8 Jun 2014

Russell, Ch.54 of Principles of Mathematics, ‘Motion’, [containing Russell’s at-at account of motion], summary notes


by Corry Shores
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[The following is summary and quotation. All boldface, underlining, and bracketed commentary are mine. Please see the original text, as I did not follow it closely. Proofreading is incomplete, so mistakes are still present.]

[Russell also gives the at-at account of motion in “Mathematics and the Metaphysicians”]


Bertrand Russell

Principles of Mathematics

Part 7: Matter and Motion

Ch.54: Motion

Brief Summary

There is no such thing as a state of motion, since there is no infinitesimal magnitude which would allow an object to be between two positions (or at two positions) at the same moment of time. An object is in motion if it is in different places at different times, and it remains in the same place at different times.





Much has been discussed on motion, and with increasing complexity especially recently [circa 1900]. But logically speaking we first need to clarify simpler matters before advancing to these more complicated issues. It seems Russell will begin with Newton, as “Newton’s scholium to the definitions contains arguments which are unrefuted, and so far as I know, irrefutable: they have been before the world two hundred years, and it is time they were refuted or accepted.” [476]

More basic than the notion of motion are the concepts of location, time, and change. [[In Russell’s definitions of these terms, note that he is not using a dialetheic logic which would permit a self-contradictory statement to be made for an object at a particular time]]

The concept of motion is logically subsequent to that of occupying a place at a time, and also to that of change. Motion is the occupation, by one entity, of a continuous series of places at a continuous series of times. Change is the difference, in respect of truth or falsehood, between a proposition concerning an entity and a time T and a proposition concerning the same entity and another time T', provided that the two propositions differ only by the fact that T occurs in the one where T' occurs in the other.

[[In a dialetheic account, perhaps we would say that change is the true contradiction between two true but contradictory propositions describing the object’s multiple positions at a single instant of time in its motion.]] For there to be change [movement for example], a certain entity must be found at different locations (found relative to a second entity) at different moments of time:

Change is continuous when the propositions of the above kind form a continuous series correlated with a continuous series of moments. Change thus always involves (1) a fixed entity, | (2) a three-cornered relation between this entity, another entity, and some but not all, of the moments of time. This is its bare minimum. Mere existence at some but not all moments constitutes change on this definition.

Russell then gives the example of pleasure. There are moments we have it and moments we do not. It changes when moving from non-existence to existence and back again. [The two entities are pleasure’s existence and its non-existence, and these are found at different moments of time. Or maybe more accurately Russell is saying that the two entities are pleasure and existence, but I do not know how to understand pleasure varying temporally in relation to existence.]

Consider pleasure, for example. This, we know, exists at some moments, and we may suppose that there are moments when it does not exist. Thus there is a relation between pleasure, existence, and some moments, which does not subsist between pleasure, existence, and other moments. According to the definition, therefore, pleasure changes in passing from existence to nonexistence or vice versâ.

Russell goes on to show how to formulate these matters in common usage, and he seems to be suggesting that you need a substance or subject whose properties vary with respect to time. But this creates a problem. There is one thing whose properties change. But would that mean we no longer have the same thing? [His solution seems to be to say that the whole thing cannot be the sum of its parts, but rather be something to which the part somehow relate. I suppose one way to think of this is that this persisting thing is what remains unchanged throughout all the changes. It seems from the beginning of the following paragraph that what remains the same are other parts which did not change, but please interpret for yourself these passages:]

we should say that colour changes, meaning that there are different colours at different times in some connection; though not colour, but only particular shades of colour, can exist. And generally, where both the class-concept and the particulars are simple, usage would allow us to say, if a series of particulars exists at a continuous series of times, that the class-concept changes. Indeed it seems better to regard this as the only kind of change, and to regard as unchanging a term which itself exists throughout a given period of time. But if we are to do this, we must say that wholes consisting of existent parts do not exist, or else that a whole cannot preserve its identity if any of its parts be changed. The latter is the correct alternative, but some subtlety is required to maintain it. Thus people say they change their minds; they say that the mind changes when pleasure ceases to exist in it. If this expression is to be correct, the mind must not be the sum of its constituents. For if it were the sum of all its constituents throughout time, it would be evidently unchanging; if it were the sum of its constituents at one time, it would lose its identity as soon as a former constituent ceased to exist or a new one began to exist. Thus if the mind is anything, and if it can change, it must be something persistent and constant, to which all constituents of a psychical state have one and the same relation. Personal identity could be constituted by the persistence of this term, to which all a person’s states (and nothing else) would have a fixed relation. The change of mind would then consist merely in the fact that these states are not the same at all times.

