1. The Atomists:

    Atomism was devised by Leucippus and his student Democritus. Democritus was born about 460 B.C., which makes him about 40 years younger than Anaxagoras, and about 10 years younger than Socrates.

    “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)

  2. Overview of atomism:

    1. Imagine each atom, taken by itself, as a Parmenidean unit. Each is indivisible. There is no differentiation between one part of an atom and another part. There is no empty space within an atom - it’s a plenum.

    2. Thus, the atomists tried to agree with Parmenides about everything except the number of real beings.

    3. Comparison with Anaxagoras and Empedocles: theirs was a qualitative pluralism. The atomists offered a quantitative pluralism.

  3. Properties of atoms:

    Each atom is uniform, homogeneous, colorless, tasteless, and indivisible. (We will inquire in a moment into precisely the sense in which atoms are indivisible.)

    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.

  4. The reality of the void:

    The void (empty space) is real, and exists just as much as the atoms do. Cf. fragments 3=A6, 5=A37, 12=A14, 16=B156, 27=A14, 42=B9.

    Fragment 16 is especially interesting here:

    There is no more reason for the “hing” {Greek den} to be than the nothing {Greek mêden, not-hing}.

    Democritus takes the Greek word for “nothing” (mêden) and subtracts the Greek word for “not” (), and ends up with the (otherwise meaningless) Greek den. He seems to be thinking that it would be unreasonable for us to deny the existence of something for which we have a word (“nothing”) when we are willing to assert the existence of what we don't even have a word for, namely, what the word “nothing” is the negation of (since “hing” isn't really a word).

  5. Democritus’s atomistic universe:

    Atoms move about in the void (empty space), collide, attach to others to form compounds. These compounds can have secondary qualities, but such qualities can be reduced to the primary qualities of their component atoms. Cf. 31=A129:

    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.

  6. Democritus vs. Anaxagoras

    1. Anaxagoras's principle of Homoiomereity tells us, e.g., that every part of sugar is sugar. That suggests that he would endorse this inference:

      Sugar is sweet; therefore, the parts of which sugar is composed are sweet.

    2. But Democritean atoms themselves do not have these secondary properties: the atoms are imperceptible. Thus, Democritus would reject this inference:

      Sugar is sweet; therefore, the atoms of which sugar is composed are sweet.

      Instead, Democritus might reason as follows:

      Sugar is sweet; therefore, the atoms composing sugar are round and good-sized.

    3. Both Anaxagoras and Democritus agree that the properties of a large-scale object are a function of the properties of its small-scale components. But Democritus is the first to see the properties of large-scale objects as emergent-not just repeating the properties of the components, but arising from them although being different from them.

  7. Appearance vs. Reality

    1. The coming together of atoms to form compounds appears to be generation (new things appear to come into existence) but it is explained away, as in Empedocles. For arrangements and clusters of atoms are not real, according to Democritus.

    2. 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.

    3. The only things that are real are the atoms, and the empty space they move about in. Cf. 42=B9:
      By convention, sweet; by convention, bitter; by convention, hot; by convention, cold; by convention, color; but in reality, atoms and void.

  8. Determinism

    The picture is entirely mechanistic. The movement of atoms is explained without recourse to reasons, motives, Mind, the Good, Love, Strife, as was common among other Presocratics. Our only fragment from Leucippus attests to this (1=B2):

    No thing happens at random but all things as a result of a reason and by necessity.

    This is causal determinism.

    1. An individual atom has no choice concerning its movements. If pushed, it moves. Its “motivational forces” are all external.

    2. The compounds of atoms don’t have any choice, either. For their movements are all a function of the movements of their component atoms. The movement of an entire system of atoms is just the sum of the movements of all of its individual component atoms.

    3. Explanations are bottom up, not top down. That is, the movements and behavior of a compound of atoms (e.g., a tree, an animal) are to be understood as the sum of the individual movements of all the atoms composing that compound. Thus, one explains why the tree or animal moves in such-and-such a way by explaining why each of its component atoms moves as it does. (It is this kind of explanation that particularly exercised Plato, who thought this idea was a colossal mistake. Cf. Phaedo 98c)

  9. Ancient vs. Modern Atomism

    This very compelling world-view has given rise to a mechanistic, deterministic, point of view that has been even more popular in modern times than it was in ancient times. (Contemporary problems about deterministic physics arising from quantum mechanics have considerably weakened the support for this point of view. The classical Newtonian view that quantum theory has replaced is basically Democritean.)

    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.

