IMAGINATION AND THE BUILDING OF MATHEMATICS
Claudio Procesi
IMAGINATION AND THE BUILDING OF MATHEMATICS
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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

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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

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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

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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.



Claudio Procesi . :

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