Showing posts with label Zeno's Paradox. Show all posts
Showing posts with label Zeno's Paradox. Show all posts

14 Jul 2017

Bergson (5.2.1-5.2.15) La pensée et le mouvant / Creative Mind, “[All reality is change and movement, which is absolutely indivisible]”, summary

 

by Corry Shores

 

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[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 1938 French edition first; then the 1946 English second, and the 1965 English third.  Section and paragraph divisions follow those in the French edition; enumerations except for chapter numbers are my own.]

 

 

 

Summary of

 

Henri Bergson

 

La pensée et le mouvant:

Essais et conférences

 

[An Introduction to Metaphysics:]

The Creative Mind

 

5. La perception du changement / The Perception of Change

 

5.2. Deuxième conférence / Second Lecture

 

5.2.1-5.2.15

[All reality is change and movement, which is absolutely indivisible.]

 

 

 

Brief summary:

All reality is change. We have an inner mental life that is a flux not of distinct states but of continuous qualitative variation. And what we experience in the world are not invariable objects undergoing a series of discrete states. Rather, there is just pure durational change itself as the only substantiality, and there are not invariable things doing the changing. This means that all change and movement is absolutely indivisible. We cannot then say that a physical motion had spatialized parts to it. The movement is entire and durational (thus non-spatial) while it is happening. Only after the motion is complete can we examine the space traveled and then, if we want, divide that traveled space. But to divide the motion up, like in Zeno’s paradoxes, is to confuse movement and duration with space, which are incommensurable. So by committing this confusion we may derive Zeno’s paradoxes of motion. The evidence for the indivisibility of motion and change is phenomenological: we experience our own bodily motions as such. It is only on account of practical needs that we artificially divide the world and our mind up into immobilities. Also, we rely heavily on our vision, which is prone to artificially carve up the world into things that subsist as they move and change. But we should instead appeal to our hearing for information about real duration and change. When we hear a melody, for example, we experience a pure, indivisible, continuous variation without an invariable object as a substrate for that variation.

 

 

 

Summary

 

5.2.1

[All change and all movement is absolutely indivisible.]

 

Bergson asks us to break our normal habits of thinking so that we may contemplate “the direct perception of change and mobility” / “la percep-| tion directe du changement et de la mobilité” (157|158 / 167 / 142). He continues, “We || shall think of all change, all movement, as being absolutely indivisible” / “Nous nous représenterons tout changement, tout mouvement, comme absolument indivisibles” (158 / 167||168 / 142, emphasis in original).

 

 

5.2.2

[The motion of our hand from point A to B is a simple movement.]

 

Bergson has us consider if we move our hand from point A to B. The movement that transpires is “simple” [which may mean that it is indivisible, or complex in a non-divisible way somehow.]

Let us begin with movement. I have my hand at point A. I move it over to point B, traversing the interval AB. I say that this movement from A to B is by nature simple.

 

Commençons par le mouvement. J’ai la main au point A. Je la transporte au point B, parcourant l’intervalle AB. Je dis que ce mouvement de A en B est chose simple.

(158 / 168 / 142)

 

 

 

5.2.3

[If our hand makes no pause in its motion, it feels unitary and indivisible to us. If it does make a pause, then it does not feel like it was one motion with distinct phases; rather, it feels like two indivisible motions.]

 

We feel this action as being singular and indivisible. Suppose we pause at an intermediary point. How would that feel to us? As one divided motion? No, it would feel like two distinct motions. So long as it is a moving without pause, it feels indivisible.

But of this each one of us has the immediate sensation. No doubt while we are moving our hand from A to B we say to ourselves that we could stop it at an intermediary point, but in that case we should not have to do with the same movement. There would no longer be a single movement from A to B; there would be, by hypothesis, two movements, with an interval. Neither from within, through the muscular sense, nor from without ||| through sight, should we still have the same perception. If I leave my movement from A to B as it is, I feel it undivided and must declare it to be indivisible.

 

Mais c’est de quoi chacun de nous a la sensation immédiate. Sans doute, pendant que nous portons notre main de A en B, nous nous disons que nous pourrions l’arrêter en un point intermédiaire, mais nous n’aurions plus affaire alors au même mouvement. Il n’y aurait plus un mouvement unique de A en B ; il y aurait, par hypothèse, deux mouvements, avec un intervalle d’arrêt. Ni du dedans, par le sens musculaire, ni du dehors par la vue, nous n’aurions encore la même perception. Si nous laissons notre mouvement de A en B tel qu’il est, nous le sentons indivisé et nous devons le déclarer indivisible.

(158 / 168 / 142|||143)

 

 

5.2.4

[Because motion traverses space, we mistakenly think that the motion is divisible like the space is. But it is not. And we cannot say that the motion ever occupies a point of space at an instant of time. For, suppose we did divide the motion into smaller and smaller intervals. We would never arrive upon a point in space or time but only upon intervals of decreasing size. So while moving, the object is always passing through points; and it only occupies a point when it stops. Nonetheless, in reality, there is no rest; there is only change and motion. What appears to be at rest is simply moving at pace with our attention and thus seems like it is not moving or changing, like two trains moving side-by-side, making each train appear stationary to the other.]

 

Bergson will now address our tendency to want to divide up the motion into parts corresponding to spatial intervals the movement traversed. {1} We can divide the spatial interval AB into as many parts as we want. Since the movement traversed this space, we can divide the movement into as many parts as we wish, corresponding to the internal spatial intervals traversed. {2} At each instant of the motion, the hand occupied a determinate point. On the basis of these points, which correlate to instants of the motion, we can divide the motion into stages corresponding to the divided spatial intervals. Bergson then begins his responses to this sort of approach of dividing motion by means of space. He first asks {a} how can movement be applied to the traversed space? {b} And note that the space is immobile, but the movement is mobile. So how do you make something immobile coincide with something mobile? {c} How can a moving object be both moving and occupy a determinate point? If it is moving, it can only be said to be passing through a point. Were it to occupy a point, it would have stopped. [Let us take note briefly of what we may consider a correction and clarification Bertrand Russell makes with regard to this idea that to occupy a point is to be at rest. (See the entries for Bertrand Russell at the Zeno’s Paradox directory.) Russell notes that it is not enough to simply occupy a point to constitute rest. You need to remain at that point as time passes. But I think if we extract some of Bergson’s insight here, we can see a problem even with Russell’s at-at definition of motion. For Russell, there are never immediately successive points, because there is always another between any two. Now suppose we begin with one motion. And suppose we divide it into smaller and smaller intervals. Would we ever arrive upon its occupation of a point during an instant using this method? No. For no matter how small the interval, in Russell’s thinking, there is always a smaller one. And since there is no smallest decomposition, you never have occupation at a point; you only have smaller and smaller passings-through of sets of points. Thus if we begin with a unitary motion, we can never decompose it into parts that simply occupy a singular position at a singular instant. So, insofar as Russell has us think of a moving object occupying a determinate point in space, he is not dealing with a part of motion but rather with the space traversed by that motion.] Bergson further clarifies this notion of unitary motion. We think of any undivided motion as a “bound”, which endures for whatever duration of the motion, which can be any size interval, no matter how small or large. As a singular bound, it is indecomposable. But, after the bound has finished, we see that it traversed an extent of space. Since space is infinitely divisible, we imagine the motion to be so as well. But we are in error as soon as we think that we can really divide the motion up this way into immobilities. Bergson then makes the remarkable claim that in reality there are no immobilities. Whatever we consider to be an immobility is really just a movement that is going at pace alongside a moving point of reference: “when two trains move at the same speed, in the ||| same direction, on parallel tracks: each of the two trains is then immovable to the travellers seated in the other” (159 / 169 / 143|||144). [I am not entirely sure how to conceive the way this transpires in life. Perhaps the idea is that suppose we have something that we think is immobile. In reality it is changing. But it is changing at pace with our flow of awareness, perhaps. Consult the quotation to follow.]

It is true that, when I watch my hand going from A to B and describing the interval AB, I say: “The interval AB can be divided into as many parts as I wish, therefore the movement from A to B can be divided into as many parts as I like, since this movement is applied exactly upon this interval.” Or again: “At each instant of its trajection, the mobile passes through a certain point, therefore one can distinguish in the movement as many stages as one likes, therefore the movement is infinitely divisible.” But let us reflect for a moment. How could the movement be applied upon the space it traverses? How can something moving coincide with something immobile? How could the moving object be in a point of its trajectory passage? It passes through, or in other terms, it could be there. It would be there if it || stopped; but if it should stop there, it would no longer be the same movement we were dealing with. It is always by a single bound that a passing is completed, when there is no break in the passage. The bound may last a few seconds, or days, months, years: it matters little. The moment it is one single bound, it is indecomposable. Only, once the passage is effected, as the trajectory is space and space is indefinitely divisible, we imagine that movement itself is indefinitely divisible. We like to imagine it because, in a movement, it is not the change of position which interests us, it is the positions themselves, the one the movement has left, the one it will take, the one it would take if it stopped on the way. We need immobility, and the more we succeed in imagining movement as coinciding with the immobilities of the points of space through which it passes, the better we think we understand it. To tell the truth, there never is real immobility, if we understand by that an absence of movement. Movement is reality itself, and what we call immobility is a certain state of things analogous to that produced when two trains move at the same speed, in the ||| same direction, on parallel tracks: each of the two trains is then immovable to the travellers seated in the other. But a situation of this kind which, after all, is exceptional, seems to us to be the regular and normal situation, because it is what permits us to act upon things and also permits things to act upon us: the travellers in the two trains can hold out their hands to one another through the door and talk to one another only if they are “immobile,” that is to say, if they are going in the same direction at the same speed. “Immobility” being the prerequisite for our action, we set it up as a reality, we make || of it an absolute, and we see in movement something which is superimposed. Nothing is more legitimate in practice. But when we transport this habit of mind into the domain of speculation, we fail to recognize the true reality, we deliberately create insoluble problems, we close our eyes to what is most living in the real.

 

Il est vrai que, lorsque je regarde ma main allant de A en B et décrivant l’intervalle AB, je me dis : « l’intervalle AB peut se diviser en autant de parties que je le veux, donc le mouvement de A en B peut se diviser en autant de parties qu’il me plaît, puisque ce mouvement s’applique sur cet intervalle. » Ou bien encore : « à chaque instant de son trajet, le mobile passe en un certain point, donc on peut distinguer dans le mouvement autant d’étapes qu’on voudra, donc le mouvement est infiniment divisible. » Mais réfléchissons-y un instant. Comment le mouvement pourrait-il s’appliquer sur l’espace qu’il parcourt ? comment du mouvant coïnciderait-il avec de l’immobile ? comment l’objet qui se meut serait-il en un point de son trajet ? Il y passe, ou, en d’autres termes, il pourrait y être. Il y serait s’il s’y arrêtait ; mais, s’il s’y arrêtait, ce n’est plus au même mouvement | que nous aurions affaire. C’est toujours d’un seul bond qu’un trajet est parcouru, quand il n’y a pas d’arrêt sur le trajet. Le bond peut durer quelques secondes, ou des jours, des mois, des années : peu importe. Du moment qu’il est unique, il est indécomposable. Seulement, une fois le trajet effectué, comme la trajectoire est espace et que l’espace est indéfiniment divisible, nous nous figurons que le mouvement lui-même est divisible indéfiniment. Nous aimons à nous le figurer, parce que, dans un mouvement, ce n’est pas le changement de position qui nous intéresse, ce sont les positions elles-mêmes, celle que le mobile a quittée, celle qu’il prendra, celle qu’il prendrait s’il s’arrêtait en route. Nous avons besoin d’immobilité, et plus nous réussirons à nous représenter le mouvement comme coïncidant avec les immobilités des points de l’espace qu’il parcourt, mieux nous croirons le comprendre. A vrai dire, il n’y a jamais d’immobilité véritable, si nous entendons par là une absence de mouvement. Le mouvement est la réalité même, et ce que nous appelons immobilité est un certain état de choses analogue à ce qui se produit quand deux trains marchent avec la même vitesse, dans le même sens, sur deux voies parallèles : chacun des deux trains est alors immobile pour les voyageurs assis dans l’autre. Mais une situation de ce genre, qui est en somme exceptionnelle, nous semble être la situation régulière et normale, parce que c’est celle qui nous permet d’agir sur les choses et qui permet aussi aux choses d’agir sur nous : les voyageurs des deux trains ne peuvent se tendre la main par la portière et causer ensemble que s’ils sont « immobiles », c’est-à-dire s’ils marchent dans le même sens avec la même vitesse. L’ «immobilité » étant ce dont notre action a besoin, nous l’érigeons en réalité, nous en faisons un absolu, et nous voyons dans le mouvement | quelque chose qui s’y surajoute. Rien de plus légitime dans la pratique. Mais lorsque nous transportons cette habitude d’esprit dans le domaine de la spéculation, nous méconnaissons la réalité vraie, nous créons, de gaieté de cœur, des problèmes insolubles, nous fermons les yeux à ce qu’il y a de plus vivant dans le réel.

(158|160 / 168||170 / 143|||144)

 

 

5.2.5

[Zeno’s paradoxes of motion confuse the space traversed, which is divisible, with the motion, which is not. For example, Achilles is said to never overtake the Tortoise, because the Tortoise is constantly creating new distances for Achilles to cross. But Achilles’ own motion was not composed of infinitely many movements that endlessly decrease in size. It was made of a series of indivisible steps.]

 

Zeno’s paradoxes of motion confuse movement with the space traversed. Or if they do not make such a simplistic equivalence, they at least treat movement as being divisible like space is, without taking into account that movement is articulated in unitary segments. [Recall the paradox of Achilles and the Tortoise (described at section II.F of this entry, and discussed by Bergson in Time and Free Will §70, Matter and Memory section 4.2.6, and Creative Evolution section 4.5.10.) The Tortoise has a head start at the beginning of its race with Achilles. For Achilles to overtake the Tortoise, he must first arrive upon the Tortoise’s starting position. But by the time he gets there, the Tortoise has already advanced a little further. So now Achilles must arrive upon the Tortoise’s new position. But again, upon arriving upon it, the Tortoise will be further. So long as the Tortoise keeps moving, it will always create a new, more distant position for Achilles to cross. Thus Achilles will never overtake the Tortoise while it keeps moving.] Bergson recalls Zeno’s paradox of Achilles and the Tortoise, where “Achilles, they say, will never overtake the tortoise he is pursuing, for when he arrives at the point where the tortoise was the latter will have had time to go further, and so on indefinitely.” Bergson notes that there have been a great variety of refutations, but he proposes a simple one. He says we need to ask Achilles after he in fact overtakes the Tortoise, how did he do it? He would say that Zeno’s description does not match the way the event transpired. He did not first arrive upon the Tortoises’ first position, then second position, and on and on. Rather, Achilles will say that he took a series of steps until finally overtaking the Tortoise. So his course was a series of indivisible actions. These are the only parts we can divide his whole course into. We should not, like Zeno, divide the acts indefinitely like spatial parts corresponding to the spatial locations of the race track, because the acts are not immobilities like space and they cannot be coordinated point-by-point with an immobility like space.

