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:
As a “bonus” link on last Friday’s potpourri I gave Sean Carroll’s recent Gifford Lectures In Natural Theology. They generally last about an hour with a Q-and-A period following. Just to further entice you, I’ll post the 2nd talk ("The Stuff of Which We Are Made") below:
"Perhaps the most surprising thing about mathematics is that it is so
surprising. The rules which we make up at the beginning seem ordinary
and inevitable, but it is impossible to foresee their consequences.
These have only been found out by long study, extending over many
centuries. Much of our knowledge is due to a comparatively few great mathematicians such as Newton, Euler, Gauss, or Riemann; few careers can have been more satisfying than theirs. They have contributed something to human thought even more lasting than great literature, since it is independent of language."
-- Edward Charles Titchmarsh, quoted in Mathematical Maxims and Minims by N. Rose
[between all the Holiday and political hoopla, blogposts here might be a little scarce the next few weeks, but I do have a new year-end book wrap over at MathTango today.]
“Gödel didn’t believe that truth would elude us. He proved that it would. He didn’t invent a myth to conform to his prejudice of the world — at least not when it came to mathematics. He discovered his theorem as surely as if it was a rock he had dug up from the ground. He could pass it around the table and it would be as real as that rock. If anyone cared to, they could dig it up where he buried it and find it just the same. Look for it and you’ll find it where he said it is, just off center from where you’re staring. There are faint stars in the night sky that you can see, but only if you look to the side of where they shine. They burn too weakly or are too far away to be seen directly, even if you stare. But you can see them out of the corner of your eye because the cells on the periphery of your retina are more sensitive to light. Maybe truth is just like that. You can see it, but only out of the corner of your eye.”
I don't have a lot of potpourri picks for you this week (over at MathTango), so I'll add this long, older pdf that seems timely, on the well-known story of Kurt Gödel and a possible loophole in Article V (perhaps) of the U.S. Constitution (a surprisingly interesting read, coming from a law review journal!): http://tinyurl.com/zs3tvb3
I wrote shortly ago that I expected Jim Propp to have an interesting take on self-referential sentences sometime this month, and that exceedingly-rich post is now up, full of fascinating ideas (...I’ll admit ‘self-reference’ isn’t everyone’s cup-of-tea of a topic, but for those it is, Jim’s thoughts and many great links and endnotes are must-reading, including Jim’s own self-referential ‘test,’ if you’re in the mood for a real mind-twister):
[One, of many, small interesting sidelights for me was discovering that Janna Levin, better known for her physics writing, had written a 2007 novel intertwining the lives of Kurt Gödel and Alan Turing.
-- p.s. here's a link to Dr. Levin's appearance at "The Moth," which Brainpickings' Maria Popova once called "the greatest story ever told on The Moth").]
Hopefully, by now I don’t even need to link to Ben Orlin’s posts, because you’re already reading him anyway. If you’re not reading Ben you should be locked up! He has a bigly (almost yuuuge) post today on the secondary math curriculum with plenty of neat ideas for consideration and discussion. They are very good, BELIEVE ME.**
This has already been, for awhile, a topic among math educators, who ought not miss Ben’s take on how to drain the math swamp and re-build it anew:
Something appropriate I s’pose that a free-spirit like Leonard Cohen should die in a year, and indeed a week, when freedom itself seems to be expiring....
[these dozen examples offer some good ones for primary/secondary students, and youngsters are especially susceptible to the following 'broken water heater' fallacy.]
#6 he calls “The Broken Water Heater Problem”… I’ve seen it in many forms, and there are a zillion ways to set it up. Hickey posts it as follows (quoting verbatim):
———————————————————
Last week, the heat in my apartment crapped out because my water heater broke.
I went to a person, showed him the water heater, he used a bunch of spare parts and then fixed it. I paid him for the repairs.
Is this person more likely:
An accountant?
…or
An accountant and a plumber?
———————————————————
Even if you’re not familiar with these type-questions, hopefully upon a moment of logical thought the reader recognizes the answer should be that the person is “an accountant.”
