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by Corry Shores
[Search Blog Here. Index tabs are found at the bottom of the left column.]
[Zeno’s Paradox, entry directory]
[Bergson’s Matter and Memory, entry directory]
[The following is summary. Boldface in quotation and bracketed commentary are my own. Proofreading is incomplete, so please forgive my typos. Citations give the pages for the 1939 French edition first; then the 2004 English. Or they will indicate the publication’s date before the page number. Paragraph enumerations and section divisions follow those in the French edition.]
Summary of
Henri Bergson
Matière et mémoire
Matter and Memory
Ch.4
De la délimitation et de la fixation des images. Perception et matière. Âme et corps.
The Delimiting and Fixing of Images. Perception and Matter. Soul and Body
4.2
Tout mouvement est indivisible
Indivisibility of Movement
Brief summary:
We often think of motion as having parts corresponding to the space it travels through and to the length of time it takes to complete. But in fact, movement is absolutely indivisible, spatially and temporally. We find evidence of its indivisibility both in our experience and in the absurdities of Zeno’s paradoxes. {1} In our experiences of our own actions, like of our hands moving from point A to point B, we feel it as one continuous, unified (although complex) action that we are aware of during a unified and flowing act of consciousness. We do not experience our hand’s motion as divisible into smaller segments, and surely not infinitely divisible. However, after the action is completed, we think abstractly about the space covered, and we understand it as having the properties of a geometrical line. We thereby conclude that our hand occupied an infinite series of points or positions in the course of its movement. We furthermore think that to each position there corresponds an indivisible instant of time when it was there. But in reality, given the indivisibility of motion, we can only say that while it is moving, it indeterminately does so as it changes place through duration. And if we artificially do make cuts in time and try to coordinate them to points in space, the best we can say is that at some moment the object is passing at some point. It never determinately occupies a point. Under this artificial view where we abstractly divide up time, we must [use paraconsistent reasoning to] claim that at some instant the object occupies not simply one point. The only way an object can determinately occupy a single position is if it stops moving. So Bergson distinguishes (under this artificialized view of the abstract time and space of movement): {1} passage at a point, from {2} halting at a point. The other evidence we have of the indivisibility of motion is the absurdity of Zeno’s paradoxes; in other words, the reason his conclusions are absurd is because he begins with the false assumption that the space or the time of motion can be thought of as being apart from the motion itself and as having the geometrical properties of a line. In “the Dichotomy” and “Achilles” arguments, the space traversed is infinitely divided; but we should have begun by assuming it cannot be, thereby avoiding the absurd conclusions. In “the Arrow” argument, the points or positions that the arrow passes through are understood spatially and thus as immobile. Zeno’s error then is that he wrongly took this also to mean that the arrow’s motion is composed of immobile poses at each point. Instead, we should have assumed either that the motion cannot be decomposed into component positions, or if we do admit of such an abstraction, we cannot say that at some instant it is determinately in (or halting at) one position; rather we should say that it is covering more than (or passing at) one position. And in “the Stadium” argument, Zeno wrongly assumes that the durations of the movements are not inherent to them but are rather determined by the objects’ speeds in relation to the relative distances covered. He also wrongly assumes that there is not one duration common to all movements, as for example the single flowing duration of an observer of the entire situation.
Summary
“Movement is indivisible; it is only the trajectory of a moving body that is divisible”
4.2.1
[It is a fact that every movement is absolutely indivisible.]
Bergson says that the following claim is not a hypothesis, but is rather a fact often mistaken for a hypothesis [or somehow confused with hypotheses related to it]: “Every movement, inasmuch as it is a passage from rest to rest, is absolutely indivisible”.
I. - Tout mouvement, en tant que passage d'un repos à un repos, est absolument indivisible.
Il ne s’agit pas ici d’une hypothèse, mais d’un fait, qu’une hypothèse recouvre généralement.
(1939: 209. Text copied from UQAC)
I. – Every movement, inasmuch as it is a passage from rest to rest, is absolutely indivisible.
This is not an hypothesis, but a fact, generally masked by an hypothesis.
(2004: 246. Text copied from Mead Project)
4.2.2
[When we move our hand from point A to B, we are conscious of a single act of a unified, indivisible motion. The spatial path the hand produced is infinitely divisible, and we are tempted to say the motion is too.]
Bergson demonstrates this by having us consider a motion of our hand, which we move from point A to B. We are consciously aware of a single act made of one continuous motion and not of a segmented set of motions. Our vision detects a line between points A and B, which are divisible, however. We are tempted at first to say that the motion is divisible like the space traversed.
Voici, par exemple, ma main posée au point A. Je la porte au point B, parcourant d’un trait l’intervalle. Il y a dans ce mouvement, tout à la fois, une image qui frappe ma vue et un acte que ma conscience musculaire saisit. Ma conscience me donne la sensation intérieure d’un fait simple, car en A était le repos, en B est le repos encore, et entre A et B se place un acte indivisible ou tout au moins indivisé, passage du repos au repos, qui est le mouvement même. Mais ma vue perçoit le mouvement sous forme d’une ligne AB qui se parcourt, et cette ligne, comme tout espace, est indéfiniment décompo- | sable. Il semble donc d’abord que je puisse, comme je voudrai, tenir ce mouvement pour multiple ou pour indivisible, selon que je l’envisage dans l’espace ou dans le temps, comme une image qui se dessine hors de moi ou comme un acte que j’accomplis moi-même.
