n e l is v e r y stable and the ability to interrupt calculations is m u c h better), and certainly I will b e looking for i m p r o v e m e n t in later releases. A n d the 1;~nd-and-Replace tool is n o t as versatile as version 2's. The p r o b l e m with the h a r d w a r e / s o f t w a r e lag is that it m a y well t a k e o u r collective i m a g i n a t i o n s o m e time to catch up with the possibilities. It is c l e a r to m e that computing with s o f t w a r e of this b r e a d t h a n d d e p t h can really revitalize m a t h e m a t i c s . When Mathematica first c a m e o u t I w a s s o m e w h a t mystified as to h o w I w o u l d m a k e g o o d use of it in n u m e r i c a l analysis, since so m a n y things w e r e built in. But it quickly b e c a m e clear t h a t the software w a s a p o w e r f u l t o o l to explain old algorithms and even to develop n e w ones. The possibilities for visualization in the a r e a s c o m m o n to the u n d e r g r a d u a t e c u r r i c u l u m are endless. A n d in m a n y r e s e a r c h areas, the w o r k e r w h o has a firm grasp o f w h a t m o d e r n s o f t w a r e c a n a c c o m p l i s h a n d h o w m u c h o r h o w little it will t a k e to c a r r y it off has a big advantage.

VOLUME19, NUMBER3, 1997 6~

Figure 4. The m e d i u m - s i z e d axis of a b o o k is unstable w h e n t h e b o o k is r o t a t e d a b o u t that axis. This image shows t h e c o u n t e r i n t u i t i v e m o t i o n that such a b o o k w o u l d follow. Wrap an elastic band around a b o o k and t r y it! This image required t h e solving of 12 s i m u l t a n e o u s differential equations, w h i c h are handled w i t h o u t difficulty. As usual, s e t t i n g up t h e model is much m o r e d i f f i c u l t than solving t h e equations, and it is this skill t h a t w e m u s t emphasize w i t h o u r students. (Joint w o r k w i t h Dan S c h w a l b e [SW])

Figure 5. This is a v i e w of x 2 y/(x 4 + y2), a d i s c o n t i n u o u s function c o m m o n l y presented in calculus courses. The f u n c t i o n a p p r o a c h e s 0 on every s t r a i g h t line into t h e origin, b u t t a k e s on d i f f e r e n t c o n s t a n t values on each parabola y -- m x 2. The singularity means t h a t it is j u s t a b o u t impossible to v i e w this surface using standard r e c t a n g u l a r coordinates, b u t t h e pinp o i n t g r a p h i c s control of M a t h e m a t i c a allows the enthusiast i c p r o g r a m m e r to g e n e r a t e a c o o r d i n a t e system based on p a r a b o l a s and ellipses, and this makes all t h e difference. (Joint w o r k with Ron Goetz [GW]).

Figure 6. A p a r a d o x in t h e hyperbolic plane. Figure 6(a) shows t h r o e s u b s e t s of t h e h y p e r b o l i c plane, A, B, and C, t h a t are congruent; thus each is a " t h i r d " of H2. In Figure 6(b) t h e v i e w p o i n t is m o v e d a little and it b e c o m e s e v i d e n t t h a t t h e d a r k e s t set, A, is c o n g r u e n t to t h e union of t h e o t h e r two! This shows that each set is also a " h a l f " of t h e h y p e r b o l i c plane. These images w e r e g e n e r a t e d by t e a c h i n g M a t h e m a t i c a a b o u t t h e free p r o d u c t E2 * ~3 and having it g e n e r a t e t h e Hausdorff p a r a d o x in t h a t group. Then some g r a p h i c s p r o g r a m m i n g effects t h e translation to t h e Klein-Fricke tessellation of t h e Poincar6 disk model. The only dlfferonce in t h e Banach-Tarski p a r a d o x in Euclidean 3-space is t h a t t h e axiom of choice must be used to g e t a fundamental domain; in H 2 w e have t h e c o n v e n i e n c e of t h e fundamental t r i a n g l e for t h e tessellation, w h i c h is t h e key to this constructive realization of a g e o m e t r i c paradox. (Joint w o r k w i t h Jan Mycielski [W].)

THEMATHEMATICALINTELLIGENCER

Figure 7. A pattern g e n e r a t e d using an idea of M . C. E s t h e r . A 1 x 1 motif is translated and rotated to fill out a 2 x 2 tile; w e use a m o t i f consisting of 7 polygons. Then t h e tile is t r a n s lated r e p e a t e d l y to fill t h e plane. It is quite difficult to figure out w h a t the proper coloring should be. We have w r i t t e n a p a c k a g e t h a t a u t o m a t e s this chore, thus allowing the g e n e r ation of m a n y beautiful p a t t e m s even using only Escher's original 5 - p i e c e motif. O u r h o p e is t h a t readers will g e n e r a t e t h e i r o w n motifs. (Joint w o r k w i t h Rick M a b r y and Doris S c h a t t s c h n e i d e r [MWS]; see also [S])

[BS] Eric Bach and Jeffrey Shallitt, Algorithmic Number Theory, Cambridge (Mass). MIT Press 1996. [BW] Joe Buhler and Stan Wagon, Secrets of the Madelung constant, Mathematica in Education and Research 5:2 (1996), 49-55. [GW] Ron Goetz and Stan Wagon, Adaptive surface plotting: A beginning, Mathematica in Education and Research 5:3 (1997), 74-83. [H] Alan Hoenig, Review of Mathematica, Mathematical Intelligencer 12:2(1990), 69-74. [HW] Leon Hall and Stan Wagon, Roads and wheels, Mathematics Magazine 65 (1992), 283-301. [SW] Dan Schwalbe and Stan Wagon, VisualDSolve: Visualizing Differential Equations with Mathematica, Springer/TELOS, New York, 1997. [MWS] Rick Mabry, Stan Wagon, and Doris Schattschneider. Automating Escher's combinatorial patterns, Mathematica in Education and Research 5:4 (1997), 38-52. IS] Doris Schattschneider, Escher's combinatorial patterns, Electronic Journal of Combinatorics, 4 (1997), #R17. [W] Stan Wagon, A hyperbolic interpretation of the Banach-Tarski Paradox, The Mathematica Journal 3:4 (1993), 58-61. [W1] Stan Wagon, The magic of imaginary factoring, Mathematica in Education and Research 5:1 (1996), 43-47. Department of Mathematics Macalester College St. Paul, MN 55105 USA e-maih [email protected]

Of course, there may well be special-purpose packages that are better at certain specific tasks, such as special-purpose high-precision number theory or group theory packages. But speaking as someone who teaches different subjects from year to year, I think M a t h e m a t i c a ' s graphics capabilities, overall elegance, and, especially, its breadth are outstanding. As for the impact of 3.0, it is quite possible that the ease of typesetting and fancy front end will be very important. When the early notebook interface came out, it was a secondary feature o f M a t h e m a t i c a , as the kernel was the heart of the product. But over time it became clear that the beautifully designed interface was having a significant impact on how people thought and communicated mathematics. Most likely the 3.0 design will do the sanle.

