At the moment I am working my way through Michel Serres’s monumental Le Système de Leibniz et ses modèles mathématiques. I am struck by how many of the ideas that have come to be thought of as characteristically Serresian are already present implicitly or (more often than not) explicitly in this 800 page doctoral thesis. If it weren’t so huge, we might think of it as the genotype which generates the phenotype of all of Serres’s subsequent writing. it holds further fascination for the Serres scholar because, constrained by the conventions of the thesis genre with its footnotes and requirement for microscopic detail, it presents us with a Serresian thought that is quite unlike any of his subsequent writing. The thesis offers an incredibly fecund incipit into Serres’s thought, fecund because it blossoms forth in multiple possibilities of thinking that we see ripen in Serres’s subsequent writing.
Here is one passage among many which I found particularly pregnant with Serres’s future thought, and which serves as a productive meditation in its own right on the relation between time and space in philosophical discourse, as well as on the concept of model in both science and philosophy. I provide my own rough and ready translation below the original:
La critique bergsonienne désignait comme erreur fondamentale la réduction du temps à l’espace; c’est, de toute évidence, le contraire qu’il faut dire. Le peu de compréhension que nous avons du temps (et, par là, des processus évolutifs) vient du fait qu’on utilise mal l’espace (qui seul, en particulier, permet de rendre rigoureuse l’idée de continuité), ou qu’on utilise toujours son contraire, c’est-à-dire la ligne. Croyant dire le temps en niant l’espace, on dit la ligne par cette négation même.
Pour prolonger alors la pensée leibnizienne en la généralisant, il faut prendre au sérieux l’idée de différentiation en temps élémentaires multiples (dans le compliqué d’une transformation quelconque) auxquels on pourrait référer l’évolution en question. Puis, considérer la ligne comme le modèle (c’est-à-dire la représentation formelle) de l’un quelconque de ces temps; enfin, projeter la multiplicité de ces lignes dans un espace de représentation. Leur ensemble définirait alors une surface compliquée, figure de l’évolution, qui comporterait des « cheminées » d’accélération forte ou de croissance infinie, des « cols » d’arrêt d’une « ascension » et de début d’une « descente », des zones de lignes stationnaires et ainsi de suite, voire des déchirures… Dès lors, la notion de progrès se régionaliserait, comme celle de décadence, d’accumulation, d’arrêt, etc.; et donc il deviendrait grossier de dire le progrès de la science, la décadence d’une civilisation, la maturité d’une situation révolutionnaire, la genèse d’une psychologie, car ce serait réduire une évolution globale à une séquence linéaire. Que nous ayons, devant les yeux, en écrivant cela, un modèle mathématique qu’ignorait Leibniz, mais que concevait Euler, montre qu’il faut enrichir nos modèles de pensée, en général d’une pitoyable pauvreté. Alors que la science met en évidence des structures ultra-fines, nous philosophons toujours à l’aide de modèles ou de schémas non affinés, au moyen de techniques de raisonnement qui, elles, n’ont guère fait de progrès.[1]
Bergson critiqued the reduction of time to space as a fundamental error; it is clear that we must say quite the opposite. The meagre understanding we have of time (and by that token of evolutionary processes as well) stems from the fact that we make bad use of space (and it is space alone that allows us to give rigour to the idea of continuity), or from the fact that we always use its opposite, namely the line. We deny space and think we are talking about time; by this very denial we speak about the line.
So in order to extend Leibnizian thought by generalising it we must take seriously the idea of differentiation into multiple elementary times (in some complex transformation) to which we could refer the evolution in question. Then we must consider the line as the model (in other words, the formal representation) of one of these times. Finally, we must project the multiplicity of these lines in a space of representation. Together they would then define a complex surface, a figure of evolution, which would contain “chimneys” of strong acceleration or infinite growth, “mountain passes” which signal the end of an “ascent” and the beginning of a “descent”, and areas with motionless lines and so forth, even tears… The notion of progress would thus become regional, like that of decadence, accumulation, cessation, etc., and so it would become crude to talk about the progress of science, the decadence of a civilisation, the maturity of a revolutionary situation, or the genesis of psychology, because this would be to reduce a global evolution to a linear sequence. The fact that as we write this we have before our eyes a mathematical model unknown to Leibniz but conceived by Euler, makes the point that we must enrich our models of thinking which are, generally speaking, lamentably poor. Whereas science noticeably employs incredibly fine-grained structures, we are still philosophising with the help of unrefined models and schemas, with techniques of thought which have hardly progressed.
One way to understand Serres’s colossal study of Leibniz is as just such a way of enriching our hitherto lamentably poor philosophical models of thinking. And one of the important ways Serres challenges and develops our impoverished philosophical models is by drawing on his (and Leibniz’s) first-rate understanding of models in mathematics.
[1] Michel Serres, Le Système de Leibniz et ses modèles mathématiques (Paris: Presses Universitaires de France, 1968) 285-6.