No title

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

More Mathematics in Its Place

The story of the Archimedes palimp­ sest, as told in "Sale of the Century,"

In his commentary (Summer 1999 , pp. 30-32 ), Edward Reed argues for less

Math. Intelligencer21 (3 ),12-15 (1999 ),

mathematics and more numeracy in

reinforces the notion that bound vol­

engineering. I would not want to quar­

umes on shelves may not be such a bad

rel with the "more numeracy" part, but

storage medium after all. Does anyone

I have a somewhat different take on

really believe that "electronic" books

the desirability of less mathematics. I

from today will be readable in 3001 ??

should say up-front that I am not an en­

The problem with an electronic for­

gineer; I am a mathematical statistician

mat is that there is no economic model

working in the areas of survey statis­

for long-term magnetic storage. Where­

tics and education statistics.

as paper works may have to be re­

My own point of view is heavily in­

printed or copied every few hundred

fluenced by a talk I heard several

years, magnetic storage has an inher­

decades ago by Paul MacCready, the

ent life on the order of 10 years and

aeronautical engineer who designed

also suffers from repeated changes in

and built the Gossamer Condor (which

data formatting. How can we make

won the Kremer Prize for the first hu­

sound archival decisions in the ab­

man-powered flight over a fixed course)

sence of a viable model for open ac­

and the Gossamer Albatross (Kremer

cessibility to the scholarly community?

Prize for human-powered flight across

Though societies-like the American

the English Channel). Before begin­

Mathematical Society will likely pro­

ning this work, Dr. MacCready had done

vide access to their publications for a

a theoretical calculation that showed

substantial time period, it seems plain

that a low-powered aircraft would have

This may

that economic concerns will eventually

to have a very large wingspan.

result in the curtailment of electronic

not seem remarkable, but several inter­

access to older material, particularly

national groups were actively pursuing

material from commercial publishers. Electronic publications have an in­

the (first) Kremer Prize with aircraft that had no hope of success. On the other

creasingly important function, but this

hand, Dr. MacCready emphasized, it is

does not mean that they will or should

not possible to design an aircraft suc­

replace all paper publications. It is rea­

cessfully with paper and pencil alone:

sonable to conclude that a role for

simulations, modeling, test flights, and

print will continue to exist in parallel

tinkering are needed. The power of

with electronic publication for many

theory and mathematics often comes

centuries.

in showing what will

not work so that

effort may be concentrated along po­

D. L. Roth

tentially successful avenues.

Caltech Library System Pasadena, CA 91 1 25

Michael P. Cohen

USA

161 5 Q Street NW (#T-1 )

e-mail: [email protected]

Washington, DC 20009-631 0

R. Michaelson

e-mail: [email protected]

USA Northwestern University Library Evanston, IL 60201 USA e-mail: [email protected]

© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 1. 2000

5

plains why the Great Wall could not be

Numeracy with Mathematics

no bridge was ever built by mathemat­

Here is my (predictable) response to Ed­

ics, a mathematician is likely to retort,

seen from the moon; here he uses

ward Reed's contribution "Less Mathe­

perhaps on the authority of figures

similar triangles (mathematics), and

matics and More Numeracy Wanted in

such as Galileo, or even Donald Duck

knowledge of the absolute size of the

Engineering" in the Summer1999 issue

in Mathmagic Land, that nothing was

Great Wall and the moon's distance

of the Math.

ever built without mathematics. I would

(could he tell the moon's distance

Professor Reed's estimation of the

guess that the professor of mathematics

without mathematics and physics?).

size of an object on Earth as perceived

took umbrage at Professor Reed's re­

Without knowing it, the Reeds are

from the Moon strikes me as an excel­

marks and got the vapors not because

standing on foundations laid by math­ ematics, but as they ignore them their

Intelligencer.

lent example of what has come to be

he objected to a reform of the engi­

called a Fermi solution to a Fermi prob­

neering curriculum that would pro­

students will ignore them even more,

lem. Here is a typical Fermi problem, as

duce numerate engineers and mini­

until one day engineering will again be

posed by the great physicist Enrico

mize mathematical irrelevancies, but

reduced to "trial and error" and recipes

Fermi to his physics class: "How many

because these remarks reflected a

("counting the eggs for the mortar")­

piano tuners are there presently in

parochial view of mathematics that

modulated by "intuitive" arguments

Chicago?" A better-known and note­

might lead students to suspect mathe­

coming from half-forgotten scientific

worthy example of a Fermi solution:

matics itself is an overrated discipline.

knowledge.

Fermi's trick of dropping confetti to es­

Well should his students appreciate

timate the energy release from the first

that were it not for mathematicians

Michael Reeken

atomic bomb explosion. An essay in

Professor Reed would today be clad in

Department of Mathematics

Hans Christian von Baeyer's book The

goatskins and crouching beneath a

Bergische Universitat-GH Wuppertal

Fermi Solution, Random House, 1993 ,

berry bush for his supper.

D-42079 Wuppertal

uses these examples to introduce read­

All in all, his piece is an entertain­

ers to some intriguing aspects of scien­

ing and extremely stimulating con­

Germany e-mail: [email protected]

tific thought. I also highly recommend

tribution to the important ongoing di­

the book by M. Levy and M. Salvatori,

alogue concerning the role of mathe­

EDITOR's NoTE: Diverse reactions to the

Why Buildings Fall Down, W.W. Norton, 199 2, as an engaging popular

matics in society and in education.

note of Professor Reed are expressed

overview of some elements of structural

Don Chakerian

want to quote one more. Apologies to

engineering and the power of what

Department of Mathematics

the writer: though he submitted his let­

Professor Reed calls "numeracy."

University of California Davis

ter for publication and gave name and

in the letters above. I nevertheless

Davis, CA 9561 6-8633

address, I was unable to reach him at

USA

the address he gave, so as to confirm

serving as a foundation for the requi­

Response to Reed

All I can do is give an excerpt, anony­

Behind such estimates there is al­ ways a mathematical principle, either

his willingness to be quoted in print.

directly pertinent to the problem, or site physics. I say "always" because, as

What Edward Reed calls "numeracy" is

mously, from

a mathematician, I adhere to a broader

rudimentary mathematical knowledge,

mathematician visiting Germany.

(apparently) an Arab

defmition of mathematics than does

a thin layer of mathematical arguments

Professor Reed. It is true that one need

which are apparently not recognized as

The letter by Edward Reed may be

not have a grasp of the detailed struc­

such. Take his remarks about building

cheering for us "underdeveloped" na­

ture of the Euclidean group of simili­

bridges. "The medieval builder," he

tions, showing us how strongly science

tudes to apply Professor Reed's thumb­

tells us, "knew that if a shape, known

is declining in the West, giving us a

nail process for earth-lunar estimation,

to us as a catenary, could be drawn so

chance of catching up ....

but anybody will concede that an im­

as to go through every stone, then this

Let me apologize for the Islamic hu­

portant mathematical principle lurks

arch would stand up." Huh? To iden­

mor. But you have earned nothing but

behind the trick, and it is this that gives

tify a shape as a catenary and to ex­

scorn

us confidence in the procedure.

plain why it has the asserted property

preparing to take up the torch of scien­

would be mathematics. Next he ex-

tific thinking from your faltering hands.

While Professor Reed asserts that

6

THE MATHEMATICAL INTELLIGENCER

from

those

nations

who

are

Opinion

The Numerical Dysfunction Neville Holmes

T

he opinions of Anatole Beck in his

article "The decimal dysfunction"

[ 1) were refreshing and interesting,

the kind described here, as a necessary basis for efforts to reverse present trends in public innumeracy.

and his discussion of enumeration and mensuration was surely important and

Enumeration

provocative. A learned and detailed ar­

A major theme of Professor Beck's ar­

gument devoted to showing as "folly"

ticle is, as its name proclaims, that dec­

the SI metric system adopted by so

imal enumeration is not the best enu­

much of the world, and soon to be

meration

adopted by the USA [2), deserves some

"appears essentially not at all in math­

system.

The

reason? Ten

The Opinion column offers

serious response. If the SI system is in­

ematics, where the natural system of

mathematicians the opportunity to

deed folly, then mathematicians every­

numeration is binary. . . . One might

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,

where have a duty to make this as

blasphemously take the importance of

widely known as possible. If it is not,

2 in mathematics as a sign that God

then a rebuttal should be published. That the only published reaction [3)

does His arithmetic in binary." By def­ inition God is omniscient, and it is blas­

should be jocular, however witty, is

phemous to imply that She has to do

therefore deplorable. Jerry King, in his

any arithmetic at all! But 1 is much

splendid The Art ofMathematics, writes,

more important in mathematics than 2

"The applied mathematician emphasizes

is, so wouldn't tallies-which indeed

and neither the publisher nor the

the application; the pure mathematician

have a long tradition [6]-be better

editor-in-chief endorses or accepts

reveres the analysis." [4, p. llO] Perhaps,

still? Mere analytical argument will not

then, neither kind of mathematician sees

settle the matter.

responsibility for them. An Opinion should be submitted to the editor-in­ chief, Chandler Davis.

simple enumeration and mensuration as

Binary enumeration, whatever its

worthy of consideration, so that both ig­

analytical virtues, is not after all prac­

nore Professor Beck's argument, and

tical. "Binary numbers are too long to

thus show themselves apathetic about

read conveniently and too confusing to

the innumeracy that "plagues far too

the eye. The clear compromise is a

many otherwise knowledgeable citi­

crypto-binary system, such as octal or

zens" [5, p.3] and about "the declining

hexidecimal." In what way, then, are

mathematical

abilities

of

American

these systems a clear improvement on the decimal? To a society now used to

[and other] students" [4, p.176). Let the shameful silence on enu­ meration-mensuration be broken by a technologist, one with a background in

decimal enumeration, any non-decimal system will be confusing. Would octal or hexidecimal be more

science,

convenient than decimal, for example

with three decades of experience in

in being more accurate or brief? Octal

engineering

and

cognitive

the computing industry, and with a life­

integers are a little longer than deci­

long interest and a decade of experi­

mal, but hexidecimal are somewhat

ence in education. This article argues

shorter. All are exact. Not much to jus­

that the SI metric system is indeed

tify change in that.

flawed, though not in the way Professor

And fractions? Different bases dif­

Beck sees it; that the way we have

fer in which denominators they handle

come to represent numerical values is

best.

even more flawed; and that the general

Along this line, it is significant to ob­

public would be best served by a re­

serve that the smaller its denominator

duced SI metric system supported by

the more used a fraction is likely to be.

an improved (SI numeric?) system for

This observation is behind the benefits

representing numeric values. If these

so often argued for the duodecimal

arguments are valid, then mathemati­

system of enumeration, which can ex­

cians everywhere have a professional

press

halves,

duty actively to promote reforms, of

sixths

exactly

thirds, and

quarters,

and

succinctly.

The

© 2000 SPRINGER-VERLAG NEW YORK, VOLUM E 22, NUMBER 1 , 2000

7

most personal of the old Imperial mea­

tion, the system is overwhelmingly su­

The most important measurements

sures were conveniently used duodec­

perior to the old humanistic systems, if

in everyday life are those of length,

imally-12 inches to the foot, 12 pen­

only because its arithmetic simplicity

weight, volume, and temperature. Here

nies to the shilling. We still have 12

and world-wide acceptance make it less

in Australia, the discouraging of the pre­

hours to the o'clock, and 12 months to

subject to cheating and misunderstand­

fixes which are not multiples of one

the year. The movement for a thorough

ing. The difficulties for adoption of the

thousand seems to have had good effect,

dozenal system is quite old-Isaac

metric system now are much fewer and

the faulty early publicity notwithstand­

Pitman tried to introduce it with his

more transient than for the illiterate and

ing [an example in 11]. This is a very

first shorthand system [7]. Full and lu­

innumerate society of Revolutionary

good thing, because now there are fewer

cid arguments for the system can be

France, where the changeover lasted for

ways to express any measurement, and

found in texts [8], and in the publica­

two generations [9, p.264].

this must greatly reduce the potential for

tions of the various Dozenal Societies.

(Of course, there is something in­

confusion. Centimetres are occasionally

(The Dozenal Society of America ad­

trinsically

vertises from time to time in The Journal of Recreational Mathematics

having a characteristic way of doing

inches when giving anyone's height),

things, and this is true of household

and the hectare seems to have replaced

valuable about a culture

used (so are, for the moment, feet and

and, going by the World Wide Web, has

measures as well as of (say) cuisine.

the acre for people of large property. But

its headquarters on the Nassau campus

But the basic vehicle of culture is lan­

of SUNY.) However, the dozenal cause

guage, and anyone truly concerned

centi- and hecto- are otherwise never seen, and the confusing deci- and deka­

now seems a hopeless one, given that

about the preservation of cultural rich­

have disappeared altogether.

decimal enumeration has taken such a

ness and variety (as surely we all

Celsius, the new temperature unit,

global hold since the First World War.

should be) would be much better em­

took over straight away, possibly be­

Even if a way could be found to con­

ployed combatting the present oligo­

cause the old scale was plainly silly and

vert to a dozenal system, doing so would

lingual rush than opposing metrica­

its cultural value slim. Oddly enough,

not compel the abandonment of the met­

tion. Languages are dying off even

the unit is almost always spoken of sim­

ric system. The SI metric standards are

faster than species!)

ply as degrees. For lengths, people seem

not inherently decimal because the ba­

The strange thing about the metric

comfortable with millimetres and me­

sic and secondary units of measurement

system, though, is that, while the basic

tres and kilometres, though in casual

could as well be used with a dozenal sys­

units

tem of enumeration as with a decimal

ones) are widely and consistently ap­

are more often used, particularly the lat­

system. Mensuration combines an enu­

plied, each of these is the basis of a be­

ter. Grams and kilograms are comfort­

merated value with a unit of measure,

wildering collection of pseudounits de­

ably used for weights, though the ab­

and a good system will provide practical

fmed through a somewhat arbitrary

breviation

and useful units of measure.

system of scaling prefixes. Not only are

preferred to the full name. For vol­

(and some of the secondary

speech the abbreviations

mil and kay

kilo (pronounced killo) is

the prefixes weird in themselves, but

umes, the use of millilitres and litres

Mensuration

they also have inconvenient abbrevia­

has completely taken over,

The many old systems were practical

tions, including highly confusable up­

again the abbreviation

and useful in respect of how quantities

per and lower cases of the same letter

as for lengths!) is often heard. The use

could then be measured (often by

(Y is yotta, y is yocto [2]), and even a letter ( t) from the Greek alphabet.

of a secondary unit, litres, for volume

some human action giving units like

though

mills (not mil

is justified by its relative brevity in

paces or bow-shots) and how stan­

Although it is averred that the prefixes

speech,

dards could be administered [9]. There

are easy to learn and use, in practice

needed. The only problem is that the

so that no abbreviation is

were different units of length for dif­

their spelling, their pronunciation, and

litre has become somewhat divorced

ferent ways of measuring them, differ­

their meanings are all confused and

from the cubic metre, and people are

confusing in popular use. And it is pop­

not always able to compare volumes in

ular use that's important.

the two units swiftly and reliably.

ent

units

of quantity

for

different

things being measured, and different units for different towns and villages.

These prefixes are really only suited

The conclusion to be drawn from

However, the old measures were prone

for use in private among consenting

the Australian experience is that, while

to being used by the powerful to ex­

adults. It took a physicist, the famous

the common metric units of measure

ploit the weak, as implied by various

Richard Feynman, to advocate the pre­

have been everywhere adopted, their

admonitions in both the Bible (for ex­

fixes be abandoned because they actu­

names have been found difficult, and

ample, Deuteronomy 25:13-16) and the

ally express scalings of the measure­

all the long ones have been abbreviated

Quran (Sura 83: 1-17).

ments being made, and because they are

in common speech, typically by elision

The worldwide metric system de­ fmes as few basic units as possible, and

"really only necessitated by the cum­

of the basic unit name. Measures, and

bersome way we name numbers." [10]

numbers, must be simple to be popular.

secondary units such as for areas and

What does the experience of Aus­

volumes are derived from the basic

tralia, a country converted to SI metrics

Emancipation

ones. Though it might spring rationally

only a few decades ago, have to tell

The challenge is to free numbers gener­

or irrationally from the French Revolu-

about the popular use of the prefixes?

ally from the thrall of technologists and

8

THE MATHEMATICAL INTELLIGENCER

mathematicians so that more of them

disguise pure numbers as measure­

However, the most popular method

become easy for people to use. A great

ments under, for example, the pseudo­

merely raises the hyphen to the super­

way to start is to get rid of the metric

unit

decibel. Eventually even the per­

script position, which doesn't actually

prefixes along the lines suggested by

centage, and its pseudosubunit the

change the sign, and certainly doesn't

Feynman, and to build on popular usage.

point or percentage point, might meet

make the distinction obvious. Some

•

Let any number to be interpreted as scaled UP by 1000 be suffixed by

k,

and let a number like lOOk be pro­ nounced •

Let any number to be interpreted as scaled DOWN by 1000 be suffixed by m'

and let a number like lOOm be pro­

nounced

•

one hundred kay.

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

their Boojum and softly vanish away. Most

importantly,

the

notation

texts even increase the problem by us­ ing a raised

+ to mark positivity [e.g.,

would allow phasing out the present

13, p. 153], thereby spreading the am­

usage in

biguity to another basic symbol.

mathematics

and science

which shows scaled numbers as ex­ pressions like 3 X 1010. This style is particularly confusing for students. Is

The ambiguity extends to the spo­ ken word. The hyphen is read out as

minus whether it is used as the nega­

it a number, or is it an expression, or

tive sign or as the subtraction symbol.

is it a calculation? A mathematician or

This is a severe problem because the

scientist may be able to see immedi­

natural word for the sign,

ately past the calculation to the num­

three syllables long, one too many for

ber it produces, but to ordinary mor­ tals the expression hides the number.

it to be popular. Words like off and short can have the right kind of mean­

Mathematicians, or at least mathemat­

ing,

ics teachers, have in this ambiguity an­

within sentences. Maybe the abbrevia­

other very good reason for adopting a

tion

scheme like that suggested above for

negative, is

but might become ambiguous

neg could be adopted.

The negative sign should be used as

representing scaled numbers.

a prefix, because it is spoken as an ad­

Adoption of these rules, and of the ex­

Representation

dinary number is its most significant

tensions they imply, is in accord with,

To confuse expressions like 3

indeed would reinforce, both the intent

with numbers is bad enough, but at least

mil two.

jective, because the left end of any or­

X 1010

end, and because the negative sign is in some ways the most significant ci­

of the SI metric standards, and the com­

elementary school children are not nor­

pher, as it completely reverses the sig­

mon sense of popular linguistic prac­

mally exposed to this particular ambi­

nificance of the value it prefixes. The

tice. Adoption of these rules would al­

guity. However, they are exposed to a

wretchedly inadequate ASCII charac­

low

and their

very similar ambiguity early in their

ter set foisted on the world by the com­

upper-case, lower-case, and Greek ab­

arithmetic education, an ambiguity that

puting industry has no suitable symbol.

breviations to be forgotten, would al­

(some say) costs the average pupil six

Selecting from what is already avail­

months of schooling, and brings some

able in T EX, a suitable symbol might be

the metric

prefixes

low common talk of numbers to be as

loose or precise as needed, and would

pupils a lifetime of innumeracy. This is

a triangle, superscripted and reduced

deliver a wider range of numbers and

the ambiguity in notations such as -1

in size to be aesthetically and percep­

quantities into common parlance and

and -15 where the role of the hyphen

tually better: v72. (A superscript vee or

understanding. Measurements outside

is ambiguous [12]. Is it the sign for the

cup could be used as an option for eas­

the scales of common usage would at

property of negativity, or is it the sym­

ier handwriting, as in V72 or u72.) The

least be recognised roughly for what

bol for the function of subtraction? A

problem is rather that of getting the

they are, if not wholly understood.

conspicuous sign is needed to stand for

symbol onto the everyday keyboard.

These rules are simple enough to be

the property of negativeness in a num­

One new symbol is not enough. The

accepted by the general public, and ex­

ber, a sign quite distinct from the sym­

fraction point needs one as much as

pressive enough to be used by scien­

bol for subtraction.

the negative sign does. The dot used in

tists and engineers, and even by math­ ematicians.

Indeed

the

notation

is

Because the present ambiguity is not

most of the world for the fraction point

overtly recognised in early schooling,

is more inconspicuous than any other

similar to the so-called engineering or

few adults are even aware of it. Per­

symbol apart from the blank space.

e-notation, but better than it because

haps mathematicians consciously dis­

Furthermore, it is used as punctuation

there are fewer ways to represent any

tinguish the two meanings given to the

in ordinary text, leading to ambiguities

particular

was

hyphen. "Unfortunately, what is clear

in particular at the end of sentences

adopted by technical people submit­

to a mathematician is not always trans­

ending in numbers. That this incon­

value.

E-notation

ting to the limitations of the printers

parent to the rest of us." [4, p.50]

spicuousness is recognised as a diffi­

that were attached to early digital com­

Particularly not to children. That this

culty is demonstrated by the common

puters, and in it 100k2 might be repre­

ambiguity is a real problem is shown

precaution of protecting the dot from

sented as 1E8 or 100E6 or 0.1E9.

by the many texts for teaching ele­

exposure by writing for example 0.1

Adoption of the notation for scaled

mentary mathematics that use tempo­

rather than .1, by the use of the comma

numbers proposed here could allow

rary notational subterfuges in an at­

instead of the dot for the fraction point

dropping of quirky notations which

tempt

in Continental Europe, and by the mis-

to

overcome

the

ambiguity.

VOLUM E 22, NUMBER 1 , 2000

9

begotten attempt by the Australian

no way in which such fractions can be

merator part, then a very convenient

Government to use the hyphen for the

either keyed directly and exactly into

and pedagogically salubrious notation

fraction point in monetary quantities

a calculation, or shown exactly on a

is provided. The symbol I would not be

when

intro­

character display. Only decimal frac­

suitable, as it is now too often used to

point symbol

tions can be keyed in directly and ex­

stand for the division function. The

decimal

currency

duced. An unambiguous

was

two and three quarters could

is needed, and with TEX the point

actly, and only decimal fractions can

number

could be contrasted with but related to

be used to display usually approximate

be keyed into a calculator as 263°4 or

the negative sign, giving numbers like

fractional results. While it is true that

2675°100 or 2675,

7t,.2 (or 7/\2 or 7n2}

a number like

�, the designers of most elec­

showing equiva­

one and two thirds has

lences which should be easy for even

in the past been representable as 1%

the elementary-school eye to see. Of

Exactness

or as 1

It is one thing to be able to express a

tronic calculators and computers have

course, a number like

two thirds could

be keyed in as 2°3 or 062°3 or 4°6, but

value unambiguously as a value, and

not provided for this kind of represen­

there is no equivalent decimal fraction.

provision of a distinctive negative sign

tation either to be displayed or to be

Numbers with decimal fractions are

and fraction point allows this. It is an­

keyed in. More than ten years ago I was

distinguished from numbers with com­

other thing to be able to express how

an observer at a meeting of senior

mon

reliable or accurate a value is. A value

mathematics teachers which agreed,

played-a number that can be exactly

fractions

when

they are dis­

can be completely reliable and accu­

without protest from any of the teach­

represented more briefly with a deci­

rate-in other words, exact-or it can

ers, that common fractions should be

mal fraction than with a common frac­

be unreliable or inaccurate to some de­

dropped from the official syllabus for

tion

gree or other. To be unambiguous

elementary schools of one of the states

Otherwise there is no mysterious dif­

about whether a value is exact or not

of Australia simply because electronic

ference to confuse the young learner.

is to tell the truth. A notation that al­

calculators don't provide for them.

will

often

be

so

represented.

lows this truth to be told would there­

It has often been remarked that the

fore be not only a public good, but a

teaching of common fractions is not

It is one kind of truthfulness to provide

mathematical

is

well done in elementary schools [15].

for exact numbers all to be repre­

From this remark it is a short step to

sented exactly. But there are two quite

If a value, like a count or a fraction,

question whether common fractions

different kinds of numbers-exact and

is exact, then its representation must

should be taught at all. The mistake

approximate-and

show plainly that it is exact. A simple

here is to suppose that decimal and

should be easily distinguished in their representation but are not. An approx­

one-"mathematics

truth, truth mathematics" [4, p.177].

