Friday, August 28, 2015
Siobhan Roberts probably didn't really get a summer vacation, being busier-than-ever with a book tour, but she reports on the mixture of "research and play" that filled the summers of some others:
Wednesday, August 26, 2015
Today, Chris Harrow called attention to a Marilyn vos Savant probability problem from earlier in the month (an inquiry from a reader at Parade Magazine) that ran as follows:
"Say four people are drawing straws for a prize. My friends and I agree that the first person to draw has a 1 in 4 chance of getting the short straw. However, if he or she does get it, the second person’s chances drop to zero. And if the first person doesn’t get the short straw, the second person’s chances increase to 1 in 3, and so on. Our disagreement: Is it better to draw first, last, or does it make any difference?"
To which Marilyn simply responded:
"The order makes no difference. Envision all four people drawing straws, but instead, not looking at them yet. At this point, each person has a straw. Does it help to be the first or last one to look? No."But Chris nicely fleshed it out a bit more as a teaching moment to make the distinction between frequentist and conditional probability; i.e. there are potentially two different probability sets here: 1) if all 4 straws are drawn and only THEN looked at, versus 2) if each straw is looked at as it is drawn:
And at the end of his post Chris makes an analogy from the straw-drawing to physics' quantum theory (or perhaps really Schrodinger's cat); I'm not sure if a physicist would be pleased with the analogy, but it's a nice thought exercise.
Monday, August 24, 2015
It's always a good weekend when I find a Raymond Smullyan book on sale for 50¢ in a local thrift store. This was a good weekend.
So we'll ease into the week with three simple quickies from the master logician (the first two I've adapted from the book, and the last one is verbatim from Smullyan):
1) $5000 is stolen from a local store. The robber or robbers make their get-away in a car. Soon thereafter 3 well-known criminals, A, B, and C, are picked up by law enforcement for questioning, and 3 facts are established:
a) NO ONE other than A, B, and/or C was possibly involved in the crime.
b) C NEVER pulls a robbery without A (and possibly others) also being involved.
c) B does NOT know how to drive.
Is A innocent or guilty?
2) A train leaves from Boston heading to New York at 2 pm., and one hour later another train leaves New York heading for Boston on a parallel track, moving the same speed as the first train. Which of the two trains will be closer to Boston when the two meet?
3) And lastly, this riddle from Smullyan:
"Those of you who know anything about Catholicism, do you happen to know if the Catholic Church allows a man to marry his widow's sister?"
1) yes A is guilty (B couldn't do it by himself, because he couldn't drive the get away car, and if he did it with C, than, by #2 A must also be a guilty party. C couldn't have acted on his own without A.
A could've acted on his own, or with B or C, or both, but in some form had to partake.
2) of course at the time they meet the two trains will be the same distance from Boston (...though a technical sort might argue that if by "meeting" one means the front of the locomotives meeting, than the tail/caboose end of the Boston train would still be "closer" to Boston).
3) a dead man (widow's spouse) can't marry anyone!
Sunday, August 23, 2015
A quick Sunday reflection this week:
"...if we take a Baconian point of view, the history of mathematics is a history of horrendously difficult problems being solved by young people too ignorant to know that they were impossible."
-- John Baez
Friday, August 21, 2015
5, 3, 11, 3, 23, 3, 47, 3, 5, 3, 101, 3, 7, 11, 3, 13, 233, 3, 467, 3, 5, 3 . . . Rowland's sequence
The always-interesting Brian Hayes takes readers on a rollicking journey with formulas created to produce prime numbers:
Both computer programmers and number theorists may find this interesting, even though in the end, Hayes admits that "It seems we are back where we began, and no closer to having a practical prime generator"...but, as he also concludes, "Along the way you may have seen something interesting, or even astonishing."
Wednesday, August 19, 2015
A post only tangential to mathematics today....
This week Wired Magazine posted a list of the "27 Best Feeds to Follow In the World of Science":
Aside from the hyperbole (as Wired is prone to) of such a claim -- finding the 27 "best" out of three-gajillion science-related feeds -- I wasn't all-that-happy with their list, despite many fine entries. So decided to see what I'd come up with attempting to pick my own 27 favorite science-oriented Web feeds (sort of a quick, initial draft, and I limited myself to science-folks who had BOTH an active twitter feed and website).
I didn't include "math" feeds, and my bias towards physics versus biology/life-sciences may be evident, but here's the eclectic list, with the twitter handles/links, followed by their associated webpages. The order is random except for lumping the 10 magazines and collaborations together at the beginning, and the 17 individuals after those (and amazingly, only one of these shows up on the Wired list -- different strokes for different folks!):
...Please let me know ASAP if you find any of these links are incorrect or broken (...and hope you find 1 or 2 here that are new to you and worthwhile).
Monday, August 17, 2015
If thinking about thinking is among your interests, a phenomenally rich (and LONG), widely-romping post from Scott Aaronson on "common knowledge," something called "Aumann's agreement theorem," Bayesian thinking, and much more here:
(it's actually from an earlier talk Scott gave at SPARC)
While this won't be everyone's cup-of-tea, and it is more epistemology-logic-cognition than it is mathematics, it is (to me) one of the most fascinating, remarkable posts I've ever read on a math-related blog! Indeed, several readings likely required to take in all the ideas Scott puts on display here.
