the mathematical intelligencer volume 22 issue 1
TRANSCRIPT
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.
Recovered Palimpsests
The story of the Archimedes palimp
sest, as told in "Sale of the Century,"
Math. Intelligencer21 (3 ), 12-15 (1999 ),
reinforces the notion that bound vol
umes on shelves may not be such a bad
storage medium after all. Does anyone
really believe that "electronic" books
from today will be readable in 3001 ??
The problem with an electronic for
mat is that there is no economic model
for long-term magnetic storage. Where
as paper works may have to be re
printed or copied every few hundred
years, magnetic storage has an inher
ent life on the order of 10 years and
also suffers from repeated changes in
data formatting. How can we make
sound archival decisions in the ab
sence of a viable model for open ac
cessibility to the scholarly community?
Though societies-like the American
Mathematical Society will likely pro
vide access to their publications for a
substantial time period, it seems plain
that economic concerns will eventually
result in the curtailment of electronic
access to older material, particularly
material from commercial publishers.
Electronic publications have an in
creasingly important function, but this
does not mean that they will or should
replace all paper publications. It is rea
sonable to conclude that a role for
print will continue to exist in parallel
with electronic publication for many
centuries.
D. L. Roth
Caltech Library System
Pasadena, CA 91 1 25
USA
e-mail: [email protected]
R. Michaelson
Northwestern University Library
Evanston, IL 60201
USA
e-mail: [email protected]
More Mathematics in Its Place
In his commentary (Summer 1999 , pp.
30-32 ), Edward Reed argues for less
mathematics and more numeracy in
engineering. I would not want to quar
rel with the "more numeracy" part, but
I have a somewhat different take on
the desirability of less mathematics. I
should say up-front that I am not an en
gineer; I am a mathematical statistician
working in the areas of survey statis
tics and education statistics.
My own point of view is heavily in
fluenced by a talk I heard several
decades ago by Paul MacCready, the
aeronautical engineer who designed
and built the Gossamer Condor (which
won the Kremer Prize for the first hu
man-powered flight over a fixed course)
and the Gossamer Albatross (Kremer
Prize for human-powered flight across
the English Channel). Before begin
ning this work, Dr. MacCready had done
a theoretical calculation that showed
that a low-powered aircraft would have
to have a very large wingspan. This may
not seem remarkable, but several inter
national groups were actively pursuing
the (first) Kremer Prize with aircraft that
had no hope of success. On the other
hand, Dr. MacCready emphasized, it is
not possible to design an aircraft suc
cessfully with paper and pencil alone:
simulations, modeling, test flights, and
tinkering are needed. The power of
theory and mathematics often comes
in showing what will not work so that
effort may be concentrated along po
tentially successful avenues.
Michael P. Cohen
161 5 Q Street NW (#T-1 )
Washington, DC 20009-631 0
USA
e-mail: [email protected]
© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 1. 2000 5
Numeracy with Mathematics
Here is my (predictable) response to Ed
ward Reed's contribution "Less Mathe
matics and More Numeracy Wanted in
Engineering" in the Summer 1999 issue
of the Math. Intelligencer.
Professor Reed's estimation of the
size of an object on Earth as perceived
from the Moon strikes me as an excel
lent example of what has come to be
called a Fermi solution to a Fermi prob
lem. Here is a typical Fermi problem, as
posed by the great physicist Enrico
Fermi to his physics class: "How many
piano tuners are there presently in
Chicago?" A better-known and note
worthy example of a Fermi solution:
Fermi's trick of dropping confetti to es
timate the energy release from the first
atomic bomb explosion. An essay in
Hans Christian von Baeyer's book The Fermi Solution, Random House, 1993 ,
uses these examples to introduce read
ers to some intriguing aspects of scien
tific thought. I also highly recommend
the book by M. Levy and M. Salvatori,
Why Buildings Fall Down, W.W.
Norton, 199 2, as an engaging popular
overview of some elements of structural
engineering and the power of what
Professor Reed calls "numeracy."
Behind such estimates there is al
ways a mathematical principle, either
directly pertinent to the problem, or
serving as a foundation for the requi
site physics. I say "always" because, as
a mathematician, I adhere to a broader
defmition of mathematics than does
Professor Reed. It is true that one need
not have a grasp of the detailed struc
ture of the Euclidean group of simili
tudes to apply Professor Reed's thumb
nail process for earth-lunar estimation,
but anybody will concede that an im
portant mathematical principle lurks
behind the trick, and it is this that gives
us confidence in the procedure.
While Professor Reed asserts that
6 THE MATHEMATICAL INTELLIGENCER
no bridge was ever built by mathemat
ics, a mathematician is likely to retort,
perhaps on the authority of figures
such as Galileo, or even Donald Duck
in Mathmagic Land, that nothing was
ever built without mathematics. I would
guess that the professor of mathematics
took umbrage at Professor Reed's re
marks and got the vapors not because
he objected to a reform of the engi
neering curriculum that would pro
duce numerate engineers and mini
mize mathematical irrelevancies, but
because these remarks reflected a
parochial view of mathematics that
might lead students to suspect mathe
matics itself is an overrated discipline.
Well should his students appreciate
that were it not for mathematicians
Professor Reed would today be clad in
goatskins and crouching beneath a
berry bush for his supper.
All in all, his piece is an entertain
ing and extremely stimulating con
tribution to the important ongoing di
alogue concerning the role of mathe
matics in society and in education.
Don Chakerian
Department of Mathematics
University of California Davis
Davis, CA 9561 6-8633
USA
Response to Reed
What Edward Reed calls "numeracy" is
rudimentary mathematical knowledge,
a thin layer of mathematical arguments
which are apparently not recognized as
such. Take his remarks about building
bridges. "The medieval builder," he
tells us, "knew that if a shape, known
to us as a catenary, could be drawn so
as to go through every stone, then this
arch would stand up." Huh? To iden
tify a shape as a catenary and to ex
plain why it has the asserted property
would be mathematics. Next he ex-
plains why the Great Wall could not be
seen from the moon; here he uses
similar triangles (mathematics), and
knowledge of the absolute size of the
Great Wall and the moon's distance
(could he tell the moon's distance
without mathematics and physics?).
Without knowing it, the Reeds are
standing on foundations laid by math
ematics, but as they ignore them their
students will ignore them even more,
until one day engineering will again be
reduced to "trial and error" and recipes
("counting the eggs for the mortar")
modulated by "intuitive" arguments
coming from half-forgotten scientific
knowledge.
Michael Reeken
Department of Mathematics
Bergische Universitat-GH Wuppertal
D-42079 Wuppertal
Germany
e-mail: [email protected] .de
EDITOR's NoTE: Diverse reactions to the
note of Professor Reed are expressed
in the letters above. I nevertheless
want to quote one more. Apologies to
the writer: though he submitted his let
ter for publication and gave name and
address, I was unable to reach him at
the address he gave, so as to confirm
his willingness to be quoted in print.
All I can do is give an excerpt, anony
mously, from (apparently) an Arab
mathematician visiting Germany.
The letter by Edward Reed may be
cheering for us "underdeveloped" na
tions, showing us how strongly science
is declining in the West, giving us a
chance of catching up ....
Let me apologize for the Islamic hu
mor. But you have earned nothing but
scorn from those nations who are
preparing to take up the torch of scien
tific thinking from your faltering hands.
Opinion
The Numerical Dysfunction Neville Holmes
The Opinion column offers
mathematicians the opportunity to
write about any issue of interest to
the international mathematical
community. Disagreement and
controversy are welcome. The views
and opinions expressed here, however,
are exclusively those of the author,
and neither the publisher nor the
editor-in-chief endorses or accepts
responsibility for them. An Opinion
should be submitted to the editor-in
chief, Chandler Davis.
The opinions of Anatole Beck in his
article "The decimal dysfunction"
[ 1) were refreshing and interesting,
and his discussion of enumeration and
mensuration was surely important and
provocative. A learned and detailed ar
gument devoted to showing as "folly"
the SI metric system adopted by so
much of the world, and soon to be
adopted by the USA [2), deserves some
serious response. If the SI system is in
deed folly, then mathematicians every
where have a duty to make this as
widely known as possible. If it is not,
then a rebuttal should be published.
That the only published reaction [3)
should be jocular, however witty, is
therefore deplorable. Jerry King, in his splendid The Art of Mathematics, writes,
"The applied mathematician emphasizes
the application; the pure mathematician
reveres the analysis." [4, p. llO] Perhaps,
then, neither kind of mathematician sees
simple enumeration and mensuration as
worthy of consideration, so that both ig
nore Professor Beck's argument, and
thus show themselves apathetic about
the innumeracy that "plagues far too
many otherwise knowledgeable citi
zens" [5, p.3] and about "the declining
mathematical abilities of American
[and other] students" [4, p.176).
Let the shameful silence on enu
meration-mensuration be broken by a
technologist, one with a background in
engineering and cognitive science,
with three decades of experience in
the computing industry, and with a life
long interest and a decade of experi
ence in education. This article argues
that the SI metric system is indeed
flawed, though not in the way Professor
Beck sees it; that the way we have
come to represent numerical values is
even more flawed; and that the general
public would be best served by a re
duced SI metric system supported by
an improved (SI numeric?) system for
representing numeric values. If these
arguments are valid, then mathemati
cians everywhere have a professional
duty actively to promote reforms, of
the kind described here, as a necessary
basis for efforts to reverse present
trends in public innumeracy.
Enumeration
A major theme of Professor Beck's ar
ticle is, as its name proclaims, that dec
imal enumeration is not the best enu
meration system. The reason? Ten
"appears essentially not at all in math
ematics, where the natural system of
numeration is binary. . . . One might
blasphemously take the importance of
2 in mathematics as a sign that God
does His arithmetic in binary." By def
inition God is omniscient, and it is blas
phemous to imply that She has to do
any arithmetic at all! But 1 is much
more important in mathematics than 2
is, so wouldn't tallies-which indeed
have a long tradition [6]-be better
still? Mere analytical argument will not
settle the matter.
Binary enumeration, whatever its
analytical virtues, is not after all prac
tical. "Binary numbers are too long to
read conveniently and too confusing to
the eye. The clear compromise is a
crypto-binary system, such as octal or
hexidecimal." In what way, then, are
these systems a clear improvement on
the decimal? To a society now used to
decimal enumeration, any non-decimal
system will be confusing.
Would octal or hexidecimal be more
convenient than decimal, for example
in being more accurate or brief? Octal
integers are a little longer than deci
mal, but hexidecimal are somewhat
shorter. All are exact. Not much to jus
tify change in that.
And fractions? Different bases differ in which denominators they handle
best.
Along this line, it is significant to ob
serve that the smaller its denominator
the more used a fraction is likely to be.
This observation is behind the benefits
so often argued for the duodecimal
system of enumeration, which can ex
press halves, thirds, quarters, and
sixths exactly and succinctly. The
© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 1, 2000 7
most personal of the old Imperial mea
sures were conveniently used duodec
imally-12 inches to the foot, 12 pen
nies to the shilling. We still have 12
hours to the o'clock, and 12 months to
the year. The movement for a thorough
dozenal system is quite old-Isaac
Pitman tried to introduce it with his
first shorthand system [7]. Full and lu
cid arguments for the system can be
found in texts [8], and in the publica
tions of the various Dozenal Societies.
(The Dozenal Society of America ad
vertises from time to time in The
Journal of Recreational Mathematics
and, going by the World Wide Web, has
its headquarters on the Nassau campus
of SUNY.) However, the dozenal cause
now seems a hopeless one, given that
decimal enumeration has taken such a
global hold since the First World War.
Even if a way could be found to con
vert to a dozenal system, doing so would
not compel the abandonment of the met
ric system. The SI metric standards are
not inherently decimal because the ba
sic and secondary units of measurement
could as well be used with a dozenal sys
tem of enumeration as with a decimal
system. Mensuration combines an enu
merated value with a unit of measure,
and a good system will provide practical
and useful units of measure.
Mensuration
The many old systems were practical
and useful in respect of how quantities
could then be measured (often by
some human action giving units like
paces or bow-shots) and how stan
dards could be administered [9]. There
were different units of length for dif
ferent ways of measuring them, differ
ent units of quantity for different
things being measured, and different
units for different towns and villages.
However, the old measures were prone
to being used by the powerful to ex
ploit the weak, as implied by various
admonitions in both the Bible (for ex
ample, Deuteronomy 25:13-16) and the
Quran (Sura 83: 1-17).
The worldwide metric system de
fmes as few basic units as possible, and
secondary units such as for areas and
volumes are derived from the basic
ones. Though it might spring rationally
or irrationally from the French Revolu-
8 THE MATHEMATICAL INTELLIGENCER
tion, the system is overwhelmingly su
perior to the old humanistic systems, if
only because its arithmetic simplicity
and world-wide acceptance make it less
subject to cheating and misunderstand
ing. The difficulties for adoption of the
metric system now are much fewer and
more transient than for the illiterate and
innumerate society of Revolutionary
France, where the changeover lasted for
two generations [9, p.264].
(Of course, there is something in
trinsically valuable about a culture
having a characteristic way of doing
things, and this is true of household
measures as well as of (say) cuisine.
But the basic vehicle of culture is lan
guage, and anyone truly concerned
about the preservation of cultural rich
ness and variety (as surely we all
should be) would be much better em
ployed combatting the present oligo
lingual rush than opposing metrica
tion. Languages are dying off even
faster than species!)
The strange thing about the metric
system, though, is that, while the basic
units (and some of the secondary
ones) are widely and consistently ap
plied, each of these is the basis of a be
wildering collection of pseudounits de
fmed through a somewhat arbitrary
system of scaling prefixes. Not only are
the prefixes weird in themselves, but
they also have inconvenient abbrevia
tions, including highly confusable up
per and lower cases of the same letter
(Y is yotta, y is yocto [2]), and even a
letter (t) from the Greek alphabet.
Although it is averred that the prefixes
are easy to learn and use, in practice
their spelling, their pronunciation, and
their meanings are all confused and
confusing in popular use. And it is pop
ular use that's important.
These prefixes are really only suited
for use in private among consenting
adults. It took a physicist, the famous
Richard Feynman, to advocate the pre
fixes be abandoned because they actu
ally express scalings of the measure
ments being made, and because they are
"really only necessitated by the cum
bersome way we name numbers." [10]
What does the experience of Aus
tralia, a country converted to SI metrics
only a few decades ago, have to tell
about the popular use of the prefixes?
The most important measurements
in everyday life are those of length,
weight, volume, and temperature. Here
in Australia, the discouraging of the pre
fixes which are not multiples of one
thousand seems to have had good effect,
the faulty early publicity notwithstand
ing [an example in 11]. This is a very
good thing, because now there are fewer
ways to express any measurement, and
this must greatly reduce the potential for
confusion. Centimetres are occasionally
used (so are, for the moment, feet and
inches when giving anyone's height),
and the hectare seems to have replaced
the acre for people of large property. But
centi- and hecto- are otherwise never
seen, and the confusing deci- and deka
have disappeared altogether.
Celsius, the new temperature unit,
took over straight away, possibly be
cause the old scale was plainly silly and
its cultural value slim. Oddly enough,
the unit is almost always spoken of sim
ply as degrees. For lengths, people seem
comfortable with millimetres and me
tres and kilometres, though in casual
speech the abbreviations mil and kay
are more often used, particularly the lat
ter. Grams and kilograms are comfort
ably used for weights, though the ab
breviation kilo (pronounced killo) is
preferred to the full name. For vol
umes, the use of millilitres and litres
has completely taken over, though
again the abbreviation mills (not mil
as for lengths!) is often heard. The use
of a secondary unit, litres, for volume
is justified by its relative brevity in
speech, so that no abbreviation is
needed. The only problem is that the
litre has become somewhat divorced
from the cubic metre, and people are
not always able to compare volumes in
the two units swiftly and reliably.
The conclusion to be drawn from
the Australian experience is that, while
the common metric units of measure
have been everywhere adopted, their
names have been found difficult, and
all the long ones have been abbreviated
in common speech, typically by elision
of the basic unit name. Measures, and
numbers, must be simple to be popular.
Emancipation
The challenge is to free numbers gener
ally from the thrall of technologists and
mathematicians so that more of them become easy for people to use. A great way to start is to get rid of the metric prefixes along the lines suggested by Feynman, and to build on popular usage.
• Let any number to be interpreted as scaled UP by 1000 be suffixed by k, and let a number like lOOk be pronounced one hundred kay.
• Let any number to be interpreted as scaled DOWN by 1000 be suffixed by m' and let a number like lOOm be pronounced one hundred mil.
• Let any number to be scaled UP twice by 1000, that is by 1000000, be suffixed by k2, and let a number like 100k2 be pronounced one hundred
kay two.
• Let any number to be scaled DOWN twice by 1000, that is by 1000000, be suffixed by m2, and let a number like 100m2 be pronounced one hundred
mil two.
Adoption of these rules, and of the extensions they imply, is in accord with, indeed would reinforce, both the intent of the SI metric standards, and the common sense of popular linguistic practice. Adoption of these rules would allow the metric prefixes and their upper-case, lower-case, and Greek abbreviations to be forgotten, would allow common talk of numbers to be as loose or precise as needed, and would deliver a wider range of numbers and quantities into common parlance and understanding. Measurements outside the scales of common usage would at least be recognised roughly for what they are, if not wholly understood.
These rules are simple enough to be accepted by the general public, and expressive enough to be used by scientists and engineers, and even by mathematicians. Indeed the notation is similar to the so-called engineering or e-notation, but better than it because there are fewer ways to represent any particular value. E-notation was adopted by technical people submitting to the limitations of the printers that were attached to early digital computers, and in it 100k2 might be represented as 1E8 or 100E6 or 0.1E9.
Adoption of the notation for scaled numbers proposed here could allow dropping of quirky notations which
disguise pure numbers as measurements under, for example, the pseudounit decibel. Eventually even the percentage, and its pseudosubunit the point or percentage point, might meet their Boojum and softly vanish away.
Most importantly, the notation would allow phasing out the present usage in mathematics and science which shows scaled numbers as expressions like 3 X 1010. This style is particularly confusing for students. Is it a number, or is it an expression, or is it a calculation? A mathematician or scientist may be able to see immediately past the calculation to the number it produces, but to ordinary mortals the expression hides the number. Mathematicians, or at least mathematics teachers, have in this ambiguity another very good reason for adopting a scheme like that suggested above for representing scaled numbers.
Representation
To confuse expressions like 3 X 1010 with numbers is bad enough, but at least elementary school children are not normally exposed to this particular ambiguity. However, they are exposed to a very similar ambiguity early in their arithmetic education, an ambiguity that (some say) costs the average pupil six months of schooling, and brings some pupils a lifetime of innumeracy. This is the ambiguity in notations such as -1 and -15 where the role of the hyphen is ambiguous [12]. Is it the sign for the property of negativity, or is it the symbol for the function of subtraction? A conspicuous sign is needed to stand for the property of negativeness in a number, a sign quite distinct from the symbol for subtraction.
Because the present ambiguity is not overtly recognised in early schooling, few adults are even aware of it. Perhaps mathematicians consciously distinguish the two meanings given to the hyphen. "Unfortunately, what is clear to a mathematician is not always transparent to the rest of us." [4, p.50] Particularly not to children. That this ambiguity is a real problem is shown by the many texts for teaching elementary mathematics that use temporary notational subterfuges in an attempt to overcome the ambiguity.
However, the most popular method merely raises the hyphen to the superscript position, which doesn't actually change the sign, and certainly doesn't make the distinction obvious. Some texts even increase the problem by using a raised + to mark positivity [e.g., 13, p. 153], thereby spreading the ambiguity to another basic symbol.
The ambiguity extends to the spoken word. The hyphen is read out as minus whether it is used as the negative sign or as the subtraction symbol. This is a severe problem because the natural word for the sign, negative, is three syllables long, one too many for it to be popular. Words like off and short can have the right kind of meaning, but might become ambiguous within sentences. Maybe the abbreviation neg could be adopted.
The negative sign should be used as a prefix, because it is spoken as an adjective, because the left end of any ordinary number is its most significant end, and because the negative sign is in some ways the most significant cipher, as it completely reverses the significance of the value it prefixes. The wretchedly inadequate ASCII character set foisted on the world by the computing industry has no suitable symbol. Selecting from what is already available in T EX, a suitable symbol might be a triangle, superscripted and reduced in size to be aesthetically and perceptually better: v72. (A superscript vee or cup could be used as an option for easier handwriting, as in V72 or u72.) The problem is rather that of getting the symbol onto the everyday keyboard.
One new symbol is not enough. The fraction point needs one as much as the negative sign does. The dot used in most of the world for the fraction point is more inconspicuous than any other symbol apart from the blank space. Furthermore, it is used as punctuation in ordinary text, leading to ambiguities in particular at the end of sentences ending in numbers. That this inconspicuousness is recognised as a difficulty is demonstrated by the common precaution of protecting the dot from exposure by writing for example 0.1 rather than .1, by the use of the comma instead of the dot for the fraction point in Continental Europe, and by the mis-
VOLUME 22, NUMBER 1 , 2000 9
begotten attempt by the Australian Government to use the hyphen for the fraction point in monetary quantities when decimal currency was introduced. An unambiguous point symbol is needed, and with TEX the point could be contrasted with but related to the negative sign, giving numbers like 7t,.2 (or 7/\2 or 7n2}
Exactness
It is one thing to be able to express a value unambiguously as a value, and provision of a distinctive negative sign and fraction point allows this. It is another thing to be able to express how reliable or accurate a value is. A value can be completely reliable and accurate-in other words, exact-or it can be unreliable or inaccurate to some degree or other. To be unambiguous about whether a value is exact or not is to tell the truth. A notation that allows this truth to be told would therefore be not only a public good, but a mathematical one-"mathematics is truth, truth mathematics" [4, p.177].
If a value, like a count or a fraction, is exact, then its representation must show plainly that it is exact. A simple number like one and two thirds is exact and, moreover, commonly useful. Yet it is nowadays almost never represented exactly. Instead some approximation like 1.667 or 1.666667 is used. There are two quite different reasons for this.
The first reason is that electronic calculators and computers, as they are almost all now designed, cannot do exact arithmetic except on a limited set of numbers. In particular, their rational arithmetic is approximate except for numbers whose denominator is a power of two. There is nothing necessary about this characteristic [14], which arose because the great limitations of early digital computers caused scientists to design an arithmetic based on semilogarithmic (wrongly calledjloating-point)
representation of numbers, an arithmetic now set in the concrete of an international standard always implemented directly in electronic circuitry.
The second reason is that, even if the arithmetic were exact for non-decimal fractions (sometimes called com
mon or vulgar fractions), there is now
10 THE MATHEMATICAL INTELLIGENCER
no way in which such fractions can be either keyed directly and exactly into a calculation, or shown exactly on a character display. Only decimal fractions can be keyed in directly and exactly, and only decimal fractions can be used to display usually approximate fractional results. While it is true that a number like one and two thirds has in the past been representable as 1% or as 1 �, the designers of most electronic calculators and computers have not provided for this kind of representation either to be displayed or to be keyed in. More than ten years ago I was an observer at a meeting of senior mathematics teachers which agreed, without protest from any of the teachers, that common fractions should be dropped from the official syllabus for elementary schools of one of the states of Australia simply because electronic calculators don't provide for them.
It has often been remarked that the teaching of common fractions is not well done in elementary schools [15]. From this remark it is a short step to question whether common fractions should be taught at all. The mistake here is to suppose that decimal and common fractions should be taught as distinct concepts. They should not. A fractional number is a fractional number, whether decimal or common. The fault is in the notation, which makes the numbers 2� and 2.75 so different in appearance. What is needed is a notational convention which makes it plain that a number like one and two
thirds is a value for which an integral part, a numerator and a denominator can be specified, and which as a spe
cial case allows certain (decimal) denominators to be left out.
The problem with representations like 1% or 1 � is that the numerator and denominator are distinguished from the integral part by typographical detail, and from each other by a symbol which implies that a calculation is to
be carried out. These representations are neither perceptually sufficient, notionally unambiguous, nor electromechanically convenient. However, if a symbol like o, distinctively pronounced say nom, were adopted as a prefix to the denominator part of a fractional number, to follow the nu-
merator part, then a very convenient and pedagogically salubrious notation is provided. The symbol I would not be suitable, as it is now too often used to stand for the division function. The number two and three quarters could be keyed into a calculator as 263°4 or 2675°100 or 2675, showing equivalences which should be easy for even
the elementary-school eye to see. Of course, a number like two thirds could be keyed in as 2°3 or 0 62°3 or 4 °6, but there is no equivalent decimal fraction. Numbers with decimal fractions are distinguished from numbers with common fractions when they are displayed-a number that can be exactly represented more briefly with a decimal fraction than with a common fraction will often be so represented. Otherwise there is no mysterious difference to confuse the young learner.
Accuracy
It is one kind of truthfulness to provide for exact numbers all to be represented exactly. But there are two quite different kinds of numbers-exact and approximate-and these two kinds should be easily distinguished in their representation but are not. An approximate value can be truthfully represented only if its representation shows plainly, not only that it is approximate, but also how approximate. In other words, the representation of an inexact value should show how accurate that value is.
If 1. 75 represents a measurement then . . . In the technological world, or in the everyday world for that matter, it is tacitly understood that it is somewhere in the range 1. 7 45 to 1. 755. The inaccuracy might spring from an unreliability in manufacture, from a limitation of a measuring tool, or from a perceived irrelevance for greater accuracy.
The representation of such measurements should show them to be measurements. Suppose a number shown with both a fraction point 6 and a scaling sign k or m but no denominator point o were treated as approximate beyond the last decimal place to
a tolerance of plus or minus half that decimal place. Then 1� would be treated as exactly one billion, while 1.o,OO� would be treated as exact only
to the last decimal place (in the range 995k2 and 1005k2), and would be a more accurate value than lL:;O� (in the range 950k2 to 1050k2). This notational convention would provide a plain and simple means for decimally inexact values of this kind to be truthfully represented.
But not all inaccuracies are of this kind. The arithmetic difference between two exact numbers 2. 75 and 1 . 75 is exactly 1. Between a measurement of 2. 75 and an exact 1. 75 it is somewhere in the range 0.95 to 1 .05, which can be shown as lL:;OkO. But between two measurements 2. 75 and 1. 75 the difference is somewhere in the range 0.9 to 1.1, which requires another notational rule to allow the value to be truthfully represented.
It seems unavoidable therefore to include in the notation a means of stating the tolerance even when that is not simply a power of 10. This returns us to the earlier theme of escaping the tyranny of base ten. More important, it allows experimental scientists the freedom to be precise in reporting the extent of the imprecision of their results, and a glance at the pages of Science will show that they value this freedom. Some symbol must support the stating of tolerances, only I would not favour the symbol ± for the purpose.
Only experience could show the level of arithmetic education at which these last notational conventions could be introduced. They would, however, be a valuable feature of any calculator and an enrichment for any talented students.
Conclusion
This article proposes, as steps necessary to reverse present trends towards popular innumeracy, that
• the adoption of SI metric basic and secondary units of measurement should be everywhere encouraged, being much better suited to popular use than the units traditionally used in the major English-speaking countries,
• the SI metric scaling system should be replaced by a simple system for representing scaled numbers, and
• traditional methods of representing
numbers are otherwise unsatisfactory and warrant being replaced.
A primary source of good advice about reform in popular usage for numbers, and measurements, and calculations should be the mathematicians, whose profession stands to gain most from wise reform, even if the choice and timing of those reforms are properly a matter for the public and its government to decide. Reforms of this kind would offer an opportunity to improve the aesthetics of mathematics generally, an aspect often considered fundamental for mathematicians [4, ch.5]. Mathematicians also have a natural responsibility for taking initiatives in promoting such reforms, and promptly introducing the teaching of them.
There is a very real danger that increasing and widening use of digital technology will prolong unthinking acceptance of a defective system for representing numbers. The essential beauty of numbers and calculation is being hidden from the vast majority of people through persistence with notational conventions whose only justification is their traditional use, and whose ugliness and unwieldiness are obscured by the familiarity engendered through imposition in elementary schools.
The opportunity is for a much better notational convention to be agreed internationally, for better electronic measurement and calculation to be enabled by that convention, and for the technology to support better the promotion of public numeracy.
REFERENCES
1 . Beck, A. (1 995) The decimal dysfunction
Math. lntelligencer 1 7( 1 ) , 5-7.
2. Jakuba, S. (1 993) Metric (Slj in Everyday
Science and Engineering, Society of
Automotive Engineers, Warrendale, PA.
3. Reingold, E. M. (1 995) A modest proposal,
Math. lntelligencer 1 7(3), 3 .
4 . King, J .P. (1 992) The Art of Mathematics,
Plenum Press, New York.
5. Paulos, J .A. (1 988) Innumeracy: Mathe
matical Illiteracy and its Consequences,
Penguin, London.
6. Menninger, K. (1 958) Zah/wort und Ziffer,
Vandenhoeck & Ruprecht, Gottingen, 2nd
edition (as Number Words and Number
Symbols by MIT Press in 1 969).
AUTHOR
NEVILLE HOLMES
School of Computing
University of Tasmania
Launceston, 7250 Australia
e-mail: [email protected]
Neville Holmes took a degree in elec
trical engineering from the University
of Melbourne, then spent two years
as a patent examiner before enlisting
in the computing industry. Since re
tiring from IBM after 30 years as a
systems engineer, he has spent 11
years lecturing at the University of
Tasmania.
7. Terry, G.S. (1 938) Duodecimal Arithmetic,
Longmans, Green and Co. , London.
8. Aitken, A.C. (1 962) The Case Against
Decimalisation, Oliver and Boyd, Edin
burgh. See also Math. lntelligencer 1 0(2),
76-77.
9. Kula, W. (1986) Measures and Men,
Princeton University Press, Princeton, NJ.
1 0. Feynman, R.P. (1 970) Letter, Scientific
American 223(5), 6 .
1 1 . Wilson, R. (1 993) Stamp Corner:
Metrication, Math. lntelligencer 1 5(3), 76.
1 2 . Hativa, N., Cohen, D. (1 995) Self learning
of negative number concepts by lower di
vision elementary students through solving
computer-provided numerical problems,
Educational Studies in Mathematics 28,
401 -431 .
1 3. Bennett, A. B. Jr. , Nelson, L. T. (1 979)
Mathematics for Elementary Teachers: A
Conceptual Approach, Wm. C. Brown,
Dubuque lA, 3rd ed. , 1 992.
1 4. Matula, D.W., Kornerup, P. (1 980)
Foundations of a finite precision rational
arithmetic, Computing, Suppl.2, 85-1 1 1 .
1 5. Groff, P. (1 994) The Mure of fractions, Int.
J. Math. Educ. Sci. Techno/. 25(4),
VOLUME 22, NUMBER 1 , 2000 11
HORST TIETZ
German History Experienced: My Studies, My Teachers 1
History as a science threatens History as memories.
During the War
My studies began with the war. For most students their studies were only an interruption of wartime service. However, I was not at risk of being called up: I was not "worthy to serve."
My life until my Abitur (high-school diploma) in Hamburg at Easter 1939 appeared to take a normal course; even during the following six months with the Reichsar
beitsdienst (Reich Labour Service) I was allowed to swim with the tide. Since, at the beginning of the war, school graduates with Abitur who wanted to study Medicine or Chemistry were granted leave for their studies, I decided on Chemistry, which did not interest me in the slightest, but which is related to Mathematics within the structure
-Alfred Heuss, 1952
of the sciences. I spent my first term in Berlin, because Hamburg was initially closed due to the expected air raids; in 1940 I was able to continue in Hamburg. When matriculating I noticed that my "blemish" had not been forgotten: there were Jews among my forefathers. Nevertheless, I was allowed to register because my father had fought in the front line during the First World War. At the university my special situation was not immediately obvious, as everyone was studying "on call," and it was assumed that the same also applied to me.
Slightly more than a dozen male and female students started studying Mathematics in Hamburg in January 1940. Our central figure was Erich Heeke (1887-1947; a student of Hilbert), one of the most fascinating personalities I have
'This article is based on a talk given at the University of Stuttgart, October 22, 1 998. The author and the Editor thank George Seligman for his advice in preparing the
present version. Much of the material appeared also in "Menschen, mein Studium, meine Lehrer" in Mitteilungen der DMV 4 (1 999), 43-52.
12 THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK
ever been privileged to meet. Many of the students and some professors wore uniforms, the air was charged with the tension of war, and civilians demonstrated their patriotic awareness by giving snappy salutes and by wearing the badges of all kinds of military and party organisations. In this martial atmosphere there was one person who-instead of raising his right arm in the Nazi salute, for which there were strict orders-entered the lecture room nodding his head silently in his friendly manner: Erich Heeke. His father charisma led to our small group organising itself as an official student group with the name "The Heeke Family"; I was only just able to prevent the others from nominating me-of all people-as the "Ftihrer" of this group.
Strangely, the veneration that Heeke enjoyed from the students was directly connected with a characteristic trait of his personality that should have baffled them-his undisguised rejection of the Nazi spirit, to which the whole of Germany was dedicated. Most people registered his behaviour by shaking their heads uncomprehendingly; but there were also a few people who looked up at him briefly with joy and amazement, and this united them for a second in conspiratorial opposition to the regime. Sometimes we students overtook the old man on the way to his lectures, and, passing him with a brisk "Heil Hitler, Herr Professor," my fellow-students raised their arms quickly in the Nazi salute. Heeke would turn round towards us with a surprised and thoughtful look, raise his hat, bow slightly, and say, "Good morning, ladies and gentlemen." Once when I accompanied him to the overhead railway I saw Heeke raise his hat respectfully to people wearing the yellow Star of David2: "For me the Star of David is a medal: the Ordrepour-le-Semite," he said quietly.
Heeke kept letters documenting the pernicious ideology of the Nazi era as curios; the two best ones hung in frames in his office. One was a complaint sent by a butcher trying to square the circle, and was addressed to the Reich Minister of Culture as a reaction to Heeke's cautioning; this letter concluded with the succinct sentence, "German scientists still do not seem to have realised that for the German spirit nothing is impossible!" The other letter was a reply by the Springer Publishing House to Heeke's query as to why the 2nd volume of Courant-Hilbert was allowed to be sold, but not the 1st. One could sense the silent cursing as they wrote, "The first volume was published in 1930, the second in 1937; in 1930 Courant was a German Jew, but in 1937 he was an American citizen." Heeke's apprehensive comment: "The fact that inhumanity is coupled with so much stupidity makes one feel almost optimistic in a dangerous way."
The most breathtaking scene occurred with one of the first air-raid warnings. The sirens suddenly started wailing during Heeke's lecture; those in uniform among the students wanted to make everybody go to the air-raid shelter, as was their duty. Heeke then said: "Do what you have to
Figure 1. Horst Tietz delivering his final lecture, 1 990.
do; I am staying here; perhaps one of them will land and take us with him .. . " Denunciation for Wehrkraftzerset
zung (undermining of military strength) could have cost him his life.
When some of my fellow-students found out how unstable the ground was on which I stood, there were heated political discussions, some human regret was expressed, but seldom any real understanding. What upset me most was the remark, "Well, in your situation you just have to think the way you do." Is it so impossible to distinguish between an attitude based on belief in justice and human dignity and one that is merely a reaction to injustice that one has experienced oneself?
Shortly before Christmas 1940 the ground was cut from beneath my feet: I was called to the university administration, where I was told that a secret ordinance of the Fiihrer instructed the university to exmatriculate people like me; the only chance of avoiding this was a petition to the "Office of the Ftihrer." Of course, I subjected myself to this procedure, which was as humiliating as it was hopeless; again, the rejection of the petition was given to me only orally: I was exmatriculated. I shall never forget the official from the administration who pressed both my hands, and with an extremely sad expression wished me "all the best, in spite of everything!" I felt completely numb, and outside I hardly noticed the shrill ringing of the two trams that almost knocked me down.