Accuracy

these

two

kinds

one and two thirds is ex­

common fractions should be taught as

act and, moreover, commonly useful.

distinct concepts. They should not. A

imate value can be truthfully repre­

Yet it is nowadays almost never repre­

fractional number is a fractional num­

sented only if its representation shows

number like

� and 2.75 so different

sented exactly. Instead some approxi­

ber, whether decimal or common. The

plainly, not only that it is approximate,

mation like 1.667 or 1.666667 is used.

fault is in the notation, which makes

but also how approximate. In other

There are two quite different reasons

the numbers 2

for this.

in appearance. What is needed is a no­

act value should show how accurate

tational convention which makes it

that value is.

The first reason is that electronic cal­

one and two

words, the representation of an inex­

If 1.75 represents a measurement

culators and computers, as they are al­

plain that a number like

most all now designed, cannot do exact

thirds is a value for which an integral

then .

arithmetic except on a limited set of

part, a numerator and a denominator

in the everyday world for that matter, it

numbers.

can be specified, and which

as a spe­ cial case allows certain (decimal) de­

is tacitly understood that it is some­

arithmetic is approximate except for numbers whose denominator is a power

nominators to be left out.

inaccuracy might spring from an unre­

In

particular, their rational

of two. There is nothing necessary about this characteristic [14], which arose be­

� is that the numerator and

The problem with representations like 1% or 1

. . In

the technological world, or

where in the range 1. 745 to 1.755. The liability in manufacture, from a limita­ tion of a measuring tool, or from a per­

cause the great limitations of early dig­

denominator are distinguished from

ital

to

the integral part by typographical de­

The representation of such mea­

design an arithmetic based on semilog­

tail, and from each other by a symbol

surements should show them to be

arithmic (wrongly calledjloating-point)

which implies that a calculation is to

measurements.

representation of numbers, an arith­

be carried out. These representations

shown with both a fraction point 6 and

metic now set in the concrete of an in­

are neither perceptually sufficient, no­

a scaling sign

ternational

tionally

tor point

computers

caused

standard

scientists

always

imple­

mented directly in electronic circuitry. The second reason is that, even if

unambiguous,

nor

electro­

mechanically convenient. However, if a symbol like

o,

distinctively

ceived irrelevance for greater accuracy.

o

k

Suppose or

m

a

number

but no denomina­

were treated as approxi­

mate beyond the last decimal place to

pro­

a tolerance of plus or minus half that

nom, were adopted as a

decimal place. Then 1� would be

com­

prefix to the denominator part of a

treated as exactly one billion, while

mon or vulgar fractions), there is now

fractional number, to follow the nu-

1.o,OO� would be treated as exact only

the arithmetic were exact for non-dec­ imal fractions (sometimes called

10

THE MATHEMATICAL INTELLIGENCER

nounced say

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 rep­ resented. But not all inaccuracies are of this kind. The arithmetic difference be­ tween two exact numbers 2. 75 and 1 . 75 is exactly 1. Between a measurement of 2. 75 and an exact 1. 75 it is some­ where 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 no­ tational rule to allow the value to be truthfully represented. It seems unavoidable therefore to in­ clude in the notation a means of stating the tolerance even when that is not sim­ ply a power of 10. This returns us to the earlier theme of escaping the tyranny of base ten. More important, it allows ex­ perimental scientists the freedom to be precise in reporting the extent of the im­ precision 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 stu­ dents.

numbers are otherwise unsatisfac­ tory and warrant being replaced. 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 tim­ ing of those reforms are properly a matter for the public and its govern­ ment to decide. Reforms of this kind would offer an opportunity to improve the aesthetics of mathematics gener­ ally, an aspect often considered fun­ damental for mathematicians [4, ch.5]. Mathematicians also have a natural re­ sponsibility for taking initiatives in promoting such reforms, and promptly introducing the teaching of them. There is a very real danger that in­ creasing and widening use of digital technology will prolong unthinking ac­ ceptance of a defective system for rep­ resenting numbers. The essential beauty of numbers and calculation is being hid­ den 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 imposi­ tion in elementary schools. The opportunity is for a much bet­ ter notational convention to be agreed internationally, for better electronic measurement and calculation to be en­ abled by that convention, and for the technology to support better the pro­ motion of public numeracy. A

This article proposes, as steps neces­ sary 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 coun­ tries, the SI metric scaling system should be replaced by a simple system for representing scaled numbers, and traditional methods of representing

NEVILLE HOLMES School of Computing University of Tasmania Launceston,

7250

Australia e-mail: [email protected]

Neville Holmes took a degree in elec­ trical engineering from the University of Melbourne, then spent two years

as a patent examiner before enlisting in the computing industry. Since re­ tiring from IBM after 30 years as a systems engineer, he has spent 11 years lecturing at the University of Tasmania.

7. Terry, G.S. (1 938) Duodecimal Arithmetic, Longmans, Green and Co. , London. 8. Aitken, A.C. (1 962) The Case Against Decimalisation,

Oliver and Boyd, Edin­

burgh. See also Math. lntelligencer 1 0(2), 76-77. 9. Kula, W. (1986) Measures and Men, Princeton University Press, Princeton, NJ. 1 0. Feynman, R.P. (1 970) Letter, Scientific American 223(5), 6.

REFERENCES Conclusion

AUTHOR

1 1 . Wilson,

1 . Beck, A. (1 995) The decimal dysfunction Math. lntelligencer 1 7( 1 ) , 5-7.

R.

(1 993)

Stamp

Corner:

Metrication, Math. lntelligencer 1 5(3), 76. 1 2. Hativa, N., Cohen, D. (1 995) Self learning

2. Jakuba, S. (1 993) Metric (Slj in Everyday

vision elementary students through solving

Automotive Engineers, Warrendale, PA.

computer-provided numerical problems,

3. Reingold, E. M. (1 995) A modest proposal,

Educational Studies in Mathematc i s 28,

and

Engineering,

Society

of negative number concepts by lower di­

of

Science

Math. lntelligencer 1 7(3), 3.

4. King, J . P. (1 992) The Art o f 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,

401 -43 1 . 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

Vandenhoeck & Ruprecht, Gottingen, 2nd

arithmetic, Computing, Suppl.2, 85-1 1 1 .

edition (as Number Words and Number

1 5. Groff, P. (1 994) The Mure of fractions, Int.

Symbols by MIT Press in 1 969).

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.

-Alfred Heuss, 1952 During the War

of the sciences. I spent my first term in Berlin, because

My studies began with the war. For most students their

Hamburg was initially closed due to the expected air raids;

studies were only an interruption of wartime service.

in 1940 I was able to continue in Hamburg. When matric­

However,

ulating I noticed that my "blemish" had not been forgotten:

I was not at risk of being called up: I was not

"worthy to serve." My life until my

there were Jews among my forefathers. Nevertheless, I was

Abitur (high-school diploma) in

allowed to register because my father had fought in the

Hamburg at Easter 1939 appeared to take a normal course;

front line during the First World War. At the university my

even during the following six months with the

Reichsar­

special situation was not immediately obvious, as every­

beitsdienst (Reich Labour Service) I was allowed to swim

one was studying "on call," and it was assumed that the

with the tide. Since, at the beginning of the war, school

same also applied to me.

graduates with Abitur who wanted to study Medicine or Chemistry were granted leave for their studies,

I decided

Slightly more than a dozen male and female students started studying Mathematics in Hamburg in January 1940.

on Chemistry, which did not interest me in the slightest,

Our central figure was Erich Heeke (1887-1947; a student

but which

of Hilbert), one of the most fascinating personalities I have

is related to Mathematics within the structure

'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 patri­ otic 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-in­ stead of raising his right arm in the Nazi salute, for which there were strict orders-entered the lecture room nod­ ding 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 undis­ guised rejection of the Nazi spirit, to which the whole of Germany was dedicated. Most people registered his be­ haviour 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 sec­ ond in conspiratorial opposition to the regime. Sometimes we students overtook the old man on the way to his lec­ tures, 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 Ordre­ pour-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 sci­ entists 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 curs­ ing 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 appre­ hensive 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 stu­ dents 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 un­ stable 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 be­ tween an attitude based on belief in justice and human dig­ nity 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 administra­ tion, 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 offi­ cial 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 (19121992), 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.

VOLUM E 22. NUMBER 1, 2000

13

cause Lenz was "not very brave." In addition, Heeke wanted

be seen. In the Department I appeared to be out of danger:

to contact van der Waerden in Leipzig; the latter was in

although I did, in fact, sometimes see Herr Blaschke and

charge of a team of mathematicians whose work had been

Herr Witt (the one an opportunist, the other politically

recognised as being important for the war, and where some

naive), they hardly took any notice of me; that suited me;

endangered mathematicians had already found refuge and

I always tried to avoid contact with a stranger, who might

been protected.

unwittingly have risked getting in trouble because of me.

I confessed to Heeke that I didn't dare go to Zassenhaus

This period saw the start of a new friendship that I owe

because he wore a Nazi badge. Heeke reassured me, "He

to Heeke. Werner Scheid, a young lecturer in neurobiology,

is someone we all trust; he only pretends to be a Nazi­

wanted to improve his understanding of the physical back­

and he does it well." And so it was; my first conversation

ground to his science and its methods; he asked Heeke how

with Hans Zassenhaus was the start of a friendship for

he could first acquire the mathematical prerequisites;

which I have been grateful all my life.

Heeke brought us into contact, and I shall never forget the

One further short episode must be mentioned: when I

warm-hearted security that I was permitted to ef\ioy in

reported to Heeke that Lenz had made his surprised stu­

Scheid's home. I assume that Heeke was also behind the

dents stand up in his lecture for a "threefold Sieg Heil for

invitation I got to teach at a very well-known private

our Fiihrer and his glorious troops," he burst out laughing:

evening school in Hamburg; although I was extremely

"Herr Lenz has been summoned to the Gestapo tomorrow!"

pleased to have received such an offer, I had to avoid the

1940 I became

exposure this would have caused. On the other hand, when

friendly with the Chemistry student Hans Leipelt. He was

the representative body of Chemistry students asked me to

1945 as a member of the "White

give an introductory course in mathematics for Chemistry

During this time before Christmas beheaded in Stadelheim in

Rose," a group of Munich students who conspired against

students, I agreed, despite many misgivings.

Hitler. Now I was studying illegally; van der Waerden did, in fact, want to take me in Leipzig, but I was unable to seize

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

this helping hand, because if I had left, by an intricacy of

rejected: "My time is too precious for that!" Zassenhaus,

the Ntimberg Laws, my father would have been obliged to

who was always ready to help his fellow human beings,

wear the Star of David.

and who had no streak of prima donna behaviour in him,

My time as an illegal student lasted for about a year and

blamed himself for this rejection: "My request was really

a half. The lectures of Heeke and Zassenhaus partially re­

immodest; his time is defmitely too precious." It was pre­

peated what I had already heard; the beginners soon no­

cious-in a different sense: a few weeks later Klaus Junge

ticed that my knowledge was more advanced and asked

was killed in action.

me to help them by organising a tutorial group. This was

The phone rang one night during the Summer Semester

1942. I was relieved to hear the familiar voice of

not unproblematic, for opposite the Department there was

of

a Gestapo office from which I had to hide my illegal presence.

Zassenhaus; however, the reason for his phone call was dis­

On the days when classes took place in the Department, I

quieting: my illegal behaviour was going to be denounced; he

had to be there in the morning before the Gestapo started

hoped that he would be able to dissuade "these people" from

work, and I was only able to go out onto the street again

doing so if he could promise them that I would not allow my­

in the evening after the start of the blackout-it was part

self to be seen in the University any more.

of the air defences that no gleam of light was permitted to

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

Figure 2. Hans Zassenhaus (1912-1992), about 1980.

14

THE MATHEMATICAL INTELLIGENCER

Heeke in Hamburg, and would help me. The aesthete Reidemeister

had, incidentally, been transferred from

Konigsberg to Marburg for disciplinary reasons as early as

mathematician there is nothing worse than not knowing

1 933, because he had complained in his lectures about the

what he ought to be thinking about," and "I wish I were

vulgar behaviour of the SA (the armed and uniformed

two puppies and could play with each other!" Choice mis­

branch of the Nazi Party).

adventures befell him; the best is the one concerning the

However, this helping hand also I was not able to grasp:

key to his letter box: he had left it behind at Harald Bofu's

shortly after my visit to Reidemeister the Gestapo acted,

house in Copenhagen, and when he arrived home he did

and on Christmas Eve 1943 my parents and I were arrested.

not want to break open the full letter box; Bohr immedi­

Before we were transported to concentration camps, I had

ately fulfilled Maak's written request to send the key to

to hack coal out of ice-coated wagons at the freight depot.

him; he sent it-in a letter!

On the march back to the prison I once saw Reidemeister,

While Maak's humour expressed itself in a roguish grin,

and was seized by a desperate hope that he would recog­

Zassenhaus liked to laugh; but humour without a problem

nize me despite my prison clothing and would be able to

was something he found difficult to accept. I told him one

tell Heeke before I sank into the inferno-but he did not

of the first post-war jokes: In Dusseldorf an old lady in a

see me.

tram asked for the Adolf Hitler Square; when the tram con­ ductor told her that it was now called Count Adolf Square,

After the War

she said with genuine sincerity, "Oh, the good man de­

My parents did not survive the concentration camps; I was

served it." While Zassenhaus was still gasping for breath

freed from Buchenwald by the Americans on April 1 2th,

from laughter, he asked, "And what did the conductor say

1945. I first made my way with difficulty to Marburg, but

then?"

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 Linear Differential Equations. Nowadays we cannot compre­ hend the situation-how hun­ grily the

emaciated

figures

with their clothes in tatters fol­ lowed the fascinating lecture in

an

atmosphere

charged

with tension. Heeke combined warmth with dignity, and thus

Zassenhaus, who thought there was no future for ivory tower mathematics in Germany, had prepared a memo­ randum for setting up a Research Institute for Practical

H eeke revived an i m age of h u m an ity that had becom e deformed d u ri n g t h e Nazi era .

Mathematics. However, since he had little hope that his plan would be realised, he was al­ ready putting out feelers in America and Britain. I was able to be of use here, and the reason was as follows: At that time democratic bodies were forming by spontaneous gen­

revived an image of humanity that had become deformed

eration, and in the expectation that a student body would

during the Nazi era. One interruption is still unforgettable:

get a response from the British military government more

it must have been in January 1946 when Heeke, who while

easily than professors suspected of being Nazis, a Central

speaking liked to look over the top of the lecture-room's

Committee of Hamburg Students

boarded-up walls and windows into the street, suddenly stopped talking with joyful surprise in the middle of a sen­

(Zentral-Ausschuss or ZA) had already been founded before my arrival in August.

It was the predecessor to the AStA (General Student

tence, put down the chalk, and with the words "I am being

Committee), and through the good offices of Herr Scheid

visited by a dear friend" hurried out into the street and em­

I soon participated in its discussions. I got the

braced the aged Erhard Schmidt; after he had fled from

ested in Zassenhaus's project, and after a lecture by him I

Berlin he had been brought to Hamburg by his pupil

was requested to canvass this idea with the English officer

ZA

inter­

Thomas von Randow (who has since become the cele­

responsible for the university. I think I did something to­

brated "Zweistein" in ZEIT magazine).

wards making a success of the project: the Institute was

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

established, but it was too late to prevent Zassenhaus from emigrating. Zassenhaus was always in a state of high mental ten­

proud when he asked me to participate; I was thus the only

sion.

student sitting together with assistants and lecturers in

Mathematics Department he mostly walked between the

On the way from the Dammtor Station to the

Heeke's own workshop.

row of trees and the curb, swinging his briefcase and chew­

I also have happy memories of other classes: Zassenhaus

ing the comer of a handkerchief; here he was not in dan­

on Space Groups, Weissinger on Integral Equations, Noack

ger of colliding with other pedestrians and of being dis­

on Kolmogoroffs Probability Calculus, and for students

tracted from his thoughts. In his lecture he revealed his

training to become teachers, Maak's lecture based on the

effervescent temperament, as if he were trying to transfer

book

Numbers and Figures by Rademacher and Toeplitz.

his high tension to his listeners. The result was remarkable

Maak listened with polite incredulity as I showed him how

speed: he got through both the volumes of Schreier-Spemer

a number-theory proof could be simplified.

in 1 1/2 terms, and he proposed that the remaining time

Maak was a person with an eccentric sense of humour:

should be devoted to Descriptive Geometry because "I can­

I have the following lovely statements from him: "For a

not 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

Figure 3. Herbert Grotzsch (1 902-1 993), about 1 960.

office shortly after the war. He was the Director of the Mathematics Department. I was able to return to him

ofthe Mathematische Annalen, which I had bor­

He had also been returning from the war: he had tried

rowed from the Department library before the bombing be­

to return to his university, Giessen, where he had taught

Volume 63

cause of Erhard Schmidt's doctoral thesis. It had accom­

until he was thrown out in 1935 for refusing to participate

panied me through the inferno. In order to "celebrate" this

in a Nazi camp for lecturers. However, the university had

event Zassenhaus gave me a large log from the trunk of a

been closed by the US military government; it was there­

tree as firewood and lent me a barrow to transport it; the

fore sensible for him to try to resume his teaching in neigh­

wheels were part of the valuable family bicycle, and to my

bouring Marburg. He was gladly accepted as a member of

horror one wheel buckled under the weight of the wood.

the staff, which had been greatly reduced in numbers: only

During these last six months together I saw

him

much

one chair and the post of a senior lecturer were already

more free from care than during the Nazi era. I learnt that,

filled; an additional chair, a lectureship and the post of a

together with some like-minded friends, he had hidden vic­

student assistant were vacant; while the post of a full as­

tims of persecution and prevented them from being caught

sistantship was blocked because the incumbent, Friedrich

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

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

February 1947 at Harald Bohr's home in Copenhagen), and

the Faculty; however, some considered Grotzsch's shabby

since Zassenhaus wanted to emigrate, I went to Marburg

clothes to be "unsuitable." I have this from the then Rector,

for the Summer Semester 1946, especially because life

the philosopher Julius Ebbinghaus.

there was easier than in the rubble wastelands of Hamburg.

Grotzsch never criticised this treatment, and he did not

Marburg had been left largely untouched by the destruc­

even seem to notice it. His poverty certainly was not detri­

tion of the war, and it thus had a strong attraction for the

mental to the effect of his personality, to his enthusiasm

streams of people returning, refugees and people displaced

in the lectures and his kindness in his contacts with his

by the war who were on the move throughout the entire

students.

country. The student body that expectantly filled the lec­

Grotzsch was widely known as a researcher. He had in­

ture rooms was accordingly very mixed: the students from

troduced his "Surface Strip Method" into the Geometric

Marburg, who still had the background of a family and had

Theory of Functions, regarding rigid conformal mappings as

just come from school, were in stark contrast to these peo­

special cases of more supple quasi-conformal mappings; then

ple, who were visibly in a terrible state. In Mathematics,

the conformal ones can often be characterised by extremal

however, they were all united by the enthusiasm of one

properties. This point of view

man, whose poverty was hardly exceeded by that of any

search for characteristic properties of certain Riemann sur­

student: Herbert Grotzsch (1902-1993).

faces and in the theory of "Teichmiiller spaces."

16

THE MATHEMATICAL INTELLIGENCER

is still fruitful today in the

In the town, which in those days was full of "charac­

while discussing or in his lectures his hands were always

ters," Grotzsch, the professor, quickly became a well­

moving, as if he wanted to explain his thoughts by means

known personality. Efforts by students-who were a little

of a virtual or real drawing.

better off-to help him here and there were rejected kindly

In the lecture on Conformal Representation, during

but firmly; it was only possible to smuggle a pair of shoes

which the lights suddenly failed, he appealed to the ability

from a US parcel into a tombola for him: visibly distressed,

of the students to think in abstractions and spoke on ln the

he went home shaking his head, but then liked wearing the

dark; nevertheless, after a few minutes had passed one

shoes instead of the clogs he had previously worn. On his

could hear the sound of the chalk on the blackboard.

way to the Mathematics Department in the Landgrafenhaus

Once I did see Grotzsch angry. In the Department library

he used to walk through the old Weidenhausen quarter and

some students were hunting insects. Extremely excited, he

drink his cup of ersatz coffee at a bakery, eat his dry roll,

closed the window with the words, "The poor creature does

and read the daily newspaper. Once he fell asleep while

not know what traps we are setting for it."

doing this and leaned on the hot stove; the sad result was

His office was in the attic of the Landgrafenhaus. A gut­

a large hole in his "good" jacket, which he had had sent to

ter ran along beneath his window, some soil had gathered

by his parents, and which he had only worn for a few

in it over the years, and a small beech tree was growing

him

days, displacing an indefmable piece of clothing from the

there. It was visible from a distance. It was his j oy, and he

war.

watered it twice a day, for which purpose he had to carry

Grotzsch lived in a tiny attic in the Galgenweg; the path

a tin can to the nearest water tap, two floors below. Once,

was so steep that when it was icy he had to slide down in

when he was away, I had the honourable task of watering

his socks.

the small tree: "But be careful not to spill any on the

Once he stood still in the middle of the Rudolphsplatz,

passers-by." When the roof was inspected the gutter was

which was busy with US vehicles, chewing at the end of his

cleaned, and the little tree disappeared. His comment: "In

short pencil and sunk deep in thought, until a friendly po­

dass die Baume nicht in den Himmel wachsen (that the trees don't grow into the sky)." In April 1948 Grotzsch

liceman took

him by the

arm

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

and led

him to

the safe foot­

Marburg they take care

The am b ience i n that period ,

so d iffic u lt to re-create and u nderstand today.

