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.
Of course, the change in title had ab
Compared to What? When my article "On the Unreasonable Effectiveness of Mathematics in Mo
solutely no effect on my remarks. Now Prof. Gelfand's remark has
lecular Biology" appeared
(The Math ematical InteUigencer 22, no. 2), I
forced me to rethink
observed with wry surprise the box
fective in biology, for rationalizing ob
containing the Wigner-Gelfand princi
servations. However, biology lacks the
inef
magnificent compression of the physi
ple, asserting "the unreasonable
Mathematics is unquestionably ef
fectiveness of mathematics in the bio
cal sciences, where a small number of
logical sciences."
basic principles allow quantitative
Perhaps the genesis of the title of
diction of many observations to high
my article would interest Prof. Gelfand
precision. A biologist confronted with a
and other readers.
large body of inexplicable observations
In 1998, The Isaac Newton Institute
does not have faith that discovering the mathematical
structure
will
for Mathematical Sciences in Cam
correct
bridge, U.K., held a programme, "Bio
make sense of everything by exposing
molecular Function and Evolution in
the hidden underlying regularities.
the Context of the Genome Project." I
The problem is that historical acci
proposed to one of the organizers (a
dent plays much too important a role.
biologist) that I might speak at the gala
A famous physicist once dismissed my
final symposium. I made this sugges
work, saying: "You're not doing sci
tion with some diffidence-the sym
ence, you're just doing archaeology!"
posium would be attended by numer ous mathematical dignitaries, it would
I'd like to think this was unfair to me
take place in the same room where
it emphasizes a genuine and severe ob
(it certainly was to archaeologists), but
Andrew Wiles initially announced the
stacle to applications of mathematics
proof of Fermat's Last Theorem . . .
in biology.
The following exchange ensued:
Prof. Gelfand may consider it not
"What would you speak on?"
only wrong but ingenuous to believe
"I propose an echo of E.P. Wigner
that mathematics will overcome these
as my title: 'On the unreasonable inef
and perhaps other limitations. In my
fectiveness of mathematics in molecu
case, on the other hand, it was the con
lar biology.' "
viction
A prolonged and uneasy silence. Then: "But, you see, this is not quite the message that we want to send
that
mathematics will
ulti
mately succeed, that motivated my original title. What is the conclusion?
Is mathe
these people." More silence. Then:
matics effective in biology? I must fall
"Would you consider changing 'inef
back on Henny Youngman's famous re
fective' to 'effective'?" Had I the stature of a Gelfand I
sponse when asked "How's your wife?" He retorted, "Compared to what?"
should no doubt have refused, but with my eye on the opportunity I acqui
Arthur M. Lesk
esced. After all (it was easy to per
Department of Haematology
suade myself), isn't it merely a ques
University of Cambridge Clinical School
tion of whether the cup is half full or
Wellcome Trust Institute for Molecular
half empty? I thereby missed what will
4
pre
Mechanisms in Disease
surely be my only chance to have my
Wellcome/MR C Building
name linked with Wigner and Gelfand.
Hills Road, Cambridge, CB2 2'XY, U.K.
THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK
Look Again at Vilnius
It was saddening to read "Vilnius Between the Wars" (Mathematical Tourist, The Mathematical InteUigen cer 22, no. 4)-frrst because the au thors seem oblivious to the terrible or deal of Wilno during World War II, and second because they give such inade quate treatment of Antoni Zygmund, who was surely one of the great glo ries of Polish mathematics. The article starts with a casual men tion of the expansion of the Polish mathematical school "in the beginning of the 20th century." Poland gained in dependence from Russia in 1919 and lost it to the Third Reich in 1939; it was not by chance that Polish mathematics flourished precisely during that 20year period. The authors describe a historical stroll around the city of Vilnius-Wilno up to 1940, and its learned institutions, without even a single mention that Wilno was not only a Polish university town but had a long tradition as a thriv ing center of Jewish culture. In 1939 Wilno (or Vilna) was home to about 60,000 Jews, of which less than 3,000 remained after WWII. I mention these facts as significant in the assessment of Zygmund's singularity at the Uni versity of Wilno. And the years of Zygmund at Wi1no are the city's main claim to mathematical fame. Zygmund became professor and head of the Mathematics Department at Wi1no in 1930, as mentioned in the article, and the years he and his family spent there were probably the very best of his long and productive life. The impact of Zygmund's personality was strongly felt at the University. He not only put Wilno on the international map mathematically, but he, a Catholic Pole by birth, took a leading role in the academic fight for the rights of mi norities (Jewish, Lithuanians, Ukrain ians, Russians-all systematically dis criminated against in Poland), and of those striving for social justice. At that time students' associations were re quired to have "curators," that is, fac ulty sponsors, responsible for them be fore the University Senate. Zygmund
became curator of various minorities' associations and of the Association of Independent Socialist Youth (the great Polish poet Czeslaw Milosz was a member). Members of this Association were frequently jailed, and it was up to their curator to get them out. During one of the periods of repression of Polish universities by the nationalist militarist regime in power, Zygmund was dismissed from his post. Letters of protest from the world mathematical community, especially from his fa mous colleagues in Great Britain and France, succeeded in getting him rein stated. [1] During his tenure at Wilno, Zygmund was one of only two professors at the University who refused to accept the ban of Jewish students from his classes. He only mentioned this brave stance many years later, when accused of antisemitism by colleagues who were much concerned for the rights of Jewish mathematicians in the Soviet Union but not for Jewish mathemati cians persecuted elsewhere. Zygmund, international in science and cosmopolitan in culture as he was, was ardently attached to Poland, and so was his wife. They lived exile as a permanent loss, and their apartment in Chicago was kept as if they had still been in Warsaw. After the end of WWII the Zygmunds considered returning home; personal circumstances delayed the project beyond realization. Pro fessor Zygmund visited his three sis ters in Poland frequently, and kept in touch with the mathematical life of his native country, especially through his involvement with Studia Mathemat ica, which became a main journal of publication for his large (and interna tional) school. Zygmund's lasting fame is due to his deep impact in developing harmonic analysis both in the classical setting and going far beyond, to that of n-di mensional function theory and the the ory of operators, with great impact in tum on probability, partial-differential equations, and several complex vari ables. He should be remembered equally as teacher, and for his unique
eye for highly talented students. He spotted them quickly, encouraged them intensely, introduced them early to his level of research, and took them as collaborators on equal footing. When Zygmund was in his early thir ties, his frrst extraordinary student was J6zef Marcinkiewicz, a first-year stu dent at the University of Wilno. [1] The results of their nine-year-long fruitful collaboration are still central to all of harmonic analysis and its many appli cations, even more than 60 years after Marcinkiewicz's death. He was killed at age 30 during the war, probably by the Soviet Army at Katyn Forest. 1 At the out break of World War II, both Zygmund and Marcinkiewicz were mobilized into the Polish Army; while Marcinkiewicz was captured by the Soviet Army and disappeared, Zygmund, after the Polish Army's defeat, succeeded in returning to Wilno-already renamed Vilnius and in Lithuania. From there he tried to save his family and his life from the immi nent danger of Nazi invasion. The terrible months from November 1939 to March 1940, when he was try ing to secure a way of escape--a job, any job, a visa, some route to safety not yet cut by the Nazis-are summarized in the article by "Thanks to friends such as J. Tamarkin, Zygmund would move to the USA in 1940, where his mathe matical career flourished." Those in volved in saving Zygmund's life were not only his friend J. Tamarkin, but also Norbert Wiener, with whom he had already collaborated, and Jerzy Neyman, who admired him greatly. It was not easy in those days, not even for people of their standing, to help others immigrate. And when after many efforts they succeeded, it was certainly not with a flourish: Zygmund went from his sig nificant and world-recognized position at Wilno to teach for five years at Mount Holyoke College, a girls' college isolated from the mathematical cen ters, with heavy teaching of trigonom etry and analytic geometry. Not only did he never complain, but he was al ways grateful and appreciative of Mount Holyoke College. Thanks to his
'This is what Zygmund always believed most probable. The authors of 'Vilnius between the wars" state it as definite, perhaps on the basis of recently opened Soviet archives.
© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 1 , 2001
5
extraordinary scientific output, he was
Stefan Banach" (or should it be "assis
even though they refer to Zygmund's in
able to move, first to the University of
tant to Professor Banach"?), he never
troduction to Marcinkiewicz's collected
Pennsylvania in
works, of which he was the editor?
1945, and two years
worked with Banach, and he did con
later to the University of Chicago,
tinually work with Zygmund from his
Zygmund's list of honors appears in
where his impact on world mathemat
thesis on. To say that he "MIGHT
the biography by W. Zelazko contained
ics became legendary.
HAVE BECOME A FIRST-RANK MATH
in Zygmund's selected papers, which
Only those who never suffered dan
EMATICIAN [caps mine] if destiny had
the authors cite. It is too bad that
ger of persecution, or even thought
been as favorable to him as it was to
they didn't bother to cite those from
about its effects on others, can see
Zygmund" is utterly wrong-HE DID
"minor" countries like Argentina and
forced exile and dislocation of life as
BECOME
MATHE
Spain, which made Zygmund an acad emician and honorary doctor at differ
A
FIRST-RANK
an opportunistic career move. Appre
MATICIAN EVEN IF HE DIED AT 30-
ciative though Zygmund was for being
and it insults both his memory and
ent universities. From Argentina came
able to survive the war, settle down in
Zygmund's.
Marcinkiewicz is recog
A.P. Calderon, the second extraordinary student of Zygmund, and his collabora
the USA, surmount the difficulties of
nized today largely because Zygmund
exile, and continue doing mathematics
survived
more and more fruitfully, he never con
champion.
the
war
and
became
his
tor during the long second half of his ca reer. I find it ironic that the authors
the
There can be no doubt about the in
chose to omit those countries in which
University of Wilno as a career im
trinsic value of Marcinkiewicz's con
the influence of Zygmund as a teacher
provement.
tributions. But there can be no doubt
and spotter of talent was largest outside
sidered
In
being
the
expelled
from
correspondence
either that Zygmund's devotion to his
of the USA-the ones the master him
1939,
memory was unique. It may be of in
self always compared with his native Poland: small countries where mathe
between
Zygmund and Tamarkin in late
there are repeated mentions of the dis
terest to your readers to learn that in
appearance of Marcinkiewicz, as well as
1939, and while already in captivity,
matics could flourish through local tal
concern for Saks-his best friend, who
Marcinkiewicz was able to write a
ent, but only when favorable political
was ultimately killed by the Gestapo
mathematical letter to Zygmund with
conditions permitted.
Kuratowski, Sierpiff;ki, Mazurkiewicz,
some ideas on how to expand the re
and many others with whom he had lost
sult of a short note to the
contact. Many of his colleagues were
Rendus
killed during the war. When years later
Sciences (which was to be his last pub
to The Mathematical Tourist. I reproach
Cambridge University Press published
lication). In the note Marcinkiewicz
the article for displaying loss of histor
Zygmund's monumental two-volume
had sketched one of his claims to pos
ical memory prevalent in countries with
treatise
it ap
terity, the now famous Marcinkiewicz
a not-too-distant past of repression and
peared "dedicated to the memories of
Interpolation Theorem, for the so-called
exclusion. In those communities there
Trigonometric Series,
A. Rajchman and
J. Marcinkiewicz, my
teacher and my pupil."
I have been reacting to the article, re
Comptes
ally, in terms appropriate to the section
of the Paris Academie des
Mathematical Communities more than
"diagonal case." Zygmund explained
is a tendency to blame the victims, and
that result to his Chicago students and
to reproach as unpatriotic the survival
presented in his seminar the general
in exile of those forced to leave their
astonishing.
case-which is by no means an evident
homelands, while ignoring the circum
They praise his talent, and quote a re
extension-while insisting that he was
stances that often make their return dif
sult in his thesis, but then they sum up
only developing Marcinkiewicz's own
ficult or impossible.
by saying that Marcinkiewicz "spent
ideas. In
About Marcinkiewicz, the authors' portrayal
is
positively
1956, when the whole result
some time at the famous Lvov mathe
was worked out, Zygmund published a
References
matical school and THIS LED TO SEV
paper with the theorem giving his for
1 . Wirszup, I . , Antoni Zygmund, 1 900-1 992, in
ERAL PAPERS IN THE LVOV STUD/A
mer student full credit. The fame of the
The University of Chicago Record,
MATHEMATICA
result is proportional to its constant
2 1 , 1 995, pp. 1 2-1 3.
[my caps]." It is a bit
biased to say that only Lvov had a "fa
use by analysts everywhere. It is hard
mous" school of mathematics, but it is
to
bad history to conceal that the papers
body paying such homage to a long-dead
Department of Mathematics
in Studia were the product of the con
colleague. How could the authors fail to
Howard University
tinuing
Washington, DC 20059
work
realize the relation between the two greatest mathematicians associated with
USA
ever "qualified as assistant professor of
Wilno during the period of their article-
e-mail:
[email protected]
indeed
author
Cora Sadosky
with
6
the
of another example of some
Marcinkiewicz
Zygmund. If
of
think
THE MATHEMATICAL INTELLIGENCER
January
Opinion
Is Mathematics W
ith very few exceptions mathe maticians have always believed,
lowed to say that n is either prime or
composite before anyone makes such
and still believe, that mathematical
a test?
truths have a strange kind of abstract
give his game away?
If Hersh agrees, does
this not
llOut There''t
reality that is discovered, not created.
Primality is a timeless property of
In recent years a tiny minority of mav
certain integers, as independent of hu
erick mathematicians have joined the
manity as pebbles and stars. Humans
Martin Gardner
postmodem ranks of the social con
are not needed to test a pile of pebbles
structivists who see both math and sci
for primality. It can be done by mon
ence as cultural artifacts, unrelated to
keys or even mindless machines. To re
any sort of timeless truth domain.
ply to this by saying that humans are
Reuben Hersh, a distinguished mathe
necessary to have the
matician, has long defended this anti
prime and to call numbers prime is to
realist view, notably in his 1997 book
say something utterly trivial.
the international mathematical
What is Mathematics, Really? If all Hersh means is that mathe
women?) from Quasar .X9 sent us their
community. Disagreement and
matics is part of human culture, then
math textbooks, we would fmd again
of course he is right, but the statement
A = 1rr2." If the aliens haven't advanced to
The ()pinion column offers mathematicians the opportunity to write about any issue of interest to
controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-chief endorses or accepts
is vacuous.Everything humans say and
Myth
concept of a
3: "If the little green men (and
do is part of culture. Hersh obviously
plane geometry, their textbooks of
means something less trivial. He is con
course would not contain this theo
vinced that mathematicians do not dis
rem, but if they knew about circles and
cover timeless
areas how could they not discover that
theorems-theorems
true in all possible worlds. Rather they
a circle's area is
1r
r 2? Given the axioms
responsibility for them. An ()pinion
create ever-changing, uncertain con
of plane geometry, the theorem holds
should be submitted to the editor-in
jectures in much the same way that
in all possible worlds.
chief, Chandler Davis.
others create art, music, religion, wars,
Myth 4: "Mathematics possesses a
and traffic regulations. In an article in
Eureka (March 1988), he discusses
attains absolute certainty of conclu
several "myths" which he claims have
sions, given the truth of the premises."
been mistakenly defended by great mathematicians.
Can Hersh be serious when he calls this a myth?
No one can quarrel with Hersh's first myth, that Euclid put plane geom etry on
method called 'proof' ...by which one
In mathematics, unlike in
science, proof is the essence. Given the symbols, and the formation and trans
a firm formal foundation. All
theorems are tautologies. They are, as
2: "Mathematical truth or
synthetic. Even in the center of the
mathematicians today agree that he did not. Myth
formation rules of a formal system,
all
Kant was the first to say, analytic, not
knowledge is the same for everyone. It
sun, Bertrand Russell once wrote, two
does not depend on who in particular
plus two equals four.
discovers it; in fact, it is true whether or not anyone discovers it."
I once put it this way. If two di nosaurs joined two other dinosaurs in
What a strange contention! In no
even though no humans were around
the Pythagorean theorem not certain
to observe it, and beasts were too stu
culture on earth, or anywhere else, is
within the formal system of Euclidian geometry.
a clearing, there would be four there
pid to know it. Mathematical structure
was deeply embedded in the universe
Consider a heap of n pebbles. The
number is prime only if, when you take
long
before
sentient
life
evolved.
Indeed, the structure was there a mi
k each
crosecond after the big bang, and even
(k not 1 or n) , there always will be one
before the bang because there had to
away pebbles in increments of
or more pebbles left over. Are we al-
be quantum fields to fluctuate and ex-
© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 1, 2001
7
plode. Is Hersh willing to say that galaxies had a spiral structure before creatures were around to use the word "spiral"? No mathematician, Roger Penrose has observed, probing deeper into the intricate structure of the Mandelbrot set, can imagine he is not exploring a pattern as much "out there," indepen dent of his little mind and his culture, as an astronaut exploring the surface of Mars.
To imagine that these awesomely complicated and beautiful patterns are not "out there," independent of you and me, but somehow cobbled by our minds in the way we write poetry and compose music, is surely the ultimate in hubris. "Glory to Man in the high est," sang Swinburne, "for Man is the master of things."
In the light of today's physics the en tire universe has dissolved into pure mathematics. The cosmos is made of molecules, in tum made of atoms, in tum made of particles which in tum may be made of superstrings. On the pre-atomic level the basic particles and fields are not made of anything. They can be described only as pure mathe matical structures. If a photon or quark or superstring isn't made of mathe matics, pray tell me what it is made of?
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Mathematical Olympiad Challenges Titu Andreescu, American Mathematics Competitions, University of Nebraska, Lincoln, NE Riizvan Gelca, University ofMichigan, Ann Arbor, MI This is a comprehensive collection of problems written by two experienced and well-known mathematics educators and coaches of the U.S. International Mathematical Olympiad Team. Hundreds of beautiful, challenging, and instructive problems from decades of national and international competitions are presented, encouraging readers to move away from routine exer cises and memorized algorithms toward creative solutions and non-standard problem-solving techniques. The work is divided into problems clustered in self-contained sections with solutions provid ed separately. Along with background material, each section includes representative examples, beautiful diagrams, and lists of unconventional problems. Additionally, historical insights and asides are presented to stimulate further inquiry. The emphasis throughout is on stimulating readers to find ingenious and elegant solutions to problems with multiple approaches. Aimed at motivated high school and beginning college students and instructors, this work can be used as a text for advanced problem-solving courses, for self-study, or as a resource for teach ers and students training for mathematical competitions and for teacher professional develop ment, seminars, and workshops. From the foreword by Mark Saul: "The book weaves together Olympiad problems with a com mon theme, so that insights become techniques, tricks become methods, and methods build to
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8
THE MATHEMATICAL INTELLIGENCER
7/00 Promotion #YI097
I $29.95 $59.95
XIAORONG HOU, HONGBO Ll, DONGMING WANG, AND LU YANG
"Russian Killer" No. 2: A Challenging Geometric Theorem
with Human and Machine Proofs
� •
n February 1998 Sergey Markelov [7] from the Moscow Center for Continuous Mathematics Education sent a set of five geometric theorems to Dongming Wang for test ing the capability of his GEOTHER package [8], with the aim of presenting a chal lenge to computer provers to prove really hard theorems. These theorems have been
used to prepare the Moscow team for the all-Russia school mathematics Olympiad, and are called kiUers to analytic ways ofgeometric problem-s olv ing. They can be proved in geometric ways, but no analytic proof could be found even by expert geometers. Let us call these five theorems the Russ ian kiUers for short. After a quick look at the five killers, Wang was convinced that some of them can be proved by GEOTHER in princi ple. For experimental purposes, he took the second of the killers which is stated below. This killer is very easy to ex plain and to understand, and it provides a beautiful repre sentation of the area of an arbitrary quadrilateral in terms of its four sides and four internal angles. Theorem. Let ABCD be an arbitrary quadrilateral with s ides !ABI = k, !Bel = l, ICDI = m, IDAI = n, and intern al angles 2a, 2b, 2c, 2d at v ertices A, B, C , D respectiv ely; and let S be th e area of th e quadrilateral. Then (k + l +m + n)2 48 = cot a + cot b + cot c + cot d (l +n- k - m)2 tan a + tan b + tan c +tan d' The beauty of the expression lies partially in the sepa ration of the sides and internal angles. The theorem gen eralizes the well-known Brah maguptaformula for the case where the quadrilateral is inscribed in a circle. After a few trials, Wang announced a machine proof of the theorem using Wu's method [11] in GEOTHER in Aprill998; this proof requires heavy polynomial computations. Meanwhile, he posted the theorem to several colleagues, so liciting other machine or human proofs. Soon after that,
Hongbo Li announced another machine proof using Clifford algebra formalism, followed by the third machine proof given by Lu Yang using complex numbers, both in May 1998. The proofs of Li and Yang are short and took only a few seconds of computing time. Finally in later May 1998, Xiaorong Hou discovered an elegant and short geometric proof of the the orem. This proof reflects the common features of traditional geometric proofs, in which one may just have to be inspired. This article collects the four proofs, together with an al ternative approach using rules of trigonometric functions provided by an anonymous referee. Our purpose is twofold: on the one hand, different kinds of possibly new proofs of a difficult geometric theorem are presented that have clear interest for geometers. On the other hand, the proofs demonstrate the power, capability and features of auto mated deduction methods and tools-which reduce quali tative difficulty to quantitative complexity instead of rely ing on individual ingenious ideas-for proving hard theorems: the Chinese provers against the Russian killers! D
Fig. 1
B
© 2001 SPRINGER-VERLAG NEW YORK. VOLUME 23. NUMBER 1, 2001
9
We look for the advantages and disadvantages of machine proofs versus human proofs. Proving mathematical theorems automatically has been an active area of research with appealing prospects for ed ucation. It has been advocated by many mathematicians and computer scientists. For geometry in particular, there are several very successful algebraic approaches including the well-known method of Wu [11]. We may also mention a recent interesting article about RENE by Ekhad [ 1] who has predicted the future of plane geometry (around 2050! ). The present case study not only provides another example for one to see the machine power and intelligence against human ingenuity, but also contributes to our understand ing about effective algorithms and software tools for au tomated theorem proving. A Traditional Geometric Proof In this section we give a geometric proof of the theorem by the first author. The non-trivial ideas and special tech niques used in the proof, besides their own value, may be illuminating as a contrast to machine proving, which is au tomatic, straightforward, and fast. Let 8 = {a, b, c, d) and
T = k + l +2 m + n ' t = l + n -2 k - m .' f3 I cot e. a= I tan(}, =
liE8
B Fig. 2
Corollary 2.
+ lcnl
V!FI+ IB'£1 -Icnl =!FBI- ViEI + IB'£1-Icnl T- CVtRI + lcnl) = t, IB'C'I + IDA'I = ICEI+ IDAI - IFB'I = !Bel - IBEI + InAI - IFB' 1 = ciBC1 + InAI) - r = t, Vt'B'I
=
=
so A'B'C'D has an inscribed circle. Moreover,
liA'B'C'D Corollary 3.
Proof Let
=
liES
=
T2 t2 73- -;; ·
STEP 1. We first note the following lemma, which is known
and can be easily proved.
8
= I!_ ' {3
I
=
(cot a
II AB;
So FBEB' has an inscribed circle. The details of construc tion are given in step 5.
STEP 3. In what follows, let liABcD or liABc denote the area of quadrilateral ABCD or triangle ABC. We have the fol lowing corollaries, of which the first two follow immedi ately from Lemma 1.
10
1,FBI -
THE MATHEMATICAL INTELLIGENCER
_
(cot
a + cot b) · T {3
2
.
b·'
a·f3 sin(a+ b) · sin(a+ d) · t
·
a·f3·y
sin(c
f3
which yields five points B', E, F, A', C', such that
T2 fiFBEB' = 73'
..!.sin d·cos
(tan a + tan b) · (cot a+ cot d)
c
STEP 2. Let ABCD be an arbitrary quadrilateral. Without loss of generality, we assume that iBCI + IDAI > W11 + ICDI and IBCI 2:: foAl. Then one can construct a diagram as in Fig. 2,
Corollary 1.
a
b) . t
·
t · T · sin
·sin 2c ·(cot c +cot d) · t · T ·sin
·
2c
a·f3
+ b) · sin(c + d) t · T ·
a·f3·y sin( 7T - a - d) · sin( 7T- a
= fiFAA'B'·
a·f3·y
STEP 4. Let I be the intersection Fig. 2. By Corollary 3, we have fiABCD = fiABEI + fi!ECD =
2a
T
(tan + tan b)
+ cot b) · T
FB' II AD, EB' II CD, B'C' II BC, A'B' IFBI + �B'I = iBEI + IFB'I = T.
(tan a+tan
=
= Vt'B'I · IFB'I · sin 2a
liEcC'B' = IB'C'I · IEB'I
Lemma 1. If ABCD has an inscribed circle, then pv 1 AD
liFAA'B'
Vt'B'I
'Y =
we have
The proof of the theorem consists of the following five steps.
,
liFAA'B' = liEcC'B'·
The formula to be proved becomes
s
f._ a
- b) · t ·T
point of B'E and A'D in
(fiFBEB' - fiFAIB) + fi!ECD
= fiFBEB'- (fiFAA'B' + fiB'A'I) + fi!ECD = fiFBEB'- (fiECC'B'- fi!ECD) - fiB'A'I = fiFBEB'- fiA'B'C'D·
The following theorem is therefore established. Theorem 1.
liABcD = liFBEB'- liA'B'C'D·
STEP 5. Construct the diagram in Fig. 3 according to the steps detailed below. It is a simple exercise to verify that the con structed diagram satisfies the requirements given in step 2.
•
•
•
• • • • •
Draw the inscribed circle of the three sides AB, BC , D A . Draw the tangent line GH to the circle, parallel to line CD . This produces a point G on the segment BC and a point H on the segment DA. Mark off a segment BP of length T and a segment BQ of length CW1J +!BGj + jGHj + �Aj)/2 on line BA from point B. Draw the parallel to line QH through point P, intersect ing line BH at point B'. Draw the parallel to line DC through point B', intersect ing line BC at E. Draw the parallel to line DA through point B', intersect ing line BA at F. Draw the parallel to line AB through point B' , intersect ing line DA at A' . Draw the parallel to line BC through B' , intersecting line CD at C' .
This completes the proof of the theorem. The geometric constructions used to reduce the prob lem in the above proof are clearly crucial and well thought out. It is not trivial to figure out such constructions and proofs even for experts. The reader is urged to work out his own geometric proofs. A Machine Proof Using Wu's Method
In contrast with the geometric proof presented in the pre ceding section, the machine proof explained below is straightforward. Instead of ingenious ideas, we simply write a short natural specification of the theorem, apply a geometry theorem prover-Wprover in GEOTHER [8] developed on the basis of Wu's method [ 11), and let the machine do the computation and proving. According to the geometric hypotheses, we have the following relations:
h1 = 28- kn sin 2a- lm sin 2c = 0, h2 = 28 - kl sin 2b- mn sin 2d = 0, h3 = k2 + n2- 2kn cos 2a- l2 - m2 + 2lm cos 2c = h4 = k2 + l2- 2kl cos 2b - m2 - n2 + 2mn cos 2d = h5 = sin (a + b + c + d) = 0.
Let e
= (a, b, c, d) as before, and Xo
=
sin 0,
Yo
=
cos 0,
0, 0,
0 E e.
By expanding the sine and cosine of double angles and sum of angles with simple substitution, the above h1, . . . , h5
will become expressions in Xa, Clearly,
l, m, n.
ho = � Thus
h1 =
..., xd, Ya, ...Yd and 8, k,
+ Jlo - 1 = o,
0, . . . , h5
=
e
E e.
0, h a = 0, . . . , hd = 0
(H)
constitute the hypothesis of the theorem, and the conclu sion to be proved is
Yo +tdL, Xo= . O IeEe Xo IeEe Yo
g =8-T2 1L
The statement of the theorem implies that the denomina tors do not vanish (e.g., x0 * 0 and y0 * 0 for every e E e). So we only need to prove that (H) implies that the nu merator g* of g is 0. For this purpose, let us simply apply Wu's method of automated geometry theorem proving [11). Without loss of generality, take n = 1. Then g* becomes a polynomial consisting of 91 terms. With respect to the variable ordering
Yb < xb < Yc < Xc < Yd < xd < Ya < Xa < l < k < m < 8, the set of hypothesis-polynomials h1, . . . , h5, ha .. . , hd may be easily triangularized by variable elimination into an "almost equivalent" set (called a quasi-characteristic set) of 8 polynomials:
C2 = he, C3 = h ,d � + b X d b b Y � X d b- YY Y 2( 2YEYcYdXc + 2YY 2 + ) X d - YcYdXc 4y� E Ya YbYcXbXc 4YY b Y c X a X b c + 2yElfc + 2yEy� + 2�y� YE- ic -y�, C5 = l5Xa + YY a bXcXd + YaYY a Y c X d b- YX b cX,d b X d c + YaYY + C5 = I�+ (YY b Y � X d c YY b X c d YbY�d Y�XbXcXd + Y�YdXb- YcYdX)b l - yy b X � c +YcYdXb + YbYdXc + - YcYhb YbYcY�Xd Y�XbXcX,d c7 = hm + ( 2YlE - l- 2!/a + 1 )k- l2 + 1, c8 = 8- Y�blk- yx d �, C1 C4
= =
h b,
where
/5 h
= YbYcYd- YbXcXd YdXbXc- YcXbX,d
3YEYcYdXb- YX E bXX c d + Y�YdXc + 3yY � X c d - YbYdXc- 2YY b X c d - YY c X d b + 4yy E x � x b x c d - 4yy E y � x d b- 4yy � y � x d c- 4y�y�xd + Y�YdXb- Y�XbXcXd + 3yblfcYdXc + 3yby�x,d h 2lic - l - 2y� + 1. =
=
During the computation, a polynomial factor l2 - 1 is re moved. The "almost equivalence" means that (H) is equiv alent to c1 = 0, . . . , c8 = 0 under the subsidiary condition that (D)
F Fig. 3
A
B
With this condition assumed, proving the theorem is re duced to verifying whether the pseudo-remainder R of g* with respect to [c1 , .. . , c8] is identically equal to 0. The remainder R is indeed 0: the verification is not easy and takes about 3 20 seconds of CPU time in Maple V.3 on an Alpha station. Some of the polynomials occurring are very
VOLUME 23, NUMBER 1, 2001
11
large. R is obtained successively as follows: Compute first
tations are performed with polynomials. It is known that
the pseudo-remainder
geometric problems may also be formalized in other al
R8 of g* with respect to c8 in S, then the pseudo-remainder R7 of R8 with respect to c7 in m, and so forth. R is the last pseudo-remainder R1 of Rz with re spect to c1 in xb. Here, S, m, . . . , xb are the leading vari ables of c8, c7, . . . , c1, respectively. Let us use an index triple [t v 8] to characterize an ar bitrary polynomial P, where t is the number of terms in P, v the leading variable of P, and 8 the degree of P in v. The reduction of g* to 0 using c8 , . . . , c1 may be sketched as
computations have to be carried out according to the
follows:
proving.
g*
�
�
�
=
!
(93 S 1] � [106 m 2]� [680 k 2]
gebras. In the next proof, Clifford algebra is used to rep resent geometric relations, where each algebraic expres sion has a clearer geometric meaning. In this case, rules in Clifford algebra. The reader is referred to Chapter 1 of
[3] for a geometric introduction to Clifford algebra, [2, 5, 6, 9) for some recent developments on Clifford
and to
algebra approaches for automated geometric theorem
A, B, C , D be considered as vectors from k = B - A, I = C - B, m = D - C , and n = A - D are also v�c!ors. Their correspond Let the points
the origin to the points. Then,
1
[4529 Xa 2]� [6541 Ya 6]� [19013 Xd 9], [3432 Xa 2] � [5221 Ya 6] � [17586 Xd 9], [2540 Xa 2] � [3690 Ya 4]� [11066 Xd 8], [1015 Xa 2] � [1543 Ya 2] � [6276 Xd 7], (4034 Xa 2] � [6067 Ya 6] � [17813 Xd 9] (687 Xc 9], [210 Xc 8], (549 Xc 9], (656 Xc 8], [327 Xc 9], [803 Xc 9], (697 Xc 9], [647 Xc 9], (524 Xc 9], [688 Xc 9], [667 Xc 9], [420 Xc 9], [697 Xc 9], (688 Xc 9], [283 Xc 9], [684 Xc 9], [432 Xc 9], (622 Xc 8], [523 Xc 8], [554 Xc 9], (549 Xc 9], [549 Xc 9], (544 Xc 9], [376 Xc 9], (732 Xc 8], (699 Xc 8], [696 Xc 8], [711 Xc 8], [602 Xc 7], [799 Xc 9], [810 Xc 9], [790 Xc 9], (558 Xc 9], (556 Xc 9], (549 Xc 9], (494 Xc 9], [437 Xc 8], [313 Xc 8], [165 Xc 6], [425 Xc 7], [641 Xc 7], (649 Xc 7], [621 Xc 7], [157 Xc 8], (308 Xc 9], [665 Xc 9], [545 Xc 9], [796 Xc 9], (780 Xc 9], (804 Xc 9]
ing unit vectors are denoted by Let
ll
geometrically the oriented parallelogram formed by two
[505 Xc 7], (582 Xc 9], [693 Xc 9], (519 Xc 9], [538 Xc 9], [730 Xc 8], [646 Xc 7], [218 Xc 8], [170 Xc 7], [293 Xc 7], [444 Xc 9], [718 Xc 9],
Then the hypotheses of the theorem may be expressed as follows. •
•
•
k +I +- m +n = 0. � = kk , I� li, m = �m, n = nii. Angle constraints: iik = -Za, ki = -zb, I m = - Zc, m ii
=
Unit magnitude constraints:
1,
Quadrilateral constraint:
Length constraints:
braic condition can be interpreted (automatically) in most cases, and we do not enter into the details of interpreta
• •
•
nn = 1.
