ussell, the most self-consciously philosophical of these
mathematicians, has devoted his full dialectic skill. The definition has
at least the merit of being free from certain arbitrary psychologizing
that has vitiated many earlier attempts at the problem. Mr. Russell
claims for it "(1) that the formal properties which we expect cardinal
numbers to have result from it; (2) that unless we adopt this definition
or some more complicated and practically equivalent definition, it is
necessary to regard the cardinal number of a class as indefinable"
(_loc. cit._, p. 4). That the definition's terms, however, are not
without obscurity appears in Mr. Russell's struggles with the zigzag
theory, the no-class theory, etc., and finally in his taking refuge in
the theory of "logical types" (_loc. cit._, Vol. III, Part V. E.),
whereby the contradiction that subverted Frege and drove Mr. Russell
from the standpoint of the _Principles of Mathematics_ is finally
overcome.

The second of Mr. Russell's claims for his definition adds nothing to
the first, for it merely asserts that unless we adopt some definition of
the cardinal number from which its formal properties result, number is
undefined. Any such definition would be, _ipso facto_, a practical
equivalent of the first. We need only consider whether or not the
formal properties of numbers clearly follow from this definition.

Mr. Russell's own experience makes us hesitate. When he first adopted
this definition from Frege, he was led to make the inference that the
class of all possible classes might furnish a type for a greatest
cardinal number. But this led to nothing but paradox and contradiction.
The obvious conclusion was that something was wrong with the concept of
class, and the obvious way out was to deny the possibility of any such
all-inclusive class. Just why there should be such limitation, except
that it enables one to escape the contradiction, is not clear from Mr.
Russell's analysis (cf. Brown, "The Logic of Mr. Russell," _Journ. of
Phil., Psych., and Sci. Meth._, Vol. VIII, No. 4, pp. 85-89).
Furthermore, to pass to the theory of types on this ground is to give up
the value of the first claim for the definition (quoted above), since
the formal properties of numbers now merely follow from the definition
because the terms of the definition are reinterpreted from the
properties of number, so that these properties will follow from it. The
definition has become circular.