Expectations and Discoveries... and a Passing

This Sunday's reflection from Freeman Dyson (in "Dreams of Earth and Sky"):

"A typical proton-proton collision in the LHC will produce a large spray of secondary particles, and the collisions are occurring at a rate of millions per second. The machine must automatically discard the vast majority of the collisions, so that the small minority that might be scientifically important can be precisely recorded and analyzed, The criteria for discarding events must be written into the software program that controls the handling of information. The software program tells the detectors which collisions to ignore. There is a danger that the LHC can discover only things that the programmers of the software expected. The most important discoveries may be things that nobody expected. The most important discoveries may be missed."

                                                                      __  __  __

....on a side note, Oliver Sacks died this morning; I don't consider that a sad note -- his life and contributions were so glorious, and his approach to death so thoughtful and intrepid, that death is more like a punctuation mark at the end of another inspiring sentence from his rich, productive life and writings.
There will no doubt be profuse tributes/obits in coming days. This from the NY Times:

http://tinyurl.com/pxg3ak7

 

What Did You Do On Summer Vacation?

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:

http://www.newyorker.com/tech/elements/cogito-ergo-summer-the-beauty-of-summer-science

Drawing Straws with Chris and Marilyn

via WikimediaCommons

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:

https://casmusings.wordpress.com/2015/08/26/marilyn-vos-savant-and-conditional-probability/

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.

50-Cent Puzzles....

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?"

answers below:
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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!

Youth!

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

Producing Those Pesky Persnickety Primes...

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:

http://bit-player.org/2015/pumping-the-primes

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."

Following Science on the Web

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":
http://www.wired.com/2015/08/the-new-cultural-literacy-science-feeds/

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!):

@QuantaMagazine          https://www.quantamagazine.org/
@NautilusMag                 http://nautil.us/
@AeonMag                      http://aeon.co/magazine/
@NPRscience                  http://www.npr.org/sections/science/
@FQXi                             http://fqxi.org/community
@CosmosMagazine         https://cosmosmagazine.com/
@AmSciMag                   http://www.americanscientist.org/
@NYTScience                 http://www.nytimes.com/section/science
@RadioLab                      http://www.radiolab.org/
@WorldSciFest                http://www.worldsciencefestival.com/

@Tom_Siegfried              https://www.sciencenews.org/blog/context
@ivanoransky                   http://retractionwatch.com/
@lpachter                          https://liorpachter.wordpress.com/
@skdh                               http://backreaction.blogspot.com/
@StartsWithABang          https://medium.com/starts-with-a-bang
@SteveStuWill                 http://www.stevestewartwilliams.com/
@ProfSimons                    http://blog.dansimons.com/
@cfchabris                        http://blog.chabris.com/
@JimalKhalil                    http://www.jimal-khalili.com
@SeanMCarroll                http://preposterousuniverse.com/
@EdYong209          http://phenomena.nationalgeographic.com/blog/not-exactly-rocket-science/ 
@IanSample                     http://www.theguardian.com/profile/iansample
@CarlZimmer                   http://phenomena.nationalgeographic.com/blog/the-loom/
@Horganism                     http://blogs.scientificamerican.com/cross-check/
@NotEvenWrong             http://www.math.columbia.edu/~woit/wordpress
@JenLucPiquant               http://blogs.scientificamerican.com/cocktail-party-physics/
@DanAriely                      http://danariely.com/

...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).

Not-so-common Common Knowledge

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:

http://www.scottaaronson.com/blog/?p=2410
(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.

Lies, Damned Lies, Statistics?

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.

Have you....

More importantly, have you seen this:

http://mathwithbaddrawings.com/2015/08/12/oliver-sacks-knows-what-it-means-to-teach/

Ben Orlin's tribute today to Oliver Sacks... not much math, but well worth passing along.
(...p.s.:  do not 'medicalize' your students!)

Tom Siegfried Worth Reading, et. al.

via WikimediaCommons

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:

https://www.sciencenews.org/blog/context/science-heroic-tragic-statistical-flaw
https://www.sciencenews.org/blog/context/top-10-ways-save-science-its-statistical-self

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!



 

"What Is A Mathematical Description..."

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.

“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?"

(From Stewart Shapiro, Thinking About Mathematics, 2000; also his paper “Mathematics and Reality” in Philosophy of Science 50:4 [December 1983].)

Sequence Man, Neil Sloane

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:

https://www.quantamagazine.org/20150806-neil-sloane-oeis-interview/

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!]

Math People; Music People... Passion

via WikimediaCommons

Music major, teacher, and recording artist Marcus Miller downplays the sometimes vaunted connection between mathematical and musical abilities while admitting that he himself is someone who "feels" both math and music. At one point he writes, "For me, the deepest, most enshrined, and magical place of intersection between math and music is passion — the cross streets of enthusiasm and curiosity."
See what else he has to say about the math/music junction:
https://medium.com/bright/how-i-use-music-to-inspire-math-students-c2ce7a5d8b14

Another worthwhile read, is this older Marcus du Sautoy piece (via Plus Magazine) on "The Music of the Primes":
https://plus.maths.org/content/music-primes

And then there is this fun Evelyn Lamb piece from last year on tuning pianos (or, sadly, failing to):
http://blogs.scientificamerican.com/roots-of-unity/the-saddest-thing-i-know-about-the-integers/ 

Tiling the Plane... Again!

Well, this is sort of neat news: discovery of a new (15th) convex pentagon that successfully tiles the plane:

https://plus.google.com/u/0/100003628603413742554/posts/JC8hqyUHuDE

 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???

Math, Music... Pornography

Today's Sunday reflection, from Michael Harris in "Mathematics Without Apologies":

"Cognitive scientist Steven Pinker has a more functional point of view. What pure mathematics and music have most clearly in common is that they both fit Pinker's definition of 'cheesecake,' 'unlike anything in the natural world because it is a brew of megadoses of agreeable stimuli which we concocted for the express purpose of pressing our pleasure buttons. Pornography is another pleasure technology... [T]he arts are a third.' Pure mathematics would appear to be a fourth, and what it shares with the arts, and with music in particular, is that, unlike cheesecake or even pornography, its variety is literally limitless."