I need not recall the arguments of Zeno of Elea. They all involve the confusion of movement with the space covered, or at least the conviction that one can treat movement as one treats space, divide it without taking account of its articulations. Achilles, they say, will never overtake the tortoise he is pursuing, for when he arrives at the point where the tortoise was the latter will have had time to go further, and so on indefinitely. Philosophers have refuted this argument in numerous ways, and ways so different that each of these refutations deprives the others of the right to be considered definitive. There would have been, nevertheless, a very simple means of making short work of the difficulty: that would have been to question Achilles. For since Achilles finally catches up to the tortoise and even passes it, he must know better than anyone else how he goes about it. The ancient philosopher who demonstrated the possibility of movement by walking was right: his only mistake was to make the gesture without adding a commentary. Suppose then we ask Achilles to comment on his race: here, doubtless, is what he will answer: “Zeno insists that I go from the point where I am to the point the tortoise ||| has left, from that point to the next point it has left, etc., etc.; that is his procedure for making me run. But I go about it otherwise. I take a first step, then a second, and so on: finally, after a certain number of steps, || I take a last one by which I skip ahead of the tortoise. I thus accomplish a series of indivisible acts. My course is the series of these acts. You can distinguish its parts by the number of steps it involves. But you have not the right to disarticulate it according to another law, or to suppose it articulated in another way. To proceed as Zeno does is to admit that the race can be arbitrarily broken up like the space which has been covered; it is to believe that the passage is in reality applied to the trajectory; it is making movement and immobility coincide and consequently confusing one with the other.”

 

Je n’ai pas besoin de vous rappeler les arguments de Zénon d’Élée. Tous impliquent la confusion du mouvement avec l’espace parcouru, ou tout au moins la conviction qu’on peut traiter le mouvement comme on traite l’espace, le diviser sans tenir compte de ses articulations. Achille, nous dit-on, n’atteindra jamais la tortue qu’il poursuit, car lorsqu’il arrivera au point où était la tortue, celle-ci aura eu le temps de marcher, et ainsi de suite indéfiniment. Les philosophes ont réfuté cet argument de bien des manières, et de manières si différentes que chacune de ces réfutations enlève aux autres le droit de se croire définitives. Il y aurait eu pourtant un moyen très simple de trancher la difficulté : c’eût été d’interroger Achille. Car, puisqu’Achille finit par rejoindre la tortue et même par la dépasser, il doit savoir, mieux que personne, comment il s’y prend. Le philosophe ancien qui démontrait la possibilité du mouvement en marchant était dans le vrai : son seul tort fut de faire le geste sans y joindre un commentaire. Demandons alors à Achille de commenter sa course : voici, sans aucun doute, ce qu’il nous répondra. « Zénon veut que je me rende du point où je suis au point que la tortue a quitté, de celui-ci au point qu’elle a quitté encore, etc. ; c’est ainsi qu’il procède pour me faire courir. Mais moi, pour courir, je m’y prends autrement. Je fais un premier pas, puis un second, et ainsi de suite : finalement, après un certain nombre de pas, j’en fais un dernier par lequel j’enjambe la tortue. J’accomplis | ainsi une série d’actes indivisibles. Ma course est la série de ces actes. Autant elle comprend de pas, autant vous pouvez y distinguer de parties. Mais vous n’avez pas le droit de la désarticuler selon une autre loi, ni de la supposer articulée d’une autre manière. Procéder comme le fait Zénon, c’est admettre que la course peut être décomposée arbitrairement, comme l’espace parcouru ; c’est croire que le trajet s’applique réellement contre la trajectoire ; c’est faire coïncider et par conséquent confondre ensemble mouvement et immobilité. »

(160|161 / 170||171 / 144|||145)

 

 

5.2.6

[Under this erroneous understanding that confuses movement with traversed space, we hold two incompatible metaphysical assumptions: {1} motion is fundamentally a passage between points, and thus it is fundamentally a mobility transpiring between immobilities, and {2} motion is decomposable into nothing more than positions dividing passages (into immobilities), and it is recomposable by taking all these positions together.]

 

Bergson says that this method of decomposition that confuses motion with traversed space is our usual way of understanding movement. It treats motion as if it were constituted by immobilities (spatial positions). And we think we can reconstitute movement by adding all the immobilities together. But at the same time that we decompose movement into positions, we assume as well that there is a passage between any two positions. Yet, how do we conceptualize this passage? By saying it is composed of yet more positions. We want to be able to have both concepts, that there is a mobile passage and that the passage is made of immobile positions. But as we can see, they are conceptually irreconcilable. [Bergson again ends with the notion that immobility is perhaps not even real.]

But that is precisely what our usual method consists in. We argue about movement as though it were made of immobilities and, when we look at it, it is with immobilities that we reconstitute it. Movement for us is a position, then another position, and so on indefinitely. We say, it is true, that there must be something else, and that from one position to another there is the passage by which the interval is cleared. But as soon as we fix our attention on this passage, we immediately make of it a series of positions, even though we still admit that between two successive positions one must indeed assume a passage. We put this passage off indefinitely the moment we have to consider it. We admit that it exists, we give it a name; that is enough for us: once that point has been satisfactorily settled we turn to the positions preferring to deal with them alone. We have an instinctive fear of those difficulties which the vision of movement as movement would arouse in our thought; and quite rightly, once we have loaded movement down with immobilities. If movement is not everything, it is nothing; and if to begin with we have sup- || posed that immobility can be a reality, movement will slip through our fingers when we think we have it.

 

Mais en cela consiste précisément notre méthode habituelle. Nous raisonnons sur le mouvement comme s’il était fait d’immobilités, et, quand nous le regardons, c’est avec des immobilités que nous le reconstituons. Le mouvement est pour nous une position, puis une nouvelle position, et ainsi de suite indéfiniment. Nous nous disons bien, il est vrai, qu’il doit y avoir autre chose, et que, d’une position à une position, il y a le passage par lequel se franchit l’intervalle. Mais, dès que nous fixons notre attention sur ce passage, vite nous en faisons une série de positions, quittes à reconnaître encore qu’entre deux positions successives il faut bien supposer un passage. Ce passage, nous reculons indéfiniment le moment de l’envisager. Nous admettons qu’il existe, nous lui donnons un nom, cela nous suffit : une fois en règle de ce côté, nous nous tournons vers les positions et nous préférons n’avoir affaire qu’à elles. Nous avons instinctivement peur des difficultés que susciterait à notre pensée la vision du mouvement dans ce qu’il a de mouvant ; et nous avons raison, du moment que le mouvement a été chargé par nous d’immobilités. Si le mouvement n’est pas tout, il n’est rien ; et si nous avons d’abord posé que l’immobilité peut être une réalité, le mouvement | glissera entre nos doigts quand nous croirons le tenir.

(161|162 / 171||172 / 145)

 

 

5.2.7

[In fact, all real change is indivisible. We erroneously think that change can be divided into states because indivisible change in the world is practically coordinated with the indivisible change of our inner experience. When our actions need to intervene in the world or when for other practical reasons we coordinate events in the world with events in our minds, we artificially designate stable states in the changes outside and inside us.]

 

Bergson says that this applies to any sort of change whatsoever: “All real change is an indivisible change” / “Tout changement réel est un changement indivisible”. Bergson then explains how it is that we come to erroneously understand real change as being divisible. [His first idea here might be the following, but check the quotation below. We suppose there is some change happening in the world. Let us imagine that we are heating water to a boil. So out in the world there is continuous qualitative change of the water. At the same time, our mind is in a flux of consciousness. These two fluxes flow together in simultaneity (see for example Ch.3 of Duration and Simultaneity.) Now, as the water comes to boil, we get anxious because it might overboil, so we lift the pan off the fire. We then want to say that there were distinct states of the water, namely,  it being below boiling, it being at boiling, and it cooling below boiling. These states correspond to states of our mind: our waiting for it to boil, our alarm when it boils, and our relief when it cools. But in reality, Bergson seems to be saying, the water was never in some determinate, stable state. It was always changing. And our mind was not in any determinate, stable state. It was always a flux of consciousness. But we came to regard the water having states corresponding to the appropriate actions we made with regard to it, and these actions that divided the change serve to mark off seemingly distinct and stable inner states of our experience of the change. So it is simply for practical reasons (our needing to boil the water but not too much) that cause us to delineate states of change in the world and in our inner life. But these immobilities are not really inherent to the world and our mind.]

I have spoken of movement; but I could say the same for any change whatever. All real change is an indivisible change. We like to treat it as a series of distinct states which form, as it were, a line in time. That is perfectly natural. If change is continuous in us and also in things, on the other hand, in order that the uninterrupted change which each of us calls “me” may act upon the uninterrupted change that we call a “thing,” these two changes must find themselves, with regard to one another, in a situation like that of the two trains referred to above. We say, for example, that an object changes color, and that change here consists in a series of shades which would be the constitutive elements of change and which, themselves, would not change. But in the first place, if each shade has any objective existence at all, it is an infinitely rapid oscillation, it is change. And in the second place, the perception we have of it, to the extent that it is subjective, is only an isolated, abstract aspect of the general state of our person, and this state as a whole is constantly changing and causing this so-called invariable perception to participate in its change; in fact, there is no perception which is not constantly being modified. So that color, outside of us, is mobility itself, and our own person is also mobility. But the whole mechanism of our perception of things, like the mechanism of our action upon things has been regulated in such a way as to bring about, between the external and the internal mobility, a situation comparable to that of our two trains, – more complicated, perhaps, but of the same kind: when the || two changes, that of the object and that of the subject, take place under particular conditions, they produce the particular appearance that we call a “state.” And once in possession of “states,” our mind recomposes change with them. I repeat, there is nothing more natural: the breaking up of change into states enables us to act upon things, and it is useful in a practical sense to be interested in the states rather than in the change itself. But what is favourable to action in this case would ||| be fatal to speculation. If you imagine a change as being really composed of states, you at once cause insoluble metaphysical problems to arise. They deal only with appearances. You have closed your eyes to true reality.

 

J’ai parlé du mouvement ; mais j’en dirais autant de n’importe quel changement. Tout changement réel est un changement indivisible. Nous aimons à le traiter comme une série d’états distincts qui s’aligneraient, en quelque sorte, dans le temps. C’est naturel encore. Si le changement est continuel en nous et continuel aussi dans les choses, en revanche, pour que le changement ininterrompu que chacun de nous appelle « moi » puisse agir sur le changement ininterrompu que nous appelons une « chose », il faut que ces deux changements se trouvent, l’un par rapport à l’autre, dans une situation analogue à celle des deux trains dont nous parlions tout à l’heure. Nous disons par exemple qu’un objet change de couleur, et que le changement consiste ici dans une série de teintes qui seraient les éléments constitutifs du changement et qui, elles, ne changeraient pas. Mais, d’abord, ce qui existe objectivement de chaque teinte, c’est une oscillation infiniment rapide, c’est du changement. Et, d’autre part, la perception que nous en avons, dans ce qu’elle a de subjectif, n’est qu’un aspect isolé, abstrait, de l’état général de notre personne, lequel change globalement sans cesse et fait participer à son changement cette perception dite invariable : en fait, il n’y a pas de perception qui ne se modifie à chaque instant. De sorte que la couleur, en dehors de nous, est la mobilité même, et que notre propre personne est mobilité encore. Mais tout le mécanisme de notre perception des choses, comme celui de notre action sur les choses, a été réglé de manière à amener ici, entre la mobilité externe et la mobilité intérieure, une situation comparable à celle de nos deux trains, - plus compliquée, sans doute, mais du même genre : quand les deux changements, celui de l’objet et celui du sujet, ont lieu dans ces conditions particulières, | ils suscitent l’apparence particulière que nous appelons un « état ». Et, une fois en possession d’ « états », notre esprit recompose avec eux le changement. Rien de plus naturel, je le répète : le morcelage du changement en états nous met à même d’agir sur les choses, et il est pratiquement utile de s’intéresser aux états plutôt qu’au changement lui-même. Mais ce qui favorise ici l’action serait mortel à la spéculation. Représentez-vous un changement comme réellement composé d’états : du même coup vous faites surgir des problèmes métaphysiques insolubles. Ils ne portent que sur des apparences. Vous avez fermé les yeux à la réalité vraie.

(162|163 / 172||173 / 146|||147)

 

 

5.2.8

[We can know that change is indivisible, because when we witness it, we experience it as such. Also, all is change, and there are no invariable objects doing those changes.]

 

Bergson offers as proof that whenever we directly view change or a movement, we have a feeling of absolute individuality. Bergson now makes the point that although there are changes, that is all there is. There are no invariable objects undergoing the changes. Even in movements there are no invariable objects that are doing the moving.

I shall not press the point. Let each of us undertake the experiment, let him give himself the direct vision of a change, of a movement: he will have a feeling of absolute indivisibility. I come then to the second point, closely allied to the first. There are changes, but there are underneath the change no things which change: change has no need of a support. There are movements, but there is no inert or invariable object which moves: movement does not imply a mobile.

Je n’insisterai pas davantage. Que chacun de nous fasse l’expérience, qu’il se donne la vision directe d’un changement, d’un mouvement : il aura un sentiment d’absolue indivisibilité. J’arrive alors au second point, qui est très voisin du premier. Il y a des changements, mais il n’y a pas, sous le change­ment, de choses qui changent : le changement n’a pas besoin d’un support. Il y a des mouvements, mais il n’y a pas d’objet inerte, invariable, qui se meuve : le mouvement n’implique pas un mobile.

(163 / 173 / 147, italics in the original, boldface mine)

 

 

5.2.9

[Our vision distinguishes things that come to appear as invariable objects persisting through their changes. But our sense of hearing only gives us the changes. For example, when we hear a melody, the “object” here is the melodic change itself.]

 

Bergson says that we develop the erroneous notion of invariable objects subsisting through change from our primary reliance on vision. Our sight separates things in our visual field, and we come to take those distinguished things as invariable figures. Sight thus prepares the sense of touch for our interactions with the world. But the other senses are better able to give us the world in changefulness. When we hear a melody, we do not perceive an invariable object that is changing but rather the change “is the thing itself”. We know this is so, because if you stop the melody at one point or at another, it is a different thing in each case. So there is no one melody that subsists throughout its changes. And we should not think of the melody being composed of the notes pictured in the score. For this is to understand it as a visualizable object with spatial properties.

It is difficult to picture things in this way, because the sense ‘par excellence’ is the sense of sight, and because the eye has developed the habit of separating, in the visual field, the relatively invariable figures which are then supposed to change place without changing form, movement is taken as super-added to the mobile as an accident. It is, in fact, useful to have to deal in daily life with objects which are stable and, as it were, responsible, to which one can address oneself as to persons. The sense of sight contrives to take things in this way: as an ad- || vance-guard for the sense of touch, it prepares our action upon the external world. But we already have less difficulty in perceiving movement and change as independent realities if we appeal to the sense of hearing. Let us listen to a melody, allowing ourselves to be lulled by it: do we not have the clear perception of a movement which is not attached to a mobile, of a change without anything changing? This change is enough, it is the thing itself. And even if it takes time, it is still indivisible; if the melody stopped sooner it would no longer be the same sonorous whole, it would be another, equally indivisible. We have, no doubt, a tendency to divide it and to picture, instead of the uninterrupted continuity of melody, a juxtaposition of distinct notes. But why? Because we are thinking of the discontinuous series of efforts we should be making to recompose approximately ||| the sound heard if we were doing the singing, and also because our auditory perception has acquired the habit of absorbing visual images. We therefore listen to the melody through the vision which an orchestra-leader would have of it as he watched its score. We picture notes placed next to one another upon an imaginary piece of paper. We think of a keyboard upon which some one is playing, of the bow going up and down, of the musicians, each one playing his part along with the others. If we do not dwell on these spatial images, pure change remains, sufficient unto itself, in no way divided, in no way attached to a “thing” which changes.