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

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

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.

have collapsed: he was tirelessly active and could be con­

When he said goodbye he forbade anyone to send him

sulted at any time. In the loud, spirited discussion held in

letters with a mathematical content: "The censors must

the Saxony dialect he was thrilling to listen to; his eyes,

consider mathematics to be a secret language, and that is

which were emphasised by the powerful lenses of his

mortally dangerous in a dictatorship, " and here he was al­

glasses, flashed with high spirits and intellectual joy. His

luding to the fate of Fritz Noether, who had been executed

stereotype "nota bene consultation!" was a motto with

in Russia as a spy, because he had received money owed

which he told students to come to see him. Everything was

to him by someone in Germany.

important! Mathematical errors were discussed until in­

Political arguments played a greater role in Marburg than

teresting fallacies appeared: paths towards solutions were

in Hamburg. There survival was all-important. However,

discussed in detail. If the arguments were too long-winded

Marburg was essentially undamaged, and middle-class life

during the practice classes he would call out, "Ladies and

outwardly fairly intact.

gentlemen! You are all thinking much too much!" But if the

Former officers were noticeable here because of their

path a student had chosen was superior to his own he

distinctly brisk behaviour; of them the physicists said,

would exclaim, "You have beaten me!"

"When 'magnitudes of higher order' are mentioned they

An unforgettable sentence from his profound thinking:

click their heels." When one fellow-student was drunk he

"Ladies and gentlemen. The main problem of mathematics

had boasted that he had been an officer in the SS, and I ex­

is: The proof is given-the theorem is to be found." Also

plained to him that I did not wish to have any further con­

his stirring explanation of the principle of Bolzano: "Think

tact with him, and why; very much later, I learnt that he­

of a fmite interval and then infmitely many points within

in the meantime a school headmaster-had described me

it! The mere consideration of this tells you that there must

as "his friend."

be a terrible crush, there must be a point at which some­

The political discussions among us students were often

thing terrible happens! And look: a point of this kind is an

violent. During my last visit to Halle, Grotzsch reminded

accumulation point." He always thought geometrically:

me of an argument of this kind, during which a chill had

VOLUME 22, NUMBER 1 , 2000

17

run down his spine: when a fellow-student had defended

stable, and it was very embarrassing for me when I, as a

his enthusiasm for the Nazi state with the words "and, any­

Full Professor, visited him, the Associate Professor to

way, I wasn't sent to a concentration camp," I had merely

whom I owed so much, on his 80th birthday. Krafft's lecture style was eccentric: clearing his throat

countered, "Why not?!" This was the time of the denazification courts. In them

and growling contentedly, he turned his back on his lis­

denazification was carried out in the style of courts exer­

teners, began to write on the board with his left hand and

cising civil and criminal jurisdiction. As there were not

continued with his right hand without his handwriting

enough politically irreproachable lawyers, I was offered the

changing in the slightest; if you were sitting opposite him

post of Public Prosecutor. Although in this time of need

at the desk he would write upside down, and his mirror

and with a future full of question marks this position with

writing was even as fast as his normal writing. He was full

the rank of an

Oberregierungsrat (senior civil servant) had

of calculating tricks, which he often made up himself, and

a fairy-tale aspect for me, I did not consider the offer for

he enlivened his lectures and seminars to an extraordinary degree by his human and mathematical originality. At that

one minute. I have reported about Grotzsch in detail because his hu­

time he was working tirelessly on a translation and revi­

maneness brings out so well the brittleness of the ambi­

sion of Tricomi's

ence in that period, which is so difficult to re-create and

cal counterpart to the older work that he had written to­

understand today. The two other men, the Full Professor

gether with Robert Konig.

Kurt Reidemeister and the Associate Professor Maximilian

During this time of hunger Krafft and Grotzsch gave superhuman service!

Krafft-who later supervised my doctoral thesis-were personalities of a different kind.

EUiptic Functions; this was the analyti­

Mter the currency reform in 1948 the vacant chair was

Following the lead of his friend, Rector Ebbinghaus,

given on a temporary basis to Hans-Heinrich Ostmann, who

who wanted to denazify the university, Reidemeister

was an expert on the Additive Theory of Numbers. He

(1893-1971) became more interested in politics than math­

taught in Marburg from 1948 until 1950 and then moved to

ematics.

the Free University in Berlin. At the end of the war Ostmann

When

the

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

philosopher Ebbinghaus

and

the

The social task of mathematician s : t o make M athematics palatable to non- mathematician s .

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

had settled in Oberwolfach and earned his

living from the fees he

charged as a consul­ tant to people squaring the circle, trisecting the angle, and the like. He

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

one day with a problem about ellipses that her uncle was

coaching in the subjects Greek, Latin and Mathematics­

unable to solve; he had impatiently sent her to me: "Go and

abbreviated to Gre-La-Ma; this well-known character used

see Tietz, he's got a feeling for trivial things!"

to cycle through the town wearing a blind person's arm­

Krafft (1889-1972) was an awkward person who was al­

band. She appeared at every possible lecture, and even

ways "against" everything: he had not got on with the

once at the Landgrafenhaus, where in front of the surprised

Nazis-it is said that in Bonn he did not become Hausdorffs

Ostmann she unrolled a 10-metre-long roll of paper in the

successor because he did not want to do any political ser­

hallway with a deft movement, and announced that this

vice on the weekend-and after the war he missed no op­

was "the prime number formula." Ostmann fled, without

portunity in my presence to make critical comments about

having collected his fee.

Jews. This odd nonconformism I found impressive rather

Wolfgang Rothstein (1910-1975) came to Marburg from

than offensive. I would take his part if he was having dif­

Wiirzburg in 1950, so the lectureship was finally filled. My

ficulties with someone. It was only in the oral part of my

wife and I became friends with him and his family, and it

doctoral examination in 1950 that I couldn't restrain my­

means much to us that we were able to continue this friend­

self any longer. In actuarial theory Krafft asked annoying

ship later in Miinster and then in Hannover.

questions; the last was, "How do insurance companies pro­ tect themselves against too unfavourable insurance poli­

Brief Episode in Physics

cies?" My reply: "Through preselection by a doctor; how­

I have jumped ahead: these events took place shortly be­

ever, that makes sense only for life insurance; the sick are

fore I left Marburg for Braunschweig in 1951; but my State

not accepted in order to avoid having to pay too early." He

Examination in 1947 with its consequences requires a few

was not satisfied: "It is also sensible in the case of pension

comments. In Germany one could no longer obtain a doc­

insurance: the healthy are not accepted so that they do not

torate without passing such a final examination. (It is said

have to be paid for too long a period." I exploded in front

that a candidate in the State Examination was once told,

of the whole Faculty: "That may be an Aryan method, Herr

"Herr Doktor, you have failed.") Three subjects were re­

Professor, I do not know it!" However, our relationship was

quired; I had chosen Mathematics, Physics, and Chemistry.

18

THE MATHEMATICAL INTELLIGENCER

Figure 4. Erich HOckel (1 896-1979) with Horst Tietz, 1 949.

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 (18961980), 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 1947 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

Social task of mathematicians: to make Mathematics palatable to non-mathematicians.

it made me aware of the

A U THOR

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

HORST nETZ

of thanks we found the statement that he did not mind writ­

Roddinger Strasse 31

ing the thesis for a student, but that he hated then having

30823 Garbsen

to explain it to him as well! During his last few months his illness gradually weakened him;

Germany

nevertheless he still

e-mail: [email protected]

worked intensively almost until the very end. Almost: dur­ ing the last few weeks he was only still able to read, de­

Horst Tietz was born in 1 921 in Hamburg, to a family of promi­

tective novels and the Bible. Allow me to mention two more mathematicians who be­

nent wood merchants. The Nazis expropriated their business

long here: both came from the Hamburg background and

and ended by killing most of them. The reader may be amazed,

had obtained their doctorates with Heeke: Heinrich Behnke

as the Editor is, that Horst Tietz, after tribulations and losses

and Hans Petersson, whom I got to known in 1956 when I

which if anything are understated in this memoir, was able to

was given a lectureship in Mtinster. They were Directors

spend the rest of a long and creative life as a mathematician

of the two Mathematics Departments, and despite (or per­

in Germany. The academic community in which he had ap­ prenticed as an outcast now honors him; today he is a re­

haps because of?) their spatial closeness-the Directors'

spected Emeritus Professor of the University of Hannover. He

rooms were opposite each other in a narrow corridor-one

is left with a wry streak of gallows humor. perhaps, but

could not call the atmosphere friendly. The difference in

uncowed.

their physical size was enough to cause tension. Behnke (1898-1979) was a huge person with a Renaissance­ like 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

around 1960. The stark difference between two opposite

President had finished their speeches of congratulation, the

temperaments with the same interests became almost

man being celebrated heaved himself up to the lectern, al­

painful. They were speaking about the training of teachers,

though this was not on the programme, with the words:

which Behnke did with great success, while Reidemeister

"Herr Senator, Herr Prasident! When I think back to my

did not get beyond reflecting on the problem. Coming from

youth I have to say: your predecessors, gentlemen, those

Reidemeister even friendly words sounded ironic, Behnke

were real men! They drove with coach and four . . . !" The

felt he was being attacked, replied more and more agitat­ and fmally left the room; when I accompanied

remainder of what he said was lost in the general cheer­

edly,

ing. Hans Petersson ( 1902-1984), a wiry, almost delicate

Reidemeister to his hotel he said with great agitation, "Herr

man, continued Heeke's modular research most inten­

Behnke thinks that I am criticising him; but I actually ad­

sively, and in 1958 he revised and published Heeke's works

mire him! How can one make oneself understood?"

together with the unforgettable Bruno Schoeneberg. I should like to return to Reidemeister once more. He

Conclusion

has always fascinated me; it was all the more painful to

I wanted to describe my meetings with personalities who

recognise the tragedy that he apparently only seldom suc­

have influenced my life and show how different from to­

ceeded in conveying his intellectual riches to other people.

day our world half a century ago actually was. I also wanted

How much he suffered under this became clear during his

with gratitude to bring to life the memory of people who

visit to Behnke, his friend from student days, in Miinster

were not only important scientists but also-Menschen.

20

THE MATHEMATICAL INTELLIGENCER

M a the rn a tica l l y Bent

Col i n Ad am s , E d it o r

Into Thin Air I

The proof is in the pudding.

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.

was up above the Lickorish Ridge,

They found him at the bottom of the

having traversed the difficult Casson­

Euler Face. Everyone had said that

Gordan Step, and was resting on a

there was no way up Euler, but McLuten

small Lenuna on the North Face of the

couldn't be dissuaded. He left three

Poincare Co[\jecture. As far as

ABO's

I knew,

no one had been up this high before,

with

no

means

of

and I felt I had a good chance of find­

It wasn't but ten years later that

ing a route all the way to the top. I was

Wiles made the summit. But Wiles pre­

still breathing hard and the adrenaline

pared. For seven years, he prepared.

was pumping through me. Those last

He knew the Euler Face was insanity.

fifty feet had been treacherous.

With this channing tale we inaugurate

behind,

support.

A few

He

came

up

Taniyama-Shimura,

a

times, my logic had slipped, and I had

route that had been championed by

barely managed to grab a handhold

Ribet. And he did it alone.

and then scramble onto solid footing.

It made Wiles an instant celeblity.

know Colin Adams through his career

But now that I was up here, the view

He had tackled the big one. He had

was incredible. The sky was an unnat­

proved no mountain was invincible.

in research and teaching, and may

ural blue.

a new column. Many of our readers

have enjoyed his article in The Mathematical Intelligencer

17 (1995),

But that wasn't why he had done it. No,

As I sucked air, I looked out into

that's not why any of us did it.

the distance. The Mathematical Range

And here I was, three quarters of the

stretched beneath me. Poking through

way up Poincare. One of the largest

41-51. Alongside this professional

the clouds were some of the peaks

unconquered peaks in the world. One

activity, he has been appearing in skits

upon which I had first tested my met­

of the few remaining giants of mathe­

no. 2,

and parodies, sometimes in the persona of Mel Slugbate; you may have seen

tle. Point Set Topology looked so tiny

matics. Who would have thought that

in comparison to where I sat now, but

I would have a shot?

at the time it had been a struggle. And

The wind was picking up a bit and

him, for instance, at meetings of the

there was Teichmiiller Theory. I would

Mathematical Association of America

have never made it up that slag heap if

Suddenly a head bobbed up at the

and the American Mathematical

it weren't for McLuten. I was so naive

edge of the lenuna. I jumped back. It

then. So many mistakes. McLuten must

was Politnikova. She pulled herself up

Society. Having enjoyed these, you may well sha1·e my pleasure at the

wispy clouds scudded by.

have saved my rear a dozen times. If it

over the edge, and lay there, trying to

weren't for him, I would be lying at the

catch her breath.

prospect of a column under his

bottom of some crevasse, crumpled up

"What the hell are you doing here?''

direction. Only I advise you not

on some counterexample to a laugh­

I exclaimed. Politnikova waved me off

to think you

know what

to expect.

-Chandler Davis

as she gasped for breath. Not a lot of

able conj ecture. McLuten

had

seemed

invincible

oxygen up here.

then. He'd climbed all kinds, the big

"Did you follow me up Geometriza­

ugly granite slabs that rose up out of

tion Co[\jecture Ridge? Nobody knew

the undulating planes of geometry, the

I was even consideling it."

treacherous ice-covered theorems that kept us all in awe of algebra, and the

Politnikova pulled off her goggles and sat up, still gasping.

crumbly rocks of the Analysis Range,

"Relax, relax," she said in her thick

where one false step could bling a

Russian accent. "I did not come up the

And

Geometrization Co[\jecture Ridge. I fol­

Column editor ' s address: Colin Adams,

McLuten had the look, too; the glizzled

lowed Poenaru's Route up the Clasp Trail and then over the Haken Ice Field."

mountain

down

upon

you.

Department of Mathematics Williams College,

visage that resembled the crags and

Williamstown, MA 01 267 USA

rocks he confronted daily, his eyes al­

e-mail: [email protected]

ways focused on the next challenge.

"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

fronting. In that split second, we both

date equipment, technology from the

knew that our dream of conquering

50's. I am using the latest technology.

Poincare that day was gone. But all

Makes a difference."

that was suddenly irrelevant. Now it

"I can't believe this. I get up this

was a question of survival.

"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

"We don't have a chance in hell if

perched moments before, and the face

"And what is so wrong with me, huh?"

we stay here on this lemma. There is­

was smooth as ice. No lemma, no

"You know perfectly well what I

n't going to be a lemma in another two

corollary, not a handhold to be had.

high on Poincare only to find you."

mean. I was going to do it on my own." "Oh, yes, sure," said Politnikova

minutes," I screamed.

"Throw your

rope over the side, and if we can make

"We will not be getting up there that way," said Politnikova.

smiling. "You would have no trouble

it down to Bing's Theorem, we can hide

single-handedly climbing those logical

behind that." I flipped Politnikova's

outcroppings up there." She pointed al­

rope onto a piton I had already ham­

I stood wearily, feeling the bruises

most straight up.

mered into the rocks, clipped her on

and scrapes. "We should head down,"

"Nope," I agreed. "This face is offi­ cially a dead end, starting today."

"Well, I hadn't figured it all out yet."

the line and shoved her over the edge,

I said, "before any other arguments col­

"Yes, but two could work together

before she could stop me. Then I

lapse."

A little

clipped on and jumped out into space.

combinatorics, I'm good at combina­

We zinged down the rope, burning

look so sad. We were higher up there than anyone else has ever been."

to get around those problems.

Politnikova stood

slowly.

"Don't

A little geometry. You are good

glove leather, until we hit the end of

at geometry. And bingo, we are there."

the rope. Up above you could hear the

"Yeah?" I said. "No one will believe

"Well, I suppose you have a point,"

roar. When we hit the bottom of the

it anyway. There isn't a trace of where

I said reluctantly. "Maybe we could

rope, we just unclipped and started

we were."

work together."

rolling down the slope. All those hard­

"Yes, but what matters is what we

earned steps for naught, I thought as I

know, not what others think Hey, you

torics.

Politnikova began to smile, but the smile froze as she jerked her head up.

careened downward. I rolled to a halt

come down to my tent, and I give you

"Do you hear that?" she said, terror in

20 feet from Bing's Theorem, battered

some very good vodka. "

her face. I pulled my hood away from

but in one piece. I glanced up at where

I laughed. In a place where every

my ear, and cocked my head to the

the lemma had been only to see it dis­

ounce counts for survival, only Politni­ kova would bring vodka.

side. In the distance, I could hear it, a

appear entirely in the torrent of argu­

slight rumble, but it was growing fast.

ments that were cascading down upon

"Sure," I said. I took one last look

"Oh, no," I said, "avalanche!"

us. Politnikova grabbed my hood and

toward the peak, enshrouded in clouds now, not even visible anymore.

When I had been down at base

pulled me toward Bing's Theorem. We

camp, I had seen how precariously bal­

managed to duck behind it just as the

"We have vodka, and we talk," she

anced the various arguments were that

avalanche reached us. Huddled there,

said, "and maybe we figure out some

made up this face of the Poincare

we saw several years' worth of mathe­

other route to the top. Maybe we use hy­

A little bit of a shift here

matics slam past. It only lasted another

perbolic 3-manifold theory. Thurston

or there, and the whole mountainside

minute, and then it was all gone. We

knows what he is doing. We do that, too."

could come down in your face. And

both sat in stunned silence and then

that was the reality we were con-

Politnikova turned to me.

Conjecture.

"Sure," I shrugged, "Why not?" We started down the mountain.

MOVING? We need your new address so that you do not miss any issues of

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22

THE MATHEMATICAL INTELLIGENCER

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

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

bd

bdh + e

bdhi + eg + ei

abd abdh + ae + ce abdhi + (a + c)e(g + i) + f

l

, (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

det(x)

3

3

THE MATHEMATICAL INTELLIGENCER

=

I I sgn(w) w(1T), w

(2)

7T

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 fo denote the network shown in Figure 2. Direct inspection shows that fo is totally connected.

COROLLARY 4. For any choice of positive weights w(e), the weight matrix x(f0, w) is totally positive. It turns out that this construction produces all totally positive matrices; this result is essentially equivalent to

To illustrate Lemma 6, consider the special case n 3. The network r0 is shown in Figure 1; its essential edges have the weights a, b, . . . , i. The weight matrix x(f0, w) is given in (1 ). Its initial minors are given by the monomials =

!:. 1,1 = d., !:.2,1 = fld, !:.3, 1 = a.bd, Figure 2. Planar network ro·

A. Whitney's Reduction Theorem [46] and can be sharp­ ened as follows. Call an edge of r0 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.

!:. 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 un­ derlined. To complete the proof of Theorem 5, it remains to show that every totally positive matrix x has the form x(f0, w) for some essential positive weighting w. By Lemma 6, such an w can be chosen so that x and x(f0, w) will have the same initial minors. Thus, our claim will follow from Lemma 7.

A weighting w of r0 is essential if w(e) =I= 0 for any essen­ tial edge e and w(e) = 1 for all other edges.

LEMMA 7. A square matrix x is uniquely determined by its initial minors, provided all these minors are nonzero.

THEOREM 5. The map w � x(f0, w) restricts to a bijec­ tion between the set of all essential positive weightings of r0 and the set of all totally positive n X n matrices.

Proof Let us show that each matrix entry Xij ofx 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 ob­ tained from t:. by deleting this row and this column. Then, t:. = t:. 'xij + P, where P is a polynomial in the matrix en­ tries 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 oc­ curs 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 pos­ itive matrices by n2-tuples of positive reals, providing a par­ tial 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 mi­ nor. It is desirable to find efficient criteria for total posi­ tivity that would only check a small fraction of all minors.

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 sev­ eral 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 r0 are related to the n2 initial mi­ nors of the weight matrix x = x(f0, w) by an invertible monomial transformation. Thus, an essential weighting w of r0 is uniquely recovered from x. Proof The network r0 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 con­ necting 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 cov­ ered by this family of paths. Note that among these edges, there is always a unique uppermost essential edge e(t:.) (in­ dicated by the arrow in Figure 2). Furthermore, the map t:. � e(t:.) is a bijection between initial minors and essen­ tial 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

EXAMPLE 8. A 2 X 2 matrix x=

[ : :]

has en - 1 = 5 minors: four matrix entries and the deter­ minant 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 ma­ trices 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.

Figure 3. Initial minors.

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.

and

Xi(t)

X(1)(t)

In this article, we shall only consider invertible totally non­

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 posi­ tive) matrices in G. The structural theory of these matri­ ces begins with the following basic observation, which is

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. 10.

Combining this proposition with the foregoing results, we will prove the following theorem of Whitney

[46].

Every invertible to­ tally non-negative matrix is the limit of a sequence of to­ tally positive matrices.

THEOREM

1 1 . (Whitney's theorem).

tEi+ l ,i

=I+

(xi(t))T

=

=

1, . .

.

, n and t =I= 0, let

(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 (i 1 . . . . , it) in the alphabet =

.stl

an immediate corollary of the Binet-Cauchy formula.

PROPOSITION

I+

(the transpose of xi(t)). Also, for i

Theorems of Whitney and Loewner negative n

=

=

{1, .

we defme the

.

.

,n-

1,

(!), . . . , @, 1,

. . .

,n-

(3)

1 },

product map Xi : (C\( O JY � G by (4)

(Actually,

xiCt1. . . . , tt) is well defmed as long as the right­

hand side of (4) does not involve any factors of the

form XC1J(O).) To illustrate, the word

i

=

CD 1
to

G>o in G. (The condition of 1 1 can, in fact, be lifted.)

Thus, G,0 is the closure of invertibility in Theorem

Proof

First, let us show that the identity matrix

the closure of G>o· By Corollary

I

=

I lies in

4, it suffices to show that

limN->oo x(f0, WN) for some sequence of positive weight­

ings WN of the network r 0· Note that the map

(J) � x(fo, (J))

is continuous and choose any sequence of positive weight­ ings that converges to the weighting wa defmed by w0(e)

=

0) for all horizontal (resp. slanted) edges e. Clearly, I, as desired. To complete the proof, write any matrix x E G?.o as x limN->oo x x( f0, wN), and note that all matrices x · x(f0, wN) are totally positive by Proposition 10. 0 1 (resp.

x(f0,

w0)

=

=

·

The following description of the multiplicative monoid

G?.o was first given by Loewner [32] under the name [46] .

"Whitney's Theorem"; it can indeed be deduced from

THEOREM 12 (Loewner-Whitney theorem). Any

invert­ ible totally non-negative matrix is a product of elemen­ tary 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 ma­

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)

26

=I+

tEi,i+l

=

THE MATHEMATICAL INTELLIGENCER

0

0

0

1

0

0

1

0

0

0

Xi(t 1 , . . . , tt) as the weight

tary Jacobi matrix is the weight matrix of a "chip" of one of the three kinds shown in Figure edges but one have weight weight

i

t.

4. In each "chip, " all 1; the distinguished edge has

Slanted edges connect horizontal levels

i

and

+ 1, counting from the bottom; in all examples in Figure 4, i 2. The weighted planar network (f(i), w( t 1 , . . . , 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 w(t1 , . . . , tt)).

the weight matrix x(f(i),

In particular, the network (f0, w) appearing in Figure 2 and Theorem 5 (more precisely, its equivalent defor­ mation) corresponds to some special word imax of length n2 ; instead of defining imax formally, we just write it for n 4: =

trices. Let EiJ denote the

1

We will interpret each matrix

matrix of a planar network. First, note that any elemen­

0

0 0

1

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 bi­ jection between n2-tuples ofpositive real numbers and to­ tally positive n x n matrices. We will prove the following refinement of Theorem which is a reformulation of its original version

THEOREM 1 4 . Every matrix x E G ,0 can be written X Ximax Ctl, . . . , tn2 ), for some t l , . . . , tnz 2:: 0. =

12,

[32].

as

(Since x is invertible, we must in fact have tk > 0 for 1)/2 < k :=::; n(n + 1)/2 (i.e., for those indices k for which the corresponding entry of imax is of the form @).)

n(n Proof

The following key lemma is due to Cryer [9] .

