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.
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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 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 :>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
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1 23 , Birkhauser, Boston 1 994, pp. 531 -568.
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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..