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.
employ SPRT or (DD) with thresholds
Letter to the Editor
P
chosen to maximize the reward rate: atti Wilger Hunter's article on Abra
fraction of
nicely illustrates how a mathematician
correct
can be stimulated by, and respond to, challenges from beyond mathematics
RR=
responses)
(RR)
per se. Your readers may not know that
(Average
Wald's sequential probability ratio test
time between
(SPRT), which was independently dis
responses).
covered by George Barnard in the
U.K.
[1] and used by Turing's group in their code-breaking work at Bletchley Park, also illustrates unexpected applica tions of existing mathematics. In the 1960s psychologists, led by Stone and Laming [2], proposed that people responding to stimuli in highly constrained choice tasks with only two alternatives, do so by accumulating ev idence and responding when a thresh old is crossed, just as in SPRT. Subse quently, Ratcliff [3] used a constant drift-diffusion process, the continuum limit of SPRT and perhaps the simplest stochastic differential equation, dx =
A dt + c d W,
(DD)
cally, reaction-time distributions and error rates. (Here term and process
c
A
denotes the drift
the variance of the Wiener
W.) Moreover, recent neural
recordings from oculomotor brain areas of monkeys performing choice tasks has shown that firing rates of groups of neurons selective for the "chosen" of the two
Since the numerator (1 - Error Rate) and denominator of (RR) are simple expressions of the drift rate variance
c,
A,
noise
and threshold for (DD), it is
an exercise in calculus to compute op timal thresholds and derive an "optimal performance curve" relating reaction time to error rates. This appears to be the first theoretical prediction of how best to solve the well-known speed-ac curacy tradeoff: it is not optimal to try to be always right, since that makes re action times too long; nor is it good simply to go fast, since then error rates are too high. We are currently assessing the abil
to fit human behavioral data-specifi
alternatives rise toward a
threshold that signals the onset of mo tor response in a manner that seems to match sample paths of (DD) [4].
As pointed out in [5], this suggests an
ity of human subjects to achieve this theoretical
optimum
performance.
While some of our subjects (Princeton undergraduates)
appear
more
con
cerned to be correct than to be fast, the overall highest-scoring group indeed lies close to the optimal perforn1ance curve, although slightly on the conser vative (high-threshold) side. Tests are planned with monkeys in which direct neural recordings will also be made. Did the subconscious, with the help of evolution, discover SPRT long be fore Wald and Barnard? Stay tuned. REFERENCES
intriguing possibility. SPRT is the opti
[ 1 ] Barnard, G. A Sequential tests in industrial
mal decision-maker, in the sense that,
statistics. J. Roy. Statist. Soc. Suppl. 8:
for a predetermined error rate, it mini
1 -26, 1 946. DeGroot, M. H. A conversa
mizes the expected time required to
tion with George A Barnard. Statist. Sci. 3:
make a decision among all possible
4
(Expected
ham Wald in the Winter 2004 issue
1 96-2 1 2, 1 988.
tests. (Human reaction times also in
[2] Stone, M. Models for choice-reaction tirne.
clude durations required for sensory and
Psychometrika 25: 251 -260, 1 960. Larning,
motor processing, and these must be al
D. R. J. Information Theory of Choice-Reac
lowed for in interpreting behavioral
tion Times. Acadernic Press, New York. 1 968.
data.) Thus, if one wishes to optimize
[3] Ratcliff, R. A theory of rnernory retrieval.
one's overall performance in completing
Psych. Rev. 85: 59-1 08, 1 978. Ratcliff, R . ,
a series of trials, one would do well to
Van Zandt, T., and McKoon, G. Connec-
THE MATHEMATICAL INTELLIGENCER © 2005 Springer Science+Busrness Medra, Inc.
tionist and diffusion models of reaction time. Psych. Rev. 1 06 (2): 261 -300, 1 999.
[4] Roitman, J. D. and Shadlen, M. N. Re sponse of neurons in the lateral interparietal area during a combined visual discrimina tion reaction time task. J. Neurosci. 22 ( 1 ) : 9475-9489, 2002. Ratcliff, R , Cherian, A , and
Segraves,
M.
A
comparison
of
macaque behavior and superior colliculus neuronal activity to predictions from mod els of two choice decisions. J. Neurophys iol. 90: 1 392-1 407, 2003.
[5] Gold, J. 1., and Shadlen, M. N. Banburis mus and the brain: Decoding the relation ship between sensory stimuli, decisions, and reward. Neuron 36: 299-308, 2002. Philip Holmes Program in Applied and Computational Mathematics and Center for the Study of Brain, Mind and Behavior Princeton University e-mail:
[email protected] Rafal Bogacz
in the arts have no mathematical train ing (much less mathematical interest). When confronted with something, even something beautiful, that one doesn't understand, there are two com mon human reactions. One is admira tion and wonderment, and a desire to learn more about it. The second is to belittle and denigrate the work so as not to have to admit one's ignorance. There is only a fine line between this latter attitude and outright hostility, and the line is easily crossed. I am afraid that the second reaction is by far the most common one in the art world. Perhaps the most egregious example of this is the January 21, 1998, review by New York Times art critic Roberta Smith of a wonderful Escher exhibition at the National Gallery of Art, where the reviewer's overt hostility cul minated in her statement, " . . . one won ders if Fascism, which Escher detested, hadn't also contaminated his art."
Department of Computer Science
Steven H. Weintraub
University of Bristol
Department of Mathematics
Bristol, UK
Lehigh University
e-mail: r.bogacz@bristol . ac . uk
Bethlehem, PA 1 801 5-31 74
Jonathan Cohen Department of Psychology and Center for the
USA e-mail:
[email protected] Study of Brain, Mind and Behavior Princeton University
Where are the Women?
e-mail:
[email protected] Joshua Gold Department of Neuroscience University of Pennsylvania e-mail: jigold@mail. med . upenn. edu
Is Escher's Art Art?
I
n his review of M. C. Escher's Legacy: A Centennial Celebration, Helmer
Aslaksen writes, "It is also important to realize that arts specialists do not share our fascination with Escher. Many of them simply don't consider him to be an artist!" This is sad but true, and I am afraid there is a very simple explanation for this. Although Escher was not a math ematician, his art has deep mathemat ical ideas, as some of the articles about H. S. M. Coxeter in the same issue of the Intelligencer, which mention Cox eter's and Escher's relationship, make clear. On the other hand, many people
I
am a junior at St. Cloud State Uni versity in Minnesota. While studying to become a mathematics educator, I came across The Mathematical Intel ligencer, vol. 25, no. 4 (Fall 2003). I think The Intelligencer will be a good resource for me as a future educator. However, I was sorry to see that at most one of the nine articles was writ ten by a female. Traditionally, math is thought of as consisting mostly of men. I think it is important that students see that females are as prominent in the field as males. As Ian Law said in "Adopting the Principle of Pro-Feminism" in the book Readings for Diversity and Social Justice (see p. 254), many men think they need to be "dominating the airspace mak ing sure it is [their] voice and views that get heard." The ideas of males as domi nant and females as subordinate need to be challenged. Another article in the san1e book, "Feminism: A Movement to End Sexist Oppression" by bell hooks,
emphasizes that overcoming the thought of men dominating women "must be solidly based on a recognition of the need to eradicate the underlying cultural bi ases and causes of sexism and other group oppression" (p. 240). This image of male dominance is given to readers when they see an issue in which no woman has a voice. Also, having more female authors will help provide female role models, which will help inspire fe male students in their love of math and encourage them to pursue it. Christina Green 1 303 Roosevelt Road St. Cloud, MN 56301 USA e-mail: grch01
[email protected] The Editor Replies:
The exact number of women authors in the issue you chanced to read first is zero. This is low, for us: many issues be fore and since it have numerous women authors (though I note that vol. 26, no. 2 again has none-sorry). It is Intelli gencer policy to encourage participation by mathematicians of whatever sex, whatever nation, whatever background. The policy has been stated in print be fore, and your letter is a welcome occa sion to state it again. As you say, we try to give women a voice. We also try to spread awareness of their achievements; and we provide a forum for discussion of ways to re move the barriers to their full partici pation in the profession. I must say, though, that I hope it was inadvertent that you said women are now equally prominent in mathematics. So far, no. We observe that more than half of the best mathematics is done by men, and we ask, are women being dis couraged from studying it? are they be ing eliminated by unfair grading? are they being refused jobs at the level they have earned? We fmd that all of these deterrents sometimes operate, and we struggle to eliminate them. In order to do it effectively, we need to acknowl edge the nature of the imbalance. I hope that as an educator you will help more girls become enthusiastic about mathematics. (Don't feel bad if you engage some boys too.)
© 2005 Spnnger Sc1ence+Bus1ness Media, Inc , Volume 27, Number 1, 2005
5
ERIC GRUNWALD
Eponymphomania "But if the arrow is straight And the point is slick, It can pierce through dust no matter how thick. " -Bob Dylan [ 1]
he Mathematical Intelligencer is full of delightful surprises. Eric C. R. Hehner, in his paper "From Boolean Algebra to Unified Algebra" [2], claims that terminology that honors mathematicians is sometimes wrongly attributed, is used deliberately to lend respectability to an idea, and even when the intention is genuinely to honor the eponymous person, the effect is to make the mathematics forbidding and inaccessible.
As I perused Hehner's paragraph (I use this term de scriptively, not honorifically), I found myself in general agreement with him, with perhaps one or two caveats. Per
Queries with disjunction are first converted to disjunc tive nor-mal form (disjunction of conjunctions) . . . [5] These gnomic utterances raised the following important research questions:
sonally, I would preserve Abelian groups: decent mathe
(a) Why ARE certain WORDS written in upper case for no
matical jokes are rare, and "What's purple and commutes?
APPARENT reason? This is surely much more off
A commutative grape" seems to lose something in the trans
putting than any mere use of an honorific name. I feel
lation. I would also vote in favour of topological spaces
I'm being yelled at by NAND and NOR, and I dislike
whose points are hausdorff from one another (and salads whose ingredients are waldorf). And I certainly advocate
them already. (b) Why is something written "#" called the "Peirce
arrow"
that we continue to remember Norbert Wiener for his sem
rather than the "Peirce sharp sign" or the "Peirce waffle
inal invention of the schnitzel. But the biggest exception to
iron"? Further internet research revealed that this sym
Hehner's generally sensible rule should surely be made for
bol appears variously as "#",
an eponymous term of astonishing beauty to be found to
is a bitter controversy amongst logicians, or whether it's
wards the end of his paragraph. It appears that there ex ists something called the Peirce arrow.
a consequence of the inadequacy of my computer (run
As Bob Dy
pect that unless Mr. Peirce was an extremely poor archer
Mr. Peirce's arrow is surely worth keeping.
lan pointed out, it penetrates dust no matter how thick. It
"D", and "!". Whether this
ning on low-octane Windows he probably meant "!", so
66) I don't know, but I sus
I'll go with that one.
is inspiring: just as Sir Karl Popper regarded Darwin's evo
(c) If we are going to be told that the Peirce arrow is not an
lutionary theory as a "metaphysical research program," so
antique car!, why aren't we told that Sheffer's stroke is
the Peirce arrow was a metaphysical research program for
not a medical condition!? Or that it's not a sexual tech
me. I determined to find out more about Mr. Peirce and his
nique!? In fact the author told us (the Sheffer stroke is
arrow. Googling intrepidly through hundreds of thousands
not a medical condition!)! (the Sheffer stroke is not a
of references, using both Dylan's and Hehner's spellings, I
sexual technique!). Or should that read "the Sheffer
uncovered these three pearls:
stroke is (a medical condition)! (a sexual technique)!"? Or should I actually have written "(the author told us the
x
=
y - Sheffer stroke, NAND; x # y - Pierce arrow,
NOR.
[3]
6
told us the Sheffer stroke is not a sexual technique!)"? (d) It's good to know that the Peirce arrow can be used to
. . . The questions also refer to "Sheffer's stroke" and "Pierce arrow" (not an antique car!) operators
Sheffer stroke is not a medical condition!)! (the author
. . . [4]
THE MATHEMATICAL INTELLIGENCER © 2005 Spnnger Scrence+Busrness Medra, Inc
generate a disjunction of conjunctions. But if you want a term truly guaranteed to put off any aspiring student,
"disjunctive nor-mal form" must be it: unlike the erudite
AU T H O R
readers of this journal, the student might become dan gerously disoriented when trying to distinguish between a disjunction of conjunctions and a conjunction of dis junctions. After
all, when Polonius so wisely advised "(a
[6],
borrower)! (a lender) be" junctively or disjunctively?
was he speaking con
As with so much of the the
oretical output of that particular author (for example his paper To Be or Not To Be? The Law of the Excluded Middle [7]), I can't fully get to grips with it, so giving my pen a long, lingering, disjunctive gnaw I must pass on. Let's give two cheers for the Peirce arrow. It may be elit
ERIC GRUNWALD
ist and off-putting. It may use an author's prestige to lend
Perihelion Ltd.
respectability to an unremarkable idea. It may, for all I
1 87 Sheen Lane
know, be attached to the wrong bloke altogether. But it's
London SW14 8LE
beautiful. It's poetic. It inspires research. And unlike other
UK
rival names, it doesn't give my eyes disjunctivitis. So please
e-mail:
[email protected] don't shoot the arrow away. You may chuck away at a stroke all reference to Sheffer. I would shed no tears at the
Eric Grunwald received his doctorate in mathematics from Ox
demise of Banach spaces or Sylow's theorem. But Peirce's
ford. Since then he has been employed in the chemical, en
arrow deserves to thrive, along with all the other beautiful
ergy, and health-care industries, and has become expert in
terms that enrich mathematics: wonderful expressions like
advising organizations on
Weyl integrals, Killing fields, the Gordan knot, the Roch
not found anyone else in the field of future thinking who knows
their future planning. He has, sadly,
group, Jordan delta functions, Plateau's plane, Taylor cuts,
much about mathematics; if there are others, he would like to
the Schur Certainty Principle, Abel-Baker-Chasles-Lie sym
meet them.
bols, and, since I'm feeling rather eponymous just now, the Grunwaldian or recursive citation.
[8]*
REFERENCES
[5] iptps03 .cs. berkeley .edu/final-papers/result_caching. pdf
[ 1 ] Bob Dylan, Restless Farewell, 1 964.
[6] W. Shakespeare, Hamlet, act 1 , scene 3, 1 601 .
[2] E. C. R. Hehner, "From Boolean Algebra to Unified Algebra," The
[7] W. Shakespeare, private communication. [8] E. J. Grunwald, "Eponymphomania," The Mathematical lntelligencer,
Mathematical lntelligencer, vol. 26, no. 2 , 2004.
[3] http://rutcor.rutgers.edu/pub/rrr/reports2000/32 .ps. Rutcor Research
vol. 27, no. 1 , 2005.
Report
[9] K. D0sen, "One More Reference on Self-Reference," The Mathe
[4] web.fccj.org/�1 dap991 1 /COT1 OOOUpdate. html
matical lntelligencer, vol. 1 4, no. 4, 4-5, 1 992.
'As a true Hehnerian, or eponymous honorific, the term "Grunwaldian" not only attempts to add weight to a pointless concept, it is also elegantly misattributed. The ngorous process of peer review through which this paper was extruded has revealed that the recursive citation has appeared previously in the literature [9].
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[email protected] © 2005 Spnnger Sc1ence+Bus1ness Media, Inc., Volume 27, Number 1 , 2005
7
W. M. PRI ESTLEY
Plato and Anaysis he Statesman, a late work of Plato's, begins with a playful allusion to mathematics. The setting is an ongoing inquiry ostensibly intended to complete the delineation of the true natures of the Sophist, the Statesman, and the Philosopher, but more basic philosophical issues are raised as well. As the scene opens we find Socrates thanking Theodorus, an elderly mathematician, for having brought
[SocRATES] I owe you many thanks, indeed, Theodorus, for
to Athens with him his young student Theaetetus and an
the acquaintance both of Theaetetus and of the Stranger.
unnamed philosopher visiting from Elea, the Greek town
[THEODORUS] And in a little while, Socrates, you will owe
in southern Italy that is home to Zeno and his paradoxes.
me three times as many, when they have completed for
The "Eleatic Stranger"-the appellation given this name
you the delineation of the Statesman and of the Philoso
less visitor in older translations of Plato-may suggest to
pher, as well as of the Sophist.
0
us the archetypal masked man who descends upon the ac
[SocRATES] Sophist, statesman, philosopher!
tion from nowhere to round up the outlaws and establish
Theodorus, do my ears truly witness that this is the es
order.
my dear
timate formed of them by the great calculator and geo
Sure enough, the Stranger has already gone after the Sophist earlier in the day, using a dichotomizing technique
metrician? [THEODORUS] What do you mean, Socrates?
that closely resembles the modern analyst's bisection
[SocRATES] I mean that you rate them all at the same
method of successive approximations. In the words of a
value, whereas they are really separated by an interval,
modern commentator [P2, p. 235], he "first offers six dis
which no geometrical ratio can express.
tinct routes for understanding the [S]ophist, by systemati
[THEODORUS] By Ammon, the god of Cyrene, Socrates,
cally demarcating specific classes within successively
that is a very fair hit; and shows that you have not for
smaller, nested ... classes of practitioners; these subclasses
gotten your geometry. I will retaliate on you at some
are then identified as the [S]ophists." Then, following a
other time .
. . . (Statesman 257a-b)
lengthy discussion to introduce a "change of coordinates," the Stranger resumes his search and finally obtains neces
What is the "hit" by Socrates that provokes Theodorus's
sary and sufficient conditions to characterize the slippery
oath? Some commentators on Plato say that Socrates is
Sophist. Socrates expresses delight.
alluding to the existence of incommensurables in geome
Here, in Benjamin Jowett's nineteenth-century transla tion, are the opening lines that follow in the
8
Statesman.
THE MATHEMATICAL INTELLIGENCER © 2005 Spnnger Scrence+Busrness Medra, Inc.
try, something that Plato was fond of mentioning in other contexts. Thus, Socrates would seem to be implying that
IIOAITIKOI. TA TOT �IAAOfOT IlPOl:OIJA
�OKPATH�,
8EO�OPO�,
;a'ENO�,
�OKPATH� 0 NEOTEPO�.
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8EO. Taxa M ye, cJ �wKpaTEf, o¢etA�UEt� Tav"i I "i J S;o \ I \ I
TTJ� Tpt7rAautav, E7rEtuav TOV TE 7rOAtrtKov a7repyaJ
uwvral UOt Kat TOll if>tAOUoif>ov,
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*
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ot TV np.fi 7rAEOll [aAA�I\wv] acf>EuTaUlV � KaTa T�V avaA.oy{av T�v T�� vwr/pa� rEXIITJf. \
8EO. Eo ye v� Tov �p.ITepov fJeov, cJ �wKpaTe�, ,A
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Only a third part of our task is done: nay, not a third, for the States man rises above the Sophist in value and
the Philo sopher above the Statesman in more than a geo metrical ratio.
'9
Tov .n.p.p.wva, Kat utKatw�, Kat 1ravu p.ev ouv !J.VTJ!J.OVt"i ' • ' , ' ' l c' Kw� E7rE7rATJc;a� p.ot TO' 7rEpt' TOU� Aoytup.ou� ap.ap�
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,
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cJ Elve, p.TJ8ap.ror Ct7r0Kap.y� xapt(op.evo�, aA.A.' \
"i
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s;;.
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\
Jl
Ec;TJ�, EtTE Tov 7T'OI\trtKOV avopa 7rpOTEpov ELTE TOV ¢tA.ouo¢ov 7rpoatpe'i, 1rpoeA.op.evo� 8tlteA.Oe. � -E •
I , ' I " t TauT., W.,. 8EOuwpe, s;;. 7rOLTJTEOV' E7rEL7rfEp a1rac; �
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.
" EDUARDO L. ORTIZ
ALLAN PINKUS
Department of Mathematics
Department of Mathematics Technion
Imperial College London, South Kensington Campus United Kingdom
Haifa, 32000 Israel
e-mail:
[email protected] e-mail:
[email protected] Eduardo L Ortiz did his doctoral work under the supervision of
Allan Pinkus, a native of Montreal, did his undergraduate work at
London, SW7 2f.Z.
Mischa Collar in Buenos Aires, and subsequently went to Dublin for research under Corneli us Lanczos. Since
1 963 he has been at
Imperial College London, where he is now Professor. He has writ
under Samuel Karlin's supervision. Since
1 977 he has been at the
Technion. His research interests center on approximation theory.
ten prolifically on functional analysis and its applications and on
He was for ten years an Editor-in-Chief of t he Journal of Approx
history of mathematics. He has held visiting positions at Harvard
imation Theory.
and the universities of Orleans and Rouen.
30
McGill University and his doctoral work at the Weizmann Institute
THE MATHEMATICAL INTELLIGENCER
[M33] Sur les problemes mixtes dans l'espace heterogene, Equation de Ia chaleur a n di
[2] S. Bernstein, Sur les recherches recentes
[7] A. Korn, U ber Minimalflachen, deren Rand
relatives a Ia meilleure approximation des
kurven wenig von ebenen Kurven abwe
mensions, C. R. Acad. Sci. Paris, 1 99 (1 934),
functions continues par des polyn6mes,
ichen Abhdl. Kg/. Akad. Wiss., Phys-math,
821 -824.
Proceedings of the Fifth International Con
Berlin, (1 909), 1 -37.
[M34] Functional Methods for Boundary Value
(Cambridge,
[8] R. von Mises and H . Pollaczek-Geiringer,
Problems (in Russian), Works of the 2nd All
22-28 August 1 91 2), E. W. Hobson and
Praktische Verfahren der Gleichungsaufl6-
Union Mathematical Congress, Leningrad,
A. E. H. Love, eds . , Cambridge, 1 91 3, Vol.
sung, Zeitschrift fur Angewandte Mathe
Leningrad-Moscow, 1 (1 935), 31 8-337 .
I, 256-266.
matik und Mechanik, 9 (1 929), 58-77 and
gress of Mathematicians,
[M35] General problems o f stability o f motion,
[3] S. N. Bernstein, Sur l 'ordre de Ia meilleure
by A. Lyapounov, (in Russian), Ch. H. Muntz,
approximation des functions continues par
ed. , Leningrad-Moscow, 1 935.
les polyn6mes de degre donne, Mem. Cl.
quoted
Sci. Acad. Roy. Be/g . , 4 ( 1 9 1 2), 1 -1 03.
Sozialerziehung der Durerschule Hochwald hausen, Hochhausmuseum and Hohha
[M36] Zur Theorie der Randwertaufgaben bei hyperbolischen Gleichungen, Prace Mat.
[4] E. L. Ortiz, "Canonical polynomials in the
Fiz. , (Gedenkschrift fur L. Lichtenstein), 43
Lanczos' Tau Method, " B. P. K. Scaife,
(1 936) , 289-305.
ed. , Studies in Numerical Analysis, New
[M37] Les lois fondamentales de l'hemody namique, C. R. Acad. Sci. Paris , 280 (1 939),
York, 1 97 4, 73-93, on 75.
REFERENCES
[ 1 ] K.
Die
subibliotek, Lauterbach, 1 986, p. 1 5. [1 0] Der
Jude,
Judischer
Verlag,
Berlin,
[1 1 ] E. C. Titchmarsh, The Theory of the Rie
of Science and Learning and the Migration
mann Zeta-Function, Oxford, 1 951 , p. 28.
of Scientists in the late 1 930s," Panel 's
[1 2] D. Amira, La Synthese Projective de Ia
Chairman's lecture, Proceedings of the
Geometrie Euclidienne, ltine and Shoshani,
1 13th annual meeting of the American His
Tel-Aviv, 1 925.
sogenannter willkurlicher
torical Association, Washington, 93 (1 999),
die
Funktionen einer reellen Veranderlichen, Sitzungsberichte der Akademie zu Berlin,
1 -28.
[1 3] G. G. Lorentz, Mathematics and Politics in the Soviet Union from 1 928 to 1 953, Jour
[6] Ch. Muntz, Wir Juden, Oesterheld and Co. ,
1 885, 633-639 and 789-805.
r
Karl-August Helfenbein,
analytische
U ber
Weierstrass,
Darstellbarkeit
in
1 9 1 6-1 928.
[5] E. L. Ort1z, "The Society for the Protection
600-602 .
1 52-164. [9] L. Butschli, HochwaldhauserDiary, 39, 39a.;
Berlin, 1 907.
nal of Approximation Theory, 1 1 6 (2002),
1 69-223.
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© 2005 Spnnger Sc1ence+Bus1ness Med1a, Inc., Volume 27, Number 1 , 2005
31
KELLIE 0. GUTMAN
Quando Che 'l Cubo
e
n the history of mathematics, the story of the solution to the cubic equation is as convoluted as it is significant. When I first read an account of it in William Dun ham's Journey Through Genius 1 in 2000, I was captivated by the personalities, the intrigues, and the controversies that were part of mathematics in sixteenth-century Italy. For those unfamiliar with it, the story runs as follows:
In the early 1500s, the mathematician Scipione del Ferro of the University of Bologna discovered how to solve a de pressed cubic-one without its second-degree term-but in the style of the day he kept his discovery to himself. On his deathbed in 1526 he divulged the solution to his student Antonio Fior. 2 Eight years later Niccolo Fontana, known as "Tartaglia" ("Stutterer"), hinted that he knew how to solve cubics that were missing their linear term. Fior publicly challenged Tartaglia to a contest in February of 1535, sending him a set of thirty depressed cubics to solve. At first Tartaglia was stumped, but with the deadline approaching, he fig ured out how to solve depressed cubics, thus winning the challenge. In Milan, the mathematician/physician Gerolamo Car dano heard about Tartaglia's grand accomplishment. For several years, he pleaded with Tartaglia to tell him his se cret. Finally in 1539, Tartaglia traveled to Milan from Venice and told Cardano the solution, but made him swear never to publish it. With continued research, Cardano figured out how to re duce a general cubic to a depressed one, thus completely solving the classical problem of the cubic. Then his assis tant Lodovico Ferrari extended this string of discoveries by solving fourth-degree problems, but both men refrained
from publishing their results because they were based on Tartaglia's solution. On a hunch, Cardano and Ferrari traveled to Bologna in 1543 to look at the papers of Fior's master, Scipione del Ferro, who they must have reasoned also knew the solu tion to depressed cubics. They found Scipione's original al gorithm and it was identical to Tartaglia's. Finally, Cardano felt released from his oath to Tartaglia Giving full credit to both Scipione and Tartaglia, he published the solution to the depressed cubic, his own solution to the general cubic, and Ferrari's solution to the quartic, in 1545, in a huge tome, Ars Magna. This widely dispersed work is con sidered by many to be the first book ever written entirely about algebra In it, Cardano devoted little space to the solution of the quartic, because a fourth power was considered a mean ingless concept, not corresponding to any physical object. Tartaglia was enraged. The following year, in his own book Quesiti et inventioni diverse, Tartaglia presented his version of a long conversation between himself and Car dano from their encounters six years earlier, in which he made it clear that his "invention" was not to be disclosed. He then presented his solution in a poem, saying this was the easiest way for him to remember it. *
*
*
' Dunham, William. Journey Through Genius . New York: John Wiley & Sons, Inc., 1 990. 2Many of the historical facts came from the MacTutor History of Mathematics archive of the School of Mathematics and Statistics, University of St Andrews, Scotland Created by John J. O'Connor and Edmund F. Robertson http://www-history.mcs.st-andrews.ac. uk/history/index.html
32
THE MATHEMATICAL INTELLIGENCER © 2005 Springer Scrence + Busrness Medra, Inc
Quando che'l cubo3 Quando che'l cubo con le cose appresso Se agguaglia a qualche numero discreto Trovar dui altri differenti in esso. Dapoi terrai questo per consueto Che'l lor produtto sempre sia eguale AI terzo cubo delle cose neto, El residuo poi suo generate Delli lor lati cubi ben sottrati Varra la tua cosa principale. In el secondo de cotesti atti Quando che'l cubo restasse lui solo Tu osservarai quest'altri contratti, Del numer farai due tal part'a volo Che l'una in l'altra si produca schietto El terzo cubo delle cose in siolo Delle qual poi, per commun precetto Torrai li lati cubi insieme gionti Et cotal somma sara il tuo concetto. El terzo poi de questi nostri conti Se solve col secondo se ben guardi Che per natura son quasi congionti. Questi trovai, e non con passi tardi Nel mille cinquecente, quatro e trenta Con fondamenti ben sald'e gagliardi
thus propelling the poem forward. This form is extraordi narily well-known by Italians. *
*
In the early sixteenth century, algebra was rhetorical that is, variables, the equal sign, negative numbers, and the concept of setting something equal to zero did not exist. Everything was described solely through words. Instead of writing "X3 + mx = n" one would write cuba con cosa ag guaglia ad un numero or "cube and thing are equal to a number." It was a cumbersome system, and calculations and proofs were difficult to follow. When I saw Tartaglia's poem for the first time in early 2004, I was so taken with it that I had to translate it, but I soon found myself faced with a dilemma. Either I could translate it literally as he wrote it, and have it be as obscure as his was (and it is obscure), or I could do a modern trans lation and essentially say, "This is what he meant, though it is not what he said." The second way would make it very clear for today's reader. Neither of these felt quite right to me. Instead, I decided to bridge the two worlds of Renais sance mathematics and modern mathematics, attempting to retain the poem's ancient flavor along with its terza rima, but using variables where Tartaglia used only words. Because the vast majority of Italian words end in an un stressed syllable, it is natural to have iambic lines of po etry with eleven syllables. It is slightly more difficult in Eng lish. In my translation I have used an alternating pattern of masculine rhymes, with the stress and rhyme on the final syllable, and feminine rhymes, which rhyme on the stressed penultimate syllable. *
*
*
When X Cubed
Nella citta dal mar' intorno centa. Any Italian who encountered this poem would have im mediately recognized it as being written in the celebrated form known as terza rima, invented by Dante Alighieri and used in his masterwork, La Divina Commed·ia. Like Dante, Tartaglia wrote in Italian, which was the language of liter ature, not Latin, which was the main language of science: this was because Tartaglia did not know Latin. Terza rima is made up of eleven-syllable, or hendecasyllabic, lines. Each line is iambic with five stressed and six unstressed syllables. It is an especially fitting form for a poem about cubic equations because there are two sets of threes con tained in it: the poem is written in tercets, or three-line stan zas, and all the rhymes, except at the start and finish of the poem, come in triplicate, with the center line of each ter cet rhyming with the outer lines of the tercet following it,
*
When x cubed's summed with m times x and then Set equal to some number, a relation Is found where r less s will equal n. Now multiply these terms. This combination rs will equal m thirds to the third; This gives us a quadratic situation, Where r and s involve the same square surd. Their cube roots must be taken; then subtracting Them gives you x; your answer's been inferred. The second case we'll set about enacting Has x cubed on the left side all alone. The same relationships, the same extracting: ----- -
-----
3Tartaglia. Niccol6, Ouesiti et inventioni diverse de N 1ccol6 Tartalea Bris01ano. [Stampata in Venetia per Venture Rotflnelli, 1 546.] Quesito XXXI II I. Fatto personalmente dalla eccellentia del medesimo messer H1eronimo Cardano 1n Millano in casa sua adi. 25. Marzo.1 539 "Quando chel cubo con le cose apresso . . . " - begins leaf 1 23 recto . . . Nella citta dal mar' intorno centa " - ends leaf 1 23 verso (Also reproduced on the following Web site: http· IIdigi lander. libero. itlbasecinqueltartaglia/eq uacu bica. htm)
© 2005 Spnnger Sc1ence+ Bus1ness Media, Inc , Volume 27, Number 1 , 2005
33
Seek numbers r and s, where the unknown rs will equal m-on-3 cubed nicely, And summing r and s gives n, as shown. Once more the cube roots must be found concisely Of our two newfound terms, both r and s, And when we add these roots, there's x precisely.