(1939: 209-210. Text copied from UQAC)
Here, for example, is my hand, placed at the point A. I carry it to the point B, passing at one stroke through the interval between them. There are two things in this movement: an image which I see, and an act of which my muscular sense makes my consciousness aware. My consciousness gives me the inward feeling of a single fact, for in A was rest, in B there is again rest, and between A and B is placed an indivisible or at least an undivided act, the passage from rest to rest, which is movement itself. But my sight perceives the movement in the form of a line AB which is traversed and this line, like all space, may be indefinitely divided. It seems then, at first sight, that I may at will take this movement to be multiple or indivisible, according as I consider it in space or in time, as an image which takes shape outside of me or as an act which I am myself accomplishing.
(2004: 246. Text copied from Mead Project)
4.2.3
[Were a moving object to occupy a determinate position, it would actually not be in motion; for, objects in motion occupy space indeterminately. We must distinguish occupation at a point, which is what happens when a moving object stops its motion, from passage at a point, which happens when a moving object is still in motion in a certain area. Motion is not divisible like space is. (Instead, we must think of a moving object always as going beyond any of the points it is located at during some ((small)) interval of time.)]
[His next points are very important, but I may not restate them perfectly well. He writes: “Yet, when I put aside all preconceived ideas, I soon perceive that ... even my sight takes in the movement from A to B as an indivisible whole, and that if it divides anything, it is the line supposed to have been traversed, and not the movement traversing it”. Maybe the idea here is that the line traversed by the hand is a mathematical abstraction projected upon the real space of the motion, and that we did not directly perceive it; we rather only perceived the motion itself as it was happening. His next point is the important one here. He seems to distinguish occupation at a position (or point) and passage at (or through or across) a position or point: “It is indeed true that my hand does not go from A to B without passing through the intermediate positions, and that these intermediate points resemble stages, as numerous as you please, all along the route ; but there is, between the divisions so marked out and stages properly so called, this capital difference, that at a stage we halt, whereas at these points the moving body passes.” It is still not perfectly clear to me what it means to pass at a point rather than to halt at a point. To halt at the point means to determinately occupy it, with that occupation being no different than were it at rest at that point. The only way I have so far to understand passage at a point is that it is not determinately occupying it like it would at rest, meaning perhaps that it is both there and not there at the same time, or at least that it is there and near there at the same time. Does this mean that at a certain moment it occupies a fuzzy or blurry region where no precise location can be specified? In that case, passing at a point would still mean residing throughout a small surrounding region, or in other words, occupying many positions at once. Bergson next writes: “Now a passage is a movement and a halt is an immobility. The halt interrupts the movement ; the passage is one with the movement itself.” It is clear how the halt interrupts the movement. But what does it mean for the passage being one with the movement? If he means “passage” as the total passage, that would not seem interesting to say (‘the movement from A to B is the passage from A to B’), and it would not make “passage” be parallel to “halt”, which happens at a point. So perhaps he means “passage” here as passage at a point or at least passage in the region of some point. Were that the case, then it would mean that passage at (or around) a point is one with the whole movement itself, and this is a much more interesting claim, although not one I fully comprehend. It would suggest that although we might want to say the object at a certain time is at a certain location, it would be more accurate to say that it is passing that location, and that passage is not distinguishable or disentangleable from the whole motion enveloping that part. Let us pursue this concept. Is this so, because ... {1} The whole movement occupies an indivisible block of time, and thus we cannot pinpoint some instant where it is at some location. Rather, during a block of time it is within a block of space. We might call it the “block-block” theory of motion, in keeping with Russell’s “at-at” theory. (This interpretation stays closely to the wording “the passage is one with the movement itself”. But it seems odd to say that the duration of time admits of no partition where the position of the object cannot be narrowed down. If we watch an ant move across a sidewalk while cars drive down the adjacent road, surely we can say that after so many cars go by, the ant is still on the first half of the sidewalk, and after so many more go by, it is on the other half. In other words, even if I see the motion of the ant as unified, it is not so obvious to me that the space-time locations cannot at least be distinguished relative to one another. And surely we are not saying that during a certain passage of time, it is always in all locations all throughout that time. The object is in some region at one phase and another region at another phase. So I am not convinced yet that Bergson is making a generalized block-block conception.) (Or is the passage the same as the whole movement for the reason that ...) {2} Because the object is always in motion, it cannot be determinately located at any point; for, then it is at rest. It must instead, at some instant, be indeterminately at some point. One reason this could be is that at some instant, it is located at more that one point, or it is located between successive points in an infinitesimal interval. He next writes: “When I see the moving body pass any point, I conceive, no doubt, that it might stop there;” (here we are thinking of the motion in its present activity, and we think at any moment it might stop at its given location) “and even when it does not stop there, I incline to consider its passage as an arrest, though infinitely short, because I must have at least the time to think of it;” (and even though its motion does not pause, we think that we might have caught it at an infinitely brief part of its movement where it occupies a specific position. But I do not know what he means by “because I must have at least the time to think of it.” Perhaps he means that at some moment we think of it being there, and since that thought occupies a moment we might assume the motion we observe also can be contained in that moment of thought. But I doubt that is the meaning, however;) “but it is only my imagination which stops there, and what the moving body has to do is, on the contrary, to move.” (This might mean that as we are watching the movement, we imagine its continuously varying location in space, and since our imagination of that movement can pause, we assume the movement can be understood as having very brief pauses where it occupies a determinate position, corresponding to where our imagination stops following the progress.) “As every point of space necessarily appears to me fixed, I find it extremely difficult not to attribute to the moving body itself the immobility of the point with which, for a moment, I make it coincide ; it seems to me, then, when I reconstitute the total movement, that the moving body has stayed an infinitely short time at every point of its trajectory.” (So because we imagine these positions during its motion, we then think we can reconstitute the motion by connecting the fabricated points or positions.) “But we must not confound the data of the senses, which perceive the movement, with the artifice of the mind, which recomposes it. The senses, left to themselves, present to us the real movement, between two real halts, as a solid [247|248] and undivided whole. The division is the work of our imagination, of which indeed the office is to fix the moving images of our ordinary experience, like the instantaneous flash which illuminates a stormy landscape by night.” (These passages reiterate the point that the senses perceive the unified whole of the motion, but the imagination is what artificially creates the cuts dividing it. The interesting part here is the vibrant metaphor of the flash of lightning giving us a snapshot of a dark stormy scene being like the way the imagination makes instantaneous immobile snapshots of activity).]
