Priest (12.3) In Contradiction, ‘The Hegelean Account of Motion’, summary

 

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. Proofreading is incomplete so mistakes are still present.]

Graham Priest

In Contradiction:
A Study of the Transconsistent

Part III. Applications

Ch.12. The Metaphysics of Change II: 
Motion

12.3 The Hegelean Account of Motion

Brief Summary:

In the Hegelean account of motion, we would think of there being a spread of moments to which the object occupies a spread of spaces. This duration is quite small and tight around a certain time point. Hegel’s view is that the position of the moving object is indiscernable at some moment, and so could be at one of many places in the same tiny moment.

Summary

 

 

Previously Priest examined the orthodox, Russellean, ‘at-at’, cinematic account of change. We found that it leads to the strange conclusion that motion is comprised of no more than states of rest. Now Priest will examine alternate accounts of motion, in particular Hegel’s.

[M]otion itself is contradiction’s immediate existence. Something moves not because at one moment of time it is here and at another there, but because at one and the same moment it is here and not here . . .
[Hegel (1840), vol. 1, ch. 1, sect. C4., quoted in Priest 175]

Hegel means that although an moving object will be at different places at different times, it is necessary as well that at specific times it be in different places.

Hegel is not denying that if something is in motion it will be in different places at different times. Rather, the point is that this is not sufficient for it to be in motion. It would not distinguish it, for example, from a body occupying different places at different times, but at rest at each of these instants. What is required for it to be in motion at a certain time is for it both to occupy and not to occupy a certain place at that time.
[175]

This account has not been well received, because it defines motion by means of contradiction. It is also not clear how exactly to relate this account with our more scientific and mathematical methods for calculating motion. However, the orthodox account of motion seems to be built into the formulas of calculus.

Thus, an equation of motion, x = f(t), just seems to encode the idea of the occupation of different places at different times: it merely records the correlation. By contrast, Hegel’s view seems to have no bearing on the matter.
[176]

[Hegel reasoning for this seems to be that the object is at a single position at a single time, but near it are positions and times so close that we are unable to localize the body.]

The reason is roughly as follows. Consider a body in motion—say, a point particle. At a certain instant of time, t, it occupies a certain point of space, x, and, since it is there, it is not anywhere else. But now consider a time very, very close to t, t'. Let us suppose that over such small intervals of time as that between t and t' it is impossible to localise a body. Thus, the body is equally at the place it occupies at t', x' (≠x). Hence, at this instant the body is both at x and at x' and, equally, not at either. This is essentially why Hegel thought that motion realises a contradiction.
[176]

Hegel also explains why we cannot localize the positions in an instant. It is because they fall along a continuum, and neighboring points along a continuum merge.

Hegel gives a reason why a moving body cannot be localised. The reason derives from his view of the continuum. Essentially, it is that in a continuum distinct points themselves merge. Thus, the reason why we cannot localise a body to t is just that t itself is not ‘‘localisable’’. As he puts it,
{quoting Hegel (1940)}

[W]hen . . . we admit that time and space are continuous, so that two periods of time or points of space are related to one another as continuous, they are, while being two, not two, but identical . . . [M]ovement means to be in this place and not to be in it, and thus to be in both alike; this is the continuity of space and time which first make motion possible. 
{end quote}

And again:
{quoting Hegel (1930)}

[When a body is moving] there are three different places: the present place, the place about to be occupied and the place that has just been vacated; the vanishing of the dimension of time is paralysed. But at the same time there is only one place, a universal of | these places, which remains unchanged throughout all the changes; it is duration existing immediately in accordance with its Notion, and as such it is Motion.
{Priest 176-177, quoting Hegel, firstly “Hegel (1840), vol. I, pp. 273, 273–4 of the translation;” and secondly “ Hegel (1830), p. 43 of the translation. The italics are original.”}

Hegel held some interesting ideas that we might want to further pursue, for example the 18th century notion of the variable point and the contradictory unity of the discrete and the continuous. But for now, Priest will formulate Hegel’s main insight as “the Spread Hypothesis”. [177]

Spread Hypothesis
[heading is bold in Priest’s text]

A body cannot be localised to a point it is occupying at an instant of time, but only to those points it occupies in a small neighbourhood of that time.
[177]

Although this might seem like a strange concept to use in physics, we already know that strange things happen at the scale of Planck’s constant. [177]

Priest will articulate the spread hypothesis using the tense logic semantics he previously described. [177]

[In the following, Priest will first formulate the Russellean at-at account, which holds that a moving body cannot be in two places at the same time. The formulations he will give are basically saying that if a moving object is found in its mathematically determined location at a specific time, then this is true, but if it is not there, it is false. Recall that v is the function that assigns truth/falsity values (0/1 values) to the given proposition (stating the object’s position). The body is called b. The proposition has the structure ‘b is at point x’, which can be expressed as the relation Bx. The function determining the objects position is x = f(t). We might read the two formulas as (1a) the statement ‘b is at point r’ is true (at time t) if r equals the value that the function produces for that given time. And, (1b): the statement ‘b is at point r’ is false (at time t) if r does not equal the value that the function produces for that given time.] The following formulation we will call the “Russellean state description”:

Now, consider a body, b, in motion. Again to keep things simple, let us suppose that it is moving along a one dimensional continuum, also represented by the real line. Let us write Bx for ‘b is at point x’. Let us also suppose that each real, r, has a name, r. This assumption is innocuous. It could be avoided by talking in terms of satisfaction rather than truth. I make it only to keep the discussion at the propositional level. Let the motion of b be represented by the equation x = (t). Then the evaluation, v, which corresponds to this motion according to the Russellean account, is just that given by the conditions:

[177]

Priest draws a diagram to depict it. As we can see, only the proper time/place coordinate for f(t) is true.