My despairing parents and I clung to the hope that Heeke might be able to give us some advice. In his private flat I had a conversation with him which I remember to this day as one of my most valuable experiences because of its openness and kindness. The concrete decision was that I should attend his lectures illegally; this also went without saying for the lecturer Hans Zassenhaus (1912-1992), as well as the theoretical physicist Wilhelm Lenz, with whom, however, Heeke wanted to speak himself, be-
2This badge, inscribed "Jude," had to be worn "clearly visible" on every Jew's clothing after 1 938.
VOLUME 22. NUMBER 1, 2000 13
cause Lenz was "not very brave." In addition, Heeke wanted
to contact van der Waerden in Leipzig; the latter was in
charge of a team of mathematicians whose work had been
recognised as being important for the war, and where some
endangered mathematicians had already found refuge and
been protected.
I confessed to Heeke that I didn't dare go to Zassenhaus
because he wore a Nazi badge. Heeke reassured me, "He
is someone we all trust; he only pretends to be a Nazi
and he does it well." And so it was; my first conversation
with Hans Zassenhaus was the start of a friendship for
which I have been grateful all my life.
One further short episode must be mentioned: when I
reported to Heeke that Lenz had made his surprised stu
dents stand up in his lecture for a "threefold Sieg Heil for
our Fiihrer and his glorious troops," he burst out laughing:
"Herr Lenz has been summoned to the Gestapo tomorrow!"
During this time before Christmas 1940 I became
friendly with the Chemistry student Hans Leipelt. He was
beheaded in Stadelheim in 1945 as a member of the "White
Rose," a group of Munich students who conspired against
Hitler.
Now I was studying illegally; van der Waerden did, in
fact, want to take me in Leipzig, but I was unable to seize
this helping hand, because if I had left, by an intricacy of
the Ntimberg Laws, my father would have been obliged to
wear the Star of David.
My time as an illegal student lasted for about a year and
a half. The lectures of Heeke and Zassenhaus partially re
peated what I had already heard; the beginners soon no
ticed that my knowledge was more advanced and asked
me to help them by organising a tutorial group. This was
not unproblematic, for opposite the Department there was
a Gestapo office from which I had to hide my illegal presence.
On the days when classes took place in the Department, I
had to be there in the morning before the Gestapo started
work, and I was only able to go out onto the street again
in the evening after the start of the blackout-it was part
of the air defences that no gleam of light was permitted to
Figure 2. Hans Zassenhaus (1912-1992), about 1980.
14 THE MATHEMATICAL INTELLIGENCER
be seen. In the Department I appeared to be out of danger:
although I did, in fact, sometimes see Herr Blaschke and
Herr Witt (the one an opportunist, the other politically
naive), they hardly took any notice of me; that suited me;
I always tried to avoid contact with a stranger, who might
unwittingly have risked getting in trouble because of me.
This period saw the start of a new friendship that I owe
to Heeke. Werner Scheid, a young lecturer in neurobiology,
wanted to improve his understanding of the physical back
ground to his science and its methods; he asked Heeke how
he could first acquire the mathematical prerequisites;
Heeke brought us into contact, and I shall never forget the
warm-hearted security that I was permitted to ef\ioy in
Scheid's home. I assume that Heeke was also behind the
invitation I got to teach at a very well-known private
evening school in Hamburg; although I was extremely
pleased to have received such an offer, I had to avoid the
exposure this would have caused. On the other hand, when
the representative body of Chemistry students asked me to
give an introductory course in mathematics for Chemistry
students, I agreed, despite many misgivings.
Klaus Junge, Germany's great chess hope, was also one
of the students attending Zassenhaus's lectures in 1941. It
hurt Zassenhaus when his request for a game of chess was
rejected: "My time is too precious for that!" Zassenhaus,
who was always ready to help his fellow human beings,
and who had no streak of prima donna behaviour in him,
blamed himself for this rejection: "My request was really
immodest; his time is defmitely too precious." It was pre
cious-in a different sense: a few weeks later Klaus Junge
was killed in action.
The phone rang one night during the Summer Semester
of 1942. I was relieved to hear the familiar voice of
Zassenhaus; however, the reason for his phone call was dis
quieting: my illegal behaviour was going to be denounced; he
hoped that he would be able to dissuade "these people" from
doing so if he could promise them that I would not allow my
self to be seen in the University any more.
Mter a day of agonised waiting he phoned again: he had
been able to avert the danger. He added, laughing, that
Heeke, when he heard that I was no longer able to come
to his Theory of Numbers, had stopped this course in the
middle of the semester and returned the lecture fee to the
students! Zassenhaus himself offered to help me with study
of the literature, which was all I could now do, and invited
me to his home for a working afternoon once a week. These
afternoons-we had, among others, worked through both
volumes of van der Waerden's Modern Algebra, and I have
saved to this day three copy-books full of exercises-were
rays of light in an everyday life that was becoming more
and more hopeless. They ended in July 1943, when the sec
ond devastating bombing raid on Hamburg left my parents
and me without a roof over our heads.
From Marburg, where we fled, I wrote to Heeke, and he
replied immediately that I should introduce myself to Kurt
Reidemeister, who had completed his doctorate under
Heeke in Hamburg, and would help me. The aesthete
Reidemeister had, incidentally, been transferred from
Konigsberg to Marburg for disciplinary reasons as early as 1933, because he had complained in his lectures about the vulgar behaviour of the SA (the armed and uniformed branch of the Nazi Party).
However, this helping hand also I was not able to grasp: shortly after my visit to Reidemeister the Gestapo acted, and on Christmas Eve 1943 my parents and I were arrested. Before we were transported to concentration camps, I had to hack coal out of ice-coated wagons at the freight depot. On the march back to the prison I once saw Reidemeister, and was seized by a desperate hope that he would recognize me despite my prison clothing and would be able to tell Heeke before I sank into the inferno-but he did not see me.
After the War
My parents did not survive the concentration camps; I was freed from Buchenwald by the Americans on April 12th, 1945. I first made my way with difficulty to Marburg, but then to Hamburg, because there the university already started to function again on November 6th, 1945.
Erich Heeke, although he was mortally ill, lectured on
mathematician there is nothing worse than not knowing what he ought to be thinking about," and "I wish I were two puppies and could play with each other!" Choice misadventures befell him; the best is the one concerning the key to his letter box: he had left it behind at Harald Bofu's house in Copenhagen, and when he arrived home he did not want to break open the full letter box; Bohr immediately fulfilled Maak's written request to send the key to him; he sent it-in a letter!
While Maak's humour expressed itself in a roguish grin, Zassenhaus liked to laugh; but humour without a problem was something he found difficult to accept. I told him one of the first post-war jokes: In Dusseldorf an old lady in a tram asked for the Adolf Hitler Square; when the tram conductor told her that it was now called Count Adolf Square, she said with genuine sincerity, "Oh, the good man deserved it." While Zassenhaus was still gasping for breath from laughter, he asked, "And what did the conductor say then?"
Zassenhaus, who thought there was no future for ivory tower mathematics in Germany, had prepared a memorandum for setting up a Research Institute for Practical
Linear Differential Equations. Nowadays we cannot comprehend the situation-how hungrily the emaciated figures with their clothes in tatters followed the fascinating lecture in an atmosphere charged
Heeke revived an image of
human ity that had become
deformed d u ring the Nazi
Mathematics. However, since he had little hope that his plan would be realised, he was already putting out feelers in America and Britain. I was able to be of use here, and the reason was as follows: At that
with tension. Heeke combined era. warmth with dignity, and thus revived an image of humanity that had become deformed during the Nazi era. One interruption is still unforgettable: it must have been in January 1946 when Heeke, who while speaking liked to look over the top of the lecture-room's boarded-up walls and windows into the street, suddenly stopped talking with joyful surprise in the middle of a sentence, put down the chalk, and with the words "I am being visited by a dear friend" hurried out into the street and embraced the aged Erhard Schmidt; after he had fled from Berlin he had been brought to Hamburg by his pupil Thomas von Rand ow (who has since become the celebrated "Zweistein" in ZEIT magazine).
In addition, Heeke introduced his studies of modular forms in a special lecture; his announcement on the notice board contained the words "Adults only," and I was very proud when he asked me to participate; I was thus the only student sitting together with assistants and lecturers in Heeke's own workshop.
I also have happy memories of other classes: Zassenhaus on Space Groups, Weissinger on Integral Equations, Noack on Kolmogoroffs Probability Calculus, and for students training to become teachers, Maak's lecture based on the book Numbers and Figures by Rademacher and Toeplitz. Maak listened with polite incredulity as I showed him how a number-theory proof could be simplified.
Maak was a person with an eccentric sense of humour: I have the following lovely statements from him: "For a
time democratic bodies were forming by spontaneous gen
eration, and in the expectation that a student body would get a response from the British military government more easily than professors suspected of being Nazis, a Central Committee of Hamburg Students (Zentral-Ausschuss or ZA) had already been founded before my arrival in August. It was the predecessor to the AStA (General Student Committee), and through the good offices of Herr Scheid I soon participated in its discussions. I got the ZA interested in Zassenhaus's project, and after a lecture by him I was requested to canvass this idea with the English officer responsible for the university. I think I did something towards making a success of the project: the Institute was established, but it was too late to prevent Zassenhaus from emigrating.
Zassenhaus was always in a state of high mental tension. On the way from the Dammtor Station to the Mathematics Department he mostly walked between the row of trees and the curb, swinging his briefcase and chewing the comer of a handkerchief; here he was not in danger of colliding with other pedestrians and of being distracted from his thoughts. In his lecture he revealed his effervescent temperament, as if he were trying to transfer his high tension to his listeners. The result was remarkable speed: he got through both the volumes of Schreier-Spemer in 1 1/2 terms, and he proposed that the remaining time should be devoted to Descriptive Geometry because "I cannot yet do that myself." We obtained the book by Ulrich
VOLUME 22, NUMBER 1 , 2000 15
Graf, fetched dusty drawing-boards from the Department
cellar, and started working with a will and with pleasure.
The last task was to draw the central projection of a cube
in general position. When Zassenhaus stopped beside me
on his walk of inspection between the rows of drawing stu
dents I asked, "Dr. Zassenhaus, is that general enough?"
and received from him the reassurance: "It's great-in fact
it's hardly recognisable."
He seemed always to be happily grappling with intuition.
On one occasion he was trying to make it intuitively clear
to us that the punctured sphere and the circular disk are
homeomorphic; when I interrupted his complicated argu
ments with the remark that one only needed to draw the
hole apart, he hesitated for a while and fmally replied: "But
then the hole must be large enough."
I believe that this wrestling with intuition provided a
constant impulse for his thoughts: the immense spectrum
of problems he tackled can perhaps be understood when
one considers this wrestling-both directly and indirectly
as a sharpening of the methods that he had created.
Hans Zassenhaus and I ran into each other at the post
office shortly after the war. He was the Director of the
Mathematics Department. I was able to return to him
Volume 63 ofthe MathematischeAnnalen, which I had bor
rowed from the Department library before the bombing be
cause of Erhard Schmidt's doctoral thesis. It had accom
panied me through the inferno. In order to "celebrate" this
event Zassenhaus gave me a large log from the trunk of a
tree as firewood and lent me a barrow to transport it; the
wheels were part of the valuable family bicycle, and to my
horror one wheel buckled under the weight of the wood.
During these last six months together I saw him much
more free from care than during the Nazi era. I learnt that,
together with some like-minded friends, he had hidden vic
tims of persecution and prevented them from being caught
by the Nazis. He reproached me for not coming to him in
time to ask him to help my family. This was no empty state
ment; he had done so much; I refer to the book by his sis
ter Hiltgunt Zassenhaus: Ein Baum bliiht im November (A
Tree Blossoms in November), Hoffman & Campe, 1974.
Since Heeke was not able to lecture any more (he died in
February 1947 at Harald Bohr's home in Copenhagen), and
since Zassenhaus wanted to emigrate, I went to Marburg
for the Summer Semester 1946, especially because life
there was easier than in the rubble wastelands of Hamburg.
Marburg had been left largely untouched by the destruc
tion of the war, and it thus had a strong attraction for the
streams of people returning, refugees and people displaced
by the war who were on the move throughout the entire
country. The student body that expectantly filled the lec
ture rooms was accordingly very mixed: the students from
Marburg, who still had the background of a family and had
just come from school, were in stark contrast to these peo
ple, who were visibly in a terrible state. In Mathematics,
however, they were all united by the enthusiasm of one
man, whose poverty was hardly exceeded by that of any
student: Herbert Grotzsch (1902-1993).
1 6 THE MATHEMATICAL INTELLIGENCER
Figure 3. Herbert Grotzsch (1902-1993), about 1960.
He had also been returning from the war: he had tried
to return to his university, Giessen, where he had taught
until he was thrown out in 1935 for refusing to participate
in a Nazi camp for lecturers. However, the university had
been closed by the US military government; it was there
fore sensible for him to try to resume his teaching in neigh
bouring Marburg. He was gladly accepted as a member of
the staff, which had been greatly reduced in numbers: only
one chair and the post of a senior lecturer were already
filled; an additional chair, a lectureship and the post of a
student assistant were vacant; while the post of a full as
sistantship was blocked because the incumbent, Friedrich
Bachmann, was to go to a chair in Kiel but needed first to
get his denazification in Marburg. The 44-year-old Grotzsch
had to make do with the student assistant's post. It was
only in 194 7 that he was nominally appointed associate pro
fessor-his duties and his salary did not change. Efforts to
correct this embarrassing situation were, in fact, made in
the Faculty; however, some considered Grotzsch's shabby
clothes to be "unsuitable." I have this from the then Rector,
the philosopher Julius Ebbinghaus.
Grotzsch never criticised this treatment, and he did not
even seem to notice it. His poverty certainly was not detri
mental to the effect of his personality, to his enthusiasm
in the lectures and his kindness in his contacts with his
students.
Grotzsch was widely known as a researcher. He had in
troduced his "Surface Strip Method" into the Geometric
Theory of Functions, regarding rigid conformal mappings as
special cases of more supple quasi-conformal mappings; then
the conformal ones can often be characterised by extremal
properties. This point of view is still fruitful today in the
search for characteristic properties of certain Riemann sur
faces and in the theory of "Teichmiiller spaces."
In the town, which in those days was full of "charac
ters," Grotzsch, the professor, quickly became a well
known personality. Efforts by students-who were a little
better off-to help him here and there were rejected kindly
but firmly; it was only possible to smuggle a pair of shoes
from a US parcel into a tombola for him: visibly distressed,
he went home shaking his head, but then liked wearing the
shoes instead of the clogs he had previously worn. On his
way to the Mathematics Department in the Landgrafenhaus
he used to walk through the old Weidenhausen quarter and
drink his cup of ersatz coffee at a bakery, eat his dry roll,
and read the daily newspaper. Once he fell asleep while
doing this and leaned on the hot stove; the sad result was
a large hole in his "good" jacket, which he had had sent to
him by his parents, and which he had only worn for a few
days, displacing an indefmable piece of clothing from the
war.
Grotzsch lived in a tiny attic in the Galgenweg; the path
was so steep that when it was icy he had to slide down in
his socks.
Once he stood still in the middle of the Rudolphsplatz,
which was busy with US vehicles, chewing at the end of his
short pencil and sunk deep in thought, until a friendly po
liceman took him by the arm and led him to the safe foot
while discussing or in his lectures his hands were always
moving, as if he wanted to explain his thoughts by means
of a virtual or real drawing.
In the lecture on Conformal Representation, during
which the lights suddenly failed, he appealed to the ability
of the students to think in abstractions and spoke on ln the
dark; nevertheless, after a few minutes had passed one
could hear the sound of the chalk on the blackboard.
Once I did see Grotzsch angry. In the Department library
some students were hunting insects. Extremely excited, he
closed the window with the words, "The poor creature does
not know what traps we are setting for it."
His office was in the attic of the Landgrafenhaus. A gut
ter ran along beneath his window, some soil had gathered
in it over the years, and a small beech tree was growing
there. It was visible from a distance. It was his joy, and he
watered it twice a day, for which purpose he had to carry
a tin can to the nearest water tap, two floors below. Once,
when he was away, I had the honourable task of watering
the small tree: "But be careful not to spill any on the
passers-by." When the roof was inspected the gutter was
cleaned, and the little tree disappeared. His comment: "In
Marburg they take care dass die Baume nicht in den
Himmel wachsen (that the trees don't grow into the sky)."
path. It was certainly not
only mathematics but also
his malnutrition that were
the cause of his "switching
off': of his meagre food ra
tion stamps he sent part to
his parents in Crirnmitschau
The ambience i n that period ,
so d ifficult to re-create and
u nderstand today.
In April 1948 Grotzsch
was offered the chair at the
University of Halle (then in the
Russian occupation zone),
and left behind him an as
tounded Faculty, but many
grateful students! From him and tried to obtain the missing vitamins with the aid of fish
paste and other stamp-free articles.
Without Grotzsch the teaching of mathematics would
have collapsed: he was tirelessly active and could be con
sulted at any time. In the loud, spirited discussion held in
the Saxony dialect he was thrilling to listen to; his eyes,
which were emphasised by the powerful lenses of his
glasses, flashed with high spirits and intellectual joy. His
stereotype "nota bene consultation!" was a motto with
which he told students to come to see him. Everything was
important! Mathematical errors were discussed until in
teresting fallacies appeared: paths towards solutions were
discussed in detail. If the arguments were too long-winded
during the practice classes he would call out, "Ladies and
gentlemen! You are all thinking much too much!" But if the
path a student had chosen was superior to his own he
would exclaim, "You have beaten me!"
An unforgettable sentence from his profound thinking:
"Ladies and gentlemen. The main problem of mathematics
is: The proof is given-the theorem is to be found." Also
his stirring explanation of the principle of Bolzano: "Think
of a fmite interval and then infmitely many points within
it! The mere consideration of this tells you that there must
be a terrible crush, there must be a point at which some
thing terrible happens! And look: a point of this kind is an
accumulation point." He always thought geometrically:
they had learnt not only the best mathematics, but he had
also shown them by his example how one can fmd hope
within oneself in times of need.
When he said goodbye he forbade anyone to send him
letters with a mathematical content: "The censors must
consider mathematics to be a secret language, and that is
mortally dangerous in a dictatorship," and here he was al
luding to the fate of Fritz Noether, who had been executed
in Russia as a spy, because he had received money owed
to him by someone in Germany.
Political arguments played a greater role in Marburg than
in Hamburg. There survival was all-important. However,
Marburg was essentially undamaged, and middle-class life
outwardly fairly intact.
Former officers were noticeable here because of their
distinctly brisk behaviour; of them the physicists said,
"When 'magnitudes of higher order' are mentioned they
click their heels." When one fellow-student was drunk he
had boasted that he had been an officer in the SS, and I ex
plained to him that I did not wish to have any further con
tact with him, and why; very much later, I learnt that he
in the meantime a school headmaster-had described me
as "his friend."
The political discussions among us students were often
violent. During my last visit to Halle, Grotzsch reminded
me of an argument of this kind, during which a chill had
VOLUME 22, NUMBER 1 , 2000 1 7
run down his spine: when a fellow-student had defended his enthusiasm for the Nazi state with the words "and, anyway, I wasn't sent to a concentration camp," I had merely countered, "Why not?!"
This was the time of the denazification courts. In them denazification was carried out in the style of courts exercising civil and criminal jurisdiction. As there were not enough politically irreproachable lawyers, I was offered the post of Public Prosecutor. Although in this time of need and with a future full of question marks this position with the rank of an Oberregierungsrat (senior civil servant) had a fairy-tale aspect for me, I did not consider the offer for one minute.
I have reported about Grotzsch in detail because his humaneness brings out so well the brittleness of the ambience in that period, which is so difficult to re-create and understand today. The two other men, the Full Professor Kurt Reidemeister and the Associate Professor Maximilian Krafft-who later supervised my doctoral thesis-were personalities of a different kind.
Following the lead of his friend, Rector Ebbinghaus, who wanted to denazify the university, Reidemeister (1893-1971) became more interested in politics than mathematics. When the philosopher Ebbinghaus and the
stable, and it was very embarrassing for me when I, as a Full Professor, visited him, the Associate Professor to whom I owed so much, on his 80th birthday.
Krafft's lecture style was eccentric: clearing his throat and growling contentedly, he turned his back on his listeners, began to write on the board with his left hand and continued with his right hand without his handwriting changing in the slightest; if you were sitting opposite him at the desk he would write upside down, and his mirror writing was even as fast as his normal writing. He was full of calculating tricks, which he often made up himself, and he enlivened his lectures and seminars to an extraordinary degree by his human and mathematical originality. At that time he was working tirelessly on a translation and revision of Tricomi's EUiptic Functions; this was the analytical counterpart to the older work that he had written together with Robert Konig.
During this time of hunger Krafft and Grotzsch gave
superhuman service!
Mter the currency reform in 1948 the vacant chair was given on a temporary basis to Hans-Heinrich Ostmann, who was an expert on the Additive Theory of Numbers. He taught in Marburg from 1948 until 1950 and then moved to the Free University in Berlin. At the end of the war Ostmann
had settled in Ober-Germanist Mitzka had a fist-fight in public on the street, it led to a slander suit in which Reidemeister appeared offering to testify; he was outraged when
The social task of mathematicians:
to make Mathematics palatable to
non- mathematicians.
wolfach and earned his living from the fees he charged as a consultant to people squaring the circle, trisecting the angle, and the like. He
the court decided not to swear him in. The Reidemeisters had a niece living with them. She was
to take her Abitur in Marburg. This young lady visited me one day with a problem about ellipses that her uncle was unable to solve; he had impatiently sent her to me: "Go and see Tietz, he's got a feeling for trivial things!"
Krafft (1889-1972) was an awkward person who was always "against" everything: he had not got on with the Nazis-it is said that in Bonn he did not become Hausdorffs successor because he did not want to do any political service on the weekend-and after the war he missed no opportunity in my presence to make critical comments about Jews. This odd nonconformism I found impressive rather than offensive. I would take his part if he was having difficulties with someone. It was only in the oral part of my doctoral examination in 1950 that I couldn't restrain myself any longer. In actuarial theory Krafft asked annoying questions; the last was, "How do insurance companies protect themselves against too unfavourable insurance policies?" My reply: "Through preselection by a doctor; however, that makes sense only for life insurance; the sick are not accepted in order to avoid having to pay too early." He was not satisfied: "It is also sensible in the case of pension insurance: the healthy are not accepted so that they do not have to be paid for too long a period." I exploded in front of the whole Faculty: "That may be an Aryan method, Herr Professor, I do not know it!" However, our relationship was
18 THE MATHEMATICAL INTELLIGENCER
continued doing this business in Marburg, and this made him a victim of "Gre-La-Ma"! This was a retired female grammar-school teacher who, in the newspapers, advertised coaching in the subjects Greek, Latin and Mathematicsabbreviated to Gre-La-Ma; this well-known character used to cycle through the town wearing a blind person's armband. She appeared at every possible lecture, and even once at the Landgrafenhaus, where in front of the surprised Ostmann she unrolled a 10-metre-long roll of paper in the hallway with a deft movement, and announced that this was "the prime number formula." Ostmann fled, without having collected his fee.
Wolfgang Rothstein (1910-1975) came to Marburg from Wiirzburg in 1950, so the lectureship was finally filled. My wife and I became friends with him and his family, and it means much to us that we were able to continue this friendship later in Miinster and then in Hannover.
Brief Episode in Physics
I have jumped ahead: these events took place shortly before I left Marburg for Braunschweig in 1951; but my State Examination in 194 7 with its consequences requires a few comments. In Germany one could no longer obtain a doctorate without passing such a final examination. (It is said that a candidate in the State Examination was once told, "Herr Doktor, you have failed.") Three subjects were required; I had chosen Mathematics, Physics, and Chemistry.
Figure 4. Erich HOckel (1 896-1979) with Horst Tietz, 1949.
The chemists not infrequently profited from my mathe
matics, and in a number of their papers I was thanked for
my "valuable advice." In chemistry, as they said, I led "a
meagre footnote existence," until I adopted their principle:
"One must not only lay eggs, one must also cluck!", and be
gan submitting papers of my own. But this did not enable
me to pass a chemistry lab test. That was a catastrophe,
capped by my attaching the Bunsen burner to the water
tap. It was thus like a message from heaven when I learnt
on the same day that Applied Mathematics had been des
ignated an examination subject. Although I knew nothing
about it, I put my name down for the examination!
Krafft examined the two Mathematics parts with Grotzsch
as the second member of the examining committee. The
theoretician Erich Hiickel examined Physics, with the
newly appointed Professor for Experimental Physics
Wilhelm Walcher as the other member.
Immediately after my State Examination, Hiickel (1896-
1980), who, as Head of the Section for Theoretical Physics,
held the post of Associate Professor, and until then had no
staff of his own, gave me the post of Auxiliary Assistant
that Walcher had obtained for him in a hard fight. Walcher's
enterprise benefitted not only the Physics Department.
Sometimes he would travel to Wiesbaden to negotiate for
money, and on his return his colleagues would be stand
ing on the station platform, hoping that he had also brought
something for them.
When Walcher was Dean he was once talking with Krafft
and Reich in front of the University building; I passed by
with Ostmann and in a loud voice parodied the title of a
novel by Graham Greene that was famous at that time: "Der
Reich, der Krafft und die Herrlichkeit."
The marvellous Mardi Gras parties in the Physics De
partment are unforgettable. At the first party in 1949 I found
Ruckel in a vine arbour; when he blissfully asked, "Tietz,
where are we here?" I was able to enlighten him: "In your
own office, Herr Professor!" However, this evening showed
me that my Physics colleagues did not take me very seri
ously: during the polonaise at midnight I switched on a
lamp that my wife had sewed into the rear seam of my
trousers; then Hans Marschall, the Assistant of Siegfried
Flugge, the nuclear physicist called out behind me: "Tietz
has confused optics with acoustics." People also talke(t of
the "Tietz Effect": When I entered the Physics Department
downstairs the fuses blew upstairs.
Nowadays the name Erich Hiickel is known to every
body in chemistry; that was not the case in those days, al
though the roots of his HMO Theory (for Hiickel Molecular
Orbits) already stretched back 20 years; this theory per
mits the calculation of the binding energies of organic com
pounds by methods from quantum theory. As a chemist his
elder brother Walter was better known in Germany. It was
in the summer of 194 7 that I was sitting in Hiickel's office
and heard searching footsteps, and the knocking on and
rattling of locked doors in the hallway of the Department
that was enjoying its after-lunch siesta; fmally, there was
also a knock on my door. An American officer entered, in
troduced himself as a physicist, and asked about the physi
cists in Marburg. The names I gave him elicited "I don't
know him" over and over, until I mentioned Hiickel's name,
which brought a radiant, "Ah! The famous Hiickel!" When
I told Hiickel about this visit, he dismissed it with the
words: "He means Walter," and could not be persuaded dif
ferently, though I stressed that the American had asked
about physicists.
Hiickel put a lot of work into his lectures, but they did
not fascinate people: nervousness led him into mistakes in
calculations and slips of the tongue. Nevertheless the lec
tures were popular: watching his difficulties made our own
seem bearable. In those days the human involvement of a
lecturer was still the surest medium in the teaching and
learning process, before education policy transferred the
task of understanding from the person learning to the per
son teaching.
Hiickel experienced phases of scientific productivity in
a state of exhilaration: he was unable to sleep for days and
kept awake by drinking huge amounts of coffee; afterwards
he often sank into a state of depressive exhaustion with
serious attacks of migraine.
His wife Annemarie, the daughter of the Nobel prize
winner Richard Zsigmondy, was the exact opposite to him:
she was bursting with the pressure of her talents, and her
violin-playing, in particular, often stretched her husband's
nerves to the breaking-point; then he would sit at his desk
with earplugs, which made conversation with him rather
difficult for me sitting next to him. These hours of work
ing together at the huge Napoleonic desk with the view of
Marburg Castle are among the most valuable memories in
my life! A close intimacy developed from this, and in his
autobiography he writes, "Tietz became my most faithful
helper and best friend." At the celebration on the occasion
of Hiickel's lOOth birthday, American researchers stressed
that Linus Pauling's Nobel Prize for Chemistry should
really have been awarded to Hiickel.
Looking back on my period with physics I can say that
VOLUME 22. NUMBER 1 . 2000 19
it made me aware of the Social task of mathematicians:
to make Mathematics palatable to non-mathematicians.
The More Recent Past
In 1993 my friend and colleague Heinrich Wefelscheid (b. 1941) and I ef\ioyed the warm hospitality of Frau Lieselotte Zassenhaus in Columbus, Ohio. We had been commis
sioned by the German Research Society to sift through the
extensive unpublished scientific work of her husband and
prepare it for transportation to the Mathe-matics archives of the University Library in Gi:\ttingen. This last meeting
with the great intellect was moving. In an undated speech of thanks we found the statement that he did not mind writing the thesis for a student, but that he hated then having to explain it to him as well! During his last few months his
illness gradually weakened him; nevertheless he still
worked intensively almost until the very end. Almost: during the last few weeks he was only still able to read, detective novels and the Bible.
Allow me to mention two more mathematicians who be
long here: both came from the Hamburg background and had obtained their doctorates with Heeke: Heinrich Behnke
and Hans Petersson, whom I got to known in 1956 when I
was given a lectureship in Mtinster. They were Directors of the two Mathematics Departments, and despite (or per
haps because of?) their spatial closeness-the Directors' rooms were opposite each other in a narrow corridor-one
could not call the atmosphere friendly. The difference in their physical size was enough to cause tension.
Behnke (1898-1979) was a huge person with a Renaissancelike manner. The marvellous scene at the celebrations for the golden jubilee of his Habilitation, which were held in
Hamburg in 1974, is memorable. When the Senator and the President had finished their speeches of congratulation, the man being celebrated heaved himself up to the lectern, although this was not on the programme, with the words:
"Herr Senator, Herr Prasident! When I think back to my youth I have to say: your predecessors, gentlemen, those
were real men! They drove with coach and four . . . !" The
remainder of what he said was lost in the general cheer
ing. Hans Petersson (1902-1984), a wiry, almost delicate
man, continued Heeke's modular research most intensively, and in 1958 he revised and published Heeke's works
together with the unforgettable Bruno Schoeneberg. I should like to return to Reidemeister once more. He
has always fascinated me; it was all the more painful to
recognise the tragedy that he apparently only seldom succeeded in conveying his intellectual riches to other people.
How much he suffered under this became clear during his visit to Behnke, his friend from student days, in Miinster
20 THE MATHEMATICAL INTELLIGENCER
A U THOR
HORST nETZ Roddinger Strasse 31
30823 Garbsen
Germany
e-mail: [email protected]
Horst Tietz was born in 1 921 in Hamburg, to a family of promi
nent wood merchants. The Nazis expropriated their business
and ended by killing most of them. The reader may be amazed,
as the Editor is, that Horst Tietz, after tribulations and losses
which if anything are understated in this memoir, was able to
spend the rest of a long and creative life as a mathematician
in Germany. The academic community in which he had ap
prenticed as an outcast now honors him; today he is a re
spected Emeritus Professor of the University of Hannover. He
is left with a wry streak of gallows humor. perhaps, but
uncowed.
around 1960. The stark difference between two opposite temperaments with the same interests became almost
painful. They were speaking about the training of teachers, which Behnke did with great success, while Reidemeister
did not get beyond reflecting on the problem. Coming from Reidemeister even friendly words sounded ironic, Behnke
felt he was being attacked, replied more and more agitat
edly, and fmally left the room; when I accompanied Reidemeister to his hotel he said with great agitation, "Herr
Behnke thinks that I am criticising him; but I actually admire him! How can one make oneself understood?"
Conclusion
I wanted to describe my meetings with personalities who
have influenced my life and show how different from to
day our world half a century ago actually was. I also wanted
with gratitude to bring to life the memory of people who
were not only important scientists but also-Menschen.
M athernatica l l y Bent
The proof is in the pudding.
With this channing tale we inaugurate
a new column. Many of our readers
know Colin Adams through his career
in research and teaching, and may
have enjoyed his article in The
Mathematical Intelligencer 17 (1995), no. 2, 41-51. Alongside this professional
activity, he has been appearing in skits
and parodies, sometimes in the persona
of Mel Slugbate; you may have seen
him, for instance, at meetings of the
Mathematical Association of America
and the American Mathematical
Society. Having enjoyed these, you
may well sha1·e my pleasure at the
prospect of a column under his
direction. Only I advise you not
to think you know what to expect.
-Chandler Davis
Column editor's address: Colin Adams,
Department of Mathematics Williams College,
Williamstown, MA 01 267 USA
e-mail: [email protected]
Col i n Ad am s , E d itor
Into Thin Air
I was up above the Lickorish Ridge, having traversed the difficult Casson
Gordan Step, and was resting on a small Lenuna on the North Face of the Poincare Co[\jecture. As far as I knew, no one had been up this high before, and I felt I had a good chance of finding a route all the way to the top. I was still breathing hard and the adrenaline was pumping through me. Those last fifty feet had been treacherous. A few times, my logic had slipped, and I had barely managed to grab a handhold and then scramble onto solid footing. But now that I was up here, the view was incredible. The sky was an unnatural blue.
As I sucked air, I looked out into the distance. The Mathematical Range stretched beneath me. Poking through the clouds were some of the peaks upon which I had first tested my mettle. Point Set Topology looked so tiny in comparison to where I sat now, but at the time it had been a struggle. And there was Teichmiiller Theory. I would have never made it up that slag heap if it weren't for McLuten. I was so naive then. So many mistakes. McLuten must have saved my rear a dozen times. If it weren't for him, I would be lying at the bottom of some crevasse, crumpled up on some counterexample to a laughable conjecture.
McLuten had seemed invincible then. He'd climbed all kinds, the big ugly granite slabs that rose up out of the undulating planes of geometry, the treacherous ice-covered theorems that kept us all in awe of algebra, and the crumbly rocks of the Analysis Range, where one false step could bling a mountain down upon you. And McLuten had the look, too; the glizzled visage that resembled the crags and rocks he confronted daily, his eyes always focused on the next challenge.
I missed him. But he wasn't the kind that could ever be satisfied with all that he had accomplished. Had to go after the big one, the one they call Fermat.
They found him at the bottom of the Euler Face. Everyone had said that there was no way up Euler, but McLuten couldn't be dissuaded. He left three ABO's behind, with no means of support.
It wasn't but ten years later that Wiles made the summit. But Wiles prepared. For seven years, he prepared. He knew the Euler Face was insanity. He came up Taniyama-Shimura, a route that had been championed by Ribet. And he did it alone.
It made Wiles an instant celeblity. He had tackled the big one. He had proved no mountain was invincible. But that wasn't why he had done it. No, that's not why any of us did it.
And here I was, three quarters of the way up Poincare. One of the largest unconquered peaks in the world. One of the few remaining giants of mathematics. Who would have thought that I would have a shot?
The wind was picking up a bit and wispy clouds scudded by.
Suddenly a head bobbed up at the edge of the lenuna. I jumped back. It was Politnikova. She pulled herself up over the edge, and lay there, trying to catch her breath.
"What the hell are you doing here?'' I exclaimed. Politnikova waved me off as she gasped for breath. Not a lot of oxygen up here.
"Did you follow me up Geometrization Co[\jecture Ridge? Nobody knew I was even consideling it."
Politnikova pulled off her goggles and sat up, still gasping.
"Relax, relax," she said in her thick Russian accent. "I did not come up the Geometrization Co[\jecture Ridge. I followed Poenaru's Route up the Clasp Trail and then over the Haken Ice Field."
"But everyone's tlied that route. That's where Fourke disappeared."
© 1999 SPRINGER· VERLAG NEW YORK, VOLUME 22, NUMBER 1 , 2000 21
"Yes, but Fourke was using out-of
date equipment, technology from the
50's. I am using the latest technology. Makes a difference."
"I can't believe this. I get up this
high on Poincare only to find you."