 

On a de la peine à se représenter ainsi les choses, parce que le sens par excellence est celui de la vue, et que l’œil a pris l’habitude de découper, dans l’ensemble du champ visuel, des figures relativement invariables qui sont censées | alors se déplacer sans se déformer : le mouvement se surajouterait au mobile comme un accident. Il est en effet utile d’avoir affaire, tous les jours, à des objets stables et, en quelque sorte, responsables, auxquels on s’adresse comme à des personnes. Le sens de la vue s’arrange pour prendre les choses de ce biais : éclaireur du toucher, il prépare notre action sur le monde exté­rieur. Mais déjà nous aurons moins de peine à percevoir le mouvement et le changement comme des réalités indépendantes si nous nous adressons au sens de l’ouïe. Écoutons une mélodie en nous laissant bercer par elle : n’avons-nous pas la perception nette d’un mouvement qui n’est pas attaché à un mobile, d’un changement sans rien qui change ? Ce changement se suffit, il est la chose même. Et il a beau prendre du temps, il est indivisible : si la mélodie s’arrêtait plus tôt, ce ne serait plus la même masse sonore ; c’en serait une autre, égale­ment indivisible. Sans doute nous avons une tendance à la diviser et à nous représenter, au lieu de la continuité ininterrompue de la mélodie, une juxtapo­sition de notes distinctes. Mais pourquoi ? Parce que nous pensons à la série discontinue d’efforts que nous ferions pour recomposer approximativement le son entendu en chantant nous-mêmes, et aussi parce que notre perception auditive a pris l’habitude de s’imprégner d’images visuelles. Nous écoutons alors la mélodie à travers la vision qu’en aurait un chef d’orchestre regardant sa partition. Nous nous représentons des notes juxtaposées à des notes sur une feuille de papier imaginaire. Nous pensons à un clavier sur lequel on joue, à l’archet qui va et qui vient, au musicien dont chacun donne sa partie à côté des autres. Faisons abstraction de ces images spatiales : il reste le changement pur, se suffisant à lui-même, nullement divisé, nullement attaché à une « chose » qui change.

(163|164 / 173||174 / 147|||148)

 

 

5.2.10

[Science decomposes things down into smaller and smaller particles understood increasingly in terms of their movements and vibrations. When we see something, we might think it is an invariable object. But that image is composed of tiny dots of color, each itself a light frequency and thus a movement.]

 

Bergson returns to sight to make the point that it is not necessary for us to only see invariable objects. Science decomposes things down into smaller and smaller things, understood more in terms of their motion than their substantiality. Bergson then notes that when we see something moving, the parts of our vision are colored spots, which are made of a “series of extremely rapid vibrations”. [I am not certain, but perhaps he is referring to something like the frequency of the light waves. See 165 / 175 / 148.]

 

 

5.2.11

[Change itself is the only thing with any substantiality in the sense of being something that endures. There are no other invariabilities. We see this with our inner life, which is a pure continuous flux like a melody. We are mistaken to think that there is a series of discrete and distinct psychic states modifying an invariable ego as substrate.]

 

Bergson says that we most encounter the “substantiality” of change in our inner life. [By substantiality he does not mean the invariable objects but change itself being the only thing that could qualify as a substrate, because change itself is the only thing that endures.] He says that certain theories of personality assume two things about consciousness, both being incorrect, and as a result they encounter a number of conceptual problems. The first incorrect assumption is that we have “a series of distinct psychological states, each one invariable”. [He also says that this series of states is thought to produce  variations of the ego by their very succession”. It is not stated whether or not the ego’s variations are a series of discrete invariable states, but I would guess they would be, as that is the form the cause takes.] The second incorrect assumption is that we have an invariable ego which “would serve as support” for the psychic variations. He first wonders how the unity of the ego could meet up with the multiplicity of the psychic variations. He next wonders how either one could have duration. Consider the flux of psychic states. Here change is superadded (s’y surajoute). [I am not sure what is meant here. I am going to guess it is the following, but please consult the text. There is only flux, which means at any present there is a variation. But what it is varying against is not present. In order to say there was change, we need to add to the present variation the past and/or future situations in order to say that a change transpires. Thus it is hard to say how the flux has duration, because in itself it is only a present variation and not something that itself endures.] Now consider the invariable ego. The ego is made of elements which do not change. [I am not sure what those elements are. Perhaps they are psychic structures of some kind.] So if its elements are invariable, how can it be said to endure? [Apparently only variable things can endure, but I am not sure why. Perhaps invariable things like structures are not temporal but some sort of idealized abstraction.] Bergson then claims that neither assumption is true. There is only the continuously varying “melody” of our inner life, and this is what constitutes our personality. [In other words, we have no stable ego.]

But nowhere is the substantiality of change so visible, so palpable as in the domain of the inner life. Difficulties and contradictions of every kind to which the theories of personality have led come from our having imagined, on the one hand, a series of distinct psychological ||| states, each one invariable, which would produce the variations of the ego by their very succession, and on the other hand an ego, no less invariable, which would serve as support for them. How could this unity and this multiplicity meet? How, without either of them having duration – the first because change is something superadded, the second because it is made up of elements which do not change – how could they constitute an ego which endures? But the truth is that there is neither a || rigid, immovable substratum nor distinct states passing over it like actors on a stage. There is simply the continuous melody of our inner life, – a melody which is going on and will go on, indivisible, from the beginning to the end of our conscious existence. Our personality is precisely that.

 

Mais nulle part la substantialité du changement n’est aussi visible, aussi palpable, que dans le domaine de la vie intérieure. Les difficultés et contra­dictions de tout genre auxquelles ont abouti les théories de la personnalité viennent de ce qu’on s’est représenté, d’une part, une série d’états psycholo­giques distincts, chacun invariable, qui produiraient les variations du moi par leur succession même, et d’autre part un moi, non moins invariable, qui leur servirait de support. Comment cette unité et cette multiplicité pourraient-elles se rejoindre ? comment, ne durant ni l’une ni l’autre – la première parce que le changement est quelque chose qui s’y surajoute, la seconde parce qui elle est faite d’éléments qui ne changent pas – pourraient-elles constituer un moi qui dure ? Mais la vérité est qu’il n’y a ni un substratum rigide immuable ni des états distincts qui y passent comme des acteurs sur une scène. Il y a simple­ment la mélodie continue de notre vie intérieure, – mélodie qui se poursuit et se poursuivra, indivisible, du commencement à la fin de notre existence consciente. Notre personnalité est cela même.

(165-166 / 175-176 / 148-149)

 

 

5.2.12

[This indivisible continuity of change that we directly experience as being fundamental to our awareness is real duration or time proper. But we normally conceive time erroneously as being spatial, like the note symbols of a melody juxtaposed simultaneously side-by-side on the musical staff.]

 

Bergson has been describing an “indivisible continuity of change”. He claims that “this is precisely what constitutes true duration.” Many might object that this notion of duration is “something inexpressible and mysterious.” But in fact this duration is “the clearest thing in the world” [because we experience it directly every moment of our consciousness.] We can think of real duration as time, so long as we qualify that it is “time perceived as indivisible.” [Now, time is often understood as being a matter of succession.] But although duration is time, we should not think of it as succession. Succession is understood as a matter of moments coming before and after, each being set beside the other. [Bergson might here, when mentioning simultaneity, be noting that this setting beside one another of moments makes them simultaneous when really they are not. See the quote below.] When we hear a melody, we have the impression that it cannot be broken up. So we should not think of the melody being broken up into notes set side-by-side and thus not in terms of this spatialized sort of succession. By putting the note symbols into the space of the musical staff, we might erroneously come to believe that the melody itself is something composed of spatially simultaneous parts. Bergson then acknowledges that we normally live our lives in the spatialized time. But real duration is something going on deep inside us. His last point might be that real duration is a single, universal time that happens exclusively in the present and that makes possible the extending changes we notice inside and outside us.]

This indivisible continuity of change is precisely what constitutes true duration. I cannot here enter into the detailed examination of a question I have dealt with elsewhere. I shall confine myself therefore to saying, in reply to those for whom this “real duration” is something inexpressible and mysterious, that it is the clearest thing in the world: real duration is what we have always called time, but time perceived as indivisible. That time implies succession I do not deny. But that succession is first presented to our consciousness, like the distinction of a “before” and “after” set side by side, is what I cannot admit. When we listen to a melody we have the purest impression of succession we could possibly have, – an impression as far removed as possible from that of simultaneity, – and yet it is the very continuity of the melody and the impossibility of breaking it up which make that impression upon us. If we cut it up into distinct notes, into so many “befores” and “afters,” we are bringing spatial images into it and impregnating the succession with simultaneity: in space, and only in space, is there a clear-cut distinction of parts external to one another. I recognize moreover that it is in spatialized time that we ordinarily place ourselves. We have no interest in listening to the uninterrupted ||| humming of life’s depths. And yet, that is where real duration is. Thanks to it, the more or less lengthy changes || we witness within us and in the external world, take place in a single identical time.

 

C’est justement cette continuité indivisible de changement qui constitue la durée vraie. Je ne puis entrer ici dans l’examen approfondi d’une question que j’ai traitée ailleurs. Je me bornerai donc à dire, pour répondre à ceux qui voient dans cette durée « réelle » je ne sais quoi d’ineffable et de mystérieux, qu’elle est la chose la plus claire du monde : la durée réelle est ce que l’on a toujours appelé le temps, mais le temps perçu comme indivisible. Que le temps impli­que la succession, je n’en disconviens pas. Mais que la succession se présente d’abord à notre conscience comme la distinction d’un « avant » et d’un « après » juxtaposés, c’est ce que je ne saurais accorder. Quand nous écoutons une mélodie, nous avons la plus pure impression de succession que nous puissions avoir, – une impression aussi éloignée que possible de celle de la simultanéité, – et pourtant c’est la continuité même de la mélodie et l’impos­sibilité de la décomposer qui font sur nous cette impression. Si nous la découpons en notes distinctes, en autant d’ « avant » et d’« après » qu’il nous plaît, c’est que nous y mêlons des images spatiales et que nous imprégnons la succession de simultanéité : dans l’espace, et dans l’espace seulement, il y a distinction nette de parties extérieures les unes aux autres. Je reconnais d’ailleurs que c’est dans le temps | spatialisé que nous nous plaçons d’ordinaire. Nous n’avons aucun intérêt à écouter le bourdonnement ininterrompu de la vie profonde. Et pourtant la durée réelle est là. C’est grâce à elle que prennent place dans un seul et même temps les changements plus ou moins longs aux­quels nous assistons en nous et dans le monde extérieur.

(166-167 / 176-177 / 149-150)

 

 

5.2.13

[Only change is real. There are no invariable things doing the changing.]

 

So both internal and external reality is “mobility itself”. There is change without things doing the changing.

Thus, whether it is a question of the internal or the external, of ourselves or of things, reality is mobility itself. That is what I was expressing when I said that there is change, but that there are not things which change.

 

Ainsi, qu’il s’agisse du dedans ou du dehors, de nous ou des choses, la réalité est la mobilité même. C’est ce que j’exprimais en disant qu’il y a du changement, mais qu’il n’y a pas de choses qui changent.

(167 / 177 / 150)

 

 

5.2.14

[Some might feel dizzy when contemplating the idea that reality has no fixed points of immobility. But we can obtain a sense of orientation by seeing that change itself is the most substantial and durable thing possible.]

 

Many will object to the notion that all is change, because they will not be able to deal with the disorienting feeling. Bergson assures such people that they can ground their sense of reality in change itself, which is “the most substantial and durable thing possible”.

Before the spectacle of this universal mobility there may be some who will be seized with dizziness. They are accustomed to terra firma; they cannot get used to the rolling and pitching. They must have “fixed” points to which they can attach thought and existence. They think that if everything passes, nothing exists; and that if reality is mobility, it has already ceased to exist at the moment one thinks it, – it eludes thought. The material world, they say, is going to disintegrate, and the mind will drown in the torrent-like flow of things. – Let them be reassured! Change, if they consent to look directly at it without an interposed veil, will very quickly appear to them to be the most substantial and durable thing possible. Its solidity is infinitely superior to that of a fixity which is only an ephemeral arrangement between mobilities. I have come, in fact, to the third point to which I should like to draw your attention.

 

Devant le spectacle de cette mobilité universelle, quelques-uns d’entre nous seront pris de vertige. Ils sont habitués à la terre ferme ; ils ne peuvent se faire au roulis et au tangage. Il leur faut des points « fixes » auxquels attacher la pensée et l’existence. Ils estiment que si tout passe, rien n’existe ; et que si la réalité est mobilité, elle n’est déjà plus au moment où on la pense, elle échappe à la pensée. Le monde matériel, disent-ils, va se dissoudre, et l’esprit se noyer dans le flux torrentueux des choses. – Qu’ils se rassurent ! Le changement, s’ils consentent à le regarder directement, sans voile interposé, leur apparaîtra bien vite comme ce qu’il peut y avoir au monde de plus substantiel et de plus durable. Sa solidité est infiniment supérieure à celle d’une fixité qui n’est qu’un arrangement éphémère entre des mobilités. J’arrive ici, en effet, au troisième point sur lequel je voulais attirer votre attention.

(167 / 177 / 150)

 

 

5.2.15

[We need to change our notion of the past, which says that just the present is real and that the past can only survive when remembered in the present.]

 

Bergson says that if we accept that “change is real and even constitutive of reality”, we will need to understand the past differently than we normally do. Usually we think that only the present exists, and the past can only survive when it is remembered in the present. In a sense, memory would be “storing them [certain parts of the past] away in a kind of box.” Bergson says this is a “profound mistake” (167-167 / 177-178 / 150-151).

 

 

[Sections 5.2.16-5.2.23 are excluded from this summary.]

 

 

 

 

 

Bergson, Henri. 1938 [3rd edition, 1990]. La pensée et le mouvant: Essais et conférences. Paris: Quadridge / PUF.

Available online at:

http://catalogue.bnf.fr/ark:/12148/cb372379203

Another version available at:

http://classiques.uqac.ca/classiques/bergson_henri/pensee_mouvant/pensee_mouvant.html

 

Bergson, Henri. 1946. The Creative Mind. English translation by Mabelle L. Andison. Westport, Connecticut: Greenwood.

 

Bergson, Henri. 1965. An Introduction to Metaphysics: The Creative Mind. English translation by Mabelle L. Andison. Totowa, New Jersey: Rowman & Allanheld.

 

 

 

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30 May 2017

B. Russell (ch5) Our Knowledge of the External World, “The Theory of Continuity”, summary

 

by Corry Shores

 

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[The following is summary. My comments are in brackets. Boldface and underlining in quotations are my own. Please forgive my distracting typos: proofreading is incomplete.]