Toutefois, en écartant toute idée préconçue, je m’aperçois bien vite que je n’ai pas le choix, que ma vue elle-même saisit le mouvement de A en B comme un tout indivisible, et que si elle divise quelque chose, c’est la ligne supposée parcourue et non pas le mouvement qui la parcourt. Il est bien vrai que ma main ne va pas de A en B sans traverser les positions intermédiaires, et que ces points intermédiaires ressemblent à des étapes, en nombre aussi grand qu’on voudra, disposées tout le long de la route ; mais il y a entre les divisions ainsi marquées et des étapes proprement dites cette différence capitale qu’à une étape on s’arrête, au lieu qu’ici le mobile passe. Or le passage est un mouvement, et l’arrêt une immobilité. L’arrêt interrompt le mouvement ; le passage ne fait qu’un avec le mouvement même. Quand je vois le mobile passer en un point, je conçois sans doute qu’il puisse s’y arrêter ; et lors même qu’il ne s’y arrête pas, j’incline à considérer son passage comme un repos infiniment court, parce qu’il me faut au moins le temps d’y penser; mais c’est mon imagination seule qui se repose ici, et le rôle du mobile est au contraire de se mouvoir. Tout point de l’espace m’apparaissant nécessairement comme fixe, j’ai bien de la peine à ne pas attribuer au mobile lui-même l’immobilité du point avec lequel je le fais pour un moment coïncider ; il me semble alors, quand je reconstitue le mouvement total, que le mobile a stationné un temps infiniment court à tous les points de sa trajectoire. | Mais il ne faudrait pas confondre les données des sens, qui perçoivent le mouvement, avec les artifices de l’esprit qui le recompose. Les sens, laissés à eux-mêmes, nous présentent le mouvement réel, entre deux arrêts réels, comme un tout solide et indivisé. La division est l’œuvre de l’imagination, qui a justement pour fonction de fixer les images mouvantes de notre expérience ordinaire, comme l’éclair instantané qui illumine pendant la nuit une scène d’orage.
(1939: 210-211. Text copied from UQAC)
Yet, when I put aside all preconceived ideas, I soon perceive that I have no such choice, that even my sight takes in the movement from A to B as an indivisible whole, and that if it divides anything, it is the line supposed to have been traversed, and not the movement traversing it. It is indeed [246|247] true that my hand does not go from A to B without passing through the intermediate positions, and that these intermediate points resemble stages, as numerous as you please, all along the route ; but there is, between the divisions so marked out and stages properly so called, this capital difference, that at a stage we halt, whereas at these points the moving body passes. Now a passage is a movement and a halt is an immobility. The halt interrupts the movement ; the passage is one with the movement itself. When I see the moving body pass any point, I conceive, no doubt, that it might stop there; and even when it does not stop there, I incline to consider its passage as an arrest, though infinitely short, because I must have at least the time to think of it; but it is only my imagination which stops there, and what the moving body has to do is, on the contrary, to move. As every point of space necessarily appears to me fixed, I find it extremely difficult not to attribute to the moving body itself the immobility of the point with which, for a moment, I make it coincide ; it seems to me, then, when I reconstitute the total movement, that the moving body has stayed an infinitely short time at every point of its trajectory. But we must not confound the data of the senses, which perceive the movement, with the artifice of the mind, which recomposes it. The senses, left to themselves, present to us the real movement, between two real halts, as a solid[247|248] and undivided whole. The division is the work of our imagination, of which indeed the office is to fix the moving images of our ordinary experience, like the instantaneous flash which illuminates a stormy landscape by night.
(2004: 246-248. Text copied from Mead Project)
4.2.4
[Again, we mistake the line traced by the object with the motion that did the tracing. The motion is not divisible into immobile spatial points.]
[Bergson restates these notions: the line that we think the moving object draws is divisible and composed of points which are immobile, but the movement itself is not made of such rests and never determinately occupied these points.]
Nous saisissons ici, dans son principe même, l’illusion qui accompagne et recouvre la perception du mouvement réel. Le mouvement consiste visiblement à passer d’un point à un autre, et par suite à traverser de l’espace. Or l’espace traversé est divisible à l’infini, et comme le mouvement s’applique, pour ainsi dire, le long de la ligne qu’il parcourt, il paraît solidaire de cette ligne et divisible comme elle. Ne l’a-t-il pas dessinée lui-même ? N’en a-t-il pas traversé, tour à tour, les points successifs et juxtaposés ? Oui sans doute, mais ces points n’ont de réalité que dans une ligne tracée, c’est-à-dire immobile ; et par cela seul que vous vous représentez le mouvement, tour à tour, en ces différents points, vous l’y arrêtez nécessairement; vos positions successives ne sont, au fond, que des arrêts imaginaires. Vous substituez la trajectoire au trajet, et parce que le trajet est sous-tendu par la trajectoire, vous croyez qu’il coïncide avec elle. Mais comment un progrès coïnciderait-il avec une chose, un mouvement avec une immobilité ?
