The real difficulty lies in the concept of the class. Dogmatic realism
is prone to find here an entity for which, as it is obviously not a
physical thing, a home must be provided in some region of "being." Hence
arises the realm of subsistence, as for Plato the world of facts
duplicated itself in a world of ideas. But the subsistent realm of the
mathematician is even more astounding than the ideal realm of Plato, for
the latter world is a prototype of the world of things, while the world
of the mathematician is peopled by all sorts of entities that never were
on land or sea. The transfinite numbers of Cantor have, without doubt,
a definite mathematical meaning, but they have no known representatives
in the world of things, nor in the imagination of man, and in spite of
the efforts of philosophers it may even be doubted whether an entity
correlative to the mathematical infinite has ever been or can ever be
specified.

Mr. Russell now teaches that "classes are merely symbolic" (_Sci. Meth.
in Phil._, p. 208), but this expression still needs elucidation. It
does, to be sure, avoid the earlier difficulty of admitting "new and
mysterious metaphysical entities" (_loc. cit._, p. 204), but the
"feeling of oddity" that accompanies it seems not without significance.
What can be meant by a merely symbolic class of similar classes
themselves merely symbolical? I do not know, unless it is that we are to
throw overboard the effort aimed at arbitrary and creative definition
and proceed in simple inductive and interpretative fashion. With classes
as entities abandoned, we are left, until we have passed to a new point
of view as to arithmetical entities, in the position of the intelligent
ignoramus who defined a stock market operation as buying what you can't
get with money you never had, and selling what you never owned for more
than it was ever worth.

The situation seems to be that we are now face to face with new
generalizations. Just as number symbols arose to denote operations gone
through in counting things when attention is diverted from the
particular characteristics of the things counted, and remained a symbol
for those operations with things, so now we are becoming self-conscious
of the character of the operations we have been performing and are
developing new symbols to express possible operations with operations.
The infinity of the number series expresses the fact that it is possible
to continue the enumerating process in