Wednesday, June 1, 2016

The Fortituous End of Kant

In grad school I took a class that surveyed Renaissance music history. In one of the few actual seminars I ever took, we sat around one large table and discussed all the readings and music in question. One day as we went through some score by Josquin des Prez (I think it was Josquin), I made a remark about some pattern in the counterpoint that I thought was especially interesting and elegant. One of my classmates, however, felt very strongly that the composer should take no credit for the combination, that the pattern was the result of pure happenstance. “You can’t say that,” he urgently argued in my direction. “That’s just fortituous.” Not fortuitous, mind you: fortituous. He was so adamant about my being wrong that he pressed his point for another minute or so and repeated his unintentional neologism at least four more times. Fortituous! Fortituous! Fortituous! The word has stuck with me ever since as one of the fortituous benefits of an otherwise rather dull class.

Kant’s Critique of Teleological Reason is, if I understand him correctly (although I think there’s only about a 50% chance that I understand him at even 50% capacity), a book about whether elements of nature exist by design or fortituously. (Yes, the new beautiful new word comes in adverbial form as well.) The punch line of the book is that God can only be proven in human thinking by examining the moral law. But along the way to that point, Kant examines how to tell if a given thing in nature exists for some further end. Rivers bring us food, for instance. So are they there because of us? Are we the purpose of rivers? In Aristotelian terms, are we the final cause of rivers? In theological terms, did God create rivers for humankind’s benefit? Kant denies it by pointing out that nothing says any humans have to live by any given river. The river’s benefits are, to call on my old colleague’s linguistic turn once more, merely fortituous.

Kant’s other examples resonated a lot with other things I’ve been thinking about lately. He says that the mathematical structure known as a parabola has uses (describing the effect of gravity on a cannonball, for instance) that have nothing to do with its existence. We can have no possible justification, says Kant, in claiming that God made math and parabolas so that one day warriors would use their arcane properties to launch projectiles accurately. I’ve been studying calculus lately (and actually learning it this time!), and I had been thinking something along the very same lines. The way a derivative comes about is stunning. But derivatives weren’t created by God in order to describe the way water drains from a cone. (Or at least it would be presumptuous to say so.) In the same light, their way of torturing middle-aged, independent scholars whose multi-page calculations lead to something completely different from the answer in the back of the book is just a fortituous by-product; they don’t exist for the purpose of my self-educational pursuits.

Here’s another example from my recent real life. The ancient Greeks had several ways of locating a number between two other numbers in some mathematically meaningful or beautiful relationship. They called these middle numbers “means.” (Or I should say that translators indicate that the ancient Greeks would have called them “means” if those ancient Greeks had spoken modern English.) Calculating the average (adding the two original numbers and dividing by 2) is only one of the means they studied: the arithmetic mean. They also recognized what they called a harmonic mean. When a harmonic mean is interpolated between two other numbers, the differences between mean and extremes have the same ratio as the two original numbers. For example, 12 is the harmonic mean of 10 and 15. 10:12:15. The difference between 12 and 10 is 2; the difference between 12 and 15 is 3. So the differences show a 2:3 ratio, as do the two extremes, 10 and 15.

The crazy thing is, where numbers represent string lengths, these two patterns – the harmonic mean and the arithmetic mean – when applied to two numbers in a 2:3 ratio, produce major and minor chords, respectively. Now the Greeks had no interest in three-part harmony from two thousand years in the future. And the English composers of the late fourteenth century who popularized using thirds in harmonies had little chance of being experts in ancient mathematics (although not zero chance). In any case, it seems absurd to say that they started writing major and minor triads because of their knowledge of Greek means. Somehow it seems more reasonable to think that arithmetic and harmonic means came to the ken of ancient humanity so that one glorious day, western composers would begin to favor harmonies that display those relationships. But Kant reigns in my Platonic predestinarianism.

Just the other day, I learned that the periods of the orbits of Venus and Earth have a ratio very nearly 13 to 8, and that a regular tracing of the line between the two planets creates a picture of a five-petaled flower. So is that beautiful picture the reason for the relationship between the orbits? Kant would say it was presumptuous to believe so.

On the other hand, he says it would be presumptuous to believe it definitely wasn’t the reason, either.

I wonder how that student would pronounce “presumptuous”?

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