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.
Letter to the Editor
Patti Wilger Hunter's article on Abraham Wald in the Winter 2004 issue
nicely illustrates how a mathematician
can be stimulated by, and respond to, challenges from beyond mathematics per se. Your readers may not know that Wald's sequential probability ratio test
(SPRT), which was independently discovered 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 applications 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 evidence and responding when a threshold is crossed, just as in SPRT. Subsequently, Ratcliff [3] used a constant drift-diffusion process, the continuum limit of SPRT and perhaps the simplest stochastic differential equation,
dx = A dt + c dW, (DD)
to fit human behavioral data-specifically, reaction-time distributions and error rates. (Here A denotes the drift term and c the variance of the Wiener process 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 alternatives rise toward a threshold that signals the onset of motor response in a manner that seems to match sample paths of (DD) [4].
As pointed out in [5], this suggests an
intriguing possibility. SPRT is the optimal decision-maker, in the sense that,
for a predetermined error rate, it minimizes the expected time required to make a decision among all possible tests. (Human reaction times also include durations required for sensory and motor processing, and these must be allowed for in interpreting behavioral data.) Thus, if one wishes to optimize one's overall performance in completing a series of trials, one would do well to
4 THE MATHEMATICAL INTELLIGENCER © 2005 Springer Science+Busrness Medra, Inc.
employ SPRT or (DD) with thresholds chosen to maximize the reward rate:
RR=
(Expected
fraction of correct
responses)
(Average time between
responses).
(RR)
Since the numerator (1 - Error Rate) and denominator of (RR) are simple expressions of the drift rate A, noise variance c, and threshold for (DD), it is an exercise in calculus to compute optimal 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-accuracy tradeoff: it is not optimal to try to be always right, since that makes reaction times too long; nor is it good simply to go fast, since then error rates are too high.
We are currently assessing the ability of human subjects to achieve this theoretical optimum performance. While some of our subjects (Princeton undergraduates) appear more concerned to be correct than to be fast, the overall highest-scoring group indeed lies close to the optimal perforn1ance curve, although slightly on the conservative (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 before Wald and Barnard? Stay tuned.
REFERENCES [ 1 ] Barnard, G. A Sequential tests in industrial
statistics . J. Roy. Statist. Soc. Suppl. 8:
1 -26, 1 946. DeGroot, M. H. A conversa
tion with George A Barnard. Statist. Sci. 3: 1 96-2 1 2, 1 988.
[2] Stone, M. Models for choice-reaction tirne.
Psychometrika 25: 251 -260, 1 960. Larning,
D. R. J. Information Theory of Choice-Reac
tion Times. Acadernic Press, New York. 1968.
[3] Ratcliff, R. A theory of rnernory retrieval.
Psych. Rev. 85: 59-1 08, 1 978. Ratcliff, R . ,
Van Zandt, T., and McKoon, G. Connec-
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
Department of Computer Science
University of Bristol
Bristol, UK
e-mail: r.bogacz@bristol .ac .uk
Jonathan Cohen
Department of Psychology and Center for the
Study of Brain, Mind and Behavior
Princeton University
e-mail: [email protected]
Joshua Gold
Department of Neuroscience
University of Pennsylvania
e-mail: jigold@mail . med . upenn . edu
Is Escher's Art Art?
In 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 mathematician, his art has deep mathematical ideas, as some of the articles about H. S. M. Coxeter in the same issue of the Intelligencer, which mention Coxeter's and Escher's relationship, make clear. On the other hand, many people
in the arts have no mathematical training (much less mathematical interest).
When confronted with something, even something beautiful, that one doesn't understand, there are two common human reactions. One is admiration 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 culminated in her statement, " . . . one wonders if Fascism, which Escher detested, hadn't also contaminated his art."
Steven H. Weintraub
Department of Mathematics
Lehigh University
Bethlehem, PA 1 801 5-31 74
USA
e-mail: [email protected]
Where are the Women?
I am a junior at St. Cloud State University in Minnesota. While studying
to become a mathematics educator, I came across The Mathematical Intelligencer, 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 written 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 making sure it is [their] voice and views that get heard." The ideas of males as dominant 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 biases 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 female 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 before and since it have numerous women authors (though I note that vol. 26, no. 2 again has none-sorry). It is Intelligencer policy to encourage participation by mathematicians of whatever sex, whatever nation, whatever background. The policy has been stated in print before, and your letter is a welcome occasion 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 remove the barriers to their full participation 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 discouraged from studying it? are they being 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 acknowledge 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 descriptively, not honorifically), I found myself in general agreement with him, with perhaps one or two caveats. Personally, I would preserve Abelian groups: decent mathematical jokes are rare, and "What's purple and commutes? A commutative grape" seems to lose something in the trans
lation. I would also vote in favour of topological spaces whose points are hausdorff from one another (and salads
whose ingredients are waldorf). And I certainly advocate that we continue to remember Norbert Wiener for his seminal invention of the schnitzel. But the biggest exception to Hehner's generally sensible rule should surely be made for an eponymous term of astonishing beauty to be found towards the end of his paragraph. It appears that there exists something called the Peirce arrow.
Mr. Peirce's arrow is surely worth keeping. As Bob Dylan pointed out, it penetrates dust no matter how thick. It is inspiring: just as Sir Karl Popper regarded Darwin's evo
lutionary theory as a "metaphysical research program," so the Peirce arrow was a metaphysical research program for me. I determined to find out more about Mr. Peirce and his
arrow. Googling intrepidly through hundreds of thousands of references, using both Dylan's and Hehner's spellings, I uncovered these three pearls:
x = y - Sheffer stroke, NAND; x # y - Pierce arrow, NOR. [3]
. . . The questions also refer to "Sheffer's stroke" and "Pierce arrow" (not an antique car!) operators . . . [4]
6 THE MATHEMATICAL INTELLIGENCER © 2005 Spnnger Scrence+Busrness Medra, Inc
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:
(a) Why ARE certain WORDS written in upper case for no APPARENT reason? This is surely much more offputting than any mere use of an honorific name. I feel I'm being yelled at by NAND and NOR, and I dislike them already.
(b) Why is something written "#" called the "Peirce arrow"
rather than the "Peirce sharp sign" or the "Peirce waffleiron"? Further internet research revealed that this symbol appears variously as "#", "D", and "!". Whether this is a bitter controversy amongst logicians, or whether it's a consequence of the inadequacy of my computer (running on low-octane Windows 66) I don't know, but I suspect that unless Mr. Peirce was an extremely poor archer
he probably meant "!", so I'll go with that one. (c) If we are going to be told that the Peirce arrow is not an
antique car!, why aren't we told that Sheffer's stroke is not a medical condition!? Or that it's not a sexual technique!? In fact the author told us (the Sheffer stroke is not a medical condition!)! (the Sheffer stroke is not a sexual technique!). Or should that read "the Sheffer stroke is (a medical condition)! (a sexual technique)!"? Or should I actually have written "(the author told us the Sheffer stroke is not a medical condition!)! (the author told us the Sheffer stroke is not a sexual technique!)"?
(d) It's good to know that the Peirce arrow can be used to 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 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
borrower)! (a lender) be" [6], was he speaking conjunctively or disjunctively? As with so much of the theoretical 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 elitist and off-putting. It may use an author's prestige to lend respectability to an unremarkable idea. It may, for all I know, be attached to the wrong bloke altogether. But it's beautiful. It's poetic. It inspires research. And unlike other rival names, it doesn't give my eyes disjunctivitis. So please don't shoot the arrow away. You may chuck away at a stroke all reference to Sheffer. I would shed no tears at the demise of Banach spaces or Sylow's theorem. But Peirce's arrow deserves to thrive, along with all the other beautiful terms that enrich mathematics: wonderful expressions like Weyl integrals, Killing fields, the Gordan knot, the Roch group, Jordan delta functions, Plateau's plane, Taylor cuts, the Schur Certainty Principle, Abel-Baker-Chasles-Lie symbols, and, since I'm feeling rather eponymous just now, the
Grunwaldian or recursive citation. [8]*
REFERENCES [ 1 ] Bob Dylan, Restless Farewell, 1 964.
[2] E. C . R. Hehner, "From Boolean Algebra to Unified Algebra," The
Mathematical lntelligencer, vol. 26, no. 2 , 2004.
[3] http:/ /rutcor.rutgers.edu/pub/rrr/reports2000/32 .ps. Rut cor Research
Report
[4] web.fccj.org/�1 dap991 1 /COT1 OOOUpdate. html
AU T H O R
ERIC GRUNWALD Perihelion Ltd.
1 87 Sheen Lane London SW14 8LE
UK
e-mail: [email protected]
Eric Grunwald received his doctorate in mathematics from Ox
ford. Since then he has been employed in the chemical, en
ergy, and health-care industries, and has become expert in
advising organizations on their future planning. He has, sadly,
not found anyone else in the field of future thinking who knows
much about mathematics; if there are others, he would like to
meet them.
[5] iptps03 .cs. berkeley .edu/final-papers/result_ caching. pdf
[6] W. Shakespeare, Hamlet, act 1 , scene 3, 1 601 .
[7] W. Shakespeare, private communication.
[8] E. J. Grunwald, "Eponymphomania," The Mathematical lntel ligencer,
vol. 27, no. 1 , 2005.
[9] K. D0sen, "One More Reference on Self-Reference," The Mathe
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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© 2005 Spnnger Sc1ence+Bus1ness Media, Inc., Volume 27, Number 1 , 2005 7
W. M. PRIESTLEY
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
to Athens with him his young student Theaetetus and an
unnamed philosopher visiting from Elea, the Greek town
in southern Italy that is home to Zeno and his paradoxes.
The "Eleatic Stranger"-the appellation given this name
less visitor in older translations of Plato-may suggest to
us the archetypal masked man who descends upon the ac
tion from nowhere to round up the outlaws and establish
order.
Sure enough, the Stranger has already gone after the
Sophist earlier in the day, using a dichotomizing technique
that closely resembles the modern analyst's bisection
method of successive approximations. In the words of a
modern commentator [P2, p. 235], he "first offers six dis
tinct routes for understanding the [S]ophist, by systemati
cally demarcating specific classes within successively
smaller, nested ... classes of practitioners; these subclasses
are then identified as the [S]ophists." Then, following a
lengthy discussion to introduce a "change of coordinates,"
the Stranger resumes his search and finally obtains neces
sary and sufficient conditions to characterize the slippery
Sophist. Socrates expresses delight.
Here, in Benjamin Jowett's nineteenth-century transla
tion, are the opening lines that follow in the Statesman.
8 THE MATHEMATICAL INTELLIGENCER © 2005 Spnnger Scrence+Busrness Medra, Inc.
[SocRATES] I owe you many thanks, indeed, Theodorus, for
the acquaintance both of Theaetetus and of the Stranger.
[THEODORUS] And in a little while, Socrates, you will owe
me three times as many, when they have completed for
you the delineation of the Statesman and of the Philoso
pher, as well as of the Sophist.
[SocRATES] Sophist, statesman, philosopher! 0 my dear
Theodorus, do my ears truly witness that this is the es
timate formed of them by the great calculator and geo
metrician?
[THEODORUS] What do you mean, Socrates?
[SocRATES] I mean that you rate them all at the same
value, whereas they are really separated by an interval,
which no geometrical ratio can express.
[THEODORUS] By Ammon, the god of Cyrene, Socrates,
that is a very fair hit; and shows that you have not for
gotten your geometry. I will retaliate on you at some
other time . . . . (Statesman 257a-b)
What is the "hit" by Socrates that provokes Theodorus's
oath? Some commentators on Plato say that Socrates is
alluding to the existence of incommensurables in geome
try, something that Plato was fond of mentioning in other
contexts. Thus, Socrates would seem to be implying that
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Only a third part of our task is done: nay, not a third, for the Statesman rises above the Sophist in value and the Philosopher above the Statesman in more than a geometrical ratio.
The beginning of Plato's Politikos-Politicus in Latin, Statesman in English.
these three types of individuals have incommensurable
natures.
Another reading suggests itself.
Existence Questions
Socrates might be referring to the classical geometric ver
sion of what is now familiar to us as the Archimedean property. To see why this is plausible, though, requires a
digression. In the modem setting of ordered fields, the
property states that if a and b are positive elements, then
there exists some natural number n such that na exceeds
b. In modem terms, a is not infinitely small (i.e., not an "in
finitesimal") relative to b. "Every little bit counts," as Paul
Halmos once quipped. In a complete ordered field, this
property is a simple consequence of (primarily) the com
pleteness axiom, for if the elements of the set { na) were all
bounded above by b, then a contradiction follows in an el
ementary way as soon as this axiom is invoked. The
Archimedean property is the main feature of standard, as
opposed to non-standard, treatments of modem analysis.
Infinitesimals and their reciprocals (infinitely large num
bers) do not exist among the positive elements of the "stan
dard" real number system.
How could Plato (427-347 BCE) have known of the prop
erty bearing the name of Archimedes, who was born later?
The answer is that this property, whose eponymous name
is relatively recent-0. Stolz's 1883 paper [St] may mark its
first appearance-was familiar to mathematicians about a
hundred years before Archimedes (287-212 BCE) drew at
tention to its fundamental role. In fact, the property (see
[Bo2, p. 129]) seems to have been introduced by Eudoxus,
who studied, perhaps briefly, in Athens as a penniless youth
and later, having become a noted mathematician and as
tronomer, returned with his own students around 365 BCE
to meet with Plato on more nearly equal terms. The dates
of Eudoxus's life are uncertain, although it is now thought
© 2005 Spnnger Science+ Bus1ness Media, Inc .. Volume 27, Number 1. 2005 9
[Gu, p. 447] that he was born about 395 BCE and outlived
Plato by a few years.
Eudoxus, according to tradition, formulated the subtle
definition that vastly extended the applicability of the the
ory of proportions by making it possible to decide when
two ratios are the same, even when each of them is a ratio
of incommensurable magnitudes. Some commentators
read the language used in Parmenides 140c as indicating
Plato's awareness of the theory of Eudoxus.
It was after Eudoxus's return to Athens that Plato com
pleted his Theaetetus-Sophist-Statesman trilogy, which is
intimately concerned with ontology, that is, with general
philosophical questions regarding existence.
I hold that the definition of being is simply power. - The Eleatic Stranger (Sophist 247e)
As we shall see, the Stranger's association of being with
power has something to do with the existence of incom
mensurables.
Plato's Academy, founded in Athens around 385 BCE, had
exceptional scholars to reflect upon philosophical and
mathematical questions. Aristotle (384-322 BCE) joined
Plato's Academy at an early age-about the time of Eu
doxus's return-and would eventually rival, if not surpass,
his teacher. And until his death around 369 BCE, there was
Theaetetus, who may have been the main proponent of an
alternative treatment of ratios that I shall mention later. In
Plato's dialogue dedicated to him it is implied that Theaete
tus, whose investigations advance as smoothly as "a stream
of oil that flows without a sound," was first to prove that
all positive integers for which there exist rational square
roots must be perfect squares.
Plato acted in the important role of catalyst for the math
ematical investigations of others, but he, like Aristotle
(whose interest in logic far exceeded his interest in math
ematics), seems to have proved no new theorems of his
own. Of Plato's contemporaries, Theaetetus and Eudoxus
contributed most heavily to the material collected around
300 BCE in Euclid's Elements.
Arithmetization: Dedekind and Eudoxus
(and Plato?)
While Benjamin Jowett, at Oxford, was busily translating
and analyzing Plato's dialogues (the first edition of Jowett's
massive project appeared in 1871), a remarkable new
movement in mathematics was developing on the Conti
nent. In 1858 Richard Dedekind (1831-1916) realized that,
in a sense, the key to the modem "arithmetic" foundations
of real analysis had been in Eudoxus's hands some 2200
years earlier.
What needed to be done to obtain a purely numerical
theory, Dedekind saw, is to retain Eudoxus's insight, but
to remove all reference to the geometric magnitudes whose
existence the Greeks took for granted. In fact, as Dedekind
remarked later in a letter to Rudolf Lipschitz, the Euclid
ean theory of ratios cannot encompass the complete sys
tem of real numbers required by modem analysis because
1 Q THE MATHEMATICAL INTELLIGENCER
only algebraic numbers can result from Euclidean con
structions. (See [Fe, p. 132] and [De, pp. 37-38].)
As is now well known, Dedekind [De, p. 15] declared
that a real number is defined-or "created"-by a cut (Schnitt), by which he meant, essentially, a partition of the
rational numbers into a pair of nontrivial segments. He first
showed [De, pp. 13-14] that there exist infinitely many cuts
not produced by rational numbers by giving a clever proof,
using the well-ordering principle, of Theaetetus's result that
square roots of non-square positive integers are irrational.
He then observed that the expected algebraic properties
(and ordering) of the real number system can be made to
follow from properties of the integers by defining arith
metic operations (and an order relation) on cuts in a nat
ural way. More importantly, he showed how the complete
ness property of the real number system flows smoothly
from these considerations. Dedekind used the word continuity (Stetigkeit) to describe the crucial property that is
more commonly called completeness or connectedness by
mathematicians today.
Dedekind [De, p. 22] insisted that such a "theorem" as
V2 V3 = V6 can be given a "real proof' only after we at
tach to these symbols their appropriate numerical mean
ings in terms of cuts (see [Fo1]). By going far beyond the
Greeks in appealing to infinite processes, Dedekind melded
the discrete and the continuous, freeing analysis to be de
veloped independent of its geometric origins. In fact, much
of geometry could now be made dependent upon analysis
by identifying geometric points inn-dimensional space with
n-tuples of real numbers. The foundations of mathematics
thus began to shift decisively from geometry toward arith
metic and set theory, to which Dedekind and his great
friend Georg Cantor (1845-1918) began to devote much at
tention. While it may be too much to call Dedekind "the
West's first Modernist" [Ev, p. 30], he certainly helped to
foster a movement that is about as close to a paradigm shift
as the history of mathematics can provide. Dedekind withheld publication of his radically modem
ideas until he realized in 1872 that other mathematicians
(he names Heine, Cantor, Tannery, and Bertrand) were also
ready to face squarely the ontological question of the ex
istence of "real" numbers [De, p. 3]. In 1887 Dedekind ac
knowledged his ancient source by writing that Euclid's
Book V sets forth "in the clearest possible way" his own
conception that an irrational number-if it is presented as
a ratio of magnitudes-can be defined by the specification
of all rational numbers that are greater and all those that
are smaller [De, pp. 39-40].
Intriguingly, "the Great and the Small" happens to be a
phrase used by Aristotle (Metaphysics 987-988) to refer
to an idea, apparently puzzling to Aristotle, whose impor
tance Plato emphasized in lectures given late in life. This
has led some scholars to speculate, once Dedekind's ideas
had become well understood, that Plato in his later years
might have been thinking along similar lines. Interest in
such speculation has been heightened by the juxtaposition
of two curious facts: ( 1) Plato wrote on more than one oc
casion that some things cannot be expressed in writing and
might be more accurately conveyed only through the (still
imperfect) give-and-take of (oral) dialectic; and (2) "the
great and the small" is a phrase used by Plato (in Statesman 283e, for example) but is seemingly never applied
anywhere in his writings in the manner described by Aris
totle [Sa, p. 96]. According to Aristotle, Plato asserted,
among other things, that numbers come "from participa
tion of the Great and the Small in Unity." I shall suggest
below what Plato might have meant by this cryptic pro
nouncement.
No one would suggest, of course, that an ancient Greek
could have foreseen all of our present basis for real analy
sis. A sea change (see [Cr]) had to occur before modem
mathematicians even began to look for a numerical, as op
posed to geometrical, underpinning to their discipline. The
"arithmetization of analysis" did not take place until the late
nineteenth century with the amalgamation of results of
Cauchy, Balzano, Weierstrass, Dedekind, Heine, Borel,
Cantor et al. [Bo2, p. 560ff.]
Before considering what Plato's ideas might have to do
with those of modem mathematical analysis, however, let
us return to Socrates and Theodorus.
Flatland in 350 acE In Euclid V a ratio (logos) is described as "a sort of relation
in respect of size between two magnitudes of the same
kind." Two geometric magnitudes are then said to have a
ratio if and only if some (positive, integral) multiple of each
exceeds the other. Thus, for example, one cannot speak in
Euclidean geometry of the ratio of a line segment to a square
because no (finite) number of copies of a given line seg
ment can make up an area exceeding that of a given square.
Nor can one speak of the geometrical ratio of a square to a
cube, for these are likewise not "of the same kind."
By now it must be clear what this has to do with
Socrates's "hit" in the opening lines of the Statesman. The
Sophist, Statesman, and Philosopher represent three types
whose relative statures differ greatly. In the opening ex
change of the Sophist the first two are spoken of as "ap
pearances" or, in Comford's translation, "shapes" that the
philosopher can assume in the eyes of others. As the dia
logue reveals, Plato sees the devious Sophist, with his pen
chant for demagoguery, as essentially nothing in compari
son to the noble Statesman, who will himself cut a small
figure when placed alongside the truly wise Philosopher. If
two magnitudes were said to be separated by an interval
that no geometrical ratio can express, a geometer like
Theodorus would immediately infer that the larger magni
tude exceeds the smaller by every (finite) multiple. In
other words, given the context, Socrates's remark implies
that the worth of one statesman exceeds that of any (arbi
trarily large) number of sophists; similarly, one philosopher
is worth more than any number of statesmen. A sophist is
thus infinitesimal in comparison to a statesman, who is him
self infinitesimal in comparison to a philosopher.
Was Plato thinking of the sophist, statesman, and
philosopher as analogous to one-, two-, and three-dimen
sional magnitudes, respectively, perhaps along the lines of
the beings brought to life in Edwin Abbott's Victorian ro
mance, Flatland [Ab]? It might have been the other way
around. Was Abbott (1838-1926), whose field was classics
and who introduces his own "Stranger" to lead a playful di
alogue about "Spaceland" [Ab, p. 65], borrowing from Plato
the idea of one-dimensional and two-dimensional beings?
A classical analogy between persons and magnitudes does
suggest itself, for in the Greek of Plato's day the word for
magnitude referred not only to line segments, rectangles,
cubes, etc.; it also, as Salomon Bochner points out [Bol, pp.
278-79], carried an older connotation (circa 775 BCE) from
Homer:
. . . [T]he Greeks did not have real numbers but, in its
place, a notion of "magnitude" [megethos] . In Homer this
noun still means: personal greatness or stature (of a hero,
say); and it is remarkable that for instance in the French
noun grandeur and the German noun Grosse the two
meanings of personal greatness and of mathematical
magnitude likewise reside simultaneously.
Perhaps there is evidence in Plato's other writings to
suggest that he might have been in the habit, as some of us
are today, of thinking of narrow-minded people as being in
some sense "one-dimensional." In the Republic (587d) we
find the remark, accompanied by an obscure explanation,
that the philosopher is 729 times happier than the tyrant.
But 729 is the cube of 9; this seems to hint at the "three-di
mensionality" of the "solid" philosopher and the relative
shallowness of the tyrant.
What is Analysis?
Whatever one intends by the meaning of a proposition, it
surely involves the collection of statements implied by that
proposition in some universe of discourse reflecting a con
text either explicitly given or implicitly understood. The
completeness axiom, for example, states that, given a bounded, nonempty set, there exists a least upper bound. If we were asked what this "really" means, we might reply
that in the context of an ordered field it means a host of con
sequences-that the system is unique up to isomorphism,
that the real numbers naturally form a connected topologi
cal space, that every non-empty convex subset is an inter
val, that there exists a point common to a collection of
nested, bounded, non-empty closed intervals, etc. [01].
Some 700 years after Plato's death the mathematician
Pappus of Alexandria described a "method of analysis" dat
ing from Plato's time (see [Kat, pp. 184-5]) that seems to
flow from this observation about meaning. To test the truth
of a proposition, Pappus says, deduce implications from it.
Should one deduce an implication that is self-evidently true,
then a synthetic proof-as in Euclidean geometry-is said
to be obtained if the steps in this deduction can be reversed
so as to obtain the given proposition as a logical conse
quence of self-evident truths. Pappus's usage of the term
analysis is criticized by Wilbur Knorr [Kn2, pp. 354-360].
Stephen Menn [Me, p. 194] remarks that the neo-Pla
tonists are conscious that they are speaking metaphorically
© 2005 Spnnger Sc1ence+Bus1ness Med1a, Inc . . Volume 27, Number I, 2005 1 1
• 1'0/NTL.fND
0 SI'AC£LAND
"Fie, fie, how franticly I square my talk!" Title page of Flatland, embellished by art work of "A. Square"-the pseudonym of Edwin Abbott Abbott.
in extending the term analysis from geometry to philoso
phy. The word, coming from the Greek analyein, meaning
"to break up," is never used by Plato in his writings [Me, p.
196], but Aristotle uses it, and-more importantly-Aristo
tle describes Plato as ever watchful to see whether an ar
gument is proceeding to or from first principles [Me, p. 193].
The word analysis has been used in different ways over
the centuries, but today mathematicians use it, of course,
to refer to the modem branch of mathematics whose first
principles involve the notions of number and limit.
Plato's early dialogues typically recount how Socrates
disproves the unexamined philosophical assertions of oth
ers by deducing from them absurd implications (reductio ad absurdum). Socrates repeatedly claims to know only
how to examine carefully the assertions of others and not
how to advance a thesis of his own. This "ignorance of
knowledge" pervades the early dialogues, where philo
sophical questions are raised and conventional answers
found wanting. Perhaps the object is to point us in the right
direction by examining in which way(s) our approxima
tions miss the mark. One is reminded of the remark by John
von Neumann [vN] near the end of The Mathematician that
truth is much too complicated to allow anything but ap
proximations.
In his so-called middle period Plato begins to apply to
philosophy something like the method of analysis described
12 THE MATHEMATICAL INTELLIGENCER
by Pappus. In Books II-X of the Republic he has Socrates
no longer criticize his interlocutors' ideas, but instead, as a
modem commentator puts it [P2, p. 972], to proceed
in a spirit of exploration and discovery, proposing bold
hypotheses and seeking their confirmation in the first in
stance through examining their consequences. He often
emphasizes the tentativeness of his results, and the need
for a more extensive treatment.
In Plato's later writings the role of Socrates is dimin
ished. The Eleatic Stranger and Timaeus, a Pythagorean
(apparently fictional characters, both), are introduced to
discuss ontology and cosmology-philosophical subjects
not associated with the historical Socrates. The self-criti
cal analysis in the Parmenides seems to hint at Plato's need
for a new voice. Here we are cautioned to pay close at
tention in discussions to the implications of the negation of the proposition in question. If one of these should be
false, then as logicians from Aristotle onward would em
phasize, the original proposition is true by reductio ad absurdum, provided that we accept the law of the excluded
middle. Plato notes what is less often emphasized, that in
this way we uncover sufficient conditions for a proposition
to hold. (If not-p implies q, then not-q is a sufficient con
dition for p.) Thus, with foresight, we can use the analysis
of implications to determine sufficient as well as necessary
conditions for a proposition to hold. This observation must
have been quite surprising when first noted.
About mathematicians we hear that they move "down
ward" in deducing theorems from accepted hypotheses,
while philosophers should, as well, learn to move "upward"
from hypotheses to an ontological level at which the hy
potheses themselves are seen to be justified. What do we
think about this today? When the axioms for a complete
ordered field were "justified" on the basis of the existence
of Dedekind cuts satisfying these axioms, was it mathe
matics or philosophy that was done?
The power of analysis had been strikingly felt (around 430
BCE, as dated by Knorr [Kn1, p. 40]) when the Pythagorean
presumption that every ratio can be expressed as a ratio of
(whole) numbers was tested and proved false by reductio ad absurdum. Aristotle indicates (Prior Analytics i23),
perhaps too laconically, that this follows from the simple
fact that an odd number cannot be even. Most beginning
students of mathematics today know how to use this fact
to deduce that V2 is irrational.
Here is a less familiar proof of this ancient result: If the
ratio of the diagonal of a square to its side were express
ible in lowest terms as (a + b )Ia, a ratio of positive inte
gers with b < a, then (a + b )2 = 2a2 by the Pythagorean
theorem. But this implies that (a - b)2 = 2b2, so the origi
nal ratio is expressible in strictly lower terms as (a - b )/b, which is a contradiction. The algebra here may at first ap
pear contrived, but the geometry behind it is natural-as
van der Waerden [vdW, p. 127] explains-and would prob
ably have been familiar to Theaetetus. It was well known
in Plato's time, and soon thereafter codified (Euclid X,
Proposition 2), that if the Euclidean algorithm never comes
to an end when applied to two line segments, then the seg
ments are incommensurable.
Number and Measure in Ancient and
Modern Mathematics
Can the square root of two be expressed in terms of ratios
of integers? Theodorus, along with latter-day Pythagoreans,
would have said no because there is no single ratio that can
measure it. Our answer today, of course, is yes, it can be
measured precisely in terms of a cut in the set of all ratio
nal numbers. Would Plato's colleagues, including Eudoxus
and Theaetetus, agree with us? It may be useful to consider
the barriers to the ancients' taking our point of view.
In classical times the word number was restricted to
"positive, whole number": a number is "a multitude com
posed of units" according to Euclid VII. The unit itself was
not considered to be a number because it is not a multi
tude, and the unit chosen in practice might be different in
different contexts, depending upon whether one were mea
suring length or measuring area, for example. One speaks
of ratios of numbers, however, just as one speaks of ratios
of geometric magnitudes. As Plato and his colleagues were
acutely aware, there are more of the latter than of the for
mer-a fact from which some might infer that geometry is
a "higher science" than arithmetic.
The ancient Greeks would have spoken of the ratio of
the diagonal to the side of a square rather than the "square
root of two," which only later denotes a numerically mea
sured quantity. Their problem was to come to grips with
such ratios in the first place-and once this was done, to
check, for example, that the ratio of diagonal to side in one
square is the same as the ratio of diagonal to side in an
other. But how can our numerical understanding of ratio
possibly be extended to incommensurable magnitudes?
Here is Eudoxus's definition of proportionality ("sameness
of ratio") from Euclid V, given two pairs of magnitudes,
each of which is assumed to have a ratio:
Magnitudes are said to be in the same ratio, the first to
the second and the third to the fourth, when, if any equi
multiples whatever are taken of the first and third, and
any equimultiples whatever of the second and fourth, the
former equimultiples alike exceed, are alike equal to, or
alike fall short of, the latter equimultiples respectively
taken in corresponding order.
The convoluted phrasing may remind us of our first en
counter with Cauchy's epsilon-delta definition of a limit.
Augustin-Louis Cauchy (1789-1857) is sometimes called the
nineteenth-century Eudoxus, for giving precise numerical
significance to a subtle concept essential to future
progress-although Dedekind is thought by some to be
even more deserving of this title.
To see what is going on here, let us consider a famous
case that Euclid left for Archimedes to study. Suppose that
the first and second magnitudes are the area A and the
square of the radius r2 of a circle, while the third and fourth
are the circumference C and the diameter D. Eudoxus's def
inition given above says that the ratio A : r2 is the same as
C : D if and only if, for arbitrary natural numbers m and n,
(1) nA >mr � nC>mD (2) nA = mr2 � nC = mD (3) nA < mr2 � nC < mD.
In proofs involving proportionality (such as Euclid V,
Proposition 8) Euclid assumes what Archimedes later (see
[Di, p. 146 and p. 43lff.]) states explicitly as an axiom: that
if two magnitudes are unequal, then some integral multiple
of their difference (the magnitude by which one exceeds the
other) exceeds either. Perhaps Euclid's readers are expected
to infer that equality of a pair of like magnitudes should be
understood to mean that their difference has no ratio to ei
ther member of the original pair-thus offering justification,
if needed, for the familiar fact that ratios of like magnitudes
are generally unchanged by the inclusion or exclusion of por
tions of their boundaries. Euclid's silence on this issue makes
it difficult to determine whether he would consider condi
tions (1), (2), and (3) to be independent.
Theaetetus must have been among those who first
thought deeply about how to treat equality in this setting, for
Plato pictures him (Theaetetus 155c) as being remarkably
concerned, even as a youth, with the problematic nature of
this seemingly transparent notion. Historians tend to credit
© 2005 Spnnger SC1ence+Bus1ness Med1a, Inc., Volume 27, Number 1, 2005 13
Eudoxus with the approach finally adopted, which (as clar
ified by Archimedes) is beautiful in its simplicity. Properties
following from inequality are postulated explicitly, so that
"being equal" (in the case of continuous magnitudes) really
means "not being unequal." Here we have an inspired use of
litotes, the rhetorician's term for the expression of an affrr
mative by the negative of its opposite.
By taking this indirect approach to the notion of "equal
ity" of continuous magnitudes, the Greeks were able to handle many limits to their apparent satisfaction without be
coming embroiled in (modern) concerns about existence
and uniqueness (and precise definitions) of limits. The
Greek method of exhaustion typically proves equality of
areas or of solids by simply showing that the assumption
of inequality leads to contradiction. If exhaustion is taken
to mean elimination (of possible answers that are too large
or too small), then the similarity of this method to the
rhetorician's artful use of litotes becomes quite clear.
Eudoxus's brilliant use of number to clarify the notion
of exactness in proportions is sometimes said [Bo2, p. 88)
to have led to "Platonic reform" in mathematics, but its ef
fect upon Plato himself has drawn surprisingly little atten
tion. The Greek word for proportion ( analogia) has a
broader meaning that encompasses analogy as well, and
Plato may have tried to extend Eudoxus's method to this
wider realm. As we shall see in the next section, Plato re
gards the rhetorical treatment of "being" and "non-being"
as worthy of serious attention having to do, among other
things, with the demonstration of "exactness itself."
Let us first recall, however, the oft-noted connection be
tween Eudoxus's condition specifying sameness of ratios
of geometric magnitudes and our modern condition for
equality of real numbers. This becomes apparent as soon
as we identify the ratios A : r2 and C : D with the real num
bers that we conventionally symbolize by Ali!- and CID. Re
quirements (1), (2), and (3) can then be interpreted to say
that the numbers Ali!- and C/D are equal if and only if the
condition that Ali!- is greater than (respectively, less than, or equal to) an arbitrary rational number min implies that
C/D is greater than (respectively, less than, or equal to) min as well. Eudoxus's condition is thus closely related to
the modern criterion that real numbers are equal if and only
if there is no rational number lying in between. It is easy
for a modern analyst to see how Dedekind's contemplation
of the condition of Eudoxus in Euclid V might have led him
to consider the reification of cuts.
When Archimedes demonstrated (by the method of ex
haustion) in his Measurement of the Circle [Di, Chap. VI]
that A is equal to the area of a right triangle with legs C
and r, it became easy to prove that A : r2 is indeed the same
as C : D. Archimedes went on to show how one can effi
ciently compute ratios, both greater and smaller, that ap
proximate C : D to any accuracy desired, and in fact, he
proved that
223 : 71 < C : D < 22 : 7.
In an analysis class today it might be anachronistic, but cer
tainly not hyperbolic, to say that Archimedes gave us an ef-
1 4 THE MATHEMATICAL INTELLIGENCER
fective algorithm to construct the Dedekind cut corre
sponding to 7T. Nevertheless, the "ratio of the circle to its
diameter" seems not to have achieved firm status as a "num
ber" until the advent of Indo-Arabic decimal fractions, il
lustrating the vast difference between our modern numeri
cal outlook and the geometrical framework it replaced. This
famous ratio was not called 7T before 1706, by which time
it was already known to many decimal places [Bo2, p. 442].
Expressing the Inexpressible: Is Non-Being a Form
of Being?
Did Plato inquire in what sense we can ascribe real nu
merical significance to a ratio of incommensurables? The
answer implicit in Eudoxus's-or Dedekind's-approach is
surprising today, and would have been startling in Plato's
Academy in 350 BCE. We cannot say, for example, what the
numerical ratio of the diagonal to the side of a square is,
except to say what it is not. And this, the pair of segments
consisting of rational numbers greater and smaller, re
spectively, as Dedekind has taught us, is all we need to deal
effectively with it as a number. While superficially similar,
perhaps, to the reification of "negative space" in art,
Dedekind's insight is considerably deeper. Non-being in the universe of rational numbers, when specified by such a dyad of segments, constitutes being in the real numbers. The phrase "shadowy forms," which Dedekind used
to describe integers defined by his new approach to num
ber theory [De, p. 33], seems even more apt to describe ir
rational numbers defined by cuts.
We would know a lot about how much Plato anticipated
Dedekind's approach if the lengthy discussion in the
Sophist of the tricky relation between "being" and "non
being" had ever been specialized to ratios of whole num
bers. The Eleatic Stranger, however, pores over this puzzle
only in the most general terms before finally concluding
(Sophist 258) that Non-Being is-as Otherness-a Form.
Later on in the sequel, however, the Stranger remarks that
. . . just as with the sophist we compelled what is not
into being, . . . so now we must compel the more and the
less, in their turn, to become measurable . . . in relation
to the coming into being of what is due measure. (Statesman 284b)
In context the practical point of this remark seems to
be that, for example, we should not simply say of a politi
cian that he is being too heavy-handed or too wishy-washy,
but, more importantly, we should recognize and affirm the
existence of the precise "mean" attitude ("the Good") that
he should try to attain in the case at hand. The ostensible
theme of the Statesman, after all, is the delineation of the
character of the true political leader.
But the Stranger remarks immediately (284d) that he
may someday require this notion of a mean for the demon
stration of exactness itself. Perhaps this is a hint as to the
content of a projected dialogue entitled the Philosopher, a
sequel to the Statesman that Plato never wrote. Could this
remark be based upon an underlying assumption by Plato
that the coming into being of a number ("due measure") is
coextensive with the specification of all ratios greater and
smaller? And is Plato here metaphorically associating the
existence of the Good with the existence of a point of unity
or harmony that brings opposing tendencies into proper
balance?
This speculation encounters a problem. Plato's commit
ment to such a metaphor should force him as well to as
sociate Evil with the dyad of the Great and Small that comes
into being simultaneously with the due measure, or "Unity,"
of the Good. Aristotle (Metaphysics 988a14-15) maintains,
however, that Plato did indeed assign "the causation of
Good and Evil" to these very elements. Aristotle's words
here do not carry all the moral connotations of their coun
terparts in English. "Good" here means (the character of
being) "in good condition." "Evil" has the opposite mean
ing, and is thus naturally associated with all ratios greater
or smaller than what is due. For Plato, ignorance-lack of
knowledge of what is right-is indeed the cause of evil.