Aumann’s Theorem predicts that all "rational disagreements" should "terminate in common knowledge of complete agreement." But of course that doesn't happen so much in real life, and in one passage (that reminds me of so much stuff on the internet ;-)) Aaronson writes,
"You could say that the 'failed prediction' of Aumann’s Theorem is no surprise, since virtually all human beings are irrational cretins, or liars. Except for you, of course: you’re perfectly rational and honest. And if you ever met anyone else as rational and honest as you, maybe you and they could have an Aumannian conversation. But since such a person probably doesn’t exist, you’re totally justified to stand your ground, discount all opinions that differ from yours, etc."Anyway, give it a gander; you'll probably know before you're half-way through it if it's the sort of mind-stretching thought-exercise that strikes your fancy or not. (I suspect I may still be re-reading it a week from now, trying to better grasp parts!) The piece also contains some key links to related material.
Sunday, August 16, 2015
A li'l statistics lesson passed along by K.C. Cole in "The Universe and the Teacup":
"In "Strength In Numbers," mathematician Sherman Stein offers the case of the men's support group that wanted to demonstrate how badly women treat the male sex. As supporting evidence, the group pointed out that more than half of the women on death row had murdered their husbands, while only a third of the men on death row had murdered their wives. What the group neglected to mention, says Stein, was that there were a total of seven women on death row. And 2,400 men."
....and on a related note, see my review of Gary Smith's "Standard Deviations" newly up at MathTango.
Wednesday, August 12, 2015
More importantly, have you seen this:
Ben Orlin's tribute today to Oliver Sacks... not much math, but well worth passing along.
(...p.s.: do not 'medicalize' your students!)
Monday, August 10, 2015
H/T to Sue VanHattum for recently pointing out these two spot-on Tom Siegfried pieces that I totally missed in Science News a month ago:
Siegfried astutely summarizes the statistical problems plaguing science for a long time, that are just recently getting addressed (at one point calling statistics "addictive poison" to science) -- the first article fleshes out the problem, the second offers 10 suggestions.
Siegfried is always worth reading and I need to get into a better habit of checking out his "Context" pieces for Science News.
As long as we're talking statistics and worthwhile reads I'll recommend two current books:
"Standard Deviations" (newly available in paperback) by Gary Smith is one of the most fun math reads I've seen for awhile; a statistical playground (but never too technical) with a fire-hose of examples, one-after-another-after-another, that will enlighten and entertain you (eventually I'll have some sort of review up at MathTango, but don't wait for the review, get this book!).
And I'll also go ahead and suggest, sight unseen, Tyler Vigen's new book, "Spurious Correlations" based on his fun site of the same name.
Wish statistics had been this much fun when I was young!
Sunday, August 9, 2015
I've taken this Sunday reflection verbatim from a prior Futility Closet posting of the same:
“The class was looking at an oscilloscope and a funny shape kept forming at the end of the screen. Although it had nothing to do with the lesson that day, my friend asked for an explanation. The lab instructor wrote something on the board (probably a differential equation) and said that the funny shape occurs because a function solving the equation has a zero at a particular value. My friend told me that he became even more puzzled that the occurrence of a zero in a function should count as an explanation of a physical event, but he did not feel up to pursuing the issue further at the time.(From Stewart Shapiro, Thinking About Mathematics, 2000; also his paper “Mathematics and Reality” in Philosophy of Science 50:4 [December 1983].)
“This example indicates that much of the theoretical and practical work in a science consists of constructing or discovering mathematical models of physical phenomena. Many scientific and engineering problems are tasks of finding a differential equation, a formula, or a function associated with a class of phenomena. A scientific ‘explanation’ of a physical event often amounts to no more than a mathematical description of it, but what on earth can that mean? What is a mathematical description of a physical event?
"What right do we have to presume that the natural world will hew to mathematical laws? And why does the universe oblige us so graciously by doing so? Repeatedly, mathematicians have developed abstract structures and concepts that have later found unexpected applications in science. How can this happen?"
Friday, August 7, 2015
Fascinating new Erica Klarreich piece (interview) in Quanta on Neil Sloane and the OEIS, Online Encyclopedia of Integer Sequences, which is now home to more than a quarter million number sequences contributed by mathematicians the world over:
Turns out that when Neil isn't curating (along with a set of editors) number sequences he's writing guidebooks about rock-climbing in New Jersey! Yet that is one of the less interesting of all the many great tidbits in the article.
[By the way, since my previous post was about math and music, worth noting that all the sequences in the OEIS can (as an option on the site) be put to music and played!]
Wednesday, August 5, 2015
See what else he has to say about the math/music junction:
Another worthwhile read, is this older Marcus du Sautoy piece (via Plus Magazine) on "The Music of the Primes":
And then there is this fun Evelyn Lamb piece from last year on tuning pianos (or, sadly, failing to):
Tuesday, August 4, 2015
Well, this is sort of neat news: discovery of a new (15th) convex pentagon that successfully tiles the plane:
If you don't know the history of this problem you can check out the Wikipedia article that is linked to in the above piece. Following additional findings after Martin Gardner originally drew attention to the problem back in the 1970s, the number of such tiling forms had stood at 14 for 30 years!
How many more are there???