Computer, Mathematics and Art
Jean-François COLONNA
jean-francois.colonna@polytechnique.edu
CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, France
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Abstract: Mathematics could be seen as a "simple" mind game hardly more useful in everyday life than chess.
But their "formidable efficiency" as the language with which are written the laws of Nature could be an evidence they are the Reality.
Thus, Mathematics would contain all works of art past, present and future, but also their creators.
CHESS AND MATHS:
Prior to study concretely the links between Mathematics and Art,
it must be first recalled what they are. Purists define them as a set of abstract
symbols, manipulation rules and axiomatic statements regarded as true (and
obvious) out of which are demonstrated theorems (that is to say new true statements).
The progress (that is to say, the accumulation of new truths) are made in
general thanks to problems posed to the community of mathematicians by one of its
members. Most of the time, if not always, these issues are abstract
and often incomprehensible to ordinary mortals and without apparent connection with
reality. A recent example, which made the headlines, was the demonstration
of Fermat's Last Theorem, completed in 1994 by Andrew Wiles after more than
three centuries of trials and errors. And despite the immensity of the undertaking
and the success met, here is a result of little practical usefulness [1] although
the tortuous path for obtaining it was incredibly rich in developments and chance
encounters. As presented, Mathematics would then seem to be a "simple"
mind game hardly more useful in everyday life than chess. Yet for two thousand
years and still more from the seventeenth century with Galileo, Mathematics
is regarded as the language with which are written the laws of Nature. They are
then, next to the microscope and the telescope, a revolutionary "observation
instrument" which reveals to us every day new and mysterious aspects of our Universe.
MATHEMATICS IS THE REALITY:
But these successes only make their profound nature
and their "formidable efficiency" (Eugene Wigner) more mysterious. What are Mathematics? Two
seemingly irreconcilable answers can be formulated: either they are "only" the fruit
of our minds, or they exist independently from us. Say differently: Is the
mathematician like Molière who devised Monsieur Jourdain in Le Bourgeois Gentilhomme,
or as Christopher Columbus who discovered America? Is the mathematician a creator
(that is to say the one that pulls out of nothing) or an explorator (the one that
travels observing)? Let us examine these two positions [2]. In the first case,
Mathematics (our Mathematics!) are the brainchild of mathematicians [3]; they can be
seen then as a language intended for the compression of the regularities observed
in the nature [4]. In the second case, Mathematics is independent of us,
so they exist outside of our time and our space, but where do they reside
and what are they made of? This question seems quite puzzling, even crippling,
but no more than to know where our universe is and what is it made of! It is even
possible to go further (that is my point of view) and to consider that our Reality
is a mathematical structure (among an infinite number of others...) inside of which
self-conscious sub-structures have emerged (us!). And so everything is "simple":
Mathematics describe well the Reality because the latter is mathematical. So far
two approaches complementary and seemingly disjointed, those of Art and Science [5],
were necessary to his knowledge; but if Reality is so, then the boundary
blurs, disappearing and our computers are unaware [6] sub-structures that help
us to gradually lift a "corner of the veil".
ART AND COMPUTER:
But before examining
the consequences of all this, let's describe briefly the concrete and pragmatic
roles of the computer and Mathematics in the artistic field [7]. Appeared in the forties,
the computer was quickly found in a strong position in all human activities and the
constant progress in hardware and software areas have only amplified this phenomenon.