"I hold that the definition of Being is simply
dynamis." (Sophist 247e)
The Greek word dynamis is usually translated power. We
get our words dynamic and dynamite from it, and dyne comes from it too, although this is a unit of force, rather
than power, in physics. The association of "being" with
"power" is one of the Stranger's most striking observations.
We might be tempted today to say that a real number ex
ists simply by virtue of its power to cut the rationals in two.
The way the Stranger "cuts out" the Sophist by conjoining
appropriate properties (or by taking intersections of sets,
as we might describe it now) may be intended to suggest
that Platonic Forms exist by virtue of their power to com
bine.
Near the beginning of the Theaetetus-Sophist-Statesman trilogy, Plato has Theaetetus introduce the word dynamis with some fanfare, giving it a special mathematical
meaning by associating it with the simplest irrationalities
like the one we call V2. Such a usage would be consistent
with modem terminology when we speak of both squares
and square roots as "powers," and many commentators
have understood the word dyna.mis to refer here to such
a "quadratic surd." This view, however, has been challenged
in more recent years. In [Kn1, pp. 65-69] Wilbur Knorr sum
marizes the arguments on both sides and then defends his
claim that dynam·is can only mean "square," concluding
that its use as "square root" is not required by the text and
does not contribute to our understanding of the Theaetetus.
It is still possible, however, that the mathematical usage
of dynamis as "square root" might contribute to our un
derstanding of the famous line in the Sophist. It seems in
disputable that the Stranger's "Being is dynamis" is in
tended to mean "Being is power"-that is, that "being" is
coextensive with possessing the capacity to affect another
or to be affected. Plato might have foreseen, however, that
some of his readers would still associate dynamis with the
mathematical meaning imprinted upon it earlier in the tril
ogy. The deftness of Plato's writing, with his occasional
penchant for puns and wordplay, sometimes enables him
to appeal in the same words to readerships of very differ
ent sophistication. Plato left no doubt that he expected his
most serious readers to be serious students of contempo
rary mathematics (Republic, Book VII). What would "Be
ing is (something like) a quadratic surd" suggest to such
readers?
The historical figure Theaetetus himself is thought to
have been familiar with the information stored within our
modem continued fraction representation of quadratic
surds. One way we prove their irrationality today is to ob
serve that their continued fractions are periodic and thus
unending, and Theaetetus probably knew something equiv
alent to this method (see [Kat, p. 80]). Fowler [Fo2] argues
that the mathematicians of Plato's Academy were deeply
concerned with such analysis (anthyphairesis).
Irrational Exuberance?
A possible connection between square roots and "the dyad
of the Great and Small" (Aristotle's phrase to describe
Plato's conception) was put forward in 1926 by the philoso
pher A. E. Taylor, who called attention to the familiar con
tinued fraction for V2: 1
1 + ------1
2 + ----,..1-2 + --
2 + . . .
Taylor observed that the convergents 1, 1 + 1h 1 + 1/(2 + 1/2), . . . are (as Theaetetus almost certainly knew) alter
nately less and greater than their limit of V2, and Taylor
identifies this "endlessness" with the dyad of the Great and
Small. As evidence Taylor cites Aristotle, who heard Plato
lecture for twenty years, and who implies (Physics A 192a)
that Plato identifies "the Great and Small" with the non-be
ing mentioned by the Stranger in the Sophist [Ta, pp.
510-11].
Taylor is convinced that Plato is close to Dedekind's sub
tle idea that conveys real numerical meaning to measure
ments of ratios in the case of incommensurables. The con
nection goes something like this. Dedekind's pair of
segments of rationals is identical, says Taylor, with the dyad
of the Great and the Small, which Aristotle says is the "non
Being" in Plato's Sophist, which-as suggested above-is
really Being in disguise, defining Plato's "due measure" of
the quantity in question. Taylor thus refers exuberantly to
Plato as "the first thinker who had formed the concept of
a 'real' number" [Ta, p. 513].
Few mathematicians would grant so much. For one
thing, Plato does not discuss the accompanying algebraic
structure that modem analysts expect numbers to enjoy.
The Greeks did not associate ratios with our conception
of common fractions, so while it is natural to "compound"
(multiply) ratios, it is not so natural to add them. Could
Plato possibly have taken addition of ratios for granted? In
the Greater Hippias (303c) we find the remark that when
© 2005 Spnnger Sc1ence+ Bus1ness Media, Inc , Volume 27, Number 1 , 2005 1 5
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Title page o f Dedekind's famous monograph o n the natural numbers. The Greek motto "Man eternally arithmetizes" invites comparison with
the phrase "God eternally geometrizes" commonly (though perhaps mistakenly [Fo2, p. 293]) ascribed to Plato. Dedekind viewed mathemat
ics as the science of number and, unlike Plato, viewed number as a free creation of the human mind.
the sum of two numbers is even, the numbers themselves
may be both even or both odd; but "when each of [the un
specified things] is inexpressible, then both together may
be expressible, or possibly inexpressible." Some commen
tators (see [Knl, p. 278]) interpret this to mean that Plato
was aware of the possibility that the sum of irrationals may
be rational. Moreover, Karl Popper [Po, pp. 250-253] gives
reasons for speculating that Plato must have conjectured,
in effect, that 7T = V2 + V3. Archimedes, however, whose
calculations with a 48-sided polygon circumscribing a cir
cle effectively disproves this, makes no mention of such a
cof\iecture. Whatever Plato thought about the possibility
of adding ratios, it would be a very long time indeed be
fore negative numbers and zero would be accepted as
"real."
16 THE MATHEMATICAL INTELLIGENCER
Creation versus Discovery
What should we make of the startling thesis that Plato sub
stantially anticipated Dedekind's definition of a real num
ber? In arriving at this thesis, A. E. Taylor seems to gloss
over apparent disparities between Plato's writings and Aris
totle's account of Plato's teachings. Such disparities have
led some scholars to suggest that Plato left significant oral
remarks unwritten. Others, such as Kenneth Sayre, resist
speculating about Plato's "unwritten teachings" and un
dertake instead a more careful reading of his extant work
Sayre discusses the interplay of ideas of Plato, Eudoxus,
and Dedekind in [Sa, p. lOOff. ] .
Whatever similarities w e may see, however, in their
views regarding real numbers, we can hardly fail to notice
the marked difference between Dedekind and Plato re-
garding ontology. According to Ferreir6s [Fe, p. 134],
Dedekind was always convinced that in mathematics we
create notions and objects. Here Dedekind writes of our
power to create the reals from the rationals:
We are of divine stock and there is no doubt that we have
creative power not only in material things (railways,
telegraphs) but in particular in spiritual things . . . . I pre
fer to say that I create something new (different from the
cut) . . . . We have every right to adjudge ourselves such
creative powers.
-Letter to H. Weber in 1888, translated by Artmann
[Ar, p. 129]
Plato takes almost the opposite point of view regarding
creation and discovery. In the Philebus, beginning at 16c,
Plato has Socrates speak in praise of a "divine method" of
dialectic by which we ordinary human beings might dis
cover the existence of (permanent, unchanging) things-a
method perhaps intended [Sa, p. 133] to be the reverse
counterpart of the (divine) method of creation by which
the system of Platonic Forms is originally composed.
Dedekind cuts, of course, are still considered somewhat
esoteric outside the world of mathematics, while Plato's
late works are not well known outside philosophical cir
cles. This leaves a small group of readers who might try to
formulate and defend a carefully considered version of Tay
lor's thesis.
Whatever the final verdict, at least we surely find
phrases from Plato that suggest Dedekind cuts or other no
tions of a limit. Why do such phrases recur, particularly in
Plato's late dialogues? How did Plato's evolving ontology
draw him toward mathematical ideas that were difficult and
"modern" two millennia later?
Cuts, or Transition Points, as
Expressions of Limits
Limits are crucial to modern analysis, but Dedekind's point
of view sometimes enables us to get the job done without
mentioning them. A Riemann integral, for example, is noth
ing but the Dedekind cut defined by all the lower sums and
all the upper sums off on [a, b ] . Thus, the number that Rie
mann denoted by fif(x)d:x: is as simple (or as complicated)
as the number denoted by V2 would be to Theaetetus or
the number denoted by 1T would be to Archimedes. In each
case it is the transition point between the (rational) num
bers that are too great and those that are too small to re
flect its due measure. (Surprisingly, one can also avoid
mentioning limits in defining the derivative, by using ap
propriate transition points instead. See [Ma] or [Prl] for de
tails.)
In the Parmenides (156d) Plato discusses briefly the
idea of a transition point in time such as the instant be
tween the states of rest and motion, but the Greek word
for limit (peras ), as we would expect, is never suitably de
fined in a modern sense. From its opposite (apeiron), which seems to refer to indeterminacy in some sense, we
can gather that peras has to do with specifying something
precisely or exactly. When Plato uses peras in the Parmenides he seems to mean the defining edge, in a spatial
sense, of a thing or construction; but in the Philebus he
seems to mean the sorts of ratios that fix, in relations to
one another, apportionments (today we might say convex combinations) of opposites. The conflation, under the
rubric of limits, of such geometrical and numerical con
ceptions as these is not unlike what we do in calculus
classes today.
When Plato speaks-alas, too vaguely-of how "due
measure" relates to the More and the Less, he may be close
to the key idea behind our modern understanding of lim
its, but only so long as we are working in one dimension.
It is awkward, as Marsden and Weinstein [Ma, p. 180] note,
to move to a discussion of limits in higher dimensions with
out giving up transition points in favor of something like
Cauchy's conception.
One wishes that Eudoxus could have returned to Athens
sooner to give Plato an earlier start on these new ideas.
Plato was in his seventies when he wrote the Philebus. The
extent to which he might have seen a connection between
one-dimensional cuts and higher-dimensional limits seems
beyond conjecture.
Forms and Sensibles
Nowadays, mathematicians generally regard the "dual na
ture" of a set as requiring little or no explanation. Mathe
maticians have no problem-and they expect their students
to have no problem-in thinking of a set S contained in a
universal set U both as a plurality of points and as a unit
unto itself, that is, as a "point" in the power set of U. In
Plato's time, however, the unity-in-plurality or one/many problem engendered a degree of interest that may remind
us of the modern fascination with the wave-particle dual
ity of quantum mechanics. A Form was to be conceived of
simultaneously as a unit, an indivisible member of the
"world of being," and yet also as a plurality by virtue of its
capacity at any particular moment to manifest itself as a
multitude of sensibles in the "world of becoming."
A natural correspondence between Forms and sets
seems to suggest itself. In the Republic [P3, p. 213] Plato
speaks of the distinction
between the multiplicity of things that we call good or
beautiful or whatever it may be and, on the other hand,
Goodness itself or Beauty itself and so on. Correspond
ing to each of these sets of many things, we postulate a
single Form or real essence, as we call it. . . . Further,
the many things, we say, can be seen but are not objects
of rational thought; whereas the Forms are objects of
thought, but invisible. (Republic 507b)
The "multiplicity of things" contains, however, only things
in our sensory world, and all such things are in flux. Some
thing that is beautiful today may fade tomorrow, giving a
time-dependence to the "set" of beautiful things. The Form
of Beauty, on the other hand, is conceived to be a fixed ob
ject of real knowledge and therefore unchanging in time.
© 2005 Spnnger Sc1ence+BuS1ness Media, Inc., Volume 27, Number 1, 2005 1 7
The Stranger [Pl, p. 408] lauds the ability of "the Philoso
pher, pure and true" to
. . . see clearly one form peiVading a scattered multitude,
and many different forms contained under one higher form;
and again, one form knit together into a single whole and
peiVading many such wholes; and many forms existing
only in separation and isolation. This is the knowledge of
classes which determines where they can have commu
nion with one another and where not. (Sophist 253d)
No modern analyst could read this passage without men
tally picturing Venn diagrams of sets illustrating the notions
of union, intersection, set inclusion (or the notion of one
propositional function implying another), and disjoint or
overlapping sets (or the notions of inconsistency or con
sistency of propositional functions).
In his late writings Plato begins to picture philosophical
knowledge as a great web of connections accessible to the
intellect, yet mediated by the senses. Plato's late interest
in sensibles is a departure from his earlier view of the
senses as an impediment to knowledge, their fallibility be
ing a main source of false opinion. His early writings in
troduce us to a realm of changeless Forms in which So
cratic ideals (Virtue, Justice, etc.) float in splendid isolation
above the changing world of the senses. This is why many
readers still associate Platonism with a vague or mystical
conception of otherworldly existence that is cut off from,
and perhaps disdainful of, the transitory experiences of
everyday life. Plato's less-familiar late ontology, however,
is concerned with the problem of the interaction of these
discrete, fixed Forms with themselves and with our flow
ing, continuous world. " [T]he thesis of radical separation
[of Forms and sensibles] is expressly rejected in the Parmen ides, is absent in the Statesman, and is replaced in the
Philebus by a contrary theory" [Sa, p. 255] .
Accounting for a causal interaction of Forms with sen
sibles would seem to require something like the notion of
a limit. A modern analyst might envisage a simplex whose
extreme points represent the Forms and whose interior
points represent states of possible sense experience. Since
each interior point is expressible as a unique convex com
bination of extreme points, we might think of measuring
how much a state "participates" (to use Plato's term) in a
Form, or in each of a collection of Forms, by the relative
sizes of its barycentric coordinates with respect to them.
In the Philebus, as already remarked, Plato speaks of some
thing like a convex combination of opposites, and says in
effect (at 24d, for example) that varying combinations re
flect the varying degrees of extremes that we momentarily
sense. As time goes on, these "coordinates" continually
change because the objects of sense perception are in con
tinual flux, yet the Forms remain our fixed points of refer
ence on the boundary. The world of becoming is thus rep
resented by the interior points, while on the boundary lies
the world of being that holds sway above the flux. Perhaps
we might push the analogy further to obseiVe that, just as
1 8 THE MATHEMATICAL INTELLIGENCER
the boundary points are limits of interior points, we can
know the Forms as limits of our possible sense experiences.
Needless to say, Plato, whose mathematics counte
nanced no simplex more complicated than a tetrahedron,
could barely begin such an analogy. In fact, an unembel
lished simplex model fails because not all the Forms are
independent-Three-ness implies Oddness, for example.
Another shortcoming is that some Forms have opposites
(Hot/Cold, for example) and there is no naturally distin
guished vertex in a simplex that is opposite a given vertex.
If we wish to indulge ourselves in modeling Plato's late on
tology by using ideas from modern mathematics, we should
begin with something else-a Hilbert cube, perhaps.
In fact, Plato never wrote down a systematic and co
herent account of his ontology. Here is Benjamin Jowett's
judgment: "At the time of his death he left his system still
incomplete; or he may be more truly said to have had no
system, but to have lived in the successive stages or mo
ments of metaphysical thought which presented them
selves from time to time" [Pl, p. 558] .
Nevertheless, Plato's final thoughts on the greatest Form
of all may best reveal why we see traces of limits and cuts
in his late writings.
Mathematics and "the Good"
In the Philebus Plato allows Socrates to take center stage
for the last time to discuss the Good, although "his manner
is more like that of the Eleatic visitor than of the ironic
Socrates we know" [Gu, p. 197]. This very late dialogue is
sometimes seen, because of a remark made by Aristotle
(Nicomachean Ethics 1 172), as written in part to counter
the philosophical views of Eudoxus, who claimed that the
Good consists in pleasure. Mter highlighting the notion of
a mixed state-a conception that, ironically, may owe much
to the mathematical views of Eudoxus-Socrates finally
concludes (Philebus 65-67) that the Good cannot be cap
tured in one Form, but is a mixture of three-Beauty, Pro
portion, and Truth-and is located "ten thousand times
nearer" to Wisdom than to Pleasure [Pl, p. 629].
On the Good is, in fact, the title of Plato's enigmatic fi
nal lecture (or series of lectures), which he is thought to
have given about the time he reached the age of eighty. Ac
cording to the testimony of some of those in attendance,
Plato surprised and confused his listeners by speaking
mainly about mathematics. How could Plato have failed to
make himself understood? Presumably, he used mathe
matical imagery intended to evoke the euphoric idea of the
Good, which he had in his writings described as lying be
yond knowledge, beauty, and being, and yet the cause of
all these. Could Plato have been trying to explain how the
Forms might be approached through limits? His thesis on
this occasion was "that Limit is the Good, a Unity" ac
cording to one translation of words in a contemporary re
port [Gu, p. 424] . Others, however, read the same words
quite differently [Ga, p. 5] .
While we may never know many details of Plato's last lec
ture (see the beginning of [Ga] for a summary of what we
now know), we can speculate more confidently about why
his last writings contain suggestions of Dedekind cuts. Forms
can have opposites whose mixtures we sense, but sensible
instances of the Good are given by certain precise appor
tionments that justly balance these opposites-those result
ing, for example, in the divisions of the Pythagorean musical
scale (Philebus 26a). The essence of the Good thus involves
its power to cut, in exactly the right places, each of these con
tinua joining opposites. "The mean or measure is now made
the first principle of good," as Jowett puts it [P1, p. 558].
Although the Form of the Good itself remains nebulous
in the writings of Plato, he seems to suggest that Wf' can
approximate its sensible manifestations arbitrarily closely
in much the same way that Theaetetus approximates \12. It was Plato's need to make sense of "good" measurements
on a continuum of possibilities that took him down the path
that Dedekind would explore so much more fully some
2200 years later.
But far more excellent, in my opinion, is the serious
treatment of these things, the treatment given when
one practices the art of diaJectj.c. Discerning a kin
dred soul, the dialecti.cian plants and sows speeches
infused with insight, speedles that are capable of de
fending themselves and the one who plants them, and
that are not barren but have a seed from which there
grow up different speeches in different characters.
Thus the seed is made immortal and he who has it
is granted well-being in the fullest measure possible for mankind. (PfuJedrus 276e-277a, translated by
Mitchell H. Miller, Jr., from Plato's 'Parmenides, ' Princeton University Press, 1986, p. vii.)
Continuing the Conversation
Seventeen-year-old Mark Kac experienced something like
an epiphany at the beginning of his calculus class at the
University of Lwow in 1931. Kac was expected to be fa
miliar with Dedekind cuts, which he had never heard of,
and a young junior assistant named Marceli Stark recom
mended something for him to read.
So I went home and read, and as I read, thf' beauty of
the concept hit me with a force that sent me into a state
of euphoria. When, a few days later, I rhapsodized to
Marceli about Dedekind cuts-in fact, I acted as if I had
discovered them-his only comment was that perhaps I
had the makings of a mathematician after all [Kac, p. 32] .
Like Kac, I remember the thrill that I felt as a college
student upon first understanding Hermann Weyl's succinct
explanation of the relation between Dedekind cuts and the
condition of Eudoxus [We, p. 39] . The present article, how
ever, derives more from a reconsideration of the following
remark of mine:
What was a limit, before it was given a name? Before
Cauchy gave a precise significance to the notion, the an
swer might have been an oracular utterance alluding to
the Greek method of exhaustion, like "that which re
mains when everything to which it is not equal is elimi
nated." If anything was ever airy nothing, this is it [Pr2,
p. 18].
Here I was close to existence problems that challenged
Plato, but I knew nothing then of what he had said about
such things in his late writings. The refutation of a single
wrong opinion, as Socrates discovers in Plato's early writ
ings, can leave us still in the dark, but the elimination of
all wrong answers, as Socrates finally learns from the
Eleatic visitor, can carry us to the threshold of enlighten
ment. Thus we have the power, so to speak, to tum wrong
answers inside out and make them tell the truth. When first
brought to light in Plato's Academy, this subtle observation
must have generated great excitement.
In modem mathematical analysis this kind of argu
ment is familiar and its value can scarcely be overstated.
If all possibilities for the answer to a problem should lie
in a one-dimensional continuum, the key is often simply
the existencf' and uniqueness of the "right answer" after
the elimination of all numbers that are too great or too
small. Devoting a little time in an analysis class to study
ing questions deliberately framed in this fashion can lead
both to an understanding of real numbers through
Dedekind cuts and to a quick grasp of "one-dimensional"
limits. The Eleatic visitor, thanks to Dedekind's kindred
insight, should be a stranger to mathematicians no
longer.
Epilogue
A discussion of possible connections between the ideas of
Dedekind, Eudoxus, and Plato might help to re-stimulate
interest among analysis instructors and their students in
the topic of Dedekind cuts and to arouse more interest in
the history and philosophy of mathematics. See [Kat] and
[Sh] , for example. Such a discussion could involve classi
cists, historians, philosophers, and mathematicians in a
richly collaborative endeavor that would be valuable in it
self. Among these groups are a multitude of fine scholars,
and some of them-including Chandler Davis, Hardy Grant,
Mitchell H. Miller, Jr. , and Jan Zwicky-have generously
given me much help and encouragement.
This paper is dedicated to the memory of Hugh Harris Cald
well, Jr. , whose philosophy classes introduced me both to
Plato and to Dedekind.
REFERENCES [Ab] E. Abbott, Flatland, A Romance of Many Dimensions, sixth ed ,
Dover, New York, 1 952.
[Ar] B. Artmann, Euclid- The Creation of Mathematics , Springer-Ver
lag, New York, 1 999.
© 2005 Spnnger Sc1ence+ Business Med1a, Inc , Volume 27. Number 1 . 2005 19
[Bo1 ] S. Bochner, The Role of Mathematics in the Rise of Science,
Princeton Univ. Press, Princeton, 1 966.
[Bo2] C. B. Boyer, A History of Mathematics, rev. by U. Merzbach, Wi
ley, New York, 1 991 .
[Cr] A. W. Crosby, The Measure of Reality: Quantification and Western
Society, 1250-1 600, Cambridge Univ. Press, Cambridge UK, 1 997.
[De] R. Dedekind, Essays on the Theory of Numbers, trans. W. W. Be
man, Open Court, Chicago, 1 901 .
[Di] E. J. Dijksterhuis, Archimedes. Princeton University Press, Prince
ton, 1 987.
[Eu] Euclid, Euclid's Elements, trans. T. L. Heath, Green Lion Press,
Sante Fe, NM, 2002.
[Ev] W. R. Everdell, The First Moderns, University of Chicago Press,
Chicago, 1 997.
[Fe] J . Ferreir6s, Labyrinth of Thought: A History of Set Theory and its
Role in Modern Mathematics, Birkhauser, Boston , 1 999.
[Fo 1 ] D. Fowler, Dedekind's Theorem: v2 x V3 = v6, Amer. Math.
Monthly 99 (1 992), 725-733,
[Fo2] D. Fowler, The Mathematics of Plato's Academy, second ed. ,
Clarendon Press, Oxford, 1 999.
[Ga] K. Gaiser, Plato's Enigmatic Lecture 'On the Good, ' Phronesis 25
(1 980), 5-37.
[Gu] W.K.C. Guthrie, A History of Greek Philosophy, Vol. V, The Later
Plato and the Academy, Cambridge University Press, Cambridge UK,
1 978.
(Kac] M . Kac, Enigmas of Chance, Harper & Row, New York, 1 985.
[Kat] V.J. Katz, A History of Mathematics, second ed. , Addison-Wes
ley, Reading MA, 1 998.
[Kn1 ] W. R. Knorr, The Evolution of the Euclidean Elements, Reidel,
Dordrecht, The Netherlands, 1 975.
[Kn2] W. R . Knorr, The Ancient Tradition of Geometric Problems,
Boston, Birkhauser, 1 986.
[Ma] J. Marsden and A. Weinstein, Calculus Unlimited, Benjamin Cum
mings, Menlo Park, CA, 1 981 .
[Me] S. Menn, Plato and the Method of Analysis, Phronesis 57 (2002),
1 93-223.
[01] J. M. H. Olmsted, The Real Number System, Appleton-Century
Crofts, New York, 1 962.
[P1 ] Plato, The Dialogues of Plato, Vol. I l l , trans. B. Jowett, Clarendon
Press, Oxford, 1 953.
[P2] Plato, Complete Works, ed. J. M. Cooper, Hackett Publishing, In
dianapolis, 1 997.
[P3] Plato, The Republic of Plato, trans. F. M. Cornford, Clarendon
Press, Oxford, 1 948.
[Po] K. R . Popper, The Open Society and Its Enemies, fourth ed , Prince
ton University Press, Princeton, 1 963.
[Pr1] W. M. Priestley, Review of Calculus Unlimited, Math. lntelligencer
4 (1 982), 96-97.
20 THE MATHEMA11CAL INTELLIGENCER
AU T H O R
W. M. PRIESTLEY Department of Mathematics and Computer Science
University of the South
Sewanee, TN 37383
USA
e-mail: [email protected]
William McGowen Priestley, known as "Mac," graduated from
the University of the South and returned there to teach in 1 967.
He received his Ph.D. at Princeton with a thesis directed by
Edward Nelson. He and his wife Mary, a botanist and now cu
rator of the Sewanee Herbarium, have raised three children in
Sewanee. His persistent efforts to put together a one-semes
ter calculus course for humanities majors have led to a dis
tinctive textbook, Galculus: A Liberal M (Springer, 1 998).
[Pr2] W. M . Priestley, Mathematics and Poetry: How Wide the Gap?
Math. lntelligencer 12 (1 990), 1 4-19 .
[Sa] K . M . Sayre, Plato 's Late Ontology, Princeton University Press,
Princeton , 1 983.
[Sh] S. Shapiro, Thinking about Mathematics, Oxford University Press,
Oxford, 2000.
[St] 0. Stolz, Zur Geometrie der Allen, insbesondere uber ein Axiom
des Archimedes, Mathematischen Annalen 22 (1 883), 504-51 9.
[Ta] A. E. Taylor. Plato: The Man and His Work, Methuen and Co. , Lon
don, 1 926.
[vdW] B. L. van der Waerden, Science Awakening, trans. A. Dresden,
Wiley, New York, 1 963.
[vN] J. von Neumann, The Mathematician, pp. 227-234 of Mathemat
ics: People, Problems, Results, Vol. I , ed. D. M. Campbell, J . C. Hig
gins, Wadsworth, Belmont CA. 1 984.
[We] H. Weyl, Philosophy of Mathematics and Natural Science, Prince
ton University Press, Princeton, 1 949.
l*@ii•i§i•@hi%11fJ.i,ir:iii,hitfj Marjorie Senechal , Ed itor I
Herman Muntz: A Mathematician's Odyssey Eduardo L. Ortiz and Allan Pinkus
This column is a forum for discussion
of mathematical communities
throughout the world, and through all
time. Our definition of "mathematical
community" is the broadest. We include
"schools" of mathematics, circles of
correspondence, mathematical societies,
student organizations, and informal
communities of cardinality greater
than one. What we say about the
communities is just as unrestricted.
We welcome contributions from
mathematicians of all kinds and in
all places, and also from scientists,
historians, anthropologists, and others.
Please send all submissions to the
Mathematical Communities Editor,
Marjorie Senechal, Department
of Mathematics, Smith College,
Northampton, MA 01 063 USA
e-mail: senechal@minkowski .smith.edu
In 1885 Weierstrass [ 1 ] proved that
every continuous function on a com
pact interval can be uniformly approxi
mated by algebraic polynomials. In other words, algebraic polynomials are
dense in C[a,b] (for any -x < a < b < + x ). This is a theorem of major im
portance in mathematical analysis and
a foundation for approximation theory.
One of the first outstanding gener
alizations of the Weierstrass Theorem
is due to Ch. H. Muntz, who answered
a col\iecture posed by S. N. Bernstein
in a paper [2] in the proceedings of the
1912 International Congress of Mathe
maticians held at Cambridge, and in his
1912 prize-winning essay [3]. Bernstein
asked for exact conditions on an in
creasing sequence of positive expo
nents an, so that the system {X"" ]� =O is
complete in the space C[0, 1 ] . Bernstein
himself had obtained some partial re
sults. On p. 264 of [2] Bernstein wrote
the following: "It will be interesting to
know if the condition that the series
:k llan diverges is necessary and suffi
cient for the sequence of powers
{X"" )� =O to be complete; it is not cer
tain, however, that a condition of this
nature should necessarily exist."
It was just two years later that
Muntz [M7] was able to confirm Bern
stein's qualified guess. What Muntz
proved is the following:
Theorem. The syste:m {x"o, x"', X"', . . . ] , where 0 ::::: ao < a1 < a2 < . . . , is complete in C[0, 1 ] if and only if ao =
0 and
1 I - = X. n = l a,
Today there are numerous proofs and
generalizations of this theorem, widely
known as the "Muntz Theorem." In fact
a quick glance at Mathematical Reviews, that is, at papers from 1940,
shows nearly 150 papers with the name
Muntz in the title. All these articles
mention Muntz's name in reference to
the above theorem, except one refer
ring to his thesis. Muntz's name with
his theorem appears in numerous
books and papers. In addition there are
Muntz polynomials, Muntz spaces,
Muntz systems, Muntz-type problems,
Muntz series, Muntz-Jackson Theo
rems, and Muntz-Laguerre filters. The
Muntz Theorem is at the heart of the
Tau Method and the Chebyshev-like
techniques introduced by Cornelius
Lanczos [4] . In other words, Muntz has
come the closest a mathematician can
get to attaining a little piece of immor
tality.
Notwithstanding, a quick search of
the mathematical literature will also
show that essentially nothing is known
about Muntz, the person and the math
ematician. The purpose of this paper is
to try to redress this oversight. Muntz's
life, mathematically and otherwise,
was an illuminating and dramatic jour
ney through the first half of the twen
tieth century. It is unfortunate that it
was not a more pleasant journey.
Early Years (1884-1 9 1 4)
Herman Muntz1 (officially named
Chaim) was born in .t6dz on August
28, 1884. Muntz's family was bourgeois
and Jewish, though not religious. At
that time .t6dz was a part of "Congress
Poland" under Russian rule. It was an
important industrial city at the western
boundary of this area. In the last
decades of the nineteenth century,
when Muntz was born, it had a vibrant
Jewish community, mainly engaged in
textiles and other related trades, as
well as in business in general. In offi
cial documents, Muntz's father is de
scribed as "in trade," with the sugges
tion that he was an estate agent. The
Eduardo L. Ort1z thanks the Royal Society, London, for its financial support while researching this paper.
1 The file on Muntz preserved at the Society for the Protection of Science and Learning, now at the Bodleian
Library, Oxford, provided a valuable start in the search for other sources included. On the Muntz files there
see Ortiz [5].
22 THE MATHEMATICAL INTELLIGENCER © 2005 Spnnger Sc1ence+ Bus1ness Med1a. Inc.
Herman MOntz
family name was spelt in the German
manner rather than the more common
Mine. Herman was the eldest of five
children, all of whom (except for the
youngest) were sent to study at Ger
man and Swiss universities. The tur
bulent economic times were such that
the family was generally, though not al
ways, comfortably well off. A notice
able decline was associated with the
depression of the late 1920s. Muntz
started his studies at the Hohere Gewerbeschule in L6dz, the top tech
nical high school in that city, with a
bias toward textiles, textile machinery,
and chemistry. He was fluent in Polish
and had a reasonably good command
of German and Russian.
In 1902 Muntz went to Berlin to
study at the Friedrich-Wilhelms-Uni
versitat, generally referred to as the
University of Berlin (called Humboldt
Universitat Berlin since 1948), where
he studied mathematics, the natural
sciences, and philosophy. In 1906 he
earned his matriculation degree. He
named Frobenius, Knoblauch, Landau,
Schottky, and Schwarz as his teachers,
singling out Frobenius and Schwarz as
his main influences.
From 1906 to 1910 Muntz was in
Berlin, where he worked, wrote, and
studied. In 1912 he married Magdalena
(Magda) Wohlman who was from the
area of Zlotk6w near Poznan, an area
of Poland under German control. Magda
had come to Berlin to study biology.
While the marriage would remain child
less, it was, by all accounts, an unusu-
ally harmonious union. During this early
period Miintz was involved in the pri
vate teaching of mathematics. Money
was always a pressing problem. For
much of his life Muntz remained en
gaged in pedagogy in one form or an
other, "teaching elementary and higher
mathematics, partly in private schools
and partly as a private tutor."2
Mi.intz was an intellectual who was
intensely interested in philosophy, po
etry, art, and music. He was especially
taken with Goethe, but more particu
larly with Nietzsche's philosophy,
which was to have a profound influ
ence on him. He attended university
lectures by the philosopher Alois Riehl,
and he seems to have written a thesis
on Nietzsche.
In these years he also became in
terested in a reassessment of Jewish
culture and the position of Jews in so
ciety. In 1907 he published a 124-page
book called Wir Juden (We Jews) [6],
dedicated to Friedrich Nietzsche and
showing the influence of his Also sprach Zamthustm. The book con
cerns the need for a basic reform of the
Jewish people in the post-orthodox pe
riod, and a reconsideration of the po
sition of Jews in society.
Mi.intz discussed in detail what he
called the "new Jew," and the contribu
tion Jewish people had made and could
make to humanity. He characterized
Jews not as a pure race but as a diver
sity of many peoples, emphasizing the
past and present connections between
Jews and a variety of other people. The
book aspired to help the young Jewish
generation of the time to achieve reli
gious and political self-definition. It em
braced a view of Zionism not uncommon
at the time, in which socialist viewpoints
were discernible. There were remarks in
Miintz's text that are very much race
based, which may make it discomforting
to read today. But they should be un
derstood in the context of the era. The
book was advertised in the Berlin Jew
ish/Zionist weekly Jiidische Rundschau in its list of "Zionistische Literatur."
These advertisements continually mis
spelt the author's name as "Miintzer,"
which might be considered as a measure
of the perceived importance of the book
Aside from a mathematical text men
tioned later, this was the only book by
Miintz that was ever published. How
ever, we have found various items of
correspondence indicating that he also
wrote at least three other (non-mathe
matical) texts. All written from about
1911 to 1924, they were: Ober Ehe und Treue (On marriage and fidelity); a book
about the Psalms; and Der Judische Staat (The Jewish state). The three
manuscripts were sent to different pub
lishers, but for a variety of reasons, in
cluding the war and lack of paper, none
seems to have been published. How
ever, parts of the last-named book ap
peared as articles in a journal.
Despite these varied activities,
Muntz's main focus in the period 1906-
1910 was his mathematical studies, un
der the supervision of Hermann Aman
dus Schwarz. His first results were of
a geometric character, having to do
with rational tetrahedra. However, he
soon began to produce results on the
main topic of his doctoral dissertation,
namely, minimal surfaces defined by
closed curves in space, that mathe
matically involved the approximate so
lution of non-linear partial differential
equations. On October 1, 1910, Muntz
was awarded a doctorate, Dr. Phil.,
magna cum laude. His official review
ers were Schwarz and Schottky. His
dissertation, under the title "On the
boundary-value problem of partial dif
ferential equations of minimal sur
faces," was published in Grelle's jour
nal [M1] . This work is still occasionally
referenced.
In this thesis Mi.intz studied the
Plateau problem in some detail. He
used potential theory and the method
of successive approximation, two tools
he would return to in subsequent pa
pers. When Muntz was near the end of
this dissertation work, Schwarz ad
vised him that Arthur Korn, who was
working in the same area, had submit
ted for publication a paper on the sub
ject of his thesis, which was later pub
lished [7) . In his Grelle paper Muntz
acknowledged Korn's work Although
their results had a common ground, the
techniques used and the final results
were sufficiently different to merit in-
·-- - --- ---
2Muntz to Geheeb, March 1 , 1 91 4. Archtve of the Ecole d 'Humantte
© 2005 Spnnger Sc1ence+Bus•ness Media. Inc , Volume 27. Number 1 . 2005 23
dependent publication. Muntz seems to
have been the last of Schwarz's doc
toral students. Other doctoral students
of Schwarz included Leopold Fejer,
Ernst Zermelo, Paul Koebe, and Leon
Lichtenstein. The latter became a close
friend of Muntz.
In late 191 1 Muntz went to Munich
to give a lecture at the seminar of Fer
dinand von Lindemann. He was also ac
cepted into Aurel Voss's circle. These
were two of the three mathematics pro
fessors at the Karl Ludwig-Maximilians
Universitat in Munich; the third was Al
fred Pringsheim. The Muntzes decided
to move to Munich primarily on the ba
sis of this visit, which seemed to open
some opportunities. But they were also
undoubtedly influenced by the fact that
two of Muntz's brothers were also then
residing in Munich.
Miintz's aim, and that of any young
aspiring mathematician in Germany at
this stage of his career, was to secure a
position as a "Privatdozent." The next
stage was to gain a Habilitation and
eventually an academic position at a uni
versity. At that time (and the same is es
sentially true today) the Habilitation was
necessary for a professorship, and a pro
fessorship is what Miintz wanted then
and throughout his life. According to his
correspondence, Miintz, who was not
the only candidate, obtained the support
of the three mathematics professors. It
seems, however, that there were also
what he termed some "strange regula
tions," and serious formal problems. The
matter dragged on. In the end, Miintz
was unsuccessful in gaining the dozent
position.
While in Munich Muntz was again
earning his living privately as a teacher
at various levels. His wife also worked
part-time and there was some financial
help from the family. Muntz attended
lectures and seminars given by von Lin
demann and Voss and was actively en
gaged in mathematics research. From
1912 to 1914 he published four papers
in the field of modem projective geom
etry and the axiomatics of geometry,
two of which appeared in Mathematische Annalen. His 1912 paper on the con
struction of geometry on the basis of
only projective axioms was read by
Voss at a meeting of the Bavarian Acad
emy. In 1913 he published two notes in
Comptes Rendus in connection with the
use of iterative techniques for the solu
tions of algebraic equations. It is very
possible that Miintz was the first to de
velop an iterative procedure for the de
termination of the smallest eigenvalue
of a positive definite matrix. It certainly
predates the more generally quoted re
sult of R. von Mises of 1929 [8]. In 1914
he published an additional two papers
on approximation theory. The first is a
note on properties of Bernoulli polyno
mials published in Comptes Rendus. The other is the paper in which the
Muntz Theorem appeared. This last
work was written as a contribution to
the Festschrift in honour of his teacher
Hermann Schwarz's 70th birthday.
In this period reference is already
made in Muntz's correspondence to se
rious problems in one of his eyes. Eye
problems would plague him through
out his life.
Boarding Schools and Martin
Buber ( 1 9 1 4-1 9 1 9)
In early 1914, probably through social
ist and feminist common friends,
Muntz started a correspondence with
the pedagogue Paul Geheeb, who ran a
boarding school called the Odenwald
schule near Heppenheim in southern
Hessen. Muntz moved to Geheeb's
school in 1914 as a mathematics
teacher, with the understanding that he
would be able to devote a considerable
amount of his time to his mathemati
cal research. It was agreed that he
would have at most three hours of
teaching a day. This was to be the first
time he taught very young children.