LEMMA 1 5 . 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 - 1 J(x) > 0. Let 11iJ(x) [resp. 11 ii ',jj' (x)] denote the minor of x obtained by deleting the row i and the column j (resp. rows i and i', and columns j and j'). Then, for any 1 :=::; i < i' :=::; n and 1 :=; j < j' :=::; n, one has

as an immediate consequence of Jacobi's formula for mi­ nors 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 pos­ itive 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) con­ verges; then, the standard compactness argument will im­ ply that the sequence of vectors (t�N)' . . . , t�'P) contains a converging subsequence, whose limit (t 1 , . . . , tn2) will provide the desired factorization x ximaxCt1 , . . . , tn2). To see that (sN) converges, we use the explicit formula =

=

�

=

SN

=

�

+

11 [l,i],[l,i] (XN) 11 [l,i- l],[ l,i - l] (XN)

1 11 [1,i - 1 ] U{i+ 1],[1,ij (XN) + 11 [1,i],[l,i- l]U{i+ 1j(XN) I 11 [1,i], [ 1,i] (XN) i=1

=l iL

(to prove this, compute the minors on the right with the help of Lindstrom's lemma and simplify). Thus, sN is ex­ pressed 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 ex­ ample, 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�p­ arately 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 mi­ nor 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 run­ ning example, there are nine chamber minors (the total number is always n2), namely X3 1 , X32, X12, X13, l123,12, l113,12, 1113,23, l112,23, and 11123,123 det(x). =

=

16. Every double wiring diagram gives rise to the foUowing criterion: an n X n matrix is totaUy pos­ itive if and only if aU its n2 chamber minors are positive.

THEOREM

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.

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 coef­ ficients in the numerator and denominator are positive). THEOREM

1 7.

Two double wiring diagrams are called isotopic if they have the same collections of chamber minors. The termi­ nology 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 dif­ fer 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 var­ ious submatrices of an n X n matrix. The identities corre­ sponding to the top two "moves" in Figure 9 are special in­ stances of the classical Grassmann-Pliicker relations (see,

VOLUME 22, NUMBER 1 , 2000

27

z

s

_______..

_______..

X; (t)

x, (t)

_______..

-

-

X(D(t) Figure 6. Double wiring diagram.

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 sub­ traction-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 re­ lated by a sequence of local 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 combinato­ rial 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 mi­ nors 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 de­ rived 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 ap­ pear in the corresponding total positivity criterion. We suggest the following refinement of Theorem 17. CONJECTURE 19. Every minor of a generic square ma­ trix can be written as a Laurent polynomial with non­ negative 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

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 dia­ gram 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 mono­ mial 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 di­ agram. 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 ) p2 - pq + q2). ficients (e.g., think of (p3 + q3)/(p + q 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 dia­ grams 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 re­ sist mentioning the related problem involving the Somos5 sequences [19]. (We thank Richard Stanley for telling us about them.) These are the sequences a 1 , a2, . . . in which any six consecutive terms satisfy this relation: (6) Each term of a Somos-5 sequence is obviously a subtrac­ tion-free rational expression in the first five terms a1, . . . , a5. It can be shown by extending the arguments in [ 19, 35] 123, 1 23 ==��====���======��======

3 1

�������==�����-- � 2

2

======��======��====��� � 0,0 Figure 5. Planar network r(i).

28

THE MATHEMATICAL INTELLIGENCER

Figure 7. Chamber minors.

3

l

3

123, 123

====�-r==========��r====== 3

==

1

==���--��-7��-r���� 2 2 �--�====����= 1

1 Figure B. Lexicographically minimal a1agram.

3

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 in­ volving 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 coeffi­ cients 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 ob­ tain a family of bijective parametrizations of the set G>o of all totally positive matrices; this family will include the pa­ rametrization 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 dig­ its 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 b y the involutions s1, . . . , Sn - 1 (ad­ jacent transpositions) subject to the relations sisi SjSi for

jl

li

li - ;:::: 2, and siSjSi SjSiSj for - 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 num­ ber 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 dou­ ble wiring diagrams are precisely the shuffles of two re­ duced words for wo, in the barred and unbarred entries, re­ spectively; equivalently, these are the reduced words for the element (Wo, wo) of the Coxeter group Sn X Sn. =

=

=

=

=

22. A word i in the alphabet .71 (see (3)) is called afactorization scheme if it contains each circled en­ try @ 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 par­ ticular, i consists of n2 entries. DEFINITION

To illustrate, the word i = 2 1 ® 2 I CD 2 1 @, appear­ ing 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. gABC

=

X

B_ _

_ c __

:vc� X V( y )(i_ B

c

B

.....

�-z-�

...

AXD

.....

:VC z :>
...

B

V( z� D

Figure 9. Local "moves."

a = xll

b

=

C =

d

=

e =

f

=

9 =

Xt2 X21

X22

X23

X32

X33

6.23. 13

n

=

c

=

D

=

6. 1 3, 1 3

E

=

6. 13 . 1 2

F

=

G

=

6. 1 3,23

6. 1 2,13 6. 1 2 , 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 = (i 1 , . . . , 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.

Theorem 23 suggests an alternative approach to total positivity criteria via the following factorization problem: for a given factorization scheme i, fmd the genericity con­ ditions on a matrix x assuring that x can be factored as

X = Xi(t 1 , . . . , tn') = Xi1(t1} · ·xinz(tn2),

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 factoriza­ tion schemes, this translates into any two factorization schemes being connected by a sequence of local transfor­ mations of the form

· · ·i j i· · · ---

�

· · ·j i j· · · , l i - jl = - - l i - jl =

1, 1,

(7)

or of the form (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 1 3, 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 o f these relations are the following. First, for i = 1, . . . , n - 1 and j = i + 1, we have

and compute explicitly the factorization parameters tk as functions of x. Then, the total positivity of x will be equiv­ alent 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 para­ meters 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 mono­ mials 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)- 1 wo[WoXT] _ .

x' =

[

- 1X21 -1 -1 X11X12 X21 X-121 X22 det(x) - 1

X31 X13

x' =

[ ][ ][ ] [ ][ ][ ] 1

t1

t2

0

1 0

0

1

0

t3

t4

1

=

1

0

t2

0

1

t4

t].

1

0

t3

0

1

Also, for any i and j such that li - jl following relation associated with (7):

=

.

1, we have the

xi(t1)xj(t2)xi(t3) = xj(t l.)xi(t2)xj(t3), xi(t 1)xj(t2)xj( t3) = xj(t l.)xi(t2)xj(t3), where

t2 =

T,

One sees that in the commutation relations above, the for­ mulas expressing the tfc in terms of the t1 are indeed sub­ traction-free. 0

30

THE MATHEMATICAL INTELLIGENCER

]

and

where

The proof of this relation (which is the only nontrivial re­ lation associated with (8)) amounts to verifying that

(10)

Here, xT denotes the transpose of x, and w0 is the permu­ tation 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

�

, t3 t t1 - 4 , t2' = T, r

(9)

�13 12 X13 �23, 12 1

X13

�12 13 X31 �12,23 X33�12,12 - det(x) �23, 12 �12,23 �

�12,23

1

X31 �

�23,12

�23.23

det(x)

The following theorem provides an alternative explana­ tion for the family of total positivity criteria in Theorem 16. [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 fac­ torization scheme. Then, the parameters t 1 , . . . , tn' ap­ pearing 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). THEOREM 24

In [15], we explicitly describe the monomial transfor­ mation in Theorem 24, as well as its inverse, in terms of the combinatorics of the double wiring diagram.

Double Bruhat Cells

nonvanishing of all "antiprincipal" minors

Our presentation in this section will be a bit sketchy; de­

and

[15]. 23 provides a family o f bijective (and biregu­

tails can be found in Theorem

lar) parametrizations of the totally positive variety

G>o by

n2-tuples of positive real numbers. The totally non-nega­ tive 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(t i , . . . , tm) with positive t1, . . . , tm, and i has the smallest number of uncircled en­ tries among all words with this property. Then, the sub­ word of i formed by entries from { 1, . . . , n - 1 } (resp. from { 1 , . . . , n - 1 }) is a reduced word for some permu­ tation 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

(u, v).

Let

G�8

C

G20

25, we say that x is of type

denote the subset of all totally non­

negative matrices of type (u,

v) ;

G2o

thus,

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,

afactorization scheme circled entry CD exactly

let us call a word i in the alphabet s1

of type (u, v)

if it contains each

once, and the barred (resp. unbarred) entries of i form a reduced word for

C(u)

+

C(v) + n.

u

(resp.

v);

in particular, i is of length

26

[ 15].

Comparing Theorems

26 and 23,

we see that

that is, the totally positive matrices are exactly the totally

(w0, w0).

We now show that the splitting of G2o into the union of

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. varieties

lower-triangular) matrices in

G.

Recall (see, e.g.,

that each of the double coset spaces B\GIB and

[1, §4])

B -\G/B_

has cardinality n!, and one can choose the permutation ma­ trices

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 in­

.l(x) = 0

.l(x) -=/=

0, for some collection of minors .1. (See [ 15, Proposition 4. 1] or [ 1 6] .) In particular, the open double Bruhat cell awo,Wo is given by equalities of the form

and/or

,n-

1, . . .

ll [l ,iJ . [n - i+ l,nJ(x)

1.

(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 sim­

ple test for total positivity of a totally non-negative matrix.

[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 ,iJ(X) -=/= O for i = 1, . . . , n. COROLLARY 28

The results obtained above for

G��wo = G>o (as well as

their proofs) extend to the variety G�8 for an arbitrary pair

u, v E Sn. In particular, the factorization v) (or rather their uncircled parts) can be visualized by double wiring diagrams of type (u, v) in the of permutations

schemes for (u,

same way as before, except now any two pseudolines in­

at most once, and the lines are permuted "accord­ u and v." Every such diagram has C(u) + C(v) + n

tersect ing to

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 modifica­ tion of the twist map

x � x'.

The details can be found in

( 15]. If the double Bruhat cell containing a matrix not specified, then testing

x

xEG

is

for total non-negativity be­

comes a much harder problem; in fact, every known crite­ rion involves exponentially many (in

n)

minors. (See

[8]

for related complexity results.) The following corollary of

[ 10] was given by Gasca and Pefta [24].

An invertible square matrix is totally non-negative if and only if all its minors occupying sev­ eral initial rows or several initial columns are non-neg­ ative, and all its leading principal minors are positive. THEOREM 29.

This criterion involves

(11) non-negative matrices of type

for i =

THEOREM 27

a result by Cryer

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 ofpositive real numbers and totally non-negative matrices of type (u, v).

THEOREM

d[n -i+ l,n] , [l, iJ (X)

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 in­ tensively studied by Gantmacher and Krein trix is

oscillatory

[20, 22]. A ma­

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 fol­

lowing theorem provides several equivalent characteriza­ tions of oscillatory matrices; the equivalence of (a)-(c) was proved in

[22], and the rest of the conditions were given in

[17]. [ 1 7,22]. For an invertible totally non-neg­ ative n X n matrix x, the following are equivalent: (a) x is oscillatory; (b) xi, i+l > 0 and Xi + l,i > O for i = 1, . . . , n - 1; THEOREM 3 0

VOLUME 2 2 , NUMBER 1 , 2000

31

(c) (d)

xn - 1 is totally positive; x is not block-triangular (cf Figure 1 1); *

*

0 0 0

*

*

*

*

*

*

*

0 0 0

*

*

*

*

*

*

*

*

*

*

0 0

*

*

*

*

*

*

*

*

0 0

*

*

*

*

*

*

*

*

0 0

*

*

*

Figure 1 1 . Block-triangular matrices.

x can be factored as x xi(t1 , . . . , t1), for positive t 1 , . . . , 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. =

(e)

=

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

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 (1 1), 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 denote the subse­ quence of jn - 1 constructed as follows. First, j 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 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 1

1

I

ACKNOWLEDGMENTS

We thank Sara Billey for suggesting a number of editorial improvements. This work was supported in part by NSF grants DMS-96255 1 1 and DMS-9700927.

REFERENCES 1 . J.L. Alperin and R.B. Bell, Groups and Representations, S pri n ger­ Verlag, New York, 1 995.

A U T H O R S

SERGEY FOMIN

ANDREI ZELEVINSKY

Department of Mathematics

Department of Mathematics

University of Michigan Ann

Arbor, Ml 48109 USA

Northeastern University Boston, MA 021 1 5 USA

e-mail: [email protected]

e-mail: [email protected]

Sergey Fomin is a native of St. Petersburg, Russia. He de­

Andrei Zelevinsky grew up in Moscow, Russia. He received

cided he wanted to be a mathemat ician at the age of 1 1 and

his Ph . D . from Moscow State University in 1 978, moved to

became addicted to combinatorics at the age of 1 6. A stu­ dent of Anatoly Vershik, he received his adva nced deg ree s

from St. Petersburg State Univ ersity . From 1 992 to 1 999, he

was at MIT. He has also held since 1 991 an appo i ntm ent at the St. Petersburg Institute

for Informatics and Automation.

His main research interest is combinatorics and its applica­ tions in representation theory, algebraic geometry, t heoretical computer science, and other areas of mathematics.

t he West in 1 990, and

has been a professor at Northeastern

since 1 99 1 . His research i nterests

tions, and algeb raic and polyhedral combinatorics. He enjoys

t raveli ng , seein g new places , a nd maki ng friends with fellow mathematicians worldwide. As a young student m aki n g his

first steps in math emati cs in the Soviet Union that was, he never dreamed that this road would eventually take him to so many wonderful places.

32

THE MATHEMATICAL INTELLIGENCER

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VOLUME 22. NUMBER 1 , 2000

33

1.5ffli•i§uflhfii*J.Irri,pi.ihi£j

Exact Thought in a Demented Time: Karl Menger and his Viennese Mathematical Colloquium 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.

M a rj o r i e Senechal ,

Ed itor

O

ne 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 Er­ gebnisse 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.

Please send all submissions to the Mathematical Communities Editor,

Marjorie Senechal, Department of Mathematics, Smith College, Northampton, MA 01 063, USA

e-mail: [email protected]

34

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,

THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER· VERLAG NEW YORK

I

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 (1 841 -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, econ­ omists Jevons and Walras hit indepen­ dently on this idea, which necessitated a complete rethinking of classical the­ ory. Like some of his Austrian contem­ poraries-Ludwig Boltzmann, for in­ stance, 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 as­ sociate 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 in­ troduced the archduke to Moritz Szeps, in whose j ournal Rudolph pub­ lished 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 al­ ready achieved world-wide renown and could count on some brilliant dis­ ciples, like Friedrich Wieser and Eugen Bohm-Bawerk, to carry on with the theoretical work. He himself concen­ trated 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 so­ ciety. In contrast, Carl Menger believed in universal economic laws, ultimately grounded in the psychology of human needs. His methodological individual­ ism was a fitting expression of the pre­ vailing mood in fin-de-siecle Vienna. Every afternoon, he resided in a cof­ fee-house where he discussed the is­ sues with the leading social scientist and law professor of Vienna. This hap­ pened 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 even­ tually had used a lottery windfall to fi­ nance 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 lead­ ing liberal deputy of the Reichstag (the Austrian parliament). Karl Menger was born on January 12, 1902. His father had recently be­ come a member of the Herrenhaus­ a 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 in­ struction. 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 gott­ lose Komoedie (the godless comedy­ in contrast with Dante's Die gotUiche

Arthur Schnitzler (1 862-1931) 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 1 900, 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 fearless

as

there

•

•

ever

•

as honest and 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

was able to seek the advice of the fore­

defining curves as one-dimensional

centers on the medieval Pope John

most playwright in town. The com­

continua. Continua had been defmed

who, as legend has it, turned out to be

ments were negative, alas, and the god­

by Cantor and Jordan already. What re­

a woman called Joan.

less comedy came to naught.

But

mained was to defme their dimension. Menger hit on the idea of proceeding

Young Menger's classmate Heinrich

Arthur Schnitzler kept notes on his

happened to be the son of Arthur

meetings with Karl Menger, and traced

inductively, assigning dimension - 1 to

Schnitzler, the most famous Viennese

in his diary a dramatic turn of events.

the empty set and defming a set

It may have been the

S to

It began in an unheated classroom

be at most n-dimensional if each of its

shared burden of descending from cul­

of the University of Vienna. The time:

points admits arbitrarily small neigh­

tural

and

March 1921, during the worst inflation

borhoods with whose frontiers

Heinrich together-an Oedipus com­

of Austrian history. Karl Menger had

at most

plex was not unheard-of in the Vienna

enrolled in theoretical physics-this

tion.

of those days. Through Heinrich, Karl

was the heyday of the Einstein fer­

Menger showed this solution to his

vor-but was not satisfied with what

friend Otto Schreier, who was already

the Physics Department had to offer,

in his second year at the University.

author of his age. heroes

that

drew

Karl

(n -

S

has

I )-dimensional intersec­

and drifted towards the Institute of

Schreier could fmd no flaw in Menger's

Mathematics. A newly appointed pro­

ideas, but quoted both Hausdorff and

fessor there, 42-year-old Hans Hahn,

Bieberbach who said the problem was

had just announced his first seminar.

intractable. Menger, however, was con­

It dealt with the concept of curves.

vinced that "one should never reason

Menger had barely entered his second

that an idea is too simple to be correct."

semester, but decided to give it a try.

He told Hahn one hour before the sec­ ond seminar that he could solve the

to sell the library when he was twenty, Karl

who had hardly looked Hahn went right to the heart of the up when I entered, became more and pr-oblem. Everyone, he began, has an mm-e attentive as I went on . . At the ·intuitive idea of curves; . . . But any­ end, after some thought . . . he nodded one who 1vanted to make the idea pre­ rather encouragingly and I l.eft. cise, Hahn said, would encounter The chronicles of mathematics re­ great difficulties. . . . At the end of the port other breakthrough discoveries seminar we should see that the prob­ by mere teenagers. What makes this lem was not yet solved. I was com­ case so special is that Menger used pletely enthralled. And when, after only the material covered in one sem­ that short introduction, Hahn set out inar talk. to develop the principal tools-the ba­ A few weeks later, disaster struck. Weakened by malnutrition and long sic concepts of Cantor's point-set the­ ory, all totally new to me-l followed working hours in unheated libraries, with the utmost attention. Karl Menger fell prey to tuberculosis­ called Morbus Viennensis at the time. Curves to Glory

problem. Hahn,

.

Karl Menger (1902-1985) inherited from his father

a

positivistic,

individual-centered

world view and a huge private library. Obliged Menger held on to the philosophy books. It

Hahn was well placed to discuss the

In the impoverished capital of an am­

may have been this heritage which immu­

curve problem. Fired up by Peano's and

putated state, this illness was claiming

nized him against the lure of Wittgenstein.

Hilbert's constructions of space-filling

thousands of victims. The chronicles of

Indeed, Austrian philosophers had antici­

curves, he had shown what became

mathematics are filled not only with

pated parts of the Tractatus; for instance,

known as the Hahn-Mazurkiewicz the­

precocious talents but also with pre­

Fritz Mauthner, who was just as sure as the

orem: every compact, connected, lo­

mature deaths-Schreier, for instance,

young Wittgenstein of having solved all philo­

cally connected set (a full square, for in­

was to die at twenty-eight, after brilliant

sophical problems, or Adolf Stoehr, the suc­

stance) was the continuous image of an

work in group theory. Stlicken by tu­

cessor to

interval.

berculosis-like Niels Hendrik Abel­

left the seminar in a daze. Like everyone else I used the word "curve". Should I not be able to spell out artic­ ulately 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.

his ideas in feverish haste-like Evariste

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

I

36

THE fv'.ATHEMATICAL INTELUGENCER

Galois-and deposited them in a sealed envelope at the Viennese Academy of Science before entering a sanatorium lo­ cated 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

be solved. There is only incorrect considera­ tion of the world."

nineteen-year-old Menger jotted down

This solution consisted essentially in

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-year­ old 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 clas­ sic Grundziige, which included a wealth of revisions. At the same time, his passion for philosophy asserted it­ self. He had come to believe that the recent work by L.E.J. Brouwer on "in­ tuitionistic" 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 fa­ ther. 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 an­ other, 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 tem­ peramental, began to get tense. In part this was due to Menger's disagree­ ments with Brouwer's anti-French views, in part to his impatience with Brouwer's legalistic mind and his oc­ casional obscurity. These differences were exacerbated by a priority dispute. The young Russian mathematician Pavel Urysohn had developed a di­ mension theory quite similar to Menger's, at about the same time, be­ fore perishing in a drowning accident. Brouwer edited the posthumous publi­ cations of Urysohn, stressing their link with a note written by himself in 1913 which contained already some essen­ tial 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 over­ looked.) Menger, who had originally known neither of Brouwer's nor of Urysohn's work, felt that his contribu­ tion was misrepresented. He was par­ ticularly incensed that Brouwer had in­ cluded 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 di­ mension. 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 po­ litical parties-the Social Christians, who ruled the country, and the Social Democrats, with their unsinkable ma­ jority in Red Vienna-seemed to have arrived at a stable balance. The eco­ nomic situation had greatly improved, inflation was stopped, and a program of sweeping social reforms was under­ way. But the two parties' private armies still paraded through the streets, and soon after Menger's return to Vienna the political truce was shat­ tered. In July 1927, a jury acquitted mil­ itant 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 sup­ pressed the outburst brutally, and more than eighty people were left dead in the streets. This explosion of irra­ tionality left a lasting mark on the young republic. Many Austrians con­ cluded 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 col­ lected his topological results in a book, Dimensionstheorie. And he accepted the invitation, by Hahn and Moritz Schlick, to join the hand-picked philo­ sophically-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 en­ gagement of Neurath and Hahn. Soon he asked to be listed, not as a member, but as someone close to the group. In 1932, Menger published his sec­ ond book, Kurventheorie, which con­ tained, among other things, his uni­ versal curve: not only can every curve be embedded in 3-space, but there ex­ ists in 3-space one curve such that every curve can be topologically em­ bedded in it (this curve, in fractal the­ ory, became known as the Menger sponge). And as a by-product of study­ ing branching points of curves, Menger proved his celebrated n-arc (alias Max­ Flow, 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 con­ necting 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 connect­ edness 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 be­ fore 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 be­ came 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 time­ hallowed statement that "a point is that which has no part." Not surprisingly, Menger quickly be­ came popular with his students, who were barely younger. In spite of being eternally busy, he was easy to ap­ proach. 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 per­ sonality, appeared as an almost super­ human embodiment of mathematical discipline. The students found it obvi­ ously easier to ask Karl Menger to di­ rect a mathematical Colloquium. This Colloquium had a flexible agenda in­ cluding 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 lim­ ited 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 Am­ sterdam 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 increas­ ingly from the Vienna Circle to the Colloquium, his friend and protege Kurt Godel moved with him Godel had en­ tered 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 founda­ tion 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 ax­ ioms. 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 rela­ tionships grew out of the lectures Tarski gave to the Colloquium in February 1930, the first between Tarski and the philosopher Rudolf Carnap, and the sec­ ond between Tarski and Kurt Godel, who, after hearing Tarski lecture, had asked Menger to arrange a meeting. .