The problem is, often people DON'T really give a moment of ‘logical thought,’ instead jumping to the conclusion that it's more likely (more probable) that the individual must have plumbing skills, and thus must be “an accountant and a plumber.” But of course ANYone who is ‘an accountant and a plumber’ is automatically ‘an accountant’ — in Venn diagram terms, the circle of accountants includes wholly within it the circle of ‘accountants and plumbers.’
Again, I just like this because it once more demonstrates how easily words and language often short-circuit, rather than aid, our logic. [In fact, I dare say our latest election is clearly another example of mere visceral words/language overriding reason.]
A substitute ‘Sunday reflection’ today, because of my experience with a tweet last evening….
I’ve always loved the above photo (brings me a smile everytime) of a 10-yr.-old Terry Tao with Paul Erdös, which I dropped into a tweet last night — the look on the faces, the synchronous postures, the concentration, the shared passion… It originally showed up on the ’Net in 2013 courtesy of Terry himself, and went somewhat viral at the time within math circles:
(...and a h/t to Patrick Honner who was the first to bring it to my own awareness, back then)
Is that not great! I thought it was iconic by now and viewed by everyone multiple times. But a few folks (responding to the tweet) apparently were not familiar with it, and while usually recognizing Erdös, were unsure of the youngster involved. So just to clear any confusion, am posting the little bit of history that I know (thanks again to Dr. Tao for sharing it originally -- apparently the pic comes from the University of Adelaide in Australia at an awards ceremony of the Australian Mathematics Competition,1985).
As some responders have noted, two people, bridging a large generation gap, one originally Hungarian, one originally Australian, who both came to America, and in different manners contributed so much to our joy of mathematics, and continue to delight us, in both death and life.
p.s... among Erdös' many noted linguistic idiosyncracies, he referred to young children as "epsilons," and as one of the commenters to Terry's original post says, he had finally found an interesting "epsilon." ;-)
Perhaps this test and its implications, by a man who treated "reasoning as an enigma" is even extra pertinent in a week that people are engaged in especially important political decision-making! From his 2003 obituary: “His aim was to reveal a surprising phenomenon—to show that thinking was not what psychologists including himself had taken it to be.” [If by any chance you're not familiar with the task, the article contains an interactive video you can play with. And it's a fun test to run by students and adults at various levels.] Also, as with many puzzles I enjoy, language or words, in addition to strict abstract reasoning, potentially come into play here, as the article indicates. [ADDENDUM: just realized, this article is a RE-run of a piece Nautilus ran in May 2015]
"The only thing necessary for the triumph of evil is for good men to do nothing." -- Edmund Burke
No math today, just blather about this craziest election of my lifetime….
Months ago, I surmised that if by some deranged alignment of planetary forces Donald Trump actually won the Republican nomination I expected a response from STEM people as never before seen in a presidential campaign: 100s or 1000s of scientists signing open letters from various organizations/societies and independent groups to denounce a Trump candidacy. There have been a trickle of such efforts and certainly individual STEM folks have voiced their concern via Twitter, Facebook, Google, etc… but the large-scale outpouring I envisioned hasn’t materialized.
Many feel uncomfortable or even constrained (sometimes contractually) from involving themselves publicly in politics; it is not a customary activity for the science crowd, and perhaps many conclude they’d only be preaching to the choir anyway (…and, as unfathomable as it seems, some scientists even support the absurdity that is Donald Trump).
But something is seriously wrong when folks who call themselves patriotic, or religious, or God-fearing, or simply concerned about the future, say they are voting for Donald Trump, as if wearing blinders. Gullibility and timidity of citizenry during the rise of German Fascism led of course to unprecedented human tragedy. I’m a bit ashamed by the lack of concerted, organized response from the STEM community to a narcissistic authoritarian in our midst — with demagogic speeches and political rallies reminiscent of Jim Jones’ assemblies. And please spare me your objections to the German Fascist analogies (they ARE apt, and I don't doubt for a second that if Hitler rose from the dead to campaign across America today, 30%+ of current voters would back him).