Inequality constraints:
Area constraint: As
Let •
=
s
1 -2 (m
Let
R
x;
a)
r=
Direct computation of the pseudo-remainder
on our machines without this technique is still not possi ble. With some thought and reasoning, the machine proof may be considerably simplified by using different formu lations. This can be seen from the formulation and proofs using complex numbers in the last section below.
A Machine Proof Using Clifford Algebra In the previous machine proof, geometric relations are ex pressed as polynomial equations with lengths of segments and sines and cosines of angles as variables, and compu-
12
THE MATHEMATICAL INTELLIGENCER
= 1,
mm
=
k, 1, m , ii * 0, Za, Zb, Zc, Zd * 0. LlADc + LlAcB, we have
k 1\ I)
1
4cn m - mn + Ik mn + Ik - kl.
=
Ze - 1 , Ze
and r =
/ eEe tan 8ll
-
kl).
= ll tan x for
8 E e.
+1
1
4 r z/I
Rll.
Then
+ 4t21I
/eEe
tan 81l.
So the conclusion to be proved has the form
However, this is already within the reach of a PC Pentium nowadays. The splitting technique in the reduction is due
ll
then
= 4(T2!{3 - t2/
In the above proof of the theorem, we made no attempt to use special techniques to simplify the algebraic formu
= 1,
=
tan 81l =
by using the same method.
lation, so the algebraic computations are very heavy.
1\ n +
kk
Trigonometric transformations: Let tan xll any scalar
some degenerate cases in which the geometric theorem may be false or meaningless. Moreover, whether the theo
S
= 4SI1; then s = nm -
tion. In general, the subsidiary condition corresponds to
rem is true in a special or degenerate case can be checked
-
-zd.
Therefore, the theorem is proved to be true under the sub sidiary condition (D). The geometric meaning of the alge
8 E e.
e28rr,
Ze =
SI1
[10].
and n.
unit vectors. Let
1 0 }.
to Wu
k, I, m,
be a unit bivector of the plane, which represents
[j
=
s-
r =
0.
The proof of the theorem now proceeds in a way simi lar to that in Wu's method. The hypothesis-expressions are first triangularized and then used to reduce the conclusion expression to
0.
The triangulation and reduction process,
which is described in
[5, 6]
tion-solving method,
is however different.
and is called a
vectorial equa
To make triangulation simple, we choose the following basic variables together with an ordering:
k <
n
<
< tan all < l < Zb < tan bll < m --:.._ Zc -:=: tan ell < zd < tan dll < k < I < m < ii < k < I < m < n < &
Za
The triangulation process for the hypothesis-expressions then consists in solving vectorial equations. The computa tions are quite simple and can be done by hand. We omit the details of triangulation and present the result as follows:
Za Z b- I tan biJ= +I' Zb
( :) ( )
)
m2=k2+l2+n2-kl z+ -kn z+ a i b Za b I _, + ln zZ a +_ b ZaZb Zc= z- .!!:._+_!!:__ Im, Zb ZZ a b I' _ m , tan cii= z- .!!:._+ __!!:__ - m \ ftz-.!!:._ + __!!:_+ Zb ZZ a b v \ Zb ZZ a b l zd= n- .!!:._+ _ Im, Za ZZ a b I' l l tan dii= n- .!!:._+ _ - m \ li n- .!!:._+ _ +m , Za ZZ Za ZaZb V\ a b
)
(
(
(
)
(
j= _....!_ k Zb ' m= -�+
(
n=-zak,
)(
)
)
)(
: b+n Z)a kjm,
I= _.l_ k Zb ' +n Za k, m= -k � b n= -n zk a ,
) (
) (
)
> r: = (k+1+m+n)"2/ (1/tan(ai) +1/tan(bi)
> (tan(ai)+tan(bi)+tan(ci)+tan(di)): > tan(ai):=(za-1) I (za+1):
> tan(bi) :=(zb-1)/{zb+1):
eqn(m):=m"2-(k"2+1"2+n"2-k*1*{zb+1/zb)
> -k*n*{za+1/za)+l*n*(za*zb+1/(za*zb))) : > tan(ci):=(1-k/zb+n/(za*zb)-m)/{1-k/zb > +n/(za*zb)+m):
> tan(di):=(n-k/za+1/(za*zb)-m)/(n-k/za +1/(za*zb)+m):
> s:=k*n*{za-1/za)+k*1*(zb-1/zb) -1*n* > (za*zb-1/(za*zb)): >
>
S
=
11nAB+ 11Bcn, S= 11ABc+11cnA, S= 11ABE- 11ncE,
cos 20=
> +1/tan(ci)+1/tan(di))+(k-1+m-n)"2/
>
Machine Proofs Using Complex Numbers
Assume that ABCD is not a parallelogram; otherwise, the statement is trivial. So, we may let lines AD and BC inter sect at E. The area S may be expressed in three forms:
e2i0
Finally we need to verify whether the conclusion-ex pression g can be reduced to 0 by the above triangularized sequence of expressions. A computer algebra system is re quired to carry out this computation. The following Maple session shows how g is reduced to 0 by simple substitution and pseudo-division.
>
Thus the theorem is proved to be true under some non degeneracy conditions. The above computation was done with Maple V .5 on a Pentium Pro 200MHz with 64M memory. The automated proof above clearly involves much less algebraic computation than that in the previous section, but it requires operations and properties from advanced Clifford algebra. The proofs in the following section are rendered even simpler by more specialized techniques with the knowledge of complex analysis.
In order to reduce the number of variables and the com putational complexity, let us denote by u0, where i= v:::.::l Then, we have
)
:
(
0
!1= ht= 2S-kn sin 2a- lm sin 2c= 0, !2= h2= 2S-kl sin 2b- mn sin 2d= 0, k2 m2 = !3= 28cot 2c+cot 2d O. cot 2a+cot 2b
I = kn za- ....!_ + kl zb- ....!_ -zn zz a b- __ · ZaZb Za Zb
s
855
> st:=time() :prem(g,eqn(m) ,m);time()-st;
which are equivalent to the following equations:
k=kk,
(
nops(g);
.094
za- _ I _ tan ail= _ +I'
(
>
g:=s-r:
st:=time():g:=numer(g):time()-st; 782
t (u+ o �0).
sin 20
. u0+I , uo I
cot 0=�
tan
---
=
-t (u-o �0).
(J I .Uu=-�--+ Uo I
ll
for 0 E e. Note that U,a ub, U,c ud are not all independent; in fact, uu a u b cu= d I because 2a+2b+ 2c+ 2d= 2'lT. Hence,
u= d
I . a UUbUc
---
For brevity and without loss of generality, we may let
k= I, S = iF. Substituting all the above expressions into the polyno mials j3, !2, fi and removing all the extra/trivial factors, re spectively, we obtain
� I) h1= m2(uu � u+ � �- uu � u � �u+ (uu � u� u�+ I)u� � +4F(uu � �- I)u,� h2= -lu(a u� I) � I)uc+mn(u�uu � - 4FUU a bUc, a c. h3= lmua(U� I)+ n(u�- I)uc+4FuU Doing the same substitution and simplification for the conclusion-polynomial
T2 t2 s-+ /3
a'
VOLUME 23, NUMBER 1, 2001
13
we have g
= 1 - 2(1 + m)(l + n)p 1 + P2 + P3
P1 = Uc + Ub + Ua, UbUc + UaUc + UaUb, P2 P3 = U�U�U� + UaUbU� + UaU�Uc
Ua, Ub, Uc,
S=
l, m, n,
F.
(n sin 81 + k sin 84)/2 = -i(2u1u�u�ui - ufu�u� - uiu�ui- uiu�ui - u�u�u� + ui + u� + u� + ui - 2)/[4(u1u�- 1)(u�ui - 1)].
The identity is then easily proved by substitution of the above expressions with simplification. All the computa
tions for the proof can be done in less than one CPU sec
Now, what we need is only to verify whether the fol
V. It is not surprising that we have arrived at such a short
ond in Maple
lowing holds
proof with little computation. For the proof process has in
h1 = 0, h2 = 0, h3 = 0 :::::) g = 0, where
8E8 ,
8'
and
�
When expanded, g is a polynomial of 84 terms in
u�u§ + 1 , 2 2 U2U3- 1
cot
=
+ U UbUc.
.
=�
1 tan0= --
- 2(1 + m) (l + n) (2 + P2)UaUbUc 2 2 + (t + m2 + n + 2ln + 2m)(1 + P2 + P3) - 4F(1 - P2 + P3 - 2u�u�u�),
where
c
cot
Ua, ub, Um
volved the referee's ingenious use of trigonometric-func
F are considered as independent pa
l, m, n as dependent variables. This may be done by computing frrst the pseudo-remainder ii2 of h2 with respect to h3 in l, and then the pseudo-remainders ofg with respect to h 3 in l, ii2 in n and h1 in m successively. The fi rameters and
tion relations; the referee's identity-proving power .is brought into full play. A method for proving geometric theorems involving trigonometric identities similar to those used in the above proofs may be found in
nal pseudo-remainder was found to be 0, so the theorem is proved. With the pseudo-division function in Maple V . 3 ,
the computation of the pseudo-remainders took about 3 seconds on a Pentium 200MHz, while the whole program, including substitution, simplification and pseudo-division, may be
run in less than 6 seconds of CPU time.
[4].
Remark. Finding machine proofs for the other four Russian killers is also an attractive challenge. In fact, D. Wang has found a simple proof (in less than one second of comput ing time) for the fifth killer, and L. Yang has given a ma chine proof for the third killer. The details will be reported
Finally, we present an alternative proof using rules of
elsewhere.
trigonometric functions, which is due to an anonymous ref eree. Let
jACI = 1
and
Acknowledgments
lh = LCAD, fh = LACD, fh = LACE, 84 = LCAB. Denote
ei1V2
by
-t (
uk for k = 1, ..., 4.Then
cating the killers and for interesting discussions on these beautiful geometric theorems, and the anonymous referee
) ) -t (u� - :�}
ei0k 1 :S k :S 4; = 8k = ok sin 2b =sin( fh + 84) . . . u�ui - 1 , =Sill fh COS 84 + COS 83 Sill 84 = -� 2 U32 U42 sin 2d = sin( 81 + fh) . ufu�- 1 . . =Sill 81 COS 82 + COS lh Sill 82 = -� 2u2u 22
sin
The authors wish to thank Sergey Markelov for communi
1
for suggesting the alternative approach explained in the last section. This work has been supported partly by CAS, CNRS, and the Chinese National
REFERENCES
[ 1 ] Ekhad, S.B.: Plane geometry: An elementary school textbook (ca. 2050 AD). The Mathematical lntelligencer 21/3: 64-70 (1 999).
·
[2] Fevre, S., Wang, D.: Proving geometric theorems using Clifford al gebra and rewrite rules. In: Proc. CADE-1 5 (Lindau , Germany, July
It follows that ur(u� - 1) sin 82 81 = Cui- 1)u� n = --m= = 4 4 U 1U 2 - 1 ' sin 2d uiu� - 1 ' Sin 2d u§(u�- 1) . (u� - 1)u� sin 83 sin 84 k= = 4 4 sin 2b U3U4 - 1 ' sin 2b u�u� - 1 ' {= --i eib = ei(7T-03-o4)12 = __ U3U4 ' i ; eid = eiC o2)/2 = _ sin
5-1 0, 1 998), LNAI1421, pp. 1 7-32 (1 998). [3]
7r- fi1-
14
THE MATHEMATICAL INTELLIGENCER
_
. u§u�cot b =t
2
2 U3U4 +
U1U2
1
1
,
Hestenes, D., Sobczyk, G . : Clifford algebra to geometric calculus. D. Reidel, Dordrecht Boston (1 984).
[4] Gao, X.-S.: Transcendental functions and mechanical theorem proving in elementary geometries. J. Automated Reasoning 6:
=
. uru� + 1 , cot a = � 2 2 U1U4 - 1
973 Project "Mathematics
Mechanization and Platform for Automated Reasoning."
403-41 7 (1 990).
[5] Li, H.: Vectorial equations solving for mechanical geometry theo rem proving. J. Automated Reasoning 25: 83- 1 2 1 (2000).
[6] Li, H . , Cheng, M . : Clifford algebraic reduction method for auto
mated theorem proving in differential geometry. J. Automated
Reasoning 21: [7]
1 -21 (1 998).
Markelov , S.: Geometry solver. E-mail communication of February 1 1 , 1 998 from (
[email protected]) .
[8] Wang, D . : GEOTHER : A geometry theorem prover. In: Proc.
CADE-1 3 (New Brunswick, USA, July 3D-August 3, 1 996), LNAI 1 1 04,
pp. 1 66-1 70 (1 996).
[9) Wang, D.: Clifford algebraic calculus for geometric reasoning with application to computer vision. In: Automated deduction in geom etry (Wang, D., ed. ) , LNAI1 360, pp. 1 1 5- 1 40 (1 997).
[1 0] Wu, W.-t . : Some recent advances in mechanical theorem-proving of geometries. In: Automated theorem proving: After 25 years (Bledsoe, W.W., Loveland, D.W., eds.), Contemp. Math. 29, pp.
235-241 (1 984).
[1 1 ) Wu, W.-t.: Mechanical theorem proving in geometries: Basic prin ciples.
Springer, Wien, New York (1 994).
A U T H O R S
XIAORONG HOU
Chengdu
HONGBO Ll
Institute of Computer Applications Academia Sinica
Institute
of Systems Science
Academia Sinica
Chengd u 610041
Beijing 1 00080
China
China
Institute of Computer Applications, and has been at the
Hongbo U got his doctorate at Peking University in 1 994. He
Laboratory for Automated Reasoning and Programming at that
theorem-proving, hyperbolic geometry, Clifford algebra , and
Institute since 1 994; he has been professor there since 1 997.
computer vision. In 1 998, he won the Qiushi Excellent Youth
He has been working on symbolic computation, automated
Scholar Award of Hongkong.
Xiaorong Hou received his MSc in mathematics at the Chengdu
is now a professor of mathematics, specializing in automated
theorem-proving, and constructive real algebraic geometry.
DONGMING WANG
Laboratoire d'lnformatique de
LU YANG
Paris 6
Universite Pierre et Marie Curie -CNRS
4 place Jussieu
75252 Paris Cedex 05
Chengdu Institute of Computer Applications Academia Sinica
Chengdu 61 0041 China
France e-mail:
[email protected]
Lu Yang received his diploma from Peking University in 1 959. His research interests include automated theorem - proving,
Dongming Wang has been a senior researcher at CNRS since
symbolic computation,
1 992. Before that, he was an assistant professor at Johannes
received the Natural Science Prize of Academia Sinica in 1 995.
and intelligent software technology . He
Kepler University in Austria after getting his PhD from
He is now a professor at Academia Sinica and concurrently
Academis Sinica. Antedating his academic interest in symbol ic
at Guangzhou University, and
computation and automated reasoning, he has an amateur in
Informatics at Peking University.
is
Chair
of
the Department of
terest in Chinese seal cutting and writing poetry.
VOLUME 23, NUMBER 1 , 2001
15
i,i,@jj:l§rr6'h¥11+J·'I,'I,p!,hii¥J
Between Discovery and Justification
Marjorie Senec h al , Editor
j
With some notable exceptions, mathe
and second, to generate debate on the
maticians-and mathematics-stayed
issues posed by his title. The responses
on the fringes of the so-called "Science
by Mary Beth Ruskai and Michael
Wars" that raged in academic circles in
Harris, which comprise this issue's col
Europe and the United States in the
umn, were solicited in hopes of start
late twentieth century. For those who
ing a discussion that may continue in
missed them, "Science Wars" was a
these pages from time to time.
catchy phrase for a wide-ranging con
We can agree at the outset that in
troversy, among social scientists, his
different times and places scientific re
torians of science, scientists, philoso
search has been (and continues to be)
phers, literary critics, and others, on
directed,
the nature of science (as opposed to
sticks, toward the solution of problems
the science of nature). To oversimplify,
by
society's
carrots
and
that society deems important. The
the central question was whether, or to
Science Wars debated the deeper and
what extent, science and mathematics
more difficult question of the ways in
are "constructed" by the societies in
which "science itself"-whatever that
which they are practiced. Most sci
may mean-is influenced by society
entists and mathematicians became
and culture.
aware of the fray only through Alan
To help prepare this introduction to
Sakal's famous hoax, a parody of ex
the responses to Graham, I turned to
treme
an article written over 20 years ago by
social
constructivists
which
This column is a forumfor discussion
a journal innocently
of mathematical communities
straight scholarship, or by reading Paul
throughout the world, and through all
Gross and Norman Levitt's uninten
in what today would be considered at
tional parody of the "extreme science
tenuated form. Hubbard pointed out
time. Our definition of "mathematical community" is the broadest. We include
published
as
side," Higher Superstition. Now, in the early twenty-first cen
the
biochemist
Ruth Hubbard,
in
which she raised some of these issues
that our consciousness is shaped by the place and time in which we live,
"schools" of mathematics, circles of
tury, the debates are cooling down:
and that science is the picture of na
correspondence, mathematical societies,
most warriors have come to realize
ture that scientists construct as they
student organizations, and informal communities of cardinality greater
that all sides have much to contribute
"transform the seamless unity of na
and no side has a monopoly on arro
ture into a carefully patterned patch
gance. This may be a good time to look
work quilt." She points out that "it is
than one. Uthat we say about the
again at the lasting questions. Is math
important to be aware of the elements
communities is just as unrestricted.
ematics a "locality" inhabited by eter
of both patchwork and patterning: for
nal truths, or do culture and society
both involve choices that are far from
mathematicians of all kinds and in
subtly influence our subject as well as
arbitrary . . . the way in which the
how and why we practice it? Such
patches are selected-the unconscious
all places, and also from scientists,
questions have no easy answers-per
decisions
historians, anthropologists, and others.
haps there are no definitive answers at
ground and what gets pulled into the
all-but it is important, for our self-un
foreground-and the way in which
We welcome contributionsfrom
on
what
remains
back
derstanding and for the picture of the
they are stitched together, are deter
world of mathematics that we transmit
mined only in part by the explicit pos
to our students, to discuss them.
tulates and rules of the 'scientific
I invited Loren Graham to con
method.' For they, as well as our total
tribute his paper "Do Mathematical
selective mind-set are social products
Equations Display Social Attributes?"
that depend on who we are and where
Please send all submissions to the
to the Mathematical Communities col
and when. Our scientific reality, then,
Mathematical Communities Editor,
umn
like all reality, is a social construct by which I don't mean to say that I am
Marjorie Senechal,
Department
(Vol. 22 , no. 3, 31-36) for two rea
sons: first, to bring his thoughtful and
of Mathematics, Smith College,
lucid study of the philosophical impli
an 18th-century idealist and believe
Northampton, MA 01 063, USA;
cations of the equations of general rel
that there is nothing out there. I am
e-mail:
[email protected]
ativity to the readers of this journal,
quite sure that there is something out
16
THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK
tions---may illuminate different aspects
there, but I believe that what we see
thus even more threatening-question
out there, and our interpretation of it,
of what it is that mathematicians actu
of the object we are studying. Insisting
depend on the larger social context
ally do. The old, inward-looking, arro
on their logical "equivalence" and ig
that determines what we actively no
gant
noring their differences masks their dif
tice and accept as real."
"Mathematics is what mathematicians
ferent mathematical and even philo
But is this what mathematicians do?
do," is not an acceptable reply. (Is it
sophical implications. Conversely, our
At first sight the answer would seem
merely coincidental that, in the "West,"
awareness of context may lead us
to be no, Graham's article and Ruskai's
extreme forms of abstract art, abstract
to those different proofs or equations.
response notwithstanding. The equa
music, and abstract mathematics all
tions for Einstein's theory of general
were touted early in the twentieth cen
Hence, Graham argues, it is fruitful to
relativity and the theory of quantum
tury and became problematic by the
of scientific and mathematical theories,
mechanics play the dual roles of math
century's end?)
and "to deny the existence and signifi
twentieth-century
definition,
think of social influences as "attributes"
ematical theory and description of an
We all were taught, and most of us
cance of these social attributes in the
external reality. It is this second role
continue to teach, the crucial distinc
body of science and mathematics would
that Graham and Ruskai emphasize
tion between mathematical discovery
be misleading." Harris broadens the
here. Graham notes that Fock's frame
and mathematical justification.
Dis
spectrum from discovery to justification
work for general relativity "had to be
coveries, we tell our students, can be
still further, noting that "mathemati
compatible with that of Einstein, but
inspired in an endless variety of ways,
cians, and scientists for that matter,
carry different philosophical implica
from a careful reading of a proof to a
judge our peers not by the truth of their
tions"; that is, Fock's equations had to
dream. But once discovered, a mathe
work but by how interesting it is."
be logically equivalent to Einstein's,
matical statement can only be consid
Of course, any definition of "inter
ered to be true if it has been justified
esting" is subjective. To me, a piece of
but the universe that they describe is
mathematics is interesting if it explains
not the same. And it is surely the de
by rigorous proof. Moreover, we be
scriptive role that Ruskai is referring
lieve, and the editors of mathematics
diverse phenomena, or gives me a
to when she remarks that "to my mind,
journals vigilantly reinforce this belief,
fresh perspective on something I had
the most convincing verification of
that we know what rigor is: deductions
thought I understood, or opens up new
quantum theory is not the individual
from explicit assumptions by estab
vistas; "interesting" may mean some
microscopic experiments but the fact
lished rules of inference. Yet-and
thing very different to you. But how
that no other theory can even come
here
close to explaining so many diverse
agree-a view of mathematics as dis
cide to give it any weight at all we find
Graham,
Harris,
and
Ruskai
ever we defme "interesting", if we de
covery and justification alone is nar
that the boundaries between mathe
some sense, mathematicians too
row and sterile. It obscures the fact
matics and the other sciences, and
try to fit pieces of jigsaw puzzles to
that much of our creative energy lies
other forms of knowledge, has sud
gether, to use Ruskai's apt metaphor.
in a broad middle ground excluded by
denly blurred.
As Martin Gardner notes in an article
the discovery/justification dichotomy.
phenomena " Or is it?
In
this issue
Graham, Ruskai, and Harris have given us a lot to think about. The
to
Graham), "With very few exceptions
ferent ways of understanding a theo
Mathematical InteUigencer welcomes
mathematicians have always believed,
rem--different proofs, different equa-
your response.
in
(not
in
response
For example, we all know that dif
and still believe, that mathematical truths have a strange kind of abstract reality that is discovered, not created." But whether we believe that mathe matical truths (and mathematical ob jects) are "out there" or are products of the human mind, mathematicians agree that truths exist in some sense (2
+2
=
4 everywhere), and they agree with the spirit,
if not the
letter, of the fust half
of Kronecker's famous remark that
Although only 3, Adam knows all about cancer.
"God made the natural numbers; all
He's got it. Luckily, Adam
else is the work of man. " As Harris
has
points out, "this situation is not an ob
doctors and scientists are
stacle to mathematics, much less ra tionality. The real absurdity is to claim otherwise." The discussion in these pages, then, is not the ancient and un ending conundrum of Platonism, but the
somewhat more
tractable-and
St. Jude Children 's
Research Hospital,
where
making progress on his disease. To learn how you can help, cal l :
1-800-877-5833.
j,itfflj.i§,Fhl£119.'1.1rrlll,hhfj
I
Marjorie Senec h a l , Ed itor
)
t is impossible to address the ques
hard-pressed to figure out who is the
tion posed by Loren Graham's title,
mainstream and who is the margin in
"Do Mathematical Equations Display
this debate, and (2) the primary moti
Social Attributes?", without bearing in
vation for hidden variables theories
Contexts of Justification
mind the current state of the debate on
seems to be the desire to preserve de
the relation between science and soci
terminism (and "objective reality") at
ety. As Graham's first paragraph makes
all costs-probably the strongest ar
Michael Harris
clear, his article was written on the
gument I've yet seen in favor of the so
(fortunately peaceful) fringes of the
cial constructivist view of science.
Science Wars, in which arguments for
I erijoyed Sakal's hoax when it came
and against the notion of social influ
to light in 1996, but it has become de
ence on science tend to be framed in
pressingly clear that its main legacy h�
the
been reinforcement of the low level of
crudest
possible
terms.
Even
Graham feels compelled to include a
debate characteristic of the Science
ritual affirmation of his belief in "the
Wars, especially since the publication
existence of objective reality. " A de
of his book with Bricmont [4]. For in
bate in which one has to talk that way
stance, researchers from the GERSULP
to be taken seriously obviously took a
(Groupe d'Etude et de Recherche sur la
wrong turn a long time ago [ 1 ] .
Science de l'Universite Louis Pasteur, in
Graham's article defmes what he
Strasbourg) could witness a marked de
This column is a forum for discussion
means by "social attributes" implicitly,
terioration in their longstanding col
of mathematical communities
through examples. He does hint, how
laboration with scientists from this
throughout the world, and through all time. Our definition of "mathematical
utes" are pervasive, or even universal,
University after the publication of Im postures intellectueUes. A case in point:
in science. I don't think it detracts in
A scientist, who for years had graciously
ever, that, in his view, "social attrib
community" is the broadest. We include
the least from Graham's contribution
allowed some of the GERSULP's gradu
"schools" of mathematics, circles of
to ask whether the Marxist influence
ate students to carry out fieldwork in
correspondence, mathematical societies, student organizations, and informal
on Fock's relativity theory is of a kind
his laboratory, wrote a very aggressive
with the philosophical motivations
letter commenting on the paper writ
behind the recent revival of hidden
communities of cardinality greater
variables theories
in quantum me
ten by the last student intern. In this
than one. What we say about the
chanics. Hidden variables theories as
to the Sokal-Bricmont book, implying
communities is just as unrestricted.
an alternative to "quantum orthodoxy"
that papers
are a recurrent Science Wars theme.
showed how right these authors were
N. Higher Superstition, by S. Goldstein in an article in The Flight from Science and Reason, and by Alan
to defend science against unqualified
tack on the exact sciences whatsoever.
Sakal's coauthor Jean Bricmont in a
A lot of energy on the part of the mem
We welcome contributionsfrom
mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.
They are defended by P. Gross and
Levitt in
Mathematical Communities Editor, Marjorie Senechal,
Department
like the one
at hand
criticism. But a conscientious scrutiny of the student's paper showed no at
separate publication [2]. As a mathe
bers of the GERSULP is being spent
matician, I can only be cheered by the
these days to try to cope as best they
et al. [3]
can with these new prejudices. Scien
may have constructed a consistent de
tists in Strasbourg now tend to be a pri
terministic account of quantum me
ori suspicious of whatever they receive
news that Diirr, Goldstein,
Please send all submissions to the
letter, the scientist explicitly referred
chanics, and the fact that it comes at
from philosophically or sociologically
the cost of adding variables that can
oriented researchers, whereas in the
never be measured is the least of my
past many of them were quite open
concerns. As a mathematician, fur
minded [5].
of Mathematics, Smith College,
thermore, the question is strictly none
Northampton, MA 01 063, USA;
of my business, but it does seem to me
in the context of a more general ques
e-mail:
[email protected]
that (1) sociologists of science will be
tion, namely: Can interaction with so-
18
THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK
I prefer to frame Graham's question
ciologists (and sociologically-minded historians and philosophers) help sci
More interestingly, one can ask what Tr
ics; they seem to think their job is to
was before the formal
explain mathematical truth. Edinburgh
entists understand, not only incidental
definition of real numbers. To assume the
sociologist David Bloor and philosopher
features of their subject (the "context
real numbers were there all along, wait
purposes as an irresponsible relativist
of Platonism [9). Dedekind wouldn't
and a moderate realist, respectively [ 17],
of discovery" as Sokal would put it), but even what is seen as the core of the subject ("context of justification")?
kind of object
ing to be defined, is to adhere to a form
have agreed [10]. In a debate marked
Philip Kitcher, cast for Science Wars
have both attempted to develop empiri
Replace "scientists" by "mathemati
by the accusation that "postmodern"
cians" in the above question (the rela
writers deny the reality of the external
edge [18]. (Knowledge and truth are not
tion between the two is by no means
world, it is a peculiar move, to say the
synonyms but they are on the same
clear), and it seems to me that the an
least, to make mathematical Platonism
wavelength [ 19].) They have their own
swer is an unequivocal "yes." I would
a litmus test for rationality [ 1 1] . Not
(very different) reasons, but in so do
even go further (down this particular
that it makes any more sense simply to
ing I'm convinced they have missed the
cist accounts of mathematical knowl
philosophical gangplank) and argue
declare Platonism out of bounds, like
that such interaction is unavoidable if
point of mathematics. As is typical in
Jean-Marc Levy-Leblond, another con
such discussions, their examples are
mathematicians ever hope to under
tributor to
Impostures Scientifiques,
drawn either from mathematical logic
stand how we actually do see our dis
who calls Steven Weinberg's gloss on
or from mathematics no more recent
cipline, not to mention how we "ought"
Sokal's comment "une absurdite, tant
than the 19th century. If the sociolo
to see it. Indeed, a particularly regret
il est clair que la signification d'un con
gist, at least, had done some field work,
table feature of the Science Wars is
cept quelconque est evidemment af
he couldn't have helped observing that
that quite a lot of non-mathematicians,
fectee par sa mise en oeuvre dans
un
what mathematicians seem to value
but hardly any mathematicians, are at
contexte nouveau" [l2] ! Now I fmd it
most are "ideas" (not necessarily of the
tempting to instruct the general public
hard
a
Platonic variety); the most respected
on how mathematics is to be under
straight face, and my personal prefer
mathematicians are those with strong
ence is to regard the formula Tr =
pher assures us, is philosophically in
mathematics, for instance, is summa
� as a creation rather than a dis
covery.
corre
defensible; Sokal and Bricmont add
rized in the sentence "A mathematical
spond to the familiar experience that
doesn't change, even if
that "intuition cannot play an explicit
there is something about mathematics,
role in the reasoning leading to the ver
the idea one has about it may change"
and not just about other mathemati
(p. 263). This claim, referring to a
ification (or falsification) of these the
"crescendo of absurdity" in Sokal's orig
away with saying "Evidemment" [ 13] !
independent of the subjectivity of in dividual scientists" [20].
stood. The Sokal-Bricmont philosophy of
constant like
Tr
inal hoax in Social
to
defend
Platonism
But Platonism does
with
cians, that precisely doesn't let us get
Text, was in turn crit
This experience is clearly captured by
icized by anthropologist Joan Fqjimura,
Alain Connes, a self-avowed Platonist,
Impostures
in his dialogue with neurobiologist
in an article translated for
"intuition." Now intuition, the philoso
ories, since this process must remain In barring intuition from the context
of justification, Sokal and Bricmont are
scientijiques [6], the most extended
J.-P. Changeux, who (to oversimplify)
using the language of truth (and falsity).
response
Sokal
expects to find mathematical struc
Bricmont book Most of Fqjimura's ar
tures in the brain [ 14]. I don't think
In fact, truth is a secondary issue in
ticle consists of an astonishingly bland
Connes (or Roger Penrose, another
prove true theorems, but this is hardly an adequate or even useful description
in
France
to
the
mathematics. Of course we want to
account of the history of non-euclidean
prominent Platonist) is confused about
geometry, in which she points out that
reality, and I have a hard time imagin
of our objective. Mathematicians, and
the ratio of the circumference to the
ing a neuronal representation that does
scientists for that matter, judge our
diameter depends on the metric. Sokal
justice to the concept of Tr. But the on
peers not by the truth of their work but
and Bricmont know this, and Fqji
tological issues are far from settled,
by how interesting it is [21]. The differ
mura's remarks are about as helpful
and while there is no reason to assume
ence between the true and the interest
as FNs referral of Quine's readers
they will ever be settled, the important
ing even has a market value. Professor
to Hume (p. 70). Anyway, Sokal ex
point is that this situation is not an ob
Ed Fredkin of Carnegie-Mellon once of
plicitly referred to "Euclid's pi", pre
stacle to mathematics, much less to ra
fered to give you $100,000---almost
sumably to avoid trivial objections
tionality [ 15]. The real absurdity is to
enough to buy a bachelor's degree at a
like Fqjimura's-wasted effort on both
claim otherwise.