(1939: 211. Text copied from UQAC)
We discover here, at its outset, the illusion which accompanies and masks the perception of real movement. Movement visibly consists in passing from one point to another, and consequently in traversing space. Now the space which is traversed is infinitely divisible ; and as the movement is, so to speak, applied to the line along which it passes, it appears to be one with this line and, like it, divisible. Has not the movement itself drawn the line ? Has it not traversed in turn the successive and juxtaposed points of that line ? Yes, no doubt, but these points have no reality except in a line drawn, that is to say motionless; and by the very fact that you represent the movement to yourself successively in these different points, you necessarily arrest it in each of them ; your successive positions are, at bottom, only so many imaginary halts. You substitute the path for the journey, and because the journey is subtended by the path you think that the two coincide. But how should a progress coincide with a thing, a movement with an immobility ?
(2004: 248. Text copied from Mead Project)
4.2.5
[We want to think that there is a continuous correlation between the geometricized spatial trail of the moving object and the temporal duration of that motion. But duration is something alive and active in the present, and what happens in that duration is not determined by past moments, just as the motion is not either. So in truth, duration is not continuously correlated with the spatialized trail. That furthermore means that we cannot correlate the indivisible points in the geometricized trail with indivisible instants in the duration of the motion. (Instead, we should see the duration of the motion as the living present of that movement, which has no parts to be divided.)]
[We now get a clear statement from Bergson that there can be no real indivisible instants. Let us go part by part. “What facilitates this illusion is that we distinguish moments in the course of duration, like halts in the passage of the moving body.” (The illusion I think is the illusion that motion is made of the spatial points that the imagination projects upon it.) “Even if we grant that the movement from one point to another forms an undivided whole, this movement nevertheless takes a certain time;” (I am not sure why it is formulated this way as if there is a conceptual tension between a movement being whole and it lasting a duration of time, but that is the point here;) “so that if we carve out of this duration an indivisible instant, it seems that the moving body must occupy, at that precise moment, a certain position, which thus stands out from the whole.” (So since we have established a correlation between the temporal and spatial extents of the motion, and since this correlation is continuous with the motion, we think that if we pinpoint some precise indivisible moment in time, we will find the object at some precise spatial location. Note here that he portrays this understanding of the instant as seeing it as standing outside the flow of the motion’s duration. In other words, the temporality of the motion is such that it is internally “organized” or integrated, and any attempt to pinpoint a precise moment would be to extract it from that internal integration somehow and thus to no longer make it a real part of that duration.) “The indivisibility of motion implies, then, the impossibility of real instants;” (Here he is saying that if the motion is indivisible, this means it must be both spatially and temporally indivisible. I am not entirely sure I understand why, but it might be because the correlation between time and space is continuous, and we are assigning them both the same properties of a geometrical continuum like a line. For the same reason it never occupies a determinate point of space it also does not occupy a determinate point in time. If that is the case, then perhaps he is giving a “block-block” theory of motion where the best we can say is that a unified act of motion involves a block of time and a block of space, but no further determinations can be made whatsoever regarding more precise locations and phases within those blocks;) “and indeed, a very brief analysis of the idea of duration will show us both why we attribute instants to duration and why it cannot have any.” (He continues:) “Suppose a simple movement like that of my hand when it goes from A to B. This passage is given to my consciousness as an undivided whole.” (This is the same example before of the hand moving from A to B, only now there is emphasis on our consciousness of this passage.) “No doubt it endures ; but this duration, which in fact coincides with the aspect which the movement has inwardly for my consciousness, is, like it, whole and undivided.” (So the movement of the hand is undivided, and so too is the duration of its motion, corresponding to the undivided duration of our consciousness of the motion.) “Now, while it presents itself, qua movement, as a simple fact, it describes in space a trajectory which I may consider, for purposes of simplification, as a geometrical line;” (we have encountered this notion many times. The hand traces a path in space, and that traced path can be understood as a geometrical line;) “and the extremities of this line, considered as abstract limits, are no longer lines, but indivisible points.” (The idea here seems to be that the traced line has ends to it, which as terminations of the line, are not lines but are rather points.) “Now, if the line, which the moving body has described, measures for me the duration of its movement, must not the point, where the line ends, symbolize for me a terminus of this duration ?” (He seems to be saying that since we coordinate the spatial extent with the temporal extent, and since the spatial ends are terminations of the line, they must correspond to temporal termination points to the duration.) “And if this point is an indivisible of length, how shall we avoid terminating the duration of the movement by an indivisible of duration ?” (The spatial terminating point is indivisible, so the temporal terminating point must be also ((or else the spatial and temporal extents would not be correlated)).) “If the total line represents the total duration, the parts of the line must, it seems, correspond to parts of the duration, and the points of the line to moments of time.” (Since the whole temporal and spatial extents correlate, and because that correlation is continuous, that means any part of the one must correspond to its counterpart in the other. It is not stated here if the parts should be understood as intervals or as points, but it would seem to apply to both sorts.) “The indivisibles of duration, or moments of time, are born, then, of the need of symmetry; we come to them naturally as soon as we demand from space an integral presentment of duration.” (I think he is saying that since we make this strict correlation between space to time, that means so long as we think space is made of indivisible points, we must also conceive there being indivisible instants). “– But herein, precisely, lies the error. While the line AB symbolizes the duration already lapsed of the movement from A to B already accomplished, it cannot, motionless, represent the movement in its accomplishment nor duration in its flow.” (I am not entirely sure here, but the idea might be the following. We said before that the line traced can only be understood in this spatial way after the motion is completed and the line is finished. It was not clear why, but that was the claim. Thus this spatial line can only be thought of as corresponding to a finished duration and not to the duration as it is happening. I am still not sure why this is exactly, beside simply appealing to the claim that the motion is indivisible in action but divisible in completion. But why? Does it have something to do with indeterminacy of where it is going or whether it will continue or not? Is it because the motion is in the present ((and is open to unpredictable variation) but the spatial and temporal trail are in the past, which is fixed and determined?) “And from the fact that this line is divisible into parts and that it ends in points, we cannot conclude either that the corresponding duration is composed of separate parts or that it is limited by instants.” (Here he seems to be calling into question the correlation between time and space. He seems to be saying that time must be understood as duration in his sense rather than in the sense of a geometrical sort of line, and thus we cannot say that duration is made of indivisible parts. But most interesting here is to say that the duration does not terminate at instants. Does he mean that it does not terminate? Does he mean that it terminates, but in an indeterminate way?)]
