Biblioteca digitale Progetto Agorà
This is not an essay but rather a personal testimony about the experience
of doing
Mathematics.
Kronecker is reported to have said about it: “God gave us the numbers,
the rest
has been made by man”. Thus Mathematics, according to this view,
is a pure
fantasy world, constructed using, as raw material, the natural num-
bers or,
after Cantor opened up the doors of a new “paradise”, the more
impalpable
“sets”.
While debating this point of view it may be illustrative to regard the
words we use
to describe the world of Mathematics and our own research.
We discover theorems, construct examples, attack and sometimes solve
problems, state
conjectures, open new fields, close lines of investigation pro-
ving conjectures,
design strategies and perform computations. Thus I believe
we think of our world as
both existing and to be constructed, so that it is
determined in the very moment in
which it is explored. In this way Mathe-
matics is a deep part of the very structure
of human thought, at the individual
as well as the collective level, growing as long
as human life does.
We are living a new golden age of Mathematics; the depth and richness
of today’s
research is unmatched in all history, though we can identify in our
past decisive
moments in which this research has been determined and deeply
transformed.
We are blessed by the fact that a (very small) fraction of our research
has
technological applications and our needs are limited. Our work being
produ-
cing ideas, we need only some supply of pens and paper to help us
remember
and to perform complicated computations, many books to transmit
our
science.
The new toy, the only expensive one we may need, the electronic com-
puter, is still
in a state of infancy as far as our work is concerned and we are
not yet able to
judge its future impact on our creations.
The mathematician has disguised himself in the past as magician, astro-
logist,
theologian, engineer, astronomer etc. in order to be able to support his
research.
I like the idea of a magician since this captures the unique amazing crea-
tivity of
our work. I cannot resist to remember a little girl, daughter of a
colleague, who explained her father’s profession as being that of a
“mathe-
magician” or even another little girl, “Alice in Wonderland”, who was in
fact
created by a mathematician.
The intellectual building we have by now constructed is extremely rich
and complex,
not completely explored by any single scientist but living
through the many
continuous exchanges of ideas and techniques between the
various schools and
individuals.
Our research we term as deep or shallow, exciting, challenging, fertile
sometimes
even wild, or dull, routine, sterile ; its results being beautiful, fasci-
nating,
impressive, amazing, incredible or boring, obvious, standard,
technical,
uninteresting (sometimes even false).
Most of it is filtered, polished, transformed and then either incorporated
in the
great building or forgotten as marginal or irrelevant. Of course this is
not a
unidirectional process, lost and forgotten work is sometimes brought
back into light
although often it is just independently rediscovered. Clearly
thus logical coherence
is only the barely necessary condition for accepting a
finished product, not even
strictly needed for a stimulating project. Some and
not all of the most beautiful
achievements of last century geometry have been
given a rigorous foundation only
now, a hundred years later, the incredible
intuition of our forefathers being
matched finally by all the necessary technical
tools.
The final rules for evaluating the quality of our research belong more to
apparently
less rational categories like beauty, harmony, completeness, depth.
The gems of
mathematical research are usually those theorems that, behind a
great simplicity and
fundamental nature of their statement hide a profound
and complex world of
structures and techniques. Especially number theory
has taught us that simple basic
statements may be the hardest to prove, like for
instance the famous unsolved
Goldbach’s conjecture: “Every even integer can
be expressed as the sum of two prime
numbers”, a statement which can be
easily explained in elementary school.
The possible mathematical theories are infinite but only very few are
selected as
being interesting and worth of investigation, thus the tautological
aspect of
mathematics belongs mostly to its form and not to its content.
Our projects are measured in time sometimes by the centuries or millen-
nia. The
problem of squaring the circle, which puzzled the Greeks, was
solved (with a
negative answer) by Lindemann one hundred years ago; after
the discovery by Cardano and Scipione
Del Ferro of the formulas for solving
equations of dregree 3 and 4 it took about
300 years to prove that a similar
solution for the higher degrees does not exist;
the discovery of non Euclidean
geometries while clarifying the work of Euclid
opened the way to an approach
to geometry which is essential for Einstein’s relativity. A rich legacy of
unsolved problems mostly on the nature of prime numbers, has been left to us
in the
last 400 years; we have no idea when (or if) will they be solved,
although we
usually take an optimistic point of view: maybe in 200, 2000
years when our
understanding of arithmetic will be different from today or
maybe even today
somewhere somebody is forging the tools which will bend
and conquer these apparently
impenetrable problems.
