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Letters to the Editor

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.

More Visible Sums

2un

1

·

2 + 2

·

3 +

·· ·

+ n(n + 1)

simultaneously. I recently gave the

Dr. Giorgio Goldoni: A visual proof for

following at an on-site teacher train

n squares and for the sum of the first n factorials of or der two, Math. Intell., vol. 24, no. 4 (2002) 67-69.

the sum of the first

Here I would like to propose an al ternative proof to deduce the formu las for

Sn

Figure 1. Continues on next page.

4

=

in the article by

I was much interested

THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK

=

12 + 2 2 +

· ··

+ n2, and

ing course in Aomori (in northern Japan).

1 +2+

···

+n

1

= 2 · n(n

+ 1) tn (triangular number). It is easy to t1 + see that tn + tn-l = n2. Let Un t2 + · · · + tn, i.e., 2un = 1 2 + 2 · 3 + · · + n(n + 1). Un may be called a Put

=

=

·

·

Figure 1. (Continued)

tetrahedral number. Then we see eas

a couple of opposite edges. This means

ily that

that

Sn

=

1 2 + 22 + · · · + n2 = t1 + t1 + t2 + t2 + t3 + · · + tn-2 + tn-1 2(t1 + t2 + · · · + tn) + tn-1 + tn

=

Un

n

L

k�l

k · (n + 1

- tn

= 2un -

t · n(n

+ 1)

(1)

On the other hand, an arrangement of

1

= 2 · n(n +

1

1)2 +

1

2 · n(n +

1

= 2 · n(n + 1 ) (n +

k)

= 2 · n(n +

·

=

-

3un

1 )2 - Sn

(2)

2un

=

1) 2), or

(1/3) n(n + 1)(n + 2). ·

Thanks to my colleague H. Yamai

(1) and (2) as simulta neous equations in Sn and Un, we have

for the figures. Note the related letter

finally

1999, no. 4, 106.

Considering

of I. Konstantinov in

Nauka i Zizn',

balls in a regular tetrahedron of side length

n

is decomposed into the sum

1 X n, 2 X (n- 1), . , (n- 1) X 2, n X 1,

of rectangles of sides

3 x (n- 2), .

.

when we cut it along planes parallel to

3sn

= n(n +

1) 2 1 2

1

2 · n(n +

= - · n(n +

1)

1)(2n + 1) and

Sin Hitotumatu, Prof. Emeritus, Kyoto University c/o Department of Information Sciences, Tokyo Denki University, Hatoyama Campus, Saitama, Japan 350-0394

VOLUME 25, NUMBER 3, 2003

5

FRI EDRICH L. BAUER

Why Legendre Made a Wrong Guess About 1r(x), and How Laguerre's Continued Fraction for the Logarithmic Integral Improved It

�.......

arl Friedrich Gau5 j , in 1792, when he was 15, found by numerical evi dence that 1r(x), the number of primes p such that p

oo.

Lx _!!±_ _ 1n t 2

are asymptotically equal: li(x)

�

1r(x) for x - -'>

oo.

From the

asymptotic expansion of the logarithmic-integral function li(x)

=

___3!_ 1n x

+

1! . x

(1n x)2

+

5084 7478 with a computational error, while the correct

Table 2

(Table 4)

F. Gauss conjectured in 1792 that the prime-counting

cated a decrease in A(x) with growing x. Further exten computers. More important, a method of calculating A(x)

1.020426

function 1T(x) and the logarithmic integral

Lehmer (1914) up to 107, the values given in Table 3 indi sions of the tables up to 109 took place after the advent of

=

strongly supports the conjecture, implicitly made by Cheby shev, that limx-w' A(x)

X

the world record as

of March 2001. It shows for growing x a clear decrease of

Another result of Rosser and Schoenfeld is very useful:

X -----,- :s 1T(X) :s

11'(1021) and 11'(1022) were calculated in October and De

� (ln x)3 +

Table 3

+

...

kl· ·X (1n x)k+1

+ 0

(

X (1n x)k+2

)

5 . 105

38102.89

41 538

12.037171

13.1 22363

1.085 1 93

1 .0901 5

1 . 1 06

72382.41

78 498

1 2.739178

1 3.81 55 1 1

1.076332

1 .08449

xh(x)

148 933

1 3.428857

1 4.508658

1 .079800

1. 08041

1T(x)

lnx

137848.73

5 . 1 06

x/ln x

2. 1 06

324150.19

3485 1 3

1 4.346667

15.424948

1 .078281

3 . 106

201151.62

216 816

13.836617

14.914123

1.077506

1.07787

1 . 107

620420.69

664579

15.047120

16.118096

1.070976

8

THE MATHEMATICAL INTELLIGENCER

X

A(x)

comes li(x)

xlln x for x �

imply the fundamental �

1r(x) (and also for x�

oo,

by Richard Crandall and Carl Pomerance, Springer, New

and the conjecture would

oo,

prime number theorem

York,2001,and the Internet home page of Chris K. Caldwell,

for x �

much better approximation to the prime-counting function

X

ln x

�

--

1r(x)

=

_x_, In x- 1

([email protected])) fail to mention that x/(ln x-1) is a

oo

1r(x) than xlln x,and a much simpler one-suitable for pocket

calculators -compared to li(x) if the higher accuracy li(x) of

the first term of the

fers is not needed. In fact,the accuracy of xl(ln x-1) is im

Laguerre continued fraction; see below).

proved for growing values of x; 1022/(ln 1022-1) has a rela

Chebyshev, around 1850,came very close to a proof of

tive error of about (ln 1022r2

the prime number theorem,and Georg Friedrich Bernhard

Riemann (1826-1866), in 1859 studying the ?-function, made important further contributions, but it took almost

50 more years, until 1896 when Charles-Jean de la Vallee

li(x)

+ O(x ·e:xp(- AVlllX ))

term was improved to O(x a =

=

1 5• Meanwhile,the error

1n x -2

=

li(x)

+ O(x112

•

ln x). In 1976,

Schoenfeld showed that under the Riemann hypothesis 1 -87T

=

·x112 ln x •

for x

2:

·

n-1

-

which is the contraction of the continued fraction of

showed that the Riemann hypothesis is equivalent to the far

:-s

4 J 1 II��

�

· · ·-l

t' f3 -i by Korobov and Vinogradov in 1958. But in

l1r(x) -li(x)l

=

·e:xp(-A(ln x)"'(ln ln x)f3)) with

1901,the Danish mathematician Helge von Koch (1870-1924) tighter error estimate 1r(x)

further improvement possible.

xl(ln x- 1) is the first term of the continued fraction that

li(x)

for some positive constant A; a suitable 1 value was determined in 1963 by Arnold Walfisz to be A

·10-4. Table 5 shows that

1886):

independently gave a rigorous proof,together with a rather =

However, there is a

=

was given in 1885 by Edmond Nicolas Laguerre (1834-

Poussin (1866-1962) and Jacques Hadamard (1865-1963)

weak error estimation 1r(x)

3.9

the values for 104 and above are lower bounds.

Nielsen (1906) for the logarithmic-integral function li(x)

=

2657.

:xl_ �-lln1xl_ �-l1n2 xl_ � -l:xl

ll

-

Many eminent mathematicians,among them Edmund Lan

·

·.

Table 5 shows that

dau (1877-1938), Atle Selberg (*1917), and Paul Erdos

(1913-1996), dealt with the prime number theorem,finally

X

reducing the proof to an elementary level free of function

ln x-1--- lnx-3

1

theory, but much more complicated.

-�

-

n ll ln x

'

the second term of the Laguerre continued fraction,has for

Under the influence of the glory of Analytic Number The

x

ory,there was not too much interest left in simple numerical

=

1022 a relative error of about (1n 1022)-3

201467286691248261498 still gives much better accuracy,

with

k

=

X

1 . 1 08

x/ln x 542868 1 .02

in many practical =

the relative error being about 10-11. But computation of

1. But most books dealing with the subject

Table 4

small

cases. Computation is simple. (No question: li(1022)

�)),is obviously of the same order as

recent literature,an exception is the book

sufficiently

·10-6,

the asymptotic expansion of the logarithmic-integral function

·(1/(1-

be

7.7

which

as (xlln x)

should

=

questions. In fact,the approximation xl(ln x-1),rewritten

(in the Prime Numbers

li(1022) is much more cumbersome.) Table 5 shows fur-

1T(x)

5 761 455

x/'lT(x)

In x

1 7.356727

A (x)

1 8.420681

1 .063954

1 . 1 09

48254942.43

50 847 534

1 9.666637

20.723266

1 .056629

1 . 1 010

43429448 1 .90

455 052 51 1

2 1 .975486

23.025851

1 .050365

1 . 1 Q11

3948131653.67

4 1 1 8 054 8 1 3

24.28331 0

25.328436

1 .045126

1 . 1 Q12

361 91 206825.27

37 607 912 01 8

26.5901 49

27.631021

1 .040872

1 . 1 013

334072678387.1 2

346 065 536 839

28.896261

29.933606

1 .037345

. 1 014

3 1 02 1 034421 66.08

3 204 941 750 802

31.201815

32.236191

1 .034376

1 . 1 015

28952965460216.79

29 844 570 422 669

33.506932

34.538776

1 .031 844

1 . 1 016

271 43405 1 1 89532.39

279 238 341 033 925

35.811701

36.84 1 36 1

1 .029660

1 . 1017

2554673422960304.87

2 623 557 157 654 233

38.1 1 61 89

39.143947

1.027758

1

1 . 1018

241 2747121 6847323.76

24 739 954 287 740 860

40.420447

41.446532

1.026085

1 . 1 019

228576043 1 06974646.1 3

234 057 667 276 344 607

42.72451 4

43.7491 1 7

1.024603

1 . 1 020

2 1 7 1 47240951 62591 38.26

2 220 8 1 9 602 560 9 1 8 840

45.028421

46.051 702

1.023281

1 . 1 021

20680689614440563221 .48

21 1 27 269 486 01 8 731 928

47.332193

48.354287

1.022094

1 . 1 022

197406582683296285295.97

201 467 286 689 3 1 5 906 290

49.635850

50.656872

1.021022


9

Table

IS. Continued fraction approximations to the prime-counting function n(x) x

X

__ _ _

1

In x-

'll{x)

ln x -

1

X

-

-

1 lnx-3

-0.35

3.68

1 01

- 'll{x)

li(x) -

'll{x)

'll{x) 2

4 25

102

2.74

8. 53

5

1 03

1 .27

8.93

10

1 68 1 229

1 04

- 1 1 .02

1 3.34

17

1 05

-79.90

27.59

38

9 592 78 498

106

-467.55

99.50

1 30

107

-3120.03

232.18

339

664 579

108

-211 5 1 . 1 9

296.76

754

5 761 455

1 09

-145991 .55

-531 .83

1 701

50 847 534

1 010

-1 040539.71

-8896.92

3 1 04

455 052 511

1011

-763851 2.27

-57738.46

1 1 588

4 1 1 8 054 813

1012

-5771 8368.01

-385385.30

38263

37 607 9 1 2 018

1013

-44667661 8.38

-2599887.97

1 08971

346 065 536 839

3 1 4890

3 204 941 750 802

1014

-35271 1 5021 .36

-1 7666487.47

1 Q15

-28336573668.95

- 1 221 29383.48

1 052619

29 844 570 422 669

3214632

279 238 341 033 925

1 Q1 6

-231 0828031 05.06

-863688021 .83

1017

-1 9091 90842201 .98

-6236796467. 1 2

7956589

2 623 557 1 57 654 233

21949555

24 739 954 287 740 860

1 018

-15955501 820884.84

-45888167744.56

1019

-134 70567387421 6.1 7

-343541 1 53401.90

99877775

234 057 667 276 344 607

222744643

2 220 8 1 9 602 560 91 8 840

1020

-1 1 476285471 86596.81

-261 3363726855.1 4

1021

-9857223 1 08746375.71

-201 63286970669.57

597394254

21 1 27 269 486 0 1 8 731 928

-8529084465601 2772.24

- 1 57576742975045.01

1 932355208

201 467 286 689 3 1 5 906 290

1022

thermore, that the values forx/(lnx- 1) are upper bounds forx

=

101 up tox

=

103, lower bounds forx = 104 up to

X= 1Q22.

183-256, 281-397 (1896). Paul Erdos, On a new method in eleme ntary number theory which leads to an elementary proof of the prime number theorem, Proc. Nat/.

Likewise, the values for

(

1 /1l lnx-1- lnx-3

x

Charles-Jean de Ia Vallee-Poussin, Ann. Soc. Sci. Bruxelles 20,

)

Acad. Sci. U.S.A. 35, 374-384 (1949). A U THOR

are upper bounds forx = 102 up tox = 108, lower bounds forx

=

109 up tox = 1022. Note: the error changes sign be

tweenx = 3

· 108 andx=

4

·

108; in this range the error is

pretty small, smaller than the error in li(x). Conclusion

For many applications, xl(lnx-1), which produces with little additional computational effort a much better ap proximation than xlln x, can be recommended-it gives three correct decimals for x 2:: 1012. With some extra ef fort, xl(lnx -1-

1-) may be used-it gives five corlnx- 3

rect decimals forx 2:: 109•

Besides, it would be highly interesting to obtain rigor ous Chebyshev-type bounds C', C" for C'

:::;

I '/ lnx-1

1T(X)

X

:::; C".

FRIEDRICH L. BAUER

Nordliehe Villenstrasse 19

D-82288 Kottgeisering Germany

Friedrich L. Bauer, born in Regensburg in 1 924, studied math ematics, physics, astronomy, and logic at Munich University

REFERENCES

after the War. He became internationally known as an inno

Pafnuty Lvovich Chebyshev, Sur Ia fonction qui determine Ia totalite des

vator in computer hardware and software and numerical analy

nombres

premiers, Oeuvres I, 27-48 (1851).

Pafnuty Lvovich Chebyshev, Memoire sur les nombres premiers, Oeu vres I, 49-70 (1854).

10

THE MATHEMATICAL INTELUGENCER

sis. In particular, he played a key role in creating ALGOL 60. He set up the collection on Computer Science at the Deutsches Museum in Munich. He is now Professor Emeritus.

