On Wednesday I mentioned a couple of tangential-mathematical matters recurring in my mind lately. Today some completely non-mathematical nostalgia I’m reminded of as, prior to even stumbling into the White House, CrookedDonald continues to commit potentially impeachable offenses.
In olden days I enjoyed some political fiction, and two of the best pieces in that genre from the 1960s were, “Seven Days In May” by Fletcher Knebel and Charles Bailey II (made into a major movie, screenplay by Rod Serling), and “Night of Camp David” a solo effort by Knebel.
The first was a thrilling novel about an attempted military coup of the U.S. Government (by a General at odds with the President), and the second one about a mentally-ill President who needed to be removed from office, is particularly interesting. Needless to say, I think these make for entirely timely reading today, despite their 50+ years-age. Many details and technology will have changed of course in the last 50 years, but human nature has NOT changed, and that is a lot of what these novels recount. Fiction sometimes has an uncanny way of presaging life events.
Anyway, if you enjoy novels you might want to give these a look, or perhaps search for the movie version of the first.
I’ve referenced “critical thinking” lately in both blogs, so figured I’d end the week with a simple example from Daniel Levitin’s timely book (“A Field Guide To Lies”) that I’ve also been touting. Many readers will already be familiar with this probability case in one form or another, but still instructive, especially for those unfamiliar. Levitin sets it up as a street vendor playing a betting game with passersby, more-or-less like this:
The vendor has 3 cards he shows you, one that is white on both sides, one that is red on both sides, and one with a red side and a white side. (And you can play the game as many times as you like.)
Again, the cards are:
Red White Red
——— ———- -———
Red White White
He puts the cards in a hat, pulls one out and places it on the table showing a red side; then says, “I’ll bet you $5 the other side is red.” Should you take the bet? The passerby may think, “well, sure, I have a 50/50 chance, since 1 card (the white/white) is eliminated, I’m looking at either the red/white or the red/red card. So there’s a 50% chance the other side is white, and even if I lose I can play the game again, and maybe get my money back.”
Seems common-sensical, but it is critically wrong.
There are actually THREE ways a red side can be showing: one way from the Red/White card and two different ways from the Red/Red card. Thus, there is a 2/3 (not 1/2) chance the other side is red — and of course the same probabilities hold if the shown side is white and the vendor says he’ll bet the other side is white. The vendor (no surprise) has a distinct advantage.
Anyway, it’s a simple little gambit that will fool a lot of people, and yet once explained, with just a bit more critical thought, the situation is clearcut.
Probability is a particularly rich area for such ‘critical thinking’ examples, that sometimes even fool professional mathematicians. Is it any wonder that the rest of us are so easily blind-sided.
Of course confusion with numbers seems a far cry from confusion with politicians' rhetoric, but in both cases there is the need for care and clarity of thought in evaluating.
…that’s what it feels like to some of us in America these days as Jan. 20th approaches (Presidential inauguration). Given the inauspicious November turn of events two thoughts keep returning to me:
1) One is how prescient, in some sense, Kurt Gödel was to predict back in 1936 the likelihood of America becoming a dictatorship — i.e., that our Constitution in no way precluded it (…many of us have said this in the last couple decades, but Gödel was noting it 80 years ago!). No one knows for sure, what “flaw” Gödel perceived in our Constitution, but several scholars believe it was the ‘self-referential’ amending-power given in Article V — basically that since Article V says the Constitution can be amended, then Article V itself can be amended, and could be amended in a manner to say that some section or aspect of the Constitution can no longer be amended, creating certain despotic powers. Ta dahhh, something like this is almost occurring in the state of North Carolina already where the GOP is subverting democracy in unprecedented ways. And other states with gerrymandered voting districts and Republican legislatures may well insidiously do the same, because power (and money) corrupts and absolute... oh, nevermind.
…Somewhere Gödel is nodding his head knowingly.
2) All of this handwringing comes about as we witness the incredible dwindling of “critical thinking” among both leaders and constituencies, simultaneous with the rise of out-and-out lies and propaganda as a norm… and short-term, there is alarmingly little that can be done about it. Longer term more of the electorate needs to be educated in “critical thinking,” but that requires significant time. I’ve already mentioned one current book that attempts such a job: Daniel Levitin’s “A Field Guide to Lies.”
Some other books I’ve previously blogged about with a statistics or numbers focus on critical thinking are:
Gary Smith’s “Standard Deviations”
Charles Wheelan’s “Naked Statistics”
Jordan Ellenberg’s “How Not To Be Wrong”
…and a related, broader, more academic favorite of mine is:
Noson Yanofsky’s “The Outer Limits of Reason”
I also mentioned a bit ago that many popularizations of General Semantics teach critical thinking in regards to language use (which is probably even more important than the way numbers and statistics are ill-used), including some quite old volumes:
But truthfully, it's pie-in-the-sky thinking to hope the masses read such books and take them to heart, when they find thin-skinned orange men so much more appealing. Oy veyyyy! (And it will take decades to train new generations in critical thinking). More likely we’ll just muddle forward from bad to worse to worser!, until, as with past situations, something wakes us, shakes us, to the gravity of the situation.
For unfortunately, the fault is not merely with puppet Donald Trump, 'the fault is in ourselves':
Ohhh, and Happy New Year everybody! But seriously, brace yourself.It should be a wonderful year ahead for White Nationalists, anti-semites, the KKK, and fans of Russia (for the rest of us, maybe not-so-much).
Enough of my rant though, I'll leave you with one of Keith Olbermann's:
"The more we ourselves are enraptured by the beauties of mathematics, the more we regret that we can bring so few people to share our pleasure. But at least those of us in the school of abstract mathematics have one consolation: as we make our presentations clearer and more transparent, they automatically become easier to understand. Bear in mind that four hundred years ago, arithmetic was a difficult art. So great an educator as Melancththon [a sixteenth century scholar who reformed German education] did not trust the average student to penetrate the secrets of fractions. Yet now every child in elementary school must master them. Perhaps eventually the beauties of higher mathematics... will be accessible to every educated person."