"And what is so wrong with me, huh?" "You know perfectly well what I
mean. I was going to do it on my own." "Oh, yes, sure," said Politnikova
smiling. "You would have no trouble
single-handedly climbing those logical
outcroppings up there." She pointed almost straight up.
"Well, I hadn't figured it all out yet."
"Yes, but two could work together
to get around those problems. A little combinatorics, I'm good at combina
torics. A little geometry. You are good at geometry. And bingo, we are there."
"Well, I suppose you have a point,"
I said reluctantly. "Maybe we could work together."
Politnikova began to smile, but the
smile froze as she jerked her head up. "Do you hear that?" she said, terror in
her face. I pulled my hood away from
my ear, and cocked my head to the side. In the distance, I could hear it, a slight rumble, but it was growing fast.
"Oh, no," I said, "avalanche!" When I had been down at base
camp, I had seen how precariously balanced the various arguments were that made up this face of the Poincare
Conjecture. A little bit of a shift here
or there, and the whole mountainside
could come down in your face. And that was the reality we were con-
fronting. In that split second, we both
knew that our dream of conquering
Poincare that day was gone. But all that was suddenly irrelevant. Now it
was a question of survival.
"We don't have a chance in hell if
we stay here on this lemma. There isn't going to be a lemma in another two
minutes," I screamed. "Throw your
rope over the side, and if we can make
it down to Bing's Theorem, we can hide behind that." I flipped Politnikova's
rope onto a piton I had already ham
mered into the rocks, clipped her on the line and shoved her over the edge,
before she could stop me. Then I clipped on and jumped out into space.
We zinged down the rope, burning
glove leather, until we hit the end of
the rope. Up above you could hear the
roar. When we hit the bottom of the
rope, we just unclipped and started
rolling down the slope. All those hardearned steps for naught, I thought as I
careened downward. I rolled to a halt
20 feet from Bing's Theorem, battered but in one piece. I glanced up at where
the lemma had been only to see it disappear entirely in the torrent of argu
ments that were cascading down upon
us. Politnikova grabbed my hood and pulled me toward Bing's Theorem. We managed to duck behind it just as the
avalanche reached us. Huddled there,
we saw several years' worth of mathematics slam past. It only lasted another minute, and then it was all gone. We
both sat in stunned silence and then
Politnikova turned to me.
MOVING? We need your new address so that you
do not miss any issues of
"We were lucky to be alive. Thank
god for Bing's Theorem."
"Yup," I said. I knew Bing's Theorem
would hold, if anything would.
I looked up to where we had been
perched moments before, and the face was smooth as ice. No lemma, no
corollary, not a handhold to be had.
"We will not be getting up there that
way," said Politnikova.
"Nope," I agreed. "This face is offi
cially a dead end, starting today." I stood wearily, feeling the bruises
and scrapes. "We should head down,"
I said, "before any other arguments col
lapse." Politnikova stood slowly. "Don't
look so sad. We were higher up there
than anyone else has ever been."
"Yeah?" I said. "No one will believe
it anyway. There isn't a trace of where
we were." "Yes, but what matters is what we
know, not what others think Hey, you
come down to my tent, and I give you some very good vodka."
I laughed. In a place where every
ounce counts for survival, only Politni
kova would bring vodka. "Sure," I said. I took one last look
toward the peak, enshrouded in clouds now, not even visible anymore.
"We have vodka, and we talk," she said, "and maybe we figure out some
other route to the top. Maybe we use hy
perbolic 3-manifold theory. Thurston
knows what he is doing. We do that, too." "Sure," I shrugged, "Why not?" We
started down the mountain.
THE MATHEMATICAL INTELLIGENCE&.
22 THE MATHEMATICAL INTELLIGENCER
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SERGEY FOMIN AND ANDREI ZELEVINSKY
Tota Positivity : Tests and Parametrizations
Introduction
A matrix is totally positive (resp. totally non-negative) if
all its minors are positive (resp. non-negative) real num
bers. The first systematic study of these classes of matri
ces was undertaken in the 1930s by F. R. Gantmacher and
M. G. Krein [20-22], who established their remarkable spec
tral properties (in particular, an n X n totally positive ma
trix x has n distinct positive eigenvalues). Earlier, I. J.
Schoenberg [41 ] had discovered the connection between
total non-negativity and the following variation-dimin
ishing property: the number of sign changes in a vector
does not increase upon multiplying by x.
Total positivity found numerous applications and was
studied from many different angles. An incomplete list in
cludes oscillations in mechanical systems (the original mo
tivation in [22]), stochastic processes and approximation
theory [25, 28], P6lya frequency sequences [28, 40], repre
sentation theory of the infinite symmetric group and the
Edrei-Thoma theorem [ 13, 44], planar resistor networks
[ 1 1 ] , unimodality and log-concavity [42], and theory of im
manants [43]. Further references can be found in S. Karlin's
book [28] and in the surveys [2, 5, 38].
In this article, we focus on the following two problems:
1. parametrizing all totally non-negative matrices
2. testing a matrix for total positivity
Our interest in these problems stemmed from a surpris
ing representation-theoretic connection between total
positivity and canonical bases for quantum groups, dis
covered by G. Lusztig [33] ( cf. also the surveys in [31 ,
34]). Among other things, he extended the subject by
defining totally positive and totally non-negative elements
for any reductive group. Further development of these
ideas in [3, 4, 15, 17] aims at generalizing the whole body
of classical determinantal calculus to any semisimple
group.
As often happens, putting things in a more general per
spective shed new light on this classical subject. In the next
two sections, we provide self-contained proofs (many of
them new) of the fundamental results on problems 1 and
2, due to A. Whitney [46], C. Loewner [32], C. Cryer [9, 10],
and M. Gasca and J. M. Pefta [23]. The rest of the article
presents more recent results obtained in [ 15]: a family of
efficient total positivity criteria and explicit formulas for
expanding a generic matrix into a product of elementary
Jacobi matrices. These results and their proofs can be gen
eralized to arbitrary semisimple groups [4, 15], but we do
not discuss this here.
Our approach to the subject relies on two combinator
ial constructions. The first one is well known: it associates
a totally non-negative matrix to a planar directed graph
with positively weighted edges (in fact, every totally non
negative matrix can be obtained in this way [6]). Our sec
ond combinatorial tool was introduced in [ 15]; it is a par
ticular class of colored pseudoline arrangements that we
call the double wiring diagrams.
© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 1 , 2000 23
3 3
2 2
1 1 Figure 1 . A planar network.
Planar Networks
To the uninitiated, it might be unclear that totally positive
matrices of arbitrary order exist at all. As a warm-up, we
invite the reader to check that every matrix given by [ d dh dhi l bd bdh + e bdhi + eg + ei ,
abd abdh + ae + ce abdhi + (a + c)e(g + i) + f
(1)
where the numbers a, b, c, d, e,j, g, h, are i are positive, is
totally positive. It will follow from the results later that
every 3 X 3 totally positive matrix has this form.
We will now describe a general procedure that produces
totally non-negative matrices. In what follows, a planar
network (f, w) is an acyclic directed planar graph r whose
edges e are assigned scalar weights w( e). In all of our ex
amples (cf. Figures 1 , 2, 5), we assume the edges of f di
rected left to right. Also, each of our networks will have n
sources and n sinks, located at the left (resp. right) edge
of the picture, and numbered bottom to top.
The weight of a directed path in f is defmed as the prod
uct of the weights of its edges. The weight matrix x(f, w)
is an n X n matrix whose (i, J)-entry is the sum of weights
of all paths from the source i to the sinkj; for example, the
weight matrix of the network in Figure 1 is given by (1).
The minors of the weight matrix of a planar network
have an important combinatorial interpretation, which can
be traced to B. Lindstrom [30] and further to S. Karlin and
G. McGregor [29] (implicit), and whose many applications
were given by I. Gessel and G. X. Viennot [26, 27] .
In what follows, 111,J(x) denotes the minor of a matrix
x with the row set I and the column set J.
The weight of a collection of directed paths in f is de
fmed to be the product of their weights.
LEMMA 1 (Lindstrom's Lemma). A minor 111,J of the
weight matrix of a planar network is equal to the sum of
weights of all collections of vertex-disjoint paths that con
nect the sources labeled by I with the sinks labeled by J.
To illustrate, consider the matrix x in (1). We have, for
example, /123,23(x) = bcdegh + bdfh + fe, which also
equals the sum of the weights of the three vertex-disjoint
path collections in Figure 1 that connect sources 2 and 3
to sinks 2 and 3.
Proof It suffices to prove the lemma for the determinant
of the whole weight matrix x = x(f, w) (i.e., for the case
I = J = [ 1, n]). Expanding the determinant, we obtain
24 THE MATHEMATICAL INTELLIGENCER
det(x) = I I sgn(w) w(1T), w 7T
(2)
the sum being over all permutations w in the symmetric
group Sn and over all collections of paths '7T = ( '7T1, . . . , 1Tn)
such that '7Ti joins the source i with the sink w( i). Any col
lection '7T of vertex-disjoint paths is associated with the
identity permutation; hence, w( 1T) appears in (2) with the
positive sign. We need to show that all other terms in (2)
cancel out. Deforming f a bit if necessary, we may assume
that no two vertices lie on the same vertical line. This
makes the following involution on the non-vertex-disjoint
collections of paths well defmed: take the leftmost com
mon vertex of two or more paths in '7T, take two smallest
indices i and j such that '7Ti and '7TJ contain v, and switch
the parts of '7Ti and '7TJ lying to the left of v. This involution
preserves the weight of '7T while changing the sign of the
associated permutation w; the corresponding pairing of
terms in (2) provides the desired cancellation. D COROLLARY 2. If a planar network has non-negative
real weights, then its weight matrix is totally non-nega
tive.
As an aside, note that the weight matrix of the network
3--___::'-c--��"" 3 2 2
(with unit edge weights) is the "Pascal triangle"
1 0 0 0 0
1 1 0 0 0
1 2 1 0 0
1 3 3 1 0
1 4 6 4 1
which is totally non-negative by Corollary 2. Similar argu
ments can be used to show total non-negativity of various
other combinatorial matrices, such as the matrices of q-bino
mial coefficients, Stirling numbers of both kinds, and so forth.
We call a planar network f totally connected if for any
two subsets I, J C [1 , n] of the same cardinality, there ex
ists a collection of vertex-disjoint paths in f connecting the
sources labeled by I with the sinks labeled by J.
COROLLARY 3. If a totally connected planar network has positive weights, then its weight matrix is totally positive.
For any n, let f o denote the network shown in Figure 2.
Direct inspection shows that f o is totally connected.
COROLLARY 4. For any choice of positive weights w( e),
the weight matrix x(f 0, w) is totally positive.
It turns out that this construction produces all totally
positive matrices; this result is essentially equivalent to
Figure 2. Planar network r o·
A. Whitney's Reduction Theorem [46] and can be sharpened as follows. Call an edge of r 0 essential if it either is slanted or is one of the n horizontal edges in the middle of the network. Note that f0 has exactly n2 essential edges. A weighting w of r 0 is essential if w( e) =I= 0 for any essential edge e and w( e) = 1 for all other edges.
THEOREM 5. The map w � x(f0, w) restricts to a bijec
tion between the set of all essential positive weightings
of r 0 and the set of all totally positive n X n matrices.
The proof of this theorem will use the following notions. A minor t:.1,J is called solid if both I and J consist of several consecutive indices; if, furthermore, I U J contains 1, then t:.1,J is called initial (see Fig. 3). Each matrix entry is the lower-right corner of exactly one initial minor; thus, the total number of such minors is n2.
LEMMA 6. The n2 weights of essential edges in an es
sential weighting w of r 0 are related to the n2 initial mi
nors of the weight matrix x = x(f 0, w) by an invertible
monomial transformation. Thus, an essential weighting
w of r 0 is uniquely recovered from x.
Proof The network r 0 has the following easily verified property: For any set I of k consecutive indices in [1, n], there is a unique collection of k vertex-disjoint paths connecting the sources labeled by [1 , k] (resp. by I) with the sinks labeled by I (resp. by [1, k]). These paths are shown by dotted lines in Figure 2, for k = 2 and I = [3, 4]. By Lindstrom's lemma, every initial minor t:. of x(f0, w) is equal to the product of the weights of essential edges covered by this family of paths. Note that among these edges, there is always a unique uppermost essential edge e(t:.) (indicated by the arrow in Figure 2). Furthermore, the map t:. � e(t:.) is a bijection between initial minors and essential edges. It follows that the weight of each essential edge e = e( t:.) is equal to t:. times a Laurent monomial in some initial minors t:.' , whose associated edges e(�') are located below e. D
Figure 3. Initial minors.
To illustrate Lemma 6, consider the special case n = 3.
The network r 0 is shown in Figure 1; its essential edges have the weights a, b, . . . , i. The weight matrix x(f 0, w) is given in (1 ). Its initial minors are given by the monomials
!:.1,1 = d., !:.2,1 = fld, !:.3,1 = a.bd,
!:.1,2 = dll, !:.12,12 = dfl_, !:.23, 12 = b[Lde,
!:.1,3 = dhi., !:.12,23 = degh, !:.123, 123 = dtif,
where for each minor t:., the "leading entry" w(e(t:.)) is underlined.
To complete the proof of Theorem 5, it remains to show that every totally positive matrix x has the form x(f 0, w) for some essential positive weighting w. By Lemma 6, such an w can be chosen so that x and x(f 0, w) will have the same initial minors. Thus, our claim will follow from Lemma 7.
LEMMA 7. A square matrix x is uniquely determined by
its initial minors, provided all these minors are nonzero.
Proof Let us show that each matrix entry Xij of x is uniquely determined by the initial minors. If i = 1 or j = 1, there is nothing to prove, since Xij is itself an initial minor. Assume that min( i, J} > 1. Let t:. be the initial minor whose last row is i and last column isj, and let t:.' be the initial minor obtained from t:. by deleting this row and this column. Then, t:. = t:.'xij + P, where P is a polynomial in the matrix entries xi'j' with (i' , j') =I= (i, J) and i' :s; i and j' :S j. Using induction on i + j, we can assume that each xi, r that occurs in P is uniquely determined by the initial minors, so the same is true for Xij = (t:. - P)lt:.'. This completes the proofs of Lemma 7 and Theorem 5. D
Theorem 5 describes a parametrization of totally positive matrices by n2-tuples of positive reals, providing a partial answer (one of the many possible, as we will see) to the first problem stated in the Introduction. The second problem-that of testing total positivity of a matrix-can also be solved using this theorem, as we will now explain.
An n X n matrix has altogether CZ:) - 1 minors. This makes it impractical to test positivity of every single minor. It is desirable to find efficient criteria for total positivity that would only check a small fraction of all minors.
EXAMPLE 8. A 2 X 2 matrix
x = [: :] has en - 1 = 5 minors: four matrix entries and the determinant t:. = ad - be. To test that x is totally positive, it is enough to check the positivity of a, b, c, and !:.; then, d =
(t:. + bc)/a > 0.
The following theorem generalizes this example to matrices of arbitrary size; it is a direct corollary of Theorem 5 and Lemmas 6 and 7.
THEOREM 9. A square matrix is totally positive if and
only if all its initial minors (see Fig. 3) are positive.
This criterion involves n2 minors, and it can be shown that this number cannot be lessened. Theorem 9 was proved by M. Gasca and Pefta [23, Theorem 4.1 ] (for rec-
VOLUME 22. NUMBER 1 , 2000 25
tangular matrices); it also follows from Cryer's results in [9] . Theorem 9 is an enhancement of the 1912 criterion by M. Fekete [ 14], who proved that the positivity of all solid minors of a matrix implies its total positivity.
Theorems of Whitney and Loewner
In this article, we shall only consider invertible totally nonnegative n X n matrices. Although these matrices have real entries, it is convenient to view them as elements of the general linear group G = GLn(C). We denote by G?.o (resp. G>o) the set of all totally non-negative (resp. totally positive) matrices in G. The structural theory of these matrices begins with the following basic observation, which is an immediate corollary of the Binet-Cauchy formula.
PROPOSITION 10. Both G?_o and G>o are closed under matrix multiplication. Furthermore, if x E G?.o and y E G>o, then both xy and yx belong to G>o.
Combining this proposition with the foregoing results, we will prove the following theorem of Whitney [46].
THEOREM 1 1 . (Whitney's theorem). Every invertible totally non-negative matrix is the limit of a sequence of totally positive matrices.
Thus, G,0 is the closure of G>o in G. (The condition of invertibility in Theorem 1 1 can, in fact, be lifted.)
Proof First, let us show that the identity matrix I lies in the closure of G>o· By Corollary 4, it suffices to show that I = limN->oo x(f 0, WN) for some sequence of positive weightings WN of the network r 0· Note that the map (J) � x(f o, (J)) is continuous and choose any sequence of positive weightings that converges to the weighting wa defmed by w0( e) =
1 (resp. 0) for all horizontal (resp. slanted) edges e. Clearly, x(f 0, w0) = I, as desired.
To complete the proof, write any matrix x E G?.o as x =
limN->oo x · x(f 0, wN), and note that all matrices x · x(f 0, wN) are totally positive by Proposition 10. 0
The following description of the multiplicative monoid G?.o was first given by Loewner [32] under the name "Whitney's Theorem"; it can indeed be deduced from [46] .
THEOREM 12 (Loewner-Whitney theorem). Any invertible totally non-negative matrix is a product of elementary Jacobi matrices with non-negative matrix entries.
Here, an "elementary Jacobi matrix" is a matrix x E G that differs from I in a single entry located either on the main diagonal or immediately above or below it.
Proof We start with an inventory of elementary Jacobi matrices. Let EiJ denote the n X n matrix whose ( i,J}entry is 1 and all other entries are 0. For t E C and i = 1, . . . , n - 1, let
Xi(t) = I + tEi,i+ l =
26 THE MATHEMATICAL INTELLIGENCER
1
0
0
0
0 0
1
0 1
0 0
0
0
0
1
and
Xi(t) = I + tEi+ l,i = (xi(t))T
(the transpose of xi(t)). Also, for i = 1, . . . , n and t =I= 0, let
X(1)(t) = I + (t - 1)Ei,i•
the diagonal matrix with the ith diagonal entry equal to t and all other diagonal entries equal to 1. Thus, elementary Jacobi matrices are precisely the matrices of the form x i(t), Xi(t), and XC1J(t). An easy check shows that they are totally non-negative for any t > 0.
For any word i = (i1. . . . , it) in the alphabet
.stl = { 1 , . . . , n - 1, (!), . . . , @, 1, . . . , n - 1 }, (3)
we defme the product map Xi : (C\(OJY � G by
(4)
(Actually, xiCt1. . . . , tt) is well defmed as long as the righthand side of ( 4) does not involve any factors of the form XC1J(O).) To illustrate, the word i = CD 1 <ID 1 gives rise to
We will interpret each matrix Xi(t1, . . . , tt) as the weight matrix of a planar network. First, note that any elementary Jacobi matrix is the weight matrix of a "chip" of one of the three kinds shown in Figure 4. In each "chip," all edges but one have weight 1 ; the distinguished edge has weight t. Slanted edges connect horizontal levels i and i + 1, counting from the bottom; in all examples in Figure 4, i = 2.
The weighted planar network (f(i), w(t1 , . . . , tt)) is then constructed by concatenating the "chips" corresponding to consecutive factors xik (tk), as shown in Figure 5. It is easy to see that concatenation of planar networks corresponds to multiplying their weight matrices. We conclude that the product xi(t1 , . . . , tt) of elementary Jacobi matrices equals the weight matrix x(f(i), w(t1 , . . . , tt)).
In particular, the network (f 0, w) appearing in Figure 2 and Theorem 5 (more precisely, its equivalent deformation) corresponds to some special word imax of length n2; instead of defining imax formally, we just write it for n = 4:
imax = (3, 2, 3, 1, 2, 3, (!), @, @, @, 3, 2, 3, 1, 2, 3).
In view of this, Theorem 5 can be reformulated as follows.
THEOREM 13. The product map Ximax restricts to a bijection between n2-tuples of positive real numbers and totally positive n x n matrices.
We will prove the following refinement of Theorem 12, which is a reformulation of its original version [32] .
THEOREM 14. Every matrix x E G,0 can be written as X = XimaxC tl , . . . , tn2 ), for some tl, . . . , tnz 2:: 0.
(Since x is invertible, we must in fact have tk > 0 for n(n - 1)/2 < k :=::; n(n + 1)/2 (i.e., for those indices k for which the corresponding entry of imax is of the form @).)
Proof The following key lemma is due to Cryer [9] .
LEMMA 15. The leading principal minors 11[l,k], [l,kl of a matrix x E G "" 0 are positive for k = 1, . . . , n.
Proof Using induction on n, it suffices to show that 11[1,n- l], [ l,n- 1J(x) > 0. Let 11iJ(x) [resp. 11ii',jj' (x)] denote the minor of x obtained by deleting the row i and the column j (resp. rows i and i' , and columnsj andj'). Then, for any 1 :=::; i < i' :=::; n and 1 :=; j < j' :=::; n, one has
as an immediate consequence of Jacobi's formula for minors of the inverse matrix (see, e.g., [7, Lemma 9.2.10]). The determinantal identity (5) was proved by Desnanot as early as in 1819 (see [37, pp. 140-142]); it is sometimes called "Lewis Carroll's identity," due to the role it plays in C. L. Dodgson's condensation method [ 12, pp. 170-180].
Now suppose that 11n,n(x) = 0 for some x E G2:0. Because x is invertible, we have 11i,n(x) > 0 and 11n,J(x) >
0 for some indices i, j < n. Using (5) with i' = j' = n, we arrive at a desired contradiction by
D
We are now ready to complete the proof of Theorem 14. Any matrix x E G2:0 is by Theorem 1 1 a limit of totally positive matrices XN, each of which can, by Theorem 13, be factored as XN = Ximax (t�N)' . . . , t�lfJ) with all t�N) positive. It suffices to show that the sequence SN = I�� 1 tkCN) converges; then, the standard compactness argument will imply that the sequence of vectors (t�N)' . . . , t�'P) contains a converging subsequence, whose limit (t1 , . . . , tn2) will pro-vide the desired factorization x = ximaxCt1, . . . , tn2). To see that (sN) converges, we use the explicit formula
� 11[l,i] , [l ,i] (XN) SN = L 11 i=l [l ,i- l], [ l,i-l] (XN)
+ I1 11[1,i-1 ]U{i+ 1 ], [ 1,ij(XN) + 11[1,i], [l ,i- l]U{i+ 1 j(XN)
i=1 11[1,i], [ 1,i](XN)
(to prove this, compute the minors on the right with the help of Lindstrom's lemma and simplify). Thus, sN is expressed as a Laurent polynomial in the minors of XN whose denominators only involve leading principal minors 11[l,k] , [l,kJ· By Lemma 15, as XN converges to x, this Laurent polynomial converges to its value at x. This completes the proofs of Theorems 12 and 14. D
Double Wiring Diagrams and Total
Positivity Criteria
We will now give another proof of Theorem 9, which will include it into a family of "optimal" total positivity criteria that correspond to combinatorial objects called double wiring diagrams. This notion is best explained by an example, such as the one given in Figure 6. A double wiring
diagram consists of two families of n piecewise-straight lines (each family colored with one of the two colors), the crucial requirement being that each pair of lines of like color intersect exactly once.
The lines in a double wiring diagram are numbered s�parately within each color. We then assign to every chamber of a diagram a pair of subsets of the set [1 , n] = { 1, . . . , n}: each subset indicates which lines of the corresponding color pass below that chamber; see Figure 7.
Thus, every chamber is naturally associated with a minor 11r,J of an n X n matrix x = (Xij) (we call it a chamber minor) that occupies the rows and columns specified by the sets I and J written inside that chamber. In our running example, there are nine chamber minors (the total number is always n2), namely X31, X32, X12, X13, l123,12, l113,12, 1113,23, l112,23, and 11123,123 = det(x).
THEOREM 16. Every double wiring diagram gives rise to the foUowing criterion: an n X n matrix is totaUy positive if and only if aU its n2 chamber minors are positive.
The criterion in Theorem 9 is a special case of Theorem 16 and arises from the "lexicographically minimal" double wiring diagram, shown in Figure 8 for n = 3.
Proof We will actually prove the following statement that implies Theorem 16.
THEOREM 1 7. Every minor of a generic square matrix can be written as a rational expression in the chamber minors of a given double wiring diagram, and, moreover, this rational expression is subtraction:free (i.e., all coefficients in the numerator and denominator are positive).
Two double wiring diagrams are called isotopic if they have the same collections of chamber minors. The terminology suggests what is really going on here: two isotopic diagrams have the same "topology." From now on, we will treat such diagrams as indistinguishable from each other.
We will deduce Theorem 17 from the following fact: any two double wiring diagrams can be transformed into each other by a sequence of local "moves" of three different kinds, shown in Figure 9. (This is a direct corollary of a theorem of G. Ringel [39]. It can also be derived from the Tits theorem on reduced words in the symmetric group; cf. (7) and (8) below.)
Note that each local move exchanges a single chamber minor Y with another chamber minor Z and keeps all other chamber minors in place.
LEMMA 18. Whenever two double wiring diagrams differ by a single local move of one of the three types shown in Figure 9, the chamber minors appearing there satisfy the identity AC + ED = YZ.
The three-term determinantal identities of Lemma 18 are well known, although not in this disguised form. The last of these identities is nothing but the identity (5), applied to various submatrices of an n X n matrix. The identities corresponding to the top two "moves" in Figure 9 are special instances of the classical Grassmann-Pliicker relations (see,
VOLUME 22, NUMBER 1 , 2000 27
z s _______..
-
_______.. _______.. -
X; (t) x, (t) X(D(t)
Figure 4. Elementary "chips."
e.g., [ 18, (15.53)]), and were obtained by Desnanot alongside (5) in the same 1819 publication we mentioned earlier.
Theorem 17 is now proved as follows. We first note that any minor appears as a chamber minor in some double wiring diagram. Therefore, it suffices to show that the chamber minors of one diagram can be written as subtraction-free rational expressions in the chamber minors of any other diagram. This is a direct corollary of Lemma 18 combined with the fact that any two diagrams are related by a sequence oflocal moves: indeed, each local move replaces Y by (AC + BD)/Z, or Z by (AC + BD)/Y. D
Implicit in the above proof is an important combinatorial structure lying behind Theorems 16 and 17: the graph tPm whose vertices are the (isotopy classes of) double
wiring diagrams and whose edges correspond to local moves. The study of tPn is an interesting problem in itself. The first nontrivial example is the graph ¢3 shown in Figure 10. It has 34 vertices, corresponding to 34 different total positivity criteria. Each of these criteria tests nine minors of a 3 X 3 matrix. Five of these minors [viz., x31, x13, ll23,12, ll12,23, and det(x)] correspond to the "unbounded" chambers that lie on the periphery of every double wiring diagram; they are common to all 34 criteria. The other four minors correspond to the bounded chambers and depend on the choice of a diagram. For example, the criterion derived from Figure 7 involves "bounded" chamber minors ll3,2, ll1,2, ll13,12, and ll13,23· In Figure 10, each vertex of ¢3 is labeled by the quadruple of "bounded" minors that appear in the corresponding total positivity criterion.
We suggest the following refinement of Theorem 17.
CONJECTURE 19. Every minor of a generic square matrix can be written as a Laurent polynomial with nonnegative integer coefficients in the chamber minors of an arbitrary double wiring diagram.
Perhaps more important than proving this conjecture would be to give explicit combinatorial expressions for the
Figure 5. Planar network r(i).
28 THE MATHEMATICAL INTELLIGENCER
Figure 6. Double wiring diagram.
Laurent polynomials in question. We note a case in which the conjecture is true and the desired expressions can be given: the "lexicographically minimal" double wiring diagram whose chamber minors are the initial minors. Indeed, a generic matrix x can be uniquely written as the product Ximax (t1, . . . , tnz) of elementary Jacobi matrices (cf. Theorem 13); then, each minor of x can be written as a polynomial in the tk with non-negative integer coefficients (with the help of Lindstrom's lemma), whereas each tk is a Laurent monomial in the initial minors of x, by Lemma 6.
It is proved in [ 15, Theorem 1 . 13] that every minor can be written as a Laurent polynomial with integer (possibly negative) coefficients in the chamber minors of a given diagram. Note, however, that this result combined with Theorem 17, does not imply Conjecture 19, because there do exist subtraction-free rational expressions that are Laurent polynomials, although not with non-negative coef
ficients (e.g., think of (p3 + q3)/(p + q) = p2 - pq + q2). The following special case of Conjecture 19 can be de
rived from [3, Theorem 3.7.4].
THEOREM 20. Conjecture 19 holds for all wiring diagrams in which all intersections of one color precede the intersections of another color.
We do not know an elementary proof of this result; the proof in [3] depends on the theory of canonical bases for quantum general linear groups.
Digression: Somos sequences The three-term relation AC + BD = YZ is surrounded by some magic that eludes our comprehension. We cannot resist mentioning the related problem involving the Somos-5 sequences [19] . (We thank Richard Stanley for telling us about them.) These are the sequences a1, a2, . . . in which any six consecutive terms satisfy this relation:
(6)
Each term of a Somos-5 sequence is obviously a subtraction-free rational expression in the first five terms a1, . . . , a5. It can be shown by extending the arguments in [ 19, 35]
123,123 ==��====���======��====== 3
1
�������==�����--� 2 2
======��======��====��� � 3 0,0
Figure 7. Chamber minors.
123,123 l ======�-r==========��r====== 3 3 1
==���--��-7��-r������ 2 2
�--�====����====��� 1 1 3 Figure B. Lexicographically minimal a1agram.
that each an is actually a Laurent polynomial in a1, . . . , a5. This property is truly remarkable, given the nature of the recurrence, and the fact that, as n grows, these Laurent polynomials become huge sums of monomials involving large coefficients; still, each of these sums cancels out from the denominator of the recurrence relation an+5 =
(an+1an+4 + an+zan+a)/a.n. We suggest the following analog of Cof\iecture 19.
CONJECTURE 21. Every term of a Somos-5 sequence is a Laurent polynomial with non-negative integer coefficients in the first five terms of the sequence.
Factorization Schemes
According to Theorem 16, every double wiring diagram gives rise to an "optimal" total positivity criterion. We will now show that double wiring diagrams can be used to obtain a family of bijective parametrizations of the set G>o of all totally positive matrices; this family will include the parametrization in Theorem 13 as a special case.
We encode a double wiring diagram by the �ord of length n(n - 1) in the alphabet { 1, . . . , n - 1, 1, . . . , n - 1 ) obtained by recording the heights of intersections of pseudolines of like color (traced left to right; barred digits for red crossings, unbarred for blue). For �xam_p�, the diagram in Figure 6 is encoded by the word 2 1 2 1 2 1 .
The words that encode double wiring diagrams have an alternative description in terms of reduced expressions in the symmetric group Sn. Recall that by a famous theorem of E. H. Moore [36], Sn is a Coxeter group of type An-1 ; that is, it is generated by the involutions s1, . . . , Sn- 1 (adjacent transpositions) subject to the relations sisi = SjSi for
_B _X_c __ .....
�-z-� :vc� ... AXD
B X c :VC z :><I .....
V( y )(i_ ... _A_X_n_ B B
V(z� D
Figure 9. Local "moves."
li - jl ;:::: 2, and siSjSi = SjSiSj for li - jl = 1 . A reduced word for a permutation w E Sn is a word j = (j1, . . . , jt) of the shortest possible length l = f( w) that satisfies w = Sj1• "Sit· The number f(w) is called the length of w (it is the number of inversions in w ) . The group Sn has a l.IDique element wo of maximal length: the order-reversing permutation of 1, . . . , n; it gives f(w0) = G).
It is straightforward to verify that the encodings of double wiring diagrams are precisely the shuffles of two reduced words for wo, in the barred and unbarred entries, respectively; equivalently, these are the reduced words for the element ( Wo, wo) of the Coxeter group Sn X Sn.
DEFINITION 22. A word i in the alphabet .71 (see (3)) is called afactorization scheme if it contains each circled entry @ exactly once, and the remaining entries encode the heights of intersections in a double wiring diagram.
Equivalently, a factorization scheme i is a shuffle of two reduced words for Wo (one barred and one unbarred) and an arbitrary permutation of the entries Q), . . . , @. In particular, i consists of n2 entries.
To illustrate, the word i = 2 1 ® 2 I CD 2 1 @, appearing in Figure 5 is a factorization scheme.
An important example of a factorization scheme is the word imax introduced in Theorem 13. Thus, the following result generalizes Theorem 13.
a = x l l
b = Xt2 C = X21 d = X22 e = X23 f = X32 9 = X33
gABC
n = 6.23. 13 c = 6.13,23 D = 6. 1 3, 1 3 E = 6. 13 . 1 2 F = 6. 12 , 13 G = 6.12 , 1 2
Figure 10. Total positivity criteria for GL_a.
VOLUME 22, NUMBER 1 , 2000 29
THEOREM 23 [15] . For an arbitrary factorization scheme i = (i1, . . . , in2), the product map Xi given by (4) restricts to a bijection between n2-tuples of positive real numbers and totally positive n X n matrices.
Proof We have already stated that any two double wiring diagrams are connected by a succession of the local "moves" shown in Figure 9. In the language of factorization schemes, this translates into any two factorization schemes being connected by a sequence of local transformations of the form
· · ·i j i · · · � · · ·j i j· · · ,
or of the form
- - - - - - li - jl = 1,
li - jl = 1, (7)
(8)
where (a,_!>) is any pair of symbols in .s!l different from ( i, i ± 1) or (i, i ± 1). (This statement is a special case of Tits's theorem [45], for the Coxeter group Sn X Sn X CS2)n.)
In view of Theorem 13, it suffices to show that if Theorem 23 holds for some factorization scheme i, then it also holds for any factorization scheme i' obtained from i by one of the transformations (7) and (8). To see this, it is enough to demonstrate that the collections of parameters { tk) and {t'k) in the equality
Xi1(t1) - - ·xin' (tn2) = Xi1(ti) · · ·xi;,z (t�2)
are related to each other by (invertible) subtraction-free rational transformations. The latter is a direct consequence of the commutation relations between elementary Jacobi matrices, which can be found in [ 15, Section 2.2 and ( 4. 17) ] . The most important of these relations are the following.
First, for i = 1, . . . , n - 1 and j = i + 1, we have
where
, t3t4 t - t2' = T, 1 - r,
The proof of this relation (which is the only nontrivial relation associated with (8)) amounts to verifying that [ 1 t1 ] [ t2 0 ] [ 1 0]
= [ 1 0] [ t2 0 ] [ 1 t4]
. 0 1 0 t3 t4 1 t]. 1 0 t3 0 1
Also, for any i and j such that li - jl = 1, we have the following relation associated with (7):
where
xi(t1)xj(t2)xi(t3) = xj(tl.)xi(t2)xj(t3), xi( t1)xj( t2)xj( t3) = xj( tl.)xi( t2)xj( t3),
t2 = T,
One sees that in the commutation relations above, the formulas expressing the tfc in terms of the t1 are indeed subtraction-free. 0
30 THE MATHEMATICAL INTELLIGENCER
Theorem 23 suggests an alternative approach to total positivity criteria via the following factorization problem: for a given factorization scheme i, fmd the genericity conditions on a matrix x assuring that x can be factored as
X = Xi(t1, . . . , tn') = Xi1(t1} · ·xinz(tn2), (9)
and compute explicitly the factorization parameters tk as functions of x. Then, the total positivity of x will be equivalent to the positivity of all these functions. Note that the criterion in Theorem 9 was essentially obtained in this way: for the factorization scheme imax, the factorization parameters tk are Laurent monomials in the initial minors of x ( cf. Lemma 6).
A complete solution of the factorization problem for an arbitrary factorization scheme was given in [ 15, Theorems 1.9 and 4.9] . An interesting (and unexpected) feature of this solution is that, in general, the tk are not Laurent monomials in the minors of x; the word imax is quite exceptional in this respect. It turns out, however, that the tk are Laurent monomials in the minors of another matrix x' obtained from x by the following birational transformation:
x' = [xTwo]+wo(x1)- 1wo[WoXT]_ . (10)
Here, xT denotes the transpose of x, and w0 is the permutation matrix with 1's on the antidiagonal; finally, y = [Y] - [Y]o[Y] + denotes the Gaussian (LDU) decomposition of a square matrix y provided such a decomposition exists.