Jaques Hadamard, Oeuvres I, 189-210 (1896). Helge von Koch, Math. Annalen 55 (1902), 441-464. Edmond Nicolas Laguerre, Sur Ia reduction en fractions continues d'une

Bernhard Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen GroBe, Werke, 136-144. J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for

fonction que satisfait a une equation ditterentielle lineaire du premier

some functions of prime numbers, Illinois J. Math. 6 (1962), 64-94.

ordre dont les coefficients sont rationelles, J. Math. Pures et Appl.

Lowell Schoenfeld, Sharper bounds for the Chebyshev functions O(x)

(4) 1 (1885).

Edmund Landau, Vorlesungen Ober Zahlentheorie, S. Hirzel, Leipzig

and 1/J(x). II. Math. Camp. 30 (1976), 337-360. Atle Selberg, An elementary proof of the prime number theorem for arithmetic progressions, Ann. Math. (2) 50, 305-313 (1949).

1927. Adrien-Marie Legendre, Th8orie des nombres. 2nd edition, 1798, No.

James Joseph Sylvester, On arithmetical series, Collected Works Ill, 573-587 (1892).

394-401.

cKichan

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11

Opinion

Mathematics and War: An Invitation to Revisit Bernheim Booss-Bavnbek and Jens H0yrup

The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-in-chief endorses or accepts responsibility for them. An Opinion should be submitted to the editor-in chief, Chandler Davis.

P

hysicists, chemists, and biologists have a tradition of discussing meta-aspects of their subject, including the military use and misuse of the knowledge they produce. Concerns of the latter kind are rare among mathe maticians. No rule without exceptions. During the Vietnam war, a number of appeals were circulated among US mathemati cians (with reverbations in particular in France and Japan and at the ICM in Moscow in 1966 and Nice in 1970) not to engage in war-related work One such appeal was published in the Notices of the AMS in January 1968. Grothendieck's resigning from mathe matics fell in the context of this debate. [Godement 1978], not really debate but politico-economical analysis, was writ ten from a mathematician's perspec tive even though it dealt with scientific research in general. [Gross 1978] was shorter but concentrated on mathe matics. In the new context of the euro-mis sile controversy of the early 1980s, mil itary research came into the focus of debate in universities of West Ger many. [BooB & H0yrup 1984] was a product of this new discussion con centrated on mathematics; the broader discussion is reflected in [Tschimer & Gobel (eds.) 1990]. The "Forum on Mil itary Funding of Mathematics" pub lished in the Mathematical Intelli gencer (1987), no. 4, reflects problems arising for the US mathematical com munity from the "Strategic Defense Ini tiative" in the same phase. See also [Davis 1989]. Some more publications

followed, mainly with historical em phasis. As warfare is now again becoming an all-too-obvious aspect of our world and of "Western" policies, the time seems ripe for taking up the issue anew. Just after the Kosovo war, Zen tralblattjur Didaktik der Mathematik dedicated an issue to it: vol. 98, no. 3 (June 1998). On August 29-31, 2002, 42 mathematicians, historians of mathe matics, military historians and ana lysts, and philosophers gathered in the historic military port of Karlskrona, to discuss four questions:1 • To what extent has the military played an active part throughout his tory, and in particular since World War II, in shaping modem mathe matics and the careers of mathe maticians? • Are mathematical thinking, mathe matical methods, and mathemati cally supported technology2 about to change the character and perfor mance of modem warfare, and if so, how does this influence the public and the military? • What were, in times of war, the eth ical choices of outstanding individu als like the physicist Niels Bohr and the mathematician Alan Turing? To what extent can general ethical dis cussions provide guidance for work ing mathematicians? • What was the role of mathematical thinking in shaping the modem in ternational law of war and peace? Can mathematical arguments sup port actual conflict resolution?

A shorter version of the present paper has appeared as "Feature - M athematics and War" in the European Mathematical Society Newsletter 46

(December 2002), 20-22.

1We use the opportunity to thank Maurice and Charlyne de Gosson and the Blekinge Institute of Technology and its Mathematics Department for organizing this conference, supported by Stig Andur Pedersen of The Dan ish Network for History and Philosophy of Mathematics (MATH NET) and Reiner Braun of The International Net work of Engineers and Scientists for Global Responsibility (INES). From the conference, a kind of enlarged pro ceedings will appear as Bernheim BooB-Bavnbek & Jens Hoyrup (eds.), Mathematics and War. Basel & Boston:

Birkhauser, 2003. M uch of what is said in the following draws on this volume. On the theme in general, see

also [BooB & Hoyrup 1 984], [Epple & Remmert 2000], [Godement 1 994 and 2001], [Meigs 2002], and [The

AMRC Papers].

2This "broad concept" of mathematics is the one that serves in the following; it also embraces computers and computer science.

12

THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK

"Noli turbare circulos meos": When Archimedes's city was conquered in spite of his astounding mathematical engineering, he pretended that he had only done pure mathematics. This anecdote has remained popular since Roman Antiquity. The mosaic, "Death of Archimedes," is now known to be a seventeenth-century forgery. [Source: Stiidtische Galerie, Frankfurt-am-Main, Germany.]

Perspectives from Mathematics All mathe!l_laticians know the ta1es, re

in logistics, which is likely to have been

law, that origin had a1ready been left

much more important from the military

behind, and the theory was linked in

point of view. Mathematics served as a

stead to the philosophica1 discussion

have- heard about. early modem ballis

toolbox, and military officers may have

of loca1 motion-and was largely irrel

tics and fortification mathematics and

been the largest group that received genera] mathematica1 training; but the

evant for the firing of guns because of the influence of air resistance, as

the iiitportance of trigonometry for

involvement of mathematics as a gen

pointed out explicitly by Ga1ileo.

navigation. All these cases of mathe matics. being -rrnplicated. in conquest,

era] endeavour with the military in

Even to this rule there is an excep

stitution was not very intimate, and

tion. That part of the Sumero-Babylon

liaf>Ie or not, about Archimedes and his defence of Syracuse. They may a1so

·

warfare, or preparation for war have one thing in common: that which was

combined with technica1 and military

specifica1ly military applications had

ian legacy which is most spoken of in

no independent role as a shaping force

genera1 histories

for mathematics. Tartaglia's composi

namely the invention and implementa

knack was a1most exclusively already

tion of straight lines and circles in ba1-

existing mathematics. In this respect

listics was clearly inspired from gun

such examples do not differ from the

nery and the war against the Turks. When Galileo introduced the parabolic

use of simple accounting mathematics

of mathematics

tion of the place va1ue system-may be

a child of war. In c. 207 4 BCE, King

Shulgi organized a military reform in the Sumerian Empire, and the next


13

There is no known picture of Turing during the wartime period, but this photograph shows Alan Turing (at left) with his athletic club in 1946. At this point he was engaged in designing a digital computer at the National Physical Laboratory, London. This design used his wartime knowledge of electronic technology to put his 1936 theory of the universal machine into a practical form. The codebreaking machinery at Bletchley Park, although very advanced, had never actually used Turing's fundamental idea of the universal machine and the stored program, but as soon as the war ended Turing set to work to bring it to reality.

year an administrative reform (seem ingly organized under the pretext of a state of emergency but soon made per manent) enrolled the larger part of the working population in quasi-servile labour crews and made overseer scribes accountable for the perfor mance of their crews calculated in ab stract units worth %0 of a working day (12 minutes) and according to fixed norms. In the ensuing bookkeeping, all work and output therefore had to be calculated precisely and converted into these abstract units, which asked for multiplications and division en masse. Therefore, a place value system with base 60 was introduced for inter mediate calculations. 3 Its functioning presupposed the use of tables of mul-

tiplication, reciprocals, and technical constants, widely taught in school; though the basic idea had been "in the air" for some centuries, implementa tion awaited decisions made at the level of the state and firmly enforced. Then, as in many later situations, only war provided the opportunity for such social willpower. Apart from that, the conclusion stands that the military had little influ ence on mathematics. As already men tioned, the employment of forti:fication mathematicians and the teaching of naval and artillery officers did serve mathematics by providing job oppor tunities and a market for mathematics text books (copiously decorated with military symbols).

All this changed in the twentieth century. The new relationship can be said to have started around the :First World War, and to reach full develop ment during the Second World War. During World War I, two important new military technologies depended on mathematics in the making: sonar, and aerodynamics. They were so im pressive that Emile Picard, in spite of his own patriotism (which non-French cannot help seeing as pure chauvin ism), feared that young mathemati cians might opt in future for applied mathematics only [Proc. . . . 1920: xxviii]. In general, however, the imme diate role of the pure sciences, mathe matical and otherwise, was that of providing manpower that could be con-

3Because it was a floating-point system with no indication of absolute place, it could be used only for intermediate calculations-just like the slide rule of engineers in quite recent times. Since intermediate calculations have not survived, the exact dating of the implementation can only be inferred by indirect arguments. See, e.g., [Hoyrup 2002: 314].

14


When listening to a music CD, we enjoy and do not think of the military origin of the coding involved. Similarly, mathematicians going to the wonderful Mathematical Research Institute Obetwolfach enjoy the ambience and have no reason to worry about the fact that the institute originated as a m ilitary research institution in 1944-apparently well planned for the purpose though too late to have effect.

verted into first-class creative engi

Aerodynamics of course survived, but

without counting costs and benefits,

neers-not restricted to applying a set

only as one current among others.

made it possible to boost a development

of standard rules but able to implement

In World War II, the organization of

theoretical knowledge and make it

science intended to support the war ef

which

otherwise

might

have

taken

decades5-and perhaps, in cases like

function in practice; this was also the

fort was a major concern of both Axis

DDT and atomic reactors, might have

role of most of the mathematicians ac

and Allied powers; mathematical tech

been stopped at an early stage when as

tually involved in the war effort (if they

nologies

sociated problems became visible.

were not, as in France, sent into the

During WWII, mathematicians

trenches). Nobody will claim that math

computer, the Bomb) can be argued to have been war-decisive; computers, nu

large numbers were recruited, many of

ematics was in any way decisive for the

clear energy, jet propulsion-all math

them to teach sailors and air-crew

outcome of the war, nor that WWI ap

ematically constructed and computed

members basic trigonometry (etc.), but

plications of mathematics left impor

for the war-have changed our world

many also to serve as top-level creative

tant traces in the post-war world (civil

beyond recognition since 1945. Admit

engineers. Afterwards, the latter have

aviation still belonged to the future).

tedly, all of these build on pre-war theo

often tended to regard what they did

retical insights4; some of them (comput

dismissively ("I did not write one line that was publishable," as J. Barkley

Picard's wonies proved unfounded.

(radar, sonar, the decipher

in

Mainstream mathematics soon reverted

ers, jet engines) had pre-war prototypes

to the pre-War model, even more swiftly

for the machines which reached com

Rosser [1982; 509f] summarizes one re

than the precariously erected organiza

pletion during the war; but in all cases

action), perhaps because puzzle-solv

tion of planned science was dismantled.

the war, by making available huge means

ing with no further theoretical impact

4At times these had been obtained in contexts fully detached from every technical application. N. Wiener and E. Hopf had calculated the radiation equilibrium at stel

lar surfaces, but their theory could be applied to the expanding surface of the exploding bomb [Wiener 1964: 142�. A. A. Markov had investigated his eponymous processes as pure mathematics and illustrated the applicability of the concept on linguistic material [Youschkevitch 1974: 129]; in the Manhattan Project they turned out to be relevant for solving diffusion equations and for describing nuclear branching processes.

5The parallel to the invention of the place value system in Sumer is striking. In that case, parallel processes not furthered by a military government indeed took a much longer time: in China the unfolding took more than a millennium, in India it never really took place before the "Indian" system was brought back from abroad.


15

did not look important in the mathe

matical tools (ballistic computation,

matician's hindsight; this assessment

modelling, . .

Target selection and order of battle, scheme

) from the creation

of modern air raid build-up. In World War II,

notwithstanding, what was done de

of new mathematical insights (se

the destruction of a major composite target

pended critically on mathematical in

quence analysis, Monte Carlo simula

might require the deployment of a thousand

genuity and training. A striking exam

tion, . . . ). As a rule but not consis

bombers. Nowadays a similar task may be ef

ple is

0. R. Frisch and R. Peierls's

.

tently, the former type is the chore of

fected by, say, 29 heavy bombers (lower level

mathematicians who are paid by or

of the above schematized front view of an at

sential questions surrounding the con

connected to the military institution

tack-heights are indicated in kilometres).

struction of a uranium bomb in March

itself; new insights directed toward

But these have to be supported by another

1940 and their "back-of-an-envelope"

military goals are more likely to come

set of 275 fighters and ground attack fight

discovery that its critical mass was so

from mathematicians inspired by the

ers ("SEAD package") to suppress enemy air

military's problems but bound less

defence (bottom). Higher up, 24 "Intelligence

closely to the military institutions.

Surveillance-Reconnaissance" (ISR) aircraft

mathematical formulation of the es

small that military use was feasible. 6 In some cases, of course, the solv ing of problems defined by the war did

•

Second, we should remember that

guide the action of the lower levels; and on

have important theoretical impact: we

mathematical research

top, dozens of ISR spacecrafts participate.

all know about the emergence of com

more than the production of theorems

Much more than informatics is thus involved

consists in

theory,

of presumed military use. Several in

in the support of the mission itself. As shown

Monte Carlo simulation, operations re

stitutions (Suss's original plan for the

in the table at the bottom of the diagram, the

search, and statistical quality control.

