A discussion of chapter 20 in Mortimer Jerome Adler's book, Aristotle for Everybody: Difficult Thought Made Easy. New York, Touchstone, 1978 [1997].
The concept of infinity fascinated ancient Greek thinkers, beginning with the predecessors of Aristotle, a group that includes the Greek atomists Leucippus and Democritus. They initially set forth "the theory of atoms" (171). The early atomists reasoned that every natural object is constituted of minute particles that are unseen, but material. They believed these particles also were kept apart from one another by a matterless void. Moreover, these atoms were supposed to be extremely small and indivisible (172).
The alpha in the Greek word ἄτομος (atomos/atomus) is privative, so it negates the tomos part of the word. LSJ demonstrates that when ancient writers use ἄτομος substantively, the word may refer to an indivisible element (something that is uncuttable). In the philosophy of Democritus, atoms are particles that act as building blocks of matter; every material object is presumably composed of atoms. Nothing is more minute than atoms in Democritus' system: nothing is more fundamental in terms of matter.
LSJ reminds us that the distinction between atoms and molecules was unknown in antiquity, so the early atomists were not able to make this distinction. See also the interesting usage of ἄτομος in 1 Corinthians 15:51-52.
Cambridge Greek Dictionary: ά-τομος ov adj. [τομή] 1 (of a meadow) uncut, unmown S.2 impossible to cut or divide (into smaller parts); (of a line, a magnitude) indivisible Arist.; (of education, into different branches of knowledge) PL; (of things, into genus or species) Arist.; (of the Forms) Arist. || neut.sb. indivisible thing Arist.
3 || neut.pl.sb. atoms Democr. Arist.; indivisible terms or
definitions Arist.
4 (of differences in character) extremely small Plu.
It appears that Democritus reckoned atoms vary only in their "size, shape, and weight" (172). He envisioned them incessantly moving and thought an infinite number of atoms exist. But Adler reports that Aristotle objected to the existence of atoms for two basic reasons: a) an atom cannot be indivisible if it's a solid parcel of matter that lacks interior empty space. Either an atom has interior empty space, which would mean it's not a parcel of matter or the atom has no space inside of it, which would mean the matter of an atom is continuous. The continuity of matter would imply atoms are not uncuttable (172). In either case, the atom as conceived by Democritus would not exist.
Adler illustrates continuous matter with a matchstick, a piece of wood that one can divide into tinier and tinier pieces ad infinitum. For instance, someone can break a matchstick in two, which then produces two separate parcels of matter. Now those two pieces of matter can each be subdivided to infinity which illustrates that if a parcel of matter is continuous, it's "infinitely divisible." On the other hand, if matter is not continuous (i.e., discrete), then one cannot subdivide it. So Aristotle bases one of his objections to atomism on the fact that the early atomists contended atoms are uncuttable units of matter with no void inside them. He reasoned that any object with no empty space inside it, is continuous and can be divided ad infinitum; on the other hand, if the atom has empty space within it, then an atom is not a unit of matter. Either way, it would seem that the atomists' reckoning would be wrong. Aristotle thus employed the reductio ad absurdum form of argumentation to undermine the atomists' claims.
Aristotle based his second objection to atoms on the concept of infinity. While the atomists contended that an infinite number of atoms exist, Aristotle evidently rejected the idea that an actual quantitative infinite can obtain: he makes a distinction between potential infinity and actual infinity. For instance, whole numbers can be added successively one after the other, but there is no last number in the series of whole numbers added successively. Furthermore, Aristotle insisted that division is another example of a potential infinity: to divide numbers ad infinitum might take endless time to do, but one would never reach the terminus of an infinite series of whole numbers by carrying out the operation of division. Dividing whole numbers is another example of potential, not actual infinity. Aristotle likewise employed this line of argumentation to counter the early Greek atomists. See Adler, pages 173-174.
On the other hand, Aristotle evidently affirms that time is infinite since he believes the world is eternal in the sense that it never began nor will it ever end. Yet he accepts infinite time because it does not exist "at any one moment" (173). Infinite time is another instance of potential infinity, it is not actual insofar as more time always could be added to the series of infinite time. Hence, Aristotle renounced the atomists' suggestion that an actual infinite number of atoms exist. He thought no actual quantitative infinite could exist: no infinite world, no actually infinite space, and no infinite number of atoms that exist with other material things. For the notable student of the Academy, only potential infinites exist.
