* And surely there were some people who liked Nader best and preferred Bush to Gore, or who liked Bush best and preferred Nader to Gore, but my imagination is not strong enough to understand what sort of people these could possibly be, so I’m going to assume their numbers are too small to materially affect the computation.
* I’ll concede it’s not clear Ralph Nader actually worries about this.
* To be precise, Mill was actually talking about the closely related “single transferable vote” system.
* But not any more—in a narrowly decided referendum, Burlington voters repealed instant-runoff voting in 2010.
* Of course, there are lots of assumptions in place here, most notably that the jurors’ judgments are arrived at independently from each other—surely not quite right in a context where the jurors confer before voting.
* The version of the fifth postulate I’ve written here is actually not Euclid’s original, but a logically equivalent version, originally stated by Proclus in the fifth century CE and made popular by John Playfair in 1795. Euclid’s version is a bit longer.
* The eponym for Tom Lehrer’s song “Lobachevsky,” surely the greatest comic musical number of all time about mathematical publishing.
* This is not supposed to be immediately obvious, but it’s not hard to convince yourself it’s true—I highly recommend getting out a tennis ball and a Sharpie and checking for yourself!
* The painters didn’t develop, or need, a formal geometric theory of the projective plane, but they understood how it translated into brushstrokes on the canvas, which was enough for their purposes.
* Some historians trace the current hypermathematization of economics back this far, saying that the habit of axioms passes from Hilbert to economics through Wald and the other young mathematicians in 1930s Vienna, who combined a Hilbertian style with strong applied interests: see E. Roy Weintraub’s How Economics Became a Mathematical Science, where this idea is fully worked out.
* It’s probably not a coincidence that Peano was yet another devotee of artificial languages constructed on rational principles: he created his own such language, Latino Sine Flexione, in which he wrote some of his later mathematical works.
* Ted Chiang’s 1991 short story “Division by Zero” contemplates the psychological consequences suffered by a mathematician unfortunate enough to uncover such an inconsistency.
* If we’re to be precise, Russell was not a formalist, like Hilbert, who declared that the axioms were just strings of symbols with no defined meaning; rather, he was a “logicist,” whose view was that the axioms were actually true statements about logical facts. Both groups shared a vigorous interest in figuring out which statements could be deduced from the axioms. The extent to which you care about this distinction is a good measure of whether you would enjoy going to graduate school in analytic philosophy.
* They really do this!
* From Poincaré’s essay “Mathematical Creation,” highly recommended reading if you care about mathematical creativity, or for that matter any kind of creativity.
* Though: Amir Alexander, in his book Infinitesimal (New York: FSG, 2014) argues that in the 17th century, it was the pure formalist position, represented by classical Euclidean geometry, that was allied with rigid hierarchies and Jesuitical orthodoxy, while the more intuitive and less rigorous pre-Newtonian theory of infinitesimals was tied to a more forward-looking and democratic ideology.
* One voting system to which Arrow’s Theorem doesn’t apply is “approval voting,” in which you don’t have to declare all your preferences; you just vote for as many of the people on the ballot as you want, and the candidate who gets the most votes wins. Most mathematicians I know consider approval voting or its variants to be superior to both plurality voting and IRV; it has been used to elect popes, secretaries-general of the United Nations, and officials of the American Mathematical Society, but never yet government officials in the United States.
* Roosevelt’s view that analytic “book-learning” stands in opposition to virility is expressed more directly by Shakespeare, who in the opening scene of Othello has Iago derisively call his rival Cassio “a great arithmetician . . . That never set a squadron in a field / Nor the division of a battle knows / more than a spinster.” This is the point in the play where every mathematician in the audience figures out Iago is the bad guy.
* Ashbery starts the second and final section of “Soonest Mended” with the lines “These then were some hazards of the course / Yet though we knew the course was hazards and nothing else”: Ashbery, who lived in France for a decade, certainly means the English word’s sense of danger to be closely followed by the echo of the French hasard, which means “chance,” fitting the poem’s overall atmosphere of rigorous uncertainty. Pascal would have called the gambling games he discussed with Fermat jeux de hasard, and the word’s ultimate origin is the Arabic word for dice.
* There are other, more sophisticated reasons to be skeptical about Silver’s approach, though these weren’t dominant among the Washington press corps. For instance, one could follow R. A. Fisher’s line and say that the language of probability is inappropriate for one-off events, and applies only to things like coin flips that can in principle be repeated again and again.
* To be precise, it was his final prediction that got all the states right; on October 26, he had everything correct except Florida, where polls swung from leaning Romney to just about even in the last two weeks of the campaign.
* Of course, if you wanted to set this up correctly, you’d have to modify the coin flip to give a greater chance of winning to the candidate who appears to be slightly ahead, etc., etc.
* In the end he didn’t succeed at either task; the Poincaré Conjecture was eventually proved by Grigori Perelman in 2003.
How Not to Be Wrong : The Power of Mathematical Thinking (9780698163843) Page 52