In a letter to Geheeb written by
Mario Jona, who interviewed him for
the position, there is the following pas
sage:3 "He [Muntz] is perfectly aware
of what he is worth and shows it, which
face to face is not so unpleasant as in
writing. As it was I imagined him from
his letter to be much more terrible. He
is short, pleasant and with a very seri
ous appearance and sometimes a little
clumsy in politeness, . . . For him the
most important thing is his scientific
work. He is in a period of important sci-
3Jona to Geheeb, March 1 0, 1 91 4. Archive of the Ecole d'Human1te.
24 THE MATHEMATICAL INTELLIGENCER
entific activity, but would like also to
work in a school like ours if he also has
time to work for himself."
Geheeb was a liberal humanist, pro
feminist, and much opposed to anti
Semitism. He and his schools hold a
special place in the history of progres
sive education in Germany. At one of
his earlier schools, in Wickersdorf, he
had established the first co-educa
tional boarding school in Germany. His
wife, Edith Cassirer, was a progressive
young teacher, the daughter of the
wealthy Berlin Jewish industrialist
Max Cassirer. With his father-in-law's
financial backing, Geheeb founded the
Odenwaldschule in 1910. It was a large
boarding school with modem or spe
cially modernized buildings. Co-educa
tion, an emphasis on physical educa
tion, and flexibility in the curriculum
were among its innovations. The new
school was run with a fair amount of
self-government. The teachers, and es
pecially Geheeb, supposedly guided
rather than led. The students were
called Kameraden, "comrades," and
the teachers Mitarbeiter, "co-work
ers." In 1914, there were 68 full-time
students, many of whom were children
of the liberal, affluent German intelli
gentsia. The children of Thomas Mann
and of other noted writers and artists
were among the pupils and were not
necessarily easy to handle. Much has
been written about this school and
Geheeb. The school survived both wars
and exists today, but the Geheebs left
in 1934 when the influence of Nazi ac
tivists reached the school, and they
moved to Switzerland. There, he and
his wife established a school of a re
lated character: the Ecole d'Humanite. According to some, Muntz included,
life in Odenwaldschule seemed anar
chic on occasion. Muntz and Geheeb
parted ways in the summer of 1915.
Nonetheless Muntz kept in touch with
some of the school's faculty and re
mained on speaking terms with
Geheeb. Muntz then found a similar po
sition at another school, Durerschule,
which does not exist today, in Hoch
waldhausen also in Hessen. Muntz
seems to have enjoyed his teaching,
and developed very definite opinions
on the teaching of mathematics and
science to younger children. Another
teacher who joined him at the Dur
erschule was his friend and brother-in
law Herman Schmalenbach, married to
his sister Sala, who later became a Pro
fessor of Philosophy at the University
of Basel.
The war was having its impact.
Muntz was an "alien," with Hessian res
idency but no German citizenship, and
he was generally restricted in his trav
els. This prevented a move to Heidel
berg planned in 1915. In a letter dated
August 1917, Muntz wrote that he had
to stay in Hessen to avoid difficulties
with the authorities. However, as an
"alien" he did not take part in the war.
Although happy at the school, he was
forced to leave after an open meeting
in 1917 where the headmaster, G. H.
Neuendorff, called him a "little Polish
Jew." See Butschli [9].
Many pupils, especially the Jews, also
did not return to Dtirerschule after the
holiday. Muntz felt he had a responsi
bility for some of these children and de
cided to return as a private scholar to
Heppenheim where he had friends, but
not to Odenwaldschule. With his wife,
he managed a small boarding house for
students: a Schiilerpensionat. Despite his many obligations and
worries, Muntz still managed to carry
on with his mathematics research. Dur
ing this period he published five more
papers, concerning problems in pro
jective geometry, and the solution of al
gebraic equations and algebraic eigen
value problems.
While still at the Odenwaldschule,
Muntz had begun to correspond with
Martin Buber, the enlightened and
broad-minded philosopher, Zionist
thinker, and writer, who was then in
Berlin. Buber was the spiritual leader
of an entire generation of German
speaking Jewish intellectuals. He ad
hered to a form of tolerant utopian so
cialism he called "Hebrew humanism."
In 1915 Muntz helped Buber find a
house4 in the town of Heppenheim,
where Buber and his family lived from
1916 until 1938. Buber then left for
Palestine to take a chair in Social Phi
losophy at the Hebrew University, and
subsequently had a distinguished ca
reer there. During the First World War
the two families kept in close contact
and exchanged fairly intense and in
teresting correspondence.
In 1915 Buber founded and co
edited a journal called Der Jude [ 10]
that for eight years was the most im
portant organ of German-reading Jew
ish intellectuals. In a letter dated in No
vember of that year, Buber invited
Muntz to become one of his collabora
tors on this journal. He wrote, "You are,
of course, amongst the first whom I am
asking to participate."" Muntz wrote 18
articles and notes for this journal,
some quite lengthy, under the pseudo
nym of Herman Glenn. It is an indica
tion of the way in which Muntz's con
tributions were valued that in the very
first issue of Der Jude, the first article
was signed by Buber, while the second
was signed by Glenn (Muntz).
GoHingen and Berlin (1 91 9-1 929)
Around 1919 or 1920 Muntz seems to
have had a nervous breakdown and
was placed at a sanatorium in Gander
sheim (now called Bad Gandersheim)
near Gi:ittingen. We do not know ex
actly how long Muntz was in the sana
torium. The few letters available from
this period are rather bleak. In a letter
to Buber in September 1923 Muntz re
called that he suffered a personal col
lapse in 1919-1920 and said he learnt
from the experience to look at things
from a distance, and in "this way they
are no longer dangerous to me."6
Toward the end of 1920 Mtintz and
his wife moved to his wife's family farm
in Poland to recuperate for some eight
to ten months. Letters show that during
this period the Muntzes, together with
his wife's brothers, considered emi
grating to Palestine. But the economic
situation there was far from encourag
ing and the idea was dropped. As he re
cuperated, Muntz took up mathematics
again and from the farm traveled to
Warsaw to attend seminars and to lec
ture on his research. This activity is re-
fleeted in a number of publications in
the journal of the then recently founded
Polish Mathematical Society.
In October 1921 the Muntzes re
turned to Germany, moving into a
boarding house in Gi:ittingen. At that
time the Schmalenbachs also lived in
that city. It is not clear how the
Muntzes supported themselves in Gi:it
tingen-probably again through pri
vate teaching and with the help of their
family. While in Gi:ittingen Muntz did
considerable mathematical research.
During this time he wrote eleven pa
pers, published between 1922 and
1927. They cover a number of topics,
including integral equations, the nbody problem, summability, Plateau's
problem, and quite a few papers on
number theory, possibly under the in
fluence of Edmund Landau. One of his
results from this period is quoted in
Titchmarsh [ 1 1 ] .
By this time Muntz seems to have
made a name for himself within both
mathematical and Jewish/Zionist cir
cles. He was a member of the editorial
board of the mathematics and physics
section of the short-lived journal
Scripta Universitatis founded by Im
manuel Velikovsky in Jerusalem. The
one and only issue of the mathematics
and physics section was edited by Ein
stein and published in 1923. He also co
operated with Hertz, Kneser, and Os
trowski in a German translation of
some lectures of Levi-Civita [M21 ] .
In April 1924 the Muntzes moved to
Berlin, while maintaining scientific
contacts in Gi:ittingen. Muntz also re
turned to writing on Jewish matters.
He sent contributions to Der Jude, some of which were excerpts from the
third of his unpublished books on Jew
ish matters.
Muntz, who never did much collab
orative research, did have at least one
student in this period. Divsha Amira
(nee Itine) from Palestine was a
geometer who officially obtained her
doctorate from the University of
Geneva in 1924. She worked with
Muntz whilst residing in Gi:ittingen. In
1925 she published a memoir [ 12] on a
�� - - - -� - ------�--- �---------�-----------
4Today called The Martin Buber House, it is home to the Internal Council of Christians and Jews.
5Buber to Muntz, November 1 1 , 1 91 5, Buber Archives, JNUL, Jerusalem.
6Muntz to Buber, September 18 , 1 923, Buber Archives, JNUL, Jerusalem
© 2005 Spnnger SC1ence+Bus1ness Media, Inc .. Volume 27, Number 1 , 2005 25
projective synthesis of Euclidean
geometiy. Mtintz had attempted a con
struction of algebraic Euclidean geome
tiy using what he called Basisfiguren. In her memoir Divsha extended Mtintz's
ideas to the general Euclidean plane,
considering, instead, sets of straight
lines. She discussed elementary con
structions, congruence axioms, and the
axiomatic construction of geometiy. Di
vsha was generous in her remarks to
Mtintz and to his research. Although not
her formal thesis supervisor, Mtintz
clearly was her mentor. Divsha's hus
band Bel\iamin, who also obtained his
doctorate from the University of Geneva,
was a student of Edmund Landau.
The first meeting of the board of
governors of the new Hebrew Univer
sity in Jerusalem took place in April of
1925. At that meeting it was decided to
establish an institute devoted to re
search in pure mathematics, staffed by
one professor and two assistants. The
board of governors also authorized the
President and Chancellor to offer Ed
mund Landau, then still in Gottingen,
the professorship in pure mathematics.
The involvement of Landau in the He
brew University started well before the
First World War, and lasted into the
1930s. He had the major say on who
would be appointed in mathematics. At
a meeting of the board of governors in
September 1925, Landau was asked to
draw up plans for the establishment of
a mathematical institute to be opened
as soon as funding became available.
At the suggestion of Landau, it was de
cided to appoint Benjamin Amira as the
first assistant.
In October of 1925 Mtintz was in
Berlin and was busy trying to find a po
sition, either in or outside Germany. The
creation of the Hebrew University un
doubtedly interested him as a mathe
matician, as someone without proper
employment, and as a Zionist. Mtintz
saw himself as the professor and thus
the director of this new mathematical
institute. In a rather manipulative way,
he used his connections, particularly
Schmalenbach's everlasting good dispo
sition towards him. He asked his well
positioned brother-in-law to contact Bu
ber, Landau, and Courant on his behalf.
Schmalenbach reported that Courant
was emphatic in stating that he had no
doubts as to Mtintz's qualifications,
which he exhibited in his papers and in
his lectures at the meetings of the local
mathematical society. However, he in
dicated that not having a Habilitation
was a serious drawback The Jerusalem
matter was resolved negatively in early
November. In retrospect, it seems Mtintz
had misread the entire situation. At this
early stage in 1925 Landau probably was
saving the professorship for himself. In any case, the academic leadership of the
Hebrew University was looking for an
established star to take up the profes
sorship-or at least for someone with a
Habilitation and also a chair somewhere
else. They were looking indeed for a per
son like Landau, who did move to
Jerusalem with his family for the initial
academic year of 1927-28. However, for
various reasons things did not work out
and he returned to Gottingen the fol
lowing year.
Throughout this period Muntz was
constantly seeking an academic ap
pointment, while at the same time at
tempting to obtain his Habilitation. In
1925 Voss and von Lindemann recom
mended him for a Habilitation in
Giessen, but nothing came of it. That
same year it appears he was recom
mended, by A A Fraenkel, for a posi
tion at the University of Cairo. Again
he was unsuccessful.
As we said, according to Courant,
the fact that Mtintz had not been given
the Habilitation in Gottingen was not
as a consequence of his lack of qualifi
cations. There were other reasons. On
the one hand there was the question of
his origin, which he did not try to hide.
On the other hand there was Gottin
gen's hierarchy. As in the cases of
Bernays, Hertz, and E. Noether, if he
were not to be called by a university,
Gottingen would feel morally obliged
to provide for his maintenance. Muntz
had heard essentially the same from
Hilbert years earlier. To this Muntz jus
tifiably complained that he was in an
impossible situation. If he had a guar
anteed position then he would have no
problem being given Habilitation, but
without the Habilitation it was almost
7MOntz to Buber, October 30, 1 927, Buber Archives, JNUL, Jerusalem.
26 THE MATHEMATICAL INTELLIGENCER
impossible to obtain a position. His
case was not unique.
One source of income for Mtintz we
have identified is the Jahrbuch iiber die Fortschritte der Mathematik (FdM).
This annual review, published from 1869
until the end of the Second World War,
was in the format adopted by the Zen
tralblatt fur Mathematik, and later
shared by Mathematical Reviews, ex
cept that it appeared each year as a sin
gle volume. As a consequence of the
First World War, the work on the annu
als was severely backlogged and re
mained so for many years thereafter. A
count of the reviews shows Mtintz wrote
nearly 800 reviews for the FdM, mainly
during the mid-1920s. He was still regis
tered among the journal's regular re
viewers up to 1929, when he had already
left Germany. Reviewers were paid 1
Reichsmark per review. The average
salary at the time seems to have been
about 120 Reichsmark per month.
The reviews by Mtintz cover an ex
tensive mathematical, as well as linguis
tic, area. Besides the languages he was
brought up in, namely Polish, German,
and Russian, Mtintz reviewed papers in
English, French, Italian, Dutch, and
Swedish. The topics, besides function
theory and differential equations, were
probability theory, fluid mechanics, the
theory of electricity and magnetism, in
cluding its geophysical applications, nu
merical methods of calculation, and the
history of mathematics.
At the end of 1927 Muntz wrote to
Buber that for several months he had
been the professional scientific collab
orator of Einstein, and added, "This, of
course, compensates me a great deal
for what has been in Germany an al
most impossible situation, as the offi
cial professionals are more 'official'
than 'professional.' " 7 He was probably
alluding to the fact that he had been
unable to obtain a Habilitation. It is not
clear when Muntz started to work with
Einstein and for how long this collab
oration continued. From his wife's cor
respondence we learn he met Einstein
socially in January, 1927. For much of
the time that Mtintz was working
with/for Einstein, another subsequently
well-known mathematician, Cornelius
Lanczos, did so too. Both seem to have
been supported by grants from a fund,
the Notgemeinschaft Deutscher Wissenschaftler, supporting prom1smg
"young" scientists. Muntz published no
joint papers with Einstein, but Ein
stein's archive has extensive corre
spondence between Muntz and Ein
stein on a range of mathematical ideas.
Moreover Mtintz and Lanczos are men
tioned and thanked in two of Einstein's
papers on distant parallelism.
In describing to his sister his work
under Einstein in September of 1927,
Muntz indicated that it was "running
'normally' and for reasons of conve
nience I have 'submitted' myself; for in
these fields it is he who is the extraor
dinary master while I am only the 'tech
nical' assistant. Nevertheless I am very
happy to be working with him." On
more than one occasion, however, Ein
stein politely expressed reservations
regarding Muntz's work. Einstein
sometimes indicated that he did not be
lieve there were sufficient reasons for
Muntz's assumptions, or he did not re
gard Muntz's reasoning as being justi
fied, or he did not think that Muntz's
arguments made "any obvious experi
mental-physical sense."
Toward the end of 1928 Muntz was
again considering the possibility of tak
ing a chair outside Germany. However,
he purposely kept these discussions
from many of his close friends and col
leagues, including Lichtenstein and
Einstein,8 which suggests that he still
expected that they might be able to
help him find a job within Germany.
Leningrad (1 929-1 937)
In May of 1929 Muntz finally obtained
an academic appointment, something
for which he had yearned for many
years. He was invited to fill the posi
tion of Professor of Mathematics and
Head of the Chair of Differential Equa
tions at the Leningrad State University.
In Leningrad Muntz was also put in a
group of "exceptional scientists," and
given a "personal salary." From 1933 he
is listed as Head of the Chair of Dif
ferential and Integral Equations.
In a letter written a few years later,
Muntz stated that in 1927 he had been
offered the Lobachevsky Chair in
Kazan, and during the technical period
of waiting was offered the Chair for
Higher Analysis in Leningrad. Accord
ing to Muntz, he "exchanged" the chair
in Kazan with that in Leningrad (ini
tially offered to Bernstein), while his
friend N. G. Chebotarev took the Kazan
Chair. We have found no direct docu
mentation to support this claim, but
the fact is that Chebotarev became pro
fessor at Kazan University in 1928 af
ter having been offered posts at both
Kazan and Leningrad.
G. G. Lorentz, who was an under
graduate at the time, recalled [ 13] that
in 1930, shortly after his arrival in
Leningrad, Muntz was called upon to
present a lecture sponsored by the
Leningrad Physical and Mathematical
Society on the so-called crisis of the ex
act sciences. The subject was the foun
dational debate in mathematics, and
Hilbert's attack on the intuitionism of
Brouwer and W eyl. Muntz was an ideal
candidate to deliver the lecture be
cause of his research background on
the foundations of mathematics; and
having recently arrived from Germany,
he was perceived as the carrier of the
latest advances on this controversy.
The lecture was well attended. Of
course Muntz stated that the crisis was
only in the foundations and did not in
any way affect the work of most math
ematicians. However, because of an
underlying power struggle between N.
M. Gunter, V. I. Smirnov and Ya. V. Us
penskyi, on the Society's traditional
side, and L. A Lefert and E. S. Rabi
novich, of an alternative young Com
munist league, the meeting turned
rowdy and undisciplined. The Leningrad
Physical and Mathematical Society sub
sequently ceased to exist in its previous
form, being amalgamated into a new
organization under Rabinovich.
The row does not seem to have af
fected Mtintz's subsequent career. From
1931 Mtintz was also in charge of math
ematical analysis at the Scientific and
Research Institute in Mathematics and
Mechanics (Nauchny'i Jssledovatelski'i Institul Mathematiki i Mehanik� or NI
IMM) at Leningrad State University. Fur
thernlOre, in 1932, Mtintz's position in
Russia must have been quite firm, for
he was given the singular distinction of
being sent to the International Con
gress of Mathematicians in Zurich as
one of the Soviet Union's four official
delegates. The other three were Cheb
otarev, representing Kazan State Uni
versity, who gave a plenary lecture on
Galois Theory (on the occasion of the
centenary of the death of Galois), the
famous topologist P. S. Aleksandrov
from Moscow State University, who
talked about Dimension Theory, and E. Ya. Kol'man, a mathematician and a
member of the Communist Academy in
Moscow. The Academy was an institu
tion created in 1918 which had been
given the task of developing Marxist
views in the fields of philosophy and
science. Kol'man, the ideologist in this
delegation, gave two talks, the first
about quaternions, and the second about
the foundations of differential calculus
in the works of Karl Marx. Mtintz, rep
resenting NIIMM at Leningrad State Uni
versity, read a paper on Boundary Value
Problems in Mathematical Physics
[M29].
While Muntz had been unable to ob
tain his Habilitation in Germany, he
was far more successful in Russia. In
1935, at the recommendation of
Leningrad State University, VAK, the
committee that gave these higher (or
second) doctorates in Russia, awarded
Muntz a higher degree without requir
ing the submission of a written thesis.
Muntz would later write that he had
been awarded an honorary doctorate,
and that could be one possible inter
pretation of this degree. In sum, with
out doubt Muntz held a senior position
at Leningrad State University and had
the respect of his colleagues. He had
fulfilled his ambition.
Muntz was active administratively,
pedagogically, and mathematically. In
a later letter to Einstein he wrote about
working on a uniform theory of the so
lutions of non-stationary boundary
value problems in homogeneous and
non-homogeneous spaces. However he
also talked about the heavy teaching
and administrative load, and the un
fortunate state of his eyes that hin
dered him greatly. In 1934 he published
8"E1nstein (as well as Lichtenstein) as well as the rest of the 1ns1der world shall not learn anything of th1s." Muntz to Schrnalenbach, Decernber 5, 1 928.
© 2005 Spnnger SC1ence+ Bus1ness Med1a, Inc , Volume 27, Number 1 , 2005 27
a textbook on Integral Equations [M32], which is still sometimes refer
enced, and in 1935 he edited a Russian
edition of Lyapounov's important
monograph on General Problems of Stability of Motion [M35]. The fact that
Muntz was given this task, of historical
as well as scientific importance, is an
other indication of the high regard in
which he was held. He also wrote some
half-dozen research papers, mainly on
boundary-value problems, integral
equations, and Mathematical Physics.
Furthermore, he was asked to write a
review of his own work for the Second
All-Union Mathematical Congress held
in Leningrad in 1934. The latter was a
definite honour, awarded at a time
when he began to recover from further
eye problems.
While in the Soviet Union Muntz kept
a low, neutral profile vis-a-vis internal
politics. Although he kept his German
citizenship, obtained in 1919, at some
stage he was given a "former foreignern
status within the Soviet Union. He also
traveled abroad widely in the company
of his wife, visiting Finland, Germany,
Switzerland, and Poland. Generally vis
its were vacations or had to do with
mathematics research or meetings, but
sometimes they were motivated by his
and his wife's health.
While visiting Berlin on vacations,
by March 1930 Muntz9 was again work
ing under Einstein. His work related to
a question "of compatibility of partial differential equations" which Einstein
indicated had been solved by Cartan,
in a wonderful fashion, but had not yet
been published. He said to Muntz, "You
will take pleasure from it."
Magda had suffered a cerebral thrombosis in 1934. Miintz himself, as
has been mentioned, suffered from se
vere problems in one eye, and around
1934, at the beginning of term, he suf
fered damage in the retina of the other
eye. This kept him from his academic
duties for several months. As he began
to recover, his department provided a
secretary to help him with his research,
and when he was able to teach, his stu
dents helped him by writing on the
blackboard before each lecture the for
mulae he needed.
9Letter to his sister Sala. March 28. 1 930.
28 THE MATHEMATICAL INTELLIGENCER
F. I. Ivanov, who was a graduate stu
dent in the 1930s, recalls that Muntz
conducted a seminar in Mathematical
Physics. Ivanov remembers Muntz as
being very actively occupied with sci
ence, but both accessible and sociable.
He writes that Muntz was stout, with
light-grey hair and very strong glasses
because of his poor eyesight. For this
reason Muntz's wife would bring him
to the seminar.
According to Muntz, he also helped,
sometimes directly and sometimes in
directly, mathematicians from Central
Europe obtain positions in the Soviet
Union. He said that the appointments
of S. Cohn-Vossen, a former collabora
tor of Hilbert who died of pneumonia
shortly after arriving in Moscow, and
of the number theorist A. W alfisz, who
went to Tbilisi, were the result of his
suggestions. With others, he helped
both A I. Plessner and Stefan Bergman
obtain positions.
In October 1937 Miintz was expelled
from the Soviet Union without, ac
cording to him, any apparent reason.
This was part of a wide movement
sweeping the country, which turned
the tide against foreigners, including
teachers and engineers. It even af
fected those, like Muntz, who had
helped develop areas of academic ac
tivity in the Soviet Union. The Muntzes
were given a few weeks to leave.
Sweden (1 937-1 956)
Leaving the Soviet Union was a shock
to Muntz, who was then 53 years old.
He had lost the position he had worked
most of his life to obtain. The Muntzes
were permitted to take their personal
possessions with them, but no financial
recompense was offered for his eight
years as a professor at Leningrad State
University.
Travelling long distances was risky
because of his wife's thrombosis. The
Muntzes left Russia and first went to
Tallinn in Estonia. According to Muntz,
the Mathematics-Mechanics Faculty of
the Technical University in Tallinn of
fered him a visiting professorship for
the spring semester. But neither Ger
man nor Russian was an acceptable
teaching language- ministry regula-
A portrait of Herman Muntz during his years
in Sweden.
tions required lectures to be given in
Estonian-so this was not a possibil
ity. In February of 1938, the Muntzes
moved to Sweden, where they later re
quested political asylum.
Immediately after being expelled
from Russia, Muntz asked for help in ob
taining an academic position from a
number of colleagues and former teach
ers. He also asked family and personal
friends for financial help. Miintz's file at
the Society for the Protection of Science
and Learning and files on him in other
scientific refugee organizations indicate
that he contacted, among others, Har
ald Bohr, Einstein, Landau, Levi-Civita,
Volterra, Weyl, and Courant.
Finding an academic position for a
scientist of Muntz's age was not easy.
Technically he was not a refugee from
Hitler's Germany, having left in 1929.
This put him outside the purview of re
lief organizations such as the Notgemeinschaft Deutscher Wissenschaftler im Ausland. Furthermore, he was al
ready in Sweden, a relatively safe
place. At this late date, competition for
academic positions in Europe and in
the United States was fierce and in
volved a large number of parameters to
overcome the resistance and anti-for
eign feeling, often as intense as the gen
erosity of those many prepared to help.
Seniority in the field, age of the candidate, area of research, even personality, without leaving aside the strength of his personal network of scientific contacts, were among these parameters.
While Mi.intz was referred to as a "mathematician of the highest rank" in the dossiers of agencies dealing with displaced scientists, he was not considered a star. There were other drawbacks in Muntz's prospects for employment. Crucially, Einstein had concerns regarding Muntz's personality, dating from the late 1920s. In a letter to a colleague, Einstein explained his reservations in terms of what he regarded as Mi.intz's inability to submit his ideas to a proper level of critical analysis and his previous mental illhealth. Einstein thought that, in a period of general distress, he should reserve his influence for more clear-cut cases. An imbalance in his personality, probably associated with his nervous breakdown of 20 years earlier, was now a serious drawback in Muntz's prospects for employment.
Mi.intz resented Einstein's attitude and especially Einstein's suggestion (in 1938!) that Muntz should look for work in his native Poland. Muntz indicated that he was now in exile from Poland, Germany, and the Soviet Union, and in Germany he had not been forgiven for his former cooperation with Einstein. The tremendous competitiveness for jobs in America at the time may not have been entirely clear to Muntz, as the job description of his aspirations suggests. In a lost letter, Einstein may have pointed to the shortage of openings. Only Muntz's reply to this letter is available. Muntz responded that it was painful and unjust after so many years "to conclude that my present fate is to be judged only by the statistics of supply and demand." Fair enough, but the job market was clearly not in Einstein's hands.
In Sweden, as previously in Germany, Muntz had less success in penetrating official academic circles than in the Soviet Union. However, from the first days of his arrival there he had the support of Professor Gi:iran Liljestrand, chairman of the funding committee for
exiled intellectuals, who helped him financially in 1939 and 1940. A number of distinguished Swedish academics also offered their help and friendship, among them Professors Folke K.-G. Odqvist, a mechanics specialist, Hugo Valentin, a physicist, Marcus Ehrenpreis, a pediatrician surgeon, and David Katz, a psychologist.
Initially, in 1940-42, he received research grants from the Karolinska Inst'itutet for work on mathematical problems related to haemodynamics, the study of the dynamics of blood flow, which involved the solution of complex non-linear partial differential equations. Research on this subject was carried out at the Maria Hospital, in Stockholm, in collaboration with a young medical doctor, Dr. A. Aperia. Muntz published a note on haemodynamics in Cornptes Rendus, submitted by Hadamard, that appeared in the February 1939 issue. In 1942 Aperia died and this research came to an abrupt end.
Although he was on good terms with some leading Swedish scientists and intellectuals, Muntz was not able to forge a working contact with the small but active Swedish mathematical community ofthe time. Nevertheless, for some years he remained interested in various mathematical problems. In correspondence with Einstein and others he indicated he was interested in problems of integral equations, turbulence, knot theory, actuarial mathematics, and of course haemodynamics. However, possibly due to the severity of his circumstances, no scientific papers of Mi.intz have come to light from this last period.
While in Stockholm Muntz returned to private teaching, and the couple moved to an apartment in Solna, a pleasant district of Stockholm. They had a telephone in their name, which suggests that their financial circumstances had improved. It seems that they did have some outside financial resources. Muntz later received a small pension from the Warburgfonden, a foundation controlled by the "Mosaic" community in Sweden.
His wife Magdalena died of hemiplegia on January 19, 1949. Muntz became a Swedish citizen in 1953 and
1 00dqvist is wrong in this po1nt, Muntz was granted Swedish Citizenship 1n 1 953. R1ksark1vet, Stockholm.
died on April 17, 1956, at the age of 71 . He was blind for the last few years of his life. But for an obituary in Svenska Dagbladet, the leading Swedish newspaper, written by Odqvist, his death passed almost unnoticed to the mathematical community of Sweden and the rest of the world. In this obituary Professor Odqvist summarized the last years of Muntz's life in a short but poignant paragraph: "Herman Muntz is dead. In spite of the fact that he lived in Sweden for 18 years, the last five years10 as a Swedish citizen, there are probably not many Swedes outside his nearest circle of acquaintances, that knew that we had among us a mathematician of international fame who was thrown up on our calm shore by the storms of the times, his life saved but with his scientific activities broken." He ended the obituary with these words: "Herman Muntz lived in an exceptionally harmonious marriage and his wife Magda meant much to him, not in the least in order to keep his floating spirit down to earth. After her death in 1949 he only seldom saw his friends and he went every day to her grave in the Jewish cemetery with fresh flowers as long as he could. Now he is gone. Let this be a modest flower of memory from his Swedish friends. May his memory be blessed."
Acknowledgements
A paper such as this could not have been written without the help of many, many people and of various institutions. While it is impossible to name them all here we hope to acknowledge them by name at a later opportunity.
HERMAN MUNTZ: LIST OF MATHEMATICAL
PUBLICATIONS
[M 1 ] Zum Randwertproblem der partiellen Dif
ferentialgleichung der Minimalflachen, J.
Reine Angew. Math. , 139 ( 19 1 1 ), 52-79.
[M2] Aufbau der gesamten Geometrie auf
Grund der projektiven Axiome allein, MUnch
ener Sitz . , (1 91 2) , 223-260.
[M3] Das Euklidische Parallelenproblem, Math.
Ann . , 73 (1 9 1 3), 241 -244.
[M4] Das Archimedische Prinzip und der Pas
calsche Satz, Math. Ann. , 74 (1 9 1 3) ,
301 -308.
© 2005 Spnnger Sc1ence+Bus1ness Med1a. Inc . . Volume 27. Number 1. 2005 29
[M5] Solution directe de !'equation seculaire et
de quelques problemas analogues tran
scendants, C. R. Acad. Sci. Paris , 1 56
(1 9 1 3) , 43-46.
(M6] Sur Ia solution des equations seculaires et
des equations integrales, C. R. Acad. Sci.
Paris , 1 56 ( 1 9 1 3) , 860-862.
[M7] Ober den Approximationssatz von Weier
strass, in H. A. Schwarz-Festschrift, Berlin,
1 91 4 , 303-3 1 2 .
[M8] Sur une propriete des polyn6mes de
Bernoulli, C. R. Acad. Sci. Paris , 158 (1 9 1 4) ,
1 864-1866.
[M9] Ein nichtreduzierbares Axiomensystem
der Geometrie, Jber. Deutsch. Math. Verein,
23 (1 9 14), 54-80.
[M 1 0] Approximation willkurlicher Funktionen
durch Wurzeln , Archiv Math. Physik, 24
(1 9 1 6) , 31 0-316.
[M1 1 ] Zur expliziten Bestimmung der Haupt
achsen quadratischer Formen und der
Eigenfunktionen symmetrischer Kerne, Gott.
Nachr. (1 9 1 7) , 1 36-140.
(M1 2] On projective analytical geometry (in Pol
ish and German), Prac. Mat. -Fiz. , 28 (1 9 1 7) ,
87-1 00.
[M 1 3] The problem of principal axes for quad
ratic forms and symmetric integral equations
(in Polish and German), Prac. Mat. -Fiz. , 29
(1 9 18), 1 09-1 77.
A U T H O R S
�-..
• '> w:·�>> . � ..,. , . � .. .,
"
(M 1 4] A general theory for the direct solution
of equations (in Polish), Prac. Mat.-Fiz. , 30
( 191 9), 95-1 1 9.
[M1 5] Die Ahnlichkeitsbewegungen beim allge
meinen n-K6rperproblem, Math. Z. , 1 5
(1 922), 1 69-1 87 .
[M1 6] Allgemeine independents Aufl6sung der
lntegralgleichungen erster Art, Math. Ann . ,
87 (1 922), 1 39-149.
[M1 7] Beziehungen der Riemannschen ?-Funk
lion zu willkurlichen reellen Funktionen, Mat.
Tidsskrift 8, (1 922). 39-47.
[M 1 8] Absolute Approximation und Dirichletsches
Prinzip, G6tt. Nachr. , 2 (1 922), 12 1-1 24.
(M 1 9] Allgemeine Begrundung der Theorie der
h6heren ?-Funktionen, Abhdl. des Sem.
Hamburg, 3 (1 923), 1 -1 1 .
[M20] Der Summensatz von Cauchy in beliebi
gen algebraischen Zahlkorpern und die
Diskriminante derselben, Math. Ann . , 90
(1 923), 279-291 .
[M21 ] Fragen der klassischen und relativistis
chen Mechanik. Vier Vortrage gehalten in
Spanien in January 192 1 , by T. Levi-Civita;
authorized translation by P. Hertz, H. Kneser,
Ch. H. Muntz, and A. Ostrowski , pp. vi +
1 1 0, J. Springer, Berlin, 1 924.
[M22] Umkehrung bestimmter Integrals und
absolute Approximation, Math. Z. , 21 (1 924),
96-1 1 0.
[M23] Ober den Gebrauch willkurlicher Funk
tionen in der analytischen Zahlentheorie,
Sitzungsberichte der Berliner Math.
Gesellschaft, 24 (1 925), 8 1-93.
[M24] Die L6sung des Plateauschen Problems
uber konvexen Bereichen, Math. Ann . , 94
(1 925), 53-96.
[M25] Zur Gittertheorie n-dimensionaler Ellip
soids, Math. z. , 25 (1 926), 1 50-1 65.
[M26] Zum Plateauschen Problem. Erwiderung
auf die vorstehende Note des Herrn Rad6,
Math. Ann. , 96 (1 927), 597-600.
[M27] Ober die Potenzsummation einer En
twicklung nach Hermiteschen Polynomen,
Math. Z. , 31 (1 929), 350-355.
[M28] Sur Ia resolution du problems dynamique
de l 'elasticite, C. R. Acad. Sci. Paris , 194
(1 932), 1 456-1 459.
[M29] Ober die L6sung einiger Randwertauf
gaben der mathematischen Physik, Ver
handlungen des lnternationalen Mathe
matiker-Kongress ZOrich 1932, Dr. Walter
Saxer, ed. , Zurich, 1 932, 1 09-1 10 .
[M30] lntegralgleichungen der Elastodynamik,
Rec. Math. Moscou, 39, 4 (1 932), 1 1 3-132.
[M31 ] Zum dynamischen Warmeleitungsprob
lem , Math. Z. , 38, 3 (1 934), 323-337.
[M32] Integral Equations, Vol. I, Volterra 's Lin
ear Equations, (in Russian), 330 pages,
Leningrad, 1 934.
EDUARDO L. ORTIZ
Department of Mathematics
Imperial College London, South Kensington Campus
London, SW7 2f.Z.
ALLAN PINKUS
Department of Mathematics
Technion
United Kingdom
e-mail: [email protected]
Eduardo L Ortiz did his doctoral work under the supervision of
Mischa Collar in Buenos Aires, and subsequently went to Dublin
for research under Cornelius Lanczos. Since 1 963 he has been at
Imperial College London, where he is now Professor. He has writ
ten prolifically on functional analysis and its applications and on
history of mathematics. He has held visiting positions at Harvard
and the universities of Orleans and Rouen.
30 THE MATHEMATICAL INTELLIGENCER
Haifa, 32000 Israel
e-mail: [email protected]
Allan Pinkus, a native of Montreal, did his undergraduate work at
McGill University and his doctoral work at the Weizmann Institute
under Samuel Karlin's supervision. Since 1 977 he has been at the
Technion. His research interests center on approximation theory.
He was for ten years an Editor-in-Chief of the Journal of Approximation Theory.
[M33] Sur les problemes mixtes dans l 'espace
heterogene, Equation de Ia chaleur a n di
mensions, C. R. Acad. Sci. Paris , 199 (1 934),
821 -824.
[M34] Functional Methods for Boundary Value
Problems (in Russian), Works of the 2nd All
Union Mathematical Congress, Leningrad,
Leningrad-Moscow, 1 (1 935), 31 8-337 .
[M35] General problems of stability of motion,
by A. Lyapounov, (in Russian) , Ch. H. Muntz,
ed. , Leningrad-Moscow, 1 935.
[M36] Zur Theorie der Randwertaufgaben bei
hyperbolischen Gleichungen, Prace Mat.
Fiz. , (Gedenkschrift fur L. Lichtenstein), 43
(1 936) , 289-305.
[M37] Les lois fondamentales de l 'hemody
namique, C. R. Acad. Sci. Paris , 280 (1 939),
600-602 .
REFERENCES
[ 1 ] K. Weierstrass, Uber die analytische
Darstellbarkeit sogenannter willkurlicher
Funktionen einer reellen Veranderlichen,
Sitzungsberichte der Akademie zu Berlin,
1 885, 633-639 and 789-805.
[2] S. Bernstein, Sur les recherches recentes
relatives a Ia meilleure approximation des
functions continues par des polyn6mes,
Proceedings of the Fifth International Con
gress of Mathematicians, (Cambridge,
22-28 August 1 91 2) , E. W. Hobson and
A. E. H . Love, eds . , Cambridge, 1 91 3 , Vol.
I, 256-266.
[3] S. N. Bernstein, Sur l 'ordre de Ia meilleure
approximation des functions continues par
les polyn6mes de degre donne, Mem. Cl.
Sci. Acad. Roy. Be/g . , 4 ( 1 9 1 2) , 1 -1 03.
[4] E . L. Ortiz, "Canonical polynomials in the
Lanczos' Tau Method, " B. P. K. Scaife,
ed. , Studies in Numerical Analysis , New
York, 1 97 4, 73-93, on 75.
[5] E. L. Ort1z, "The Society for the Protection
of Science and Learning and the Migration
of Scientists in the late 1 930s," Panel 's
Chairman's lecture, Proceedings of the
1 13th annual meeting of the American His
torical Association, Washington, 93 (1 999),
1 -28.
[6] Ch. Muntz, Wir Juden, Oesterheld and Co. ,
Berl in , 1 907.
[7] A. Korn, Uber Minimalflachen, deren Rand
kurven wenig von ebenen Kurven abwe
ichen Abhdl. Kg/. Akad. Wiss., Phys-math,
Berl in , (1 909), 1 -37.
[8] R . von Mises and H . Pollaczek-Geiringer,
Praktische Verfahren der Gleichungsaufl6-
sung, Zeitschrift fur Angewandte Mathe
matik und Mechanik, 9 (1 929), 58-77 and
1 52-164.