German

puter

science,

information

Oberwolfach

Institute

in

average precision of bombing and firing is

This time, nothing was dismantled

1944, the American Mathematics Re

commonly characterized by the Circular Er

after the war (though many mathe

search Center in Wisconsin) exem

ror Probable (CEP), that is, the radius of a disc

maticians hurried to get out of military

age) 50% of the shots hit while 50% fall out

on. In the slightly longer run (a decade

a two-way chain, which grosso modo works as follows:7 A core group of highly

side. (Kolmogorov's approach was more so

or so), civil re-application of the new

skilled mathematicians familiar with

phisticated). The table shows how CEP has

mathematical war techniques trans

the direct problems of the military em

decreased dramatically in aerial bombing

formed them and accelerated their de

ployer (efficiency of bombing, con

over the last 60 years and how the efficiency

velopment: only the war effort had al

trolled

bacteriological

of a bomber increased correspondingly. The

lowed the creation of the first costly

agents, better radar detection and

table gives the calculated number of bombs

computers, but only commercial use

avoidance of enemy detection,

research)-the Cold War was already

allowed mass production, open com

plify an efficient model,

spread

of

or

around the goal point within which (on aver

required for "destroying" (i.e., hitting once) a

whatever) find out which of these can

20m

x

30m object. So, the dramatic decrease

ef

be approached mathematically, un

of CEP has to be paid for by dramatic in

forts, and reduction of costs. (Actually,

dertake an initial translation, and di

crease of support craft. However, air raids

stored-program

rect the translated problems to other

are still the cheapest way of waging punitive

ENIAC reached the working stage only

experienced mathematicians who are

war (forbidden by international law, but prac

after the war, though at first in military

well informed and centrally located

tised), inflicting huge economic losses on the

contexts). We may add that getting rid

within the mathematical milieu; these

enemy at extremely low operational costs.

of insistence on immediate applicabil

parcel out the questions into problems

petition,

intensive

development

computers

like

the

ity ("better a fairly satisfactory answer

which colleagues may take up as

now than the really good answer two

mathematically interesting,

years after defeat") gave space for

even without knowing that they enter

was implemented too late to become

fruitful interaction between theoretical

into a network of military relevance;

efficient; we know that it has func

understanding and applications in (for

once such questions have been an

tioned in the US. We know less about

instance) computer science.

swered, the same chain functions

the organization of military mathemat

backwards, reassembling the answers

ical research in the late Soviet Union,

and channelling the global solution to

but it appears that there, as in produc

In discussing mathematical research for

military

purposes,

both

perhaps

during

the employer. (Evidently, mathemat

tion and research in general, the civil

World War II and in recent decades, we

ics of civilian relevance can be created

should differentiate several situations

this way too, if the funding is there.)

and

and problems. •

First, we must distinguish the appli

military

domains

were

more

sharply separated than in the West.

This is only one among several mod

We may sum up in some general ob

some

els. We know that it was planned to

servations on "the perspective from

times repetitive) of existing mathe-

function in World War II Germany but

mathematics":

cation (sometimes creative,

6See [Gowing 1964; 40-43, 389-393] and [Dalitz & Peierls 1997: 277-282]. This latter volume presents Peierls as a physicist, but his actual chair was in "applied math ematics"; even a "broad concept" of mathematics does not free us from delimitation problems. 7Concerning the Oberwolfach Institute, this structure follows (conventional whitewashing notwithstanding) from analysis of the material presented by H. Gericke [1984], ct. [H0yrup 1986]; on the same institution, see also [Remmert 1999]. For the Wisconsin Institute, see [The AMRC Papers].

16


ISR spacecraft

3 6000

-

(24)

GPS:

Key to the scheme of modern air raid build-up- Precision Bombing

COM: 10+

Acronyms AWACS

Airborne warning and control system for air

IR-NRO: 3

1000

and combat control B-

Long-range bomber with weapon payload

850

of more than 1 0 tons

MET: 10

".,-

COM

Military communication I signals intelli-

LM-NRO: 600

2

gence spacecraft E-8

Joint surveillance and targeting attack radar system JSTARS

EA-68

"Prowler" carrierborne radar jammer

EC-130

"Compass Call" communication jammer

F-

100

Fighter and fighter ground attack aircraft

ISR aircraft

GPS

U2: 20

Global positioning system navigation

5

satellite IR-NRO

AWACS: 12

Infra-red (US} National Reconnaissance Office space-

10

craft

RC-135: 5 E-8:

2

ISR

Intelligence, surveillance, reconnaissance package

10

LM-NRO

SEAD package

F-15: 51

National Reconnais-

EC-130: 6

sance Office space-

EA-68: 37 F-16: 157 F-117:

5

Imaging radar (US}

craft MET

Weather satellite

RC-135

"Rivet Joint" signals intelligence gathering

24

aircraft SEAD

Suppressing enemy air defence package

U-2

- •

u

ll

..

�I

Mathematical war research has re sulted in certain fundamental theo retical innovations. It is striking, how ever, that all of these appear to depend on an exceptional mathe-

Circular Error (CEP) Table War

WWII

Korea Vietnam

� •

w

Gulf Kosovo

•

matician. The names of Turing, von Neumann, Shannon, Wald, and Pon tryagin may suffice to make the point. However, the utility of mathematics for the treatment of military prob-

Optical spy plane

CEP[ m ]

1100 330 130 70 13

#bombs

9140 823 128 38 2

lems does not depend critically on the presence of an exceptional math ematician. Mathematicians in large numbers have proved themselves unexpectedly able to function as ere-


17

•

•

ative mathematical engineers, in the

bombs provided with guidance sys

of fragmentation bombs on human

sense explained above.

tems); delivery systems (including for

bodies was to be predicted but hu

This ability has depended in large part on their capacity to become fa

instance aeroplanes provided with

manitarian concerns prohibited test

electronic countermeasure circuitry);

ing on pigs, mathematical simulation

miliar with methods and approaches

the reconnaissance, control, and com

of various mathematical disciplines

munication interface ("to ensure that

and to synthesize these. The survival

the right forces are at the right spot at

of the unity of mathematics is thus

the right moment, and with the right

made more acceptable to the public by the presentation of warfare as

information

enemy"

precise and hence "more rational

mathematical journals then in tech

Svend Bergstein); and, across all of

and clean." Although that aspect of the matter is not much discussed in

about

the

nical application.

these, high-speed cryptography. The

It should not be forgotten that the

improvement

data-transmission

the public sphere, this increased

traditional application of the toolbox

technologies is of general importance

precision of weapons (which is real)

of already existing mathematics goes

strongly made by Colonel Svend Berg lated, no more today than in the times

•

of the Prussian military thinker Carl

preached German invincibility by presenting the Wehrmacht as "Fast as German greyhounds, tough as

even more sophisticated mathematics

German lederhosen, hard as Krupp

than the transmission. 9 Similarly, the strategic planning of

steel," mathematics presents mod em warfare as "fast by avionics, pre cise by GPS, safe by optimized op erations planning."

depends on mathematical calcula tion; even the dismantling of weapons

many unpredictable external factors involved, but also we see those aspects of human behaviour which are most atavistic and contrary to reason-in Bergstein's view due especially to the prevalence of stress and sleep depri vation during combat. Nevertheless, mathematics has be come an integral and even essential part of modem warfare. (This does not major expense of the military appara

presupposition for their transmission

the possible use of weapons systems

von Clausewitz: not only are there too

mean that mathematics has become a

tion of mathematics. Whereas Hitler

nowadays often asks for the use of

the point was

stein that actual war cannot be calcu

depends essentially on the applica

creation of data is not only an evident

but is, in itself, something which

Military Perspectives

At the conference,

of

for many of these questions, but the

cent mathematical research.

•

ical representations of the task to be performed may serve to make the

negotiations was analyzed mathe

soldier see it as mere manipulation

matically. Fortunately, nobody im

of symbols and thus to eliminate the

plemented the strategy suggested by

need for appeals to atavistic in

the naive versions of such planning

stincts-say, seeing a village to be

games-to make a nuclear first strike

bombed as triangles in a computer

and promise help to the SOo/o-annihi

game may facilitate the killing. (Evi

lated enemy if no counterattack were made. l0

metres already has much the same

Perhaps unexpected by civilians

effect.)

ics performed by mathematically

We list various aspects of this role

trained independent personnel and not by the active warriors is manda

of mathematics as discussed at the conference and elsewhere. 8

tory if strategic gains and losses are to be assessed realistically; leading

Mathematics serves in managing the

Similarly, certain uses of mathemat

lizing disequilibrium in the SALT

alysts, simple accounting mathemat

use costly resources more efficiently.)

•

systems without the risk of destabi

but emphasized by some military an

tus-mathematics is a cheap way to

•

Ideologically, the waging of war is

demonstrated ad oculos, if not in the

on, now at the level created by re

•

was employed. •

dently, being at a height of 5 kilo

Utility is one thing, backfiring an

other. First, seeing war as "more ratio nal and clean" may affect (and often ap pears to affect) not only the public but also the political planners. The planners may be unmoved by the devastating ef fect on the victims; still, they don't want

institution. Purchases of weapons sys

officers, like all of us, are easy vic

tems are planned, war-games and lo

tims of self-deceiving optimism and

to be misled into recklessly engaging

gistics are calculated.

pessimism

their armed forces in operations and wars that are less easily won than pre

Weapons and weapons systems are optimized and their efficiency during

action is enhanced. This regards munitions (including missiles and

•

according

to

circum

stances. At the opposite end of the scale, mathematics may also be an indis pensable tool. Thus, when the effect

dicted by the machine-rational percep tion of the character of war. Less dangerous for planners but just

8Evidently it is difficult to find any technology created during the last decades which is not somehow driven by mathematics. The list discusses such facets of the mat ter as go beyond what holds for any practice that involves computers or microelectronics.

91nterestingly, the analysis of damage to the intestine of a wounded soldier by magnetic resonance imaging (MRI) and the localization of enemy ground forces by syn thetic aperture radar (SAR) build on the same mathematics- both, indeed, by cleverly arranged rapid repetition squeeze out of a "short antenna" as much information as could be gained from an extended antenna without advanced mathematics [Schempp 1998: 44 and passim]. 1 0This does not disprove the utility of game-theoretical modelling, only the belief that human behaviour is always adequately described by the "rational economic man." Actually, sociobiological models of the same mathematical type show that the survival of the species is better guaranteed if egoistic suboptimizing is punished. The fear that the enemy might not accept the kind offer but take "irrational" revenge was exactly what made nuclear deterrence work, thus saving our species during the Cold War.

18


Demolished Varadin Danube River Bridge in Novi Sad, Yugoslavia. The destruction of the bridges across the important international water way Danube, some of them in the North of Yugoslavia and hence far away from Kosovo where the Yugoslav military operational capability was to be hindered, was unlawful by The Geneva Protocol I. The justification given is that this kind of warfare is, after all, cost-efficient in human lives, even for the target population-as illustrated by the undamaged blocks of flats standing near to the crushed bridge (carpet bombing in Europe being unacceptable). The mysterious health problems of NATO soldiers who participated in the Gulf and Kosovo wars and the dramatic increase in cancer rates in Iraq tell us that other damage that does not show up on photographs may turn up in medical statistics. An even greater cost of this high-precision warfare supported by mathematics is the very introduction of the concept of justified punitive wars without bloodshed. This creates invincibility illusions and talks people into accepting war. [Source: NATO Crimes in Yugoslavia. Ministry of Foreign Affairs, Belgrade, 1 999.]

as thr�atening to victims is the relative inexpensiveness of present-day mathe matically supported asymmetric war fare for the attackers-if the subjuga tion of Serbia in the Kosovo war cost only $7 billion, that is, $700 per Yu goslav capita, the temptation is great to solve all similar problems in a similar way. The moment such a war turns out to . involve the use of ground forces, costs of course explode, and we are brought back to the situation discussed in the previous paragraph. Another feature of the mathemati zation of warfare is the transformation of the "Krupp model" into an "infinite Krupp model"; this feature regards sym metric situations rather than the field

of easy asymmetric wars and updated "gunboat diplomacy." War and pre pared war is always between two (pos sibly more) parts-Clausewitz would speak of a Zweikampj, a duel, which has now become a "duel of systems." In the nineteenth century, Friedrich Krupp would first develop nickel-steel armour that could resist existing shells, then chrome-steel shells that could pierce this armour, then high-carbon armour plate that resisted these, then cap-shot shells that could break this plate-and that was the end of it. In the duel between ground-to-air missiles and aeroplanes, no physical limit pre vents an ever-ongoing sophistication and an ensuing arms race iterating ad

infinitum. Capshot shells were and re main extremely expensive; so are stealth bombers and fighters-but such measures as depend solely on so phistication of software and hardware have neither budgetary nor intellectual definitive bounds. Processes depending on physics and chemistry may have natural boundaries. Those depending solely on mathematics seem to have none. The virtual absence of limits en hances the stress on both sides, and thus the speed and instability of such a race. Ethics

Mathematics, according to a familiar view, is a neutral tool. As once formu-


19

lated by Jerzy Neyman, "I prove theo rems, they are published, and after that I don't know what happens to them." This is certainly an important fea ture of the mathematical endeavour, and not only for theorems and theorem production. Also the teaching of math ematics, the production of high-level general mathematical competence in the population, is a precondition not only for the waging of modem war but also for the functioning of our whole technological society (quite apart from the cultural value we suppose it to pos sess). But the title "mathematics and war"

implies ethical dilemmas. We start by looking at the actual ethical choices of some well-known figures.

Kolmogorov's Theory of Firing, four front pages 1 942, 1 945, 1 948, 2002. The work by Kolmogorov and collaborators on Firing The ory was interesting enough to be translated

•

•

Laurent Schwartz used his high aca demic prestige to make his resis tance to the French and American wars in Algeria and Vietnam more ef fective; he saw no connection be tween his work in mathematics and his political commitment. Niels Bohr, on becoming aware of the German nuclear bomb project, sup ported the competing Anglo-Ameri can project; later discovering the dangers that were to arise from

by the RAND Corporation in 1 948 and is still regarded as fundamental in quite recent US military education. It is one of the apparent paradoxes of the relation between mathe matics and warfare that an earlier paper by Kolmogorov on the same topic was published in 1 942 for everybody to read (including the enemy). Could it be that this early paper (in contrast to the 1 945 book) was too mathe matical and too general to inform military practice directly?

Leading Polish army officers were present at a ceremony in 2001 when a memorial plaque was unveiled at the tomb of the cryptologist Mar ian Rejewski (1 905-1980). The photo shows some generals, together with Rajewski's daughter and the President of the Polish Mathematical Society. Not many mathematicians have experienced similar honours in life or posthumously. As a mathematics student, Rejewski had been recruited in 1 929 by the Cipher Bureau of the General Staff of the Polish Army. Rejewski then created a mathematical method for breaking the German Enigma code of that time. Long before their competitors, the Polish Cipher Bureau officers realized the potential of mathemat ics in cryptological research. [Source: Polish Mathematical Society, c/o Prof. K. Ciesielski, Jagiellonian University, Krakow.]