    Their impetus did not come from physical inquiries, but from the logical and metaphysical positions of Parmenides and Zeno. As Barnes says (Presocratics, p. 346: “the first atoms came from Elea.” Atoms were postulated in response to the Eleatic view that a truly real entity must be one and indivisible. So we must ask: In what sense were Democritus’s atoms indivisible? Democritus might have meant either of the following:

    1. It is physically impossible to divide an atom.

    2. It is logically or conceptually impossible to divide an atom.

      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.

    Opinion is divided on this issue.

    1. In favor of (a) are

      1. Burnet (EGP, p. 336):
        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.

      2. KRS, p. 415:
        [An atom] is presumably only physically, not notionally, indivisible, since for example atoms differ in size.

    2. In favor of (b) are

      1. Guthrie (vol. 2, p. 396):
        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.

      2. Furley: I will give a quick sketch of the case he makes for (b).

    Furley’s argument for theoretically indivisible Democritean atoms:

    1. Aristotle says that atoms were postulated to meet (what he called) Zeno’s “Dichotomy Argument.” This would be either the paradox of the race course, or the paradox of plurality.

    2. But, as we have seen, both of these arguments of Zeno’s are meant to show that infinite divisibility (whether physical or theoretical) leads to absurd results. Hence,

    3. The atomists would not be meeting Zeno’s argument unless they conceived of atoms as both physically and theoretically indivisible. Furley (p. 510):
      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.

    Furley’s conclusion is supported by further evidence from Aristotle, who claims that atomism conflicts with mathematics (De Caelo 303a20):

    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.

    Objections to theoretically indivisible Democritean atoms.

    1. According to Simplicius, Democritus thought that atoms had size and shape:
      5=A37: For some of them are rough, some are hooked, others concave and others convex, while yet others have innumerable other differences.

      27=A14: These atoms, which are separate from one another in the infinite void and differ in shape and size and position and arrangement, move in the void . . . .

    2. But it is hard to see how someone could conceive of atoms as having size and shape, and still being theoretically indivisible. For it would seem that, for any size x, we can always think of something that is only half that size: we can always divide x by 2.

  10. Atomism and Mathematics

    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.

    1. The Weyl Tile Argument:

      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 that clearly shows what is problematical about atomistic geometry. (Note that the name “Weyl” is pronounced “vile.” Hence this is jokingly called the “Weyl tile” argument.) Here is the argument (vile indeed!):

      Consider any geometrical figures (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 side is some whole number of atoms long. How long is the diagonal of the square? (Weyl suggested that we think of each atom as a tile, so that the length of a side that is n atomic units long is represented by a row of n tiles lying along the side. Our question then becomes: how many tiles lie along 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:

      \[c^2 = a^2 + b^2\]

      We solve for c by taking the square root of each side of the equation:

      \[c=\sqrt{a^2 + b^2}\]

      In the case in which c is the diagonal of a square, the sides a and b are equal. So in the case of a square, we have a + b = a + a = 2a. So a2 + b2 = a2 + a2 = 2a2. That gives us:


      But √xy = √xy. Applying that to our equation above, we get:

      \[c = \sqrt{a^2}\sqrt{2}\]

      Since √a2 = a, this means that:

      \[c = a\sqrt{2}\]

      But this is an irrational number (since √2 is irrational, and the product of an integer and an irrational number is itself irrational). And notice that this is true irrespective of the size of a. So the situation does not change if we suppose that the sides of the square are composed of a very large number of very small “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 = a2. This means that there is no integer, c, such that the diagonal of a square is c space atoms long if its side is some integer n space atoms long. To put it another way, the diagonal and the side of a square cannot both be measured atomistically.

      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?”

    2. Zenos Paradox of the Arrow:

      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, t1, it’s in one place, p1; at some later moment, t2, it’s in another place, p2. But if you pick any time ti that falls between t1 and t2, the arrow is either still at p1 or already at p2. It never moves from p1 to p2, because the space from p1 to p2 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.)

  11. Do atoms have shape?

    Finally, let us consider Democritus’s idea that atoms have shape:

    1. Democritus did not think that atoms merely had magnitude. He thought that they had different sizes, and shapes. And this seems to conflict with the idea that there are atomic sizes. For how could one atom be larger than another unless one of them were either larger than (or smaller than) the atomic size?

    2. Perhaps Democritus thought that there was a smallest size atom, and the size of that atom was the atomic unit of measurement. But if that atom has a shape, the view seems to unravel. Cf. Furley (p. 521):
      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).

    3. Furley concludes that Democritus did, indeed, think of his atoms as being both theoretically indivisible and differing in shape, and that his view was therefore internally inconsistent.

    4. For more on this interpretation, see Guthrie, vol. 2, Appendix, pp. 503-7.

      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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