Mathematics and Representations

Jean-François COLONNA
CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, France

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[en français/in french]

For two thousand years and more from the seventeenth century with Galileo, Mathematics is seen as the language with which are written the laws of Nature. They are then, next to the microscope and the telescope, a revolutionary "instrument of observation" that reveals every day new and mysterious aspects of our Universe. These successes only make more mysterious their deep nature and the cause of their "extremely effectiveness" (Eugene Wigner). A possible explanation could be that Mathematics is independent of us, that they exist outside our space-time and that our reality is "just" a mathematical structure (among countless others...) inside which emerged autonomous self-conscious sub-structures (we!).

To make progress in our knowledge of the Universe, we must do Mathematics: juggling with formulas, solving equations,... All this is generally abstract, but nothing forbids us to use the sense that evolution has given us and particularly the vision; our eyes are made to be surprised, and today the computer allows us a new experimental approach: the virtuality. A so-called virtual experiment will consist of the in silico study of a model of a certain system (elementary particles to the Universe through the car...). After very heavy computations, raw results are mountains of numerical values that are a priori useless without further processing, a "staging".

The translation from numbers to pictures is far to be a childish coloring. As a matter of fact, many characteristics of the mathematical "objects" that are manipulated will gang up to prevent any simple and objective representation. Thus, they often have no natural images (properly stupid questions then arise and, for example: what is the color of the numbers?), the spaces they inhabit are generally (much...) more than bidimensional, the time (in the sense of dynamics) is intimately related to space, etc... Experience in this area shows that it is very easy to hide what is present and to exhibit what is not, voluntarily or involuntarily...

But Mathematics is also a tool for artistic creation. The concept of potential work of art emerges: the work of art is no longer an image or a sound, but simply the underlying mathematical model. We can be assured that we are not at the end of the road of discoveries, as was the case in past years with the fractal geometry of Benoît Mandelbrot... So what scientific and artistic surprises await us, which will open new horizons to us?

Copyright (c) Jean-François Colonna, 2011-2015.
Copyright (c) CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2011-2015.