prestigious private university-if you
sides [7]. If one insists on making triv
This leaves the "context of discov
ial objections, one might recall that the
ery vs. context ofjustification" in a cu
interesting
theorem that "' is transcendental can
rious
mathematicians
[22]. Whereas any beginner can program
be stated as follows: the homomor
have largely given up worrying [ 16]
the computer to prove a true theorem
Q [X] � R taking X to "' is in jective. In other words, "' can be iden tified algebraically with X, the variable par exceUence [8].
about the nature of what it is they need
after a single lesson [23].
phism
limbo,
since
were first to make a computer prove an theorem
in
mathematics
to justify. This doesn't bother philoso
On the other hand, it is surprising to
phers and philosophically-minded so
see just how little we seem to be con
ciologists concerned with mathemat-
cerned with "truth" these days. Math-
VOLUME 23, NUMBER 1 , 2001
19
ematicians rarely discuss foundational issues any more, so it was significant that an article by Arthur Jaffe and Frank Quinn, reaffirming the impor tance of rigorous proof in the current context of strong interaction between physics and mathematics, provoked no fewer than 16 responses by eminent mathematicians, physicists, and histo rians. No two of the positions ex pressed were identical, which already should suggest caution in laying down the law on rationality, as Sokal and Bricmont (and Levy-Leblond) seem in clined to do. Remarkably, almost none of the responses had much to say about "truth."[24] "Truth" was central, predictably, only to the responses of Chaitin and Glimm. Chaitin's branch of mathematics treats "truth" as a techni cal term, without metaphysical conno tations, and Chaitin's claim to have "found mathematical truths that are true for no reason at all" suggests that it may be harder than Fredkin suspects to know just when to award his prize. Glimm's brand of truth is quite the op posite: it "lies not in the eye of the be holder, but in objective reality. . . . It is thus reproducible across barriers of distance, political boundaries and time" [25]. Turning to the introduction to the book Quantum Physics, by Glimm and Jaffe , one fmds the unusual assertion that "mathematical analysis must be included in the list of appro priate methods in the search for truth in theoretical physics." I can't help thinking it's not a coincidence that both Bricmont and Sakal are amply represented in the Glimm-Jaffe bibli ography. Ironically, given the starting point of Graham's exchange with Sakal, the substance of the Jaffe-Quinn article, and the subsequent debate in the Bulletin of the AMS, is precisely the context of justification, specifically the extent to which physical intuition should be admitted as an alternative to rigorous proof [26]. (Admitted by whom? That is a question for the soci ologists!) Fredkin's theorem-proving machine may share the Sokal-Bricmont commitment to truth as that-which-is to-be-justified, but what are we to make of Thurston's emphasis on the "continuing desire for human under-
20
THE MATHEMATICAL INTELLIGENCER
standing of a proof, in addition to knowledge that the theorem is true" [27]? We know what he means, as we know what Robert Coleman means, when, having discovered a gap in Manin's proof of Mordell's col\iecture over function fields, he nevertheless writes, "I believe that all this is testi mony to the power and depth of Manin's intuition" [28]. Is Coleman try ing to slip a counterfeit coin between the context of discovery and the con text of justification? Do these offhand comments touch on something gen uine and profound about mathemat ics? Or is it just my indoctrination that makes me think so? Let's try a thought experiment. How do we know Wiles's proof of Fermat's Last Theorem, completed by Taylor and Wiles, is correct? Although this particular theorem, better publicized than any in history, has been treated with unusual care by the mathematical community, whose "verdict" is devel oped at length in a graduate textbook of exceptionally high quality, I'd guess that no more than 5% of mathemati cians have made a real effort to work through the proof [29]. Some scientists (and some mathematicians as well) apparently view Wiles and his proof as an "anachronism" [30]. The general public is not entirely convinced. Why are we? Can a sociologist study this question without knowing the proof? Can mathematicians pose the question in terms sociologists would find mean ingful? Knowing the truth of the mat ter is obviously of no help, and rela tivism is not the issue: it's not clear what kind of "reality" would be rele vant to settling the question, but the fact that no one has found a coun terexample is certainly not a good candidate. Few of us would choose to treat our belief that Wiles proved Fermat's last theorem as "a mythical and false ide ology" [31] , but is it possible that our attempts to justify this belief always in volve an element of self-delusion? And how are we to convince a skeptical outsider that this is not the case? The only reasonable answers that come to mind are empirical in nature, and specifically historical and sociological, rather than philosophical [32]. We
would have to pay attention to the ql,les tion of how knowledge is transmitted among mathematicians. Fermat's Last Theorem provides a particularly good test case. Wiles's proof generated an un precedented [33] number of reports, sur vey articles, colloquium talks, working seminars, graduate courses, and mini conferences, as well as books, newspa per and magazine articles, television reports, and other forms of communi cation with non-mathematicians. Not to mention the spate of announcements, designed to impress public policy-mak ers and the public at large, citing Wiles's work as proof that mathemat ics "has never been healthier." Has anyone been keeping track of all these incitements to belief formation, chec� ing them for contamination by myth and false ideology? Studying questions like these pro vides a second answer to the thought experiment proposed above, comple mentary to the answer we would nat urally provide based on our experience as mathematicians, and potentiallyjust as interesting. These two answers are not in competition, much less on op posite sides of a battlefield. Arriving at the second answer would be the work of sociologists. For this, full coopera tion with mathematicians would be necessary. I hope such cooperation is still possible. Acknowledgments I am indebted to many people for help in formulating and clarifying the ideas developed in this note, and in the arti cle from which it was extracted (see note [4]); I am especially grateful to Marie-Jose Durand-Richard, Catherine Goldstein, Catherine Jami, Patrick Petit jean, and Jim Ritter, co-organizers of the 1997 IHP workshop on the Sokal affair. REFERENCES
[1 ]
The mathematics department may be the only spot on campus where belief in the reality of the external world is not only op tional but frequently an annoying distrac tion.
[2] Higher Superstition,
p. 261 , note 9; Gold
stein, "Quantum Philosophy: the Flight from Reason in Science" in Gross, Levitt, Lewis, The Flight from Science and Reason,
pp.
[4] The avoidance of serious debate in
marks are contained in his article "Sakal's
is one of the main themes of Yves
Hoax," in the N e w York Review o f Books,
Jeanneret's L 'affaire Sakal ou Ia querel/e
August 8, 1 996.
des impostures
(Presses Universitaires de
[1 3) Metaphors from virtual reality may help
France, 1 998), the best book I've seen on the Sokal affair and its ramifications in France. This theme is also addressed in
MICHAEL HARRIS
VII
Cedex
example, Connes writes, "Je pense que
le mathematician developpe un 'sens', ir
which can be viewed at
http://www.math .jussieu. fr/-harris. Most
roouctible a Ia vue, a l'ou'ie et au toucher,
of the present article was extracted from
aussi contraignante mais beaucoup plus
this review.
stable que Ia realite physique, car non lo
qui lui permet de percevoir une realite tout
calises dans l'espace-temps" (p. 49). In
and Strasbourg mathematician Norbert
fact, the debate is even more complex,
05
France
Schappacher for providing this informa
because Changeux often comes across
tion.
as a social constructivist, though one who
[6] Impostures scientifiques
e-mail:
[email protected]
[1 4) Matiere a penser. Odile Jacob, 1 989. For
[5] I thank Josiane Olff-Nathan of GERSULP
2 place Jussieu
75251 Paris
here.
my unpublished review of Fashionable N onsense,
Universite de Paris
Michael Harris studied at Harvard
(henceforth IS},
sees society as materialized in the human
sous Ia direction de Baudouin Jurdant,
brain; thus he sees mathematical objects
Paris: Editions La Decouverte, 1 998. Also
as
published as a special issue of the journal
(PhD 1 977 with Barry Mazur), and taught at Brandeis University before moving to Paris. Among his research
"
'representations culturelles' suscepti
bles de se propager, se fructifier et de pro liferer et d'etre transmises de cerveau a
Alliage. [7]
contributions, the most recent is "On the geometry and cohomology of
p. 38, footnote 26. Weinberg's re
[ 1 2] IS,
Science Wars literature-on both sides
There are many circles in Euclid, but no
cerveau. "
pi, so I can't think of any other reason for
[1 5] See Barry Mazur's astonishing "Imagining
Sokal to have written "Euclid's pi," unless
Numbers (particularly Y-15)," particu
some simple Shimura varieties" (with Richard Taylor). He has been a visi
this anachronism was an intentional part
larly for (among other things) an attempt
of the hoax. Sakal's full quotation was "the
to get beyond sterile ontological debates.
tor at the Institute for Advanced Study, Bethlehem University in the
thought to be constant and universal, are
West Bank, the Tata Institute, the Steklov Institute, and other institu
now perceived in their ineluctable his
quirement in any university with which I am
toricity. " But there is no need to invoke
familiar. One would think this fact would
tions. In 1 986-1 989 he helped orga
non-Euclidean geometry to perceive the
be of interest to sociologists of science,
nize a program of scientific coopera
historicity of the circle, or of pi: see
but I have not seen it addressed in the lit
tion between US and Nicaraguan
Catherine Goldstein's "L'un est !'autre:
erature. As a graduate student at Harvard,
universities. He organized with five historians of science a workshop at
pour une histoire du cercle," in M. Serres,
I saw foundations actively discussed only
Bordas,
in the graffiti in the men's room on the sec
[8] This is not mere sophistry: the construc
[ 1 7] What they really think hardly matters. The
title "La guerre des sciences n'aura
tion of models over number fields actually
"strong program" of Bloor and Barry
pas lieu."
uses arguments of this kind. A careless
Barnes is criticized at length in the philo
Photo credit: Emiliano Harris.
construction of the equations defining
sophical "intermezzo" of FN ; but Sokal
7T
of Euclid and the G of Newton, formerly
Elements d'histoire des sciences,
the lnstitut Henri Poincare in 1 997 on
1 989, pp. 1 29-149.
the subject of this article, under the
Fashionable Nonsense
(henceforth FN)
In public, at least. A course on founda tions of mathematics is not a core re
ond floor of the science library.
modular curves may make it appear that
"agrees with nearly everything" in Kitcher's
1r
attempt to "occupy middle ground" in
is included in their field of scalars.
[9) Unless you claim, like the present French 1 HH 25. These two books, together with
[1 6)
Minister of Education, that real numbers
Koertge, op. cit. [2] [1 8] D. Bloor, Knowledge and Social Imagery
exist in nature, while imaginary numbers
(U. of Chicago press, 1 991 ), chapters 5-8;
1r
P. Kitcher, The Nature of Mathematical
by Sokal and Bricmont, first published in
were invented by mathematicians. Thus
French as Impostures intellectuelles, and A
would be a physical constant, like the
Knowledge
(N. Koertge, ed.) are
mass of the electron, that can be deter
1 984).
House Built on Sand
(Oxford
University
Press,
the canonical texts of what passes for the
mined experimentally with increasing ac
pro-science camp of the Science Wars.
curacy, say by measuring physical circles
discourse. Thurston's extended response
[3] DOrr, D., S. Goldstein and N. Zanghi,
with ever more sensitive rulers. This sort
to the Jaffe-Quinn article, in BAMS, April
"Quantum equilibrium and the origin of ab
of position has not been welcomed by
1 994, discussed below does refer to truth,
solute uncertainty,"
most French mathematicians.
but he seems more interested in knowl
(1 992):
843-907;
Goldstein and V.
J.
Statistical Phys. 67
also
DOrr
Zanghi,
D.,
S.
"Quantum
Mechanics, randomness, and determinis tic reality" Phys. Letters; 1 72 (1 992): 6-1 2.
[1 0) Cf. M. Kline, Mathematics: The Loss of Certainty,
p. 324.
[1 1 ) Compare Morris Hirsch's remarks in BAMS,
April 1 994.
[ 1 9] One might also say they share a common
edge and especially in understanding. [20] Chapter 3 of Kitcher's op. cit. [1 8] is de voted to a refutation of Kantian or Platonist intuition as a means to mathematical knowl-
VOLUME 23. NUMBER 1, 2001
21
edge, and what we mean when we use the
Corry, "The Origins of Eternal Truth in
temative (or complementary) standa�d of
word informally is presumably even less de
Modern Mathematics," Science in Context
justification to rigorous proof. This point is
fensible. The quotation is from Fashionable
1 0 (1 997), 297-342. Corry is so intent on
a cliche of the popular literature on chaos,
developing his theme (that "the idea of eter
and it is repeated in the article by Amy
[21 ] As Barry Mazur reminded me, "interest" in
nal mathematical truth . . . has not itself
Dahan-Dalmedico and Dominique Pestre
this context is generally used as shorthand
been eternal") that he completely misses
for an intellectual criterion, such as
the near absence of the word "truth" from
Nonsense,
pp. 1 43-44, note 1 83.
in Impostures Scientifiques (pp. 95-96). [27] Thurston, op. cit. [25]. Thurston's com
"enhancement of understanding" (see
the debate, claiming with no attempt at jus
ment referred to the computer-assisted
Thurston's comments, quoted below).
tification that "the eternal character of
proof of the Four Color Theorem, and
Such a criterion is by nature not well-de
mathematical truth" was "implicit at the very
echoes Deligne's remarks on the same
fined, yet we have the sense that we know
least" in the Jaffe-Quinn proposal.
topic, quoted by Ruelle in Chance and
[25] Glimm goes on to say that mathematical
what it means.
pp. 3-4.
Chaos,
[22] Actually a "major mathematical discovery."
truth is to be compared with the stronger
(The author obtained this information
standard of truth in science, "the agree
conjecture
November 24, 1 997 from Fredkin's web
ment between theory and data." The
L 'Enseignement Math.
page on Radnet. This web page has ap
whole discussion can be found in Bulletin
parently been discontinued and the infor
of the AMS,
mation regarding the prize has not been
July 1 993 and April 1 994.
[26] There is a similar irony in the discussion of chaos in FN (pp. 1 34-1 46). As in sim
reconfirmed since 1 997 .)
[28] Coleman, "Manin's proof of the Mordell over
function
fields,"
36 (1 990), p. 393.
[29] The text book is G. Cornell, J. H. Silverman, G. Stevens, eds. : Modular Forms
and
Fermat's
Last
Theorem,
Springer-Verlag (1 997). If I believe, or u �
ilar discussions in The lntelligencer, the
derstand, or have some meaningful rela
says something similar in IS (p. 39), and
non-post-modern
position emphasizes
tion to the proof, it's mainly because I have
[23] This point is hardly novel; Levy-Leblond Dieudonne distinguished further between
the continued role of proofs in the theory
been collaborating with Richard Taylor to
"math8matiques vides" and "mathema
of non-linear dynamical systems, insists
generalize parts of the argument to auto
tiques significatives" (quoted in Dominique
on the determinism inherent in dynamical
morphic forms of higher dimension.
Lambert,
systems, and points to quotations from
[30] John Horgan, "The Death of Proof, "
"L'incroyable
efficacite
des
mathematiques," La Recherche, janvier
Maxwell and Poincare to argue that no
Scientific American,
1 999, p. 50). Truth is also not what inter
conceptual revolution has taken place.
92-1 03.
October 1 993, pp.
(See also Higher Superstition , p. 92 ft.,
[31 ] As in David Bloor: "What if scientists need
verite qui inspire Ia philosophie, mais des
and Bricmont's contribution to The Flight
to believe a mythical and false ideology,
categories comme celles d'lnteressant, de
from Science and Reason.)
This is all rea
because they would lose motivation with
Remarquable ou d'lmportant qui decident
sonable (though I would need to be a his
out it?" Social Studies of Science 28/4
de Ia reussite ou de l'echec." This quota
torian to explain why a few late-1 9th-cen
(August 1 998), p. 658.
tion is taken from Qu'est-ce que Ia philoso
tury quotations are hardly decisive in
[32] As Thurston wrote, Wiles's proof-still in
phie? (1 991 ), p. 80, one of the books sub
discussions of conceptual revolutions),
complete at the time-"helps illustrate how
jected to extensive ridicule by Sokal and
but once again the interesting point has
mathematics evolves by rather organic psy
Bricmont.
been missed. Namely, in the wake of
chological and social processes." Thurston,
[24] The debate provoked by the Jaffe-Quinn
chaos, computer modeling ("experimental
op. cit.
article is taken up in a recent article by Leo
mathematics") is being proposed as an al-
ests Deleuze and Guattari: "ce n'est pas Ia
[33] Or so I would assume.
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22
THE MATHEMATICAL INTELLIGENCER
[25]
l,�@ii • f§rr@h(¥119.111J,II!Ihh¥J
G Response to Graham: the Quantum View
M a rj o rie Senechal , Ed itor
raham's article discusses the pos sibility
of social influence
or below a certain standard amount.
on
But this already assumes something
I will not
about the markings on the stick. Even
comment directly on his example: rel
with accurate calibration, a low read
mathematical
Mary Beth Ruskai*
I
equations.
ativity is far from my specialty. The his
ing could result from not inserting the
tory of quantum theory provides ample
stick properly. Before adding oil, one
evidence that different people with dif
often inserts the stick a second time or
ferent views may formulate a theory in
(perhaps not trusting the gas station at
quite
tendant) checks oneself. 1 In science
different mathematical terms.
The issue is to what extent this implies
this is referred to as reproducibility:
that science is shaped by social and
experimental result is not accepted un
cultural factors. The development of
less it can be repeated by an indepen
quantum theory, from its inception to
dent observer.
an
the most recent novel experiments ,
However, reproducibility of single
provides fertile territory for discussing
experiments is not enough. For exam ple,
this question. Before doing so, I would like to comment on the distinction between
G.P. Thompson's observation of dif
fraction rings due to the transmission of electrons through a
thin metal foil ap
I prefer the term "verifica
of mathematical communities
pears to give unequivocal evidence that
tion" for the latter. Such a term sup
only when one remembers the many
throughout the world, and through all
poses the existence of an external
other experiments which seem to offer
objective reality. Without such an as
convincing evidence that electrons are
This column is a forum for discussion
time. Our definition of "mathematical community" is the broadest. We include
the context of "discovery" and of "jus tification".
electrons are waves. 2 Problems arise
sumption it makes no sense even to
particles. Verification of a theory re
discuss
quires what one might term "consis
the
concept
of
science.
"schools" of mathematics, circles of
Histories of science often focus on the
tency"
correspondence, mathematical societies,
role of
Reproducibility guards against the sub
student organizations, and informal communities of cardinality greater
a small number of critical ex
periments. However, it is important to
as
well
as
reproducibility.
jectivity or bias of a single observer.
remember these are only part of the
Consistency is more complex. At the
legacy and that virtually all experi
simplest level, it requires verification by
than one. What we say about the
ments have multiple interpretations
different types of experiments as well as
communities is just as unrestricted.
and/or hidden assumptions.
independent observation. Modem sci
We welcome contributions from
ex
ence often requires complex experi
tremely simple process of using a dip
Consider,
for
example,
the
stick to check the oil level in a car en
ments with sophisticated instruments
all places, and also from scientists,
gine (one of the few measurements not
and rely for their interpretation on the
historians, anthropologists, and others.
yet replaced by an electronic device
assumption of the validity of a large
with digital readout). The results may
body of scientific evidence and theories.
seem unambiguous: the oil level on the
Thus, even experimental observations
stick tells how much the total is above
can be viewed as human constructs.
mathematicians of all kinds and in
which permit only indirect observation
*Partially supported by National Science Foundation Grant DMS-97-06981 and Army Research Oftice Grant
DAAG55-98- 1 -037 4 1 1f one continues scrutinizing this experiment, questions arise that may seem similar to those of quantum the ory. Before starting. oil is wiped off the dipstick, so that the stale of the oil reservoir is necessarily changed by
Please send all submissions to the
the measurement. However, unlike quantum theory, in which all information about the initial state may be lost,
Mathematical Communities Editor,
with careful procedures a good estimate of the initial state can still be made. Nevertheless, even in such a rudi
Marjorie Senechal,
Department
mentary situation, the observer can not be completely separated from the experiment.
of Mathematics, Smith College,
2The more familiar Davidson-Germer experiment in which electrons incident on a nickel crystal show diffrac tion patterns was the first to confirm the wave-like properties of electrons. However that evidence is not un
Northampton, MA 01 063, USA;
equivocally in favor of waves because the full set of experimental data contains evidence consistent with par
e-mail:
[email protected]
ticle properties in addition to the peak of electron emission associated with diffraction.
© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 1 , 2001
23
My own view of physics is that it is like a giant mosaic or jigsaw puzzle. When key pieces are found, sections fits together so well that the result seems evident. But many different pieces have similar shapes. Thus, it is possible to find edges that seem to match and build up promising aggre gates which eventually break down. The "old quantum theory," developed in the first two decades of the previous century, fits this metaphor. One had bits and pieces of the puzzle Einstein's description of the photo electric effect, de Broglie waves, the Bohr atom--each piece explained physical phenomena in strikingly con vincing ways, yet gaps and inconsis tencies remained in the larger puzzle. In 1925 and 1926, two theories were proposed-the matrix mechanics of Heisenberg, and the wave mechanics of Schrodinger-whose mathematical for mulation was so different that one could hardly expect both to be valid. Yet, within a few months of Schrodinger's second paper, a link was found by at least three people independently, Eckart, Pauli, and Schrodinger.3 Subsequently, Dirac [4) and von Neumann [31) formulated quan tum mechanics in terms of abstract Hilbert spaces, in which the Heisenberg and Schrodinger pictures are but two of many possible equivalent representa tions. It is worth noting that both Heisenberg and Schrodinger's original formulations required some intuitive leaps over logical obstacles-Heisenberg required that "matrices" satisfy commu tation relations which can not hold in finite dimensions, while Schrodinger replaced the second derivative of time in the wave equation by a first deriva tive with a pure imaginary coefficient. As far as I am aware, there is no ev idence that their different approaches were motivated by political or social
factors.4 However, they had very differ ent attitudes toward the then-emerging Nazi philosophy. (See, e.g., [3,21,27].) Heisenberg, if not an active proponent, was at least sufficiently partisan to be comfortable as head of German sci ence under Hitler; Schrodinger, if not a courageous opponent, was unwilling to make the accommodations neces sary to remain in Austria. Suppose, for the sake of argument, that Heisenberg and Schrodinger's5 different formula tions of quantum theory were moti vated by opposing political views. What difference could it have made? The two theories were still equivalent. Indeed, if they had proposed inequiva lent theories, one would have been verified and the other discredited by subsequent experiments-and the out come would not have depended on the political views of the proposer. It is often said that mathematics is the language of science. But it would be more accurate to say that mathe matics is a family of related languages. The languages of algebra, analysis, geometry, etc. often give very different insights into a problem. Barry Simon's views on p. xv of the Introduction to [28) describe one such situation. I was partly motivated by a not atyp ical functional analyst's suspicion of probability as nothing more than a subset of functional analysis with strange names . . . any statement in probability theory has a translation into functional analysis. But this somehow misses the point. Like any other foreign language, probability theory is structured around its own natural thought patterns and so is critical to a mode of thinking; but more prosaically, certain exceed ingly natural constructions of prob ability theory look ad hoc and un-
natural when viewed as functidnal analytic constructions. . . . I speak probability with a marked func tional analysis accent: lecturing in Zurich, I couldn't help feeling that I was speaking not hoch probability theory but only a kind of Schweitzer probability. Simon was referring to the spectacular progress in constructive quantum field theory which followed Nelson's syn thesis of several earlier ideas into what became known as "Euclidean quantum field theory." Many such examples can be found. In 1982 Donaldson used ideas from classical field theory to solve problems in 4-dimensional topol ogy. The recognition that what alge braic topologists called a "connection" on fibre bundles was but another name for a gauge field,6 and that this seem ingly abstract language was important in physics, led to advances in both fields. Inspired by electric-magnetic duality, Seiberg and Witten's insightful work in 1994 led to another impressive series of rapid advances, in which long proofs of Donaldson were dramatically shortened and long-standing questions resolved [ 1 7). Hilbert space is the language of quantum mechanics. Although other languages can give useful insights, they can also mislead. In particular, experi ence with classical waves is helpful in understanding some aspects of super positions; however, attempts to give too literal an interpretation to the no tion of "wave function" have also been a major source of confusion. There was at least one other math ematical approach which appeared in the development of quantum theory. In 1925, C. Lanczos [18] tried to formulate quantum theory in terms of integral, 7 rather than differential, equations, but
3Aithough this is generally attributed to Schri:idinger, there is clear evidence of independent discovery by Eckart [7,29], and [33] contains the text and interpretation of a letter from Pauli to Jordan. 4There are, however, historians who have argued that that both Heisenberg and Schri:idinger were infiuenced by the Weimar culture to find an indeterministic compo nent in physics. See [1 1 , 1 4]. 5Schr6dinger's biographer raises another social element. Moore [2 1 ] devotes considerable attention to Schri:idinger's sexual philandering, and even raises the possi bility that the psychological impact of a liaison during the Christmas holidays at the end of 1 925 may have stimulated the creative period of 1 926 in which wave me chanics was developed. 6Aithough this relationship was known to a few experts, prior to Donaldson's work algebraic topology was generally regarded as an abstract area of mathematics with little relevance to physics. In fact, I recall an anecdote from the early 1 970's in which an advisor told an enthusiastic graduate student that the outstanding problem in algebraic topology was "finding an application." 7See [32] for a summary of this work and a response by Lanczos 35 years later.
24
THE MATHEMATICAL INTELLIGENCER
was unable to specify the kernels needed
to
complete
his
theory.
However, his approach did lead quite
A Modern View of EPR for pedestrians*
naturally to operators which satisfied
Suppose that every morning when
Heisenberg's commutation relations.
you log onto your computer the
In view of that, one wonders why
his
work had so little impact.8 Was it be
lar internet provider seem to be paired up in a similar way. When two members of a pair choose the same box, one, and only one, wins.
It appears that the internet entre
screen shows three boxes
preneur is sending out paired mes
cause physicists were less comfortable
sages
with the dialect of operator theory
boxes are complementary--e.g., if
programmed
so
that
the
differential
flashing on and off with the words
your boxes are coded W W L, your
equations? Because Lanczos's equa
CLICK ME. You can not proceed to
partner's will be L L W.
based
on
integral
than
tions required singular kernels which
check your e-mail (or do anything
However, one astute pair notices
seemed inconsistent with Heisenberg's
else) until you choose one of the
something curious. On the days
diffe-rent boxes,
discrete matrices? Or because Dirac's
boxes. As soon as you click, the
when they choose
delta function was not yet the familiar
other two boxes disappear and the
both win 3/8 of the time and both
tool needed to complete the picture?
remaining box changes to either
Despite the overwhelming experi
I Win I
or
ILosel
indicating that
lose
3/8 of the time; only 114 of the
time does one win and the other
mental evidence in support of quan
your "frequent web buyer" account
lose with different boxes. Yet, an el
tum mechanics, some of its implica
has won or lost 500 points.
ementary calculation shows that
tions made many reluctant to accept
You choose at random but, in
it. Difficult questions about its inter
the hope of finding a better strat
probabilities should be both win 114
pretation led to controversies, some
egy, keep careful notes of your
and both lose 114 of the time.
of which are still debated. The best re
choice and the result. The game ap
seems to eliminate the complemen tary box hypothesis.
with
complementary
pairs,
the
This
"God
pears fair, in the sense that you win
does not play dice with the universe"
50% of the time; however, no strat
What other explanations are pos
is "Indeed, She does not." For quan
egy appears. In January, you attend
sible? The internet provider (located in Kansas) may be sending entan
sponse to
Einstein's
famous
tum theory, although it has a proba
the annual AMS meeting where
bilistic aspect, is not described by the
you meet a colleague from the op
gled pairs of polarized photons.
classical random probabilities associ
posite coast who uses the same in
Clicking on box A, B, C chooses one
ated with dice games. Quantum states
ternet provider and has kept simi
among polarization filters set at 120°
are based upon superpositions which
lar records. You compare notes
angles. If the photon passes through
give rise to non-classical correlations
and discover an amazing coinci
the filter and hits the detector you
which permit experimental tests of
dence. On those days when you
win. The observed results are con
some of quantum theory's most per verse features.9 When "gedanken ex
both choose the same box one
sistent with the predictions of quan
wins and the other loses. Further
periments,"
investigation
such
Podolsky-Rosen and
as
the
(EPR)
"Schrodinger's
Einstein
experiment
cat"
assumed
to
be
that
other
mathematicians using this particu-
tum theory [18, 1 9,30]. Finding any
other explanation poses a challenge, as discussed in [30].
paradox,
were first suggested, the outcomes were
reveals
obvious.
"inspired by Nelson's 1 980's version (24) using "game cards" from a fast food chain.
However, modern technology has per mitted similar experiments to be per
cepted, particularly by philosophers,
in the puzzle, than by experimental ev
formed, and, in every case, the out
despite the fact that the experimental
idence, which left room for alternative
the
evidence was much less compelling.
theories of gravitation. One attempt at
predictions of quantum theory and
come
Indeed, the initial verifications, such as
a modified theory is the search for a
seem
is to
consistent preclude
with
"hidden
vari
the red shift and deflection of light
"fifth force" in the 1980's [30]. Several
1 ables." 0 Any alternative to quantum
around the sun, were based on mea
theory must be consistent with these
surements of distinctions so fine that
periments seemed to support gravita
experiments as well as replacing an
they were the same order of magnitude
tional corrections. But the need for
impressively successful framework.
extremely careful and reproducible ex
as the error bounds. Scientists may
consistency, as well as reproducibility,
By contrast, Einstein's theory of
have been influenced more by the ele
proved crucial-some experiments in
general relativity was more readily ac-
gance with which the theory filled gaps
dicated repulsive deviations from the
81t should be noted that Lanczos's work preceded that of Schrbdinger, who cited it, but seems to have misinterpreted a critical feature [32). See also [6) .
9See Faris [8,9] for brief accounts of the important differences between quantum theory and classical probability, especially pp. 205-206 of [8]. See Mermin (1 9,20) for a very readable discussion of the EPR experiment, Wick [33) for a overview of the entire history of paradox and controversy, and Bell's original papers in (1 ].
10'fo be precise, these experiments only preclude local hidden variables. They are most important as evidence that quantum theory, with or without hidden variables, requires non locality or "spooky action at a distance."
VOLUME 23. NUMBER 1 . 2001
25
conventional theory, while others sug gested attractive forces. One of the most extensive attempts at an alternative to quantum theory is E. Nelson's "stochastic mechanics." In 1966, Nelson, building on ideas of Bohm and Feynes, proposed an alter native theory based on classical me chanics and a type of Brownian motion arising from interaction with a back ground field. For the next 20 years, Nelson [22,23] continued to develop this remarkable theory. However, in 1984, at a conference in Boulder, he reported that "Markovian stochastic mechanics is untenable as a realistic physical theory" [24]. Compatibility with the results of the increasingly convincing EPR experiments required a non-Markovian diffusion process. 1 1 In the jigsaw metaphor, Nelson's the ory is a good example of an alternative whose pieces fit together and whose edges can even be attached to part of the puzzle, but still does not quite fit. 12 Adding a non-Markov process may ex tend this arm of the puzzle, but gaps are likely to remain unfilled, and the fi nal resolution must come from exper iments. This assumes, of course, that some experiment can always be found to distinguish between mathematically inequivalent theories. One might also ask, instead, what it might mean if this assumption were false. This question is discussed further within the partic ular context of Bohmian mechanics in Appendix B. One can identify several periods in the development of quantum theory. In the first, physicists tried to come to grips with a new theory with unfamil iar properties. A mathematical frame work which satisfied most physicists was found more readily than answers
to many of the perplexing questions which arose. This led to a period of about 50 years during which the Copenhagen Interpretation based on "complementarity" pushed fundamen tal questions under the rug. While some 13 have derided this as a period of "cognitive repression" [ 14], I think it was natural that most physicists ig nored foundational questions while they learned the language needed to explore their strange new land. In the past 20 years sophisticated experi mental tools, such as ion traps and scanning tunneling microscopes, have given physicists new ways to observe quantum phenomena. For many this was the icing on the cake of a mature theory whose status was secure; for others, it opened the door to new in sights and more ways to examine quan tum paradoxes. We are now entering a new era, in which quantum phenom ena are no longer viewed as perverse, but as the potential source of exciting new practical tools for computation and communication. (See, e.g., [12] for an enthusiastic summary of this view point, and [25,26] for comprehensive treatments of quantum computation and communications.) These stages seem a necessary part of our development in understanding the physics of a world far from our or dinary experience. What is not so clear is whether different cultural and social influences could have length ened or shortened some of them sig nificantly. It should be noted that even though studying the foundations of quantum mechanics has long been far from the mainstream, it has never been suppressed. The papers of Bohm, Bell, et al were published in reputable journals, and heads of lab-
oratories which had the neces,sary equipment agreed to perform the ex periments proposed by people like Clauser and Shimony. 14 Delicate experiments, such as those done by the laboratory groups as sociated with Aspect, Leggett, and Zeilinger are useful in clarifying some critical aspects of quantum theory. However, to my mind, the most con vincing verification of quantum theory is not the individual microscopic ex periments, but the fact that no other theory can even come close to ex plaining so many diverse phenomena (including macroscopic as well as mi croscopic phenomena); literally, "from atoms to stars" [16]. Few jigsaw puz zles fit together so neatly. We a,re forced to overcome the biases arising from our experience with the familiar macroscopic world of classical me chanics despite the challenge of re solving all questions about the founda tions of quantum theory. In the end, quantum theory remains a human con struct subject, in principle, to social forces. But it is a theory so remarkable, so different from ordinary experience, that it transcends social and cultural forces. Afterword: It may be worth men tioning the personal bias I bring to these matters. As a student I was suf ficiently fascinated with the mathe matics and foundations of quantum mechanics to seek out a postdoctoral position in Geneva, then a major cen ter in research on the foundations, based on Jauch and Piron's ap proaches. However, I found the for malism of Yes-No experiments too far from the physics to which quantum theory was applied. I realized that my interest lay more with the mathemati-
1 1To be precise, it is not the EPR probability distribution that require the process to be non-Markovian but compatibility of that process with the assumption of "active locality." Mhough the EPR experiments seem to imply some form of passive non-locality" in the sense of correlations between distant events, this does not necessarily imply that action at one point can instantaneously affect the probabilities of events at an arbrtrarily distant location. Nelson regarded active locality (which precludes such effects) as es sential to a realistic physical theory. Others disagree; in particular, some advocates of Bohmian mechanics accept active non-locality and even regard rt as an asset. As far as I
am aware, a non-Markovian form of stochastic mechanics has not been developed. For more information on the subtle issues of active and passive localrty see [1 0].