20


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II�)I>.TE/ThCf80 AI t II � :1' a! .a> • a � • men); or one may, like Bohr, use one's 0 "t'. • • * t ., � � particular standing and insight to mod IC 1" "1: � • � ftt t .. ' � 1"o • .&. tt· • 4IIJ . t erate, warn, or point to alternatives. . IC � • ' * .. = It lv J:. . The situation of the sceptic is less ft 111 ..... 111.. -.,.- " . ' D -e clear-cut. Very few of us are in a situ 0 .. ' * . ' .... . fi . Ill ation (the situation, say, of von Neu l, "' · t · • �· 0 • a ' mann and Pontryagin) where nobody "' 0 • • \.... a ' :JL 1: . i'f i -; "1: l: ., • • else could do what we are doing; these 1\ fl � - N � .. • few may influence matters directly by t � J,: 11• JJ I! 1'.> co ' 0 � 1fi *' b ft! Je deciding to cooperate or not to coop II It· 0 If 1: 11• .,, � I 1: IC erate, but they remain exceptions. l:: l:: � '· � :'if � � ; Most mathematicians, if they choose --. "' :,... It 1l: li. t.' ·"' .. 1t: 't' . ,. .. not to cooperate with the military in 0 t[) "' 81 • ' b 111 mathematics research and teaching, :X: · -r It"' � * 0 .a> 1C . '1: " "ft. • will have little effect, and little of what ' 11.. .. . . • • � . L "' most mathematicians do in research or "l' "" ;. "-' I> JM � teaching is directed toward a specific application. Deciding to abstain from working with a particular discipline be cause it seems "corrupt" is most futile. Giving up mathematics is giving up not Translation from the Japanese of the Ogura text: only military applications but anything mathematics can be used for-and Today is a critical time to fight the Greater East Asia War with all whatever cultural value we may as the might of our whole nation. In particular, contemporary war be cribe to mathematics. ing a war of science, the responsibility of those w ho study science However, the practice of the math and technology is very heavy. What should mathematical research and ematician consists in more than the ab education be, to meet Japan's grave needs today? I have tried in this stract production and dissemination of book to discover a correct guiding principle for this serious task in theorems. Any mathematician is in a historical perspective. particular situation, and in any partic ular situation there are specific condi tions and specific room for decisions. The first page of Ogura's 1 944 book on Mathematics in Wartime One may, for instance, widen one's Kinnosuke Ogura (1 885-1 962) was an excellent Japanese mathematician, struggling for the own insight and global understanding modernization and democratization of his home country. However, during the Greater East of the role of mathematics, and try to Asia War he was trapped in the mobilization of the Tennoist aggresson against China. Above, share it with students, colleagues, and the first page of Ogura's 1 944 book on Mathematics in Wartime. As the translation below it the public-or one may choose to re shows, he is writing a bellicose appeal to mobilize mathematics for the Tennoist victory. Af main (and leave others) blissfully ig ter 1 945, lending of extant library copies was banned, and the book was silently excluded norant. One may be a teacher within a from Ogura's Collected Works in 8 volumes.

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23

matter can be found, however, in [BooB & H0yrup 1984]. What can be said in general is that the supposed neutrality of mathemat ics per se does not entail the neutral ity of these ethical choices. An Enlightenment Perspective?

The Enlightenment believed that rea son might serve general progress; Jean Jacques Rousseau and Jonathan Swift pointed out that too often reason is used in the service of purely technical rationality, and for purposes of subop timization, with morally and physically disfiguring effects. According to Daniel Defoe's Robinson Crusoe, "Reason is the Substance and Original of the Math ematicks." Where does mathematics stand with regard to disfigurement and progress today? Much of what was said above points to disfigurement. Most alarming are probably not the actual uses but the ideological veil of rationality, clean ness, and "surgical accuracy" which is derived from the mathematization of warfare. By generalization one might claim that this applies not only to the military aspects of modem technical society but to the technically rational society as a whole. However, one of the ways in which mathematics serves the military points in the opposite direction: the sober minded elimination of self-deceiving optimism and pessimism which can be provided by mathematical reasoning and calculation. Mathematics-based reason at its best should allow us also in larger scale to unlearn conventional wisdom, to undermine facile indoctri nation, to distinguish the possible from glib promises. It might help us, if not to fmd any absolutely best way-this is too much to expect from reasoned analysis-then at least to evade the worst. If reason is really "the Substance and Original of the Mathematicks," mathematics might serve to make clear to us that war is fundamentally irra tional and unreasonable not only in commonplace ideological generality but in specific detail. If mathematics is not able to do such things, then its presumed cultural value might be noth ing but a convenient excuse for ruth less technical suboptimization.

24


HVGO NIS

G R. O T I I

D E

I V R E B E L L I A

p

c

c

A

I T R.

L I B R

I s

E S,

In quibus ju s Naturre & Gentium ) item juris publici przcipua explicantur. B D I T I O tum

N O V A

A N N o TAT u

Atllllrii _

Ex poftrema ejus ante obitum cura multo nunc auetior.

Accet'ferunt &: A N N o T A T A

in Bpiftolatn Pauli

ad Philemonem.

A

:M S

T

'B ._ 1> A M t ,

Apud I 0 H A N N B M B L A E V.. M D

C X L V t.

Title page of the second edition of Grotius's De Jure Belli Ac Pacis. As indicated by the armil lary sphere, the publisher was also engaged in mathematical publishing. Historians of inter national law credit Hugo Grotius with the creation of modern international law, as in partic ular established in the Peace Treaty of Westphalia of 1 648 and the Charter of the United Nations of 1 945, and trace the origins of it back to patterns of mathematical thinking of strik ing public appeal in Grotius's time. Military analysts of our own time blame the striking pub lic appeal of mathematics-supported modern warfare for undermining international law.

Admittedly, technical rationality prevails over reason for the moment, both in the general political arena and in the uses to which mathematics is put. Mathematical theories are ethically neutral, it has been argued. Mathematics as a social undertaking is ethically am biguous: responsibility, whether they acknowledge it or not, remains with

its practitioners, disseminators, and users. REFERENCES

BooB, B ern h eim , &

Jens H0yrup, 1984. Von

Mathematik und Krieg. Ober die Bedeutung von ROstung und milita rischen Anforderun gen fO r die Entwicklung der Mathematik in Geschichte und Gegenwart. (Schriftenreihe

A U T H O R

-- , 2001. "Postface. Science, technologie, armament", pp. 377-465 in idem, Analyse mathematique, vol. II. Berlin etc. : Springer.

Gowing, Margaret, 1964. Britain and Atomic Energy 1 939- 1945. London: Macmillan & Co

Ltd. Gross, Horst -Eckart, 1978. "Das sich wandel nde Verhii.ltnis von Mathematik und Produk tion," pp. 226-269 in P. Plath & H. J. Sand kOhler (eds), Theorie und Labor. Dialektik als Programm

Koln:

der Naturwissenschaft.

Pahi-Rugensetin Verlag. H0yrup, Jens, 1986. Review of H. Gericke

BERNHELM BOOSS·BAVNBEK

JENS HSYRUP

Department of Mathematics and Physics

SectiOn for Philosophy and Science

1984. Zentralblatt fOr Mathematik 565, 7.

Roskilde University

Studies

H0yrup, Jens, 2002. Lengths, Widths, Sur

Postboks 260

Roskilde University

faces: A Portrait of Old Babylonian A lgebra

Postboks 260

and Its Kin . (Studies and Sources in the His

Denmark

4000 Roskilde

e-mail: [email protected]

Denmark

4000 Roskilde

tory of Mathematics and Physical Sciences). New York: Springer. Meigs, Montgomery C. , 2002. Slide Rules and

e-mail: [email protected] Bernheim Booss- Bavnbek has written pro lifically on mathematics, especially global

Submarines. American Scientists and Sub Trained first as a physicist, Jens H0yrup

surface Warfare in World War II. Honolulu,

analysis of partial-differential equations,

has been since 1 973 a specialist in con

Hawaii:

and applications. A recent publication i s

ceptual and cultural history of mathemat

Reprinted from the 1990 edition.

University Press of

the

Pacific.

Proc. . . . 1920: Comptes Rendus du Congres

Elliptic Boundary Problems for Dirac Op

ics, and a faculty member at Roskilde Uni

erators (Birkhauser, 1 993). He also has a

versity. He has recently published Human

International des Mathematiciens (Strasbourg,

Sciences: Reappraising the Humanities

22-30 Septembre 1920). Toulouse, 1921.

continuing professional interest in the so cial context of mathematics. Readers may recall his contentious article "Mem ories and Memorials" in The lntelligencer 1 7 (1 995), no. 2, 1 5-20.

through History and Philosophy (SUNY

Remmert, Volker, 1999. "Vom Umgang mit der

Surfaces: A Portrait of Old Babylonian Al

tut im 'Dritten Reich. ' " 1 999. Zeitschrift fUr

Press , 2000), and Lenghts - Widths

Macht. Das Freiburger Mathernatische lnsti

gebra and its IVn (Springer, 2002).

Sozialgeschichte des 20. und 2 1 . Jahrhun derts 14, 56-85.

Rosser, J. Barkley, 1 982. "Mathematics and Mathematicians in World War II." Notices of the American Mathematical Society 29: 6,

Wissenschaft und Frieden, Nr. I ). Marburg:

angewandter Mathematik': Kriegsrelevante

Bund demokratischer Wissenschaftler. Some

mathematische Forschung in Deutschland

what updated English translation as pp.

wahrend des II. Weltkrieges", vol. I, pp.

Resonance Imaging. Mathematical Founda

225-278, 343-349 in Jens H0yrup, In Mea

258-295

tions and Applications. New York, Wiley.

in

Doris

Kaufmann

(ed.) ,

509-515. Schempp, Walter Johannes, 1998. Magnetic

sure, Number, and Weight. Studies in Math

Geschichte der Kaiser-Wilhelm-Gesellschaft

ematics and Culture . New York: State Uni

im Nationalsozialismus. Bestandsaufnahme

Madison Wisconsin Collective.

versity of New York Press, 1994.

und Perspektiven der Forschung. Two vol

Wisconsin: Science for the People, 1973.

Dalitz, Richard H., & Sir Rudolf Ernst Peierls

umes, Gbttingen: Wallstein Verlag.

The AMRC Papers. By Science for the People,

Madison,

Tschirner, Martina, & Heinz-Werner Gobel

(eds), 1997. Selected Scientific Papers of Sir

Gericke, Helmuth, 1984. "Das Mathematische

Rudolf Peierls. With Commentary. Singapore

Forschungsinstitut Oberwolfach," pp. 2 3-39

and London: World Scientific Publishing, Im

in Perspectives in Mathematics. Anniversary

Phillips-Universitat Marburg. 50 Jahre nach

perial College Press.

of Oberwolfach 1984. Basel: Birkhii.user.

Beginn des II. Weltkrieges. Marburg: Eigen

Davis, Chandler, 1989. "A Hippocratic oath for

Godernent, Roger, 1978. "Aux sources du

mathematicians?", pp. 44-47 in Christine

modele scientifique americain 1-111". La Pen

Keitel (ed.), Mathematics, Education, and

see 201 (Octobre 1978), 33-69, 203 (Fevrier

Society. Science and Technology Education.

1979), 95-1 22, 204 (Avril 1979), 86-110.

Document Series No. 35, Paris: UNESCO. Epple, Moritz, & Volker Remmert, 2000. " 'Eine ungeahnte Synthese zwischen reiner und

--

, 1994. "Science et defense. Une breve

histoire du sujet 1 . " Gazette des Mathemati

ciens 61, 2-60. (Part II has not appeared).

(eds), 1 990. Wissenschaft im Krieg- Krieg in

der Wissenschaft. Ein Symposium an der

verlag AMW. Wiener, Norbert, 1964. I Am a Mathematician: The Later Life of a Prodigy. Cambridge,

Mass.: M.I.T. Press. Youschkevitch, Alexander A , 1974. "Markov," pp. 124-130 in Dictionary of Scientific Biog

raphy, vol IX. New York: Scribner.


25

M a thern ati c a l l y B e n t

Co l i n Adam s ,

E d itor

The Three Little P igs The proof is in the pudding.

Opening a copy of The Mathematical Intelligencer you may ask yourself

uneasily, "What is this anyway-a mathematical journal, or what?" Or you may ask, "Where am !?" Or even "Who am !?" This sense of disorienta tion is at its most acute when you open to Colin Adams's column. Relax. Breathe regularly. It's mathematical, it's a humor column, and it may even be harmless.

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

Colin Adams

O

nce upon a time there were three little pigs who lived with their mother, a mathematician at Nebraska. The first little pig adored number the ory. His goal in life was to prove the Swine-erton-Dyer Conjecture. The sec ond little pig was fascinated by ques tions about complexity. Her goal in life was to prove that Pig = NPig. The third little pig loved algebraic geometry. He wanted nothing more than to prove the Hog Conjecture. One day, their mother said, "I have taught you everything I know. It is now time for you to go to graduate school." So each pig packed a small bag filled with favorite math books and slung it on a stick, and all three set off to make their way in the mathematical world. The first little pig did a Ph.D. at the University of Illinois and obtained a post-doc at Columbia. In his first year there, he announced a proof of the Swine-erton-Dyer Conjecture. Hugo C. Wolfe was visiting Columbia from Berkeley at the time, and after devour ing the preprint, he rushed down the hall and banged on the little pig's door. "Little pig, little pig," he roared. "It is I, Hugo Wolfe, from Berkeley. Open the door and let me in." The little pig, who had recently grown a fashionable goatee, replied "Not by the hair on my chinny chin chin," as he cowered inside his office. For he was afraid of Hugo C. Wolfe, as was everybody else in mathematics. "Then I will huff and puff and fmd a hole in your proof. " And Wolfe huffed and he puffed and he came up with a hole in the first lemma.

