by Corry Shores

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[The following is summary. My own comments are in brackets, but please consult the original text, as I am not a logician. All boldface and underlining are my own.]

Graham Priest

*In Contradiction: A Study of the Transconsistent*

Part III. Applications

Ch.12. The Metaphysics of Change II:

Motion

12.2 The Orthodox Account of Change

Brief Summary:

Priest is providing a dialetheic account of motion and change, meaning that motion is understood as being in a state of change involving a true contradiction (being in one place and not in that place at the same time). The orthodox view, articulated by Russell in his ‘at-at’ account of motion, (also the cinematic model), says that there are no such moments of being in two places at once. The problem with Priest’s account is it requires a non-standard logic that we might not be familiar with. The problem with Russell’s account is that it makes the strange claim that motion is made up of nothing more than a sum of resting states. While Russell might think this solves Zeno’s paradox, it only presents us with the paradoxical conclusion that a going-somewhere results from a conglomerate of going-nowheres. Thus Russell’s account is highly unsatisfactory.

Summary

Priest will discuss the orthodox account of motion, which is considered such because it is the received view, although not all philosophers endorse it. He draws from Russell’s formulation in *Principles of Mathematics* where Russell gives his ‘at-at’ account of change an motion. [See that chapter on motion, or see his “Mathematics and the Metaphysicians” where he formulates it again.]

It was formulated clearly and precisely by Russell, according to | whom motion consists merely in the occupation of different places at different times. As he puts it, [block quoting Russell]

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. [end block quote]

Thus, what it is for something to be in motion at an instant is simply that it is found at different places at arbitrarily close instants.

(Priest pp. 172-173, citing Russell [1903] section 447. Note, in my edition this quote is in section 446)

Even though this is the received account of motion, it faces some difficult challenges that call it into question even though they do not entirely knock it down. “It is certainly not the universal panacea for the discomforts people have felt about change that those such as Russell hoped it would be.” (173)

But all that Russell really accomplishes is saying that there is no intrinsic state of motion. However, instantaneous velocity is not an intrinsic state of motion. It is extrinsic, because it involves the differential relation between instants near the limit [in the infinitesimal view that I take, I would formulate this as, it is the extrinsic relation between two infinitesimally-near values, but it is intrinsic in the sense that these two values are separated by no other value and are taken together at once.]

First, it follows from the definition that there is no such thing as an intrinsic state of motion. If one had a body in motion and took, as it were, a logical ‘‘picture’’ of it at an instant, the picture obtained would be no different from one of the same body at the same place, but at rest. Of course, an object in motion can have an instantaneous non-zero velocity, but it would be wrong to think that this differentiates it intrinsically from a static body. For to say that it has an instantaneous velocity at t0 is just to say that df/dt ≠ 0 at t = t

_{0}, where f is the functional specification of position with respect to time. But this is just to say thatAnd the quantifier ‘lim’ quantifies, in effect, over all instants around t0. Hence instantaneous velocity is essentially relational. Russell, in fact, points out that there is no such thing as an intrinsic state of change, and even revels in it: [block quoting Russell]

[Zeno’s arrow argument] denies that there is such a thing as a state of motion . . .

This has usually been thought so monstrous a paradox as scarcely to deserve serious attention. To my mind, I confess, it seems a very plain statement of a very elementary fact, and its neglect has, I think, caused the quagmire in which the philosophy of change has long been immersed . . .

Change does not involve a state of change.

[Priest p.173, quoting Russell (1903) p.351, 350, xxxiii. Note, see especially ch.42 or also see his “Mathematics and the Metaphysicians”]

But this is just the cinematic account of change [discussed previously in section 11.2]. It does not account for motion, since the object is always at rest.