Ce qui facilite ici l’illusion, c’est que nous distinguons des moments dans le cours de la durée, comme des positions sur le trajet du mobile. À supposer que le mouvement d’un point à un autre forme un tout indivisé, ce mouvement n’en remplit [211|212] pas moins un temps déterminé, et il suffit qu’on isole de cette durée un instant indivisible pour que le mobile occupe à ce moment précis une certaine position, qui se détache ainsi de toutes les autres. L’indivisibilité du mouvement implique donc l’impossibilité de l’instant, et une analyse très sommaire de l’idée de durée va nous montrer en effet, tout à la fois, pourquoi nous attribuons à la durée des instants, et comment elle ne saurait en avoir. Soit un mouvement simple, comme le trajet de ma main quand elle se déplace de A en B. Ce trajet est donné à ma conscience comme un tout indivisé. Il dure, sans doute; mais sa durée, qui coïncide d’ailleurs avec l’aspect intérieur qu’il prend pour ma conscience, est compacte et indivisée comme lui. Or, tandis qu’il se présente, en tant que mouvement, comme un fait simple, il décrit dans l’espace une trajectoire que je puis considérer, pour simplifier les choses, comme une ligne géométrique ; et les extrémités de cette ligne, en tant que limites abstraites, ne sont plus des lignes mais des points indivisibles. Or, si la ligne que le mobile a décrite mesure pour moi la durée de son mouvement, comment le point où la ligne aboutit ne symboliserait-il pas une extrémité de cette durée ? Et si ce point est un indivisible de longueur, comment ne pas terminer la durée du trajet par un indivisible de durée ? La ligne totale représentant la durée totale, les parties de cette ligne doivent correspondre, semble-t-il, à des parties de la durée, et les points de la ligne à des moments du temps. Les indivisibles de durée ou moments du temps naissent donc d’un besoin de symétrie; on y aboutit naturellement dès qu’on demande à l’espace une représentation intégrale de la durée. Mais voilà précisément l’erreur. Si la ligne AB symbolise la durée écoulée du mouvement [212|213] accompli de A en B, elle ne peut aucunement, immobile, représenter le mouvement s’accomplissant, la durée s’écoulant ; et de ce que cette ligne est divisible en parties, et de ce qu’elle se termine par des points, on ne doit conclure ni que la durée correspondante se compose de parties séparées ni qu’elle soit limitée par des instants.
(1939: 211-213. Text copied from UQAC)
What facilitates this illusion is that we distinguish moments in the course of duration, like halts in the passage of the moving body. Even [248|249] if we grant that the movement from one point to another forms an undivided whole, this movement nevertheless takes a certain time ; so that if we carve out of this duration an indivisible instant, it seems that the moving body must occupy, at that precise moment, a certain position, which thus stands out from the whole. The indivisibility of motion implies, then, the impossibility of real instants ; and indeed, a very brief analysis of the idea of duration will show us both why we attribute instants to duration and why it cannot have any. Suppose a simple movement like that of my hand when it goes from A to B. This passage is given to my consciousness as an undivided whole. No doubt it endures ; but this duration, which in fact coincides with the aspect which the movement has inwardly for my consciousness, is, like it, whole and undivided. Now, while it presents itself, qua movement, as a simple fact, it describes in space a trajectory which I may consider, for purposes of simplification, as a geometrical line; and the extremities of this line, considered as abstract limits, are no longer lines, but indivisible points. Now, if the line, which the moving body has described, measures for me the duration of its movement, must not the point, where the line ends, symbolize for me a terminus of this duration ? And if this point is an indivisible of length, how shall we avoid terminating the duration of the movement by an indivisible of duration ? If [249|250] the total line represents the total duration, the parts of the line must, it seems, correspond to parts of the duration, and the points of the line to moments of time. The indivisibles of duration, or moments of time, are born, then, of the need of symmetry; we come to them naturally as soon as we demand from space an integral presentment of duration. – But herein, precisely, lies the error. While the line AB symbolizes the duration already lapsed of the movement from A to B already accomplished, it cannot, motionless, represent the movement in its accomplishment nor duration in its flow. And from the fact that this line is divisible into parts and that it ends in points, we cannot conclude either that the corresponding duration is composed of separate parts or that it is limited by instants.
(2004: 248-250. Text copied from Mead Project)
“Zeno transfers to the moving body the properties of its trajectory: hence all the difficulties and contradictions”
4.2.6
[Zeno’s paradoxes of motion involve a confusion of the space travelled with the motion itself and its real duration. In “the Dichotomy” and “the Achilles” arguments, the space traversed is infinitely divided, forgetting that motion itself cannot be. In “the Arrow” argument, the immobility of the points of the space travelled are wrongly thought to mean that the arrow’s motion is composed of immobile poses at each point. And in “the Stadium” argument, we wrongly think that the duration of the movement should be calculated based on the relative spatial distances covered, and we forget there is one duration that comprehends all the motions involved.]
