) odd numbers is (
), then the sum of the first
odd numbers must necessarily be
. We assume that this law will hold however great
may be, or, in other words, must always hold; and our mathematical law is established. In short, mathematics aspires to give us general laws in place of particular facts, or, again, to proceed from the particular to the general. A search for generality and unity constitutes the leit-motif of mathematics.[136] A few specific examples may make this point clearer.
Suppose we start from the natural sequence of integers and labour under the impression that these integers alone constitute true numbers. We should notice that if we divided one of our numbers, say the number 10, by the number 5, we should obtain one of our integers, namely, 2. On the other hand, if we divided 10 by 4, the result would not yield one of our integers. Accordingly, we should be compelled to state that the division of one number by another might or might not yield a true number. In order to re-establish generality, mathematicians were compelled to assume the existence of fractional numbers, just as actual or true as the integers. Thanks to this introduction it became possible to assert that the division of one number by another would always yield a true number. Now suppose we attempted to extract square roots. We should find that whereas the square roots of some of our true numbers were themselves true numbers (i.e.,