REFERENCES

[AW] Victor Adamchik and Stan Wagon, ~-: A 2000-year old search changes direction, Mathematica in Education and Research 5:1 (1996), 11-19. [BBBP] David H. Bailey, Jon Borwein, Peter Borwein, and Simon Plouffe, The quest for pi, Mathematical Intelligencer 19:1 (1997), 50-57. [BBP] David H. Bailey, Peter Borwein, and Simon Plouffe, On the rapid computation of various polylogarithmic constants, Mathematics of Computation (to appear). [BBP1] David Borwein, Jonathan Borwein, and Christopher Pinner, Convergence of Madelung-like lattice sums, forthcoming.

VOLUME 19, NUMBER 3, 1997

67

Theory of Moves by Steven J. Brams CAMBRIDGE, UK: CAMBRIDGEUNIVERSITYPRESS, 1993, 260 PP. US $18.95, ISBN 0-52145-867-6

Game Theory and Strategy by Philip D. Straffin WASHINGTON, DC: MATHEMATICALASSOCIATIONOF AMERICA 1993, 200 PP. US $33.95, ISBN 0-88385-637-9

REVIEWED BY MARC KILGOUR

hY do people do what they do? The objective of the social sciences is to answer this question, despite its hopeless breadth. To narrow the task, we can ask instead: Why do human decisions so often lead to unfortunate, unexpected, and unintended results, even when the decision-maker is careful, reasonable, and informed? One useful insight into this version of the question is that sometimes consequences depend on the choices of more than one decisionmaker--and when there are too many cooks, what might have been a tasty broth can become indigestible. John yon Neumann and Oskar Morgenstern may well have had thoughts like this in mind as they collaborated, during World War II, on Theory of Games and Economic Behavior (TGEB) [6], a work now regarded as one of the greatest intellectual achievements of the twentieth century, in any field. One of their fundamental conclusions was that when decision-makers interact, rationality itself may not be well-defmed, and the meaning of "optimal choice" may be far from obvious. Although TGEB raised many more questions than it answered--it included satisfying definitions of optimal choice in only a few important cases, such as two-person zero-sum games--its publication was hailed like the discovery of a new continent. One contemporary review [1] called it the "foundation for an entirely new science." Von Neumann and Morgenstern inspired generations of mathematicians

W

68

and social scientists, especially economists, in the quest to understand human decision-malting better, and thereby to improve it. In 1994, exactly haft a century after the publication of TGEB, the Nobel Memorial Prize in Economic Science was awarded to three gametheorists, John Nash, Reinhard Selton, and John Harsanyi, "for their pioneering analysis of equilibria in the theory of non-cooperative games." [5] The Royal Swedish Academy of Sciences noted explicitly that the award was made fifty years after the publication of von Neumann and Morgenstern's "monumental study." The recognition, most commentators agreed, was long overdue. Yet, for all its renown, Theory of Games and Economic Behavior is not often read these days. One reason is that TGEB is not a compendium of knowledge--this is unsurprising, of course, as its contribution was to pioneer the study of interacting decisions, rather than, like Euclid, to organize and complete a substantial body of existing work. But TGEB is also not particularly insightful as a guide to modem game theory, nor is it even an efficient introduction. This is surprising, particularly because the general outline of the theory remains very much as von Neumann and Morgenstern would have expected, and the vocabulary still follows TGEB in large measure. For example, games are still organized according to von Neumann and Morgenstern's primary classification, non-cooperative (How should independent decision-makers select their actions?) versus cooperative (Given that all decision-makers have agreed to cooperate, how should the benefits of cooperation be shared among them?). But if the ends are the same, the means are certainly different. The routes that von Neumann and Morgenstern identified as most likely to lead to their objective turned out to be short and fraught with difficulty, and are now largely abandoned. While their fundamental principle, that the interaction of decision-makers changes everything, is now almost a commonplace, their ideas about how to analyse this interaction have been largely superseded. For example, the concept of equi-

THE MATHEMATICALINTELLIGENCER9 1997SPRINGER-VERLAGNEW YORK

librium, rightly recognized by the Royal Swedish Academy of Sciences as the "principal aspect" of non-cooperative game theory, was not mentioned by von Neumann and Morgenstern; it did not appear in print until 1950. True, TGEB established that an entirely satisfactory theory of two-person zerosum games can be based on the Minimax Theorem, which was in fact discovered and proven by von Neumann during the 1920s. It also solved extensive-form games of perfect information (another earlier idea), and showed how to convert games from extensive form to strategic form. But to read TGEB gives no hint at all about most of modern non-cooperative game theory, and can even be misleading. There is a suggestion, for instance, that an understanding of two-person non-zero-stun games would follow from a good theory of three-person zero-sum games. This idea was, to be blunt, fruitless---no useful insights have been obtained from any connection between these two classes of games, nor are the two classes now seen as particularly close. On the other hand, the idea of Nash equilibrium, published only six years after the first edition of TGEB, is now the foundation of an enormous superstructure of concepts, including Selten's refinements integrating dynamics, and Harsanyi's extensions incorporating uncertainty. For this work, these three became Nobel laureates. The two books under review provide another perspective on Theory of Games and Economic Behavior. Both are short, both are well-written, and neither is a difficult read for the mathematically sophisticated. They both present mathematical ideas clearly but often avoid rigorous or general formulations, and they both provide a wealth of enlightening applications to the social sciences. But their aims are completely different: Game Theory and Strategy recapitulates developments within game theory as set out by von Neumann and Morgenstern, whereas Theory of Moves calls for a fundamentally different way of representing and analysing the interaction of decision-makers. Game Theory and Strategy is designed as the text for an "interdisciplinary division" course in game theory, so