Completing the S quare m
I
X
The final case is easy to assess: Look closely at the second case I mention It's so alike that I shall not digress. These things I've quickly found, they're my invention, In this year fifteen hundred thirty-four, While working hard and paying close attention,
Figure 1 . A version of AI-Khwarizmi's completion of the square. Mov ing left to right, the equation can be read directly off the diagram.
Surrounded by canals that lap the shore. So what exactly is Tartaglia saying? He's saying that when ,i3 + mx = n, two other numbers, r and s, can be found such that r - s = n and rs = (m/3)3. Mathematicians of his day knew that when they were told the values of a product and a difference (or sum) of two unknown numbers, they had what I have called a "quadratic situation" (there was no such thing as a quadratic equation). They had an algorithm, which was tricky but manageable, to fmd the solutions to such sit uations. In fact, because they didn't recognize negative num bers, they had a set of variants of what we would think of as one single thing, namely the quadratic formula. Using the applicable variant, one could solve for r and s. Next, Tartaglia is telling his readers to take the cube roots of the numbers r and s, and to subtract the cube root of s from that of r. This will be x, the solution to the given cubic. He then moves on, in the fourth stanza, to what was con sidered a different situation, when .i3 = mx + n, and he gives the solution again. The third case, when .i3 + n = mx, he says, in the seventh stanza, is almost exactly like the second, and so he leaves that for the reader to figure out. He con cludes with a flourish by claiming credit for the discovery, and telling his readers he found the solution in Venice. *
*
*
Tartaglia discovered his solution by thinking about an actual physical cube. To him, and most likely to Scipione as well, the solution to a problem involving a cubic was em bodied in a real cube. Seven hundred years earlier, in Bagh dad, Al-Khwarizmi (from whose name comes the word "al gorithm") thought about a square when working on problems involving quadratics. He came up with a formula for "completing the square" to solve such problems. An equation of the type x2 + mx = n can be pictured by first drawing a square of side x (see Figure 1). Next make two congruent rectangles of length x and width m/2, and attach them to two adjacent sides of the square. The di mensions m/2 and x are picked for very good reasons two rectangles of this size together make up an area of mx, to add to the original square of the area x2, and these three together have a joint area of n, giving x2 + mx = n.
34
THE MATHEMATICAL INTELLIGENCER
The picture looks like a square cardboard box from above, with two adjacent flaps open. It calls out for one other square, of side length m/2, to be drawn in, in order to complete the larger square. Let's call the side of this new big square t, and the side of the new little square u. When we combine the area n with the area u2, which is (m/2)2, we get the area of the larger square, t2• The square root of this square area-that is, the square root of n + (m/2)2gives us the side length t. But t is equal to x + m/2, so x equals Vn + (m/2)2 - m/2. Thus by completing the square, Al-Khwarizmi solved the quadratic. In a similar fashion to Al-Khwarizmi, Tartaglia envisioned "completing the cube" to solve the depressed cubic. He took Al-Khwarizmi's drawing into a third dimension (Fig. 2). With an equation of the form .i3 + mx = n, he started by imagining a cube of side x (this corresponded to the square of side x in two dimensions). He then looked for analogous volumes to play the role of the two rectangles flanking the square of side x, but since he was in three di mensions he instead imagined three slabs. Each had one side of length x, and two other sides of unknown lengths, which we will call t and u. These three slabs fit neatly
Completing the Cube
Figure 2. Tartaglia's completion of the cube. Once again the equa tion can be read directly off the diagram.
x3 + 3tux
3tux
I
I
x3 + mx
n
This is a breakthrough moment for Tartaglia, because it tightly connects the unknowns, t and u, with the knowns, m and n:
3tu = m,
Figure 3. Like a Necker cube, this picture flips between two inter pretations. In the intended interpretation, one sees three slabs, each of volume
tux,
of side
In the other interpretation (and this came as a complete
u.
swirling counter-clockwise around a (missing) cube
and lovely surprise to me) one sees a cube of side u sitting nestled in one corner of a cutaway cube of side t, and thanks to the colors
This is very promising, but he is not there yet, because he doesn't know how to solve these equations for t and u in terms of m and n. As he considers these equations, however, Tartaglia sees that he has a situation that comes very close to being a quadratic in t and u, but just misses-namely, he has a product and a difference involving t and u, but one of them involves their cubes. Thus provoked, Tartaglia has another in sight. He gives names to the two cubic volumes, calling t3 "r" and ua "s, " knowing that in this way he will obtain a genuine quadratic situation (involving a difference and a product) with his new variables r and s. Now his equations are
painted on the large cube's walls, one cannot help "seeing" (though
tux, once counter-clockwise about the little cube of side u. they are missing) the three slabs of volume
around the cube of side x, thus giving him a larger cube of side t, but (as before) with one crucial piece missing. In or der to complete the larger cube, Tartaglia added one last cube of side u (corresponding to the little square of side u that completed Al-Khwarizmi's square; Fig. 3). Each of the three slabs has sides of length t, u, and x, and so the total volume of the slabs is 3tux. Now the vol umes of the two interior cubes are x3 and u3, so the total volume of the big cube is .x3 + 3tux + u3 , but of course it is also t:1. In symbols,
.i3 +
1· - s = n rs = (m/3)3 .
again swirling
3tux + u:l = (l.
We can imagine Tartaglia striving to imagine the di mensions of a physical cube that would represent the so lution to an actual depressed-cubic problem posed by his challenger Fior. In Al-Khwarizmi's quadratic, the value of u is known instantly without calculation. But in the case of the cubic, things are not so simple, because one doesn't know the value of either t or u. In the realm of all possible cubes, Tartaglia needed to find the one cube with the ex act dimensions that satisfy his problem. He had to imagine the lengths u and t both changing (the overall cube grow ing and shrinking, and also the cube of side x changing size because it is determined by t and u, its side being t - u) . It seemed as if the search for the proper cube could only be carried out by trial and error, without any formula, and thus it was not really a mathematical solution. At this point, though, rather than giving up, Tartaglia has a brilliant insight. Looking at his equation (above), he re alizes that if he merely moves u3 to the right side, it will give him a new equation that precisely embodies Fior's de pressed cubic _Tl + m:J.: = n, with 3tu playing the role of m and t3 - u3 playing the role of n.
The last equation is an immediate consequence of the def inition of r and s. From 3tu = m it follows that tu m/3, and thus, cubing both sides, t3u3 = (m/3)3. Now he is operating in familiar territory. He can easily find his quadratic by eliminating r as follows: r = n + s and therefore rs = s(n + s) , giving =
s2
+
ns = (m/3)3.
Tartaglia has at last come full circle. Mter starting out with Al-Khwarizmi's model of completing the square in or der to come up with his own model of the cubic, he now applies Al-Khwarizmi's square-completing method to solve this quadratic for r and s; having gotten those, he can then take their cube roots to obtain the values of t and u. Then he merely subtracts u from t, and x has been found. *
*
*
When Cardano published Ars Magna, rather than giving a general proof, he illustrated the solution to this particu lar cubic: ;il + 6x 20. Following the poem's directions, here is how it is solved. =
.i3 + 6x
=
20
r - s = 20 rs = (6/3)3 = 23 = 8 r = 20 + s and therefore s(20 s2 + 20s = 8 s2 + 20s - 8 = 0.
+
Using the quadratic formula to solve for
s) =
8
s, we get
s = c - 20 ± V4oo + 32)/2 = - 10 ± v'i08 = v1o8 - 10 r = s + 20 v'i08 + 10. =
..
© 2005 Springer Sc•ence+Bus1ness Med1a. I n c Volume 27, Number 1 . 2005
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Numerically,
r=
AU T H O R
20.3923 and
s
=
.3923.
Then, taking these numbers' cube roots,
x = Vr - Vs X = 2. 73205 - . 73205 X = 2.
6x
If we plug this back into the original equation x3 + = 20, we find that it is correct: 8 + 12 20. The method works, although it must be admitted that it makes it look fortuitous that the answer is a simple integer.
=
KELLIE 0. GUTMAN *
*
*
75 Gardner Street West Roxbury, MA 021 32-4925
Finding a solution by radicals to the cubic was a monu mental accomplishment. However, it led to a thorny ob stacle: in the case of a cubic equation that had only one real root (back then, mathematicians would have said the equation had only one root at all, for no one suspected that all cubics have three roots), the algorithm always yielded that root. By contrast, in the case of a cubic that had three real roots, the algorithm seemed to yield nonsense. Even if the three real roots were already known, it led to expres sions featuring negative numbers under the square-root sign, a situation that Cardano dubbed the casus irre ducibilis, reflecting the fact that Renaissance mathemati cians were not comfortable with negative numbers, let alone their square roots. The Bologna mathematician Rafael Bombelli took Car dana's casus irreducibilis very seriously and tried to make sense of the square roots of negative numbers. He figured out how to do the four standard arithmetical operations not only with negative numbers but also with their "imaginary" square roots, and shortly before his death in 1572, he pub lished a book on this topic titled Algebra, in which he pre sented an early symbolic notation system. Although he never found out how to take cube roots of complex num bers in general, he was able to determine the complex cube root called for by Cardano's algorithm in one specific case, and he showed that the two imaginary contributions to the fmal answer canceled each other out, leading to a purely real root. More details of Bombelli's work will be found in a recent scholarly article in this journal by Federica LaNave and Barry Mazur; see vol. 24, no. 1 (2002), 12-2 1 . Despite this accomplishment, Cardano's formula pro vided Bombelli with only one of the equation's three roots, and it took another 40 years until Fran0, it is true with probability 1 that for all sufficiently large n, the d-di mensional IDLA blob of wdnd particles will contain every point in a ball of radius ( 1 - E)n, and no point outside of a ball of radius (1 + E)n. To be more specific, we could hope to define inner and outer error terms such that, with probability 1 , the blob lies between the balls of radius n {lj(n) and n + 80(n). In a subsequent paper [10], Lawler proved that these 1 could be taken on the order of n 13• Most recently, Blachere [3] used an in duction argument based on Lawler's proof to show that these error terms were even smaller, of logarithmic size. The precise form of the bound changes with dimension; when d = 2 he shows that Mn) O((ln n ln(ln n)) 112) and 80(n) = O((ln n)2). Errors on that or der were observed experimentally by Moore and Machta [ 12]. So how does the random walk-based IDLA relate to the deter ministic rotor-router? I start drawing the connection with one key fact. =
It's Abelian!
Here's a possibly unexpected property of the rotor-router model: it's Abelian. There are several senses in which this is true. Most simply, take a state of the ro tor-router system-a set of occupied sites and the directions all the rotors point-and add one bug at a point Po (not necessarily the origin now) and let it run around and find its home P1 . Then add another at Q0 and let it run until it stops at Q 1 . The end state is the same as the result of adding the two bugs in the opposite order. This relies on the fact that the bugs are indistinguishable. Consider the (next-to-)simplest case, in which the paths of the P and Q bugs cross at ex actly one point, R. If bug Q goes first instead, it travels from Q0 to R, and then follows the path the P bug would have, from R to P1 . The P bug then goes from P0 to R to Q1. At the place where their paths would first cross, the bugs effectively switch identities. For paths
whose intersections are more compli cated, we need to do a bit more work, but the basic idea carries us through. Taking this to an extreme, consider the "rotor-router swarm" variant, where traffic is still directed by rotors at each lattice site, but any number of bugs can pass through a site simulta neously. The system evolves by choos ing any one bug at random and moving it one step, following the usual rotor rule. Here too the final state is inde pendent of the order in which bugs move; read on for a proof. To create our rotor-router picture, we can place three million bugs at the origin simul taneously, and let them move one step at a time, following the rotors, in what ever order they like. In fact, even strictly following the rotors is unnecessary. The rotors con trol the order in which the bugs depart for the various neighbors, but in the end, we only care about how many bugs head in each direction. Imagine the following set-up: we run the original rotor-router with three mil lion bugs as first described, but each time a bug leaves a site, it drops a card there which reads "I went North," or whichever direction. Now forget about the bugs, and look only at the collec tion of cards left behind at each site. This certainly determines the final state of the system: a site ends up oc cupied if and only if one of its neigh bors has a card pointing toward it. Now we could re-run the system with no rotors at all. When a bug needs to move on, it may pick up any card from the site it's on and move in the in dicated direction, eating the card in the process. No bug can ever "get stuck" by arriving at an occupied site with no card to tell it a way to leave: the stack of cards at a given site is just the right size to take care of all the bugs that can possibly arrive there coming from all of the neighbors. (There is, however, no guarantee that all the cards will get used; left-{)vers must form loops.) A version of this "stacks of cards" idea appeared in Diaconis and Fulton's original paper, in the proof that the random-walk version is likewise Abelian-i.e., that their prod uct operation is well defined. If the bugs are so polite as to take the cards in the cyclic N-W-S-E order
© 2005 Spnnger Science+ Business Media, Inc., Volume 27, Number 1 , 2005
59
in which they were dropped, then we simulate the rotor-router exactly. If we start all the bugs at the origin at once and let them move in whatever order they want-but insist that they always use the top card from the site's stack we get the rotor-router swarm variant above; QED. Rotor-Roundness
Now let me outline a heuristic argu ment that the rotor-router blob ought to be round, letting the Lawler-Bram son-Griffeath paper do all the heavy lifting. I'd like to say that, for any c < 1, the n-bug rotor-router blob contains every lattice site in the disk of area en-as long as n is sufficiently large. My strat egy is easy to describe. Just as we did four paragraphs ago, think of each lat tice site as holding a giant stack of cards: one card for each time a bug de parted that site while the n-bug rotor router blob grew. Now we start run ning IDLA: we add bugs at the origin, one at a time, and let them execute their random walks. But each time a bug randomly decides to step in a given direction, it must first look through the stack of cards at its site, find a card with that direction written on it, and destroy it. As long as the randomly walking bugs always find the cards they look for, the IDLA blob that they generate must be a subset of the rotor-router blob whose growth is recorded in the stacks of cards. This key fact follows directly from the Abelian nature of the models. So the central question is, how long will this IDLA get to run before a bug wants to step in a particular direction and fmds that there is no correspond ing card available? Philosophically, we expect the IDLA to run through "almost all" the bugs without hitting such a snag: for any c < 1, we expect en bugs to ag gregate, as long as n is sufficiently large. If we can show this, we are certainly done: the rotor-router blob contains an IDLA blob of nearly the same area, which in turn contains a disk of nearly the same area, with probability one. To justify this intuition, we clearly need to examine the function d(iJ) which counts the number of depar-
60
THE MATHEMATICAL INTELLIGENCER
tures from each site. This is a nonneg ative integer-valued function on the lattice which is almost harmonic, away from the origin: the number of depar tures from a given site is about one quarter of the total number of visits to its four neighbors.
d(iJ)
=
± (d(i + 1,J} + d(i - 1,J} + d(i,j + 1) + d(i,j - 1)) - b(iJ)
Here b(iJ) = 1 if (i,J} is occupied and 0 otherwise, to account for the site's first bug, which arrives but never departs. When (i,J} is the origin, of course, the right-hand side should be increased by the number of bugs dropped into the system. Matthew Cook calls this the "tent equation": each site is forced to be a little lower than the average of its neighbors, like the heavy fabric of a cir cus tent; it's all held up by the circus pole at the origin-or perhaps by a bundle of helium balloons which can each lift one unit of tent fabric, since we do not get to specify the height of the origin, but rather how much higher it is than its surroundings. For the rotor-router, the approxima tion sign above hides some rounding er ror, the precise details of which encap sulate the rotor-router rule. For IDLA, this is exact if we replace d by a, the ex pected number of departures, and re place b by b, the probability that a given site ends up occupied. (The results of [9] even give an approximation of d.) Now, I'd like to say that at any par ticular site, the mean number of de partures for an IDLA of en bugs (for any c < 1 and large n) should be less than the actual number of departures for a rotor-router of n bugs. If so, we'd be nearly done, with a just a bit of easy calculation to show that the Vd-sized error term at each site in the random walk is thoroughly swamped by the (1 - c)n extra bugs in the rotor-router. But this begs the question of show ing that the rotor-router's function d and the IDLA's function d are really the same general shape. Their difference is an everywhere almost-harmonic func tion with zero at the boundary-but to paraphrase Mark Twain, the difference between a harmonic function and an almost-harmonic function is the differ ence between lightning and a lightning bug.
