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THE NATURE OF MATHEMATICS 87 

by such a phrase as : " Socrates was a mortal and the husband 
of a scold." If and this is the characteristic of modern logic 
we notice that the notions of variable and function (corre- 
spondence, relation) which appeared first in a special form in 
mathematics, are fundamental in all the things which are the 
objects of our thought, we are led to replace the particular con- 
ceptions in a proposition by variables, and thus see more clearly 
the structure of the proposition. Thus: "a; is a y and has 
the relation R to z, a member of the class u " gives the general 
form of a multitude of propositions, of which the above is a 
particular case, the above proposition may be true, but it is 
not a judgment of logic, but of history or experience. The 
proposition is false if " Kant " or " Westminster Abbey " is 
substituted for "Socrates": it is neither if "x" a sign for a 
variable, is, and then becomes what we call a " prepositional 
function" of x and denote it by "x" or "fa." If more 
variables are involved, we have the notation " (x,y,z, . . . ) 
implies, for all values of the variables, i^(x,y,z, . . . ). The 
structure of the propositional functions involves only such ideas 
as are fundamental in logic, like implication, class, relation, the 
relation of a term to a class of which it is a member, and so 
on. And, of course, mathematics uses the notion of truth. 

When we say that " 1 + 1 = 2," we seem to be making a 
mathematical statement which does not come under the 
above definition. But the statement is rather mistakenly 
written : there is, of course, only one whole class of most