Kurt Godel (1906-1 978). In Menger's posthu­ mous Reminiscences of the Vienna Circle and the Mathematical Colloquium

slim, unusually quiet young man

"he was a

. • • •

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 1 954, 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 di­ rectly, he did provide him with a math­ ematical setting in which to develop them. Godel participated in the Collo­ quium 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 appearphilosophically 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 Wald, 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

1932, as the economic and po­

tempts by deputies to meet again. Its

litical situation in Vienna deteriorated,

anti-socialist measures became increas­

spring of

the students faced increasing fmancial

ingly brazen, and provoked a short but

difficulties. Menger understood this and

murderous bout of civil war in February

hit upon a novel fund-raising source.

1934. As a result, the Social-Democrats

he later explained,

were banned.

As 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 neverfound 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

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

plan with Hahn, who suggested the physicist Hans Thirring, who in turn led

Franz Alt (born 1 910) received his Ph.D. in

them to the chemist Hermann Mark To­

mathematics from Karl Menger, who asked

gether they outlined several series of

him to look after the Colloquium during his

lectures, the first of which had the gen­

frequent stays abroad. But on March 1, 1938,

eral title "Crisis and Reconstruction in

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,

needed") had to write to Menger, "Until now

Menger reported, every seat in the au­

I always closed my letters expressing my

ditorium was taken. Mark opened the

hope to see you again soon in Vienna. At pre­

first

sent I have to hope to find some way to get

series

with

"Classical

Physics,

Shaken by Experiments," Thirring con­

together with you over there." In the US, Alt

tinued

the

contributed to the development of the com­

with

"The

Changes

of

Conceptual Frame of Physics," followed

puter, working at the Computing Laboratory

by Hahn on the "Crisis of Intuition."

in

Nobeling gave the fourth lecture, on

Standards, and the American Institute of

chemical engineer.

"The Fourth Dimension and the Curved

Physics in New York. He was a founding

mathematics at the University of Vienna in

Aberdeen,

the

National

Bureau

of

Olga Taussky-Todd (1 906-1 995) was born in Olomouc (today, Czechia), a daughter of a She began studying

Space," and Menger fmished the series

member and a president of the Association

1 925 and became one of the most active

with "The New Logic." Menger's lecture

for Computing Machinery, and the first edi­

members of the Colloquium. Olga's thesis

was the first popular presentation of

tor of Advances in Computers.

was on class fields and group theory, but she

Godel's results.

later steeple-chased through a vast number

Thus, although Menger was barely

of topics. She worked for a spell in Gottingen,

older than this followers, his role was

editing Hilbert's Zah/bericht for his complete

almost fatherly. Responsibility for his

Nazi

small group hung heavily on his shoul­

Austrian

ders, especially after his own mentor

think of nothing better than to

threats

and

the

works, and returned in 1 933 for a couple of

could

years to Vienna, supported by a small stipend

turn for

funded by the series of public lectures

Hahn died unexpectedly in summer

help to Mussolini. The Austrian parlia­

("rather elegant affairs") organised by Hahn

1934. The future began to look very

ment, in a rather remarkable instance

and Menger ("very enterprising people"). She

bleak for Vienna's mathematicians and

of befuddlement, managed to eliminate

then taught at Bryn Mawr and Girton College,

chancellor

terrorism, Dollfuss

philosophers. Menger, who had never

itself. Because of a ballot hanging in

Cambridge. In 1938, she married the British

shared Hahn's willingness to engage in

the balance, the first president of the

mathematician John Todd, and during the

political action, now greatly missed

house (a kind of speaker), who was

war years turned to applied mathematics.

this "tireless and effective speaker for

prevented by office from casting a

After the war, both John and Olga held dis­

progressive

time

vote, stepped down. Not to be outdone,

tinguished positions in the US, eventually set­

when such speakers were permitted to

the second president (who belonged,

tling down at Caltech. When she was given,

raise their voices was gone.

of course, to the opposite camp) did

in 1 963, the Woman of the Year Award by the

In Berlin, Hitler had swept to power;

the same. In the heat of the moment,

Los Angeles Times, she noted gratefully that

the annexation of his native Austria,

the third president followed suit. No

"none of my colleagues could be jealous

where he had many supporters, stood

one was left to chair the session. The

(since they were all men)." (Photograph cour­

at the top of his program. Faced with

Social-Christian government quelled at-

tesy of E. Hlawka.)

40

causes."

But

THE MATHEMATICAL INTELUGENCER

the

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

(1 902-1 994),

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

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 C ech 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,

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

While the political situation in Austria made it extremely difficult to concen­ trate on pure mathematics, socio-po­ litical problems and questions ofethics imposed themselves on everyone al­ most every day. In my desirefor a com­ prehensive world view I asked myself whether some answers might not come through exact thought.

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."

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. Nobeling 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

plicitly advised him against submitting

was unavoidable, given his father Carl

the paper to its journal, the Zeitschrift

and his uncle Anton. When, on return­

fur NationalOkonomie.

1923, Karl

Mayer's assistant Oskar Morgen­

Menger had proceeded with the revised

stern was outraged by this further proof

edition of his father's magnum opus, he

of his professor's ineptitude. Morgen­

ing from the sanatorium in

had made contact with Austrian econo­

stern, who was of Menger's age, be­

mists.

longed to the fourth generation, to­

This so-called third generation

(the first wa'l made up of Carl Menger

gether with his friend Kurt Haberler and

alone)

the future Nobel Prize winner Friedrich

was

dominated

by

Joseph

Schumpeter and Ludwig von Mises (the

von Hayek Like Menger, he had found

brother of the applied mathematician

support from the Rockefeller Founda­

and philosopher Richard von Mises).

tion. During his extensive travels, he had

Significantly, neither of the two held a

been most impressed by a meeting with

him the

chair. The professors in the economics

Edgeworth, which instilled in

department were not in the same league.

unshakeable conviction that economists

This was

Most of the discussions took place out­

needed mathematical tools.

side of the university, in circles, private

not then a fashionable view, and the

seminars, and coffee houses.

Austrian marginalists, in particular, had

Twenty-year-old Karl Menger had

traditionally

shunned

mathematics.

written an essay "On the role of uncer­

Morgenstern must have found in Karl

tainty in economics" dealing with the

Menger an answer to his prayers; al­

Berlin, a son of an illegitimate daughter of the

two-hundred-year-old

though his own mathematical training

Prussian Emperor Frederic I, Morgenstern

paradox. Suppose that a casino offers

was not substantial, he quickly estab­

studied economics in Vienna. His friend, the

the following game: you throw a coin

lished contact.

economist Haberler, told him that he should

repeatedly, until "heads" comes up for

His zeal was unbounded.

Within a few years, he became man­

always sign his name as "Oskar Morgen­

the first time; if this happens on the

aging

of the

stern, Aryan" to avoid the rejection of his pa­

St. Petersburg

nth throw, you receive

editor

Zeitschrift fur

Oskar Morgenstern (1902-1977). Born in

2n dollars. Of

NationalOkonomie, where he had the

pers by anti-semitic colleagues. Soon con­

course you will have to pay some ad­

satisfaction of publishing Menger's es­

vinced of the importance of mathematics for

mission fee. How much should you be

say. Morgenstern had, by then, suc­

economics, he suffered from his lack of

willing to pay? The first answer com­

ceeded Hayek as director of the small

knowledge in the field. "I was an idiot not to

ing to mind is: anything less than the

Institute

for

Konjunkturforschung

have studied mathematics at least as a side­

expected value of the gain. But this

(Business Cycle Research) in Vienna.

line at the university of Vienna, instead of this

value is infinitely large. Indeed, the

This was a paradoxical appointment,

silly philosophy." He tried to make up for this

probability that "heads" comes up for

for Morgenstern's main work so far

by taking lessons from Alt and Wald. In 1 935,

the first time at the nth throw is

had been about the impossibility of

and this yields a payoff of

2-n, 2n. Hence

he wrote in his diary: "Again a mathematical

economic predictions: he claimed that

lesson. Now we are already into differentia­

one should be willing to stake all one's

the interdependence of economic de­

tion. Wald thinks that in one year I am going

possessions to be admitted to the

cision-makers defeated forecasts and

to be advanced enough to understand nearly

game. But no reasonable person is pre­

prevented equilibrium. But Cech, who

everything

pared to do so. Bernoulli proposed an

often took part in the Colloquium and

Morgenstern did not become a mathemati­

ingenious solution to the paradox: util­

had proposed a notion of dimension

cian, but a gold mine for mathematicians, in­

ity does not grow linearly with the gain,

which was to supplant, in some re­

spiring Menger, Wald, von Neumann, Shubik,

but

spects, the one by Menger and Urysohn,

and Schotter.

logarithmically.

Menger

recog­

nised, however, that this, and indeed

drew Morgenstern's attention to the

any other

minimax theorem proved by John von

unbounded utility function,

would only lead to a similar paradox. Menger's essay, which went on to

in

mathematical

economics."

lead to an infinite regress. But von

Neumann several years before: for

Neumann's minimax theorem offered a

zero-sum

solution: it consisted in throwing a

games with two players,

discuss how individuals differ in their

equilibrium was consistent with per­

coin. This gave Morgenstern pause.

evaluation of how much to pay for the

fect foresight.

Keener than ever on mathematical

chance to gain an amount D with prob­

In

order

to

make

his

point,

ability p, was to inspire many mathe­

Morgenstern had used in many papers

matical economists,

including John

and lectures (including talks in the

von Neumann, Kenneth Arrow, and

Vienna Circle and the Colloquium) the

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 stu­

Paul Samuelson. But when, after his re­

example

and

turn from Amsterdam, Menger lec­

dents Alt and Wald, who both were job­

Professor Moriarty-that mathemati­

less, and with no prospects in the face

of

Sherlock

Holmes

tured on the topic to the Economic

cian gone wrong-whose attempts to

Society, its president Hans Mayer ex-

of ever-rising anti-semitism. Their eco­

outguess each other apparently had to

nomic plight led to a breakthrough in

42

THE MATHEMATICAL INTELLIGENCER

function, and Wald laid the foundations

Walras, and later Cassels, had estab­

did more than provide his two tutors

for general equilibrium theory.

lished a system of as many equations

nated by economics. Soon Alt wrote a

Equilibrium in a

lieved to ensure the existence of a

paper on the measurability of the utility

Collapsing Country

unique equilibrium. Menger, of course,

Menger had arranged for Wald to

knew that it was not sufficient.

economics. Indeed, Oskar Morgenstern

as there were unknowns. This was be­

with pocket money. Both became fasci­

coach not only Morgenstern, but also

Schlesinger was the first to lecture

Karl Schlesinger, a banker. Schlesinger

in the Colloquium on the fundamental

was attracted by the so-called imputa­

equations

tion problem, central to the theories of

stressing that all the factors which oc­

of

Walras

and

Cassels,

Carl Menger and Leon Walras: how do

curred in the equations had to be

the prices of the products determine

"scarce" in the sense that they were en­

the prices of the factors of production?

tirely used up in the production; for if

(This problem was the reverse of that

there remained a surplus, that factor

studied by classical economists, who

would cost nothing. However, whether

assumed that the prices of the factors

a surplus remains or not depends on

determined the prices of the products.)

the

production

process.

This

led

John Von Neumann (1903-1 957). "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 i n 1 937, 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

Schlick's Assassination. After the death of Hahn and the exile of Neurath in 1 934, the Vienna

the table, while making rapid and audible

circle lost the third of its founders in June 1936, when the philosopher Moritz Schlick was

progress. . . . " In the thirties, von Neumann

shot on the steps of the University. Aristocratic Schlick had been bom in Ber1in {sad, but true,

frequently visited Vienna, where he had many

as he said to Menger) and was "extremely refined, sometimes introverted." "Serenity is our

discussions with Menger and Godel (a histo­

duty" was Schlick's motto. A few weeks before being murdered, Schlick told Menger that he

rian described him as a member in pectore

had been threatened for years by a paranoiac who had been in and out of mental institu­

of the Viennese Colloquium). But his clos­

tions. The police had assigned him a bodyguard for some time; but as an actual assault had

est Viennese collaborator became Oskar

never taken place, he did not dare tum to the police again. Schlick added with a forced smile:

Morgenstern, whom he met only later, in

"I fear that they begin to think it is I who am mad." The psychopathic killer was Johann NelbOck,

Princeton. According to Morgenstern, John

who had studied philosophy and mathematics, and had written his thesis on "The meaning

von Neumann was amazed at the primitive

of logic in empiricism and positivism" under Schlick's supervision. Nelbock had felt thwarted

state of mathematics in economics; he held

by Schlick, both in his love for Sylvia Borowicka (another student of philosophy) and in his

that if all economics texts were buried and

career; but at the trial, he managed to persuade the jury that he had killed the free-thinker

dug up one hundred years later, people

Schlick for ideological reasons. He was sentenced to 10 years and released right after the

would think that what they were reading had

Anschluss,

been written in the time of Newton.

maxims alien and pernicious to the people, had rendered a service to National Socialism."

having pointed out that "his deed, the elimination of a teacher spreading Jewish

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 depar­ ture, Waid proved the existence of a unique positive solution-an equilib­ rium. 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 produc­ tion 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 follow­ ing period. John von Neumann had proceeded to prove the existence of an equilibrium solution by a generalisa­ tion of Brouwer's fixed point theorem, underscoring the connections with his own minimax result. His Colloquium paper became a milestone in econom­ ics-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 math­ ematical economics which he planned to bring out in the following, ninth, vol­ ume 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, em­ broiled in his Abyssinian fiasco and badly needing allies against the League of Nations, had decided to stop an­ noying 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 to­ tal isolation, at long last decided to tum to his own people for support, and

44

THE MATHEMATICAL INTELLIGENCER

organised a plebiscite, firmly expect­ ing 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 over­ whelming 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 assassina­ tion of Moritz Schlick, he gratefully ac­ cepted 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 "po­ litically undesirable" in Vienna. He quickly obtained a position as lecturer in Princeton, but found his new col­ leagues 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 lat­ ter 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 man­ aged 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 fer­ vent 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 travel­ ling 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 mo­ mentum 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 (in­ troducing, 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 commu­ nity, but he did not share the tremen­ dous success of some fellow emigrants like John von Neumann, Stanislaw Ulam, and Abraham Wald. During the war, Menger published little, not because his work was clas­ sified 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 dis­ cuss critically the usual approaches to­ wards 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 con­ tinues to mar the classroom experi­ ences 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 .

Circle Collection val. 1 3, Kluwer, Dordrecht

A U T H O R S

(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

KARL SIGMUND

LOUISE GOLLAND

Networking Services & Information

lnst�ut

fOr Mathematik

Universitat Wien

Technologies

The University of Chicago

1090 Vienna

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

Chicago IL 60637

Austria

nomics aspect. For a start see E. Craven,

e-mail: [email protected]

e-mail: [email protected]

The emigration of Austrian economists,

Louise Galland studied mathematics

Karl Si gm u nd , a former ergodic-the­

Hist.

of Political Economics

1 8 (1 989),

1 -32, as well as M. Dore, P. Chakravarty,

at the Illinois Institute of Technology,

orist turned biomathematician, has

and R. Goodwin (eds), John von Neumann

where she was inspired by the lec­

written a popular book (The Games

and

tures of M enger. She received her

of Ute.

evolutionary

(1 989); and in particular the articles by K.J.

Ph.D. in history from the University o f

Chicago . special izing in t he history of is an

independent

Penguin)

on

Modern

Economics,

Oxford

UP

game theory. This is his second

Arrow, Von Neumann and the Existence

fntelfigencer art i cle on mathemati­

Theorem

cians in the Vienna Circle. He admits

for General

Equilibrium

(pp.

scholar in the history of mathematics

he is hopeless at the Austrian national

P.A. Samuelson, A Revisionist View of von Neumann ' s Growth Model

and astronomy, while cont inu ing to

sports, skiing and waltzing, but he

(pp.

tries to make up for it by d evoted ly

Neumann and Karl Menger ' s Mathematical

science.

She

work for the Un iversity.

frequenting coffee houses.

1 5-28);

1 00-124);

and

L.F.

Punzo, Von

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, 730-76 1 , and: Ethics and

For further material, see K. Menger, Reminis­ cences

of the

Mathematical

Vienna

Circle

Colloquium,

and

the

Vienna Circle

Collection vol. 20, Kluwer, Dordrecht (1 994);

the Excluded Middle: Karl Menger and

K. Menger, Selected Papers in Logic and

Social Science in Interwar Vienna, Isis 89

Foundations, Didactics, Economics, Vienna

{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 1 980 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 ques­

to a different conclusion. The amount of light that falls on

tion of starting somewhat mathematically. With the right

a linear meter at a distance 2x from a star is one-half the

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

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.

with one dimension less, there might be planets and stars,

The figure resembles a production of that well-known

but they would have to be disks of two-dimensional mat­

toy, the spirograph, in which gears laid on a sheet of pa­

ter. In laying out the basic picture I followed informal prin­

per roll around each other.

ciples of simplicity and similarity. Other things being equal,

one of the gears might trace such a figure. Are the two­

A pencil inserted in a hole in

a feature in the planiverse should be as much like its coun­

dimensional orbits spirograph figures? Probably not. They

terpart in our universe as possible, but not at the cost of

look like epicycles, the paths that early astronomers

simplicity within the two-dimensional realm. The simplest

thought might explain the looping orbits of Mars and

two-dimensional analog of a solid sphere is a disk.

Jupiter in an Earth-centered system! (It is tempting to con­

What sort of orbits would the planets follow? In our own universe, Newtonian mechanics takes its particular form

clude that what goes around comes around.) Encouraged by such speculations, I begin to develop the

A planet circling

impression that such a universe might actually exist. It would

a star, for example, "feels" an attraction to that star which

be completely invisible to us three-dimensional beings, wher­

from the inverse-square law of attraction.

varies inversely with the square of the distance between

ever it might be. But places, even imaginary ones, need names.

the two objects. The same reason in the planiverse leads

What could a two-dimensional universe be, but the Planiverse?

46

THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK

I I

----:I

�F� - - - - - - - - - - - - - - - - - - meter

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 sev­ eral speculations, including the drawing of a two-dimen­ sional fish shown in Figure Three below. Gardner wrote back, saying that he not only found the planiverse a delightful place, he would devote a forthcom­ ing 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 col­ umn 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,

a paper symposium with colleague Richard Lapidus, a physi­ cist 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 satisfy­ ing 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 2. Orbit of a two-dimensional planet.

Figure 3. Two-dimensional fish.

VOLUME 22. NUMBER 1 , 2000

47

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

Figure 4. Planiversal vehicle.

profound. Closed curves lurked everywhere.

picked it up with the glee reserved for UFO reports and es­

of two-dimensional grains and pebbles in which any pocket

caped lions. There followed a rush of magazine and news­

of water fmds itself permanently trapped within the closed

paper articles, as well as television stories publicizing our

circle of surrounding stones. The water cannot percolate, as

Consider, for example, Ardean soil, a mechanical mixture

two-dimensional world. In particular, a piece in

Newsweek

magazine caught the attention of publishers.

our groundwater does, up or down. It is trapped, at least un­ til the soil is mechanically disturbed. Consider also the sim­

In the midst of a series of papers on programming logic,

ple matter of Yendred attempting to lift a two-dimensional

I was suddenly face to face with a big writing job. There

plank on the Ardean surface. The plank, the ground, and

were contracts with Poseidon Press (Simon & Schuster) in

Yendred himself would form a simple closed curve, and the

the US, with Pan/Picador in England, and with McClelland

air trapped inside the enclosed space would become in­

& Stewart in Canada. I viewed these new responsibilities

creasingly rarefied. The plank would seem to get heavier and

with irritation. It was assuredly fun to think about the plani­

heavier. Perhaps readers can imagine themselves to be

verse, but my research came first. And was I not in danger

Ardeans lifting such a plank If you were Yendred, what tech­

of being regarded as a nut-case? The media were no help.

nique would you adopt to make it easier?

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

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

to weave together all the scientific and technical elements

string! Yendred's father, who takes him fishing near the be­

that had emerged from the Planiverse Project. But a com­

ginning of the book, never has trouble with tangled lines,

pendium of these speculations, no matter how wild or en­

for knots in two-space are impossible. Moreover, sailing re­

tertaining, would surely prove a dry read. It would have to

quires nothing more than a mast!

be a work of fiction, set in the planiverse itself. There would

Yendred sets out on his quest shortly after the fishing

be a planet called Arde, a disc of matter circling a star

trip with his father. His home, like all Ardean homes, is un­

called Shems. There would be a hero named Yendred (al­

derground. The surface of Arde must be left as pristine as

most my name backwards) and his quest for the third di­

possible. There are travelling plants and periodic rains

mension or, at least, a spiritual version of it. Yendred is

which make temporary rivers, basically floods. Any surface

convinced that the answer to his quest lies on the high

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

Figure 5. Yendred, a typical Ardean.

48

THE MATHEMATICAL INTELLIGENCER

line. Yendred passes over numerous houses, apartment

Figure 6. Ardean sailing vessel.

buildings, and factories, marked only by the exit or entrance

three dimensions. It must be given some thickness, of

mensional ants. Overhead pass delivery balloons, each with

plates to simulate the restriction of no sideways movement.

its cargo of packages. Balloon drivers adjust to near-neutral

I have often wondered whether we could build a car with

buoyancy, then take great hops over their fellows.

a one-inch thick steam engine mounted underneath. Think

of fellow citizens bent on private tasks like so many two-di­

Access to underground structures is managed by swing­

course, and it must also be enclosed between two parallel

of the additional room that would provide!

stairs. Although some of the larger structural beams are

Ardean technology is a strange mixture of advanced and

held together by pegs, the fastener of choice is glue. Wires

primitive machines. Although steam engines are the main

(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 bat­ tery-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 thou­ sand individuals inhabit its lone continent. Consequently, the Ardeans have no great demand for power machines, the steam engine sufficing for most needs, such as eleva­ tors and factories. Readers might be able to figure out the operation of an Ardean steam engine from the accompa­ nying 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 reser­ voir above the piston and escapes when the piston com­ pletes its travel to the head of the "cylinder." Interestingly, almost any two-dimensional machine can also be built in

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

(0 and 1)

the same binary principles

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 put is a

1

signal if and only if exactly one in­

1.