12Wick [30] reports (p. 79 and pp. 1 72-1 73) unpublished work by Wallstrom which raises another experimental objection to Nelson's theory and a possible modifica
tion which would reconcile the difficulties. 1 3Aithough Keller [1 5] never uses the term "hidden variables," her assertion that "knowability" must be relinquished would appear to imply a belief in some form of hid den variables. In view of this, it is curious that this essay, originally published in 1 979, was reprinted in 1 985 without change, desprte widely publicized and impressive experiments of Aspect, et al in 1 982 agreeing with the predictions of traditional quantum theory [33]. To retain belief in hidden variables one now had to give up an other traditional belief, locality. The notion of physical science implies a subject whose "cognitive development" requires the development of an understanding consis tent with the results of experiments. It would be interesting to know why Keller did not even comment on the issue of locality or the EPR experiments. 14See the chapter on "Testing Bell" in [33].
26
THE MATHEMATICAL INTELLIGENCER
cal problems which arose from exam
the insights of gender can play a role
of theory connecting phenomena with
ining physical models within the con
analogous to that of a mathematical
the output from modern devices.
text of Hilbert space and operator al
language such as geometry.
gebras that are naturally associated
It is harder to see a role for gender
Appendix B. Some thoughts on
with Dirac and von Neumann's ap
in a field like quantum theory. Does
proach. I do not believe that a focus on
this mean that quantum theory is "gen
Nelson's
the mathematical equations implies a
der-free"? I think this question misses
"Bohmian mechanics" are theories
Bohmian mechanics
stochastic mechanics and
lack of appreciation for the founda
the point. Those who see a role for gen
which assume that particles have tra
tional puzzles. Rather, I believe that
der do so in the expectation that
jectories; they differ from classical me
some of the confusion arises from try
women's experiences will provide dif
chanics by introducing a probabilistic
ing to force quantum phenomena into
ferent and valuable insights. However,
a cognitive framework developed from
quantum phenomena are far from
a different set of experiences, and that
every physicist's personal experiences.
component. In Nelson's approach, the
trajectories are random. In Bohm's the trajectories are deterministic; all the
mathematical concepts are needed to
All physicists, regardless of their gen
randomness is in the initial conditions.
supplement that framework My views
der or ethnicity, must enter a strange
Both theories are designed to agree
may not satisfy others; but they are not
land and learn its customs and lan
with quantum mechanics for single
those of one who has ignored founda
guage in order to understand quantum
time distributions.
tional issues.
theory and develop an intuition for its
The advocates of Bohmian me
phenomena. Scientific inquiry and the
chanics claim [2,5] that even joint time
Acknowledgments
language of mathematics provide an
distributions or multi-particle experi
It is a pleasure to thank Professors Eric
opportunity for people to go beyond
ments can not distinguish it from stan
Carlen, William Faris, Chris Fuchs,
their ordinary experience. To insist
dard quantum theory. Their argument
Stephen Fulling, Chris King, and Alan
relies on the fact that both Nelson's
C. King for reference [ 13) and to S.
that gender play a special role in such inquiries confines women instead of aUowing them the freedom to learn new languages and explore the uni verse.
Fulling for [30]. I owe a particularly
The cultural milieu in which boys
Sokal for comments on earlier drafts. I am grateful to C. Fuchs for bringing references [ 1 1, 14] to my attention, to
theory and Bohmian mechanics imply that the Schrodinger equation holds. However, there is more to quantum theory than the Schrodinger equation,
which yields only wave mechanics. 1 5
large debt to W. Faris for clarifying a
often begin the study of science with
In the Nelson and Bohm theories, the
number of issues related to both
more exposure to the phenomena of
position operator plays a special role and the equivalence between position
stochastic mechanics and
classical physics may be changing-ir
Bohmian mechanics. Any errors or
respective of whether or not girls are
and momentum representations is lost.
misinterpretations remain my own re
encouraged to engage in comparable
More to the point, there are other ob
Nelson's
sponsibility. Moreover, I should em
activities. The days when children
servables, such as spin and angular
phasize that the opinions and biases
could take apart a radio or tinker with
momentum. I am not convinced that no
expressed here are entirely my own
a car engine to see how it works are
experimental distinction is possible us
and do not necessarily reflect the
rapidly disappearing. Circuits and vac
ing these other observables. (See [30]
uum tubes have long been replaced by
for a lively discussion of this issue and
views of Professor Faris. Appendix A. Gender Issues
much smaller electronic devices, and
[ 13] for a recent proposal of such an
modern car engines use sophisticated
experiment.)
My own involvement in the "science
fuel injection devices. Neither girls nor
Precisely because of the role played
wars" began in 1986 with a letter to the
boys will have-as I did-the opportu
by trajectories and position, the unitary
editor in the A WM Newsletter, and my
nity to learn to read a scale and observe
equivalence of different representations
interest has remained focused on gen
thermal expansion when my mother
is lost. However, the concepts of unitary
der issues. Some social constructivists
checked my temperature when I was
gates and outcome measurements are
ill. All they see is a digital readout from
essential to quantum computation. Su
ity might also provide special insights
a mysterious device. The playing field
perposition gives quantum computers
affecting the development of science.
may be leveled because modern tech
their massively parallel power; and uni tary transformation provides a natural
have suggested that gender or ethnic
In those cases where gender has been
nology has removed many phenomena
shown to have an impact, the most
from our direct experience and obser
tool for (reversible) gates. But these
convincing examples have been in sub
vation. This raises new pedagogical
tools only push the computational com
specialties of biomedical research,
challenges: it may be necessary to
plexity problems from the computa
psychology, or anthropology, where
teach children to explore and observe
tional phase to the output phase.
gender is directly relevant to the sub
at the most basic level before they can
quite remarkable that algorithms can be
ject. In these situations, I believe that
recognize the necessity of a long chain
developed that leave the computer in a
It is
1 5See [32] for an explicit discussion of the impossibility of deriving Heisenberg's hypothesis about transition probabilities from Schrbdinger's theory.
VOLUME 23, NUMBER 1, 2001
27
state upon which a suitable measure ment yields useful information. Even if quantum computation is consistent with Bohmian mechanics (which im plies reversible trajectories), the key concepts rely upon the standard Hilbert space formalism. It is difficult to imag ine how the recent advances in quan tum computation and communication could have occurred without the in sights obtained from the usual Hilbert space view. Some advocates of quantum com puting believe (e.g., [ 12]) that it will yield new information about the foun dations of quantum mechanics. Thus far, the early experiments have been too close to those of EPR type to pro vide a basis for distinguishing the usual Dirac/von Neumann theory from Bohmian or stochastic mechanics. However, the development of larger multi-bit devices may eventually pro vide such tools. As I wrote this final section, I real ized that I was describing a situation re markably similar to Graham's example. The equivalence of position, momen tum, and other representations plays an important role in the Hilbert space ap proach to quantum theory developed by Dirac and von Neumann. Bohmian me chanics singles out a particular repre sentation with special properties. What is missing is an explicit acknowledg ment of political or cultural motivation for that view. Although there are certainly situa tions in which a particular representa tion or coordinate system may give use ful insights, that is quite different from an insistence on using that representa tion exclusively. Quantum mechanics not only explained the puzzles of the early decades of this century, it also predicted the results of experiments not performed until many years later and explained phenomena not anticipated. I find this far more convincing than a sub sequent theory that is merely consistent with results known at its inception. It is curious that its proponents assert the impossibility of experimental verifica tion as a virtue rather than seeking new phenomena to explain or test their the ory. Whether or not the Bohmian view is useful, this seems to place it outside the realm of physics.
28
THE MATHEMATICAL INTELLIGENCER
REFERENCES
[1 ] J.S. Bell, Speakable and unspeakable in quantum mechanics
A U T H O R
(Cambridge Univer
sity Press, 1 987). [2] K.
Berndl, M. Daumer, D. Durr, S.
Goldstein, and N. Zangh, "A SuNey of Bohmian Mechanics" II Nuovo Cimento
1 1 08,
737-750(1 995).
[3] J. Bernstein and D. Cassidy, "Bomb Apologetics: Farm Hall, August, 1 945" 32-36 (August, 1 995).
Phys. Today 48(8),
[4] P.A.M . Dirac, The Principles of Quantum (Oxford, 1 930).
Mechanics
MARY
"Quantum equilibrium and the origin of ab solute uncertainty" J. Stat. Phys. 61, 843-
BETH RUSKAI
Department of Mathematics
[5] D. Durr, S. Goldstein, and N. Zangh,
University of Massachusetts Lowell Lowell, MA 01 854
907 (1 992).
USA
[6] C. Eckart, "The solution of the Problem of
e-mail:
[email protected]
the Simple Oscillator by a Combination of the Schrodinger and Lanczos Theories" Proc. NAS
12, 473-476 (1 926).
M . Beth Ruskai got her Ph.D. at Wisconsin in 1 969. She has been at
[7] C. Eckart, "Operator Calculus and the So
her present University most of the
lution of the Equations of Quantum Dy
years since, but with numerous tem
namics" Phys. Rev. 28, 71 1 -726 (1 926).
porary positions, including Geneva,
[8] W. Faris, "Shadows of the Mind: A Search
the University of Oregon, MIT, the
for the Missing Science of Consciousness"
University of Michigan, Case Western
1 996).
Reserve, Mittag-Leffler, and Paris IX
[9] W. Faris, "Review of Roland Omnes, The
(Dauphine). She is known for her work
AMS Notices 43(2), 203-208 (Feb. ,
Interpretation
of Quantum
AMS Notices
43(1 1 ) , 1 328-1 339 (Nov.,
Mechanics"
on operator-theoretic problems aris ing
from
quantum
mechanics
among others, the "Ruskai-Sigal the
1 996). [1 0] W. Faris "Probability in Quantum Mechanics"
orem" that an ion with sufficiently large excess of electrons can not
Appendix to [33]. [1 1 ] P. Forman, "Weimar Culture, Causality,
have a bound state. In recent years
and Quantum Theory, 1 91 8-1 927: Adap
she works especially on quantum in
tation by German Physicists and Mathe
formation. She is also known for her
maticians to a Hostile Intellectual Environ
off-duty writing on gender and sci
ment," in Historical Studies in the Physical
ence, and other social issues of the
Sciences,
3,
R. McCorrnmach (U.
ed.
profession. Her outdoor enthusiasms are gardening, hiking, and most es
Pennsylvania Press, 1 97 1 ) pp. 1 -1 1 5. [1 2] C.A. Fuchs, "The Structure of Quantum
pecially SNOW.
Information," available at http://www.its. caltech.edu/-cfuchs/lwl.html or directly from the author at
[email protected]. [1 3] P. Ghose, "The Incompatibility of the de Broglie-Bohm theory with Quantum Mechanics" posted at xxx.lanl.gov/abs/ quant-ph/0001 024. [1 4] P.
A.
Hanle,
"Indeterminacy
Before
on Gender and Science
(Yale University
Press, 1 985). [1 6] E. Lieb, "The Stability of Matter: From Atoms to Stars" Bull. AMS 22, 1 -49 (1 990).
[1 7] D. Kotschick "Gauge Theory is Dead!
Heisenberg: The Case of Franz Exner and
Long Live Gauge Theory!" AMS Notices
Erwin Schrodinger," in Historical Studies
42(3), 335-338 (Mar., 1 996).
in
the
Physical Sciences
10, ed. R.
McCormmach, L. Pyenson, and R. S. Turner
(Johns
Hopkins
U.
Press,
Baltimore, 1 979), pp. 225-269. [1 5] E.F.
Keller, "Cognitive Repression
[1 8] C. Lanczos, Z. FOr Physik 35
[ 1 9] N. D. Mermin, "Is the Moon Really There When Nobody Looks? Reality and the Quantum Theory" Physics Today 38(6),
in
Contemporary Physics" Am. J. Phys 48(8) 7 1 8-721 (1 979); reprinted in Reflections
38-47 (April, 1 985). [20] N. D. Mermin, Boojums All the Way Through (Cambridge University Press, 1 990).
The M IT Pres [2 1 ] W. Moore. Schr6dinger: Life and Thought (Cambridge University Press, 1 989).
[22] E. Nelson, Dynamical Theories of Brownian Motion (Princeton University Press, 1 967).
[23] E. Nelson, Quantum Fluctuations (Prince ton University Press. 1 985). [24] E.
Nelson. "Quantum Fluctuations-an
Introduction" in Mathematical Physics VII,
W.E. Brittin, K.E. Gustafson, W. Wyss,
eds, pp. 509-51 9 (North Holland, Amster dam. 1 984).
I.L. Chuang,
[25] M.A. Nielsen and
Quantum
Computation and Quantum Information
(Cambridge University Press, in press) . [26] J. Preskill, lecture notes on Theory of Quantum
Information
and
Quantum
Computation available at http://theory.cal
tech.edu/preskill/ph229.
Women Becoming Mathematicians C reating a Professional Identity in Post-World War I I America Margaret A. M. Murray "A sophisticated, schol ar l y,
and readable study - this is
without a doubt the best book
yet written on American
M AT H [ M A T I C l A N S
women mathematician s . ·
[27] R. Rhodes, The Making of the Atomic Bomb (Simon and Schuster, 1 988).
[28] B. Simon, The P(h Euclidean Quantum Field Theory (Princeton University Press,
1 974).
- Ann Hibner Kobl itz, Women's Stud ies Program, Arizona State U n i versity 304 pp . ,
23 illus. $29.95
[29] K. R. Sopka, Quantum Physics in America: The years through
[30] M.O.
Scully,
1 935 (AlP, 1 988).
"Do
Bohm
Trajectories
Always Provide a Trustworthy Physical Picture
of
Particle
Motion?"
Physica
To order call SOo-356-0343 (US & Canada) or 617· 625-8569. Prioes subject to change without notioe.
Scripta T76 4 1 -46 (1 998).
[3 1 ] 8. Schwarzschild "From Mine Shafts to Cliffs-the 'Fifth Force' Remains Elusive"
Phys. Today 41(7), 2 1 -24 (July, 1 988).
[32]
J . von Neumann tion
of
Mathematical Founda
Quantum
Mechanics
(English
translation, Princeton University Press, 1 955). [33] B.
L. van der Waerden, "From Matrix
Mechanics and Wave Mechanics to Unified Quantum Mechanics" in The Physicists Conception of Nature,
J. Mehra, ed., pp.
276--293 (D. Reidel Publishing, Dordrecht,
Holland, 1 973); reprinted in AMS Notices 44(3), 323-328 (1 997).
[34] D. Wick, The Infamous Boundary (Birk hauser, 1 995).
VOLUME 23, NUMBER 1, 2001
29
N.G. KHIMCHENKO
From the " Last I nterview" with A. N . Kol mogorov I, for one, have followed all my life the precept that truth is sacred, that it is our duty to seek it out and to defend it, regardless of whether it is pleasant or not. A.N.K., 1984
While I pursued a fairly wide range of practical mathematical applications, and at times obtained useful results, I remain predominantly a pure mathematician. I admire those mathematicians who became significant in technology; I fully recognize the importance of computers and cybernetics for the future of mankind; nonetheless I feel that pure mathematics in its traditional form has not yet ceded its deserved place of honor among the sciences. The only thing that could kill it would be too sharp a division of mathematicians into two tendencies: those who cultivate the newest abstractfacets without account for their ties to the real world which bred them, and those who busy themselves with "applications, " neglecting the need for in-depth analysis. A.N.K., 1963
I consider my scientific career, in the sense of getting new results, to be completed. This saddens me, but I yield to the inevitable. In recent years I have been directing my energies elsewhere, on textbooks for secondary schools and books for the mathematically talented. Ifeel the desire to participate in this project with the vigor of youth. A.N.K., 25 April 1986 (His next-to-last birthday.) The great Russian mathematician Andrei Nikolaevich Kolmogorov was open and outgoing with friends, but rarely granted interviews; few direct records of conversations with him survive. Fortunately, the film-maker Aleksandr Nikolaevich Marutyan, in planning his successful 1983 film "Stories on Kolmogorov," tape-recorded long, wide-ranging conversations in which he explored areas of potential use for the film. After Kolmogorov's death in 1987, the unique interest of these tapes was recognized by V.M. Tikhomirov, with whom they had been left. The enormous task of transcription of the fragmentary (sometimes incomprehensible) materials was undertaken by Natal'ya Grigor'evna Khimchenko. The full printed text she prepared circulated privately, and recently became available in the book Yavlenie Chrezvychainoe (Extraor dinary phenomenon) devoted to Kolmogorov. l For presentation to the general reader, it seemed ap propriate to sift the materials and put them in some kind of order. Fortunately for the mathematical public, N.G. Khimchenko returned to her labors, editing and organizing the text and translating it into English. Thanks also to 1 Edited by V.M. Tikhomirov. FAZIS, MIROS, Moscow, 1 999. See pp. 1 83-21 4.
30
THE MATHEMATICAL INTELLIGENCER ©
200 1
SPRINGER-VERLAG NEW YORK
V.M. Tikhomirov and Ya. G. Sinai for advice, to Smilka Zdravkovska for great help in the fmal editing-and of course to A.N. Marutyan for conducting the interview in the first place. In the original interview, Marutyan often doubled back on a topic several times. The editing process, aimed at uni fying and at reducing repetition, sometimes juxtaposed passages from different points in the tapes.-Editor's Note A.N.
Marutyan: Andrei Nikolaevich, I wanted to ask how you
began your journey into mathematics, what influenced your choice of direction, your choice of specialty in mathematics. A.N. Kolmogorov: It seems to me that for young mathe maticians the most common scenario does not involve afree choice of direction, but rather an attempt to solve concrete problems presented by the older generation. This is the norm. My first works in trigonometric series were all of this nature. Ideas of undertaking, let us say, a reconstruction of an entire branch of science arise at a later time. In my case, it was not that much later; namely, when I was investigat ing the basic tenets of probability theory, I aspired from
View(s) of Andrei N. Kolmogorov by the famous portraitist Dmitrii I. Gordeev. (Used by permission of A.N. Shiryaev.)
the start to build a more logical system of the concepts of this whole science. This was in the early 1930s, when I was around 30 years old. Generally mathematicians start from a certain catalogue of existing problems of interest to a given mathematical cir cle-the Luzin school in Moscow, for instance. And the young people struggle over the solutions. When they can not solve one problem, they take up another. There are al ways plenty of those closest unsolved problems to choose from. At times an excessive insistence on solving precisely one stated problem is quite detrimental to a mathemati cian's career. The majority of mathematicians begin their work under someone's tutelage; the supervisor has many such unsolved problems, and his task becomes one of matching the young students with the most suitable prob lems. If success is long in coming, then he may suggest a switch to some other similar problem.
M.: How do you face the fact that you have worked all your life in a field which most people do not understand? K.: Calmly . . . I think that the achievements of mathemat ics prove useful to mankind, while to us, mathematicians, they bring such inner satisfaction! It is a perfect solution, to have such a peaceful coexistence. A friend of mine, a pure humanist, used to say that to him mathematicians were like useful domesticated animals [Marutyan laughs], that they had to be treated in a utilitar ian way. To him, all true cultural values were humanistic; but, technology being necessary, mathematics is necessary, so we have to give mathematicians what they need to stay alive and keep going. M.: Have you ever felt jealous of someone accomplishing the same task you were working on, or doing it more elegantly? K.: No.
VOLUME 23, NUMBER 1, 2001
31
M.: Why? Because you didn't see them as rivals, or are you simply free of such a complex?
K.: Probably the latter. If a problem is solved-this is good and I am simply glad to know it. I understand that what I am saying sounds like copy-book morality. But looking back, I really do not recall such a situation of rivalry. Upon discovering that something I had been working on had al ready been done. I would feel a relief of sorts: Thank God, it's done! I am not boasting. M.: You have probably felt that some of your abilities sur pass those of your colleagues?
K.: Well, in some cases yes, in others no. Of course, when it does happen, the feeling is a pleasant one, I guess. M.: Have you ever encountered ill-will because you were generally quicker, deeper, more talented?
K.: No, I must say, I don't think I have. M.: Would you say this is because of the fairness and ob jectivity of mathematicians?
K.: Mathematics is generally a reasonably objective science. The potential of a new idea to go far and to solve existing problems becomes evident fairly quickly. Mathematics is a tremendously pleasant field of work precisely because real progress is not lost, and is in the vast majority of cases ac knowledged in time-to a greater degree than in any other sphere of human activity. M.: But was this on your mind when you were choosing mathematics?
A.N. Kolmogorov at the blackboard.
K.: That it would be a calmer life? [Laughs.] No, I was not thinking that. My final decision to opt for mathematics as my main field came fairly late, at a time when I had already
Especially, I was always extremely interested in work ing in general education.
produced works of my own. At that point it had become
I studied in an altogether extraordinary school, founded
the clearly logical path. [Pause. ] Until then I had been turn
by two dedicated women, Repman and Fedorova. And one
ing over several possibilities for myself. My first serious
wish that I have always felt is to concentrate on realizing
adult idea of a future career was forestry. I was also seri
a somehow
ously interested in history.
when my greatest ambition was to be director of a school-
A.N. Kolmogorov with a hiking party. V.N. Tikhomirov is the man on the right.
32
THE MATHEMATICAL INTELLIGENCER
ideal school. There was a rather long period
I don't mean specially a mathematical school. This was the influence of my own experience as a student. The Repman high school was set up by a group of Moscow intellectuals specially for their own children. [Pause. ] I finally decided to become a mathematician only when I became convinced that that would
suit me, whereas I had
no idea whether anything else would work or not. M.: You didn't want to risk trying out as a school director, eh?
K.: Right. But I was secretary of a school soviet for two and a half years. At Potylikh. Did you know about that episode in my life, the Potylikhin school? M.: But Andrei Nikolaevich, that Potylikhin school was so
experimental. . . . Were you also involved in the experiments?
K.: I wasn't involved, I was carrying out the experiments:
the so-called Dalton plan. To this day I believe there was a great deal of good in them. The scheme was, the teacher of each school subject, say mathematics, if there were 5 hours for it [per week] , would
tell about the subject, in an entertaining way, with demon
strations-but only for one of those hours. The remaining time the students would follow a monthly schedule of tasks: look at such-and-such a book, read such-and-such, solve such-and-such problems, find such-and-such a rela tionship and represent it graphically. M.: What are your interests outside of mathematics?
K.: If I were to rank them, then after mathematics comes my interest in educating the young, in all fields.
As to prosody [the study of verse forms] , which even among humanistic fields is a very special nook-that I re
Kolmogorov in his study.
gard as another branch of my scientific work. I am taken seriously there. I even served as
opponent
[external ex
aminer] of the doctoral dissertation of the philologian Gasparov. My works are published;
Zhirmunski'i valued
Once I was returning from Rome to Moscow, and found myself in the same car with Cardinal Wyszyriski, who was
them, and among foreign experts, Jacobson. Among other interests, let me call them
K.: No, I was attracted by art itself, but I rarely got in step, so to say, with the latest trends in our intelligentsia.
consumer inter
returning from Rome to Cz�stochowa. And in the same
ests (where I do not produce anything myself), I would name
two-person
music and also early pictorial arts, especially Russian.
Poznanski.
An indelible impression was made on me by my adven
Christianity
compartment We and
talked about
with
for
me
hours
was in
Archbishop
German
non-religious
about
humanism-of
tures in Russia's North. I would set off on such voyages,
Thomas Mann, say. . . . But the Archbishop, naturally, took
having found the location of old wooden churches in
the position that true humanism not based on religion is
Grabar's
History of Russian Art. From one church to the
impossible; I, naturally, set about trying to prove to him
next I would travel alone, sometimes on foot, sometimes
that on the contrary, faith in the eternal is not necessary
by rowboat, sometimes on board ship. The project would
for a positive human philosophy. . . .
lead me to priests, who would often put me up for the night.
much all the way from Rome to Cz�tochowa, where
This went on pretty
Traveling with a companion was also a favorite pursuit.
though he hadn't reached Warsaw he had to get off together
In these activities people trust each other and fully open up. I traveled with Dima Amol'd2 for 20 days, with Igor'
with Wyszynski. Right at the end of the conversation, my Archbishop took a tape recorder out from under the seat
Zhurbenko. Earlier, it would be with my own age group, of
[Marutyan laughs] and said, "Herr Professor, I hope you
ten with Gleb Seliverstov. He was one of my closest friends.
have no objection to my having recorded our instructive
One of my first works on trigonometric series was done
exchange." [Both laugh. )
jointly with him; he was something of a mathematician.
M.: So, a tricky agent, I'd say!
A few people
were
really close
to
me:
Volodya
Tikhomirov, Igor' Zhurbenko; and Il'dar lbragimov, my
K.: I told him I had no objection; I don't think I had been
saying anything terrible . . . . And we parted friends.
friend in Leningrad.
M.: Whereas I am openly taking a recording, Andrei
M.: You were probably also attracted by the art world?
Nikolaevich.
21J.I. Arnol'd.
VOLUME
23,
NUMBER 1 , 2001
33
What attracted you in peopl�onunon interests, or were
-�
you intrigued by people from spheres distant from yours?
K.: No, in most cases I was drawn to people with similar interests to my own. M.: How important in your life was your friendship with Pavel Sergeevich Aleksandrov?
K.: Very important. Although the difference in age between us was only seven years, in 1929 when we grew close this
\
was still noticeable. I was the junior in our relationship, a sort of protege or ward. And this tinge of patronage per sisted all our life, completely accepted by me. Have you seen the little note Pavel Sergeevich wrote for my jubilee? M.: Yes, certainly, and also the memoir he wrote.
K.: That little note about me he wrote less than a year before his death. You remember how he speaks there of this friend
ship which in the whole 53 years was always unclouded.
M.: People say of him, and he has said himself, that at the
beginning of his mathematical career there was a time when things didn't go well, he got no results, and he was
)
of a mind to give up mathematics.
K.: Yes, that's true. Nikolai Nikolaevich Luzin, the teacher of both of us, took great pleasure in constructing hy
I,
potheses, which sometimes worked out and sometimes didn't, about who in his inunediate circle of students should work on which problem and would get somewhere. In the case of Pavel Sergeevich, Nikolai Nikolaevich got the no tion-how, I don't know-that a famous problem for which there was then no known avenue of approach, the Problem of the Continuum, was going to be solved by Pavel
A portrait that hangs in Komarovka. (Artist unknown.)
Sergeevich Aleksandrov. With great insistence, and with
dents would take turns presenting papers, but the instruc
the great persuasiveness he possessed, he planted this no
tor would speak more than the other participants. The stu
tion in Pavel Sergeevich's soul, where it led to such a cri
dents with whom one eventually undertakes individual
sis as to make him decide to leave mathematics.
work tend to come from these seminars.
M.: So you and Pavel Sergeevich had difficult relations with
M.: Do you take naturally to collective work, or does it take
Luzin. . . .
away the joy of your own creativity?
K.: Yes. M.: But just when you were beginning your independent
K.: It need not. Maybe, with the critical problem of over
specialization, we must find new forms of collective work,
work, you left the Luzin entourage and the two of you
finding ways somehow to divide up a problem and solve it
founded a new circle.
piecemeal. Among my
To what extent was your work with graduate and un
students, by the way, are
masters
of organizing collective work, with the ability to divide an
dergraduate students a creative process?
area of research into pieces, overcoming the difficulties
K.: Well, even when a seminar covers elementary mater
that this entails by constant contact, distributing the work
ial, and the teacher is already familiar with the solutions
among close colleagues: Izrail' Moiseevich Gel'fand, and
to all the problems, one has the challenge of putting one
now Vladimir Igorevich Arnol'd.
self on the same level as the student.
M.: Have you been successful in such collective work, or
M.: What role have your students played in your life?
do you have more of a "loner style"?
K.: A very significant one, certainly. Emotionally, of course
K.: No, compared with them I did not find it easy to orga
some were more important than others. Some are very
nize
close to me personally.
out. . . . I must say that when I wasn't working entirely on
All my years of active work at the university would in
large groups. When I try to think of my work I carried
my own I did best in a collaboration of
two. This worked
clude two hours a week of some required course. I con
with Pavel Sergeevich Aleksandrov, with Boris Vladimiro
ducted many of these core classes: Theory of functions of
vich Gnedenko, with various collaborators.
a real variable, Functional analysis, Differential equations,
Pavel Sergeevich and I collaborated intensively only once,
Theory of probability. This was a conunon distribution of
on a fairly narrow question of topology; after that, only oc
his works and he in mine, but
work for all our professors: one such required course, and
casionally. I was interested in
a two-hour special course in lecture format addressing re
a really shared project occurred only that one time.
cent work, including one's own. Then normally there would
The first people with whom I worked closely and suc cessfully were Dmitri! Evgen'evich Menshov-on trigono-
be one or two seminars a week, in which about ten stu-
34
THE MATHEMATICAL INTELLIGENCER
Here is the list of Kolmogorov's students,* as found in V.M. Tikhomirov's article in
Mathematics
(ed.
S.
Golden Years of Moscow
Zdravkovska and P.L. Duren),
American Mathematical Society, 1993, pp.
125-127.
A.M. Abramov (education)
P. Martin-Lef (complexity) A.V. Martynov (probability) R.F. Matveev (stochastic processes) Yu.T. Medvedev (mathematical logic) L.D. Meshalkin (probability, ergodic theory)
V.M. Alekseev (classical mechanics)
V.S. Mikhalevich (probability)
A.M. Arato (probability)
M.D. Millionshchikov (turbulence)
V.I. Arnol'd (superpositions, classical mechanics)
AS. Monin (oceanology, turbulence)
E.A. Asarin (complexity)
S.M.
G.M. Bavli (probability)
A.M. Obukhov (atmospheric physics, turbulence)
G.l. Barenblatt (hydrodynamics)
Yu.S. Ochan (set theory)
L.A. Bassalygo (information theory)
Yu.P. Ofman (complexity)
ikol'skll (approximation theory)
Yu. K. Belyaev (stochastic processes)
B. Penkov (probability)
V.I. Bityuskov (probability)
A.A. Petrov (probability)
E.S. Bozhich (mathematical logic)
M.S. Pinsker (information theory)
L.
. Bol'shev (mathematical statistics)
A.V. Prokhorov (study of prosody)
A.A. Borovkov (probability)
Yu.V. Prokhorov (probability)
A.V. Bulinskii (stochastic processes)
Yu.A. Rozanov (stochastic processes)
N.A. Dmitriev (stochastic processes)
M. Rozenblat-Rot (stochastic processes)
R.L. Dobrushin (probability)
B.A. Sevast'yanov (stochastic processes)
AN. Dvoichenkov (theory of functions)
A.N. Shiryaev (stochastic processes)
E.B. Dynkin (stochastic processes)
F.l. Shmidov (theory of functions)
V.D. Erokhin (approximation theory)
Ya.G. Sinai (ergodic theory)
M.K. Fage (functional analysis)
S.Kh. Sirazhdinov (probability)
S.V. Fomin (ergodic theory)
V.M. Tikhomirov (approximation theory)
G.A. Gal'perin (dynamical systems)
A.
I.M. Gel'fand (functional analysis)
V.A.
B.V. Gnedenko (probability)
I.Ya. Verchenko (theory of functions)
O.S. Ivashev-Musatov (theory of functions)
V.G. Vinokurov (probability)
. Tulal"kov (theory of functions) spenskll (mathematical logic)
AT. Kondurar' (theory of functions)
V.G. Vovk (complexity)
M.V. Kozlov (stochastic processes)
A.M. Yaglom (turbulence)
V.V. Kozlov (probability)
B.M. Yunovich (theory of functions)
V.P. Leonov (probability)
V.
