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



“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.









[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.





[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.






[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.





[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.





[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.





[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.





[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.





[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.





[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.






[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.





[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.





[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.





[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.






[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.






[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.”





[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.





[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.





[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.






[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.





[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.





[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.





[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.





[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.





[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.


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





[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.





[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?





[{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.





[{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.





[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.





[{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.





[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.





[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.






[{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.





[(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.”)




[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.





[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.







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.

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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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