Written: Oct. 20, 2019
Way back in the mid 1960s, when I was in ninth grade, we still learned Plane Geometry the traditional way, essentially following (a watered-down, but methodologically isomorphic) version of Euclid's 2300-year-old Elements. This awful text, with its false claims to absolute truth, was almost as influential as the bible, and many generations of students shed tears trying to "prove" (often intuitively obvious) propositions, by a step-by-step "deductive" process. On the other hand, quite a few otherwise very smart people (Spinoza, Hobbes, ... ) loved it, and used its (apparent!) air of absolute certainty, adapting Euclid's style, to claim that their philosophical theses are just as absolutely certain.
But (most) philosophers are basically good guys, and I forgive them. On the other hand, absolute monarchs, and other tyrants, are not! What I found out from Amir Alexander's fascinating new book Proof! (How the World Became Geometrical) was how Euclidean Geometry was abused by them to "prove" their inherent superiority. The `canonical example', narrated at length in the book, was Louis XIV, who commissioned a very elaborate geometrical garden, the famous Jardins du Château de Versailles, to prove his point. It was a proof by analogy. Just as the theorems of Euclidean Geometry are absolutely certain, it is absolutely certain that someone who can build such an elaborate and spectacular garden has a divine right to his position.
As Alexander describes in his gripping style, the Sun King's "proof" was borrowed by quite a few other absolute monarchs, and colonial invaders (for example the British in India) to construct gardens, and later whole cities, inspired by Versailles, to "prove" that their occupation is true a priori and just as justified as the Pythagorean theorem.
In the climax of the book, it is described that even our beloved capital city, Washington DC, was designed, by French-born-and-educated architect Pierre L'Enfant, in the style of Versailles, slightly tweaked to make it more egalitarian, but essentially to make a similar point, the superiority of the American democracy.
Not everyone agreed with L'Enfant's design, and one strong opponent was no other than Thomas Jefferson, who much preferred the "Babylonian" (I guess it was an allusion to Hammurabi's invention of streets), or "Cartesian" Manhattan lattice grid. This plan was so abhorrent to L'Enfant that he forgot his usual smooth manners, to criticize Jefferson's plan so strongly, something that got him fired. But someone else, whose name was George Washington, did like L'Enfant's plan, and the latter's design was completed by other people.
This got me thinking, that myself, I am siding with Jefferson, and I like a "Babylonian" city rather than a "Greek" (Geometrical) city. More important, I like Babylonian (algorithmic, concrete, experimental ...) mathematics much more than Greek (deductive,abstract, axiomatic, ...) mathematics. And indeed, it looks like, that very slowly things are moving that way, since very soon computers will do all the heavy lifting, and they don't have our human hang-ups.
I really liked Amir Alexander's ``Proof!'' since it supplied a new- much more convincing- proof why "Greek" mathematics should be chucked. Before my only reason was that "I don't like it", but now I can say, thanks to Amir, "it was used by tyrants to `prove' their superiority". We now know that Louis' `proof' was flawed, and he is not any better than you or me.
The next-in-line to be chucked is a much worse tyrant than our dear Louis quatorze. It is the notion of "absolute certainty", so dear to pure mathematicians. Nothing is absolutely certain, and the few things that are nearly so, say the Pythagorean theorem and even Fermat's Last Theorem and The Four Color Theorem, are (almost) absolutely certain because they are trivial, or nearly so. All non-trivial results (e.g. the Collatz property) would (probably) only have, at best, semi-rigorous proofs, and more often, only empirical and/or heuristic proofs. But the whole notion of proof will lose its centrality, and the keyword algorithm will inherent the earth. Amen.