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: dzrlib@library.caltech.edu

R. Michaelson

Northwestern University Library

Evanston, IL 60201

USA

e-mail: rmichael@nwu.edu

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: mcohen@cpcug.org

© 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: reeken@math.uni-wuppertal .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: Neville.Holmes@utas.edu.au

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: tietz@math.uni-hannover.de

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: Colin.C.Adams@williams.edu

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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Springer-Verlag �ew York, Inc.,

Journal Fulfillment Services

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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: fomin@math.lsa.umich.edu

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: l-golland@uchicago.edu

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: ksigmund@esi.ac.at

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: akd@csd.uwo.ca

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

I@.JMMfij.l§,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

domenicano@axscaq.aquila.infn.it

Istvan Hargittai

Institute of General and

Analytical Chemistry

Budapest Technical University

H - 1 521 Budapest;

Hungary

e-mail: hargittai.aak@chem.bme.hu

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.

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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: ROWE@mat.mathematik.uni­

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: kia@oregano.unh.edu

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: lee@research.bell-labs.com

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: r.j.wilson@open.ac.uk

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

1 529 1 8th Street NW

Washington, D.C. 20036 USA

e-mail : ffasanelli@juno.org

Leonardo

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Pacioli