In the special cases n = 2 and n = 3, the transformation x � x' is given by
and
x' =
[ -1 - 1 x' =
X1 1X12 X21 - 1 X12
- 1 ] X21 X22 det(x)- 1
� �12 13 1 X31 X13 X31 �12,23 X31 �13 12 X33�12,12 - det(x) �
X13 �23,12 �23,12 �12,23 �23,12 1 � �23.23
X13 �12,23 det(x)
The following theorem provides an alternative explanation for the family of total positivity criteria in Theorem 16.
THEOREM 24 [15] . The right-hand side of (10) is well defined for any x E G>o; moreover, the "twist map" x � x' restricts to a bijection of G>o with itself.
Let x be a totally positive n X n matrix, and i a factorization scheme. Then, the parameters t1 , . . . , tn' appearing in (9) are related by an invertible monomial transformation to the n2 chamber minors (for the double wiring diagram associated with i) of the twisted matrix x' given by (10).
In [15] , we explicitly describe the monomial transformation in Theorem 24, as well as its inverse, in terms of the combinatorics of the double wiring diagram.
Double Bruhat Cells
Our presentation in this section will be a bit sketchy; details can be found in [15] .
Theorem 23 provides a family of bijective (and biregular) parametrizations of the totally positive variety G>o by n2-tuples of positive real numbers. The totally non-negative variety G20 is much more complicated (note that the map in Theorem 14 is surjective but not injective). In this section, we show that G20 splits naturally into "simple pieces" corresponding to pairs of permutations from Sn.
THEOREM 25 [15] . Let x E G20 be a totally non-negative matrix. Suppose that a word i in the alphabet s1 is such that x can be factored as x = Xi(ti, . . . , tm) with positive t1, . . . , tm, and i has the smallest number of uncircled entries among all words with this property. Then, the subword of i formed by entries from { 1, . . . , n - 1 } ( resp. from {1 , . . . , n - 1 }) is a reduced word for some permutation u ( resp. v) in Sn. Furthermore, the pair ( u, v) is uniquely determined by x (i.e., does not depend on the choice of i).
In the situation of Theorem 25, we say that x is of type ( u, v ). Let G�8 C G20 denote the subset of all totally nonnegative matrices of type ( u, v ) ; thus, G2o is the disjoint union of these subsets.
Every subvariety G�:8 has a family of parametrizations similar to those in Theorem 23. Generalizing Defmition 22, let us call a word i in the alphabet s1 afactorization scheme of type ( u, v) if it contains each circled entry CD exactly once, and the barred (resp. unbarred) entries of i form a reduced word for u (resp. v); in particular, i is of length C(u) + C(v) + n.
THEOREM 26 [ 15] . For an arbitrary factorization scheme i of type ( u, v ), the product map Xi restricts to a bijection between (C(u) + C(v) + n)-tuples of positive real numbers and totally non-negative matrices of type (u, v).
Comparing Theorems 26 and 23, we see that
(11)
that is, the totally positive matrices are exactly the totally non-negative matrices of type (w0, w0).
We now show that the splitting of G2o into the union of varieties G';,� is closely related to the well-known Bruhat decompositions of the general linear group G = GLn. Let B (resp. B-) denote the subgroup of upper-triangular (resp. lower-triangular) matrices in G. Recall (see, e.g., [1 , §4]) that each of the double coset spaces B\GIB and B-\G/B_ has cardinality n!, and one can choose the permutation matrices w E Sn as their common representatives. To every two permutations u and v we associate the double Bruhat cell au,v = BuB n B_vB-; thus, G is the disjoint union of the double Bruhat cells.
Each set au,v can be described by equations and inequalities of the form .l(x) = 0 and/or .l(x) -=/= 0, for some collection of minors .1. (See [ 15, Proposition 4. 1] or [16] .) In particular, the open double Bruhat cell awo,Wo is given by
nonvanishing of all "antiprincipal" minors ll[l,iJ . [n-i+ l,nJ(x) and d[n -i+ l,n], [ l,iJ (X) for i = 1, . . . , n - 1.
THEOREM 27 (15] . A totally non-negative matrix is of type ( u, v) if and only if it belongs to the double B�hat cell au,v.
In view of (11), Theorem 27 provides the following simple test for total positivity of a totally non-negative matrix.
COROLLARY 28 [23] . A totally non-negative matrix x is totally positive if and only if d[l,i], [n-i+l ,nJ(x) -=/= 0 and d[n-i+l ,nJ, [ l,i J(X) -=/= Ofor i = 1, . . . , n.
The results obtained above for G��wo = G>o (as well as their proofs) extend to the variety G�8 for an arbitrary pair of permutations u, v E Sn. In particular, the factorization schemes for ( u, v) (or rather their uncircled parts) can be visualized by double wiring diagrams of type ( u, v) in the same way as before, except now any two pseudolines intersect at most once, and the lines are permuted "according to u and v." Every such diagram has C(u) + C(v) + n chamber minors, and their positivity provides a criterion for a matrix X E au,v to belong to G�8- The factorization problem and its solution provided by Theorem 24 extend to any double Bruhat cell, with an appropriate modification of the twist map x � x'. The details can be found in ( 15].
If the double Bruhat cell containing a matrix x E G is not specified, then testing x for total non-negativity becomes a much harder problem; in fact, every known criterion involves exponentially many (in n) minors. (See [8] for related complexity results.) The following corollary of a result by Cryer [ 10] was given by Gasca and Pefta [24].
THEOREM 29. An invertible square matrix is totally non-negative if and only if all its minors occupying several initial rows or several initial columns are non-negative, and all its leading principal minors are positive.
This criterion involves 2n+l - n - 2 minors, which is
roughly the square root of the total number of minors. We do not know whether this criterion is optimal.
Oscillatory Matrices
We conclude the article by discussing the intermediate class of oscillatory matrices that was introduced and intensively studied by Gantmacher and Krein [20, 22]. A matrix is oscillatory if it is totally non-negative while some power of it is totally positive; thus, the set of oscillatory matrices contains G>o and is contained in G2o- The following theorem provides several equivalent characterizations of oscillatory matrices; the equivalence of (a)-( c) was proved in [22], and the rest of the conditions were given in [17] .
THEOREM 30 [ 17,22]. For an invertible totally non-negative n X n matrix x, the following are equivalent: (a) x is oscillatory; (b) xi,i+l > 0 and Xi+ l,i > Ofor i = 1, . . . , n - 1;
VOLUME 2 2 , NUMBER 1 , 2000 31
(c) xn-1 is totally positive; (d) x is not block-triangular (cf Figure 1 1);
* * 0 0 0
* * 0 0 0
* * * * *
* * * * *
* * * * *
Figure 1 1 . Block-triangular matrices.
* * * * *
* * * * *
0 0 * * *
0 0 * * *
0 0 * * *
(e) x can be factored as x = xi(t1 , . . . , t1), for positive t1 , . . . , t1 ang a word i that contains every symbol of the form i or i at least once; (f) X lies in a double Bruhat cell au,v, where both u and v do not fix any set { 1 , . . . , i } , for i = 1, . . . , n - 1 .
Proof Obviously, (c) => (a) => (d). Let us prove the equiv
alence of (b), (d), and (e). By Theorem 12, x can be rep
resented as the weight matrix of some planar network f(i) with positive edge weights. Then, (b) means that sink i + 1 (resp. i) can be reached from source i (resp. i + 1), for
all i; (d) means that for any i, at least one sink j > i is
reachable from a source h :::; i, and at least one sink h :::; i is reachable from a source j > i; and (e) means that f(i) contains positively and negatively sloped edges connecting
A U T H O R S
SERGEY FOMIN
Department of Mathematics
University of Michigan
Ann Arbor, Ml 48109 USA
e-mail: [email protected]
Sergey Fomin is a native of St. Petersburg, Russia. He de
cided he wanted to be a mathemat ician at the age of 1 1 and
became addicted to combinatorics at the age of 1 6. A stu
dent of Anatoly Vershik, he received his advanced degrees
from St. Petersburg State University . From 1 992 to 1 999, he
was at MIT. He has also held since 1 991 an appo intment at
the St. Petersburg Institute for Informatics and Automation.
His main research interest is combinatorics and its applica
tions in representation theory, algebraic geometry, theoretical
computer science, and other areas of mathematics.
32 THE MATHEMATICAL INTELLIGENCER
any two consecutive levels i and i + 1. These three state
ments are easily seen to be equivalent.
By Theorem 27, (e) <=> (f). It remains to show that (e) =>
(c). In view of Theorem 26 and (11), this can be restated
as follows: given any permutation j of the entries 1, . . . , n - 1, prove that the concatenation jn - 1 of n - 1 copies of
j contains a reduced word for w0. Let j 1 denote the subse
quence of jn- 1 constructed as follows. First, j 1 contains all
n - 1 entries of jn-1 which are equal to n - 1. Second, j ' contains the n - 2 entries equal to n - 2 which interlace
the n - 1 entries chosen at the previous step. We then in
clude n - 3 interlacing entries equal to n - 3, and so forth.
The resulting WOrd j I Of length m will be a reduced WOrd
for Wo, for it will be equivalent, under the transformations
(8), to the lexicographically maximal reduced word (n - 1,
n - 2, n - 1, n - 3, n - 2, n - 1, . . . ). 0
ACKNOWLEDGMENTS
We thank Sara Billey for suggesting a number of editorial
improvements. This work was supported in part by NSF
grants DMS-9625511 and DMS-9700927.
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VOLUME 22. NUMBER 1, 2000 33
1.5ffli•i§uflhfii*J.Irri,pi.ihi£j Marj o r i e Senechal , Ed itor I
Exact Thought in a Demented T ime: Karl Menger and h is Viennese Mathematica l Col loquium Louise Galland and Karl Sigmund
This column is a fornm for discussion
of mathematical communities
throughout the world, and through all
time. Our definition of "mathematical
community" is the broadest. We include
"schools" of mathematics, circles of
correspondence, mathematical societies,
student organizations, and informal
communities of cardinality greater
than one. What we say about the
communities is just as unrestricted.
We welcome contributions from
mathematicians of all kinds and in
all places, and also from scientists,
historians, anthropologists, and others.
Please send all submissions to the
Mathematical Communities Editor,
Marjorie Senechal, Department
of Mathematics, Smith College,
Northampton, MA 01 063, USA
e-mail: senechal@minkowski .smith.edu
One evening in 1928, a group of
students from the University of
Vienna gathered at Karl Menger's
apartment to discuss current topics in
mathematics. This was the beginning
of what became the famous Mathe
matical Colloquium (Mathematisches KoUoquium), which met regularly dur
ing the academic year from 1928 to 1936.
The notes that Menger took during the
sessions grew into the Ergebnisse eines mathematischen Kolloquiums; it is a
telling footnote to twentieth-century
history that no complete copies of the
first edition survived at the Univer
sity of Vienna. More happily, the Ergebnisse was republished in 1998; we
hope that our retelling of the story will
help to call attention to it although, as
Franz Alt says of the Ergebnisse itself,
we can offer "only a pale reflection of
what it meant to be present at the
Colloquium meetings, to experience
the give and take, the absorbing inter
est, the earnest or sometimes hu
mourous exchanges of words."
Today the Colloquium is receiving
increasing attention from mathemati
cians and historians of mathematics,
attention that is sure to grow with the
republication of the Ergebnisse, as
many important concepts of twentieth
century mathematics were formulated
and discussed in the Colloquium. Our
focus here will be on the remarkable
mathematical community that the
Colloquium sustained for a few bright
years before it was dispersed around
the world by fascist terror. Though
many of its participants met again later
in their lives, the Colloquium never re
sumed, and had no direct successor.
Mathematics may be eternal, but math
ematical communities are even more
fragile than mathematicians.
The Viennese Enlightenment
Some Viennese hold that their home
town became the Capital of Music be
cause there was so little else to do.
Counterreformation, absolutism, and,
34 THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER· VERLAG NEW YORK
after the Napoleonic Wars, a half-cen
tury of political reaction weighed heav
ily on free enterprise, free trade, and
free thought. An all-pervading censor
ship left no room for intellectual dis
cussions: if you met friends, it had bet
ter be for a musical soiree. But in the eighteen-sixties, Vienna
became, almost overnight, a Capital of
Literature, Thought, and Science. Inner
unrest and military defeats had forced
the Emperor Francis Joseph to accept
a liberal constitution. Almost instantly,
Viennese intellectual life made up for
the centuries of repression. This sud
den blossoming ended, even more sud
denly, in the nineteen thirties. The cul
tural effervescence, which some called
the "gay apocalypse" and others the
"golden autumn," lasted only for a few
decades: Freud in psychology, Boltz
mann in physics, Kokoschka, Schiele,
and Klimt in painting, Mahler and
Schoenberg in music, Otto Wagner in
architecture, Popper and Wittgenstein
in philosophy, Schumpeter and Hayek
in economics, and many others.
Several of these people enter our story.
Two generations were enough to
cover the whole period. The economist
Carl Menger (1841-1920) shaped the
beginning, and his son, the mathemati
cian Karl Menger (1902-1985), wit
nessed the end. This story deals with
the younger Menger, but it is worth
spending a few lines on the father.
Founding Father
Carl Menger was the son of a landowner
in the Polish part of the Habsburg em
pire. He studied law in Prague and
Vienna, got his degree in Cracow, and en
tered civil service just when a tri
umphant liberalism reshaped the monar
chy. Having to write reports on market
conditions, Carl wished to understand
what makes prices change, and in 1871
he published a path-breaking book,
Grundsiitze der Volkswirtschajtslehre (Principles of National Economics).
He thus became the founding father of
Cart Menger (1841-1920) contributed to a
revolution in economic analysis, but steered
clear of mathematical models-in part, no
doubt, because he had never been taught
even the rudiments of calculus. This defi·
ciency remained a hallmark of Austrian mar
ginalists. "Though their instinct was very
good," wrote a historian of economics, "their
mathematical equipment was not up to what
was required." Kart Menger often pondered
''the psychological problem . . . why such em·
inent minds as the founders (and perhaps
also several younger members) of the
Austrian School were, as mature men, un
successful in their self-study of analysis."
(The "younger members" may have included
Morgenstern.) The culprit, according to Karl
Menger, was the confusing notation used In
calculus texts, especially for variables and
functions.
Austrian Marginalism, an economic school of world-wide influence, and not at all a marginal Austriacism. In his book, Carl Menger derived economic value from individual human wants (rather than from some inherent quality of the goods, or from the working hours spent on them). In the same year, economists Jevons and Walras hit independently on this idea, which necessitated a complete rethinking of classical theory. Like some of his Austrian contemporaries-Ludwig Boltzmann, for instance, Gregor Mendel, or Ernst Mach-Carl Menger had managed to jump, almost out of the blue, into the forefront of research. His work earned him, at the age of 33, a position as associate professor at the University of Vienna. From 1876 to 1878, he was the tutor of Crown Prince Rudolph, the emperor's only son, and travelled
with him through England, France, Germany, and Switzerland. Rudolph, a talented youth avidly espousing new ideas, remained passionately devoted to the liberal cause, even after Carl Menger returned to the University, this time as full professor. Menger had introduced the archduke to Moritz Szeps, in whose journal Rudolph published anonymous articles attacking Viennese anti-semitism, corruption in the administration, and even imperial foreign policy. Eventually, his father put an end to it. A few years later, worn down by the narrow-mindedness of court life, the crown prince capped his scandals by committing double suicide with an 18-year-old society girl.
Carl Menger, by that time, had already achieved world-wide renown and could count on some brilliant disciples, like Friedrich Wieser and Eugen Bohm-Bawerk, to carry on with the theoretical work. He himself concentrated on highly publicized polemics against the German economists of his day, who claimed that their science could at best undertake the historical study of economic events in a given society. In contrast, Carl Menger believed in universal economic laws, ultimately grounded in the psychology of human needs. His methodological individualism was a fitting expression of the prevailing mood in fin-de-siecle Vienna. Every afternoon, he resided in a coffee-house where he discussed the issues with the leading social scientist and law professor of Vienna. This happened to be none other than his brother Anton Menger, an eccentric firebrand who had been ejected from school for insubordination, had turned into an apprentice mechanic, and eventually had used a lottery windfall to finance his studies. Anton was an ardent social utopian and fought a lifelong crusade for reforming private law. The third of the Menger brothers, Max, did not attain academic distinction, but he was for more than thirty years a leading liberal deputy of the Reichstag (the Austrian parliament).
Karl Menger was born on January 12, 1902. His father had recently become a member of the Herrenhausa life peer. He retired from teaching one year later, to the regret of many
students, in order to devote himself solely to research. With such a father and such uncles, and a mother who was a successful novelist, Karl must soon have felt the urge to make a name for himself, or more precisely, a first name-an initial, in fact. And it is likely that this pressure to succeed did not relax at school: two of Menger's schoolmates, namely Wolfgang Pauli and Richard Kuhn, were heading for Nobel Prizes.
Karl Menger was a brilliant pupil, as his school certificates show, shining most brightly in Catholic religious instruction. Like many a schoolboy of his time, he set out to write a play-it must have looked like the quickest way to fame. Karl's religious instructor would have been dismayed to learn that the play was intended to deal the Church a devastating blow. The title was Die gottlose Komoedie (the godless comedyin contrast with Dante's Die gotUiche
Arthur Schnitzler (1862-1931) was undlsput·
edly the leading author of fln-de-slecle
Vienna. "When I see a talent blossoming, like
yours," wrote Theodor Herzl, the Zionist
leader, "I am as happy as with the carnations
in the garden". Schnitzler used the stream·
of-consciousness technique years before
James Joyce, and his erotic comedy Der
Reigen, written in 1900, was deemed so
shocking that it took twenty years before it
was produced-and when it was it caused a
major public uproar. Sigmund Freud saw in
Schnitzler his "double" and called him an
"explorer of the psyche • • • as honest and
fearless as there ever was." Arthur
Schnitzler's diary reports Kart Menger's me
teoric rise to mathematical prominence.
VOLUME 22. NUMBER 1, 2000 35
Komodie, the Divine Comedy). The plot centers on the medieval Pope John who, as legend has it, turned out to be a woman called Joan.
Young Menger's classmate Heinrich happened to be the son of Arthur Schnitzler, the most famous Viennese author of his age. It may have been the shared burden of descending from cultural heroes that drew Karl and Heinrich together-an Oedipus complex was not unheard-of in the Vienna of those days. Through Heinrich, Karl
Karl Menger (1902-1985) inherited from his
father a positivistic, individual-centered
world view and a huge private library. Obliged
to sell the library when he was twenty, Karl
Menger held on to the philosophy books. It
may have been this heritage which immu
nized him against the lure of Wittgenstein.
Indeed, Austrian philosophers had antici
pated parts of the Tractatus; for instance,
Fritz Mauthner, who was just as sure as the
young Wittgenstein of having solved all philo
sophical problems, or Adolf Stoehr, the suc
cessor to Mach's chair in philosophy.
Mauthner described traditional philosophy as
word fetishism and attacked it in a three-vol
ume Critique of Language culminating in the
prescription of silence; Stoehr wrote that "nonsense cannot be thought, it can only be spoken . • . " And Karl's father had noted in
1867 already: "There is no metaphysics . . . .
There is no riddle of the world that ought to
be solved. There is only incorrect considera
tion of the world."
36 THE fv'.ATHEMATICAL INTELUGENCER
was able to seek the advice of the foremost playwright in town. The comments were negative, alas, and the godless comedy came to naught. But Arthur Schnitzler kept notes on his meetings with Karl Menger, and traced in his diary a dramatic turn of events.
It began in an unheated classroom of the University of Vienna. The time: March 1921, during the worst inflation of Austrian history. Karl Menger had enrolled in theoretical physics-this was the heyday of the Einstein fervor-but was not satisfied with what the Physics Department had to offer, and drifted towards the Institute of Mathematics. A newly appointed professor there, 42-year-old Hans Hahn, had just announced his first seminar. It dealt with the concept of curves. Menger had barely entered his second semester, but decided to give it a try.
Curves to Glory
Hahn went right to the heart of the pr-oblem. Everyone, he began, has an ·intuitive idea of curves; . . . But anyone who 1vanted to make the idea precise, Hahn said, would encounter great difficulties . . . . At the end of the seminar we should see that the problem was not yet solved. I was completely enthralled. And when, after that short introduction, Hahn set out to develop the principal tools-the basic concepts of Cantor's point-set theory, all totally new to me-l followed with the utmost attention.
Hahn was well placed to discuss the curve problem. Fired up by Peano's and Hilbert's constructions of space-filling curves, he had shown what became known as the Hahn-Mazurkiewicz theorem: every compact, connected, locally connected set (a full square, for instance) was the continuous image of an interval.
I left the seminar in a daze. Like everyone else I used the word "curve". Should I not be able to spell out articulately how I used the word? After a week of complete engrossment in the problem I felt I had arrived . . . at a simple and complete solution.
This solution consisted essentially in
defining curves as one-dimensional continua. Continua had been defmed by Cantor and Jordan already. What remained was to defme their dimension. Menger hit on the idea of proceeding inductively, assigning dimension - 1 to the empty set and defming a set S to be at most n-dimensional if each of its
points admits arbitrarily small neighborhoods with whose frontiers S has at most (n - I)-dimensional intersection.
Menger showed this solution to his friend Otto Schreier, who was already in his second year at the University. Schreier could fmd no flaw in Menger's ideas, but quoted both Hausdorff and Bieberbach who said the problem was intractable. Menger, however, was convinced that "one should never reason that an idea is too simple to be correct." He told Hahn one hour before the second seminar that he could solve the problem. Hahn, who had hardly looked up when I entered, became more and mm-e attentive as I went on . . . At the end, after some thought . . . he nodded rather encouragingly and I l.eft.
The chronicles of mathematics report other breakthrough discoveries by mere teenagers. What makes this case so special is that Menger used only the material covered in one seminar talk.
A few weeks later, disaster struck. Weakened by malnutrition and long working hours in unheated libraries, Karl Menger fell prey to tuberculosiscalled Morbus Viennensis at the time. In the impoverished capital of an amputated state, this illness was claiming thousands of victims. The chronicles of mathematics are filled not only with precocious talents but also with premature deaths-Schreier, for instance, was to die at twenty-eight, after brilliant work in group theory. Stlicken by tuberculosis-like Niels Hendrik Abelnineteen-year-old Menger jotted down his ideas in feverish haste-like Evariste Galois-and deposited them in a sealed envelope at the Viennese Academy of Science before entering a sanatorium located on a mountaintop in near-by Styria. In eerie peace, surrounded by sn·angers each fighting a private battle with death. Menger found plenty of time to study, to read, and to think.
During his stay at the alpine retreat, which lasted more than one year, both his 80-year-old father and his 50-yearold mother passed away. They were not to witness their son's heady ascent.
When Karl Menger returned to the university, completely recovered, he had developed a full-fledged theory of curves which almost inunediately earned him his doctorate with Hahn. He also supervised the publication of the second edition of his father's classic Grundziige, which included a wealth of revisions. At the same time, his passion for philosophy asserted itself. He had come to believe that the recent work by L.E.J. Brouwer on "intuitionistic" set theory, with its insis-
Luitzen Egbertus Jan Brouwer (1881-1966).
"His hollow-cheeked face," as Menger wrote,
"faintly resembling Julius Caesar's, was ex
tremely nervous with many lines that perpet
ually moved • • • • Outward intensity in speech
and movement and action was the hallmark
of his personality." Dominated by a streak of
mysticism, Brouwer saw human society as a
dark force enslaving the individual, and lan
guage as a means of domination. His feuds
with David Hilbert and the French mathe
matical establishment became legendary.
After a good start, relations between
Brouwer and Menger became increasingly
bitter. Yet, in each of the Colloquium's main
topics, Brouwer's work turned out to be fun
damental, be it topology, mathematical logic,
or mathematical economics (the fixed-point
theorem).
tence on constructive proofs, was a counterpart to Mach's positivism, which had so deeply influenced his father. Soon, armed with a Rockefeller scholarship, Menger travelled to the Netherlands. Brouwer was, of course, a leading destination for topologists, and there seems to have been a kind of conduit from Vienna to Amsterdam, which was used, at one time or another, by Weitzenbock, Hurewicz, and Vietoris.
Soon Menger was offered a job as assistant to Brouwer. But after a good start, the relations between the two men, both of whom were highly temperamental, began to get tense. In part this was due to Menger's disagreements with Brouwer's anti-French views, in part to his impatience with Brouwer's legalistic mind and his occasional obscurity. These differences were exacerbated by a priority dispute. The young Russian mathematician Pavel Urysohn had developed a dimension theory quite similar to Menger's, at about the same time, before perishing in a drowning accident. Brouwer edited the posthumous publications of Urysohn, stressing their link with a note written by himself in 1913
which contained already some essential ideas. (So did a short, even earlier passage in Poincare's Dernieres
Pensees, and a much older remark by the Bohemian priest Bernhard Bolzano in his posthumous Paradoxes of
Infinity, which Hahn, the editor of that volume, had unaccountably overlooked.) Menger, who had originally known neither of Brouwer's nor of Urysohn's work, felt that his contribution was misrepresented. He was particularly incensed that Brouwer had included in Urysohn's memoir a reference to his 1913 paper without marking it as an editor's addition. Brouwer, who had proved in that paper that dimension was a topological invariant, was infuriated in his turn when Menger stated bluntly that if dimension were not invariant under homeomorphism, this would be a worse blow to homeomorphism than to dimension. Karl Menger's position in Amsterdam became extremely difficult.
By a stroke of good fortune, Hahn was able to arrange for the return of his favorite student. Kurt Reidemeister, the
young German associate professor of geometry in Vienna, had accepted a chair in Konigsberg. Karl Menger, barely twenty-five, was appointed to succeed him.
The Glow of Red Vienna
"I personally was a rather untypical Viennese," Menger wrote much later, "and deeply and openly loved the Vienna of 1927." The two leading political parties-the Social Christians, who ruled the country, and the Social Democrats, with their unsinkable majority in Red Vienna-seemed to have arrived at a stable balance. The economic situation had greatly improved, inflation was stopped, and a program of sweeping social reforms was underway. But the two parties' private armies still paraded through the streets, and soon after Menger's return to Vienna the political truce was shattered. In July 1927, a jury acquitted militant right-wingers who had fired into a socialist parade, killing two workers and a child. An angry crowd set fire to the Palace of Justice. Police suppressed the outburst brutally, and more than eighty people were left dead in the streets. This explosion of irrationality left a lasting mark on the young republic. Many Austrians concluded that it was better not to engage in political activity at all, rather than to risk bedlam again. Others joined the ranks of the street fighters, including those of the Nazis.
Menger, on his appointment, had embarked on an ambitious program of lecture courses covering all aspects of geometry in the widest possible sense-Euclidean, affme, projective, but also differential and set-theoretic (today's general topology). He collected his topological results in a book, Dimensionstheorie. And he accepted the invitation, by Hahn and Moritz Schlick, to join the hand-picked philosophically-oriented Vienna Circle of mathematicians and philosophers. The Vienna Circle, so famous today, was only one of many intellectual circles that flourished in Vienna at the time, anticipating in a sense the Internet's discussion groups. Menger played an important if somewhat junior role. However, he did not share the infatua-
VOLUME 22, NUMBER 1 , 2000 37
tion (as he called it) of Schlick and Waismann with the remote figure of Wittgenstein, and he felt uneasy with the outspoken social and political engagement of N eurath and Hahn. Soon he asked to be listed, not as a member, but as someone close to the group.
In 1932, Menger published his second book, Kurventheorie, which contained, among other things, his uni
versal curve: not only can every curve be embedded in 3-space, but there exists in 3-space one curve such that every curve can be topologically embedded in it (this curve, in fractal theory, became known as the Menger
sponge). And as a by-product of studying branching points of curves, Menger proved his celebrated n-arc (alias MaxFlow, Min-Cut) theorem: If A and B are two disjoint subsets of a graph, each consisting of n vertices, then either there exist n disjoint paths, each connecting a point in A to a point in B, or else there exists a set of fewer than n vertices which separates A and B. Today, Menger's theorem is considered as the fundamental result on connectedness of graphs; but when Menger told his result to the Hungarian Denes Konig, who at the time was writing an encyclopedic work on graph theory, he was met with open disbelief. Konig told Menger, on taking leave from him that evening, that he would not go to bed before he had found a counterexample. Next morning he met him with the words: "A sleepless night!"
Another significant advance took place when Menger developed, in his course on projective geometry in 1927/28, an axiomatic approach for the operations of joining and intersecting. His so-called algebra of geometry became one of the first formulations of lattice theory, and was applied by John von Neumann in his subsequent work on continuous geometries. Menger himself used it to explicate the timehallowed statement that "a point is that which has no part."
Not surprisingly, Menger quickly became popular with his students, who were barely younger. In spite of being eternally busy, he was easy to approach. The full professors seemed, in contrast, almost like remnants from another age; Wirtinger was deaf,
38 THE MATHEMATICAL INTELLIGENCER
Furtwangler was lame, and Hahn, with his booming voice and crushing personality, appeared as an almost superhuman embodiment of mathematical discipline. The students found it obviously easier to ask Karl Menger to direct a mathematical Colloquium. This Colloquium had a flexible agenda including lectures by members or invited guests, reports on recent publications and discussions on unsolved problems. To some extent, the topics that the group discussed reflected Menger's own interests, but they were not limited to them. In its initial year (1928/ 29), the main themes were topology (including curve theory and set theory) and geometry. Even in its first year, the Colloquium speakers included foreign visitors: M. M. Biedermann from Amsterdam and W. L. Ayres from the United States. (The other speakers were Menger, Hans Hornich, Helene Reschovsky, Georg Nobeling, and Gustav Bergmann). Vienna was a mathematical attractor at that time, and Menger's curve theory and its related theory of dimensions had earned him an international reputation, in the United States as well as in Europe.
As Menger's interest shifted increasingly from the Vienna Circle to the Colloquium, his friend and protege Kurt Godel moved with him. Godel had entered the university in 1924, and Menger met him first as the youngest and most silent member of the Vienna Circle. In 1928, Godel started working on Hilbert's program for the foundation of mathematics, and in 1929 he succeeded in solving the first of four problems of Hilbert, proving in his Ph.D. thesis (under Hans Hahn) that first-order logic is complete: any valid formula could be derived from the axioms. That summer, Menger traveled to Poland to visit the Warsaw topologists and was so impressed by the logicians he met there that he invited Tarski to visit Vienna and the Colloquium. Two lasting and significant professional relationships grew out of the lectures Tarski gave to the Colloquium in February 1930, the first between Tarski and the philosopher Rudolf Carnap, and the second between Tarski and Kurt Godel, who, after hearing Tarski lecture, had asked Menger to arrange a meeting.
Kurt Godel (1906-1978). In Menger's posthu
mous Reminiscences of the Vienna Circle
and the Mathematical Colloquium "he was a
slim, unusually quiet young man . • • • He ex
pressed all his insights as though they were
matters of course, but often with a certain
shyness and a charm that awoke warm per
sonal feelings." Olga Taussky-Todd remem
bered that "he was very silent. I have the im·
pression that he enjoyed lively people, but did
not like to contribute to nonmathematical
conversations." With Menger, however, he
spoke a great deal about politics. During his
later years at the Institute for Advanced
Study, Godel struck up a close friendship
with Albert Einstein (who claimed that he only
went to his office in order to walk back home
with Godel) and produced signal contribu
tions to both set theory and relativity. But af
ter Einstein's death in 1954, he started "en
tangling himself' (as Menger had always
feared) and turned into the tragic Princeton
recluse who ultimately starved himself to
death.
Although there is no indication that Menger influenced Godel's ideas directly, he did provide him with a mathematical setting in which to develop them. Godel participated in the Colloquium fully and shared his ideas in a way that was not duplicated later. Not only did he become a co-editor of Ergebnisse, he also made frequent and significant comments in Colloquium discussions (these are available with background commentary both in their original German and in English in the
first volume of the Complete Works.)
It was to the Colloquium that Godel
first presented his famous incomplete
ness theorem. Alt recalls, "There was
the unforgettable quiet after Godel's
presentation ended with what must be
the understatement of the century:
'That is very interesting. You should
publish that.' " Godel soon took a hand
in running the Colloquium and editing
its Ergebnisse.
Menger, then visiting Rice Univer
sity in Texas, immediately grasped the
importance of Godel's results and in
terrupted his lecture series to report
on them. From then on he never tired
of broadcasting the achievement of the
Colloquium's new star. For a proof of
the self-consistency of a portion of
mathematics, in general a more in
clusive part of mathematics is neces
sary. This result is so fundamental
that I should not be surprised if there
were shortly to appear philosophically
minded non-mathematicians who
will say that they never expected any
thing else.
Godel's brilliance may have dis
couraged others from venturing into
his field. He never had a student or a
co-worker. But in spite of his intro
verted character, he needed interested
company and competent stimulation,
and this was amply provided by Hahn
and Menger. "He needed," as Menger
wrote, "a congenial group suggesting
that he report his discoveries, remind
ing and, if necessary, gently pressing
him to write them down." More than half
of GOdel's published work appeared
within a few years in the Monatshefte or
the Ergebnisse, sometimes as a direct
answer to a question by Menger or
Hahn.
Menger was particularly fond of
Godel's results on intuitionism, which
vindicated his own "tolerance princi
ple." Specifically, Godel managed to
prove that intuitionist mathematics is
in no way more certain, or more con
sistent, than ordinary mathematics.
This was a striking result. Since intu
itionists do not accept many classical
proofs, the theorems of intuitionistic
number theory obviously constitute a
proper subset of the theorems of clas
sical number theory. But Godel
showed that a simple translation trans-
forms every classical theorem into an
intuitionistic counterpart: classical
number theory appears here as a sub
system of intuitionistic number theory.
Menger brought Oswald Veblen to the
Colloquium when Godel lectured on
this result. Veblen, who had been
primed by John von Neumann, was
tremendously impressed by the talk
and invited GOdel to the Institute for
Advanced Study during its first full
year of operation: a signal honour that
proved a blessing for Godel's later life.
The participants of the Colloquium
were mostly students or visitors. The
eminent visitors included, in addition to
Tarski for several extended stays, W. L.
Ayres, G. T. Whyburn, Karel Borsuk,
Norbert Wiener, M. H. Stone, Eduard
Cech, and John von Neumann. Heinrich
Grell, a student of Ernrny Noether, gave
a series of talks on ideal theory and the
latest results of Noether, Artin, and
Brandt. Among the foremost regulars
were Hans Hornich, Georg Nobeling,
Franz Alt, and Olga Taussky. Hornich
was Menger's first student, writing his thesis on dimension theory, and eking
out his life as the librarian of the
Institute. N obeling was a brilliant young
topologist from Germany; he ran the
Colloquium while Menger was away in
1930/31. Alt wrote his thesis on curva
ture in metric spaces-an aspect that
Menger, who was bent on developing
geometry without the help of coordi
nate systems, felt particularly challeng
ing. Olga Taussky, who had written her
thesis on class fields under Furtwan
gler, became increasingly attracted by
Menger's investigations of metrics in
abstract groups.
And then there was Abraham W ald,
a Romanian born in the same year as
Menger, but a late bloomer by contrast.
His appearance at the Institute had been
erratic until 1930, when he started in
earnest. He began by solving a problem
suggested by Menger (an axiomatization
of the notion of "betweenness" in met
ric spaces). From then on he kept ask
ing for more, contributing prodigiously
to the Colloquium and soon becoming a
co-editor of the Ergebnisse. In 1931, he
obtained his doctorate, having taken
only three courses. His main interest,
at first, was differential geometry in
metric spaces. In particular, he sue-
Abraham Wald (1902-1950). As son of an or
thodox rabbi, Wald could not enroll in the gym
nasium because he would not attend school
on the Sabbath. He thus came late to Vienna
University, "a small and frail figure, obviously
poor, looking neither old nor young, strangely
contrasting with the lusty undergraduates."