 

 

Summary of

 

Bertrand Russell

 

Our Knowledge of the External World: As a Field for Scientific Method in Philosophy

 

Ch.5

“The Theory of Continuity”

 

 

Brief summary:

Philosophers who think that a continuous medium cannot be made of a dense continuum of discrete parts probably are unaware of the mathematical discoveries which show this to be unproblematic. Mathematical continua like mathematical time and mathematical space have “compactness” (or “density”), meaning that between any two terms there is always another. This is compatible with the philosophical insight into the notion of continuous motion that the object which has moved from one location to another has traversed the infinity of intermediary points along the dense spatial continuum of its movement. Thus it never jumps over any points or intervals. This cannot mean that the intervals involve limits that are immediately next to one another. Rather, the continuity consists instead in the fact that between any two there is always another, and thus there is never a next one. This thus means there are not infinitesimally small intervals as smallest ones, because there should always be another division between any two. With regard to motion, this continuity holds for the coordination of time and space, in what is called Russell’s “at-at” account of motion: the moving body occupies a certain position at some instant, and another position at another instant, and between any two different positions at their given instants, there are an infinity (a dense continuity) of more intervening positions at their instants that the moving body occupies. In other words, no matter where you point to within the space that the moving body traversed, it occupied that place some time during its motion, and thus the object never skips any points or intervals. Furthermore, that spatio-temporal continuity is enough to account for the motion itself. Objections to this view, including Bergson’s, say that motion cannot be decomposed in this way into discrete states. By showing the inadequacy of such objections to the mathematical view, Russell argues that we must side with the mathematical account of motion and change.

 

 

 

 

 

Summary

 

5.1

[There is a philosophical problem with continuity, namely that space and time are thought to be continuous but also composed of points or instants. Yet continuity is thought to be lost if they are ultimately composed of discrete parts.]

 

Russell claims that the theory of continuity “is, in most of its refinements and developments, a purely mathematical subject” and “not, strictly speaking, a part of philosophy” (129). However, the logical basis of the theory of continuity is something that falls under philosophy, and it is this logical basis that Russell will examine here. Russell then explains what the [philosophical] problem of continuity is. On the one hand, we regard space and time as being composed of points and instants, while on the other hand space and time have to have the property of continuity which is lost when they are decomposed into such points and instants. Russell then gives examples in how this problem has played out in philosophy.

The theory of continuity, with which we shall be occupied in the present lecture, is, in most of its refinements and developments, a purely mathematical subject—very beautiful, very important, and very delightful, but not, strictly speaking, a part of philosophy. The logical basis of the theory alone belongs to philosophy, and alone will occupy us to-night. The way the problem of continuity enters into philosophy is, broadly speaking, the following: Space and time are treated by mathematicians as consisting of points and instants, but they also have a property, easier to feel than to define, which is called continuity, and is thought by many philosophers to be destroyed when they are resolved into points and instants. Zeno, as we shall see, proved that analysis into points and instants was impossible if we adhered to the view that the number of points or instants in a finite space or time must be finite. Later philosophers, believing infinite number to be self-contradictory, have found here an antinomy: Spaces and times could not consist of a finite number of points and instants, for such reasons as Zeno’s; they could not consist of an infinite number of points and instants, because infinite numbers were supposed to be self-contradictory. Therefore spaces and times, if real at all, must not be regarded as composed of points and instants.

(129)

 

 

5.2

[The problem of continuity remains even if we discard points and instants, so we will begin with accounts containing these concepts.]

 

Russell will show that the problem of continuity remains even if we discard the points and instants as independent entities. So he will keep them in our analysis of the problem (130).

But even when points and instants, as independent entities, are discarded, as they were by the theory advocated in our last lecture, the problems of continuity, as I shall try to show presently, remain, in a practically unchanged form. Let us therefore, to begin with, admit points and instants, and consider the problems in connection with this simpler or at least more familiar hypothesis.

(130)

 

 

 

5.3

[We have the feeling that were space or time ultimately composed of points or instants, then the transitions between them would be discontinuous, and thus ultimately they would not be continuous things. This results from our failure to properly intuit what mathematicians tell us about continuity and infinity.]

 

In a future chapter, Russell will show that “the positive theory of the infinite” does away with the conceptual difficulties that arise with regard to infinite numbers as involved in continuity. Nonetheless, we still might have the intuition that no matter how many points there are, the transitions between them will discontinuous. Russell thinks that this is not a correct conception. He believes that it results from that fact that we have not properly intuited what mathematics tells us about continuous series.

The argument against continuity, in so far as it rests upon the supposed difficulties of infinite numbers, has been disposed of by the positive theory of the infinite, which will be considered in Lecture VII. But there remains a feeling—of the kind that led Zeno to the contention that the arrow in its flight is at rest—which suggests that points and instants, even if they are infinitely numerous, can only give a jerky motion, a succession of different immobilities, not the smooth transitions with which the senses have made us familiar. This feeling is due, I believe, to a failure to realize imaginatively, as well as abstractly, the nature of continuous series as they appear in mathematics. When a theory has been apprehended logically, there is often a long and serious labour still required in order to feel it: it is necessary to dwell upon it, to thrust out from the mind, one by one, the misleading suggestions of false but more familiar theories, to acquire the kind of intimacy which, in the case of a foreign language, would enable us to think and dream in it, not merely to construct laborious sentences by the help of grammar and dictionary. It is, I believe, the absence of this kind of intimacy which makes many philosophers regard the mathematical doctrine of continuity as an inadequate explanation of the continuity which we experience in the world of sense.

(130)

 

 

5.4

[While the mathematical theory’s notions of points and instants may not correspond to real physical entities, its conceptualization of continuity does correspond to physical reality.]

 

Russell then outlines what he will present. He will discuss the philosophically relevant notions in the mathematical theory of continuity. He will deal then with points and instants. But that does not mean he thinks that the points and instants as understood in this mathematical way also have some physical reality. Nonetheless, he does think that “the continuity of actual space and time may be more or less analogous to mathematical continuity” (131). Russell claims that when we understand the mathematical notion of continuity, “certain characteristics of space and time, previously very hard to analyse, are found not to present any logical difficulty” (131). After learning this mathematical theory, we will return to what our senses tell us about continuous change.

In the present lecture, I shall first try to explain in | outline what the mathematical theory of continuity is in its philosophically important essentials. The application to actual space and time will not be in question to begin with. I do not see any reason to suppose that the points and instants which mathematicians introduce in dealing with space and time are actual physically existing entities, but I do see reason to suppose that the continuity of actual space and time may be more or less analogous to mathematical continuity. The theory of mathematical continuity is an abstract logical theory, not dependent for its validity upon any properties of actual space and time. What is claimed for it is that, when it is understood, certain characteristics of space and time, previously very hard to analyse, are found not to present any logical difficulty. What we know empirically about space and time is insufficient to enable us to decide between various mathematically possible alternatives, but these alternatives are all fully intelligible and fully adequate to the observed facts. For the present, however, it will be well to forget space and time and the continuity of sensible change, in order to return to these topics equipped with the weapons provided by the abstract theory of continuity.

(130-131)

 

 

5.5

[In mathematics, continuity is something found in sequentially ordered series. This ordered arrangement is the essence of continuity.]

 

Russell then begins to detail the features of continuity as it is understood in mathematics. He says that continuity only applies to a sequentially ordered series of terms. As the order is essential to the continuous series, we must turn to nature of this ordered arrangement to analyze their continuity.

Continuity, in mathematics, is a property only possible to a series of terms, i.e. to terms arranged in an order, so that we can say of any two that one comes before the other. Numbers in order of magnitude, the points on a line from left to right, the moments of time from earlier to later, are instances of series. The notion of order, which is here introduced, is one which is not required in the theory of cardinal number. It is possible to know that two classes have the same number of terms without knowing any order in which they are to be taken. We have an instance of this in such a case as English husbands and English wives: we can see that there must be the | same number of husbands as of wives, without having to arrange them in a series. But continuity, which we are now to consider, is essentially a property of an order: it does not belong to a set of terms in themselves, but only to a set in a certain order. A set of terms which can be arranged in one order can always also be arranged in other orders, and a set of terms which can be arranged in a continuous order can always be arranged in orders which are not continuous. Thus the essence of continuity must not be sought in the nature of the set of terms, but in the nature of their arrangement in a series.

(131-132)

 

 

5.6

[Mathematical continua like mathematical time and mathematical space have “compactness”, meaning that between any two terms there is always another. It cannot be known with certainty if physical time and space continua have compactness.]

 

[Russell first notes some more complex mathematical notions that will not concern us here. Apparently there are different degrees of continuity, with the lowest degree of continuity being the one that concerns us here philosophically.] The sort of continuity that we will examine is called “compactness,” where there is always another term between any given two (132). [It seems like what I have elsewhere seen termed “dense continuum” See here and here. In the following I may use density or compactness interchangeably.] Russell claims that mathematical space and time have this mathematical property of compactness, but we cannot be sure if physical time and space do. [Making that determination would require empirical study, but probably no empirical study could be precise enough to make such a fine determination.]

Mathematicians have distinguished different degrees of continuity, and have confined the word “continuous,” for technical purposes, to series having a certain high degree of continuity. But for philosophical purposes, all that is important in continuity is introduced by the lowest degree of continuity, which is called “compactness.” A series is called “compact” when no two terms are consecutive, but between any two there are others. One of the simplest examples of a compact series is the series of fractions in order of magnitude. Given any two fractions, however near together, there are other fractions greater than the one and smaller than the other, and therefore no two fractions are consecutive. There is no fraction, for example, which is next after 1/2: if we choose some fraction which is very little greater than 1/2 say 51/100, we can find others, such as 101/200, which are nearer to 1/2. Thus between any two fractions, however little they differ, there are an infinite number of other fractions. Mathematical space and time also have this property of compactness, though whether actual space and time have it is a further question, dependent upon empirical evidence, and probably incapable of being answered with certainty.

(132)

 

 

5.7

[What we might find particularly difficult is using the mathematical account of the compactness of continuity to explain real physical motion.]

 

There is the difficulty of somehow imagining an infinities of terms between any two. “But when these difficulties have been solved, the mere compactness in itself offers no great obstacle to the imagination” (133). Russell then notes that this idea of compactness does not seem to us intuitively speaking to work well in the case of motion. So first he will examine the mathematical account of motion to show the logical possibility of compactness in it. [I do not follow how he makes his next points, so I will quote later. He says that the physical world might not correspond exactly with this mathematical account, because this account might oversimplify the structures of physical reality. He then claims that “what actually occurs must be capable, by a certain amount of logical manipulation, of being brought within the scope of the mathematical account.” Here I cannot not discern why this must be so, but perhaps it becomes more evident later. Maybe he means that no matter how much they are conceptually incompatible, some degree of contortion will bring them into alignment. He also seems to say that an analysis of the real physical circumstances of motion will raise the same problems that are raised by the mathematical account. But I am not sure if I am reading that part correctly. Finally he seems to be saying that we should put aside for now whether or not the mathematical account corresponds accurately to real physical motion and instead think about how it could be possible for this account to make a formal statement about motion in general. I quote, as I am unsure:]

In the case of abstract objects such as fractions, it is perhaps not very difficult to realize the logical possibility of their forming a compact series. The difficulties that might be felt are those of infinity, for in a compact series the number of terms between any two given terms must be infinite. But when these difficulties have been solved, the mere compactness in itself offers no great obstacle to the imagination. In more concrete cases, however, such as motion, compactness becomes much more repugnant to our habits of thought. It will therefore be desirable to consider explicitly the mathematical account of motion, with a view to making its logical possibility felt. The mathematical account of motion is perhaps artificially simplified when regarded as describing what actually occurs in the physical world; but what actually occurs must be capable, by a certain amount of logical manipulation, of being brought within the scope of the mathematical account, and must, in its analysis, raise just such problems as are raised in their simplest form by this account. Neglecting, therefore, for the present, the question of its physical adequacy, let us devote ourselves merely to considering its possibility as a formal statement of the nature of motion.

(133)

 

 

5.8

[The important philosophical insight into the notion of continuous motion is that the object which has moved from one location to another has traversed the infinity of intermediary points along the dense spatial continuum of its movement. Thus it never jumps over any points or intervals.]

 

[Russell then gives an intuitive, philosophical account of continuous motion. He says the insight tells us that if something has moved continuously, then it has occupied all points in between. And if we examine any two points within that range of motion, the object will have covered the infinity of points between them. Let me note something here that I think is important for this discussion. Although we might in fact have this insight, there is a danger in thinking that there is nothing more than this with regard to the spatial determinations in continuous physical movement. If what gives the motion its continuity is simply the object’s occupation of all the infinity of intermediating points, then all we have so far are fixed positions where no movement can be said to have taken place. Russell notes how in continuous motion, there are no spatial jumps. But I remind us that there is also no movement anyway in this account. So the idea of jumping – or perhaps alternatively, of ‘sliding’ – never comes into play, because these spatial determinations do not involve any sort of motion in the first place.]

In order to simplify our problem as much as possible, let us imagine a tiny speck of light moving along a scale. What do we mean by saying that the motion is continuous? It is not necessary for our purposes to consider the whole of what the mathematician means by this statement: only part of what he means is philosophically important. One part of what he means is that, if we consider any two positions of the speck occupied at any two instants, there will be other intermediate positions occupied at intermediate instants. However near together we take the two positions, the speck will not jump suddenly from the one to the other, but will pass through an infinite number of other positions on the way. Every distance, however small, is traversed by passing through all the infinite series of positions between the two ends of the distance.

(133)

 

 

5.9

[One way we can know that there are not immediate next positions is that were it so, two objects moving at different speeds down the same continuum of space would hold the same next positions at the same next instants. But this is absurd, because the faster one should always hold a further position.]

 

Russell then notes how we might imagine this continuity. We might think that the speck in motion always goes from one position to its immediate neighbor. But this is incorrect, because under this conception, any posited next point would have still more points before it, and more before them, and so on. [What is odd here is that Russell next says that were we to have the conception that there are next moments, we will encounter Zeno’s paradoxes. But it would seem to me that the opposite is the case (or perhaps I should say, it might apply to the paradox of Achilles, but not to the paradox that says motion can never begin). When we claim that there is always another point between a beginning point and some further point in the motion, it would seem impossible for the object to even begin move in the first place. For, there is never a next point in space it moves into and thus there is never a next moment of its motion. So I do not understand at all how his conception avoids the paradoxes. But he says we discuss it in a forthcoming chapter, so let us leave it for now.] Russell then notes a problem with the idea that there are next points and moments. [Suppose you have Achilles and the Tortoise both starting down the race track at the same time. We know that Achilles is faster. However, begin at the first instant of the motion coming right after the instant where they are at the starting position. If there is a next spatial point for Achilles, it would be the same next spatial point for the Tortoise, because they are on the same track. This means means that instant-for-instant, Achilles and the Tortoise match each other position-for-position. Thus we can infer that they move at the same speed. But that contradicts the fact that we know Achilles is much faster. Russell says the problem here is the assumption that there are nexts, so we must reject that assumption. But how does the dense continuum do any better? How does something move to another position if there is never a next one anyway? I can see how we make the claim that an object which has moved has crossed the infinity of intervening points. But I do not see how simply taking note of that accounts for how it moves from one point to the next. Instead, the way I can see this working is if we keep our original assumption that there are next moments and next positions, and then say that in one temporal instant, Achilles covers more points than the Tortoise. (This might be something like François Évellin’s  proposal.) The main objection I would think for this would be that it is physically and logically impossible to be in more than one position at one instant, for the object would both be and not be in some particular location at some particular time, which is absurd. (This assumes that to be in one position means not not be in the others.) But this is absurd only if we insist on the physical world and its dynamics being of such a nature that our propositional descriptions of it would conform to the laws of classical logic. But how can we be sure that physical reality must have these restrictions, especially when they lead to the confusing and bizarre claims Russell is making in this essay? (See especially Graham Priest’s discussion of the problems with the classical logic involved in Russell’s account of motion: In Contradiction 12.2. Also see Priest’s spread hypothesis solution: In Contradiction 12.3. Perhaps the idea would be the following. A paraconsistent reasoning would say that it is true that being in one position means not being in others, but motion is a situation where objects both are in a position and not in that position, because they are moving to another position.]

