SÉMINAIRE DE L'ÉCOLE
Jeudi 26 mars 1998, 16h
Capi CORRALES
Université de Madrid
SPACE:
1. An interval between points or objects; a limited
portion of extension; distance; area.
2. The abstract possibility
of extension; that which is characterized by illimitable dimension;
continuous boundless extension in all directions.
(Collier's Dictionary)
SPACE & TIME:
Terms used in philosophy to describe the structure
of nature. They are sometimes described as containers in which
all natural events and processes occur, and sometimes, as relations which
connect such events.
(Collier's Encyclopedia)
The second sentence in the last definition tells us that Space and Time are at times thought of as "containers", and at times thought of as "relations". Curiously, these two words, "container" and "relations" describe, respectively, the idea of space held in eighteenth century and contemporary mathematics.
In the first part of this talk we will follow the path taking us from one notion to the other, from a container-space to space as a byproduct of relations among objects.
Once we have mathematical spaces wel defined (Hausdorff, 1914), in the second part of the talk we will follow a little of the process of abstraction carried on by mathematicians in order to construct structures with which to handle the different mathematical spaces.
Our only luggage in this voyage will consist of the paintings done during the different periods through which our trip will take us, paintings which will be used as illustrations to help us understand better the shift in the mathematical point of view taking place at each step of the process.
We will finish by describing three examples of the very rich fruits produced by the use of these tools: the cubists in painting, and, in mathematics, the classification of the arabescs of the palace of Alhambra in Granada, and Andrew Wiles' proof of Fermat's last theorem.