Chapter 4
Mathematics and Society in the 20th Century1
Benoit B. Mandelbrot
Mathematics education and research are two separate crafts,
but--for practical as well as intellectual reasons--it is best if
they know each other. In particular, it is very important for
mathematics educators to have a broad and balanced view of
the way research mathematicians perceive their craft. They
must realize that the perception has kept changing throughout
history and never as sharply as in the 20th century. This
chapter's goal is to recount a few highly significant features
of the strife that came in the preceding hundred years. Mathematics
ended that century in great spirit and in a state of great
vigor, renewed collegiality and marvelous diversity.
But in the 1960s and 1970s, the representatives of the profession
described the flow of 20th century mathematics as
that of a single majestic river whose irresistible course was
not touched by historical accident but had been preordained
by inner logic. It necessarily proceeded inevitably and inexorably
towards increasingly general, structural, or fundamental
notions--which happened to be increasingly abstract. In
the spirit of "the end of history," the descriptions never referred
to the past or the messiness of Earth.
The majestic flow in question was unflinchingly understood
to be leaving aside many people (including myself), and
innumerable topics that concern either the foundations (logic)
or the applications. We were told that much of what looks like
mathematics is not really mathematics, even though the distinction
may not be obvious to the outsider.
The position I am about to describe is starkly different. I
believe and I hope to convince you that mathematics is not the
conservatives' ivory tower. It is a very big house on a rolling
terrain, with many doors, windows open to many horizons
and bridges to many other houses.
1Adapted from an invited address "What will remain of 20th century formal
science" at the Europe¨aisches Forum 1992, held in Alpbach, Austria.
This text remains self-contained and preserves some of its original flavor, in
part by repeating some points that were already made elsewhere in this book
but bear emphasis.
It need not be the Queen of Court Etiquette in Science looking
down on most of her subjects from an ivory tower up on
a high hill. It deserves to be the beloved Queen of all the
Scientists' Hearts, and of the Soul of Science, the only noncontrived
link that could prevent various parts from scattering
away from one another.
Compared to the conservative view of mathematics, mine
is far broader and far more strongly linked to other human
activities. It is also a more diverse and lively subject. In particular,
it is attractive to persons who are not professional research
mathematicians, a category that includes students and
most teachers of mathematics. My strong opinions represent
a minority view, but one that is increasingly widely shared
and I have no doubt will prevail.
In any event, my interpretations and opinions are neither
capricious nor based on idle rumor or anecdote, but on
widely ranging reading, active and uninterrupted participation
in events that occurred in the USA and France over fifty
years, and reports by an uncle who was a prominent mathematician
in Paris and Houston and participated in the immediately
preceding thirty-five year period.
I see mathematical science as a very broad enterprise that
shelters many diverse topics, ranging from the very concrete
to the very abstract. This view is well represented by a simile
I heard used by Hermann Weyl (1885­1955). He compared
mathematics to the delta of a great river, one made of many
streams: they may vary in their width and the speed of the
flow through them; nevertheless, all are always a part of the
system, and no individual stream is permanently the most important.
This simile represented the mood of mathematics
close to the year 1900--and also, for that matter, its mood
near the year 1800. More importantly, mathematics has been
changing so fast for a decade or so that I feel that Weyl's simile
became applicable again in the year 2000.
But the resemblances between these snapshots taken centuries
apart certainly do not imply that mathematics is unchanging,
something outside ordinary history. In mathemat-
29
M.L. Frame & B.B. Mandelbrot
Washington D.C. Mathematical Association of America, 2002
Fractals, Graphics, & Mathematics Education
30 Chapter 4. Mathematics and Society in the 20th Century
ics, as in every other aspect of human life, the 20th century
gave us an example of something starkly different: a rocky
history and continuing conflict. Mathematics was not ruled
by its own determinism; it did not evolve separately from every
other aspect of human knowing and feeling; it has on the
contrary been profoundly affected by endless external vicis-
situdes.
The words profoundly affected by must not be misunderstood
as meaning enslaved by. Of all the triumphs of humanity,
the discovery and the development of mathematics is
perhaps the greatest kind. A field's importance to the overall
human experience is necessarily reflected by the role that internal
logic has upon its development; nevertheless, strife has
been present in mathematics since the Ancient Greeks. We
shall see this when this story ends by mentioning the longstanding
conflict between the traditions of Plato, the ideologue,
and Archimedes, the experienced scientist. Like every
individual human activity, mathematics very much participates
in general history, politics, demography, and technology,
and it is heavily influenced by the idiosyncrasies of a
few key people. Let me give some examples from this cen-
tury.
Around 1920, a group of Polish mathematicians collected
around a very forceful man named Waclaw Sierpinski (1882­
1969). They chose to concentrate on a field that was not practiced
much in the reigning intellectual capitals, and founded
a very abstract new branch often called Polish mathematics.