[9] L. Butschli, HochwaldhauserDiary, 39, 39a.;
quoted in Karl-August Helfenbein, Die
Sozialerziehung der Durerschule Hochwald
hausen, Hochhausmuseum and Hohha
subibliotek, Lauterbach, 1 986, p. 1 5.
[1 0] Der Jude, Judischer Verlag, Berl in ,
1 9 1 6-1 928.
[1 1 ] E. C. Titchmarsh, The Theory of the Rie
mann Zeta-Function, Oxford, 1 951 , p. 28.
[1 2] D. Amira, La Synthese Projective de Ia
Geometrie Euclidienne, ltine and Shoshani,
Tel-Aviv, 1 925.
[1 3] G. G. Lorentz, Mathematics and Politics in
the Soviet Union from 1 928 to 1 953, Jour
nal of Approximation Theory, 1 1 6 (2002),
1 69-223.
r
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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 Genius1 in 2000, I was captivated by the personalities, the
intrigues, and the controversies that were part of mathematics in sixteenth-cen-
tury 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 depressed 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 figured out how to solve depressed cubics, thus winning the challenge.
In Milan, the mathematician/physician Gerolamo Cardano heard about Tartaglia's grand accomplishment. For several years, he pleaded with Tartaglia to tell him his secret. 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 reduce a general cubic to a depressed one, thus completely solving the classical problem of the cubic. Then his assistant 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 solution to depressed cubics. They found Scipione's original algorithm 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 considered 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 meaningless 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 Cardano 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 Mac Tutor 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
Nella citta dal mar' intorno centa.
Any Italian who encountered this poem would have immediately 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 literature, 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 contained in it: the poem is written in tercets, or three-line stanzas, and all the rhymes, except at the start and finish of the poem, come in triplicate, with the center line of each tercet rhyming with the outer lines of the tercet following it,
3Tartaglia. Niccol6, Ouesiti et inventioni diverse de N1ccol6 Tartalea Bris01ano.
[Stampata in Venetia per Venture Rotflnelli, 1 546.]
thus propelling the poem forward. This form is extraordinarily well-known by Italians.
* * *
In the early sixteenth century, algebra was rhetoricalthat 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 agguaglia 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 translation 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 Renaissance 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 unstressed syllable, it is natural to have iambic lines of poetry with eleven syllables. It is slightly more difficult in English. 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
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:
----- - -----
Quesito XXXII I 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· I I digi 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.
The final case is easy to assess: Look closely at the second case I mentionIt'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,
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 situations. In fact, because they didn't recognize negative numbers, 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 rand 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 considered 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 concludes 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 embodied in a real cube. Seven hundred years earlier, in Baghdad, Al-Khwarizmi (from whose name comes the word "algorithm") 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 dimensions m/2 and x are picked for very good reasonstwo 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
Completing the Square m I
X
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.
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)2-gives us the side length t. But t is equal to x + m/2, so x equals V n + ( 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 dimensions 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.
3tux
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, swirling counter-clockwise around a (missing) cube
of side u. In the other interpretation (and this came as a complete
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
painted on the large cube's walls, one cannot help "seeing" (though
they are missing) the three slabs of volume tux, once again swirling
counter-clockwise about the little cube of side u.
around the cube of side x, thus giving him a larger cube of side t, but (as before) with one crucial piece missing. In order 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 volumes 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 + 3tux + u:l = (l. We can imagine Tartaglia striving to imagine the di
mensions of a physical cube that would represent the solution 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 exact dimensions that satisfy his problem. He had to imagine the lengths u and t both changing (the overall cube growing 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 realizes that if he merely moves u3 to the right side, it will give him a new equation that precisely embodies Fior's depressed cubic _Tl + m:J.: = n, with 3tu playing the role of m and t3 - u3 playing the role of n.
x3 + 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,
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 insight. 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
1· - s = n rs = (m/3)3.
The last equation is an immediate consequence of the definition 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 order 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 particular 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 + s) = 8 s2 + 20s = 8 s2 + 20s - 8 = 0.
Using the quadratic formula to solve for 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. Inc . . Volume 27, Number 1 . 2005 35
Numerically,
r = 20.3923 and s = .3923.
Then, taking these numbers' cube roots,
x = Vr - Vs X = 2. 73205 - . 73205 X = 2.
If we plug this back into the original equation x3 + 6x = 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.
* * *
Finding a solution by radicals to the cubic was a monumental accomplishment. However, it led to a thorny obstacle: 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 expressions featuring negative numbers under the square-root sign, a situation that Cardano dubbed the casus irreducibilis, reflecting the fact that Renaissance mathematicians were not comfortable with negative numbers, let alone their square roots.
The Bologna mathematician Rafael Bombelli took Cardana'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 published a book on this topic titled Algebra, in which he presented an early symbolic notation system. Although he never found out how to take cube roots of complex numbers 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), 12-21 .
Despite this accomplishment, Cardano's formula provided Bombelli with only one of the equation's three roots, and it took another 40 years until Fran<;ois Viete figured out how to find the other two real roots, and then a further 300 years until mathematicians penetrated the mystery of the casus irreducibilis and finally understood why complex numbers were needed to express the real roots to cubic equations through radicals.
When Ferrari based his solution of the quartic equation on that of the cubic, just as Tartaglia had based his solution of the cubic on that of the quadratic, it seemed as if this clever method could go on indefinitely: lower the de-
36 THE MATHEMATICAL INTELLIGENCER
AU T H O R
KELLIE 0. GUTMAN
75 Gardner Street
West Roxbury, MA 021 32-4925
USA
e-mail: [email protected]
Kellie Gutman has studied mathematics and audiology-and,
since 1 999, poetry. One piece of mathematical research she
wrenched into poetic form, quite impressively; see The Math
ematical lntelligencer 23 (2001 ), no. 3, 50. Wrth her husband,
Richard Gutman, she is co-owner for 25 years of a company
specializing in audio-visual presentations for museum installa
tions; co-au1hor of two books; and parent of Lucy.
gree of an equation by one, and use this new equation's formula to help solve the original. But when mathematicians tried to solve the quintic equation in this way, they hit a brick wall. It wouldn't yield.
For the next 250 years, mathematicians struggled to solve quintics by radicals. Finally in 1 799, Paolo Ruffini, another mathematician/physician, wrote a book Teoria Generate delle Equazioni, offering a proof that fifth-degree equations-indeed, all equations of degree greater than four-were in general unsolvable by radicals; but almost no one accepted his claims. Twenty-two years later the distinguished French mathematician Cauchy wrote to Ruffini, praising his proof, but few people agreed with Cauchy. In a few years, however, Niels Henrik Abel in 1825 and Evariste Galois in 1830 published works on the unsolvability of the quintic equation and equations of higher order, and their discoveries, which were centered on the symmetry groups of the roots, were widely accepted.
For the thousand or so years between the destruction of the Library of Alexandria and the Renaissance, European mathematics, with a few notable exceptions, had made slow progress. But the Italian mathematicians who worked on solving the cubic initiated a series of events that led to the use of negative numbers, complex numbers, powers and dimensions higher than the third, and symbolic algebra, with its highly efficient system of symbol manipulation. This work, spanning roughly one hundred years, reinvigorated mathematics and led directly to many of the discoveries of the modem era.
M a them a tic a l l y Bent
The proof is in the pudding.
Opening a copy of The Mathematical
Intelligencer you may ask yourself
uneasily, "What is this anyway-a
mathematical journal, or what?" Or
you may ask, "Where am I?" Or even
"Who am I?" This sense of disorienta
tion is at its most acute when you
open to Colin Adams's column.
Relax. Breathe regularly. It's
mathematical, it 's a humor column,
and it may even be harmless.
Column editor's address: Colin Adams,
Department of Mathematics, Bronfman
Science Center, Williams College,
Williamstown, MA 0 1 267 USA
e-mail: [email protected]
C o l i n Adam s , Ed itor
Tria l and Error Colin Adams
Please be seated, Mr. Phipps. Actually, it's Dr. Phipps. Really? Are you a medical doctor? No. But I have a Ph.D. That means
I have a doctorate. So I should be addressed appropriately.
And tell the court, Mr. Phipps, do you often insist on being called doctor?
I prefer to be called that. Perhaps because of some insecurity
on your part? A need to assert your authority through a title?
I want my students to know I am in control.
And do you take some pleasure in that control, Mr. Phipps?
Objection, this is an irrelevant line of questioning.
Sustained. I retract the question. Now tell me
Mr. Phipps, were you the instructor for Math 105 Multivariable Calculus at Freedmont College this last fall?
I was the professor for that course, yes.
And was there a student named Jeffrey Foible in that class?
Yes, there was. And do you see him here today in
the courtroom? Yes, he is sitting over there next to
his mother. And can you tell us how Jeffrey did
in your class? He received a C + . Was h e close to a B - ? Yes. But h e was clearly in the C +
range. I see. And how many students were
in the course, Mr. Phipps? About 1 50. 150? With that many students, is it
difficult to keep track of individual students, and how they are doing in the course?
I have teaching assistants.
I see. And so they keep track of the individual students, relieving you of the necessity to do so?
No, I pay attention, too. I review the grades on the homework and I do all the grading on the exams.
Really? How much time does that take?
On the two midterms, I spent about twelve hours grading each of them. Then the final took longer. About twenty hours.
Twenty hours? That's a tremendous amount of time to be sitting, staring at student work. It must be exhausting. Is it hard to keep up your concentration that long?
I take breaks. You mean like getting a drink of
juice, using the bathroom, maybe watching a little TV?
Yes, that's right. And of course, you aren't going to
complete twenty hours of grading in a day. I suppose it is stretched over a period of several days. How many days did it take you on the exam for this course?
As I remember it, about three days. I see. That's quite a bit of time. So
you might finish for one day, and then have a 12- or 15-hour break before resuming.
Yes. Now, correct me if I'm wrong. After
all, you're the math teacher. But 20 hours means 1200 minutes. Divide by 150 and that means an average of 8 minutes per exam.
Yes, that's right. And how many pages were there on
the final? Eight pages. So, a minute per page. Yes. And how many problems are on a
page? About four, if each part of a multiple-
part question is counted separately. So 15 seconds a problem. About that, yes. And do you give partial credit?
© 2005 Spnnger Sc1ence+Bus1ness Media, Inc , Volume 27, Number 1 , 2005 37
Oh, yes.
Do you feel that in the 15 seconds
you apportion to a given problem, you
can fairly determine the appropriate
amount of partial credit?
Some problems take less than 15
seconds and some take more. When I
need to think about how much to give,
I take longer.
Oh, and so some of the time when
you are grading, you aren't thinking at
all. Kind of on automatic pilot.
There are some problems where
there isn't much partial credit to give.
Either they have the answer right or
they do not.
And do you consider yourself a con
sistent grader?
I try.
Do you know a Ms. Elaine Plepp
meyer?
I believe she was a student in the
course.
Good for you, Mr. Phipps. It must
not be easy getting to know all the
students.
I do my best.
And can you tell me, is she seated
in the courtroom right now?
Well, urn, I'm not sure.
But I thought you knew her.
Is that her over there, seated by
Foible?
No that is Jeffrey Foible's sister.
How about the blonde woman by
the door?
No, that is not her.
Then I am not sure.
In fact, Mr. Phipps, Elaine Plepp
meyer is not here today. However I do
have her final exam right here. Can you
verify that this is her final exam from
the course?
Well, yes, it does appear to be her
exam.
I would like to enter this as Exhibit
A. Can you tell the court, please Mr.
Phipps, what grade appears at the top
of the exam?
It is a B - .
And the numerical score?
An 81.
I see, and can you verify for the
court that here I have a copy of Mr.
Foible's fmal exam?
Yes, that appears to be the exam.
I would like to enter this as Exhibit
38 THE MATHEMATICAL INTELLIGENCER
B. Can you tell the court what grade
and score is on this exam?
It is a C + , with a score of 79. Interesting. Not so different, is it?
Now tell me Mr. Phipps, and please feel
free to consult your roll book, but other
than this difference in their final ex
ams, how close were the grades for
these two students coming into the
final?
Well, Foible did slightly better on
the homeworks, and Pleppmeyer did
slightly better on the two midterms.
They were both borderline between a
B and a C.
So you would characterize their per
formance up to the final as being es
sentially equivalent as far as their ulti
mate grade is concerned?
Yes, that is true.
Oh, dear, so it seems that this minor
two-point difference between their
scores on the final made the difference
between Jeffrey Foible receiving a B
and a C.
Actually, a B - and a C + .
Whatever. I t is a big difference, Mr.
Phipps. Perhaps the difference be
tween getting into law school and not
getting into law school?
I don't know about that. I just give
students the grades they deserve.
Do you have any idea of the poten
tial earning power that a lawyer has,
Mr. Phipps? Do you realize that this
tiny two-point difference may have
cost Mr. Foible five million dollars over
his lifetime?
That isn't my concern.
Well, perhaps it should be, Mr.
Phipps, because if you made a mistake
in grading the exam, that could be a
very costly mistake.
What do you mean?
Please open the exam to page 4. Let's take a look at problem 1 1. Could
you please read the problem to the
court?
It says, "Find the volume of the
tetrahedron in the first octant of space
bounded by the three coordinate
planes and the plane x+2y +3z = 6." Thank you, Mr. Phipps. Perhaps you
could explain the problem to us. After
all, we are not all experts in calculus,
as you are.
Well, the three coordinate planes
are the xy, the yz and the xz plane,
which are three orthogonal planes in
tersecting pairwise along the coordi
nate axes.
I'm sorry. I must be slower than
your students. Orthogonal? Intersect
ing pairwise?
Umm. It's like the corner of a box.
Three planes meeting perpendicularly.
Then a fourth plane slices through and
cuts off the corner of the box.
Thank you. That makes a bit more
sense. Now you say the plane x+2y
+3z = 6. So x+2y +3z = 6 is a plane.
Yes, that is the equation of a plane.
Oh, so it's not itself a plane. It is the
equation of a plane. That seems like an
important distinction.
If you are going to be picky, then
yes, it would be slightly more correct
to say that x+2y +3z = 6 is the equa
tion of a plane, rather than a plane
itself.
Well, perhaps when five million dol
lars is at stake, it would pay to be picky.
Now, as I understand it, the tilted plane
intersects the x, y, and z axes at 6, 3,
and 2, respectively. Is that correct?
Yes.
Then the base is a right triangle with
the two edges at the right angle of
lengths 6 and 3. So it has area 9. That is correct.
And the height of the tetrahedron is
2, so it appears you want the students
to apply the famous formula for the
volume of a tetrahedron, which is one
third of the area of the base times the
height. In this case, we obtain 6. Did I
do that right?
That is the correct answer, but no,
I did not want them to use the formula
for the volume of a tetrahedron. The
whole point of the class is to learn cal
culus. They were supposed to create a
double integral that gives the volume
of the solid.
You mean as in this solution given
by Ms. Elaine Pleppmeyer, which we
see on this overhead slide.
Yes, that is what I intended. Only the
upper limit of integration on that outer
integral should be 6, not 3.
I see, and that's why you took off
two points out of the fifteen possible.
Yes, that's right.
Seems fair enough.
It is a standard amount I took off for
a mistake like that.
Very good. Now, shall we look at
Jeffrey Foible's solution? I have it on
this next overhead slide. Well, look at
that. He appears to have the correct so
lution 6. And he appears to have done
it using the exact method I described.
Can you tell us how many of the fifteen
points you took off, and what it is you
wrote on the exam?
Umm, it looks as though I took off
5 points. And I wrote in the margin,
"Use calculus to solve these problems,
not memorized formulas."
So, correct me if I misunderstand.
Ms. Pleppmeyer got the answer wrong,
and you deducted two points, and Mr.
Foible got the answer right, and you de
ducted five points.
Well, yes. But he didn't use calculus.
Oh, and can you point me to where
on the exam it says that all problems
must be solved using calculus?
That was implied.
Oh, I see. Implied?
I did put on the front of the exam,
"In all cases, grade is determined by the
instructor."
I see. And tell me, Mr. Phipps, if you
wrote on the exam, "In all cases, grade
to be determined by size of student's
butt," would that then make it accept
able to determine the grade in that
manner?
Objection, your honor.
Sustained. The jury is instructed to
ignore the word "butt. " Strike it from
the record.
Let me rephrase the question. Are
instructors not bound by some code of
ethics? Should it not be the case that
if students get the right answer, they
should get the points they deserve?
This was a calculus course. Stu
dents were supposed to learn calculus.
We had done problems like that on
the homework They knew they were
supposed to do the problem using
calculus.
And because Jeffrey Foible had this
additional knowledge, because he had
taken the time in his previous mathe
matical career to learn this formula, to
remember this formula, you deducted
5 points?
I don't believe he knew the formula.
What do you mean?
I think he coped it from Karen
Lapala's paper.
What makes you think that?
Karen Lapala is an A student. She is
the only other student in the entire
class who used the formula to solve
that problem. I think Foible didn't
know how to do that problem, looked
at her paper, and wrote down the an
swer. I think he cheated.
Really? And do you know if Jeffrey
Foible was sitting anywhere near Ms.
Lapala during the exam?
I can't be sure, but I have a vague
recollection he was sitting near her.
Really? If you are right, you must
have one of the most prodigious mem
ories known to humankind. Let's see.
With 150 people sitting in the audito
rium, each person sitting next to say,
two other people, how many pairs does
that make? That's approximately 150
pairs that you would need to remem
ber. You looked out at that mass of
people and immediately memorized
150 pairs. Is that the case?
No, I don't remember who everyone
was sitting next to, but I think Foible
was sitting next to Lapala.
Forgive me if I seem skeptical that
you could possibly remember that. But
let me ask you this. Might your suspi
cion that Jeffrey Foible cheated off
Ms. Lapala have influenced how many
points you took off on his exam? Or
perhaps the better question is, how
many points did you take off on this
problem on Ms. Lapala's exam?
Urn, I think I took off 3.
Three? But that isn't fair. After all,
she and Mr. Foible had the same an
swer. Why should she lose only 3
points when he lost 5? For all you
know, she could have cheated off
Mr. Foible.
She had a picture of the tetrahe
dron.
But nowhere in the problem did it
say to draw the tetrahedron. Why are
you giving points for it?
It demonstrates that she understood
what was going on. I gave partial credit
for that.
Come now, Mr. Phipps, do you re
ally expect the jury to believe that by
drawing random pictures on an exam,
pictures that were not asked for, a stu
dent can receive points?
Well, yes, because . . .
If I draw a picture of a person for
my Abnormal Psychology essay ques
tion on schizophrenia, should I get par
tial credit?
Of course not, but . . .
The truth, Mr. Phipps, is that you
didn't take off more points for Mr.
Foible because he failed to draw a
tetrahedron. You took off the extra
points because you had it in for him.
Had it in for him? I barely knew him.
My client recalls waking up one day
in lecture, and you fixing him with a
particularly malevolent stare. It was at
this moment that he realized that you
had it in for him.
I didn't have it in for him. And even
if I did, I couldn't have graded his exam
more harshly. I grade blind.
Do you mean to tell us that you
grade your students' exams with your
eyes closed?
No, of course not. I mean that I flip
over the cover sheet with the student's
name on it before I ever start grading.
Then when I grade a page, I have no
idea which student is which. I cannot
be influenced by my impressions of
their abilities.
Well, isn't that a clever idea. So
you never know who it is that you are
grading?
That's right.
But I suppose that once in a while,
there is a student who has very dis
tinctive handwriting, and sometimes,
you are aware of who it is.
Maybe once in a while.
Perhaps, that was the case with Jef
frey Foible?
I don't believe so.
I would like everyone to direct your
attention to the screen at right. On the
overhead I have a sample of Jeffrey
Foible's handwriting. Note how the c's
have an unusually sharp curvature.
And notice the angle between the two
lines making up the x. That angle is
27.3245 degrees, approximately. How
ever the national average is 29.2234.
Am I right, Mr. Phipps, that x's come
up a lot on your exams?
Yes, they do. But you can't believe
that I would be able to recognize
© 2005 Spnnger Sctence+ Bus1ness Medta, Inc., Volume 27, Number 1 , 2005 39
Foible's handwriting from these minor
variations?
Perhaps not consciously, Mr.
Phipps, but the human mind is an in
credibly intricate and subtle device. It
is capable of much more than we give
it credit for. Do you know that they say
we use less than 10% of our possible
brain capacity? What do you think the
rest of our brain is doing?
I have no idea.
No, it does not appear that you do.
Thank you, Mr. Phipps. You may step
down.
Members of the jury, this case now
falls into your capable hands. And from
my point of view, it is a relief to see it
there. Because I believe you under
stand the critical importance of this
trial.
Take a look at Jeffrey Foible, sitting
there. Look at him. There sits a boy
who was one of the best and the bright
est, on the verge of manhood, ready to
embark on his future. But what future
is that? What future is left to him now?
Who was it that destroyed his con
fidence, that crushed his dreams, that
thwarted him from following his right
ful path? I think you know the answer
to that. It was the defendant, Mr.
Phipps.
Fifteen seconds! That's how long
Mr. Phipps put into scoring each prob-
40 THE MATHEMATICAL INTELLIGENCER
lem. Fifteen seconds. That's the amount
of time he bequeathed, in his magna
nimity, to determining the difference
between the B grade that would get
Jeffrey into law school and the C grade
that shuts him out forever. Fifteen sec
onds. Does that sound fair to you? No,
I'm betting it doesn't.
Now, of course, this case isn't just
about compensating Jeffrey Foible for
the suffering he has incurred. It is not
about one student and the disastrous
results of his professor's incompe
tence.
No, it is about how systematically,
across this country, faculty destroy
their student's hopes and dreams. It is
about how the whim or mood of a pro
fessor can change the course of a stu
dent's life. How insults exchanged in a
heated department meeting can gener
ate emotions that alter the distribution
of points on an exam. How a poorly di
gested bean burrito can cause a stu
dent to be barred from the career to
which he has always aspired.
What happens when you put ab
solute power into the hands of a
despot? When checks and balances
don't exist? When one person has the
unbridled authority to capriciously de
termine the fates of others?
I am not just asking you to chastise
Mr. Phipps. I am asking you to send a
message to all the teachers out there.
To tell them that the age of tolerance
for their misdeeds is over, once and
for all.
Each and every one of you members
of the jury knows what I am talking
about. Because each and every one of
you knows that when you were a stu
dent, you were unfairly graded. You did
not receive the points you deserved,
perhaps because your teacher had a
head cold, or even worse, because your
teacher didn't like the way you looked.
Do the right thing. Do it for the millions
of students who have gone before. Do
it for the millions of students yet to
come. And do it for Jeffrey Foible.
Thank you for your attention.
Doug Phipps jerked awake, his heart
pounding. Throwing off the covers, he
leaped out of bed and flicked on the
desk lamp. He rifled through the pile of
papers strewn about the desk until he
found Foible's exam.
Flipping it open to the fourth page,
he crossed out his written remarks and
the circled - 5, replacing them with a
check mark Then he flipped the exam
closed, turned out the light, and settled
back into bed. But sleep eluded him, and
four hours later, as light began to stream
through his windows, he looked forward
to the day's grading with dread.
ULRICH DAEPP, PAUL GAUTHIER, PAMELA GORKIN, AND GERALD SCHMIEDER
A ice i n Switzerland · The Life and Mathemat ics of A ice Roth
lice Roth, now well known for her work in rational approximation theory,
was the first woman to win the Silver Medal at the Eidgenossische Tech-
nische Hochschule (ETH) in Zurich, Switzerland and the second woman to
obtain a Ph.D. in mathematics there. (The next female recipient of the Ph.D.
in mathematics at the ETH would appear over 20 years
later.) Though Roth published her thesis in 1938, it would
be many years before her work was rediscovered and ap
preciated. And it would be 35 years before she was able to
concentrate on her research again, but then-at the age of
66-her influential work in the field of rational approxi
mation and function theory would become the focus of her
life and remain so until the day she died.
Three of her discoveries were outstanding for complex
approximation theory: her celebrated "Swiss cheese," be
fore which the very raison d'etre of qualitative approxi
mation was in doubt; her extension of approximation the
ory from bounded to unbounded sets; and her discovery of
fusion, making it possible to simultaneously approximate
two different functions on two different sets by a single
function, even though these sets may overlap.
Dubbed "Alice in Switzerland" by her friend and co
author, Paul Gauthier, she was agreed to be a remarkable
woman. Why? We tum to the life and work of this woman
who, in all areas of her life, was always too early to bene
fit from her talent, determination, and strong will.
The Early Years
Alice Roth was born on February 6, 1905, in Bern, Switzer
land. Her father, Conrad Roth, was then the director of the
Gaswerk und Wasserversorgung der Stadt Bern (Gas and
Water Supply for the City of Bern), a prestigious and in
fluential post that he was elected to when only 27 years of
age.
Conrad Roth came from a family of boatmen, but his fa
ther died when Conrad was just ten years old. He appren
ticed four years as a mechanic, took evening courses on the
© 2005 Spnnger Science+ Bus1ness Media, Inc., Volume 27, Number 1, 2005 41
Alice's baptism at the Gaswerk Bern.
side, and attended the Technikum Wi nterth:u.r where he
earned a diploma as a machine technician. Alice's mother,
Marie Landolt, was a daughter of the mayor of Zi.ilich-Enge
(before it was incorporated into the city of Zi.irich). Marie
Roth-Landolt was described as a warm and loving woman
who kept house perfectly and was an excellent hostess,
skills practiced in tum by Alice. Alice's parents married in 1902, and one year later they had a son, Conrad. Alice was
born two years later. Four years later tht:> family was com
pleted with the arrival of a second son, Waltt>r.
In 1911 Alice's father took a new position in the man
agement (at the national level) of gasworks and coal sup
ply. This required the family to move to Zi.irich, where Marie
Roth-Landolt had grown up. Alice, who started her school
ing in Bern, continued her education in Zurich, but changed
schools once more when her family moved into their newly
built house in Zollikon. The house, situated in a suburban
community of Zurich and overlooking the lake, must have
been a peaceful place for a young girl to grow up. Alice's
older brother Conrad studied forestry at the ETH in Zi.irich,
and received an assistantship there while writing his doc
toral dissertation. He eventually took a position as forPstry
superintendent in the canton Aargau. Alict:>'s younger
brother, Walter, did an apprenticeship as a bookselkr in
Zurich and later emigrated to Rio de Janeiro in Brazil where
he opened a bookstore.
After finishing the mandatory school years in Zollikon,
Alice commuted daily to Zi.irich where she attended the
Gymnasialabteilung der Hoheren Tochterschule der Stadt Zi.iri.ch. Schooling in Switzerland is the responsibility of the
canton, but preparatory schools for the university (Gymnasien) were only open to boys. Some of the larger Swiss
cities had excellent schools for girls and, fortunately, the
Hohere Tochterschule der Stadt Zurich was one of them.
As Alice Roth mentions, she had excellent teachers at this
school. Her mathematics teacher, Prof. Dr. William Brun
ner, instilled in her the desire to continue with her study
of mathematics. He resigned (to become professor of as
tronomy at the ETH and director of the Federal Observa
tory in Zurich) just a few years later. At his retirement from
the Hohere Tochterschule he was thanked:
. . . he worked at our school for many years (since 1908). In hin1 we had a truly outstanding mathematics teacher. In a
way fuat only a few teachers are able, he made his subject
one that is, in general, difficult to grasp-understandable
and easily comprehensible to girls. [ 17, 1925/'26 p. 13]
The curriculum at the school was typical for a Realgymnasium at this time in Switzerland. Latin took up the
most weekly hours of any single subject. Mathematics,
physics, chemistry, and other sciences were balanced witl1
German and two modern foreign languages (French and ei
ther English or Italian). In the spring of 1924, Alice passed
tht:> Matura , the examination that entitled her to admission
to a university.
Graduate Years- ETH and Silver Medal
Alice Roth's goal was clear: she wanted to study mathe
matics. Her mother, a practical woman, had nothing against
this study but wanted her daughter first to learn the basics
of household management. (This request was most likely
influenced by a national trend that led, in some places, to
the introduction of mandatory domestic training for girls
[ 12, pp. 361-:365] .) Thus it was that Alice went to Schloss Ralligen, on the shore of Lake Thun. This medieval castle
was home of a Haushaltungsschule for well-to-do daugh
ters. During that year, Alice also took courses in needle
work and dressmaking, and helped in the family household.
She then spent the summer semester of 1925 at the Urt'i
versiUit Zilrid�, where she prepared for her entrance to
The Roth family house in Zollikon. Alice with her mother.
42 THE MATHEMATICAL INTEUIGENCER
Schloss Ralligen, 2004.
the department for Fachlehrer in Ma,thematik und Physik
of the ETH in Zurich. In the fall, she entered the ETH. From
1925 to 1929 her major field of study was mathematics,
physics her first minor, and astronomy her second.
The ETH is Switzerland's premier university for the sci
ences and technology. It was very much dominated by men;
there were very few female students and there were no
women on the faculty. (In 1910, a woman wrote her Habil
itation in mineralogy, but she died in 1916. For the forty
years following, no other woman was on the faculty at the
ETH [31 , p. 163] .) Nevertheless, Alice Roth did very well.
Her grades for the diploma exam and on her Diplmnnrbeit
were exceptionally high. Though Alice was surrounded by
men, most of her lasting friendships wpre with women very
much like herself. One of these was her colleague at the
ETH, Hanna Bretscher-Greminger. In HJ30 Alice completed
her Diplmrwrbeit (Master's thesis), entitlPd Ausdeh rrung des Wwiw·stra.ss 'schen Approxima/.ionssatze,-; a.uf das kmnple.r:e Geb·iet und a,uf ein u nendl iches Interval! (Ex
tension of Weierstrass's Approximation Theorem to the
complex plane and to an infinite intt>rval), under the di
rection of Professor George P6lya.
Following the completion of her diploma as Pachlehrer
in mathematics and physics, Roth spent 10 years as sub
stitute teacher and Hiifslehrer at various middle schools in
Zurich and St. Gallen. She taught mathematics, physics,
Alice Roth during her graduate school years.
arithmetic, geometry, bookkeeping, business mathematics,
zoology, and anthropology. All but one of the schools (the
Freies Gymnasium in Zurich, where she was Hiifslehrer for
mathematics, April-December 1939) were girls' schools.
This was hardly a coincidence as we learn from the fol
lowing quote.
True coeducation, in the sense that at all levels a mixed
student body is taught by a mixed teaching staff, hardly
exists anywhere in Switzerland. As long as female teach
ers are almost completely excluded from the coeduca
tional higher level Primar-schule, the co-educational
Sekundarschule, and the Gymnasien, and as long as they
are relegated almost exclusively to being in charge of
girls' schools, in the interest of working women, we will
hardly be prepared to tum existing girls' schools into co
educational schools. [ 12, p. 399]
Her primary place of employment was at the school she
had attended, which was now renamed Tdchterschule der Sta.d/. Zurich. In April 1930, Alice Roth was elected Hilfs
lehnrrin fiir Buchhaltung, Rechnen, Mathema,tik und Geometri.e (temporary teacher of accounting, arithmetic,
mathematics, and geometry) in the Abteilung I. This part
of the school had more than 600 students-all girls-of
whom slightly less than half were in the Gymna,sium, and
a faculty of 36 male and 28 female teachers. This was the public school for academically inclined girls in Zurich. It
had high standards and an exceptional faculty. Many of the
faculty members had connections to the two universities
in the city, the ETH and the Universitat Ziirich.
With respect to the teaching staff, the following tradition
can be noted with satisfaction and approval: Time and
time again, the school gave young academics the oppor
tunity to fulfill their teaching obligations while preparing
themselves for work at the universities or to teach Uni
versity Extension classes. [30, p. 5 1 ]
Alice Roth seems to have been well integrated into the
faculty. She participated in social and recreational func
tions of the school. In the winter of 1932-33 she led 36 girls
to a ski camp in Arosa. One of her companions on this trip
(and the main leader) was Dr. Alfred Aeppli, a long-time
mathematics teacher at the school and a former Ph.D. stu
dent of Professor P6lya at the ETH. The following year, Al
ice Roth again helped lead a ski vacation week, this time
with Dr. Boller, another mathematics teacher and former
Ph.D. student of Professor P6lya. Alice Roth's colleague
and friend, Hanni Bretscher, was also a substitute teacher
at this school. After completing this stage in her education
and until the age of 35, Alice Roth still lived under her par
ents' roof.
In looking at the education of her colleagues, it must
havP seemed to Alice that she would be able to obtain a
satisfying permanent position teaching mathematics only if
she earned a doctoral degree. In fact, in the academic year
1932-33, of the 42 teachers at the Tochterschule, 32 had a
© 200!1 Spnngf!r Sr.•ence+ Bus1ness Mecha, Inc . Volume 27, Number 1. 2005 43
Ph.D. (1 1 of these were women). The remaining ten taught
non-academic subjects, except for one Latin teacher.
Roughly half of the Hiifslehrer had Ph.D.'s and, in light of
the above quote, we suspect that many of them were in the
process of earning one. Since Alice was an excellent stu
dent at the ETH, loved mathematics, and had a good rela
tionship with her diploma advisor, it is not surprising that
she decided to continue her studies.
Thus, while still teaching, she worked on her disserta
tion in function theory with P6lya once again as advisor.
Roth completed her thesis in 1938, becoming the second
woman to earn a Ph.D. in mathematics at the ETH. 1
Alice Roth's thesis, Approximationseigenschajten und Strahlengrenzwerte meromorpher und ganzer Funktionen (Properties of approximations and radial limits of
meromorphic and entire functions), was recognized as ex
cellent. In it, she answered a question suggested by P6lya
and Szego [20, volume 2, p. 33] , and much more. As Pro
fessor Heinz Hopf, in his report as co-referee, wrote,
In my opinion, both the main theorem, which is pre
sented first and which in such a nice and simple manner
characterizes radial limits, as well as the approximation
theorems I just described, which indicate the role of
fairly general point sets in the theory of analytic func
tions, are new, interesting, and important; and I consider
Fraulein Roth's achievement of having discovered and
proved these theorems truly laudable. The presentation
is also clear and lucid. I therefore recommend the work
for acceptance as a dissertation. [ 15]
Roth's thesis was singled out as worthy of special recog
nition by her advisor and Hopf. The university had a prize
that would allow them to recognize Roth's work, but it was
a prize that had never been awarded to a woman: the ETH
Silver Medal.
The medal has a curious past. In the beginning of the
school's history, prize questions were posed every year and
a sum was paid to students who answered them or made
significant progress toward a solution. At some point, it was
believed that a medal would be a more suitable reward.
On August 10, 1866, the Swiss school board decided to
replace the hitherto existing monetary prizes by medals.
The medal will be cast in gold and silver. The gold medal
is intended only for solutions of prize questions that are
in every respect worthy of the attribute "outstanding,"
while the silver medal together with a correspondingly
higher or lower additional payment will be provided for
the main prize or secondary prize. This modus operandi
was first used during the year under review. [3, p. 6]
The prize questions were no longer posed, and the medal
and the money of the Kern Stijtung were used to reward
outstanding Diplomarbeiten or dissertations. Over the
The Silver Medal of the ETH with the inscription: "FRAULEIN ALICE
ROTH VON KESSWIL THURGAU."
years, less than 1% of these were so honored. Incidentally,
in our studies of the Berichte des eidgenossischen Poly
technikums (later ETH), we were unable to locate an in
stance between 1870 and 1940 in which a student was
awarded a gold medal.
On July 14, 1938, the Conference of the Department of
Mathematics and Physics requested that a Kern prize of 400
francs plus the silver medal be awarded to Alice Roth for
an excellent doctoral thesis. Records of the deliberation
and decision appeared in Protokoll des Schweizerischen Schulratesfiir das Jahr 1938 [21 , pp. 310-3 1 1 ] , from which
we now quote:
In the spring of 1930, Fri. Roth was granted a diploma for
Subject Teacher in Mathematics with a grade point aver
age of 5.43; she received a grade of 5. 75 for her thesis. Her
doctoral thesis was examined by Professors Polya and
Hopf and was judged as outstanding. By motion of the pres
ident it was decided [that] . . . For the outstanding doctoral
thesis a sum of Fr. 400.-from the Kern foundation and the
silver medal of the ETH will be given to Fri. Dip!. Fachl.
Mathern. A. Roth of Kesswil (Thurgau). [21, S. 310-311 ]
For a young mathematician the atmosphere at the ETH
was exciting. During her studies, Roth took a course from
Wolfgang Pauli. Rolf Nevanlinna and Lars Ahlfors were
both at the ETH when she was completing her Diplomarbeit. George D. Birkhoff also appeared for a brief visit, and
Roth told the following story about his visit: "P6lya said
that if I sat next to Birkhoff and was a pleasant dinner com
panion, Birkhoff might help me get a job in the United
States. So I did, and P6lya got a job in the United States."
All the same, P6lya was a hero for Alice Roth. Unfortu
nately for her, he left the ETH for the United States the
same year she left Zurich. P6lya eventually took a position
at Stanford, where Roth and he were to meet again in the
early 1970s. They maintained contact over the years, and
met whenever P6lya and his wife visited Zurich, which was
quite frequently. Several of their letters to each other have
survived, and P6lya's final letter to Roth reached her shortly
before her death. (See [ 1 ] for more about P6lya.)
' Elsa Frenkel, who worked in geodesics, was the first woman to earn a Ph.D. in mathematics at the ETH. The Referent and Koreferent of her thesis were A. Wolfer
and A. Einstein, respectively [7, p. 31 ] .
44 THE MATHEMATICAL INTELLIGENCER
The Next 30 Years- Humboldtianum
Curiously, after her success at the ETH, in 1940 Alice took
a position at a private gymnasium in Bern, the Institu/. Humboldtianum, where she became Hauptlehrerin fur Darstellende Geometrie, Mathematik und Physik. This po
sition required her to work more hours than at a state
school, and she was less well-paid than her colleagues at
state schools. According to one student,
. . . in fact, the wages of a teacher at the Humboldtianum
are lower than the wages of a teacher at a public Gymnasium. The fact that the number of weekly hours of
teaching is at least 28, while teachers at a public gym
nasium teach at most 25 hours per week should also not
be overlooked. Therefore, it can be said that teachers in
private schools are rather badly off financially.