L.A. Levin (complexity)
I.G. Zhurbenko (probability)
AI. Mal'tsev (mathematical logic)
V.M. Zolotarev (probability)
. Zasukhin (stochastic processes)
·Someone whose name does not appear on this list of formal advisees, but who in effect was Kolmogorov's last student and is often listed
as
such, is the fre
quent Mathematical lntelligencer contributor, Alexander Shen.
metric and orthogonal series-and Aleksandr Yakovlevich
often in science: a problem is solved in a roundabout way,
Khinchin-on application of function-theoretic methods,
and only later is a more direct approach discovered.
especially trigonometric series and orthogonal functions,
M.:
to important aspects of probability theory. This led to joint
would you name as the greatest contributors to the
publications with both of them. The collaboration with
progress of science?
Among
twentieth-century mathematicians,
whom
Khinchin was the most productive of all, because we got
reaUy important results: we found the criteria for conver gence of random series, and conditions for applicability of the law of large numbers, and more. Then I should mention the relatively brief period of
In the country house where P.S. Alexandrov and
working with Arnol'd. The so-called Hilbert 13th problem
A.
should really be counted as a joint result of the two of us.
many years, there was a seminar room with a black
The decisive step was taken by Arnol'd, although all the
board. Several years after Kolmogorov's death it had
foundation for this and related problems was laid by me. The particular problem stated by Hilbert was settled by Arnol'd; soon after, I was able to find a much simpler al
. Kolmogorov held their famous gatherings for
not been erased, and still had in his hand, in English,
the motto
MEN ARE CRUEL, BUT MAN IS KIND.
ternative solution, but it was published later. This happens
VOLUME 23, NUMBER 1, 2001
35
flights of fancy not always directly related to anything in the real world. Can you explain your point of view on this?
K.: [Hesitates.] The essential thing is, what is an applied mathematician? There is really no separate science of ap plied mathematics. An applied mathematician is a mathe matician who knows how to apply ordinary mathematics to real problems. Thus a real applied mathematician is in terested in the real problems of some related field. [Pause. ] H e effectively ceases t o be a pure mathematician. M.: So it all depends on the nature of the problem he is solving?
K.: Yes, and an applied mathematician working on some thing like ocean hydrodynamics is treating the study of
ocean with mathematical tools. M.:
the
This leads into another question: theoretical physics. I
know that often mathematicians are very skeptical about theoretical physicists, because they apply mathematics in a "dirty" way. You personally, when you worked on turbu lence-how far were you from the real problems of physics?
K.: I would first like to reply to your reference to "dirty" mathematics.
You see, mathematicians always want
mathematics to be as pure as possible, in the sense of A.N. Kolmogorov at the seashore.
being rigorous, proof-oriented. But generally the most in teresting problems brought before us are not tractable in this manner. Then it is very important for the mathe
K.: [Pause.] Hilbert, of course . . . . Hadamard. . . . After that
matician himself to be able to find, not a rigorous, but an
it gets more difficult. . . .
effective treatment of the problem. Anyway for me this
M.: Did you pursue administrative jobs, or did they over
was always the way: if I am studying turbulence, then I
take you?
am studying
K.: I never pursued them. In some cases there was a feel
do not work, then I look into experimental materials,
ing of duty, a belief that if I took on a task it would be bet ter done.
In the case of my deanship, for example.
turbulence. If purely mathematical methods
seek to discover in them a thread of coherence, and then proceed to make rigorous mathematical deductions, but
speculative assumptions.
M.: Do you think you were successful in this?
starting from such entirely
K.: To a degree. For our Department, at least, this was one
for one value most highly this type of applied mathe-
of the better periods. I never approached administrative duties with revulsion. When Otto Yulievich Schmidt asked me to join the Presidium of the Academy of Sciences, as the academic secretary of the physics and mathematics section, in 1939, I was straightaway interested. The underlying reason for Schmidt's bringing me in was this. The physics and math ematics section didn't give much administrative role to mathematics (there are many more physics institutions, and their material resources are much greater). At that time there were two physicists with every claim to leadership of all of physics: Kapitsa and Ioffe. So Schmidt took it into his head to set between them a very young mathematician! And it really didn't work out too badly. [Marutyan laughs. ] When the astronomers had t o select a location for a large geophysical observatory somewhere in the South, I visited the proposed sites-in private capacity, just as a tourist to get a better perspective on them. My Academy work was only three years, 1939-42. M.: Andrei Nikolaevich, you have said several times, if I understand correctly, that the distinction between pure and
applied mathematics is not at all sharp. But in the popular image, applied mathematics is especially computers and computation, whereas pure mathematics is more abstract
36
THE MATHEMATICAL INTELLIGENCER
The bust at Kolmogorov's grave in Novodevichi Cemetary.
I
matician, who essentially ceases to be a mathematician
to some scientific lecture or reading something new, you in
and simply solves problems of physics-if possible by
evitably try to figure: perhaps I could do something here?
rigorous "pure" deduction, but if that doesn't work, then
M.: Andrei Nikolaevich, in the days of Archimedes or even
by introducing assumptions.
Newton, the study of the surrounding world was accessi
M.: So you favor flexibility of thought?
ble to any educated person.
K.: And where possible, participating in
experiment with
M.:
K.: That is true. M.: But now it is not sufficient simply to be educated. One
the physicists.
In the course of solving a problem, have you ever
thought about whether it was important? Did you have
may have understanding, and not that full, in a single field
of theory. Do you feel that this condition is a natural result
global goals?
of progress, or is it a stage through which we will pass and
K.: No. To be sure, in the overall planning of one's work, in
perhaps once more return to some kind of common un
choosing which new books to read, in combing the scientific
derstanding?
journals to decide which works need closer study-there this
K.: Are you familiar with Shklovskii's The
rational element is present, must be present. But on the other
and the Mind?
Universe, Life,
hand, spontaneous sparking of interest in a hypothesis which
M.: Yes.
just leaps into one's head can often be crucial. M.: Has intuition played a role in your thought process, and
K.: He maintains that the development of every culture, if it
to what degree?
what might befall humankind now-culminates in a stage of
is not aborted by some catastrophic events-and we all know
K.: Of course, a very important role. This is very common
loss of interest in technology. Perhaps he really is right.
for a mathematician.
M.: What does "loss of interest in technology" mean? You
M.: How do you work, Andrei Nikolaevich?
mean that people occupy themselves more with humanis
K.: Real scientific work? Usually it goes as follows: You
tic problems?
read books, you prepare your own lectures with some new
K.: Not really humanistic problems. But it must be possi
variations, and suddenly, an unexpected idea springs up
ble to return to a more basic and child-like
if the problem
Do you know the German writer Hesse?
out of the soil of this everyday work: what
joy in living.
at hand can be solved entirely differently from known ap proaches? Another way becomes vaguely visible . . . . Then for a mathematician, and probably for any other scientist,
A U T H O R
it is overwhelmingly important to set aside everything else and simply think, think unrestrictedly on this new way which has just appeared. Fortunately I usually had the op portunity to do this. There's a mathematician named Boris Nikolaevich Delone. You must have heard a lot about him. M.: Yes, sure.
K.: When he would speak to students and they would ask him what is the essence of a scientist's creative work, he would answer like this: "Suppose you are in a mathe matical Olympiad. They give you four hours to solve five problems (that's how it goes in the Olympiad). So you
N.G. KHIMCHENKO
devote around an hour to a problem. Now imagine a prob
Mathematics Department
lem which would take you not one hour to solve, but
Moscow State University
some idea of what a real scientist does." Now no doubt
Moscow 1 1 9899
Boris Nikolaevich was greatly exaggerating. For me, any
e-mail:
[email protected]
(say)
5000 hours of constant thought! Then you'll get
Russia
way, in the making of all of my scientific discoveries, such utter concentration, excluding all else, would last
Natal'ya Grigor'evna Rychkova (married name Khimchenko)
sometimes a week, maybe sometimes two weeks, but no
has been an associate professor in the Chair of Probability
more.
Theory
M.: What considerations led you sometimes to divert your
ment since 1 964. She worked under the supervision of A.N.
at
the Moscow State University Mathematics Depart
attention sharply into new areas?
Kologorov on mathematical problems of linguistics and poetry.
K.: I do not believe you put the question correctly, because the various fields of mathematics in which I have worked
Currently
works left by Kolmogorov. Both interests are exemplified in the
have usually led directly into one another, so that passing
article "Analysis of the rhythm of Russian verse and probabil
from one to another was
natural.
is much occupied with recovering and editing
ity theory" prepared from an unpublished manuscript of his,
In principle, you see-not with a conviction that I will un doubtedly achieve something, but out of general curiosity
aU mathematics more or less interests me.
she
published in Teoriya Veroyatnostei i ee Primeneniya 44 (1 999), 4 1 9-431 .
When listening
VOLUME 23, NUMBER 1. 2001
37
M.: Yes. K.: In Das Glasperlenspiel, Hesse depicts such a society, and quite brilliantly, I would say. A society which has lost interest in technological progress. M.: What role has chance played in your life? [Both laugh.] After all, you worked on stochastic processes. K.: I would be hard pressed to say. On the whole I believe that in a slightly different time, with a different form, still es sentially what I was able to contribute to science would have been done if the distribution of roles had been different. M.: In other words, if you had been surrounded by other people, worked in different circumstances . . . ? K.: It is likely that the objective outcome would have been more or less the same.i
Andrei Nikolaevich, you know there will have to be music in this film I'm making. . . . K.: Yes, certainly. M.: And I'd like the music heard in the film to be some thing close to you. Do you have some favorite pieces? K.: I hope there will be a place in the film where you tell about the musical evenings at Komarovka for our friends. Pavel Sergeevich and I would regularly have a good many guests for those occasions. At that point in the film I would like you to play Bach's Concerto for Two Violins. M.: That was the favorite piece of you and Pavel Sergeevich? K.: I think we had that in common. We would listen often to Mozart's G minor Symphony. M.:
S P R I N G E R F O R M AT H E M AT I C S ROBIN WILSON, The Open University, Oxford, England
PETER HILTON, State University of New York, Binghamton, New York; DEREK HOLTON, University of Otago, Dunedin,
STAMPING THROUGH MATHEMATICS
New Zealand; JEAN PEDERSEN, Santa Clara U niversity, Santa Clara, CA
In this unique book, you will find a full, rich color,
MATHEMATICAL VISTAS
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From a Room with Many Windows
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of which is illustrated with mathematical figures, people, and content. This is the perfect gift book for anyone interested in stamps, or the surprising use of mathematics in the real world. The author is widely known for his column on stamps in the magazine,
The Mathematical lntelligencer.
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the interest of bright people in mathematics. The
book consists of nine related mathematical essays
which will intrigue and inform the curious reader. This book is a sequel to the authors· popular
book, Mathematical Reflections, and can be read
independently. 2001/APPROX.
JAMES J, CALLAHAN, Smith College, Northampton, MA
THE GEOMETRY OF SPACETIME An Introduction to Special and General Relativity
In 1905, Albert Einstein
The goal
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- special relativity- to explain some of the most troubling problems in current physics concerning electromagnetism and motion. Soon afterwards,
Hermann !v!inkowski recast special relativity essentially
344
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38
THE MATHEMATICAL INTELLIGENCER
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A Story of a Pai nting
Wojbor A. Woyczynski The lUldistinguished low-sltiDg building of the A.M. Obukhov Institute of Atmospheric Physics of the Russian Academy at 3 Pyzhevski'i per. in Moscow, which I visited in January 2000, is dwarfed by the massive, ornate, and marble-clad edifice of the Ministry of Medium Machine Construction across the street. The Ministry's name is a euphemistic cover for the bureaucracy that long supervised the Russian nuclear weapons program. My friend and host, Valeri'i Isakevich Klyatskin, a mathematical physicist, tells me that in 1937 the Institute's mul tilevel basement contained the laboratory of Academician Igor Vasilevich Kurchatov, the father of the Soviet atomic bomb, and it is here that the first chain reaction
in Russia was achieved. Curiously, there is no plaque
commemorating that event.
A second-story office, obviously little used, contains a second surprise, an oil painting of A.
. Kolmogorov.
Valeri! explains to me that the portrait was painted in 1966 by Dmitrii Gordeev, who was then Kolmogorov's student. Gordeev was a teacher at the time in the well-lawwn school for mathematically talented children l'tlll
in Moscow by Kolmogorov, and the painting was intended to hang in the school building. However, Kolmogorov disliked the painting, which shows him
in an idiosyncratic knuckle-cracking gesture well remembered by all
those who knew him. He vetoed the proposal. Gordeev's wife, who worked at the Institute, mentioned to several yotiDg turbulence researchers there that the painting was looking for a home. They put together 40 rubles to purchase it. (A professor's monthly salary was 105 rubles.) Since then Gordeev's career as a painter blossomed, and his portraits of many m::Yor figures are widely known. Yost Hall
220
Case Western Reserve University Cleveland, OH
441 06-7058
USA e-mail: waw @ po.cwru . ed u
VOLUME
23,
NUMBER 1 , 2001
39
I,�Mj,i§,Fh1¥11@i§#fii,i,i§,id
This column is devoted to mathematics forjun. What better purpose is there for mathematics? To appear here, a theorem or problem or remark does not need to be profound (but it is allowed to be); it may not be directed
only at specialists; it must attract and fascinate. We welcome, encourage, and frequently publish contributions
Alexander S h e n , E d itor
The Importance of Being Formal K.S. Makarychev and Yu . S. Makarychev
from readers-either new notes, or replies to past columns.
T
I
(say) A's cards does not prevent him having any specific card, because each of the cards
0, . . . , 5
occurs on every
row: Sum
Possible combinations
0 1 2 3 4 5 6
{0,2,5}, {0,3,5 }, {0,4,5}, {0, 1,2}, {0, 1,3}, {0,1,4}, {0, 1,5},
{0,3,4}, { 1,2,5}, { 1 ,3,5}, { 1 ,4,5}, {2,4,5} {0,2,3}, {0,2,4},
{ 1,2,4} { 1,3,4} {2,3,4} {2,3,5} {3,4,5} { 1,2,3}
C cannot name any of A's cards. C cannot name any of B's cards. If C has any other card instead of 6 , the situation is similar: cir Thus
he following problem (suggested by A. Shapovalov) was given to the
participants
in
the
Olympiad in spring
Moscow
Math
2000.
For the same reason
cular shift of the cards does not change anything.
The deck of cards contains seven However, the problem is subtler cards labeled 0,1,2,3,4,5,6. The cards than the organizers realized. To ex are shuffled and distributed among plain why, let us consider the follow three people A, B, C. A and B receive ing "solution." IfA holds {p, q,r}, he says three cards each; the remaining card to B, "If you don't have card p, then I have cards {p , q ,r )." Likewise B, hold is given to C. Show that A and B can exchange information about their ing {u,v,w }, says to A, "If you don't have cards (ensuring that B knows A's u then I have {u,v,w } ." Each of them cards and vice versa), speaking in the knows that the other's hypothesis is presence of C, in such a way that C true and so has been informed of the stiU cannot name any card (other other's cards. But if their hands were than his own) and say whether A or exchanged (so that A held {u,v,w } and B held {p , q ,r}) , both hypotheses would B has it. be false and therefore both statements
The organizers of the Olympiad con
would be true. So
C, if he is
allowed to
sidered this problem as well posed,
draw conclusions only from the state
having in mind the following solution:
ments and not from the knowledge
Each of the players A and B declares the sum modulo 7 of the three cards he has.
not exclude either scenario,
The sum of the declared cards will be
you agree that something is wrong
(0 + 1 + 2 + . . . + 5 + 6) minus C's card, so A and B will
would not have enough information to
needed by
A and B to assert them, can hence
does not know the location of any card. Is this solution "good"? Probably
the sum of all cards
know C's card and thus each other's cards. Please send all submissions to the
with it In fact
if
A held
{u,v,w},
he
say truthfully to B, "If you don't have p then I have {p, q,r} " (for he would not
To show that the solution is valid, it
know which three of the four cards
remains to show that this does not give
it fail to satisfy the conditions stated?
Information Transmission , Ermolovoi 1 9,
his own). Assume for example that
What is a "good" solution anyway?
K-51 Moscow GSP-4, 1 01 447 Russia;
has the card with
e-mail:
[email protected]
ing table shows that fixing any sum of
Mathematical Entertainments Editor, Alexander Shen, Institute for Problems of
C the
location of any card (except for
6
C
on it. The follow
hidden from him B held). But how does We see that to make the problem
clear we need a formal definition.
© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 1 , 2001
It 41
turns out that there are several natural definitions which are not equivalent. Let us describe two of them briefly. (1) A solution is a pair of algorithms A and B that prescribes the behavior of A and B during the game: what to say given the cards and any messages received from the other player. We re quire that for any configuration of cards, the protocol of exchange to gether with A's cards determines B's cards uniquely and vice versa. On the other hand, for any configuration and for each card x other than C's card, there should exist another configura tion that produces the same protocol but gives x to a different player. (2) A solution is a rule that says what statements A and B may make on see-
A Bet With Leonid Levin T
he axiom of choice is a well known source of paradoxes. Here is one more suggested by Leonid Levin from Boston University (e-mail:
[email protected]). Let us begin with a standard con struction. We call two reals x,y E [0,1) equivalent if (x - y) is a fmite decimal fraction, i.e., if (x - y) 10" is an inte ger for some positive k E N. Select one element from each equivalence class. Let A be the set of all selected ele ments. Then the sets A+f (mod 1) for all (finite decimal) fractions j E [0, 1) are disjoint and cover [0,1). If we con·
42
THE MATHEMATICAL I NTELUGENCER
ing their cards. A statement determines a subset of the set of all configurations. We require that A's statement together with B's cards determine A's cards and vice versa, but that A's and B's state ments together with C's card do not de termine the position of any other card. These definitions can be applied not only to this game but also to other sim ilar games, and one can prove that de finition (1) is always stronger. The in tended solution of the Olympiad problem satisfies both defmitions (1) and (2), but the second one, the "bad" one, satisfies only (2). For the same problem with only five cards instead of seven, there is no solution according to definition ( 1), but the "bad" solution still works for definition (2).
We know of another natural defmi tion intermediate in strength between (1) and (2). What is the moral of the story? The importance of being formal is well un derstood in logic-and in computa tional cryptography, which, by the way, is essentially the way our prob lem is solved. But this problem illus trates that even in very simple cases the absence of a formal defmition can lead to ambiguity-which is better avoided, especially in a math competi tion problem!
vert [0,1) into a circle, all A+f are im ages of A under rotation. Letfi andf2 be two different (finite decimal) fractions. Consider the fol lowing bet: we pick at random some x E [0,1) (by rolling a die for each digit). If x E A+fi (mod 1) then I pay you $1; if x E A + !2 (mod 1) then you pay me $1; otherwise (if x is outside both sets) nothing happens. This is a fair bet since A+fi and A + !2 differ only by a rotation. To make the bet at tractive to you I will even pay you $2 against your $1. If you agree to play this game, I pro pose we make many bets simultane ously. For any finite decimal fraction! E (0, 1), letf' denote the fraction!' E [0,1) that is the fractional part of 10!, i.e., f' = lOf mod 1. (For example, if f = 0.502 1 thenf' = 0.021, and if j = 0.6 thenf' = 0.) Let us make (J, f')-bets for all frac tionsjat the same time. (Note that each
individual bet is good for you, so the whole game should be also good.) If you agree with me, we start play ing immediately before you realize the consequences. Imagine what happens if (say) x E A + 0.057 (mod 1). Then you win the (0.057,0.57)-bet and get $2, but at the same time you lose the (0.0057, 0.057)-bet, the (0. 1057,0.057)-bet, the (0.2057,0.057)-bet, etc. (altogether 10 bets). So in fact you pay $10 and get only $2. Ifx E A + 0, you win no bets and lose 9 bets, so the situation is even worse. Note that only two people participate in the game, and for each x only finitely many bets (at most 1 1) are implemented, so this scheme does not resemble pyra mid schemes where infinitely many par ticipants pay one another in such a way that each has positive balance. Doesn't charity make donors richer? [and the recipients more miserable? Ed.]
PO Box 65 Moscow 1 27322 Russia e-mail:
[email protected]
LAN WEN
Sem anti c Parad oxes as Eq u ati o n s1 By a paradox we mean generally an argument that leads to contradiction for no clear reason. The most ancient and most influential paradox in history is perhaps the paradox of the Liar. Here is a well-known version of it:
The Liar Paradox.
"This sentence is false."
If it is true then it is false, and if it is false then it is true. This popular argument is quite short. So we repeat the argument with some commonly understood explanation in serted: If it is true then what it says should be the case and hence it is false. If it is false then what it says should be negated and hence it is true. The argument leads to contradiction. It is unclear at first glance what goes wrong. A number of theories have been proposed in the literature to resolve the Liar paradox, no tably the hierarchy theory of language of Tarski [ 1 1], which separates sentences into different levels, and the truth value gap theory of Kripke [5], which adopts three-valued logic. Nice accounts can be found in [1], [2], [4], [5], [6], [7], [9], [10], [11]. In this paper I present a different solu tion to the Liar paradox. It is not hierarchic, and adopts the classical two-valued logic. The main observation is this: Main Observation. There is an assumption implicitly used in the Liar argument. With this assumption un covered, and announced explicitly in front, the Liar ar gument wiU be found to be a normal ''proof by contra diction", but not at aU paradoxical.
This is supported by a "Three Cards paradox" I found recently, which uncovers the connection between Liar-like paradoxes and inconsistent Boolean systems. To make my
point precise I need to introduce some basic notions into our ordinary language, such as "sentence given", "sentence unknown", "sentence equation", and "sentence solution", for sentences. Note that analogous notions are standard in elementary algebra, for numbers. When these notions be come available for sentences, we can solve not only para doxes of the Liar type, but also some others. LOb's para dox will be an example. After giving a thorough exposition of the ideas, I will proceed in the last section of this paper to a more formal treatment. The Liar Paradox as an Equation
First note that what the Liar paradox displays is not a sin gle object, but a relation between two. I hinted at this by using quotation marks in stating it. From the argument we can see that the paradox needs to use the term "This sen tence" to refer to the statement "This sentence is false". Indeed, the word "it" printed above in italic appears six times and serves as a link: In first, second, fourth, and fifth occurrences it stands for "This sentence is false" (four words), but the third and sixth times it stands for "This sentence" (two words) (take a few seconds to verify this). Since the relation is "refer to", we may call it a "referential relation." In symbols, if we use A to abbreviate "This sen tence", F to abbreviate "is false", and ": =" to abbreviate "refer to", then what the Liar paradox displays is not a sin gle object AF, which reads "This sentence is false", but a referential relation
A : =AF, which reads " "This sentence" refers to "This sentence is false" ". Some people take the referential relation to be straight equality A =AF. All conclusions of the present pa per hold automatically under such a stronger identification, but it is clearer to keep the referential relation less special. A further observation is that this relation should better be considered a "presumed" one but not a "verified" one.
1Work supported partially by NSF of China and Qiu Shi Science & Technologies Foundation.
© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 1, 2001
43
The term "This sentence", denoted by A, is like a pronoun that can be, but is perhaps not yet, specified to be some specific sentence. This is like an unknown x in algebra, which can be, but is perhaps not yet, specified to be some specific number, that is, some given. Thus the Liar refer ential relation should better be regarded as a "referential equation," and written as
sentence says is the case, hence the first sentence shotlld be true, which contradicts that the first sentence is false. This way we have run out of possibilities with contradic tions everywhere. This is a new paradox I discovered just recently. Both the statement and the argument are of the same type as the Liar. In symbols, it is written as
{
X : = XF. Of course, to regard a relation as an equation does not exclude the possibility that the relation might be verified later, hence it is not a loss but just a precaution. Also note that, though the notions of "given", "unknown", and "equa tion" are standard for numbers in elementary algebra, they are not customarily applied to sentences in our ordinary language. The Three Cards Paradox: The Secret of the Liar
To present my solution to the Liar paradox, perhaps the best way is first to present a new paradox, the Three Cards paradox (the device of using cards is inspired by the fa mous Jourdain's Cards paradox, which involves two cards [2]). Uncovering the secret of it leads directly to the solu tion to the Liar paradox.
The Three Cards Paradox. Consider three cards with the following three sentences: The sentence on the second card is true, and the sen tence on the third card is false. Either the sentence on the first card is false, or the sentence on the third card is true. The sentences on the first and second cards are both true. This paradox looks fancier than the Liar (and Jourdain), what with the logical connectives "and" and "or". The ar gument must be more complicated, but may yield insight. Assume the first sentence is true. Then what it says should be the case, which means the second sentence is true and the third sentence is false. Hence what the third sentence says should be negated, which means that either the first sentence is false, or the second sentence is false. Putting these together, we have that either the second sen tence is true and the first sentence is false, or the second sentence is both true and false. But this "or" is impossible, because we adopt the classical two-valued logic. Thus this "either" must hold. That is, the second sentence is true and the first sentence is false. This contradicts the assumption that the first sentence is true at the beginning. Thus the first sentence must be false. Then what the second sentence says is the case, hence the second sentence is true. Moreover, what the third sen tence says is not the case, hence the third sentence is false. In summary, the first and third sentence are each false, but the second sentence is true. In particular, what the first
44
THE MATHEMATICAL INTELLIGENCER
X : = IT !\ ZF, Y : = XF V zr,
Z : = XT !\ IT,
where !\ stands for "and", and V stands for "or". But where does this paradox come from? Let me reveal the secret: In writing such a complicated "paradoxical" ar gument, I had in mind the following Boolean system.
The Boolean Model for the "Three Cards. " The Boolean
{
system
x
=
yz,
x + z, z =xy
y=
has no solution. Proof Assume there exists a solution. That is, assume there are three givens x, y, and z that satisfy the system. This will lead to a contradiction. Assume x = 1. Then, by equation 1, y = 1 and z = 0. By equation 3, z 0 yields either x = 0, or y = 0. Putting these together, we have either y = 1 and x = 0, or y = 1 and y = 0. But this "or" is impossible. Thus this "either" must hold. That is, y = 1 and x = 0. This contradicts the assumption x = 1 at the beginning. Thus x = 0. Then z =0 from equation 3, and y = 1 from equation 2. In summary, x = z = 0, but y = 1. However putting x = z = 0 and y = 1 into equation 1 yields a contradiction. This proves there are no three givens x, y and z that sat isfy the system. That is, the system has no solution. The reader may have noticed a clear resemblance be tween the Boolean system and the Three Cards paradox. The difference is clear too: In their statements, one has a phrase "has no solution", the other does not; in their argu ments, one has a standard frame of proof by contradiction, that is, the "head" "Assume there is a solution, we derive the following contradictions" and the "tail" "This contra diction proves there is no solution", while the other does not. In fact my analysis of Three Cards was just translation of the Boolean problem into ordinary language, only I cut off the phrase "has no solution" from the statement, and cut off the head and the tail from the argument. As ex pected, the normal Boolean proof became a mysterious ar gument that leads to contradiction apparently with no rea son, that is, a "paradox." However, removing the head does not affect the argu ment. Because the head "Assume there exists a solution" is merely an announcement for the assumption. The actual use of this assumption takes place not in the head, but in =
the body of the Boolean proof. Removing the tail does not affect the argument either, because the argument has fm ished already. Thus the solution to the Three Cards para dox must be (informally) this: It is (the translation of) the assumption of existence of a solution that causes contradiction in the Three Cards paradox. The assumption is tacit and goes unnoticed. It is believed traditionally that Liar-like paradoxes are logically different from Boolean problems. It is believed that in Boolean "proofs by contradiction" one assumes ex istence of solution and hence derives contradiction, but in Liar-like paradoxes one does not assume anything, except some basic rules of language and logic, hence contradic tions must have some deep, yet unknown cause in our lan guage or logic. The Three Cards paradox shows this is not the case.
An Informal Solution to the Liar Paradox
The Liar paradox involves the same secret. Indeed, the Liar paradox corresponds in the same way to the following Boolean model.
The Boolean Model for the Liar. x = x has no solution.
The Boolean equation
Proof Assume there exists a solution, that is, assume there is a given x that satisfies the equation. We derive the following contradiction. If x = 1 ("If it is true"), then x = 0 ("then it is false"). And if x = 0 ("And if it is false"), then x = 1 ("then it is true"). This contradiction proves that there is no given x that satisfies the equation. That is, the equation has no solution. We have enclosed the Liar argument in the parentheses for comparison. It is indeed the translation of the Boolean proof, with the head and tail removed. (Here by translation I mean logically, but not historically. Historically, the Liar paradox is perhaps 2500 years older than Mr. Boole.) Thus the solution to the Liar paradox is informally this:
one that corresponds to the algebraic term "equation" could be called "referential equation" (I have already in formally called the Liar relation X : = XF a "referential equa tion"), and this reduces to establishing the notion " : = " or "refer to", for sentences. The one that corresponds to the algebraic term "given" could be called "sentence given", which should be established in a way that is compatible with the notions of truth and "refer to". We need also some related notions such as "sentence unknown" and "sentence solution", to a "referential equation". While the ideas are very natural, the formal work involved is deferred to the last section of this paper. Assuming this formal work done, I can state the formal solution to the Liar paradox as follows:
Solution to the Liar paradox (Formal version). It is the assumption of existence of a sentence given that satis fies the Liar equation X : = XF that causes contradiction in the Liar paradox. In other words, there can be no sen tence given that says, of itself, that it is false. Thus the solution to the ancient Liar paradox is simply the negation of the original Liar relation, with only one word "given" put in! This sounds like cheating. But actu ally this is the right conclusion, expressed in the new ter minology. The Huge Class of Liar-like Paradoxes
The reader can create a huge class of "Liar-like paradoxes," corresponding in this way to Boolean systems that have no solutions. The Liar paradox, the Three Cards paradox, and Jourdain's paradox are just the three simplest examples in the class. The number of sentences or cards involved can be arbitrarily large. Without the help of Boolean theory, we would not suspect there is such a huge class of "paradoxes" in ordinary language. All Liar-like paradoxes involve the same secret, and can be solved the same way. Criteria in Boolean algebra that determine which Boolean systems have no solution become automatically criteria in ordinary language that determine paradoxical referential systems of sentences. The Truth-teller
Solution to the Liar paradox (Informal version).
It is (the translation of) the assumption of existence of a so lution that causes contradiction in the Liar paradox.
Thus a referential system of sentences is paradoxical if the corresponding Boolean system has no solution. But what if the Boolean system does have solution? Here is such a problem, known as the Truth-teUer.
The Formal Solution to the Liar Paradox
The above solution to the Liar paradox is informal. What it needs is how to translate the term "existence of solution" from Boolean algebra into our ordinary language. The term "existence" needs no translation. It is a universal term used in many disciplines. The term "solution" reduces to two other terms, "given" and "equation" (in algebra a solution is just a given that satisfies an equation), which need some preparation. There are not yet corresponding notions for sentences in our ordinary language. We need first to es tablish these notions before we can do the translation. The
The
Truth-teUer."This sentence is true."
Again, this is understood as not merely a single object, but a relation between two. More precisely, the Truth teller gives a referential relation X : = XT. The corre sponding Boolean equation is hence x = x, which cer tainly has solutions. Thus Boolean diagnosis reveals nothing wrong. But in some sense something is wrong with the Truth teller. A diagnosis for Truth-teller is given below in the last
VOLUME
23,
NUMBER 1 , 2001
45
section about the formal work According to the diagno sis, the Truth-teller equation has solutions respecting some interpretations of "refer to", but no solution re specting some other interpretations. This fact is not per ceivable by Boolean diagnosis. Boolean diagnosis is coarse. If it says fine", things may not be really fine, as the Truth-teller shows. (But if it says "ill", things must be seri ous. Contradictions that appear in Liar-like arguments are of serious Boolean nature.) An Application to Lob's Paradox
Establishing the notions of "sentence given" and so on does more than just solve Liar-like paradoxes. The well known Lob's paradox, which is not Liar-like, is an example.