Simulation with Constant Error
After I wrote the preceding section, I learned of a brand-new result of Joshua Cooper and Joel Spencer. It doesn't tum my hand-waving into a genuine proof, but it gives me hope that doing so is within reach. Their paper [4] con tains an amazing result on the rela tionship between a random walk and a rotor-router walk in the d-dimensional integer lattice ll_d. Generalizing the rotor-router bugs above, consider a lattice ll_d in which each point is equipped with a rotor that is to say, an arrow which points towards one of the 2d neighboring points, and which can be incremented repeatedly, causing it to point to all 2d neighbors in some fixed cyclic order. The initial states of the rotors can be set arbitrarily. Now distribute some finite number of bugs arbitrarily on the points. We can let this distribution evolve with the bugs following the rotors: one step of evolution consists of every bug incre menting and then following the rotor at the point it is on. (Our previous bugs were content to stay put if they were at an uninhabited site, but in this ver sion, every bug moves on.) Given any initial distribution of bugs and any ini tial configuration of the rotors, we can now talk about the result of n steps of rotor-based evolution. On the other hand, given the same initial distribution of bugs, we could just as well allow each bug to take an n-step random walk, with no rotors to influence its movement. If you believe my heuristic babbling above, then it is reasonable to hope that n steps of ro tor evolution and n steps of random walk would lead to similar ending dis tributions. With one further assumption, this turns out to be true in the strongest of senses. Call a distribution of bugs "checkered" if all bugs are on vertices of the same parity-that is, the bugs would all be on matching squares if ll_d were colored like a checkerboard. Theorem (Cooper-Spencer). There is an absolute constant bounding the divergence between the rotor and 'ran dom-walk evolution of checkered dis tributions in ll_d, depending only on
Figure 6. The greedy sand-pile with three million grains.
Figure 7. A non-greedy sand-pile. Here the dominant color is yellow, which again indi cates the maximal stable site, now with three grains. It is hard to see the interior pix els colored black, indicating sites which were once filled but are now empty, impossi ble in the greedy version.
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the dimension d. That is, given any checkered initial distribution of a fi nite number of bugs in 7l_d, the differ ence between the actual number of bugs at a point p after n steps of ro tor-based evolution, and the expected number of bugs at p after an n-step random walk, is bounded by a con stant. This constant is independent of the number of steps n, the initial states of the rotors, and the initial dis tribution of bugs. I am enchanted by the reach of this result, and at the same time intrigued by the subtle "checkered" hypothesis on distributions. (Not only initial dis tributions: since each bug changes par ity at each time step, a configuration can never escape its checkered past.) The authors tell me that without this assumption, one can cleverly arrange squadrons of off-parity bugs to reorient the rotors and steer things away from random walk simulation. Thus the rotor-router deterministi cally simulates a random walk process with constant error-better than a sin gle instance of the random process usually does in simulating the average behavior. Recall that we saw a similar outcome in one dimension, with the goldbugs. There are other results which like wise demonstrate that derandomizing systems can reduce the error. Lionel Levine's thesis [ 1 1 ] analyzed a type of one-dimensional derandomized aggre gation model, and showed that it can compute quadratic irrationals with constant error, again improving on the Vn-sized error of random trials. Joel Spencer tells me that he can use an other sort of derandomized one-di mensional system to generate binomial distributions with errors of size In n instead of Vn. Surely the rotor-router should be able to cut IDLA's already logarithmic-sized variations down to constant ones. Right? Coda: Sandpiles
All of the preceding discussion ad dresses the overall shape of the rotor router blob, but says nothing at all about the compelling internal structure that's visible when we four-color the points according to the directions of
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the rotors. When we introduced the function d(i,J), counting the number of departures from the (i,J) lattice site, we were concerned with its approxi mate large-scale shape, which exhibits some sort of radial symmetry. The di rection of the rotor tells you the value of d(i, J) mod 4, and the symmetry of these least significant bits of d is an en tirely new surprise. I can't even begin to explain the fine structure-if you can, please let me know! But I can point out a surprising connection to another discrete dynam ical system, also with pretty pictures. Consider once again the integer points in the plane. Each point now holds a pile of sand. There's not much room, so if any pile has five or more grains of sand, it collapses, with four grains sliding off of it and getting dumped on the point's four neighbors. This may, in tum, make some neigh boring piles unstable and cause further topplings, and so on, until each pile has size at most four. Our question: what happens if you put, say, a million grains of sand at the origin, and wait for the resulting avalanche to stop? I won't keep you hanging; a picture of the resulting rub ble appears as Figure 6. Pixels are col ored according to the number of grains of sand there in the final configuration. The dominant blue color correspond ing to the largest stable pile, four grains. (This makes some sense, as the interior of such a region is stable, with each site both gaining and losing four grains, while evolution happens around the edges.) This type of evolving system now goes under the names "chip-firing model" and "abelian sandpile model"; the adjective abelian is earned because the operations of collapsing the piles at two different sites commute. In full generality, this can take place on an ar bitrary graph, with an excessively large sand-pile giving any number of grains of sand to each of its neighbors, and some grains possibly disappearing per manently from the system. Variations have been investigated by combina torists since about 1991 [2]; they adopted it from the mathematical physics community, which had been developing versions since around 1987
[ 1,5]. This too was a rediscovery, as it seems that the mechanism was first de scribed, under the name "the proba bilistic abacus," by Arthur Engel in 1975 in a math education journal [7,8]. I couldn't hope to survey the current state of this field here, or even give proper references. The bulk of the work appears to be on what I think of as steady-state questions, far from the effects of initial conditions: point-to point correlation functions, the distri bution of sizes of avalanches, or a mar velous abelian group structure on a certain set of recurrent configurations. Our question seems to have a dif ferent flavor. For example, in most sandpile work, one can assume with out loss of generality that a pile col lapses as soon as it has enough grains of sand to give its neighbors what they are owed, leaving itself vacant. The ver sion I described above is what I'll call a "greedy sandpile," in which each site hoards its first grain of sand, never let ting it go. The shape of the rubble in Figure 6 does depend on this detail; Figure 7 is the analogue where a pile collapses as soon as it has four grains, leaving itself empty. Most compelling to me is the fine structure of the sandpile picture. I'm amazed by the appearance of fractalish self-similarity at different scales de spite the single-scale evolution rule; I think this is related to what the math ematical physics people call "self-or ganizing criticality," about which I know nothing at all. But personally, in both pictures I am drawn to the eight petalled central rosette, the boundary of some sort of phase change in their internal structures. Bugs in the Sand
So what is the connection between the greedy sandpile and the rotor-router? Recall the swarm variant of rotor router evolution: we can place all the bugs at the origin simultaneously, and let them take steps following the rotor rule in any order, and still get the same final state. Since we get to choose the order, what if we repeatedly pick a site with at least four bugs waiting to move on, and tell four of them to take one step each? Regardless of its state, the rotor
directs one to each neighbor, and we mimic the evolution rule of the greedy sandpile perfectly. If we keep doing this until no such sites remain, we re alize the sandpile final state as one step along one path to the rotor-router blob. Note, in particular, that the n-bug rotor-router blob must contain all sites in the n-grain greedy sandpile. Surely it should therefore be possible to show that both contain a disk whose radius grows as Vn. More emphatically, the sandpile per forms precisely that part of the evolution of the rotor-router that can take place without asking the rotors to break sym metry. If we define an energy ftmction which is large when multiple bugs share a site, then the sandpile is the lowest-en ergy state which the rotor-router can get to in a completely symmetric way. When we invoke the rotors, we can get to a state with minimal energy but without the a priori symmetry that the sandpile evolution rule guarantees. And yet, empirically, the rotor-router final state looks much rounder than that of the sandpile, whose boundary has clear horizontal, vertical, and slope ::+:: 1 segments. At best, this only hints at why the sandpile and rotor-router internal structures seem to have something in common. For now, these hints are the best I can do.
mutative algebra and algebraic geometry, II
Acknowledgments
Thanks most of all to Jim Propp, who introduced me to this lovely material, showed me much of what appears here, and allowed and encouraged me to help spread the word. Fond thanks to Tetsuji Miwa, whose hospitality at Kyoto University gave me the time to think and write about it. Thanks also to Joshua Cooper, Joel Spencer, and Matthew Cook, for sharing helpful comments and insights.
(Turin, 1 990). Rend. Sem. Mat. Univ. Po litec. Torino 49 (1 991 ), no. 1 , 95-1 1 9 (1 993).
[7] Engel, Arthur. The probabilistic abacus. Ed. Stud. Math. 6 (1 975), 1 -22.
[8] Engel, Arthur. Why does the probabilistic abacus work? Ed. Stud. Math.
7
(1 976),
59-69. [9] Lawler, Gregory; Bramson, Maury; Grif feath, David. Internal diffusion limited ag gregation. Ann. Probab. 20 (1 992), no. 4, 21 1 7-21 40. 1 0. Lawler, Gregory. Subdiffusive fluctuations
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[ 1 ] Bak, Per; Tang, Chao; Wiesenfeld, Kurt. Self-organized criticality. Phys. Rev. A (3) 38 (1 988), no. 1 , 364-374.
Ann. Probab. 23 (1 995), no. 1 , 7 1 -86.
1 1 . Levine, Lionel. The Rotor-Router Model. Harvard University senior thesis. Preprint
[2] Bjorner, Anders; Lovasz, Laszlo; Shor, Pe
arXiv:math.C0/0409407 (September 2004).
ter. Chip-firing games on graphs. European
1 2. Moore, Christopher; Machta, Jonathan. In
J. Combin. 1 2 (1 99 1 ) , no. 4 , 283-291 .
[3] Blachere, Sebastien. Logarithmic fluctua tions for the Internal Diffusion Limited Ag gregation. Preprint arXiv:rnath.PR/01 1 1 253 (November 2001).
ternal diffusion-limited aggregation: paral lel algorithms and complexity. J. Statist. Phys. 99 (2000), no. 3-4, 661 -690.
1 3. Witten, T. A . ; Sander, L. M. Diffusion limited aggregation. Phys. Rev. B (3)
[4] Cooper, Joshua; Spencer, Joel. Simulat
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(1 983), no. 9, 5686-5697.
ing a Random Walk with Constant Error.
1 4. Winkler, Peter. Mathematical Puzzles: A
Preprint arXiv:math C0/0402323 (Febru
Connoisseur's Collection. A K Peters Ltd,
ary 2004); to appear in Combinatorics,
Natick, MA, 2003.
Probability and Computing.
[5] Dhar, Deepak. Self-organized critical state of sandpile automation models. Phys. Rev.
The Broad Institute at MIT
Lett. 64 (1 990), no. 1 4, 1 61 3-1 61 6.
320 Charles Street
[6] Diaconis, Persi; Fulton, William. A growth
Cambridge, MA 021 41
model, a garne, an algebra, Lagrange in
USA
version, and characteristic classes. Com-
e-mail:
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BRUNO DURAND, LEONID LEVI N, AN D ALEXANDER SHEN
Local R u l es an d G l o bal Ord e r , O r Aperi od i c Ti l 1 n g s
an local rules impose a global order? If yes, when and how? This is a philo sophical question that could be asked in many cases. How does local interaction of atoms create crystals (or quasicrystals) ? How does one living cell manage to develop into a pine cone whose seeds form spirals (and the number of spirals usually is a Fibonacci number)? Is it possible to program locally connected computers in such a way that the net work is still functional if a small fraction of the nodes is corrupted? Is it possible for a big team of people (or ants), each trying to reach private goals, to behave reasonably? These questions range from theology to "political sci ence" and are rather difficult. In mathematics the most prominent example of this kind is the so-called Berger the orem on aperiodic tilings (exact statement below). It was proved by Berger in 1966 [ 1 ] . 1 In 1971 the proof was sim plified by Robinson [7], who invented the well-known "Robinson tiles" that can tile the entire plane but only in an aperiodic way (Fig. 1 ). Since then many similar constructions have been in vented (see, e.g., [3, 6]); some other proofs were based on different ideas (e.g., [4]). However, we did not manage to fmd a publication which provides a short but complete proof of the theorem: Robinson tiles look simple, but when you
00 0 00 0
Fig. 1 . The Robinson tiles [reflections and rotations are allowed].
start to analyze them you have to deal with many technical details. ("This argument is a bit long and is not used in the remainder of the text, so it could be skipped on first read ing," says C. Radin in [6] about the proof.) It's a pity, however, to skip the proof of a nice theorem whose statement can be understood by a high school stu dent (unlike the Fermat Theorem, you don't even need to know anything about exponentiation). We try to fill this gap
1 1n tact, the motivation at that time was related to the undecidability of a specific class of first-order formulas, see [2] .
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and provide a simple construction of an aperiodic tiling with a complete proof, making the argument as simple as possible (at the cost of increasing the number of tiles). Of course, simplicity is a matter of taste, so we can only hope you will find this argument simple and nice. If not, you can look at an alternative approach in [5]. Definitions
Let A be a finite (nonempty) alphabet. A configuration is an infinite cell paper where each cell is occupied by a let ter from A; formally, the configuration is a mapping of type 7L2 ----> A. A local rule is an arbitrary subset L C A4 whose elements are considered as 2 X 2 squares: (!l'spnnger-ny.com • WRITE to Spri nger-Verlag New York, Order Dept. 57805, PO Box 2485, Secaucus, NJ 07096-2485
Springer www.springer-ny.com
V1SJT your
Inc.,
local sCientific bookstore or urge your librarian to order for your
department. Pnce� subject to change wtthour notice.
Please mention 57805 when ordering to guarantee listed prices. PromotiOn S7H05
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D av i d E .