Menger recalls his "unmistakable Hungarian
accent" and adds, "It seemed to me that Wald
had exactly the spirit which prevailed among
the young mathematicians who gathered to
gether about every other week in our
Mathematical Colloquium." In his last years in
Vienna, Wald did path-breaking work in what
is today general equilibrium theory, publish
ing two of his pioneering papers in the
Ergebnisse. The third, "Wald's lost paper," has
become somewhat of a legend among math
ematical economists. In the US, he quickly be
came professor at Columbia University and
contributed fundamentally to mathematical
statistics, in particular, statistical decision
functions and sequential analysis. His work
remained classified during the war. Wald and
his wife died in a plane crash in India.
ceeded in introducing a notion of sur
face curvature in metric spaces (which
reduced to Gaussian curvature for sur
faces in Euclidean space), and he
showed that every compact convex
metric space admitting such a curva
ture at every point is congruent to a
two-dimensional Riemann surface.
In 1929, the economic recession had
reached Austria with full force. In 1931,
the largest bank went broke. Unem
ployment reached record heights. By the
VOLUME 22, NUMBER 1 , 2000 39
spring of 1932, as the economic and po
litical situation in Vienna deteriorated,
the students faced increasing fmancial
difficulties. Menger understood this and
hit upon a novel fund-raising source. As
he later explained, Vienna was teem
ing with physicians and engineers,
lawyers and JYUblic servants, business
men and bankers, seriously interested
in the ideas and the philosophy of sci
ence-! have never found the like any
where else. It occurred to me that many
of these people might be wiUing to pay
a relatively high admission to a series
of interesting lectures on basic ideas of
science and mathematics; and the re
ceipts might subsidize the research of
young talents. Menger discussed his
plan with Hahn, who suggested the
physicist Hans Thirring, who in turn led
them to the chemist Hermann Mark To
gether they outlined several series of
lectures, the first of which had the gen
eral title "Crisis and Reconstruction in
Franz Alt (born 1910) received his Ph.D. in
mathematics from Karl Menger, who asked
him to look after the Colloquium during his
frequent stays abroad. But on March 1, 1938,
Alt (who was described by Olga Taussky-
the Exact Sciences." Tickets cost as Todd as •a man helpful whenever help was
much as for the Vienna Opera, and,
Menger reported, every seat in the au
ditorium was taken. Mark opened the
first series with "Classical Physics,
Shaken by Experiments," Thirring con
tinued with "The Changes of the
Conceptual Frame of Physics," followed
by Hahn on the "Crisis of Intuition."
Nobeling gave the fourth lecture, on
"The Fourth Dimension and the Curved
Space," and Menger fmished the series
with "The New Logic." Menger's lecture
was the first popular presentation of
Godel's results.
Thus, although Menger was barely
older than this followers, his role was
almost fatherly. Responsibility for his
small group hung heavily on his shoul
ders, especially after his own mentor
Hahn died unexpectedly in summer
1934. The future began to look very
bleak for Vienna's mathematicians and
philosophers. Menger, who had never
shared Hahn's willingness to engage in
political action, now greatly missed
this "tireless and effective speaker for
progressive causes." But the time
when such speakers were permitted to
raise their voices was gone.
In Berlin, Hitler had swept to power;
the annexation of his native Austria,
where he had many supporters, stood
at the top of his program. Faced with
40 THE MATHEMATICAL INTELUGENCER
needed") had to write to Menger, "Until now
I always closed my letters expressing my
hope to see you again soon in Vienna. At pre
sent I have to hope to find some way to get
together with you over there." In the US, Alt
contributed to the development of the com
puter, working at the Computing Laboratory
in Aberdeen, the National Bureau of
Standards, and the American Institute of
Physics in New York. He was a founding
member and a president of the Association
for Computing Machinery, and the first edi
tor of Advances in Computers.
Nazi threats and terrorism, the
Austrian chancellor Dollfuss could
think of nothing better than to turn for
help to Mussolini. The Austrian parlia
ment, in a rather remarkable instance
of befuddlement, managed to eliminate
itself. Because of a ballot hanging in
the balance, the first president of the
house (a kind of speaker), who was
prevented by office from casting a
vote, stepped down. Not to be outdone,
the second president (who belonged,
of course, to the opposite camp) did
the same. In the heat of the moment,
the third president followed suit. No
one was left to chair the session. The
Social-Christian government quelled at-
tempts by deputies to meet again. Its
anti-socialist measures became increas
ingly brazen, and provoked a short but
murderous bout of civil war in February
1934. As a result, the Social-Democrats
were banned.
With Austria's left wing repressed,
the Nazis felt that their hour had come,
and attempted a coup in July 1934.
They failed ignominously, but not be
fore assassinating Dollfuss in his chan
cellery. His successor Schuschnigg
made pathetic attempts to copy fascist
Olga Taussky-Todd (1906-1995) was born in
Olomouc (today, Czechia), a daughter of a
chemical engineer. She began studying
mathematics at the University of Vienna in
1925 and became one of the most active
members of the Colloquium. Olga's thesis
was on class fields and group theory, but she
later steeple-chased through a vast number
of topics. She worked for a spell in Gottingen,
editing Hilbert's Zah/bericht for his complete
works, and returned in 1933 for a couple of
years to Vienna, supported by a small stipend
funded by the series of public lectures
("rather elegant affairs") organised by Hahn
and Menger ("very enterprising people"). She
then taught at Bryn Mawr and Girton College,
Cambridge. In 1938, she married the British
mathematician John Todd, and during the
war years turned to applied mathematics.
After the war, both John and Olga held dis
tinguished positions in the US, eventually set
tling down at Caltech. When she was given,
in 1963, the Woman of the Year Award by the
Los Angeles Times, she noted gratefully that
"none of my colleagues could be jealous
(since they were all men)." (Photograph cour
tesy of E. Hlawka.)
fashions, rallies, parades, and intern
ment camps, in the vain hope of con
solidating his regime, but it was obvious
that Hitler-who, for the moment, was
busy with purging his party, re-arming
Germany, and persecuting Jews
would be back. "Viennese culture," in
Menger's words, "resembled a bed of
delicate flowers to which its owner re
fused soil and light while a fiendish
Karl Popper (1902-1994), who studied
physics and psychology but also attended
the Mathematics Colloquium, recalled years
later: "Maybe the most interesting of all these
people was Menger, quite obviously a genius,
bursting with ideas . • . Karl Menger was a
spitfire (feuerspriihend)." He characterized
Menger's pamphlet on ethics as "one of the
few books trying to get away from that silly
verbiage in ethics." Menger recollected that
Popper "tried to make precise the idea of a
random sequence, and thus to remedy the
obvious shortcomings of von Mises's defini
tion of collectives. I asked him to present the
important subject in all details to the Mathe
matical Colloquium. Wald became greatly in
terested and the result was his masterly pa
per on the self-consistency of the notion of
collectives." Popper was looking for more
(namely the construction of finite random
like sequences of arbitrary length): "I dis
cussed the matter with Wald, with whom I
became friendly, but these were difficult
times. Neither of us managed to return to the
problem before we both emigrated, to differ
ent parts of the world."
neighbour was waiting for a chance to
ruin the entire garden."
Ethics and Economics
The Vienna Circle was now regarded as
a leftist conspiracy. Schlick was vehe
mently criticized for refusing to dis
miss his Jewish assistant. Nazi agita
tion was rife among the students, and
street fights often forced the closing of
the University. Still, the Circle kept on
meeting, as did the Colloquium. It fell
to Menger, who as professor had a key
to the Mathematics Institute, to let the
members in. An eerie feeling must have
reigned among the small group, lost in
the huge, empty building, while out
side, fascist Heimwehr battled with il
legal stormtroopers. Even within the
Colloquium group, there were dissen
sions: Nobeling, who because of his na
tionality had lost his position as assis
tant in Vienna, decided to pursue his
career in Nazi Germany, to Menger's
dismay.
Still, as late as 1934/35 the Collo
quium continued to attract foreign vis
itors, Leonard Blumenthal and Eduard
Cech among them. In that year, the
philosopher Karl Popper gave a talk at
the Colloquium in which he "tried to
make precise the idea of random se
quence and thus to remedy the obvi
ous shortcomings of von Mises's defi
nition of Collectives," and Friedrich
Waismann, Schlick's assistant, pre
sented a report on the definition of num
ber according to Frege and Russell. We
may regret-especially because Godel,
Tarski, and Menger were involved
that the details of the discussion were
not recorded.
Menger wrote many years later,
While the political situation in Austria
made it extremely difficult to concen
trate on pure mathematics, socio-po
litical problems and questions of ethics
imposed themselves on everyone al
most every day. In my desire for a com
prehensive world view I asked myself
whether some answers might not come
through exact thought.
To any member of the Vienna Circle, it
was obvious that value judgements
could not be grounded on objective
facts. But Menger was looking for a
theory of ethics-a general theory of
relations between individuals and
groups based on their diverse demands
on others. Within a few months, partly
spent at a mountain resort, he wrote a
booklet on Morality, Decision and·
Social Organisation, meticulously es
chewing all value judgments on social
norms, but investigating the possible
relationships between their adher
ent..<;-enumerating, for instance, all
possible types of cohesive groups.
"Menger's reconfigured ethics," as
Robert J. Leonard, the historian of
game theory, wrote recently, "was
above all an analysis of social order.
. . . The study of ethics should concern
only the social structures yielded by
combining individuals with different
ethical positions, and not pronounce
ments about the intrinsic value of their
stances." In a way, this was a transfer
of the tolerance principle from logic to
ethics.
N obeling wrote in one of his last let
ters to Menger that "the whole formu
lation of the question fills me with
loathing," whereas Veblen politely
evinced "doubt whether there is scope
in this field for a mathematician of
your prowess." The book's unusual
style-part letters to a friend, and part
Platonic dialogue-and its avoidance
of any commitment clashed with the
mood of the time. In retrospect, the
thirties seem the worst moment to ap
ply "social logic" to ethics. Applica
tions to economics turned out to be
much more acceptable. Menger had
anticipated them when he wrote that
"similar groups [based on individual
decisions] might also be formed ac
cording to . . . political or economic cri
teria. Groups of the last kind might not
be irrelevant in theories of economic
action." As pointed out by Leonard in
painstaking detail, this remark was not
lost on economists, and Menger's ut
terly original study of social combina
torics was to play a major role in the
birth of game theory.
Today, Menger's role in mathemati
cal economics may be seen as one of
his most original contributions. Topol
ogy and mathematical logic would
have flourished in Vienna even without
him, but not the mathematics of social
and economic problems. That such
VOLUME 22. NUMBER 1, 2000 41
problems would attract his attention was unavoidable, given his father Carl and his uncle Anton. When, on returning from the sanatorium in 1923, Karl Menger had proceeded with the revised edition of his father's magnum opus, he had made contact with Austrian economists. This so-called third generation (the first wa'l made up of Carl Menger alone) was dominated by Joseph Schumpeter and Ludwig von Mises (the brother of the applied mathematician and philosopher Richard von Mises ). Significantly, neither of the two held a chair. The professors in the economics department were not in the same league. Most of the discussions took place outside of the university, in circles, private seminars, and coffee houses.
Twenty-year-old Karl Menger had written an essay "On the role of uncertainty in economics" dealing with the two-hundred-year-old St. Petersburg
paradox. Suppose that a casino offers the following game: you throw a coin repeatedly, until "heads" comes up for the first time; if this happens on the nth throw, you receive 2n dollars. Of course you will have to pay some admission fee. How much should you be willing to pay? The first answer coming to mind is: anything less than the expected value of the gain. But this value is infinitely large. Indeed, the probability that "heads" comes up for the first time at the nth throw is 2-n,
and this yields a payoff of 2n. Hence one should be willing to stake all one's possessions to be admitted to the game. But no reasonable person is prepared to do so. Bernoulli proposed an ingenious solution to the paradox: utility does not grow linearly with the gain, but logarithmically. Menger recognised, however, that this, and indeed any other unbounded utility function, would only lead to a similar paradox.
Menger's essay, which went on to discuss how individuals differ in their evaluation of how much to pay for the chance to gain an amount D with probability p, was to inspire many mathematical economists, including John von Neumann, Kenneth Arrow, and Paul Samuelson. But when, after his return from Amsterdam, Menger lectured on the topic to the Economic Society, its president Hans Mayer ex-
42 THE MATHEMATICAL INTELLIGENCER
plicitly advised him against submitting the paper to its journal, the Zeitschrift
fur NationalOkonomie.
Mayer's assistant Oskar Morgenstern was outraged by this further proof of his professor's ineptitude. Morgenstern, who was of Menger's age, belonged to the fourth generation, together with his friend Kurt Haberler and the future Nobel Prize winner Friedrich von Hayek Like Menger, he had found support from the Rockefeller Foundation. During his extensive travels, he had been most impressed by a meeting with Edgeworth, which instilled in him the unshakeable conviction that economists needed mathematical tools. This was not then a fashionable view, and the Austrian marginalists, in particular, had traditionally shunned mathematics. Morgenstern must have found in Karl Menger an answer to his prayers; although his own mathematical training was not substantial, he quickly established contact. His zeal was unbounded. Within a few years, he became managing editor of the Zeitschrift fur
NationalOkonomie, where he had the satisfaction of publishing Menger's essay. Morgenstern had, by then, succeeded Hayek as director of the small Institute for Konjunkturforschung
(Business Cycle Research) in Vienna. This was a paradoxical appointment, for Morgenstern's main work so far had been about the impossibility of economic predictions: he claimed that the interdependence of economic decision-makers defeated forecasts and prevented equilibrium. But Cech, who often took part in the Colloquium and had proposed a notion of dimension which was to supplant, in some respects, the one by Menger and Urysohn, drew Morgenstern's attention to the minimax theorem proved by John von Neumann several years before: for zero-sum games with two players, equilibrium was consistent with perfect foresight.
In order to make his point, Morgenstern had used in many papers and lectures (including talks in the Vienna Circle and the Colloquium) the example of Sherlock Holmes and Professor Moriarty-that mathematician gone wrong-whose attempts to outguess each other apparently had to
Oskar Morgenstern (1902-1977). Born in
Berlin, a son of an illegitimate daughter of the
Prussian Emperor Frederic I, Morgenstern
studied economics in Vienna. His friend, the
economist Haberler, told him that he should
always sign his name as "Oskar Morgen
stern, Aryan" to avoid the rejection of his pa
pers by anti-semitic colleagues. Soon con
vinced of the importance of mathematics for
economics, he suffered from his lack of
knowledge in the field. "I was an idiot not to
have studied mathematics at least as a side
line at the university of Vienna, instead of this
silly philosophy." He tried to make up for this
by taking lessons from Alt and Wald. In 1 935,
he wrote in his diary: "Again a mathematical
lesson. Now we are already into differentia
tion. Wald thinks that in one year I am going
to be advanced enough to understand nearly
everything in mathematical economics."
Morgenstern did not become a mathemati
cian, but a gold mine for mathematicians, in
spiring Menger, Wald, von Neumann, Shubik,
and Schotter.
lead to an infinite regress. But von Neumann's minimax theorem offered a solution: it consisted in throwing a coin. This gave Morgenstern pause. Keener than ever on mathematical methods, he asked Menger to provide him with tutors who could remedy his own lack of expertise. Menger was happy to oblige, and sent his forn1er students Alt and Wald, who both were jobless, and with no prospects in the face of ever-rising anti-semitism. Their economic plight led to a breakthrough in
economics. Indeed, Oskar Morgenstern did more than provide his two tutors with pocket money. Both became fascinated by economics. Soon Alt wrote a paper on the measurability of the utility
John Von Neumann (1903-1957). "He darted
briefly in our domain and it has not been the
same since," said Paul A. Samuelson, the first
Nobel Prize winner in economics. Logicians,
quantum physicists, meteorologists, or com
puter scientists could say the same. John von
Neumann's paper on equilibrium theory,
which was published in the Ergebnisse in
1937, had been conceived almost ten years
earlier, during a seminar on economics which
he attended in Berlin. An eye-witness re
membered that "von Neumann got very ex
cited, wagging his finger at the blackboard,
saying, 'but surely you want inequalities, not
equations, there.' It became difficult to carry
the seminar to conclusion because von
Neumann was on his feet, wandering around
the table, while making rapid and audible
progress . . . . " In the thirties, von Neumann
frequently visited Vienna, where he had many
discussions with Menger and Godel (a histo
rian described him as a member in pectore
of the Viennese Colloquium). But his clos
est Viennese collaborator became Oskar
Morgenstern, whom he met only later, in
Princeton. According to Morgenstern, John
von Neumann was amazed at the primitive
state of mathematics in economics; he held
that if all economics texts were buried and
dug up one hundred years later, people
would think that what they were reading had
been written in the time of Newton.
function, and Wald laid the foundations for general equilibrium theory.
Equilibrium in a
Collapsing Country
Menger had arranged for Wald to coach not only Morgenstern, but also Karl Schlesinger, a banker. Schlesinger was attracted by the so-called imputation problem, central to the theories of Carl Menger and Leon Walras: how do the prices of the products determine the prices of the factors of production? (This problem was the reverse of that studied by classical economists, who assumed that the prices of the factors determined the prices of the products.)
Walras, and later Cassels, had established a system of as many equations as there were unknowns. This was believed to ensure the existence of a unique equilibrium. Menger, of course, knew that it was not sufficient.
Schlesinger was the first to lecture in the Colloquium on the fundamental equations of W alras and Cassels, stressing that all the factors which occurred in the equations had to be "scarce" in the sense that they were entirely used up in the production; for if there remained a surplus, that factor would cost nothing. However, whether a surplus remains or not depends on the production process. This led
Schlick's Assassination. After the death of Hahn and the exile of Neurath in 1 934, the Vienna
circle lost the third of its founders in June 1936, when the philosopher Moritz Schlick was
shot on the steps of the University. Aristocratic Schlick had been bom in Ber1in {sad, but true,
as he said to Menger) and was "extremely refined, sometimes introverted." "Serenity is our
duty" was Schlick's motto. A few weeks before being murdered, Schlick told Menger that he
had been threatened for years by a paranoiac who had been in and out of mental institu
tions. The police had assigned him a bodyguard for some time; but as an actual assault had
never taken place, he did not dare tum to the police again. Schlick added with a forced smile:
"I fear that they begin to think it is I who am mad." The psychopathic killer was Johann NelbOck,
who had studied philosophy and mathematics, and had written his thesis on "The meaning
of logic in empiricism and positivism" under Schlick's supervision. Nelbock had felt thwarted
by Schlick, both in his love for Sylvia Borowicka (another student of philosophy) and in his
career; but at the trial, he managed to persuade the jury that he had killed the free-thinker
Schlick for ideological reasons. He was sentenced to 10 years and released right after the
Anschluss, having pointed out that "his deed, the elimination of a teacher spreading Jewish
maxims alien and pernicious to the people, had rendered a service to National Socialism."
VOLUME 22, NUMBER 1 , 2000 43
Schlesinger to propose a system of equations and inequalities in lieu of Cassels's system of equations. Taking this new system as his point of departure, W aid proved the existence of a unique positive solution-an equilibrium. This was a giant step forward. Both Morgenstern and Menger grasped the significance of Wald's result and did their best to spread the news.
John von Neumann had passed through Vienna a few times during the thirties, usually on the way to or from his native Budapest. When he was told of Wald's breakthrough, he published in the Ergebnisse his own analysis of a model for an expanding economy. It transpired that he had grasped the role of inequalities in models for production at an even earlier date, and had lectured on it in Princeton, apparently without impressing the economists. In his dynamical model, he described a closed production loop: the supply is the output of the preceding period, and the demand is the input of the following period. John von Neumann had proceeded to prove the existence of an equilibrium solution by a generalisation of Brouwer's fixed point theorem, underscoring the connections with his own minimax result. His Colloquium paper became a milestone in economics-at least half a score of Nobel Prizes drew on it.
This paper was the last article that appeared in the Ergebnisse. Wald had finished a further manuscript on mathematical economics which he planned to bring out in the following, ninth, volume of the series. It contained a proof of the existence of an equilibrium in a pure exchange economy, again based on Brouwer's fixed-point theorem. But this paper was never to appear, and the manuscript vanished in the turmoil of the times.
Hitler had struck Mussolini, embroiled in his Abyssinian fiasco and badly needing allies against the League of Nations, had decided to stop annoying the Nazis with his protection of Austrian sovereignty. The pressure from Germany now became overwhelming. In March 1938, the chancellor Schuschnigg, whose diplomatic efforts had led to total isolation, at long last decided to tum to his own people for support, and
44 THE MATHEMATICAL INTELLIGENCER
organised a plebiscite, firmly expecting a vote for independence. Hitler must have expected that outcome too, and launched his troops to prevent it. Pleasantly surprised to fmd welcoming crowds, he annexed Austria on the spot. The plebiscite, now phrased in Hitler's own terms, brought an overwhelming majority in favour of the Anschluss, probably due only in small part to the offices of Gobbels and Gestapo.
Menger watched this catastrophe, which he had seen coming for years, from abroad. In 1935, he had married his long-time sweetheart Hilda Axamit, a student of actuarial mathematics, and in the following year his son Karl Jr. had been born. Convinced of the hopelessness of his situation in Vienna, and deeply shocked by the assassination of Moritz Schlick, he gratefully accepted an offer from the University of Notre Dame. Morgenstern happened also to be in the US at the time of Austria's annexation, and soon learned that he was now blacklisted as "politically undesirable" in Vienna. He quickly obtained a position as lecturer in Princeton, but found his new colleagues as unwilling as the Austrian economists to engage in mathematics. Fortunately, the Institute for Advanced Study was only a short walk away.
Immediately after the Anschluss, in a cable sent from South Bend, Menger resigned from his professorship in Vienna. In part through his efforts, Alt and Wald were able to escape. (The latter lost all but one of his relatives in the holocaust). Karl Schlesinger had committed suicide on the day Hitler's troops entered town. Kurt Godel managed to leave Austria in the fall of 1938
for a visit to Princeton, and spent the spring of 1939 with Karl Menger at Notre Dame. But then, in spite of Menger's fervent pleas, he insisted on returning to Nazi Vienna, although Hitler, who had marched into Czechoslovakia, was now obviously preparing for war against Poland. Menger's feelings for Godel were irremediably upset. But Godel, forever secretive, had left a wife back home and wanted to fetch her. In Vienna, thugs mistook him for Jewish and knocked his glasses off in the street. More threateningly still, the Wehrmacht
deemed him fit for duty. Eventually, the GOdels managed against all odds to leave the German Reich and reach the safety of Princeton in 1940, after travelling around a world already torn by war. Asked by Morgenstern how things were back in Vienna, Godel replied that "the coffee was wretched."
Menger's career lost some of its momentum after emigration. His attempts at reconstructing something like the Colloquium or the Circle at Notre Dame did not live up to his expectations. He kept producing first-class research (introducing, for instance, fuzzy metrics, probabilistic geometry, and what has become known as Menger algebras), he had outstanding co-workers such as Bert Schweizer and Abe Sklar, and he certainly held a respected rank within the American mathematical community, but he did not share the tremendous success of some fellow emigrants like John von Neumann, Stanislaw Ulam, and Abraham W ald.
During the war, Menger published little, not because his work was classified like that on computers or the bomb, but because his enormous teaching load made research almost impossible. Menger was engaged in the mathematical training of Navy cadets, an experience that induced him to discuss critically the usual approaches towards teaching calculus, and to devise some more transparent notations. But Menger's crusade did not vanquish the inertia of tradition, and what he termed the "x-itis" of calculus curricula continues to mar the classroom experiences of students today. The textbook on calculus published by Menger in 1955 soon vanished from the market.
After the war, the University of Vienna did not invite Menger to return. As father of four children (of which three were US-born), he could hardly be expected to live in a devastated town. Or could he? Tactful authorities decided it was better not to ask After all, Menger had resigned voluntarily, and there was a cable to prove it.
REFERENCES
An excellent biographical introduction is
Seymour Kass (1 996), Karl Menger, Notices of
the AMS 43, 558-561 .
A U T H O R S
LOUISE GOLLAND Networking Services & Information
Technologies
The University of Chicago
Chicago IL 60637
e-mail: [email protected]
Louise Galland studied mathematics
at the Illinois Institute of Technology,
where she was inspired by the lec
tures of Menger. She received her
Ph.D. in history from the University of
Chicago . special izing in the history of
science. She is an independent
scholar in the history of mathematics
and astronomy, while continuing to
work for the University.
For further material, see K. Menger, Reminis
cences of the Vienna Circle and the
Mathematical Colloquium, Vienna Circle
KARL SIGMUND lnst�ut fOr Mathematik
Universitat Wien
1090 Vienna
Austria
e-mail: [email protected]
Karl Sigmund , a former ergodic-the
orist turned biomathematician, has
written a popular book (The Games
of Ute. Penguin) on evolutionary
game theory. This is h is second
fntelfigencer art icle on mathemati
cians in the Vienna Circle. He admits
he is hopeless at the Austrian national
sports, skiing and waltzing, but he
tries to make up for it by devoted ly
frequenting coffee houses.
Collection vol. 20, Kluwer, Dordrecht (1 994);
K. Menger, Selected Papers in Logic and
Foundations, Didactics, Economics, Vienna
Circle Collection val . 1 3, Kluwer, Dordrecht
(1 979); and the recent reprinting of the
Ergebnisse eines mathematischen Kol/o
quiums (eds. E. Dierker and K. Sigmund,
Springer, Wien 1 998), with contributions
from G. Debreu, · K. Sigmund, W. Hildebrand, R. Engelking, J .W. Dawson, Jr . .
and F . Alt.
For more on the Vienna Circle, see K. Sigmund:
A philosopher's mathematician-Hans Hahn
and the Vienna Circle, Mathematical fnteffi
gencer 1 7 (4), 1 6-29 (1 995). The authorita
tive biography on G6del is by J.W. Dawson,
Jr. Logical Dilemmas: the fife and work of
Kurt Godel, Peters, Mass. (1 997).
There is an enormous literature on the eco
nomics aspect. For a start see E. Craven,
The emigration of Austrian economists,
Hist. of Political Economics 1 8 (1 989),
1 -32, as well as M. Dore, P. Chakravarty,
and R. Goodwin (eds), John von Neumann
and Modern Economics, Oxford UP
(1 989); and in particular the articles by K.J.
Arrow, Von Neumann and the Existence
Theorem for General Equilibrium (pp.
1 5-28); P.A. Samuelson, A Revisionist
View of von Neumann 's Growth Model
(pp. 1 00-124); and L.F. Punzo, Von
Neumann and Karl Menger's Mathematical
Colloquium, (pp. 29-68). Karl Menger's
contribution to game theory is highlighted
in R .J . Leonard's essays: From Parlor
Games to Social Science, J. of Economic
Literature 23, 730-761 , and: Ethics and
the Excluded Middle: Karl Menger and
Social Science in Interwar Vienna, Isis 89
{1 998), 1 -26.
VOLUME 22, NUMBER 1 , 2000 45
A. K. DEWDNEY
The P an iverse Project : Then and Now
• s a two-dimensional universe possible, at least in principle? What laws of physics � might work in such a universe ? Would life be possible? It was while pondering such
imponderables one steamy summer afternoon in 1980 that I came to the sudden con
clusion that, whether or not such a place exists, it would be possible to conduct a
gedanken experiment on a grand scale. It was all a question of starting somewhat mathematically. With the right basic assumptions (which would function like axioms), what logical consequences might emerge?
Perhaps the heat was getting to me. I pictured my toy universe as a balloon with an infinitesimal (that is to say, zero-thickness) skin. Within this skin, a space like ours but with one dimension less, there might be planets and stars, but they would have to be disks of two-dimensional matter. In laying out the basic picture I followed informal principles of simplicity and similarity. Other things being equal, a feature in the planiverse should be as much like its counterpart in our universe as possible, but not at the cost of simplicity within the two-dimensional realm. The simplest two-dimensional analog of a solid sphere is a disk.
What sort of orbits would the planets follow? In our own universe, Newtonian mechanics takes its particular form from the inverse-square law of attraction. A planet circling a star, for example, "feels" an attraction to that star which varies inversely with the square of the distance between the two objects. The same reason in the planiverse leads
46 THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK
to a different conclusion. The amount of light that falls on a linear meter at a distance 2x from a star is one-half the light that reaches the square at a distance x from the star. (see Figure 1); correspondingly, attraction is proportional to the inverse first power of the distance
The resulting trajectory is not a conic section, but a wildly weaving orbit, as in Figure 2.
The figure resembles a production of that well-known toy, the spirograph, in which gears laid on a sheet of paper roll around each other. A pencil inserted in a hole in one of the gears might trace such a figure. Are the twodimensional orbits spirograph figures? Probably not. They look like epicycles, the paths that early astronomers thought might explain the looping orbits of Mars and Jupiter in an Earth-centered system! (It is tempting to conclude that what goes around comes around.)
Encouraged by such speculations, I begin to develop the impression that such a universe might actually exist. It would be completely invisible to us three-dimensional beings, wherever it might be. But places, even imaginary ones, need names. What could a two-dimensional universe be, but the Planiverse?
X
2x Figure 1 . The law of gravity.
In a fit of scientific irresponsibility I sent a letter to Martin Gardner, then author of the Mathematical Garnes column for Scientific American magazine. I included several speculations, including the drawing of a two-dimensional fish shown in Figure Three below.
Gardner wrote back, saying that he not only found the planiverse a delightful place, he would devote a forthcoming column to it. His column, which appeared in July, 1980, lifted our speculations about two-dimensional science and technology to a new level by bringing it to the attention of
a much wider public. Among those who read Gardner's column were not only scientists and technologists, but average readers with novel and startling contributions of their own.
I left for a sabbatical at Oxford that summer, hoping to work on the theory of computation and hoping also to get away from the planiverse project, which was claiming more and more of my time. I stayed in an abbey in the village of Wytham, near Oxford. There was leisure not only to work on the logical design for an entirely new way to compute things, but the opportunity to work on the Planiverse Project,
Figure 2. Orbit of a two-dimensional planet.
I I
----: I
�F� - - - - - - - - - - - - - - - - - - -meter
a paper symposium with colleague Richard Lapidus, a physicist at the Stevens Institute of Technology in New Jersey. Our symposium had contributions from around the world on everything from two-dimensional chemistry and physics to planetary theory and cosmology. There was, moreover, a section devoted to technology, wherein the only feasible two-dimensional car ever designed appeared for the first time. It had no wheels, but was surrounded by something like a tank tread that ran on disk-bearings. The occupants got in and out of the vehicle by unhooking the tread.
The Planiverse Project was now proceeding at a satisfying rate. I assumed that within a few years it would die away to nothing. We would have had our fun, no harm done.
But a press release, written by a journalist at my home institution in the fall of 1981, changed all that. Wire services
Figure 3. Two-dimensional fish.
VOLUME 22. NUMBER 1 , 2000 4 7
Figure 4. Planiversal vehicle.
picked it up with the glee reserved for UFO reports and es
caped lions. There followed a rush of magazine and news
paper articles, as well as television stories publicizing our
two-dimensional world. In particular, a piece in Newsweek
magazine caught the attention of publishers.
In the midst of a series of papers on programming logic,
I was suddenly face to face with a big writing job. There
were contracts with Poseidon Press (Simon & Schuster) in
the US, with Pan/Picador in England, and with McClelland
& Stewart in Canada. I viewed these new responsibilities
with irritation. It was assuredly fun to think about the plani
verse, but my research came first. And was I not in danger
of being regarded as a nut-case? The media were no help.
One interviewer asked, "So, Professor Dewdney. Are you
saying the Earth is flat after all?" (He was serious!)
The writing job, as I finally came to view it, would have
to weave together all the scientific and technical elements
that had emerged from the Planiverse Project. But a com
pendium of these speculations, no matter how wild or en
tertaining, would surely prove a dry read. It would have to
be a work of fiction, set in the planiverse itself. There would
be a planet called Arde, a disc of matter circling a star
called Shems. There would be a hero named Yendred ( al
most my name backwards) and his quest for the third di
mension or, at least, a spiritual version of it. Yendred is
convinced that the answer to his quest lies on the high
Figure 5. Yendred, a typical Ardean.
48 THE MATHEMATICAL INTELLIGENCER
plateau of Arde's lone continent (a requirement of two
dimensional plate tectonics).
All the elements of our earlier speculations now fell
more or less into place. Think for a moment of even the
humblest respects in which a two-dimensional existence
on the "surface" of Arde might differ from our own.
The Jordan curve theorem's implications for Arde were
profound. Closed curves lurked everywhere.
Consider, for example, Ardean soil, a mechanical mixture
of two-dimensional grains and pebbles in which any pocket
of water fmds itself permanently trapped within the closed
circle of surrounding stones. The water cannot percolate, as
our groundwater does, up or down. It is trapped, at least un
til the soil is mechanically disturbed. Consider also the sim
ple matter of Y endred attempting to lift a two-dimensional
plank on the Ardean surface. The plank, the ground, and
Yendred himself would form a simple closed curve, and the
air trapped inside the enclosed space would become in
creasingly rarefied. The plank would seem to get heavier and
heavier. Perhaps readers can imagine themselves to be
Ardeans lifting such a plank If you were Y endred, what tech
nique would you adopt to make it easier?
But for every disadvantage of life in two dimensions,
there seems to be an equal and opposite advantage. Bags
and balloons are trivial to make-from single pieces of
string! Yendred's father, who takes him fishing near the be
ginning of the book, never has trouble with tangled lines,
for knots in two-space are impossible. Moreover, sailing re
quires nothing more than a mast!
Y endred sets out on his quest shortly after the fishing
trip with his father. His home, like all Ardean homes, is un
derground. The surface of Arde must be left as pristine as
possible. There are travelling plants and periodic rains
which make temporary rivers, basically floods. Any surface
structure would either disrupt the delicate one-dimensional
ecology or be swept away, in any case. A simple pole stuck
in the ground would become a dam which could never with
stand the force of kilometers of water that would rapidly
build up behind it.
In the Ardean cities which Yendred must walk through
(or over) on his travels to the high plateau, we encounter the
acme of two-dimensional infrastructure. There is no skyline,
only the typical Ardean surface periodically marred by traf
fic pits. If an eastbound Ardean should happen to encounter
a westbound colleague, one of them must lie down and let
the other walk over him/her. Elaborate rules of etiquette dic
tate who must lie down and who proceed, but in an urban
context there is no time for niceties. Whenever a westbound
group of Ardeans encounters a west-pit, they descend the
stairs, hook up an overhead cable and wait. At the sound of
a traffic gong, an eastbound group marches across the cable.
What would be a tightrope act in our world amounts to lit
tle more than a springy walk in two dimensions for the east
bounders. West-pits and east-pits alternate so that neither di
rection has an advantage over the other.
From a privileged view outside the Planiverse, the "sky
line" of an Ardean city resembles an inverted Earth-city sky
line. Yendred passes over numerous houses, apartment
Figure 6. Ardean sailing vessel.
buildings, and factories, marked only by the exit or entrance of fellow citizens bent on private tasks like so many two-dimensional ants. Overhead pass delivery balloons, each with its cargo of packages. Balloon drivers adjust to near-neutral buoyancy, then take great hops over their fellows.
Access to underground structures is managed by swingstairs. Although some of the larger structural beams are held together by pegs, the fastener of choice is glue. Wires (yes, the Ardeans have electricity) run only short distances, from batteries to appliances. Electrical distribution is out of the question since power lines would trap everyone within their homes. Reading by the feeble glow of a battery-powered lamp, an Ardean might reach for his favorite book, reading text that resembles Morse Code, one line per page. This demands a highly concentrated prose style that is more suggestive than comprehensive.