[following into the next paragraph]

Thus we may say that a term changes, when it has a fixed relation to a collection of other terms, each of which exists at some part of time, while all do not exist at exactly the same series of moments.

In this next paragraph, Russell goes on to wonder if the universe itself changes [for its parts are in constant change, and presumably every part will change at some point in its existence. Please interpret these passages for yourself, but perhaps he is saying that we cannot think of the whole as the sum of the parts (for this would mean the universe does not persist); instead, we must think of the ‘whole’ as a class concept to which particulars (contained under it) relate.]

Can we say, with this | definition, that the universe changes? The universe is a somewhat ambiguous term: it may mean all the things that exist at a single moment, or all the things that ever have existed or will exist, or the common quality of whatever exists. In the two former senses it cannot change; in the last, if it be other than existence, it can change. Existence itself would not be held to change, though different terms exist at different times; for existence is involved in the notion of change as commonly employed, which applies only in virtue of the difference between the things that exist at different times. On the whole, then, we shall keep nearest to usage if we say that the fixed relation, mentioned at the beginning of this paragraph, must be that of a simple class-concept to simple particulars contained under it.


Russell notes how change has traditionally been conceived as a substance whose accidents alter. Russell rejects this. He thinks that change happens because terms change in relation to moments of time. [Russell goes on to describe the existence of terms in a way that is very similar to Spinoza’s notion of modal essence and existence. For Spinoza, a mode’s essence is eternal, but at a particular moment of duration, it also has existence. When it dies, it loses existence but maintains its essence. (See for example Deleuze’s discussion of Spinoza’s correspondence with Blyenbergh.) Compare this idea to what Russell has to say about the existence and being of entities.]

The notion of change has been much obscured by the doctrine of substance, by the distinction between a thing’s nature and its external relations, and by the pre-eminence of subject-predicate propositions. It has been supposed that a thing could, in some way, be different and yet the same: that though predicates define a thing, yet it may have different predicates at different times. Hence the distinction of the essential and the accidental, and a number of other useless distinctions, which were (I hope) employed precisely and consciously by the scholastics, but are used vaguely and unconsciously by the moderns. Change, in this metaphysical sense, I do not at all admit. The so-called predicates of a term are mostly derived from relations to other terms; change is due, ultimately, to the fact that many terms have relations to some parts of time which they do not have to others. But every term is eternal, timeless and immutable; the relations it may have to parts of time are equally immutable. It is merely the fact that different terms are related to different times that makes the difference between what exists at one time and what exists at another. And though a term may cease to exist, it cannot cease to be; it is still an entity, which can be counted as one, and concerning which some propositions are true and others false.


Russell then addresses fictional events. He says that it is possible for a fictional event to take place at a time without actually existing at that time. [see pp.478-479]

But these matters do not concern our mathematical discussions here. [479]


[Russell discusses some complexities regarding how to conceptualize the notion of occupying a place at a time, and he concludes:]

mathematically, the whole requisite conclusion is that, in relation to a given term which occupies a place, there is a correlation between a place and a time.


[[[Note that in the previous chapter, Russell asserts that one piece of matter cannot occupy two difference places at the same moment. He does not explain why. In my assessment, it is because he uses a classical logic which does not allow for true contradiction, like with dialetheic logic. From the prior chapter:

The most fundamental characteristic of matter lies in the nature of its connection with space and time. Two pieces of matter cannot occupy the same place at the same moment, and the same piece cannot occupy two places at the same moment, though it may occupy two moments at the same place. That is, whatever, at a given moment, has extension, is not an indivisible piece of matter: division of space always implies division of any matter occupying the space, but division of time has no corresponding implication. (These properties are commonly attributed to matter: I do not wish to assert that they do actually belong to it.) By these properties, matter is distinguished from whatever else is in space.