Lob's paradox (1955). Let A be any sentence. Let B be the sentence: "If this sentence is true, then A." Then a contradiction arises. In fact, B asserts: "If B is true, then A." Now consider the following argument: Assume B is true. Then, by B, since B is true, A is true. This argument shows that, if B is true, then A. But this is exactly what B asserts. Hence, B is true. Therefore, by B, since B is true, A is true. Thus, every sen tence is true, which is impossible. This is the argument of Lob's paradox, quoted from [8]. Observe that the argument is based on a referential rela tion. Indeed, in the statement, the letter B denotes the sen tence "If this sentence is true, then A", and so does the let ter B at the beginning of the argument. But right next to that, inside the quotation marks, the same letter B denotes merely the term "this sentence". Thus the paradox uses "this sentence" to refer to "If this sentence is true, then A". In symbols, this is the referential equation
x := cx� A) . Here is the solution of Lob's paradox. With the notions of "sentence given" and so on at hand, it reduces, as do the Liar-like paradoxes, to a proof by contradiction. The se mantic statement is that LOb's equation has no solution. The proof is exactly LOb's argument. Formal Work
This section contains the formal work The main issue is to establish formally the notions of "refer to" and "sentence given". Then the notions of "sentence unknown", "sentence equation", and "sentence solution" will follow. Another is sue is to make precise the correspondence between Liar like paradoxes and inconsistent Boolean systems. I also in clude some different interpretations for the notion of "refer to", which are not essential to the Liar paradoxes, but es sential to explaining the Truth-teller. I start with the standard sentential logic, denoted by L. The notions of sentence, truth value, sentence connectives --, ("not"), 1\ ("and"), V ("or"), � ("if . . . then"), � ("if and only if"), etc., are standard, for instance from [3] or [8]. I use capital letters to denote sentences. With these con nectives several sentences could combine into a longer sen-
46
THE MATHEMATICAL INTELLIGENCER
tence. As convention I will not write long Boolean exp:r;:es sions below, but shorter ones such as (A /\ B) V (C /\ D) instead. Subscripts could be resorted to if needed. I use T(A) to denote the truth value of a sentence A. Thus T(A) is an element of the two-valued Boolean algebra; it is either 1, meaning A is true, or 0, meaning A is false. The truth value of a long sentence is determined in the natural way by the truth values of its component sentences. For instance, the truth value of the sentence (A 1\ B) V (C 1\ D) is T(A)T(B) + T(C)T(D).
To study our paradoxes I follow Kripke [5] by extend ing L to a language :£ by adding to L a truth predicate "is true", denoted by T, together with its negation "is false", denoted by F. T and F are supposed to possess the fol lowing natural properties: Tl. AT � --..AF. T2. (A 1\ B)T � (AT 1\ BT). T3. (A V B)T � (AT V BT), where A, B are sentences in :£. All this is standard. Now I introduce a relation REfer to, denoted by " : = ", between sentences of :£. It is to obey the following axiom.
Axiom of truth-REference.If a sentence A REfers to a sen tence S, then A is true if and only if S. In symbols, if A : = S, then AT � s.
This axiom restates formally the two rules used in the argument of the Liar paradox: If a sentence A is true, what it "says", or what it "REfers to", which is S, should "be the case". And conversely, if a sentence A is false, what it "says", or what it "REfers to", which is S, should "be negated". We will see below that it is this axiom that trans fers Liar-like arguments into Boolean arguments. Like many other axiomatically defmed objects, the no tion of "REfer to" is to be without properties other than the axiom which defmes it. To distinguish it from the usual verb "refer to" in the dictionary, I have written its first two letters in capital. It hence allows many interpretations, not just different words such as "says that" or "refers to". I name two, one very fine, and one very coarse.
Example A. Here ": =" is interpreted as the "quotation mark
name." That is, a sentence enclosed within two quotation marks, and only this sentence, is considered to REfer to the sentence with the quotation marks removed. A classi cal example of Tarski is that "It is snowing" is a true sen tence if and only if it is snowing [1 1, P.156]. This interpre tation is very fme. For instance, the interpreted relation is not reflexive, nor symmetric, nor transitive. To admit such an interpretation one has to extend :£ further because, for a sentence A in :£, "A" is not in :£. This is compatible with Tarski's hierarchy theory of language.
Example B. �.
Here ": =" is interpreted as the biconditional
sentence solution of a
set of sentence givens is called a
�
REferential system if, for every equation of the system, re
A for
placing uniformly the unknowns by these sentence givens,
any A. Thus, under this interpretation of "REfer to", the
the obtained sentence given on the left-hand side of the
relation
This interpretation is very coarse. Since A
A for any A, the axiom gives immediately that AT
�
truth predicate T has no effect. This is often done in math
equation REfers to the obtained sentence given on the
ematics.
right.
Now
I proceed to the central issue, "sentence given".
Definition. Sentence givens of 5£ are defmed recursively: G 1. Any sentence of L is a sentence given of 5£. G2. AT is a sentence given of 5£ if and only if A is a sen tence given of 5£. Likewise AF.
G3. If B : = A, then B is a sentence given of 5£ if and only
A special feature of REferential equations that is not
shared by the usual algebraic equations is that whether or
not a REference system has a solution may depend on in terpretations of "REfer to". For instance, the Truth-teller equation
tion is so strict that only "AT", but not A, can REfer to AT.
However, the same equation
if A is a sentence given of 5£.
G4. If A is part of a sentence B, then replacing A with a sentence with the same 5£-givenness does not change the
from Example B above, any sentence given A in this case Now I make precise the correspondence between
G5. Only sentences that reduce to sentences of the orig procedures
X : = XT does have solutions In fact, as seen
under the interpretation of biconditional.
satisfies A � AT and hence is a solution of Truth-teller.
5£-givenness of B. inal sentential logic
X : = XT has no solution under the interpretation
of Tarski's "quotation mark name", because this interpreta
L
in finitely many steps of reduction
REferential systems and Boolean systems. A REferential equation gives rise to a Boolean equation as follows.
G1 through G4 are sentence givens of 5£.
Replace ": = " by Then we defme a
sentence unknown X of 5£
to be a
place-holder that can be occupied by sentence givens
"=",
replace respectively 1\ and
V by
Boolean multiplication and addition, and replace capital letters by corresponding small letters in the following way:
Replace the capital letters on the left-hand side of the equa
of 5£. Note that G3 indicates a relation between givenness and
tion by the corresponding small letters, replace the capital
REference. This is like the situation in algebra, where not
letters together with predicate T on the right-hand side of
3 and so on are givens, but
the equation also by the corresponding small letters, but
also symbols 7T and sin 7T/2 are, as long as these symbols
replace the capital letters together with F on the right-hand
only the actual numbers 1, 2,
side of the equation by the opposite of the corresponding
refer to some givens. On the other hand, "This sentence", denoted by A in the
small letters. Let us call the Boolean equation so obtained
associated
Liar paradox, is not a sentence given. Indeed, it does not
the
satisfy the above defmition for sentence givens. This is be
tion. For instance, the associated Boolean equation of the
cause, among the four reduction procedures G 1 through G4, only G3 applies. But by G3, its givenness reduces to
REferential equation
G2, back to the givenness of "this sentence" itself, or reduces, by G3 and G4, to the givenness of " "This sentence is false" is
is
X:=
the givenness of "This sentence is false". The givenness of "This sentence is false" then either reduces, by
false", and so on so forth. This will not give a sentence given of
L in fmitely many steps. Thus this A does not sat
Boolean equation of the REferential equa
V (VF 1\ WT)
(YT 1\ ZF)
x = yz + mv. The following theorem shows that this correspondence preserves solutions and hence
is
a "homomorphism." This
isfy the defmition. Likewise, the Liar sentence "This sen
justifies the claim that semantic diagnosis
tence is false" is not a sentence given either. This reminds
Boolean diagnosis.
us of the celebrated notion of
grounded sentences
of
Kripke [5]. However, they are different. For instance, for sentences in the original sentential logic
L, while the no
tion of groundedness depends on the choice of the exten sion 81 and the antiextension
S2 of the truth predicate
[5],
Having made precise the notions of "REfer to", as well as "sentence given" and "sentence unknown", I define ref erential equations and their solutions. By a
X : = (YT 1\ ZF) where
REferential
x = yz + 'i5w has a solution. In fact, the truth values of any solution ated Boolean system.
X through W are sentence unknowns of 5£.
(Remember that by convention this could be a long ex
REferential system
has a solution respecting one, not necessarily all, inter
to the REferential system form a solution to the associ
V (VF 1\ WT),
pression with subscripts.) By a
Transfer Theorem. If a REferential system X : = (YT 1\ ZF) V (VF 1\ WT) pretation of ": = ", then the associated Boolean system
the notion of givenness defined above does not.
equation I mean an expression of the form
is fmer than
we
mean a system of finitely many REferential equations.
A
Proof Let A through E be a solution of the REferential sys tem, respecting some interpretation of ": = ". Then A : = (BT 1\ CF)
V (DF 1\ ET).
VOLUME 23, NUMBER 1, 2001
47
Hence by the Axiom of truth-REference,
A U T H O R
AT � (B T A CF) V (DF A ET).
This implies by properties
Tl
through T3 that
AT � ((B A -.c) V (-,D A E))T.
This means that
T(A) = T((B A -.C) v c-.n A E)),
which is the same as
T(A) = T(B)T(C) + T(D)T(E).
Thus the truth values satisfy the associated Boolean system
x = yz which proves the theorem.
LAN WEN
School of Mathematics
+ vw,
Peking University 1 00871
Beijing
China
Acknowledgment
This paper has been written with interaction from many of my colleagues. I thank Manuel Blum, Steve Smale, Xinghua Wang, Jian Wen and Jingzhong Zhang for many good ref erences, comments, and encouragement. I thank Shiming Guo and Elliott Mendelson for many critical comments and stimulating discussions that sharpened the ideas signifi cantly. Finally, I thank the Department of Mathematics and the Department of Computer Science of City University of Hong Kong for kind hospitality.
e-mail:
[email protected]
Lan Wen was educated, and now teaches, at Peking University a campus with a beautiful lake named Not Yet Named. He did his doctoral work at Northwestern University on the shore of the beautiful Lake Michigan. His specialty is ,
dynamical systems, a beautiful field noted for dramatic fea tures like evolution and chaos.
AEFiiRENCI!S
[1] J. Barwise & J. Etchemendy, The Liar, Oxford University Press,
[7] Benson Mates, Skeptical Essays, The University of Chicago Press, 1 981 .
1 987. [2] N. Falletta, The Paradoxicon, John Wiley & Sons, Inc., 1 990.
[8] E. Mendelson, Introduction to Mathematical Logic, Third edition,
[3] H. Kahane & P. Tidman, Logic & Philosophy: A Modern
[9] R. Sainsbury, Paradoxes, Second edition, Cambridge University
Introduction.
Wadsworth Publishing Company, 1 995.
[4] R. Kirkham, Theories of Truth, MIT Press, 1 995. [5] S. Kripke, Outline of a theory of truth, The Journal of Philosophy, 72 (1 975), 690-7 1 6.
[6] R. Martin, Recent Essays on Truth and the Liar Paradox, Oxford University Press, 1 984.
48
THE MATHEMATlCAL INTELUGENCER
Wadsworth & Brooks/Cole Advanced Books & Software, 1 987. Press, 1 995. [1 0] K. Simmons, Universality and the Liar, Cambridge University Press, 1 993. [1 1 ] A Tarski, Logic, Semantics, Metamathematics, Hackett Publishing Company, Second edition, 1 983.
M athe m a t i c a l l y B e n t
C o l i n A d a m s , Ed itor
Overcoming Math Anxiety The proof is in the pudding.
Opening a copy of The Mathematical In.telligencer you may ask yourself uneasily, "What is this anyway-a mathematical journal, or what?" Or you may ask, "Where am !?" Or even "Who am !?" This sense of disorienta tion is at its most acute when you open to Colin Adams's column. Relax. Breathe regularly. It's mathematical, it's a humor column, and it may even be harmless.
Colin Adams
T
here is a crippling disease that has a vice grip on the nation. It is low ering the Gross National Product, causing whole communities to break out in hives, and convincing many peo ple to stay home with the covers over their heads. Of course, I could only be talking about Math Anxiety. This poison ivy of the soul has a long and mangy history. Isaac Newton himself had such a bad case that while he wrote the Principia Mathematica he was shaking from head to foot. Joseph Louis Lagrange was bedridden for a week before he could bring him self to write down his famous multi plier. And Evruiste Galois preferred risking his life in a duel to grappling with the mathematics that made him so nauseous. Of course, math anxiety is not the only ailment associated with mathe matics. There is math incontinence, math male pattern baldness, math itch, and math runny nose. However, today, we will focus our attention on math anxiety, leaving those other maladies to a series of articles that I am writ ing for the Notices of the American
Mathematical Society.
How do you know if you suffer from math anxiety? Here is a quick test. Check off each of the symptoms that you experience when confronted with mathematics. Symptoms of Math Anxiety.
Column editor's address: Colin Adams, Department of Mathematics, Williams College, Williamstown, MA 0 1 267 USA e-mail:
[email protected]
A. Hyperventilation. B. Holding your breath. C. Sweating profusely, while holding your breath.
D. Sweating profusely, while holding Spanier's Algebraic Topology. E. Eating other people's bag lunches. F. Uncontrollable shaking, hopping, or doing the rhumba. G. Weruing a heavy winter coat in the Math Resource Room. H. Putting pencils in your nostrils or ear holes. I. Sucking your thumb. J. Sucking your T.A.'s thumb. K. Rapid heart beat. L. Rapid pulse. M. Rapid heart beat but no pulse. N. Rapid pulse, but no heart beat. 0. No pulse or heart beat. P. K, L, but not 0. Q. P, M, N but not L. R. Not R. S. Extreme nausea, accompanied by hallucinations of large mammals lec turing you on Euclid's parallel postu late. T. The feeling that you and calculus are in a custody battle over your math ematical future and the judge has or dered you to make child support pay ments. U. The sensation that someone has poured soda water up your nose, and now expects you to thank him for do ing so. V. Dizziness, accompanied by an in ability to stand straight on an inclined plane. W. The feeling that the alphabet is endless. X. A thousand red ants are crawling over your body, biting and stinging you until you want to scream. Y. The impression that a thousand red ants are crawling over your body, bit ing and stinging you until you want to scream. Z. The feeling that you are running out of ideas, but you must complete a list. For each of the symptoms that you checked off, write down the number 6.9986. Add these numbers together. Divide by 27T. Take the natural log of
© 2001 SPRINGER·VERLAG NEW YORK, VOLUME 23, NUMBER 1 , 2001
49
the result. Add 1 . 145, and subtract 1.946. Exponentiate the result.
If you
are now sweating profusely and feel as if you
rings a bell. The student is immediately
there are still many important ques
forced to nm a maze, at the end of which
tions to pursue.
he or she is force-fed a pellet of rat food.
Will there ever be a vaccine for
fessor begins the lecture, and as soon
math anxiety? And if so, will it be one
you may have eaten bad
as he sees a student looking uncom
pink cube of sugar? Should triskadeko
tuna If so, you may be suffering from
fortable, he stops the lecture, comes
phobia be considered a type of math
gastronomic masochism. I should have
over and gives the student a warm hug.
anxiety? Are math anxiety and math
an article out on that in about a week.
He says, "Don't worry, you can do it.
phobia the same or slightly different?
Try to hang on until then.)
You're special. We're all special. Love
These are just a few of the issues addressed in my upcoming anthology,
Math Anxiety, including all of the pres
is all around, if you just let it in." The other students in the auditorium come
appearing in a special issue of the
idents from 1872 to 1891, and Teddy
over, take hands, form a giant circle
Journal of the Mathematical Psychoses Institute.
had
eaten bad tuna, you have
math anxiety. (Note:
Many
Or
presidents
suffered
from
2. The Nurturing Approach: The pro
Roosevelt, who had to wear diapers as
around the student, and sway back and
he charged up San Juan Hill, lrnowing
forth, singing songs about how great
he would need to count the enemy
Coke tastes.
once he got to the top. Sharon Stone
3. Confronting Your Fear Approach:
of those ones where you swallow a
Until this scourge can be cured, we will need dedicated facilities: ambu lances to rush those with sudden-onset
asked to give a proof of the central
The student is tied to the chair. A drill
limit theorem, as does Woody Har
want to lrnow about real anxiety? You
quarantine wings in hospitals to pre
relson. Ed Begley Jr. refuses to appear
have
vent Ebola-like epidemics. And most
in any movie involving a covariant
you real anxiety. I'm going to make you
importantly, we will need substantial
fimctor.
wish you could hide your head in a big
federal grants to support those of us
breaks into a torrential sweat when
Psychologists have settled on the following four treatments for math
math anxiety to emergency rooms staffed by ready Ph.D. math educators, .
sergeant screams in his face, "You no
idea!
I'm
going to
show
fat textbook and never come out."
who are at the cutting edge of research
4. Nature's Own Approach: The stu
in this seminal field.
anxiety:
dent is tied to a large rock and thrown
L B.F. Skinner Approach: Here, the
in a pond.
student is hooked up to an anxiety de
surface, the rock is replaced by a larger
sign. Lunches will not be lost to loga
tector-usually a rabbit taped to the
rock and the process is repeated. lfthe
rithms. Then, researchers like me will
Some day, perhaps, no one will
If the student floats to the
tremble at the sight of a percentage
student's leg. A trigonometry lecture
student does not float to the surface,
need to find other sources of support.
begins. As soon as the rabbit senses
he or she is declared cured.
But in the meantime, continue to read
anxiety on the part of the student, it
my papers on the subject.
Although much has been learned,
Errata We recently have had some trouble correctly attributing articles. In vol. 22, no.
3, misspellings occurred of the names of
Bernard Geneves and Nikolai V. Ivanov. And now in vol. 22, no. 4, we inadvertently interchanged the photographs of the authors, Oleksiy Andriychenko and Marc Chamberland (see below for the correction). Our apologies to these authors, and to the readers.
OLEKSIY ANDRIYCIIENKO
50
THE MATHEMATICAL INTELLIGENCER
MARC CHAMBERLAND
Magellan's and Elcano's Proof . . .
.
.
•
and the gap pointed out by Michael Little-Endings in the last issue of the Transactions of the SubAntarctic Mathematical Society
" . . . Leaving on March 9, 1521, Magellan steered west-southwestward. . . . Less than two months later, however, Magellan was killed in a fight with natives on Mactan Island. . . . It had been left for Elcano [originally master on the ship "Concepcion"], returning by the Cape route, to give practical proof that the Earth was round." Encyclopedia Britannica, 1999
Sasho Kalajdzievski Department of Mathematics University of Manitoba Winnipeg, R3T 2N2 Canada e-mail:
[email protected]
VOLUME 23, NUMBER 1, 2001
51
Curves in Traditional Architecture in East Asia Hiroshi Yanai
O
ne of the pleasures for people
In
travelling to East Asia may be to
by an envelope of stretched strings.
see various curves decorating tradi
Some of the procedures and equations
tional
are illustrated in the figures.
architecture.
Curves
are
ob
other cases, they are approximated
served everywhere: in gables, in eaves,
The author knows little about the
over the entrances, and sometimes in
mathematical aspects of the curves in
jacket walls of castles. In particular,
China, Korea, and in other countries
roofs play the most important role in
like Indonesia, Kazakhstan, and Thai
their appearance, as the fac;ade does in
land where beautiful and characteris
European architectures.
tic curves are also observed in archi
However, if you watch the curves
tecture. Mathematical forms are basic
very carefully, you will notice the dif
not only to architectural engineering,
ference among the curves in different
but also to the cultural connotation
regions in East Asia. The curves in
hidden in architecture.
Chinese architecture have, in general,
Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where thefamous conjecture was made, the desk where thefamous 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
stronger curvature than those in Japan.
Figures are reprinted with permission
Korean curves are somewhere in be
from
tween. One might say that they reflect
Yanai,
H.,
"Curves
of
there must be also some philosophical
the Operations Research Society of Japan
or religious connotations.
33 (1 988), No. 6 (in Japanese)
Some old procedures are known, in
Yanai, H . , "Curves in Traditional Japanese
Japan, to draw such curves, which can
Architecture,"
be translated into western mathemat
Opera-tions Research Society of Japan
ics, although many traditional archi
(1 991 ), No. 3 (in Japanese)
tects copy and modify older curves nowadays.
Communications
English)
or textbooks, Japanese curves are most
In
some cases,
Prof. Hiroshi Yanai
may follow in your tracks.
result from the process of numerical so
KEIO University
lution of the differential equation y" =
3-1 4-1 , Hiyoshi, Kohokuku
0 with boundary condition at both ends.
Please send all submissions to Mathematical Tourist Editor, Aartshertogstraat 42,
8400 Oostende, Belgium e-mail:
[email protected]
THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK
the
36
Civil Engineering," Forma Vol. 14, No. 4 (in
According to many old manuscripts frequently parabolas.
of
Yanai, H . , "Curves in Traditional Architecture and
they are formed by line segments which
52
Walls
Mathematics in Style," Communications of
a map or directions so that others
Dirk Huylebrouck,
Stone
the aesthetic senses of the nations;
Faculty of Science and Technology
Yokohama 223-8522, Japan
Xian, China
Wak:ayama, Japan Pavilions y
y(x) = --=--cosO (sinO - (LcosONL-x)) ' N L
X
where is the length of the beeline, is the number of the nodes and 0 is the angle of the inclination.
Drawing Procedure of Gable
y
y = d (1 = IJlJ, d=AD. bu. + h
where
-
h = AB,
3u)2 2h
b
B
C
II+· -b-�·1 -
B
Drawing Procedure of Stone Wall
Kanazawa, Japan
VOLUME 23. NUMBER 1 , 2001
53
The Shape of Divinity Kim Williams
Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck,
Aartshertogstraat 42,
8400 Oostende, Belgium e-mail:
[email protected]
54
T
he mathematical tourist visiting medieval cathedrals in Europe certainly has a wealth of material to sort out. We have already seen tracery [A] and hammerbeam ceilings [H3] dis cussed in these pages. I would like to add another item to the list of note worthy architectural features: the sym bolism of geometric shapes, and in par ticular, the vesica piscis. The vesica, which translated literally means "fish bladder", is also known as a mandorla, Italian for almond. The shape is created by the inter section of two circles of equal diame ter, the perimeter of each passing through the center point of the other (Fig. 1). Its frequent appearance in me dieval ornamentation, particularly in the sculpted reliefs that fill the arches over entrance portals (an element properly known as a tympanum) sig nals the importance of this shape. It is associated with the themes of ASCEN SION or ASSUMPTION, and most usu ally forms the frame around the figure of Christ, but is used as a frame for the Virgin Mary as well. It is said that orig inally the almond-like shape repre sented the cloud which carried the saintly figures into heaven, but it grad ually assumed the role of an aura or kind of "glory" [H1: 197]. The first use of the vesica in art appears in the Byzantine art of the 5th or 6th century, when it was used to represent the in carnation of Christ in the womb, in a figure known as a Platytera [H1: 337]. From there the symbol passed into western Europe in the middle ages. The Gothic may be the single most geometric style of architecture. John Harvey made a strong case that the rise of Gothic architecture was directly re lated to the new availability of Euclid's Elements, as well as Arabic astronom ical texts, in western Europe in about 1 120. "It can be no mere accident," Harvey has noted, "that this placing of the world of thought within a strictly scientific framework parallels the sud den rise of the new Gothic art and ar chitecture. [ . . . ] The study of practi cal geometry was indeed the essence of architectural design" [H2]. When we encounter a geometrical form in Gothic architecture, we are likely en countering a symbol. The iconographic
THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK
content of the vesica, created from two intersecting circles, could have been the reconciliation of opposing duals, such as terrestrial-celestial and hu man-divine into a harmonious unity [CG: v.2, 59]. The geometrical proper ties follow from the simplicity of the construction [K: 54]. The vesica is also important because it could form the basis for a cohesive proportional sys tem (Fig. 1). In Romanesque architecture, Christ is found in a vesica in an altar apse fresco of the Chapel at Berze-la-Ville, France (ca. 1 100 AD); in the tympanum over the entrance portal of Ste. Madeleine, vezelay, France (ca. 1 120 AD); and in the tympanum of the en trance portal of Sant'Trophine, Aries (ca. 1 170). In Gothic architecture, Christ appears in a vesica on the west facade of Notre-Dame-la-Grande, Poitiers (ca. 1 162), and in the tympa num of the central portal of the west fat;ade of Chartres Cathedral (ca. 1 145) (Fig. 2). The Virgin Mary appears in the vesica of the tympanum of the north entrance to the Cathedral of Santa Maria del Fiore in Florence, the so called "Porta della Mandorla," in a re-
root-S
root-4
root-3 root-2
--!-'f-++-�-
Figure 1 . The vesica piscis, and root-n con structions (drawing by S. Hisano).
Figure 2. Christ in the vesica. Central portal, west facade, Chartres Cathedral, France. Photograph by the author.
lief executed by the artist Nanni di Banco in 1418 (Fig. 3). The Virgin also appears in a vesica in the glittering glass mosaics that decorate the me dieval fru;ade of the Cathedral of Orvieto (Fig. 4). One needn't travel further than a good research library to find other me dieval examples of vesicas. The arm chair tourist will find them in many a medieval illuminated manuscript. As the Gothic age gave way to the Renaissance, ideals and concepts changed, and the vesica appears to have fallen out of favor. One late ex ample appears in the tomb slab com memorating Cosimo de' Medici, de signed by Verrocchio in 1467, and laid in the pavement under the crossing of the Basilica of San Lorenzo in Florence [W] (Fig. 5). The tomb slab is noted by historians for its abstract design, because it features a veritable vocabulary of shape, but no figurative imagery. Two vesicas of green por phyry flank a central 3:4:5 rectangle. Cosimo had wished to be eternally present at the celebration of the Eucharist, hence the location of the tomb slab at the foot of the altar (though the actual tomb is in the crypt
Figure 3 {Top). Mary in the vesica. Tympanum,
north entrance, Cathedral of Santa Maria del Fiore, Florence. Photograph by the author. Figure 4 (Bottom). Mary in the vesica. Mosaics, main facade, Cathedral of Orvieto, Italy. Photograph by the author.
Figure 5. Verrocchio's tombslab for Cosimo de' Medici, Basilica of San Lorenzo, Florence. Drawing by the author from Italian Pavements.
Patterns in Space (Houston: Anchorage Press, 1998). Reproduced by permission.
Figure 6. Christ in the vesica, intrados of the entrance portal, Certosa of Pavia, Italy. Photograph by the author.
below), and the appearance of the
I hope that other tourists will let me
vesica, the fish-shaped symbol of
hear of more examples.
Bridge Between Art and Science.
Christ. However, note that the pro REFERENCES
the "classic" vesica. Where the tradi
[A]
root-3 rectangle, Verrocchio's vesica is circumscribed by a root-2 rectangle. I do not know the reason for this. A last late example of Christ in the
vesica is found in the reliefs decorating the intrados (the inner surface) of the arch over the main entrance portal of the Certosa of Pavia (ca. 1497) (Fig. 6).
New York:
McGraw Hill, 1 991 .
portions of these vesicas differ from tional vesica is circumscribed by a
[K] Kappraff, Jay, Connections: The Geometric
Artmann,
[W] Williams, Kim, "Verrocchio's Tombslab for Benno,
"The
Cloisters
of
Cosima
de'
Medici:
Designing with
a
Hauterive," Mathematical lntelligencer, val.
Mathematical Vocabulary," in Kim Williams,
1 3 , no. 2, pp. 44-49.
ed. Nexus: Architecture and Mathematics.
[CG] Chevalier, Jean and Alain Gheerbrant, Dizionario
dei
Simboli,
2
vols.
Milan:
Fucecchio,
Florence:
Edizioni
deii'Erba,
1 996, pp. 1 91 -205.
Biblioteca Universals Rizzoli, 1 986. [H 1 ] Hall, James, Dictionary of Subjects and Symbols in Art.
London: John Murray, 1 97 4.
[H2] Harvey, John, The Medieval Architect. London: Wayland Publishers, 1 972.
Kim Williams Via Mazzini 7
Though I have personally tracked
[H3] Horowitz, David, "The English Hammer
down vesicas in Italy and France, I
beam Roof, " Mathematical lntelligencer, val.
Florence, Italy
have not yet done so in other countries.
1 8 , no. 4, pp. 61 -64.
e-mail:
[email protected]
50054 Fucecchio
The Better the Workers the Fewer It Takes Q: How many pure mathematicians does it take to unscrew a light bulb? A: None are needed! With gravity pulling steadily down, and ambient vi bration (trucks passing in the street, Brownian motion, distant earthquakes
and meteorite impacts, etc.), it will eventually come out on its own. Robert Haas 1 081 Carver Road Cleveland Heights, OH 441 1 2 USA
VOLUME 23, NUMBER 1, 2001
57
PAUL L. ROSIN
On Se r i o ' s Co n stru ct i o n s of Ova s
fter a dearth of exposition in the Middle Ages, when architectural principles were kept secret by the guilds, the Renaissance was witness to an explosion of architectural treatises and handbooks [ 11]. These were mostly written by architects, in manuscript form until the end of the fifteenth century, and provided a combination of theories, rules and patterns con
The first treatise of the Renaissance (written around
cerning all aspects of architecture. Contents range over the
1450 and published in 1485, 1 3 years after his death) was
suitability and preparation of building materials, the design
Alberti's
of plans, fa<;ades, and ornamentation, theories of beauty, and
was inspired by)
primers on geometry, to details of suitable inscriptions for
Vitruvius. Alberti recommended nine basic geometric
tombs, and methods for repelling and exterminating insects.
shapes. First there are the regular polygons: the square,
De Architectura,
hexagon, octagon, decagon, and dodecagon, all of which
The main prior work was Vitruvius's
which was (and remains) the only extant architectural trea
De Re Aedificatoria, and it followed (or at least classical buildings,
and of course
can be constructed from a circle (which he considered the
tise from Roman times. Written before 27 AD, it has, despite
ideal form [32]). In addition, there are three rectangles
its failings, enjoyed substantial popularity; numerous man
based on extensions of the square. Buildings could then be
uscript copies can be traced through Europe up until the
planned by taking these forms as the underlying compo
Renaissance [9], and more than 20 copies made in the flf
nents, and combining them to form a pleasing plan.
teenth century alone are known
[5] . According to Vitruvius
Following Alberti came a host of other authors elabo
the design of a harmonious, pleasing building requires care
rating, recodifying, and extending the previous writings. 1
ful attention to the proportions of its parts. Their sizes are
holding "an almost magical power" over the architects [32].
simple multiples of a unit length called the module.
During this period the geometry of the circle predominated,
1 For example. in the fifteenth century we have amongst others Filarete, Georgio, and Grapaldi, while those of the next century include Vignola, Palladio, and Scamozzi. As can be seen, the majority of activity took place within Italy.
58
THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK
In contrast, the ellipse "was as emphatically rejected by
son said, "The books became the architectural bible of the
High Renaissance art as it was cherished in Mannerism"
civilised world. The Italians used them, the French owed
[22]. Not only was the circle considered to have the most
nearly everything to Serlio and
perfect and beautiful form, but its preeminence over the
Flemings based their own books on his, the Elizabethans
ellipse was bolstered by arguments based on the circular movements of the human body [22].2
his books, the Germans and
cribbed from him . . . ". Of particular relevance to this paper,
Serlio agreed with Alberti on the ascendance of the circular
Eventually the fashion shifted from Alberti's restrictive
form, but extended the range to include amongst the poly
circle geometry, and the ellipse came to provide a "conve
gons the Greek Cross and the oval, thus heralding a new ap
nient form for the Baroque spirit for dynamic expression"
proach to ecclesiastical architecture
[ 19], not only for aesthetics, but also astronomy, mechan
church patterns he included one with an elliptical plan, al
ics, etc. For instance, somewhat later William Hogarth, in
though he himself did not build any elliptical churches.
The
states " . . . the oval has a noble
Consequently, in large part encouraged by Serlio's book,
simplicity in it, more equal in its variety than any other ob
many elliptical churches were built in Italy and Spain from
ject in nature3 . . . [and] is as much to be preferred to the
the sixteenth century onwards. 7 These are covered in great
Analysis of Beauty,
circle, as the triangle to the square4". An early proponent was Michelangelo, who considered the ellipse during his first project for the tomb of Julius 115, and later used it in his design for the Piazza del Campidoglio.6
Another pioneer in the use of the ellipse was Baldassare
Peruzzi, who had a predilection for difficult or eccentric
[ 18, 32]. Amongst the
detail by Wolfgang Lotz
also
[ 18] and Vincenzo Fasolo [6]; see [ 14, 19, 31]. Moreover, elliptical theatre design became
popular from the end of the seventeenth century, fust in Italy and then in France, while the ellipse also featured prominently in art [24]. Over the years the ellipse has continued to figure in ar
problems. However, he neither built nor published any re
chitectural design. For example, instances in recent build
lated material before his early death. It was left to his pupil,
ings are the striking "Lipstick Building" by Philip Johnson
Tutte l'Opere
and John Burgee, which consists of a stack of elliptical cylin
Sebastiana Serlio, to write the celebrated
d'Architettura,
which was published over the period
ders, the Tokyo International Forum designed by Rafael
1537-1575. The work incorporated and acknowledged ma
Vifioly, which contains the vast elliptical Glass Hall, and the
terial left by Peruzzi, although the accusations of whole
additions to the British Library's Reading Room by Foster
sale plagarism by many (e.g., Vasari and Cellini) appear un
and Partners (see Figure
fair. Serlio's approach to the writing of the architectural
range of scales. Figure 2 shows examples, starting with small
treatise was significantly new on several counts: [ 5, 1 1 ]
scale decorative details such as the patera, continuing to the
• •
For the first time a coordinated scheme o f architectural education was devised. It contains the earliest detailed work on the five Orders (Doric, Ionic, Corinthian, Tuscan, and composite). Serlio attempted
to
sort
out
the
discrepancies
between
Vitruvius and actual Roman monuments, and establish clear consistent rules for the Orders. •
It provided practical design patterns, and was written for the use of architects rather than perusal by wealthy pa trons and noblemen.