No matter what combination o f zeros o r ones enter this circuit along the wires labelled x and y, the same signals

Figure 8. A logic crossover.

leave the circuit along the wires bearing these labels. Readers may readily satisfy themselves that if x and y both

Earth scientists some time ago. If one sounds a note on

carry a zero (or one), for example, then both output lines

Arde, the sound wave alters as it travels. A sharp attack

will also carry this signal. But if x is one and y is zero, the

smears out in time, so that a single note of C, for example,

middle gate will output a one which will cancel the x-sig­

is heard at a distance as a glissando rising from some lower

nal in the upper gate and combine with the zero on the y­

pitch and asymptotic to C. Cosmologically speaking, Ardean scientists have much

input in the lower gate to produce a one. Fun though technology may be, it isn't until he visits the

to ponder. Like us, they wonder if their universe is closed

Punizlan Institute of Technology (PIT) that Yendred en­

like a balloon (we say it is) or open like a saddle-shaped

counters the deep scientific ideas of his time. Scientists at

space. It is apparently expanding, and the balloon analogy,

PIT have developed a periodic table of the elements based

so often used to illustrate how our own universe is appar­

on the theory that while just two electrons can occupy the

ently expanding, can be taken quite literally. A deeper ques­

frrst shell of a planiversal atom, up to six can occupy the

tion concerns the orientability of the planiverse. Perhaps

second shell. We have labelled the planiversal elements

it is really a projective plane, so that Yendred, travelling by

with the symbols of the elements from our own universe

rocket across the planiverse, might return to fmd that everyone has reversed their handedness and all Ardean

which they most resemble. Strangely, the planiversal elements quickly run out, ow­

writing appears backward.

ing to the instability of very large planiversal atoms. In the

As for space travel, another problem awaits the rocket

planiverse, one simply cannot pack as many neutrons and

voyager. There is no escape velocity in the planiverse. The

protons into a small space as one can in our universe.

amount of work required to escape the gravitational field

Consequently, nuclear forces (other things being equal)

of an isolated planet is infinite! (Try integrating

llx from 1

must act across larger distances and the nuclear compo­

to infmity.) However, if one can travel far enough to fall

nents are rather less tightly bound. Quite possibly, there is

under the gravitational influence of some other body, the

a lot more radioactivity in the planiverse than in our own.

infmite escape velocity no longer matters.

Other strange features of the planiverse include rather

The Planiverse Project had the most fun designing two­

low melting points and the strange behaviour of sound

dimensional life forms. Readers who turn back to the

waves. Low melting points might militate against the pos­

picture of the fish (Figure

sibility of life, except that chemical reactions proceed at

well-developed exoskeleton, like an insect, and with a rudi­

3) will fmd a creature with a

lower temperatures, in any event. Sound waves travel much

mentary endoskeleton, as well. The key anatomical com­

farther and have a very strange property frrst deduced by

ponent in any two-dimensional life form is the zipper or-

� H

3 9 15 25

4 10 16 26

35

36

Li

Na

K

Rb

Cs

Be

Mg X

X

X

5 11 17 18 19 20 21 31 27 28 45 40 41 42 37 1 38 e9 1

Figure 9. Planiversal table of the elements.

50

THE MATHEMATICAL INTELLIGENCER

c

Si

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

29

30

43

44

X

X

� He

6 7 8 12 13 14 22 23 24 32 47 0

F

Ne

s

Cl

Ar

X

X

46

X

Br

33

I

At

Kr

34

Xe

48

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 logi­ cal 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-di­ mensional universe to illustrate the curvature of space. For all their charm, these books have various short­ comings. Abbott made no attempt to endow his uni­ verse 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 restric­ tions of his characters' two-dimensionality, seating his characters "side by side" at a banquet, for example. Berger attempts to reconcile the two previous uni­ verses, 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 subtext) 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-of­ the-mill science fiction universe fashioned out of the whol� cloth of wish-driven imagination. The planiverse is a weirder place than that precisely because so much of it was worked out in the Planiverse Project. Reality, even the pseudo-reality of such a place, is invariably stranger than anything we merely dream up. REFERENCES Edwin A. Abbott, Flatland: A Romance of Many Dimensions. Princeton University Press, Princeton, 1 991 . Charles H. Hinton, An Episode of Flatland. Swan Sonnenschein & Co. London, 1 907. Dionys Burger, Sphere/and: A Fantasy About Curved Spaces and an Expanding Universe. Thomas Y. Crowell Company, New York, 1 965.

A. K. Dewdney. The Planiverse: Computer Contact with a Two Dimensional World. Poseidon Press (Simon & Schuster), New York, 1 984. A new edition soon by Copernicus Books (Springer Verlag),

New York, 2000.

A U T H O R

A.K. DEWDNEY Department of Computer

Science

University of Western Ontario London, Ontario N6A 587 Canada e-mail: [email protected]

A.K. (Kee) Dewdney was born in

London, Ontario, and did un­

dergraduate work there. He then did graduate work at the Universities of Waterloo and Michigan,

completing his PhD at

Waterloo in 1 97 4. His thesis, extend ing some graph-theoretic thecrerns to

h ig her 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 colum nist

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

Mfij.l§,flh£il .ilhtil [email protected]

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

D i rk H uy l e b ro u c k , Editor

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 doc­ trines (1st century B.C.-6th century A.D.). The magic was probably associ­ ated with the self-contained character of the text, which, because of the sym­ metry 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, yield­ ing a somewhat different order of words. When the text is Greek or Arabic, numerical values can be at­ tached to the letters in a straightfor­ ward manner [ 1 ] . There i s a well-preserved magic square in San Pietro ad Oratorium, a beautiful medieval church a few kilo-

I

meters south-east of the small town of Capestrano (Abruzzi, Italy). The build­ ing, as we see it now, dates from the 12th century, when a previous church, dating from the 8th century, was ex­ tensively renovated [2]. The magic square is carved in a block of lime­ stone. 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 fol­ lowing five words: ROTAS OPERA TENET AREPO SATOR. Of these, four are certainly Latin [ROTAS wheels (accusative); OPERA = work (nomi­ native or ablative); TENET = keeps; SATOR = sower (nominative)] . The remaining word, AREPO, is not Latin though it recalls the Latin word ARA­ TRO = plough (ablative). In any case, the meaning of the text remains ob­ scure. Magic squares like the one de=

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,

Figure 1 . Detail of the facade with the entrance of the church San Pietro ad Oratorium, a few

8400 Oostende, Belgium

kilometers south-east of Capestrano (Abruzzi, Italy}. The position of the block with the magic

e-mail: dirk. huylebrouck@ping. be

square is fourth from the left and seventh from the bottom. (Photographs by the authors.)

52

THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK

the four elements (earth, water, air, and fire) and the universe. REFERENCES 1 . G.R.

Cardona,

Storia

Universale

della

6-

Scrittura , Mondadori, M ilano, 1 986, pp. 6 69 and 291 . 2. M.

Moretti, Architettura

Medioevale

in

Abruzzo , De Luca, Roma, pp. 36-41 (year

of publication unknown). 3. Plato, Timaeus, LIII-LVI.

Aldo Domenicano Department of Chemistry, Chemical Engineering and Materials University of L'Aquila 1-671 00 L'Aquila

Italy [email protected] Istvan Hargittai Figure 2. The block with the magic square (in the wall it is positioned upside-down).

Institute of General and Analytical Chemistry Budapest Technical University

scribed here are often found in me­ dieval 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 popular­ ity of this magic square in the Christian tradition may have been enhanced by the Latin words PATER NOSTER, eas­ ily reconstructed using its letters [1]. ("Pater noster" (Our Father) is the be­ ginning 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 SATOR and AREPO, respectively, when read backwards. The word TENET is palindromic, as it reads the same from either end. These properties recall the duality relation­ ship of the five regular polyhedra, orig­ inating from their symmetry and the in­ terchanged roles of their vertices and faces. According to this relationship the icosahedron is the dual of the do­ decahedron, 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 cosmogo­ nic role, as they were associated with

H - 1 521 Budapest;

Hungary e-mail: [email protected]

VOLUME 22, NUMBER 1, 2000

53

MICHAEL LONGUET- HIGGINS

A Fo u rfo d Po i nt of

Con c u rre n ce Lyi ng o n th e Eu er Li ne of a Tri an g 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 twenti­ eth, largely abandoned. The present author's interest in the subject was rekin­ dled by a recent article by Hofstadter [9], which summa­ rizes the properties of the most "notable" points of a gen­ eral triangle ABC in the Euclidean plane. The cast of characters is as follows (see Fig. 1):

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 join­ ing 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 tri­ angle A 'B 'C '.

1. The circumcenter 0 of ABC is the meet of the three per­ pendicular bisectors of the sides BC, CA, and AB.

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 homo­ theticity, center M, takes A 'B 'C ' into ABC. One other notable point of ABC is sadly left out of the

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

w;t§ii;JIM

ii@ii;JIM

A

- .. -

H

B

A

_ .. 0�

_ ..

M

-

c

A'

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 contain­ ing 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 fig­ ure by considering these two segments as two of the me­ dian lines of a complete triangle. However, none of the ad­ ditional 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 in­ center I. It seemed to me that the previous cast of charac­ ters needed to be augmented. A promising candidate was a little-noticed point of concurrence related to the incen­ ter 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, be­ cause the two tangents from each vertex of the triangle are equal in length, it follows that AY · BZ · CX _____-AZ · BX · CY

=

1

B

c

A'

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 Ic. If XA, YA, and ZA denote the feet of the perpendiculars from IA, say, to the sides of ABC, it follows, by the same argu­ ment, that AXA, BYA, and CZA will meet in a point GA, and similarly for the other excenters h and Ic. 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 di­ rections. 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 pedes­ trian 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

'

and so by the converse of Ceva's theorem (see [2, p. 160]) the lines AX, BY, and CZ are concurrent. The point of con­ currence 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.

B

X

c

VOLUME 22. NUMBER 1, 2000

55

w;tgii;IIM A

; ;;

;

; ;; /

/

'

;;

/

'

'

; ' '

'

'

L

I

B

'

D

c

A'

E, '

'

,

' '

'

'

and F denote the feet of the altitudes AH, BH, and CH; see Figure 6. Then, HDCE, for example, is a cyclic quadri­ lateral; hence, LBHD = C. Therefore,

'

'

,

' '

' '

' '

'

' ' ,

,

HD = ED cot C = AB cos B cot C. I .} I ...

Because AB = 2R sin C, this gives us

lA

H = 2R( cos B cos C, cos C cos A, cos A cos B).

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 in­ side 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),

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, re­ spectively (see also [4] and [5]). What are the coordinates of the Gergonne point G? Now, from Figure 3, we see that XC = r cot(l C), and therefore 2 for X, we have 2 y = XC sin C = 2r cos (lC) 2

•

Similarly, 2 z = XB sin B = 2r cos (lB). 2 Therefore, the coordinates of G, which lies on the line AX, must also satisfy 1 + cos c

where R is the circumradius and A, B, and C are the an­ gles at the vertices of ABC. For the ortlwcenter H, let D,

1$1311;14¥

,

1 + cos B Hence, 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 D=

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.

H

56

1 (1 + {3)( 1 + y) a - {3y

THE MATHEMATICAL INTELLIGENCER

Likewise, the coordinates of the excenter IA are

IA = rA

iplijii;IJM A

( - 1, 1, 1).

In determining the coordinates of GA, 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 DA will be found to vanish in a similar way. Hence, we have proved the following theorems:

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 or­ thocenter H in the circumcenter 0. Thus, the separa­ tions of L, 0, M, 0', and H along the Euler line are in the ratios 6 : 2 : 1 : 3; see Figure 5. 1.

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:

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 me­ dian triangle (see Fig. 6). If I'A, I'B, and I'c are the excenters of the median tri­ angle and G.,.i, GiJ, and G(; are the corresponding Gergonne points, then IA.GA., I8Gf1, and leGe also pass through the orthocenter H.

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 ea + b - c), where a, b, and c are the sides of ABC. Therefore,

t

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, A YB

3. The

4.

BZe

·

AZe BXA

·

·

=

1'

M§tgam;:w #,

;

;

;

;

/

;

;

/

/

/

;

/

;

/

;

/

;

I

I

;

I

;

I

;

/

;

I

'

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], [ 1 1 ] . As before, let XA denote the point o f contact o f the excir­ cle, center IA, with the side BC opposite A, and let YB and

I;

/

'

'

'

'

' '

'

' '

' '

;

/

/

;

/

;

/

;

/

"

/

/

; 1

Is

,. 1 1 I I I

; I /

/

I I

I I I

The lines IG, IAGA, IBGB, and leGe all pass through the de Longchamps point of ABC.

Or more succinctly, 6.

CXA CYB

and by Ceva's theorem, the three lines are concurrent.

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 C1 has certain in­ teresting properties related naturally to the triangle A 1B1 C1 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.

·

'

'

,

'

'

'

'

'

' ' ' ' ' ,

I � I �

lA

VOLUME 22. NUMBER 1 , 2000

57

By similar triangles, it may be shown (see [2, pp. 161162] ) that N is collinear with I and M and that MN 2M/. Hence, N is the incenter of the triangle A 1B1C1, or N = h Now, corresponding to each excenter of ABC, say IA, we also have a Nagel point. Thus, if �. YJA, ?A are reflec­ tions of XA , YA, ZA, respectively, ?A in the midpoints A ' , B ' , C', respectively(so � = X), then A�, BYJA, C?A meet i n 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

To establish relation sider only the

(A) we need, by symmetry, to con­ x components of this equation. Thus, we

need to show only that

2R(l

-

cos B) (I

-

cos C)

i

i

i

+ 8R sin( A) sinC B) sin( C) =

i

i

However, from sinC A) = cosf CB

+

2R sin B sin C.

C) ] this last result be­ ,

comes evident. Therefore, M indeed divides the line the ratio

IN in

1:2.

In a similar way, we fmd

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,

NA

=

2R[(l + /3)(1

+

y) , -( 1 + y)(1 - a) , - ( 1

-

a)(l

+

and

where

9. NA, NB, and Nc are the three excenters of the triangle A 1B C1 . 1 An analytic proof is as follows. Our method will be to show

that

N + 21 = 3M.

(A)

A U T H O R

As earlier, the ratio of the perpendiculars from XA to the sides A C and AB is given by

y _ X AC sin C _ XB sin C _ XC sin B z XAB sin B

r cot(.!.B) sin C

2 . ' 1 r cot(-B) sin B 2

that is to say, .

Sill2(

J!... z

1

C)

2

sin2(.!.B) 2

=

1 - y 1 - f3

Hence, the trilinear coordinates of N are

N = [(1

-

{3) (1

-

y), ( 1 - y)( 1 - a), ( 1 - a)(1 - f3)Jn,

where

n is a normalizing constant, which we need to eval­ uate. Now, the trilinear coordinates of any point (x, y, z)

must obviously satisfy ax

+

cz = 2.:l,

=

2R sin A, and so forth, we find, after some use of + B + C = 7T, that

the relation A

n = 2R . r(l, 1, 1 ) of I, leads to the result

i

i

i

r = 4R sinC A) sinC B) sin( C), 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 author graduated in mathematics from Cambridge Uni­

THE MATHEMATICAL INTELLIGENCER

3 years' National Service at the

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

The same process of normalization, when applied to the coordinates

La Jolla, California 92093-0402 USA

Admiralty Research Laboratory in Teddington, where he be­

.:l = 2R2 sin A sin E sin C a

Institute of Nonlinear Science

University of California La Jolla

versity in 1 946 and did

+ by

where .:l denotes the area of ABC. Because

and

MICHAEL LONGUET·HIGGINS

faces. From 1 969 to 1 989, he was a Royal Society Research Professor at Cambridge University, commuting regularly to the I nstitute 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 .

,8) ]

and it may be verified, as before, that all three components of the equation

New York, 1 97 1 .

(B) are satisfied. Hence, M also divides the line I»VA in the ra­ tio

3. J.L. Coolidge, A Treatise on the Circle and the Sphere, Chelsea,

2 : 1, and similarly for the lines IBNB and leNc. 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 litera­ ture, not so the de Longchamps point H1. The coinci­ dence of L and H1 invites further investigation.

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), 5 1 -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.

ACKNOWLEDGMENT The author thanks an anonymous referee for helpful com­

I I . C. Nagel, Untersuchungen uber die Wichtigsten zum Dreiecke

ments.

Geh6rigen Kreise, Mohler'schen Buchhandlung im Ulm, Leipzig 1 836. 1 2. M. Simon, Uber die Entwicklung der Elementar-Geometrie im XIX

REFERENCES

1 3. G. Spieker, "Ein merkwurdiger Kreis um den Schwerpunkt des

Jahrhundert, Teubner, Leipzig, 1 906, pp. 1 24-1 41 .

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

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.

York, 1 952.

Revisit the Birth of Mathematics . . . EUCLID

w•w•tW·i· '·'

.J e re m y G ray, E d i t o r

Episodes in the Berlin· GOttingen Rivalry,

1 870- 1 9301 David E . Rowe

I

Higher mathematics at the German universities during the nineteenth cen­ tury 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 as­ piring talent not only from within the German states but from numerous other countries as well. 2 The rivalry be­ tween these two dynamos has long been legendary, yet little has been writ­ ten about the sources of the conflicts that arose or the substantive issues be­ hind 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, con­ cerns 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 better­ known figures who held academic po­ sitions in Gottingen or Berlin. As an added bonus, this leads to a very use­ ful 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 scene. Berlin's preeminent position de-

Periodization of Mathematics in GOttingen and Berlin

1801-1855

Gauss

Dirichlet

W. Weber

Steiner Jacobi

1 855-1892

Dirichlet

Kummer

Riemann

Weierstrass

Clebsch

Kronecker

Schwarz

Fuchs

Klein

1892-1917

Klein

Fuchs

H. Weber

Schwarz

Hilbert

Frobenius

Minkowski

Schottky

Runge Landau Caratheodory

rived in part from the prestige of the Prussian universities, which through­ out the century did much to cultivate higher mathematics. During the 1860s and 70s practically all the chairs in mathematics at the Prussian universi­ ties 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 tra­ dition. 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 served as a counterforce to the Berlin-

1 The following is based on a lecture delivered on 22 August 1 998 at a symposium held at the Berlin ICM.

Column Editor's address:

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

Faculty of Mathematics, The Open University,

ter 5).

Milton Keynes, MK7 6AA, England

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 1 870-by permission of SMPK Berlin - Nationalgalerie.) dominated journal founded by Crelle,

ematicians made a bid to found a na­

edited after

by Carl Wilhelm

tionwide organization. Clebsch's unex­

would have very likely been the same

Borchardt. Other leading representa­

pected death in November 1872 slowed

had not Georg Cantor persuaded Leo­

tives of this Konigsberg tradition dur­

the momentum that had been building

pold Kronecker, the most powerful

ing the 1 860s and 1870s included Otto

for this plan, but the effort was carried

and important Berlin mathematician of

Hesse,

Adolf

on by Felix Klein and other close as­

the 1 880s, to throw his support behind the venture.

Mayer.

1855

Heinrich Along

Weber, with

and

Clebsch

prevailed 20 years later, and the result

and

sociates of the Clebsch school. A meet­

Neumann they operated on the pe­

ing took place in Gottingen in 1873, but

After 1871, the Franco-Prussian ri­

riphery of Berlin and its associated

the turnout was modest and the results

valry loomed large in the minds of

Prussian network These mathemati­

disappointing. None of the prominent

many German mathematicians. That

cians had very broad and diverse in­

"Northern" German mathematicians at­

Berlin should occupy a place analo­

terests, making it difficult to discern

tended-the label "Northern" being a

gous to Paris was, for many mathemati­

striking intellectual ties. What they

euphemism within the Clebsch school

cians, merely the natural extension of

shared, in fact, was mainly a sense of

for "Prussian". Klein and his allies soon

political developments to the intellec­ tual sphere. One need only read some of

being marginalized, and they looked up

thereafter gave up this plan, as without

to Clebsch as their natural leader. With

the support of the Berliners there could

Kummer's speeches before the Berlin

the founding of the German Empire in

clearly be no meaningful German Math­

Academy-which

1871, these "Southern" German math-

ematical Society. 4 The same situation

as

its

Perpetual

Secretary he was required to deliver on

4 Further 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

mous. Later representatives-Schwarz,

Frederick the Great-in order to realize

Frobenius, Hensel, Landau, and Issai

how completely this celebrated and

Schur-saw themselves as exponents

revered mathematician identified with

of this same Berlin tradition, though

the world-historical purpose of the

they drew their main inspiration from

Prussian state and its innermost spirit,

the lecture courses of Weierstrass and

its

Kronecker.

Geist. 5 I doubt that Hegel himself

could have described that mysterious

To gain a quick, first-hand glimpse

dialectical linkage more eloquently.

of Berlin mathematics during the 1870s

This was the same Kummer who, in a

we can hardly do better than follow

letter to his young pupil Kronecker,

the letter written by Gosta Mittag­

written in 1842, urged him to attend

Leffler to his former mentor Hjalmar

Schelling's lectures in Berlin, despite

Holmgren on 19 February 1875. The

the fact that Schelling's brand of ide­

young Swede was traveling abroad on

alism

deeper

a postdoctoral fellowship that had first

Hegelian truth that "mind and being"

brought him to Paris. There he met

were

failed initially

to

grasp

one

the

and

the

same.

Charles Hermite, a great admirer of

Schelling, according to Kummer, was

German mathematical achievements

the "only world-historical philosopher

despite his limited knowledge of the

still living. "6 Kummer held the office of

language, who told him that every as­

perpetual

Berlin

piring analyst ought to hear the lec­

Academy for 14 years, 1865-1878, dur­

tures of Weierstrass. 7 Here is Mittag­

secretary

of

the

ing which time he conducted himself

Leffler's account of these and much

in a manner that won much admira­

else

tion. His role during Berlin's "golden

Prussian capital:

that

he

encountered

in

the

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-

. . . 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, ofavoiding 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 afunction 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 1 9 1 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 1 874-75; 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

arut 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

closely with Julius Plucker in Bonn, ar­

component of the Berlin-Gbttingen ri­

rived in the Prussian capital to under­

valry with his claims for the method­

take postdoctoral studies. Like nearly

ological superiority of Weierstrassian

all aspiring young Prussian mathe­

analysis over the geometric function

maticians, Klein recognized the impor­

theory of Riemann or the mixed meth­

tance of making a solid impression-in

ods of Clebsch. Still, what he wrote must

the Berlin seminar run by Kummer and

be placed in proper perspective. During

Weierstrass. Before presenting himself

Riemann's lifetime, the Gbttingen math­

as a candidate,

ematician's reputation stood very high

about five weeks to write up an im­

in Berlin, and it remained untarnished

pressive paper on a topic in his special

therefore,

he took

1866. He was elected

field of line geometry. 1 1 He then sub­

as a corresponding member of the Berlin

mitted the manuscript to Kummer,

Academy in August

1859, which gave

thereby fulfilling one of the require­

him occasion to travel to the Prussian

ments for membership in the seminar.

after his death in

capital the following month. There he

Klein's paper dealt systematically with

was welcomed by the leading Berlin

the images of ruled surfaces induced

mathematicians-Kummer, Kronecker,

by a mapping found a short time ear­

Weierstrass, and Borchardt-with open

lier by Klein's friend, Max Noether (the

arms, as his friend Dedekind, who ac­

Noether map sends the lines of a lin­

companied him on this journey, later re­

ear complex to points in complex pro­

called (Dedekind

jective 3-space ).

