George Berkeley’s 329th Birthday

The stream of mathematics-trivia tweets brought to my attention that the 12th of March, 1685 [1], was the birthday of George Berkeley, who’d become the Bishop of Cloyne and be an important philosopher, and who’s gotten a bit of mathematical immortality for complaining about calculus. Granted everyone who takes it complains about calculus, but Berkeley had the good sorts of complaints, the ones that force people to think harder and more clearly about what they’re doing.

Berkeley — whose name I’m told by people I consider reliable was pronounced “barkley” — particularly protested the “fluxions” of calculus as it was practiced in the day in his 1734 tract The Analyst: Or A Discourse Addressed To An Infidel Mathematician, which as far as I know nobody I went to grad school with ever read either, so maybe you shouldn’t bother reading what I have to say about them.

Fluxions were meant to represent infinitesimally small quantities, which could be added to or subtracted from a number without changing the number, but which could be divided by one another to produce a meaningful answer. That’s a hard set of properties to quite rationalize — if you can add something to a number without changing the number, you’re adding zero; and if you’re dividing zero by zero you’re not doing division anymore — and yet calculus was doing just that. For example, if you want to find the slope of a curve at a single point on the curve you’d take the x- and y-coordinates of that point, and add an infinitesimally small number to the x-coordinate, and see how much the y-coordinate has to change to still be on the curve, and then divide those changes, which are too small to even be numbers, and get something out of it.

It works, at least if you’re doing the calculations right, and Berkeley supposed that it was the result of multiple logical errors cancelling one another out that they did work; but he termed these fluxions with spectacularly good phrasing “ghosts of departed quantities”, and it would take better than a century to put all his criticisms quite to rest. The result we know as differential calculus.

I should point out that it’s not as if mathematicians playing with their shiny new calculus tools were being irresponsible in using differentials and integrals despite Berkeley’s criticisms. Mathematical concepts work a good deal like inventions, in that it’s not clear what is really good about them until they’re used, and it’s not clear what has to be made better until there’s a body of experience working with them and seeing where the flaws. And Berkeley was hardly being unreasonable for insisting on logical rigor in mathematics.

[1] Berkeley was born in Ireland. I have found it surprisingly hard to get a clear answer about when Ireland switched from the Julian to the Gregorian calendar, so I have no idea whether this birthdate is old style or new style, and for that matter whether the 1685 represents the civil year or the historical year. Perhaps it suffices to say that Berkeley was born sometime around this time of year, a long while ago.