• •
It was written in Italian, whereas most of the previous writings had been made in Latin. The illustrations were a substantial component of the work In contrast to Alberti's original which contained none, while Vitruvius's had been long lost, out of the Serlio's
155 pages of third book 1 18 had one or more illustrations.
This combination of practicality and ease of use made it
1). The ellipse occurs over a wide
scale of a single room (Robert Adam's hallway at Culzean Castle), and continuing beyond the scale of a single house to
John Wood's conglomeration of the Royal Crescent at Bath.
In order to build elliptical structures, a means of con structing the shape of an ellipse is required. The instruc tions found in books on building construction or technical drawing fall into three classes-geometric rules for draw ing true ellipses, mechanical devices for drawing true el lipses, and geometric rules for drawing approximations to ellipses using four or more circular arcs. The methods are generally based on the geometric properties of ellipses which have been available from various classical treatises on conics. The most complete one that has survived, by Apollonius, was republished in Venice in
1541. However, (1522)
this is predated by works such as Werner's conics and Diller's geometry
(1525). Sinisgalli asserts that
Leonardo da Vinci was the first among the moderns to draw an ellipse, in around
1510. Thus there was plenty of theory
one of the most important architectural treatises, and nu
available during the Renaissance. However, translating this
merous editions and translations were made. As Summer-
into the practicalities of architecture could be problematic.
2Actually, the same anthropomorphic argument can be applied to ellipses ( 1 2]. As seen from above, the rotation of the two arms sweeps out a pair of circular arcs reminiscent of the circles used in one of Serlio's schemes to approximate ellipses. 31n chapter 1 2 - "0f Light and Shade".
41n chapter
4- "Of Simplicity,
or Distinctness".
5This view is disputed on lack of evidence [30]. 6There are three major architectural works in Rome in which the ellipse figures prominently: the Piazza del Campidoglio, the Colosseum, and Bernini's colonnade out side St. Peter's cathedral. 7For instance, some of the sixteenth- and seventeenth-century architects who designed elliptical churches are Vignola (in 1 550 this was the first built on an oval plan), Mascherino, Volterra, Vitozzi, Bernini, Rainaldi, and Borromini.
VOLUME 23, NUMBER 1, 2001
59
ations in humidity and temperature can have an effect. A,nd finally,
if the
pen or marker is not kept perfectly perpen
dicular to the drawing surface then additional inaccuracies will occur. From the next class of ellipse constructions, drawing devices such as trammels were used in the Renaissance, and there were
many contemporary developers of ellipse artists such as Leonardo
compasses--including well-known
da Vinci, Diirer, and possibly Michelangelo [15, 25]. These enabled reasonably precise ellipses to be drawn in plans, but again could not be readily applied to the large-scale marking out of buildings. This leaves the third class of ellipse constructions which, at the expense of only approximating the shape of
Figure 1. Examples of modem elliptical buildings.
From the first class of ellipse constructions the best known is the "gardener's method" in which two pins or pegs are placed at the foci and a length of string is attached to the pegs. The pen is pulled out against the string and around, thereby drawing out the ellipse. This is adequate for drawing rough, small ellipses, but there are several dif ficulties in scaling the approach up to tackle accurate, large ellipses. First, in order to draw the full
360°
properly the
string must be looped rather than tied to the pegs. Even so, the knot in the string can interfere with the drawing process. Second, particularly with long sections of string it is difficult to maintain a constant tension, and even vari-
60
THE MATHEMATICAL INTELLIGENCER
Figure 2. The ellipse in architecture at different scales.
the ellipse, provides a more practical tool for the architect and builder. In fact, there is a double benefit, for one avoids the continually varying curvature of the ellipse. For in stance, measuring the perimeter along a section of an el lipse can only be approximated; there is no simple analyt ical solution. Also, for small ellipses such as door or window arches, precise building construction would re quire a large range of brick shapes [20]. As an indication
of the difficulty in dealing with the ellipse, Batty Langley [16] in an eighteenth-century builder's manual reckons on
an extra
500Al expense in workmanship and materials for
constructing elliptical walls as compared to straight or cir cular walls.
l t l
Although there are various possible ap
proaches to approximating the ellipse (e.g., Blackwell de scribes a polygonal approximation [2]), the most popular approach has been to use a combination of circular arcs. The advantage of working with circles is that they provide a powerful geometric tool: with circles alone one can copy
I I
I
angles, bisect angles and lines, construct perpendiculars to lines, construct equilateral triangles, etc. [4]. Moreover, they keep the construction process simple and reliable, and provide a reasonably accurate oval with a pleasing ap pearance.
In his unpublished papers Peruzzi roughly
sketched out two such constructions (types III and IV de scribed below);8 see Figure 3. In his Tutte l'Opere d'Architettura Serlio provides, in addition to a plan of an elliptical church and a method for constructing true el lipses, four construction methods for drawing ovals, namely piecewise circular approximations to the ellipse.
Serlio's four alternative constructions for approximating the ellipse from his First Book are shown in Figure
4. Each
generates
an oval consisting of four circular arcs with cen tres (±h, 0) and (0, ±k) and radii a h and b + k respec tively. Constructions II-IV have been called by Giulio Troili -
in the seventeenth century and others since by the more de
ovato longo, ovato, and ovato tondo. More quadrarc has been used in their
recently, the general term
description and analysis [8, 27]. Kitao [13] incorrectly states that constructions tion I.
II-IV are all special cases of construc
In fact, the more general construction would be one
in which an arbitrary type of triangle is used with its sides extended by an arbitrary length. This would give many pos sible alternative constructions for any desired ellipse (i.e., any semi-major and semi-minor axes
a and b). The impor
tant factor is that due to the underlying triangle all the con structions exhibit circular arcs with tangent continuity.9 This geometric constraint can be algebraically expressed as
k
h ==
_ a-b 2 k
--
a-b
- 1
Serlio's constructions I and IV contain an underlying equi lateral triangle. Both constructions II and III use an isosce
les triangle made up from half a square, but extended by different amounts,
gle, but only extended by �. z
The presence of circles, triangles, squares, etc. under-
lying the constructions is not surprising, for the combina tion of geometric primitives was a favourite device of artists and architects in the Renaissance (as well as in ear lier and later times). In ltaly the normal proportions were based on the equilateral triangle, leading to the geometric system
ad triangulum [9}.
Many examples of Numerolo
gical preoccupations ("sacred geometry") often are ex pressed in architecture, and are particularly evident in im portant buildings such as churches. Constrvctlon I
This is the only one of Serlio's constructions that enables ovals with varied eccentridties to be drawn. It consists of
two equilateral triangles whose bases are centred on the origin. Their intersections with the axes determine k, which can be expressed as
h=
.
� and w, respectively, where w is the
length of the square's side. In addition, the three-square construction described later is made up of a similar trian
Serlio's Four Ova l Constructions
scriptive tenns
Figure 3. Peruzzi's sketch containing two oval constructions.
a - b . 1'
\13 -
_ k-
v3ca - b) v'3 - 1
The radii of the circular arcs are s and increased both
a
and
b
h and
·
2h +
s.
When s is
are increased, and so it is not
81t is perhaps not coincidental that these scribbles are adjacent to his design for Palazzo Massimo. in which he uses a curved wall cleverly to combine two adjacent
extsttng sites into a single building.
9Despite Arnheim's assertion to the contrary [1 J, circular arcs can be joined easily and pleasingly as demonstrated in these oval constructions.
VOLUME 23, NUMBER 1. 2001
61
a
b Construction I
Construction II
a
a
b
b Construction Ill
Construction IV
Figure 4. Serlio's four constructions for drawing ovals.
straightforward to (geometrically) choose correct values of h and s so as to achieve specified values of a and b. In fact, the ratio is given by a
b
=
h+s . h(2 - V3) + s
a
h = k = \12
Figure 5 shows how, for a triangle with unit-length sides, increasing s from zero initially rapidly increases the circu larity of the oval, while as s increases beyond 1, the aspect ratio very slowly decreases, reaching a perfect circle in the limit. Thus not all aspect ratios can be achieved with this method: s > 0 entails � < 2 _1V3 3. 732. =
Construction II
The second construction uses three circles, each passing through its neighbouring circle(s)' centres. The centres of the circular arcs are at
h = k = !.!:._
2'
producing an approximate ellipse with the fixed aspect ratio of � = Y2 1 .414. The ratio of the radii of the arcs is Y2 - 1. Serlio considered this oval to be very similar to the shape of a natural egg. To verify this, a rough test was made in which the typical aspect ratios of the eggs of 100 randomly selected species of European bird were obtained [23]. The mean aspect ratio was found to be 1.38 (with standard deviation .09), which is indeed closer to Construction II than to Construction III or IV. Such an arrangement of circles was often used by artists in their designs. An extensive example of its use is given by Pinturicchio's frescos in the Library of Siena Cathe dral [3]. =
62
Construction Ill
The third construction uses yet another geometric primi tive as its basis: the square. This yields
THE MATHEMATICAL INTELLIGENCER
+1
�= ratio of the radii of the arcs is f. with a fixed aspect ratio of
'
�":_11
2
=
1.320, and the
Construction IV
The fmal construction was recommended by Serlio for its beauty, and is the simplest and quickest to construct. Not surprisingly, this was the ellipse approximation most used in architectural practise [ 13]. The configuration of two in tersecting circles is common in sacred geometry [ 17], where it is called the Vesica Piscis, a symbol often ap pearing in Christian iconography. As an example of its nua\b 4 3 .
1
0 .
2
4
6
Figure 5. Aspect ratio of Serlio's Construction I.
8
As
in Construction I, there is an underlying equilateral tri angle, although this is not required to be explicitly drawn to locate the arcs. The centres of these arcs are located at
h = Q:..' 3 and the flxed aspect ratio is
k=
�=
a
V3 ' 4
_3\f:i
=
1.323. Thus this
construction provides an oval very much like Construc tion III. Other Oval Constructions
Figure 6. The trinity of roots contained in the Vesica Piscis.
merical significance, its proportions contain the three ba sic roots, as shown in Figure 6. Kitao also notes the sim ple ratios present in the relationship of its parts: 1. the lengths of the arcs are 1:1, 2. the radii of the arcs are 1:2, 3. the major axis of the oval to the smaller arc radius is 1:3.
Having run through Serlio's four oval constructions, I now consider alternative approaches. Some are variations on the methods already described; others are extensions or improvements. Three-square construction
In the same way that Serlio's Construction II increased the two circles in Construction IV to three circles, consider in creasing the two squares in Construction III to three squares (Figure 7a). The same shaped underlying isosceles
k a
(a) three squares
b
(b) four circles
a
(d) Bianchi
(c) Vignola
a
a
b (f) Kitao's generalisation
(e) Langley
c
a
(g) Mott Figure 7. Alternative oval constructions.
a
(h) Hewitt's method 3
VOLUME 23, NUMBER 1 , 2001
63
triangle remains, but is extended by a shorter section, giv
3a - 2b ± V4ab - 3a2
h=
ing
4
,
k = Ya(a - 2h).
The solution becomes complex when
-"'-b > i. 3
On exami-
nation we see the cause for the breakdown at this point in
= 13-+ � v2
The fixed aspect ratio is -"' b
of the radii of the arcs is
t·
=
1.522, and the ratio
reaches a peak of
Continuing Serlio's constructions of two and three inter secting circles, I show four intersecting circles in Figure
the radii of the arcs is
.vr.c1 3
+1
=
2
=
dius of the underlying circles) and thereafter decreases to
unity again when the circles no longer overlap-see Figure
8. Thus, for any aspect ratio
<� there are two possible con
i
5
� = V35+
h = tr (where r is the ra
when
arcs is constant, namely -the only such case among the
-
5
f=�
structions. Curiously enough, the ratio of the radii of the
3 h=a' k = V3 a·'
the fixed aspect ratio is
derlying circles are moved apart. Starting in the limit from concentric circles which produce a circle, the aspect ratio
Four-circle construction
7b, giving
the unusual behaviour of the oval's aspect ratio as the un
variable aspect-ratio oval constructions analysed in this pa
1.340, and the ratio of
0.366.
per. Despite having identical aspect ratios, the ovals differ, as demonstrated in Figure 9. Compared to the oval in (a), the oval in (b) is made up with arcs of larger radii, and
Increasing from two to three circles produced an in
therefore has a squarer appearance.
crease in aspect ratio, but moving from three to four has decreased it almost back to the two-circle aspect ratio. This is due to the alteration of the position of
k at the top of the
central circle or at the intersection of the middle two cir
Bianchi's construction
Although Serlio's Construction I does allow a range of oval
eccentricities, it does not easily enable the designer to start
ber of circles used.
a and b. Later constructions overcame this instance, Paolo Bianchi's Istituzione Pratica deU' Architettura Civile in 1 766 provides the fol
Half-square triangle
lowing scheme (see Figure 7d):
cles depending on whether there are an odd or even num
by prescribing difficulty.
Just as Serlio's Construction I extends an equilateral tri angle by variable amounts to generate ovals over a range of eccentricities, the same technique applied to the isosceles triangle underlying his Constructions II and III, yields
1. mark point F such that the length Fa =
2. point h is marked such that
� = c\/2��)� +
.v"h r.c 2• + s
•
s'
Oh = � OF
k: 4 h=3 Ca - b); k =
2
of the radii of the arcs
and the ratio
A similar behaviour of
aspect ratio as a function of extension length is observed: for small values of s, minor changes cause large differences in aspect ratio.
Ob
3. an equilateral triangle is constructed, its vertices defm ing h and
b h = k = a--\12' with an aspect ratio of
For
4
V3
(a - b).
There are two intersection points; it is better to use the one closer to
(a, 0). This approach operates up until the break
down aspect ratio of
�-1
=
3.619.
Bianchi's approach provides identical ovals, but using an alternative construction, to those of the Slantz method
Vignola's construction
Kitao [13] describes a construction by Giacomo Vignola
a\b
based around the Pythagorean triangle with side-lengths 3, 4, and 5 as shown in Figure 7c:
1.
2 a k=b=3 a, h = 2' giving a fixed aspect ratio radii of the arcs as
f = %•
with the ratio of the
%· Golvin also claims this approach was
used by the Romans in their building of amphitheatres [7]. Kitao
Kitao [ 13] generalises Serlio's Construction IV by allowing
the circles to move closer or further apart (Figure 7f), re sulting in
64
THE MATHEMATICAL INTELLIGENCER
1 . 25 1.2 1 . 15 1.1 1.0 0.2
0.4
0.6
0.8
1
Figure 8. Aspect ratio of Kitao's construction (unit circle radius).
h
(b)
(a) Figure 9. Two ovals with identical aspect ratios generated using Kitao's construction.
[33] described in current engineering drawing books. Another variation producing the same oval was given by Langley [ 16], in which the equilateral triangle is drawn so that it touches the bottom of the ellipse (Figure 7e).
Thus it provides precisely the same oval as Serlio's Con struction I. Its advantage is that one starts with the desired aspect ratio of the oval.
Equilateral triangle constructions
Around 1744, motivated by his interest in astronomy and the orbits of the planets, James Stirling devised an ap proximation in which not only do the arc joins have tan gent continuity, but they also lie on the true ellipse [29]:
Robert Simpson's construction
Many oval constructions involve the use of equilateral tri angles. Serlio's I and N and Bianchi's construction have al ready been described. Two more modem examples are now shown. Mott's construction (his method number 7.8) [2 1] also involves two circles like Serlio's Construction N. However, the circles touch rather than intersect, and an equilateral triangle is drawn with its bottom comers in the circle centres (Fig. 7g), yielding
a k = V3 a. h = -· 2' 2 -
Although it is similar to Vignola's construction with the Pythagorean triangle replaced by the equilateral triangle, there is a further difference: the radius of the arc centered at k is set to k + b ; this enables a range of aspect ratios (although only [1 + -7a, V7; V3 ] [1.5774, 2.1889]) to be constructed, but the ovals are no longer tangent con tinuous. Hewitt's method 3 [ 10] is somewhat different, in that (unlike the previous examples) the equilateral triangle is centred on the semimajor axis rather than the oval centre (Figure 7h). The procedure is: =
1.
the arc centred at the origin with radius b is drawn out; it cuts the triangle at d; 2. the straight line through b and d is drawn, and cuts the triangle again at e; 3. the line through e parallel to the triangle side Oc inter sects the axes at h and k, yielding
h = (a - b) '. V3 - 1
k
V3(a - b) - V3 - 1
-
h= k
=
(a - b)(a + b + v'a2 + 6ab + b2) ; a - b + Va2 + 6ab + b2 (a - b)(a + 3b + v'a2 + 6ab + b2) . 4b
Stirling proposed to Robert Simpson that he find a geo metric construction for his algebraic description. In a man uscript Simpson provided the following solution in 1745, shown in Figure 10:
1. 2.
the semicircle baO is drawn and bisected at point C; the circle 10 with centre C and radius Cb intersects the ellipse at point I; 3. the isosceles triangle is drawn as shown; its intersection with the axes provides h and k. Stirling's approximation was recently shown to be ex tremely accurate [27]. Unfortunately, however, due to the shallow intersection of the circle and ellipse in stage 2 it is difficult to determine point I, making its geometric con struction rather impractical. In addition, Simpson asks the true ellipse to be drawn prior to the oval! Squaring of the circle
In the rather different context of sacred geometry, Lawlor [ 17] describes a construction that could lead to an oval. Figure 1 1 illustrates an attempt at squaring the circle, i.e., using only a compass and straight-edge to construct a square with perimeter equal to the circumference of a given
"'This circle is in fact the locus of arc joints satisfying the C1 constraint [27]. VOLUME 23, NUMBER 1 , 2001
65
b
b
(a)
(b)
(c)
Figure 10. Simpson's oval construction. circle. First a circle with unit radius is drawn with two half sized circles inside. Taking the top of the outer circle as centre, a circular arc is drawn tangent to both the inner circles as shown in Figure l la. The point of contact is
where
¢> =
v5/ 1 "" 1.618
is
the
Golden
Ratio
which
often appears in sacred geometry and theories of propor tions and aesthetics. The radius of the arc is thus ¢>, and the intersection of the arc with the
X axis is at x = W.
Drawing another circle centred at the origin and passing through the intersection (i.e., with radius circle whose circumference
\,I;{J) produces a
is 2 7TW "" 7.99. The perime
ter of the square circumscribing the initial circle equals eight, and thus provides a very close estimate to the final circle's circumference. Figure l la also provides a means of constructing a circular approximation to an ellipse, us ing the inner circles (radius ing at
..
a k = a·, -2 h,
.
. .
.
t) and the arcs of radius ¢>join
(xt, yt). The centres are at
the fiXed aspect ratio of the oval is ¢>, and the ratio of the radii of the arcs is 2¢>. Fidelity to the Ellipse Having defined a substantial range of oval constructions, the obvious question is: how well do they approximate the desired ellipse? The analytic solutions are complicated by the involvement of the elliptic curve and so a numerical approximation to the approximation errors is calculated
(a)
instead. The circular arcs are sampled at approximately
-- Scrlio l
I I ' I •. I .
• • •' ·.
' ' '
' '
e • 0 • e
' ' ' .. • · .
r
I I
'�
·� I �
•*
.·
·
,
·
I
1
.
a
+
't'
••••
Serlio JV
3 squares
4 circle
half-square triangle Vignola Kitao
-- Moll - - • Bianchi - - · Simp on
A
golden ratio
- optimai C J
- -......... - . ....... - - - � ... ,
.. ..
-
... .... _
_
-
(b) Figure 1 1 . Squaring of the circle.
66
Serlio l l Serlio lll
Tl-IE MATl-IEMAnCAL INTELUGENCER
--
a/b
Figure 1 2. Approximation errors of the ovals with respect to the de sired ellipse.
(a) Serlio
(b) Stirling
(c) Lockwood
(d) Walker
Figure 13. Various oval constructions showing their discrepancies against the ellipse.
equally spaced points, and at each point the distance along the normal to the ellipse is approximated. See [26, 27] for more details. Fixing b = 100 and using 1000 sample points in total, the graph in Figure 12 was generated. Although only the maximum error is displayed, the average error ex hibits a similar pattern. The error incurred by the optimal tangent-continuous (Cl) oval (which was numerically estimated by a 1D search) [27] is included for reference. We can see that Serlio's con structions do reasonably well, but are certainly not the clos-
(a)
est to the ellipse (although of course this may not reflect their aesthetic qualities). For instance, Simpson's construc tion does uniformly well and is generally superior (it was previously found to outperform most of the more modern methods too [27]). In addition, Vignola's construction does especially well. Nevertheless, the extensions of all of Serlio's constructions (i.e., half-square triangle, 3 squares, 4 circles, and l(itao's generalisation) mostly perform poorly (the ex ception is the half-square triangle construction at low ec centricities). This shows how apparently plausible con-
( b)
Figure 14. The best-fit ellipse and oval overlaid on Serlio's church plan.
VOLUME 23, NUMBER 1 , 2001
67
A U T H O R
overlaid in Figure 14a and the discrepancies at the diago nals are evident. The best-fit oval was determined by per forming an optimisation over all parameters using Powell's method, and as can be seen in Figure 14b, not only does it provide a better fit to the church's perimeter, but it is clearly Serlio's Construction IV. Acknowledgments
I would like to thank Ian Tweddle for providing a copy of his translation of Simpson's construction from the original Latin. PAUL L. ROSIN
Department of Computer Science
REFERENCES
Cardiff University
[O]
Cardiff CF24 3XF
[1]
UK
Sebastiana Serlio on Architecture, translated with commentary by Vaughan Hart and Peter Hicks. Yale University Press, 1 996.
e-mail: Paui.
[email protected]
R. Arnheim. The Dynamics of Architectural Form . University of California Press, 1 977.
[2]
W. Blackwell. Geometry in Architecture. John Wiley and Sons.
[3]
C. Bouleau. The Painter's Secret Geometry. Thames and Hudson,
search scientist at the Institute for Remote Sensing Applica
[4]
J.A. Brown. Technical Drawing. Pitman, 1 962.
tions, Joint Research Centre, lspra, Italy, and lecturer at Curtin
[5)
Paul Rosin is senior lecturer at Cardiff University. Previous posts include lecturer at the Department of lnfonmation
1 984.
Systems and Computing, Brunei University London, UK, re
1 963 .
University of Technology, Perth, Australia. His research interests include the representation, seg
W.B. Dinsmoor. The literary remains of Sebastiana Ser1io: I . The Art Bulletin, XXIV( 1 ) :55-91 , 1 942 .
[6) V.
mentation, and grouping of curves, knowledge-based vision
Fasolo. Sistemi ellitica nell'architettura. Architettura e Arti
Decorative, 7:309-324, 1 93 1 .
systems, early image representations, machine vision ap
[7) J .C .
proaches to remote sensing, and the analysis of shape in art
[8)
and architecture. A common factor of much of his research is a near obsession with the theme of the ubiquitous ellipse.
Golvin. L 'Amphitheatre Romain. Boccard, 1 988.
N.T. Gridgeman. Quadrarcs, St. Peters, and the Colosseum. The Mathematics Teacher, 63:209-2 1 5, 1 970.
[9) [1 0)
J.H. Harvey. The Mediaeval Architect. Wayland, 1 972. D.E. Hewitt. Engineering Drawing and Design for Mechanical Technicians.
structions do badly, and suggests that some care was taken in developing Serlio's original constructions. This argument is supported by the fact that Serlio's Construction I is always better than Bianchi's, and mostly better than Mott's con struction. It is interesting to note that for ovals with aspect ratios just less than two, Serlio's construction I approaches the optimal approximation. The oval approximations of ellipses with low eccen tricity are mostly good, and the errors are barely notice able. For more elongated ellipses we can see the discrep ancies more clearly, as shown in Figure 13. The first two show four-arc ovals discussed in this paper, and the supe riority of Stirling's approximation over Serlio's construc tion is obvious. The following two ovals are constructed using eight arcs-details are given in [28]. Naturally, using more arcs it is possible to improve the accuracy of the ap proximation to the ellipse, as is achieved by Walker's method. Perhaps surprisingly, several eight-arc ovals were found to be inferior to four-arc ovals, as demonstrated by Lockwood's oval; see also Rosin [27].
[1 1 )
Renaissance Architectural Treatise. [1 2)
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T.K. Kitao. Circle and Oval in the Square of Saint Peter's. New York University Press, 1 974.
[1 4)
G. Kubler and M . Soria. Art and Architecture in Spain and Portugal 1500..1 800. Penguin Books, 1 959.
[1 5) 0.
Kurz. Durer, Leonardo and the invention of the ellipsograph.
Raccolta Vinciana; Archivio Storico del Comune di Milano, 1 8: 1 5-24, 1 960.
[1 6)
B. Langley. The London Prices of Bricklayers Materials and Works. Richard Adams and John Wren, 1 750.
[1 7]
R. Lawlor. Sacred Geometry. Thames and Hudson, 1 982 .
[ 1 8)
W. Lotz. Die ovalen Kirchenraume des Cinquecento. R6misches Jahrbuch fOr Kunstgeschichte, 7 : 7-99, 1 955.
[1 9)
G.B. Milani and V. Fasolo. Le Forme Architettoniche, volume 2 . Casa Editrice Vallardi, 1 934.
[20)
C.F. Mitchell and G.A. Mitchell. Building Construction. B.T. Batsford, 1 925.
[2 1 )
68
New Haven, 1 998.
M. lliescu. Bernini's "idea del tempio". http://www.arthistory.su. se/bernini.htm, 1 992.
[ 1 3)
An Application
Serlio himself made good use of his methods of oval con struction. Figure 14 shows his plan for an oval church from the Tutte le Opere d'Architettura. The best-fit ellipse is
Macmillan, 1 975.
P. Hicks and V. Hart, editors. Paper Palaces: the Rise of the
L.C. Mott. Engineering Drawing and Construction. Oxford Univer sity Press, 2nd edition, 1 976.
[22]
E. Panofsky. Ga/i/eo as a Critic of the Arts. Martinus Nijhoff, 1 954.
[23] R.T. Peterson, G. Mountfort, and P.A.D. Hollom. Birds of Britain and Europe.
[24] I. Preussner. Jahrhunderts.
A Field Guide to
[28] P.L. Rosin and M.L.V. Pitteway. The ellipse and the five-centred
Houghton Mifflin Co, 1 993.
arch.
Ellipsen und Ovate in der Malerei des 15. und 16.
Weinheim, 1 987. Physis,
[30] W.K. West. Problems in the cultural history of the ellipse.
1 2:37 1 -404, 1 970.
for the History of Technology,
[26] P.L. Rosin. Ellipse-fitting using orthogonal hyperbolae and Stirling's oval.
CVG/P: Graphical Models and Image Processing,
[31] R. Wittkower.
60(3):209-
2 1 3, 1 998.
[32] R. Wittkower.
cular approximations to the ellipse.
Society
pages 709-71 2, 1 978.
Art and Architecture in Italy 1600-1 750.
Yale
University Press, 1 992.
[27] P.L. Rosin. A survey and comparison of traditional piecewise cir Design,
85(502), 2001 .
James Stirling: 'This about series and such things'.
Scottish Academic Press, Cambridge, UK, 1 988.
[25] P.L. Rose. Renaissance Italian methods of drawing the ellipse and related curves.
The Mathematical Gazette,
[29] I. Tweddle.
Architectural Principles in the Age of Humanism.
John Wiley and Son, 1 998.
Computer Aided Geometric
[33] F. Zozorra.
1 6(4):269-286, 1 999.
Engineering Drawing.
McGraw-Hill, 2nd edition,
1 958.
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VOLUME 23, NUMBER 1, 2001
69
I a§IH§'.ifj
.J et Wi m p , Editor
I
Inverse Problems: Activities for Undergraduates by Charles W. Groetsch
temperature gradient at the Earth's surface, and the medical inverse prob lem of interpreting tomographic scans. Groetsch argues persuasively for the value of "inverse thinking" as a part of an undergraduate mathematics cur
WASHINGTON, DC: THE MATHEMATICAL ASSOCIATION OF AMERICA, 1 999, xii+222 pp.
riculum. While forward problems dom inate the training of mathematics ma
US $26.00; ISBN 0-88385-716-2
jors, sole reliance on this direction is too limiting; viewing problems from
REVIEWED BY STEPHEN HUESTIS
the other (inverse) direction leads to a
Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited
H
richer understanding of the whole pic. ow does one defme an inverse
ture.
problem? Charles Groetsch intro
This book is not intended to be a
duces this volume with a discussion of
student's text on inverse theory. Nor is
review of a book of your choice; or, if
the difficulties of arriving at a formal
it a monograph on important concepts
definition and of recognizing a given
of inverse theory which an instructor,
you would welcome being assigned
problem
who might be ill-prepared to teach it,
a book to review, please write us,
(Groetsch uses the term "direct" for
could use to develop personal back
the latter, but "forward" seems more
ground. For that, one might refer to
telling us your expertise and your
predilections.
as
inverse
or
forward.
suggestive of the duality between the
Groetsch ( 1 ] , to which the present
two). Though there may be a certain
book can be considered a companion
arbitrariness in choosing which prob
volume. Rather, it is meant to be a
lem is "inverse" to which, in practice
source of ideas for the teacher inter
this seems not to be so problematical.
ested in incorporating inverse prob
It comprises
A collection of data is measured to
lems into the curriculum.
make quantitative inference about a
a miscellany of problems from various
physical cause. Suppose that a mathe
disciplines, some also discussed in [ 1 ] ,
matical model is available so that, were
which illustrate aspects o f inverse
this cause known, the values of the
thinking.
measurements could be calculated un
Groetsch takes a very broad view of
ambiguously. Then it is logical to call
what comprises an inverse problem,
this calculation the forward problem,
with the advantage that he can then ad
and to call the data interpretation the
dress some of the concerns of inverse
problem inverse to it.
theory from precalculus onward. To
Groetsch's interesting first chapter
Groetsch, the unknown of an inverse
provides a series of historical vignettes
problem might be a finite collection of
of familiar important problems that, in
discrete parameters, a continuous ftmc
the broadest sense, are inverse prob
tion of one or more independent vari
lems. His goal, effectively achieved, is
ables, or even the nature of a physical
to demonstrate the importance that in
law. There is not, however, universal
verse problem solutions have had in
agreement on this very inclusive defi
the development of the sciences, even
nition.
if such problems were not explicitly
As an Earth scientist, I came to in
stamped with the label "inverse the
verse theory as it is applied to geo
ory." Examples include Newton's de
physical inference, receiving my tute
duction of the inverse-square law for
lage from Robert Parker, one of its
gravity from observations of orbital
foremost practitioners. Parker devel
Column Editor's address: Department
forms, Kelvin's (incorrect) inference of
oped his ideas from his unique view that
of Mathematics, Drexel University,
the age of the Earth from a conductive
the unknown of an inverse problem is
Philadelphia, PA 1 91 04 USA.
cooling model applied to the current
some ftmction representing a parame-
70
THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK
ter distribution, often corresponding to the spatial variation of a physical prop erty in the inaccessible interior of the Earth (density, temperature, seismic velocity, etc.). As such, it is a member of an appropriately defmed infmite dimensional space of functions, and data are values of linear or nonlinear functionals defmed on this space. Parker's notion requires at least ele mentary functional analysis. Normed spaces, Hilbert spaces, and optimiza tion theory play large roles. Such top ics might be beyond the lower-division preparation of the students targeted by Groetsch. Instructors drawing ideas from Groetsch, however, would be well served by also reading Parker's [2] very accessible text. Inverse problems often do not share certain nice properties enjoyed by for ward problems. A forward problem is cast as a well-posed transformation (linear or nonlinear), mapping every member of a well-defined domain to a unique member of its range. The for ward problem is generally stable, with output depending continuously on in put. The inverse problem on the other hand, can be expected to be ill-posed. Recovery of a solution from the data is likely unstable: arbitrarily small per turbations to the data can lead to large variations in the solution. Such ex treme sensitivity becomes a practical problem when data are contaminated by random noise: the noise translates to wildly improbable oscillations in the constructed solution. Another new concern is the possible non-existence of any solution at all, if the candidate data function falls outside the range of the forward transformation. In such a case, one can conclude that the for ward problem is incorrectly posed. Perhaps a priori constraints on the do main are too stringent. Here I mention examples from my geophysical experi ence: (1) Measured gravity anomalies are to be interpreted in terms of subsur face density contrast. A data collection containing values of both signs would be incompatible with an a priori as sumption of solution positivity. Such data would force a redefmition of the domain to allow two-signed density contrasts.