Hilbert's Early Career: Encounters with Allies and Rivals David E. Rowe
Send submissions to David E. Rowe,
R ow e , E d i t o r
I
It seems to me that the mathemati cians of today understand each other far too little and that they do not take an intense enough interest in one an other. They also seem to know-so far as I can judge-too little of our classical authors (Klassiker); many, moreover, spend much effort working on dead ends. -David Hilbert to Felix Klein, 24 July 1890
P
robably no mathematician has been quoted more often than Hilbert, whose opinions and witty re marks long ago entered mathematical lore along with his legendary feats. Fame gave him a captive audience, but as the opening quotation illustrates, even before he attained that fame Hilbert had no difficulty expressing his views. When he wrote those words, in fact, he had just completed [Hilbert 1890], the first in an impressive string of achievements that would vault him to the top of his profession. Initially, he made his name as an ex pert on invariant theory, but Hilbert's reputation as a universal mathemati cian grew as he left his mark on one field after another. Yet these achieve ments alone cannot account for his sin gular place in the history of mathe matics, as was recognized long ago by his intimates ([Weyl 1932], [Blumen thal 1935]). Those who belonged to Hilbert's inner circle during his first two decades in Gottingen pointed to the impact of his personality, which clearly transcended the ideas found between the covers of his collected works (see [Weyl 1944], [Reid 1970]). Hilbert's name became attached to thoughts of fame in the minds of many young mathematicians who felt in spired to tackle one of the twenty-three "Hilbert problems." Some of these he had merely dusted off and presented
anew at the Paris ICM in 1900, but they then acquired a special fascination. As Ben Yandell puts it in his delightful sur vey, The Honors Class, "solving one of Hilbert's problems has been the ro mantic dream of many a mathemati cian" [Yandell 2002, 3].1 Hilbert's ability to inspire was clearly central to Gottingen's success, even if only a part. His leadership style fostered what I have characterized as a new type of oral culture, a highly competitive mathematical community in which the spoken word often carried more weight than the information con veyed in written texts (see [Rowe 2003b] , [Rowe 2004]). Hilbert was an unusually social creature: outspoken, ambitious, eccentric, and above all full of passion for his calling. Moreover, he was a man of action with no patience for hollow words. Thus, when in July 1890 he con veyed the rather harsh views cited in the opening quotation to Klein, he was not merely bemoaning the lack of com munal camaraderie among Germany's mathematicians; he was expressing his hope that these circumstances would soon change. At that time plans were underway to found a national organi zation of German mathematicians, the
Deutsche Mathematiker-Vereinigung,
and Hilbert was delighted to learn that Klein would be present for the inau gural meeting, which would take place a few months later in Bremen. Both knew that much was at stake; as Hilbert expressed it, "I believe that closer personal contact between math ematicians would, in fact, be very de sirable for our science" (Hilbert to Klein, 24 July 1890 [Frei 1985, 68]). Soon after his arrival in Gottingen in 1895, Hilbert put this philosophy into practice. At the same time, his opti mism and self-confidence spilled over and inspired nearly all the young peo ple who entered his circle.
Fachbereich 1 7 -Mathematik, Johannes Gutenberg University,
1 1t was Hilbert's star pupil, Hermann Weyl, who called those who actually succeeded the "honors class"; he
055099 Mainz, Germany.
also wrote that "no mathematician of equal stature has risen from our generation" [Weyl 1 944, 1 30].
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Hilbert's impact on modem mathe matics has been so pervasive that it takes a true leap of historical imagina tion to picture him as a young man struggling to find his way. Still, many of the seeds of later success were planted in his youth, just as several episodes from his early career have since be come familiar aspects of the Hilbert leg end. In [Rowe 2003a] , I describe the quiet early years he spent in Konigs berg, where he befriended two young mathematicians who influenced him more than any others, Adolf Hurwitz and Hermann Minkowski. By 1885 Hilbert emerged as one of Felix Klein's most promising proteges. In this role, he traveled to Paris to meet the younger generation of French mathematicians, especially Henri Poincare, reporting back all the while to Klein, who avidly awaited news about the Parisian math ematical scene. Here I pick up the story at the point when Hilbert returned from this first trip to Paris. Afterward he had several important encounters with the leading mathematicians in Germany. These meetings not only shed light on the contexts that motivated his work, they also reveal how he positioned him self within the fast-changing German mathematical community. Returning from Paris
After a rather uneventful and disap pointing stay in Paris during the spring of 1886, Hilbert began the long journey
home. Stopping first in Gottingen, he learned something about the contem porary Berlin scene when he met with Hermann Amandus Schwarz, the se nior mathematician on the faculty. Schwarz had long been one of the clos est of Karl Weierstrass's many adoring pupils in Berlin; yet much had changed since the days when Charles Hermite advised young Gosta Mittag-Leffler to leave Paris and go to hear the lectures of Weierstrass, "the master of us all" (see [Rowe 1998]). During the 1860s and 1870s, the Berliners had dominated mathematics throughout Prussia, with the single exception of Konigsberg, which remained an enclave for those with close ties to the Clebsch school and its journal, Mathematische An nalen (see [Rowe 2000]). However, af ter E. E. Kummer's departure in 1883, the harmonious atmosphere he had cultivated as Berlin's senior mathe matician gave way to acrimony. Weier strass, old, frail, and decrepit, refused to retire for fear of losing all influence to Leopold Kronecker, who remained amazingly energetic and prolific de spite his more than sixty years. Presumably Hilbert heard about this situation from Schwarz, who would have conveyed the essence of the situ ation from Weierstrass's perspective (see [Biermann 1988, 137-139]). If so, Hilbert would have heard how rela tions between Weierstrass and Kro necker had deteriorated mainly be cause of the latter's dogmatic views, in particular his sharp criticism of Weier strass's approach to the foundations of analysis. Only a few months after Hilbert passed through Berlin, Kro necker delivered a speech in which he uttered his most famous phrase "God made the natural numbers; all else is the work of man" ("Die ganzen Zahlen hat der liebe Gott gemacht, alles an dere ist Menschenwerk") [Weber 1893, 19]. Kronecker had never made a se cret of his views on foundations, but by the mid-1880s he was propounding these with missionary zeal. No one was more taken aback by this than H. A. Schwarz, to whom Kronecker had writ ten one year earlier:
Fig. 1 . Hilbert in the days when he was re garded as merely one of many experts on the theory of invariants.
If enough years and power remain, I will show the mathematical world
that not only geometry but also arithmetic can point the way for analysis-and certainly with more rigor. If l don't do it, then those who come after me will, and they will also recognize the invalidity of all the procedures with which the so called analysis now operates [Bier mann 1988, 138]. Weierstrass had written to Schwarz at around that time, claiming that Kro necker had transferred his former an tipathy for Georg Cantor's work to his own. And in another letter, written to Sofia Kovalevskaya, he characterized the issue dividing them as rooted in mathematical ontology: "whereas I as sert that a so-called irrational number has a real existence like anything else in the world of thought, according to Kronecker it is an axiom that there are only equations between whole num bers" (Weierstrass to Kovalevskaya, 24 March, 1885, quoted in [Biermann 1988, 137]). Whether or not Schwarz alluded to this rivalry when he spoke to Hilbert in 1886, he definitely did warn him to expect a cold reception when he pre sented himself to Kronecker (Hilbert to Klein, 9 July 1886, in [Frei 1985, 15]). Instead, however, Hilbert was greeted in Berlin with open arms, and his ini tial reaction to Kronecker was for the most part positive. Back in his native Konigsberg, Hilbert reported to the ever-curious Klein about these and other recent events. He had just completed all re quirements for the Habilitation except for the last, an inaugural lecture to be delivered in the main auditorium of the Albertina. His chosen theme was a propitious one: recent progress in the theory of invariants. Hilbert was pleased to be back in Konigsberg as a Privatdozent, even though this meant he was far removed from mathemati cians at other German universities. To compensate for this isolation, he was planning to tour various mathematical centers the following year, when he hoped to meet Professors Gordan and N oether in Erlangen. Although he had to postpone that trip until the spring of 1888, it eventually proved far more fruitful than his earlier journey to Paris. What is more, it helped him so-
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Fig.
2. Leopold
Kronecker
emerged
as
Berlin's leading mathematician during the 1880s when his outspoken views caused con siderable controversy.
lidify his relationship with Klein, who always urged young mathematicians to cultivate personal contacts with fellow researchers both at home and abroad (see for example [Hashagen 2003, 105-116, 149-162]). Within Klein's net work, the Erlangen mathematicians, Paul Gordan and Max Noether, played particularly important roles. The latter was Germany's foremost algebraic geometer in the tradition of Alfred Clebsch, and the former was an old fashioned algorist who loved to talk mathematics. Felix Klein knew from first-hand ex perience how stimulating collabora tion with Paul Gordan could be. Dur ing the late 1870s, when Klein taught at the Technical University in Munich, he took advantage of every opportunity to meet with his erstwhile Erlangen col league, who was himself enormously impressed by Klein's fertile geometric imagination. Gordan was widely re garded as Germany's authority on al gebraic invariant theory, the field that would dominate Hilbert's attention for the next five years. His principal claim to fame was Gordan's Theorem, which he proved in 1868. This states that the complete system of invariants of a bi nary form can always be expressed in terms of a finite number of such in-
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variants. In 1856 Arthur Cayley had proved this for binary forms of degree 3 and 4, but Gordan was able to use the symbolical method introduced by Siegfried Aronhold to obtain the gen eral result. During his stay in Paris, Hilbert had briefly reported to Klein about these matters (Hilbert to Klein, 21 April 1886, in [Frei 1985, 9]). There he learned from Charles Hermite about J. J. Sylvester's recent efforts to prove Gordan's Theorem using the original British techniques he and Cayley had developed. Hilbert thus became aware that the elderly Sylvester was still trying to get back into this race (see [Parshall 1989]). Presumably Hilbert thought that progress was unlikely to come from this old-fashioned line of at tack, but neither had the symbolical methods employed by German alge braists produced any substantial new results since Gordan first unveiled his theorem. Over the next two years Hilbert had ample time to master the various com peting techniques. Mter becoming a
Privatdozent in Konigsberg, he was
free to develop his own research pro gram, and his inaugural lecture on re cent research in invariant theory clearly indicates the general direction in which he was moving. Still, there are no signs that he was on the path to ward a major breakthrough. Indeed, tucked away in Konigsberg, it seems unlikely that he even saw the need to strike out in an entirely new direction in order to make progress beyond Gor dan's original finiteness theorem. That goal, nevertheless, was clearly in the back of his mind when he set off in March 1888 on a tour of several lead ing mathematical centers in Germany, including Berlin, Leipzig, and Gottin gen. During the course of a month, he spoke with some twenty mathemati cians from whom he gained a stimu lating overview of current research interests throughout the country. Al though we can only capture glimpses of these encounters, a number of im pressions can be gained from Hilbert's letters to Klein, as well as from the
Felix Christian Klein Hochzeitsbilder 1 875
Fig. 3. On leaving Erlangen for the Technische Hochschule in Munich in 1 875, Felix Klein mar ried Anna Hegel, a granddaughter of the famous philosopher.
notes he took of his conversations with various colleagues. 2 A Second Encounter with Kronecker
In Berlin, Hilbert met once again with Kronecker, who on two separate occa sions afforded him a lengthy account of his general views on mathematics and much else related to Hilbert's own research. A gregarious, outspoken man, the elder Kronecker still exuded intensity, so Hilbert learned a great deal from him during the four hours they spent together. Reporting to Klein, he described the Berlin mathemati cian's opinions as "original, if also somewhat derogatory" (Hilbert to Klein, 16 March 1888 [Frei 1985, 38]). Hilbert told Kronecker about a paper he had just written on certain positive definite forms that cannot be repre sented as a sum of squares. Kronecker replied that he, too, had encountered forms that cannot be so represented, but he admitted that he did not know Hilbert's main theorem, which dealt with the three cases in which a sum of squares representation is, indeed, al ways possible ("Bericht iiber meine Reise," Hilbert Nachlass 741). A noteworthy feature of this paper, [Hilbert 1888a], is that Hilbert actually credits Kronecker with having intro duced the general principle behind his investigation. This work lies at the root of Hilbert's seventeenth Paris problem, which also played an important role in Hilbert's research on foundations of geometry. Interestingly, a second Hilbert problem, the sixteenth, also crept into his conversations with Kro necker. It concerns the possible topo logical configurations among the com ponents of a real algebraic curve. The maximal number of such components had been established by Axel Harnack, a student of Klein's, in a celebrated the orem from 1876 (for a summary of sub sequent results, see [Yandell 2002, 276-278]). Kronecker assured Hilbert that his own theory of characteristics, as presented in a paper from 1878, enabled one to answer all questions of this type, clearly an overly optimistic assessment.
Whatever he may have thought about Kronecker's "priority claims," Hilbert stood up and took notice when his host voiced some sharp views about the significance of invariant the ory. Kronecker dismissed the whole theory of formal invariants as a topic that would die out just as surely as had happened with Hindenburg's combina torial school (which had flourished in Leipzig at the beginning of the nine teenth century, but by the 1880s had entered the dustbin of history). The only true invariants, in Kronecker's view, were not the "literal" ones based on algebraic forms, but rather num bers, such as the signature of a qua-
The o n ly true i nvariants , i n Kro necker' s view , were n u m bers , such as the signat u re of a q u ad ratic form . dratic form (Sylvester's theorem, the algebraist's version of conservation of inertia). He then proceeded to wax forth over foundational issues, begin ning with the assertion that "equality" only has meaning in relation to whole numbers and ratios of whole numbers. Everything beyond this, all irrational quantities, must be represented either implicitly by a finite formula (e.g., x2 = 5), or by means of approximations. Us ing these notions, he told Hilbert, one can establish a firm foundation for analysis that avoided the Weierstrass ian notions of equality and continuity. He further decried the confusion that so often resulted when mathemati cians treated the implicitly given irra tional quantity (say, x = v5) as equiv alent to some sequence of rational numbers that serve as an approxima tion for it. Not surprisingly, Hilbert took fairly
extensive notes when Kronecker be gan expounding these unorthodox views ("Bericht iiber meine Reise"). But he also jotted down a brief com ment made by Weierstrass that sheds considerable light on the differences between these two mathematical per sonalities. When Hilbert visited Weier strass shortly afterward, he informed him of Kronecker's comments regard ing invariant theory, including the pre diction that the whole field would soon be forgotten, like the work of the Leipzig combinatorial school. Weier strass responded by sounding a gentle warning to those who might wish to prophesy the future of a mathematical theory: "Not everything of the combi natorial school has perished," he said, "and much of invariant theory will pass away, too, but not from it alone. For from everything the essential must first gradually crystallize, and it is neither possible nor is it our duty to decide in advance what is significant; nor should such considerations cause us to demur in investigating such invariant-theo retic questions deeply" ("Bericht tiber meine Reise"). These words, with their almost fa talistic ring, probably left little impres sion on the young mathematician who recorded them. For Hilbert's intellec tual outlook was filled with a buoyant optimism that left no room for resig nation. He may not have enjoyed Kro necker's braggadocio, but he was clearly far more receptive to his pas sionate vision than to Weierstrass's more subdued outlook. Moreover, mathematically he was far closer to the algebraist than to the analyst. Even in his later work in analysis, Hilbert showed that his principal strength as a mathematician stemmed from his mas tery of the techniques of higher algebra (see [Toeplitz 1922]). True, Klein and Hurwitz had drawn his attention to Weierstrass's theory of periodic com plex-valued functions, about which he spoke in his Habilitationsvortrag shortly after returning to Konigsberg from Paris. Nevertheless, Kronecker's algebraic researches lay much closer to his heart. Soon after their encounter
2"Bericht uber meine Reise vom 9ten Marz bis ?ten April 1 888," Hilbert Nachlass 7 4 1 , Handschnftenabteilung, Niedersachs1sche Staats· und Umversitatsbibliothek Gbt· tingen.