"But there can't be a hole in the first lemma," squealed the little pig. "That is a folk lemma that has been around for ever." Wolfe laughed viciously. "You should know better than to rely on a folk lemma. " And the little pig's proof fell apart like a house made of straw, and the little pig's career did likewise. The second little pig did her Ph.D. at Wisconsin, and she received a post doc at Rice University. In her second year there, she announced a proof that Pig NPig. After reading the preprint posted on the Web, Hugo C. Wolfe im mediately flew to Houston, and arrived at the little pig's office door. "Little pig, little pig, it is I, Hugo C. Wolfe," cried Wolfe in his most intimi dating manner. "Open the door and let me in." "Not by the hair on my chinny chin chin," replied the second pig. Although she didn't have any hair on her chin, her brother, who now worked in kitchen supplies, had told her it was tradition to say this. "Then I will huff and puff and find a hole in your argument." And Wolfe huffed and he puffed and he found a hole in the second lemma. "That lemma can't be wrong," said the second pig through the door. "I found it on the Web. I'll fix the proof." But the hole could not be fixed, and her entire proof came tumbling down like a house made of sticks, and the second little pig's mathematical career did likewise. Discouraged, she opened a barbecue joint in South Houston, rev elling in the irony of it all. The third little pig received his Ph.D at Northwestern University, and took a post-doc at Michigan. After three years of work, the little pig announced a proof of the Hog Conjecture. Hugo C. Wolfe didn't even wait to read the preprint. He rushed to Michi gan, relishing the opportunity to de stroy the third pig's career as well. =

© 2003 SPRINGER-VERLAG NEW YORK, VOLUME 25,

NUMBER 3, 2003

27

Wolfe stood outside the door and

"Little pig, little pig, open the door

stroked his long nose, a malevolent

and let me in." "I go by Herbert T. Boar," replied the pig through his closed door.

added, "You don't know that the iso metry group will have a subgroup of or

smile creasing his face. Then he stalked

der p. " He was grasping at straws now,

off down the corridor.

but he was desperate. Everyone was

"Open the door, Boar, or I will huff

The little pig was awarded the Fields

and puff and create a counterexample

Medal, the award to be given at the In

that will destroy this entire department."

"As any undergradute mathematics

ternational Congress of Mathematics,

major can tell you," replied the pig, "the

"Listen Wolfe, I'm not scared of you.

taking place in Porkugal. Before the cer

Sylow subgroup theorems imply that

I know my proof is airtight. The lem

emony, each ofthe awardees was to give

since p divides the order of the isome try group, there must be a subgroup of

looking at him, some smiling.

mas are all from refereed journals and

a talk As the little pig fmished his

I checked each one of their proofs my

remarks,

self. So huff and puff away."

opened the floor to questions. Hugo C.

Wolfe's collar felt much too tight. He

Now Wolfe was greatly angered that

Wolfe stood up. "Little Pig," he said. "I

found he couldn't swallow. He felt like

the master of ceremonies

order p. "

the pig wasn't afraid of him. So he

have a question." A hush fell over the

he was on fire. Someone laughed. Wolfe

huffed and he puffed, but no coun

audience, for they knew what it meant

tried to disappear, to sink beneath the

terexample came to him. So he huffed

when Hugo C. Wolfe had a question.

and puffed some more. He tried to

"It appears to me that in Lemma 3.4,

show that the tangent bundle did not

you Dehn fill the Whitehead link com

lift

to

the universal cover.

He

chairs around him, but there was no es cape.

Everyone began laughing

and

pointing. Wolfe let out a strangled cry

at

plement to obtain a hyperbolic orb

and ran out of the auditorium, never to

tempted to prove that the metric was

ifold. But the result is not in fact an orb

be seen in mathematical circles again.

not in fact Riemannian, but still to no

ifold at all. It is a manifold." Wolfe

Eventually, the little pig became chair

avail. Finally, he banged on the pig's

grinned wickedly. The little pig smiled

of the department at Michigan, which

door once more.

back confidently. For Hugo C. Wolfe

was housed in a beautiful brick build

had fallen into his trap.

ing designed by the pig himself. He in

"Little pig, little pig," he roared. "I don't have a counterexample yet, but it's only a matter of time."

"Yes," said the pig, "but as any grad

vited his brother and sister to come live

uate student knows, a manifold is an

with him in Ann Arbor, where they cre

The little pig said, "I'm not afraid of

orbifold, just with trivial singular set."

you. Even the weakest link in my ar

Wolfe began to turn red. "Well, yes,

gument, that Dehn filling the cusped

3-

manifold yields a hyperbolic orbifold,

is safe from you."

ated the first dinner theater serving up hickory smoked barbecue with lec

I suppose that is true," he mumbled. He

tures on projective algebraic varieties.

began to sweat.

And the dinner choices were chicken,

It suddenly felt much

too hot in the room. "But anyway," he

beef, and soy meat substitute.

E XPAND YOUR MIND Four Colors Suffice

How the Map Problem

Was So lved

Robin Wilson

On October 23, 1 852,

Professor Augustus De Morgan launched one of

the most famous mathe matical con undrums i n

Robin W:il

h istory: What is t h e least

� as Solved

possible n u m ber of colors needed to fill in any map

n

( real or invented) so that neighboring counties are always colored d ifferently?

Providing a clear and elegant explanation of the problem and the proof, Robin Wilson tells how a seemingly i nnocuous question baffled g reat minds and stimu lated exciting mathe matics with far-flu n g applications.

Cloth $24.95 ISBN 0-691 - 1 1 533-8

Gamma

Exploring Euler's

Constant

Julian Havil With a foreword by Freeman Dyson Among the myriad of con stants that appear in mathe matics,

n,

e, and i are the

most famil iar. Following

closely behind is g, or

gamma, a constant that arises in many mathemati cal areas yet maintains a profound sense of mystery.

Gamma travels through

countries, centuries, lives and works, unfolding along the way the stories of some remarkable mathematics from some remarkable mathematicians.

Cloth $29.95 ISBN 0·691 -09983-9

Available from Princeton only in the U.S. and the Philippines

Princeton University Press

800-777-4726.

•

READ EXCERPTS AT MATH.PUPRESS.PRINCETO N . EDU

i,f,@j!i§ufihi¥119-'I,J,IIijihtiJ

Sem inar· Workshop in Mathematics, Yaounde, Cameroon, Decem ber

1 0- 1 5, 2001

T

Marjorie Sen ech a l ,

Ed itor

he Republic of Cameroon, a cen

ering was "Classical Analysis, Partial

tral African nation situated be

Differential Equations, and Applica

tween Nigeria, Chad, Central African

tions." The international character of

Republic, Republic of Congo, Gabon,

the meeting is indicated by the list of

Equatorial Guinea, and the Bight of Bi afra, achieved independence in 1960.

speakers from many countries of Eu rope and Africa, including:

Its capital, Yaounde, a city on seven

hills, is also the capital of the Center province (one of ten). The University of Yaounde, founded in 1961, was the

Prof. Fulvio Ricci, Scuola Normale

only one in Cameroon for 32 years.

Superiore di Pisa, Italy

Naturally, Yaounde has long been a point of convergence for all the stu dents from the high schools and lycees who knocked on the university's doors. In 1993, because the university could not handle the growing number of stu

throughout the world, and through all

Prof. Norbert Noutchegueme, Uni versite de Yaounde I, Cameroon Prof. David Bekolle, Universite de

dents, the public authorities created six universities in Cameroon, two of

Prof. Hamidou Toure, Universite de

them

Ouagadougou, Burkina-Faso

in

Yaounde.

University

of

Yaounde I, on the site of the former University of Yaounde, is composed of

ulty of Sciences, which includes the

of mathematical communities

Prof. Marco Peloso, Politecnico de Torino, Italy

Yaounde I, Cameroon

ters, and Social Sciences, and the Fac

This column is a forum for discussion

Prof. Aline Bonarni, Universite d'Or leans, France

two faculties: the Faculty of Arts, Let

Cyrille Nana

I

Department of Mathematics.

Gustavo

Garrig6s,

Universidad

Aut6noma de Madrid, Spain The workshop featured

a short

course in classical analysis, "Continu ity of Bergman projectors on tube do

The Department of Mathematics,

mains over cones, from the analytic and

headed by Professor David Bekolle, of

geometric points of view," with lectures

fers the basic specialties,

by David Bekolle, Aline Bonarni, Gus

algebra,

time. Our definition of "mathematical

analysis, and geometry. It is notewor

tavo Garrig6s, Fulvio Ricci, and Marco

community" is the broadest. We include

thy that the department has organized

Peloso. One is especially interested

"schools" of mathematics, circles of correspondence, mathematical societies,

at

here in the symmetric cones and most

Yaounde, including one on computer

several mathematics

workshops

particularly in Lorentz cones; the tech

tools in complex dynamical systems

niques used by the speakers were orig

student organizations, and informal

organized by CIMPA (Centre Interna

inal and their way of presenting this ab

communities of cardinality greater

tional de Mathematiques Pures et Ap

struse subject led us to understand

pliquees) in 1999, a workshop on math

what Bergman spaces are, the impor

than one. What we say about the communities is just

as

unrestricted.

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

ematics and fil�aria_irl �QOO, a.rHl most

tance of the Paley-Wiener theorem for

recently a GIRAGA (African Research

calculating the formula of the Bergman

Group in Algebra, Geometry, and Ap

kernel or to associate to the dyadic de

plications) workshop in September

2002. Here I describe the seminar

composition of the cone and extend

workshop held in Yaounde from De

the continuity of the Bergman projec tor, of which one of the applications is

cember 10 to December 15, 2001, in

the theorem of atomic decomposition

connection with the opening of the

of functions in the Bergman space of a

African Center for Research in Mathe

tube above the Lorentz cone. The iden

matics and Computer Science (CARIM),

tification that they established be

established in September 2000 by a de

tween the Lorentz cone and the cone

of Mathematics, Smith College,

cree of Professor Henri Hogbe Nlend, Minister of Cameroon Scientific Re

of positive-definite symmetric matrices in dimension 3 is also of interest.

Northampton, MA 01 063 USA

search.

Please send all submissions to the Mathematical Communities Editor, Marjorie Senechal,

Department


It is important to note that in sev

The theme of this international gath-

eral of these lectures, the speakers es-

© 2003 SPRINGER-VERLAG NEW YORK, VOLUME 25,

NUMBER 3, 2003

29

tablished connections between classi cal analysis and signal-processing and

to general relativity; Gustavo Garrig6s,

ter his first lecture, "I have found the

and Hamidou Toure, the group on nu

exposition, all the time." The zeal of the

image-processing, or between classical

and applications to signal-processing;

tions. To this effect, Bonami showed,

merical analysis of partial-differential

analysis and partial-differential equa starting with Fourier analysis, the con nection which exists between the sam

pling theorem used in the atomic de

demand sources." Fulvia Ricci said af

the group on the theory of wavelets

equations (homogenization, numerical

solutions) and applications to hydrol

ogy and petroleum research.

participants very attentive during my auditors attracted the attention of the

lecturers who were interested in the

participants. Thus Gustavo Garrig6s af

ter his first lecture declared, "I appreci

In the different working groups,

ate the hospitality of the people here.

rem, well known in signal-processing.

students, researchers, engineers, and

having to give it in English, my French

method of wavelets in image compres

ated debates and exchanges among the

composition of functions in Bergman

space and Shannon's sampling theo

Gustavo

Garrig6s

spoke

about

the

sion. Fulvio Ricci showed that Fourier

analysis permits the solution of certain

partial-differential equations; he gave some

there were presentations by doctoral

students. All the presentations gener participants. What interested us most

was the possibility for our students to

discuss mathematics with high-level re

My course went very well; I truly regret

is very bad, I can't even write in French.

If I tried to write phonetically, everyone would have been scandalized."

At the end of the work, there was a

round-table,

a

discussion

forum

be

a priori estimates for elliptic op

searchers. It was clear from these dis

tween mathematicians and engineers on

ities to the Schrodinger equation. In

different participants was complete. As

to petroleum research, to hydrology, and

erators and applied Strichartz inequal

cussions that the contact between the

addition, Norbert Noutchegueme talked

Aline Bonami said, "We are not isolated

ory of general relativity, and Hamidou

people pose many questions, and we are

about Einstein's equations and the the

here, not at all; the contact is good, the

Toure discussed applications to hy

getting to know them. When we were in

To give a larger number of par

the thesis students. Here it is different,

drology and petroleum research. ticipants an

opportunity to speak,

Poland, we could not even meet with

the people greet you and get to know

the organizers set up four parallel

you. " Also, the dedication with which

the classical analysis group; Norbert

was noted. As Gustavo Garrig6s af

working groups. Fulvio Ricci directed

Noutchegueme, the group on partial differential equations with applications

the participants followed the course

firmed, "After a half-hour course, the

people are interested in wavelets; they

the theme, "Application of mathematics

to signal-processing and image-process ing," presided over by Minister Henri

Hogbe

Nlend,

internationally

Cameroon mathematician.

known

We received impressions and advice

from several speakers, and the global

point of view of a participant from

Burkina-Faso.

Interview with Marco Peloso

CYRILLE NANA: Is this the first time you

have been in Africa?

Group shot of the participants. Jean-Luc Dimi (Congo-Brazzaville} and Gustavo Garrig6s (Spain} are squatting in the front. Behind Garrig6s is David Bekolle (Cameroon}; continuing to the right are Hamidou Toure (Burkina Faso}, Henri Hogbe Nlend (Minister of Research, Cameroon}, Aline Bonami (France}, Blaise Some (Burkina Faso}; behind Hamidou and Bonami are Marco Peloso and Fulvio Ricci (Italy}.