(2) Potential field data are some times downward-continued toward the source, through the source-free region. This is an unstable, roughening process, inverse to the stable forward problem of upward continuation. If we plan to continue downward by a specified dis tance, through what we believe to be the source-free region, then certain functions yield no solutions, lacking sufficient smoothness to be in the range of the forward upward continuation op erator. Discovering conditions which must be met by data, in order to guar antee solution existence, is an impor tant part of the complete formal analy sis of any inverse problem. Arguably the most severe difficulty which can be encountered in inverse theory, not shared by the forward problem, is the possibility of solution nonuniqueness. In the theoretical (but unattainable in practice) situation of perfect data (complete and accurate), certain inverse problems can be shown to have a unique solution; others are fundamentally nonunique. A goal of formal inverse theory is a characteri zation of the complete solution set. In practice, data are incomplete, comprising only a fmite collection of numbers from which we cannot hope to learn which member of an infinite dimensional domain is the true solu tion. Nonuniqueness is inevitable; data are consistent with an infinite number of different solutions, possibly with tremendous variation in form. We must abandon the hopeless goal of recover ing the single true solution: the data do not specify it. Yet, they certainly contain some information about the solution set, and the inverse problem becomes one of inferring such information. One approach is the construction of some extremal solution, which is the simplest possible by some chosen mea sure of complexity. Such a solution might be as smooth as possible, thereby avoiding any artificial features of rapid variation not dictated by the data. The conclusion to be drawn is not that an ex tremal solution is correct, but instead that the true solution can be no simpler. Alternatively, we might seek prop erties common to all solutions without constructing any particular solution. In linear theory, the famous Backus-
Gilbert method is an example of such an approach, wherein certain weighted averages are constructed that are shared by all solutions. Another exam ple which has seen much geophysical application is the bounding of various solution properties-e.g., placing a greatest lower bound on the uniform norm of the solution. Notice that both extremal solution construction and bounding of solution properties are op timization problems that might rely on methods, such as linear programming, which are more advanced than the background of the students targeted by Groetsch, in their first two years of un dergraduate training. It is Groetsch's position that the teaching of mathematics, even at the lower division, can be significantly en riched by viewing problems with in verse, as well as forward thinking. By taking a broader view of the purview of inverse theory than does Parker, he is able to propose activities which stimulate students to appreciate and investigate some of these concerns even before their exposure to calculus. Following the introductory chapter, Groetsch divides his book into four subsequent chapters that follow the student's progress through the first years of an undergraduate curriculum. Each chapter comprises a set of six in dependent modules, each of which be gins with a brief description of a par ticular forward problem, followed by a collection of activities designed to prompt investigation into the rela tion between the forward and inverse problem. Activities are of six types: Questions, to be answered in discus sion form; mathematical Exercises which are illustrative, but routine; more challenging Problems; Calcula tions, which require numerical explo ration using a graphical calculator; computer-based Computations of a more intensive numerical nature (Groetsch supplies ail Appendix of MATLAB scripts which are useful for some of these); and occasional open-ended Projects, inviting the most flexible de gree of student exploration. For example, in the precalculus chapter we see simple projectile mo tion, with the associated inverse range problem, and an elementary geophysi-
VOLUME 23, NUMBER 1 , 2001
71
cal problem of recovering the position
tinuously on the data. Or, it might be
of a buried point mass, using one or
explored numerically by developing
erage difficult to interpret. (For this rea
more surface gravity measurements.
an approximate solution construction
out of favor with many geophysicists.)
son, Backus-Gilbert theory has fallen
The calculus chapter contains prob
method, then applying it to a data set
In an example, Groetsch observes a
lems as diverse as the forward/inverse
without and with the addition of ran
significant difference between a partic ular solution value and the associated,
problem pair for the density and cen
dom noise. In each module, activities
troid functions of a nonhomogeneous
exploring such issues are nicely tied
universally valid solution average. He at
bar, and the continuously compounded
together in a sensible development.
tributes this to the sparseness of the
data set, but fails specifically to look
interest model with variable interest
Construction of the true solution
rate, to be recovered from the value
would seem to be the ultimate goal in
deeper into the nature of the weighting
history. Examples from the chapter on
any practical inverse problem. When the
function he has constructed, and in gen
inverse problems in differential equa
data function is known completely and
eral at what Backus-Gilbert analysis is
tions include the problem of recover
accurately, analysis might then show
really giving (and not giving).
ing resistance laws from the shapes of
that the forward problem is invertible,
Because this book is neither a text
equitemporal curves for descent down
with a formal solution construction
on inverse theory, nor a tutorial on
inclined planes, and parameter estima
method. So, for example, Groetsch de
practical inversion, these shortcom
tion (mass, spring constant, damping
votes one module to the application of
ings in no way detract from its value.
coefficient) for a one-dimensional dy
the Laplace transform in a problem re
namic system. The fmal chapter, tapping linear al
lating weir notch shape to flow rate us
ing Torricelli's law. The convolution the
tors involved in the first years of t�e
gebra, begins with a module casting
orem for Laplace transforms allows a
highly recommended even to teachers
It is a rich source of ideas for instruc
mathematics curriculum, and can be
the general problem of solving systems
formal inversion of flow rate function
who didn't realize they were looking
of linear equations in the language of
for notch shape. Application to the prac
for such a book!
inverse theory. Solution nonunique
tical case is problematical, however, be
ness or nonexistence are, of course,
cause the inversion is a deconvolution,
REFERENCES
the problems of under- or overdeter
whose instability Groetsch has us ex
(1] Groetsch, C.W.,
mined systems. Instability for invert
plore in several activities.
ible matrices is illustrated using the
Inverse Problems in the
Mathematical Sciences,
Where Groetsch provides the least
familiar concept of the condition num
guidance is in inference issues for real
ber. The chapter ends with a module
problems with incomplete, inaccurate
Vieweg, Braun
schweig, 1 993. [2] Parker, R.L.,
Geophysical Inverse Theory,
Princeton University Press, Princeton, 1 994.
underdeter
data. As discussed above, when data
mined problem of inferring informa
don't specify a single true solution, we
Department of Earth and Planetary Sciences
tion about the unlmown density struc
can still ask what they do tell us.
University of New Mexico Albuquerque, NM 871 31
on
the
fundamentally
disc,
Construction of some particular solu
knowing only its total mass and mo
tion is of little value. In one module,
USA
ment of inertia. Here, for the first time,
Groetsch develops an algorithm, the
e-mail:
[email protected]
ture
of a radially symmetric
Lagrange multipliers are introduced to
algebraic
impose auxiliary constraints; the no
which is a geometrically based itera
reconstruction
technique,
tion of Lagrange multipliers is a tool
tion method for tomographic recon
crucial to the optimization-oriented ap
struction. He admits that such prob
proach to inverse theory championed
unique object satisfying the data. The
lems
are
undetermined,
without a
by Parker. In addition, this problem is of special interest as a relative of the
algorithm puts out a solution, but its
geophysical inverse problem of recov
value is not justified. How different
ering quantitative information about a
might the true solution be?
planet's radially stratified density struc
Groetsch does deal with the problem
The Development of Prime Number Theory: From Euclid to Hardy and Littlewood by Wladystaw Narkiewicz
ture from the astronomically measur
of finding properties common to all so
able mass and moment of inertia.
lutions compatible with the data when
BERLIN: SPRINGER, 2000.
Groetsch also gives plenty of activi
there is nonuniqueness. For the mass
xii + 448 pp.
ties in which an assumed data set is
/moment-of-inertia problem for a radi ally
nonuniqueness. He might then propose
the Backus-Gilbert formalism which,
an activity of fully characterizing the so
for linear problems, constructs certain
The title of this volume could mislead
lution set, or one of investigating how a
weighted averages shared by all solu
the potential reader. It misled me. I ex
more complete data set can eliminate
tions.
An average is designed to attempt
pected a largely historical narrative
the nonuniqueness. Instability might be
to estimate the value of the solution at
outlining the principal achievements in
shown by demonstrating that the size
a chosen point, but the shape of the
explorations of the primes. The author
of the solution does not depend con-
weighting function often makes the av-
quickly disabuses us of this notion,
THE MATHEMATICAL INTELLIGENCER
disc,
he
introduces
US $ 94.00, ISBN 3-540-66289-8
limited enough to illustrate solution
72
symmetric
SPRINGER MONOGRAPHS IN MATHEMATICS
REVIEWED BY GERALD L. ALEXANDERSON
however, when he says in the preface
(The answer is not known.) The cast
most of us probably don't know many
(do only reviewers read prefaces?):
of characters even for the variants of
proofs. Here we can contrast the proof
"This is not a historical book since we
Euclid's proof is impressive: Hermite
refrain from giving biographical details
and Stieltjes; for variants of Euler's
in this development and we do not dis
proof: Sylvester, Kummer, and Thue.
cuss the questions concerning why
about history and biography, he does
each particular person became inter
generously supply footnotes
of the people who have played a role
In spite of the author's disclaimer
giving
of Hadamard with that of de la Vallee
Poussin,
for
example.
Hadamard's
proof was simpler, something admitted
by de la Vallee-Poussin: "I proved for
s) + f3i.
the first time . . . that the function ?C does not have roots of the form
1
Mr. Hadamard, before knowing about
ested in primes, because, usually, ex
minimal
act answers to them are impossible to
about those
obtain. Our idea is to present the de
dates, as well as lists of principal ap
orem in a simpler way. " There is ex
velopment of the theory of the distrib
pointments. Occasionally he cannot re
ution of prime numbers in the period
tensive discussion of Landau's work in
sist the temptation to add something to
simplifying the proof, and descriptions
starting in antiquity and concluding at
a biographical footnote, such as his
the end of the first decade of the 20th
biographical
information
cited: birth and death
my research, also found the same the
of elementary proofs (not using ana
comment on Paul Erdos, "Authored
lytic means) beginning with the fa mous Selberg-Erdos proof.
century." This too is slightly mislead
more than 1300 papers in number the
ing, though the author goes on to ex
ory, combinatorics and analysis, which
plain that he does indeed move beyond
seems to be a world record," or on
While these variations on the PNT
theme will be of considerable interest
1910, though with less detail. This
Alfred Pringsheim, "Father-in-law of the
book is no general historical narrative,
writer Thomas Mann." The footnotes
easily read by the casual reader inter
are more than usually rewarding. Philip
sections more rewarding. There is no
ested in the primes. It's chock full of
J. Davis wrote a whole book motivated
shortage of amazing results.
mathematics. And throughout the au
by the question of the origin of Pafnuti
thor exhibits a dazzling display of
Chebyshev's first name
scholarship.
here Narkiewicz worries in a footnote
The bulk of the text, as one would expect, given the topic and the time
frame, is devoted to the Prime Number
Theorem (PNT), both the work leading
(The Threaff);
about the various transliterations of Chebyshev from the Cyrillic ! He comes
to readers, I suspect that many readers
like myself will find some of the other
In Chapter 1, for example, we learn that x12 + 488669 assumes composite values for all integers x satisfying lxl < 616980. (K McCurley, 1984) Is that widely known? Does everyone remember the polyno
up with fifteen different spellings and
mial in
tells where they appear. Those cited are
positive values are prime at nonnega
26 variables, degree 25, whose
up to it over centuries and the work fol
again stellar: Poincare, Dickson, Hardy
lowing, principally efforts to provide
and Wright, Sierpiriski, Markov, Niven,
easier or different proofs. The Riemann
Cesaro, Ribenboim, and Landau, among
prime can be obtained in this way?
?-function plays a central role.
others. The author prefers Cebysev.
number of variables is
The book starts off, naturally, with
proposition 20 of the ninth book of
tive values of the variables and every (Jones, Sato, Wada, Wiens,
1976) The
convenient,
The first chapter deals with early
given our alphabet! The latter result de
questions about the primes: Are there in
rives from the work of J. Robinson, M.
Euclid: there are infinitely many prime
numbers. But not content with that
finitely many? What is the sum of the
proof, the author proceeds to give
mulas giving prime numbers? Chapter 2
reciprocals of the primes? Are there for theorem
on
the attempts to get closer and closer to
eleven more proofs of the theorem,
deals
primes in arithmetic progressions. From
the proof in Euclid; proofs suggested
there on the concentration is on 11(x),
by Euler's early explorations of his
with Chapter 3 on Chebyshev's theorem,
large even integers can be expressed as
Chapter
the sum of a prime and a product of at
product formula; those derived from
Dirichlet's
on Hilbert's Tenth Problem. In Chapter 6 there is a fairly extensive account of
gathered together by type: variants of
certain sequences of pairwise coprime
with
Davis, H. Putnam, and Yu. Matijasevic
4 on the Riemann ?-function
the Goldbach Col\iecture, right down to and beyond Chen's result
(1966), that
and Dirichlet series, and Chapter 5 on
most two primes.
Chapter 6 gives information on what fol
it from cover to cover, but many will
Few will pick this book up and read
one involving Fermat numbers, attrib
the Prime Number Theorem (PNT).
lowed the proofs of Hadamard and de
fmd it a useful source of information
one derived from topology. Not con
la Vallee-Poussin, with Landau's ap
to use when teaching a number theory
the col\iectures of Hardy and Little
browsing. There's no end to the star
positive integers (the most famous the uted by Hurwitz to P6lya); and even thor describes at some length work
proach to the PNT, Tauberian methods,
prompted by this proof, questions such
wood, among other things.
tent with Euclid's simple proof, the au
as the following: if
While most sections of the book con
class. And it's a great text for casual tling results proved about primes-or
{an} is a sequence of primes, a1 = 2, a1, a , . . . , an- 1 al 2 ready defined, then let an be the largest prime divisor of Pn = 1 + a 1 a2 . . . an-1. Does this sequence contain all
casional proof, the sections on the PNT
not anticipate using it as a textbook
are considerably more detailed and
"Exercises" may be a misnomer; many
proofs are actually given at length.
of them look like hard problems.
primes? (The answer is no.) Does it
Anyone interested in number theory has
contain all sufficiently large primes?
probably seen a proof of the PNT, but
sist of accounts of results with the oc
in
some
cases
only
col\iectured.
Though there are exercises, I would
The author's command of English is
excellent. The text reads very well and
VOLUME 23, NUMBER 1 , 2001
73
the number of misprints and errors is
vehicle to teach or re-teach a variety of
requires an entire chapter because the
relatively small, given the extent of the
mathematical concepts: ingenious log
limit cannot be computed separately
coverage. Most errors are trivial and
arithms,
beautiful
on each side of the parentheses. Here
most unimportant. (Einar Hille's death
geometry, fascinating number theory,
Maor provides another excellent re
date is clearly wrong, for example, and
and mind-bending complex function
fresher course, this time on the bino
there are other little problems of that
theory. Maor is one of a small handful
mial formula, which in turn leads to a
sort.) The cutoff point (the Hardy
of mathematical authors who create a
Martin Gardner-like subject: Pascal's
Littlewood conjectures) can be frus
satisfying blend of historical anecdote
fascinating symmetric triangle of coef
trating. While the author has tried to
and mathematical information that ap
ficients.
bring up to date certain problems, oc
peals to amateur and expert alike.
casionally this falls short of bringing us
useful
calculus,
Following this is a delicious list of
The first three chapters give us a
curious numbers related to e. For ex
completely up to the present. For ex
fresh look at logarithms, that primor
ample, what x do you think yields the
ample, the conjecture (which would
dial ooze out of which e arises.
maximum possible value of
have implied the Riemann Hypothesis)
= In s xA(n) :::; 0, where the Liouville function A(n) ( - 1)0Cn) and ncn) is the number of prime factors of n counting multiplicities, was correctly disproved by C.B. Haselgrove (1958), but R.S. Lehman (1960) did more than confirm
that L(x)
=
Haselgrove's
computations.
Lehman actually found the first coun terexample. (Haselgrove only proved
1
Chapter
covers Napier's life and
Chapter
5
y = xllx?
paints the pre-calculus
200 B.C., comprised of
his fetal logarithmic table. Maor ex
landscape, circa
plains the value of finding numbers
well-understood straight-lined geomet
with a common base, which can be ex
ric figures. On the horizon are curved
pressed as a function of their expo
figures, waiting to be measured. But,
nents. It was a little tough getting
alas, calculus was not to be. The jump
through the description of Napier's
from a geometric way of thinking to an
table without an illustration.
algebraic one was too much of a Grand
2, Maor introduces other
Canyon even for Archimedes, who
men who invented, improved, or pro
held the crown of mathematics longer
In chapter
moted the use of logarithms. Unfortu
than anyone before or since. It's amaz
found the least coun
nately, history often attaches only one
ing to think that calculus was that
terexample. Though I am sure the au
name to a great invention. Maor con
close to our grasp, yet had to wait al
thor was aware of these points, his
tinues
most
chronological time frame limited his
Napier while throwing some crumbs to
exposition on this point, and, I'm sure,
Burgi, Briggs, and others.
that counterexamples exist.) Further, Tanaka
(1980)
this
tradition
by
crediting
2,000 years to be discovered (in
vented?).
In chapter 6, Viete and Wallis relate
on others as well. But as he said in
Maor's book has peripheral sections
a finite quantity to an infinite series of
his preface when he pointed out the
scattered throughout, when there is
multiplied quantities, thus uncovering
time limitation, "The following years
more to be said about a subject that is
one of those mathematical gems that
brought a great leap forward in our
related to e, but doesn't warrant a
ject. Maor uses this as a lead-in to the
knowledge of prime numbers deter
whole chapter. The first is a refresher
debunk the myth of math as a dry sub
mined by the birth of new powerful
course on using logarithm and antilog
acceptance of infinite processes, nec
methods, but this should be, possibly,
arithm tables. This scary section will
essary for the calculus.
a subject of another book" We can
turn any Luddite into a computer geek.
only hope that the author is hard at
Maor gets these math class flashbacks
ideas in chapter
work on the sequel.
out of the way early-stick with him!
much about the hyperbola. It doesn't
Chapter
Department of Mathematics & Computer Science Santa Clara University Santa Clara, CA 95053-0290
3 connects money and e, a
e: The Story of a Number by Eli Maar
are
some
7.
pretty
amazing
I'd never thought
have the intricacy of a lemniscate. It's
surprise for those of us anxious to read
not a nice neat closed figure. It's a de
about
and
creasing function and so we wouldn't
seashells. Kind of like a meeting be
e's
role
in
sunflowers
want it to describe our retirement sav
tween yuppies and hippies, e is an un
ally. It has no beginning and no end.
likely thread between nature's poetic
USA
There
ings over time. Its hard to plot manu
subtlety and humans' harsh acquisi
Yet the desire to "square" (measure the
tiveness.
area of) the hyperbola led to interest
Chapters
4 through
10 are a Trojan
Horse, used to sneak a history of cal
PRINCETON, NJ: PRINCETON UNIVERSITY PRESS
culus into a book on e. These seven
chapters describe the fascinating per
ing
spin-off
discoveries,
course, have to do with e. Calculus
then
that,
evolved
of
from
Descartes's and Fermat's determina
1 998, 232 pp.
sonalities, controversies, and versatil
tions of particular cases to Newton's
US $1 4.96, paperback, ISBN: 0691 058547
ity associated with calculus, a tool that
and Leibniz's general algorithms, de
REVIEWED BY MELISSA HOUCK
is to e what lasers are to an eye with
signed to handle any case that might
I
n this delightfully thought-provoking yet digestible book, Maor uses e as a
74
THE MATHEMATICAL INTELLIGENCER
cataracts.
Chapter
4 gets to the crux of e by
computing the limit of
(1
+
1/n)n. This
come along. The case
llx,
however,
steadfastly
to
conform.
Chapter
8
refused introduces
the
missing
link-Mercator's
t - t2/2 + &/3
-
series:
t414
+ . . .
log(l +t) =
Chapter 9 brings to life the battle over kudos for inventing calculus. The average calculus course would be more memorable if students were introduced to the subject via a lively account of the priority dispute between Newton and Leibniz. Maor describes the different perspectives each had on the subject Newton: physics, Leibniz: philosophy and how this drove their approaches. In historical retrospect, having these two divergent-seeming paths both converge on the same method was almost neces sary for the universal appreciation ofthe subject. "The Evolution of a Notation" is pre sented next as an aside. Maor stresses how essential an effective notation is. Perhaps the most taken-for-granted aspect of mathematics (and, quite frankly, the most boring), notation can stymie mathematical progress (e.g., ro man numerals) or grease the tracks. As Maor demonstrates, Leibniz's nota tion was butter compared to Newton's "pricked letters." Chapter 10 fmally describes the connection between calculus and e. The beauty of Maor's discourse on e is that he presents e's meaning in several different ways: geometrically-as a key player in the hyperbola's area; ap plied-as in the solution to a para chutist's problem; naturally-as it de scribes the relationship between pitch and frequency. And, in this chapter, al gebraically-as a function that equals its own derivative. Geometry is meant to be generously illustrated, and Maor doesn't let us down. He uses several advanced mathe matical concepts in chapter 1 1-polar coordinates, mapping, invariance, cur vature-to describe the beautiful loga rithmic spiral and its narcissism: It is its own evolute, just as ec is its own deriv ative. If this geometric chapter is indica tive of what Maar's book Trigorwmetric Delights is like, then I'm sold. Next is an interlude about a ficti tious meeting between Bach and Johann Bernoulli, where they discuss music scales. It is fascinating that these two overlapped in their time on earth, studied complementary sub-
jects, and probably did affect one an other, albeit indirectly. Those who pos sess a strong correlation between their mathematical and musical neurons will enjoy this passage. Another interlude-this time it's the sunflowers and seashells we've been waiting for. And some Escher litho graphs as the cherry on top. Again, Maor isn't stingy with the pictures. This chapter should suffice as the an swer to the question: "Why do I need to study math?" Most students will not grow up to engineer great structures or invent the next computing device. But everyone should have a chance to mar vel over nature and its unexpected connection with mathematics. Ah, the stuff of mental blocks: hy perbolic functions. In chapter 12, Maor transforms these from dryly memo rized, funny-sounding (sinh, cosh) cal culus 101 mysteries to a river of geo metric and analytic meaning. Maar uses the hanging chain (catenary), (f!C + e-x)/2, to take us to these tran scendental, underrated analogies of trigonometric functions. Another interlude, this time on the analogies between x2 + y2 = 1 and x2 y2 = 1. The circle and hyperbola look so unalike but the identities within them are close (identical, in some cases). A topic that lends itself to both number theory and geometry is always enlightening. The properties of most numbers pale in comparison to e. Maor could have elaborated on e's role in three key integrals that he discusses all too briefly in this next interlude. Probability Theory, Exponential Integrals and Laplace Transforms are all sophisti cated concepts that should not have daunted the author, who has a gift for bringing sophisticated concepts down to earth. I think a deeper handling of these would have shed light on e's util ity as well as its beauty. In chapter 13, Maar does elaborate on one of the formulas he discusses in the previous interlude: eia:. The author gives a very satisfying account of Euler bring ing e into the complex (as in i) world. Kudos to an author who not only dares to address complex number theory, but who does so in a way that preserves the -
nature of the beast. Maor points out that
ix describes 4 cycles, not numbers in the usual sense. Maar derives Euler's for mulas connecting trigonometry with ex ponentials in a way that is surprisingly easy to understand. Chapter 14 continues Maar's won derfully accessible account of complex number theory. He boils i down so that it is seen as a compact way of intro ducing rotation into analysis, rather than as a hard-to-handle blockade be tween problem and solution. Maar ap plies complex function theory to e", and shows us how to map equidistant lines to exponentially increasing circles, thus circling back (literally) to Napier's loga rithms. Maor describes i as the least in teresting constant in the equation em + 1 = 0. After reading his chapters on imaginary numbers, I would have to dis agree. He points out some differences between complex and real numbers in math: Exponentials become periodic; logarithms become multivalued; circu lar (sin, cos) and hyperbolic (sinh, cosh) functions become directly related. Best of all: ii is multivalued and real! Perhaps i is uninteresting to Maor, but he writes an interesting account of it. Another of those unexpected con nections is the following interlude this time between integers and tran scendentals (namely e!). Maor gives a typically lucid explanation of primes and the prime number theory, high lighting e's prominent role. The last chapter summarizes e's place in mathematical history. Maor keeps the discourse lively by contrast ing e with 1r, that other transcendental wonder, and by including a brief synopsis of the different categories of numbers-rational, irrational, alge braic, transcendental, etc.-showing how e fits into a teeny tiny hole in the number line. Having read Maor's To Infinity and Beyond, I was sure that e, The Story of a Number would not disappoint me. I was right. Now its time to pick up a copy of Trigonometric Delights. 232 Yale Road Wayne, PA 1 9087 USA e-mail:
[email protected]
VOLUME 23, NUMBER 1, 2001
75
Equations and Inequalities: Elementary Problems and Theorems in Algebra and Number Theory by Jifi Herman, Radan Kucera, and Jaromir Si'mSa
equations:
thors state (without proof) a valuable
q =/=
transformation theorem: if
X� + X� = 464 XI + X2 = 4.
1 is
real, then there exists a polynomial
Q(x) of degree m and a real number d
One can eliminate one of the variables,
such that
but then one is faced with the problem
S(q,n)
=
of solving a quartic equation. Rewrite
d + Q(n)qn.
the equations using symmetric func
The authors show how the polynomial
tions:
u1 - 5uYu2 + 5uw�
determined coefficients.
MATHEMATICS
£TI = 4,
There is a valuable discussion of
vii + 344 pp.
polynomials and their properties. One
US $69.95 ISBN 0-387-989420
common practice in proving an iden
REVIEWED BY JET WIMP
l
an example, consider the simultaneous
call the above a finite q-series. The au
Q may be found by the method of un
CANADIAN MATHEMATICAL SOCIETY BOOKS IN
NEW YORK: SPRINGER PUBLISHING,
m. A special function specialist would
tity is to show the quantity in question
=
464
This leads rapidly to the solutions x1
2 ± v2, x2
=
=
2 + v2. (It turns out that
MAPLE can solve the original equations,
satisfies a polynomial equation with
but it is not difficult to devise similar
have a long-standing addiction to
one or more known roots, and to rule
problem books, and have over a
out the presence of the other roots. For
equations solvable this way which will
dozen of them on my bookshelf. Some present a concatenation of assorted problems, organized roughly by type,
�3V2i + 8 - �3v2i - 8 =
with little explanatory material and no
denote
exposition of systematic techniques.
Elementary algebra gives
An
example
is
the
recent
book
Berkeley Problems in Mathematics, edited by de Souza and Silva, Springer,
1998.
Some-and the present book is
an example-are much more tech
the
left
hand
side
A3 + 15A - 16 =
Thus A is a zero F(x) = x3 + 15x zero
cripple computer algebra systems.)
Section 6 of this Chapter deals with
instance, to prove
the solution of irrational algebraic
1,
by
A.
arcane equations considered here is
Vx+1 + � + Vx+3 = 1.
0.
of the polynomial 16.
equations, and introduces a variety of techniques. A typical example of the
x= 1
of this polynomial.
As might be expected, computer alge
is also a
bra systems fail on these sorts of equa
We
tions.
find
nique-oriented. They explain how large
F(x)l(x - 1) = x2 + x +
classes of problems may be attacked,
mial which has only complex roots.
and the problems themselves are much
Thus
16, a polyno
A = 1, as required.
more like the exercises found in texts.
The discussion of symmetric poly
Of the latter type, this book is about
nomials is unusual and fascinating. I
the most successful I have seen, and
wish I had been aware of some of this
the editors of the series, Jonathan and
material when attempting to solve
2, Algebraic Inequalities, is
Chapter
one of the most illuminating studies of the subject I have ever seen. Again, symmetry becomes a powerful tool. If one is attempting to show that
fi:x1>X2, . . . , Xn) > 0
f is symmetric in all its variables,
Peter Borwein, are to be congratulated
some equations arising from the trans
and
for bringing it before a large public in
formation theory of hypergeometric
then it is no loss of generality to assume
a superbly smooth translation.
functions. The following result, which
XI 2: x2 2:
The book is divided into four parts: Algebraic Identities and Equations; Al gebraic Inequalities; Number Theory; and Hints and Answers. Each chapter presents an assortment of techniques for dealing with the problems in that area. Chapter
1 discusses fmite sums and
combinatorial identities. As might be
has generalizations to more than two
(The proof of this and its extensions is in van der Waerden's algebra book.)
rem is sufficient to prove the formula
The authors give a nice little table ex
in question. One unusual aspect of this
pressing powers
chapter is the attention devoted to
and
76
THE MATHEMATICAL INTELLIGENCER
The same technique will work to
prove that
xf + x� in terms of £TI a(a - b)(a - c) + b(b - c)(b - a) + c(c - a)(c - b) 2: 0 . . . , 5, and a sim
u2 for n = 1, 2,
ilar table for the case of three vari
· · ·
This assumption
for an arbi ple, suppose we wish to show that trary polynomial F(x1,x2) symmetric 1oo < 3100 (a iOO + b ioo + l� c in the variables x1 and x2 there exists (a + b + c) a unique polynomial H(y1,y2) such for a, b, c, positive. Assume a 2: b 2: c. that We get F(x1,x2) = H(avr2), (a + b + c) Ioo ::; (3a) Ioo < 3 100 (a !OO + b iOO + c lOO . £TI = X1 + X2, £T2 = X1X2. )
imaginative use of the binomial theo
S(q,n) = P(l)q + P(2)q2 + P(3)q2 + + P(n)qn, where P(x) is a polynomial of degree
Xn.
variables, is invaluable:
suspected, for many such sums the
sums of the form
. . . 2:
can have a magical efficacy. For exam
ables. Symmetric functions are very useful in solving simultaneous nonlin ear algebraic equations when the equa tions are symmetric in the variables. As
for positive
a, b, c, since the left-hand
side above may be written
a(a - b) (a - c) + (c - b) [c(c - a) - b(b - a)],
a ::::; b ::::;
x = 1, y = 2, or x = 0, 0). The chapter closes with a dis
c
multiple and greatest common divisor,
(SOLUTIONS:
guarantees that both the term in brack
prime numbers,
function. The number of bizarre results
y =
As an exercise, the authors ask the
that may be proved using very ele
nomials
reader to demonstrate an intriguing ho
mentary
Eisenstein irreducibility criterion.
mogeneous nonnegative lower bound
boundless. I will whet the reader's ap
for the difference between the arith
petite by giving just a few of them:
and the assumption that
0<
ets and the first term are positive.
metic and the geometric mean:
a + b . ;:; - v ab 2 (a - b)2(a + 3b)(b + 3a) > � ��� � --� - 8(a + b)(a2 + 6ab + b2 ) --
between
22n + 1
means
is
apparently
and
numbers. ii) Let
the results that struck my fancy was
� +V'bc +� - 1 2 ::::; -(a + b + c). 3 The third chapter, Number Theory, was, to me, a revelation. The chapter is self-contained, with a treatment of basic concepts such as divisibility, the Euclidean algorithm, least common
a, b, and Va + Vb be posi
tive rational. Then tional.
generalizations of Cauchy's inequality receive a sound treatment also. Among
cannot be expressed as a dif
ference of fifth powers of two natural
--
•
Inequalities
techniques
i) Any prime number of the form
--
--
primality, Euler's
iii) The number by
13 .
Ya and Vb are ra
260 + 730 is divisible
cussion of the factorizability of poly and
a
statement
of
the
The fourth chapter has the solu tions for the exercises. Sometimes these are only sketched, which is ap propriate.
After
all, in a problem
book, the reader has the responsibil ity for doing at least some of the work. I have nothing but praise for this book, and I can't imagine a working mathematician who wouldn't want to own it. As a matter of fact, my unwill ingness to surrender the book to some
Congruences receive a thorough
one else prompted this review.
treatment, including systems of con gruences and nonlinear congruences. The discussion of diophantine equa tions is extensive and systematic, and the equations treated include many strange nonlinear diophantine equa tions, for instance
1 + x + x 2 + x3 = 2Y.
Department of Mathematics and Computer Science Drexel University Philadelphia, PA 1 9904 USA e-mail:
[email protected]
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VOLUME 23, NUMBER 1 , 2001
77
4jfi i.MQ.ip.i§i ..
Robin Wilson
Romanian Mathematics
IMatematicii
n 1895 the monthly periodical Gazeta made its first appear ance. Founded in order "to improve the knowledge of mathematics of high school students," but also featuring orig inal papers in mathematics, it has since appeared without a break, and at one time had over 120,000 subscribers. Romanian stamps were issued to commemorate its 50th and 100th an niversaries. One of the 1945 stamps featured "the four pillars of Gazeta Matematicii," Ion Ionescu, Andrei loachimescu, Gheorghe 'fi leica and Vasile Cristescu, while the centenary stamp depicted Ion Ionescu alone, the "spiritus rector" of the Gazeta who ran it for over fifty years. Gheorghe Titeica, the only pure mathematician among the "four pillars," made distin-
guished contributions to differential geometry. The Romanian mathematical educa tion system was established in 1898 by the Minister of Education, the mathe matician Spiro Haret. With his support and encouragement, Gazeta Matematicii initiated a series of mathematical text books that became of great importance in secondary schools for many years. Romania has always been at the forefront of mathematical competj tions at high-school level. Since 1897 it has regularly organized national com petitions, and in 1959 launched the flrst International Mathematical Olympiad. Forty-one such Olympiads have now taken place in over twenty countries, and Romania has played host on no fewer than four occassions.
Gazeta Matematka
Please send all submissions to
Jon lonescu
the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics,
The Open University, Milton Keynes, MK7 6AA, England e-mail:
[email protected]
80
THE MATHEMATICAL INTELUGENCER 0 2001 SPRINGER·VERLAG NEW YORK
The "four pillars of the Gazeta"
Gheorghe Tilelca
Spiru Haret