But at this point imagination suggests that we may describe the continuity of motion by saying that the speck always passes from one position at one instant to the next position at the next instant. As soon as we say this or imagine it, we fall into error, because there is no next point or next instant. If there were, we should find Zeno’s paradoxes, in some form, unavoidable, as will appear in our next lecture. One simple paradox may serve as an illustration. If our speck is in motion along the scale throughout the whole of a certain time, it cannot be at the same point at two consecutive instants. But it cannot, from one instant to the next, travel further than from one point to the next, for if it did, there would be no instant at which it was in the positions intermediate between that at the first instant and that at the next, and we agreed that the continuity of motion excludes the possibility of such sudden jumps. It follows that our speck must, so long as it moves, pass from one point at one instant to the next point at the next instant. Thus there will be just one perfectly definite velocity with which all motions must take place: no motion can be faster than this, and no motion can be slower. Since this conclusion is false, we must reject the hypothesis upon which it is based, namely that there are consecutive points and instants. Hence the continuity of motion must not be supposed to consist in a body’s occupying consecutive positions at consecutive times.

(134)

 

 

 

5.10

[We might be tempted to think that the infinite divisibility terminates at infinitesimal distances. For, this could resolve the paradox by having nexts but without a finite divisible distance between them. Russell argues that then it is not infinitely divisible, so there cannot be infinitesimals.]

 

Russell then claims that we now might fall to the temptation to think of there being next points with infinitesimal distances between them. [Thus they would be successive and contiguous with no intervening points.] Russell reminds us that infinite divisibility has no end to it, and so there would never be any ultimate smallest parts; for, even those, under our assumption, would have to be divisible. [I thought that the infinitesimal conception that Russell rejects also somehow does not see the divisions as requiring some procedure that can only be imagined as occurring in some finite duration of time, but rather as being already accomplished in a sense, as an actual infinity rather than a potential infinity. See Deleuze’s discussion of this in his course lecture of 10-03-1981. I wonder if this issue could be conceived in the following way. Suppose you have an extent of space. And suppose also we want to say it is composed of an infinity of points, as Russell does. The space between those infinity of points cannot be finite, or else they would add up to an infinite extent. (We are assuming our extent is finite.) So somehow we must think of the extent as being composed of just the infinity of points and not some finite space between them. But the length of a point is zero, and an infinity of zeros will not add up to one. Suppose instead we see the extent of space as being composed of an infinity of divisions that are already there and that ultimately terminate such that there are contiguous points with only an infinitesimal amount of distance between them. The sum of the line is thus the sum of an infinity of infinitesimal distances. For some reason, it does not seem inconceivable for me that a finite space can be composed of an infinity of infinitesimals, for we might think that somehow the infinite smallness of the parts is counter-balanced by the infinite greatness of their number and thereby constitute a finite extent. However, it does seem inconceivable to me that a finite extent is made up of a sum of an infinity of dimensionless points (which would come to zero) or of an infinity of finite fractions (which would come to infinity). In other words, Russell seems to want us to think of spatial composition in the following way. We begin with an extent, say 1 meter. We can divide it up into two halves, which total the whole. We can halve each half, to give us four fourths, or a whole. Thus any division we make will still constitute a whole. So if we just stick with the basic idea here, we would say that our divisions are interminable, and each time they produce a finite distance, but that does not mean we have an infinity of finite distances. It rather means that we can make any and as many divisions we want, and each time we get smaller fractions that total a finite whole. But what Russell is doing here is decomposition of something given, which even he admits will never produce ultimate components.  I am concerned with explaining not how the extent can be endlessly decomposed but rather how it could have its finite, extensive composition in the first place. So I think Russell’s conception only works if we say that after a finite number of divisions we have finite fractions of the space. But it does not work if we say there are an infinity of points or divisions, because then we have an infinity of finite parts.  He wants us to think of infinite divisibility as being a matter of making any finite division we could possibly want to make (and we have an infinity of options). But then we are not dealing with infinity but rather with arbitrarity. He needs to explain not how many-many, very-very tiny finite fractions compose a whole but rather how an infinity of them do. Otherwise the divisibility is not infinite but rather just a huge finite. So, his complaint is that the infinitesimal is not compatible with the notion of infinite divisibility, because between any two points should be another. (This I think is possibly a misconception of the actual infinity involved in the notion of the infinitesimal.) But as far as I can see, it is rather his notion of all spatial divisions being of finite magnitude that is not compatible with the notion of infinite divisibility.]

The difficulty to imagination lies chiefly, I think, in keeping out the suggestion of infinitesimal distances and times. Suppose we halve a given distance, and then halve the half, and so on, we can continue the process as long as we please, and the longer we continue it, the smaller the resulting distance becomes. This infinite divisibility seems, at first sight, to imply that there are infinitesimal distances, i.e. distances so small that any finite fraction of an inch would be greater. This, however, is an error. The continued bisection of our distance, though it gives us continually smaller distances, gives us always finite distances. If our original distance was an inch, we reach successively half an inch, a quarter of an inch, an eighth, a sixteenth, and so on; but every one of this infinite series of diminishing distances is finite. “But,” it may be said, “in the end the distance will grow infinitesimal.” No, because there is no end. The process of bisection is one which can, theoretically, be carried on for ever, without any last term being attained. Thus infinite divisibility of distances, which must be admitted, does not imply that there are distances so small that any finite distance would be larger.

(135)

 

 

5.11

[We should think that there is always a smaller finite distance, but never that there is possibly a smaller distance that is smaller than any givable finite one (that is to say, an infinitesimal distance).]

 

[I do not know with certainty what Russell’s next point is. It might be the following, but please consult the quotation. Russell wants to explain how one might mistakenly think that something like the infinitesimal exists. The infinitesimal is something smaller than any givable finite distance. To understand how we might mistakenly think that such a infinitesimal distance might be allowable under our assumptions regarding division, we begin with the insight that there is always a distance shorter than some given finite distance. Under this view, we should interpret it to mean there is always a finite distance smaller than some other finite distance. We make a mistake, however, if we say that there is always a distance that is smaller  than any given finite distance. This could mean that there is always either a smaller finite or a smaller infinitesimal distance. Here is the quotation:]

It is easy, in this kind of question, to fall into an elementary logical blunder. Given any finite distance, we can find a smaller distance; this may be expressed in the ambiguous form “there is a distance smaller than any finite distance.” But if this is then interpreted as meaning “there is a distance such that, whatever finite distance may be chosen, the distance in question is smaller,” then the statement is false. Common language is ill adapted to expressing matters of this kind, and philosophers who have been dependent on it have frequently been misled by it.

(135)

 

 

5.12

[The continuity of motion consists in the fact that a moving body never jumps over any points or gaps in space. For, the moving body occupies a certain position at some instant, and another position at another instant, and between any two different positions at their given instants, there are an infinity of more intervening positions at their instants that the moving body occupies. In other words, no matter where you point to within the space that the moving body traversed, it occupied that place some time during its motion.]

 

[Russell then on this basis of potential infinite divisibility tries to give an account of motion. It consists of the fact that it traverses the infinity of intermediating points. The insight here seems to be that continuity of motion is understood as the object never jumping over any point. As we can see, this accounts for the continuity of motion, but I do not think it is sufficient to explain how the object makes a transition from place to place, as we noted in the comments to 5.9. And I also do not know how to understand the infinity of divisions that must always yield a finite distance, as we noted in 5.10.] [Note Russell’s definition of rest here: “Rest consists in being in the same position at all the instants throughout a certain finite period, however short; it does not consist simply in a body’s being where it is at a given instant.” This seems correct, and it may not go against Bergson’s definition of rest, which is halting at point (see Matter and Memory section 4.2.3) rather than passing at a point. For Bergson, there is only one sort of halting at a point, and that is the object at the beginning and the end of its motion. In between, it is passing through points. As such, it seems for Bergson that we cannot even conceptualize whether or not it is occupying the points in between, because it cannot be a matter of occupation, and he does not think that there are indivisible instants of motion (see section 4.2.5). In order to make the comparison, we would have to wonder, were Bergson to consider the artificial mode of analysis of motion that designates points in time and space of the motion, what would he say about the object’s location at time t, as measured by a clock? He would not say that the object is mobile at that point. Does it mean that it is at that point, or also moving beyond it into another point? That is not made clear. So with regard to Russell’s analysis, he wants us to think that an object in motion occupies singular, determinate positions at determinate times, but that, even though there is no intrinsic difference between a moving object at that position and a resting object at that position, we can still know if it is moving if in another instant (no matter how near) it is in another place. My sense is that Bergson’s complaint would simply be that we are making determinations that are not related to the motion in itself as it happened by rather with space and time determinations that are in a way external to the motion, because they are abstract and conceptual.]

In a continuous motion, then, we shall say that at any | given instant the moving body occupies a certain position, and at other instants it occupies other positions; the interval between any two instants and between any two positions is always finite, but the continuity of the motion is shown in the fact that, however, near together we take the two positions and the two instants, there are an infinite number of positions still nearer together, which are occupied at instants that are also still nearer together. The moving body never jumps from one position to another, but always passes by a gradual transition through an infinite number of intermediaries. At a given instant, it is where it is, like Zeno’s arrow;2 but we cannot say that it is at rest at the instant, since the instant does not last for a finite time, and there is not a beginning and end of the instant with an interval between them. Rest consists in being in the same position at all the instants throughout a certain finite period, however short; it does not consist simply in a body’s being where it is at a given instant. This whole theory, as is obvious, depends upon the nature of compact series, and demands, for its full comprehension, that compact series should have become familiar and easy to the imagination as well as to deliberate thought.

(136)

 

 

5.13

[A more mathematical explanation would understand continuous motion in the following way: the position of a moving body must be a continuous function of time. This can be shown by designating an instant of continuous motion at which a particle is in a certain location. We then consider a spatio-temporal interval enveloping that point. There are two conditions for saying the motion was continuous at that first point: {1} If there is a shorter, internal interval where the particle is found, and {2} if condition 1 holds no matter how small we make the interval, in other words, if there is a dense continuum of positions surrounding the one in question.]

 

Russell will now state this notion of continuous motion in mathematical terminology: “the position of a moving body must be a continuous function of time” (136). [I may not be following this part well, but the idea seems to be the following. Suppose at some point of time a continuously moving object is found at a certain spatial location. This means that for any of the positions near that point the object was found at a different time. Now let us walk through his illustration. We begin by considering a particle in  motion that at time t is found at point P.

continuity motion p diagram full.y

Now we select an portion of the particle’s motion, P1P2, which includes P.

continuity motion p diagram full.z

We are supposing that the particle’s motion is continuous at time t and thus also at point P. Russell then says that this means we should be able to find two instants, t1, t2, where the particle is still found between P1P2. That so far does not seem too important. But then he says that this will hold no matter how small we make the interval between P1P2. Perhaps this could be understood another way. Suppose the situation fulfills these criteria. That means there is always a dense continuum of movement locations around point P at time t. For, no matter how small the interval, the speck was found in locations between that interval, surrounding point P. He says that when this is so, motion is continuous at time t. And, if we can say that the motion is continuous at all times, then the motion is continuous in its entirety. Russell then explains how this criteria would not be met. Suppose that the point jumps from P to some more distant point Q.

continuity motion p diagram full

This means there is not a dense continuum of positions and thus not continuous motion. (I would think that the Q could be placed inside the interval P1P2 and it still would be shown discontinuous, but I am not sure. Suppose Q is placed close to P. We would still be able to find two instants, t1, t2, close to when the object is near P1 and P2,where the particle lies between P1 and P2. The idea here might be that while it fulfills this criterion, it would not follow the next criterion that this holds no matter how small the interval P1P2.) Russell emphasizes that he has defined continuity without the notion of the infinitesimal. (But perhaps we can call that into question. The interval P1P2 will always have a finite distance, no matter how small we make it. So the motion as far as we describe it will always be understood as spanning a finite gap, even if small, and thus we never arrive at the continuity of the motion, even though it is supposedly implied by having us think that the continuity must be occurring within the small intervals. That is not something shown. We must take it on faith, because no matter how small the segment, there is always a finite gap within which, for all we know, the motion was discontinuous.)] [Let me state this one last way. For Russell, the object moves through every point. But there is never a next point. Thus the distance between every point is a finite distance. Now, if objects move from point to point, and if there is always a finite distance between points, no matter how near, then the object must on some scale jump across a finite distance. Thus he has not succeeded at accounting for the continuity of motion, although he has accounted perhaps for the density or compactness of continuous space.]

What is required may be expressed in mathematical language by saying that the position of a moving body must be a continuous function of the time. To define accurately what this means, we proceed as follows. Consider a particle which, at the moment t, is at the point P.

continuity motion p diagram full

Choose now any small portion P1P2 of the path of the particle, this portion being one which contains P. We say then that, if the motion of the particle is continuous at the time t, it must be possible to find two instants t1, t2, one earlier than t and one later, such that throughout the whole time from t1 to t2 (both included), the particle lies between P1 and P2. And we say that this must still hold however small we make the portion P1 P2. When this is the case, we say that the motion is continuous at the time t; and when the motion is continuous at all times, we say that the motion as a whole is continuous. It is obvious that if the particle were to jump suddenly from P to some other point Q, our definition would fail for all intervals P1 P2 which were too small to include Q. Thus our definition affords an analysis of the continuity of motion, while admitting points and instants and denying infinitesimal distances in space or periods in time.

(136-137)

 

 

 

5.14

[Other philosopher’s, including Bergson, have tried to give a non-infinitesimal account of continuous motion (but these accounts fail because they are ignorant of the mathematical analysis of continua).]

