Report: International workshop

Bergsonian « duration » and its mathematical ranges


29 Sunday, September 

University of Tokyo, Hongo Campus, Faculty of Law&Letters, Bldg. 1, Room 215

Organizer: Takuya Nagano

(This workshop is supported by: the Grant-in-Aid for Scientifc Research (C)

under Grant No.JP17K02200 from JSPS (Japan Society for the Promotion of Science) )


Session 1

1. On the Problem of Multiplicity in Early Bergson: from Time and Free Will to Matter and Memory

    Ryusuke Okajima (Keio University)

2. La structure bergsonienne et géométrique de la mémoire  

    Sébastien Miravete (Université Toulouse Jean-Jaurès (France))

Session 2

1.The non-local character of duration as a factor of indeterminacy

    Élie During (Université Paris Nanterre (France))

2. La durée bergsonienne entre l’occurrent et le continuant : son statut dans les Cours au Collège de France 1902-1903

    Takuya Nagano (Kumamoto National Institute of Technology) 

General Discussion


    This International workshop was organized as a part of the research project of Takuya Nagano, supported by the Grant-in-Aid for Scientific Research. The research is about the development of the relation between Bergson’s philosophical theory of duration and his conception about mathematical objects or concepts. For this workshop we chose the researchers who had interpreted Bergsonian theory of duration connecting it in their manner to his thought about mathematical objects. Discussing with them, we attempted to obtain the fruitful suggestions for forthcoming improvement of the researches of this domain. We abbreviate as follows the contents of presentations and discussions in order. 


    Ryusuke Okajima examined the development of concepts of the multiplicity from Time and Free Will (TFW) to Matter and Memory. He confirmed that there are two kinds of multiplicity in TFW; the numerical multiplicity and the qualitative multiplicity. Criticizing the position of Miravete, who had interpreted the duration in TFW as a sort of number distinct from the space, Okajima indicated that the qualitative multiplicity which characterizes the duration in TFW is not numerical enough, for its components are too heterogeneous to be numerical unities, whose components are mutually homogeneous. Such a homogeneity is brought, according to Okajima, by the multiplicity of the vibrations constituting physical world in Matter and Memory. He calls it “sensible multiplicity”. For him it is the sensible multiplicity which really deserves the name of numerical multiplicity. Miravete replied to this indicating that the notion of the duration as a kind of number is elaborated in TFW. Indeed, in many passages, Bergson uses the word "number"(nombre) and the components are never absolutely heterogeneous (the fundamental idea of “nuance” would show it). 


    Sébastien Miravete indicated that in Matter and Memory, Bergson tried to found, in the function of perception and memory, certain dynamical geometric structures, which remain the same despite the variation and diversity of their materials. Though he seems to suggest a certain kind of intimacy between such structures and mathematical equivalence class, he insisted on the dynamical nature of the former Bergsonian structure. From his interpretation an intense discussion took place with Elie During. Miravete read a dynamical structure in the whole of the memory as function (mémoire). For him the virtuality of memory indicated by Bergson consists principally in the potentiality of such a dynamical structure of memory. Then each memory as a content (souvenir) and the plane constituted by the whole of mnemic content has no “virtuality” (in Deleuze’s sense) in itself. In other words, the planes of consciousness in pure memory appear successively, one after the other. They do not coexist. Contrary to this view, During claimed that memories “in themselves”, which cannot be reduced to discrete duplicates of perceptual episodes stored up in memory, enjoy a non-local (and thus virtual) mode of existence at various degrees of “resolution” in the psychological unconscious. As such, their unity is distributed across a multiplicity of coexisting planes which consciousness actualizes in turn in its acts of remembering. Such a multiplicity is qualitative in nature. The element of virtuality affecting the unity of what we like to refer as “a particular memory” should not be any more mysterious than what is already involved in any durational reality, namely: the virtual continuation of the past within the present. Here we are dealing with a virtual contamination of each particular memory by the past as a whole. An unconscious elaboration of memories is constantly going on, and it is partly independent from our actively trying to recollect them. 