]]] Russell now will consider the nature of motion. A moving object cannot be in two places at the same time [see the above discussion on his unsupported assertion of this. A dialetheic logic applied to motion, as in Graham Priest’s analysis (see chapters 11 and 12 of his In Contradiction) , would say that an object is found in two places at the same time, that ‘A is there now’ and ‘A is here now’, using Russell’s formulation.]

A simple unit of matter, we agreed, can only occupy one place at one time. Thus if A be a material point, “A is here now” excludes “A is there now”, but not “A is here then”. Thus any given moment has a unique relation, not direct, but viâ A, to a single place, whose occupation by A is at the given moment; but there need not be a unique relation of a given place to a given time, since the occupation of the place may fill several times.

[Note, in accordance with Leibniz’ Law of Continuity, we can say there is a moment when a moving object is in a state of transition (status transitus) moving from movement to rest. So in the same moment an object can be both in motion and at rest.] In defining motion and rest, Russell seems to be saying that if the object in two moments is in two different places, then it is in motion. If it is in the same place, it is at rest. But it cannot be in two places at the same time.

A moment such that an interval containing the given moment | otherwise than as an end-point can be assigned, at any moment within which interval A is in the same place, is a moment when A is at rest. A moment when this cannot be done is a moment when A is in motion, provided A occupies some place at neighbouring moments on either side. A moment when there are such intervals, but all have the said moment as an end-term, is one of transition from rest to motion or vice versâ. Motion consists in the fact that, by the occupation of a place at a time, a correlation is established between places and times; when different times, throughout any period however short, are correlated with different places, there is motion; when different times, throughout some period however short, are all correlated with the same place, there is rest.

Russell then gives logical/mathematical formulation for defining movement and rest.

We may now proceed to state our doctrine of motion in abstract logical terms, remembering that material particles are replaced by many-one relations of all times to some places, or of all terms of a continuous one-dimensional series t to some terms of a continuous three-dimensional series s. Motion consists broadly in the correlation of different terms of t with different terms of s. A relation R which has a single term of s for its converse domain corresponds to a material particle which is at rest throughout all time. A relation R which correlates all the terms of t in a certain interval with a single term of s corresponds to a material particle which is at rest throughout the interval, with the possible exclusion of its end-terms (if any), which may be terms of transition between rest and motion. A time of momentary rest is given by any term for which the differential coefficient of the motion is zero. The motion is continuous if the correlating relation R defines a continuous function. It is to be taken as part of the definition of motion that it is continuous, and that further it has first and second differential coefficients. This is an entirely new assumption, having no kind of necessity, but serving merely the purpose of giving a subject akin to rational Dynamics.


Russell now clearly states that he rejects the idea of there being a state of motion. [Nothing is in actuality in a state or process of moving from one place to another; but things do find themselves at different places at different times. There are no in-between states when the object is in-between points in space in-between moments of time (or at two places in one moment), which is a formulation that is allowed with the concept of the infinitesimal and/or also with dialetheic logic.] Russell says this creates problems when trying to state the laws of motion, which he discusses later, but he says it is necessary for us to accept these problems given Weierstrass’ reform to calculus in doing away with the concept of the infinitesimal.

in consequence of the denial of the infinitesimal, and in consequence of the allied purely technical view of the derivative of a function, we must entirely reject the notion of a state of motion. Motion consists merely in the occupation of different places at different times, subject to continuity as explained in Part V. There is no transition from place to place, no consecutive moment or consecutive position, no such thing as velocity except in the sense of a real number which is the limit of a certain set of quotients. The rejection of velocity and acceleration as physical facts (i.e. as properties belonging at each instant to a moving point, and not merely real numbers expressing limits of certain ratios) involves, as we shall see, some difficulties in the statement of the laws of motion; but the reform introduced by Weierstrass in the infinitesimal calculus has rendered this rejection imperative.



Bertrand Russell. Principles of Mathematics. London/New York: Routledge, 2010 [1st published 1903].


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