Page:IntroductionToMathematicsWhitehead.pdf/19

Mathematics as a science commenced when first someone, probably a Greek, proved propositions about any things or about some things, without specification of definite particular things. These propositions were first enunciated by the Greeks for geometry; and, accordingly, geometry was the great Greek mathematical science. After the rise of geometry centuries passed away before algebra made a really effective start, despite some faint anticipations by the later Greek mathematicians.

The ideas of any and of some are introduced into algebra by the use of letters, instead of the definite numbers of arithmetic. Thus, instead of saying that ${\displaystyle 2+3=3+2}$, in algebra we generalize and say that, if ${\displaystyle x}$ and ${\displaystyle y}$ stand for any two numbers, then ${\displaystyle x+y=y+x}$. Again, in the place of saying that ${\displaystyle 3>2}$, we generalize and say that if ${\displaystyle x}$ be any number there exists some number (or numbers) ${\displaystyle y}$ such that ${\displaystyle y>x}$. We may remark in passing that this latter assumption—for when put in its strict ultimate form it is an assumption—