Thursday, June 4, 2020

Was Berkeley Right about Mathematics?

It is pretty common in discussions of the history of mathematics for authors to mention Berkeley's attack on the calculus, usually with some derision, though in the more sophisticated authors it is a balance of derision and respect (or, at least, empathy).

For example, here is a nice summary of the episode by Penelope Maddy:
In the late 1600s, in response to a number of questions from physical science, Sir Isaac Newton and Gottfried Wilhelm von Leibniz simultaneously and independently invented the calculus. Though the scientist's problems were solved, the new mathematical methods were scandalously error-ridden and confused. Among the most vociferous and perceptive critics was the idealist Berkeley, an Anglican bishop who hoped to silence the atheists by showing their treasured scientific thinking to be even less clear than theology. The central point of contention was the notion of infinitesimals, ridiculed as 'the ghosts of departed quantities'. Two centuries later, Bolzano, Cauchy, and Weierstrass had replaced these ghosts with the modern theory of limits. (Realism in Mathematics, p. 22)
What seems funny about Berkeley's critique is the apparent confidence with which he proffered it, in contrast to the rigorous development the calculus would undergo in the following centuries. What extracts a measure of respect is the fact that, at the time he wrote, Berkeley was not wrong.

But if, on a charitable reading of Berkeley, his point wasn't forever to cast doubt on the calculus, but rather to point out that the metaphysical and logical foundations of mathematics are not more justified than the suppositions of faith, it is interesting (and somewhat amusing) to ask whether Berkeley has actually been disproved in this.

Obviously, mathematics as a science is certain. Berkeley never denied it. But that's different from saying the metaphysical and logical foundations of mathematics are certain. These latter are characterized by disputes without termination. They involve paradoxes and uncertainty. And they require the acceptance of assertions whose justification and warrant are, by the admission of philosophers and mathematicians, puzzling.

For example, the most common picture throughout the 20th century was that set theory serves as a foundation for mathematics. However, touring the landscape of views on this matter will make you dizzy.

Here are a few questions that are recognized as utterly central yet by all appearances unresolved: Is there only one correct set theory, objectively true to the exclusion of all others? What are sets? Are they physical or non-physical? If they are non-physical, how can we even possibly know about them? If they are physical, which physical things are they, and are we sure there are enough of them to serve as a foundation for mathematics? (And if there are, how could we know that?)

Of course, answers to these questions have been proposed  some would claim successfully. Maybe so. But many theologians say the same about their solutions to puzzles about the mysteries of the faith, and a comparison of their theories with those of philosophers of mathematics might well show that the former aren't any worse off than the latter.

Again, Berkeley's point is just that the theological mysteries are not more doubtful than the foundations of our beloved mathematics. He gets teased for not realizing that his objections to the calculus would be answered a hundred years later, but there are questions similar to his that are still unanswered by logicians and philosopher of mathematics. So, joke's on them!