In our continuous struggle we search (often blindly) in our minds, dream
improbable
solutions, make wild conjectures, play all the tricks of the art we
have learnt, try
all the analogies we can think of attemping to weave the new
fabrics of our research
which may consist of a patient analyzing and untan-
gling of a complex puzzle, a
tough, ruthless and exhausting fight against an
unwielding computation or a master
plan of vast and sometimes unpredictable
strategies. Always aiming at the prize of a
proof of a Theorem, the classifica-
tion of objects, the discovery of new methods
and ideas.
When some 30 years ago a new interest in the classification of finite
simple
groups arose and new sporadic groups were discovered progress came
very fast
until the biggest sporadic group appeared. With its
2 46 .3 20.5 9 .7 6 .11 2 .13
3 .17.19.23.29.31.41.47.59.71 elements (written into prime
factors) it appeared
gigantically larger than its predecessors and it was named
“the monster” (of
Fischer-Griess); after a few years the monster has been
tamed and his wonderful
properties discovered, now he is “the friendly
giant ”.
Where is the limit? We do not know and hope it does not exist, that the
power of
our mind may grow with the growth of science and meet any new
challenge. We
frantically produce hundreds of thousands of pages of research
every year, the
only tool we have to hold this together is to perform a pityless
selection and to
organize it into higher and deeper levels of abstraction. This
is a road full of
snares and pitfalls: very abstract theories, without the master-
ful strategies
of the scientists who aim far away at the solution of deep pro-
blems, are like
empty shells, fascinating mermaids who have trapped many
researchers into playing
with complex formal structures only to find them-
selves with empy hands.
Abstraction is also not for all seasons, the quest for
very general theories,
abstract constructions, unifying ideas, which was the
main trend in Mathematics
for many years gave way abruptly some 15 years
ago to a new vigorous interest for
algorithms, special examples, individual but
extremely complex structures. When a
new level of abstraction will be
needed the trend will change again, although one
should be aware that such
separations are not so neat or permanent.
The decision of whether a theory is essentially empty falls unfortunately
(or
perhaps fortunately) on us, we have no physical structures to test, experi-
ments to
perform, markets to challenge. So clearly schools develop which
keep a low level of mathematical thinking, contented of playing with axioms
and
refusing to understand the achievements outside their field but living con-
fortably
in a small area of research which slowly decays. But this experience
is certainly
not unique of the mathematical world but belongs more to the life
of Academia.
Fortunately the power of truly outstanding research is overwhelming;
when a really
amazing theory appears it usually sweeps away a lot of irrelevan-
cies.
There is a vast debate on how much is Mathematics self contained and
uniquely
determined. The evidence is contradictory, on one hand the impact
of theoretical
Physics on the developement of many mathematical theories can
hardly be ignored, but
often in the final product almost every trace of this
process is erased and fields
like functional analysis, representation theory, dif-
ferentiable manifolds (whose
history is deeply intertwined with that of relati-
vity theory or quantum mechanics)
can be presented with absolutely no refer-
ence to Physics. This is almost certainly
an ominous sign but it is a fact.
I should finally mention the connection between Mathematics and sym-
bolic
logic. We are aware of the debates and the attempts to formalize ma-
thematics.
We are now quite contented by both the successes and the failures
of this
program. We may never know whether Mathematics is consistent but
certainly it
seems that it cannot be destroyed as a creative activity since, as
Gödel taught us, undecidability, incompleteness and lack of mechanical algo-
rithms
for proving theorems make it a field completely open to human imagi-
nation.
Paradoxically until a new shake up of our foundations (which we do not
foresee in
the immediate future), our main relationship with logic is through
the use of
computers. Here comes the most immediate danger of finding our-
selves soon to
confront the physical limitations of our methods.