30

Tl-IE MATHEMATICAL INTELLIGENCER

Workshop participants with three of the lecturers: first, third, and sixth from the left are Gustavo Garrig6s, Marco Peloso, and Fulvio Ricci. MARCO PELoso: I have never been in Africa before; this is the first time I have been here. c. N.: Are you surprised to see Africans doing mathematics? M. P.: No, not at all. c. N.: What idea did you have of Africa in general? M. P. : A very poor continent where the people struggle to live and survive. Everything has gone well so far. c. N.: What do you think of the way the Seminar has been organized? M. P.: I think the organization is excel lent: 100 participants, the environment and the hall very suitable, we had the facilities that we needed; there was much discussion in the working groups; the organization was perfect. c. N.: I noticed that each of you devel oped part of a single theme: how did you arrange to do that? M. P.: I Inlls_t s�y_tha,t this_was done on purpose, but I think that there was a problem of notation. The subjects were developed in a logical order and there was coordination between the speak ers, but I think that it would have been advantageous for each speaker to write out his lecture in detail. Unfortunately, we didn't have the time to do that. c. N . : What do you think of the fact that we included applied mathematics in the seminar? M. P.: It was important to organize the

sessions for applied mathematics be cause Africa must try to develop in the applied sciences. I think also that the association of pure and applied math ematics is very useful and beneficial for both groups of researchers. Per sonally, I would like to keep up with the applications to, say, hydrology. c. N.: A propos of the young re searchers, how have you found them and what do you advise? M. P.: As far as I'm concerned, it was in teresting to listen to the young African mathematicians. I hope that the young can find this conference useful, both for learning and for meeting people. Thus you can have the chance perhaps to travel in Europe if you wish. c. N . : What areas of mathematics inter est you the most? M. P.: Harmonic analysis, real variables, Hardy spaces, convergence of singular integrals, BJ\1:0 d:ual of H1, holomor phic functions, the study of the behav ior of the atomic decomposition at the boundary, functions of several complex variables in general, and other areas. Interview with Fulvio Ricci CYRILLE NANA:

before?

Have you been in Africa

FULVIO RICCI: I was in Egypt once, but it is different here. c. N . : Do you know any African mathe maticians?

F. R.:

I know African mathematicians such as Noel Lohoue. c. N . : What do you think of the organi zation of this gathering? F. R.: The organization was done with much care, it was generous, the at mosphere very pleasant, there were enough people who followed the talks, and I see that there are many activities in this part of Africa. Moreover, I have seen young people who are very open to listening about areas in which they do not work. We had a large enough room, but perhaps the blackboard was defective; certainly this room was not made for lectures. It is good to have thought of a room with Internet access; the hotel is reasonable. Good organization! I understand that the hours were set in relation to local activities, but one gets used to it. I had feared that it would be very warm, but it is very pleasant and the air-condi tioning of the room is good too. c. N.: What areas of mathematics inter est you the most? F. R.: Harmonic analysis; I work in Eu clidean analysis, classical Fourier analysis, singular integrals, analysis on Lie groups (in particular nilpotent Lie groups, the Heisenberg group), the prob lem of resolvability of differential oper ators, questions of harmonic analysis that are related to complex analysis.


31

c. N . : To hear you speak, you do har monic analysis; how is it that here you have been speaking about Bergman spaces? F. R.: I would first like to remark that there is a part of complex analysis which is geometric, for example, func tions of several complex variables; also, Lie groups, it is analysis on prob lems having geometric aspects. I began to work on Bergman spaces in 1995, I had heard Aline Bonami and David Bekolle speak about them, and I was interested in collaborating with them. c. N.: How long have you been doing harmonic analysis? F. R.: I began in harmonic analysis when I was in America (USA). In Italy, the field was not developed in the 1970s; I received a Ph.D. in the United States and then returned to Italy. c. N.: I suppose you have had many stu dents? F. R. : I have had about a dozen students who are now professors, researchers, etc. c. N.: What advice would you give to a young person beginning research? F. R. : It's necessary that the researcher engage in the study on the intellectual level, seek out contacts with the sub ject, and imitate what is available in the literature and what others have done. Researchers must be clever enough to pursue the problem posed, and to go on from there to enlarge the field of in terest, to find connections with related problems. c. N.: In general, did the seminars in clas sical analysis interest many people? F. R. : There are relatively few partici pants who are interested in classical analysis, I understand well that there are many people in applied math for obvious reasons. c. N.: What do you think of CARIM? F. R.: CARIM is very important not only for Cameroon, but for the entire region; it would be important to make con nections with other institutes, in Eu rope in particular. c. N.: Thank you.

Brief Discussion with Gustavo Garrigus CYRILLE NANA:

Do you teach in Madrid? I don't have a per-

GUSTAVO GARRIGOS:

32


A U THOR

manent teaching post. From time to time I go to Orleans and Italy to work. N. c.: What are the domains in which you work. G. G.: I have worked on Bergman spaces for two years; wavelets, since my the sis in 1995. It was Bonami who en couraged me to study Bergman spaces and also signal-processing. c. N.: How is it that in such a short time you have been able to do so many dif ferent things? G. G.: In fact, after my thesis I had op portunities to do something else as a post-doc. I have collaborated with en gineers; the best moment, it is to do something new with other people. It is necessary to note that it is easier to talk with an analyst than with an engineer; in fact, it takes time to understand what an engineer is saying. c. N.: Thank you.

CYRILLE NANA

Departement de Mathematiques Universite de Yaounde I

B.P. 8 1 2 , Yaounde , Cameroon e-mail: [email protected]

Cyrille Nana, a student at the Univer sity of Yaounde I, was a participant in the Seminar-Workshop. He is writing his thesis on the continuity of the Bergman projector in tube domains,

Interview with Prof. Blaise Some,

and recently made a preliminary re

Universite de Ouagadougou,

port on his results on a research visit

Burkina-Faso

to the laboratory MAPMO in Orleans ,

CYRILLE NANA:

What are your impres sions of the results of this seminar? BLAISE SOME: I believe that the organi zation is good; on the level of recep tion, it was good; the lodging, good; the restaurant was good; the first day, the service was slow. The lectures concerning applica tions were good. Prof. Noutchegueme introduced things well. As a whole, the course was very good. I much appreci ated the round-table, for it permitted the public to see that Mathematics is well. The graphics were good, but there were transparencies written by hand, and that was not good. On the scientific side, it was a total success; the time was very short, how ever, so one could not go into depth on some points. c. N.: How would you advise the students who participated in the colloquium? B. s. I heard several students' work; there was one that I noticed particu larly, on finite elements. He showed that he had mastered his domain of re search. The advice for students is first to learn the literature in his domain, to have a large view of all those who have worked in that area, and to know the specialized journals. N. c.: One word about CARIM?

France, directed by Aline Bonami.

B.

s.: CARIM is a good idea as such; now it is necessary to see the texts that will defme its functioning. c. N.: Thank you.

Summary

The Seminar-Workshop, according to the opinions expressed in the inter views, went very well, on the scientific level as in all other respects. At the end, the organizing committee convened an evaluation session which was coordi nated by Professor Jean Luc Dirni of the Universite Marien Ngouabi de Braz zaville in Congo. During this session the participants brought out some weaknesses that, while not negligible, did not hinder at all the progress of the work. I remark in conclusion that during the academic year 2001-2002, David Bekolle taught a course for DEA (Diplome d'Etudes approfondies) stu dents on the theme developed in the seminar. Course notes are now in preparation which spell out in detail the results that the participants es tablished during the workshop, writ ten by themselves.

@,i,Fftj.J§!:bhifli=tft%§[email protected]§•id

The Card Game SET

Benjamin Lent Davis and Diane Maclagan

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

M i c h ael Kleber a n d Ravi Vak i l ,

T

he card game called SET1 is an ex tremely addictive, fast-paced game found in toy stores nationwide. Al though children often beat adults, the game has a rich mathematical struc ture linking it to the combinatorics of finite affine and projective spaces and the theory of error-correcting codes. Last year an unexpected connection to Fourier analysis was used to settle a basic question directly related to the game of SET, and many related ques tions remain open. The game of SET was invented by population geneticist Marsha Jean Falco in 1974. She was studying epilepsy in German Shepherds and began repre senting genetic data on the dogs by drawing symbols on cards and then searching for patterns in the data. Af ter realizing the potential as a chal lenging puzzle, with encouragement from friends and family she developed and marketed the card game. Since then, SET has become a huge hit both inside and outside the mathematical community. SET is played with a special deck of cards (Fig. 1 ). Each SET card displays a design with four attributes-number, shading, color, and shape-and each attribute assumes one of three possi ble values, given in Table 1 . Table 1

Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil,

Number:

{One, Two, Three}

Shading:

{Solid, Striped, Open}

Color:

{Red, Green, Purple}

Shape:

{Ovals, Squiggles. Diamonds}

A SET deck has eighty-one cards, one for each possible combination of at tributes. The goal of the game is to find collections of cards satisfying the fol lowing rule.

Ed itors

The SET rule: Three cards are called a SET if, with respect to each of the four attributes, the cards are either all the same or all different.

For example, Figure 2 illustrates a green SET. All cards have the same shape (ovals), the same color (green), and the same shading (solid), and each card has a different number of ovals. On the other hand, Figure 3, also green, fails to be a SET, because there are two oval cards and one squiggle card. Thus the cards are neither all the same nor all different with respect to the shape attribute. To play the game, the SET deck is shuffled and twelve cards are dealt to a table face-up (Fig. 4). All players si multaneously search for SETs. The first player to locate a SET removes it, and three new cards are dealt. The player with the most SETs after all the cards have been dealt is the winner. Occasionally, there will not be any SETS among the twelve cards initially dealt. To remedy this, three extra cards are dealt. This is repeated until a SET makes an appearance. This prompts the following SET-theoretic question. Question. How many cards must be dealt to guarantee the presence of a SET?

Figure 5 shows a collection of twenty cards containing no SETs. A brute-force computer search shows that this is as large as possible, as any collection of twenty-one cards must contain a SET. There is a wonderful geometric re formulation of this Question as fol lows. Let IF3 be the field with three el ements, and consider the vector space

Stanford University,

Department of Mathematics, Bldg. 380,

SET

is a trademark of SET Enterprises, Inc. The

Stanford, CA 94305-2125, USA

cards are depicted here with permission.


play is protected intellectual property.

SET

SET

game

Figure 1 . Typical SET cards.

© 2003 SPRINGER-VERLAG NEW YORK. VOLUME 25, NUMBER 3. 2003

33

Figure 2. A SET.

1Ft A point of IF§ is a 4-tuple of the form (x1, x2, x3, x4), where each coordinate assumes one of three possible values. Using the table of SET attributes (Table 1), SET cards correspond to points of IF§, and vice-versa. (2, 1 , 3, 2)

�

[ill .

Two Solid Purple

Squiggles

�

Under this correspondence, three cards form a SET if and only if the three associated points of IF§ are collinear. To see this, notice that if a, {3, y are three elements of IF3, then a + f3 + y = 0 if and only if a = f3 = y or { a, {3, y} { 0, 1 , 2 } . This means that the vectors a, b, and c are either all the same or all different with respect to each coordi nate exactly when a + b + c = 0. Now a + b + c = 0 in IF§ means that a - b = b - c, so the three points are collinear. Note that this argument works when IF§ is replaced by IF � for any d. From this point of view, players of SET are search ing for lines contained in a subset of IF§. We summarize this rule as follows. =

The Affine Collinearity Rule. Three points a, b, c E IF � represent collinear points if and only if a + b + c = 0.

We define a d-cap to be a subset of

IF� not containing any lines, and ask the following. Equivalent Question. What is the maximum possible size of a cap in IF§?

In this form the question was first answered, without using computers, by Giuseppe Pellegrino [ 19] in 1971. Note that this was three years before the game of SET was invented! He ac tually answered a more general ques tion about "projective SET, " which we explain in the last section.

Figure 3. Not a SET.

34


Figure 4. Can you find all five SETS? (Or all eight for those readers with black-and-white pho tocopies.)

Although SET cards are described by four attributes, from a mathematical perspective there is nothing sacred about the number four. We can play a three-attribute version of SET, for ex ample by playing with only the red cards. Or we can play a five-attribute version of SET by using scratch-and sniff SET cards with three different odors. In general, we define an affine SET game of dimension d to be a card game with one card for each point of IF�, where three cards form a SET if the corresponding points are collinear. A cap of the maximum possible size is called a maximal cap. It is natural to ask for the size of a maximal cap in IF�, as a function of the dimension d. We denote this number by ad, and the known values are given in Table 2.

6 1 1 2 ,;

a6

,; 1 1 4

The values of ad in dimensions four and below can be found by exhaustive com puter search. The search space be comes unmanageably large starting in dimension five. Yves Edel, Sandy Fer ret, Ivan Landjev, and Leo Storme re-

[]] [!!] [JJ ill] []] [!!] [JJ [)I]

Figure 5. Twenty cards without a SET.

cently created quite a stir by announc ing the solution in dimension five [6]. We shall spend some time working our way up to their solution. There are many other possible gen eralizations of the game of SET. For ex ample, we could add another color, shape, form of shading, and number to the cards, to make the cards corre spond to points of IF 4. Here, however, several choices need to be made about the SET rule. Is a SET a collection of cards where every attribute is all the same or all different, or is it a collec tion of collinear points? In IF i, there are four points on a line, so do we require three or four collinear points to form a SET ? Furthermore, if we choose the collinearity criterion, then collinearity of SET cards is sensitive to the choice of which color, shape, etc. corresponds to which element of IF 4. Because of these complications we will restrict our attention here to caps (line-free collections) in IF�. We can exhibit caps graphically us ing the following scheme. Let us con sider the case of dimension d = 2. A two-attribute version of SET may be re alized by playing with only the red ovals. The vector space IF § can be

[IT!] [ill] I D D D I l l l l l [ill OJ I � HI ill] [I] [B] � � � � l lo o o l

[[] [}] []]

5-cap

H� # #x ### �� it= * # * �� * # # lj * # * ### # * xl:lx * # # ### **# xj:jx * #x H� H� =#= l-eap

X I IX 2-cap

Figure 6. The correspondence between 2-at tribute SET and

IF3.

graphically represented as a tic-tac-toe board as in Figure 6. We indicate a sub set S of IF § by drawing an "X" in each square of the tic-tac-toe board corre sponding to a point of S. The lines con tained in S are almost plain to see: most of them appear as winning tic-tac-toes, while a few meet an edge of the board and "loop around" to the opposite edge. Check that the two lines in Fig ure 7 correspond to SETS in Figure 4. Figure 8 contains pictures of some low-dimensional maximal caps. In di mensions one through four, the caps are visibly symmetrical, and each cap contains embedded copies of the max imal caps in lower dimensions. No such pattern is visible in the diagram of the 5-cap. It is natural to ask if the maximal caps in Figure 8 are the only ones in each dimension. In a trivial sense, the answer is 'no', since we can make a new cap by permuting the col ors of an old cap. There are many other permutations of IF:� guaranteed to pro duce new caps from old. Permutations of IF� taking caps to caps are exactly

,'

, '

,

'

X , ,

'

'

V: V': ,

'

.,'-��----+----+-""x ·-, " : ,

, I

I ., '

'

... "" 1 � , .. .. ... .... (

'

-, '

I

>( >< '

,

..

