BERKELEY'S CRITICISM OF THE INFINITESIMAL

Abstract

IN his two notebooks, now known as Philosophical Commentaries, which contain the raw materials of his thought when he was a very young man, Berkeley made a number of critical remarks about mathematics. It was more than twenty-five years, however, before he developed these and published his tract The Analyst (1734). Historians of mathematics have recognised its value, but philosophers have generally supposed that he was indulging in a futile trial of strength with Newton on Newton's home ground. Berkeley attacked the logic of the method offluxions or infinitesimal calculus, holding that the infinitesimal was a zero-increment, a finite quantity of no size, that it was treated at one stage as finite and at another as zero as convenience dictated, that its effects were retained after it was made to vanish--that in fact it was self-contradictory. His two ways of bringing this out are the acme of lucidity; one concerns the fluxion of a power, the other that of a product. He deals with the fluxion of xn, using the binomial expansion. For brevity I will make his point by considering x2. When x' flows ', as he puts it, he calls the increment o. The incrementary ratio is (2x. o + 02)/10 or 2x + o. He notes that o is here supposed to be 'something '. The next step, however, is to let o become zero, so as to produce the fluxion 2x. Of this Berkeley says there is now introduced a supposition contrary to the first, namely, that there is now no increment of x (or that o is now nothing), so that it is invalid to retain the result 2x, because this was arrived at by supposing the o was something. In short, if o is something what is obtained is not 2x by 2x + o where o is not zero; while, if o equals zero, nothing at all is obtained.