‘Those who fail to learn the lessons of history are doomed to repeat it’… Trump will most likely lose this election… but there are more like him coming down the pike. That his antics and laughably-shallow “policies” appeal to so many doesn’t bode well for the future. And science is in their crosshairs. Those who dare ‘preach’ evolution or vaccination or climate mitigation or brain science or particle physics or space travel or… or… are all vulnerable (no doubt Jews, gypsies, and trade-unionists, are as well). Perhaps America’s 200+ year-old experiment in democracy and slow liberal progressivism is simply running out of steam to continue against the regressive, anti-science, anti-rational sentiment and thuggery that is creeping across the globe. Make no mistake about it though, silence is not golden; it is complicit. Admittedly, whatever the outcome of this election the Trump cult will remain, potentially sabotaging the next four years, even worse than Republicans sabotaged the last eight.
"We will not walk in fear, one of another. We will not be driven by fear into an age of unreason, if we dig deep in our history and our doctrine, and remember that we are not descended from fearful men — not from men who feared to write, to speak, to associate and to defend causes that were, for the moment, unpopular. This is no time for men who oppose Senator McCarthy's methods to keep silent, or for those who approve. We can deny our heritage and our history, but we cannot escape responsibility for the result. There is no way for a citizen of a republic to abdicate his responsibilities. As a nation we have come into our full inheritance at a tender age. We proclaim ourselves, as indeed we are, the defenders of freedom, wherever it continues to exist in the world, but we cannot defend freedom abroad by deserting it at home. The actions of the junior Senator from Wisconsin have caused alarm and dismay amongst our allies abroad, and given considerable comfort to our enemies. And whose fault is that? Not really his. He didn't create this situation of fear; he merely exploited it — and rather successfully. Cassius was right. 'The fault, dear Brutus, is not in our stars, but in ourselves.' Good night, and good luck."
ADDENDUM (11/11/16):When you turn voters against their government, as Reagan did, and against the press as Trump did, and against science, as the GOP has done, don't be surprised if democracy withers and dies.
“That’s how real science advances… Three steps forward, two steps back. Mathematicians have the luxury of living in a logical bubble, where once something is proved true, it remains true. Interpretations and proofs may change, but the theorems don’t get unproved by later discoveries. Though they may become obsolete or irrelevant to current concerns. Science is always provisional, only as good as the current evidence. In response to such evidence, scientists reserve the right to change their minds.”
As Seinfeld fans know, one of the things that made that show so enjoyable was a style they developed of weaving two (or more) disparate plots together, in a single 30-min. episode, that somehow resolved or came together at the end.
I suddenly realized that, to some degree, this is also what makes many of Evelyn Lamb’s posts for her “Roots of Unity” blog so wonderful. She’s developed a knack for bringing up multi-subjects or ideas and showing how a mathematical thread draws them together. In her newest post she weaves Ramsey Theory and birthdays into Facebook with her own little fun daily game (also, includes several excellent links):
Of course math is EVERYwhere, and so too 'interesting configurations.' Always fun to be reminded of it. Lamb writes that she wants to "share a little way in which a little bit of math enhances my life a little" ...and in so doing she enhances her readers.
“…the terminology is misleading, for it suggests that there is some greater ‘reality’ to these so-called real numbers than there is to the so-called imaginary numbers. This impression comes about, I suppose, because there is the feeling that distance measures are, in some sense ‘really’ such real-number quantities. But we do not know this. We know that these real numbers are indeed very good for describing distances and times, but we do not know that this description holds good at absolutely all scales of distance or time."We have no actual understanding of the nature of a physical continuum at a scale of, say, one googolith of a metre or of a second, for example. The so-called real numbers are mathematical constructions, which are, nevertheless immensely valuable for the formulation of the physical laws of classical physics.”
