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

310 MATHEMATICAL PHILOSOPHY
Lucretius, as already said, postulated the existence of
an infinitude of atoms. These "seeds of things"-by
whose clashings together and interlockings with one another all things (including souls) were produced, to be
sooner or later again resolved into their elements by
ceaseless hammering of atomic storms-these "seeds,"
the ultimate constituents of all the world (including
minds), were not all of them identical in shape nor in
size, though all of them were too minute to be seen singly
or to be thus apprehended by any other sense; in respect
to shape and size the atoms presented a number of varieties but only a finite number. The atoms of each
variety, it was held, constituted an infinite multitude; and
so there was some finite number of infinite classes of
atoms. The physical functions of the atoms of one class
were, in virtue of their size and shape, different from the
functions of the atoms of any other class. In respect,
however, of multiplicity, these infinite classes were equivalent-they were each of them denumerable-and each
of the classes was equivalent to the class which we today
should call their logical sum,-to the class, that is, of
all the atoms in the universe.
It is sufficiently evident that the poet's conception of
infinite multitude was identical with that now employed
by rrfathernaticians. If you will carefully scrutinize the
poem, you will discover that the same may be said of the
author's conception of infinite magnitude. Formal definition of the notion is not present. We are told, however,
that all the atoms are, in respect of size, between a finite
upper bound and a finite lower bound, and this notion
of lower bound is of critical importance-what the lower
bound is we are not told but we are told that there is
such a bound and that it is finite (not zero),-in other