Page:IntroductionToMathematicsWhitehead.pdf/21

The first point to notice is the possibilities contained in the meaning of some, as here used. Since ${\displaystyle x+2=2+x}$ for any number ${\displaystyle x}$, it is true for some number ${\displaystyle x}$. Thus, as here used, some does not exclude any. Again, in the second example, there is, in fact, only one number ${\displaystyle x}$, such that ${\displaystyle x+2=3}$, namely, only the number ${\displaystyle 1}$. Thus the some may be one number only. But in the third example, any number ${\displaystyle x}$ which is greater than ${\displaystyle 1}$ gives ${\displaystyle x+2>3}$. Hence there are an infinite number of numbers which answer to the some number in this case. Thus some may be anything between any and one only, including both these limiting cases.

It is natural to supersede the statements ${\displaystyle (2)}$ and ${\displaystyle (3)}$ by the questions:

Considering ${\displaystyle (2')}$, ${\displaystyle x+2=3}$ is an equation, and it is easy to see that its solution is ${\displaystyle x=3-2=1}$. When we have asked the question implied in the statement of the equation ${\displaystyle x+2=3}$, ${\displaystyle x}$ is called the unknown. The object of the solution of the equation is the determination of the unknown. Equations are of great importance in mathematics, and it seems as though ${\displaystyle (2')}$ exemplified a much more thorough-going and fundamental idea than the original statement ${\displaystyle (2)}$. This, however, is a complete mistake. The idea of the undeter-