They proudly proclaimed that their goal involved national
politics: they did not want the newly reestablished Poland
to become a mathematical satellite of Paris or G¨ottingen. I
know that Providence is credited with working in mysterious
ways. Yet, would anyone claim that Polish nationalism after
more than a century of partition had anything to do with the
historical determinism of mathematics? Polish mathematics
became an important force pushing towards abstraction at all
cost. Yet, by a bitter irony, some of the notions it originated
failed to become important in mathematics, but eventually became
important to physics--through fractal geometry.
My second example concerns Godfrey Harold Hardy
(1877­1947), a strong person as well as a strong and highly
inventive mind. The Poles had no strong native physics to
contend with, but British mathematics of Hardy's youth was
dominated by a form of mathematical physics that was extraordinarily
effective (the Heaviside Calculus differentiated
discontinuous functions!) but had little concern with continental
rigor. During World War I, Hardy was an outspoken
pacifist who recoiled from the practical uses of this old British
mathematics. During another War, he wrote (Hardy (1940)),
an impassioned account of his ideal of pure mathematics. For
him, good mathematics could have no bad application--for
the simple reason that it could have no application of any sort.
By another bitter irony, his best example of total inapplicability
turned out, in due time, to be essential to a problem he
would have loathed: cryptography.
A three-page review of Hardy (1940) in the famous weekly
Nature by the Nobel-winning chemist Frederick Soddy begins
"This is a slight book. From such cloistral clowning
the world sickens ... `Imaginary' universes are so much more
beautiful than this stupidly constructed `real' one, according
to the author ... Most scientists, however, still believe
that ... the real universe ... is not stupidly constructed." But
nothing can break the appeal of a tract that discriminates between
the good and the bad without hesitation. Hardy's book
remains in print and continues to this day to enchant some
of the young. But would anyone claim that Hardy's militant
anti-nationalism had anything to do with the historical determinism
of mathematics?
From ideology, let us move on to demography. The 1910s
were very cruel to French mathematics. First, Henri Poincar´e
(1854­1912) died prematurely on the operating table, then
millions of young people died in trench warfare, and finally--
perhaps worst of all--millions returned broken in health or
spirit to a country that did not dare make heavy demands on
them. As a result, the young postwar French mathematicians
of the 1920s found that the only available teachers were men
who had already been ill or old in 1914 and so did not go
to war. Some have written movingly about the hardship of
training without the usual parental supervision from slightly
older advisors, and (as may have been expected) this hardship
contributed to the emergence of several very strong personalities.
In any event, the France of the late 1920s and the 1930s
gave rise to an extremist movement calling itself Bourbaki.
But would anyone claim that a demographic unbalance in a
country with a long and glorious mathematical tradition has
anything to do with the historical determinism of mathemat-
ics?
Andr´e Weil (1906­1994), now acknowledged as the mind
behind Bourbaki, observed late in life that in his prime years,
mathematics was little influenced by physics. Was that a natural
feature of the preordained development of mathematics?
Or could it be that Weil's views were set even before a visit to
G¨ottingen in the 1920s? David Hilbert's dream Mathematics
Institute there had three parts: a very pure one that Weil worshipped,
one on numerical methods and one on mathematical
physics. In the latter part, Max Born and Werner Heisenberg
were in the process of creating quantum mechanics--but Weil
apparently did not notice.
From demography, let us move to another form of ideology.
Soviet anti-semitism treated Jewish mathematicians
harshly; Jewish physicists, less so. Hence a number of very
gifted mathematicians transferred to physics institutes, where
they were welcome. Their move contributed greatly to the
formation of the current very rigorous form of mathematical
physics. Would anyone claim that Soviet ethnic politics have
anything to do with the historical determinism of mathemat-
ics?
No one would claim that the specific historical determinism
of mathematics only reflected the intellectual moods and
Chapter 4. Mathematics and Society in the 20th Century 31
fashions that rule society at large. But it happens that a very
unusual mood prevailed early in this century, particularly in
the 1920s. One especially visible and durable effect was the
invention of the International Style in architecture, with its
heavy emphasis on structure. In Finland, the very unusual
small country where this style was born, modern architecture
merged smoothly into what came before it, without discontinuity
and without heavy dogmatism. But modern architecture
became dogmatic in Germany with the Bauhaus and in
France with Le Corbusier (1887­1965). The latter built few
houses but made many sketches (for example, his proposed
ideal improvement of Paris evokes the worst present suburbs
of Moscow). When I was young, Le Corbusier was billed as
a great intellect to whom modern architecture owed its intellectual
legitimacy. Indeed, he wrote a great deal, but I find
little in his writings beyond sophomoric trash. It may be that
Bauhaus was useful, even commercially inevitable at a certain
stage of the technology and economics of raising large
buildings, but no one ever convinced me that they were an
inevitable intellectual wave of the future.