What we have here, as the last sentence makes plain, is just the cinematic account of change, where the change in question is motion. And this particular | case of the account is no more plausible than the general form. A sequence of states, even a dense and continuous one, indistinguishable from corresponding rest-states, does not seem to be a state of motion. If God were to take temporal slices of an object at rest in different places and string them together in a continuous fashion, he would not make the object move. (173-174)

[In the next paragraph I think Priest is making the following point. For Russell, instantaneous velocity cannot be seen as a relation between different moments of time. But in a Laplacean world, we are able to determine other moments on the basis of just one. Priest does not think the world is Laplacean, but he notes it is “curious” that Russell’s theory rules it out a priori. Perhaps it is curious, because perhaps Russell is not an indeterminist. See p.174a.b]

So “a journey is not a series of states indistinguishable from states of rest, even a lot of them close together.” (174) Priest will now look at Zeno’s paradox, particularly the paradox of the arrow, as a challenge to the orthodox account. (174)

[Priest will now just describe the paradox, and later discuss answers to it. In the paradox of the arrow, the arrow moves from point A to point B from moment 1 to moment 2. But at any one instant, the arrow makes no progress. But the interval between the moments is made of such instants, so the arrow at no time moves from A to B.]

Consider a point-object in uniform motion from x to y, say the tip of an arrow. And consider an instant of its motion, t0. At t

_{0}the arrow advances not on its journey to y. (If it did make some headway, this would take time. The temporal stretch involved would not, therefore, be an instant.)Thus, at t = t. It would therefore seem that, since no progress is made in any basic part of the interval [x, y], no progress can be made in the whole. That is, the arrow never makes any progress on its journey at all. This is absurd._{0}, total progress made equals zero. But a temporal interval, [x, y], is made up of such points

[174]

The ‘received answer’ to the paradox is given by Russell, and it is thus bound up with the orthodox account of change. This account agrees with all the points mentioned above, but not the last one which says that “since no progress is made in any basic part of the interval [x, y], no progress can be made in the whole.” [Priest seems to say that one answer to this is to say that the sum of all the positions or instants is greater than these parts, because an infinity of such instants can make up a value greater than zero. I am not sure how to understand this unless we use the concept of the infinitesimal, which Russell does not use.] Yet, still we make the puzzling conclusion that going somewhere results from going nowheres.

The only possibility for avoiding the paradox is a denial of the final step. Even given that at each instant the arrow makes no progress on its journey, in the sum of all instants it does. The whole is greater than the sum of its parts. Technically, though the measure (=length) of the points traversed in an instant is zero, the measure of points traversed in a sum of instants may be nonzero (provided there are sufficiently, i.e. uncountably, many points). To deny this step is to say where the argument fails, but it is hardly to solve the paradox. For the denial of the principle involved in the final step of the argument seems just as | puzzling as the conclusion of the paradox. How can going somewhere be composed of an aggregate of going nowheres?

[174-175]

Priest then offers a mathematical solution [which I do not understand yet. The main idea seems to be that if we define the instant as an interval, we can measure it as having a non-zero value. See p.175, qtd below. The question I still have is, does this make it just a tiny finite segment, or an infinitesimal, or does that not matter?] But this mathematical solution does not help us understand very well the philosophical question of what is happening physically in such instants of motion.

One should separate here a technical mathematical question from a philosophical one. We can represent the length of a certain set of points by a measure function, s. If we define a measure function on the real line in one of the standard ways, we can show that if Z is a finite (or even a countable) set of points, σ(Z)=0; while if Z is an interval, [x, y], σ(Z)=y-x. Thus, the length of the set of points occupied at an instant (which is a singleton set) is zero; but the length of the set of points occupied in an interval of time (which is itself an interval) has non-zero measure. That one can prove a small mathematical theorem or two is one thing; but it does not ease the discomfort that one finds (or at least, that I find) when one tries to understand what is going on physically, when one tries to understand how the arrow actually achieves its motion. At any point in its motion it advances not at all. Yet in some apparently magical way, in a collection of these it advances. Now a sum of nothings, even infinitely many nothings, is nothing. So how does it do it?

[175]

Most citations from:

Priest, Graham. *In Contradiction: A Study of the Transconsistent.* Oxford/New York: Clarendon/Oxford University, 2006 [first published 1987].

Or otherwise indicated, from:

Russell, B. (1903) *Principles of Mathematics*, Cambridge University Press.

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