SOURCE: “Democritus and the Impossibility of Collision,” in Philosophy, Vol. 65, No. 252, April, 1990, pp. 212-17.

[In the following essay, Godfrey explains a modern argument concerning the mathematical indivisibility of atoms and compares it to Greek thought on the subject.]

The Abderite philosophers Leucippus and Democritus sought to solve many of the problems facing Greek thought in the fifth century b.c. by taking all things to be made up of atoms of matter moving in a void. One of the major controversies surrounding their work is whether their atomism was logical or merely physical. Did they consider their atoms to be mathematically indivisible?

An important cause of difficulty here is that Democritus seems to have been fully involved in the mathematics of his day¹ and to have been aware of the discussion of infinite divisibility and points with no magnitude, found, for example, in the paradoxes of Zeno.² It seems unlikely that such a man would cheerfully hold that his atoms could have shape without having parts and without having magnitude. The different shapes of atoms was a major part of his physical theory, which makes it difficult to see how he could have held that they were partless and thus mathematically indivisible.

On the face of it, the Abderites would have done more towards solving the logical problems of the day if the indivisibility of their atoms had been more than merely physical. It would, therefore, be convenient to have a mathematical interpretation of indivisibility which did not involve complete lack of magnitude and shape.

I hope to show that A. David Kline and Carl A. Matheson³ provide a modern and revitalized version of something very close to the very arguments which Leucippus and Democritus advanced.

THE KLINE-MATHESON ARGUMENT

Kline and Matheson argue plausibly against the naive attractions of what they call completely mechanistic physics, by attempting to show that within such a theoretical framework there would be no possibility of collision. The completely mechanistic physics would thus lack the single mechanism claimed by it, in preference to any form of action at a distance.

Their argument is simply stated in seven succinct sentences. The bulk of their article attempts to identify all logically possible responses to the simple argument and to point out reasons why these are unattractive. What matters, however, is not whether they can make these responses unattractive to those of us who are sufficiently sophisticated to take various field theories in our stride, but whether they can make them appear unattractive in terms of criteria acceptable to those naive enough to espouse a completely mechanistic physics.

If Kline and Matheson are correct and if they had presented their arguments to Leucippus and Democritus, the Abderites would have accepted the impossibility of collision and deserted their completely mechanistic physics.

THREE MEANINGS OF ‘TOUCHING’

The most contentious of the seven lines in Kline and Matheson's argument against the possibility of collision is:

‘(2) If two bodies are touching, then they either occupy adjacent points in space or they overlap.’

This omits a third possibility (which they later admit) that two bodies are touching when the distance between them is zero. It is not fair on the Abderites to omit this here because it is quite likely that they would offer this interpretation of touching in preference to the other two.

That bodies should occupy adjacent points in space is impossible for anyone who accepts Kline and Matheson's statement:

‘(3) Space is continuous’

But, although this employs the language and conceptual paraphernalia of modern mathematics, its acceptance does not require a modern mathematical training in the techniques and theory of analysis. Anyone familiar with the arguments of Zeno would have to be offered powerful counter-arguments before they would start thinking in terms of adjacent points. Only the modern codification was lacking in the fifth century understanding of these issues.⁴

Kline and Matheson argue that an adherent of a completely mechanistic physics could not defend the possibility of collision by allowing material bodies to overlap at a point since this would render completely mysterious the fact that the same bodies could not overlap further and thus pass through each other rather than collide and rebound in the manner required for a completely mechanistic physics. In connection with the Abderites this argument is unnecessary since the idea of physical objects overlapping is unacceptable to naive modern ears and we have no reason to suspect that it was less so to the Greeks.

There is no reason to think that the Abderites would have rejected the third interpretation. There was an argument current at that time that two things could not be separate without something to separate them.⁵ If two adjacent atoms were to touch and yet remain two, they would need something to keep them separate. If they were adjacent then what was between them would have to be nothing. The Abderites held that the void was nothing.⁶ There was also an idea in circulation that a geometrical point of zero magnitude was nothing.⁷ A void of zero magnitude separating the atoms would meet all requirements and explain the Abderite view that the reason why two touching atoms can be separated is that there is void between them.⁸ Thus when two atoms were ‘touching’ the distance between them would be zero.

THE TOPOLOGICAL RESPONSE

Kline and Matheson pick out, as the most interesting response to their argument, the topological response. This is the view that physical objects are topologically open and do not include their own boundaries. Thus, when two such bodies are in contact along a boundary, the boundary is not occupied by either of them. They take this response to be most interesting because, if anybody wished to make it, it would invalidate the argument against the possibility of collision. They offer three objections to this response.

THE FIRST OBJECTION

Kline and Matheson ‘find the assertion that a physical object does not include its boundary to be completely mysterious’. This claim, if seen as more than an introspective account of their personal experience, is not reasonable.

As the topological boundary of an object has zero width it is not open to inspection by the senses; so other considerations must determine what is the most attractive way to deal with this boundary. Since an atom in empty space must be surrounded by space, i.e. surrounded by nothing, and since what has zero width and zero volume is (in practical physical terms) nothing, it actually seems quite sensible to assign the boundary to empty space rather than to the atom itself.