 

Russell then notes that other philosophers who are “mostly in ignorance of the mathematician’s analysis” have proposed “other and more heroic methods” for dealing with the problems of continuous motion. He cites one case, namely Bergson, which he has address in his article “The Philosophy of Bergson”, pages 337-341 (sections 2.10-2.14.) [I think he means here that some philosophers have tried to deal with the problems of continuous motion without appealing to the concept of the infinitesimal. With regard to Russell’s reading of Bergson, I think Russell makes two related errors, intentionally or not. One error is that Russell confuses the problem of explaining the continuity of motion with explaining motion itself. They are not the same problem, and they are not the same solution, but Russell thinks they are. What I mean is the following. Russell thinks that the thing to be explained is how continuous motion does not skip any positions along its course. His solution is thus to say that the positions make a dense continuum and thus do not skip any positions. He then wants this to also suffice for a more general account of motion. He wants us to think that so long as you explain the continuity of the spatial positions, which correlate to a dense continuous sequence  of time positions, that you have thereby explained how the object moves through those positions. So we established that this explains the continuity. The insight with regard to the motion seems to be that if the space-time determinations are continuous, how could it not have been moving? We cannot have some notion of the object doing something like “teleporting” between positions, always being at rest each time, because it would have teleported to every possible intermediate position, which would seem to be no different than to actually move through those positions. Perhaps to find the problem with this conception, we should take note of something else. Russell is assuming that there are time-points, between which make temporal intervals. The question is, while the motion is happening, what constitutes the present moment? Let us consider two possibilities. {1} The present moment is an instant, at which point in time the object occupies a point in space. Suppose this is the case. Russell would say that there is no next present. There is necessarily a finite gap between one present moment and the next, meaning that any present coming after will be a finite duration away. This contradicts his claim that the motion is continuous. It only works if we see the motion after its completion and we can thereby think abstractly about its time and space dimensions. But we must also assume that the motion at some period of time actually did take place. And with Russell’s restricted field of concepts, we seem forced to see that as discontinuous motion, if we define the present as a point in time rather than an interval. {2} The present is an interval with some extent of duration. This means that in one present moment, the object is in many locations. The motion could still be continuous, but now we have the confusing notion that an object, in its present moment, occupies multiple positions which are spatially exclusive, meaning that it is presently both in a location and not in that location (this would be confusing if we used the classical logic that Russell uses). The way to make this work would seem to be to say that we must not confuse being co-present with being simultaneous (Barry Dainton seems to work more or less with such a distinction in his analysis of the specious present.) But then the notion of the present becomes unclear. Exactly how long is the present? How do these interval presents connect continuously? Does one pick up when the last one leaves off, or do they overlap? If they do not overlap, how does their transition happen without there being any discontinuity? If they do overlap, how are the same moments of a motion in two different presents? And so on. So given that the first conception of the present seems to lead to discontinuity, and since the second conception of the present leads to conceptual unclarities that Russell does not address, it would seem that he thinks the motion need not be understood as taking place in a living present. But this is one of Bergson’s main points, that it must be understood in this way. (See for example Matter and Memory section 4.2.3, section 4.2.5, and section 4.2.6, here toward the end, “concern far less the living movement itself than a dead and artificial reorganization of movement by the mind”; Creative Evolution section 4.5.2, section 4.5.5, here especially toward the middle, “In order to advance with the moving reality, you must replace yourself within it. Install yourself within change, and you will grasp at once both change itself and the successive states in which it might at any instant be immobilized. But with these successive | states, perceived from without as real and no longer as potential immobilities, you will never reconstitute movement”, section 4.5.8, here especially toward the middle, “the illusion arises from this, that the movement, once effected, has laid along its course a motionless trajectory on which we can count as many immobilities as we will. From this we conclude that the movement, whilst being effected, lays at each instant beneath it a position with which it coincides. We do not see that the trajectory is created in one stroke, although a certain time is required for it; and that though we can divide at will the trajectory once created, we cannot divide its creation, which is an act in progress and not a thing”, and Time and Free Will section §69 and section §70.)  Why does Russell not address the issue of motion happening in a real present? Does he think there is no real present moment of motion? The second error Russell makes, closely related to the first, is that he thinks Bergson’s point with the Zeno paradoxes is that because motion is spatially continuous, it cannot be made of spatial determinations. To this Russell argues that the spatial continuity of motion can be made of such determinations, when they are coordinated with time determinations so to make up a dense continuum. But this is not Bergson’s point really. Rather, Bergson’s point is that since motion is unified durationally, it is not decomposible into spatial and temporal divisions. The element of spatial continuity is not at the forefront here. At best, it is a matter of temporal continuity, but this continuity is rooted in an intertwinement of moments that is not simply that of one moment being continuous with a successor, but rather all moments somehow intertwining no matter where they are on the succession. When motion is happening, it is durational, and that durational component cannot be divided. When you do so, you obtain the absurdities of Zeno’s paradoxes. Russell would like the problem of motion to be reducible to the mathematical problem of giving a description of time-space continuity, and thus to solve it to only require a method for tracking an object through its continuous temporal-spatial variations. But unlike in Bergson’s analysis, Russell does not deal with motion in its actual present activity, nor does he offer a description of how the object makes its transition between determinate space-time locations, when, for him, it is never at two spatial locations at once and when every interval between locations is a finite extent.] [Let me also note that Bergson’s early background was in mathematics, and he was apparently quite talented at it. So it is not obvious to me that Bergson was ignorant of the mathematical account of motion. It would seem more likely to me that he rejected it for philosophical reasons. See for example the biographical information at the Stanford Encyclopedia entry for Bergson.]

Philosophers, mostly in ignorance of the mathematician’s analysis, have adopted other and more heroic methods of dealing with the prima facie difficulties of continuous motion. A typical and recent example of philosophic theories of motion is afforded by Bergson, whose views on this subject I have examined elsewhere.1

1 Monist, July 1912, pp.337-341.

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5.15

[Another objection to the mathematical account is based on feeling (or phenomenological evidence), because we seem to directly perceive solid motions that do not involve an occupation of an infinity of discrete positions.]

 

Russell next says that we might object to the mathematical account of motion on the basis of certain feelings we have, which do not constitute actual reasons. [His points seems to be the following. We compare our perception of an hour-hand with a second hand. We do not see the hour hand moving. But over time we might be able to note it occupied certain determinate positions. However, our perception of the second-hand seems to give us a direct impression of motion, and this motion does not appear to admit of successive mathematical divisions. People thus conclude that real motion is not of the sort that the mathematical account can apply to. (This by the way seems to be one of Bergson’s main arguments against the mathematical account, namely, that it goes against phenomenological evidence.)]

Apart from definite arguments, there are certain feelings, rather than reasons, which stand in the way of an acceptance of the mathematical account of motion. To begin with, if a body is moving at all fast, we see its motion just as we see its colour. A slow motion, like that of the hour-hand of a watch, is only known in the way which mathematics would lead us to expect, namely by observing a change of position after a lapse of time; but, when we observe the motion of the second-hand, we do not merely see first one position and then another—we see something as directly sensible as colour. What is this something that we see, and that we call visible mo- | tion? Whatever it is, it is not the successive occupation of successive positions: something beyond the mathematical theory of motion is required to account for it. Opponents of the mathematical theory emphasize this fact. “Your theory,” they say, “may be very logical, and might apply admirably to some other world; but in this actual world, actual motions are quite different from what your theory would declare them to be, and require, therefore, some different philosophy from yours for their adequate explanation.”

(137-138)

 

 

5.16

[We will first make a more precise statement of this objection, which we will answer from the mathematical perspective.]

 

Russell says we can reply to this objection (that the mathematical account goes against phenomenological evidence) by keeping within the mathematical perspective. First, we should more fully state the objection.

The objection thus raised is one which I have no wish to underrate, but I believe it can be fully answered without departing from the methods and the outlook which have led to the mathematical theory of motion. Let us, however, first try to state the objection more fully.

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5.17

[We obtain this intuition of motion’s indivisibility when it is going fast enough that we see it as if it were constituting a solid chunk of activity.]

 

[Russell’s next point is not entirely clear to me, but it might be the following. We want to explain how do philosophers such as Bergson arrive upon the conclusion that motion does not admit of determinate positions but is rather a solid whole? Russell’s point seems to be that when the motion is fast enough, it appears to our vision and mind to be itself undivided. (Note, the hand example is given by Bergson in Matter and Memory section 4.2.5.)]

If the mathematical theory is adequate, nothing happens when a body moves except that it is in different places at different times. But in this sense the hour-hand and the second-hand are equally in motion, yet in the second-hand there is something perceptible to our senses which is absent in the hour-hand. We can see, at each moment, that the second-hand is moving, which is different from seeing it first in one place and then in another. This seems to involve our seeing it simultaneously in a number of places, although it must also involve our seeing that it is in some of these places earlier than in others. If, for example, I move my hand quickly from left to right, you seem to see the whole movement at once, in spite of the fact that you know it begins at the left and ends at the right. It is this kind of consideration, I think, which leads Bergson and many others to regard a movement as really one indivisible whole, not the series of separate states imagined by the mathematician.

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5.18

[Russell will now give a physiological, psychological, and logical answer to this objection.]

 

Now that Russell have more fully articulated the insight behind this objection to the mathematical account of motion, he will answer it in three ways, physiologically, psychologically, and logically.

To this objection there are three supplementary answers, physiological, psychological, and logical. We will consider them successively.

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5.19

[The physiological answer involves showing that the mathematical account describes a physical situation in the world that can produce the physiological situation of our perception of it.]

 

[I may not get the point right, but it seems to be the following. To give the physiological answer, we merely need to show that the mathematical account describes a physical situation in the world which when perceived would correspond to our (visual) impressions of the motion. See the quote, because there is more to it.]

(1) The physiological answer merely shows that, if the physical world is what the mathematician supposes, its sensible appearance may nevertheless be expected to be what it is. The aim of this answer is thus the modest one of showing that the mathematical account is not impossible as applied to the physical world; it does not even attempt to show that this account is necessary, or that an analogous account applies in psychology.

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5.20

[The reason we directly perceive motion is that it is fast enough that the afterimages fill our present awareness such that we visually see it as a solid movement by means of the visual streak trailing behind it.]

 

[Russell’s next point is phenomenological. It is similar to the notion of afterimages, with the idea being that they diminish with how far into the recent past they go. To directly perceive motion, then, means that it happens fast enough that we can seem many successive positions at once. (Note, this is understood in terms of nerve impulses, but the neurophysiological explanation is quite limited, and it mixes this neurophysiological framework with a phenomenological one. It would have been better if he could have found more neurological research, but it was likely lacking at the time. Perhaps one useful recent source would be Luck and Hollingworth’s edition Visual Memory, particularly chapter 8, “Neural Mechanisms of Visual Memory: A Neurocomputational Perspective,” by Gustavo Deco and Edmund T. Rolls.)] ]

When any nerve is stimulated, so as to cause a sensation, the sensation does not cease instantaneously with the cessation of the stimulus, but dies away in a short finite time. A flash of lightning, brief as it is to our sight, is briefer still as a physical phenomenon: we continue to see it for a few moments after the light-waves have ceased to strike the eye. Thus in the case of a physical motion, if it is sufficiently swift, we shall actually at one instant see the moving body throughout a finite portion of its course, and not only at the exact spot where it is at that instant. Sensations, however, as they die away, grow gradually fainter; thus the sensation due to a stimulus which is recently past is not exactly like the sensation due to a present stimulus. It follows from this that, when we see a rapid motion, we shall not only see a number of positions of the moving body simultaneously, but we shall see them with different degrees of intensity—the present position most vividly, and the others with diminishing vividness, until sensation fades away into immediate memory. This state of things accounts fully for the perception of motion. A motion is perceived, not merely inferred, when it is sufficiently swift for many positions to be sensible at one time; and | the earlier and later parts of one perceived motion are distinguished by the less and greater vividness of the sensations.

(139-140)

 

 

5.21

[Because this mathematical account is compatible with the physiology of the situation, it is possible. But under this view, it is not necessarily true that the motion is physically composed of determinate positions. For, we assumed it to be so.]

 

Thus the mathematical account is compatible with our physiological understanding of the experience of directly perceiving motion. [The idea might be the following. The mathematical account says that the object occupies a distinct position for any instant of its motion. We perceive it being at these positions each instant. But we also have sensory memory, which means that we retain a fainter afterimage of where it was in previous instants, when it moves fast enough to fill our sensory memory. The solid streak of present image plus afterimages leads us to believe that the motion is a solid indivisible unity.  But really it is the continuous tapering where the afterimages blur together that deceives us into inferring that the motion did not involve occupying determinate locations at determinate times.] [I also may not get the next point right, but perhaps it is the following. These considerations assume that physically speaking the moving object does occupy determinate positions. It then shows how that is consistent with the physiology (and phenomenology) of the perception. But this consistency does not prove that the physical situation is the mathematical one. It only shows that it is not inconsistent with other facts and thus it only shows it to be possible, not necessary.]

This answer shows that physiology can account for our perception of motion. But physiology, in speaking of stimulus and sense-organs and a physical motion distinct from the immediate object of sense, is assuming the truth of physics, and is thus only capable of showing the physical account to be possible, not of showing it to be necessary. This consideration brings us to the psychological answer.

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5.22

[The psychological reply involves converting what our senses tell us about the physical world into logical constructions of types of physical entities whose mathematical properties correspond to their counterparts in physics, including such physical concepts as points, instants, and particles.]

 

The psychological reply is quite complicated theoretically, and much of this framework is worked out in later chapters. So Russell can only give a vague outline here. He notes that we infer the physical world from what is given in sensation. But, what is given in sensation is probably little like the world of physics. [Russell does not elaborate here on those differences. Maybe he is referring to certain limitations of our perception, like being unable to see many properties directly, as for example, temperature. I am not sure exactly what he means, but it is not hard to fill in possible illustrations.] [So given that our senses furnish us with information of a physical world which does not directly correspond to the world of sensation,] we might question whether or not we can make inferences from our senses about the physical world. Russell thinks that we can, although his reasoning was given in prior chapters. [I am not sure, but the main idea here seems to be the following. Physics uses such concepts as particles, points, and instants. Such things are not given in experience and are probably not even actually existing things. However, by examining what our senses tell us about the physical world, we can construct conceptual entities (or whatever he means by “logical constructions”) which bear mathematical properties that are shared by the particles, points, and instants of physics.  (Note. It seems odd that Russell admits these things probably do not have an actual existence. Is he saying that our account of motion should correspond not to real, actual physical things but rather just with the abstract entities used in physics to understand the actual physical world? Maybe the idea is that these artificial constructions in physics correspond accurately to real physical situations, and although they should not be taken literally, they still provide us with true intuitions of physical reality.) So supposing that we can construct logical entities from our impressions of the physical world that are mathematically equivalent to their counterparts in physics, we can then translate all the propositions of physics into propositions based on objects given in sensation.]

(2) The psychological answer to our difficulty about motion is part of a vast theory, not yet worked out, and only capable, at present, of being vaguely outlined. We considered this theory in the third and fourth lectures; for the present, a mere sketch of its application to our present problem must suffice. The world of physics, which was assumed in the physiological answer, is obviously inferred from what is given in sensation; yet as soon as we seriously consider what is actually given in sensation, we find it apparently very different from the world of physics. The question is thus forced upon us: Is the inference from sense to physics a valid one? I believe the answer to be affirmative, for reasons which I suggested in the third and fourth lectures; but the answer cannot be either short or easy. It consists, broadly speaking, in showing that, although the particles, points, and instants with which physics operates are not themselves given in experience, and are very likely not actually existing things, yet, out of the materials provided in sensation, together with other particulars structurally similar to these materials, it is possible to make logical constructions having the mathematical properties which physics assigns to particles, points, and instants. If this can be done, then all the propositions of physics can be translated, by a sort of dictionary, into propositions about the kinds of objects which are given in sensation.