-- German mathematician Wolfgang Krull (1930), quoted in Ivars Peterson's "Islands of Truth: A Mathematical Mystery Cruise"
A final book blurb before Christmas, touching upon 3 of the books I'd appended onto my longer, prior book year-end post:
a) I thought Brian Clegg’s new book, “Are Numbers Real?” would be about the Platonic/non-Platonic divide among mathematicians — a subject that interests me, though it may bore many readers! BUT, I was wrong and the volume is more an account of historical highlights in mathematics — a topic (math history) that many others find interesting, but I don’t particularly :( The second half of the book (perhaps 19th century on) however, is more interesting and meaty than the first half, and it’s a fine historical rendering, but, given other choices, I’m less inclined to recommend it as a stocking-stuffer for the Holidays, unless a math history-highlights volume is precisely what you’re looking for.
b) On-the-other-hand I’m very much enjoying Stephen Wolfram’s anecdotes and mini-bios in his new “Idea Makers,” and have no trouble recommending it for anyone who likes reading about the lives of scientists and mathematicians; a nice quick compendium, in small nuggets, of 16 varied, deceased individuals (...for those sensitive to such aspects though, I’ll warn that only one female, Ada Lovelace, is included).
c) Finally, also a BIG thumbs-up to Daniel Levitin’s latest volume, “A Field Guide to Lies” (Daniel’s earlier works on music and the brain were also good). The key here is the book’s subtitle: “Critical thinking in the information age.” Recently, I wrote about my own concerns regarding “critical thinking” and it’s important to have as much discussion/treatment of this subject as possible given the alarming degree of anti-scientific, non-critical thinking that prevails today. In fact, I'm VERY pessimistic, in the short-term, as to what can be done about societal lapses of critical thinking, but at least the discussion needs to be underway, and Levitin's treatment looks excellent. I especially like the way he has divided the topic into 3 categories (parts): 1) "Evaluating Numbers" 2) "Evaluating Words" and 3) "Evaluating the World" (about how science works).
"...math islike music. The aesthetic element in mathematics is essential, not peripheral. I’m not sure, but I think that in the minds of many people mathematics is reduced to a collection of more-or-less arbitrary facts, like the fact that the area of a circle equals pi times the square of its radius. Each of these facts, however, is like the final cadence of a symphony. It may be thrilling by itself, but it’s missing the indispensable context of “where did we start?” and “how did we get here?”This is why mathematicians insist on proving things: the proof is a whole symphony, not a single chord. Mathematicians are lauded not for stating facts, but for demonstrating their necessity, the way composers and musicians are praised for the whole course of a piece or a performance, not just its ending. When executed well, a proof has rhythm. It has themes that are developed and interwoven. It has counterpoint. It sets up expectations that are satisfied or subverted. Economy of material is valued, but not exclusively; an argument that wanders into neighboring territory, like a modulation to a neighboring key, can provide fuller appreciation of the main theme."
A quick end-of-year retrospective of some posts I had fun doing this year (more are from MathTango than Math-Frolic), in no particular order. Almost none have significant math in them, but rather touch on related subject matter:
10) Finally, I always enjoy the interviews I get to do, and this year, not counting the Donald Trumpster one, there were 5 6 (with Mircea Pitici, Samuel Hansen, Katie Steckles, Jim Propp, and Brian Hayes, ADDENDUM: Grant Sanderson now squeezed in before year-end) the links to which can be found at the main interview page:
"To leave the safe familiarity of the shore and sail off into unknown territory, that is what it is like to do mathematics... "...'doing mathematics' begins with a state of mind that allows you to
travel to a place deep inside the subconscious to open body, mind, and
spirit to the contemplation of a mathematical idea. Doing mathematics
can be a mental voyage to a place where clarity of thought and openness
to insight make it possible to see the deeper beauty of a mathematical
structure, to enter a world where triumph over a problem depends less on
conscious effort than on confidence, creativity, determination, and
intellectual rigor."
FQXi, a physics/cosmology community site, runs an essay contest each year, and this year’s theme has been announced as, “Wandering Towards a Goal – How can mindless mathematical laws give rise to aims and intentions?” Certainly a thought-provoking topic with lots of approaches. Deadline for entry (anyone is eligible) is Mar. 3, 2017:
Sometimes I wish I followed primary/secondary math education more closely than I do, there’s so much amazing stuff going on there. The resources, technologies, ideas, possibilities in secondary math today have changed SO much (for the better) since I was younger. Wish I could take it all over again!
Below is the keynote address (~50 mins.) to the recent California Mathematics Council Convention (4 presenters; all good, but Dan Meyer and Fawn Nguyen especially not-to-be-missed). If you haven’t seen it, I hope you’ll find time for it; you’ll be inspired:
A philosopher wishes to measure the height
of a certain flag pole. All he has to do so is a measuring tape, and though he tries and tries he is unable to slide the tape up the full length of the pole. Eventually, an
engineer comes by and sees the philosopher struggling. He says, "allow me," at which point he pulls the pole out
of its hole in the ground, lays it flat on the ground, and easily measures it.
“Your pole is 5.5 meters long," he announces.
“But,” says the philosopher, “I wanted the height, not the length!”
A bit more HERE, and Richard Rorty in Wikipedia. [Will just add that some of us have been making essentially these same predictions ever since the election of Ronald Reagan to a 2nd term in 1984.]
ADDENDUM: talk about prescient, how did I not think to include this classic 1976 movie clip here:
