Differences: chaos in the history of the sciences
¨ Academie Francaise, 23 quai de Conti, 75270 Paris cedex 06, CS 90618, France ° Translated by Taylor Adkins 3047 Hollywood Drive, Decatur, GA 30033, USA
Abstract. In this paper from the book Les origines de la geometrie (The origins of geometry), subtitled ¨ ¨ tiers livre des fondations (third book of foundations) (Serres, 1993, Flammarion, Paris), I argue that the history of the sciences and, in particular, the history of mathematics cannot be written using the tools and models of traditional historiography. Rather, I claim that there is a need for a science of history that takes seriously what I see as a radical contemporaneity or copresence of the archaic and the contemporary. The model of history that I propose attempts to seek a degree of congruence between a model of time that is not chronological but rather percolating and filtering the ways that the mathematical tradition is reinvented. There existsöor seems to exist öa structural similitude between mathematics itself and the form or model of historiography required to write its history.
Several sciences, several histories The history of the mathematical sciences transforms as soon as their invention is investigated, and so profoundly sometimes that it seems to change nature more than allure. In fact, it sometimes seems to follow regular lines of expansion or growth, spirals of resumptions or circles of invariance, sometimes undergoing abrupt declines, reversals, or gaps through forgetting, stabilities, or ongoing preservation ... . Ten different models of stagnation, regression, or progress, either discrete or continuous, could be composed in such a way as to lose the orientation of their development the moment the complex variety of these different fluxes, networks, or spectra is observed. We therefore doubt the meaning of the history of science: in order to begin, should we search for a science of history? Yes, and this is precisely what has slowed me down for thirty-five years. Moreover, we can neither conceive the origin without some sort of preliminary philosophy of time, nor ultimately conceive the origins of the first geometry without clarifying those of the space that it constructed. This text primarily responds to these three questions. It has taken its author his entire life to attempt to shed light on the respective answers. Geometries
Let us begin with the history of geometry: can we decide what this science designates? Do we begin with the ancient and modern measure of the arable or constructible earth, namely that of farmers or masons? The archaic figures of Pythagorean arithmetic? Those of the school of Chios? Platonic forms or ideas? Euclid's Elements ? That which remains of Archimedes or Apollonius? Cartesian representation? The descriptive blueprints of the 19th century? Non-Euclidean reconstructions? Leibniz's ``analysis ¨ situs'', the topology of Euler, Riemann, and Poincare? Hilbert's formal demonstrations? Contemporary algebraic geometry? The plans of computer programmers developing robotic movement ... ? Seen from afar, the universal almost transforms into a jungle of
sciences so different that it overwhelms the number of histories to relate them to, all divergent and enrooted in forgotten pasts. Are they thus convergent? For example, the diagonal and the squareöthe triangle and its elementsöreturn in each of the aforementioned domains, no doubt inherited from the most primitive geometries. And the universal guides us through this jungle of differences in this strange and familiar theorem that demonstrates the existence of a model of all geometry within that of Euclid, which is precisely the origin I am seeking. Although invariants, it seems these elements never refer to the same system of thought, such that it is neither a question of a...