II

Variables

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 $2+3=3+2$, in algebra we generalize and say that, if $x$ and $y$ stand for any two numbers, then $x+y=y+x$. Again, in the place of saying that $3>2$, we generalize and say that if $x$ be any number there exists some number (or numbers) $y$ such that $y>x$. We may remark in passing that this latter assumption–-for when put in its strict ultimate form it is an assumption–-is

of vital importance, both to philosophy and to mathematics; for by it the notion of infinity is introduced. Perhaps it required the introduction of the arabic numerals, by which the use of letters as standing for definite numbers has been completely discarded in mathematics, in order to suggest to mathematicians the technical convenience of the use of letters for the ideas of any number and some number. The Romans would have stated the number of the year in which this is written in the form MDCCCCX., whereas we write it 1910, thus leaving the letters for the other usage. But this is merely a speculation. After the rise of algebra the differential calculus was invented by Newton and

Leibniz, and then a pause in the progress

of the philosophy of mathematical thought occurred so far as these notions are concerned; and it was not till within the last few years that it has been realized how fundamental any and some are to the very nature of mathematics, with the result of opening out still further subjects for mathematical exploration.

Let us now make some simple algebraic statements, with the object of understanding exactly how these fundamental ideas occur.

(1) For any number $x$, $x+2=2+x$;

(2) For some number $x$, $x+2=3$;

(3) For some number $x$, $x+2>3$.

The first point to notice is the possibilities contained in the meaning of some, as here used. Since $x+2=2+x$ for any number $x$, it is true for some number $x$. Thus, as here used, any implies some and some does not exclude any. Again, in the second example, there is, in fact, only one number $x$, such that $x+2=3$, namely only the number $1$. Thus the some may be one number only. But in the third example, any number $x$ which is greater than $1$ gives $x+2>3$. 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.