'

'

,

-

' I

I

I

I I I

-

I

I '

Figure 7. This collection of points contains two lines which are indicated by dashed curves.

X

x

3-cap

x

x

X

x

4-cap

X

X

Figure 8. Low-dimensional maximal caps.

those taking lines to lines, and such a permutation is called an affine trans formation. Another characterization of affine transfonnations is that they are the permutations of IF� of the form

a(v)

=

Av + b,

where A is an invertible d X d-matrix with entries in 1Fa, b is an arbitrary vec tor of IF�, and v is a vector in IF �. We say that two caps are of the same type if there is an affine transformation taking one to the other. For example, consider the affine transformation a(x, y) = ( -x - y, - x + y - 1) taking a vector (x, y) E IF§ to another vector in IF�. Applying this to a 2-cap gives an other 2-cap of the same type. This is il lustrated in Figure 9, where we have

declared the center square of the tic tac-toe board to be the origin of 1Fl It is known that in dimensions five and below there is exactly one type of maximal cap. An affine transformation taking a cap to itself is called a sym metry of the cap. Although it is not ob vious from Figure 8, the maximal 5-cap does have some symmetries. In fact, its symmetry group is transitive, meaning that, given two points of the 5-cap, there is always a symmetry taking one

Figure 9. Two 2-caps of the same type.

VOLUME 25. NUMBER 3 . 2003

35

to the other. Michael Kleber reports that the stabilizer of a point in the 5-cap is the semidihedral group of or der 16. The symmetry group is useful for re ducing the number of cases that need to be checked in exhaustive computer searches for maximal caps, thus greatly speeding up run times. To see this idea in action, check out Donald Knuth's SET-theoretic computer pro grams [17] .

We can make some progress on com puting the size of maximal caps using only counting arguments. Proposition 1. A maximal 2-cap has four points.

Proof We have exhibited a 2-cap with four points. The proof proceeds by contradiction. Suppose that there ex ists a 2-cap with five points, x1, x2, Xs, X4, X5. The plane rF§ can be decomposed as the union of three horizontal paral lel lines as in Figure 10. Each line contains at most two points of the cap. Thus, there are two horizontal lines that contain two points of the cap, and one line, H, that con tains exactly one point of the cap. With out loss of generality, let x5 be this point. There are exactly four lines in the plane containing the point x5, which we denote H, L1, L2, Ls. This is illustrated in Figure 1 1 . Since the line H contains none o f the points x1, . . . , x4, by the pigeon-hole principle two of these points Xr and Xs must lie on one line Li. This shows that the line Li contains the points Xr, Xs and x5, which contradicts the hypothesis that the five points are a cap. 0 We can apply the method of Propo sition 1 to compute the size of a max imal cap in three dimensions.

� >diF2, respectively.

This result is due to Jennifer Key and Ernest Shult [ 16], Hall [ 10], and William Kantor [ 15]. Interestingly, the proofs use part of the classification of finite simple groups. If we actually play projective SET we want to lmow how many cards need to be dealt to guarantee a SET. Just as in the affin e case, we call a collection of points in [P>diF2 containing no three points on a projective line a cap. The problem of fmding maximal caps for projective SET was solved in 1947 by Raj Chandra Bose. In [2], he showed that the maximal caps d of [P>diF2 have 2 points. Bose's interest in this problem certainly didn't stem from SET, as the game was not to be invented for another 27 years. Rather, he was coming at it from quite another direc tion, namely, the theory of error-cor recting codes, which is the study of the flawless transmission of messages over noisy communication lines. As detailed in the book of Raymond Hill [ 14], there is a correspondence between projective caps in [P>diF2 and families of efficient codes. Specifically, if we form the ma trix whose columns are vectors repre senting the projective points of the cap, then the kernel of this matrix is a linear code with Hamming weight four. The more points the cap contains, the more "code-words" the corresponding code has, and so this naturally motivates the problem of finding maximal projective caps. Bose completely solved this prob lem when q = 2, but, as in the affine case, things become much more difficult when q 3. We denote by bd the size of a maximal projective cap in [P>diF3. The lmown values of bd are given in Table 3. =

bd

2

Acknowledgments

We have been fascinated with SET since first spending many many hours playing it when we started graduate school to gether, and have had many helpful con versations with other SET-enthusiasts. We would never have spent so many hours thinking about the "sET-Problem" without Joe Buhler's goading remark that it was shocking that two Berkeley students couldn't work out the answer. Josh Levenberg helped then by writing our original brute-force program, before we came up with a non-computer proof of "20." Galen Huntington got us think ing along the right lines with the proof of Proposition 1. Thanks also to Michael Kleber for many interesting conversa tions on this topic, only a fraction of which have occurred during the writing of this article, and for informing us of Proposition 10. More recently Mike Zabrocki's computer demonstration at FPSAC 2001 reinspired us, and Sunny Fawcett has been invaluable with inte ger programming assistance and other ideas. Finally, we are indebted to Joe Buhler, Juergen Bierbrauer, and Yves Edel for extensive comments on an ear lier draft of this paper. REFERENCES

[1] J. Bierbrauer and Y. Edel. Bounds on affine

Table 3 d

game with cards given by points of IP'51F3, there is still some interesting SET theory associated with the study of maximal projective caps in this space. In particular, the 45-point affine cap in Figure 8 was constructed by deleting a hyperplane from the 56-point projec tive 5-cap given by Hill in Figure 4 of [ 13]. Uniqueness of this affine cap was shown in [6] to be a consequence of the uniqueness of the projective cap, which in tum was demonstrated by Hill in [ 12] by means of a code-theoretic argument.

2

3

4

5

6

4

10

20

56

?

caps. J. Combin. Des . , 1 0(2): 1 1 1-1 1 5, 2002. [2] R. C. Bose. Mathematical theory of the

[5] Y. Edel. Extensions of generalized product caps. Preprint available from http://www. mathi. uni-heidel berg. de/ �yves/. [6] Y. Edel, S. Ferret, I. Landjev, and L. Storme. The classification of the largest caps in AG(5, 3). Journal of Combinator ial Theory, Series A , 99:95-1 1 0, 2002.

[7] W. Fulton and J. Harris. Representation theory, A first course. Readings in Mathe

matics. Springer-Verlag, New York, 1 99 1 . [8] H. T. Hall. Personal communication. [9] M. Hall, Jr. Automorphisms of Steiner triple systems. IBM J. Res. Develop , 4 :460-472, 1 960. [ 1 0] M. Hall, Jr. Steiner triple systems with a dou bly transitive automorphism group. J. Com bin. Theory Ser. A, 38{2) : 1 92-202, 1 985.

[1 1 ] R. Hill. On the largest size of cap in 55,3. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 54:378-384 (1 974), 1 973.

[1 2] R. Hill. Caps and codes. Discrete Math . , 22(2) : 1 1 1 -1 37, 1 978. [13] R. Hill. On Pellegrino's 20-caps in 54,3. In Combinatorics '8 1 (Rome, 198 1) , pages

433-447. North-Holland, Amsterdam, 1 983. [1 4] R. Hill. A first course in coding theory. The Clarendon Press Oxford University Press, New York, 1986. [1 5] W. M . Kantor. Homogeneous designs and geometric lattices. J. Combin. Theory Ser. A , 38( 1 ):66-74, 1985.

[1 6] J. D. Key and E. E. Shult. Steiner triple sys tems with doubly transitive automorphism groups: a corollary to the classification the orem for finite simple groups. J. Combin. Theory Ser. A , 36(1 ) : 1 05-110, 1 984.

[ 1 7] D. Knuth. Programs setset, setset-all, set set-random. Available from http://sunburn. stanford.edu/�knuth/programs.html. [1 8] Roy Meshulam. On subsets of finite abelian groups with no 3-term arithmetic progressions. J. Combin. Theory Ser. A, 71 ( 1 ) : 168-172, 1995. [1 9] G. Pellegrino. Sui massimo ordine delle calotte in 54,3. Matematiche (Catania) , 25:149-157 (1 97 1 ) , 1970. [20] J. H. van Lint and R. M. Wilson. A course in combinatorics . Cambridge University

Press, Cambridge, second edition, 2001.

symmetrical factorial design. Sankhya, 8:

The sizes in dimensions 2 and 3 are due to Bose [2], dimension 4 is due to Pel legrino [19], and dimension 5 is due to Hill [ 1 1, 12]. We note that we always have a :S b , since there is a copy of d d IF� inside [P>d1F3, so Pellegrino's result is the first proof that a4 :S 20. Even though there is no abstract SET

40


107- 1 66, 1 947. [3] A R. Calderbank and P. C. Fishburn. Max

Department of Mathematics, and Computer Science, Saint Mary's College, Moraga, CA

imal three-independent subsets of {0, 1 , 2}n

94575, USA

Des. Codes Cryptogr. , 4(3):203-21 1, 1994.

e-mail address: [email protected]

(4] C . J. Colbourn and A Rosa. Triple sys tems. Oxford Mathematical Monographs.

The Clarendon Press Oxford University Press, New York, 1 999.

Department

of

Mathematics,

Stanford

University, Stanford, CA 94305, USA e-mail address: [email protected]

HANS-JOACHIM ALBINUS

Pyth ag o ras ' s Oxe n Revi s ited e t is said that when Pythagoras discovered his famous theorem, in a right-angled

/) (/

triangle the squares of the smaller sides sum up to the square of the hypo teneuse, he sacrificed a hundred oxen to thank the gods.

There is a poem in German describing this sacrifice:

Yom Pythagoraischen Lehrsatze

Die Wahrheit, sie besteht in Ewigkeit, Wenn erst die blade Welt ihr Licht erkannt; Der Lehrsatz nach Pythagoras benannt Gilt heute, wie er galt zu seiner Zeit. Ein Opfer hat Pythagoras geweiht Den Gotten, die den Lichtstrahl ihm gesandt Es taten kund geschlachtet und verbrannt Ein hundert Ochsen seine Dankbarkeit. Die Ochsen, seit dem Tage, wenn sie wittern DaB eine neue Wahrheit sich enthtille Erheben ein unmenschliches Gebrtille; Pythagoras erftillt sie mit Entsetzen; Und machtlos sich dem Licht zu widersetzen VerschlieBen sie die Augen und erzittern. In the German vernacular there is a double meaning to the word Ochse, which constitutes the poem's essential irony:

1. an ox 2. a blockhead. 1 He

Readers unfamiliar with German perhaps are aware of one of the two known English translations, one by the American biophysicist and biologist Max Delbrtick (19061981) , 1 the other by Robert Edouard Moritz (1868-1940), an American mathematician, who had studied in Gottingen: Delbrtick's translation can be found in [7] , p. 53; Moritz's, in his book On Mathematics and Mathematicians2 ([6] , p. 309). Both translations are weak in various respects. Moritz's, for example, has lost the structure of rhyme in the first two verses, but it is in words and rhythm closer to the German text than Delbrtick's. Of course, there is no adequate trans lation for Ochsen covering both senses. It is often believed (see for example remarks by Olga Taussky ([7] , p. 53) or Mark Kac ([5] , p. 98-99) that the poem is by the German writer Heinrich Heine. But this is definitely not true. The author of the sonnet is Adelbert von Chamisso (30.01 . 1781-2 1.08. 1 838) , French by birth, whose family fled to Prussia during the French Revolution. There he grew up to become an officer in the Prussian army and later joined a circumnavigation of the globe as a natural scientist. Af ter his return to Germany he worked at the botanical gar dens in Berlin, where he became head of the herbarium. From his literary writings the most famous perhaps is the novel Peter Schlemihls wundersame Geschichte (pub lished 1814) about the fate of a man who sells his shadow.

was the son of the German historian Hans Delbruck. In 1 969 Max Delbruck (together w1th Salvador Edward Luria and Alfred Day Hershey) received the Nobel Prize

for Physiology and Medicine for his bas1c research in molecular biology and bacteriological genetics. 2Prev1ous editions titled Memorabilia Mathematlca or the Philomath's Quotation-Book.

© 2003 SPRINGER-VERLAG NEW YORK. VOLUME 25, NUMBER 3, 2003

41

Delbriick's translation

Moritz's translation

The truth: her hallmark is eternity soon as stupid world has seen her light Pythagoras' theorem today is just as right As when it first was shown to the fraternity.

Truth lasts throughout eternity, When once the stupid world its light discerns: The theorem, coupled with Pythagoras' name, Holds true today, as't did in olden times.

The Gods who sent to him his ray of light To them Pythagoras a token sacrificed: One hundred oxen, roasted, cut, and sliced Expressed his thanks to them, to their delight.

A splendid sacrifice Pythagoras brought The gods, who blessed him with this ray divine; A great burnt offering of a hundred kine, Proclaimed afar the sage's gratitude.

The oxen, since that day, when they surmise That a new truth may be unveiling Forthwith burst forth in fiendish railing.

Now since that day, all cattle [blockheads] when they scent New truth about to see the light of day, In frightful bellowings manifest their dismay;

Pythagoras forever gives them jittersQuite powerless to stem the thrust of such emitters Of light, they tremble and they close their eyes.

Pythagoras fills them all with terror; And powerless to shut out light by error, In sheer despair they shut their eyes and tremble.

