on perception. Clearly the concept "finger" is abstracted from perception; but how about the
concept "ten"? Here we may seem to have arrived at a true universal or Platonic idea. We cannot
say that "ten" is abstracted from perception, for any percept which can be viewed as ten of some
kind of thing can equally well be viewed otherwise. Suppose I give the name "digitary" to all the
fingers of one hand taken together; then I can say "I have two digitaries," and this describes the
same fact of perception as I formerly described by the help of the number ten. Thus in the
statement "I have ten fingers" perception plays a smaller part, and conception a larger part, than in
such a statement as "this is red." The matter, however, is only one of degree.
The complete answer, as regards propositions in which the word "ten" occurs, is that, when these
propositions are correctly analysed, they are found to contain no constituent corresponding to the
word "ten." To explain this in the case of such a large number as ten would be complicated; let us,
therefore, substitute "I have two hands." This means:
"There is an a such that there is a b such that a and b are not identical and whatever x may be, 'x is
a hand of mine' is true when, and only when, x is a or x is b."
Here the word "two" does not occur. It is true that two letters a and b occur, but we do not need to
know that they are two, any more than we need to know that they are black, or white, or whatever
colour they may happen to be.
Thus numbers are, in a certain precise sense, formal. The facts which verify various propositions
asserting that various collections each have two members, have in common, not a constituent, but
a form. In this they differ from propositions about the Statue of Liberty, or the moon, or George
Washington. Such propositions refer to a particular portion of space-time; it is this that is in
common between all the statements that can be made about the Statue of Liberty. But there is
nothing in common among propositions "there are two soand-so's" except a common form. The
relation of the symbol "two" to the meaning of a proposition in which it occurs is far more
complicated than the relation of the symbol "red" to the meaning of a proposition in which it
occurs. We may say, in a certain sense, that the symbol "two" means nothing, for, when it occurs
in a true state-