ABSTRACT
The philosophical theory of the Calculus has been, ever since the subject was invented, in a somewhat disgraceful condition. Leibniz himself-who, one would have supposed, should have been competent to give a correct account of his own invention-had ideas, upon this topic, which can only be described as extremely crude. He appears to have held that, if metaphysical subtleties are left aside, the Calculus is only approximate, but is justified practically by the fact that the errors to which it gives rise are less than those of observation.* When he was thinking of Dynamics, his belief in the actual infinitesimal hindered him from discovering that the Calculus rests on the doctrine of limits, and made him regard his dx and dy as neither zero, nor finite, nor mathematical fictions, but as really representing the units to which, in his philosophy, infinite division was supposed to lead.† And in his mathematical expositions of the subject, he avoided giving careful proofs, contenting himself with the enumeration of rules.‡ At other times, it is true, he definitely rejects infinitesimals as philosophically valid;§ but he failed to show how, without the use of infinitesimals, the results obtained by means of the Calculus could yet be exact, and
not approximate. In this respect, Newton is preferable to Leibniz: his Lemmas* give the true foundation of the Calculus in the doctrine of limits, and, assuming the continuity of space and time in Cantor’s sense, they give valid proofs of its rules so far as spatio-temporal magnitudes are concerned. But Newton was, of course, entirely ignorant of the fact that his Lemmas depend upon the modern theory of continuity; moreover, the appeal to time and change, which appears in the word fluxion, and to space, which appears in the Lemmas, was wholly unnecessary, and served merely to hide the fact that no definition of continuity had been given. Whether Leibniz avoided this error, seems highly doubtful: it is at any rate certain that, in his first published account of the Calculus, he defined the differential coefficient by means of the tangent to a curve. And by his emphasis on the infinitesimal, he gave a wrong direction to speculation as to the Calculus, which misled all mathematicians before Weierstrass (with the exception, perhaps, of De Morgan), and all philosophers down to the present day. It is only in the last thirty or forty years that mathematicians have provided the requisite mathematical foundations for a philosophy of the Calculus; and these foundations, as is natural, are as yet little known among philosophers, except in France.† Philosophical works on the subject, such as Cohen’s Princip der Infinitesimalmethode und seine Geschichte,‡ are vitiated, as regards the constructive theory, by an undue mysticism, inherited from Kant, and leading to such results as the identification of intensive magnitude with the extensive infinitesimal.§ I shall examine in the next chapter the conception of the infinitesimal, which is essential to all philosophical theories of the Calculus hiterto propounded. For the present, I am only concerned to give the constructive theory as it results from modern mathematics.