This view obviously rests on an assumption that empty space is fairly freely interchangeable with nothing and this assumption could lead to arguments about whether a vacuum is something or about whether nothing is something which exists. In this sort of argument it would be difficult not to confuse grammar with insight and perhaps even more difficult not to feel pressed into certain statements merely to avoid appearing to confuse grammar with insight.⁹

In their completely mechanistic physics the Abderites clearly found it attractive to hold that the universe is made up of particles of matter in a void. It was also attractive (and, in a sense, correct) to see the particles of matter as being something and the void as being nothing.¹⁰

It is even possible to see that attraction in speaking of matter as that which is and the void as that which is not.¹¹ This is not the place, however, for a full discussion of that issue.

Now a material particle is surrounded by the void, i.e. surrounded by nothing. It is, therefore, quite natural to think of the object being bounded by nothing and thus having a boundary which is void.

These remarks clearly do not constitute an argument in favour of holding physical objects to be topologically open but they show that this view is not ‘completely mysterious’. If some proponent of a completely mechanistic physics were to offer this as an argument, then people disposed to reject his views would be able no doubt to regard this as not much more than equivocation and confusion. It is no more so, however, than the view apparently assumed by Kline and Matheson, that it is natural to regard the boundary of a physical object as being part of that object.

THE SECOND OBJECTION

Kline and Matheson claim that an explanation must be provided of why all physical objects are open, why no physical objects include their boundaries. However, once someone accepts the attractiveness of the idea that a physical object should not include its boundaries (e.g. in the light of the comments above) it seems fruitless to try to distinguish between those objects which are open and those which are not. There would be nothing in their appearance to distinguish them. People would want to say pretty much the same sorts of things about them. The only way of detecting which objects were not open would be their inability to collide with other closed objects. Someone whose metaphysical leanings favoured the openness of physical objects would therefore find that the evidence of his senses, in so far as it was relevant, favoured the openness of all physical objects.¹² This would support his inclination to see objects in this way. There is nothing more objectionable in this than in any of the broad assumptions about the properties of all entities of a certain type which particle physics is bound to involve.

There would be no observable difference between a topologically open physical object and one which was exactly similar except that it was topologically closed. The topological boundary of the object has no volume and could not hold anything which could intervene in our perception of the open physical object or the closed object less its boundary.

An open physical object is not necessarily fuzzy in the way that a ball of wool is fuzzy. It could be smoother than a polished crystal ball. The object could extend exactly equally in all directions from a central point exactly as far as the boundary. Direct observation of an individual physical object would give us no cause for thinking either that it was open or that it was closed. Other considerations would have to help us decide. One such consideration might be the fact that collision and contact between physical objects is observed and from this we can deduce (by the arguments of Kline and Matheson) that for each part of the common boundary at least one of the objects in contact or collision fails to include it.

It would be rather more mysterious if rules were produced to tell us which object failed to include which parts of its boundary rather than simply accept that all physical objects are completely open in topological terms.

THE THIRD OBJECTION

Finally they state that a continuous open object split into two parts would not produce two open objects. The boundary between the two parts, having contained matter before the split, must contain matter after the split and thus one of the new physical objects would include at least part of its boundary. The consequent need for material particles to be seen as indivisible atoms is taken as making this position unattractive.

It is, in fact, difficult to see why an event which was grave enough to cause the particle to split (without moving) should not be grave enough to cause the matter (of zero mass and volume) in the boundary to move away (through zero distance).

The Greeks did take this sort of problem seriously.¹³ If, however, Leucippus and Democritus held a view similar to that pilloried by Kline and Matheson, they would have no need to consider the gravity of the events involved in atomic fission. They would have a sense in which their atoms could be mathematically indivisible whilst having shape and magnitude. Thus they would avoid most of the logical problems besetting the physics of their day and at the same time offer the only account of contact between extended material bodies which is acceptable when translated into standard modern topology.

Notes

1. When Thrasyllos arranged the works of Democritus in thirteen tetralogies, three of these were concerned with mathematics.

2. Ancient commentators were sufficiently impressed with the importance of the links between Democritus, Leucippus and Zeno that they report that Leucippus ‘heard’ Zeno.

3. ‘The Logical Impossibility of Collision’, Philosophy 62, No. 242 (October 1987), 509-15.

4. Strictly speaking there is no evidence that anyone before Aristotle hit upon the idea of the finite sum of an infinite series. If this element of our modern conceptual toolkit was absent from the fifth century, it seems to have been the only one.

5. Aristotle, De Generatione et Corruptione I.8.324b mentions this doctrine as part of the background against which the Abderite philosophy developed.

6. Cf., e.g., Aristotle, Metaphysics I.4.985b.

7. Simplicius, Physics 139, includes this as part of a long argument of Zeno.

8. Simplicius, De Caelo 242.

9. Cf. C. J. F. Williams, ‘The Ontological Disproof of the Vacuum’, Philosophy, 59, No. 229 (July 1984), 382-4.

10. Simplicius, De Caelo 294, cites Aristotle for this.

11. Cf. Aristotle Metaphysics I.4.985b.

12. Aristotle, De Generatione et Corruptione I.2.316a suggests that Democritus would have been interested in this sort of consideration.

13. Cf. the argument of Zeno already mentioned in note 7, which is concerned with the effect or lack of effect of adding a single point to an extended body and of removing a single point.