In this school it is difficult to cultivate a lively rela
tionship between students and teachers at the Gymnasium level (and also at other levels; for example, in the
trade school). Lack of time as well as stressful situations
(caused by the shorter preparation time of three years
for the Matura) prevent the realization of a relationship
beyond the traditional one of teacher as teacher and stu
dent as the subordinate. [ 16, Urs GrabPr, p. 102)
It is important to note that while students in the public
gymnasium in Bern are more or less unifornlly well-preparE>d
for study, students at the Humboldtianurn had extremely
varied background and ability. As one student wrote,
I was fascinated by the composition of the student body:
"Sons and daughters by profession," for whose parents
the private gymnasium was the last hope that their off
spring would reach an academic career; children of diplo
mats and Swiss abroad with an aura of distance around
them; young people with some sort of handicap who
were better accommodated in a private school than in a
public school; latecomers like myself who had at least
gone through an apprenticeship already and who knew
exactly what they wanted. [ 16, Rita Liitzelschwab
1951/52, pp. 106-107]
Fraulein Dr. Alice Roth with her godchild Verena Gloor at the Sech
seUiuten 1 939.
lnstitut Humboldtianum, Schlosslistrasse, Bern.
While Alice Roth appeared to be content in her work at
the Humboldtianum, in approximately 1959, she applied for
a position at the Stiidtisches Gymnasium in Bern, but her
application was unsuccessful. Roth (as well as many of her
colleagues) believed that had she not been a woman, she
would have had a chance at a better position. On the other
hand, one must note that the years in which Roth was look
ing for her teaching positions were not good years for
Switzerland.
The years 1933-1940 were labeled crisis years. In
Switzerland, there was also general unemployment and
a shortage of food. [30, p. 70]
As a teacher, Roth was very influential. She remained
close friends with some of her students long after they left
the Humboldtianum and, in some cases, until her death.
She was frequently lovingly referred to as ''Mammeli Roth" by her students, sometimes warmly referred to as
''Rotchiipph" (Little Red Riding Hood), and sometimes less
lovingly as ''Rotkappe" (Big Red Riding Hood). The students
repeatedly mention her ability and desire to explain things
many different ways. Though many students clearly ap
preciated her efforts in this direction, others report that she
would explain things several different times, in several dif
ferent ways, and often the fourth or fifth explanation was
so complicated that even those who understood at the be
ginning were confused. As one student described it,
Then there was the review in the subject of mathemat
ics. Here the "fairy-tale" atmosphere was literally not
missing, for we had ''Rotkappe, " namely Fri. Dr. Roth who
managed, not infrequently, to confuse us in the most lov
ing manner. [ 16, Erhard Erb und Gattin Vreni (Isen
schmid) 1952-54, p. 140]
In the words of a second student,
I do not know of a single Humber colleague who did not
adore Fraulein Dr. Alice Roth, our "Mammi," who un
fortunately died too early. In her lectures we started to
understand that mathematical problems could not only
be solved like this or that, but also in one way or another.
CO 2005 Spnnger Sc1ence 1 Business Med1a, Inc , Volume 27, Number 1, 2005 45
Whoever did not understand this and asked, received a number of other possibilities to choose from. Often, we feared that our dear "Mammi" would tangle up her arms while explaining-her gestures were so graphic. . . . "Mammi" always succeeded in hiding the intellectual woman behind a refreshing, natural, and cheerful personality. I always looked forward to her lectures or to a private visit with her. [ 16, Jiirg Scharer, 1965-67, preparation for agricultural studies at the ETH, pp. 157-160]
While by all accounts Roth was a much-loved, successful, and apparently happy teacher at the Humboldtianum, she also seems to have become more disillusioned with teaching as time went on. While clearly influenced by P6lya and his teaching methods, she also became frustrated by her lack of opportunity to implement them. In a letter to P6lya on the occasion of his 80th birthday (dated November 29, 1967) Roth wrote:
On the other hand, it certainly depresses me to think that I so inadequately follow your teaching methods; methods with which I have been familiar for such a long time. In particular, in more recent years I use so much time in my courses explaining important concepts and material for the exams to my students, most of whom were at most 1 1/2 years away from their examinations, that all too little time remains to correct the tendency of the less talented students to misconceive mathematics as simply an application of formulas. Unfortunately, unlike former times at the Humboldtianum, I am no longer given the opportunity to teach courses at the lower or middle level while retaining my "reduced" (but linked with so much work to correct) weekly teaching load of 24 hours. And right now, when it would be particularly important for me that the final years of my teaching be enjoyable, I am especially depressed by the disparity between effort and effect, whereby I am aware that this is not only caused by the exterior circumstances of our school, but is in some degree my own failure; on the other hand, there are, of course, several things about my school work that do not look so negative. Please excuse the personal outburst that is so completely out of place in a congratulatory letter, and that, at best, shows that aside from very successful students, you also have an aging student whose effectiveness is rather problematic. 2
It appears that during her employment at the Humboldtianum with such a heavy work load Roth had little chance to keep up with mathematics. However, as Roth put it, she always dabbled in mathematics. Friends suggest that it was the only way that she could deal with her disillusionment with her teaching at the Humboldtianum. Roth filled her life in other ways. She was an accomplished pianist, she enjoyed hiking and skiing, she took frequent trips, and as a result of her early training she was an excellent cook She was also a determined, complicated, and strong woman. She em-
Flowers for Dr. Alice Roth to celebrate 30 years at lnstitut Hum
boldtianum in 1970. On the left is colleague Hans Roder and on the
right Rektor Dr. Donald Keller.
phatically insisted on being called "Fraulein Dr. Roth" as opposed to "Frau Dr. Roth," which to a German speaker indicates that the degree is her own rather than that of her husband. Roth was a long-time friend of Marie Boehlen, a Bemese lawyer, family rights activist, suffragette, and soonto-be Grossrahn. And Roth herself was a strong supporter of women's right to vote. She often vented her frustration with a system that forced her to pay taxes, but allowed her no say in governance. Swiss women received the right to vote in 1971-the year of her retirement.
Retirement-A New Beginning
Alice Roth remained at the Humboldtianum until her retirement in 1971. It appears that shortly before retirement she had begun her transition back to work in mathematics. After announcing her plans to return to research to friends and relatives, she was told by one of them that in his field of medicine it would be impossible to return after so long an absence. Surely, most mathematicians would agree that it is impossible in the field of mathematics as well. After all, as G. H. Hardy said,
No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's gan1e . . . . Galois died at twenty-one, Abel at twentyseven, Ran1anujan at thirty-three, Riemann at forty. There have been men who have done great work a good deal later; Gauss's great memoir on differential geometry was published when he was fifty (though he had had the fundamental ideas ten years before). I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself. [ 13, p. 70]
And so Alice Roth would seem an unlikely candidate for success. Yet much had changed in the thirty years that she
·----· · ·-- · ----· -- --- -- ---- -- -
2ETH-Bibliothek Zurich, Hs 89:580/39.
46 THE MATHEMATICAL INTELUGENCER
had been teaching. In particular, Roth's area of research
begun over thirty years earlier-had become fashionable.
Alice Roth re-entered mathematical research with great
enthusiasm and pleasure. She could count on the help of
her former student at the Humboldtianum, Peter Wilker,
who had become a professor of mathematics at the Universitdt Bern. Though Wilker was in a different field of
mathematics, he helped her to obtain access to some of the
papers that she needed. A second mathematician who was
influential in Roth's new work was another student of
P6lya, Professor Albert Pfluger, who was now teaching at
the ETH.
For the first time, Roth was in the right place at the right
time. Her first work that appeared after her retirement
found its way to Paul Gauthier, then a young mathemati
cian at the Universite de Montreal. Impressed, Gauthier be
gan a correspondence with Roth. The two were highly com
patible and a close friendship developed. They began joint
work, which resulted in an invitation to lecture in Montreal.
At the age of 70, a very excited Alice Roth left for her first
mathematical trip outside Switzerland.
Roth also spent some time with her friend Hanni
Bretscher in her secluded mountain retreat in the southern
Swiss alps translating the Anhang from P6lya's book Math
ematics and Plausible Reasoning. She clearly enjoyed this,
as she describes to Albert Pfluger in a (much more upbeat)
letter dated October 31 , 1973:
We are having fun doing it, because we not only trans
late, we also discuss the mathematical content.:>
At last Alice Roth had time on her side and was able to
put her mathematical creativity to work. She was now "am chnobble" (pondering a problem) full-time, gave talks to
other mathematicians at universities, and made good
progress-at the cutting edge of contemporary mathemat
ics. But this happy last period of her life was cut short. In
1976 she began to suffer from an illness eventually diag
nosed as cancer. What did she manage to do in the very
short time she devoted to mathematics?
Approximation of All Continuous Functions
on a Closed Set
We must begin with a brief overview of complex approxi
mation theory. What follows is a "fusion" of the authors'
perspectives, but we believe it to be a good approximation
of Alice Roth's own mathematische Weltanschauung. We shall be dealing with complex-valued functions de
fined on subsets of the complex plane C All topological
notions (such as closure, etc.) are with respect to the com
plex plane except for a few instances, where we shall ex
plicitly state that we are dealing with the Riemann sphere
C We are interested in approximating a continuous func
tion on a compact set of the plane uniformly by polynomi
als in the complex variable z.
There are many sets on which polynomial approxima-
3ETH-Bibliothek Zunch, Hs 1 446: 1 75
Between Osco and Calpiogna in 1 968. Alice Roth (on the right) with
unidentified friend.
tion fails. For example, consider the functionj(z) = liz on
the unit circle. Now, suppose that for each natural number
n there is a polynomial Pn such that :j - Pn1 < lin on the
circle. Multiplying by z we have that I I - ZPn(z) l < lin for
each z on the unit circle. Applying the maximum principle
we see that the same inequality holds in the open unit disc
[D. In particular, for z = 0 we have 1 < lin, a contradiction.
Thus, polynomial approximation fails on the unit circle. We
tum now to cases in which it is possible to approximate
continuous functions by polynomials.
The most famous positive result in polynomial ap
proximation is the celebrated theorem of Karl Weierstrass
(1885), which states that on a closed and bounded inter
val of the real line, each continuous function can be ap
proximated uniformly by polynomials. In 1926, Joseph L.
Walsh [33] proved that in the Weierstrass theorem, we
may replace the closed interval by a compact Jordan arc
(homeomorphic image of a compact interval). In 1927,
Torsten Carleman [5] extended the Walsh, and hence the
Weierstrass, theorem: Let us define an unbounded Jordan
arc to be a homeomorphic image of the real line, such that
both "ends" tend to infinity. Carleman asserted that on an
unbounded Jordan arc, each continuous function can be
approximated uniformly by entire functions. To see how
Walsh's theorem follows from Carleman's theorem, take
a continuous function on a compact Jordan arc and ex
tend it to a continuous function on an unbounded arc.
Apply Carleman's theorem to obtain an approximating
function that is also entire. The polynomial required by
Walsh's theorem can now be obtained by taking partial
sums of the Taylor series that represents the entire func
tion. A particular case is that each continuous function
on the real line can be uniformly approximated by entire
functions.
In fact, this is the only case that Carleman actually
© 2005 Spnnger Sc1ence+ Bus1ness Media, Inc., Volume 27, Number I, 2005 47
proved. He left the general proof to the interested reader. In this case, it seems the reader was Alice Roth. The frequency with which Roth mentioned the work of Carleman in conversations, and the fact that she took over where Carternan left off, suggest that Carleman and his work may have been the initial source of inspiration that led to one of Alice Roth's major contributions, the extension of Runge's theorem (see below) to unbounded sets. Indeed, in the very first sentence of her dissertation, she proclaims that the theorem of Carle man is the Ausgangspunkt of her work.
It is not surprising that Roth would be attracted by Carternan's work, as his theorem has exciting consequences. For example, Carleman's theorem makes it relatively easy to construct an entire function such that the image of the real line under that function is dense in the plane: Let M denote a countable and dense subset of the complex plane, and consider a bijective mapping u : 1L � M. The function IP : � � C defined by
ip(x) = u(n + 1)(x - n) + u(n)(1 - x + n) (x E [n,n + 1), n E 7L)
connects all points of M by line segments and is obviously continuous. For f(x) = exp(x2)1P(x) there exists, by Carleman's result, an entire function F with IF(x) - f(x)l < 1 , and thus I F(x)exp( -x2) - ip(x)l < exp( -x2) for all x E �The function g(z) = F(z) exp( -z2) is entire, and we see from the last inequality that g(�) is a dense subset of the plane.
We now consider approximation by rational or meromorphic functions. In the previous paragraphs we mentioned that on a Jordan arc each continuous function can be approximated uniformly by polynomials. We also proved that polynomial approximation fails on a Jordan curve (homeomorphic image of a circle). In the same paper in which Walsh proved that one could approximate continuous functions on Jordan arcs by polynomials, he also showed that one can approximate continuous functions on Jordan curves by rational functions. A closer look at the history of rational approximation will lead us directly to Roth's work.
In 1931 , Friedrich Hartogs and Arthur Rosenthal proved the very nice result that one can approximate continuous functions on compact sets of Lebesgue measure zero by rational functions [ 14 ] .
In view of the Hartogs-Rosenthal theorem, one might believe that continuous functions on more general compact sets can always be approximated by rational functions. The next step would be to investigate nowhere dense sets of positive measure. This is precisely what Roth did in 1938, and the result was a crucial example of a compact set on which not every continuous function can be approximated uniformly by rational functions-the so-called Swiss cheese. Among counterexamples in the field of function algebras, the Swiss cheese is hard to match.
The set Roth considered is just slightly more complicated than the example that is now �nerally attributed to her: take K to be the closed unit disc ID with infinitely many
48 THE MATHEMATICAL INTELUGENCER
Figure 1 . Alice Roth's Swiss cheese.
holes tJ.1, where the tJ.1 are open discs in ID with pairwise disjoint closures such that the sum of their radii is less than 1 and the remaining set has no interior points.
To see that the Swiss cheese does what needs to be done, we show why the construction implies that there exists a function that is continuous on K but cannot be uniformly approximated by rational functions with poles off K.
Choose K so that 0 a= K and recall that the open discs tJ.1 were chosen so that the sum of their radii is less than 1 . From Cauchy's theorem, any rational function R with poles off K satisfies
t!= l R(z)dz = � L.J R(z)dz,
where all circles are positively oriented. Thus, if we can approximate a function g uniformly by rational functions with poles off K, the function must satisfy
J g(z)dz = L J g(z)dz. lz = l j <ll!.j
Now consider the function j(z) = lzl!z, which is continuous on K. For this function, flzl= l f(z)dz = 2m, while the integral 'i.1 fat>Jft:z)dz is less than 2 7T in modulus. Therefore the continuous function f cannot be uniformly approximated by rational functions with poles off K. (See Gaier [8, chapter 3, section 3] for more details.)
On an unbounded closed set, there is no hope of approximating every continuous function by rational functions, because rational functions are continuous at infinity (as functions from the Riemann sphere to the Riemann sphere). Thus, for example, it is impossible to approximate, on the entire real axis, the function sin x by rational functions. On unbounded sets, the proper generalization of rational functions are meromorphic functions, just as entire functions are the proper generalization of polynomials. In 1938, Alice Roth extended the Hartogs-Rosenthal theorem to unbounded sets by showing that, on closed sets of measure zero, one can approximate continuous functions by meromorphic functions. This result, interesting in itself, was a key ingredient in her generalization of the Runge theorem (which we will discuss below) to unbounded sets.
Approximation of All Holomorphic Functions
on a Closed Set
Runge's theorem on approximation by polynomials or ra
tional functions can be regarded as the starting point of
complex approximation. Carl Runge published it in 1885 (see [29]), the same year that the Weierstrass approxima
tion theorem appeared. When we speak of functions holo
morphic on a closed set, we will always mean holomorphic
on a neighborhood of the closed set.
Theorem 1 (Runge). Let K be a compact subset of C Then each function holommphic on K can be uniformly approximated by rational functions.
Furthermore, unifo'rrn approx·imation by polynomials is possible if and only if C\K is connected.
The following theorem of Alice Roth extends Runge's
theorem to unbounded closed sets. It gives approximations
by entire or meromorphic functions, and it was proved in
her thesis [24, Satz III and the Zusatz p. 1 10] .
Theorem 2 (Roth-Runge theorem). Let E be a closed subset of the complex plane. Then each function holomorphic on E can be uniformly approximated by .functions merommphic on C
Furthermore, U1!:_iform approximation by entire functions is possible ifC \E is connected and locally connected in C
It was Roth's discovery that the topological condition
just mentioned is sufficient for uniform approximation by
entire functions. This was also independently introduced
one year later by M. Keldysh and M. Lavrentieff. In fact, the
requirement that the complement be connected and locally
connected is also necessary for approximation of this type,
as in Arakelian's theorem (below). We now turn to an ap
plication of the Roth-Runge theorem that is related to yet
another result of Roth.
In 1925, George P6lya and Gabor Szego published their
famous exercise book [20] ; a book that was to inspire much
research in analysis in the decades to come. In her thesis
[24, §4], Roth completely answered a question arising from
[20, vol. 2, Abschnitt IV, Nr. 187, p. 33 and p. 2 12] . Let F(z) = F(rei<P) be an entire or meromorphic function
with the property that the limit (which she called the
Strahlengrenz1nert)
f(ei<P) : = lim F(rei<P) 1"------).X
exists (where oo is allowed) for all r.p. Alice Roth found3 characterization of the corresponding functions.f : iJ[]) � C
The Roth-Runge theorem can be used to obtain an ex
ample of an entire function F such that the corresponding
functionf has the property thatf(e; '�') = 1 for all r.p but F is
not constant. This is in contrast to Liouville's theorem,
which says that every bounded entire function must be con
stant. We turn to the creation of this "unusual" function F. So let E be the closed set indicate_<"! in Figure 2, and let
A = E U [ 0, 1 } . Then A� closed, and IC\A is connected and
locally connected in C Consider a function g holomor-
E
Figure 2. The closed set E.
phic on (a neighborhood of) A such that g(z) = z on E U [O J and g(l) = 2. By the Roth-Runge theorem, we can ap
proximate this function uniformly on A by entire func
tions. In particular, there exists an entire function h such
that h(z) - g(z)l < 112 on A. Now define a function F by
F(z) = (h(z) - h(O))Iz. Then F is entire, and we will show
that it has Strahlengrenzwert 1 everywhere and is non
constant.
By our construction, g(O) = 0 and on E U [ 0} we have
jh(z) - z l = h(z) - g(z)l < 112. So for z E E we have
jh(z) - z - h(O)j F(z) - 1 = lz,
jh(z) - g(z) + jh(O) - g(O)j :::; lz1
Now, for each r.p E [0,2 7T) there is some r0(r.p) such that
rei<P E E for r 2:: r0(r.p). Therefore, f(r.p) = lim,._,x F(rei'�') =
1 for all r.p. Now, if F were constant, then it would have to be the
constant function 1; in other words,
1 = F(z) = (h(z) - h(O))Iz.
In this case we have h(l) = 1 + h(O), and consequently
!h(l) - 21 = jh(O) - 1 1 2:: 1 - h(O)! > 1/2.
But we know that ih(l) - 2 = jh(l) - g( 1) < 112, and this
establishes the contradiction. Therefore, the function F is
nonconstant.
Intermezzo
While Alice Roth taught, research in her area continued.
The Swiss cheese was rediscovered by Mergelyan and was
"known affectionately as Mergelyan's Swiss Cheese." [34, p. 69] Mergelyan's name was frequently found attached to
the Swiss cheese up until about 1968 and occasionally even
later (e.g., [32] and [6]). As E. L. Stout writes in his Mathematical Review of Vitushkin's 1975 [32] paper,
It should be noted that the author attributes to Mergelian
the first example of a nowhere dense compact set E C C for which C(E) =F R(E). This is an unfortunate though com
mon error. Alice Roth gave an example of such a set in
1938 [24].
© 2005 Spnr1ger Sc1ence 1 Bus1ness Media, Inc , Volume 27. Number 1 , 2005 49
Apparently, as Roth began working her way back into mathematics, she found Zalcman's notes [34] and recognized Mergelyan's example as her own. By 1969 the error was corrected [9] , and most mathematicians felt the name "Swiss cheese" could not have been more appropriate.
Roth's past as well as future work was to have a strong and lasting influence on mathematicians working in this area. Her Swiss cheese has been modified (to an entire variety of cheeses); see e.g., Gaier [8, pp. 103-106] or Gardiner [ 10] . We now tum to Roth's fusion lemma, which appeared in her 1976 paper [27] and influenced a new generation of mathematicians worldwide.
Roth's Fusion Lemma
Let K1 and K2 be disjoint compact subsets of C, and assume that the rational functions r1 and r2 are close on some compact set K, in the sense that 1r1 - d is small on K. Is there a rational function r that approximates r1 on K1 U K and simultaneously approximates 1·2 on K2 U K? Of course, such an r cannot approximate both r1 and r2 on K much better than r-1 and r2 approximate each other on this set.
If K1 U K2 U K has a decomposition into two disjoint compact sets, with one containing K1 and the second containing K2, then this problem can be solved by Runge's theorem. But, if K is a "bridge" connecting K1 and Kz, as in Figure 3, then Runge's theorem will not answer this question for us.
Before turning to the solution, we consider the question from a real perspective. Consider the case in which we choose real intervals for the compact sets Ki> K2, and K and real functions r·1 and r-2, as in Figure 4.
The function r that we wish to find should approximate r1 on K1 U K and rz on K2 U K, and the error bounds, lh - �IK1 uK and l lr-z - �IKzuK, should both be close to lh - r2IIK· Even in this real case, it is not altogether obvious that such a rational function exists. However, the existence of such an approximation is guaranteed by Roth's fusion lemma [27].
Lemma 3 (Fusion of rational functions). Let Ki> K2, and K be compact subsets of the extended plane with K1 and K2 d·isjoint. If r-1 and r2 ar-e any two r-ational functions satisfying, for some E > 0,
hCz) - r-2(z)! < E, .for z E K,
then the1·e is a positive number a, depending only on K1 and K2, and a r-ational function r- such that for-j = 1, 2,
' I r(z) - rj(z) < aE, for z E Kj U K.
Figure 3. K is a bridge between K1 and K2•
50 THE MATHEMATICAL INTELLIGENCER
-·----·-· rl - - - - - - r2 -- r
Figure 4. Fusing r1 and r2 with r.
The fusion lemma is a powerful tool that can be used to extend approximation on compact sets to obtain results on closed sets. The proof is based on techniques used in studying smooth (not necessarily analytic) functions. The socalled Pompeiu formula for differentiable functions with compact support plays an important role. This formula is similar to Cauchy's integral formula, but there is an extra term reflecting the "extent" of non-analyticity in terms of the Cauchy-Riemann operator. The miracle that allows Alice Roth to obtain results concerning meromorphic functions from these differentiable techniques is an implicit use of a clever trick that has become more and more fashionable, namely solving a non-homogeneous Cauchy-Riemann equation. In this way, she is able to get rid of the non-analytic part that arises from the Pompeiu fommla.
Roth used her fusion lemma to prove Bishop's localization theorem on compact sets. To see how this goes, let f be a continuous function defined on a compact set E. The localization theorem of Errett Bishop [4] states that f can be approximated by rational functions if and only if for each point z E E there is a closed disc Kz centered at z such that the restriction off to E n Kz can be approximated by rational functions. Of course, one direction of this theorem is obvious. In [27] , Alice Roth showed how to derive the (interesting) half of Bishop's theorem from her fusion lemma. The idea is quite simple:
Having chosen the sets Kz as above, return to each point z E E and choose a disc kz that is again centered at z, but has half the radius of the disc Kz. Since the set E is compact, we can find finitely many of the smaller discs, kzp . . . , kzp' such that E is contained in the union of these p sets. We show how to obtain Bishop's localization theorem from the fusion lemma by using each of the larger sets, KzJ n E, to find an approximating rational function Tj and then fusing the smaller sets kzj n E to obtain one rational function approximating f on all of E.
So suppose that we have a compact subset F1 of E on which we can approximate F uniformly and we also have a disc kzJ from above. (In the first step F1 = kz1 n E and kzJ = kz2, but in the steps thereafter, F1 is simply a compact set on which uniform approximation is possible, and kzJ is a disc we have not yet "fused" into FJ.) If F1 and kzJ happen to be disjoint, Runge's theorem will imply that there is a rational function that approximates f on F1 U (kzJ n E), and the interested reader can find the details in [8, p. 1 14].
If F1 n kzj =F 0, then we must use the fusion lemma, which we do as follows:
Let SJ denote the radius of the larger disc, KzJ' and consider the three compact sets E1 = F1 n [w: lw - Zjl 2: 3sj l4 l , E2 = kzj n E = (w E E : w - zj s sj 12 ) and K = F1 n K,j = F1 n [w: w - zjl s sj). Then E1 and E2 are disjoint and, for future use, we note that F1 U (kzj n E) = E1 U
E2 U K. Given E > 0, by our assumption we can approximate our
function.fon F1 to within El4 by a rational function r1 • Now K,j was our bigger disc on which approximation was possible. Therefore, we can find a rational function r2 approximating/uniformly to within E/4 on K::j n E. Now each of these rational functions is within E/4 off on F1 n K::j = K, so r1 - r2 < El2 on K. By the fusion kmma, there is a constant a (depending on E1 and E2 alone) and a rational function r such that for m = 1,2 we haVf• r - 1"111 1 < aE on
E111 U K. Since f\ U (k::j n E) = E1 U E2 U K, we have r
.f ! s r - 1"11,' + rm -I I < aE + E on F1 U (kzj n E). Furthemwre, E is arbitrary, so we are able to approximate.( on the set F1 U (kzj n E), and the fusion of F1 and kzj n E is complete. To finish the proof, add one disc to the set obtained from the previous step and repeat the argument above until all sets have been fused.
In 1976 Roth [27], via her fusion lemma, also extended the Bishop localization lemma to unbounded sets to obtain Roth's localization theorem on closed sets. Namely, she showed that if f is a function defined on a closed set E, then f can be approximated by meromorphic functions if and only if for each point p in E, there is a closed disc K1, centered at p such that the restriction off to E n Kp can be approximated by rational functions. Again, one direction is obvious. But the other direction had profound consequences as Alice Roth noted. We present these in the following section.
Complete Solution for the Class of Plausibly
Approximable Functions
In earlier sections we attempted to approximate all functions belonging to a natural class of functions on E, namely, the class of functions continuous on E or the class of functions holomorphic on E.
Becausl:' the uniform limit of continuous functions is continuous and the uniform limit of holomorphic functions on an open set is holomorphic, it follows that if a particular function f can be approximated on a sl:'t E by entirl:' functions or by meromorphic functions having no poles on E, then nt>cessarily f is continuous on E and holomorphic on the interior of E. Hence, the most natural class of functions to try to approximate on a set E is tlw class A(E) of functions continuous on E and holomorphic on the interior of E. This class combines the attributes, continuity and holomorphy, of the two cla.•;;st>s we considered earlier, in just the right dosage. We will refer to ACE) as the cla.'>s of "plausibly" approximable functions. The most natural question on approximating a class of functions is then to dt>termine those sets E for which the plausibly approximable functions are in fact approximable. For approximation by
entire functions, this question was answered in 1964 by Norair Arakelian [2].
Theorem 4 (Arakelian). Let E be a closed set in C. The following are equivalent:
(1) Each function continuous on E and holomorphic on the interior of E can be uniformly approximated by entire functions.
(2) The complement of E in C is connected and locally connected.
In 1972, Ashot Nersessian answered this question for meromorphic approximation [ 19].
Theorem 5 (Nersessian). Let E be a closed set in C. The following are equivalent:
(1) Each function continuous on E and holomorphic on the interior of E can be uniformly approximated by functions meromorphic on C .
(2) For each closed disc K, each function continuous on E n K and holomorphic on the interior of E n K can be unUonnly approximated b:IJ rational functions.
As fundamental as these two theorems are, giving a characterization of those closed sets E on which the class of all plausibly approximable functions can be approximated, a still more fundamental question is to decide when an individual function can be approximated. Roth's localization theorem does this, and is so powerful that it yields the sufficiency (2 implies 1) in both of the preceding theorems. In the cast> of Nersessian's theorem, this is obvious.
We now derive the sufficiency in Arakelian's theorem. Suppose, then, that the complement of E in C is connected and locally connected, and let f be continuous on E and holomorphic on the interior of E. Let K be any closed disc. The complement of the compact set E n K is connected and so, by a famous theorem of Mergelyan [ 18], each function in A(E n K), in particular, the restriction off to E n K, can be uniformly approximated by polynomials. From the Roth localization theorem, for each E > 0, there is a function h meromorphic on C such that .f - hi < E/2 on E. By the Roth-Runge theorem (Theorem 2), there is an entire function g such that .h - gl < El2 on E. From the triangle inequality, If - g < E on E, which gives the sufficiency in the theorem of Arakelian.
The theorems of Arakelian and Nersessian are best possible (in the sense indicated), but for most applications the analogous Runge-type theorems of Alice Roth suffice.
In her second period of creative mathematical work, after retirement, Alice Roth extended to arbitrary plane domains her thesis results on holomorphic and meromorphic approximation on closed subsets of the plane. Over the past 30 years, attempts were made to further extend these results and in particular to Riemann surfaces. These attempts were successful with respect to extending fusion on compact sets, but attempts to approximate on closed subsets of Riemann surfaces encountered major obstacles still not resolved. On the other hand, Roth's work inspired highly successful research in potential theory. At the present time,
ID 2005 Spnnger Sc1ence • Bus1ness Media, Inc , Volume 27. Number 1, 2005 51
advances are also being made on such questions for solutions of differential operators other than the Cauchy-Riemann and Laplace operators.
Looking Back at Roth's Work and Life
Alice Roth's 1938 thesis was largely overlooked, but by 1969 mathematicians were learning that the Armenian "Swiss cheese" was, in fact, discovered by a Swiss woman. As V. P. Ravin writes in his Mathematical Review of Roth's 1973 paper Meromorphe Approximationen:
In 1938, the author published an article that contains important ideas of the later theory of approximation by rational functions. We mention only the first example of a nowhere dense compact set E c C on which not every continuous function can be approximated uniformly by rational ones (this set was named the "Swiss cheese" much later) or the formulation of questions that anticipated the later excellent work (by Arakelian, for example) on tangential approximation (in the sense of Carleman). This earlier work seems to have fallen into almost complete oblivion, a common (but unfair) punishment for anyone who dares to enter a subject too early, without waiting until it is accepted an1ong experts and has even become fashionable.
All of Alice Roth's publications are cited in our references [ 1 1,22,23,24,25,26,27,28] . A good account of her work and influence in approximation theory appears in Gaier's book [8].
As her friend and former student at the Humboldtianum, Prof. Peter Wilker, wrote in an obituary in the Bernese newspaper Der Bund on July 29, 1977,
In Switzerland, as elsewhere, women mathematicians are few and far between . . . Alice Roth's dissertation was awarded a medal from the ETH, and appeared shortly after its completion in a Swiss mathematical journal . . . . One year later war broke out, the world had other worries than mathematics, and Alice Roth's work was simply forgotten. So completely forgotten, that around 1950 a Russian mathematician re-discovered sinlilar results without having the slightest idea that a young Swiss woman mathematician had published the same ideas more than a decade before he did. However, her priority was recognized.
Alice Roth was hospitalized in Bern in 1977. Although she was very ill, she continued to work on mathematics. She completed her work on her final paper Uniform Approximation by Meromorphic Functions on Closed Sets with Continuous Extension into the Boundary in the Inselspital in Bern, and Wilker helped her with the English translation as well as the typing. As Wilker continued,
Alice Roth died a mathematician . . . As her last work lay finished before her, she was as pleased with it as she was
4Coined by Jean-Paul Berrut. Universitat Freiburg, Switzerland.
52 THE MATHEMATICAL I NTELLIGENCER
Fraulein Dr. Alice Roth, 1 975 (photo by A. Ballinger-Roth).
with the many flowers that decorated her room, and if someone had asked her whether or not an obituary for her were justified, she surely would have said, "Don't make a fuss about me-but my approximation [by] merom orphic functions on closed sets, about those you may write a story."
Her last paper arrived at the Canadian Journal of Mathematics on July 1 1, 1977. Alice Roth, "die bekannteste unbekannte Schweizermathematikerin,"4 died on July 22, 1977. She is buried in the Roth-Landolt family grave at Friedhof Bergli, in the medieval Swiss town Zofingen.
Acknowledgments
We thank Professors Jiirg Ratz and Hans-Martin Reimann for their assistance as well as encouragement. We are also particularly grateful to Judith Fahrlander for library services customized to our needs, and to the friends and family of Alice Roth, on whom we depended so much for information, recollections, and photographs: Verena and Alfred Ballinger-Roth, Deli Roth, Verena de Boer, Johanna Meyer, Roland Weisskopf, and Irene Aegerter. We thank each of these people for their generosity and willingness to assist us. We are also grateful to the Universitat Bern for its support, to the Gosteli Archiv and Angela Gastl at Archiv der ETH Zurich for their assistance, and to Professor David Coyle for carefully reading t11e manuscript and for helpful suggestions. Aside from the photo of Schloss Ralligen, all photos are in the possession of the Roth family, except the
picture with her godchild, which is from Verena De BoerGloor. We thank both parties for allowing us to reprint
them.
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und ganzer Funktionen. ETH-Bibliothek Zurich, Hs 620 : 1 07 , 9. VI I .
1 938.
[1 6] 75 Jahre Humboldtianum Bern: zum Geburtstag . Bern, 1 979.
[1 7] Jahresbericht der Hoheren Tochterschule der Stadt Zurich. Zurich,
1 925/26--1 927/28.
[ 1 8] S. N. Mergelyan. On the representation of functions by series of
polynomials on closed sets. Ooklady Akad. Nauk SSSR (N. S.) ,
78:405-408, 1 951 . English translation: Amer. Math. Soc. Transla
tion 1 953 (1 953), no. 85, 8pp.
[1 9] A. H. Nersessian. Uniform and tangential approximation by mero
morphic functions. lzv. Akad. Nauk Armyan. SSR Ser. Mat . ,
Vll :405-4 1 2 , 1 972. (Russian).
[20] G. P61ya and G. Szegb. Aufgaben und Lehrsatze aus der Analy
sis . Springer-Verlag, 1 925. 2 Bande.
[2 1 ] Protokoll des Schweizerischen Schulrates fUr das Jahr 1 938.
Zurich, 1 938. Archiv der ETH Zurich, SR 2 .
[22] A. Roth. Ausdehnung des Weierstrass'schen Approximations
satzes auf das komplexe Gebiet und auf ein unendliches Interval!.
Diplomarbeit, ETH , Abteilung fUr Fachlehrer in Mathematik u.
Physik, Zurich, 9 . November 1 929.
[23] A. Roth. Uber die Ausdehnung des Approximationssatzes von
Weierstrass auf das komplexe Gebiet. Verhandlungen der
Schweizer. Naturforschenden Gesel/schaft, page 304, 1 932.
[24] A. Roth. Approximationseigenschaften und Strahlengrenzwerte
meromorpher und ganzer Funktionen. PhD thesis, Eidgen6ssische
Technische Hochschule, Zurich, 1 938. Separatdruck aus Com
ment. Math. Helv. 1 1 , 1 938, 77-1 25.
[25] A. Roth. Sur les limites radials des fonctions entieres (presentee
par M. Paul Montel) . Academie des Sciences, pages 479-481 , 1 4
Fevrier 1 938.
[26] A. Roth. Meromorphe Approximationen. Comment. Math. Helv. ,
48: 1 51 -1 76, 1 973.
[27] A. Roth. Uniform and tangential approximations by meromorphic
functions on closed sets. Canad. J. Math. , 28(1 ) : 1 04-1 1 1 , 1 976.
[28] A Roth. Uniform approximation by meromorphic functions on
closed sets with continuous extension into the boundary. Canad.
J. Math . , 30(6) : 1 243-1 255, 1 978.
[29] C. Runge. Zur Theorie der eindeutigen analytischen Funktionen.
Acta Math. , 6:229-244, 1 885.
[30] 1 00 Jahre Tochterschule der Stadt Zurich. Schulamt der Stadt
Zurich, Zurich, 1 975.
[31 ] Verein Feministische Wissenschaft Schweiz. Ebenso neu als kuhn,
120 Jahre Frauenstudium an der Universitat Zurich . eFeF-Verlag,
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[32] A G. Vitushkin. Uniform approximations by holomorphic functions.
J. Functional Analysis , 20(2): 1 49-1 57, 1 975.
[33] J . L. Walsh. Uber die Entwicklung einer Funktion einer komplexen
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[34] L. Zalcman. Analytic Capacity and Rational Approximation. Lec
ture Notes in Mathematics, No. 50. Springer-Verlag, Berlin, 1 968.
© 2005 Springer SC1ence+ Bus1ness Media, Inc , Volume 27, Number 1, 2005 53
A U T H O R S
Left: Paul Gauthier (with hat) and Gerald Schmieder. Right: Pamela Gerkin and Ul
rich Daepp.
ULRICH DAEPP
Department of Mathematics
Bucknell University
Lewisburg, PA 1 7837
USA
e-mai l : [email protected]
Ulrich Daepp was a student at the Sttidtisches Gymnasium Bem
and could have had Alice Roth as a teacher, had they hired her
when she applied. He crossed the Atlantic Ocean in his earty twen
ties and completed his Ph.D. in algebraic geometry at Michigan
State University. He is now at Bucknell University in Pennsylvania,
where he has several joint projects with the third author, the most
important being their two children.
PAMELA GORKIN
Department of Mathematics
Bucknell University
Lewisburg, PA 1 7837
USA
e-mail: [email protected]
Pamela Gorkin received her Ph.D. from Michigan State Univer
sity. She also studied at Indiana University for one year, and it was
during this year that she first heard about the Swiss cheese and
Alice Roth. Upon completing her doctorate, she began teaching
at Bucknell University. She has been there ever since with the ex
ception of three sabbaticals, all of which have been spent at the
University of Bern, Switzerland. Her mathematical interests are pri
marily function theory and operator theory. She enjoys watching
films, learning languages, cooking, and eating.
54 THE MATHEMATICAL INTELLIGENCER
PAUL M. GAUTHIER
Department of Mathemat ics and Stat ist ics Universite de Montreal
Montreal H3C 3J7
Canada
e-mail: [email protected]
Paul Gauthier was born and raised in the United States of French
Canadian parents and, after completing his studies, chose to re
turn to the "old country" (Quebec) for his career. He has six chil
dren, and with his family has spent two years in the Soviet Union
and one year in China. His hobby is singing. Alice Roth was one
of Paul's most beloved friends and (to his surprise) he was men
tioned in her will. One of his children was named after Alice and,
amazingly, only in writing this biography did he learn that she has
the same birthday as Alice Roth.
