Logic and Lewis Carroll
That was the title of a "Books & Arts" Comment piece in Nature, November 19, 2015: "As Alice's Adventures in Wonderland reaches 150, Francine Abeles surveys its creator's wide-ranging legacy." As Abeles tells us, besides Alice and Through the Looking-glass, Charles Dodgson (Lewis Carroll was his pen name) "produced many pamphlets and ten books on mathematical topics." Many of them involved puzzles, or shortcuts to computation like an 1897 piece in Nature itself, "Brief Method of Dividing a Given Number by 9 or 11" (Nature 56 565-566). "Carroll did not influence his contemporary colleagues in the devopment of mathematical ideas. However, posthumously ..." it was recognized that he had anticipated several significant developments.
In his papers on voting theory (1874-76) "Carroll was the first to create a voting method that would achieve biproportional representation -- that is, proportionality with respect both to the population in the districts and to the apportionment of seats to the political parties in the legislature."
"Carroll's work in logic, notably the unpublished second part of his book Symbolic logic, foreshadowed results that appeared about 100 years later."
"His 'condensation' method for computing determinants sparked research that led to a formulation of the alternating sign matrix conjecture by David Robbins and Howard Rumsey in the 1980s."
Carroll's "What the Tortoise said to Achilles" (1895) raised a delicate problem about logical implication that was later considered by Bertrand Russell and others.
A nice example of one of his "knots," his name for mathematical puzzles, is "Knot II, Mad Mathesis: 'I waited for the train'" on page 76 of the 1880 Monthly Packet (of Evening Readings for Members of the English Church).
Brian Greene in The New Yorker
The "Talk of the Town" on November 30, 2015 had a "Brave New World" department with a piece by Rebecca Mead entitled "Mathphilic." The occasion was a performance of Brian Greene's multimedia work Light Falls at the opening of a two-day Institute for Advanced Study conference celebrating the 100th anniversary of Einstein's completion of the theory of General Relativity. "Greene ... narrates, telling the story of Einstein's discovery in language accessible even to audience members without a Ph.D." Mead spoke backstage with Greene about his personal history with physics and mathematics, and quotes him: "Math is hard for just about everybody. It is not what our brains evolved to do. ... You didn't have to take exponentials or use imaginary numbers in order to avoid that lion or that tiger or to catch that bison for dinner. So the brain is not wired, literally, to do the kinds of things we now want it to do." He goes on to give us a picture of how a physicist uses math. "He starts with a blank page, and he thinks of some kind of mathematical relation that may be able to describe the physics that he's thinking about, and then he starts with that equation. He thinks, O.K., with that as a starting point, let me manipulate it. And then you come to the act of inspiration, where, after manipulating the equations, you say, Wow, look at that--the pattern right there in the equations aligns with the pattern of gravity, or that pattern of motion."
Math in The New York Times
A busy week:
"The Politics of Math Education," an Op-Ed piece by Christopher J. Phillips (History, Carnegie-Mellon), ran on December 3, 2015. Prof. Phillips' thesis is that "debates about learning mathematics are debates about how educated citizens should think generally. Whether it is taught as a collection of facts, as a set of problem-solving heuristics or as a model of logical deduction, learning math counts as learning to reason. That is, in effect, a political matter, and therefore inherently contestable. Reasonable people can and will disagree about it." He briefly surveys the 1950-1980 history of the New Math (he wrote the book: The New Math, a Political History, Chicago, 2014) and concludes: "The new math's reception was fundamentally shaped by Americans' trust in federal initiatives and elite experts, their demands for local control and their beliefs about the skills citizens needed to face the problems of the modern world. Today these same political concerns will ultimately determine the future of the Common Core."
"Let Math Save Our Democracy," a December 5, 2015 opinion piece by Sam Wang (Program in Law and Public Affairs, Princeton) addresses gerrymandering and proposes a simple mathematical criterion for detecting it. As Wang explains, with electoral districts set up as they are, "It is possible for a party to win more than half of the popular vote in a state, yet control fewer than one-third of the legislative seats. This is not a theoretical problem: Precisely such a thing happened in 2012 with the congressional delegations of Pennsylvania and North Carolina." For example in the 2012 U.S. House race in Pennsylvania, Democrats won 50.5% of the vote but only 28% of the seats. To detect likely gerrymandering, Wang proposes the average-median difference: "This century-old statistic uses math that is in the Common Core standards for sixth grade" and is "simple enough that a busy judge can calculate it in the margin of a brief." Target one of the parties, and calculate its median vote share: the percentage of votes it wins in "the middle district on a list that is sorted in order of increasing party vote share." Also calculate its overall average vote share. "If the targeted voters have been packed into a few districts, they are counted in the average but have little effect on the median." Evaluate the discrepancy by comparing the spread between those two numbers to a zone of chance (very roughly speaking "the largest gap that would separate the average and median by chance.") Applied to the Pennsylvania example, "the average-median spread falls outside the zone, indicating a partisan gerrymander." Unfortunately Wang does not tell us how a sixth-grader would calculate the zone of chance, but there are more details in the paper he published on the Social Science Research Network.
For each of the 18 electoral districts in Pennsylvania, Wang recorded the 2012 "Democratic two-party share" of the vote, i.e. the percentage of votes the Democrats won when the Democrat plus Republican vote was set to 100%.
34.4, 36, 37.1, 38.3, 40.3, 40.6, 41.5, 42.9, [43.2, 43.4,] 45.2, 45.2, 48.3, 60.3, 69.1, 76.9, 84.9, 90.6.
In this tabulation, the districts are ordered by Democratic share, with italics for districts won by Republicans, and the middle two districts set off by [brackets]. The median Democratic vote share is calculated as the average of those two, i.e. 43.3%. The mean Democratic vote share (here, the average of the 18) is 51.0%. The difference is 7.7%.