Think also of physics. Having confirmed existence of the
atom in the 1900s, it went on to focus increasingly on the
search for the most fundamental structural components of
matter, increasingly tiny ones. Biology took this path later.
How was mathematics affected by the above-mentioned
politics, demographics, and general intellectual moods? I
view them all as responsible for the fact that near the middle
of our century mathematics behaved in ways totally at
variance with its mood today and its mood in 1900 or 1800.
This atypical mathematics is conveniently denoted by the
name it took in France, but the current that gave rise to Bourbaki
also affected many countries other than Britain, France
and Poland. It strongly affected the USA, with a little-known
wrinkle. One might have expected a brash new industrial
giant to favor applications, but in terms of mathematical research
the precise contrary was true. In Europe, the 19th century
had created wide-ranging establishments against which
Bourbaki could revolt. In the USA, before the arrival of
refugees from Stalin and Hitler, research mathematics was
dominated by aristocrats and anarchists, hence was very pure
(as well as outstanding on its terms). Bourbaki did not reach
the outlying countries Sweden and Finland, and there were
strong counteracting forces in Germany and Russia. In the
1960s, when Bourbaki was its strongest, it benefited from another
extraneous event: Sputnik created a period of unprecedented
economic growth in Academia, with minimal social
pressure on the sciences, and greatly increased the number
of math PhDs, including many Bourbaki products. The math
departments' balance was overwhelmed by them.
To sum up, Bourbaki found roots by selecting one of the
many components of the mathematics of 1875­1925, gathered
strength during the second quarter of our century (the
period to which the above examples refer), and took power
around 1950. During the third quarter of the century it exerted
an extraordinary degree of control. There was no disorder
in mathematics, but the field was narrowed down to a
truly extraordinary extent. At one time it seemed to reduce to
little more than algebraic topology; at a later time, to number
theory and algebraic geometry. These are extremely important
fields, to be sure, but concentration on a single field was
quite contrary to the historical tradition that I have already
mentioned and that had led Hermann Weyl to the image of
the delta of the Nile. Mathematics seemed to have reduced
itself to basically a single stream at any given time. This happened
to be the clich´e description that Herman Weyl (in a
contrasting image) applied to physics.
The Bourbaki, as has already been implied, never paid
attention to the historical accidents that contributed to their
birth; they felt themselves to be the necessary and inevitable
response to the call of history. Today, however, this call seems
forgotten, and there is wide consensus that, like new math,
"Bourbaki is dead."
Who killed Bourbaki? Throughout its heyday, my friend
Mark Kac (1914­84) and many other open-minded mathematicians
argued, in vehement speeches and articles, that
Bourbaki had misread mere accidents for the arrow of history.
But such negative criticism invariably lacked bite, and
it had no effect. My own partisan opinion is that Bourbaki's
fate was typical of many ideologies outside science.
The founders could only insure their immediate succession;
gradually, the ideological fervor weakened and the movement
continued largely by force of habit. The resulting weakening
was gradual and not obvious. But everyone noticed when
the movement was knocked down by yet another event that
had nothing to do with the historical determinism in mathematics.
This event was something I view as a return to sanity,
namely the rebirth of experimental mathematics that followed
(slowly, as we shall see) the advent of the modern computer.
From where did the computer come? From the mathematical
sciences understood in a broad sense. What relation is
there between the advent of the computer and mathematics
as narrowly reinterpreted by Bourbaki? None whatsoever.
The computer arose from the convergence of two fields that
surely belong to mathematics but were spurned by Bourbaki,
namely, logic and differential equations. We all know that one
must never rewrite history as it might have proceeded if two
crucial events had chanced to occur in the reverse order. But
in this instance the temptation is strong to air the following
conviction. Had an earlier arrival of the computer saved experimental
mathematics from falling into a century of decline,
Bourbaki might have never seemed to anyone to be an unavoidable
development. Let me elaborate on the computer's
roots.
Surprisingly, while Foundations of Analysis was (for a
while) the overall title of their treatises, the Bourbaki had
only contempt for the logical foundations of mathematics, as
in the work of Kurt G¨odel (1906­78) and Alan Turing (1912­
54). In the 1930s, Turing had phrased his model of a logical
32 Chapter 4. Mathematics and Society in the 20th Century
system in terms of an idealized computer. His Turing machine
had a very great influence on the thinking of those who
developed the actual hardware.
However, the man who made the computer into a reality
was John von Neumann (1903­57). He was not only a mathematician,
but also a physicist and an economist, and his great
breadth of interests came to include a passion to find ways to
predict the weather.