This is especially true for industry, research or everyday life [8],
but it is much less in the Art world. From the beginning of the seventies,
I designed and developed the SMC (Conversational Multimedia System) system,
a priori a system for computer aided instruction [9], who was one of the first
to allow both the synthesis and the processing of pictures (see figure 1). I realized
very quickly the potential of digital techniques in audio and visual fields. In particular,
all I announced more than twenty years ago at the conclusion of an article about
the cinema aided by computer [10] is today a reality in particular this convergence
now taking place between movies and video games: the viewer has the opportunity to
become an actor, while being able to immerse himself in realms unimaginable
yesterday. But in the world of visual arts, the evolution was not as striking,
far from it. Perhaps it is bound to certain questions: Who is the author? Where is
the work of art? What is it made of? Is it perennial? Can we protect it? Is the computer
able to create? And these questions are justified: indeed, the creation of
a work of art by means of computers, next to the creative act itself,
make use of softwares and hardwares that cannot be neutral [11]. And the work of art,
is it what appears on a screen [12] or is it the string of binary digits which represents
it in the memory of the machine or is it the set of comnands and gestures used to
bring it to life? Moreover is it perennial: will it cross the chasms of time and
will it be "readable" in several thousands of years as easily as the frescoes of
Lascaux [13]? One of the consequence of the principle of the digital representation
of information is that copies cannot be distinguished from the original: this means
that on one hand the works here loses its property of uniqueness and on other hand
it can be duplicated more or less easily [14], with or without the consent of
its author. Finally, the question of whether the machine replace or will
replace the artist is certainly the most delicate. From today, the complexity
of procedures that can be programmed may cause surprises (see figure 2),
but obviously the machines lack the will to create and the awareness of themselves,
but it is perhaps only a matter of complexity and therefore of time...
ART AND MATHEMATICS:
I have solved myself most of these problems with the concept of potential work of art.
In this context, the work of art is simply the underlying mathematical model.
Obviously, this excludes from the creative field the works of art born from
the artist's gesture, but this is not a limit for me, given the definition
that was given previously of Mathematics: if they are indeed the Reality,
they describe it in full! Let's illustrate this with an example: the one of the fractal
geometry (see figure 3). Born in the sixties with the work of Benoît Mandelbrot,
it allows in particular to describe mathematically irregular and disordered phenomena
unreachable using the euclidean geometry. In this context, I developed a
model that can produce images of cloudy mountainous landscapes, but also
to animate them: the Alps as well as the American deserts or again the Moon... But
all these images taken individually are not, in this context, the
works of art and moreover it is impossible to display them all (and even more difficult
to choose among them...), since their number is strictly astronomical; the
works of art are these equations -and these algorithms [15]-
that contain potentially all these images [16]...
Thus,
Mathematics would contain all works of art past, present and future,
but also their creators. Everything would be "written", but it doesn't matter
since we have the consciousness to exist, to be free and to create...
- [01]
The theorem itself (ie that the Diophantine equation X^n+Y^n=Z^n has no solution for n>2) has virtually
no application, unlike some tools used for his demonstration; for example new methods of encryption
using elliptic curves have emerged.
- [02]
In both cases, the mathematician is often (always?) doing Physics without knowing it.
- [03]
matching could then be that the cortical structures of the senses (especially the sight) and
of the "creation" are of the same nature and communicate with each other.
- [04]
as JPEG allows the compression of pictures.
- [05]
respectively subjective and objective.
- [06]
But for how long?
- [07]
We will ignore here all that happened before the Second World War. Therefore it will be made no
allusion to the golden ratio or again to the perspective.
- [08]
What would everyday life be without the Internet?
- [09]
It was based on a T1600 Télémécanique computer with a memory of only 32 kilobytes while occupying several cubic meters!
- [10]
La Recherche, September 1987 issue.
- [11]
For example, some features of a program can inspire an artist, allowing him
something he would not have imagined without this intellectual prosthesis. Must the authors
of these tools share the authorship of the work of art? This is the answer to this question
that has frustrated most of my collaborations with traditional artists...
- [12]
or on any media: paper or other ...
- [13]
This question relates both to coding standards (like JPEG) as well as to media-recording devices such as DVD.
To ignore it or to do nothing, implies the disappearance over time of an important part
of our artistic and scientific heritage!
- [14]
There are ways of protection, such as watermarking, but I'm not convinced of
their effectiveness and especially their neutrality towards the work of art.
- [15]
A new word for them: Argorithm, made of Art
and Algorithm?
- [16]
This design would certainly have pleased Jorge Luis Borges. He could make a sequel to his Library of Babel...
Copyright © Jean-François Colonna, 2010-2022.
Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2010-2022.