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in Berlin, Hilbert would enter Leopold Kronecker's principal research do main, the theory of algebraic number fields. The latter's sudden death in 1891 may well have emboldened Hilbert to reconstruct this entire theory six years later in his "Zahlbericht." As Hermann Weyl later emphasized, Hilbert's am bivalence with respect to Kronecker's legacy emerged as a major theme throughout his career [Weyl 1944, 613]. Like Hilbert, his two closest mathe matical friends, Hurwitz and Minkowski, also held an1bivalent views when it came to Kronecker. No doubt these were colored by their mutual desire to step beyond the lengthy shadows that Kronecker and Richard Dedekind, the other leading algebraist of the older generation, had cast. Since Dedekind had long since withdrawn to his native Brunswick, a city well off the beaten path, it was only natural that the Konigsberg trio came to regard the powerful and opinionated Berlin math ematician as their single most imposing rival. In later years, Hilbert developed a deep antipathy toward Kronecker's philosophical views, and he did not hes itate to criticize these before public au diences (see [Hilbert 1922]). Yet during the early stages of his career such mis givings-if he had any-remained very much in the background. Indeed, all of Hilbert's work on invariant theory was deeply influenced by Kronecker's ap proach to algebra. Hilbert's encounters in the spring of 1888 with Berlin's two senior mathe maticians left a deep and lasting im pression. 3 Based on the notes he took of these conversations, he must have felt particularly aroused by Kro necker's critical views with regard to invariant theory, for he surely found no solace in Weierstrass's stoic advice. Primed for action and out to conquer, Hilbert could never have contemplated devoting his whole life to a theory that might later be judged as having no in trinsic significance. Whatever prob lems he chose to work on-even those he merely thought about but never tried to solve-he always thought of them as constituting important mathe matics. What makes a problem or a
theory important? Probably Hilbert carried that question within him for a long time, though anyone familiar with his career knows the answer he even tually came up with; one need only reread his famous Paris address to see how compelling his views on the sig nificance of mathematical thought could be. From the vantage point of these early, still formative years, we can begin to picture how his larger views about the character and signifi cance of mathematical ideas fell into place. A few strands of the story emerge from the discussions he en gaged in during this whirlwind 1888 tour through leading outposts of the German mathematical community. Tackling Gordan's Problem
From Berlin, Hilbert went on to Leipzig, where he finally got the chance to meet face-to-face with Paul Gordan, who came from Erlangen. Despite their mathematical differences, the two hit
It is neither possi ble nor is it our d uty to decide i n advance what is s i g n ificant . it off splendidly, as both loved nothing more than to talk about mathematics. Hermann Weyl once described Gordan as "a queer fellow, impulsive and one sided," with "something of the air of the eternal 'Bursche' of the 1848 type about him-an air of dressing gown, beer and tobacco, relieved however by a keen sense of humor and a strong dash of wit. . . . A great walker and talker-he liked that kind of walk to which fre quent stops at a beer-garden or a cafe belong" [Weyl 1935, 203]. Having heard about Hilbert's talents, Gordan longed to make the young man's acquaintance, so much so that he wished to remain incognito while in Leipzig to take full advantage of the opportunity (Hilbert to Klein, 16 March 1888 [Frei 1985, 38]). Although originally an expert on
3He recalled this trip when he spoke about his life on his seventieth birthday; see [Reid 1 970, 202].
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THE MATHEMATICAL INTELLIGENCER
Fig. 4. Paul Gordan joined Klein on the Er langen faculty in 1 874 and remained there un til his death in 1 91 2. His star student was Emmy Noether, daughter of Gordan's col league, Max Noether.
Abelian functions, Gordan had long since focused his attention exclusively on the theory of algebraic invariants. This field traces back to a fundamen tal paper published by George Boole in 1841, as it was this work that inspired young Arthur Cayley to take up the topic in earnest [Parshall 1989, 160-166]. Following an initial plunge into the field, Cayley joined forces with another professional lawyer who be came his life-long friend, J. J. Sylvester. Together, they effectively launched in variant theory as a specialized field of research. Much of its standard termi nology was introduced by Sylvester in a major paper from 1853. Thus, for a given binary form f(x,y), a homoge neous polynomial J in the coefficients off left fixed by all linear substitutions (up to a fixed power of the determinant of the substitution) is called an invari ant of the form f In 1868 Gordan showed that for any binary form, one can always construct m invariants h, /z, . . . , Im such that every other in variant can be expressed in terms of these m basis elements. Indeed, he proved that this held generally for ho mogeneous polynomials J in the coef ficients and variables ofj(x,y) with the same invariance property (Sylvester called such an expression J a con comitant of the given form, but the term covariant soon became stan dard). In 1856 Cayley published the first
finiteness results for binary forms, but in the course of doing so he committed a major blunder by arguing that the number of irreducible invariants was necessarily infinite for forms of degree five and higher [Parshall 1989, 167-179]. Paul Gordan was the first to show that Cayley's conclusion was incorrect. More importantly, in the course of do ing so he proved his finite basis theo rem for binary forms of arbitrary de gree by showing how to construct a complete system of invariants and co variants. Two years later, he was able to extend this result to any finite sys tem of binary forms. His proofs of these key theorems, as later presented in [Gordan 1885/1887], were purely al-
Fig. 5. Otto Blumenthal, Hilbert's first biog rapher, alluded to the critical meeting when Hilbert and Gordan first met.
gebraic and constructive in nature. They were also impressively compli cated, so that subsequent attempts, in cluding Gordan's own, to extend his theorem to ternary forms had pro duced only rather meager results. Little evidence has survived relating to Hilbert's first encounter with Gor dan, but it is enough to reconstruct a plausible picture of what occurred. Gordan may have been a fairly old dog, but this does not mean he was averse to learning some new tricks. Even though he and Hilbert had divergent views about many things, they never theless understood each other well (Hilbert to Klein, 16 March 1888 [Frei 1985, 38]). Their conversations soon fo cused on finiteness results, in particu lar a fairly recent proof of Gordan's finiteness theorem for systems of bi nary forms published by Franz Mertens in Grelle's Journal [Mertens 1887]. This paper broke new ground. For unlike Gordan's proof, which was based on the symbolic calculus of Clebsch and Aronhold, Mertens's proof was not strictly constructive. Gordan and Hilbert apparently discussed it in con siderable detail, and Hilbert immedi ately set about trying to improve Mertens's proof, which employed a rather complicated induction argu ment on the degree of the forms. After spending a good week with Gordan, he was delighted to report to Klein that "with the stimulating help of Prof. Gor dan an infinite series of thought vibra tions has been generated within me, and in particular, so we believe, I have a wonderfully short and pointed proof for the finiteness of binary systems of forms" (Hilbert to Klein, 2 1 March 1888 [Frei 1985, 39]). Hilbert had caught fire. A week later, when he met with Klein in Got tingen, he had already put the finishing touches on the new, streamlined proof. This paper [Hilbert 1888b] was the first in a landslide of contributions to alge braic invariant theory that would tum the subject upside down. Between 1888 and 1890 Hilbert pursued this theme re lentlessly, but with a new methodolog ical twist which he combined with the formal algorithmic techniques em ployed by Gordan. Beginning with three short notes sent to Klein for publication
in the Gottinger Nachrichten [Hilbert 1888c] , [Hilbert 1889a] , [Hilbert 1889b], he began to unveil general methods for proving finiteness relations for general systems of algebraic forms, invariants being only a quite special case, though the one of principal interest. With these general methods, combined with the al gorithmic techniques developed by his predecessors, Hilbert was able to ex tend Gordan's finiteness theorem from systems of binary forms over the real or complex numbers to forms in any number of variables and with coeffi cients in an arbitrary field. By the time this first flurry of activ ity came to an end, Hilbert had shown how these finiteness theorems for in variant theory could be derived from general properties of systems of alge braic forms. Writing to Klein in 1890, he described his culminating paper [Hilbert 1890] as a unified approach to a whole series of algebraic problems (Hilbert to Klein, 15 February 1890 [Frei 1985, 61]). He might have added that his techniques borrowed heavily from Leopold Kronecker's work on al gebraic forms. Yet from a broader methodological standpoint, Hilbert's approach clearly broke with Kro necker's constructive principles. For Hilbert's foray into the realm of alge braic forms revealed the power of pure existence arguments: he showed that out of sheer logical necessity a finite basis must exist for the system of in variants associated with any algebraic form or system of forms. Hilbert found his way forward by noticing the following general result, known today as Hilbert's basis theo rem for polynomial ideals. It appears as Theorem I in [Hilbert 1888c] . It states that for any sequence of alge braic forms in n variables c{J1, c{Jz, 4>3, . . . there exists an index m such that all the forms of the sequence can be written in terms of the first m forms, that is, cP
=
CXlcPl
+
CX2 cP2
+ ...+
CXmcPm ,
where the ai are appropriate n-ary forms. Thus, the forms cfJ1 , c{Jz, . . . cPm serve as a basis for the entire system. By appealing to Theorem I and draw ing on Mertens's procedure for gener ating systems of invariants, Hilbert
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proved that such systems were always finitely generated. There was, how ever, a small snag. Hilbert attempted to prove Theorem I by first noting that it held for small n. He then introduced a still more general Theorem II, from which he could prove Theorem I by induction on the number of variables. If this sounds confusing, a number of contemporary readers had a similar reaction, including a few who expressed their misgivings to Hilbert about the validity of his proof. Paul Gordan, however, was not one of them. According to Hilbert, to the best of his recollection, he and Gordan had only discussed the proof of Theorem II dur ing their meeting in Leipzig (Hilbert to Klein, 3 March 1890 [Frei 1985, 64]). As it turned out, Hilbert's Theorem II, as originally formulated in [Hilbert 1888c], is false. 4 Moreover, since it was conceived from the beginning as a lemma for the proof of Theorem I, Hilbert dropped Theorem II in his de fmitive paper [Hilbert 1890] and gave a new proof of Theorem I. Nevertheless, the latter remained controversial, as we shall soon see. Today we recognize in Hilbert's Theorem I a central fact of ideal theory, namely, that every ideal of a polynomial ring is finitely generated. Thirty years later, Emmy Noether in corporated Hilbert's Theorem I (from [Hilbert 1890]) as well as his Nullstel lensatz (from [Hilbert 1893]) into an abstract theory of ideals (see [Gilmer 1981]). Her classic paper "Idealtheorie in Ringbereichen" [Noether 1921 ] is nearly as readable today as it was when she wrote it. The same cannot be said, however, for Hilbert's papers (for En glish translations, see [Hilbert 1978]). Not that these are badly written; they simply reflect a far less familiar math ematical context. In [Hilbert 1889a, 28] Hilbert hinted that much of the inspiration for both the terminology and techniques came from Kronecker's theory of module systems. When he wrote this, he knew very well that Kronecker held very neg ative views about invariant theory, making it highly improbable that he
would view Hilbert's adaptation of his ideas with approval. Indeed, Kro necker had made it plain to Hilbert that, in his view, the only invariants of interest were the numerical invariants associated with systems of algebraic equations. Still, Hilbert quickly recog nized the fertility of Kronecker's con ceptions for invariant theory. Ac knowledging his debt to the Berlin algebraist, he parted company with him by adopting a radically non-con structive approach. Ironically, the ini tial impulse to do so apparently came from his conversations with Gordan. Thus, with his early work on invariant theory Hilbert sowed some of the seeds that would eventually flower into his modernist vision for mathematics, thereby preparing the way for the dra matic foundations debates of the 1920s (see [Hesseling 2003]). Mathematics as Theology
Kronecker seems to have simply ig nored Hilbert's dramatic break through, but others closer to the field of invariant theory obviously could not afford to do so. Paul Gordan, who had initially supported Hilbert's work en thusiastically, now began to express misgivings about this new and, for him, all too ethereal approach to invariant theory. His views soon made the rounds at the coffee tables and beer gardens, and more or less everyone heard what Gordan probably said on more than one occasion: Hilbert's ap proach to invariant theory was "theol ogy not mathematics" [Weyl 1944, 140].5 No doubt many mathematicians got a chuckle out of this epithet at the time, but a serious conflict briefly reared its head in February 1890 when Hilbert submitted his definitive paper [Hilbert 1890] for publication in Mathematis che Annalen. Klein was overjoyed, and wrote back to Hilbert a day later: "I do not doubt that this is the most impor tant work on general algebra that the Annalen has ever published" (Klein to Hilbert, 18 Feb. 1890, in [Frei 1985, p. 65]). He then sent the manuscript to Gordan, the Annalen's house expert on
invariant theory, asking him to report on it. Klein, having already heard some of Gordan's misgivings about Hilbert's methods in private conversations, may well have anticipated a negative reac tion. He certainly got one. The cantan kerous Gordan forcefully voiced his objections, aiming directly at Hilbert's presentation of Theorem I, which Gor dan claimed fell short of even the most modest standards for a mathematical proof. "The problem lies not with the form," he wrote Klein, " . . . but rather lies much deeper. Hilbert has scorned to present his thoughts following for mal rules; he thinks it suffices that no one contradict his proof, then every thing will be in order . . . he is content to think that the importance and cor rectness of his propositions suffice. That might be the case for the first ver sion, but for a comprehensive work for the Annalen this is insufficient." (Gor dan to Klein, 24 Feb. 1890, in [Frei 1985, p 65]. Perhaps the misgivings come down to the non-constructivity in volved in the [implicit] use of the Ax iom of Choice. Concise modem proofs like [Caruth 1996] put the latter clearly in evidence.) Klein forwarded Gordan's report to Hurwitz in Konigsberg, who then dis cussed its contents with Hilbert. After that the sparks really began to fly. Clearly irked by Gordan's refusal to recognize the soundness of his argu ments, Hilbert promptly dashed off a fierce rebuttal to Klein. He began by reminding him that Theorem I was by no means new; he had, in fact, come up with it some eighteen months ear lier and had afterward published a first proof in the Gottinger Nachrichten [Hilbert 1888c]. He then proceeded to describe the events that had prompted him to give a new proof in the manu script now under scrutiny. This came about after he had spo ken with numerous mathematicians about his key theorem; he had also car ried on correspondence with Cayley and Eugen Netto, who wanted him to clarify certain points in the proof. Tak ing these various reactions into ac-
4See the editorial note in [Hilbert 1 933. 1 77). 5The earliest reference to Gordan's remark- ''Das ist keine Mathematik, das ist Theologie"-appears to be [Blumenthal 1 935, 394).
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THE MATHEMATICAL INTELLIGENCER
count, Hilbert had prepared a revised proof, which he had tested out in his lecture course the previous semester. Afterward he spoke with one of the au ditors in order to convince himself that the argument as presented had actually been understood. Having reassured himself that this new proof was indeed clear and understandable, he wrote it up for [Hilbert 1890]. Hilbert then con cluded this recitation of the relevant prehistory by saying that these facts clearly refuted the ad hominem side of Gordan's attack, namely his insinua tion that Hilbert's new proof of Theo rem I was not meant to be understood and that he was content so long as no one could contradict the argument. Regarding what he took to be the substantive part of Gordan's critique, Hilbert stated that this consisted mainly of "a series of very commend able, but completely general rules for the composition of mathematical pa pers" (Hilbert to Klein, 3 March 1890 [Frei 1985, 64]). The only specific crit icisms Gordan made were, in Hilbert's opinion, plainly incomprehensible: "If Professor Gordan succeeds in proving my Theorem I by means of an 'order ing of all forms' and by passing from 'simpler to more complicated forms,' then this would just be another proof, and I would be pleased if this proof were simpler than mine, provided that each individual step is as compelling and as tightly fastened" (ibid.). Hilbert then ended this remarkable repartee with an implied threat: either his paper would be printed just as he wrote it or he would withdraw his manuscript from publication in the Annalen. "I am not prepared," he intoned, "to alter or delete anything, and regarding this pa per, I say with all modesty, that this is my last word so long as no definite and irrefutable objection against my rea soning is raised" (ibid.). Certainly Klein was not accustomed to receiving letters like this one, and especially from young Privatdozenten. Yet however impressed he may have been by Hilbert's self-assurance and pluck, he also wanted to preserve his longstanding alliance with Gordan. Moreover, in view of his older friend's irascibility, Klein knew, he had to han dle the squabble delicately before it be-
came a full-blown crisis. Hilbert re ceived no immediate reply, as Klein wanted to wait until he could confer with Gordan personally. Over a month passed, with no word from Gottingen about the fate of a paper that Klein had originally characterized as one of the most important ever to appear in the pages of Die Mathematische Annalen. Then, in early April, Gordan came to Gottingen to "negotiate" with Klein about these matters, which clearly weighed heavily on the Erlangen math ematician's heart. To facilitate the process, Klein asked Hurwitz to join them, knowing that Hilbert's trusted friend would do his best to help restore harmony. Gordan spent eight days in Gottin gen, following which Klein wrote Hilbert a brief letter summarizing the results of their "negotiations." He be gan by reassuring him that Gordan's opinions were by no means as uni formly negative as Hilbert had as sumed. "His general opinion,'' Klein noted, "is entirely respectful, and would exceed your every wish" (Klein to Hilbert, 14 April 1890, in [Frei 1985, p. 66]). To this he merely added that Hurwitz would be able to tell him more about the results of their meeting. But then he attached a postscript that con tained the message Hilbert had been waiting to hear: Gordan's criticisms would have no bearing on the present paper and should be construed merely as guidelines for future work! Thus, Hilbert got what he de manded; his decisive paper appeared in the Annalen exactly as he had written it. Gordan surely lost face, but at least he had been given the opportunity to vent his views. In short, Klein's diplo matic maneuvering carried the day. Gordan knew, of course, that he was dealing with someone who had little patience for methodological nit-pick ing. He also knew that Klein consis tently valued youthful vitality over age and experience. Hilbert represented the wave of the future, and while this conflict, in and of itself, had no imme diate ramifications, it foreshadowed a highly significant restructuring of the power constellations that had domi nated German mathematics since the late 1860s.