1892, p. 554). Weier­

strass practically worshiped Riemann, calling

him,

according

to

Mittag­

Some weeks later

Kummer returned the manuscript, and Klein soon lost interest in the topic as

Leffler, an "anima candida" like no one

well as the results he had obtained. 12

else he ever knew. 1 0

In the meantime, Klein introduced him­

His

colleague

Kronecker, to be sure, had a. far less

self to Weierstrass and Kronecker,

flattering opinion of Riemann's suc­

though he otherwise kept his distance

cessor,

from their lecture halls. This aloofness,

but

he,

too,

already passed from

Clebsch,

the

scene

November

the

in

however, did not prevent him from

subter­

asking Weierstrass for his assistance in

ranean rumblings within the German

helping him cultivate contacts with his

1872.

Thus

had

mathematical community so apparent

advanced students.

in Mittag-Leffler's letter reflected not

made it plain that he could not spare

Klein no doubt

so much personal animosities directed

the

toward Riemann and/or Clebsch but

W eierstrassian

rather the way in which their work had

ground up; what mattered to him was

become bound up in an ongoing rivalry

getting to know the "inner life" of math­

time

it

would

take

analysis

to

learn

from

the

between Berlin's leading mathemati­

ematics

cians and those associated with the

many would have dared to approach

"remnants"

school.

Weierstrass this way, but the latter

Within the latter group the most visi­

willingly obliged, suggesting that Klein

of the

Clebsch

in

Berlin.

Presumably,

not

ble figure was its youngest star, an am­

seek out Ludwig Kiepert's counselP

bitious and controversial fellow named

Their meeting marked the beginning of

Felix Klein.

a lifelong friendship which both Klein

The Berlin establishment had got­

and Kiepert came to value, and for

ten a first taste of Klein during the win­

good reason: it turned out to be one of

ter semester of

1869-70 when the 20-

the few bridges connecting members

year-old Rhinelander, who had worked

of the Berlin and Gbttingen "schools."

9Quoted in (Frostman, pp. 54-55) (my translation). For a discussion of Kovalevskaya's work, see (Cooke 1 984).

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 Universitatsbibliothek 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 .

-�

ready alerted his protege to the possi­ bility 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 inti­ mately connected with line geometry, Klein and Lie soon emerged as its two stars. Although Lie was still without a doctorate--a circumstance so embar­ rassing to him that he introduced him­ self 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 suc­ cess sparked their intense collabora­ tion, which began with a sojourn in Paris during the spring of 1870 and lasted un­ til 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 fol­ lowed, however, their interests drifted apart, though they continued an avid correspondence. Returning now to our main theme, the first overt signs of struggle be­ tween Klein and Berlin came in the early 1880s when Klein was Professor of Geometry in Leipzig. Six years after Mittag-Leffler had given his Figure 2. The first page o f Klein's untitled and unpublished manuscript o n 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 1 3A.

private

description of how Berlin mathemati­ cians assessed the drawbacks of a geo­ metrically-grounded theory of com­ plex functions, this issue was taken up by Klein in a public forum. Klein's re­ marks were prompted by a priority dis­ pute

with

the

Heidelberg

analyst,

Klein made other significant con­

sonality captivated him so completely.

Lazarus Fuchs, a leading member of

tacts in Berlin, but mainly with other

As a backdrop to future events, a few

the Berlin network (he took a chair in

outsiders like the Austrian Otto Stolz,

words must be said with regard to the

Berlin in 1884).

from whom he learned the rudiments

Klein-Lie collaboration. 14 Like Klein,

As in so many priority

claims in mathematics, the issues at

of non-Euclidean geometry. By far the

Lie was an expert on Pliickerian line

stake here were far more complicated

most significant new friendship was

geometry, and thus someone Klein

than might first meet the eye. Partic­

the one Klein made with Sophus Lie,

knew by reputation beforehand. In

ularly interesting were the interna­

that Nordic giant whose ideas and per-

fact, Klein's mentor, Clebsch, had al-

tional

1 4For more on Klein and Lie, see (Rowe and Gray).

64

THE MATHEMATICAL INTELLIGENCER

dimensions

of

the

conflict,

Figure 3. A contemporary sketch of the then-new Auditorium building in Gottingen.

including the part played by Mittag­ Leffler, 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 Rendus of the Paris Academy in which he named a special class of complex functions, those invariant un­ der a group with a natural boundary cir­ cle, "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 rel­ evant "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

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 mean­ time, 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 con­ tributions failed to grasp the funda­ mental ideas, which required the notion of group actions on Riemann surfaces (see (Rowe 1992)). Klein's brilliant stu­ dent, Adolf Hurwitz, apparently enjoyed this feud, especially his mentor's at­ tacks 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 net­ work 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 mathe­ matics 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 for­ eigner, Sophus Lie, as his successor in Leipzig.16 Thereafter, both Lie and Klein were scorned by leading Berlin­ ers, particularly Frobenius. According to Frobenius, Weierstrass made it

1 5For details. see (Gray 1 986, pp. 275-31 5). 1 6For details, see (Rowe 1 988, pp. 39--40).

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 com­ mon front with his Leipzig colleagues Lie and Adolf Mayer, but Lie became increasingly wary of this manuevering aimed mainly at enhancing Klein's per­ sonal power. When Kronecker suddenly died in December 1891, Weierstrass could fi­ nally 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 at­ tack from a decade earlier, Fuchs merely added that he had nothing against Klein personally, only his per­ nicious effect on mathematical sci­ ence.l8 Thus, Schwarz got Weier­ strass'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 lim­ iting 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

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 in­ creasingly embittered by the way he was treated by his Leipzig colleagues and certain allies of Klein, who re­ garded him mainly as one of Klein's many subordinates. 2 1 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 trans­ formation groups. To clarify his rela­ tionship 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 terri­ bly good at doing. In retrospect, the de­ cisive turning point was clearly Hilbert's appointment in December 1894, a goal Klein had long been plan­ ning. Mathematically, Klein and Hilbert complemented one another beauti­ fully; moreover, both shared a strong antipathy for the Berlin establishment, which they considered narrow and au­ thoritarian. Whereas Klein tried to ad­ vance a geometric style of mathemat­ ics rooted in the work of Riemann and Clebsch, Hilbert championed an ap­ proach to abstract algebra and number theory that was largely inspired by ideas first developed by Dedekind and Kronecker. With regard to founda­ tional 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 mathe­ matical knowledge:

NUMBER

FIGURE

Arithmetic

Euclidean Geometry

Algebra

Projective Geometry

Analysis

Higher geometry

Analytical Mechanics

Geometrical Mechanics

Analytic & Differential Geometry

This bifurcated scheme, I would argue, portrays how most mathematicians in Germany saw the various components of their discipline during the late nine­ teenth 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 re­ flect the high status accorded to pure mathematics, especially number the­ ory and synthetic geometry, by many influential German mathematicians. The Berlin tradition of Kummer, Weier­ strass, 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 Weier­ strass taught geometry after Steiner's death in 1863 in an effort to sustain the geometrical component of Berlin's cur­ riculum). Still, as we have seen, pure mathe­ matics, for Weierstrass, mainly meant analysis, and the foundations of analy­ sis derived from the properties of numbers (irrational as well as ratio­ nal). 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 alge­ bra. In other words, Kronecker wished to ban from rigorous, pure mathemat­ ics 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 -·-·-·

___

1 7See (Biermann 1 988, p. 2 1 5).

1 8See (Biermann 1 988, p. 305-306). 1 9For details, see (Rowe 1 986, pp. 433-436). 20See (Young 1 g2o. p. /iii). 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

Kronecker and Weierstrass that se­

rather low rung in the purists' hierar­

came to Berlin in 1914 on a special ap­

verely paralyzed Berlin mathematics

chy of knowledge.

pointment that included membership

during the 1880s and beyond, right up until Kronecker's death.

and

which

occupied

a

Einstein's relativity theory. Einstein

From an institutional standpoint,

in the Prussian Academy. Just one year

we can easily spot other glaring con­

later the Gottingen Scientific Society

This is not the place to go into de­

trasts between Gottingen and Berlin

offered him a corresponding member­

tails about how Gottingen quickly out­

during the era 1892 to 1917. Whereas

ship, elevating him to an external mem­

stripped Berlin during the years that

Frobenius and Schwarz largely saw

ber in 1923. Ironically enough, general

followed, but we should at least notice

themselves as defenders of Berlin's

relativity

that part of this story concerns a very

purist legacy, Klein and Hilbert pro­

closely within Gottingen circles, as

different vision of this "inverted tree"

moted an open-ended interdisciplinary

well as by the Dutch community sur­

of mathematics, a vision shared by

approach that soon made Gottingen a

rounding Paul Ehrenfest, than it was in

As I have argued else­

far more attractive center, drawing in­

Einstein's immediate Berlin surround­

common outlook helps ex­

ternational talent in droves. A glimpse

ings. Klein, Hilbert, Einstein, and Weyl

of

in

were friendly competitors during the period 1915-1919 (see (Rowe 1998b)).

Klein and Hilbert. where,

this

plain how they managed to form such a

this

multi-disciplinary

style

was

followed

far

more

successful partnership in Gottingen de­

Gottingen can be captured merely by

spite their apparent differences. 22 Both

looking at some of the appointments

In Gottingen, the work of Klein and

were acutely aware of the possibilities

Klein pushed through with the support

Hilbert on GRT was supported by sev­

for establishing a linkage between the

of the Prussian Ministry and the fman­

eral younger talents, including Emmy

two principal branches of the tree.

cial assistance of leading industrial

Noether, whose famous paper on con­

Indeed, both made important contribu­

concerns: Karl Schwarzschild (astron­

servation laws grew out of these ef­

tions toward securing these ties (Klein

omy),

forts.

through his work on projective non­

Ludwig Prandtl (hydro- and aerody­

Euclidean geometry; Hilbert with his

namics),

arithmetical characterization of the

mathematics, numerical analysis).

Emil and

Wiechert Carl

(geophysics),

Runge

(applied

In

the case of Hilbert and Einstein,

we can also observe a strong affinity in their insistence on the need to uphold

Symptomatic of the Gottingen style

international scientific relations and to

by a proof of the consistency of its ax­

was an interest in physics, both classical

resist those German nationalists who

ioms). Noteworthy, beyond these con­

and modem. Arnold Sommerfeld, Max

supported the unity of Germany's mili­

tributions, was their background in

Born, and Peter Debye interacted closely

tary and intellectual interests. Thus, the

and familiarity with invariant theory, a

with Klein, Hilbert, and Minkowski, all

controversial pacifist and international­

field that was particularly repugnant to

of whom were deeply interested in

ist, George Nicolai, enlisted the support

continuum, which he hoped to anchor

22For more on the Hilbert-Klein partnership, see (Rowe 1 989).

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 normal­ ized 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, over­ came this effort by organizing a dele­ gation 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 mis­ understanding to construct differences or even contrasts according to peoples or human races . . . mathematics lrnows no races . . . . For mathematics the en­ tire 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 com­ munity was still barely formed. More striking, however, is the clash with Bieberbach's vision, which asserted that mathematical style could be di­ rectly 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 is­ sues of pure vs. applied and national allegiance vs. international coopera­ tion most certainly are not. The pres­ ent ICM in Berlin represents not only an opportunity for mathematicians to gather and celebrate recent achieve­ ments but also to reflect on the role of mathematics and its leading represen­ tatives of the not so distant past, draw­ ing whatever lessons these reflections may offer for mathematics today.

1 7 -Mathematik

Johannes Gutenberg University 55099 Mainz

e-mail: [email protected]­

Kurt-A. Biermann, Die Mathematik und ihre Dozenten

1 8 1 D-1933

an

der

Berliner

mainz.de

Universitat,

(Berlin: Akademie Verlag, 1 988).

After studying topology under Leonard

Roger Cooke, The Mathematics of Sonya

Rubin at Oklahoma, David Rowe be­

Kovalevskaya (New York: Springer-Verlag,

gan work on a second doctorate in

1 984). Richard

history of science with Joseph Dauben

Dedekind,

"Bernhard

Riemann's

at CUNY's Graduate Center. During

Lebenslauf," in H. Weber, ed. , Bernhard

the academic years 1 98H5 he was

Riemann's

Mathematische

a fellow of the Alexander Humboldt

Werke, 2nd ed. (Leipzig: Teubner, 1 892), pp.

Foundation in Gottingen, where he

Gesammelte

combed local archives studying the

539-558. Albert Einstein, "Cosmological Considerations

lives and work of Klein and Hilbert

on the General theory of Relativity," English

Since then, the Gottingen mathemat­

trans. of " Kosmologische Betrachtungen zur

ical community has been the main fo­

allgemeinen Relativitatstheorie" (191 7) in A

cus of his research. In 1 992 he was

Sommerfeld, ed., The Principle of Relativity

appointed Professor of History of

(New York: Dover. 1 952).

Mathematics and Exact Sciences at

Albrecht Folsing, Albert Einstein: A Biography, (New York: Viking, 1 997).

Mainz University. In recent years he has become increasingly interested in

Otto Frostman, "Aus dem Briefwechsel von G .

the interplay between mathematics

Mittag-Leffler," Festschrift zur Gedachtnis­

and physics, particularly relativity the­

feier fOr Karl Weierstra/3, 1 8 1 5- 1965, ed. H.

ory. Since 1 998 he has been a con­

Behnke and K. Kopfermann, Koln: West­

tributing

deutscher Verlag, 1 966, pp. 53-56.

Papers Project at Boston University.

Lars Garding, Mathematics and Mathemati­ cians: Mathematics in Sweden before 1950.

editor

with

the

Einstein

Here he is shown with his son Andy on vacation in Fife Lake, Mi chigan .

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 Alg eb ras 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). 25Quoted in English translation in (Reid 1 970, p. 1 88). 26See the portrayal of Bieberbach's political transformation in (Mehrtens 1 987).

THE MATHEMATICAL INTELUGENCER

DAVID E. ROWE Fachbereich

Germany

REFERENCES

24For more details on this collaboration, see (Pais 1 982, pp. 487-488).

68

AUT H OR

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),

Research

Community,

56-65.

Sylvester,

Felix Klein,

E. E. Kummer, " U ber die akademische Freiheit. Eine Rede, gehalten bei der U bernahme des Rektorats der O niversitat 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

1 8 76- 1 900. and E.H.

J.J.

Moore,

History of Mathematics, vol. 8 (Providence: American

Mathematical

Society/London

--

. "Der Briefwechsel Sophus Lie- Felix Beziehungen,"

NTM.

Schriftenreihe fOr Geschichte der Naturwis­ senschaften,

Science and the Life of Albert Einstein ,

Oxford: Clarendon Press, 1 982.

. "Mathematics in Berlin, 1 8 1 0-1 933" in

Mathematics in Berlin , ed. H.G.W. Begehr,

J. Thiele, Basel: Birkhauser, 1 998, pp. 9--- . "Einstein in Berlin, " in Mathematics in Kramer, N. Schappacher, and E.-J. Thiele,

(Washington: The Mathematical Association

Abraham Pais, 'Subtle is the Lord . . . ' The

--

Berlin , ed.

of America, 1 987), pp. 1 95-241 .

matica 50(1927), I-XXIII.

(Basel:

G6ttingen in the Era of Felix Klein," Isis ,

wissenschaftlichen

N. E. N6rlund, "G. Mittag-Leffler," Acta Mathe­

Scriba,

77(1 986), 442-449.

History of Mathematics, ed. Esther Phillips,

1 33-1 98.

Christoph

26.

Verlag, 1 970). David E. Rowe, " 'Jewish Mathematics' at

Klein, eine Einsicht in ihre pers6nlichen und

Kowalewsky, " Acta Mathematica 39(1923),

and

Birkhauser, 1 992), pp. 598-6 1 8.

H. Koch, J. Kramer, N. Schappacher, and E.­

Mathematical Society, 1 994). Constance Reid, Hilbert (New York: Springer­

'Deutsche Mathematik, ' " in Studies in the

G6sta Mittag-Leffler, "Weierstrass et Sonja

Rowe,

Technik und Medizin,

25(1 )

.

"Klein,

Hilbert,

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

(1 988), 37-47. --

H.G.W. Begehr, H. Koch, J.

and the G6ttingen

Mathematical Tradition," Science in Ger­ many: The Intersection of Institutional and

ematical commentary), New York: Springer­ Verlag, forthcoming. Renate Tobies and David E. Rowe, eds.

Intellectual Issues, ed. Kathryn M. Olesko

Korrespondenz

(Osiris, 5, 1 989), 1 89-2 1 3 .

Teubner Archiv zur Mathematik, Band 1 4,

--

. "Klein, Mittag-Leffler, and the Klein­

Poincare Correspondence of 1 88 1 -1 882,"

Felix

Klein-Adolf

Mayer,

(Leipzig: Teubner, 1 990). W. H. Young, "Adolf Hurwitz," Proceedings of

Karen Parshall and David E. Rowe, The

Amphora. Festschrift fur Hans Wussing, ed.

the London Mathematical Society, Ser. 2,

Emergence of the American Mathematical

Sergei Demidov, Mensa Folkerts, David E.

20(1 920), xlviii-liv.

VOLUME 22, NUMBER 1, 2000

69

ld§'h§l.lfj

J et Wi m p ,

Editor

I

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

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.

70

''

M

y first assumption about mathematics: It's something people do." This is the sentence (p. 30) that stands out as the author's credo. He elaborates on this considerably, through­ out the book, and his purpose is to use this as his philosophy of mathematics, which he aptly terms humanism. More­ over, 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] in­ sists [that] that one limited aspect is mathematics. " Indeed, such one-sided­ ness certainly seems contrary to hu­ manism, 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 impossibil­ ity of contact between the flesh and blood mathematician and the immate­ rial mathematical object. . . . " Hersh also has things to say about the effect on teaching of taking Platonism too se­ riously. 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: 'Mathe­ matics is a meaningless game.' " Put that way, math does seem cold and in-

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 for­ malism 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 exer­ cise to disallow various axioms, ex­ plore other worlds, and see what we can still salvage (also, to distinguish between constructive and non-con­ structive 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 ex­ istence 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 "phi­ losophy.'' 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, ques­ tions, 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 for­ malization and construction are es­ sential features of mathematics. But the philosophies of formalism and con­ structivism are long-standing rival

schools. It would be more productive

there were many good explanations in

I have written, for example, about "lit­

to see how formalization and con­

the "Mathematical Notes" at the end,

tle x's and y's crawling about like

struction interact than to choose one

there were still missing links (for ex­

frightened insects. " I had math dreams

and reject the other."

ample, calculus is not

Indeed, there are many passages

only distance,

and wrote math poems; I still do. Math

velocity, and acceleration), a few inac­

was always very human to me, but it

where Hersh pokes fun at rival philoso­

curacies (such as p. series and pp.

285 on Fibonacci 309-10 on distribution

wasn't, back then, · a matter of inter­

phies of math. I'm hard put to select my favorites. We might begin with two

theory),

unless

I'm misunder­

man beings. Quite the opposite; math

section titles: "Neoplatonists-Still in

standing, a couple of gross misprints

was too precious a thing for this over­

102) and "Beautiful Idea­ Didn't Work" (p. 159). We might then move on to p. 1 10: "Philosophy stu­

on p.

3 13. (In paragraph 4, line 2, x should be n, and in line 3, X should be

sensitive adolescent to even speak of.

x). There were also some passages

hanced by sharing it with others, but I

dents are supposed to read Descartes'

which seemed fallacious, or perhaps

still don't believe that this sharing is

(1637). They

Heaven" (p.

'Discourse on Method'

and,

acting mathematically with other hu­

With maturity I learned that math is en­

just confusing. Sometimes I had trou­

the only thing that makes math "a hu­

complete

ble figuring out when he was being

man endeavor. "

'Discourse' includes Descartes' mathe­

facetious and when not. While reading

In a simple, honest, and moving pas­

matical masterpiece, the 'Geometry'.

obscure passages I couldn't help think­

sage Hersh says, "this book is written

don't

realize

that

the

. . . On the other hand, mathematics

ing, in an almost fond (and I hope hu­

out of love for mathematics." Even bet­

students also are miseducated. They're

manistic!) way, "Ah, well, he is, after

ter is the preceding sentence: "In at­

supposed to know that Descartes was

all, a philosopher. . . . "

tacking Platonism and formalism and

that his 'Geometry' was part of a great

What's Meant by Humanistic?

right to do mathematics as we do." Still,

work on philosophy."

By calling math "humanistic" Hersh

I kept noticing passages where I wanted

seems to mean at least four main con­

him to be

tentions, all corollaries of math being

more sensitive and perceptive, and I

"something people do":

think there are some humanistic oppor­

a founder of analytic geometry but not

P. nism:

138, skeptically describing Plato­ "These

[mathematical]

objects

aren't physical or material. They're out­

neo-Fregeanism,

I'm

defending

our

more humanistic, meaning

side space and time. They're immutable.

(1) Math, or our body of knowledge

They're uncreated. . . . Mathematicians

of math, including what is "currently

are empirical scientists, like botanists."

fashionable" in math, changes with

devotes considerable negative energy to

time.

the "most influential living philosopher,

And formalism:

"A strong version says

that there are rules for deriving one for­

(2) The above is also a function of

mula from another, but the formulas

place-of

aren't

stances, and so on. That is, math is po­

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

culture,

societal

circum­

litical. That is, mathematicians are fallible.

(4) Mathematicians interact with

ematics [italics mine] , and the least

one another. That is, math is "some­ thing people do" On p.

For example, from pp.

170 to 176 he

W.V.O. Quine"; a lot of this is amusing and

probably

warranted.

together. 23 Hersh makes what he con­

170:

P.

"Quine's most famous bon mot is his de­ fmition of existence. 'To be

(3) The above contains mistakes.

trustworthy branch at that." All these passages are certainly re­

tunities that Hersh misses.

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

certain

siders his most controversial point.

or not to be the value of a variable.' I Or

other passages, I found myself turning

"There is no need to look for a hidden

Descartes'

the skepticism back on him. P. xii: "The

meaning or defmition of mathematics

fore I am the value of a variable.' "

freshing.

However,

reading

Meditations: 'I think, there­

book has no mathematical or philo­

beyond

social-historic-cultural

Such bantering, which Hersh often

sophical

meaning. " Indeed, "biology is destiny,"

indulges in, is fun and often well-taken.

As Hersh says, 17, "our mathematical ideas . . .

prerequisites."

Maybe

not

its

technically. But to get the flavor of the

even as regards math.

Still,

book, and virtually any paragraph in it,

on p.

quote. No, it makes no apparent literal

one needs an interest and

I

liked that particular

Quine

some back­

match our world for the same reason

sense, nor philosophical nor mathe­

ground in math, if not philosophy. On

that our lungs match earth's atmo­

matical, but it struck me as having

sphere."

some sense-psychological, perhaps,

p.

29 he reiterates, "This book aims to

be easily comprehensible to anyone. If

Still, I feel that it's not only the do­

or poetic-or "shock value"? I am cur­

some allusion is obscure, skip it. It's

ing of math with other human beings

rently trying to write a poem about it.

inessential!"

that makes it humanistic; it's also the

Another example: Hersh seems to

math itself. This I have experienced.

take a dim view of Kant's inquiry, "How

By

those

instructions,

most of the book is inessential! I feel that the book could be shorter

When ninth-grade algebra hooked me,

is mathematics possible?" "It's a futile

and better organized, perhaps with

it never occurred to me that math

question" is the title of the section in

Part II on the history of the philosophy

might

which he deals with this, and he could

of math coming first. Also, although

not be humanistic. Math struck an emotional chord in me. Elsewhere

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.

