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

308 MATHEMATICAL PHILOSOPHY
What did Lucretius mean by infinity? What did he
mean by an infinite multitude and by an infinite magnitude? No formal definition of any of these terms is to
be found in his work. It is perfectly clear, however, that,
if one had asked him whether an infinite multitude of
elements was such that it could not be exhausted by removing from it one element at a time, he would have
answered in the affirmative; and that, if one had asked
him whether the elements of an infinite multitude could
be thought of as arranged, like beads on a string, in an
endless succession of elements, he would have again answered affirmatively. In short, an infinite multitude signified for Lucretius what mathematicians now describe
as a denumerably infinite multitude or class. In his work
there is no hint or suggestion that he had any conception
or any inkling of any higher order of infinity. It is highly
probable or indeed quite certain that, owing to his lack
of mathematical discipline, such a conception, had it been
suggested, would have seemed to him unintelligible or
absurd.
It is in itself noteworthy, and if one is really to understand Lucretius it is essential to note, that, with the possible exception of time, the fundamental Lucretian infinities were not mere variables capable of increase beyond
any prescribed finite amount-they were not, that is, what
we have called dynamic infinities; on the contrary, they
were, like the infinites of Cantor, constant or static affairs; but, unlike the Cantor infinites, those of Lucretius
were composed of actual concrete things and not of abstract ones like points, for example, or pure nunibers;
thus the Lucretian infinitude of atoms, for example, was
an infinitude of material particles (taking up roorn) and
they all existed at once.