Thus the computer was born in the 1940s from a strange
combination of abstract logic and the desire to control Nature.
Eventually, the computer changed mathematics in a very
profound fashion. But for a very long time, core mathematicians
felt totally unconcerned, and viewed it with revulsion.
Because of his work on the computer, von Neumann ceased
to be accepted as a mathematician and in 1955 he decided to
resign from the Princeton Institute for Advanced Study. (He
died before his planned move to California.)
The year 1955 was also the date of publication of Fermi,
Pasta & Ulam (1955), a text that appeared only in a Los
Alamos report but was widely read and viewed as an early
masterpiece of experimental mathematics before it was actually
printed in Fermi (1965) (pp 977­988) and then in Ulam
(1974) (pp 490­501). A comment by Stanislaw Ulam (1909­
1984) informs us that the initiative for using the computer
to assist mathematical research had come from Enrico Fermi
(1901­1954), who was of course a physicist, not a mathematician.
And Ulam asserts that "Mathematics is not really
an observational science and not even an experimental one.
Nevertheless, computations were useful in establishing some
rather curious facts about simple mathematical objects." Surprised,
I reached for a more positive statement in the autobiography,
Ulam (1976), but found nothing worth quoting.
How did experimental mathematics fare during the 25
years after 1955? That period happens to end in the year
of publication of Mandelbrot (1980), and coincided with the
heyday of Bourbaki. In a near-perfect first approximation,
it saw no experimental mathematics at all. Not only was
the lead of von Neumann and Fermi not followed by mathematicians,
but their disinterest for the computer was carefully
considered, not caused by ignorance. For example, when I
was new at IBM, which I joined in 1958, opportunities to use
computers were knowingly and systematically offered to every
mathematician with a good name who could be coaxed
into the building. Not one of them paid attention to the offer.
Interest in experiment did not spread to at least some mathematicians
until my work started attracting wide attention.
In understanding the process of discovery, the slowness of
the acceptance of the computer brings up forcibly a very old
issue: the respective contributions of the tool and of its user.
Galileo wrote a whole book complaining bitterly about those
who belittled him by claiming that his discovery of sunspots
was only due to his having lived during the telescope revolution.
In fact, telescopes were widespread but useless before
one reached Galileo's steady hand and good eye. For contrast,
consider the chapter of mathematics called the global
theory of iteration of rational functions, to which the Mandelbrot
set belongs. Pierre Fatou (1878­1929) and Gaston Julia
(1893­1978) are--quite rightly--praised for developing this
chapter, and no one would dream of belittling their contributions
as being due to their having lived during the age of Paul
Montel (1876­1975). Montel was the mathematician who,
in 1912, discovered Fatou's and Julia's key tool, called the
normal families of functions. Soon afterwards he was called
into the Army, leaving behind Fatou (who was a cripple) and
Julia (who had come back from the trenches as a wounded
war hero). After World War I, Montel looked after the theory
of iteration as his baby. Today, in the noise that accompanies
changes in mathematics, those who use the computer are
treated like a Galileo and not like a Montel. That is, critics are
found to belittle their work as solely due to their living in the
computer age. If it were so, experimental mathematics would
have thrived after von Neumann and Fermi; the preceding remarks
show that it did not.
Let me summarize, make a general comment, and conclude.
One cannot disregard the lessons of history, contrary
to the belief of those who argued that the pure mathematics
of the mid-20th century was preordained by destiny. Its birth
in the 1920s was influenced by Polish and English ideology,
a demographic catastrophe in France, and the general intellectual
mood of the day; its success was hastened by a long
spurt of economic growth, and its demise was hastened by a
mere technological development. None of these events was
influenced by mathematics, none was preordained, and none
acted immediately. In any event, von Neumann's and Fermi's
lead was not followed by other mathematicians.
To conclude, "What will remain of 20th century mathematics?"
There can be no short and truthful answer, because at
this point of its history, mathematics is in healthy and constructive
turmoil. Once again, the Bourbaki utopia flourished
when every science was experiencing unprecedented growth
and minimal social pressure. It seemed that any would-be
peer group could organize itself and prosper with no hindrance
from other, equally self-interested peer groups. But
today the sciences face scarcity and strong pressure to justify
both their size and their goals, and everyone bemoans
the absence of generalists capable of representing more than
a few groups. How the effects of the resulting intractabilities
and pressures will combine with the internal logic of
mathematics, of the computer and also of today's mathematical
physics-- that thriving no man's land between theorem
proving and observation of nature--is simply beyond prediction.
Fortunately for the teachers of mathematics, they are
not asked to predict, but it is best for them to know the past,
if only to avoid being drawn to repeating its deep errors.