Publishing this, his first, paper in The lntelligencer fulfills an am
bition of many years: to catch up with another of his daughters,
who published in The lntelligencer when she was still in grade
school (see volume 1 8, no. 1 , p. 7).
GERALD SCHMIEDER
Falkultat V. lnstitut fOr Mathematik
Universitat Oldenburg
261 1 1 Oldenburg
Germany
e·mail : GSchm [email protected]
Gerald Schmieder was born in Bad Pyrmont, Germany, and stud
ied mathematics and physics in Hannover. He works on complex
approximation theory, Riemann surfaces, and geometric function
theory. His hobby is playing violin, mainly string quartets.
ll"@ii!i§j.fiii£11=tfii§#fii,j,i§.Jd M i chael Kleber and Ravi Vaki l , Editors
Gold bug Variations Michael Kleber
This column is a place for those bits of
contagious mathematics that travel
from person to person in the
community, because they are so
elegant, suprising, or appealing that
one has an urge to pass them on.
Contributions are most welcome.
Please send all submissions to the
Mathematical Entertainments Editor.
Ravi Vakil, Stanford University,
Department of Mathematics, Bldg. 380,
Stanford, CA 94305-21 25, USA
e-mail: [email protected] .edu
J im Propp bugs me sometimes. I'm usually glad when he does.
Today, Jim's bugs are trained to hop
back and forth on the positive integers: place a bug at 1, and with each hop, a bug at ·i moves to either i + 1 or i - 2. Of course, it might jump off the left edge; we put two bug-catching cups at 0 and - 1. Once a bug lands in a cup, we start a new bug at 1 .
What I haven't mentioned is how the bugs decide whether to jump left or right. We could declare it a random
walk, stepping in either direction with
probability 1/2, but the U. of Wisconsin
professor's bugs are more orderly than that. At each location i, there is a signpost showing an arrow: it can point either Inbound, toward i - 2 and the bug
cups, or Outbound, toward infinity. The bugs are somewhat contrarian, so when a bug lands at i, it first .flips the arrow to point the opposite direction, and then hops that way (Fig. 1). Add
an initial condition that all arrows begin pointing Outbound, and we have a deterministic system. Let bugs hop till they drop (Fig. 2).
Well, what happens? Or, for those
who would like a more directed question: First, show that every bug lands in a cup (as opposed to going off to infinity, or bouncing around in some bounded region forever). Second, find what fraction of the bugs end their journeys in the cup at - 1 , in the manybugs limit. Go ahead, I'll wait. Do the first ten bugs by hand and look for the beautiful pattern. You can even skip ahead to the next section and read about another related bug, one with far more inscrutable behavior, and come back and read my solution another day.
I should mention, by way of a delaying tactic, that the analysis of this
bug was done by Jim Propp and Ander Holroyd. A previous and closely related
bug of theirs, in which every third visit
to i leads to i - 1, appears in Peter
Winkler's new book Mathemal'ical Puzzles: A Connoisseur's Collection, yet another delightful mathematical of-
Figure 1 . The two bug bounces.
fering from publishers A K Peters [ 14]. The book itself is a gold-mine of the type of puzzles that I expect readers of this column would enjoy immensely.
Winkler's solutions are insightful, wellwritten, and often leave the reader with more to think about than before. The preceding is an unpaid endorsement.
Very well, enough filler; here's my answer. If you solved the problem without developing something like the
Figure 2. The bounces of (a) the first bug, (b)
the fourth bug.
© 2005 Spnnger Sc1ence+Bus1ness Media, Inc , Volume 27. Number 1. 2005 55
theory below, please do let me know
how.
Before we get to the serious work, let's answer the question, can a bug hop around in a bounded area forever? It cannot: let i be the minimal place vis
ited infinitely often by the bug-oops, half the time, that visit is followed by a trip to i - 2, a contradiction. So each bug either lands in a cup or, as far as we know now, runs off to infinity.
To motivate what follows, let's have a little inspiration: some carefully chosen experimental data. Perhaps you noticed that of the first five bugs, the cup at - 1 catches three, while the cup at 0 catches two. If that is not sufficiently suggestive, let me mention that after bug eight the score is 5 : 3, while after bug thirteen it is 8 : 5. On the speculation that this golden ratio trend continues, let us refer to the bouncing insects as goldbugs.
Now the details. Suppose that r.p isI can only imagine your surprise-a real number satisfying r.p2 = r.p + 1 ; when I care to be specific about which
root, I will write 'P± for (1 ± V5)/2. The goldbugs and signs are, in fact,
a number written in base r.p. Position i has "place value" r.pi, and the digits are conveniently mnemonic: Outbound is
0, Inbound is 1, and the bug itself is the not entirely standard digit r.p, which may appear in addition to the 0 or 1 in the "r.pi's place," where it contributes r.pi+ 1 to the total. Of course, numbers do not have a unique representation with this set-up, but that's quite deliberate: the total value is an invariant, unchanged by bug bounces.
• Bounce left: Suppose the bug arrives in position i and there is an Outbound arrow. The value of this part
of the configuration is ( r.pi X r.p) + ( r.pi X 0). After the bounce, position
i holds an Inbound arrow and the bug has bounced to position i - 2, for a total value of ( r.pi X 1) + (r.pi-2 X r.p). And these are the same,
since 'Pi- l + 'Pi = 'Pi+ 1 .
• Bounce right: Suppose the bug arrives in position i and there is an Inbound arrow. The value before the bounce is (r.pi X r.p) + (r.pi X 1) . After
the bounce, the arrow points Outbound, value zero, and the bug is at
56 THE MATHEMATICAL INTELLIGENCER
i + 1 , for a value of r.pi+2, again un
changed.
Now let's see what happens when we add a bug at 1, let it run its course, and then remove it after it lands in a bug cup. Placing a bug at 1 increases the system's value by r.p2. If it eventually lands in the cup at 0 and is then removed, the value
of the system drops by r.p, while if it lands in the cup at - 1 and is removed, the value of the system drops by r.p0
=
1. So the net effect of adding a bug at 1 , running the system, and then removing the bug is that the value of the system increases by <f!- - r.p = 1 for cup 0, and by r.p2 - 1 = r.p for cup - 1 .
Now we are well-equipped to answer the original questions, and then some.
• Can a bug run off to infinity? It can
not: if we take r.p to mean 'P+ =
1 .61803, then each bug can increase the net value of the system by at most 'P+. and the positions far to the right are inaccessible to the gold
bugs, because they would make the value of the system too high. So every bug lands in a cup.
• What is the ratio of bugs landing in the two cups? This time for r.p, think about 'P- = - .61803. Between bugs, when the system consists of just Inbound and Outbound flags, its minimum possible value is r.p1
_ + r.p3 _ +
r.p5 _ + · · · = - 1, while its maxi
mum possible value is <f!-_ + r.p4 _ +
r.p6 _ + · · · = - 'P- = .61803, which I
will write instead as 1/ 'P+ to help us
remember its sign. So the value of the system is
trapped in the interval [ - 1, 11 'P+] , and as each successive bug passes through the system, the value either increases by 1 , or else increases by 'P-, i.e. decreases by li'P+ · For the
value to stay in any bounded region, it must, in the limit, step down 'P+ times as often as it steps up, and so
the bugs land in the - 1 cup 1/r.p+ of
the time. Moreover, this approxima-
tion is very good: if a bugs have
landed at - 1 and b have landed at 0, the system's value b - a/r.p+ must lie in our interval, so lb!a - 11'P+ i < 11a. Every single approximation bla is one of the two best possible given the denominator, and that denominator grows as nlr.p+.
In fact, notice that the length of the interval [ - 1, 1lr.p+ ] is the sum of the two jump sizes, the smallest we could possibly hope for. If the value lies in the bottom 11r.p+ of the interval, it must increase by 1 , while if it lies in the top 1 of the interval, it must decrease by 11'P+ · This leaves a
single point, 1lr.p+ - 1 = - .38297, where the bug's destination cup isn't determined. But that value can be attained only with infinitely many Inbound signs: if we run a single bug
through a system with that starting value, its ending value would be either 11 'P+ or - l-and to attain those two bounds, we found above, you
must sum the infinite series of all positive or negative place values. So
for any initial configuration with only finitely many signs pointing Inbound, the configuration's numerical value alone determines the destina
tion cup of every single bug; you don't need to keep track of the arrows at all.
• If you prefer integers to irrationals, consider instead the following in
variant. Label the (bug cups and the) sites with the Fibonacci numbers (0, 1), 1 ,2,3,5,8, . . . , as in Figure 3. Give an Inbound arrow the value F; of its site, and give the bug there the value F; + 1 of the site to its right. This invariant, of course, is an appropriate linear combination of the 'P+ and 'P- ones above.
Adding a bug to the system now
increases the value by 2, but what's special is that removing the bug at either 0 or - 1 subtracts 1. So the bugs
implement an accumulator: after the
nth bug lands in its cup, we can look
u u 1)
0 I 0 I I 0 I I
(0 1 <--- ----->
2 3 <--- <--- <--- ----->
5 8 13 21 34 55 Figure 3. The signs after bug 1 1 7 passes through the system; 1 1 7 = 2 + 5 + 8 + 1 3 + 34 + 55.
Representations in base Fibonacci are not unique; ours is characterized by a lack of two con
secutive zeros.
at the signs it has left behind, and read off the number n, written in base Fibonacci!
. . . Hold the presses! Matthew Cook has pointed out to me that this point of view can be taken further. We still get an invariant if we shift all those Fibonacci labels one site to the right. Now adding a bug increases the system's total value by 1, and removing it from the right bug cup decreases the value by 1-but removing it from the left bug cup decreases the score by 0. So after n bugs pass through, the value of the Inbound arrows they leave behind counts the number of bugs that ended their journeys in the left bug cup.
Furthermore we can shift the labels right a second time. Now when the bug lands in the left cup, its value must be the - 1st Fibonacci number, before 0--which is 1 again, if the Fibonacci recurrence relation is still to hold. So with these labels, the Inbound arrows will count the number of bugs which landed in the right cup. As an exercise, decode the arrows to learn that the 117 bugs leading to Figure 3 split 72:45.
Once we know that the same set of arrows simultaneously counts the total, left-cup, and right-cup bugs, it's straightforward to see that the ratios of these three quantities are the same as the ratios of three consecutive Fibonacci numbers, in the limit.
These arrow-directed goldbugs are doing a great job of what Jim Propp calls "derandomization." It's straightforward to analyze the corresponding random walk, in which bugs hop left or right, each with probability 1/z: Let Pi be the probability that a bug at place i eventually ends up in the cup at - 1, and solve the recurrence Pi = CPi-2 + Pi+l)/2 with P-1 = 1, Po = 0-oh, and make up for one too few initial conditions by remembering that all the Pi are probabilities, so no larger than 1. Here too we get p1 = 1/'P+·
So the deterministic goldbugs have the same limiting behavior as their random-walking cousins. But if we run the experiment with n random walkers and count the number of bugs in a cup, we'll generally see variation on the or-
der of Vn around the expected number. Remember the sharp results of the 'P- invariant: n goldbugs, by contrast, simulate the expected behavior with only constant error!
There are more results on these and related one-dimensional not-very-random walks. But let's move on-these bugs long for some higher-dimensional elbow room.
The Rotor-Router
This time, we will set up a system with bugs moving around the integer points in the plane. It will differ from the above in that this will be an aggregation model. Bugs still get added repeatedly at one source, but instead of falling into sinks (bug cups), the bugs will walk around until they find an empty lattice point, then settle down and live there forever.
Generalizing the Inbound and Outbound arrows of the goldbugs, we decree that each lattice site is equipped with an arrow, or rotor, which can be
.. * ...
* * t ... t ..
�
w .ffi__
.. _., · ·i ,
•
- -t� - - - - ......
... ..� - - - - - - - --, •
it t .. I
Figure 4. A bug's ramble through some ro
tors: before and after.
rotated so that it points at any one of the four neighbors. (Propp uses the word "rotor" to refer to the two-state arrows in dimension 1 as well.) The first bug to arrive at a particular site occupies it forever, and we decree that it sets the arrow there pointing to the East. Any bug arriving at an occupied site rotates the arrow one quarter turn counterclockwise, and then moves to the neighbor at which the rotor now points-where it may find an empty site to inhabit, or it may find a new arrow directing its next step. In short, the first bug to reach ( i, J) lives there, and bugs that arrive thereafter are routed by the rotor: first to the North, then West, South, and East, in that order, and then the cycle begins with North again (Fig. 4).
Once a bug finds an empty site to inhabit, we drop a new bug at the origin, and this one too meanders through the field of rotors, both directed by the arrows and changing them as a result of its visits. Every bug does indeed fmd a home eventually, and the proof is the same as for the goldbugs: the set of all sites which a particular bug visits infinitely often cannot have any boundary.
And so we ask, as we add more and more bugs, what does the set of occupied sites look like?
Let's take a look at the experimental answer. The beautiful image you see in Figure 5 is a picture of the set of occupied sites after three million bugs have found their permanent homes . The sites in black are vacant, still awaiting their first visitor. Other sites are colored according to the direction of their rotor, red/yellow for East/West and green/blue for North/South. On the cover is a close-up of a part of the boundary of the occupied region. At http:/ /www.math. wisc.edu/-propp/ rotor-router-1.0/ you can find a Java applet by Hal Canary and Francis Wong, if you'd like to experiment yourself.
As you can see, the edge of the occupied region is extraordinarily round: with three million bugs, the occupied site furthest from the origin is at distance Y956609, while the unoccupied site nearest the origin is at distance Y953461, a difference of about 1.6106.
And the internal coloration puts on a spectacular display of both large-
© 2005 Springer Sc�ence+Business Media, Inc .. Vc>ume 27. Number 1 . 2005 57
Figure 5. The rotor-router blob after 3 million bugs.
scale structure and intricate local patterns. When Ed Pegg featured the rotor-router on his Mathpuzzle Web site, he dubbed the picture a Propp Circle, and to this day I am jealous that I didn't think of the name first: it connotes precisely the right mixture of aesthetic appreciation and conviction that there must be something deep and not fully understood at work
Recall, for emphasis, that this was formed by bugs walking deterministically on a square lattice, not a medium known for growing perfect circles.
58 THE MATHEMATICAL INTELLIGENCER
Moreover, the rule that governed the bugs' movement is inherently asymmetric: every site's rotor begins pointing East, so there was no guarantee that the set of occupied sites would even appear symmetric under rotation by 90°, much less by unfriendly angles. On the other hand, while the overall shape seems to have essentially forgotten the underlying lattice, the internal structure revealed by the colorcoded rotors clearly remembers it.
Lionel Levine, now a graduate student at U.C. Berkeley, wrote an under-
graduate thesis with Propp on the rotor-router and related models (11 ] . It contains the best result so far on the roundness of the rotor-router blob: after n bugs, it contains a disk whose radius grows as n114. Below I report on two remarkable theorems which do not quite prove that the rotor-router blob is round, but which at least make me feel that it ought to be. I have less to offer on the intricate internal structure, but there is a connection to something a bit better understood. Let's get to work
IDLA
Internal Diffusion Limited Aggregation
is the random walk-based model which
Propp de-randomized to get the rotor
router. Most everything is as above
the plane starts empty, add bugs to the origin one at a time, each bug occupies the first empty site it reaches. But in IDLA, a bug at an occupied site walks to a random neighbor, each with probability 1/4.
The idea underlying IDLA comes
from a paper by Diaconis and Fulton
[6]. They define a ring structure on the
vector space whose basis is labeled by the finite subsets of a set X equipped with a random walk. To calculate the product of subsets A and B, begin with the set A U B, place bugs at each point
of A n B, and allow each bug to execute a random walk until it reaches an outside point, which is then added to
the set. The product of A and B records the probability distribution of possible outcomes. This appears to depend on the order in which the bugs do their
random walks, but in fact it does notwe'll explore this theme soon.
Consider the special case where X is the d-dimensional integer lattice with the random walk choosing uniformly from among the nearest neighbors. Then repeatedly multiplying the singleton {0} by itself is precisely the random-walk version of the rotorrouter. A paper of Lawler, Bramson,
and Griffeath [9] dubbed this Internal Diffusion Limited Aggregation, to em
phasize similarity with the widely-studied Diffusion Limited Aggregation model of Witten and Sandler [ 13]. DLA simulates, for example, the growth of dust: successive particles wander in "from infinity" and stick when they reach a central growing blob. The resulting growths appear dendritic and fractal-like, but rigorous results are hard to come by.
In contrast, the growth behavior of
IDLA has been rigorously established. It is intuitive that the growing blob should be generally disk-shaped, since the next
particle is more likely to fill in an unoccupied site close to the origin than one further away. But the precise statement in [9] is still striking: the random walk manages to forget the anisotropy of the underlying lattice entirely!
Theorem (Lawler-Bransom-Grif
feath). Let wd be the volume of the ddimensional unit sphere. Given any E > 0, it is true with probability 1 that for all sufficiently large n, the d-dimensional 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
could be taken on the order of n113• Most recently, Blachere [3] used an induction 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 order were observed experimentally by Moore and Machta [ 12] .
So how does the random walk-based IDLA relate to the deterministic 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 rotor-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 Q1. 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 exactly 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 P 1 . 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 choosing 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 simultaneously, 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 control 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 million 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 neighbors 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 indicated 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 product 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 stackwe get the rotor-router swarm variant above; QED.
Rotor-Roundness
Now let me outline a heuristic argument that the rotor-router blob ought to be round, letting the Lawler-Bramson-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 strategy is easy to describe. Just as we did four paragraphs ago, think of each lattice site as holding a giant stack of cards: one card for each time a bug departed that site while the n-bug rotorrouter blob grew. Now we start running 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 corresponding 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 aggregate, 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 nonnegative integer-valued function on the lattice which is almost harmonic, away from the origin: the number of departures 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 circus 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 approximation sign above hides some rounding error, the precise details of which encapsulate the rotor-router rule. For IDLA, this is exact if we replace d by a, the expected number of departures, and replace 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 particular site, the mean number of departures 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 showing 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 function with zero at the boundary-but to paraphrase Mark Twain, the difference between a harmonic function and an almost-harmonic function is the difference 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] contains an amazing result on the relationship 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 rotorthat 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 incrementing 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 version, every bug moves on.) Given any initial distribution of bugs and any initial 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 rotor evolution and n steps of random walk would lead to similar ending distributions.
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 'random-walk evolution of checkered distributions 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.
© 2005 Spnnger Science+Bus1ness Media, Inc , Volume 27, Number i , 2005 61
the dimension d. That is, given any checkered initial distribution of a finite number of bugs in 7l_d, the difference between the actual number of bugs at a point p after n steps of rotor-based evolution, and the expected number of bugs at p after an n-step random walk, is bounded by a constant. This constant is independent of the number of steps n, the initial states of the rotors, and the initial distribution 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 distributions: since each bug changes parity 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 deterministically simulates a random walk process with constant error-better than a single 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 likewise demonstrate that derandomizing systems can reduce the error. Lionel Levine's thesis [ 11 ] analyzed a type of one-dimensional derandomized aggregation 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 another sort of derandomized one-dimensional 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 addresses the overall shape of the rotorrouter 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
62 THE MATHEMATICAL INTELLIGENCER
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 approximate large-scale shape, which exhibits some sort of radial symmetry. The direction 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 entirely 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 dynamical 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 neighboring 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 rubble appears as Figure 6. Pixels are colored according to the number of grains of sand there in the final configuration. The dominant blue color corresponding 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 arbitrary graph, with an excessively large sand-pile giving any number of grains of sand to each of its neighbors, and some grains possibly disappearing permanently from the system. Variations have been investigated by combinatorists 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 described, under the name "the probabilistic 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-topoint correlation functions, the distribution of sizes of avalanches, or a marvelous abelian group structure on a certain set of recurrent configurations.
Our question seems to have a different flavor. For example, in most sandpile work, one can assume without loss of generality that a pile collapses as soon as it has enough grains of sand to give its neighbors what they are owed, leaving itself vacant. The version I described above is what I'll call a "greedy sandpile," in which each site hoards its first grain of sand, never letting 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 despite the single-scale evolution rule; I think this is related to what the mathematical physics people call "self-organizing criticality," about which I know nothing at all. But personally, in both pictures I am drawn to the eightpetalled 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 rotorrouter 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 realize 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 performs precisely that part of the evolution of the rotor-router that can take place without asking the rotors to break symmetry. If we define an energy ftmction which is large when multiple bugs share a site, then the sandpile is the lowest-energy 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.
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.
REFERENCES
[ 1 ] Bak, Per; Tang, Chao; Wiesenfeld , Kurt.
Self-organized criticality. Phys. Rev. A (3)
38 (1 988), no. 1 , 364-374.
[2] Bjorner, Anders; Lovasz, Laszlo; Shor, Pe
ter. Chip-firing games on graphs. European
J. Combin. 1 2 (1 991 ) , 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) .
[4] Cooper, Joshua; Spencer, Joel. Simulat
ing a Random Walk with Constant Error.
Preprint arXiv:math C0/0402323 (Febru
ary 2004); to appear in Combinatorics,
Probability and Computing.
[5] Dhar, Deepak. Self-organized critical state
of sandpile automation models. Phys. Rev.
Lett. 64 (1 990), no. 1 4 , 1 61 3-1 61 6.
[6] Diaconis, Persi; Fulton, William. A growth
model, a garne, an algebra, Lagrange in
version, and characteristic classes. Com-
mutative algebra and algebraic geometry, II
(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 l imited ag
gregation. Ann. Probab. 20 (1 992), no. 4 ,
21 1 7-21 40.
1 0 . Lawler, Gregory. Subdiffusive fluctuations
for internal diffusion limited aggregation.
Ann. Probab. 23 ( 1 995), no. 1 , 7 1 -86.
1 1 . Levine, Lionel. The Rotor-Router Model.
Harvard University senior thesis. Preprint
arXiv:math.C0/0409407 (September 2004).
1 2 . Moore, Christopher; Machta, Jonathan. In
ternal diffusion-limited aggregation: paral
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Phys. 99 (2000) , no. 3-4, 661 -690.
1 3. Witten , T. A . ; Sander, L. M. Diffusion
limited aggregation. Phys. Rev. B (3) 27
(1 983), no. 9, 5686-5697.
1 4 . Winkler, Peter. Mathematical Puzzles: A
Connoisseur's Collection. A K Peters Ltd,
Natick, MA, 2003.
The Broad Institute at MIT
320 Charles Street
Cambridge, MA 021 41
USA
e-mail: [email protected]
© 2005 Springer Sc1ence+Bus1ness Media, Inc , Volume 27, Number 1 , 2005 63
BRUNO DURAND, LEONID LEVIN, AND ALEXANDER SHEN
Local Ru les and G lobal Order , Or Aperiod ic Ti l 1 ngs
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 network 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 science" and are rather difficult. In mathematics the most prominent example of this kind is the so-called Berger theorem on aperiodic tilings (exact statement below). It was proved by Berger in 1966 [1 ] . 1 In 1971 the proof was simplified 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 invented (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
0 0 0 0 0 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 reading," 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 student (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] .
64 THE MATHEMATICAL INTELLIGENCER © 2005 Spnnger Sc1ence+Bus1ness Med1a, Inc.
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 letter 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: <a1, a2, a:l, a4) E L is a square
� [;I;] We say that these squares are allowed by rule L. A configuration T satisfies local rule L if all 2 X 2 squares in it are allowed by L. Formally this means that
<T(i,j), T (i + 1, j), T(i,j + 1), T(i + 1, j + 1)) E L
for any i ,j E 7L. A non-zero integer vector t = (t1.t2) is aperiod of T if the t-shift preserves T, i.e.,
for any x1,x2 E 7L.
The Aperiodic Tilings theorem
Theorem (Berger): There exist an alphabet A and a local rule L such that
(1) there are tilings that satisfy L; (2) any tiling satisfying L has no period.
To prove the theorem we need some auxiliary definitions.
Substitution Mappings
A substitution is a mapping s of type A ----> A4 whose values are considered as 2 X 2 squares:
s: a � s1(a) s2(a)
s3(a) s4(a)
We say that a substitution s matches local rule L if two conditions are satisfied:
(a) all values of s belong to L; (b) taking any square from L and replacing each of the
four cells by its s-image, we get a 4 X 4 square that satisfies L (this means that all nine 2 X 2 squares inside it belong to L).
Remark. Consider a square X of any size N X N (filled with letters from A) satisfying L. Apply substitution s to each letter in X and obtain a square Y of size 2N X 2N. If the substitution s matches L, then Y satisfies L. Indeed, any
2 X 2 square in Y is covered by an image of some 2 X 2 square in X.
This is true also for (infinite) configurations: applying a substitution to each cell of a configuration that satisfies L, we get a new configuration that satisfies L (assuming that the substitution matches L).
Proposition 1. If a substitution s matches a local rule L, there exists a configuration T that satisfies L.
Proof Take any letter a E A and apply s to it. We get a 2 X 2 square s(a) that belongs to L. Then apply s to all letters in s(a) and get a 4 X 4 square s(s(a)) that satisfies L. Next is an 8 X 8 square s(s(s(a))) that satisfies L, etc. Using a compactness argument, we conclude that there exists an infinite configuration that satisfies L.
Here is a direct proof not referring to compactness. Assume that substitution s is fixed. A letter a' is a descendant of a letter a, if a' appears in the interior part of some square s(s( . . . s(a) . . . )) obtained from a. Each letter has at least one descendant, and the descendent relation is transitive (if a' is a descendant of a and a" is a descendant of a' , then a" is a descendant of a). Therefore, some letter is a descendant of itself (start from any letter and consider descendants until you get a loop). If a appears in the interior part of sCn)(a), then sCn)(a) appears in the interior part of sC2n)(a), which appears (in its tum) in the interior part of sC3n)(a), and so on. Now we get a increasing sequence of squares that extend each other and together form a configuration. (Here we use that a appears in the interior part of the square obtained from a.)
Proposition 1 is proved. Now we formulate requirements for substitution s and
local rule L which guarantee that any configuration satisfying L is aperiodic. They can be called "self-similarity" requirements, and guarantee that any configuration satisfying L can be uniquely divided (by vertical and horizontal lines) into 2 X 2 squares that are images of some letters under s, and that these pre-image letters form a configuration that satisfies L. Here is the exact formulation of the requirements:
(a) s is injective (different letters are mapped into different squares);
(b) the ranges of mappings s1.s2,s3,s4 : A ----> A (that correspond to the positions in a 2 X 2 square, see above) are disjoint;
(c) any configuration satisfying L can be split by horizontal and vertical lines into 2 X 2 squares that belong to the range of s, and pre-images of these squares form a configuration that satisfies L.
The requirement (b) guarantees that there is only one way to divide the configuration into 2 X 2 squares; the requirement (a) then guarantees that each square has a unique preimage.
Proposition 2. Assume that substitution s and local rule L satisfy requirements (a), (b) and (c). Then any configuration satisfying L is aperiodic.
Pmof Let T be a configuration satisfying L and let t = (t1 ,t2) be its period. Both t1 and t2 are even numbers. Indeed, (c) guarantees that T can be split into 2 X 2 squares,
© 2005 Spnnger SC1ence+ Bus1ness Media, Inc . Volume 27, Number 1 . 2005 65
and then (b) guarantees that the t-shift preseiVes these squares (since, say, an upper left corner of a square must go to another upper left corner).
Then (a) guarantees that pre-images of these 2 X 2 squares form a configuration that satisfies L and has period t/2. Therefore, for each periodic L-configuration with period t we have found another periodic L-configuration with period t/2. An induction argument shows that there are no periodic L-configurations.
Proposition 2 is proved. Using Propositions 1 and 2 we conclude that to prove
the Aperiodic Tilings theorem it is enough to construct a local rule L and substitution s matching L that satisfy (a), (b) and (c). This we now do.
Construction: An Alphabet
Letters of A are considered as square tiles with some drawings on them. We describe a local rule and substitution in terms of these drawings.
Each of the four sides of a tile (1) is dark or light (has one of two possible colors); (2) has one of two possible directions, indicated by
arrows; (3) has one of two possible orientations; this means that
one of two possible orthogonal vectors is fixed; we say that this orthogonal vector goes "from inside to outside". (Our drawings show the orientation by a gray shading inside.)
In this way we get three bits per side, i.e. , 12 bits for each tile. In addition to these 12 bits, a tile carries two more bits, so the size of our alphabet is 214 = 8192. These two additional bits are graphically represented as follows: we draw a cross (Fig. 2) in one of four versions (which differ by a rotation).
Fig. 2. One version of cross.
It is convenient to assign color, direction, and orientation to the segments that forming a cross. Namely, two neighboring sides of a cross are dark, the other two are light. The direction arrows go from the center outward, and the orientation is shown by a gray stripe that shows the "inside" part as indicated in the picture (gray stripes are inside the dark angle).
This will be important when we define the substitution.
Substitution
To perform the substitutions, we cut a tile into four tiles. The middle lines of the tile become sides of the new (smaller) tiles, with the same color, direction and orientation. Before cutting we draw crosses on the small tiles in such a way that the dark angles form a square as shown (Fig. 3).
66 THE MATHEMATICAL INTELLIGENCER
. . . . . . . . . . . . . . . . . . . . . . .
Fig. 3. A tile split i n four parts.
It is immediately clear that conditions (a) and (b) of Proposition 2 are satisfied. Indeed, to reconstruct tile x from its four parts, it is enough to erase some lines, and the position of a tile in s(x) is uniquely determined by the orientation of its central cross. The condition (c) will be checked later after the local rule is defined.
Local Rule
The local rule (L) is formulated in terms of lines and their crossings. There are two types of crossings that appear when tiles meet each other. First, a crossing appears at the point where corners of four tiles meet; crossing lines are formed by the tile sides. Second, a crossing appears at the middle of tile sides, where middle lines of tiles meet the tile side. First of all, the following requirement is put:
if two tiles have a common side, this shared side has the same color, direction, and orientation in both tiles.
Therefore, we can speak about the color, direction and orientation of a boundary line between two tiles without specifying which of the two tiles is considered.
We also require that
all crossings (of both types) are either crosses or meeting points. A cross is formed by four outgoing arrows that have colors and orientation as shown in Fig. 4 (up to a rotation, so there are four types of crosses). In a meeting point, two arrows of the same color, the same orientation, but opposite directions, meet "face to face," and the orthogonal line goes through this meeting point without change in color, direction, or orientation. One more restriction is put: if two dark arrows meet, then the orthogonal line goes "outward" (its direction agrees with the orientation of the arrows).
Our local rule is formulated in terms of restrictions saying which crossings are allowed when lines meet. Formally speaking, the local rule is a set of all quadruples of tiles where these restrictions are not violated. Fig. 4 shows the first type of allowed crossing, a cross, in one of four possible versions (which differ by a rotation). The second type
+ '
Fig. 4. A cross formed by outgoing arrows.
t ·········-!"·········
t Fig. 5. Arrows meet.
of allowed crossings (symbolically shown in Fig. 5) has
more variations: (a) the meeting arrows can be horizontal or vertical; (b) the vertical line can have two orientations; (c) the horizontal line can have two orientations; (d) the vertical line can have one of two colors; (e) the horizontal line can have one of two colors; and finally (f ) if two light arrows meet, the perpendicular line can go in either of two directions. So we get 2 · 2 · 2 · 2 · 3 = 48 variations in this way.
Remark. The Local rule ensures that the orientation of any horizontal or vertical line remains unchanged along the
whole line. (Indeed, the orientation does not change at
crosses or meeting points.)
Substitution and Local Rule
We have to check that the substitution matches the local
rule. Indeed, when tiles are split into groups of four, the old lines still form the same crossings as before, but new crossings appear. These new crossings appear (a) in the centers of new tiles (where new lines cross new ones) and (b) at the midpoints of sides of new tiles (where new lines
cross old ones). In case (a) we have legal crosses by defi
nition. In case (b) it is easy to see that two arrows meet
creating a legal meeting point. See Fig. 6, which shows a tile split into four tiles, with all possible meeting places of new and old lines circled. The orientation matches because the orientation of the new crosses is fixed by s; all other requirements are fulfilled, too.
Fig. 6. New lines meet old lines.
Self-similarity Condition
It remains to check condition (c) of Proposition 2. Assume that we have a configuration that satisfies the local rule.
Step 1 . Tiles are grouped by fours.
Consider an arbitrary tile in this configuration and a dark
arrow that goes outward. It meets another arrow from a
neighboring tile, and this arrow must be dark by the local
rule. These two arrows must have the same orientation, therefore we get half of a dark square (Fig. 7), not a Zshape. Repeating this argument, we conclude that tiles form
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fig. 7. Two neighbor tiles.
groups of four tiles whose central lines form a dark square (Fig. 8).
:· · · · · · · · · · · · · · · · · · : · · · · · · · · · · · · · · · · · :
f EEl ! . . . . . . . . . . . . · . . . . . . . . . . . . . . . . . . · . . . . . . . . . . . . . . . . . . .
Fig. 8. Four adjacent tiles.
Step 2. These 2 X 2 squares are aligned.
If two groups (each forming a 2 X 2 square) were wrongly
aligned, as shown in Fig. 9, then the orientation of one of the lines (in our example, the horizontal line) would change along the line (recall that all crosses have fixed orientation of lines). Therefore, 2 X 2 squares are aligned.
Fig. 9. Bad placement.
Step 3. Each group has a cross in the middle.
What can be in the group center? The middle points of the sides of the dark square are meeting points for dark arrows. Therefore, according to the local rule, an outgoing arrow should be between them. So a meeting point cannot appear in the center of a 2 X 2 group, and the only possibility is a cross.
Step 4. Uniform colors on sides.
To finish the proof that each group belongs to the range of
the substitution, it remains to show that the color, direction, and orientation do not change at the midpoint of a
side of a 2 X 2 group. This is because this midpoint is a meeting point for arrows perpendicular to the side.
Step 5. Pre-image tiles satisfy the local rule.
This is evident: the substitution adds new lines. So taking
the pre-image just means that some lines are deleted, and no violation of the local rule can happen.
The Aperiodic THings theorem is proved.
© 2005 Spnnger Sc1ence+ Bus1ness Media, Inc , Volume 27, Number 1, 2005 67
A U T H O R S
ALEXANDER SHEN
ul. Begovaya, 1 7- 1 4
Moscow, 1 25284
RusSia
e-mail: [email protected]
Alexander Shen has been since 1 982 at the
Institute for Problems of Information Trans
mission in Moscow. He is known to lntelli
gencer readers as aU1hor of several contri
butions and as former Mathematical
Entertainments Editor. He has written text
books based on courses at Moscow State
University and at the Independent Univer
sity of Moscow: see ftp.mccme.ru/usersl
shen. Some have been translated into En
gl ish: Algorithms and Programming: Prob
lems and Solutions, Birkhauser, 1 997; and
two from the American Mathematical Society.
REFERENCES
BRUNO DURAND
Laboratoire d'lnformatique Fondamentale de
Marseille
CMI, 39 rue Joliot-Curie
1 3453 Marseille Cedex 1 3
France
e-mail: [email protected]
Educated in Computer Science at the
Ecole Normale Superieure de Lyon, Bruno
Durand is now both Professor at the Uni
versite de Provence and Director of the
Laboratoire d'lnformatique Fondamentale
de Marseille. His research is on cellular au
tomata, tilings, and complexity. He is an
editor of Theoretical Computer Science.
LENOID LEVIN
Department of Computer Science
Boston University
Boston, MA 0221 5-24 1 1
USA
Leonid Levin has worked rncstly in theory of
compU1ation. He was one of the originators
of the P-NP conjecture, whose monetary
price-tag has now grown to $1 CXXXXlO, bU1
whose scientific value is not compU1able.
Readers curious aboU1 his more recent
thoughts are invited to his Web site, http://
www.cs.bu.edu/fac/lnd.
[ 1 ] R. Berger, The undecidability of the domino problem, Memoirs
Amer. Math. Soc., 66 (1 966), 1 -72.
[4] J . Kari, A small aperiodic set of Wang tiles, Discrete Math. , 160
(1 996), 259-264.
[5] Leonid A Levin, Aperiodic Tilings: Breaking Translational Symme
try, http:l/arxiv.org/abs/cs/0409024. [2] E. Borger, E. Gradel, Yu. Gurevich , The Classical Decision Problem,
Springer, 1 997. [Berger's theorem is considered in the Appendix
written by C. Allauzen and B. Durand.]
[3] B. Grunbaum, C. G. Shephard. Tilings and Patterns, Freeman, New
York, 1 986.
[6] C. Radin, Miles of Tiles, AMS, 1 999 (Student Mathematical Library,
vol. 1 ).
[7] R. Robinson, Undecidability and non periodicity of tilings in the plane,
Invent. Math . , 12 ( 1971 ) , 1 77-209.
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li,iM.�ffl!•i§u6'h£ili.Jiiijbl D i rk H uylebrouck, Ed itor I
The Home of Golden Numberism Roger Herz-Fischler
Does your hometown have any
mathematical tourist attractions such
as statues, plaques, graves, the caje
where the famous conjecture was made,
the desk where the famous initials
are scratched, birthplaces, houses, or
memorials? Have you encountered
a mathematical sight on your travels?
If so, we invite you to submit to this
column a picture, a description of its
mathematical significance, and either
a map or directions so that others
may follow in your tracks.
Please send all submissions to
Mathematical Tourist Editor,
Dirk Huylebrouck, Aartshertogstraat 42,
8400 Oostende, Belgium
e-mail: [email protected]
The expression "golden numberism"
refers to the set of claims con
cerning the purported use of the golden number (division in extreme ratio, golden section, golden ratio, . . . ) in man-made objects (art, architecture, etc.) or its purported appearance in nature (human body, plants, astronomy, etc.). If we leave aside the vague statements by Kepler, the early nineteenthcentury association of the golden number with phyllotaxis, and a few other
extremely obscure comments, then we
can state that the beginning of golden
numberism is due to one person, the German intellectual Adolph Zeising (1810-1876).