    Elie During’s presentation reveals that his criticism of Miravete derives from his own views on the coexistence of durations, beyond the particular case of human memory. According to him, genuine indeterminacy is inseparable from the way degrees of separation between durations are expressed in the form of relations of real simultaneity within the universe. The theory of relativity provides a formalization of this general situation by defining simultaneity in terms of spatially separated events in space-time, events among which no place-to-place causal connection is possible. Such a separation, which During suggests we consider from the perspective of an extended understanding of non-locality, permits temporal series to encounter in unpredictable ways. This perspective was introduced by reference to the theory of relativity, to quantum mechanics, as well to Cournot’s definition of “chance” as an expression of the interference of mutually independent causal series. During then showed how it resonated with certain holistic aspects of Bergson’s theory of duration. If the realization of a state of affairs within a duration is by itself truly unpredictable for Bergson, it is in virtue of incorporating the continuous “action” of the past as such, which functions as a temporal equivalent of Leibniz’ principle of indiscernibles, introducing a measure of difference within repetition. But the effectiveness of the past, bearing as a whole upon each present, is homologous to the way coexistent durations matter to each other—even when their coexistence is loose, involving an element of separation. This general interpretation can also draw from Bergson’s understanding of the virtual coexistence of memories. If the structural whole of the memory is dynamical and allows unpredictable creative encounters and interferences, it is because the elements of such a structure bear simultaneity relations to each other and are thus “connected” in a non-local sense. Mnemic contents do not connect through definite, traceable paths, nor are they separated by explicit borders. The coexistence of memory contents, as well as the coexistence of the planes across which they distribute themselves, is virtual. Similarly, the relations of simultaneity at a distance are virtual (and absolute) from each duration’s perspective, although they may appear to be actual (and relative) in the eye of the physicist “surveying” the entire field of coexistence from an abstract, non-situated viewpoint. The truth is that even for the physicist, simultaneity relations can only be determined in retrospect, when durations meet in a local relation of co-presence and their past coexistence takes on concrete form.


    In this way, Miravete’s and During’s respective claims seemed to rely on particular conceptions of the part-whole relation in Bergson’s theory of duration. Takuya Nagano tried in his manner to seize the nature of the part-whole relation in the philosophy of Bergson. He chose as his domain of examination the Bergson’s lectures given at the Collège de France in 1902-1903. Nagano introduced two sorts of temporal persistence, following P. Simons; on the one hand, there is “occurrent (or perdurant)” which has its temporal parts and on the other hand “continuant (or endurant)” which does not have its temporal parts. The continuant is interpreted as being found upon the occurrents so long as they form a structure of equivalent class. Bergsonian analysis of traditional philosophical concepts of temporal realities seems to present these concepts as continuants, in the sense of equivalent classes. Criticizing the deforming concept of temporal realities, Bergson indicated that the duration has no parts. In this sense the duration of Bergson is not clearly classified neither to continuant nor occurrent. The duration contracts however the other durations whose rhythms are more diluted. With this constitution, the duration described in the lectures seems to be a kind of structural whole, which forms an equivalent class contracting diluted durations. Bergson analyzed the birth of modern mechanics in this framework. The duration here is considered to contain many directions. Mathematically, these directions are described as forming a structure of equivalent class, keeping a certain quantity invariant. In this way, the duration is considered to be a structure of basis for the mathematical structure. This structure in the duration however could not be conceptualized in mathematical way.


    Unfortunately, in the second session, we did not have enough time to discuss about the presentations. But in both presentations, structural features of the duration are emphasized through the examination of Bergsonian theory of duration. According to During moreover, his perspective about coexistence of separated simultaneities is verified by some passages of “Introduction to metaphysics” of 1903. As Nagano indicated, Bergson’s lectures of almost the same period suggest a similar view to During’s conception of coexistence, concerning the directions contracted in the duration. In addition, besides the common points with the interpretation of Miravete, the notion of sensible multiplicity indicated by Okajima is obtained from coexistence described by Bergson of plural durations. In this way, we could estimate that this workshop as a whole has brought several common topics for farther development of researches.


    But actually, the discussion between Miravete and During was interrupted. We asked them to resume it, in the form of their papers. Perhaps we could place them here in the near future. And if the presentations in this workshop will be inserted in some journals, we will note their information here, in so far as it will be allowed (for a moment we place here the text of the presentation of Nagano).


    At the end of this brief report, we express our deep gratitude for the collaboration of the Society for the Study of the Philosophy of Bergson in Japan, notably for the attentive preparations for the event by Mayuko Nakahara, Akiko Nose and Hideki Mochiji, who had dedicated their precious time to it. We are also grateful for the kind support by the Project Bergson in Japan (PBJ), namely intimate advises and participations of Yasushi Hirai and Hisashi Fujita, who had accepted this report as a page of the site of their association. This workshop project was realized with these sympathetic supports by colleagues.


(Takuya Nagano)

Japanese version is here.


UPDATE(2023/01/26):A transcript of the discussion between Elie During and Sébastian Miravète on the coexistence of memory planes on the day will be available online (Bergsoniana no. 3).(Japanese Translation PDF