How suspicious is this discrepancy? To explain "zone of chance" Wang starts with an ideal example, where the two-party vote split is exactly 50-50. Then, if the districts are evenly apportioned, each district is like the toss of a coin: the probability of a 13-5 district split can be calculated and is quite small (3.2%); therefore it is quite likely that the districts were not evenly apportioned. [It is not explicitly stated, but the "zones of chance" shown in the diagram in the Times seem to correspond to 2 standard deviations from the mean.] To make a more realistic assessment, Wang uses computer simulation: he mines the total 2012 U.S. congressional vote for random "combinations of districts from around the United States that add up to the same statewide vote total [as Pennsylvania] for each party," uses the outcome of the elections to create a delegation ("like a fantasy baseball team"), millions of times. "[T]he average result of these simulations approximates a 'natural' seats-votes relationship that can be defined with mathematical rigor and exactitude." The 2012 Pennsylvania outcome is a conspicuous outlier.
"The Mathematicians Who Saved a Kidnapped N.Y.U. Computer" by James Barron (December 7, 2015) tells the story of the takeover of the Courant Institute in the spring of 1970. Students incensed by the Kent State killings overran the building and focused on Courant's CDC 6600 supercomputer, leased from the Atomic Energy Commission: they declared it captured and demanded a $100,000 ransom (to provide bail for a member of the Black Panthers). The students left after a two-day occupation but when Frederick Greenleaf and Emile Chi, then Assistant Professors, ran in to check the computer room, as Greenleaf recalls "They had knocked the doorknobs off the door so you couldn't open it." There was a small window on the door. "We could see there was an improvised toilet-paper fuse. It was slowly burning its way to a bunch of containers, bigger than gallon jugs. They were sitting on top of the computer." Barron: "Already, he said, smoke was curling under the door. He and Professor Chi grabbed a fire extinguisher in the stairwell. The only way to douse the fuse was to ain the fire extinguisher under the door. The only way to know where to aim it was to look through the window" high up in the door. "So one functioned as the eyes for the pair, sighting through the window and directing the other to point the fire extinguisher up or down or left or right." Peter Lax and Jürgen Moser also figure in the story, but Greenleak and Chi saved the day.
Amir Aczel's obituary, in the Times on December 9, 2015, contains some nice mathematical information. Aczel, as William Grimes tells it, "took readers on a mathematical mystery tour in Fermat's Last Theorem, his account of how a famous 300-year-old problem in number theory was finally solved in the 1990s, and went on to write more than a dozen popular books on intriguing scientific ideas and discoveries." Grimes walks us through the history of the Fermat problem, including the quote: "I have a truly marvelous demonstration of this proposition which this margin is too narrow to accomodate" and the chronology of Andrew Wiles' 1994 proof. Grimes quotes Aczel: "What is interesting about Fermat's Last Theorem is that it spans mathematical history from the dawn of civilization to our own time. And the theorem's ultimate solution also spans the breadth of mathematics." His most recent book, Finding Zero, A Mathematical Odyssey to Uncover the Origins of Numbers took him to Cambodia in search of a stone slab. Grimes: "Its inscriptions include the number 605, a reference to the number of years that had passed since the beginning of the Chaka era in A.D. 78. This, the first known use of zero, has strongly suggested to scholars that zero originated in Asia and was carried to the Middle East by Arab traders." Grimes quotes from a 2014 Scientific American interview of Aczel: "To me it represents something immense because it's the human understanding of the concept of nothingness. Why would nothing be a number? If it's nothing, then it shouldn't be a number, but the nothing is really very important." (The Times left out the last sentence of the interview transcript "It's the empty set.")
Update on "abc"
News of the abc conjecture was covered in this column two months ago. Meanwhile a top-tier conclave assembled in Oxford, December 7-11, to try and bring some light on the problem. "Biggest mystery in mathematics in limbo after cryptic meeting" was Davide Castelvecchi's report on the proceedings for Nature, posted on December 16, 2015. "A collective effort to scrutinize one of the biggest mysteries in mathematics has ended with a few clues but no firm answers." What is the mystery? Shinichi Mochizuki, a very well respected number theorist, posted a series of 4 papers in August, 2012 which purport to prove the "abc conjecture." A correct proof would have enormous repercussions in the field. The mystery is, how does Mochizuki's proof work? So far, nobody understands it well enough to communicate it, even to the most battle-tested experts. As Castelvecchi explains, "Mochizuki's papers, which totalled more than 500 pages, were exceedingly abstract and cryptic even by the standards of pure mathematics. That has made it tough for others to read the proof, let alone verify it. Moreover, the papers built on an equally massive body of work that he had accumulated over the years, but that few were familiar with." One source of frustration, as Mochizuki himself explained (via Skype), is that "over many years he had developed a host of tools that he thought would be useful to prove abc--but that in the end he realized he did not need all of them" and unfortunately no one in the community knows yet which are the right ones to concentrate on.
Castelvecchi reports one ray of hope: "A consensus emerged that the highlight of the workshop was a lecture on 9 December by Kiran Kedlaya, an arithmetic geometer from the University of California, San Diego. He zeroed in on a result from a 2008 paper by Mochizuki that linked the statement of the abc conjecture to another branch of maths called topology. The link was immediately recognised as a crucial step in Mochizuki's grand strategy." But "Even Kedlaya agrees that the insight needs to be followed up by many others, and by an understanding of the strategy that links those key passages to one another."
A comprehensive survey of the Oxford meeting, by Brian Conrad, is posted on the Mathbabe website.
Stony Brook University
tony at math.sunysb.edu