its mathematical prerequisites are min- less, the ideas are clearly expressed, coimal. Philip Straffm writes engagingly, gently argued, and promising. and includes enough topics and ideas to The theory of moves is not game suggest the sweep of game theory as it theory, because the structure used to exists today. Not only are the topics express the interacting decisions is well connected, they are illustrated not, in the von Neumann-Morgenstern with an impressive array of examples sense, a game. In TOM, interactions are and exercises, including applications in not described in terms of strategies, or economics, political science, psychol- complete plans of action that apply in ogy, and anthropology, with brief ex- every possible contingency, but in cursions into philosophy and biology. terms of the state of the interaction, Straffin also manages to present a few which may of course change over time. important proofs, and provides hints In a TOM model, the initial state, or staabout others, despite the constraint of tus quo, must be specified ("the startlow prerequisites. Altogether, Game ing point matters"). Players choose eiTheory and Strategy may be the best ther to move away from the current general low-level introduction to game state, or to pass; when all players pass, theory available today. It is certainly a "non-myopic equilibrium" has been worthy of its place in the MAA's highly attained. Brains supplies a model for the calculations a player might make regarded New Mathematical Library. More pertinent to the observations to decide whether to move, and idenabove, however, is that the organization tifies the resulting non-myopic equilibof Game Theory and Strategy mirrors ria. However, this model applies only that of Theory of Games and Economic when there are two players and four Behavior. Straffm divides his material states such that each player always has i~to three sections: two-person zero- exactly one move available, all moves sum games, which covers the conse- are reversible, and there are no comquences of the Minimax Theorem and mon moves. This arrangement can be some extensive-form models; two-per- conveniently described using the mason non-zero-sum games, which con- trix of a 2 x 2 game. Another fundamental difference becentrates on small (usually 2 x 2) games like Prisoners' Dilemma, and tween the theory of moves and game discusses Nash equilibrium and related theory is that the former requires only developments only briefly; and n-person ordinal information about preferences. games, which contains mainly coopera- In other words, a TOM model includes tive game theory material. The latter in- only the decision-makers' preference cludes a frank assessment of the von rankings over the states, rather than Neumann-Morgenstern cooperative so- their von Neumann-Morgenstern utililution concept, now called the stable ties. Consequently it may be much easset, that leaves no doubt as to why it has ier to develop TOM models of some sitbeen replaced by alternatives such as uations, for example in politics. Still, the core (which is mentioned in TGEB), this difference may be less essential the bargaining set, the Shapley value, than it looks: in games, pure Nash equiand the nucleolns. In summary, libria can be identified using only orStraffm's admirable text makes clear dinal preference information, and this that the goals of von Neumann and in fact is how Straffm calculates them. Morgenstern's research program remain Nonetheless, a theory of games based sensible and desirable, although most of on ordinal rather than cardinal preferthe progress toward those goals has not ence information would be insufficient from the point of view o f v o n Neumann followed the routes they foresaw. Steven Brams's Theory of Moves, on and Morgenstern, as it would not genthe other hand, is more a monograph eralize the Minimax Theorem. The theory of moves is not the only than a text. Its purpose is to describe, and argue for, an alternative way of alternative to von Neumann and Mormodelling and analysing strategic con- genstern's structures for analysing inflicts. Brams's proposal, the theory of teracting decision situations. Brams moves (TOM), is explained and illus- gives references to several other systrated only in simple contexts; nonethe- tems that are based on states rather than

strategies, and more have been catalogued elsewhere [4]. Another recent development is the system of Greenberg [3] that attempts to model rational decisions to cooperate. This link between non-cooperative and cooperative games was never really addressed by von Neumann and Morgenstern. Theory of Moves is really a proposal for a new model of interacting decision behaviour. In fact, Brains only sketches out the theory--for the most part, the book is restricted to small models (based on 2 x 2 games), and there is no discussion of computer implementation, which has proven effective for some of the alternatives (e.g., [2]). Nonetheless, Brains expresses his ideas clearly and convincingly, and makes good use of illustrative examples drawn mainly from politics. He also describes several potentially important directions for development of the theory, such as models of how relative levels of power affect behaviour in interactions. Again, he illustrates these ideas with cleverly chosen, insightful models. Is the theory of moves really different from the theory of games? Definitely, provided the question is interpreted as referring to ends rather than means. Some structures that have been proposed as alternatives to game models can in fact be formulated as games, and the theory of moves probably could be as well. But anyone who attempts to do so, warns Brains, "would have missed the point." (p. 18) The purpose of Brams's system is to model interacting decisions in a way that emphasizes long-term versus short-term thinking, and that uses the simple and parsimonious TOM structure instead of some cumbersome extensive game. Does the proposing of alternatives, such as the theory of moves, indicate that there are problems, or flaws, in the theory of games? In a word, no. The enterprise begun so decisively by von Neumann and Morgenstern continues to develop, and anyone who doubts its successes or its wide applicability need only consult Straff'm. The 1994 Nobel prize in economics was a triumph for game theory, a triumph not in any way diminished by suggestions for new ways to address related prob-

VOLUME 19, NUMBER 3, 1997

69

lems. As Brams makes clear, the need for models and mechanisms that are alternatives to game theory does not imply that there is any reason at all to doubt this "monumental intellectual achievement." (p. 18) Looldng back at Theory of Games and Economic Behavior is a useful exercise, for it makes clear the dimensions of the continent that von Neumann and Morgenstern discovered. They expressed their perspective in terms of economic decision making, comparing a "Robinson Crusoe" economy to a social-exchange economy. A rational Crusoe, they pointed out, needs to choose values for "variables which his will controls," taldng Into account only his capacities and tastes and "the unalterable physical background of the situation." (p. 12) But a participant in a social-exchange economy must deal also with variables that "reflect another person's will or intention of an economic ldnd--based on motives of the same nature as his own." For von Neumann and Morgenstern, and for many who followed them, the essence of game theory is the meaning of rationality in this context of simultaneous interactIng choices. No matter how successful the initial approach, it can hardly be surprising that variations are put forward eventually. The quotations above make clear that, for von Neumann and Morgenstern, game theory problems were static problems. Their notion that noncooperative games could be analysed in terms of strategies had many implications, including the suppression of any explicit representation of behaviour changes over time. This lack of dynamics makes it difficult or impossible to address such questions as limited knowledge, limited reasoning capacity or memory, strategic learning, or limited foresight. New structures are built with new tools; some of the most useful of these are developed within game theory as originally set out, but perhaps some, like the theory of moves, must inevitably lie outside. And has the game-theory research program helped us to understand why people do what they do, or to improve human decisions? We certainly know more about interacting decisions than

70

THE MATHEMATICALINTELLIGENCER

we did in 1944, but it is not yet clear whether, when, and how much we will benefit from this knowledge.

Calculus Lite by Frank Morgan

REFERENCES

1. A.H. Copeland, Review of Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern, Bulletin of the American Mathematical Society 51 (1945), 498-504. 2. Liping Fang, Keith W. Hipel, and D. Marc Kilgour, Interactive Decision-Making: The Graph Model for Conflict Resolution, New York, NY, USA: John Wiley and Sons (1993). 3. Joseph Greenberg, The Theory of Social Situations: An Alternative Game-Theoretic Approach, Cambridge, UK: Cambridge University Press (1990). 4. D. Marc Kilgour, Review of Theory of Moves by Steven J. Brams, Group Decision and Negotiation 4 (1995), 287-288. 5. Royal Swedish Academy of Sciences, Stockholm, Sweden, Press Release, 11 October 1994. 6. John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior (First Edition), Princeton, NJ, USA: Princeton University Press (1944). Department of Mathematics Wilfrid Laurier University Waterloo, Ontario N2L 3C5 Canada mkilgour@machl .wlu.ca

Mathematics Not A Tour of the Calculus

WELLESLEY, MS: A. K. PETERS, 1996. XlV+281 PP. US $29.95, ISBN 1-56881-037-7 REVIEWED BY JET WIMP

ow do we know that someone knows something? is an old philosophical dilemma. If I study the pronunciation of ancient Norse and, by dint of rote memorization, manage to recite some pages of The Greater Eddas, no one would grant that I understand the text. However, to up the stakes, suppose I memorize every word and symbol of the book of Rapoport and Zink 1 on p-divisible groups--not at all my m~tier--so as to reproduce any portion of the book on demand; further, suppose I, having an eidetic memory, read and commit to memory every syllable of available lecture notes, conference proceedings, and published material in the discipline, so that I can answer almost any question. Can anyone determine whether I really understand p-divisible groups? The reader may recall that this very issue of knowing what it is one utters--in the guise of the Chinese room paradox---occupies a prominent place in the attacks of John Searle and others on the Strong Form of Artificial Intelligence. Ludwig Wittgenstein 2, the great philosopher and mathematician, provided, I think, the most satisfying answer to the riddle of knowing. We know something, Wittgenstein said, when we know how to use it. I have been deluged by a spate of books on mathematics which claim to demystify mathematical arcana for the common reader. Popular mathemati-

H

by David Berlinski NEW YORK: PANTHEON BOOKS, 1995. XV1+331 PP. US $27.50, ISBN 0-679-42645-0

The World According to Wavelets by Barbara Burke Hubbard WELLESLEY, MS: A. K. PETERS, 1996. XIX+265 PP. US $34.00, ISBN 1-56881-047-4

1Rapoport, M., and Zink, Th., "Period spaces for pdivisible groups," Annals of Mathematics Studies, Princeton University Press, Princeton, N.J. (1996). 2In "On Certainty," (1969). The classical philosophers, from at least the fifth century B.C., viewed knowing as a mental faculty, related to, but different from, believing or having an opinion. Wittgenstein, and other 20th century philosophers, denied that knowing is a mental state, such as doubting something, feeling pain, or liking strawberries. What, then, is it? J. L. Austin, in a famous 1946 paper, found an infamous solution to the puzzle. To say "1 know," he asserted, is merely to engage in a social act.

cal writing is not an inherently disreputable form. Neither Stephen Hawldng nor Roger Peurose should feel the least ashamed of their recent books. 3 Other books are less successful. Concerning Constance Reid's famous biographies, Gian-Carlo Rota has deemed it a disgrace that someone who knows nothing of calculus should presume to write about the life of a mathematician, namely, Hilbert. 4 At any rate, do the survivors of these popular books really learn anything--in the Wittgensteinian sense-about the subject at hand? The issue is complicated by the fact that there are manifold manifestations of the act of u s i n g - which can include everything from the ability to conjure up detailed numerical answers for highly technical problems to framing effective social policies. The implications of true versus false knowing, even when the subject is a rather abstract one, such as mathematics, are immense, and impinge on both our quality of life and our survival as a species. I'll take one example. Last year I attended a lecture by a German mathematical demographer on the propagation of AIDS. He announced, as a conclusion of a study he had conducted using sophisticated models of disease propagation based on stochastic difference equations, that were each person in the United States to limit his or her sexual contacts to seven a year or fewer, AIDS would eventually disappear from the national scene. The magic number seven was what workers in asymptotics call the critical number 5. Should, then, the Centers for Disease Control promote as public policy a fewer-than-eight-sexualcontacts-per-year-per-person guideline for personal behavior? Clearly, anyone

malting such a risky prescription should understand something of what the demographer understands in order to discern whether or not his claim to know is valid. It is a betrayal of the academic enterprise to infLx in someone the illusion of knowing without its substance. Suppose, by recounting jocose tales about the lives of mathematicians and issuing easy tests, I turn out students convinced they know calculus when they don't. What must their reactions be when they discover--as they sooner or later must--the charade? If I turn

power; flaunting it is our only go at being powerful people. I have chosen to discuss three recent books in this review; the authors have accomplished the goal of popularization with varying degrees of success. With very few books of this type am I convinced that the act of authorship is a high-minded and selfless, as opposed to a venal, enterprise, and very few of them convince me that we need more such books. However, I was pleasantly surprised by two of these offerings. Reading Berlinski's book A Tour of the Calculus, I was first angered, then revolted, then finally weafled: the three stages of grief of the hapless reviewer. Berlinski wants to make the calculus available to everyo n e - a n y o n e who wants, simply, "a little more light shed on a dark subject." This delirious tract is the result. He apparently seeks to turn John Dewey's dictum, "we learn by doing," around. Here the motto is "we learn by feeling." We live in the age of easy access, of remote control, of user-friendly kef; consequently this nearly equation-free 6 book has a very contemporary feel. I leave it to the social critics to fret whether the movement to easy assimilation will spread to other fields: feel-good neurosurgery, feel-good disassembly of nuclear weaponry. Whenever such movements threaten, you can be certain some avid publisher will be at the threshold, book advance in hand. Berlinski's greatest friend, but ultimately his worst enemy, is metaphor. The gongorisms that saturate this book actually confound what the author claims is its central mission: to teach the novice calculus. The Berlinski rhetoric ultimately becomes suffocating. 7 The metaphors explode from all directions:

Mathematicians especially seem to believe that knowledge is the one absolute manifestation of power; flaunting it is our only go at being powerful people. out graduate students who believe that convergence studies aren't necessary when in fact they are, who believe they can appraise the asymptotic stability of, say, stochastic difference equations purely on a gut-level basis, how many will suffer the consequences? A similar gravitas prevails when the teaching medium is a book. If I write a bad research monograph, it will be forgotten. If I write a bad book claiming to teach novices something of calculus, I have ventured onto a different moral ground. And the temptation of popular writing is one to which mathematicians, perhaps more than other academics, are peculiarly susceptible. We chafe at our stenotopic existences. We want to convince others of the varieties of our knowledge, of our Renaissance sensibilities. Mathematicians especially seem to believe that knowledge is the one absolute manifestation of

3penrose, Roger, The emperor's new mind: concerning computers, mind and the laws of physics, Oxford University Press (1989). Hawking, S. W., A brief history of time, Bantam Books, NY (1988). 4in: Rota, Gian-Carlo, Kac, Mark, and Schwartz, Jacob T., Discrete thoughts: essays on mathematics, science, and philosophy, Birkh&user, Boston (1986). Sin many non-linear phenomena described by difference equations that depend on a parameter, there is a critical number such that if the parameter exceeds that number, the solution of the equation is asymptotically unstable; if the parameter is less, the solution is asymptotically stable. It has recently been suggested that urban crime is a non-linear phenomenon, and the implications of this suggestion are both interesting and profound, see the New Yorker article, "The tipping point," June 3, 1996. The idea of a critical or threshold value is also relevant to linear problems, but it occurs more dramatically in non-linear problems. 61 have heard that Stephen Hawking was informed by his publisher that each equation in the book, A brief history of time, would cost him 50,000 readers. A glance at the book shows that Hawking stood his ground. 71 thought of the ongoing column in the old New Yorker magazine: Block That Metaphor.

VOLUME19, NUMBER3, 1997 71

Speaking of limits: "To catch the shimmering river in silvery symbols, the mathematician assumes thatf(a) is negative a n d f ( b ) is positive." Spealdng of speed: "The road toward a defmition of instantaneous speed, which had that evening in Hanover seemed straight, n o w looks crooked as the very jaws of Hell." Speaking of the fundamental theorem: "To this dark Boschian landscape, the fundamental theorem of the calculus brings light, the effulgence brought about by the dramatic emptying of the conceptual arena of everything but the e s s e n t i a l . . , those functions that stand poised and brooding over all." This expositional overload implies a cynical disrespect for the subject. Calculus is uncivilly difficult and certainly can't be intrinsically interesting, the author seems to be asserting. So let's gussy it up with fine writing. This approach is so inimical to what we seek to do as teachers. Calculus, beautiful and mysterious as it is, is not necessarily difficult. My freshmen students are puzzled when I tell them this. Difficult? Memorizing nonsense syllables is difficult, I tell them. Learning calculus need not be, because calculus is so anchored in our perceptions and misperceptions of the world around us. I was particularly annoyed by Berlinski's biographical snippets. About Newton: "The tension at his mouth suggests s o m e o n e prepared to withdraw quivering in irritation from his senses." About Leibniz: "His is the face of a man who would enjoy mulled wine, poached eggs on buttered toast, a warm fire as the wind rattled the windows of a country castle, a young serving girl bending low over the plates and after dinner saying softly but without real surprise: Why, Herr Leibniz, really now, bitte!" Had Berlinski really done his homework, he could have told us some interesting things about mathematicians that were really true. He might have told us, for example, that Newton's explosive temper and dark m o o d s were most likely caused by mercury poisoning, and chemical analysis of the floor-

boards of his still extant alchemical laboratory have revealed heavy concentrations of that metal. But then, perhaps such an observation lacks poetry. I was dismayed at the author's rudimentary grasp of mathematical history. It is painful to find so little learning in a book that purports to explain an intellectual discipline. About numbers, he asks, "When did this fantastic idea come about? I have no idea. It did not occur to the ancients [sic]." Well, let's see; there were the Indians, there were the Mesopotamians, there were the Syrians, there were the Egyptians.. 9 Berlinski might try visiting Chapter 1, "Origins, The Concept of Number," in Boyer's famous b o o k s, with its chart that traces the development of the idea of n u m b e r through early civilizations. Of all the passages in the book, I found the following the m o s t mortify-

ing: RoUe's sloe-eyed mistress, her black hair spread over the muslin of their single pillow, has long since fallen asleep, a childish bubble forming on her fuU red lips; and as the moments pass in the seventeenth century and again in the twentieth, one thought engenders another, the movement of thoughts expressing an inferential chain so natural as to appear unforced, as breathing itself. The curve rises and then falls. I flushed with embarrassment (as would anyone who loves mathematics) when I read this rebarbative grunge quoted (disapprovingly) in a review in The New Scientist by The Intelligencer's own Ian Stewart. Burying mathematics under language benefits neither mathematics nor language. Berlinski says he seeks to arouse in the reader the reaction, "Now I understand." But will an inexperienced reader understand anything of calculus after wading through this 330-page book? Will the reader be able to use calculus? Will he or she be able to find the longest ladder that will fit around a hallway, or understand the construc-

tion of a hydrogen bomb 9, or the overtone structure of a violin string, or the process of eutrophication of a lake? Because such things are what people who k n o w a little calculus (it doesn't take much) can understand and can do; it is the doing that takes place in my calculus recitations that causes hands to be raised impatiently, and the distinct and busy hum that pervades the classroom at such times is the very sound of learning. Regrettably, Berlinski's readers will emerge from his verbal thickets hearing nothing. The World According to Wavelets, by Barbara Burke Hubbard, is a different sort of b o o k entirely. Hubbard is not a mathematician, and she confronts head-on the difficulties of communicating mathematical truths to non-mathematicians. She introduces the description of her task with diffidence, even apologetically. I suspect it was her active intelligence as well as her artlessness that enabled her to obtain the unstinting support and advice of leaders in the field: Yves Meyer, Ingrid Daubechies, St@hane Mallat. Hubbard is a fme writer: her prose is well-honed and precise. She has the best discussion I have ever seen of the perils of popularizing a private idiom, and her insights partially explain w h y the previous b o o k is such a mess. "Mathematicians claim that math is not a spectator sport. You cannot understand math, or enjoy it, without doing it," she says. Wittgenstein again. She recalls an enervating session she had with Bob Strichartz, w h o was trying to explain function spaces. "This is getting vaguer and vaguer to me," he groaned. "When I stick to the truth, I don't communicate, and w h e n I communicate, I stray from the truth." I recalled m y experience of trying to explain to a chemist, totally without effect, Euclid's p r o o f of the infmitude of prime numbers. There follows a courageous passage: Talking without being understood is pointless, lying is painful, so

8Boyer, Carl C., and Merzbach, Uta C., A History of Mathematics, Wiley, New York (1991). 9A chilling application of an innocent conic section: The lithium deuteride in the bomb is detonated by the pressure of light (gamma radiation) reflected from a parabolic mirror.

72

THE MATHEMATICALINTELLIGENCER

m o s t often m a t h e m a t i c i a n s abandon the attempt, i f they have the courage to try i n the f i r s t place. It would be a s h a m e to leave it at that. There exists, certainly, a l i m i t to w h a t someone w i t h o u t the right background can understand o f mathematics, but I a m convinced that we are f a r f r o m having reached that limit.

Explaining wavelets without mathematics! What an intimidating venture! But what makes Hubbard's book more successful than most of the genre is that she is not trying to teach the reader to know, i.e., use, mathematics. She is interested in enabling the reader to understand h o w wavelets have been used by others.

marily to John Hubbard, the mathematician spouse of the author. Further mathematical observations, or proofs of essential analytical results are confmed to appendices. Shaded rectangles in the first haft of the text labeled, quaintly, Beyond Plain English, warn that mathematical tools are required for a full understanding of the present material and refer to the second half of

Fortunately, Fourier lived to find a sinecure in the Bureau of Statistics in Paris, under Louis XVIII. Hubbard chronicles the development of the representation of arbitrary function by means of Fourier series. Since she is not able to assume the reader even knows what a sine or cosine is, her explanations are a little nubilous, but she is able to nicely characterize a Fourier series, and later, the Fourier transform itself, by analogies from everyday life. She talks about the fast Fourier transform and is careful to give a correct attribution of the F F r - Gauss deserves the credit, not Cooley and Tukey, as is c o m m o n l y believed. I wish all popularizers of mathematics were as diligent as this writer in limning the origins of mathematical ideas. The section on page 23, Searching for Lost Time: Windowed Fourier Analysis is an exceptional piece of informal writing about mathematics. I wish I had space to quote it all. Fourier analysis, she points out, forces us to choose between time in the original domain and frequency in the image domain. However, D. Gabor discerned in 1946 that certain everyday experiences, auditory sensations in particular, demanded a description in terms of both time and frequency. Gabor chose a technique of windowed Fourier analysis to do so, but the process had severe drawbacks:

"in c o m m o n language, as fully, clearly, and definitely as in mathematical formulae . . . "

I have w r i t t e n this book on two levels. The m a i n article c o n t a i n s no formulas . . . showing a formula tO someone who can't r e m e m b e r w h e t h e r s m e a n s s u m or integral is like a s k i n g s o m e o n e who doesn't k n o w w h i c h note on the s t a f f is B-fiat a n d w h i c h is D to a n a l y z e the score o f a f o u r - v o i c e f u g u e by Bach.

After all, she reminds us, even such a genius as Michael Faraday wrote to James Clerk Maxwell plaintively asking whether it might not be possible for the latter to communicate the truths of electromagnetism "in c o m m o n language, as fully, clearly, and definitely as in mathematical f o r m u l a e . . . " Hubbard, one way or another, provides the necessary mathematical background. She has taken a novel approach. The b o o k is divided into two parts; the first part contains an exposition, virtually without the use of detailed mathematical concepts or symbology, while the s e c o n d haft contains substantial mathematics which, however, doesn't require m u c h more than a few tools from introductory calculus. I assume the second part is due pri-

the book. Appendices discuss elementary trig, integrals, the Fourier transform, p r o o f of the sampling theorem, p r o o f of Heisenberg's uncertainty principle. Hubbard has not solved all the problems inherent in popularizing mathematics, but her b o o k presents a well-conceived, carefully dispatched exposition of a subject that seems more closely dependent on symbology than other topics in mathematics a r e - - a f t e r all, what is a Fourier transform without the integral? 1~ She has, clearly, worked very hard, and I believe the result is one of the more successful and honorable popular books on a deeply technical subject. If things seem to go wrong near the end of the non-mathematical exposition, it is not really her fault; she has embarked on a fored o o m e d enterprise. Chapter 1: Fourier Analysis, a P o e m Transforms Our World u, begins with a two-page a c c o u n t of Fourier's life. Excellent biographical writing here, compelling and, above all, informed. [Fourier w a s arrested] i n J u n e 1 795, on charges o f terrorism, w h e n he w a s roused f r o m his bed and scarcely given time to get dressed. A s he and his guards left his home, the concierge told Fourier she hoped he would soon be freed . . . the guard's reply: "You can come and get h i m y o u r s e l f - - i n two pieces."

The smaller y o u r w i n d o w , the better y o u can locate sudden changes 9 . . but the blinder you become to the lower f r e q u e n c y components o f y o u r s i g n a l . . . I f you choose a bigger w i n d o w , y o u can see more o f the low frequencies, but the worse you do at "localizing i n time."

It's impossible to pinpoint the origin of wavelets, but related ideas go back at least to the 1930's. Strangely, their first application was probably due to the French geophysicist Jean

lOSome mathematical disciplines are more amenable than others to popularization. Many concepts of abstract algebra can be easily illustrated, even those of noncommutative algebra, with tilings, for example. The Schaum's outline book, General Topology, one of the very finest pieces of mathematical exposition, illustrates nearly all its ideas, even very sophisticated ones, using examples of finite point sets. Those of us trying to lure talented graduate students into analysis should pray they read Royden before they read this book. 11This is not Hubbard's self-indulgent conceit. Maxwell called Fourier's book, La Theorie Analytique de la Chaleur, "a great mathematical poem."