Bergson claims that all of Zeno’s paradoxes of motion result from this error of confusing the properties of real motion and real duration with the properties of geometrical lines. Zeno was guided by common sense to conceive the movement in terms of the trajectory or traversed path, and he was guided by language to conceive of movement and duration in terms of space. Common sense and language, for their own purposes, treat becoming as a thing, and thereby ignore “the interior organization of movement”. In practical life, there are two facts that lead common sense to spatialize the movement: {1} the fact that every movement describes a space (by “describes” perhaps he means draws a path through or at least happens within), and {2} the fact that at every point of this described space the moving thing might stop. Zeno mistakenly holds these to be facts of motion. [Bergson then speaks of Zeno’s four arguments. They are summarized within parts II.C-E at this entry. The first one he calls “the Dichotomy”. I cannot tell if it refers to the paradox I numbered II.D or II.E. In II.D, the moving object, before reaching its destination, must first get half-way there. But before reaching the halfway point, it must get half that way (a quarter of the total distance). Since the distance is infinitely divisible, there is always a new halfway mark set away at some distance. In other words, it can never get past its starting position, because it can never reach a first halfway point. In II.E, we begin by noting that within a finite extent of motion, there are still an infinity of points for the moving body to cross. But arriving upon any point requires a finite amount of time. Thus, to cross the infinity of points within a finite range of motion will still take an infinite amount of time. Bergson describes it as: “By the first argument (the Dichotomy) he supposes the moving body to be at rest, and then considers nothing but the stages, infinite in number, that are along the line to be traversed we cannot imagine, he says, how the body could ever get through the interval between them.” (Note: the Internet Encyclopedia of Philosophy and the Stanford Encyclopedia of Philosophy list the dichotomy argument as the halving one.) Bergson’s comment here is not that we should conclude motion is impossible but rather that it cannot be constructed from a series of immobilities. (Bergson also says that this is “a thing no man ever doubted”, but Russell’s theory of motion in fact tries to construct movement this way.) Bergson says that the real question is whether or not the moving object passes through an infinity of points. He emphasizes again that the spatial trajectory is infinitely divisible but the movement (or the component movements) is not. The second Zeno argument is “the Achilles argument”. Here the Tortoise leads Achilles at the start of the race. To catch-up, Achilles needs to reach the Tortoise’s advanced position. Suppose he does. By that time, however, the Tortoise has advanced further. So long as they are both moving, the Tortoise will always lead, no matter how far ahead he is. This paradox, Bergson notes, requires that we consider the paths traversed as distinct from their movements and thereby as infinitely divisible. Rather, Achilles running is made of a number of bounds and the Tortoise a number of steps, and since Achilles’ bounds are much greater or faster, he will overtake the Tortoise. Zeno’s third argument is “the Arrow”. Here a moving arrow has a certain length, and that length equals the amount of space (the part of its path of movement) it occupies at any moment. But if it always occupies no more than its own length of space, then it is never moving (because it is never changing location). Here Bergson again reminds us that we are confusing the space of motion with the motion itself. Bergson then turns to Zeno’s fourth argument, “the Stadium”, about which Bergson writes, “which has, we believe, been unjustly disdained, and of which the absurdity is more manifest only because the postulate masked in the three others is here frankly displayed”, then in a footnote he explains what he means. (In this case, you have three objects of equal length and running on parallel tracks, moving past one another. The first one is stationary, and the other two move in opposite directions at the same speed, and coming together like so:
--AAAA-- --AAAA--
BBBB---- => --BBBB--
----CCCC --CCCC--
The conclusion in Aristotle is that “half a given time is equal to double that time” and in Bergson “a duration is the double of itself.” Think now of just B’s motion in relation to A’s. Suppose it is going one meter per second and each segment is one meter. It traveled two A segments, so it must have taken two seconds. Now consider B’s motion in relation to C’s. In the same motion, it passed four C’s. So it must have also taken four seconds to complete that same motion. (Possibly I have this wrong, but it is not very obvious how we reach the conclusion and thus how we are to portray the situation.) Now, for this to work (at least with how I set it up) and for us to follow what might be Bergson’s point here, we need to look at the assumptions that lead us to this conclusion. {1} The speeds of B and C are constant, and the lengths of A, B, and C are all the same. {2} The duration of the entire event is not absolutized; rather, it is calculated on the basis of the object’s given speed in relation to the distance traveled. {3} the spatial coordinates of the situation are not absolutized but are rather relativized, and thus the distances we use in our calculations are determined by an arbitrary selection of any two of the given bodies. Thus B can be said to move two different distances in the same stroke and thus its motion consumes two different durations. (I am ignoring other possibilities, like we compare the time of B’s movement to that of C, because I think it comes out the same but is less directly paradoxical.) From what I can tell, Bergson’s claim is that Zeno does not think of a real duration shared by all three and given directly to a consciousness aware of the entire event (I am thinking here of the simultaneity notion in Duration and Simultaneity §42 and discussed in Deleuze’s Bergsonism §76.) Bergson instead thinks the duration can be represented in terms of the space covered, as we see in our calculations. In other words, instead of the duration being understood (correctly) as part of all the motions and thus being just as indivisible as they are, duration is instead thought (incorrectly) to be a by-product of the space covered at a certain speed. (I might be misreading, so check the text below.) Bergson concludes this paragraph by reemphasizing that these paradoxes do not do justice to the lived duration of the motion and rather deal with its contorted abstraction in the mind, and he says all this discussion leads us to the conclusion that there are in fact real movements (the topic of the next section).]
