Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

312 MATHEMATICAL PHILOSOPHY
The significance of the passage and the erroneous use
it makes of the concept of infinity will be clearer to us
if we observe that the passage is a portion of an argument
by which Lucretius endeavors to prove that a finite portion of matter is not indefinitely or limitlessly divisible.
He assumed, as we have seen, that matter is composed
of invisibly small, absolutely solid particles called atoms,
or "seeds of things," these atoms being, by hypothesis,
the smallest particles capable of existing spatially separate from one another. He conceived an atom, however,
to be composed of parts, which were, of course, not separable spatially from the atom. His contention was that
among the parts of an atom there was a least part-a
part, that is, such that none of the parts was smaller.
The foregoing quotation is, as I have said, a part of the
poet's argument in behalf of this contention. Paraphrased in modern terms this portion of the argument
would run about as follows: "If among the parts composing an atom and being such that no two of them have
points in common (save points of a common surface)
there be no least part, then the atom consists of an infinite number of non-interpenetrating parts; the infinite
multitude of atoms in the universe and the infinite multitude of parts of one atom are, as multitudes, equivalent
(in the sense of one-to-one correspondence between the
atoms in the former multitude and the atom-parts in the
latter); the sum of the elements (atoms) of the multitude
of the atoms is an infinite magnitude, the total quantity
of the universe's matter; so, too, the sum of the elements
(atom-parts) of the infinite multitude of parts of one
atom is an infinite magnitude; but this latter sum is the
atom itself; hence, if there be no least part among the
parts of an atom, an atom is an infinite magnitude, and