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5.23

[We know from sense experience that motion admits of instantaneous views. When we see a motion that is fast enough to be perceived directly (and thus may have a tapering streak of afterimages trailing behind it) but that is given in more than one sensation (and so we see the whole streak occupying different places as the object moves), we can infer that our visual experience consisted of instantaneous views (corresponding to each place along the path that the streak was seen.) (And given that the streaks taper continuously) we can conclude that the moments of perception form a dense continuum. Thus the data of our senses correspond with the mathematical account.]

 

[Russell’s next point seems to be the following. We are now going to consider ways to construct from our sense data conceptual entities that match the types of objects in physics, like particles,  instants, and points. One thing we find is that it accords both with our sense experiences and with our physical models that objects admit of instantaneous states forming a compact (dense) series of moments. Next he turns to phenomenological evidence. I might have this wrong, but it seems to be the following. We noted above the visual streaks trailing behind objects moving fast enough to fill our sensory memory with afterimages. Russell has us consider a motion that is fast enough that we see the streak but is not so fast that it only is perceived in one moment (or one interval of sensory memory); rather, we see it moving for a number of moments. Russell notes that we can distinguish one phase of its motion from another (we see the streak occupying one part of the path of motion in one moment, and another part of the path in another.) Russell then concludes that each such perception of a phase of motion must be an instant that makes a dense continuum with the others. (His reasoning here is not so obvious. Barry Dainton concludes that the present, both phenomenologically and physically, is not an instant but rather takes up a duration of about a half second or so. It is also not obvious from the phenomenological evidence why the moments must make a dense continuum. Perhaps it is the continuity of the tapering streak. But why is it that we must conclude the instants make a dense continuum rather that a discrete series where the differences between them are imperceptible?) Russell thus thinks that given we can infer that the continuum of present moments of experience is a dense continuum, that it then is compatible with the mathematical account.]

Applying these general considerations to the case of motion, we find that, even within the sphere of immediate sense-data, it is necessary, or at any rate more consonant with the facts than any other equally simple view, to distinguish instantaneous states of objects, and to regard such states as forming a compact series. Let us consider a body which is moving swiftly enough for its motion to be perceptible, and long enough for its motion to be not wholly comprised in one sensation. Then, in spite of the fact that we see a finite extent of the motion at one instant, the extent which we see at one instant is different from that which we see at another. Thus we are brought back, after all, to a series of momentary views of the moving body, and this series will be compact, like the former physical series of points. In fact, though the terms of the series seem different, the mathematical character of the series is unchanged, and the whole mathematical theory of motion will apply to it verbatim.

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5.24

[Our senses often perceive continuous physical changes as discrete changes involving thresholds.]

 

[I do not follow Russell’s next point so well, but it seems to be one of the following two possibilities. Both of them take note of the fact that we often sense continuous increases of stimuli in incremental changes rather than continuous ones. His point seems either to be that {1} we sense the continuous variation without realizing it. We notice discrete changes, but we sensed unconsciously all the variations between; or {2} continuous variations in the physical world may correspond to discrete variations in our senses. I think he means the second one, with his broader point being that sometimes we need to go beyond sense data to understand the physical world. I am not sure. Here he references a Poincaré text, that itself references Fechner’s Law. Then Russell proceeds with illustrations much like from Weber’s experiments.  Suppose we are holding a weight, and a small amount more is added. If the increase is small enough, we will not notice the change. In other words, we will not be able to distinguish the two different sensations of weight. Now suppose that we add yet another small weight. Again it is too small to notice the difference. But if we add both small weights at the same time, we might notice the change. So the first is indistinguishable from the second, and the second from the third, but the first is distinguishable from the the third. Following the second interpretation, we would say that the second change is part of the continuum, but it was not discerned, and thus real physical continuous variations can sometimes go unperceived. He shows this also with color variations. (Notes on sources. The Poincaré French text can be found here, with the passages at p.29. English translation of these passages can be found at p.639 of this text here or at p.22 of Poincaré’s Science and Hypothesis, Walter Scott, 1905, available here.)]

When we are considering the actual data of sensation in this connection, it is important to realize that two sense-data may be, and must sometimes be, really different when we cannot perceive any difference between them. An old but conclusive reason for believing this was emphasized by Poincaré.1 In all cases of sense-data capable of gradual change, we may find one sense-datum indistinguishable from another, and that other indistinguishable from a third, while yet the first and third are quite easily distinguishable. Suppose, for example, a person with his eyes shut is holding a weight in his hand, and someone noiselessly adds a small extra weight. If | the extra weight is small enough, no difference will be perceived in the sensation. After a time, another small extra weight may be added, and still no change will be perceived; but if both extra weights had been added at once, it may be that the change would be quite easily perceptible. Or, again, take shades of colour. It would be easy to find three stuffs of such closely similar shades that no difference could be perceived between the first and second, nor yet between the second and third, while yet the first and third would be distinguishable. In such a case, the second shade cannot be the same as the first, or it would be distinguishable from the third; nor the same as the third, or it would be distinguishable from the first. It must, therefore, though indistinguishable from both, be really intermediate between them.

(141-142)

1 “Le continu mathématique,” Review de Métaphysique et de Morale, vol. i. p.29.

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5.25

[In the case of motion, our sense data might not present a moving object as determinately occupying a densely continuous series of positions, but this mathematical conception is not incompatible with our sense data, which gives us a dense continuum of present moments of perception of the motion.]

 

Russell then writes, “Such considerations as the above show that, although we cannot distinguish sense-data unless they differ by more than a certain amount, it is perfectly reasonable to suppose that sense-data of a given kind, such as weights or colours, really form a compact series.” [This makes things a bit confusing. He is saying that the sense-data are continuous, but we only distinguish changes at thresholds. This leads us to the first suggested interpretation above where we sense the changes but they remain unconscious. That seems odd because there would seem to be no phenomenological evidence of it. Perhaps he is confusing the continuum of physical quantities with the sense data of those quantities. Or at least he might be assuming that the one by necessity must correspond to the other, perhaps for physiological reasons, even though it is not phenomenologically evident.] [I do not follow his next points, so please read the text below. I will guess his is saying the following. We might perceive the object as occupying an interval of positions at some present instant. And thus we may say that it does not occupy a determinate position at some instant, like Russell claims. However, there are a continuum of such present instants corresponding to a continuum of perceived intervals of location. And this continuum is of the mathematical kind. Thus the mathematical definition of motion is compatible with our sense data. I quote so you can interpret:]

Such considerations as the above show that, although we cannot distinguish sense-data unless they differ by more than a certain amount, it is perfectly reasonable to suppose that sense-data of a given kind, such as weights or colours, really form a compact series. The objections which may be brought from a psychological point of view against the mathematical theory of motion are not, therefore, objections to this theory properly understood, but only to a quite unnecessary assumption of simplicity in the momentary object of sense. Of the immediate object of sense, in the case of a visible motion, we may say that at each instant it is in all the positions which remain sensible at that instant; but this set of positions changes continuously from moment to moment, and is amenable to exactly the same mathematical treatment as if it were a mere point. When we assert that some mathematical account of phenomena is correct, all that we primarily assert is that something definable in terms of the crude phenomena satisfies our formulæ; and in this sense the | mathematical theory of motion is applicable to the data of sensation as well as to the supposed particles of abstract physics.

(142-143)

 

 

5.26

[There are four questions raised by concerns regarding the insensibility of the mathematical continuum: {1} are mathematical series logically possible? {2} is it not impossible to sense such a continuum? {3} does the assumption of points and instants make the mathematical account fictitious? {4} is there any empirical evidence suggesting that the world of sense is continuous?]

 

Russell notes that there are four questions we might ask when we mistakenly think that the mathematical continuum is not sensible: {1} are mathematical series logically possible? {2} is it not impossible to sense such a continuum? {3} does the assumption of points and instants make the mathematical account fictitious? {4} is there any empirical evidence suggesting that the world of sense is continuous?

There are a number of distinct questions which are apt to be confused when the mathematical continuum is said to be inadequate to the facts of sense. We may state these, in order of diminishing generality, as follows:—

(a) Are series possessing mathematical continuity logically possible?

(b) Assuming that they are possible logically, are they not impossible as applied to actual sense-data, because, among actual sense-data, there are no such fixed mutually external terms as are to be found, e.g. in the series of fractions?

(c) Does not the assumption of points and instants make the whole mathematical account fictitious?

(d) Finally, assuming that all these objections have been answered, is there, in actual empirical fact, any sufficient reason to believe the world of sense continuous?

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5.27

[{1} If we properly understood the infinity involved in mathematical continuity, we would see that it is logically possible.]

 

We would question the logical possibility of the mathematical continuum if we misunderstand the mathematical infinite. [Russell says we examine this in the next chapters.]

Let us consider these questions in succession.

(a) The question of the logical possibility of the mathematical continuum turns partly on the elementary misunderstandings we considered at the beginning of the present lecture, partly on the possibility of the mathematical infinite, which will occupy our next two lectures, and partly on the logical form of the answer to the Bergsonian objection which we stated a few minutes ago. I shall say no more on this topic at present, since it is desirable first to complete the psychological answer.

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5.29

[{2} Our sense data of continuous variation does involve a dense continuum of divisions, only we do not notice them.]

 

[I do not follow the ideas in this paragraph very well, but maybe the points are as follows. Bergson for example says that the flux of change that we sense admits of no inherent divisions, and they can only be created by an act of the intellect that falsifies the reality of the flux. Russell will not try to show that immediate experience contradicts this claim (and of course it does not, as Bergson shows.) Instead, Russell will argue that immediate experience is incapable of proving that the flux is given without divisions. Russell seems to refute this with the insight that most likely there are divisions, but we do not notice them. It gets complicated, but he seems to make this point by working through the following distinctions and ideas. First he seems to be saying that to notice a difference between sense data would be to notice the divisions. So we notice all the different qualitative states in a continuous color change, but we may not notice all the differences between them. We instead select certain differences based on thresholds. (It is not clear to me then if the points of the divisions are the degrees of variation or the differences between them.) He then traces the confusion involved here as a confusion of ‘acquaintance’ with ‘knowledge about’. This discussion is not very clear to me, but maybe he is saying that we can be acquainted with the very tiny variations in a continuous change, but we might only have knowledge about the larger more discernible ones. That is a guess, so please read for yourself.]

(b) The question whether sense data are composed of mutually external units is not one which can be decided by empirical evidence. It is often urged that, as a [143|144] matter of immediate experience, the sensible flux is devoid of divisions, and is falsified by the dissections of the intellect. Now I have no wish to argue that this view is contrary to immediate experience: I wish only to maintain that it is essentially incapable of being proved by immediate experience. As we saw, there must be among sense-data differences so slight as to be imperceptible: the fact that sense-data are immediately given does not mean that their differences also must be immediately given (though they may be). Suppose, for example, a coloured surface on which the colour changes gradually—so gradually that the difference of colour in two very neighbouring portions is imperceptible, while the difference between more widely separated portions is quite noticeable. The effect produced, in such a case, will be precisely that of “interpenetration,” of transition which is not a matter of discrete units. And since it tends to be supposed that the colours, being immediate data, must appear different if they are different, it seems easily to follow that “interpenetration” must be the ultimately right account. But this does not follow. It is unconsciously assumed, as a premiss for a reductio ad absurdum of the analytic view, that, if A and B are immediate data, and A differs from B, then the fact that they differ must also be an immediate datum. It is difficult to say how this assumption arose, but I think it is to be connected with the confusion between “acquaintance” and “knowledge about.” Acquaintance, which is what we derive from sense, does not, theoretically at least, imply even the smallest “knowledge about,” i.e. it does not imply knowledge of any proposition concerning the object with which we are acquainted. It is a mistake to speak as if acquaintance had degrees: there is merely acquaintance and non-acquaintance. When we [144|145] speak of becoming “better acquainted,” as for instance with a person, what we must mean is, becoming acquainted with more parts of a certain whole; but the acquaintance with each part is either complete or non-existent. Thus it is a mistake to say that if we were perfectly acquainted with an object we should know all about it. “Knowledge about” is knowledge of propositions, which is not involved necessarily in acquaintance with the constituents of the propositions. To know that two shades of colour are different is knowledge about them; hence acquaintance with the two shades does not in any way necessitate the knowledge that they are different.

(143-145)

 

 

5.30

[On the basis of our sense data, we must conclude that continuous variations in the physical world are dense continua of mutually exclusive units. We know this because that is (for some reason) the only logical way to explain the composition of complex sense data.]

 

Thus we cannot prove that there is not a dense continuum of divisions in a sensible continuous change just because we do not sense them. [I do not grasp the next point, but I will guess it is the following. We acknowledge that the sense data do not preclude the claim that continuous changes involve a discrete series of non-interpenetrating and thus mutually exclusive units (divisions, points). However, one might say that the sense data do not necessitate that we arrive upon this conclusion. Russell then maybe claims that it is necessary in order to explain how sense data are complex. Russell thinks that if for example you want to account for how the visual field is complex, but you also deny it is made up discrete units, then you will encounter a contradiction. He does not explain himself here, however.]

From what has just been said it follows that the nature of sense-data cannot be validly used to prove that they are not composed of mutually external units. It may be admitted, on the other hand, that nothing in their empirical character specially necessitates the view that they are composed of mutually external units. This view, if it is held, must be held on logical, not on empirical grounds. I believe that the logical grounds are adequate to the conclusion. They rest, at bottom, upon the impossibility of explaining complexity without assuming constituents. It is undeniable that the visual field, for example, is complex; and so far as I can see, there is always self-contradiction in the theories which, while admitting this complexity, attempt to deny that it results from a combination of mutually external units. But to pursue this topic would lead us too far from our theme, and I shall therefore say no more about it at present.

(145)

 

 

5.31

[{3} If we properly define the terms “point” and “instant”, we find that they do not make the mathematical account of motion fictitious. We need to note that space and time can be absolute or relative and that the ultimate components of things in space and time may either have extension and duration or not. Also, Russell thinks that the hypothesis (that continuous motion involves a mathematical continuum) is consistent with facts and logic but not necessitated by them.]

 

[The third question was: “Does not the assumption of points and instants make the whole mathematical account fictitious?”] Russell says the question of points an instants meaning a fictitious account involves two component questions. {a} is space relative or absolute, and {b} do things occupying space and time have [ultimate or most basic] components with extension and duration or with no extension and duration? He then says these questions can take two forms: {i} is the hypothesis consistent with facts and logic? and {ii} is the hypothesis necessitated by facts and logic? In each case [I think, in {a} and {b} above and perhaps for {3} altogether] Russell thinks that yes the hypothesis is consistent with facts and logic but no it is not necessitated by them. He furthermore says that the notions of points and instants do not make the mathematical account fictitious, so long as we properly define these terms.

(c) It is sometimes urged that the mathematical account of motion is rendered fictitious by its assumption of points and instants. Now there are here two different | questions to be distinguished. There is the question of absolute or relative space and time, and there is the question whether what occupies space and time must be composed of elements which have no extension or duration. And each of these questions in turn may take two forms, namely: (α) is the hypothesis consistent with the facts and with logic? (β) is it necessitated by the facts or by logic? I wish to answer, in each case, yes to the first form of the question, and no to the second. But in any case the mathematical account of motion will not be fictitious, provided a right interpretation is given to the words “point” and “instant.” A few words on each alternative will serve to make this clear.