Les arguments de Zénon d’Élée n’ont pas d’autre origine que cette illusion. Tous consistent à faire coïncider le temps et le mouvement avec la ligne qui les sous-tend, à leur attribuer les mêmes subdivisions, enfin à les traiter comme elle. À cette confusion Zénon était encouragé par le sens commun, qui transporte d’ordinaire au mouvement les propriétés de sa trajectoire, et aussi par le langage, qui traduit toujours en espace le mouvement et la durée. Mais le sens commun et le langage sont ici dans leur droit, et même, en quelque sorte, font leur devoir, car envisageant toujours le devenir comme une chose utilisable, ils n’ont pas plus à s’inquiéter de l’organisation intérieure du mouvement que l’ouvrier de la structure moléculaire de ses outils. En tenant le mouvement pour divisible comme sa trajectoire, le sens commun exprime simplement les deux faits qui seuls importent dans la vie pratique : 1º que tout mouvement décrit un espace ; 2º qu’on chaque point de cet espace le mobile pourrait s’arrêter. Mais le philosophe qui raisonne sur la nature intime du mouvement est tenu de lui restituer la mobilité qui en est l’essence, et c’est ce que ne fait pas Zénon. Par le premier argument Ca Dichotomie) on suppose le mobile au repos, pour ne plus envisager ensuite que des étapes, en nombre indéfini, sur la ligne qu’il doit parcourir : vous chercheriez vainement, nous dit-on, comment il arriverait à franchir l’intervalle. Mais on prouve [213|214] simplement ainsi qu’il est impossible de construire a priori le mouvement avec des immobilités, ce qui n’a jamais fait de doute pour personne. L’unique question est de savoir si, le mouvement étant posé comme un fait, il y a une absurdité en quelque sorte rétrospective à ce qu’un nombre infini de points ait été parcouru. Mais nous ne voyons rien là que de très naturel, puisque le mouvement est un fait indivisé ou une suite de faits indivisés, tandis que la trajectoire est indéfiniment divisible. Dans le second argument (l’Achille), on consent à se donner le mouvement, on l’attribue même à deux mobiles, mais, toujours par la même erreur, on veut que ces mouvements coïncident avec leur trajectoire et soient, comme elle, arbitrairement décomposables. Alors, au lieu de reconnaître que la tortue fait des pas de tortue et Achille des pas d’Achille, de sorte qu’après un certain nombre de ces actes ou sauts indivisibles Achille aura dépassé la tortue, on se croit en droit de désarticuler comme on veut le mouvement d’Achille et comme on veut le mouvement de la tortue : on s’amuse ainsi à reconstruire les deux mouvements selon une loi de formation arbitraire, incompatible avec les conditions fondamentales de la mobilité. Le même sophisme apparaît plus clairement encore dans le troisième argument (la Flèche), qui consiste à conclure, de ce qu’on peut fixer des points sur la trajectoire d’un projectile, qu’on a le droit de distinguer des moments indivisibles dans la durée du trajet. Mais le plus instructif des arguments de Zénon est Peut-être le quatrième (le Stade), qu’on a, croyons-nous, bien injustement dédaigné, et dont l’absurdité n’est plus manifeste que parce qu’on y voit étalé dans toute sa franchise le postulat [214|215] dissimulé dans les trois autres1. Sans nous engager ici dans une discussion qui ne serait pas à sa place, bornons-nous à constater que le mouvement immédiatement perçu est un fait très clair, et que les difficultés ou contradictions signalées par l’école d’Élée concernent beaucoup moins le mouvement lui-même qu’une réorganisation artificielle, et non viable, du mouvement par l’esprit. Tirons d’ailleurs la conclusion de tout ce qui précède :
II. – Il y a des mouvements réels.
(1939: 213-215. Text copied from UQAC)
1 Rappelons brièvement cet argument. Soit un mobile qui se déplace avec lune certaine vitesse et qui passe simultanément devant deux corps dont l'un est immobile et dont l'autre se meut à sa rencontre avec la même vitesse que lui. En même temps qu'il parcourt une certaine longueur du premier corps, il franchit naturellement une longueur double du second. D'où Zénon conclut « qu'une durée est double d'elle-même ». - Raisonnement puéril, dit-on, puisque Zénon ne tient pas compte de ce que la vitesse est double, dans un cas, de ce qu'elle est dans l'autre. - D'accord, mais comment, je vous prie, pourrait-il s'en apercevoir ? Que, dans le même temps, un mobile parcoure des longueurs différentes de deux corps dont l'un est en repos et l'autre en mouvement, cela est clair pour celui qui fait de la durée une espèce d'absolu, et la met soit dans la conscience soit dans quelque chose qui participe de la conscience. Pendant qu'une portion déterminée de cette durée consciente ou absolue s'écoule, en effet, le même mobile parcourra, le long des deux corps, deux espaces doubles l'un de l'autre, sans qu'on puisse conclure de là qu'une durée est double d'elle-même, puisque la durée reste quelque chose d'indépendant de l'un et l'autre espace. Mais le tort de Zénon, dans tolite son argumentation, est justement de laisser de côté la durée vraie pour n'en considérer que la trace objective dans l'espace. Comment les deux traces laissées par le même mobile ne mériteraient-elles pas alors une égale considération, en tant que mesures de la durée ? Et comment ne représenteraient-elles pas la même durée, lors même qu'elles seraient doubles l'une de l'autre ? En concluant de là qu'une durée « est double d'elle-même » Zénon restait dans la logique de son hypothèse, et son quatrième argument vaut exactement autant que les trois autres.