As

The sonnet Vom Pythagordischen Lehrsatze was writ ten in August 1835 and first published in 1836 ([3] , p. 765). Cited above is the version found in a modern critical edition of Chamisso's complete works (for example [3] , p. 536; there are identical editions by Carl Hanser Verlag, Mi.inchen, 1982, and Insel-Verlag, Leipzig, 1980); see [3], p. 679-682, for remarks concerning the punctuation and or thography, 3 which was left as close as possible to the orig inal text. By the way, the source from which Chamisso was in spired to write the poem is believed to be an aphorism (no. 268, to be exact) by Ludwig Borne ( 1 786-- 1 837), who had already made use of the two meanings of Ochsen ( [3], p. 765):

Als Pythagoras seinen bekannten Lehrsatz entdeckte, brachte er den Gottern eine Hekatombe dar. Seitdem zit tern die Ochsen, sooft eine neue Wahrheit an das L icht kommt. These sentences can be found in Borne's complete works4 (for example [ 2 ] , p. 3 18); and again Moritz gives an English translation ([6] , p. 308):

After Pythagoras discovered his fundamental theorem he sacrificed a hecatomb of oxen. Since that time all dunces tremble whenever a new truth is discovered. Here he decided to use the second meaning of Ochsen. Now, how has the belief arisen that the poem Vom Pythagordischen Lehrsatze was written by Heinrich Heine

(1 797-1856), as stated in the editor's note5 to Taussky's ar ticle ([7], p. 53) and in Mark Kac's (1914-1984) autobiog raphy Enigmas of Chance ([5], p. 98-99), too? One hint could be Borne, who was at first a close friend of Heine and later, in the days of their exile in Paris, became his in timate enemy (the so-called Heine-Borne controversy). Could this be the reason that Taussky and Kac and others remembered Heine's name instead of Borne's? Heine was more popular than Chamisso or Borne. And very likely they all remembered a passage in a text Heine wrote during a visit to the Frisian island of Norderney (see for example (4], p. 71), where he mentions Pythagoras and his hundred oxen in the context of the transmigration of souls (both human and animal) and possible funny inci dents caused thereby:

Wer weijS! wer weijS! die Seele des Pythagoras ist viel leicht in einen armen Candidaten gefahren, der durch das Examen fdllt, weil er den pythagordischen Lehrsatz nicht beweisen konnte, wdhrend in seinen Herren Ex aminatoren die Seelen jener Ochsen wohnen, die einst Pythagoras, aus Freude iiber die Entdeckung seines Satzes, den ewigen Gottern geopfert hatte. A translation may read as follows:

Who knows! Who knows! Perhaps Pythagoras 's soul is now in a poor candidate who fails his examination because he is unable to prove the Pythagorean theorem, while in his examiners live the souls of those oxen which once, in the joy of the discovery of his them·em, Pythagoras had sacrificed to the eternal gods.

3A more modern spelling of the poem can be found in [1]. p. 72, of which Moritz was aware. 4Be aware that some other editions of Borne's works use a different numbering of the aphorisms and miscellanea. 5The editor's question in The Mathematica/ lntelligencer for a hint to the poem's authorship or to a reference went unanswered at the time.

42


Thus, Pythagoras appears in the works of at least three prominent contemporary poets of the early 19th century: Ludwig Borne, Adelbert von Chamisso, and Heinrich Heine! REFERENCES

[1] Ahrens, W . : Scherz und Ernst in der Mathematik. Geflugelte und ungef/Qgelte Worte. Teubner, Leipzig, 1904.

[2] Borne, Ludwig: Sarntliche Schriften , vol. 2. Melzer, Dusseldorf, 1964. [3] Charnisso, Adelbert von: Sarntliche Werke in zwei Banden , vol. 1: Gedichte- Oramatisches. Wissenschaftliche Buchgesellschaft, Darm

stadt, 1982.

Cauchy Product of Series MAURICE MACHOVER Abel, Mertens, Cauchy, Hardy Proved some theorems very arty: Sum up A,., sum up Bn,

[4] Heine, Heinrich: Sakularausgabe. Werke- Briefwechsei- Lebens zeugnisse, vo l . 5: Reisebilder / 1 824- 1 828. Akademie, Berlin, 1970.

[5] Kac, Mark: Enigmas of Chance. An Autobiography. Harper & Row,

Take the Cauchy product Cn. If the fll'St two sums converge, Will the third to limit surge'?

New York, 1985. [6] Moritz, Robert Edouard: On Mathematics and Mathematicians. Dover, New York, 1958.

Let us these four men consult.

Abel proved the first result:

[7] Taussky, Olga: From Pythagoras' Theorem Via Sums of Squares to Celestial Mechanics. The Mathematical lntelligencer, 10 (1988) , no. 1' p. 52-55.

Let the Cn sum to C,

An to A, Bn to B; Abel showed accordingly That A times B will equal C. But we would like to sum Cn

ot in the "IF" but in the "THE

A U THO R

"

Along comes Mertens with a beaut About convergence absolute:

If absolu tely sums up one

(Each still converging to its sum),

Convergence of Cn comes free, And A times B still equals C.

And furthermore great Cauchy says, Two absolute sum-ups more pays, For then the summing up to C Will also go absolutely.

HANS-JOACHIM ALBINUS

lnnenministerium Baden-WOrttemberg

And last comes Hardy with his gift.

Dorotheenstrasse 6

This time we have a little shift:

D-701 73 Stuttgart

We still assume the first two sum,

Germany

But as to Cn, we


are

mum;

Instead, assume that bounds we ken Hans-Joachim Albinus ought perhaps to have been named for his famous ancestor Decimus Clodius Albinus, claimant to the Roman imperial throne until he was killed in 1 97

A.D.

by

his rival Septimus Severus. H.-J. Albinus did a master's degree in mathematics and

On nAn and nBn. With these assumptions, Hardy says, Convergence of the Cn stays; It needn't go absolutely,

But A times B still equals C.

geography at Ruhr-Universitat Bochum, then worl<ed in com puting centres. Progressively doing more and more manage ment and less and less mathematics, he became deputy head of electronic communication and management science in the provincial lnnenministerium. His interest in history of mathematics and computing sur

Department of Mathematics St. John's University Jamaica,

NY 1 1 439 USA

e-mail: cherokeezip @ webtv .net

vived. He also loves contemporary arts, fashion design, haute cuisine, and motorboats. He lives in Leonberg, one of the "Kepler towns" he invited us to in his Mathematical Tourist ar

ticle, vol . 24, no. 3, 50-58.

VOLUME 25, NUMBER 3 . 2003

43

ip,i$•VMj.l§rr@ih$ili.IIIQt11

Regiomontanus Ottomar Gotz

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the caje 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? .(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 follow in your tracks.

Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck,

Aartshertogstraat 42,

8400 Oostende, Belgium e-mail: [email protected]

44


© 2003

''

D i rk H uylebro u c k ,

K

Ed itor

onigsberg in Bavaria" is a picturesque market town situated in southern Germany about 100 kilo meters northwest of Nuremberg. Whilst Konigsberg was a busy trading centre on an important trade route during the Middle Ages, today the town appears as a tranquil place with a well preserved medieval town-centre amidst the lovely hills of the "HaBberge." It is worth visiting for its characteristic atmosphere with timbered medieval houses still in use, a city wall fortified by towers, and the ruins of an 800-year old castle founded by the Emperor Bar barossa. It was in this lovely town on June 6th in 1436 that the most famous as tronomer and mathematician of the 1 5th century, Johannes Muller-in la tinized form called Regiomontanus was born. Living at the end of the Mid dle Ages, he was embedded in the traditional thought of a geocentric cos mology based on the authority of Aris totle's Metaphysics and the conviction of mankind's unique central place in the cosmos-in agreement with the Holy Scripture. Regiomontanus died in Rome at the age of only 40. There he had been invited to the Court of Pope Si�:tus IV for the purposes of a calen dar reform. Copernicus, who was to revolutionize cosmology, was then just 3 years old. Regiomontanus was a versatile ge nius. His scientific life spans ancient and medieval mathematics completely, including the sophisticated methods of Ptolemy in modeling the geocentric system. In addition to his theoretical abilities, he was gifted with a distinct practical sense. Nobody of the 1 5th century exploited the possibilities of book printing in science more effec tively than Regiomontanus. He had the gigantic ambition to pub lish the works of ancient mathematics and astronomy as well as his own rich work He also designed astronomical instruments. His most significant work in mathematics is a geometry of plane

SPRINGER-VERLAG NEW YORK

I

and spherical triangles supplemented by a table of sines to 7 decimal places, the "De Triangulis omnimodis," which must be regarded as the first textbook of trigonometry ever. In this work Re giomontanus introduced for the first time in the Western world the theorems of sine and cosine for spherical trian gles, setting a standard in trigonometry for centuries. Another work of his came to have extraordinary significance and wide use. It was an improved and essentially revised edition of the monumental compendium of ancient astronomy, the Almagest of Ptolemy. In his Epitoma in Almagestum Regiomontanus added new data and advanced alternative mathematical methods, which un doubtedly gave Copernicus's studies further impetus. The Epitoma became the most important astronomical text published in the 1 5th century. It was used as a textbook as well as a manual in practical astronomy for more than 100 years. Most of his works resulted from commissions he received from various eminent persons. Emperor Friedrich III requested a birth reading for his bride-at that time an honorable and serious request. For the King of Hun gary and his chancellor, the archbishop of Esztergom, he developed tables for the positions of planets and stars with an appendix containing numerical ta bles of sine and tangent. It is known that Columbus used Regiomontanus's tables on his famous voyages of dis covery. His most creative period, how ever, appears to have been the time when he was in the service of Cardinal Bessarion. He followed the Cardinal to Rome after his studies at the university of Vienna, and was a scientific advisor at his court for some years. At that time he finished the successful astronomi cal work Epitoma and cultivated a rich correspondence, from which a collec tion of 400 mathematical problems has been preserved. He gave lectures at the University of Padova. Last but not

The house where Regiomontanus was born, located on the "salt-mar

A monument in the marketplace of Konigsberg includes a statue of

ketplace" in Konigsberg in Bavaria (photographs by Ottomar Gotz}.

Regiomontanus. In the background the town hall.

Title page and last page of his sine table (from Stadtarchiv Schweinfurt}.

VOLUME 25, NUMBER 3. 2003

45

of his great plans short. He was buried in the "Gottesacker," which is the Campo Santo Teutonico, the cemetry for German citizens at St. Peter's. Vis itors will find this memorial tablet:

•

In memoriam Johannes Miiller genannt Regiomontanus Astronom-Mathematiker Wegbereiter des neuen Weltbildes * 6. VI. 1436 zu Konigsberg in Franken + 6. VII. 1476 in Rom

N u re m berg

B udapest

Venice Padova

The m a p shows the main stations in Regiomontanus's life.

least, he rediscovered an ancient man uscript of 6 books of Diophantus's fa mous Arithmetica in Venice and trans lated it from Greek into Latin. In 1471, only 5 years before his tragic death in Rome, Regiomontanus started a career as a scientific pub-

lisher and practical astronomer in Nuremberg. He established a print shop and set up an observatory, whose equipment Regiomontanus described in a letter to the mathematician Chris tian Roder of Erfurt. His sudden death cut the realization

Die Biirger seiner Vaterstadt AD. MCMLXXVI

Apart from his birthplace Konigsberg, where there is a statue of him on the marketplace, the house where he was born, and the parish church where he was baptized, the Mathematical Tourist can fmd places outside of Germany where he is memoralized. The above mentioned tablet in the Vatican's Campo Santo Teutonico, a memorial in the castle of Budapest, and a memorial tablet on the wall of the one-time castle of the archbishop of Esztergom. Finally, there is yet another memorial that can be viewed periodically by anybody-a moon crater named Regiomontanus. And everyone can find it easily by look ing at the site http://www.lunarrepub lic.com/atlas/sections/f4.shtml. Birkenstrasse 40 D-97422 Schweinfurt Germany e-mail: [email protected]

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46


The

M a the m atic a l

Tourist

The Christoffel Plaque in M onschau Helena Alexandra van der Waall and Robert Willem van der Waall

r rk H u y l e b ro u c k ,

Editor

,.�ile on holiday in Monschau WW (Germany), the authors found

themselves in front of a commemora tive plaque marking the 1 50th birthday of the mathematician Elwin Bruno Christoffel (1829-1900). That occasion stimulated them to prepare an over view of his work; of some places where he lived, worked, and taught; and of the discovery of some less known but very conclusive sources of information on that score. Introduction

The mathematics of Elwin Bruno Christoffel (born 1829, Montjoie; died 1900, StraBburg) is well known. Many people are aware of the Christoffel Darboux-Einstein summation formu lae, of Christoffel symbols, or of the Christoffel-Schwarz mapping theorem. Besides some unpublished lecture notes, Christoffel left thirty-five printed papers dealing with subjects like function the ory including conformal mapping, prop agation of electricity, Gaussian quad rature, continued fractions, dispersion of light, movements of points in peri ods, continuity conditions for differen tial equations, minimal surfaces, the ory of invariants, geodetic triangles, geometry and tensor analysis, orthog-

The memorial.

I

onal polynomials, shock waves, poten tial theory, Riemann integrals, Jacobi's theta-sequences, irrational numbers altogether, a beautiful collection of 19th-century mathematics; here one could consult his Collected Papers [8). There also exist retrospective sur veys on the mathematical works of Christoffel. To start with, see [9) and [12), about a century old. Additional material of a much later date about Christoffel's geometry can be found in [ 1 1 ) ; the "Riemann example" of a con tinuous non-differentiable function is scrutinized in depth in [4). Some mod em, very informative contributions are provided in [ 13] and [ 14). At first sight, less seems to be known about Christoffel as a person acting in his society and milieu, or about his social, mathematical, and intellec tual acquaintances in Berlin, ZUrich, and StraBburg (the places where he gave his lectures). However, as we will see in the next section, there are some illuminating sources.

The only known photograph of Christoffel.

© 2003 SPRINGER-VERLAG NEW YORK, VOLUME 25, NUMBER 3. 2003

47

The (German) caption in [6] to this photo-

View from dormer window of the building at the RurstraBe 1 , in 1 889, at the place where

graph indicates, on physiognomical grounds,

Christoffel was born.

that it might be another picture of Christoffel, an at older age.

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48


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