A Final Tour de Force
If Hilbert was scornful of Gordan's ed itorial pronouncements, this does not mean that he failed to see the larger is sue at stake. His general basis theorem proved that for algebraic forms in any number of variables there always ex ists a finite collection of irreducible in variants, but his methods of proof were of no help when it actually came to constructing such a basis. Hilbert ob viously realized that if he could de velop a new proof based on arguments that were, in principle, constructive in nature, then this would completely vi tiate Gordan's criticisms. Two years later, he unveiled just such an argu ment, one that he had in fact been seek ing for a long time. In an elated letter to Klein, he described this latest break through, which allowed him to bypass the controversial Theorem I com pletely. He further noted that although this route to his finiteness theorems was more complicated, it carried a ma jor new payoff, namely "the determi nation of an upper bound for the de gree and weights of the invariants of a basis system" (Hilbert to Klein, 5 Jan uary 1892 [Frei 1985, 77]) . When Hermann Minkowski, who was then in Bonn, heard about Hilbert's lat est triumph, he fired off a witty letter congratulating his friend back in Konigs berg: I had long ago thought that it could only be a matter of time before you finished off the old invariant theory to the point where there would hardly be an i left to dot. But it re ally gives me joy that it all went so quickly and that everything was so surprisingly simple, and I congratu late you on your success. Now that you've even discovered smokeless gunpowder with your last theorem, after Theorem I caused only Gor dan's eyes to sting anymore, it really is a good time to decimate the fortresses of the robber-knights [i.e., specialists in invariant theory] [Georg Emil] Stroh, Gordan, [Ky parisos] Stephanos, and whoever they all are-who held up the indi vidual traveling invariants and locked them in their dungeons, as there is a danger that new life will
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never sprout from these ruins again. [Minkowski 1973, 45]. Minkowski's opinions were a constant source of inspiration for Hilbert, so he probably took these remarks to heart. Indeed, this letter may well mark the beginning of one of the most enduring of all myths associated with Hilbert's exploits, namely that he single-hand edly killed off the till then flourishing field of invariant theory. As Hans Freudenthal later put it: "never has a blooming mathematical theory with ered away so suddenly" [Freudenthal 1971, 389]. Hilbert published his new results in another triad of papers for the Gdt tinger Nachrichten ([Hilbert 189 1 ] , [Hilbert 1892a], [Hilbert 1892b] ) . Nine months later he completed the manu script of his second classic paper on in variant theory [Hilbert 1893].6 He sent this along with a diplomatically worded letter to Klein, noting that he had taken pains to ensure that the presentation followed the general guidelines Prof. Gordan had recommended (Hilbert to Klein, 29 September 1892 [Frei 1985, 85]). Then, in a short postscript, Hilbert added: "I have read and thought through the manuscript carefully again, and must confess that I am very satisfied with this paper" (ibid.). Klein reassured him that "Gordan had made his peace with the newest developments," and emphasized that doing so "wasn't easy for him, and for that reason should be seen as much to his credit" (Klein to Hilbert, 7 January 1893, in [Frei 1985, p. 86]). As evidence of Gordan's change of heart, Klein men tioned his forthcoming paper entitled simply "Ober einen Satz von Hilbert" [Gordan 1892]. The Satz in question was, of course, Hilbert's Theorem I, which really had caused Gordan's eyes to sting, but not because he doubted its validity. Nor did he ever doubt that Hilbert's proof was correct; it was simply incomprehensible in Gordan's opinion. As he put it to Klein back in 1890: "I can only learn something that
is as clear to me as the rules of the mul tiplication table" (Gordan to Klein, 24 Feb. 1890, in [Frei 1985, p 65]). Hilbert had claimed that he would welcome a simpler proof of Theorem I from Gordan, and here the elderly al gorist delivered in a gracious manner. He began by characterizing Hilbert's proof as "entirely correct" [Gordan 1892, 132], but went on to say that he had nevertheless noticed a gap, in that Hilbert's argument merely proved the existence of a finite basis without ex amining the properties of the basis el ements. He further noted that his own proof relied essentially on Hilbert's strategy of applying the ideas of Kro necker, Dedekind, and Weber to in variant theory [Gordan 1892, 133]. Probably only a few of those who saw this conciliatory contribution by the "King of Invariants" were aware of the earlier maneuvering that had taken place behind the scenes. Nor were many likely to have anticipated that Gordan's throne would soon resemble a museum piece. 7 Not surprisingly, Hilbert put method ological issues at the very forefront of [Hilbert 1893], his final contribution to invariant theory. Here he called atten tion to the fact that his earlier results failed to give any idea of how a finite basis for a system of invariants could actually be constructed. Moreover, he noted that these methods could not even help in finding an upper bound for the number of such invariants for a given form or system of forms [Hil bert 1933, 319]. To show how these drawbacks could be overcome, Hilbert adopted an even more general ap proach than the one he had taken be fore. He described the guiding idea of this culminating paper as invariant the ory treated merely as a special case of the general theory of algebraic func tion fields. This viewpoint was inspired to a considerable extent by the earlier work of Kronecker and Dedekind, al though Hilbert mentioned this connec tion only obliquely in the introduction, where he underscored the close anal-
ogy with algebraic number fields [Hilbert 1933, 287] . Hilbert's introduction also contains other interesting features. In it, he set down five fundamental principles which could serve as the foundations of invariant theory. The first four of these he regarded as the "elementary propositions of invariant theory," whereas the existence of a finite basis (or in Hilbert's terminology a "full in variant system") constituted the fifth principle. This highly abstract formu lation would, of course, later come to typify much of Hilbert's work in nearly all branches of mathematics. Indeed, the only thing missing from what was to become standard Hilbertian jargon was an explicit appeal to the axiomatic method. Immediately after presenting these five propositions, he wrote that they "prompt the question, which of these properties are conditioned by the others and which can stand apart from one another in a function system." He then mentioned an example that demon strated the independence of property 4 from properties 2, 3, and 5. These fmd ings were incidental to the main thrust of Hilbert's paper, but they reveal how axiomatic ideas had already entered into his early work on algebra. 8 On September 1892, the day he sent off the manuscript of [Hilbert 1893], Hilbert wrote to Minkowski: "I shall now definitely leave the field of invari ants and tum to number theory" [Blu menthal 1935, 395]. This transition was a natural one, given that his final work on invariant theory was essentially an application of concepts from the the ory of algebraic number fields. One year later, Hilbert and Minkowski were charged with the task of writing a re port on number theory to be published by the Deutsche Mathematiker-Vereini gung. Minkowski eventually dropped out of the project, but he continued to offer his friend advice as Hilbert strug gled with his most ambitious single work, "Die Theorie der algebraischen Zahlkorper," better known simply as the
Zahlbericht. 9
6For an English translation of this and other works by Hilbert. see [Hilbert 1 978]. 7Gordan later presented a streamlined proof of Hilbert's Theorem
I
in a lecture at the 1 899 meet1ng of the DMV in Munich. Hilbert was present on that occasion (see
Jahresbericht der Deutschen Mathematiker-Vereinigung 8(1 899), 1 80. Gordan wrote up this proof soon thereafter for [Gordan 1 899] . 8For a detailed examination of Hilbert's work on the axiomatization of physics, see [Corry 2004].
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THE MATHEMATICAL INTELLIGENCER
Killing off a Mathematical Theory
nor episode at the outset of his
Thus by 1893 Hilbert's active involve ment with invariant theory had ended. In that year he wrote the survey article [Hilbert 1896] in response to a request from Felix Klein, who presented it along with several other papers at the Mathematical Congress held in Chicago in 1893 as part of the World's Columbian Exposition. Hilbert's ac count offers an interesting partici pant's history of the classical theory of invariants. At the time he wrote it, in variant theory was a staple research field within the fledgling mathematical community in the United States, which first began to spread its wings under the tutelage of J. J. Sylvester at Johns Hopkins (see [Parshall and Rowe 1994]). Hilbert briefly alluded to the contributions of Cayley and Sylvester in his brief survey, describing these as characteristic for the "naive period" in the history of a special field like alge braic invariant theory. This stage, he added, was soon superseded by a "for mal period, " whose leading figures were his own direct predecessors, Al fred Clebsch and Paul Gordan. A ma ture mathematical theory, Hilbert went on, typically culminates in a third, "crit ical period," and his account made it clear that he alone was to be regarded as having inaugurated this stage. What better time to quit the field? Hilbert realized very well that many as pects of invariant theory had only be gun to unfold, but after 1893 he was content to point others in possibly fruitful directions for further research, such as the one indicated in his four teenth Paris problem. Although he did offer a one-semester course on invari ant theory in 1897 (see [Hilbert 1993]), by this time his eyes were already on other fields and new challenges. Mathematicians are constantly look ing ahead, not backward, and by 1893 probably no one gave much thought to the events of five years earlier. Over time, Hilbert's decisive encounter with Gordan in Leipzig was reduced to a mi-
Siegeszug through invariant theory.
Otto Blumenthal, Hilbert's first biogra pher, even got the city wrong, claiming that Hilbert went to Erlangen to visit Gordan in the spring of 1888 [Blumen thal 1935, 394]. By then forty years had passed, and presumably no one, not even Hilbert, remembered what had happened. Yet his own characteriza tion of this encounter could not be more telling: it had been thanks to Gordan's "stimulating help" that he left Leipzig with "an infinite series of thought vibrations" running through his brain. Scant though the evidence may be, it strongly suggests that the week he spent with the "King of In variants" gave Hilbert the initial im pulse that put him on his way. Back in Konigsberg, he adopted several of Gor dan's techniques in his subsequent work. Numerous citations reveal that he was thoroughly familiar with Gor dan's opus, especially the two volumes of his lectures edited by Georg Ker schensteiner [Gordan 1885/1887]. That work, the springboard for many of Hilbert's discoveries, was by 1893 prac tically obsolete, though no comparable compendium would take its place. His friend Hermann Minkowski saw that this presented a certain dilemma: it was all very well to blow up the cas tles of those robber knights of invari ant theory, so long as something more useful could be built on their now bar ren terrain. Minkowski thus expressed the hope that Hilbert would some day show the mathematical world what the new buildings might look like. In the same vein, he kidded him that it would be best if Hilbert wrote his own mono graph on the new modernized theory of invariants rather than waiting to find another Kerschensteiner, who would likely leave behind too many "cherry pits" (misspelled by Minkowski as "Ker schensteine") [Minkowski 1973, 45]. Hilbert did neither, 1 0 leaving the theory of invariants to languish on its own while the Gordan-Kerschensteiner
volumes gathered dust in local li braries. Invariant theory thus entered the annals of mathematics, its history already sketched by the man who wrote its epitaph. To the younger gen eration, Paul Gordan would mainly be remembered for having once declared Hilbert's modem methods "theology." Now that he and his mathematical regime had been deposed, classical in variant theory was declared a dead subject, one of those "dead ends" ("tote Strange") that Hilbert had decried in his letter to Klein from 1890.U Al though Leopold Kronecker had pre dicted this very outcome, he would hardly have approved of the execu tioner's methods. Yet ironically, it was Hilbert's decision to move on to "greener pastures"-even more than the wealth of new perspectives his work had opened-that hastened the fulfillment of Kronecker's prophecy. LITERATURE
[Biermann 1 988] Kurt-R. Biermann, Die Math ematik und ihre Dozenten an der Berliner Uni versitat, 1 8 1 0- 1 933, Berlin: Akademie-Ver
lag, 1 988. [Blumenthal 1 935] Otto Blumenthal, "Lebens geschichte," in [Hilbert 1 935, pp. 388-429]. [Caruth 1 996] A Caruth, "A concise proof of Hilbert's basistheorem." Amer. Math. Monthly 1 03 (1 996), 1 60-1 61 . [Corry 2004] Leo Corry, Hilbert and the Ax iomatization of Physics (1898- 1 9 1 8): From "Grundlagen der Geometrie" to "Grundlagen der Physik", to appear in Archimedes: New Studies in the History and Philosophy of Sci ence and Technology, Dordrecht: Kluwer
Academic, 2004. [Fisher 1 966] Charles S. Fisher, "The Death of a Mathematical theory: A Study in the Soci ology of Knowledge, " Archive for History of exact Sciences, 3 (1 966), 1 37-1 59.
[Frei 1 985] Gunther Frei, Der Briefwechsel David Hilbert-Felix Klein (1 886- 1 9 1 8), Ar
beiten aus der Niedersachsischen Staats und Universitatsbibliothek Gbttingen, Bd. 1 9, Gbttingen: Vandenhoeck & Ruprecht, 1 985. [Freudenthal 1 98 1 ] Hans Freudenthal, "David Hilbert," Dictionary of Scientific Biography,
9For a brief account of the work and its historical reception, see the Introduction by Franz Lemmermeyer and Norbert Schappacher to the English edition [Hilbert
1 998,
xxiii-xxxvi]. 1 0Perhaps the closest he came to fulfilling Minkowski's w1sh was the
Ausarbeitung
of his
1 897 lecture course. now available in English translation in [Hilbert 1 993]
1 1 Historical verdicts with regard to the sudden demise of invariant theory have varied considerably (see [Fisher
1 966] and [Parshall 1 989]). The merits of classical in
variant theory were later debated in print by Eduard Study and Hermann Weyl. For an excellent account of this and other subsequent developments in algebra, see [Hawkins 2000].
© 2005 Spnnger SC1ence+ Bus1ness Med1a, Inc , Volume 27, Number 1 , 2005
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I 6 vols. , ed. Charles C. Gillispie, vol. 6, pp.
[Hilbert 1 890] --, " U ber die Theorie der al
Volume 1 : Ideas and their Reception, ed.
388-395, New York: Charles Scribner's
gebraischen Formen," Mathematische An
David E. Rowe and John McCleary, Boston:
nalen 36 (1 890), 473-534. , " U ber die Theorie der al
(Parshall and Rowe 1 994] Karen H. Parshall
Sons, 1 97 1 . [Gilmer 1 98 1 ] Robert Gilmer, "Commutative
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Ring Theory," in Emmy Noether. A Tribute to
gebraischen lnvarianten 1," Nachrichten der
and David E. Rowe, The Emergence of the
her Life and Work, ed. James W. Brewer and
Gesel/schaft der Wissenschaften zu G6ttin
American Mathematical Research Commu
Martha K. Smith, New York: Marcel Dekker,
gen 1 891 , 232-242. [Hilbert 1 892a] -- , " U ber die Theorie der al
nity, 1 8 76-1900. J.J. Sylvester, Felix Klein,
[Gordan 1 885/1 887] Paul Gordan, Vorlesungen
gebraischen lnvarianten I I , " Nachrichten der
vol. 8, Providence, Rhode Island: American
uber lnvariantentheorie, 2 vols., ed. Georg
Gesellschaft der Wissenschaften zu G6ttin
Kerschensteiner,
gen 1 892, 6-1 6.
1 981 , pp. 1 31 -1 43.
Leipzig:
Teubner,
1 885,
1 887.
