Atomism is the final, and most successful, attempt to rescue the reality of the physical world from the fatal effects of Eleatic logic by means of a pluralistic theory. (Guthrie, vol. 2, p. 389)
Atoms have size, shape, and (perhaps) weight. And they can move. That is, atoms have (what have come to be called) primary qualities. As for such secondary qualities as color, taste, etc., atoms do not have them - an atom cannot be yellow, or salty.
He makes sweet that which is round and good-sized; astringent that which is large, rough, polygonal, and not rounded; sharp tasting, as its name indicates, that which is sharp in body, and angular, bent and not rounded; pungent that which is round and small and angular and bent; salty that which is angular and good-sized and crooked and equal sided; bitter that which is round and smooth, crooked and small sized; oily that which is fine and round and small.
That is, the (secondary) qualities of a compound are completely determined by and reducible to the (primary) qualities of its component atoms.
Nor are the apparent properties of those compounds of atoms real. Such a compound may appear to be white, or green, but this is not so. There is nothing that is really white or green.
The only things that are real are the atoms, and the empty space they move about in. Cf. 26=B 9:
By convention, sweet; by convention, bitter; by convention, hot; by convention, cold; by convention, color; but in reality, atoms and void.
No thing happens at random but all things as a result of a reason and by necessity.
This is causal determinism.
The ancient atomists may appear to have provided a brilliant anticipation of a much later scientific theory. But is this picture accurate? Our enthusiasm for the achievements of the ancient atomists must be tempered by a closer look at the basis of their view.
If (a) is the Democritean position, then it would make sense to talk about the parts of an atom - there might even be such parts - although it would not be physically possible to separate the parts.
If (b) is what Democritus maintained, then this sort of talk makes no sense. The very idea of “splitting an atom” would represent not just a technological difficulty (or even a technological impossibility) but a conceptual absurdity.
We must observe that the atom is not mathematically indivisible, for it has magnitude; it is however physically indivisible, because, like the One of Parmenides, it contains no empty space.
[An atom] is presumably only physically, not notionally, indivisible, since for example atoms differ in size.
Democritus held that his atoms, being not only very small but the smallest possible particles of matter, were not only too small to be divided physically but also logically indivisible.
A theoretically divisible atom would not answer either of Zeno’s arguments. [The plurality paradox] would show that an atom theoretically divisible to infinity must be infinite in magnitude; and [the race course] would show that such an atom could never be traversed - that is, if one starts imagining it, one can never imagine the whole of it.
They must be in conflict with mathematics when they say there are indivisible bodies.
But an atom that is (merely) physically unsplittable would not conflict with mathematics.
If this interpretation is correct, and atoms are theoretically indivisible, then the differences between the Democritean view and modern scientific atomism are greater than the similarities.
5=A37: For some of them are rough, some are hooked, others concave and others convex, while yet others have innumerable other differences.10=A14: These atoms, which are separate from one another in the unlimited void, and differ in shape and size and position, and arrangement, move in the void . . . .
It’s possible that Democritus thought not just of matter, but also of space in an atomistic way. That is, the size of an atom would be an atomic space. In such a system, the ultimate unit of measurement would be the size of an atom. Within that framework, the very notion of half of an atomic space would be unintelligible. So, Democritus would be able to say, coherently, that an atom has size even though it is theoretically indivisible.
But if Democritus “atomized” space in this way (as he very likely did), he runs into another problem. For Euclidean geometry (in particular, the Pythagorean theorem) requires that space be continuously divisible. Hence, if atomism denies the continuity of space, it will fail to get mathematics right.
Why is this? There is a famous argument by the mathematician Hermann Weyl (the “Weyl tile” argument) that clearly shows what is problematical about atomistic geometry:
Consider any geometrical figure (e.g., squares, triangles, etc.) with straight lines as sides. The length of each side will be measured in (space) atoms, and each side will be assigned an integer as its measure. (Each side will be n atomic units long, where n is a positive integer.)
Now consider a square whose sides are some whole number of atoms long. How long is the diagonal of the square?
Since the diagonal of a square divides it into two right triangles, we get our answer from the Pythagorean theorem: the square of the hypotenuse of a right triangle = the sum of the squares of the other two sides:
c^{2}= a^{2} + b^{2}4:56 PM 7/6/02
So if a square is, e.g., 4 atoms long per side, then a = 4 and
b = 4.
So: c^{2} = 32 = 16 × 2.
So c =
4Ö2.
But this is an irrational number. So no square whose sides are 4 atoms long can have a diagonal that is any whole number of atoms long!
Note that the situation does not change if we suppose that the sides of the square are composed of a larger number of even smaller “space atoms.” Even if the sides are billions of atoms long, the length of the hypotenuse will still be an irrational number of such atoms. Let a, the number of atoms in the side of a square, be as large as you like, and let c be the number of atoms in the hypotenuse; c will still be an irrational number, for c = aÖ2.
So “atomistic geometry” seems inherently flawed. If we measure the sides of a square as a whole number (of atomic units), and we try to measure the diagonal as a whole number (of atomic units), we will never get the correct answer to the question: “How long is the diagonal of a square?”
Zeno’s argument that an (apparently) moving arrow is really at rest throughout its flight seems easy to evade if one insists that space is continuous (and hence infinitely divisible). But an atomist who insists on theoretically indivisible atoms seems bound to deny that space is infinitely divisible. And Zeno’s Arrow Paradox poses an especially troubling problem for such an atomist.
For how will the arrow (or any object, in fact) move through an atomic space? Since the space cannot be divided, the tip of the arrow must advance from one end of the space to the other without ever having occupied any of the intervening space. At one moment, t_{1}, it’s in one place, p_{1}; at some later moment, t_{2}, it’s in another place, p_{2}. But if you pick any time t_{i} that falls between t_{1} and t_{2}, the arrow is either still at p_{1} or already at p_{2}. It never moves from p_{1} to p_{2}, because the space from p_{1} to p_{2} is atomic and therefore cannot be divided.
Although we cannot, of course, be certain that Zeno intended his Arrow Paradox specifically against the atomists, it constitutes a formidable objection to an “atomic” conception of space.
(Nevertheless, physicists are still enamored of the idea that space and time come in discrete “quanta” which cannot meaningfully be further sudivided, even conceptually. If you want proof, check out this New York Times article of December 7, 1999.)
Democritus’ atoms had many variations in shape and size. There seems to be an inescapable contradiction here. If we take together a smaller atom and a larger one, we can always distinguish in the larger one that part which is covered by the smaller and that which is not. Even within the limits of a single atom, supposing it to be of a complex shape (say hook-shaped), we can always distinguish one part of the shape from another (say the hook from the shaft).
For an opposing view, cf. Barnes, Presocratics, 352-360. Barnes considers the idea that Democritean atoms are theoretically indivisible, in three different senses: conceptually, geometrically, and logically indivisible. He argues that the available texts do not adequately support the idea that atoms are theoretically indivisible, and concludes that the case has not been proven either way.
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