Infinitesimals revisited

Please enjoy this free extended excerpt from Lectures on the Philosophy of Mathematics, published with MIT Press 2021, an introduction to the philosophy of mathematics with an approach often grounded in mathematics and motivated organically by mathematical inquiry and practice. This book was used as the basis of my lecture series on the philosophy of mathematics at Oxford University.

Infinitesimals revisited

For the final theme of this chapter, let us return to the infinitesimals. Despite the problematic foundations and Berkeley's criticisms, it will be good to keep in mind that the infinitesimal conception was actually extremely fruitful and led to many robust mathematical insights, including all the foundational results of calculus. Mathematicians today routinely approach problems in calculus and differential equations essentially by considering the effects of infinitesimal changes in the input to a function or system.

For example, to compute the volume of a solid of revolution y = f(x) about the x-axis, it is routine to imagine slicing the volume into infinitesimally thin disks. The disk at x has radius f(x) and infinitesimal thickness dx (hence volume πf(x)²dx), and so the total volume between a and b, therefore, is

Another example arises when one seeks to compute the length of the curve traced by a smooth function y = f(x). One typically imagines cutting it into infinitesimal pieces and observing that each tiny piece is the hypotenuse ds of a triangle with infinitesimal legs dx and dy. So by an infinitesimal instance of the Pythagorean theorem, we see

from which one “factors out” dx, obtaining

and therefore the total length of the curve is given by

Thus, one should not have a cartoon understanding of developments in early calculus, imagining that it was all bumbling nonsense working with the ghosts of departed quantities. On the contrary, it was a time of enormous mathematical progress and deep insights of enduring strength. Perhaps this situation gives a philosopher pause when contemplating the significance of foundational issues for mathematical progress — must one have sound foundations in order to advance mathematical knowledge? Apparently not. Nevertheless, the resolution of the problematic foundations with epsilon-delta methods did enable a far more sophisticated mathematical analysis, leading to further huge mathematical developments