[Hilbert 1 892b]
[Gordan 1 892]
, " U ber einen Satz von
and E.H. Moore, History of Mathematics,
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--
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Springer Verlag, 1 970.
gebraischen lnvarianten Ill," Nachrichten der
[Rowe 1 998] --, "Mathematics in Berlin,
Hilbert," Mathematische Annalen 42 (1 892),
Gesel/schaft der Wissenschaften zu G6ttin
1 81 0-1 933," in Mathematics in Berlin, ed.
1 32-1 42.
gen 1 892, 439-449.
--
[Gordan 1 899]
--
, "Neuer Beweis des
[Hilbert 1 893]
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H.G.W. Begehr, H. Koch, J . Kramer, N.
, " U ber die vollen lnvari
Hilbertschen Satzes uber homogene Funk
antensysteme," Mathematische Annalen 42
tionen," Nachrichten der Gesellschaft der
(1 893), 31 3-373; reprinted in [Hilbert 1 933,
Wissenschaften zu G6ttingen 1 899, 240-
242.
287-344] . [Hilbert 1 896]
Schappacher,
and
E.-J.
Thiele,
Basel:
Birkhauser, 1 998, pp. 9-26. (Rowe 2000]
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, "Episodes in the Berlin
Gbttingen Rivalry, 1 870-1 930," Mathemati --
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cal lntelligencer, 22(1 ) (2000), 60-69.
[Hashagen 2003] Ulf Hashagen, Walther von
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Oyck (1856-1934). Mathematik, Technik und
pers Read at the International Mathematical
tingen: A Sketch of Hilbert's Early Career,"
TH
Congress Chicago 1893, New York: Macmil
Mathematical lntelligencer,
Munchen, Boethius, Band 47, Stuttgart:
lan, 1 896, 1 1 6-1 24; reprinted in [Hilbert
44-50.
Wissenschaftsorganisation
an
der
Franz Steiner, 2003.
1 933, 376-383].
[Hawkins 2000] Thomas Hawkins, Emergence
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[Rowe 2003a]
--
[Rowe 2003b]
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, "Neubegrundung der
, "From Konigsberg to Gbt
--
25(2)
(2003),
. "Mathematical Schools,
Communities, and Networks," in Cambridge
of the Theory of Lie Groups. An Essay in the
Mathematik. Erste Mitteilung," Abhandlun
History of Science, val. 5, Modern Physical
History of Mathematics, 1 869-1 926. New
gen aus dem Mathematischen Seminar der
and Mathematical Sciences, ed. Mary Jo
York: Springer-Verlag, 2000.
Hamburgischen Universitat,
[Hesseling 2003] Dennis E. Hesseling, Gnomes in the Fog. The Reception of Brouwer's In tuitionism in the 1920's, Basel: Birkhauser,
1:
1 57-1 77;
reprinted in [Hilbert 1 935, 1 57-1 77]. [Hilbert 1 932/1 933/1 935]
--
, Gesammelte
Abhandlungen, 3 vols., Berlin: Springer, 1 932-
1 935.
2003. [Hilbert 1 888a] David Hilbert, " U ber die Darstel
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Nye, Cambridge: Cambridge University Press, pp. 1 1 3-1 32. [Rowe 2004]
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, "Making Mathematics in an
Oral Culture: Gbttingen in the Era of Klein and Hilbert," to appear in Science in Con
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, Hilbert's Invariant Theory
text, 2004.
lung definiter Formen als Summe von For
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(Toeplitz 1 922] Otto Toeplitz, "Der Alge
menquadraten," Mathematische Annalen 32
Groups: History, Frontiers, and Applications,
braiker Hilbert," Die Naturwissenschaften 1 0,
(1 888), 342-350; reprinted in [Hilbert 1 933,
ed. Robert Hermann, Brookline, Mass . : Math
1 54-1 6 1 ] .
Sci Press, 1 978. , " U ber die Endlichkeit des
[Hilbert 1 993] --, Theory of Algebraic Invari
lnvariantensystems fUr binare Grundformen,"
ants, trans. Reinhard C. Lauenbacher, Cam
[Hilbert 1 888b]
--
Mathematische Annalen 33 (1 889), 223-
226; reprinted in [Hilbert 1 933, 1 62-1 64]. [Hilbert 1 888c] braischen
--
, "Zur Theorie der alge
Gebilde
1,"
Nachrichten
der
bridge: Cambridge University Press, 1 993. [Hilbert 1 998]
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, The Theory of Algebraic
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necker," Jahresbericht der Deutschen Math ematiker-Vereinigung 2(1 893), 5-1 3.
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Die
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senschaften 20 (1 932), 57-58; reprinted in [Weyl 1 968, vol. 3, 346-347].
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[Mertens 1 887] Franz Mertens, Journal fUr die
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223-230.
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Nachrichten
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[Parshall 1 989], Karen H. Parshall, "Toward a
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ory," in The History of Modern Mathematics:
Natick, Mass. : A K Peters, 2002.
82
THE MATHEMATICAL INTELLIGENCER
i;i§lh§l,'tJ
O s m o Peko n e n , Ed itor
I
Statistics on the Table: The History of Statistical Concepts and Methods Stephen M. Stigler CAMBRIDGE, MASS , HARVARD UNIVERSITY PRESS PAPERBACK 2002 (first pnnllng 1 999) 5 1 0 PAGES US $ 1 9.95 ISBN 0-674-00979-7
REVIEWED BY IVO SCHNEIDER
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T
he book consists of 22 chapters, all of which except the first were pre viously printed as articles in reviewed journals or books. The chapters are distributed in five parts with the titles: I. Statistics and Social Science, II. Gal tonian Ideas, III. Some Seventeenth Century Explorers, IV. Questions of Discovery, and V. Questions of Stan dards. One could argue about the choice of these titles, their order, or their interrelations. In a strictly histor ical account one would begin with part III; part I is as much concerned with economic as with social questions, and Galtonian ideas are very much related to social science. In short, the selection arrangement of these 22 articles is less stringent than in a book which treats a topic systematically or strictly chrono logically. Trying to get the original publica tions in order to find out about the changes made for the sake of this book (which according to the sample I could check are small), I learned that many of these articles are not easily avail able. The book contains 21 of the 38 ar ticles and books concerning the history of statistics that Stigler published be tween 1973 and 1997, including his monograph from 1986, The History of
Statistics-the Measurement of Un certainty before 1 900. So one advan Column Editor: Osmo Pekonen, Agora Center, University of Jyvaskyla, Jyvaskyla, 40351 Finland e-mail:
[email protected] tage of the book is to make available some of Stigler's publications which are otherwise not easy to get. From a historical point of view the most interesting question is: How does
Statistics on the Table relate to Stigler's History of Statistics, which appears to
claim to cover the history of statistics, at least for the time before 1900? First, Stigler justifies the new book with the argument that, because sta tistical thinking and so statistical con cepts permeate the whole range of hu man thought, statistics in historical accounts is practically "never covered completely." Statistics on the Table represents, according to Stigler, "only a small selection of the possible themes and topics," but it treats, however in completely, one of the most important aspects of statistical work: statistical evidence in the form of data and their interpretation for the solution of a problem. For many, this project in such a general formulation represents the whole of statistical science, so it is not surprising that Statistics on the Table and The History of Statistics have sev eral things in common. Chapter 1 deals with the Karl Pearson of 1910/1 1 and so its material is not contained in The History of Statistics, which ends with 1900. It deals with the effect or non-effect of parental alcoholism on the alcoholism of the offspring. Pear son's vote for non-effect in the light of data collected in the Galton Laboratory met with considerable resistance and criticism due to different factors and interests, one of them being the tem perance movement of the time. What Pearson expected, or rather requested, from his critics was "statistics on the table," data which could confirm the position of his opponents and so dis prove him. Interestingly, Pearson's re quest was not seen by economists like Keynes, one of his opponents, as the appropriate method to deal with the controversy. Having shown in this way that "statistics on the table" was still far from being generally accepted as the method of handling questions of this kind, Stigler goes back in time in order to reconstruct the way in which collections of data were used before
1910.
© 2005 Spnnger Sc1ence+ Bus1ness Med1a, Inc , Volume 27, Number I , 2005
83
Starting with an evaluation of Quetelet's statistical work in chapter 2, which is less detailed than the chapter on Quetelet in his History, Stigler de votes the next two chapters to the sta tistical work of the economist Jevons, who is mentioned several times in the History but without any further evalu ation of his statistics. Stigler's interest in Jevons is motivated by the effect of his statistical work in overcoming the typical mid-nineteenth-century separa tion of statistics, understood as data collection, from the interpretation of the data, especially in the social and economic domain. A chapter on the work of the econ omist and statistician Francis Ysidro Edgeworth ends the first of the five parts of Statistics on the Table. Edge worth does not figure prominently in the history of statistics, because his statistical ideas were dispersed over a great many not easily digestible arti cles. So Stigler, like a Robin Hood of the history of statistics, takes away from the rich in reputation in order to give to the historically neglected Edge worth, whom he had honoured already in his History with a full chapter, indi cating that in his eyes Edgeworth was as important in the development of sta tistics as Galton and Karl Pearson. The five chapters devoted to "Gal tonian ideas" in part II. differ from the Galton chapter in the History by em phasizing different topics and the im pact of Galton and his methods. So whereas Galton's work on fingerprints is only mentioned without any further analysis in the History, Stigler devotes the whole of chapter 6 of Statistics on the Table to it, including the accep tance of fingerprints as evidence in court. Galton's and his contempo raries' contribution to "stochastic sim ulation," with a set of special dice used for the generation of half-normal vari ants, plays no role whatsoever in the History. Regression is the topic of the next two chapters, of which only the second deals with Galton's contribu tion to it, whereas chapter 8, "The his tory of statistics in 1933," hints at the more subtle aspects of regression visi ble in Hotelling's devastating review of Horace Secrit's book The Triumph of Mediocrity in Business from 1933, the
84
THE MATHEMATICAL INTELLIGENCER
year which Stigler likes to consider as the proper starting point of mathemat ical statistics. Stigler's inclination to act occasionally as an agent provoca teur might in this case be seen at odds with his History, which should be con sequently " The history of non-mathe matical statistics. Chapter 10 then discusses the relatively early use (com pared with other social sciences) of statistical techniques in psychology, which according to Stigler is due to ex perimental design. The following three chapters deal with publications of the second half of the 17th century. It is not clear to me why the first two of them belong in Sta tistics on the Table. Nor do I under stand the use of the word "probability" in the context of Huygens's tract on the treatment of games of chance or the correspondence between Pascal and Fermat common to these chapters. Pascal, Fermat, and Huygens were all well aware of contemporary concepts of probability, though none of these concepts appears in their treatment of games of chance. In chapter 13, how ever, a contemporary concept of prob ability is treated, as used by John Craig in order to determine the trustworthi ness of statements concerning histori cal events, or in Craig's term the "his torical probability," which increases in proportion to the number of witnesses in favour of it and decreases in pro portion to the time elapsed since the event. A function representing the de pendence of Craig's historical proba bility on time and distance is tested by putting on the table the data of Laplace's birth and death found in 65 books of the 19th century. Of the six articles making up "Ques tions of discovery," chapter 18, which treats the history of the so-called Cauchy distribution, has no relation to any concrete set of data put on the table, whereas the other five chapters do. The first chapter of this part is de voted to eponymy, the practice in the scientific community of affixing the name of a scientist to a discovery, the ory, etc. as a reward for scientific ex cellence in the relevant field. Such a re ward presupposes distance in time and place from the work honoured by eponymy. Accordingly, it cannot be ex"
pected that scientific discoveries are named after their original discoverer, or, to formulate it more aphoristically as a law of eponymy, "no scientific dis covery is named after its original dis coverer." Since Stigler sees the sociol ogist Robert K. Merton as the originator of this so-called law, and since Stigler does not want to begin with a counterexample for the validity of this law, the title-giving eponymy of this law is "Stigler's law of eponymy." For the joke's sake it does not matter that this is not an eponymy proper, which would have demanded that the community of sociologists after an ap propriate period of time ought to have accepted it. A proper eponymy namely the affixing of the names of Gauss and sometimes Laplace, but not the name of de Moivre, the real "dis coverer," to the normal distribution is discussed on the basis of 80 books published between 1816 and 1976. The discussion showed that at least in this case, the eponymy was awarded only after considerable time by the scien tific community. Seen with eyes ac customed to eponymic practice, the next chapter (15), which answers the question "Who discovered Bayes's the orem" with Nicholas Saunderson as the most probable candidate, appears as Stigler's next attempt to avoid a coun terexample to the law of eponymy, be cause most people believe that Bayes discovered Bayes's theorem. A prob lem not touched so far when dealing with discoveries was treated by Thomas S. Kuhn, who pointed to si multaneous discovery of the "same" thing by several people. In his discus sion of Kuhn, Yehuda Elkana pointed to difficulties inherent in the concept of "sameness. " Difficulties of this kind are the topic of chapter 16, which de scribes the first steps made by Daniel Bernoulli and Euler concerning the theme of maximum likelihood. Sameness plays no important role in the next chapter, dealing with the claims of Legendre and Gauss con cerning the method of least squares. The data of the French meridian arc measurements and their interpolation by Gauss in 1 799 are interpreted as in conclusive for Gauss's claim to have devised and applied the method of
least squares at that time or even be fore. Again Stigler's social attitude as the Robin Hood of the history of sta tistics becomes evident when he states that, despite Gauss's undisputed mer its in developing algorithms for the computation of estimates, it was Le gendre "who first put the method within the reach of the common man." However, Gauss's contribution to the method of least squares is seen much more positively in this article than in Stigler's History. The last chapter (19) in this part is mainly concerned with a paper of Karl Pearson and his collaborators from 1913, in which Pearson fitted a quasi-in dependent model to the data of incom plete contingency tables, testing the fit by a chi-square test, which, as was rec ognized by R. A Fisher, used the wrong number of degrees of freedom. But the concept of degrees of freedom had been introduced only in 1922 by Fisher, who had not seen that for the special class of tables considered by Pearson the use of the correct number of degrees of free dom would not have changed Pearson's conclusions. The last three chapters are sub sumed under the title "Statistics and Standards." In the first the observation that many of the most powerful statis tical methods, like the method of least squares, are originally connected with the determination of standards like the standard meter, is interpreted not as ac cidental but as a consequence of the purpose of a standard to measure, count, or compare as accurately as pos sible. In a second part Stigler describes how the fact that in experimental sci ence no absolute accuracy can be achieved, that every measurement is in evitably connected with error and so with uncertainty, eventually led to the creation of standards of uncertainty, standards in statistics like standard er ror curves or standard deviations. The last chapter (22), written to gether with William H. Kruskal, is re lated to standards in statistics in that the first part of it answers the question when and why the normal distribution was called "normal." The other parts of the chapter are concerned with the am biguity of the words "normal" and "nor mality," exemplified in paragraphs de-
voted to the terms "normal equations" in connection with the method of least squares, "normality in medicine," and "normal schools" in the educational system. Stigler maintains that there is a mutual dependence between the use of "normal" in science and in the realm of public discourse. Before this last chapter I found my favourite "The trial of the Pyx," which is a test to control the quality and cor rectness of the coin production at the Royal Mint for more than seven cen turies. Stigler interprets the trial of the Pyx as "a marvellous example of a sam pling inspection scheme for the main tenance of quality." It is amusing to read his report of the most famous master of the Royal Mint, Isaac New ton. Stigler amasses arguments for scepticism concerning Newton's hon esty as master of the Mint, thus dis qualifying Stigler forever as a member of the invisible college of Newtonians. However, perhaps concerned about his good relations to highly regarded British institutions, he finds on the ba sis of the research work of others "no grounds for believing that he 1 SUPz ErJJ(fz,r).
Finally, K. Seip's Theorem states the following: Let 0 < p < oo and - 1 < a
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