The population of Arde is not great. Only a few thousand individuals inhabit its lone continent. Consequently, the Ardeans have no great demand for power machines, the steam engine sufficing for most needs, such as elevators and factories. Readers might be able to figure out the operation of an Ardean steam engine from the accompanying illustration alone.
A boiler converts water into steam, and when a valve opens at the top of the boiler, the steam drives a piston to the right. However, this very motion engages a series of cams that close the valve. The steam then enters a reservoir above the piston and escapes when the piston completes its travel to the head of the "cylinder." Interestingly, almost any two-dimensional machine can also be built in
three dimensions. It must be given some thickness, of course, and it must also be enclosed between two parallel plates to simulate the restriction of no sideways movement. I have often wondered whether we could build a car with a one-inch thick steam engine mounted underneath. Think of the additional room that would provide!
Ardean technology is a strange mixture of advanced and primitive machines. Although steam engines are the main
Figure 7. A steam engine.
VOLUME 22, NUMBER 1 , 2000 49
power source, rocket planes travel from city to city, while
space satellites orbit overhead. It is absurdly easy to make
space stations airtight. Any structure that contains at least
one simple closed curve is automatically airtight.
And of course, there are computers! These operate on
the same binary principles (0 and 1) as our own do. Ardean
technologists had a difficult time developing the appropri
ate circuits, however, owing to the impossibility of getting
wires to cross each other. One brilliant engineer finally hit
on the idea of a "logic crossover." Symbolically rendered
below, this circuit consists of three exclusive-or gates, each
transmitting a logic 1 signal if and only if exactly one in
put is a 1. No matter what combination of zeros or ones enter this
circuit along the wires labelled x and y, the same signals
leave the circuit along the wires bearing these labels.
Readers may readily satisfy themselves that if x and y both
carry a zero (or one), for example, then both output lines
will also carry this signal. But if x is one and y is zero, the
middle gate will output a one which will cancel the x-sig
nal in the upper gate and combine with the zero on the y
input in the lower gate to produce a one.
Fun though technology may be, it isn't until he visits the
Punizlan Institute of Technology (PIT) that Yendred en
counters the deep scientific ideas of his time. Scientists at
PIT have developed a periodic table of the elements based
on the theory that while just two electrons can occupy the
frrst shell of a planiversal atom, up to six can occupy the
second shell. We have labelled the planiversal elements
with the symbols of the elements from our own universe
which they most resemble.
Strangely, the planiversal elements quickly run out, ow
ing to the instability of very large planiversal atoms. In the
planiverse, one simply cannot pack as many neutrons and
protons into a small space as one can in our universe.
Consequently, nuclear forces (other things being equal)
must act across larger distances and the nuclear compo
nents are rather less tightly bound. Quite possibly, there is
a lot more radioactivity in the planiverse than in our own.
Other strange features of the planiverse include rather
low melting points and the strange behaviour of sound
waves. Low melting points might militate against the pos
sibility of life, except that chemical reactions proceed at
lower temperatures, in any event. Sound waves travel much
farther and have a very strange property frrst deduced by
� H 3 Li 9 Na 15
K 25 Rb
35 Cs
4 Be 10 Mg 16 X 26 X
36 X
37 138 e9 140 X X X X Figure 9. Planiversal table of the elements.
50 THE MATHEMATICAL INTELLIGENCER
17 X 27 X 41 X
18
Figure 8. A logic crossover.
Earth scientists some time ago. If one sounds a note on
Arde, the sound wave alters as it travels. A sharp attack
smears out in time, so that a single note of C, for example,
is heard at a distance as a glissando rising from some lower
pitch and asymptotic to C.
Cosmologically speaking, Ardean scientists have much
to ponder. Like us, they wonder if their universe is closed
like a balloon (we say it is) or open like a saddle-shaped
space. It is apparently expanding, and the balloon analogy,
so often used to illustrate how our own universe is appar
ently expanding, can be taken quite literally. A deeper ques
tion concerns the orientability of the planiverse. Perhaps
it is really a projective plane, so that Yendred, travelling by
rocket across the planiverse, might return to fmd that
everyone has reversed their handedness and all Ardean
writing appears backward.
As for space travel, another problem awaits the rocket
voyager. There is no escape velocity in the planiverse. The
amount of work required to escape the gravitational field
of an isolated planet is infinite! (Try integrating llx from 1 to infmity.) However, if one can travel far enough to fall
under the gravitational influence of some other body, the
infmite escape velocity no longer matters.
The Planiverse Project had the most fun designing two
dimensional life forms. Readers who turn back to the
picture of the fish (Figure 3) will fmd a creature with a
well-developed exoskeleton, like an insect, and with a rudi
mentary endoskeleton, as well. The key anatomical com
ponent in any two-dimensional life form is the zipper or-
� He 5 6 7 8
c 0 F Ne 1 1 12 13 14 Si s Cl Ar 19 20 21 22 23 24
X X X X X Br Kr 28 29 30 31 32 33 34 X X X X X I Xe 42 43 44 45 46 47 48 X X X X X At Rn
Other Attempts at Two-Dimensional Universes
The Planiverse has had a long evolutionary history, marked by previous books on two-dimensional worlds. The first of these was Flatland, written in 1884 by Edwin A. Abbott, an English clergyman. Some years later, in 1907, Charles Hinton, an American logician, wrote An Episode of Flatland, which reorganised Abbott's tabletop world into the somewhat more logical disk planet that he called Astria. Much later, in 1965, Dionys Burger, a Dutch physicist, published Sphere
land, which attempted to reconcile Abbott's and Hinton's worlds and then to use the resulting two-dimensional universe to illustrate the curvature of space.
For all their charm, these books have various shortcomings. Abbott made no attempt to endow his universe with coherent physics. His beings float about in two-space with no apparent mode of propulsion. Being geometrical figures, they have no biology at all. Hinton's universe is rather more like the planiverse, his planet being a disk But Hinton, immersed in a sort of socialist Utopian fantasy, keeps forgetting the restrictions of his characters' two-dimensionality, seating his characters "side by side" at a banquet, for example. Berger attempts to reconcile the two previous universes, but he is really after just an expository vehicle to illustrate various ideas about space and physics.
gan, two strips of interdigitating muscle that meet to form a seam. Just inside the fish's bony jaws, for example, the muscles which crush and chew the prey also part to admit its fragments into a digestive pouch. Because portions of the two muscles are always in contact, structural integrity is maintained. The fragments are enclosed in a pocket that travels along the seam from front to back
Yendred, after many adventures, fmally reaches the high plateau and meets the mysterious Drabk, an Ardean who has developed the ability to leave the planiverse entirely and move "alongside" it, so to speak Since The Planiverse
is about to re-appear, I will not give the plot away, but I had better mention the deus ex machina that makes it all possible: In the book a class project results in a program called 2DWORLD that simulates a two-dimensional world, including a disk-shaped planet the students call Astria. Imagine the student's surprise when 2DWORLD turns out to be a sophisticated communication device which, by a Theory of Lockstep, begins to transit images of an actual two-dimensional universe, including a planet called Arde and a being called Yendred!
When The Planiverse first appeared 16 years ago, it caught more than a few readers off guard. The line between willing suspension of disbelief and innocent acceptance, if it exists at all, is a thin one. There were those who wanted to believe (despite the tongue-in-cheek sub text) that we had actually made contact with a two-dimensional world called Arde.
It is tempting to imagine that those who believed, as well
as those who suspended disbelief, did so because of the persuasive consistency in the cosmology and physics of this infmitesimally thin universe, and in its bizarre but oddly workable organisms. This was not just your run-ofthe-mill science fiction universe fashioned out of the whol� cloth of wish-driven imagination. The planiverse is a weirder place than that precisely because so much of it was worked out in the Planiverse Project. Reality, even the pseudo-reality of such a place, is invariably stranger than anything we merely dream up.
REFERENCES
Edwin A. Abbott, Flatland: A Romance of Many Dimensions. Princeton
University Press, Princeton, 1 991 .
Charles H. Hinton, An Episode of Flatland. Swan Sonnenschein & Co.
London, 1 907.
Dionys Burger, Sphere/and: A Fantasy About Curved Spaces and an
Expanding Universe. Thomas Y. Crowell Company, New York, 1 965.
A. K. Dewdney. The Planiverse: Computer Contact with a Two
Dimensional World. Poseidon Press (Simon & Schuster), New York,
1 984. A new edition soon by Copernicus Books (Springer Verlag),
New York, 2000.
A U T H O R
A.K. DEWDNEY
Department of Computer Science
University of Western Ontario
London, Ontario N6A 587
Canada
e-mail: [email protected]
A.K. (Kee) Dewdney was born in London, Ontario, and did un
dergraduate work there. He then did graduate work at the
Universities of Waterloo and Michigan, completing his PhD at
Waterloo in 1 97 4. His thesis, extending some graph-theoretic
thecrerns to higher dimensions, did not concern computers, and
Dewdney was pleased to discover that Oike much of discrete
mathematics) it counted as computer science and brought him
close to theory of computation. His service as columnist for
Scientific 1\mertcan tended to crowd out his other activities, but in recent years he can follow his many interests, having taken
early retirement at UWO and no longer having those Scientific
Arnertcan monthly deadlines. Among his books is A Mathematical
Mystery Tour ry.Jiley, 1 999), which seeks to answer the notori
ous question, Is mathematics discovered or created?
VOLUME 22, NUMBER 1 , 2000 51
[email protected]§,flh£ili.ilhtil D i rk H uylebrouck, Editor I
Alphabetic Magic Square in a Med ieval Church Aldo Domenicano and
Istvan Hargittai
Does your home town have any
mathematical tourist attractions suck
as statues, plaques, graves, the cafe
where the famous conjecture was made,
the desk where the famous initials
are scratched, birthplaces, houses, or
memorials? Have you encountered
a mathematical sight on your travels?
If so, we invite you to submit to this
column a picture, a description of its
mathematical significance, and either
a map or directions so that others
may follow in your tracks.
Please send all submissions to
Mathematical Tourist Editor,
Dirk Huylebrouck, Aartshertogstraat 42,
8400 Oostende, Belgium
e-mail: dirk. huylebrouck@ping. be
A lphabetic magic squares, often consisting of a square array of let
ters symmetrical with respect to both diagonals, occur frequently in the Christian and Islamic tradition. Their origin probably dates back to the neopythagorean and neoplatonic doctrines (1st century B.C.-6th century A.D.). The magic was probably associated with the self-contained character of the text, which, because of the symmetry of the array, can be read both horizontally and vertically, starting from either the top left or the bottom right corner of the square. The text can also be read as a boustrophedon, yielding a somewhat different order of words. When the text is Greek or Arabic, numerical values can be attached to the letters in a straightforward manner [1 ] .
There is a well-preserved magic square in San Pietro ad Oratorium, a beautiful medieval church a few kilo-
meters south-east of the small town of Capestrano (Abruzzi, Italy). The building, as we see it now, dates from the 12th century, when a previous church, dating from the 8th century, was extensively renovated [2]. The magic square is carved in a block of limestone. This is inserted upside-down in the facade (Figure 1), and is thus likely to have originated from the previous building.
The 5 X 5 array of letters is shown in Figure 2. The text consists of the following five words: ROTAS OPERA TENET AREPO SATOR. Of these, four are certainly Latin [ROTAS = wheels (accusative); OPERA = work (nominative or ablative); TENET = keeps; SATOR = sower (nominative)] . The remaining word, AREPO, is not Latin though it recalls the Latin word ARATRO = plough (ablative). In any case, the meaning of the text remains obscure. Magic squares like the one de-
Figure 1 . Detail of the facade with the entrance of the church San Pietro ad Oratorium, a few
kilometers south-east of Capestrano (Abruzzi, Italy}. The position of the block with the magic
square is fourth from the left and seventh from the bottom. (Photographs by the authors.)
52 THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK
Figure 2. The block with the magic square (in the wall it is positioned upside-down).
scribed here are often found in medieval religious buildings. However, the earliest example is from Pompeii, i.e., before A.D. 79, when Pompeii was destroyed by an eruption of Mt. Vesuvius. This suggests the possibility of a pre-Christian origin. The popularity of this magic square in the Christian tradition may have been enhanced by the Latin words PATER NOSTER, easily reconstructed using its letters [1 ] . ("Pater noster" (Our Father) is the beginning of the Lord's Prayer.) These words can be identified twice, reusing the unique letter N. The unused letters A and 0 can be taken as standing for alpha (the beginning) and omega (the end).
The symmetry properties of the 5 X 5 array cause the words ROTAS and OPERA to become SA TOR and AREPO, respectively, when read backwards. The word TENET is palindromic, as it reads the same from either end. These properties recall the duality relationship of the five regular polyhedra, originating from their symmetry and the interchanged roles of their vertices and faces. According to this relationship the icosahedron is the dual of the dodecahedron, the octahedron is the dual of the cube, and the tetrahedron is the dual of itself. The regular polyhedra were first described by Plato [3], and
were certainly known to his followers. They played a fundamental cosmogonic role, as they were associated with
the four elements (earth, water, air, and fire) and the universe.
REFERENCES
1 . G.R. Cardona, Storia Universale della
Scrittura, Mondadori, Milano, 1 986, pp. 66-69 and 291 .
2. M. Moretti , Architettura Medioevale in
Abruzzo, De Luca, Roma, pp. 36-41 (year
of publication unknown).
3. Plato, Timaeus, LII I-LVI.
Aldo Domenicano
Department of Chemistry, Chemical
Engineering and Materials
University of L'Aquila
1-671 00 L'Aquila
Italy
Istvan Hargittai
Institute of General and
Analytical Chemistry
Budapest Technical University
H - 1 521 Budapest;
Hungary
e-mail: [email protected]
VOLUME 22, NUMBER 1, 2000 53
MICHAEL LONGUET -HIGGINS
A Fou rfo d Po i nt of Concurrence Lyi ng on the Eu er Li ne of a Triang e
• n mathematics, it occasionally happens that a subject thought to be completely worked
~ out yields a surprising new result, indicating some possibly deeper relationships still
to be discovered. Such may have occurred with the geometry of the triangle in the
Euclidean plane-a subject inaugurated by Greek geometers, given new life by Euler
and other celebrated mathematicians in the eighteenth and nineteenth centuries\ and, since the middle of the twentieth, largely abandoned.
The present author's interest in the subject was rekindled by a recent article by Hofstadter [9], which summarizes the properties of the most "notable" points of a general triangle ABC in the Euclidean plane. The cast of characters is as follows (see Fig. 1):
1 . The circumcenter 0 of ABC is the meet of the three perpendicular bisectors of the sides BC, CA, and AB.
1 For a history of the subject before 1 900, see Simon [1 2]; also the bibliographies
cited by Vigarie [14]. Some excellent historical notes will be found in [2] and [3].
54 THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK
2. The orthocenter H is the meet of the three altitudes; that is, the lines through a vertex, say A, and perpendicular to the opposite side BC.
3. The median point M is the meet of the three lines joining the vertices A, B, and C to the midpoints A', B', and C' of the opposite sides.
4. The nine-point center 0' is the circumcenter of the triangle A 'B 'C '.
It was noted by Euler [7] that the three points 0, M, and H are collinear and spaced in the ratios 1:2; see Figure 1. The fourth point 0' must lie on the same line; for a homotheticity, center M, takes A 'B 'C' into ABC.
One other notable point of ABC is sadly left out of the
w;t§ii;JIM A
.. - - - _ .. -- _ .. 0� M
H
B A' c
above scheme; unlike the others, it does not lie on the Euler line. This is
5. The incenter I, the center of the circle touching the sides of ABC internally.
On the other hand, I does lie on a line though M containing two other notable points; see Figure 2. One of these is the incenter I' of the triangle A 'B 'C ' (also called the Spieker point; see [2], [3] , and [ 13]). The other notable point is the Nagel point of ABC, which will be defmed later.
The four collinear points IMI 'N are also spaced in the ratio 2 : 1 : 3. Indeed, the similarly between the segments OMO 'H and IMI 'N has led Hofstadter [9] to extend the figure by considering these two segments as two of the median lines of a complete triangle. However, none of the additional points in the scheme appears to be related directly to the original triangle ABC.
Motivated by Hofstadter's article [9], I began to explore some typical triangles, with ruler and compass, in the hopes of finding any unsuspected relationships involving the incenter I. It seemed to me that the previous cast of characters needed to be augmented. A promising candidate was a little-noticed point of concurrence related to the incenter and defined as follows; see Figure 3. Let X, Y, and Z de
note the points of contact of the incircle with the three sides BC, CA, and AB, respectively, of the triangle. Of course X, Y, and Z may be constructed to be the feet of the perpendiculars from I to the three sides of ABC. Then, because the two tangents from each vertex of the triangle are equal in length, it follows that
AY · BZ · CX _AZ
_·_B_X_
· -C
-Y =
1'
and so by the converse of Ceva's theorem (see [2, p. 160]) the lines AX, BY, and CZ are concurrent. The point of concurrence is called the Gergonne point of the triangle ABC,
after J. D. Gergonne who apparently was the first to notice it.2 We shall denote it by G. One can easily verify that G
2J. D. Gergonne (1 771-1 859) was the founder and first editor of Annales de Mathematiques (GAM.), Paris, 1 81 0-1 832.
ii@ii;JIM A
B A' c
lies neither on the Euler line OMO 'H nor on the Spieker line IMI 'N. Indeed, relatively few properties of G seem to have been unearthed.
In addition to the incircle, the three sides of ABC are also touched by three excircles with centers IA, IB, and I c.
If XA, YA, and Z A denote the feet of the perpendiculars from IA, say, to the sides of ABC, it follows, by the same argument, that AXA, BYA, and CZA will meet in a point GA, and similarly for the other excenters h and I c. In this way, to each excenter there corresponds a Gergonne point, as in Figure 4.
Upon constructing experimentally the four Gergonne points G, GA, GB, and Gc, the author noticed that the four lines IG, IAGA, IBGB, and leGe appeared to pass through the same point L (see Fig. 4). Moreover, L and H were equidistant from the circumcenter 0, but in opposite directions. Thus, L seemed to lie on the Euler line. Could this really be true?
Altogether, four different proofs of this conjecture have been constructed so far. The first proof made rather pedestrian use of cartesian coordinates having one vertex A as the origin and a side AB as one axis. The proof involved some heavy algebra. The second proof, also algebraic, used symmetrical, areal coordinates, with advantage. The third proof was, desirably, more geometrical and used the prop-
13tdii;IIM A
B X c
VOLUME 22. NUMBER 1, 2000 55
w;tgii;IIM
'
; ;
' ' ', '
' '
; ; ; ; ;
/ '
/ /
; ; ; '
' '
'
' ' ', '
' ' ' '
' ' '
,, I .} I
... lA
' ' L I
erties of two pencils of rays having the same cross-ratio, but with one ray in common. The proof is too long to be given here. The fourth and shortest proof goes as follows.
The trilinear coordinates of a point P in the plane of a triangle ABC may be defined [4, 10] as the lengths (x, y, z)
of the perpendiculars from P to the three sides BC, CA, and AB. By convention, x, y, and z are positive when P lies inside the triangle.
Thus, the incenter I, being equidistant from the three sides, has coordinates
I = r(l, 1 , 1), (1)
where r is the radius of the incircle. For the circumcenter
0, since LBOC = 2 X LBAC, it will be seen that
0 = R( cos A, cos B, cos C),
where R is the circumradius and A, B, and C are the angles at the vertices of ABC. For the ortlwcenter H, let D,
1$1311;14¥
H
56 THE MATHEMATICAL INTELLIGENCER
A
B D A' c
E, and F denote the feet of the altitudes AH, BH, and CH;
see Figure 6. Then, HDCE, for example, is a cyclic quadrilateral; hence, LBHD = C. Therefore,
HD = ED cot C = AB cos B cot C.
Because AB = 2R sin C, this gives us
H = 2R( cos B cos C, cos C cos A, cos A cos B).
The coordinates of the point L, which is the reflection of H in 0 (i.e., L = 2 X 0 - H), can therefore be written
L = 2R( a - {3y, {3 - ya, y - a/3), (2)
where a, {3, and y stand for cos A, cos B, and cos C, respectively (see also [4] and [5]).
What are the coordinates of the Gergonne point G? Now, from Figure 3, we see that XC = r cot(lC), and therefore 2 for X, we have
Similarly,
y = XC sin C = 2r cos2( lC) 2 •
z = XB sin B = 2r cos2( lB). 2
Therefore, the coordinates of G, which lies on the line AX, must also satisfy
Hence,
1 + cos c 1 + cos B
G = [(1 + /3)(1 + y), (1 + y)(l + a), (1 + a)(1 + {3)]g, (3)
where g is a normalizing constant. Therefore, in order to prove the collinearity of I, G, and L, we have only to show that the determinant
1 D = (1 + {3)(1 + y)
a - {3y
1 (1 + y)(1 + a)
{3 - ya
1
(1 + a)(I + {3)
a - {3y
vanishes. However, upon adding the elements of the third row to those of the second, we see that each term becomes equal to 1 + a + {3 + y, bringing the first two rows of D into proportions, so D = 0. This proves that IG passes through L.
Likewise, the coordinates of the excenter IA are
IA = rA ( - 1, 1, 1).
In determining the coordinates of G A, we have only to re
place B and C by 7T - B and 7T - C, respectively, also x by
-x. Hence,
GA = [ - (1 - /3)(1 - y), (1 - y)(l + a), (1 + a)(1 - f3)]gA,
where gA is another constant. The corresponding determi
nant D A will be found to vanish in a similar way. Hence,
we have proved the following theorems:
1. The jour lines IG, IAGA, IBGB, and leGe aU meet in a point L.
2. L lies on the Euler line and is the reflection of the orthocenter H in the circumcenter 0. Thus, the separations of L, 0, M, 0', and H along the Euler line are in the ratios 6 : 2 : 1 : 3; see Figure 5.
Note a corollary. From Figure 5 we see that ML = 2 X MH. Now, consider the homotheticity in which points of
the plane are first reflected in the median point M and then
enlarged by a factor of 2. All transformed points being de
noted by the suffix 1 , the orthocenter H1 of the triangle
A1B1C1 is coincident with L. But A, B, C are the midpoints
of the sides of A1B1C1, so ABC is the median triangle of
A1B1C1• Substituting ABC for A1B1C1, we have the follow
ing theorems:
3. The orthocenter of a triangle ABC is collinear with the incenter I' of the median triangle (i.e. , the Spieker point of ABC) and the Gergonne point G' of the median triangle (see Fig. 6).
4. If I' A, I 'B, and I' c are the excenters of the median triangle and G.,.i, GiJ, and G(; are the corresponding Gergonne points, then IA.GA., I8Gf1, and leGe also pass through the orthocenter H.
When I wrote to my long-time friend and colleague
H. S.M. Coxeter about these results, he pointed out that the
French geometer G.A.G. de Longchamps (1842-1906) had
shown [6] that the orthocenter H1 of A 1B1 C 1 has certain in
teresting properties related naturally to the triangle A 1B 1 C 1
but having no obvious connection with I or G; see [ 1 ] , [8].
The point H1 has been called the de Longchamps point of
ABC; see [1] and [5]. Thus, theorems 1 and 2 can be stated
alternatively as
5. The lines IG, IAGA, IBGB, and leGe all pass through the de Longchamps point of ABC.
Or more succinctly,
6. L coincides with H1.
The question now arises: Are there any other fourfold
concurrencies analogous to the one through L? There is a somewhat similar situation involving the
Nagel point N mentioned earlier. The Nagel point of the
triangle ABC may be defined in the following way [2], [ 11 ] . As before, let XA denote the point of contact of the excir
cle, center IA, with the side BC opposite A, and let YB and
iplijii;IJM A
Ze be defmed similarly. Then, the three lines AXA, BYB, and
CZe are concurrent in the point N; see Figure 7. To prove this, note that the length of the tangent CXA
from C to the excircle center IA is tea + b - c), where a, b, and c are the sides of ABC. Therefore,
cxA = CYB = tea + b - c),
which is equal to the length of the tangent BX from B to
the incircle, center I. In fact, XA, YB, and Ze are the re
flections of X, Y, and Z in the midpoints A', B', and C' of
the three sides of ABC. Hence, as before,
AYB · BZe · CXA = 1
AZe · BXA · CYB '
and by Ceva's theorem, the three lines are concurrent.
M§tgam;:w
'
I ; /
; ; ;
; ;
; ;
/ /
' ' ' '
/
' ' '
/
' '
/
' '
Is #, ,. 1 1
; I I " ;
// I
; 1 I ;
; / ; /
; / ; / ; /
; ; /
; / ; I
I /
/
I
/ I
I
I I
I
I I
' ' ' , ' ' ' ' ' ' ' ' ,' I
� I � lA
VOLUME 22. NUMBER 1 , 2000 57
By similar triangles, it may be shown (see [2, pp. 161-162]) that N is collinear with I and M and that MN = 2M/. Hence, N is the incenter of the triangle A1B1C1, or N = h
Now, corresponding to each excenter of ABC, say IA, we also have a Nagel point. Thus, if �. YJA, ?A are reflections of XA, YA, ZA, respectively, ?A in the midpoints A', B', C', respectively( so � = X), then A�, BYJA, C?A meet in the Nagel point NA, say; and similarly for NB and Nc.
We can now prove two theorems somewhat analogous to Theorems 1 and 2, namely
7. The four lines IN, lANA, IBNB, and IcNc all meet in the
median point M. 8. M divides each of IN, lANA, IBNB, and IcNc in the ra
tio 1 : 2.
Hence,
9. NA, NB, and Nc are the three excenters of the triangle
A1B1C1.
An analytic proof is as follows. Our method will be to show that
N + 21 = 3M. (A)
As earlier, the ratio of the perpendiculars from XA to the sides AC and AB is given by
r cot( .!.B) sin C y _ XAC sin C _ XB sin C _ 2 . - - - - 1 ' z XAB sin B XC sin B r cot( -B) sin B
that is to say,
J!... z
. 2( 1 C) Sill 2 1 - y
sin2( .!.B) = 1 - f3 2 Hence, the trilinear coordinates of N are
2
N = [(1 - {3)(1 - y), (1 - y)(1 - a), (1 - a)(1 - f3)Jn,
where n is a normalizing constant, which we need to evaluate. Now, the trilinear coordinates of any point (x, y, z)
must obviously satisfy
ax + by + cz = 2.:l,
where .:l denotes the area of ABC. Because
.:l = 2R2 sin A sin E sin C
and a = 2R sin A, and so forth, we find, after some use of the relation A + B + C = 7T, that
n = 2R.
The same process of normalization, when applied to the coordinates r(l, 1, 1) of I, leads to the result
r = 4R sinCiA) sinCiB) sin(iC),
which, of course, may be proved independently. Now, the distance of the median point M from the side
BC is one-third of the height of the altitude AD. Hence, the coordinates of M are
M = :!:.R (sin B sin C, sin C sin A, sin A sin B). 3
58 THE MATHEMATICAL INTELLIGENCER
To establish relation (A) we need, by symmetry, to consider only the x components of this equation. Thus, we need to show only that
2R(l - cos B) (I - cos C) + 8R sin(iA) sinCiB) sin(iC) = 2R sin B sin C.
However, from sinCiA) = cosfiCB + C)] , this last result becomes evident. Therefore, M indeed divides the line IN in the ratio 1:2.
In a similar way, we fmd
NA =
2R[(l + /3)(1 + y), -(1 + y)(1 - a), - (1 - a)(l + ,8)]
and
where
A U T H O R
MICHAEL LONGUET·HIGGINS Institute of Nonlinear Science
University of California La Jolla
La Jolla, California 92093-0402 USA
The author graduated in mathematics from Cambridge Uni
versity in 1 946 and did 3 years' National Service at the
Admiralty Research Laboratory in Teddington, where he be
came interested in various geophysical phenomena. He re
turned to Cambridge to take a Ph.D. in geophysics in 1 952 .
He has published extensively on topics in fluid dynamics, par
ticularly on surface waves and ripples, wave breaking and
sound generation in the ocean, oceanic Rossby waves, shore
line processes, and bubble dynamics. He has also contributed
to the statistical theory of Gaussian and non-Gaussian sur
faces. From 1 969 to 1 989, he was a Royal Society Research
Professor at Cambridge University, commuting regularly to the
Institute of Oceanographic Sciences in Surrey. Following "re
tirement," he has been at the University of California, San
Diego. Ever since constructing models of all the concave uni
form polyhedra in the 1 940s and early 1 950s, he has retained
an interest in pure geometry. His hobbies include the design
and demonstration of mathematical toys. He has four children
and seven grandchildren .
and it may be verified, as before, that all three components of the equation
(B)
are satisfied. Hence, M also divides the line I»VA in the ratio 2 : 1 , and similarly for the lines IBNB and leN c.
We have shown, then, that the median point M is also a fourfold point of concurrence, lying on the Euler line of the triangle ABC. However, some qualitative differences between L and M may be noted:
1. In the concurrence through M, the median point divides each of the segments in the simple ratio 2:1, whereas in the concurrence through L, the ratios of the segments are neither simple nor equal.
2. The geometries of the median point M and of the Spieker and Nagel points have been well explored in the literature, not so the de Longchamps point H1. The coincidence of L and H1 invites further investigation.
ACKNOWLEDGMENT
The author thanks an anonymous referee for helpful comments.
REFERENCES
1 . N. Altshiller-Court, "On the de Longchamps circle of a triangle,"
Am. Math. Monthly 33 (I 926), 638-375.
2. N . Altshiller-Court, College Geometry, Barnes and Noble, Inc . New
York, 1 952.
3. J.L. Coolidge, A Treatise on the Circle and the Sphere, Chelsea,
New York, 1 971 .
4. H.S.M. Coxeter, The Real Projective Plane, 2nd ed. , Cambridge
University Press, Cambridge, 1 955.
5 . H.S.M. Coxeter, "Some applications of trilinear coordinates," Linea_r
A/g. Appl. 226-228 (1 995), 375-388.
6. G. de Longchamps, "Sur un nouveau cercle remarquable," J. Math.
Speciales (1 886) 57-50, 83-87, 1 00-1 04, and 1 25-1 28.
7 . L. Euler, "Solutio facilis problematum quorumdam geometricorum
difficillimorum, "Novi Comment" Acad. Imp. Sci. Petropolitanae I I (1 765, published 1 767), 1 03-1 23. For an English abstract by J.S.
Mackay, see Proc. Edin Math. Soc. 4 (1 886), 51-55.
8. A. Gob, "Sur Ia droite et le cercle d'Euler," Mathesis (1 889)
Supplement, 1-2.
9. D .R . Hofstadter, "Discovery and dissection of a geometric gem,"
Geometry Turned On!, ed. by J.R. King and D. Schattschneider,
Mathematical Association of America, Washington, DC, 1 997, pp.
3-1 4.
1 0. W.P. Milne, Homogeneous Coordinates, Edward Arnold, London,
1 924.
I I . C. Nagel, Untersuchungen uber die Wichtigsten zum Dreiecke
Geh6rigen Kreise, Mohler'schen Buchhandlung im Ulm, Leipzig 1 836.
1 2 . M. Simon, Uber die Entwicklung der Elementar-Geometrie im XIX
Jahrhundert, Teubner, Leipzig, 1 906, pp. 1 24-1 41 .
1 3 . G. Spieker, "Ein merkwurdiger Kreis um den Schwerpunkt des
Perimeters des geradlinigen Dreiecks als Analogen des Kreises der
neun Punkte," Grunert's Arch. 51 (1 870), 1 0-1 4.
1 4. E. Vigarie, "La bibliographie de Ia geometrie du triangle," C.R. Fr.
Avance. Sci. 2 (1 895), 50-63.
Revisit the Birth of Mathematics . . .
E U C L I D
w•w•tW·i· '·' .J e remy G ray, Editor I
Ep isodes in the Berlin· GOttingen Rivalry, 1 870- 1 9301
David E . Rowe
Column Editor's address:
Faculty of Mathematics, The Open University,
Milton Keynes, MK7 6AA, England
Higher mathematics at the German universities during the nineteenth century was marked by rivalry between major centers. Among these, Berlin and Gottingen stood out as the two leading institutions for research-level mathematics. By the 1870s they were attracting an impressive array of aspiring talent not only from within the German states but from numerous other countries as well. 2 The rivalry between these two dynamos has long been legendary, yet little has been written about the sources of the conflicts that arose or the substantive issues behind them. Here I hope to shed light on this rivalry by recalling some episodes that tell us a good deal about the forces that animated these two centers. Most of the episodic information I will draw upon, little of it widely known, concerns the last three decades of the nineteenth century. It will be helpful to begin with a few remarks about the overall development of mathematics in Germany, so I will proceed from the general to the specific. In fact, we can gain an overview of some of the more famous names in German mathematics simply by listing some of the betterknown figures who held academic positions in Gottingen or Berlin. As an added bonus, this leads to a very useful tripartite periodization (see table, top of next column)
The era of Kummer, Weierstrass, and Kronecker-the period from 1855 to 1892 in Berlin mathematics-has been justly regarded as one of the most important chapters in the history of 19th-century mathematics.3 Still, it is difficult from today's perspective to appreciate the degree to which Berlin dominated not only the national but also the international mathematical
Periodization of Mathematics in GOttingen and Berlin
1801-1855 Gauss
W. Weber
1855-1892 Dirichlet
Riemann
Clebsch
Schwarz
Klein
1892-1917 Klein
H. Weber
Hilbert
Minkowski
Runge
Landau
Caratheodory
Dirichlet
Steiner
Jacobi
Kummer
Weierstrass
Kronecker
Fuchs
Fuchs
Schwarz
Frobenius
Schottky
rived in part from the prestige of the Prussian universities, which throughout the century did much to cultivate higher mathematics. During the 1860s and 70s practically all the chairs in mathematics at the Prussian universities were occupied by graduates of Berlin, several more of whom also held positions outside Prussia. Berlin's dominance was reinforced by the demise of Gottingen as a major center following Dirichlet's death in 1859 and Riemann's illness, which plagued him throughout most of the 1860s and eventually led to his death in 1866. Mterward, Richard Dedekind, who spent most of his career in the relative isolation of Brunswick, was the only major figure whose work revealed close ties with this older Gottingen tradition.
By 1870 a rival tradition with roots in Konigsberg began to crystallize around Alfred Clebsch, who taught in Gottingen from 1868 to 1872. Together with Carl Neumann, Clebsch founded Die Mathematischen Annalen, which
scene. Berlin's preeminent position de- served as a counterforce to the Berlin-
1The following is based on a lecture delivered on 22 August 1 998 at a symposium held at the Berlin ICM.
I wish to thank the symposium organizers, Georgia Israel and Eberhard Knobloch, for inviting me.
2For the case of North Americans who studied in Giittingen and Berlin, see (Parshall and Rowe, 1 994, chap
ter 5).
3For an overview, see (Rowe 1 998a); the definitive study of mathematics at Berlin University is (Biermann 1 988).
60 THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK
Figure 1 . The drama of Berlin. In this contemporary painting by Adolph Menzel, the assembled citizenry hails King Wilhelm on his departure
for the battlefront in the Franco-Prussian War. One of the spoils of the war he won five years earlier was the annexation of Hannover to
Prussia, which led to the Prussianization of its university in Gtittingen. (Abriese Konig Wilhelms I. zur Armee am 31 Juli 1870-by permission
of SMPK Berlin-Nationalgalerie.)
dominated journal founded by Crelle,
edited after 1855 by Carl Wilhelm
Borchardt. Other leading representa
tives of this Konigsberg tradition dur
ing the 1860s and 1870s included Otto
Hesse, Heinrich Weber, and Adolf
Mayer. Along with Clebsch and
Neumann they operated on the pe
riphery of Berlin and its associated
Prussian network These mathemati
cians had very broad and diverse in
terests, making it difficult to discern
striking intellectual ties. What they
shared, in fact, was mainly a sense of
being marginalized, and they looked up
to Clebsch as their natural leader. With
the founding of the German Empire in
1871, these "Southern" German math-
ematicians made a bid to found a na
tionwide organization. Clebsch's unex
pected death in November 1872 slowed
the momentum that had been building
for this plan, but the effort was carried
on by Felix Klein and other close as
sociates of the Clebsch school. A meet
ing took place in Gottingen in 1873, but
the turnout was modest and the results
disappointing. None of the prominent
"Northern" German mathematicians at
tended-the label "Northern" being a
euphemism within the Clebsch school
for "Prussian". Klein and his allies soon
thereafter gave up this plan, as without
the support of the Berliners there could
clearly be no meaningful German Math
ematical Society. 4 The same situation
prevailed 20 years later, and the result
would have very likely been the same
had not Georg Cantor persuaded Leo
pold Kronecker, the most powerful
and important Berlin mathematician of
the 1880s, to throw his support behind
the venture.