 

[In the next formulation Priest describes the Hegelean account of motion. It seems he is saying that we need to think not just of single time points but as well time points surrounding in a set of time points here called θ_t. It also seems to be saying that although there are different instants in this set which correspond to different locations, if these instants are included in set θ_t, which surrounds specific point t, then it is true that the object is in these other locations. Specifically the formulations might be read (2a) the proposition ‘b is at location r’ is true if within the spread of moments around t (that is, in set θ_t), there is at least one time point which when used in the function produces that value for r. And (2b) the proposition ‘b is at location r’ is false if within the spread of moments around t (that is, in set θ_t), there is at least one time point which when used in the function produces a value that is not r. So while θ_t might be the set of time points around t, there is also the resulting ‘spread’ of spatial locations corresponding to all those time points. This set Priest calls Σ_t. In the diagram we see how this spatial spread matches the temporal spread, and that only those falling within those spread are true. Another concept Priest uses is ‘degenerate’ which seems to mean that a set of locations corresponds to a single time point and not a set of time points, but please consult the text to be sure, p.178.]

The appropriate state description for the Hegelean account will, of course, be different, incorporating, as it does, the spread hypothesis. In accordance with the hypothesis, there is an interval containing t, θ_t (which may depend not only on t but also on f) such that, in some sense, if t' ∈ θ_t, b’s occupation of its location at t' is reproduced at t. I suggest that a plausible formal interpretation of this is that the state description of b at t is just the ‘‘superposition’’ of all the Russellean state descriptions, v_t', where t' ∈ θ_t. More precisely, it is the evaluation, v, given by the conditions

Let us call this the Hegelean state description of the motion. Suppose we write Σ_t for the spread of all the points occupied at t, i.e., for {f(t') | t' ∈ θ_t}. If Σ_t is degenerate, that is if Σ_t={f(t)}, then the Hegelean state description is identical with the Russellean one. If it is not, then, as may easily be seen, the condition on the righthand side of (2b) is satisfied by all r, and we may depict the Hegelean state description as follows

[178]

[Priest then discusses the contradiction that would arise if Σ_t were not degenerate, which I think means that there are many locations corresponding to just one time point, but I am not sure. Perhaps the contradiction he describes is that for one time point, the object is in many places, but that means it is both in one certain such place and not in it during the same instant. But suppose that the object remains in one position within Σ_t. This does not lead to a contradiction and in fact describes a state of rest. It is even compatible with the Russellean description of rest (being at the same place throughout different times). However, even with all points in Σ_t being the same, there can still be a contradiction. This would happen if the moments surrounding very near the given time point (or θ_t) extend beyond the scope of , and thus the object will be again both in one location in that temporal spread and not in that location. But, since the temporal spread θ_t is very brief, “this unstable state of affairs can never last for very long.” He then introduces the idea of the derivative df/dt. It seems like he is saying that since the object does not move far enough to register an assignable value greater than 0, and thus finitely speaking does not change assignable locations, hence making it not a contradiction. But please consult p.179 and check.]

As the picture shows, if Σ_t is not degenerate, then at t a number of contradictions are realised. For all r ∈ Σ_t, 1 ∈ v_t(Br ∧ ¬Br). Σ_t may be degenerate for one of two reasons. The first is that θ_t may itself be degenerate. That is, θ_t={t}. The other is that, though θ_t is not degenerate, f is constant over it. Now θ_t is not, in general, degenerate (or the Hegelean account collapses into the Russellean one). It is quite plausible to suppose that its length depends on the velocity of b, so that the faster b is going the more difficult it is to ‘‘pin it down’’. At any rate, provided θ_t is non-degenerate, if b satisfies the Russellean conditions of motion at | t (namely that at arbitrarily close points of time it is to be found elsewhere), then contradictions will be realised at t. If, on the other hand, a body occupies the same spot at all times in θ_t, St will be degenerate and no contradiction will be realised. It is possible (for all I have said so far) for a body to satisfy the Russellean conditions for rest, that is, to occupy the same place over a period of time, and yet for a contradiction to be realised during that time. This will happen at t if θ_t extends beyond this period of constant position. But since θ_t is very small (maybe in the order of Planck’s constant?) this unstable state of affairs can never last for very long. We might even suppose that if df /dt=0 then θ_t is degenerate. Now, if f is constant for a period around t, then df /dt=0 at t. In this case, therefore, no contradiction is realised at t.
[178-179]

[Priest finishes by noting that we might need to know whether or not  θ_t extends beyond t or if t is the least upper bound of θ_t. This is a problem, because it might imply there can be backwards causation. Priest provides a solution. Please see page 179 for details.]

 

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:

 

Hegel, G.W.F. (1830) Philosophy of Nature, English translation by A. V. Miller, Clarendon Press, 1970.

Hegel, G.W.F. (1840) Lectures on the History of Philosophy, English translation by E. S. Haldane, Kegan Paul, 1892.