…although that was quickly followed by John McGowan saying he would take over administration of the blog for the interim (honestly, I had always thought McGowan was the principal administrator of this blog, which is among my favorites, so I’m not clear how big a change this is, or how mistaken I was?):
3) Jason Rosenhouse covered a lot more than just math at his long-running Evolution Blog, and I always found him to be one of the clearest (and perhaps under-appreciated) explicators of math and science out there. So, quite sad to see him calling it quits after more than 10 years of elucidation:
This isn't for the ones who blindly follow
Jingoistic bumper stickers telling you
To love it or leave it and you'd better love Jesus
And get out of the way of the Red, White and Blue
This isn't for the ones who buy their six-packs
At the 7-Eleven where the clerk makes change
Whose accent makes clear he sure ain't from here
They call him a camel jockey instead of his name
No, this is for the ones who stand their ground
When the lines in the sand get deeper
When the whole world seems to be upside down
And the shots being taken get cheaper, cheaper
This isn't for the ones who would gladly swallow
Everything their leader would have them know
Bowing and kissing while the truth goes missing
"Bring it on," he crows, puttin' on his big show
This isn't for the man who can't count the bodies
Can't comfort the families, can't say when he's wrong
Playing 'I'm the decider' like some sort of Messiah
While another day passes and a hundred souls gone
No, this is for the ones who stand their ground
When the lines in the sand get deeper
When the whole world seems to be upside down
And the shots being taken get cheaper, cheaper
This is for the ones that I see above me
Three little stars in a great big sky
Light for the world and hope for the weary, they try
This isn't for the ones with their radio signal
Calling for bonfires and boycotts, they rave
Exhorting their listeners to spit on the sinners
While counting the bucks of advertising, they'll say
This isn't for you and you know who you are
So just do what you want 'cause I know that you can
But I gotta be true to myself and to you
So on with the song, I don't give a damn
ICYMI, I enjoyed this little logic brain-twister from Alex Bellos in The Guardian yesterday. So as not to copy it verbatim, I’ll entirely re-word it below if you want to try it out, but then you can go here to see his original statement of it, with the neat solution:
Alan, Bob, and Carl play checkers among themselves, with the rule that the winner of each game keeps playing, while the loser awaits for another turn to play after each game. When they are done for the day they have each played the following numbers of games:
Alan played 10 Bob played 15 Carl played 17 Question: Who lost the 2nd game?
"So I would say that like Turing, I am absolutely struck with the power of mathematics, and that's why I'm a theoretical physicist. If I want to answer questions, I love that we can all share the mathematical answers. It's not about me trying to convince you of what I believe or of my perspective or of my assumptions. We can all agree that one plus one is two, and we can all make calculations that come out to be the same, whether you're from India or Pakistan or Oklahoma, we all have that in common. There's something about that that's deeply moving to me and that makes mathematics pure and special."
-- Janna Levin (interviewed in Krista Tippett's "Einstein's God")
“I miss mathematicians! They have such a good time with each other. They are like eager children swarming into the same old playground and creating ever-new uses for the same old equipment.“
This morning's 'Sunday reflection' comes from Morris Kline (in "Mathematics and the Search For Knowledge"):
"Mathematicians had given up God; so it behooved them to accept man, and this is what they have done. They have continued the development of mathematics and the search for laws of nature, knowing that what they produced was not the design of God but the work of man. Their past successes helped them to retain confidence in what they were doing, and fortunately, hosts of new successes greeted their efforts. What has preserved the life of mathematics was the powerful medicine humans had themselves concocted -- the enormous achievements in celestial mechanics, acoustics, hydrodynamics, optics, electromagnetic theory, and engineering, and the incredible accuracy of its predictions. Thus, mathematical creation and application to science have continued at an even faster pace."
In a new Wired article, Adam Kucharski explains why poker may be more difficult/interesting to AI researchers than either chess or Go, where all strategic information is right in front of the players:
He quotes chess master Garry Kasparov (who lost to IBM's Deep Blue computer in 1997), saying that computers play games like chess and Go "like a machine." And then writes further,
"Kasparov hoped that games such as poker would be different. You cannot win by following a fixed set of rules because some cards are hidden, and your information is imperfect. The same is true of many other situations in life, from negotiations to auctions and trading."
Kucharski reports that the latest poker-playing robots "are revealing new and innovative ways of juggling risks and making decisions with imperfect information" and "The world's top poker bots have taught themselves to bluff, feign aggression and even manipulate their opponents."
One successful poker bot from Canada that Kucharski cites (and that progressively learns "by playing billions of simulated games") is "Cepheus" (specifically for a limit version of Texas hold 'em):