After 1871, the Franco-Prussian ri
valry loomed large in the minds of
many German mathematicians. That
Berlin should occupy a place analo
gous to Paris was, for many mathemati
cians, merely the natural extension of
political developments to the intellec
tual sphere. One need only read some of
Kummer's speeches before the Berlin
Academy-which as its Perpetual
Secretary he was required to deliver on
4Further details on this early, abortive effort to found a national organization of mathematicians in Germany can be found in (Tobies and Rowe 1 990, pp. 2G-23, 59-72).
VOLUME 22, NUMBER 1, 2000 61
ceremonial occasions like the birth of
Frederick the Great-in order to realize
how completely this celebrated and
revered mathematician identified with
the world-historical purpose of the
Prussian state and its innermost spirit,
its Geist.5 I doubt that Hegel himself
could have described that mysterious
dialectical linkage more eloquently.
This was the same Kummer who, in a
letter to his young pupil Kronecker,
written in 1842, urged him to attend
Schelling's lectures in Berlin, despite
the fact that Schelling's brand of ide
alism failed to grasp the deeper
Hegelian truth that "mind and being"
were initially one and the same.
Schelling, according to Kummer, was
the "only world-historical philosopher
still living. "6 Kummer held the office of
perpetual secretary of the Berlin
Academy for 14 years, 1865-1878, dur
ing which time he conducted himself
in a manner that won much admira
tion. His role during Berlin's "golden
age of mathematics" bore a strong re
semblance to that played by Max
Planck after 1900. Indeed, Planck's
worldview (about which see (Heilbron
1986)) had much in common with
Kummer's belief in the harmony of
Prussia's intellectual, spiritual, and po
litical life.
Kummer is, of course, mainly re
membered today for his daring new the
ory of ideal numbers, which served as
the point of departure for Dedekind's
ideal theory; we also think of him in
connection with Kummer surfaces, spe
cial quartics with 16 nodal points. But
he founded no special school; his im
pact was clearly more diffuse than that
of his colleagues Weierstrass and
Kronecker. Still, he embodied for many
the heart and soul of the Berlin tradi
tion, and like his colleagues he instilled
in his students the same sense of lofty
ideals-the purity and rigor for which
the Berlin style was soon to become fa-
mous. Later representatives-Schwarz,
Frobenius, Hensel, Landau, and Issai
Schur-saw themselves as exponents
of this same Berlin tradition, though
they drew their main inspiration from
the lecture courses of Weierstrass and
Kronecker.
To gain a quick, first-hand glimpse
of Berlin mathematics during the 1870s
we can hardly do better than follow
the letter written by Gosta Mittag
Leffler to his former mentor Hjalmar
Holmgren on 19 February 1875. The
young Swede was traveling abroad on
a postdoctoral fellowship that had first
brought him to Paris. There he met
Charles Hermite, a great admirer of
German mathematical achievements
despite his limited knowledge of the
language, who told him that every as
piring analyst ought to hear the lec
tures of Weierstrass. 7 Here is Mittag
Leffler's account of these and much
else that he encountered in the
Prussian capital:
. . . With regard to the scientific
aspect I am very satisfied with my
stay in Berlin. Nowhere have I
found so much to learn as here.
Weierstrass and Kronecker both
have the unusual tendency, for
Germany, of avoiding publications
as much as possible. Weierstrass,
as is known, publishes nothing at
all, and Kronecker only results
without proofs. In their lectures
they present the results of their re
searches. It seems unlikely that the
mathematics of our day can point
to anything that can compete with
Weierstrass's function theory or
Kronecker's algebra. Weierstrass
handles function theory in a two
or three-year cycle of lecture
courses, in which, starting from
the simplest and clearest founda
tional ideas, he builds a complete
theory of elliptic functions and
their applications to Abelian func
tions, the calculus of variations,
etc. 8 What is above all characteris
tic for his system is that it is com
pletely analytical. He rarely draws
on the help of geometry, and when
he does so, it is only for illustra
tive purposes. This appears to me
an absolute advantage over the
school of Riemann as well as that
of Clebsch. It may well be that one
can build up a completely rigorous
function theory by taking the
Riemann surfaces as one's point of
departure and that the geometrical
system of Riemann suffices in or
der to account for the presently
known properties of the Abelian
functions. But [Riemann's ap
proach] fails on the one hand when
it comes to recovering the proper
ties of the higher-order transcen
dentals, whereas, on the other
hand, it introduces elements into
function theory which are in prin
ciple altogether foreign. As for the
system of Clebsch, this cannot even
deliver the simplest properties of
the higher-order transcendentals,
which is quite natural, since
analysis is infinitely more general
than is geometry.
Another characteristic of Weier
strass is that he avoids all general
definitions and all proofs that con
cern functions in general. For him
a function is identical with a power
series, and he deduces everything
from these power series. At times
this appears to me, however, as an
extremely difficult path, and I am
not convinced that one does not in
general attain the goal more easily
by starting, like Cauchy and
Liouville, with general, though of
course completely rigorous defini
tions. Another distinguishing char
acteristic of Weierstrass as well as
Kronecker is the complete clarity
5Another sterling example from Kummer's Breslau period is his lecture on academic freedom (Kummer 1 848) delivered in the midst of the dramatic political events of
1 848.
6Kummer to Kronecker, 1 6 January 1 842, published in (Jahnke 191 0, pp. 46-48).
7For a brief account of Mittag-Leffler's career, see (Garding 1 998, pp. 73-84) .
8Mittag-Leffler took Weierstrass's standard course on elliptic functions during the winter semester of 1 87 4-75; he also was one of only three auditors who attended
his course that term on differential equations. During the summer semester of 1 875, he followed Weierstrass's course on applications of elliptic functions to geometry
and mechanics. This information can be found in (Nbrlund 1 927, p. vii), along with the claim that Mittag-Leffler was offered a Lehrstuhl in Berlin in 1 876. Presumably
this story stemmed from Mittag-Leffler himself, and while difficult to refute, its implausibility is so apparent that we may safely regard this as a Scandinavian legend. A
similar conclusion is reached in (Garding 1 998, pp. 75-76).
62 THE MATHEMATICAL INTELUGENCER
a rut precision of their proofs. By the
same token, both have inherited
from Gauss the fear of any kirui of
metaphysics that might attach to
their fundamental mathematical
ideas, arui this gives a simplicity
and naturalness to their deduc
tions, which have hardly been seen
heretofore presented so systemati
cally arui with the highest degree of
precision.
In respect to form, Weierstrass's
manner of lecturing lies beneath all
criticism, arui even the least im
portant French mathematician,
were he to deliver such lectures,
would be considered completely in
competent as a teacher. If one suc
ceeds, however, after much difficult
work, in restoring a lecture course
of Weierstrass to the form in which
he originally conceived it, then
everything appears clear, simple,
and systematic. Probably it is this
lack of talent which explains why
so extremely few of his many stu
dents have uruierstood him thor
oughly, arui why therefore the liter
ature dealing with his direction of
research is still so insignificant.
This circumstance, however, has not affected the nearly god-like rev
erence he enjoys in general.
Presently there are several young
arui diligent mathematicians in
whom Weierstrass places the high
est hopes. At the top of the list as
"the best pupil that I have ever had" he places the young Russian
Countess Sophie v. Kovalevskaya,
who recently took her doctorate
in absentia from the faculty in
Gottingen on the basis of two works
that will soon appear in Grelle; one
on partial differential equations,
the other on the rings of Saturn. 9
Clearly, these views reflect more than just one man's opinion. Mittag-
Leffler put his finger on an important component of the Berlin-Gbttingen rivalry with his claims for the methodological superiority of Weierstrassian analysis over the geometric function theory of Riemann or the mixed methods of Clebsch. Still, what he wrote must be placed in proper perspective. During Riemann's lifetime, the Gbttingen mathematician's reputation stood very high in Berlin, and it remained untarnished after his death in 1866. He was elected as a corresponding member of the Berlin Academy in August 1859, which gave him occasion to travel to the Prussian capital the following month. There he was welcomed by the leading Berlin mathematicians-Kummer, Kronecker, Weierstrass, and Borchardt-with open arms, as his friend Dedekind, who accompanied him on this journey, later recalled (Dedekind 1892, p. 554). Weierstrass practically worshiped Riemann, calling him, according to MittagLeffler, an "anima candida" like no one else he ever knew. 10 His colleague Kronecker, to be sure, had a. far less flattering opinion of Riemann's successor, Clebsch, but he, too, had already passed from the scene in November 1872. Thus the subterranean rumblings within the German mathematical community so apparent in Mittag-Leffler's letter reflected not so much personal animosities directed toward Riemann and/or Clebsch but rather the way in which their work had become bound up in an ongoing rivalry between Berlin's leading mathematicians and those associated with the "remnants" of the Clebsch school. Within the latter group the most visible figure was its youngest star, an ambitious and controversial fellow named Felix Klein.
The Berlin establishment had gotten a first taste of Klein during the winter semester of 1869-70 when the 20-year-old Rhinelander, who had worked
9Quoted in (Frostman, pp. 54-55) (my translation). For a discussion of Kovalevskaya's work, see (Cooke 1 984).
closely with Julius Plucker in Bonn, arrived in the Prussian capital to undertake postdoctoral studies. Like nearly all aspiring young Prussian mathematicians, Klein recognized the importance of making a solid impression -in the Berlin seminar run by Kummer and Weierstrass. Before presenting himself as a candidate, therefore, he took about five weeks to write up an impressive paper on a topic in his special field of line geometry. 1 1 He then submitted the manuscript to Kummer, thereby fulfilling one of the requirements for membership in the seminar. Klein's paper dealt systematically with the images of ruled surfaces induced by a mapping found a short time earlier by Klein's friend, Max Noether (the Noether map sends the lines of a linear complex to points in complex projective 3-space ). Some weeks later Kummer returned the manuscript, and Klein soon lost interest in the topic as well as the results he had obtained. 12
In the meantime, Klein introduced him
self to Weierstrass and Kronecker, though he otherwise kept his distance from their lecture halls. This aloofness, however, did not prevent him from asking Weierstrass for his assistance in helping him cultivate contacts with his advanced students. Klein no doubt made it plain that he could not spare the time it would take to learn W eierstrassian analysis from the ground up; what mattered to him was getting to know the "inner life" of mathematics in Berlin. Presumably, not many would have dared to approach Weierstrass this way, but the latter willingly obliged, suggesting that Klein seek out Ludwig Kiepert's counselP Their meeting marked the beginning of a lifelong friendship which both Klein and Kiepert came to value, and for good reason: it turned out to be one of the few bridges connecting members
of the Berlin and Gbttingen "schools."
10(Mittag-Leftler 1 923, p. 1 91 ). Mittag-Leffler's remark was undoubtedly the source E. T Bell drew upon for the title ("Anima Candida") of the chapter on Riemann in
his popular but idiosyncratic Men of Mathematics. 1 1The manuscript can be found in Klein Nachlass 1 3A, Handschriftenabteilung, Niedersachsische Staats-und Universitatsbibl iothek Gbttingen. According to the dating
in Klein's hand at the top, he began to write the paper on 5 September 1 869 and completed it on 1 5 October 1 869.
121 have found no traces of this original study in Klein's published work, although there are several references to Noether's mapping, which is related to the famous
line-sphere map investigated by Sophus Lie soon thereafter. Erich Bessel-Hagen later added a note to the unpublished manuscript relating that, according to Klein,
Kummer returned the manuscript to him after a few weeks without any comments and apparently unread ("anscheinend ungelesen").
13This story is recounted in (Kiepert 1 926, p. 62).
VOLUME 22, NUMBER 1, 2000 63
t:�-.. .,. ,./,• ., ,,•, •' / $', I t' .
s· . tf . - �
Figure 2. The first page of Klein's untitled and unpublished manuscript on line geometry, writ
ten in autumn 1869. Klein submitted this work to Kummer as his ticket for admission to the
Berlin Mathematical Seminar, which Sophus Lie also attended that semester. (But, according
to Erich Bessel-Hagen's report many years later, Kummer seems never to have read the man
uscript.} Niedersachsiche Staats- und Universitatsbibliothek Gottingen, Cod. Ms. F. Klein 13A.
Klein made other significant contacts in Berlin, but mainly with other outsiders like the Austrian Otto Stolz, from whom he learned the rudiments of non-Euclidean geometry. By far the most significant new friendship was the one Klein made with Sophus Lie, that Nordic giant whose ideas and per-
14For more on Klein and Lie, see (Rowe and Gray).
64 THE MATHEMATICAL INTELLIGENCER
sonality captivated him so completely. As a backdrop to future events, a few words must be said with regard to the Klein-Lie collaboration. 14 Like Klein, Lie was an expert on Pliickerian line geometry, and thus someone Klein knew by reputation beforehand. In fact, Klein's mentor, Clebsch, had al-
ready alerted his protege to the possibility of meeting Lie personally in Berlin, and in October 1869 they greeted each other at a meeting of the Berlin Mathematics Club. Before long they were getting together nearly every day to discuss mathematics. Since Kummer's seminar theme concerned the geometry of ray systems, a topic intimately connected with line geometry, Klein and Lie soon emerged as its two stars. Although Lie was still without a doctorate--a circumstance so embarrassing to him that he introduced himself as Dr. Lie anyway-the Norwegian's brilliant new results dazzled Klein, who was six years younger. At the time, Lie's German was minimal, so Klein offered to present his work to the members of the Berlin seminar. Kummer was duly impressed by Lie's mathematics as well as Klein's presentation of it, and this success sparked their intense collaboration, which began with a sojourn in Paris during the spring of 1870 and lasted until Klein's appointment as Professor
Ordinarius in Erlangen in the fall of 1872. Lie even accompanied Klein when he moved from Gottingen, all the while discussing with him the ideas that soon appeared in Klein's famous "Erlangen Program." During the years that followed, however, their interests drifted apart, though they continued an avid correspondence.
Returning now to our main theme, the first overt signs of struggle between Klein and Berlin came in the early 1880s when Klein was Professor of Geometry in Leipzig. Six years after Mittag-Leffler had given his private
description of how Berlin mathematicians assessed the drawbacks of a geometrically-grounded theory of complex functions, this issue was taken up by Klein in a public forum. Klein's remarks were prompted by a priority dispute with the Heidelberg analyst, Lazarus Fuchs, a leading member of the Berlin network (he took a chair in Berlin in 1884). As in so many priority claims in mathematics, the issues at stake here were far more complicated than might first meet the eye. Particularly interesting were the international dimensions of the conflict,
Figure 3. A contemporary sketch of the then-new Auditorium building in Gottingen.
including the part played by MittagLeffler, who was then busy plotting to launch Acta Mathematica (see (Rowe 1992)). The episode began innocently enough in 1881 when Henri Poincare published a series of notes in the Comptes Rend us of the Paris Academy in which he named a special class of complex functions, those invariant under a group with a natural boundary circle, "Fuchsian fimctions." Klein soon thereafter entered into a semi-friendly correspondence with Poincare, from which he quickly learned that the young Frenchman was quite unaware of the relevant "geometrical" literature, including Schwarz's work, but especially Klein's own.15
Before long Poincare found himself in the middle of a German squabble that he very much would have liked to avoid. Quoting a famous line from Goethe's
1 5For details. see (Gray 1 986, pp. 275-31 5). 16For details, see (Rowe 1 988, pp. 39--40).
Faust, he wrote Klein that "Name ist Schall und Rauch" ("name is but sound and smoke"). Nevertheless, he found himself forced to defend his own choice of names in print, while hoping he could placate Klein by naming another class of automorphic functions after the Leipzig mathematician. In the meantime, Klein and Fuchs exchanged sharp polemics, Klein insisting that the whole theory of Poincare had its roots in Riemann's work, and that Fuchs's contributions failed to grasp the fundamental ideas, which required the notion of group actions on Riemann surfaces (see (Rowe 1992)). Klein's brilliant student, Adolf Hurwitz, apparently enjoyed this feud, especially his mentor's attacks which reminded him of a favorite childhood song: "Fuchs, Du hast die Funktion gestohlen I Gieb sie wieder her." Fuchs and the Berlin establish-
ment were, of course, not amused at all, and neither was Klein.
Over the next ten years, Klein launched a series of efforts, nearly all of them futile, to make inroads against the entrenched power of the Berlin network In 1886 he finally managed to gain a foothold in Prussia when he was called to Gottingen. But in the meantime, the former Hannoverian university had become Prussianized, and after the death of Clebsch in 1872 its mathematics program was dominated by H.A. Schwarz, Weierstrass's leading pupil. Both Schwarz and his teacher were incensed that Klein had managed to engineer the appointment of a foreigner, Sophus Lie, as his successor in Leipzig.16 Thereafter, both Lie and Klein were scorned by leading Berliners, particularly Frobenius. According to Frobenius, Weierstrass made it
VOLUME 22, NUMBER 1 , 2000 65
known that Lie's theory would have to be junked and worked out anew from scratchP Klein sought to make a common front with his Leipzig colleagues Lie and Adolf Mayer, but Lie became increasingly wary of this manuevering aimed mainly at enhancing Klein's personal power.
When Kronecker suddenly died in December 1891, Weierstrass could finally retire in peace-they had been archenemies through the 1880s-and this led to a whole new era in German mathematics. It opened with a series of surprising events. First, Klein was vehemently rejected by the Berliners, including Weierstrass and Helmholtz, who characterized him as a dazzling charlatan. Still stinging from Klein's attack from a decade earlier, Fuchs merely added that he had nothing against Klein personally, only his pernicious effect on mathematical science.l8 Thus, Schwarz got Weierstrass's chair, and Kronecker's went to Frobenius, while in the midst of these appointments Klein tried to get Hurwitz for Gottingen, even though the faculty placed Heinrich Weber first on its list of leading candidates. In this episode, Klein's strategy was to rely on Friedrich Althoff, the autocratic head of Prussian university affairs, to reach over Weber and appoint Hurwitz, who was second on the list. The idea backfired, leaving Klein in a state of despair, largely due to his loss of face in the faculty, which had witnessed how Schwarz once more won his way against Klein, even though Schwarz was now sitting in Berlin.19 Gottingen's informal policy limiting the number of Jews on the faculty to one per discipline may have been the decisive factor that prevented Hurwitz's appointment. Strangely enough, after Hurwitz's death in 1919 the fallacious story circulated that he had turned down the call to Gottingen in 1892 out of a sense of loyalty to the ETH.20
As it turned out, the decisive year 1892 was nothing short of a fiasco for
17See (Biermann 1 988, p. 2 1 5) .
18See (Biermann 1 988, p. 305-306).
1 9For details, see (Rowe 1 986, pp. 433-436).
20See (Young 1 g2o. p. /iii).
Klein. Following his futile efforts on behalf of Hurwitz, the alliance with Lie, whom he wanted to appoint to the board of Mathematische Annalen, fell apart completely. Lie had been under stress practically from the moment he came to Leipzig as Klein's successor in 1886. At the same time, he grew increasingly embittered by the way he was treated by his Leipzig colleagues and certain allies of Klein, who regarded him mainly as one of Klein's many subordinates. 21 By late 1893, the whole mathematical world knew about Lie's displeasure when he published a series of nasty remarks in the preface to volume three of his work on transformation groups. To clarify his relationship with Klein he wrote: "I am not a student of Klein's nor is the opposite the case, even if it comes closer to the truth" (Lie 1893, p. 17).
So, Klein had to regroup his forces and try again, something he was terribly good at doing. In retrospect, the decisive turning point was clearly Hilbert's appointment in December 1894, a goal Klein had long been planning. Mathematically, Klein and Hilbert complemented one another beautifully; moreover, both shared a strong antipathy for the Berlin establishment, which they considered narrow and authoritarian. Whereas Klein tried to advance a geometric style of mathematics rooted in the work of Riemann and Clebsch, Hilbert championed an approach to abstract algebra and number theory that was largely inspired by ideas first developed by Dedekind and Kronecker. With regard to foundational issues, on the other hand, Hilbert's ideas clashed directly with the sceptical views Kronecker had championed in Berlin. In a schematic fashion, we may picture Klein and Hilbert as universal mathematicians whose strengths were mainly situated on the right and left sides, respectively, in the following hierarchy of mathematical knowledge:
NUMBER
Arithmetic
Algebra
Analysis
Analytical Mechanics
Analytic & Differential
Geometry
FIGURE
Euclidean Geometry
Projective Geometry
Higher geometry
Geometrical Mechanics
This bifurcated scheme, I would argue, portrays how most mathematicians in Germany saw the various components of their discipline during the late nineteenth century. I've pictured the "tree of mathematics" turned upside down so that its roots (number and figure) appear at the top. This is meant to reflect the high status accorded to pure mathematics, especially number theory and synthetic geometry, by many influential German mathematicians. The Berlin tradition of Kummer, Weierstrass, and Kronecker clearly favored that branch of mathematics derived from the concept of number, but the tradition of synthetic geometry tracing back to Steiner also played a major part in the Berlin vision (thus Weierstrass taught geometry after Steiner's death in 1863 in an effort to sustain the geometrical component of Berlin's curriculum).
Still, as we have seen, pure mathematics, for Weierstrass, mainly meant analysis, and the foundations of analysis derived from the properties of numbers (irrational as well as rational). He thus drew a reasonably sharp line (indicated by - · - · - · above) that excluded geometrical reasonings from real and complex analysis, whereas his colleague Kronecker drew an even sharper line (marked above as ___ ) that excluded everything below algebra. In other words, Kronecker wished to ban from rigorous, pure mathematics all use of limiting processes and, along with these, the whole realm of mathematics based on the infmitely small. This, of course, was one of the
21 Lie's difficulties in Leipzig were compounded by a variety of other factors, including jealousy aroused by the publications of Wilhelm Killing on the structure theory of
Lie algebras, about which see (Rowe 1 988, pp. 41-44). For a detailed account of Killing's work, see (Hawkins).
66 THE MATHEMATICAL INTELLIGENCER
Ruditorium (Rkademisches Viertel) Figure 4. Students outside the Auditorium building. The woman is believed to be Grace Chisholm, whose 1894
Gottingen doctorate was the first to a woman anywhere in Prussia, and who went on to a distinguished ca
reer in analysis.
main sources of the conflict between
Kronecker and Weierstrass that se
verely paralyzed Berlin mathematics
during the 1880s and beyond, right up
until Kronecker's death.
This is not the place to go into de
tails about how Gottingen quickly out
stripped Berlin during the years that
followed, but we should at least notice
that part of this story concerns a very
different vision of this "inverted tree"
of mathematics, a vision shared by
Klein and Hilbert. As I have argued else
where, this common outlook helps ex
plain how they managed to form such a
successful partnership in Gottingen de
spite their apparent differences. 22 Both
were acutely aware of the possibilities
for establishing a linkage between the
two principal branches of the tree.
Indeed, both made important contribu
tions toward securing these ties (Klein
through his work on projective non
Euclidean geometry; Hilbert with his
arithmetical characterization of the
continuum, which he hoped to anchor
by a proof of the consistency of its ax
ioms). Noteworthy, beyond these con
tributions, was their background in
and familiarity with invariant theory, a
field that was particularly repugnant to
Kronecker and which occupied a
rather low rung in the purists' hierar
chy of knowledge.
From an institutional standpoint,
we can easily spot other glaring con
trasts between Gottingen and Berlin
during the era 1892 to 1917. Whereas
Frobenius and Schwarz largely saw
themselves as defenders of Berlin's
purist legacy, Klein and Hilbert pro
moted an open-ended interdisciplinary
approach that soon made Gottingen a
far more attractive center, drawing in
ternational talent in droves. A glimpse
of this multi-disciplinary style in
Gottingen can be captured merely by
looking at some of the appointments
Klein pushed through with the support
of the Prussian Ministry and the fman
cial assistance of leading industrial
concerns: Karl Schwarzschild (astron
omy), Emil Wiechert (geophysics),
Ludwig Prandtl (hydro- and aerody
namics), and Carl Runge (applied
mathematics, numerical analysis).
Symptomatic of the Gottingen style
was an interest in physics, both classical
and modem. Arnold Sommerfeld, Max
Born, and Peter Debye interacted closely
with Klein, Hilbert, and Minkowski, all
of whom were deeply interested in
22For more on the Hilbert-Klein partnership, see (Rowe 1 989).
Einstein's relativity theory. Einstein
came to Berlin in 1914 on a special ap
pointment that included membership
in the Prussian Academy. Just one year
later the Gottingen Scientific Society
offered him a corresponding member
ship, elevating him to an external mem
ber in 1923. Ironically enough, general
relativity was followed far more
closely within Gottingen circles, as
well as by the Dutch community sur
rounding Paul Ehrenfest, than it was in
Einstein's immediate Berlin surround
ings. Klein, Hilbert, Einstein, and Weyl
were friendly competitors during the
period 1915-1919 (see (Rowe 1998b)).
In Gottingen, the work of Klein and
Hilbert on GRT was supported by sev
eral younger talents, including Emmy
N oether, whose famous paper on con
servation laws grew out of these ef
forts.
In the case of Hilbert and Einstein,
we can also observe a strong affinity in
their insistence on the need to uphold
international scientific relations and to
resist those German nationalists who
supported the unity of Germany's military and intellectual interests. Thus, the
controversial pacifist and international
ist, George Nicolai, enlisted the support
VOLUME 22, NUMBER 1 , 2000 67
of both Hilbert and Einstein for these causes.23 Scientifically, perhaps the most important link joining Hilbert and Einstein came through one of Hilbert's many doctoral students, an Eastern European Jew named Jacob Grommer. Grammer's name appeared for the first time in Einstein's famous 1917 paper introducing the cosmological constant and his static, spatially-closed model of the universe (Einstein 1917). Soon thereafter Grommer joined Einstein and worked closely with him until 1929 when he apparently left Berlin-the longest collaboration Einstein had with anyone.24
Relations between Gottingen and Berlin mathematicians largely normalized after World War I, but they heated up again in 1928 when Brouwer and Bieberbach sought to boycott the Bologna ICM. As is well lrnown, Hilbert, who was then on the brink of death from pernicious anemia, overcame this effort by organizing a delegation of German mathematicians to attend the congress. What he said when he addressed the delegation on the 2nd of September 1928 was not recorded in the Congress Proceedings, but it can be found, scratched in Hilbert's hand, among his unpublished papers. There one reads these words: [Bologna Rede] "It is a complete misunderstanding to construct differences or even contrasts according to peoples or human races . . . mathematics lrnows no races . . . . For mathematics the entire cultural world is one single land."25
These views were very different from the ones held earlier by Eduard Kummer, admittedly during an era when the German mathematical community was still barely formed. More striking, however, is the clash with Bieberbach's vision, which asserted that mathematical style could be directly understood in terms of racial types. That story leads, of course, into the complex and messy problematics of mathematics during the Nazi era and its historical roots-a topic I can only mention here.26 Nevertheless, I hope these glimpses into the mathematical
life of Germany's two leading research centers have conveyed a sense of the clashing visions and intense struggles that took place behind the scenes. The setting may be unfamiliar, but the issues of pure vs. applied and national allegiance vs. international cooperation most certainly are not. The present ICM in Berlin represents not only an opportunity for mathematicians to gather and celebrate recent achievements but also to reflect on the role of mathematics and its leading representatives of the not so distant past, drawing whatever lessons these reflections may offer for mathematics today.
REFERENCES
Kurt-A. Biermann, Die Mathematik und ihre
Dozenten an der Berliner Universitat,
181D-1933 (Berlin: Akademie Verlag, 1 988).
Roger Cooke, The Mathematics of Sonya
Kovalevskaya (New York: Springer-Verlag,
1 984).
Richard Dedekind, "Bernhard Riemann's
Lebenslauf, " in H. Weber, ed. , Bernhard
Riemann's Gesammelte Mathematische
Werke, 2nd ed. (Leipzig: Teubner, 1 892), pp.
539-558.
Albert Einstein, "Cosmological Considerations
on the General theory of Relativity," English
trans. of "Kosmologische Betrachtungen zur
allgemeinen Relativitatstheorie" (191 7) in A Sommerfeld, ed., The Principle of Relativity
(New York: Dover. 1 952).
Albrecht Folsing, Albert Einstein: A Biography,
(New York: Viking, 1 997).
Otto Frostman, "Aus dem Briefwechsel von G .
Mittag-Leffler," Festschrift zur Gedachtnis
feier fOr Karl Weierstra/3, 1815-1965, ed. H. Behnke and K. Kopfermann, Koln: West
deutscher Verlag, 1 966, pp. 53-56.
Lars Garding, Mathematics and Mathemati
cians: Mathematics in Sweden before 1950. History of Mathematics, val. 1 3 , (Providence,
R. I ./Landon: American Mathematical Society/
London Mathematical Society, 1 998).
Jeremy J . Gray, Linear Differential Equations
and Group Theory from Riemann to Poincare
(Basel: Birkhauser, 1 986).
Thomas Hawkins, "Wilhelm Killing and the
Structure of Lie Algebras, " Archive for History
of Exact Sciences 26(1 982), 1 27-192.
John L Heilbron, The Dilemmas of an Upright
23Qn Einstein's alliance with Hilbert, see (Fblsing 1 997, p. 466).
24For more details on this collaboration, see (Pais 1 982, pp. 487-488).
25Quoted in English translation in (Reid 1 970, p. 1 88).
26See the portrayal of Bieberbach's political transformation in (Mehrtens 1 987).
68 THE MATHEMATICAL INTELUGENCER
AUT H OR
DAVID E. ROWE
Fachbereich 1 7 -Mathematik
Johannes Gutenberg University
55099 Mainz
Germany
e-mail: [email protected]
mainz.de
After studying topology under Leonard
Rubin at Oklahoma, David Rowe be
gan work on a second doctorate in
history of science with Joseph Dauben
at CUNY's Graduate Center. During
the academic years 1 98H5 he was
a fellow of the Alexander Humboldt
Foundation in Gottingen, where he
combed local archives studying the
lives and work of Klein and Hilbert
Since then, the Gottingen mathemat
ical community has been the main fo
cus of his research. In 1 992 he was
appointed Professor of History of
Mathematics and Exact Sciences at
Mainz University. In recent years he
has become increasingly interested in
the interplay between mathematics
and physics, particularly relativity the
ory. Since 1 998 he has been a con
tributing editor with the Einstein
Papers Project at Boston University.
Here he is shown with his son Andy
on vacation in Fife Lake, Michigan .
Man: Max Planck as Spokesman for German
Science (Berkeley: University of California
Press, 1 986).
E. Jahnke, et al. , eds., Festschrift zur Feier des
100. Geburtstages Eduard Kummers (Leipzig:
Teubner, 1 91 0) .
Ludwig Kieper!, "Personliche Erinnerungen an
Karl Weierstrass," Jahresbericht der Deut-
schen Mathematiker-Vereinigung, 35(1926),
56-65.
E. E. Kummer, "Uber die akademische Freiheit .
Eine Rede, gehalten bei der Ubernahme des
Rektorats der Oniversitat Breslau am 1 5.
Oktober 1 848, " in A Weil , ed. , Ernst Eduard
Kummer, Collected Papers, vol. I I (Berlin:
Springer-Verlag, 1 975), pp. 706-71 6.
Sophus Lie, Theorie der Transformationsgrup
pen, val. 3 (Leipzig: Teubner), 1 893.
Herbert Mehrtens, "Ludwig Bieberbach and
'Deutsche Mathematik, ' " in Studies in the
History of Mathematics, ed. Esther Phillips,
(Washington: The Mathematical Association
of America, 1 987), pp. 1 95-241 .
G6sta Mittag-Leffler, "Weierstrass et Sonja
Kowalewsky, " Acta Mathematica 39(1923),
1 33-1 98.
N. E. N6rlund, "G. Mittag-Leffler," Acta Mathe
matica 50(1927), I-XXI I I .
Abraham Pais, 'Subtle is the Lord . . . ' The
Science and the Life of Albert Einstein,
Oxford: Clarendon Press, 1 982.
Karen Parshall and David E. Rowe, The
Emergence of the American Mathematical
Research Community, 1876-1900. J.J. Sylvester, Felix Klein, and E.H. Moore,
History of Mathematics, vol. 8 (Providence:
American Mathematical Society/London
Mathematical Society, 1 994).
Constance Reid, Hilbert (New York: Springer
Verlag, 1 970).
David E. Rowe, " 'Jewish Mathematics' at
G6ttingen in the Era of Felix Klein," Isis ,
77(1 986), 442-449.
-- . "Der Briefwechsel Sophus Lie- Felix
Klein, eine Einsicht in ihre pers6nlichen und
wissenschaftlichen Beziehungen," NTM.
Schriftenreihe fOr Geschichte der Naturwis
senschaften, Technik und Medizin, 25(1 )
(1 988), 37-47.
-- . "Klein, Hilbert, and the G6ttingen
Mathematical Tradition," Science in Ger
many: The Intersection of Institutional and
Intellectual Issues, ed. Kathryn M. Olesko
(Osiris, 5, 1 989), 1 89-2 1 3 .
-- . "Klein, Mittag-Leffler, and the Klein
Poincare Correspondence of 1 881-1 882,"
Amphora. Festschrift fur Hans Wussing, ed.
Sergei Demidov, Mensa Folkerts, David E.
Rowe, and Christoph Scriba, (Basel:
Birkhauser, 1 992), pp. 598-6 1 8.
-- . "Mathematics in Berlin, 18 10-1 933" in
Mathematics in Berlin, ed. H.G.W. Begehr,
H. Koch, J. Kramer, N. Schappacher, and E.
J . Thiele, Basel: Birkhauser, 1 998, pp. 9--26.
-- . "Einstein in Berlin, " in Mathematics in
Berlin , ed. H.G.W. Begehr, H. Koch, J .
Kramer, N . Schappacher, and E.-J . Thiele,
Basel: Birkhauser, 1 998, pp. 1 1 7-1 25.
David E. Rowe and Jeremy J . Gray, Felix Klein:
The Evanston Colloquium Lectures,
"Erlangen Program, " and Other Selected
Works (English language edition of Klein's
most famous works with historical and math
ematical commentary), New York: Springer
Verlag, forthcoming.
Renate Tobies and David E. Rowe, eds.
Korrespondenz Felix Klein-Adolf Mayer,
Teubner Archiv zur Mathematik, Band 1 4,
(Leipzig: Teubner, 1 990).
W. H. Young, "Adolf Hurwitz," Proceedings of
the London Mathematical Society, Ser. 2,
20(1 920), xlviii-liv.
VOLUME 22, NUMBER 1, 2000 69
ld§'h§l.lfj J et Wi m p , Editor I
Feel like writing a review for The
Mathematical Intelligencer? You are
welcome to submit an unsolicited
review of a book of your choice; or, if
you would welcome being assigned
a book to review, please write us,
telling us your expertise and your
predilections.
Column Editor's address: Department
of Mathematics, Drexel University,
Philadelphia, PA 1 91 04 USA.
What is Mathematics, Really� by Reuben Hersh
OXFORD: OXFORD UNIVERSITY PRESS, 1997. 384 PP.