VOLUME19, NUMBER3, 1997 73

Morley, who in 1975 investigated their use as a tool for oil prospecting. The application to the problem of frequency-time decomposition is probably due to Yves Meyer. Meyer knew of the limitations of Fourier analysis and had also heard of the little waves, ondelettes in French. He was aware that with them it was possible to decompose signals simultaneously by time and by frequency. In 1985 Meyer began working with a physicist, Alex Grossman in Marseille, who had worked with Morley; from there on the field expanded explosively. Hubbard has intriguing observations about the sociology of wavelet research, and the same observations probably apply to any new field of knowledge whose moment time and social forces have deemed to arrive. By and large, those working with wavelets were applied scientists, but scientists from vastly disparate areas. They neither read each other's papers nor understood each other disciplines. Consequently early wavelet research was inchoate and fragmentary. Only when a common language was developed (and, I would add, when mathematicians began to get into the act) did the field burgeon into its present dimensions and began to bestow its benefits on the researchers--the greatest of which was the dispensation to attend wavelet conferences (this is a wavelet in-joke). The occasions of conferences on the subject have become arenas of cultural cross-fertilization, where researchers from various countries and various disciplines can meet convivially. The benefits to the world scientific community of this intermingling of sensibilities was as great as the benefits accruing from the research itserf. Hubbard barges ahead courageously, expounding the properties and applications of wavelets without a single equation. She compares wavelets to the Fourier transform, discusses orthogonality, wavelet transforms, multiresolution, the fast wavelet transform, Daubechies wavelets (which are not

constructed analytically, rather, they are constructed using iterative algorithms), multiwavelets, the Heisenberg uncertainty principle. Unfortunately, as the material develops, Hubbard's insistence on keeping the text untainted by mathematics begins to impose on any reader an enormous conceptual burden. The mathematician will develop a ravenous hunger for mathematics, real mathematics, like a midnight Big Mac attack in a foreign city; on the other hand, the non-mathematician will feel tossed helplessly about by a tsunami of jargon, which he or she may learn to reproduce in a measure sufficient to impress listeners at cocktail parties, but which doesn't indicate true understanding. One can only communicate so much about mathematics without using mathematics, as Hubbard herself undoubtedly would concede. I applauded her gumption, but lamented its inevitable outcome: the duel between mathematical concepts and mathematical equations is a straw man. The two are the same. As I pointed out in another review, to a great extent the formulas we use and our derivations of them are the mathematics. Hubbard's epistemological apparatus collapses somewhere around page 48, during an internecine attack on the Heisenberg uncertainty principle. There is an interesting discussion in Chapter 4, Applications. I and a lot of other people have grown weary of the wavelet salesmen who extol endlessly the advantages of using wavelets for data compression. This application, which seems to promise the most stunning benefits of wavelet technology, has been used to justify the most intransigent and misdirected license to abstraction. To rephrase Marie Antoinette, "Oh, Applications. What crimes of Abstraction have been committed in your name? "12 I was glad to find out in this chapter that Ingrid Daubechies herself is a little less sanguine about the prospect of using wavelets to create a savory soup out of stones:

I f you j u s t buy a commercially available image compressor you can get a factor of lO to 12 [in data compression], so we're doing better than that. However, people in research groups who fine-tune the Fourier transform techniques in the commercial image compressors claim they can also do something on the order of 35. So it's not really that we can beat the existing techniques. I do not think that image compression--for instance, television image compression--is really the place where wavelets will have the greatest impact. Frank Morgan's Calculus Lite, is the most interesting book of the three. Last week a colleague, after teaching a debilitating section of trailer calculus, staggered into my office to announce, "Goddamit! We can only make this stuff so simple!" And how simple is that? I wondered. Frank Morgan has the answer. This is bare bones, no frills, bargain basement calculus. The book, only 250 pages, furnishes a dramatic contrast to those biceps-building tomes we have all servilely lugged up the steps to our classrooms. "This lean text," the author announces, "covers single-variable calculus in 250 pages by 1) getting right to the point, and stopping there; 2) introducing some standard preliminary topics, such as trigonometry and limits, by using them in the calculus." The first chapter treats the derivative: instantaneous velocity; chain, product, quotient rules; sines, cosines and their derivatives; maxima and minima and applications; exponentials and logarithms; curve-sketching; anti-differentiation; differentiability and continuity. Chapter II is the integral: area and the Riemann integral; fundamental theorem of calculus; substitution; trig functions and their inverses; volume, arc-length; partial fractions, integration by parts. Chapter III treats infmite series and Chapter IV differential equa-

12At a lecture on hypergeometric functions I attended at the University of Edinburgh, the speaker, after foisting on us a colossal equation, announced grandly, "This result has applications in particle physics." He then added, roguishly, "Of course, every result has applications in particle physics--the field is in such disarray." I seemed to be the only one in the audience who was amused.

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tions. There are many, many carefully graded exercises, and answers to oddnumbered exercises. I found the claim in statement 2) a tad dishonest. We all know that the concept of limit is the major conceptual Rubicon of the calculus. Morgan never really defmes a limit, he just sort of slips it in, hoping, I suppose, the student won't notice its presence sufficiently to b e c o m e alarmed by it. It first appears in the definition of a derivative, "seeing what limiting value you get as At goes to 0," and then, there it is, hm.

at---~0

Also unprecedented is the author's use of the ];~-notation. He discusses sums after the integral. The following sentence

The estimate given by the s u m of the areas of the rectangles is Y~f(x)Ax, which is called the R i e m a n n s u m is the ftrst deployment of the ~-conventjon. Note particularly the eccentric deficiency of subscripts. Equally curious is the fact that in the accompanying illustrations---the standard rectangularization of the graph of the function f ( x ) - - n e i t h e r f(x) n o r ha; is identified. The author's habit of using n o n c e n o t a t i o n - - n o t a t i o n used before it is defined, if it is ever defined--is sure to set the purists yelping. However, as I teach m o r e and more marginally qualified students, I'm less and less sure about w h a t t h e y actu-

ally need. Do they need to be learned and meticulous about the use o f sume and subscripts, or do they n e e d rather to be skilled manipulators o f integrals? Do the two even have anything to do with each other? I have no feeling for w h e t h e r Morgan's a p p r o a c h might work, but haven't w e tried, with s o m e of our m o r e r e f r a c t o r y clay, a l m o s t everything else? At least once a y e a r I'm c o n f r o n t e d with the task o f teaching science and near science majors, n o n e of w h o m will ever use calculus in a primary way, the rudiments o f the subject. This att e m p t to g r o u n d mathematical ideas in s e n s o r y perceptions c o m m o n to everyone--slow, fast, get close to, stay a w a y f r o m - - j u s t might w o r k for such students. And then it might not; it might result in a mess, with students leapfrogging each other to the dean's office to bay their complaints. I'm thankful that conditions don't exist now that prevailed in the universities in the Middle Ages, which were in the thrall of the student guilds. If students objected to a teacher's classroom presentations, they boycotted his lectures, thereby destroying his professional career. 13 Morgan's treatment of the rudiments is, I believe, rock-bottom basic: this, to address my friend's anguish, is as simple as calculus can be made. The illustrations, if under-labelled, are at least u n e n c u m b e r e d by patches of color. (It is probably indelicate of me

to suggest that furnishing calculus texts with brilliantly colored teaching aids is like giving the blind gaily decorated canes.) Furthermore, the b o o k is impeccably t y p e s e t - - a feature that is more crucial to a text's success than most of us realize. I do have, however, a single recommendation for the author. The title, "Calculus lite," needs some work. We have a freshman calculus section at Drexel called the Gold Team. The euphemism no longer fools anyone; the team is dross, having washed out of the usual freshman sequence, and the freshmen consigned to it feel stigmatized. While not as shaming as "Calculus for the anencephalic," the title "Calculus lite" still suggests a not fully functional clientele. I would suggest something both ambiguous and exalted, "Calculus without limits," say. I have a considerable curiosity about this book, and w o n d e r about Morgan's personal success in teaching from such a regimen. Apparently, the book grew out of a National Science Foundation site for Research Experiences for Undergraduates, headquartered at Williams College. I might even like to try the book, pending the formation of a Platinum Team where I teach. Department of Mathematics and Computer Sciences Drexel University Philadelphia, PA 19104 USA

13Further, a professor was obriged, at the beginning of the term, to deposit money in escrow in a bank; any dereliction in his duties--departing from the established syllabus, for instance--was punished by fines executed against the escrowed funds. I found this an intriguing idea.

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VOLUME 19, NUMBER3, 1997

75

Symmetry and Perception:

ogos of Rotational Point-Groups

Induce the Feeling of

Motion 0 ~

n addition to being aesthetically pleasing, the symmetric design of decorations can induce the feeling of motion or the feeling of stopping motion (see the References). Polar one-dimensional space-group border decorations (frieze patterns) can direct the movement of people in underpasses or airline terminals. Two-dimensional space-

group patterns of rotational symmetry only have been suggested for decorating dance halls; those containing symmetry planes have been suggested for decorating the sites of serious meetings. Glide-reflection may induce the feeling of confusion. In this article we suggest that point-groups also have the capability of inducing a feeling of motion, and that certain symmetries in company logos may be better suited to convey the essence of company activities than others. First, let us consider a four-bladed propeller (Figure 1). It has four-fold rotational symmetry and no symmetry plane. Having rotational symmetry only corresponds to its function, as do the rotational symmetries of other rotating parts in machinery, such as propellers, turbine wheels, windmills, or children's pinwheels. Logos themselves do not rotate physically, but they may best convey the essence of the company's activities if their symmetries induce consistent feelings in observers. Thus a railway company, or travel companies in general, may be best represented by a logo with rotational symmetry only, and even more specifically, by two-fold rotational symmetry. There is always motion, and the motion is back and

Figure 1. Four-bladed propeller displayed in front of the Budapest Technical Museum. (All photographs in this article are by the authors.)

9 1997 SPRINGER-VERLAG NEW YORK, VOLUME 19, NUMBER 3, 1997

Figure 2.

Sampler of Iogos of transportation companies (all of two-fold rotational symmetry)

THEMATHEMATICALINTELLIGENCER

Bank of Austria (Vienna)

Novi Sad Bank (Yugoslavia)

Osterreichische Verkehrakreditbank (Linz, Austria)

Bank in Stockholm (Sweden)

Banca Popolare di Ancona (Rome)

Sicilcassa (Palermo, Italy)

Korea Housing Bank (Seoul)

Israeli credit card logo

Banco Mello (Portugal)

Chase Manhattan Bank

Bank in Tokyo (Japan)

Rabobank (Holland)

American Service Bank

Bank in Illinois

Frost Bank (Austin, Texas)

Figure 3. Sampler of bank Iogos

forth: the train is taking you there and bringing you back, again and again. Our sampler of examples in Figure 2 includes logos of railway companies and other transportation companies, such as subways, tourist bureaus, bus companies, and expediters. Of course, we are not suggesting that a transportation company with a logo containing mirror planes would per-

form its function any worse. We are suggesting, though, that a logo of only rotational symmetry conveys the essence of transportation companies better than a logo with mirror planes. Banks very frequently have logos of rotational symmetry only and no symmetry planes. A sampler of examples is shown in Figure 3. Here the abstraction is of even higher

VOLUME 19, NUMBER 3, 1997

57

Reynolds Aluminum Recycling (Honolulu)

Recycling (Washington, DC)

New Hampshire recycling

Bottles recycling (Italy) Figure 4. Sampler of recycling Iogos

degree, as banks and other fmancial institutions do not represent or perform physical motion. Yet turning around money is characteristic of them, and this activity may be the reason, if only subconsciously, why logos with rotational symmetry come to them so naturally. By the same token, we would suggest mirror-symmetric logos for insurance companies, health care services, retirement systems, and any other organizations where mobility is less desirable. We are not suggesting any rigorous correspondence between the symmetries of logos and the activities of the companies they represent, but there seems to be some correlation. Note also that the logos of transportation companies, displayed in Figure 2, are invariably of two-fold symmetry, yet the bank logos have no such characteristic number and show diversity in their rotational symmetries. This again seems natural, as there is a definite two-way directionality in the activities of transportation companies but a multiplicity of possibilities in directionality of bank activities. Our third and final category is recycling logos. They are, again, of only rotational symmetry, in keeping with the process of recycling--that is, turning around the wastes and producing new materials. Although three-fold rotational symmetry is the most common, there is a variety in rotational symmetries. The variety of design is less than for banks, in keeping with the international and less competitive character of recycling. REFERENCES 1. A.V. Shubnikov and V.A. Koptsik, Symmetry in Science and Art, Plenum Press, New York (1974). [Russian original: Simmetriya v nauke i iskusstve, Nauka, Moscow (1972)]. 2. I. Hargittai and M. Hargittai, Symmetry: A Unifying Concept, Shelter Publications, Bolinas, California (1994). 3. I. Hargittai and M. Hargittai, Symmetry through the Eyes of a Chemist, Second Edition, Plenum Press, New York (1995).

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THE MATHEMATICALINTELUGENCER

University of Toronto recycling

Recycle Hawaii (Honolulu)

~--"l~..liil,Bio'J[.1Mi[~lill R o b i n

Stamps of Unusual Shape II

Wilson

I

In the previous Stamp Comer, we presented a range of triangular and quadrilateral stamps. We now illustrate some other stamps whose shapes are uncomm o n - i n particular:

Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics, The Open University, Milton Keynes, MK7 6AA, England

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THE MATHEMATICALINTELLIGENCER9 1997 SPRINGER-VERLAGNEW YORK