The Early Mathematical Manuscripts of Leibniz (Classic Reprint)

Excerpt from The Early Mathematical Manuscripts of Leibniz

A Study of the early mathematical work of Leibniz seems to be of importance for at least two reasons. In the first place. Leibniz was certainly not alone among great men in presenting in his early work almost all the important mathematical ideas contained in his mature work. In the second place, the main ideas of his philosophy are to be attributed to his mathematical work, and not vice versa. The manuscripts of Leibniz, which have been preserved with such great care in the Royal Library at Hanover, show, perhaps more clearly than his published work, the great importance which Leibniz attached to suitable notation in mathematics and, it may be added, in logic generally. He was, perhaps, the earliest to realize fully and correctly the important influence of a calculus on discovery. The almost mechanical operations which we go through when we are using a calculus enable us to discover facts of mathematics or logic without any of that expenditure of the energy of thought which is so necessary when we are dealing with a department of knowledge that has not yet been reduced to the domain of operation of a calculus. There is a frivolous objection raised by philosophers of a superficial type, to the effect that such economy of thought is an attempt to substitute unthinking mechanism for living thought. This contention fails of its purpose through the simple fact that this economy is only used in certain circumstances. In no science do we try to make subject to a mechanical calculus any trains of reasoning except such that have not been the object of careful thought many times previously. Not only so, but this reasoning has been universally recognized as valid, and we do not wish to waste energy of thought in repeating it when so much remains to be discovered by means of this energy. Since the time of Leibniz, this truth has been recognized, explicitly or implicitly, by all the greatest mathematical analysts.