"We're here to worship the

What is startling is that persistent er­

question itself," and question-worship­

ror is often knowingly used by dynami­

wrote,

However, is "futile" the same as "un­

ping seems to be one of the things

cists. Green [ 1 ] carried out famous stud­

worthy"? Just because we don't know

mathematicians are destined to do. We

ies of the "standard map" in the early

how to answer, or pose, a question

could fare worse.

1980s. This area-preserving map was the simplest example for showing regions of

doesn't mean that question is merely "amusing," as Hersh says. Kant's ques­ tion seems interesting, in part because

Department of Mathematics and Computer

"chaos." The phase portrait was known to have an infinity of unstable saddle pe­

Science

it leads to other questions (such as

Drexel University

riodic orbits and to display instability,

"How can mathematics

Philadelphia, PA 1 9 1 04

and yet the pictures one obtained of

not be possi­

ble?" "What do we mean by possible?"

chaotic regions were repeatable and

USA

and of course, "What is mathematics?"). "This much is clear," says Hersh (p. 21): "Mathematics

seemingly machine-independent. Green remarked that the errors in the iterative

Dynamical Systems and Numerical Analysis

process of the standard map were along

is possible. . . . 'What is happening can happen.' " True, but that says only that it can happen, not why or how it happens.

by A. M. Stuart and

although the orbits were not repeatable

Hersh's attitude here brings to mind a

A. R. Humphries

the regions took on similar shapes as

the boundaries of the chaotic regions and not transverse to it.

boundary changes were limited. Similar

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

observations can be made of the various

CAMBRIDGE: CAMBRIDGE UNIVERSITY PRESS, 1 998 US $64.95, ISBN-0-521 -49672-1 (HARDBACK)

chaotic

US $39.95, ISBN 0-521 -64563-8 (PAPERBACK)

Really- Really?

attractors,

for

example

the

Henon, Lorenz, and Rossler attractors.

REVIEWED BY DAVID ARROWSMITH

So the dynamicist is often provided with the clearest reasons for ignoring dy­

spirit of math.

What is Mathematics,

This meant that

A

study at the interface between

namical error.

two areas often provides some

Another classic case is the control

surprises. Interdisciplinary studies are

of chaos (see Ott, Grebogi, and Yorke

Hersh begins the Preface to his book by

now all the rage.

of

[2]), where a chaotic orbit is used to

connecting it with his own odyssey, in

course, that deep knowledge and even

sample a chaotic region until the orbit

a beautifully honest way: "Forty years

language in one area can provide un­

arrives sufficiently close to the target,

ago, as a machinist's helper, with no

expected approaches and spin-offs in

say an unstable periodic point. A new

thought that mathematics could be­

the other. There have been many such

regime is then imposed to obtain a lo­

come my life's work, I discovered the

cases

systems

cal control to target. The accuracy of

classic,

over the last few decades: algebraic

the chaotic orbit is not essential pro­

What is Mathematics? by

involving

The hope is,

dynamical

Richard Courant and Herbert Robbins.

and number-theoretic techniques in

vided it does the job of numerically

They never answer their question; or

dynamical systems; nonlinear analysis

sampling the target region. In fact,

rather, they answered it by

of heart arrhythmias; the use of non­

many attractors that we refer to as

showing what mathematics is, not by telling

linear modelling to provide effective

"chaotic" are not known to be chaotic

what it is. After devouring the book with

control of lasers. So we should expect

but are numerically chaotic. The orbits

wonder and delight, I was still left ask­

good things when dynamical systems

are restricted to a compact region, they

ing, 'But what is mathematics, really?' "

and numerical analysis come together.

exhibit local sensitive dependence on

book "with wonder and delight," am

In some senses they are inextrica­ bly linked. At its base level, a dynami­

initial conditions, and the regions seem

left asking, "But what is mathematics,

So now I, after "devouring" Hersh's

to be sampled by many "dense" orbits.

cal system is an iterative process, and

It is with this dysfunctional view of

really?" Surely the credo quoted at the

whenever we consider iterations in­

numerical experiments that I come to

beginning of this review-"mathemat­

volving real numbers, numerical simu­

review the book "Dynamical Systems

ics is something people do"-was not

lations inevitably involve a truncation

and Numerical Analysis." One's frrst as­

intended by Hersh as a defmition.

of the real numbers to a fmite number

sumption is that the book will be two

of decimal

Thus numerical

sections sewn together-well, there are

all people do. Also, mathematicians do

problems immediately arise: what is

two sections, but the good news is that

other things besides math.

the significance of the repeated round­

they are interwoven! Certainly, the in­

off in iterations? Conversely, numeri­

troduction is reassuring to the dynami­

Mathematics is not something that

Hersh certainly brings us closer to

places.

knowing what mathematics is, and he

cal analysis has its own dynamical al­

cal systems specialist. The examples

thoroughly describes and makes us be­

gorithms. It is surprising, given this

and the language are totally recognized

lieve in the ubiquitous and welcome as­

intimacy, that experimentalists in dy­

at once, and the belief starts to grow

pect of humanism in mathematics,

namical systems will often seek nu­

that this is the book for the dynamicist

even if the question "What is mathe­

merical solutions and not worry too

who is ignorant of NA. The frrst few

matics?" remains. The poet Anne Sexton

much about the errors that arise!

chapters cover all the basic ingredients

72

THE MATHEMATICAL INTELLIGENCER

of dynamical systems, limit sets, stabil­

stable and stable manifolds of simple

ity and bifurcation, period doubling,

differential systems and how the dis­

What other motivation would lead so

chaos, invariant manifolds, attractors,

cretized

many of us, professional and amateur,

system

has

approximating

to a large number of mathematicians.

global features. The first chapter covers

manifolds. These observations are then

to spend immense amounts of time

maps and the second ordinary differen­

extrapolated to more general results

worrying about whether we could color

tial equations. The section titles are vir­

where appropriate. Similarly, the au­

maps with four colors, when any car­

tually identical except for the necessary

thors

consider near preservation of

tographer could have told us that with

change from area-preserving maps to

limit-set behaviour for contractive sys­

the possible exception of the Red Sea,

Hamiltonian systems.

tems and gradient systems. This devel­

bodies of water are colored blue? And,

opment terminates in clean statements

furthermore, that it would require con­

The book is strewn with examples to test the reader's understanding. The remaining six chapters are de­ voted

to

numerical

methods.

The

about upper semi-continuity properties

siderably less effort to provide each

for the distance between an attractor

map-maker with a couple of extra bot­

and its multi-step approximants.

tles of colored ink Mathematicians may not all phrase their ambitions at

Runge-Kutta and multi-step methods

The book fmishes by addressing the

are treated first. The discussion in­

corresponding numerical problems of

the

volves truncation error, order, and fi­

Hamiltonian systems where the dis­

when faced with intellectual moun­ tains they have a strong urge to climb

level of Hilbert's program, but

The chapter

cretizations have to take a symplectic

ends with stiff systems and stability.

form to retain the conservational as­

them. When the mountains look like

Crucially, the authors do not abandon

pects of the dynamics.

molehills, mathematicians are tempted

nite-time convergence.

the style of the first two chapters and

The real strength of this book is that

to make remarks like the famous com­

retain this reader's confidence. To re­

the numerical analysis is described by

ment of Minkowski about the Four­

inforce this, the fourth chapter is enti­

authors who are sympathetic to the

Color Problem, that the problem had

tled "Numerical methods as dynamical

qualitative aspects of dynamical sys­

not been solved because no first-rate

3

tems and therefore make the numeri­

mind had attacked it. But each attack

are revisited. It is shown that various

cal

that fails increases the value of the

Lipschitz

than most of the traditional texts I have ' seen on Numerical Analysis.

systems." The techniques of chapter conditions

allow

Runge­

Kutta methods to be viewed as dy­

medicine

much more palatable

prize. Rudolf Fritsch is Professor of Math­ ematics Education at the University of

namical systems. The authors discuss structural assumptions for this to oc­

REFERENCES

Munich. The book grew out of his

cur; linear decay, one-sided Lipschitz

[1 ] Green, J. M . , A method for determining sto­

thoughts "about how one could make

(6), 1 1 83-1 201 1 979. ' [2] Ott, E., C. Grebogi, and J. A. Yorke,

Four-Color Problem more accessible

Controlling Chaos, Phys. Rev. Lett. 64 (1 1 ) ,

wife Gerda provided the historical

1 1 96-1 1 99, 1 990.

(first) chapter of the book

conditions,

dissipative systems, and

gradient systems. The dynamical systems theory comes back with full force from chapter

5 on­

wards-numerical techniques go global!

chastic transition. Journ. Math. Phys. 20

is the geometri­

mathematical

workings

of the

to students and professors alike." His

This begins with an introduction to

One of the great benefits of global dy­ namical systems theory

the

the mid-nineteenth-century mathemati­

School of Mathematical Sciences

cal insight that it affords and the ability

Queen Mary & Westfield College

cians who first considered a question

it gives us to see key orbital features that

University of London

from a student who had successfully

give the system its prime characteris­

Mile End Road, London E1 4NS

colored a map

tics-for example, the types and struc­

England

England with four colors and won­

ture of attractors that can occur.

e-mail: D. K.Arrowsmith@qmw .ac. uk

of the counties of

dered whether this could be done with

The asymptotic behavior of a dy­

all maps. Next, the volume introduces

namical system is given by its u.rlimit

Arthur Kempe, a lawyer and fine ama­

sets. Thus it is necessary to know the

teur mathematician whose fame rests

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-

The Four-Color Theorem by Rudolf and Gerda Fritsch

1879, one of the

most clever and insightful incorrect proofs in the history of mathematics. It then introduces Percy Heawood

Translated by Julie Peschke

who, as a young man in

NEW YORK: SPRINGER-VERLAG, 1 998

1890, not only

demolished Kempe's "proof' as one re­

US $29.00, ISBN 0-387-98497-6

sult in a tremendously impressive pa­

REVIEWED BY KENNETH APPEL

W

upon publishing, in

per, but also generalized the problem from its setting in the plane to arbitrary

hen asked why he wanted to

surfaces and proved the appropriate

climb Mount Everest,

sufficient conditions in every setting

George

Mallory replied, "Because it is there."

except the one in which the question

That response made him a soul-mate

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 sub­ ject 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 al­ most 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 computa­ tions were feasible. We are introduced to one of the great amateur mathe­ maticians 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 sub­ jects 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 partici­ pated in the fmal trip to the summit along with John Koch, who was a grad­ uate 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 de­ veloped in the search for a proof. Chapter 2 presents a careful intro­ duction to the topological background that permits a precise statement of the four-color theorem. Chapter 3 intro­ duces the inductive plan of attack and points out some of the standard sim­ plifications. In Chapter 4 the problem is phrased in the language of graph the-

74

THE MATHEMATICAL INTELLIGENCER

ory. In most of the presentation, as pre­ viously indicated in Chapter 2, the au­ thors use the word graph for what is usually known as a plane drawing of a planar graph. Dual graphs, which per­ mit one to rephrase the problem in a form much easier to work with, are in­ troduced 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 intro­ duced by vertex inflation. This re­ viewer prefers the approach of bring­ ing in the dual somewhat earlier, for reducing the degrees of regions to ob­ tain a triangulation of the dual by adding edges to regions of degree higher than three seems much more in­ tuitive. 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 ver­ sion 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 de­ scribes 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 defini­ tion 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 al­ gorithms that provide unavoidable sets (not consisting entirely of reducible configurations). In an apparent misun­ derstanding, the authors refer to Appel's discharging algorithm. No such beast ever existed. Indeed, the dis­ charging procedure used in the origi­ nal proof of the four-color theorem can best be thought of as a sequence of dis­ charging algorithms, each algorithm designed to overcome a flaw in its pre­ decessor, and was the major contribu-

tion of the joint provers of the theo­ rem. The error is understandable, since the proof of the theorem that was pub­ lished 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 mathe­ matically capable undergraduate. Both the mathematical and historical mate­ rial is clearly written and the English translation (by Julie Peschke) pre­ serves the clarity of the Fritsches' ex­ position. University of New Hampshire Kingsbury Hall Durham, NH 03824 USA e-mail: [email protected]

Special 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

O

ccasionally 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 reveal­ ing 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.

analytic continuation. Also clearly pre­ sented are newer developments, such

6. The material is important to a broad spectrum of readers.

5. The material has applications to a wide range of both mathematical

gamma and q-beta functions and theta

as the connections of contiguous rela­

functions are introduced. Applications

tions with the recent summation meth­

to sums of squares are given. Here the

ods of R. W. Gosper, and H. Wilf and

work of S. Ramantijan, M. D. Hirschqom, and S. C. Milne might have been men­

D. Zeilberger.

4 on Bessel functions is

Chapter

and scientific disciplines.

4.

q-integrals is the best I have read. The q­

tioned. Some of the fundamental theo­

The level of exposition is accessi­

shorter than one might expect. Besides

rems on basic hypergeometric series are

ble to beginning graduate students

many standard theorems, monotonic­

presented. Previously,

and to some well prepared under­

ity properties are also established.

Andrews's

Chapter 2

in

The Theory of Partitions

5 provides an excellent,

was the best introduction to q-series

well-motivated introduction to orthogo­

before embarking on the more ambi­

further important facets of the

nal polynomials, with the Chebyshev

subject.

polynomials and trigonometric func­

tious treatise, Basic Hypergeometric Series, by G. Gasper and M. Rahman.

Chapter

graduates.

3. Challenging exercises bring out

2. The book deals with special func­ tions. 1. The book generates

e-special-ly

outrageous puns from at least one

tions as the motivating forces. Gaussian

Now I will tell my students to read first

quadrature,

Chapter 10 of the book under review.

continued fractions,

and

Partition analysis, the topic of Chap­

moment problems are shown to be nat­ ural outgrowths of the general theory. Special instances of orthogonal poly­

reviewer.

ter 1 1, has not been treated in text form since P. A. MacMahon developed it in

nomials are the topic of Chapter 6, with

his

the Jacobi polynomials playing the lead­

indeed partition analysis has been un­

ters are delineated below, it will be

ing role.

the past few decades, or­

fairly neglected. A number of elemen­

made manifest that all of the first nine

thogonal polynomials have had promi­

tary theorems about generating func­

listed criteria are satisfied. However, if

nent applications in combinatorics, and,

tions for certain partition functions are

mathematicians

in particular, matching polynomials are

derived in an easy, painless way. The

As the topics in the book's 12 chap­

had

followed

Paul

Turan's advice that the functions ad­ dressed in this treatise be called

extensively examined in

this

chapter.

7 offers some beautiful

Chapter

use­

ful functions, then criterion number 1

In

topics wherein orthogonal polynomi­ als arise. The positivity of coefficients

would likely not be satisfied.

famous

Combinatory Analysis;

chapter ends with proofs of Ramanu­ jan's congruences modulo

5 and 7 for

the ordinary partition function. The powerful

method

of Bailey

Many books in analysis and special

in the power series expansions of cer­

pairs has also not been heretofore ex­

functions have sections or chapters on

tain rational functions, a particularly

amined at any length in textbooks.

the gamma function.

the

favorite topic of one of the authors, is

Several applications are given; in par­

present authors' beginning chapter on

nicely introduced. The positivity of

ticular, the second of

the gamma and beta functions is espe­

certain

proofs of the Rogers-Ramantijan iden­

However,

polynomials,

trigonometric,

both the

ordinary

cially elegant. Kummer's Fourier ex­

and

pansion of log f(x), Dirichlet's multi­

Master Theorem,

MacMahon

ple integral, Gauss and Jacobi sums

proof of the irrationality of

(the finite field analogues of, respec­

also presented.

and F.

Beukers's

In the past couple decades,

tively, the gamma and beta functions),

�{3) are

L. J. Rogers's

tities is presented in Chapter 12. The prerequisites for reading in real and complex analysis.

Selberg's

this

book are sound undergraduate courses

In particu­

lar, uniform processes should have been

and the p-adic gamma function are

integral and various generalizations

mastered.

some of the topics not found in most

and analogues have been the focus of

amount of the book is not dependent on

treatises even with a chapter or large section on the gamma function.

However,

a

considerable

much research. Many proofs are ex­

knowledge of complex analysis; for ex­

traordinarily difficult and lengthy, but

ample, little is used in Chapters 10-12.

8 gives a very accessible in­

The book contains 440 well selected

series.

troduction to this important area; no

exercises, virtually none trivial and all

They contain an enormous wealth of

text had heretofore attempted to give

interesting.

material and these functions permeate

such an introduction. The beautiful

In conclusion, it carmot be overem­

later chapters as well. Not to deprecate

proofs and extensions of K. Aomoto and

phasized how the authors' rich histor­

other writers, many accounts of hy­

G. W. Anderson are given. A fmite field

ical knowledge generates a fuller un­

pergeometric functions are clearly and

analogue is also established.

The heart of the book is the next two chapters on hypergeometric

Chapter

Chapter

logically presented but lack the per­

derstanding and appreciation of their

9 deals with ultraspherical

spicacity of the present authors, whose

polynomials

narration is replete with insights, mo­

with representation theory.

and

their

connections

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

tivation, and history. For example, on

Chapter 10 is a beautiful and su­

pages 126 and 127, we learn that in his

premely well-motivated treatment of q­

obtained

proof of one of the fundamental qua­

series. The chapter begins with a dis­

Indeed this treatise is

dratic transformations, Gauss demon­

cussion of the binomial theorem and

should become a classic. Every stu­

strated a grasp of the principle of

its

dent, user, and researcher in analysis

q-analogue.

The

introduction

to

from

other

papers

cited.

special and

VOLUME 22, NUMBER

1 , 2000

75

will want to have it close at hand as

voted to high-dimensional integration.

lish conformance to within a fmite error

she/he works.

The third chapter deals with breaking

threshold.

the curse of dimensionality for integra­

Other chapters deal with linear equa­

Department of Mathematics

tion by settling for stochastic assur­

tions, integral equations, nonlinear op­

University of Illinois

ance. The fourth chapter is on comput­

timization,

1 409 West Green St.

ing

computation with noisy information.

Urbana, IL 6 1 801 , USA

mathematical fmance.

high-dimensional

integrals

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

T

his 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

•

and

A typical prob­

There is a brief history of informa­ tion-based complexity and a bibliogra­

106 float­

ing-point operations for a single evalu­ ation of the integrand. The fmance com­ munity

long

believed

that

phy containing over

400 papers and

books. Those who are interested in numeri­

these

cal analysis and scientific computing

integrals should be evaluated using

can benefit from reading this book

Monte Carlo. Experiments conducted

Scientists and engineers from other dis­

at Columbia University showed that

ciplines, such as networking, might be

low-discrepancy methods from number

interested as well. Internet is changing

theory beat Monte Carlo by one to three

the world communication. However, the

orders of magnitude. These results ap­

fundamental problem of flow control re­

parently contradict the conventional

mains unsolved, and that severely hin­

wisdom that low-discrepancy methods

ders or limits its applications such as

were not good for problems of high di­

Internet Telephony. The flow control

mensions. The fifth chapter describes

problem can be formulated as a mathe­

the computational complexity of path

matical programming problem based on

integration. The problem of path inte­

the Internet traffic information, which is

gration occurs in many areas, including

partial, contaminated, priced, and highly

quantum physics, chemistry, fmancial

dynamic. Would Internet researchers

mathematics, and the solution of partial

benefit by a study of this theory? Would information-based complexity-theorists

differential equations. The complexity of ill-posed prob­ lems

is

discussed

in

chapter

six.

Practical examples of ill-posed prob­

be interested in broadening their scope of study and in looking at this or other real-world problems?

lems occur in remote sensing and im­

Since the primary goal of this book

age processing. It is well-known that

is exposition, many technical details

ill-posed problems can be solved, if the

are omitted. Those who want to know

residual is used to measure the quality

more about the theory could consult

of the approximation. Suppose, how­

the references listed below and pro­

ever, that the quality of the approxi­

vided in the book, which survey the

mation is measured by how close it is

major advances since

to the true solution. Then ill-posed

namic field.

1988 in this dy­

problems are unsolvable in the worst case setting, even for a large error

REFERENCES

threshold. The concept of well-posed­

[1 ] J. F. Traub, G. W. Wasilkowski, H. Woz ni­

computational com­

ness on the average is introduced, and

akowski, Information-Based Complexity.

plexity of problems of numerical

it is shown that if a problem is well­

analysis?

posed on the average, then it is solv­

New York: Academic Press (1 988). [2] J. F. Traub, H. Woz niakowski, Information­

How do problems suffer from the

able on the average.

mation-based complexity include: •

linear programming,

lem involves a few hundred dimensions, and requires on the order of

Complexity and Information

for

What is the

curse of dimensionality?

Verification and testing are essential for the reliability of numerical software.

This book is a guide to the numer­

For problems of numerical analysis that

based complexity: new questions for math­ ematicians. Mathematical lntelligencer 1 3

(1 99 1 ) , no. 2 , 34-43. [3] J. F. Traub, A G. Werschulz, Unear ill-posed

ous papers that study these and other

involve functions, apparently one has to

problems are solvable on the average for

problems. The presentation is informal

test the code that implements these

all Gaussian measures. Mathematical lntel­

but with motivation and insight. The first two chapters are an intro­

functions an infinite number of times to

ligencer 1 6 (1 994), no. 2, 42-48.

guarantee conformance. Two chapters

duction to information-based complex­

of this book report the study of the com­

ity. It uses a simple example of integra­

plexity of verification and implementa­

Murray Hill

tion to explain the main ideas of the

tion testing. For stochastic assurance, a

NJ 07974 USA

theory. The next three chapters are de-

fmite number of tests suffices to estab-

e-mail: [email protected]

76

THE MATHEMATICAL INTELLIGENCER

Bell Laboratories

k1fii,i.M$•iQ:I§I

R o b i n Wi l son

Renaissance Art Florence Fasanelli and Robin Wilson

Alberti

Brunelleschi

I

A

n outstanding example of a Re­ naissance man, Leon Battista Al­ berti (1404-72) introduced one-point perspective construction in its mathe­ matical form. The first written exposi­ tion of "painter's perspective" appears in his Della pittura [On Painting]. There, Alberti systematically ordered vi­ sual reality in geometric terms, combin­ ing 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 de­ sign the cupola of the Cathedral in Florence. Brunelleschi, also known as the founder of "painter's perspective," worked out linear perspective in rela­ tion to geometric-optical principles in a practical mode. Alberti, inspired by his techniques, wrote down the mathemat­ ical rules and filled out the scheme. Piero della Francesca (c. 1412-92) found a perspective grid especially ap­ plicable 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 (1472), from the Brera Gallery in Milan. It is rationally con­ structed with overarching symbolism;

the solidity of the egg (symbolizing the four elements of the universe) is in per­ fect 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 sec­ ond 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 per­ spective 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: [email protected]

Leonardo

Piero

Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics,

_TH ME T I C A GEOMETAIA PAOPOATIOz � < w a < " " ::>
:3 _, w a

The Open University, Milton Keynes, MK7 6AA, England e-mail: [email protected]

80

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Pacioli