What is of particular interest to the mathematical tourist is that the origin of golden numberism is associated with a particular time and place. Zeising's father was a court musician in the tiny dukedom of Anhalt-Bernburg. This dukedom, as well as the other An
halts-the various pieces of which will bring joy to any map colourist or
topologist interested in non-connected surfaces-were contained in what is now the German Land of SachsenAnhalt. Because of his father's occupation, Zeising was born in Ballenstedt, the location of the summer palace, but a visit there made it clear that he is now
completely unknown to local officialdom.
Where Zeising is known to some extent-though not in connection with the golden number-is in the city of Bernburg, which is some 75 kilometres northwest of Leipzig. Despite the ravages caused by events of the last sixty years, Bernburg, situated on the Saale river, remains a charming city. There are two edifices in Bernburg that should probably be jointly designated as the birthplace of golden numberism.
The first is the building formerly oc
cupied by the Karls-Gymnasium. In
1842, after having completed a doctorate with a specialty in Hegelian philosophy, Zeising became a full-time professor at that institution, and it was
while he taught there that a combination of readings in philosophy and
other fields inspired him to think of the golden number as an inherent rule of nature.
The other building of interest is the Bernburg castle. Zeising was a leader
of the liberal left during the German revolution of 1848-1849, and he was elected to the first Landtag, which met in the castle. Perhaps Zeising thought
about the golden number during some
long-winded speeches, but more im
portantly, the castle represented polit
ical power. By 1851 the power was firmly in the hands of a very reac
tionary group, and Zeising was pen
sioned off in December 1852. Using this money he went to Leipzig to do further research, and in 1854 he published his book, An Exposition of a New The
ory of the Proportions of the Human
Body, Based on a Previously Unrec
ognized Fundamental Morphological
Law which Permeates all of Nature
and Art, Together with a Complete
Comparative Overview of Previous
Systems.
In the period from 1855 to 1858 Zeising continued to publish articles and a booklet dealing with the golden number. Some of these were of a popular nature, in particular his articles "Hu
mans and Leaves" and "Face Angles," which were published in the widely read science magazine Die Natur. His book and these articles ensured that his theory became widely known, and
by the time of his death in 1876 golden numberism was widespread in Germany. Philosophers debated the foundations of his theory, and authors suggested the use of the golden number in such fields as typography and fashion.
The polymath Gustave Fechner, inspired by Zeising's claims concerning
the Sistine Madonna, started scientific investigations, which in tum laid the
foundations of the field of experimen
tal aesthetics. By the end of 1855 Zeising had
moved to Munich, where he became
© 2005 Spnnger Science+Bus1ness Media, Inc., Volume 27, Number 1 , 2005 69
�but8fr .ft4rl��mufium ht b« 9unftt1Jellf 1 84 l -- 1882
Fig. 1 . The Karls-Gymnasium, Bernburg, Germany.
active in literacy circles and wrote novels and plays. He also wrote on philosophy-his 1855 Aesthetics, which presented an overview of the systems of Hegel and his followers, was very highly regarded-and politics. After a hiatus of ten years Zeising again wrote
Fig. 2. Bernburg Castle.
70 THE MATHEMATICAL INTELLIGENCER
on the golden number, but, aside from an article on the Cologne Cathedral, these works were of a cultural or philosophical nature. A particular honour came in 1856: his admission to the Deutsche Akademie der Naturforscher.
There have been many interesting
twists and turns in the development of the myth of golden numberism. Thus the association of the golden number with virtually all the pyramids of Egypt except the Great Pyramid was made by Friedrich Rober in 1855, independent of Zeising. On the other hand the first example of golden numberism in English dates from 1866 and deals with the golden number and the Great Pyramid-but in a manner not equivalent to that of Rober-and again this was independent of Zeising's writings.
Mter having entered France and the English-speaking world from Germany in the early part of the twentieth century, golden numberism spread rapidly. By a careful examination of sources, it is possible to trace the path travelled by the golden number myth. Aside from the topics of phyllotaxis and the Great Pyramid, virtually everything that has been written on the subject can ultimately be traced back to the influence of Zeising. The next time the reader hears another "historical" claim concerning the golden number he or she might care to glance at the accompanying photographs of the Gymnasium and the Bernburg castle, and remember where the myth started.
I consider Zeising the most intellectual of authors on the subject of the golden number. Unlike others, he attempted to present a true foundationin his case philosophical-for his arguments. Not that I fmd him convincing! I am fascinated by the "sociology of mathematical myths" and the history of ideas. Thus my most recent work, Adolph Zeising, started out as a few paragraphs and then an appendix to my forthcoming book The Golden Number. In Adolph Zeising I trace the spread of golden numberism from Nees von Essenbeck in 1852 through 1876, the year of Zeising's death. The Golden Number will present a complete discussion of golden numberism, including the historical, sociological, and philosophical aspects. Parts of the story can be found in my other writings listed below.
Biographical Notes
Ebersbach's book discusses the period (1835-1852) when Zeising lived in Bernburg. In particular, there are sev-
eral references to Zeising's role in
the 1848-1849 revolution. The pho
tographs (1864 for the castle and the
early part of the twentieth century for
the Gymnasium) were taken from
[Erfurth, p. 63) and [Schulgemeinschaft
Carolinum und Friederiken-Lyzeum,
p. 142] respectively. These photographs
are reproduced with the kind permission
of the Mittledeutsche Verlag (Halle).
Herz-Fischler, R. "How to Find the "Golden
Number' Without Really Trying . " Fibonacci
Quarterly 1 9 (1 981 ) , pp. 406-4 1 0.
mid. Waterloo, Wilfrid Laurier University
Press, 2000.
Herz-Fischler, R. Adolph Zeising (1 810-1876):
Herz-Fischler, R. A Mathematical History of
Division in Extreme and Mean Ratio . Wa
terloo, Wilfrid Laurier University Press.
1 987. Re-printed as A Mathematical His
tory of the Golden Number. New York,
Dover, 1 998.
The Life and Work of a German Intellectual.
Ottawa: Mzinhigan Publishing, 2004.
Herz-Fischler, R. The Golden Number. To ap
pear. Ottawa: Mzinhigan Publishing, 2005.
Schulgemeinschaft Carolinum und Friederiken
Lyzeum. Geschichte der h6heren Schulen zu
Bernburg: Friederikenschule von 181 0 bis
1 950, Kar/sgymnasium von 1 835- 1944,
Karls-Realgymnasium von 1 853-1945. Mu
nich: Wedekind, 1 980.
Herz-Fischler, R . "A 'Very Pleasant' Theorem."
REFERENCES
College Mathematics Journal 24 (1 993), pp.
31 8-324.
Ebersbach, V. Geschichte der Stadt Bernburg,
vol. 1 . Dessau: Anhaltische Verlagsgesell
schaft, 1 998.
Erfurth, H. Gustav V61ker/ing & die altesten Fo
tografien Anhalts . Dessau: Anhaltische Ver
lagsgesellschaft, 1 991 .
Herz-Fischler, R. "The Golden number, and Di
vision in Extreme and Mean Ratio." in Com
panion Encyclopedia of the History and
Philosophy of the Mathematical Sciences,
London, Routledge, 1 994, pp. 1 576-1 584.
340 Second Avenue
Ottawa, K1 S 2J2
Canada
Herz-Fischler, R. The Shape of the Great Pyra- e-mail: [email protected]
Mathematics and Culture Mathematics and Culture Michele Emmer, University of Rome 'La Sapienza', Italy (Ed.)
This book stresses the strong links between mathematics and culture, as mathematics links theater, literature, architecture, art, cinema, medicine but also dance, cartoon and music. The articles introduced here are meant to be interesting and amusing starting points to research the strong connection between scientific and literary culture. This collection gathers contributions from cinema and theatre directors,
musicians, architects, historians, physicians, experts in computer graphics and writers. In doing so, it highlights the cultural and formative character of mathematics, its educational value. But also its imaginative aspect: it is mathematics that is the creative force behind the screenplay of films such as A Beautiful Mmd, theater plays like Proof, musicals like Fermat's Last Tango, successful books such as Simon Singh's Fermat's Last Theorem or Magnus Enzensberger's The Number Devil.
2004/352 PP., 54 ILLU S./HARDCOVER/$59.95/ISBN 3-540-01 770-4
Springer www.springer-ny.com
Mathematics, Art, Technology and Cinema Michele Emmer, University of Rome 'La Sapienza', Italy; and Mirella Manaresi, University of Bologna, Italy (Eds.)
This book is about mathematics. But also about art, technology and images. And above all, about cinema, which in the past years, together with theater, has discovered mathematics and mathematicians. The authors argue that the discussion about the differences between the so-called rwo cultures of science and humanism is a thing of the past. They hold that both cultures are truly
L--"""'-'li..o..-.....J linked through ideas and creativity, not only through technology. In doing so, they succeed in reaching out to non-mathematicians, and those who are not particularly fond of mathematics. An insightful book for mathematicians, film lovers, those who feel passionate about images, and those with a questioning mind.
2003/242 PP./HARDCOVER/$99.00/ISBN 3-540-00601 -X
EASY WAYS TO ORDER: CALL Toll-Free 1 -800-SPRINGER • WEB www.springcr-ny.com E-MAIL order>(!l'spnnger-ny.com • WRITE to Spri nger-Verlag New York, Inc., Order Dept. 57805, PO Box 2485, Secaucus, NJ 07096-2485 V1SJT your 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.
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© 2005 Spnnger Setence+Bus1ness Med1a. Inc . Volume 27. Number 1, 2005 71
D avid E . R owe , E d i t o r I
H i l bert's Early Career: Encounters with A l l ies and Rivals David E. Rowe
Send submissions to David E. Rowe,
Fachbereich 1 7 -Mathematik,
Johannes Gutenberg University,
055099 Mainz, Germany.
It seems to me that the mathematicians of today understand each other far too little and that they do not take an intense enough interest in one another. 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
Probably no mathematician has been quoted more often than
Hilbert, whose opinions and witty remarks 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 expert on invariant theory, but Hilbert's reputation as a universal mathematician grew as he left his mark on one field after another. Yet these achievements alone cannot account for his singular place in the history of mathematics, as was recognized long ago by his intimates ([Weyl 1932], [Blumenthal 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 inspired 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 survey, The Honors Class, "solving one of Hilbert's problems has been the romantic dream of many a mathematician" [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 conveyed 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 conveyed the rather harsh views cited in the opening quotation to Klein, he was not merely bemoaning the lack of communal 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 organization of German mathematicians, the Deutsche Mathematiker-Vereinigung, and Hilbert was delighted to learn that Klein would be present for the inaugural 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 mathematicians would, in fact, be very desirable 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 optimism and self-confidence spilled over and inspired nearly all the young people who entered his circle.
1 1t was Hilbert's star pupil, Hermann Weyl, who called those who actually succeeded the "honors class"; he
also wrote that "no mathematician of equal stature has risen from our generation" [Weyl 1 944, 1 30].
72 THE MATHEMATICAL INTELLIGENCER © 2005 Spnnger Science+ Busrness Medra, Inc.
Hilbert's impact on modem mathematics has been so pervasive that it takes a true leap of historical imagination 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 become familiar aspects of the Hilbert legend. In [Rowe 2003a] , I describe the quiet early years he spent in Konigsberg, 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 mathematical 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 himself within the fast-changing German mathematical community.
Returning from Paris
After a rather uneventful and disappointing stay in Paris during the spring of 1886, Hilbert began the long journey
Fig. 1 . Hilbert in the days when he was re
garded as merely one of many experts on the
theory of invariants.
home. Stopping first in Gottingen, he learned something about the contemporary Berlin scene when he met with Hermann Amandus Schwarz, the senior mathematician on the faculty. Schwarz had long been one of the closest 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 Annalen (see [Rowe 2000]). However, after E. E. Kummer's departure in 1883, the harmonious atmosphere he had cultivated as Berlin's senior mathematician gave way to acrimony. Weierstrass, old, frail, and decrepit, refused to retire for fear of losing all influence to Leopold Kronecker, who remained amazingly energetic and prolific despite his more than sixty years.
Presumably Hilbert heard about this situation from Schwarz, who would have conveyed the essence of the situation from Weierstrass's perspective (see [Biermann 1988, 137-139]). If so, Hilbert would have heard how relations between Weierstrass and Kronecker had deteriorated mainly because of the latter's dogmatic views, in particular his sharp criticism of Weierstrass's approach to the foundations of analysis. Only a few months after Hilbert passed through Berlin, Kronecker 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 andere ist Menschenwerk") [Weber 1893, 19]. Kronecker had never made a secret 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 written one year earlier:
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 socalled analysis now operates [Biermann 1988, 138].
Weierstrass had written to Schwarz at around that time, claiming that Kronecker had transferred his former antipathy 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 assert 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 numbers" (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 presented 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 initial 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 requirements 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 mathematicians 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-
© 2005 Spnnger Sc1ence+Bus1ness Media, Inc Volume 27, Number 1 2005 73
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 network, 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 oldfashioned algorist who loved to talk mathematics.
Felix Klein knew from first-hand experience how stimulating collaboration with Paul Gordan could be. During the late 1870s, when Klein taught at the Technical University in Munich, he took advantage of every opportunity to meet with his erstwhile Erlangen colleague, who was himself enormously impressed by Klein's fertile geometric imagination. Gordan was widely regarded as Germany's authority on algebraic 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 binary form can always be expressed in terms of a finite number of such in-
74 THE MATHEMATICAL INTELLIGENCER
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 general 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 attack, but neither had the symbolical methods employed by German algebraists produced any substantial new results since Gordan first unveiled his theorem.
Over the next two years Hilbert had ample time to master the various competing techniques. Mter becoming a
Privatdozent in Konigsberg, he was free to develop his own research program, and his inaugural lecture on recent 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 toward 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 Gordan'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 leading mathematical centers in Germany, including Berlin, Leipzig, and Gottingen. During the course of a month, he spoke with some twenty mathematicians from whom he gained a stimulating overview of current research interests throughout the country. Although we can only capture glimpses of these encounters, a number of impressions 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 1875, 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 occasions 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 mathematician'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 represented 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, always possible ("Bericht iiber meine Reise," Hilbert Nachlass 741).
A noteworthy feature of this paper, [Hilbert 1888a], is that Hilbert actually credits Kronecker with having introduced 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 Kronecker. It concerns the possible topological configurations among the components 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 theorem from 1876 (for a summary of subsequent 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 theory. Kronecker dismissed the whole theory of formal invariants as a topic that would die out just as surely as had happened with Hindenburg's combinatorial school (which had flourished in Leipzig at the beginning of the nineteenth 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 numbers, such as the signature of a qua-
The on ly true
invariants , i n
Kronecker' s view,
were numbers ,
such as the
s ignature of a
quadratic form . dratic form (Sylvester's theorem, the algebraist's version of conservation of inertia). He then proceeded to wax forth over foundational issues, beginning 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. Using these notions, he told Hilbert, one can establish a firm foundation for analysis that avoided the Weierstrassian notions of equality and continuity. He further decried the confusion that so often resulted when mathematicians treated the implicitly given irrational quantity (say, x = v5) as equivalent to some sequence of rational numbers that serve as an approximation for it.
Not surprisingly, Hilbert took fairly
extensive notes when Kronecker began expounding these unorthodox views ("Bericht iiber meine Reise"). But he also jotted down a brief comment made by Weierstrass that sheds considerable light on the differences between these two mathematical personalities. When Hilbert visited Weierstrass shortly afterward, he informed him of Kronecker's comments regarding invariant theory, including the prediction that the whole field would soon be forgotten, like the work of the Leipzig combinatorial school. Weierstrass responded by sounding a gentle warning to those who might wish to prophesy the future of a mathematical theory: "Not everything of the combinatorial 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-theoretic questions deeply" ("Bericht tiber meine Reise").
These words, with their almost fatalistic ring, probably left little impression on the young mathematician who recorded them. For Hilbert's intellectual outlook was filled with a buoyant optimism that left no room for resignation. He may not have enjoyed Kronecker's braggadocio, but he was clearly far more receptive to his passionate 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 mastery of the techniques of higher algebra (see [Toeplitz 1922]). True, Klein and Hurwitz had drawn his attention to Weierstrass's theory of periodic complex-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 41 , Handschnftenabteilung, Niedersachs1sche Staats· und Umversitatsbibliothek Gbt·
tingen.
© 2005 Spnnger Sc1ence+Bus1ness Media, Inc , Volume 27, Number 1, 2005 75
in Berlin, Hilbert would enter Leopold theory important? Probably Hilbert Kronecker's principal research domain, 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 ambivalence with respect to Kronecker's legacy emerged as a major theme throughout his career [Weyl 1944, 613].
Like Hilbert, his two closest mathematical 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 mathematician as their single most imposing rival. In later years, Hilbert developed a deep antipathy toward Kronecker's philosophical views, and he did not hesitate to criticize these before public audiences (see [Hilbert 1922]). Yet during the early stages of his career such misgivings-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 approach to algebra.
Hilbert's encounters in the spring of 1888 with Berlin's two senior mathematicians left a deep and lasting impression. 3 Based on the notes he took of these conversations, he must have felt particularly aroused by Kronecker'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 intrinsic significance. Whatever problems he chose to work on-even those he merely thought about but never tried to solve-he always thought of them as constituting important mathematics. What makes a problem or a
carried that question within him for a long time, though anyone familiar with his career knows the answer he eventually came up with; one need only reread his famous Paris address to see how compelling his views on the significance 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 significance of mathematical ideas fell into place. A few strands of the story emerge from the discussions he engaged 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
I t is neither
possi ble nor is
it our duty to
decide i n advance
what is s ign 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 onesided," 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 frequent 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].
76 THE MATHEMATICAL INTELLIGENCER
Fig. 4. Paul Gordan joined Klein on the Er
langen faculty in 1874 and remained there un
til his death in 191 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 fundamental 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 became his life-long friend, J. J. Sylvester. Together, they effectively launched invariant theory as a specialized field of research. Much of its standard terminology was introduced by Sylvester in a major paper from 1853. Thus, for a given binary form f(x,y), a homogeneous 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 invariant 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 invariant can be expressed in terms of these m basis elements. Indeed, he proved that this held generally for homogeneous polynomials J in the coefficients and variables ofj(x,y) with the same invariance property (Sylvester called such an expression J a concomitant of the given form, but the term covariant soon became standard).
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 + CX2cP2 + . . . + 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
© 2005 Spnnger SCience+ Bus1ness Media, Inc , Volume 27, Number 1 , 2005 77
proved that such systems were always finitely generated. There was, however, 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 during 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 defmitive 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 incorporated Hilbert's Theorem I (from [Hilbert 1890]) as well as his Nullstellensatz (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 English translations, see [Hilbert 1978]). Not that these are badly written; they simply reflect a far less familiar mathematical 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 negative views about invariant theory, making it highly improbable that he
4See the editorial note in [Hilbert 1 933. 1 77).
would view Hilbert's adaptation of his ideas with approval. Indeed, Kronecker 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 recognized the fertility of Kronecker's conceptions for invariant theory. Acknowledging his debt to the Berlin algebraist, he parted company with him by adopting a radically non-constructive approach. Ironically, the initial 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 dramatic foundations debates of the 1920s (see [Hesseling 2003]).
Mathematics as Theology
Kronecker seems to have simply ignored Hilbert's dramatic breakthrough, 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 enthusiastically, 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 approach to invariant theory was "theology 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 Mathematische Annalen. Klein was overjoyed, and wrote back to Hilbert a day later: "I do not doubt that this is the most important 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 reaction. He certainly got one. The cantankerous Gordan forcefully voiced his objections, aiming directly at Hilbert's presentation of Theorem I, which Gordan 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 formal rules; he thinks it suffices that no one contradict his proof, then everything will be in order . . . he is content to think that the importance and correctness of his propositions suffice. That might be the case for the first version, but for a comprehensive work for the Annalen this is insufficient." (Gordan to Klein, 24 Feb. 1890, in [Frei 1985, p 65]. Perhaps the misgivings come down to the non-constructivity involved in the [implicit] use of the Axiom 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 discussed its contents with Hilbert. After that the sparks really began to fly. Clearly irked by Gordan's refusal to recognize the soundness of his arguments, 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 earlier 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 manuscript now under scrutiny.
This came about after he had spoken with numerous mathematicians about his key theorem; he had also carried on correspondence with Cayley and Eugen Netto, who wanted him to clarify certain points in the proof. Taking these various reactions into ac-
5The earliest reference to Gordan's remark- ''Das ist keine Mathematik, das ist Theologie"-appears to be [Blumenthal 1 935, 394).
78 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
© 2005 Spnnger Sc1ence+ Bus1ness Media, Inc , Volume 27, Number 1 . 2005 79
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 Gdttinger Nachrichten ([Hilbert 1891 ] ,
[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-
6For an English translation of this and other works by Hilbert. see [Hilbert 1 978].
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
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].
80 THE MATHEMATICAL INTELLIGENCER
Killing off a Mathematical Theory
Thus by 1893 Hilbert's active involvement 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 account offers an interesting participant's history of the classical theory of invariants. At the time he wrote it, invariant 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 algebraic invariant theory. This stage, he added, was soon superseded by a "formal period," whose leading figures were his own direct predecessors, Alfred Clebsch and Paul Gordan. A mature mathematical theory, Hilbert went on, typically culminates in a third, "critical 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 aspects of invariant theory had only begun 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 fourteenth Paris problem. Although he did offer a one-semester course on invariant theory in 1897 (see [Hilbert 1993]), by this time his eyes were already on other fields and new challenges.
Mathematicians are constantly looking 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-
nor episode at the outset of his Siegeszug through invariant theory. Otto Blumenthal, Hilbert's first biographer, even got the city wrong, claiming that Hilbert went to Erlangen to visit Gordan in the spring of 1888 [Blumenthal 1935, 394]. By then forty years had passed, and presumably no one, not even Hilbert, remembered what had happened. Yet his own characterization 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 Invariants" gave Hilbert the initial impulse that put him on his way. Back in Konigsberg, he adopted several of Gordan's techniques in his subsequent work. Numerous citations reveal that he was thoroughly familiar with Gordan's opus, especially the two volumes of his lectures edited by Georg Kerschensteiner [Gordan 1885/1887]. That work, the springboard for many of Hilbert's discoveries, was by 1893 practically 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 castles of those robber knights of invariant theory, so long as something more useful could be built on their now barren 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 monograph 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 "Kerschensteine") [Minkowski 1973, 45].
Hilbert did neither, 10 leaving the theory of invariants to languish on its own while the Gordan-Kerschensteiner
volumes gathered dust in local libraries. Invariant theory thus entered the annals of mathematics, its history already sketched by the man who wrote its epitaph. To the younger generation, 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 invariant 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 Although Leopold Kronecker had predicted this very outcome, he would hardly have approved of the executioner'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, 1810-1933, 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 (1898-1918): 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 (1 886-1918) , 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 81
I 6 vols. , ed. Charles C. Gillispie, vol. 6, pp.
388-395, New York: Charles Scribner's
Sons, 1 97 1 .
[Gilmer 1 981 ] Robert Gilmer, "Commutative
Ring Theory," in Emmy Noether. A Tribute to
her Life and Work, ed. James W. Brewer and
Martha K. Smith, New York: Marcel Dekker,
1 981 , pp. 1 31 -1 43.
[Gordan 1 885/1 887] Paul Gordan, Vorlesungen
uber lnvariantentheorie, 2 vols., ed. Georg
Kerschensteiner, Leipzig: Teubner, 1 885,
1 887.
[Gordan 1 892] -- , "Uber einen Satz von
Hilbert," Mathematische Annalen 42 (1 892),
1 32-1 42.
[Gordan 1 899] -- , "Neuer Beweis des
Hilbertschen Satzes uber homogene Funk
tionen," Nachrichten der Gesellschaft der
Wissenschaften zu G6ttingen 1 899, 240-
242.
[Hashagen 2003] Ulf Hashagen, Walther von
Oyck (1856-1934). Mathematik, Technik und
Wissenschaftsorganisation an der TH
Munchen, Boethius, Band 47, Stuttgart:
Franz Steiner, 2003.
[Hawkins 2000] Thomas Hawkins, Emergence
of the Theory of Lie Groups. An Essay in the
History of Mathematics, 1 869-1926. New
York: Springer-Verlag, 2000.
[Hesseling 2003] Dennis E. Hesseling, Gnomes
in the Fog. The Reception of Brouwer's In
tuitionism in the 1920's, Basel: Birkhauser,
2003.
[Hilbert 1 888a] David Hilbert, "Uber die Darstel
lung definiter Formen als Summe von For
menquadraten," Mathematische Annalen 32
(1 888), 342-350; reprinted in [Hilbert 1 933,
1 54-1 61 ] .
[Hilbert 1 888b] -- , "Uber die Endlichkeit des
lnvariantensystems fUr binare Grundformen,"
Mathematische Annalen 33 (1 889), 223-
226; reprinted in [Hilbert 1 933, 1 62-1 64].
[Hilbert 1 888c] -- , "Zur Theorie der alge
braischen Gebilde 1 , " Nachrichten der
Gesellschaft der Wissenschaften zu G6ttin
gen 1 888, 450-457; reprinted in (Hilbert
1 933, 1 76-183].
[Hilbert 1 889a] -- , "Zur Theorie der alge
braischen Gebilde I I , " Nachrichten der
Gesellschaft der Wissenschaften zu G6ttin
gen 1 889, 25-34; reprinted in [Hilbert 1 933,
1 84-1 91 ] .
[Hilbert 1 889b] -- , "Zur Theorie der alge
braischen Gebilde I l l , " Nachrichten der
Gesellschaft der Wissenschaften zu G6ttin
gen 1 889, 423-430; reprinted in [Hilbert
1 933, 1 92-1 98].
82 THE MATHEMATICAL INTELLIGENCER
[Hi lbert 1 890] --, "Uber die Theorie der al
gebraischen Formen," Mathematische An
nalen 36 (1 890), 473-534.
[Hilbert 1 891 ] -- , "Uber die Theorie der al
gebraischen lnvarianten 1," Nachrichten der
Gesel/schaft der Wissenschaften zu G6ttin
gen 1 891 , 232-242.
[Hilbert 1 892a] -- , "Uber die Theorie der al
gebraischen lnvarianten I I , " Nachrichten der
Gesellschaft der Wissenschaften zu G6ttin
gen 1 892, 6-1 6.
[Hilbert 1 892b] -- , "Uber die Theorie der al
gebraischen lnvarianten I l l , " Nachrichten der
Gesel/schaft der Wissenschaften zu G6ttin
gen 1 892, 439-449.
[Hilbert 1 893] -- , "Uber die vollen lnvari
antensysteme," Mathematische Annalen 42
(1 893), 31 3-373; reprinted in [Hilbert 1 933,
287-344] .
[Hilbert 1 896] -- , "Uber die Theorie der al
gebraischen lnvarianten," Mathematical Pa
pers Read at the International Mathematical
Congress Chicago 1893, New York: Macmil
lan, 1 896, 1 1 6-1 24; reprinted in [Hilbert
1 933, 376-383].
[Hi lbert 1 922] -- , "Neubegrundung der
Mathematik. Erste Mitteilung," Abhandlun
gen aus dem Mathematischen Seminar der
Hamburgischen Universitat, 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.
[Hilbert 1 978] -- , Hilbert's Invariant Theory
Papers, trans. Michael Ackermann, in Lie
Groups: History, Frontiers, and Applications,
ed. Robert Hermann, Brookline, Mass . : Math
Sci Press, 1 978.
[Hilbert 1 993] --, Theory of Algebraic Invari
ants, trans. Reinhard C. Lauenbacher, Cam
bridge: Cambridge University Press, 1 993.
[Hilbert 1 998] -- , The Theory of Algebraic
Number Fields, trans. lain T. Adamson, New
York: Springer-Verlag, 1 998.
[Mertens 1 887] Franz Mertens, Journal fUr die
reine und angewandte Mathematik 1 00(1 887),
223-230.
[Minkowski 1 973] Hermann Minkowski, Briefe
an David Hilbert, eds. L. ROdenberg und H.
Zassenhaus, New York: Springer-Verlag,
1 973.
[Noether 1 92 1 ] Emmy Noether, "ldealtheorie in
Ringbereichen," Mathematische Annalen 83
(1 921 ) , 24-66.
[Parshall 1 989], Karen H. Parshall, "Toward a
History of Nineteenth-Century Invariant The
ory," in The History of Modern Mathematics:
Volume 1 : Ideas and their Reception, ed.
David E . Rowe and John McCleary, Boston:
Academic Press, 1 989, pp. 1 57-206.
(Parshall and Rowe 1 994] Karen H. Parshall
and David E. Rowe, The Emergence of the
American Mathematical Research Commu
nity, 1876-1900. J.J. Sylvester, Felix Klein,
and E.H. Moore, History of Mathematics,
vol. 8, Providence, Rhode Island: American
Mathematical Society, 1 994.
(Reid 1 970] Constance Reid, Hilbert. New York:
Springer Verlag, 1 970.
[Rowe 1 998] --, "Mathematics in Berlin,
1 81 0-1 933," in Mathematics in Berlin, ed.
H.G.W. Begehr, H. Koch, J . Kramer, N.
Schappacher, and E.-J. Thiele, Basel:
Birkhauser, 1 998, pp. 9-26.
(Rowe 2000] -- , "Episodes in the Berlin
Gbttingen Rivalry, 1 870-1 930," Mathemati
cal lntelligencer, 22(1 ) (2000), 60-69.
[Rowe 2003a] -- , "From Konigsberg to Gbt
tingen: A Sketch of Hilbert's Early Career,"
Mathematical lntelligencer, 25(2) (2003),
44-50.
[Rowe 2003b] -- . "Mathematical Schools,
Communities, and Networks," in Cambridge
History of Science, val. 5, Modern Physical
and Mathematical Sciences, ed. Mary Jo
Nye, Cambridge: Cambridge University Press,
pp. 1 1 3-1 32.
[Rowe 2004] -- , "Making Mathematics in an
Oral Culture: Gbttingen in the Era of Klein
and Hilbert," to appear in Science in Con
text, 2004.
(Toeplitz 1 922] Otto Toeplitz, "Der Alge
braiker Hilbert," Die Naturwissenschaften 1 0 ,
(1 922) 73-77 .
(Weber 1 893] Heinrich Weber, "Leopold Kro
necker," Jahresbericht der Deutschen Math
ematiker-Vereinigung 2(1 893), 5-1 3.
[Weyl 1 932] Hermann Weyl, "Zu David Hilberts
siebzigsten Geburtstag, " Die Naturwis
senschaften 20 (1 932), 57-58; reprinted in
[Weyl 1 968, vol. 3, 346-347].
(Weyl 1 935] --, "Emmy Noether," Scripta
Mathematica, 3 (1 935), 201-220; reprinted
in [Weyl 1 968, vol. 3, 435-444].
[Weyl 1 944] -- , "David Hi lbert and his Math
ematical Work," Bulletin of the American
Mathematical Society, 50, 61 2-654; reprinted
in [Weyl 1 968, vol. 4, 1 30-1 72] .
(Weyl 1 968] -- , Gesammelte Abhandlun
gen, 4 vols . , ed. K. Chandrasekharan, Berlin:
Springer-Verlag, 1 968.
(Yandell 2002] Ben H. Yandell, The Honors
Class. Hilbert's Problems and their Solvers,
Natick, Mass. : A K Peters, 2002.
i;i§lh§l,'tJ Osmo Pekonen , Ed itor I
Feel like writing a review for The
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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
The 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. Galtonian Ideas, III. Some SeventeenthCentury Explorers, IV. Questions of Discovery, and V. Questions of Standards. One could argue about the choice of these titles, their order, or their interrelations. In a strictly historical 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 chronologically.
Trying to get the original publications 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 available. The book contains 21 of the 38 articles and books concerning the history of statistics that Stigler published between 1973 and 1997, including his monograph from 1986, The History of Statistics-the Measurement of Uncertainty before 1900. So one advantage 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 statistical thinking and so statistical concepts permeate the whole range of human 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 incompletely, 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 several 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. Pearson'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 temperance 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 disprove him. Interestingly, Pearson's request 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 devotes the next two chapters to the statistical work of the economist Jevons, who is mentioned several times in the History but without any further evaluation of his statistics. Stigler's interest in Jevons is motivated by the effect of his statistical work in overcoming the typical mid-nineteenth-century separation 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 economist and statistician Francis Ysidro Edgeworth ends the first of the five parts of Statistics on the Table. Edgeworth does not figure prominently in the history of statistics, because his statistical ideas were dispersed over a great many not easily digestible articles. 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 Edgeworth, whom he had honoured already in his History with a full chapter, indicating that in his eyes Edgeworth was as important in the development of statistics as Galton and Karl Pearson.
The five chapters devoted to "Galtonian ideas" in part II. differ from the Galton chapter in the History by emphasizing different topics and the impact 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 acceptance of fingerprints as evidence in court. Galton's and his contemporaries' contribution to "stochastic simulation," with a set of special dice used for the generation of half-normal variants, 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 contribution to it, whereas chapter 8, "The history of statistics in 1933," hints at the more subtle aspects of regression visible 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 mathematical statistics. Stigler's inclination to act occasionally as an agent provocateur might in this case be seen at odds with his History, which should be consequently "The history of non-mathematical statistics. " Chapter 10 then discusses the relatively early use (compared with other social sciences) of statistical techniques in psychology, which according to Stigler is due to experimental 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 Statistics on the Table. Nor do I understand 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, however, a contemporary concept of probability is treated, as used by John Craig in order to determine the trustworthiness of statements concerning historical events, or in Craig's term the "historical probability," which increases in proportion to the number of witnesses in favour of it and decreases in proportion to the time elapsed since the event. A function representing the dependence of Craig's historical probability 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 "Questions 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 devoted to eponymy, the practice in the scientific community of affixing the name of a scientist to a discovery, theory, etc. as a reward for scientific excellence in the relevant field. Such a reward 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 discovery is named after its original discoverer." Since Stigler sees the sociologist 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 appropriate period of time ought to have accepted it. A proper eponymynamely the affixing of the names of Gauss and sometimes Laplace, but not the name of de Moivre, the real "discoverer," to the normal distributionis 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 scientific community. Seen with eyes accustomed to eponymic practice, the next chapter (15), which answers the question "Who discovered Bayes's theorem" with Nicholas Saunderson as the most probable candidate, appears as Stigler's next attempt to avoid a counterexample to the law of eponymy, because most people believe that Bayes discovered Bayes's theorem. A problem not touched so far when dealing with discoveries was treated by Thomas S. Kuhn, who pointed to simultaneous discovery of the "same" thing by several people. In his discussion of Kuhn, Yehuda Elkana pointed to difficulties inherent in the concept of "sameness. " Difficulties of this kind are the topic of chapter 16, which describes 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 concerning the method of least squares. The data of the French meridian arc measurements and their interpolation by Gauss in 1799 are interpreted as inconclusive for Gauss's claim to have devised and applied the method of
least squares at that time or even before. Again Stigler's social attitude as the Robin Hood of the history of statistics becomes evident when he states that, despite Gauss's undisputed merits in developing algorithms for the computation of estimates, it was Legendre "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-independent model to the data of incomplete contingency tables, testing the fit by a chi-square test, which, as was recognized 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 freedom would not have changed Pearson's conclusions.
The last three chapters are subsumed under the title "Statistics and Standards." In the first the observation that many of the most powerful statistical methods, like the method of least squares, are originally connected with the determination of standards like the standard meter, is interpreted not as accidental but as a consequence of the purpose of a standard to measure, count, or compare as accurately as possible. In a second part Stigler describes how the fact that in experimental science no absolute accuracy can be achieved, that every measurement is inevitably connected with error and so with uncertainty, eventually led to the creation of standards of uncertainty, standards in statistics like standard error curves or standard deviations.
The last chapter (22), written together with William H. Kruskal, is related 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 ambiguity of the words "normal" and "normality," 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 correctness of the coin production at the Royal Mint for more than seven centuries. Stigler interprets the trial of the Pyx as "a marvellous example of a sampling inspection scheme for the maintenance of quality." It is amusing to read his report of the most famous master of the Royal Mint, Isaac Newton. Stigler amasses arguments for scepticism concerning Newton's honesty as master of the Mint, thus disqualifying 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 basis of the research work of others "no grounds for believing that he <Newton) took advantage of this knowledge for illegal personal gain."
Nearly all the 22 articles of this collection display Stigler's wit and humour; they are good to read. They do not make up a coherent story or history, but most of them contain, encapsulated in some special question, the whole of statistics.
Muenchner Zentrum fUr Wissenschafts- und
Technikgeschichte
Deutsches Museum
80306 Muenchen, Germany
e-mail: IVo [email protected]
Theory of Bergman Spaces Boris Korenblum, Haakan
Hedenmalm, and Kehe Zhu
HEIDELBERG, SPRINGER-VERLAG 2000. 286 PAGES, 2 ILLUS €59 50
REVIEWED BY DAVID BEKOLLE
Let D denote the unit disk of the complex plane. For 0 < p :s; oc and
- 1 < a :S oo, the Bergman space A€ of
D is the closed subspace of the Lebesgue space Lg : = LP(D, (1 - lzl2)"'dxdy (z = x + iy) consisting of holomorphic functions. For a = 0, we write AP = A{;. Intensive research on the theory of Bergman spaces has been carried on since the early 1970s. The start followed the "essential" completion of the theory of Hardy spaces on D. In particular, A� is a closed subspace of the Hilbert space a. We call Bergman projector and we denote by Pa the orthogonal projector of the Hilbert space L� onto its closed subspace A�. It is well known that Pa is the integral operator defined on L� by the Bergman kernal Ba(z,w) : = Ca(l - ZW)-(l+a).
One of the first fundamental results in the theory of Bergman spaces was established in 1984 independently by F. Forelli and W. Rudin [7] and E. M. Stein [10] . According to this result, for 1 :S p < oc and - 1 < a < oc, the Bergman projector Pa extends to a bounded projector of L€ to A€ if and only if p > 1. For an account of the results obtained in the 1970s and 1980s, see the book of K. Zhu Operator Theory in Bergman Spaces [ 11 ] .
The aim of the present book by Korenblum, Hedenmalm, and Zhu is to present some deep results obtained in the 1990s in function theory and in operator theory in Bergman spaces on the unit disk, namely:
1. K. Seip's geometric characterizations of interpolation and sampling sequences for A€;
2. the discovery by H. Hedenmalm of contractive zero divisors for A� and its implementation for A€ (p =t- 2) by P. Duren, D. Khavinson, H. S. Shapiro, and C. Sundberg;
3. outstanding results related to the "curiously resistant" characterization problem of zero sequences of A€ functions;
4. other striking results on the biharmonic Green function due to H. Hedenmalm, P. Duren, D. Khavinson, H. S. Shapiro, and C. Sundberg, and on invariant subspaces of A€ due to A Aleman, A. Borichev, H. Hedenmalm, S. Richter, S. M. Shimorin, and C. Sundberg.
