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. 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.