US $35.00, ISBN 01 951 1 3683
REVIEWED BY MARION COHEN
'' My first assumption about math-ematics: It's something peo-
ple do." This is the sentence (p. 30) that stands out as the author's credo. He elaborates on this considerably, throughout the book, and his purpose is to use this as his philosophy of mathematics, which he aptly terms humanism. Moreover, he believes that humanism is not, by and large, taken into account by philosophers who advocate other philosophies of math-that in fact, "mainstream" philosophies (especially taken literally and exclusively) tend to work against humanism.
"Humanism sees," he states on p. 22, "that constructivism, formalism, and Platonism [the three schools of thought that he singles out] each fetishizes one aspect of mathematics, [and each] insists [that] that one limited aspect is
mathematics. " Indeed, such one-sidedness certainly seems contrary to humanism, as well as to common sense.
For example, re Platonism: On p. 12, Hersh objects to "the strange parallel existence of two realities-physical and mathematical; and the impossibility of contact between the flesh and blood mathematician and the immaterial mathematical object. . . . " Hersh also has things to say about the effect on teaching of taking Platonism too seriously. P. 238: "Platonism can justify the belief that some people can't learn math."
Re formalism: P. 7, "The formalist philosophy of mathematics is often condensed to a short slogan: 'Mathematics is a meaningless game.' " Put that way, math does seem cold and in-
70 THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK
human. For Hersh, viewing math as "something people do," it is pertinent that the way people do math is usually not via formalism-that theorems are thought of first and then proven. He also sees social dangers in taking formalism too seriously-in particular, he blames it for the educational disaster of The New Math.
Re constructivism: It's "ignored by most of the mathematical world" (p. 138) and sometimes by Brouwer, the "master constructivist" himself. And although it seems an interesting exercise to disallow various axioms, explore other worlds, and see what we can still salvage (also, to distinguish between constructive and non-constructive proofs), one doesn't have to, and constructivism doesn't encompass all of math.
Moreover (p. 22), "none of the three [philosophies] can account for the existence of its rivals." Thus Hersh views over-emphasis on any of the three as distasteful, inadequate, even politically dangerous, because rt virtually ignores what mathematicians (and maybe other scholars) actually do.
More About Philosophy
"Humanistic" and "mathematics" is what made me want to read and review this book. I (ahem) forgot about "philosophy.''
Philosophy was one of the required undergraduate courses that just never got on my wavelength. I was always thinking, "Why are the various theories considered contradictory? Can't they all just be there, treated as ideas, questions, gropings, different sides to the story?"
I felt affirmed to see some of these questions echoed in Hersh's book. P. 30, for example: "Simplicity . . . goes with single-mindedness . . . both formalization and construction are essential features of mathematics. But the philosophies of formalism and constructivism are long-standing rival
schools. It would be more productive
to see how formalization and con
struction interact than to choose one
and reject the other."
Indeed, there are many passages
where Hersh pokes fun at rival philoso
phies of math. I'm hard put to select
my favorites. We might begin with two
section titles: "Neoplatonists-Still in
Heaven" (p. 102) and "Beautiful Idea
Didn't Work" (p. 159). We might then
move on to p. 1 10: "Philosophy stu
dents are supposed to read Descartes'
'Discourse on Method' (1637). They
don't realize that the complete
'Discourse' includes Descartes' mathe
matical masterpiece, the 'Geometry'.
. . . On the other hand, mathematics
students also are miseducated. They're
supposed to know that Descartes was
a founder of analytic geometry but not
that his 'Geometry' was part of a great
work on philosophy."
P. 138, skeptically describing Plato
nism: "These [mathematical] objects
aren't physical or material. They're out
side space and time. They're immutable.
They're uncreated . . . . Mathematicians
are empirical scientists, like botanists."
And formalism: "A strong version says
that there are rules for deriving one for
mula from another, but the formulas
aren't about anything . . . . "
The observation on p. 149 is per
ceptive and important: "Far from a
solid foundation for mathematics, set
theory/logic is now a branch of math
ematics [italics mine], and the least
trustworthy branch at that."
All these passages are certainly re
freshing. However, reading certain
other passages, I found myself turning
the skepticism back on him. P. xii: "The
book has no mathematical or philo
sophical prerequisites." Maybe not
technically. But to get the flavor of the
book, and virtually any paragraph in it,
one needs an interest and some back
ground in math, if not philosophy. On
p. 29 he reiterates, "This book aims to
be easily comprehensible to anyone. If
some allusion is obscure, skip it. It's
inessential!" By those instructions,
most of the book is inessential!
I feel that the book could be shorter
and better organized, perhaps with
Part II on the history of the philosophy
of math coming first. Also, although
there were many good explanations in
the "Mathematical Notes" at the end,
there were still missing links (for ex
ample, calculus is not only distance,
velocity, and acceleration), a few inac
curacies (such as p. 285 on Fibonacci
series and pp. 309-10 on distribution
theory), and, unless I'm misunder
standing, a couple of gross misprints
on p. 313. (In paragraph 4, line 2, x
should be n, and in line 3, X should be
x). There were also some passages
which seemed fallacious, or perhaps
just confusing. Sometimes I had trou
ble figuring out when he was being
facetious and when not. While reading
obscure passages I couldn't help think
ing, in an almost fond (and I hope hu
manistic!) way, "Ah, well, he is, after
all, a philosopher . . . . "
What's Meant by Humanistic?
By calling math "humanistic" Hersh
seems to mean at least four main con
tentions, all corollaries of math being
"something people do":
(1) Math, or our body of knowledge
of math, including what is "currently
fashionable" in math, changes with
time.
(2) The above is also a function of
place-of culture, societal circum
stances, and so on. That is, math is po
litical.
(3) The above contains mistakes.
That is, mathematicians are fallible.
( 4) Mathematicians interact with
one another. That is, math is "some
thing people do" together.
On p. 23 Hersh makes what he con
siders his most controversial point.
"There is no need to look for a hidden
meaning or defmition of mathematics
beyond its social-historic-cultural
meaning." Indeed, "biology is destiny,"
even as regards math. As Hersh says,
on p. 17, "our mathematical ideas . . .
match our world for the same reason
that our lungs match earth's atmo
sphere."
Still, I feel that it's not only the do
ing of math with other human beings
that makes it humanistic; it's also the
math itself. This I have experienced.
When ninth-grade algebra hooked me,
it never occurred to me that math
might not be humanistic. Math struck
an emotional chord in me. Elsewhere
I have written, for example, about "lit
tle x's and y's crawling about like
frightened insects." I had math dreams
and wrote math poems; I still do. Math
was always very human to me, but it
wasn't, back then, · a matter of inter
acting mathematically with other hu
man beings. Quite the opposite; math
was too precious a thing for this over
sensitive adolescent to even speak of.
With maturity I learned that math is en
hanced by sharing it with others, but I
still don't believe that this sharing is
the only thing that makes math "a hu
man endeavor. "
In a simple, honest, and moving pas
sage Hersh says, "this book is written
out of love for mathematics." Even bet
ter is the preceding sentence: "In at
tacking Platonism and formalism and
neo-Fregeanism, I'm defending our
right to do mathematics as we do." Still,
I kept noticing passages where I wanted
him to be more humanistic, meaning
more sensitive and perceptive, and I
think there are some humanistic oppor
tunities that Hersh misses.
For example, from pp. 170 to 176 he
devotes considerable negative energy to
the "most influential living philosopher,
W.V.O. Quine"; a lot of this is amusing
and probably warranted. P. 170: "Quine's most famous bon mot is his de
fmition of existence. 'To be is to be the
value of a [bound] variable.' I This has
the merit of shock value. I In the Old
Testament, Yahweh roars 'I am that I am.'
Must we construe this as: '1, the value of
a variable, am the value of a variable"? I Or Hamlet's 'to be the value of a variable
or not to be the value of a variable.' I Or
Descartes' Meditations: 'I think, there
fore I am the value of a variable.' "
Such bantering, which Hersh often
indulges in, is fun and often well-taken.
Still, I liked that particular Quine
quote. No, it makes no apparent literal
sense, nor philosophical nor mathe
matical, but it struck me as having
some sense-psychological, perhaps,
or poetic-or "shock value"? I am cur
rently trying to write a poem about it.
Another example: Hersh seems to
take a dim view of Kant's inquiry, "How
is mathematics possible?" "It's a futile
question" is the title of the section in
which he deals with this, and he could
be right, in the sense that the question
VOLUME 22, NUMBER 1 , 2000 71
will very possibly never be answered
or even posed in rigorous terms.
However, is "futile" the same as "un
worthy"? Just because we don't know
how to answer, or pose, a question
doesn't mean that question is merely
"amusing," as Hersh says. Kant's ques
tion seems interesting, in part because
it leads to other questions (such as
"How can mathematics not be possi
ble?" "What do we mean by possible?"
and of course, "What is mathematics?").
"This much is clear," says Hersh (p.
21): "Mathematics is possible . . . .
'What is happening can happen.' "
True, but that says only that it can
happen, not why or how it happens.
Hersh's attitude here brings to mind a
disgruntled non-math major taking a
required math course and muttering,
"Who cares? Why bother with such
things?" To say, "It works. Why ask
why?" seems the very opposite of the
spirit of math.
What is Mathematics,
Really- Really?
Hersh begins the Preface to his book by
connecting it with his own odyssey, in
a beautifully honest way: "Forty years
ago, as a machinist's helper, with no
thought that mathematics could be
come my life's work, I discovered the
classic, What is Mathematics? by
Richard Courant and Herbert Robbins.
They never answer their question; or
rather, they answered it by showing
what mathematics is, not by telling
what it is. After devouring the book with
wonder and delight, I was still left ask
ing, 'But what is mathematics, really?' "
So now I, after "devouring" Hersh's
book "with wonder and delight," am
left asking, "But what is mathematics,
really?" Surely the credo quoted at the
beginning of this review-"mathemat
ics is something people do"-was not
intended by Hersh as a defmition.
Mathematics is not something that
all people do. Also, mathematicians do
other things besides math.
Hersh certainly brings us closer to
knowing what mathematics is, and he
thoroughly describes and makes us be
lieve in the ubiquitous and welcome as
pect of humanism in mathematics,
even if the question "What is mathe
matics?" remains. The poet Anne Sexton
72 THE MATHEMATICAL INTELLIGENCER
wrote, "We're here to worship the
question itself," and question-worship
ping seems to be one of the things
mathematicians are destined to do. We
could fare worse.
Department of Mathematics and Computer
Science
Drexel University
Philadelphia, PA 1 9 1 04
USA
Dynamical Systems and Numerica l Analysis by A. M. Stuart and
A. R. Humphries
CAMBRIDGE: CAMBRIDGE UNIVERSITY PRESS, 1 998
US $64.95, ISBN-0-521 -49672-1 (HARDBACK)
US $39.95, ISBN 0-521 -64563-8 (PAPERBACK)
REVIEWED BY DAVID ARROWSMITH
A study at the interface between
two areas often provides some
surprises. Interdisciplinary studies are
now all the rage. The hope is, of
course, that deep knowledge and even
language in one area can provide un
expected approaches and spin-offs in
the other. There have been many such
cases involving dynamical systems
over the last few decades: algebraic
and number-theoretic techniques in
dynamical systems; nonlinear analysis
of heart arrhythmias; the use of non
linear modelling to provide effective
control of lasers. So we should expect
good things when dynamical systems
and numerical analysis come together.
In some senses they are inextrica
bly linked. At its base level, a dynami
cal system is an iterative process, and
whenever we consider iterations in
volving real numbers, numerical simu
lations inevitably involve a truncation
of the real numbers to a fmite number
of decimal places. Thus numerical
problems immediately arise: what is
the significance of the repeated round
off in iterations? Conversely, numeri
cal analysis has its own dynamical al
gorithms. It is surprising, given this
intimacy, that experimentalists in dy
namical systems will often seek nu
merical solutions and not worry too
much about the errors that arise!
What is startling is that persistent er
ror is often knowingly used by dynami
cists. Green [1] carried out famous stud
ies of the "standard map" in the early
1980s. This area-preserving map was the
simplest example for showing regions of
"chaos." The phase portrait was known
to have an infinity of unstable saddle pe
riodic orbits and to display instability,
and yet the pictures one obtained of
chaotic regions were repeatable and
seemingly machine-independent. Green
remarked that the errors in the iterative
process of the standard map were along
the boundaries of the chaotic regions
and not transverse to it. This meant that
although the orbits were not repeatable
the regions took on similar shapes as
boundary changes were limited. Similar
observations can be made of the various
chaotic attractors, for example the
Henon, Lorenz, and Rossler attractors.
So the dynamicist is often provided with
the clearest reasons for ignoring dy
namical error.
Another classic case is the control
of chaos (see Ott, Grebogi, and Yorke
[2]), where a chaotic orbit is used to
sample a chaotic region until the orbit
arrives sufficiently close to the target,
say an unstable periodic point. A new
regime is then imposed to obtain a lo
cal control to target. The accuracy of
the chaotic orbit is not essential pro
vided it does the job of numerically
sampling the target region. In fact,
many attractors that we refer to as
"chaotic" are not known to be chaotic
but are numerically chaotic. The orbits
are restricted to a compact region, they
exhibit local sensitive dependence on
initial conditions, and the regions seem
to be sampled by many "dense" orbits.
It is with this dysfunctional view of
numerical experiments that I come to
review the book "Dynamical Systems
and Numerical Analysis." One's frrst as
sumption is that the book will be two
sections sewn together-well, there are
two sections, but the good news is that
they are interwoven! Certainly, the in
troduction is reassuring to the dynami
cal systems specialist. The examples
and the language are totally recognized
at once, and the belief starts to grow
that this is the book for the dynamicist
who is ignorant of NA. The frrst few
chapters cover all the basic ingredients
of dynamical systems, limit sets, stabil
ity and bifurcation, period doubling,
chaos, invariant manifolds, attractors,
global features. The first chapter covers
maps and the second ordinary differen
tial equations. The section titles are vir
tually identical except for the necessary
change from area-preserving maps to
Hamiltonian systems.
The book is strewn with examples
to test the reader's understanding.
The remaining six chapters are de
voted to numerical methods. The
Runge-Kutta and multi-step methods
are treated first. The discussion in
volves truncation error, order, and fi
nite-time convergence. The chapter
ends with stiff systems and stability.
Crucially, the authors do not abandon
the style of the first two chapters and
retain this reader's confidence. To re
inforce this, the fourth chapter is enti
tled "Numerical methods as dynamical
systems." The techniques of chapter 3 are revisited. It is shown that various
Lipschitz conditions allow Runge
Kutta methods to be viewed as dy
namical systems. The authors discuss
structural assumptions for this to oc
cur; linear decay, one-sided Lipschitz
conditions, dissipative systems, and
gradient systems.
The dynamical systems theory comes
back with full force from chapter 5 on
wards-numerical techniques go global!
One of the great benefits of global dy
namical systems theory is the geometri
cal insight that it affords and the ability
it gives us to see key orbital features that
give the system its prime characteris
tics-for example, the types and struc
ture of attractors that can occur.
The asymptotic behavior of a dy
namical system is given by its u.rlimit
sets. Thus it is necessary to know the
extent of the difference between the be
havior of the limit sets of the underly
ing system and their numerical approx
imations. This problem is immediately
recognizable to the dynamicist. The in
troduction of spurious periodic solu
tions (which have no corresponding or
bit in the original system) is of key
interest. Theorems are given which de
scribe the bifurcation of spurious solu
tions at hyperbolic fixed points. Nice
motivational examples transfer to
global considerations by looking at un-
stable and stable manifolds of simple
differential systems and how the dis
cretized system has approximating
manifolds. These observations are then
extrapolated to more general results
where appropriate. Similarly, the au
thors consider near preservation of
limit-set behaviour for contractive sys
tems and gradient systems. This devel
opment terminates in clean statements
about upper semi-continuity properties
for the distance between an attractor
and its multi-step approximants.
The book fmishes by addressing the
corresponding numerical problems of
Hamiltonian systems where the dis
cretizations have to take a symplectic
form to retain the conservational as
pects of the dynamics.
The real strength of this book is that
the numerical analysis is described by
authors who are sympathetic to the
qualitative aspects of dynamical sys
tems and therefore make the numeri
cal medicine much more palatable
than most of the traditional texts I have
seen on 'Numerical Analysis.
REFERENCES
[1 ] Green, J. M . , A method for determining sto
chastic transition. Journ. Math. Phys. 20 (6), 1 1 83-1 201 ' 1 979.
[2] Ott, E., C. Grebogi, and J. A. Yorke,
Controlling Chaos, Phys. Rev. Lett. 64 (1 1 ) ,
1 1 96-1 1 99, 1 990.
School of Mathematical Sciences
Queen Mary & Westfield College
University of London
Mile End Road, London E1 4NS
England
e-mail: D. K.Arrowsmith@qmw .ac. uk
The Four-Color Theorem by Rudolf and Gerda Fritsch
Translated by Julie Peschke
NEW YORK: SPRINGER-VERLAG, 1998 US $29.00, ISBN 0-387-98497-6
REVIEWED BY KENNETH APPEL
When asked why he wanted to
climb Mount Everest, George
Mallory replied, "Because it is there."
That response made him a soul-mate
to a large number of mathematicians.
What other motivation would lead so
many of us, professional and amateur,
to spend immense amounts of time
worrying about whether we could color
maps with four colors, when any car
tographer could have told us that with
the possible exception of the Red Sea,
bodies of water are colored blue? And,
furthermore, that it would require con
siderably less effort to provide each
map-maker with a couple of extra bot
tles of colored ink Mathematicians
may not all phrase their ambitions at
the level of Hilbert's program, but
when faced with intellectual moun
tains they have a strong urge to climb
them. When the mountains look like
molehills, mathematicians are tempted
to make remarks like the famous com
ment of Minkowski about the Four
Color Problem, that the problem had
not been solved because no first-rate
mind had attacked it. But each attack
that fails increases the value of the
prize.
Rudolf Fritsch is Professor of Math
ematics Education at the University of
Munich. The book grew out of his
thoughts "about how one could make
the mathematical workings of the
Four-Color Problem more accessible
to students and professors alike." His wife Gerda provided the historical
(first) chapter of the book
This begins with an introduction to
the mid-nineteenth-century mathemati
cians who first considered a question
from a student who had successfully
colored a map of the counties of
England with four colors and won
dered whether this could be done with
all maps. Next, the volume introduces
Arthur Kempe, a lawyer and fine ama
teur mathematician whose fame rests
upon publishing, in 1879, one of the
most clever and insightful incorrect
proofs in the history of mathematics.
It then introduces Percy Heawood
who, as a young man in 1890, not only
demolished Kempe's "proof' as one re
sult in a tremendously impressive pa
per, but also generalized the problem
from its setting in the plane to arbitrary
surfaces and proved the appropriate
sufficient conditions in every setting
except the one in which the question
had originally been asked. After
VOLUME 22. NUMBER 1 , 2000 73
demonstrating this initial brilliance, Bishop Heawood showed incredible persistence; his last paper on the subject was published almost 60 years later.
We are introduced to George Birkhoff, who, one assumes, even Minkowski would admit had a first-rate mind. In 1913, Birkhoff generalized Kempe's techniques in a way that led to the eventual solution of the problem. We meet Heinrich Heesch, who spent almost 40 fruitful years (1937-1976) working on the problem. Starting in the 1950s he recast Birkhoffs ideas into a form amenable to computation and, with his student Karl Durre, showed that the needed reducibility computations were feasible. We are introduced to one of the great amateur mathematicians of our century, Professor (of French literature) Jean Mayer of Universite Paul Valery in Montpellier, who in the 1960s and 1970s made many contributions to both of the subjects treated in Heawood's paper. Finally we are introduced to Kenneth Appel and Wolfgang Haken, who started from a very high base camp with stronger computers and participated in the fmal trip to the summit along with John Koch, who was a graduate student at the University of Illinois when Appel and Haken were colleagues there.
Mter this first historical chapter, the Fritsches present the topological results that enable the problem to be phrased in terms of graph theory, and then explain, in considerable detail, the ideas involved in the proof.
Someone with no other knowledge of efforts to prove the theorem might be misled by the fact that the material presented shows a rather straight path from the work of Kempe to the final success. Actually, as can be seen from other books on the subject, there were many other powerful techniques developed in the search for a proof.
Chapter 2 presents a careful introduction to the topological background that permits a precise statement of the four-color theorem. Chapter 3 introduces the inductive plan of attack and points out some of the standard simplifications. In Chapter 4 the problem is phrased in the language of graph the-
74 THE MATHEMATICAL INTELLIGENCER
ory. In most of the presentation, as previously indicated in Chapter 2, the authors use the word graph for what is usually known as a plane drawing of a planar graph. Dual graphs, which permit one to rephrase the problem in a form much easier to work with, are introduced at the end of Chapter 4.
The argument up to this point is a careful presentation of the ideas of Kempe's paper as simplified somewhat by Joseph Story, the editor of the American Journal of Mathematics, in a paper inunediately following Kempe's in the second volume of the journal. Thus, the idea of cubic map is introduced by vertex inflation. This reviewer prefers the approach of bringing in the dual somewhat earlier, for reducing the degrees of regions to obtain a triangulation of the dual by adding edges to regions of degree higher than three seems much more intuitive. Support for this view grows when we notice that Story was so busy simplifying the cubic map part of Kempe's paper that he never observed that the proof was fallacious.
In Chapter 5, the combinatorial version of the four-color theorem, i.e., the statement in terms of vertex coloring of the dual graph of the original map, is given, and configurations and rings are defmed. Chapter 6 introduces Kempe chains and Kempe's argument (stated combinatorially). It also describes the argument of Birkhoff to show the reducibility of the diamond of vertices of degree 5, from which Heesch's C-reducibility is derived. This is followed by a more precise definition of the types of reducibility studied by Heesch.
In Chapter 7, the problem of fmding unavoidable sets is described, followed by examples of simple discharging algorithms that provide unavoidable sets (not consisting entirely of reducible configurations). In an apparent misunderstanding, the authors refer to Appel's discharging algorithm. No such beast ever existed. Indeed, the discharging procedure used in the original proof of the four-color theorem can best be thought of as a sequence of discharging algorithms, each algorithm designed to overcome a flaw in its predecessor, and was the major contribu-
tion of the joint provers of the theorem. The error is understandable, since the proof of the theorem that was published never describes the algorithm, but only provides an argument that the set obtained is unavoidable.
The book accomplishes its major purpose of providing an introduction to the concepts involved in the proof of the four-color theorem to a mathematically capable undergraduate. Both the mathematical and historical material is clearly written and the English translation (by Julie Peschke) preserves the clarity of the Fritsches' exposition.
University of New Hampshire
Kingsbury Hall
Durham, NH 03824
USA
e-mail: [email protected]
Specia l Functions by George F Andrews, Richard
Askey, and Ranjan Roy
ENCYCLOPEDIA OF MATHEMATICS AND ITS
APPLICATIONS, #71
CAMBRIDGE: THE UNIVERSITY PRESS. 1 999.
xvi + 64 pp. US $85.00, ISBN 0-521 -62321 -9
REVIEWED BY BRUCE BERNDT
Occasionally there is published a mathematics book that one is
compelled to describe as, well, let us say, special. Special Functions, by Andrews, Askey, and Roy, is certainly one of those rare books. What makes a tome special? At the risk of revealing to readers that the reviewer has frittered away some of his evening time watching the beginning portions of the Late Show, we offer the Top Ten criteria for determining if a book is special or not.
10. The book contains material not found in any other book of the same sort.
9. The authors' insights and special
expertise are pervasive. 8. The contents are placed in an his
torical perspective to give readers a better understanding of the subject.
7. The contents generate further
research.
6. The material is important to a
broad spectrum of readers.
5. The material has applications to a
wide range of both mathematical
and scientific disciplines.
4. The level of exposition is accessi
ble to beginning graduate students
and to some well prepared under
graduates.
3. Challenging exercises bring out
further important facets of the
subject.
2. The book deals with special func
tions.
1. The book generates e-special-ly
outrageous puns from at least one
reviewer.
As the topics in the book's 12 chap
ters are delineated below, it will be
made manifest that all of the first nine
listed criteria are satisfied. However, if
mathematicians had followed Paul
Turan's advice that the functions ad
dressed in this treatise be called use
ful functions, then criterion number 1
would likely not be satisfied.
Many books in analysis and special
functions have sections or chapters on
the gamma function. However, the
present authors' beginning chapter on
the gamma and beta functions is espe
cially elegant. Kummer's Fourier ex
pansion of log f(x), Dirichlet's multi
ple integral, Gauss and Jacobi sums
(the finite field analogues of, respec
tively, the gamma and beta functions),
and the p-adic gamma function are
some of the topics not found in most
treatises even with a chapter or large
section on the gamma function.
The heart of the book is the next two
chapters on hypergeometric series.
They contain an enormous wealth of
material and these functions permeate
later chapters as well. Not to deprecate
other writers, many accounts of hy
pergeometric functions are clearly and
logically presented but lack the per
spicacity of the present authors, whose
narration is replete with insights, mo
tivation, and history. For example, on
pages 126 and 127, we learn that in his
proof of one of the fundamental qua
dratic transformations, Gauss demon
strated a grasp of the principle of
analytic continuation. Also clearly pre
sented are newer developments, such
as the connections of contiguous rela
tions with the recent summation meth
ods of R. W. Gosper, and H. Wilf and
D. Zeilberger.
Chapter 4 on Bessel functions is
shorter than one might expect. Besides
many standard theorems, monotonic
ity properties are also established.
Chapter 5 provides an excellent,
well-motivated introduction to orthogo
nal polynomials, with the Chebyshev
polynomials and trigonometric func
tions as the motivating forces. Gaussian
quadrature, continued fractions, and
moment problems are shown to be nat
ural outgrowths of the general theory.
Special instances of orthogonal poly
nomials are the topic of Chapter 6, with
the Jacobi polynomials playing the lead
ing role. In the past few decades, or
thogonal polynomials have had promi
nent applications in combinatorics, and,
in particular, matching polynomials are
extensively examined in this chapter.
Chapter 7 offers some beautiful
topics wherein orthogonal polynomi
als arise. The positivity of coefficients
in the power series expansions of cer
tain rational functions, a particularly
favorite topic of one of the authors, is
nicely introduced. The positivity of
certain polynomials, both ordinary
and trigonometric, the MacMahon
Master Theorem, and F. Beukers's
proof of the irrationality of �{3) are
also presented.
In the past couple decades, Selberg's
integral and various generalizations
and analogues have been the focus of
much research. Many proofs are ex
traordinarily difficult and lengthy, but
Chapter 8 gives a very accessible in
troduction to this important area; no
text had heretofore attempted to give
such an introduction. The beautiful
proofs and extensions of K. Aomoto and
G. W. Anderson are given. A fmite field
analogue is also established.
Chapter 9 deals with ultraspherical
polynomials and their connections
with representation theory.
Chapter 10 is a beautiful and su
premely well-motivated treatment of q
series. The chapter begins with a dis
cussion of the binomial theorem and
its q-analogue. The introduction to
q-integrals is the best I have read. The q
gamma and q-beta functions and theta
functions are introduced. Applications
to sums of squares are given. Here the
work of S. Ramantijan, M. D. Hirschqom,
and S. C. Milne might have been men
tioned. Some of the fundamental theo
rems on basic hypergeometric series are
presented. Previously, Chapter 2 in
Andrews's The Theory of Partitions
was the best introduction to q-series
before embarking on the more ambi
tious treatise, Basic Hypergeometric
Series, by G. Gasper and M. Rahman.
Now I will tell my students to read first
Chapter 10 of the book under review.
Partition analysis, the topic of Chap
ter 1 1, has not been treated in text form
since P. A. MacMahon developed it in
his famous Combinatory Analysis;
indeed partition analysis has been un
fairly neglected. A number of elemen
tary theorems about generating func
tions for certain partition functions are
derived in an easy, painless way. The
chapter ends with proofs of Ramanu
jan's congruences modulo 5 and 7 for
the ordinary partition function.
The powerful method of Bailey
pairs has also not been heretofore ex
amined at any length in textbooks.
Several applications are given; in par
ticular, the second of L. J. Rogers's
proofs of the Rogers-Ramantijan iden
tities is presented in Chapter 12.
The prerequisites for reading this book are sound undergraduate courses
in real and complex analysis. In particu
lar, uniform processes should have been
mastered. However, a considerable
amount of the book is not dependent on
knowledge of complex analysis; for ex
ample, little is used in Chapters 10-12.
The book contains 440 well selected
exercises, virtually none trivial and all
interesting.
In conclusion, it carmot be overem
phasized how the authors' rich histor
ical knowledge generates a fuller un
derstanding and appreciation of their
subject. Some additional papers, espe
cially from the past couple of decades,
might have been mentioned, but gen
erally most of these references can be
obtained from other papers cited.
Indeed this treatise is special and
should become a classic. Every stu
dent, user, and researcher in analysis
VOLUME 22, NUMBER 1 , 2000 75
will want to have it close at hand as
she/he works.
Department of Mathematics
University of Illinois
1 409 West Green St.
Urbana, IL 6 1 801 , USA
Complexity and Information by Joseph F. Traub and Arthur G.
Werschulz
CAMBRIDGE UNIVERSITY PRESS, CAMBRIDGE, 1 998, 1 39
pp., $1 9.95, PAPERBACK, ISBN 0-521 -48506- 1 , $54.95,
HARDBACK, ISBN 0-521 -48005-1
REVIEWED BY DAVID LEE
This expository book is on the com
putational complexity of continu
ous problems. Consider a typical prob
lem: the numerical solution of a partial
differential equation. The initial and
boundary conditions are given by real
functions. Since these functions can
not be read into a digital computer, the
computer input is a discretization of
the function, and hence the computer
has only partial information of the math
ematical problem. Furthermore, the
computer information is typically con
taminated by roundoff errors. Finally, it ·
can be expensive to obtain the function
evaluations. Information-based com
plexity is the study of computational
complexity of problems for which the
information is partial, contaminated,
and priced.
Typical questions studied by infor
mation-based complexity include:
• What is the computational com
plexity of problems of numerical
analysis?
• How do problems suffer from the
curse of dimensionality?
This book is a guide to the numer
ous papers that study these and other
problems. The presentation is informal
but with motivation and insight.
The first two chapters are an intro
duction to information-based complex
ity. It uses a simple example of integra
tion to explain the main ideas of the
theory. The next three chapters are de-
76 THE MATHEMATICAL INTELLIGENCER
voted to high-dimensional integration.
The third chapter deals with breaking
the curse of dimensionality for integra
tion by settling for stochastic assur
ance. The fourth chapter is on comput
ing high-dimensional integrals for
mathematical fmance. A typical prob
lem involves a few hundred dimensions,
and requires on the order of 106 float
ing-point operations for a single evalu
ation of the integrand. The fmance com
munity long believed that these
integrals should be evaluated using
Monte Carlo. Experiments conducted
at Columbia University showed that
low-discrepancy methods from number
theory beat Monte Carlo by one to three
orders of magnitude. These results ap
parently contradict the conventional
wisdom that low-discrepancy methods
were not good for problems of high di
mensions. The fifth chapter describes
the computational complexity of path
integration. The problem of path inte
gration occurs in many areas, including
quantum physics, chemistry, fmancial
mathematics, and the solution of partial
differential equations.
The complexity of ill-posed prob
lems is discussed in chapter six.
Practical examples of ill-posed prob
lems occur in remote sensing and im
age processing. It is well-known that
ill-posed problems can be solved, if the
residual is used to measure the quality
of the approximation. Suppose, how
ever, that the quality of the approxi
mation is measured by how close it is
to the true solution. Then ill-posed
problems are unsolvable in the worst
case setting, even for a large error
threshold. The concept of well-posed
ness on the average is introduced, and
it is shown that if a problem is well
posed on the average, then it is solv
able on the average.
Verification and testing are essential
for the reliability of numerical software.
For problems of numerical analysis that
involve functions, apparently one has to
test the code that implements these
functions an infinite number of times to
guarantee conformance. Two chapters
of this book report the study of the com
plexity of verification and implementa
tion testing. For stochastic assurance, a
fmite number of tests suffices to estab-
lish conformance to within a fmite error
threshold.
Other chapters deal with linear equa
tions, integral equations, nonlinear op
timization, linear programming, and
computation with noisy information.
There is a brief history of informa
tion-based complexity and a bibliogra
phy containing over 400 papers and
books.
Those who are interested in numeri
cal analysis and scientific computing
can benefit from reading this book
Scientists and engineers from other dis
ciplines, such as networking, might be
interested as well. Internet is changing
the world communication. However, the
fundamental problem of flow control re
mains unsolved, and that severely hin
ders or limits its applications such as
Internet Telephony. The flow control
problem can be formulated as a mathe
matical programming problem based on
the Internet traffic information, which is
partial, contaminated, priced, and highly
dynamic. Would Internet researchers
benefit by a study of this theory? Would
information-based complexity-theorists
be interested in broadening their scope
of study and in looking at this or other
real-world problems?
Since the primary goal of this book
is exposition, many technical details
are omitted. Those who want to know
more about the theory could consult
the references listed below and pro
vided in the book, which survey the
major advances since 1988 in this dy
namic field.
REFERENCES
[1 ] J. F. Traub, G. W. Wasilkowski, H. Wozni
akowski, Information-Based Complexity.
New York: Academic Press (1 988).
[2] J. F. Traub, H. Wozniakowski, Information
based complexity: new questions for math
ematicians. Mathematical lntelligencer 1 3
(1 991) , no. 2 , 34-43.
[3] J. F. Traub, A G. Werschulz, Unear ill-posed
problems are solvable on the average for
all Gaussian measures. Mathematical lntel
ligencer 1 6 (1 994), no. 2, 42-48.
Bell Laboratories
Murray Hill
NJ 07974 USA
e-mail: [email protected]
k1fii,i.M$•iQ:I§I Robin Wi l son I
Renaissance Art Florence Fasanelli and Robin Wilson
Alberti
Brunelleschi
Piero
Please send all submissions to
the Stamp Corner Editor,
Robin Wilson, Faculty of Mathematics,
The Open University, Milton Keynes,
MK7 6AA, England
e-mail: [email protected]
A n outstanding example of a Renaissance man, Leon Battista Al
berti (1404-72) introduced one-point perspective construction in its mathematical form. The first written exposition of "painter's perspective" appears in his Della pittura [On Painting].
There, Alberti systematically ordered visual reality in geometric terms, combining geometry, placement, and optics to create apparent 3-dimensional space on a 2-dimensional surface.
Alberti dedicated Della pittura to his close friend the artisan-engineer Filippo Brunelleschi (1377-1446). In 1417, Brunelleschi won the competition to design the cupola of the Cathedral in Florence. Brunelleschi, also known as the founder of "painter's perspective," worked out linear perspective in relation to geometric-optical principles in a practical mode. Alberti, inspired by his techniques, wrote down the mathematical rules and filled out the scheme.
Piero della Francesca (c. 1412-92) found a perspective grid especially applicable to his own investigations of Euclidean solid geometry, and wrote De
perspectiva pingendi [On the Perspec
tive of Painting]. The picture on the stamps is his last painting, Madonna and
Child with Saints (14 72), from the Brera Gallery in Milan. It is rationally constructed with overarching symbolism;
80 THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK
the solidity of the egg (symbolizing the four elements of the universe) is in perfect mathematical perspective. The two other titles on the stamps are Piero's LibeUus de quinque corporibus regu
laribus [Book on the Five Regular
Solids] (1480s) and the renowned De
divina proportione [The Divine Propor
tion] (1509) by Piero's friend Fra Luca Pacioli (c. 1445--1514); a mathematician and expositor, Pacioli is the monk second from the right in the painting.
A close friend of Pacioli's was Leonardo da Vinci (1452-1519). It was probably Pacioli who taught Leonardo mathematics, and it was for Pacioli's De divina proportione that Leonardo made his unsurpassed woodcuts of polyhedra. Leonardo explored perspective perhaps more thoroughly than any other Renaissance painter; in his Trattato della pittura [Treatise on
Painting], he warns, "Let no one who is not a mathematician read my work."
Florence Fasanelli
Mathematical Association of America
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Leonardo
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Pacioli