Read The Nature of Mathematics Online

48 THE NATURE OF MATHEMATICS 

of great numbers sometimes an infinity of facts of nature 
may be expressed very concisely or even embodied in a single 
expression. The essence of the formula is that it is an expres- 
sion of a constant rule among variable quantities. These ex- 
pressions "constant" and "variable" have come down into 
ordinary language. We say that the number of miles which a 
certain man walks per day is a "variable quantity"; and we 
do not mean that, on a particular day, the number was not 
fixed and definite, but that on different days he walked, 
generally speaking, different numbers of miles. When, in 
mathematics, we speak of a " variable," what we mean is that 
we are considering a class of definite objects for instance, the 
class of men alive at the present moment and want to say 
something about any one of them indefinitely. Suppose that 
we say : " If it rains, Mr. A will take his umbrella out with 
him " ; the letter " A " here is what we call the sign of the 
" variable." We do not mean that the above proposition is 
about a variable man. There is no such thing ; we say that 
a man varies in health and so in time, but, whether or not 
such a phrase is strictly correct, the meaning we would have to 
give the phrase " a variable " in the above sentence is not one 
and the same man at different periods of his own existence, but 
one and the same man who is different men in turn. What we 
mean is that if "A" denotes any man, and not Smith or Jones 
or Robinson alone, then he takes out his umbrella on certain 
occasions. The statement is not always true ; it depends on A. 
If " A " stands for a bank manager, the statement may be true ; 
if for a tramp or a savage, it probably is not. Instead of "A," 
we may put " B " or " C " or " X " ; the kind of mark on 
paper does not really matter in the least. But we attach, by 
convention, certain meanings to certain signs ; and so, if we 
wrote down a mark of exclamation for the sign of a variable, 
we might be misunderstood and even suspected of trying to be 
funny. We shall see, in the seventh chapter, the importance 
of the variable in logic and mathematics. 

" Laws of nature " express the dependence upon one another 
of two or more variables. This idea of dependence of variables 
is fundamental in all scientific thought, and reaches its most 
thorough examination in mathematics and logic under the name 
of " functionality." On this point, we must refer back to the