The book under review is welcome and will be very useful-among many
© 2005 Spnnger Sc1ence+ Bus1ness Med1a, Inc , Volume 27, Number 1, 2005 85
reasons, because it includes a self-contained proof of K. Seip's geometric characterizations of interpolation and sampling sequences for Bergman spaces A g. Recall that a sequence r = {zj}j of distinct points of D is an interpolation sequence for AR (0 < p < oo) if, for every sequence {w1)1 of complex numbers satisfying the condition
LCl - lzJI2?+ajwjf < 00, j
there exists a function! E Ag such that f(z1) = w1 for all j. A geometric characterization of interpolation sequences for Hardy spaces was obtained in 1958 by L. Carleson [5] for p = oo. For general p E (O,oo] see, e.g., the books [6] and [8].
Following the theorem of ForelliRudin and Stein stated above, interpolation sequences for Ag were first studied by Eric Amar [ 1 ] , and it is unfair that his name is not quoted in this book regarding results of Chapter 4.
To prove his theorems, K. Seip relies heavily
• on a fundamental paper of B. Karenblum [9] and
• on earlier results of A. Beurling [3] on interpolation for the Banach space of functions of exponential type :S a and bounded on the real line.
Seip's characterizations use notions of density inspired by Beurling and Karenblum. One of these notions of density, denoted D1(f)), is defmed as follows. Let r = {z1)1 be a separated (with respect to the Bergman distance) sequence in D, and let r E Cf, 1). We set
D(f,r) LllogjzJI : _!_ < lzJ I < r) . 2
J
log (-1 ) 1-r
For every z E D, we form a new sequence
r z : = { _1Z..Li _-__ z_ }j· - ZjZ
The upper Seip density of r is defined by
D1(f) = limsupr-->1 SUPzErJJ(fz,r).
Finally, K. Seip's Theorem states the following: Let 0 < p < oo and - 1 < a <
86 THE MATHEMATICAL INTELLIGENCER
oo, let r be a sequence of distinct points of D. Then the following conditions are equivalent:
1. r is an interpolating sequence for A&'; 2. r is separated and D1(f) < " ; 1 •
The proof of this theorem sheds light on Korenblum's difficult paper [9].
The book under review is directed at graduate students and new researchers in the field. It will be very useful for senior researchers as well. At the end of each chapter, various exercises are proposed to the reader. Many open problems are also stated. For a more recent report on open problems on zero sequences, invariant subspaces, and factorization of functions in A2, the interested reader may consult the report by Aleman, Hedenmalm, and Richter [2].
As a minor point (compared to the quality and quantity of the book as a whole), the reviewer mentions the falsity of the proof of Theorem 1.21, page 21, which identifies the dual space of A&' (0 < p :S 1, - 1 < a < oo) with the Bloch space. The statement is correct, but its proof should be corrected in the second edition of the book. For a correct proof, see [4] .
REFERENCES
[1 ] Amar, E. Suites d'interpolation pour les
classes de Bergman de Ia boule et du
polydisque de en. Can. J. Math. 30 (1 978),
71 1 -737.
[2] Aleman, A, H. Hedenmalm, and S.
Richter. Recent progress and open prob
lems in the Bergman space (preprint).
[3] Beurling, A The Collected Works of Arne
Beurling by L. Carleson, P. Malliavin, J .
Neuberger, and J . Wermer. Vol. 2, Har
monic Analysis, Boston, Birkhauser (1 989),
341 -365.
[4] Bekolle, D. Bergman spaces with small ex
ponents. Indiana Univ. Math. J. 49 (3)
(2000), 973-993.
[5] Carleson, L. An interpolation problem for
bounded analytic functions. Amer. J. Math.
80 (1 958), 921 -930.
[6] Duren, P. Theory of HP spaces, Academic
Press, New York (1 970).
[7] Forelli, F. and W. Rudin. Projections on
spaces of holomorphic functions in balls,
Indiana Univ. Math. J. 24 (1 974), 593-602.
[8] Garnett, J. B. Bounded Analytic Functions,
Academic Press, New York (1 981) .
[9] Koren blum, B. An extension of the Nevan
linna theory. Acta. Math. 1 35 (1 975), 1 87-
219 .
[1 0] Stein, E. M. Singular integrals and estimates
for the Cauchy-Riemann equations. Bull.
Amer. Math. Soc. 79 (1 973), 440-445.
[1 1 ] Zhu, K. Operator Theory in Function
Spaces. Marcel Dekker, New York (1 990).
Faculte des Sciences
Universite de Yaounde I
B.P. 81 2
Yaounde
Cameroon
e-mail: [email protected]
Gamma: Exploring Euler's Constant Julian Havil
PRINCETON, PRINCETON UNIVERSITY PRESS 2003
XXIII + 266 PAGES. US $29.95. ISBN 0-691 -09983-9
REVIEWED BY GERALD L. ALEXANDERSON
This is a Golden Age-well, at least it is for students and those of us
who love to read about mathematics outside our own area of expertise. In my youth we could choose from the books of E. T. Bell, What Is Mathematics? by Courant and Robbins, some of P6lya's books, and Rademacher and Toeplitz. There were others, but the list was short. Today the catalogues of Springer, Cambridge, Princeton, Oxford, the AMS, and the MAA overflow with general books, accessible to students and mathematical amateurs, and on a wide variety of subjects. Now we even see books coming out on specific numbers, notably Eli Maor's e: The Story of a Number (Princeton University Press, 1998), Paul Nahin's An Imaginary Tale/The Story of v=I (Princeton University Press, 1998), David Blatner's The Joy of Pi (Walker, 1999), Charles Seife's Zero: The Biography of a Dangerous Idea (Penguin, 2000), Hans Walser's The Golden Section (MAA, 2002), and Mario Livia's The Golden Ratio: The Story of ¢, the World's Most Astonishing Number (Broadway, 2003). And here we have Havil's book on the Euler-Mascheroni constant. Given the plethora of inter-
esting numbers, this series could go on for some time.
Of course, numbers don't get very much more interesting than Euler's number 'Y· I was dubious when I read G. J. Chaitin's article, "Thoughts on the Riemann Hypothesis" in the Winter, 2004, issue of this magazine (vol. 26, no. 1) in which he included Havil's book in his list of recent books on the Riemann Hypothesis. I checked Harold Edwards's review of the RH books by Derbyshire, du Sautoy, and Sabbagh in the same issue and noted that he did not see fit to include Gamma in his review. At the time I was familiar with the three reviewed by Edwards and didn't see the relevance of Havil's book in this context. But I see that a case can clearly be made for its inclusion, as Chaitin does.
Havil is obviously enthusiastic about his subject and remarkably erudite. There are many references to developments in mathematics that are related to Euler's constant, often quite recent results. He obviously watches the literature. Most of the connections have to do with problems that at some level involve either natural logarithms or partial sums of the harmonic series.
He looks at ways of calculating 'Y to great accuracy. The problem is nontrivial, for y is defined as limn__,x (Hn -ln n), where Hn = 1 + t + f + i + . . . + _1_, Both Hn and ln n grow without n bound, but they grow very slowly, so just calculating the difference for larger and larger n is not efficient.
What do we know about y? First we learn what we don't know-whether it is irrational, for example. Unlike constants such as 1T and e, where questions of whether they are irrational and transcendental were settled before the 20th century, at the beginning of the 21st we still don't have this information about y. We do learn that from an EulerMaclaurin expansion we get
n 1 1 1 r = I - - ln n - - + --
k � l k 2n 12n2
1 1 -- + -- + 120n4 252n6
One might ask how Lorenzo Mascheroni (better known for his proof that a geometric construction
possible with straightedge and compass can be carried out with the compass alone) got his name attached to y. He approximated it to 32 decimal places, only the first 19 of which were correct! In 1962 Donald Knuth computed 'Y to 1271 decimal places, and in 1999 it was calculated to 108,000,000 decimal places. Writing 'Y as a continued fraction, we find that the convergent 323007/559595 differs from 'Y by 1.025 X 10- 12. Again using continued fractions, Thomas Papanikolaou showed that if 'Y were to be a fraction, its denominator would be greater than 10242080, perhaps providing a hint that 'Y is irrational.
In demonstrating these and many other facts about y, we're led on an illuminating tour of Bernoulli numbers; the Basel problem; f(x), Euler's gamma function; Stirling's approxima-
N u mbers don 't
get m uch more
interest ing than
Euler 's number y. tion formula; and much, much else. And if that weren't enough, in a series of appendices we find a concise introduction to complex function theory.
Sometimes the mathematical statements are sturming and leave one wondering how things seemingly so disparate can be related. In the chapter about appearances of harmonic series, the author talks about musical tones, of course, then describes surprising results on the infrequency of record rainfalls, for example. Next he provides an economical test for destruction of beams to check their strength and breaking points. Then there's a question of sending Jeeps across the desert, and problems of card sorting, Hoare's Quicksort algorithm, the maximum possible overhang of playing cards placed on the edge of a table, and so on and so on. Some of these are fairly well known, though others were to me quite new and surprising. On logarithms, he describes clearly Benford's now well-known but surprisingly recent law on the lack of uniform distribution of digits in collec-
tions of data (like baseball statistics, geographic areas, street addresses, death rates, and such).
Just as Benford's Law says that l's appear more often than 2's as leading digits, 2's more often than 3's, and so on, for a descending curve of frequencies, for me the chapters of the book were rather the opposite, increasing in interest as I went along. I found the beginning material on the history of logarithms rather heavy going. Perhaps it's just too familiar, but there's also the problem that however valuable logarithms were in their infancy, the calculations are not likely to be exciting to the modern reader. Havil quotes Laplace, however: " . . . by shortening the labors, [logarithms] doubled the life of the astronomer." The middle chapters have many cormections to interesting problems; the latter ones make cormections to number theory and succeed in making these quite clear in spite of the details being considerably more mathematically challenging to the reader.
The historical references are charming, and many of the quotes are fascinating and, to me, unfamiliar. Here are a few examples:
• Leo Tolstoy: "A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the denominator the smaller the fraction."
• Heinrich Hertz: "One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them."
• Krzysztof Maslanka: "We mayparaphrasing the famous sentence of George Orwell-say that 'all mathematics is beautiful, yet some is more beautiful than the other. ' But the most beautiful in all mathematics is the Zeta function. There is no doubt about it."
• Paul Erdos (paraphrasing Einstein): "God may not play dice with the Universe, but there's something strange going on with the prime numbers!"
• David Hilbert (in response to a question of which mathematical problem
© 2005 Springer Science+Bus1ness Med1a, Inc., Volume 27, Number 1 , 2005 87
was the most important): "The problem of the zeros of the Zeta function, not only in mathematics, but absolutely most important!"
• This is followed by the following from Morris Kline: "If I could come back after five hundred years and find that the Riemann Hypothesis or Fermat's last 'theorem' was proved, I would be disappointed, because I would be pretty sure, in view of the history of attempts to prove these conjectures, that an enormous amount of time had been spent on proving theorems that are unimportant to the life of man."
The last is identified as the response in an interview. The book in which it appeared is identified correctly and the date is correct, but the interviewer and editors of the book are not identified anywhere. The interviewer happened to be this reviewer.
These and many other sections raise a question. There's a fine line between a book meant largely for entertaining and educating the generally informed but non-specialist reader, and a scholarly book that carefully documents statements made in the text. The quotations above appear without specific references. Even more frustrating to this reader is the lack of specific references to theorems and articles cited. On page 1 13, the author cites a remarkable result of de la Vallee-Poussin of 1898, which he calls "baffling," that "if we divide an integer n by all integers less than it and average the deficits of each quotient to the integer above it, the answer approaches 'Y as n ---> oo. " As an example he points out that _1_ Ig�gg cl lOOOOl - 10000) gives 10000 r� 1 I r r 0.577216 . . . . He then adds that the re-sult remains true if the divisors are those in any arithmetic sequence or if they are only the prime divisors. Someone interested in pursuing this further is in for a literature search, for no specific references are given, and de la Vallee-Poussin, though listed in the index, is not listed in the set of references at the end. Because of the age of the work, Mathematical Reviews and MathSciNet will be of no help. And even Zentralblatt would fail if it hadn't
88 THE MATHEMATICAL INTELLIGENCER
recently incorporated the earlier material from the Jahrbuch. So tracking down something like this is not trivial. On page 1 15, Havil cites a 1990 result of "Maier and Pommerance," in a wellknown paper on gaps between primes (Trans. Amer. Math. Soc. 322 (1) (1990) 201-237), but he does not give this reference, and a reader not knowing something of the result would have a problem: there are lots of Maiers in mathematics-this one happens to be Helmut Maier-and Havil misspells the name of the second author. (The second author is, of course, Carl Pomerance.) The result, which Havil justifiably calls stupendous, is that
lim [ CPn + 1 - Pn) (log log log PnY 7 n->x [(log Pn)(log log Pn)(log log log log Pn)]
2: 4ey!c,
where c = 3 + e-c and Pn is the nth prime. (Incidentally, as stated in the book, the right-hand side is incorrect; it should be 4e�'/c.) I conclude that even in a general book of this sort, some endnotes pointing the reader to the sources would be helpful.
Now, at the risk of appearing petty I'll bring up something even less important but still something of a problem. Recently I reviewed another Princeton University Press book and commented that the use of both serif and sans serif type faces gives "the pages rather a strange look" Another reviewer was less generous and said it looked like an explosion in a font factory. There's no such problem here. The type is dignified and carefully set, to the point where the typesetter has fastidiously made sure that in every differential, the "d" is in roman and the "x" or "y" is in italics! But one could have hoped that the Press would bring on board a sufficient number of proofreaders and fact checkers that we would not find formulas misstated, misleading, or not corresponding to the accompanying figure (see pages 44, 7 4, 94, 100). The errors cited are easily corrected, but when one fmds these slips from time to time, it makes a person wonder what might be wrong in the material that is not so easily checked and with which one is not familiar. Diacrit-
ical marks are only sometimes present (never, apparently, on Erdos and, in the case of the oft-cited de la ValleePoussin, almost always with a grave accent instead of the correct acute accent). Authors are not the best people for fmding these things; they have spent too much time with the manuscript. It is clear from the text that this author regards Euler as a mathematical hero, yet in citing Euler's Introductio, probably his masterpiece, the title is misspelled and the date of publication is wrong (page 15). Other authors' names are misspelled-Montucla, Bernoulli (and why does Jacob Bernoulli have separate entries in the index?). There are the usual typographical errors and little slips in grammar. Those one expects. But some of the errors I have cited are disturbing to the careful reader, and a press with the prestige of Princeton should not be placing something like this in print without more careful checking.
For all that, this is a most enjoyable book-full of good historical asides, truly beautiful bits of mathematics, and clear exposition. I highly recommend it-but I hope that subsequent editions (including the paperback) can include some corrections and more references.
Department of Mathematics and Computer
Science
Santa Clara University
Santa Clara, CA 95053-0290
USA
e-mail: [email protected]
Schliisseltechnologie Mathematik Einbl icke in aktuelle Anwendungen der Mathematik Hans Josef Pesch
STUITGART/LEIPZIG/WIESBADEN: B.G. TEUBNER
VERLAG.
1 AUFLAGE 2002
1n the series Mathematik fur lngenieure und Naturwissenschaftler. 1 85 PAGES. € 22 90 ISBN 3-51 9-02389-X
REVIEWED BY GERHARD BETSCH
The title means what the author claims: Mathematics is a key
technology of our future. It permeates more and more our everyday life. Nevertheless, the number of students of mathematics and of subjects with a strong mathematical background is decreasing-at least in Germany. How can we change this development?
The intended readership consists of high school graduates, high school teachers, and freshmen in subjects with a strong mathematical "flavour" or "vein." But people from all professions with a sufficient interest in mathematics will profit considerably from this book
I give a sketch of the contents.
• John Bernoulli's problem of the brachistochrone (1696) and the Fermat Principle. The isoperimetric problem and its roots in antiquity. More generally: Calculus of variations, its origin and development.
• Problems of optimal control. Pontrjagin's Maximum Principle. Differential Games. The key role of modem numerical mathematics in solving control problems.
• Problems and methods of Optimal Control and Numerical Analysis in connection with the development of the ISS (International Space Station). Return of a space shuttle into the atmosphere. Experiments in space; the Genesis mission. Lagrange points in gravitational fields.
• Automatic steering of aircraft. Compensation, or avoiding of "microbursts."
• Optimal planning of robots. Optimal control of chemical processes. Control problems in economics. Computation in real time.
Features of this book are an abundance of solid historical information, and very informative sections on technical problems involved. The text is supported by very instructive highquality illustrations.
For readers who do not skip the formulas, the author offers carefully selected (non-trivial) problems.
The author is a professor of engineering mathematics in the University of Bayreuth (Germany). The book belongs to a series of textbooks for future engineers and scientists.
A translation of this work into English would be desirable.
Furtbrunnen 1 7
7 1 093 Wei I im Sch6nbuch
Germany
e-mail: Gerhard . Betsch@t -online.de
Sync-How Order Emerges from Chaos in the Universe, Nature, and Daily L ife Steven Str-ogatz
QG, NEW YORK, HYPERION BOOKS, 2004, $US 1 4 95
ISBN 078688721 4
REVIEWED BY A . W . F . EDWARDS
''Being obliged to stay in my room for several days," wrote the
feverish Christiaan Huygens in February 1665, "I have noticed an admirable effect which no-one could ever have imagined. It is that my two newly-made [pendulum] clocks hanging next to each other and separated by one or two feet keep an agreement so exact that the pendulums always oscillate together without variation. After admiring this for a while I finally realised that it occurs through a kind of sympathy: mixing up the swings of the pendulums I found that within half an hour they always return to consonance."
A third of a millennium later the designers of London's new Millennium Bridge for pedestrians-claimed as the world's flattest suspension bridgewere treated to another "admirable effect" which perhaps they should have imagined. As the enthusiastic crowd crossed it on opening day its imperceptible swaying motion caused walkers to adopt an unconscious sailor's roll so as to keep their balance. But of course they did so all together and, as with a child on a swing, positive feedback did the rest. The bridge was closed. Brian Josephson, Nobel Laureate in Physics in 1973, was first with the explanation.
Both these stories, and many more, are relayed in Sync, an eloquent grand tour of synchronous behaviour in physical, biological, and human systems. Divided into three parts, Living Sync, Discovering Sync, and Exploring Sync, it would have been easier reading if Discovering Sync had come first. Sympathetic pendulums and swaying bridges are more accessible images than firing brain cells and coupled oscillators.
The physical chapters in Discovering Sync are particularly rewarding, partly because they are lighter on autobiographical detail. Strogatz almost makes quantum theory and Josephson junctions comprehensible.
Living Sync is burdened with a somniferous chapter "Sleep and the daily struggle for sync" about circadian rhythms and experiments in which people spent long periods isolated from any information about the passage of time (but no mention of the Polar Eskimos and their dayless winters).
In Exploring Sync the author dwells on the inappropriateness of linear models in human affairs-hardly a new thought, but one which needs constant repetition at a time when universities, at least in Europe, are increasingly afflicted by intervention based on the bureaucratic assumption that each is merely the sum of its parts.
A penultimate chapter, "Small world networks," is an interesting account of the properties of communication networks and how their structure-varying between regular and random-influences the path-lengths between typical nodes. One almost expects a further digression into population genealogies and Bayesian probability networks, but it is hard to fit network stories into a book on synchrony.
There is no mathematics. "To convey the vitality of mathematics to a broad spectrum of readers, I've avoided equations altogether, and rely instead on metaphors and images from everyday life to illustrate the key ideas." With considerable success.
Gonville and Caius College
Cambridge CB2 1 TA, U .K.
e-mail: awfe@cam .ac.uk
© 2005 Spnnger Sc1ence+Bus1ness Medta, Inc . Volume 27, Number 1 , 2005 89
Mathematics Unl im ited-2001 and Beyond
dents to learn more and focus on what further away from the relevant probare really the difficult issues rather lems in the natural sciences and soci-
Bjorn Engquist and
Wilfried Schmid, editors
NEW YORK, SPRINGER-VERLAG, 2001 1 237 PAGES,
HARDCOVER US $59.95 ISBN: 3-540-6691 3-2
REVIEWED BY PETER W. MICHOR
Expectations were high at the turn of the century for publications that
would describe the state of mathematics by looking into the future. Hilbert's problems at the Paris congress set the example. This book is the contribution from Springer-Verlag.
This is an anthology of 63 articles including five interviews, by a range of well-known mathematicians and other scientists. Among these articles one finds overviews over particular fields stressing open problems, descriptions of neglected themes, and essays on the relation between mathematics and society. Two themes that are interconnected in many ways in many of the articles are "applications" and "computing."
Let me start by quoting. 1
The authors remember from their high school days how they had to learn computing sines and cosines from tables. The calculator was there, actually we all had our own, but the educational programs had not yet adapted to the new technology. Basically, the same effect hits the universities when we ignore the existence of Matlab, Mathematica, Maple, and similar software, which solves virtually any exercise in basic calculus and linear algebra. Sometimes it appears that many teachers in mathematics regard such software as a threat towards their profession. That is in our view a tragic misunderstanding; proper application of software would allow us to increase the level of calculus, by enabling the stu-
than wasting their time on repetitive trivialities. [ . . . j It is the author's opinion that the undergraduate education at most universities is out of phase with the modern professional application of mathematical models. [ . . . } Not only pure mathematicians seem to neglect the importance of doing computer based mathematics and the need to adapt the education accordingly. Also in classical subjects, like physics and the geosciences, the role of mathematics and computers are kept at a moderate level with little impact on the culture or courses. Some trivial observations explain the slow progress at incorporating modern computing tools. Students are sent like ping-pong balls between university buildings. Each building has its own traditional culture and theories-and its own budget that must be protected. Each building gets its share of courses in a program, and the professors in the building put in much effort to preserve the traditions of their particular subject. The result is a set of 'pure' subjects and strong conservatism-two characteristics that are not well correlated with a multidisciplinary and rapidly developing technological world.
In basic mathematics, it seems that we really teach our students something that can be compared to doing astronomy without telescopes, or doing biology without microscopes. Nobody should come out of basic mathematics instruction at universities without fluency in at least one general-purpose computer algebra program. The following quotation reinforces this opinion2:
Do you have a message that you would like university mathematicians to hear? Of course, there is a very clear message. I am worried that mathematicians are moving further and
1 Langtanger, Tveito: How should we prepare the students of Science and Technology, p. 8 1 2 .
2Mathematics: From the outside looking i n . Achim Sachem interviewed b y V.A. Schmidt.
3After the "Golden Age": What next? Lennart Carleson interviewed by Bjorn Enquist.
4Bailey, Borwein: Experimental mathematics: recent developments and future outlook, p. 55.
5H. Cohen: Computational aspects of number theory, pp. 301-330.
90 THE MATHEMATICAL INTELLIGENCER
ety. In our core scientific engineering areas, we are continuously involved with mathematics, but there are very few mathematicians working at the labs. When we go to the universities, we find that even there the mathematicians are not dealing with the questions we confront. Why? A mathematician's response to a very practical problem is often that it is too complicated. A mathematician will often want to ignore a particular constraint and limit the discussion to a specific problem that captures the essence. This approach is not helpful at all to the engineers. Furthermore, mathematicians seem to feel that there are enough beautiful problems within mathematics.
Also Lennart Carleson3 stresses the importance of maintaining "all the contacts with the neighboring applications, not only with computer science but also with physics and scientific computation and chemistry and biology and so on."
Research in pure mathematics can benefit a lot from using computers. Consider the following remarkable formula4 "whose formal proof requires nothing more sophisticated that freshman calculus:
"' 1 ( 4 2 1T = :;?;0 16k 8k + 1
-8k + 4
1 1 ) --- ---8k + 5 8k + 6
This formula was found using months of PSQL computations, after corresponding but simpler n-th digit formulas were identified for several other constants, including log(2)."
This theme is taken up in another article, 5 which describes in 22 gems the uses of computational techniques and of computer experiments in number theory, in particular in class field theory, and in arithmetic geometry, e.g.,
for the construction of tables of elliptic cuiVes of given conductors. This article concludes with 12 challenges for the twenty-first century, including the Birch and Swinnerton-Dyer conjecture, which is also one of the seven millennium problems.
Only 1 1 of the 63 articles are devoted to pure mathematics alone. One of them6 describes an intriguing new countable class (ff of complex numbers called periods which contains all algebraic numbers. The elementary definition of a period says that its real and imaginary parts are given by the values of absolutely convergent integrals of rational functions with rational coefficients, over domains in !Rn given by polynomial inequalities with rational coefficients. The number 7T, logarithms of algebraic numbers, and values of Riemann's zeta function at integers :::::2 are periods, whereas e, 117T, and Euler's constant y are conjectured not to be periods. It is still an open problem to exhibit at least one number which is not a period. The question is to find an algorithm whether or not two periods are equal, and the conjecture is that one may pass between two integral representations of a period by using only additivity of the integrals, the change of variables formula, and the fundamental theorem of calculus. Periods have also an important role in the theory of £-functions and motives.
This book gives a wide overview of different aspects of the possible future development of pure mathematics, also by posing conjectures, on wideopen fields in applied mathematics and other sciences where mathematics plays or should play an important role, and on questions of education and the usefulness of mathematicians for answering questions in engineering and natural, economic, and social sciences. It is very interesting reading. I hope that it will have some impact on the way in which we educate young mathematicians so that they will be able not only to push the frontiers in mathematics itself in the future (where the accomplishments and prospects are bright enough) but also to answer expectations from outside mathematics.
6M. Kontsevich, D. Zagier: Periods, pp. 771-808.
Fakultat fUr Mathematik
Universitat Wien
Nordbergstrasse 1 5
1 090 Vienna
Austria
e-mail: peter.michor@esi .ac.at
The Mathematical Century by Piergiorgio Odifreddi
Arturo Sangalli, translator;
Foreword by Freeman Dyson
PRINCETON, PRINCETON UNIVERSITY PRESS 2004.
xx + 204 pages. US $27.95 ISBN 0-691 -09294-X
REVIEWED BY GERALD L. ALEXANDERSON
To choose and explain to a general audience the thirty most important
mathematical problems solved in the twentieth century takes courage. Professor Odifreddi has done it here with remarkable clarity and elegance. He recognizes the difficulty of the task, particularly if one tries to do it in 180 pages. The challenges are: (1) the abstraction of modem mathematics and the difficulty of explaining the meaning of the theories to the non-specialist; (2) the vast amount of mathematics produced, particularly in the second half of the century; and (3) the fragmentation of mathematics into subfields. These difficulties do not deter him, and he exhibits an amazing grasp of the various streams of modem mathematics. Further, he largely succeeds in showing how the various problems are related.
The author is never without opinions. On computers: "As is often the case with technology, many changes are for the worse, and the mathematical applications of the computer are no exception. Such is, for example, the case when the computer is used as an idiot savant, in the anxious and futile search for ever larger prime numbers. The record holder at the end of the twentieth century was 26,972.593 - 1, a number that is approximately 2 million digits long." On mathematics: " [A majority of the sub fields of mathematics] are no more than dry and atrophied
twigs, of limited development in both time and space, and which die a natural death." He claims that the discipline "has clearly adopted the typical features of the prevailing industrial society, in which the overproduction of low-quality goods at low cost often takes place by inertia, according to mechanisms that pollute and saturate, and which are harmful for the environment and the consumer. The main problem with any exposition of twentieth-century mathematics is, therefore . . . to separate the wheat from the chaff, burning up the latter and storing the former away in the barn."
When readers recover from trying to decide whether their own contributions would be burned or stored, they can go on to Chapter 1, a brilliant essay on the foundations of mathematics, before they explore the thirty problems. Here the author shows his own mathematical predilection: mathematical logic. In the seventeen pages of this introductory chapter the author takes us from Pythagoras to Leibniz, to Frege and Cantor, Russell, Zermelo, and Fraenkel, then on to Grothendieck, Godel, Bourbaki, Eilenberg and Mac Lane, Church and Rosser, Kleene and Scott. It's a fast tour but leaves the reader with a good idea of how to relate sets, functions, categories, and lambda calculus, a treatment that even someone not much interested in foundations can enjoy. This chapter is something of a tour de force.
Now we come to his choice of problems. With such a list one can never please everyone. He has relied strongly on Hilbert's famous list of twenty-three problems described at the 1900 Paris Congress, as well as those problems solved by Fields Medalists or by winners of the Wolf Prize. This is probably as rational a plan as any in searching for the "top 30" problems, but it also raises questions. Wiles's solution of the Fermat problem is included in spite of the fact that Wiles did not receive a Fields Medal because his age exceeded by a year the traditional cut-off age of 40. There is another question we could raise about using the Fields Medals as a criterion: some have suggested that
© 2005 Spnnger Science+Bus1ness Media, Inc , Volume 27, Number 1 , 2005 91
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92 THE MATHEMATICAL INTELLIGENCER
certain branches of mathematics have been favored by Fields committees and some others have been ignored.
Choosing the most important thirty inevitably raises the "41st chair" question. L'Academie Fran<;aise chooses for membership the "40 immortals," and that makes people wonder who would occupy the 4 1st chair if there were one. From each of the past four centuries there were intellectual luminaries who did not make it: Descartes, Rousseau, Zola, and Proust are in that august company. So what are the problems that might have occupied the 31st or 32nd slots? Or should others replace some of those already on the list? That could provide mathematical dinner party conversation for years.
Let's look at the problems that did make the author's list. They're divided into three categories:
Pure Mathematics
1. Analysis: Lebesgue Measure (1902); 2. Algebra: Steinitz Classification of
Fields (1910); 3. Topology: Brouwer's Fixed-Point
Theorem (1910); 4. Number Theory: Gelfond Trans
cendental Numbers (1929); 5. Logic: Godel's Incompleteness
Theorem (1931); 6. The Calculus of Variations: Doug
las's Minimal Surfaces (1931); 7. Analysis: Schwartz's Theory of Dis
tributions (1945); 8. Differential Topology: Milnor's Ex
otic Structures (1956); 9. Model Theory: Robinson's Hyper
real Numbers (1961); 10. Set Theory: Cohen's Independence
Theorem (1963); 1 1 . Singularity Theory: Thorn's Classi
fication of Catastrophes (1964); 12. Algebra: Gorenstein's Classifica
tion of Finite Groups (1972); 13. Topology: Thurston's Classification
of 3-Dimensional Surfaces (1982); 14. Number Theory: Wiles's Proof of
Fermat's Last Theorem (1995); and 15. Discrete Geometry: Hales's Solu
tion of Kepler's Problem (1998).
Applied Mathematics
1. Crystallography: Bieberbach's Symmetry Groups (1910);
2 . Tensor Calculus: Einstein's Gen-
eral Theory of Relativity (1915) ; 3. Game Theory: Von Neumann's
Minimax Theorem (1928); 4. Functional Analysis: Von Neu
mann's Axiomatization of Quantum Mechanics (1932);
5. Probability Theory: Kolmogorov's Axiomatization (1933);
6. Optimization Theory: Dantzig's Simplex Method (1947);
7. General Equilibrium Theory: the Arrow-Debreu Existence Theorem (1954);
8. The Theory of Formal Languages: Chomsky's Classification (1957);
9. Dynamical Systems Theory: The KAM Theorem (1962); and
10. Knot Theory: Jones Invariants (1984).
Mathematics and the Computer
1. The Theory of Algorithms: Turing's Characterization (1936);
2. Artificial Intelligence: Shannon's Analysis of the Game of Chess (1950);
3. Chaos Theory: Lorenz's Strange Attractor (1963);
4. Computer-Assisted Proofs: The Four-Color Theorem of Appel and Haken (1976); and
5. Fractals: The Mandelbrot Set (1980).
One could easily argue about which problems belong where in the list (knot theory as part of applied mathematics?), but the author describes the distinction he makes between pure and applied mathematics thus: "Mathematics, like the Roman god Janus, has two faces. One is turned inward, toward the human world of ideas and abstractions, while the other looks outward, at the physical world of objects and material things. The first face represents the purity of mathematics, where the attention is unselfishly focused on the discipline's creations, seeking to know and understand them for what they are. The second face constitutes the applied side of mathematics, where the motives are interested, and the aim is to use those same creations for what they can do."
Overall the exposition is extraordinarily fine. The context of a problem is set and the result is explained in terms as simple as the subject allows. Contributors at all levels of the solution are introduced, and if any were awarded a
Fields Medal or a Wolf Prize, that information is included. The language at times is almost poetic. Not having available to me the original Italian edition, I cannot say whether the elegance of the language is primarily the contribution of the author or of the translator. Of course, occasionally one becomes aware that it is a translation. For example, Bishop Berkeley's classic description of infinitesimals as "ghosts of departed quantities" when passed from English to Italian and back to English becomes "ghosts of deceased quantities." It's not as good. Nor is it even correct!
There are a few hints that English is not the native language of the author or the translator, but they are rare: "a regular polyhedra" (page 72), "the best of the two" (page 88). The description of the Mobius strip with a top spinning on the surface (page 78) is a delightful device for explaining orientation, but it could be expressed with less ambiguity. It could have been made more clear by inserting parenthetically what is meant by "traveling once along the strip" or even by drawing a suitable figure. The Mordell Conjecture is credited to Leo Mordell in various references and in the index when surely "Louis Mordell" was intended. The author consistently attributes A. 0. Gelfond's work to Gelfand. Hassler Whitney's name is usually spelled correctly but is misspelled on page 70. David Rodney Heath-Brown may be called "Roger" by his friends, but for the rest of us it looks odd. I would have referred to I. R. Shafarevich, not "Igor." Perhaps the author is closer to some of these giants in mathematics than this reader. But this is quibbling when so much of the text is full of interesting insights and so eloquently expressed.
A person could ask about the intended audience for the book The author divides the references into two parts: "for general readers" and "for advanced readers." When he says "general readers" he doesn't mean someone without any mathematical background. Though the author does a masterful job of describing difficult mathematics in accessible terms, still, sentences like the following are not for the faint of heart: " . . . W eil proposed his own conjecture, a version of the
© 2005 Spnnger Sc1ence+Business Med1a, Inc , Volume 27, Number 1 , 2005 93
Riemann hypothesis for multidimensional algebraic manifolds over finite fields, which became known as the Weil conjecture. It was proved in 1973 by Pierre Deligne [whose] proof was the first significant result obtained through the use of an arsenal of extremely abstract techniques in algebraic geometry (such as schemas and l-actic cohomology) introduced in the 1960s by Alexandre Grothendieck. . . . "
As a lagniappe, the author includes four open problems for the 21st century:
(1) Arithmetic: The Perfect Numbers Problem (300 Be);
(2) Complex Analysis: The Riemann Hypothesis (1859);
(3) Algebraic Topology: The Poincare Conjecture (1904); and
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( 4) Complexity Theory: The P = NP Problem (1972).
The last three are on the Clay Institute's list of million dollar Millennium Prize Problems. In addition to the other Clay Institute problems, there are a few others one might have expected to see-the Goldbach Conjecture, the Twin Primes Conjecture, for example. They're famous and they're still attracting people to work on them.
In addition to the short list of suggested readings and a very helpful index, the author includes chronological lists in a concluding chapter: Hilbert Problems, Fields Medalists, Wolf Prize winners, Turing Awardees, Nobel Laureates-but only those cited in the text. It would be interesting to see com-
plete lists, if only to see what the author has not found to be worthy of inclusion. For example, the first Fields Medalist on the list is Jesse Douglas, who won in 1936 and is cited earlier in the text as the first Fields Medalist, when Lars Ahlfors, who also won that year, was probably the first, at least alphabetically. His contributions just didn't make it into the book
We should, however, be thankful for what we get, a truly gripping account of big problems of the twentieth century.
Department of Mathematics & Computer
Science
Santa Clara University
Santa Clara, CA 95053-0290
USA
e-mail: [email protected]
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94 THE MATHEMATICAL INTELLIGENCER
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The Ph i lamath' s A lphabet-G
Galois: Abel's work on the unsolvability of the general quintic equa
tion was continued by the brilliant young French mathematician Evariste Galois (1811-1832), who determined (in terms of the so-called Galois group) which equations can be solved by radicals. Galois had a short and turbulent life, being sent to jail for political activism. He died tragically in a duel at the age of 20, having sat up the previous night writing out his mathematical achievements for posterity. Gauss: Carl Friedrich Gauss (1777-1855) presented the first satisfactory proof of the fundamental theorem of al-
Galois
Gauss
Please send all submissions to
the Stamp Corner Editor,
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MK7 6AA, England
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gebra (that every polynomial equation has a complex root) and made the first systematic study of the convergence of series. In number theory he initiated the study of congruences and proved the law of quadratic reciprocity. He also showed that a regular n-sided polygon can be constructed with straight-edge and compasses whenever n is a Fermat prime (such as 17, as shown on the stamp). Gazeta mathematica: The Romanian monthly Gazeta mathematica was first published in 1895. With its aim of developing the mathematical knowledge of high-school students, it has had an enormous influence on mathematical life in Romania for many decades. Gerbert: Gerbert of Aurillac (938-1003) trained in Catalonia and was probably the first to introduce the Hindu-Arabic numerals to Christian Europe, using an abacus that he had designed for the purpose. He was crowned Pope Sylvester II in 999. Goldbach's conjecture: In 1742 Christian Goldbach wrote to Leonhard
Gazeta matematica
Gerbert
96 THE MATHEMATICAL INTELLIGENCER © 2005 Spnnger Sc1ence+Bus1ness Med1a, Inc
Euler conjecturing that every even integer (> 2) can be written as the sum of two prime numbers. Although this remains unresolved, a partial result of Chen Jing-Run (1966), shown on the stamp, implies that every sufficiently large even integer can be written as the sum of a prime number and a number with at most two factors. Gregorian calendar: The Julian calendar of 45 BC had 3651/4 days, which was eleven minutes too long. In 1582, Pope Gregory XIII issued an edict that corrected the over-long year by removing three leap days every 400 years, so that 2000 was a leap year but 2 100, 2200 and 2300 are not. The Gregorian calendar was quickly adopted by the Catholic world, and other countries eventually followed suit: Germany in 1700, Britain and the American colonies in 1752, Russia in 1917, and China in 1949.
Goldbach's conjecture
Gregorian calendar