(1939: 215. Text copied from UQAC)
The arguments of Zeno of Elea have no other origin than this illusion. They all consist in making time and movement coincide with the line which underlies them, in attributing to them the same subdivisions as to the line, in short in treating them like that line. In this confusion Zeno was encouraged by common sense, which usually carries over to the movement the properties of its trajectory, and also by language, which always translates movement and duration in terms of space. But common sense and language have a right to do so [250|251] and are even bound to do so, for, since they always regard the becoming as a thing to be made use of, they have no more concern with the interior organization of movement than a workman has with the molecular structure of his tools. In holding movement to be divisible, as its trajectory is, common sense merely expresses the two facts which alone are of importance in practical life: first, that every movement describes a space ; second, that at every point of this space the moving body might stop. But the philosopher who reasons upon the inner nature of movement is bound to restore to it the mobility which is its essence, and this is what Zeno omits to do. By the first argument (the Dichotomy) he supposes the moving body to be at rest, and then considers nothing but the stages, infinite in number, that are along the line to be traversed we cannot imagine, he says, how the body could ever get through the interval between them. But in this way he merely proves that it is impossible to construct, a priori, movement with immobilities, a thing no man ever doubted. The sole question is whether, movement being posited as a fact, there is a sort of retrospective absurdity in assuming that an infinite number of points has been passed through. But at this we need not wonder, since movement is an undivided fact, or a series of undivided facts, whereas the trajectory is infinitely divisible. In the second argument (the Achilles) movement is [251|252] indeed given, it is even attributed to two moving bodies, but, always by the same error, there is an assumption that their movement coincides with their path, and that we may divide it, like the path itself, in any way we please. Then, instead of recognizing that the tortoise has the pace of a tortoise and Achilles the pace of Achilles, so that after a certain number of these indivisible acts or bounds Achilles will have outrun the tortoise, the contention is that we may disarticulate as we will the movement of Achilles and, as we will also, the movement of the tortoise : thus reconstructing both in an arbitrary sway, according to a law of our own which may be incompatible with the real conditions of mobility. The same fallacy appears, yet more evident, in the third argument (the Arrow) which consists in the conclusion that, because it is possible to distinguish points on the path of a moving body, we have the right to distinguish indivisible moments in the duration of its movement. But the most instructive of Zeno's arguments is perhaps the fourth (the Stadium) which has, we believe, been unjustly disdained, and of which the absurdity is more manifest only because the postulate masked in the three others is here frankly displayed.1 Without entering on a dis- [252|253] cussion which would here be out of place, we will content ourselves with observing that motion, as given to spontaneous perception, is a fact which is quite clear, and that the difficulties and contradictions pointed out by the Eleatic school concern far less the living movement itself than a dead and artificial reorganization of movement by the mind. But we now come to the conclusion of all the preceding paragraphs: [253|254]
II. There are real movements.
(2004: 250-253. Text copied from Mead Project)
1 We may here briefly recall this argument. Let there be a moving body which is displaced with a certain velocity, and which passes simultaneously before two bodies, one at rest and the other moving towards it with the same velocity [252|253] as its own. During the same time that it passes a certain length of the first body, it naturally passes double that length of the other. Whence Zeno concludes that ‘a duration is the double of itself.’ A childish argument, it is said, because Zeno takes no account of the fact that the velocity is in the one case double that which it is in the other. – Certainly, but how, I ask, could he be aware of this ? That, in the same time, a moving body passes different lengths of two bodies, of which one is at rest and the other in motion, is clear for him who makes of duration a kind of absolute, and places it either in consciousness or in something which partakes of consciousness. For while a determined portion of this absolute or conscious duration elapses, the same moving body will traverse, as it passes the two bodies, two spaces of which the one is the double of the other, without our being able to conclude from this that a duration is double itself, since duration remains independent of both spaces. But Zeno's error, in all his reasoning, is due to just this fact, that he leaves real duration on one side and considers only its objective track in space. How then should the two lines traced by the same moving body not merit an equal consideration, qua measures of duration ? And how should they not represent the same duration, even though the one is twice the other ? In concluding from this that ‘a duration is the double of itself,’ Zeno was true to the logic of his hypothesis; and his fourth argument is worth exactly as much as the three others.
(2004: 252-253. Text copied from Mead Project)
Texts:
Bergson, Henri. 1939 [this one 3rd edn. 1990]. Matière et mémoire: Essai sur la relation du corps à l'esprit. Paris: Quadridge / Presses Universitaires de France.
PDF available online at:
http://catalogue.bnf.fr/ark:/12148/cb37237615p
PDF of 1903 edition at:
http://www.archive.org/details/matireetmmoiree01berggoog
Text copied from 1939 edition at:
http://classiques.uqac.ca/classiques/bergson_henri/matiere_et_memoire/matiere_et_memoire.html
Bergson, Henri. 2004 [says originally published by George Allen & Co., Ltd., London, 1912. But there is 1911 edition (below)]. Matter and Memory. Translated by Nancy Margaret Paul & W. Scott Palmer. Mineola, New York: Dover.
PDF of 1911 edition [8th printing 1970] at:
http://www.archive.org/details/mattermemory00berg
Text copied from 1911 edition at:
https://brocku.ca/MeadProject/Bergson/Bergson_1911b/Bergson_1911_toc.html
[chapter 4:]
https://brocku.ca/MeadProject/Bergson/Bergson_1911b/Bergson_1911_04.html
.