(145-146)

 

 

5.32

[Having the notions of points and instants in the mathematical account does not necessarily mean that it is fictitious.]

 

Mathematics assumes an absolute conception of space and time where there are entities called  ‘points’ and ‘instants’ that are occupied by things in space and time. Some mathematicians see these assumptions as convenient fictions. Russell sees no evidence in favor or against this view. These assumptions are consistent with the facts. But the facts are also consistent with the assumption that there are no “spatial and temporal entities over and above things with spatial and temporal relations”. And by Occam’s razor, we would say that these entities are superfluous and best left out. [Russell then makes a distinction that I do not follow well. His conclusion will be that we should leave open the possibility that points and instants exist over and above things. He distinguishes refusing to assume points and instants from denying their existence. But I cannot tell how that distinction fits in with the relational theory. Is he saying that one of these options is part of that theory? Can one take the relational theory and also deny their existence? His complaint is that to deny them is to add a dogmatic element. Maybe this is his way to keep these items in the mathematical theory without leading necessarily to the conclusion that the theory is fictitious. But I cannot follow it. Please see for yourself.]

Formally, mathematics adopts an absolute theory of space and time, i.e. it assumes that, besides the things which are in space and time, there are also entities, called “points” and “instants,” which are occupied by things. This view, however, though advocated by Newton, has long been regarded by mathematicians as merely a convenient fiction. There is, so far as I can see, no conceivable evidence either for or against it. It is logically possible, and it is consistent with the facts. But the facts are also consistent with the denial of spatial and temporal entities over and above things with spatial and temporal relations. Hence, in accordance with Occam’s razor, we shall do well to abstain from either assuming or denying points and instants. This means, so far as practical working out is concerned, that we adopt the relational theory; for in practice the refusal to assume points and instants has the same effect as the denial of them. But in strict theory the two are quite different, since the denial introduces an element of unverifiable dogma which is wholly absent when we merely refrain from the assertion. Thus, although we shall derive | points and instants from things, we shall leave the bare possibility open that they may also have an independent existence as simple entities.

(146-147)

 

 

5.33

[Ideas that we established in chapter 4 regarding points and instants also show that the mathematical account of motion can use the notions of points and instants without it thereby using fictions.]

 

Russell now concerns himself with the question of whether objects in space and time consist of parts without extension or duration, that is, “of elements which only occupy a point and an instant”. Physics understand things as consisting of elements “which occupy only a point at each instant, but persist through time”. In chapter 4, Russell explained that the persistence of the parts may not be an actual persistence but is rather a “logical construction”. [I do not follow his next points. He again will conclude that the mathematical account of motion can use the notions of points and instants without it thereby using fictions. Since he builds from ideas explained in a prior chapter that I have not yet read, I will refrain from summarizing and butchering the meaning. See the quotation below.]

We come now to the question whether the things in space and time are to be conceived as composed of elements without extension or duration, i.e. of elements which only occupy a point and an instant. Physics, formally, assumes in its differential equations that things consist of elements which occupy only a point at each instant, but persist throughout time. For reasons explained in Lecture IV., the persistence of things through time is to be regarded as the formal result of a logical construction, not as necessarily implying any actual persistence. The same motives, in fact, which lead to the division of things into point-particles, ought presumably to lead to their division into instant-particles, so that the ultimate formal constituent of the matter in physics will be a point-instant-particle. But such objects, as well as the particles of physics, are not data. The same economy of hypothesis, which dictates the practical adoption of a relative rather than an absolute space and time, also dictates the practical adoption of material elements which have a finite extension and duration. Since, as we saw in Lecture IV., points and instants can be constructed as logical functions of such elements, the mathematical account of motion, in which a particle passes continuously through a continuous series of points, can be interpreted in a form which assumes only elements which agree with our actual data in having a finite extension and duration. Thus, so far as the use of points and instants is concerned, the mathematical account of motion can be freed from the charge of employing fictions.

(147)

 

 

 

5.34

[{4} We cannot know if the sense world is continuous, because our senses are unable to discriminate with infinite precision the tiny changes involved in dense continuous changes. Even when things seem continuous, for all we know, they are composed of finite jumps that are too small to be perceived.]

 

We turn now to the fourth question, which asks if there is any reason to think that the world of sense is continuous. Russell says no. Our abilities to discriminate variations is “not infinitely precise”. So we cannot directly sense such a dense continuum of variations. At best, we can experimentally determine when we miss these variations, as by the Weber/Fechner sorts of experiments. For all we know, what the sensory world presents as continua could really be composed instead of finite jumps that are too small to be perceived.

(d) But we must now face the question: Is there, in actual empirical fact, any sufficient reason to believe the [147|148] world of sense continuous? The answer here must, I think, be in the negative. We may say that the hypothesis of continuity is perfectly consistent with the facts and with logic, and that it is technically simpler than any other tenable hypothesis. But since our powers of discrimination among very similar sensible objects are not infinitely precise, it is quite impossible to decide between different theories which only differ in regard to what is below the margin of discrimination. If, for example, a coloured surface which we see consists of a finite number of very small surfaces, and if a motion which we see consists, like a cinematograph, of a large finite number of successive positions, there will be nothing empirically discoverable to show that objects of sense are not continuous. In what is called experienced continuity, such as is said to be given in sense, there is a large negative element: absence of perception of difference occurs in cases which are thought to give perception of absence of difference. When, for example, we cannot distinguish a colour A from a colour B, nor a colour B from a colour C, but can distinguish A from C, the indistinguishability is a purely negative fact, namely, that we do not perceive a difference. Even in regard to immediate data, this is no reason for denying that there is a difference. Thus, if we see a coloured surface whose colour changes gradually, its sensible appearance if the change is continuous will be indistinguishable from what it would be if the change were by small finite jumps. If this is true, as it seems to be, it follows that there can never be any empirical evidence to demonstrate that the sensible world is continuous, and not a collection of a very large finite number of elements of which each differs from its neighbour in a finite though very small degree. The continuity of space and time, the infinite number of [148|149] different shades in the spectrum, and so on, are all in the nature of unverifiable hypotheses—perfectly possible logically, perfectly consistent with the known facts, and simpler technically than any other tenable hypotheses, but not the sole hypotheses which are logically and empirically adequate.

(147-149)

 

 

5.35

[(Given the nature of continuity in sensation,) we cannot know whether our sense data are discontinuous or if there is no lower limit to the duration and extension of single sense datum.]

 

[The next ideas are quite complicated, and I cannot summarize them well. I will guess they are the following. Russell will define a relational theory of instants. I do not grasp this at all. But it defines an instant as a group of events that are simultaneous with each other but not simultaneous with any event outside the group. Then he writes, “if our resulting series of instants is to be compact, it must be possible, if x wholly precedes y, to find an event z, simultaneous with part of x, which wholly precedes some event which wholly precedes y.” I do not grasp this. For one thing, we began by saying that in one instant, the events are not simultaneous with any others outside the group. Then with the lettering notation, it seems he is saying that instant x does in fact have events that are simultaneous with another instant z. (Perhaps those other events are not outside the first group somehow.) At any rate, this seems to amount to a description of the density of instants, but somehow it is a “relational” theory of instants. (Perhaps the idea is that any instant needs to be defined in terms of relations rather than with fixed points in an absolutized dimension of time, like his other account is. I am guessing.) I also do not follow his next point. (I must guess. The sense I am getting is that the continuity is defined by overlaps. And for that reason, given any one sense datum, it overlaps with another, which somehow involves overlaps with infinitely more, given the way the continuity is structured.) At the end, Russell concludes that we cannot know whether our sense data are discontinuous or if there is no lower limit to the duration and extension of single sense datum. See the quotation:]

If a relational theory of instants is constructed, in which an “instant” is defined as a group of events simultaneous with each other and not all simultaneous with any event outside the group, then if our resulting series of instants is to be compact, it must be possible, if x wholly precedes y, to find an event z, simultaneous with part of x, which wholly precedes some event which wholly precedes y. Now this requires that the number of events concerned should be infinite in any finite period of time. If this is to be the case in the world of one man’s sense-data, and if each sense-datum is to have not less than a certain finite temporal extension, it will be necessary to assume that we always have an infinite number of sense-data simultaneous with any given sense-datum. Applying similar considerations to space, and assuming that sense-data are to have not less than a certain spatial extension, it will be necessary to suppose that an infinite number of sense-data overlap spatially with any given sense-datum. This hypothesis is possible, if we suppose a single sense-datum, e.g. in sight, to be a finite surface, enclosing other surfaces which are also single sense-data. But there are difficulties in such a hypothesis, and I do not think that these difficulties could be successfully met. If they cannot, we must do one of two things: either declare that the world of one man’s sense-data is not continuous, or else refuse to admit that there is any lower limit to the duration and extension of a single sense-datum. I do not know what | is the right course to adopt as regards these alternatives. The logical analysis we have been considering provides the apparatus for dealing with the various hypotheses, and the empirical decision between them is a problem for the psychologist.

(149-150. Note: in the Routledge Classics epub edition that I copy the text from, there is a discrepancy between the texts. The part in the above 1915 text that reads “I do not know what | is the right course to adopt as regards these alternatives” is missing in the Routledge version, and in its place is the sentence, “The latter hypothesis seems untenable, so that we are apparently forced to conclude that the space of sense-data is not continuous; but that does not prevent us from admitting that sense-data have parts which are not sense-data, and that the space of these parts may be continuous.”)

 

 

5.36

[The counter-claim to the mathematical account is that change and motion cannot be decomposed into states. The support for this claim is that when you dissect a complex whole into its constituent parts, you remove the parts from their relations to each other and to the whole. Doing so changes the nature of the parts. Russell notes that there is no obvious way to understand why this would be so. Thus the logical answer to objections to the mathematical account would be that the counter-claim is inadequately supported.]

 

Russell turns now to the logical answer to the objections leveled at the mathematical account of motion. Bergson (and others) claims that motion is not divisible into a series of states. Russell then says: “This is part of a much more general doctrine, which holds that analysis always falsifies, because the parts of a complex whole are different, as combined in that whole, from what they would otherwise be.” [I am not sure what he means, but it sounds similar to the idea in Bergson that duration is not homogeneous, so were you to divide it, you would get parts that are different in kind from each other and with the whole. It is a continuous but heterogeneous multiplicity.] Russell then claims this insight is not easily made clear. [Russell then goes on to try to give it some more precise meaning. His manner of doing so seems quite foreign to Bergson’s point, and even Russell admits at the end that this conception is so obviously false that probably Bergson and the other philosophers taking a similar view did not mean it. By the end of the paragraph, we seem to rest on the conclusion that the concept is too vague to deal with effectively. I suppose we are to furthermore conclude that Bergson’s support for his claim that change cannot be decomposed into states must be seen as inadequate given it is too hard to conceptualize. What Russell seems to be doing is the following. He says that the Bergsonian approach claims that when you dissect the parts of change and motion, you separate the parts from their relations with each other and with the whole, and thereby you change the nature of the parts. But I do not quite get how the father son example illustrates that. Let me quote:]

(3) We have now to consider the logical answer to the alleged difficulties of the mathematical theory of motion, or rather to the positive theory which is urged on the other side. The view urged explicitly by Bergson, and implied in the doctrines of many philosophers, is, that a motion is something indivisible, not validly analysable into a series of states. This is part of a much more general doctrine, which holds that analysis always falsifies, because the parts of a complex whole are different, as combined in that whole, from what they would otherwise be. It is very difficult to state this doctrine in any form which has a precise meaning. Often arguments are used which have no bearing whatever upon the question. It is urged, for example, that when a man becomes a father, his nature is altered by the new relation in which he finds himself, so that he is not strictly identical with the man who was previously not a father. This may be true, but it is a causal psychological fact, not a logical fact. The doctrine would require that a man who is a father cannot be strictly identical with a man who is a son, because he is modified in one way by the relation of fatherhood and in another by that of sonship. In fact, we may give a precise statement of the doctrine we are combating in the form: There can never be two facts concerning the same thing. A fact concerning a thing always is or involves a relation to one or more entities; thus two facts concerning the same thing would involve two relations of the same thing. But the doctrine in question holds that a thing is so modified by its relations that it cannot be the same | in one relation as in another. Hence, if this doctrine is true, there can never be more than one fact concerning any one thing. I do not think the philosophers in question have realized that this is the precise statement of the view they advocate, because in this form the view is so contrary to plain truth that its falsehood is evident as soon as it is stated. The discussion of this question, however, involves so many logical subtleties, and is so beset with difficulties, that I shall not pursue it further at present.

(151)

 

 

5.37

[We must thus reject the hypothesis that motion is indecomposable into a dense continuum of discrete parts and instead accept the mathematical conception which says it is decomposable into a dense continuum of instants where the object occupies determinate locations along a dense spatial continuum. ]

 

[Russell then seems to say that we have given enough reason to reject the claim that continuous change and motion are not decomposable into a dense continuum of states. Thus we must conclude that it is so decomposable. This means that change and motion are analyzable. But that analysis is not complete if it only breaks things down into smaller changes or motions. Rather, the analyses must reach terms which are not changes but which are “related by relations of earlier and later”. Yet, if we decompose continuous changes like motions into parts with finite duration, we will have smaller motions rather than terms which are not motions. Thus our analysis must go to instants without duration. This analysis thus brings us to a conception of motion which is the mathematical conception. We have seen that this mathematical conception is consistent will all facts (either physical, physiological, or psychological).]

When once the above general doctrine is rejected, it is obvious that, where there is change, there must be a succession of states. There cannot be change—and motion is only a particular case of change—unless there is something different at one time from what there is at some other time. Change, therefore, must involve relations and complexity, and must demand analysis. So long as our analysis has only gone as far as other smaller changes, it is not complete; if it is to be complete, it must end with terms that are not changes, but are related by a relation of earlier and later. In the case of changes which appear continuous, such as motions, it seems to be impossible to find anything other than change so long as we deal with finite periods of time, however short. We are thus driven back, by the logical necessities of the case, to the conception of instants without duration, or at any rate without any duration which even the most delicate instruments can reveal. This conception, though it can be made to seem difficult, is really easier than any other that the facts allow. It is a kind of logical framework into which any tenable theory must fit—not necessarily itself the statement of the crude facts, but a form in which statements which are true of the crude facts can be made by a suitable interpretation. The direct con- | sideration of the crude facts of the physical world has been undertaken in earlier lectures; in the present lecture, we have only been concerned to show that nothing in the crude facts is inconsistent with the mathematical doctrine of continuity, or demands a continuity of a radically different kind from that of mathematical motion.

(151-152)

 

 

 

 

Text:

Russell, Bertrand. (1915). “The Theory of Continuity.” In Our Knowledge of the External World: As a Field for Scientific Method in Philosophy, pp.129-152. Chicago/ London: Open Court.

PDF available at:

https://archive.org/details/ourknowledgeofex00inruss

Text copied from a Routledge Classics 2009 edition (any discovered discrepancies are changed to the 1915 version and are noted after quotations).

 

 

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