Speaking of Leakers...

Nice explication of the Ellsberg paradox with a simple card gambit from Presh Talwalkar. I like the way this little problem interweaves risk (or ambiguity) aversion, uncertainty, and probability all together in one presentation, and doesn't require any deep math to comprehend:

http://mindyourdecisions.com/blog/2013/07/31/ellsberg-paradox-facing-the-uncertain/#.UfjmHFOhRo4

(p.s., this paradox indeed comes from Daniel Ellsberg, who before his fame as the 'Pentagon Papers' leaker, was an economic theorist specializing in decision theory)

A Choice Posting

If set theory is definitely NOT your cup-o-tea then stop here and just move along to LOLCats or elsewhere.
But otherwise, read on....

According to RJ Lipton, Gregory Moore's "Zermelo's Axiom of Choice" (from Dover) is quite a page-turner and beach-read -- 450 scintillating pages-worth! -- in fact, Lipton calls it "hard to put down" ;-) His latest post at "Gödel's Last Letter" is a glance at the Axiom of Choice based on Moore's volume, and worth a gander... if, you so choose:

http://rjlipton.wordpress.com/2013/07/30/making-choices/

(image via Norvy/WikimediaCommons)

1, or No, 2, or Wait, 3, or....

As long-time readers know, self-referential conundrums are among my favorites, and recently CSE blog cited an old one from Martin Gardner:

The number of 1′s in this paragraph is ___; the number of 2′s is ___; the number of 3′s is ____; and the number of 4′s is ___.

And more general bonus follow-up:

The number of 1′s in this paragraph is ___; the number of 2′s is ___; ….(and so on) and the number of N’s is ___.

Happy paragraph parsing! Answers supplied here:

http://mathfactor.uark.edu/2012/04/hr-cardcolm/

Euler... Thinker Extraordinaire

1) Probably any mathematician would rank Leonhard Euler as one of the greatest mathematicians of all time (he pretty much shows up on anyone's top five list). And Professor William Dunham reveres Euler as the very top of the heap. In this video he entertainingly makes the convincing case for Euler -- Dunham talks for about an hour, followed by 30 mins. of question-answers -- for such a long presentation it maintains interest well:

You'll learn what i^i equals (and how Euler computed it), as well as Euler's better-known contributions, all accomplished in an era long before computers and calculators.
One odd, quirky side-note: it turns out that of the top 5 mathematical formulas/results (as voted on somewhere) 3 come from Euler and the other 2 from Euclid... just an interesting twist that mathematicians with 2-syllable names beginning with "Eu" are responsible for so much powerful math! (...uhh, in the future, please just call me Eushek).




2) No promises that it'll get you thinking as sharply as Euler, but the below post offers a nice primer on mathematical thinking/proof (based on logic and abstract reasoning), which I think may be similar to the approach Keith Devlin takes in his Coursera MOOC course on the same subject (but someone feel free to correct me if the comparison is off-base).

http://amininima.wordpress.com/2013/07/25/how-to-prove-somethin/

"Well-nigh Significant"

P-values are often the brunt of criticism by the statistically-oriented… and here's a chuckle-worthy post indicating why, and how the astute researcher may work around them:

http://mchankins.wordpress.com/2013/04/21/still-not-significant-2/

Your Lovers Have More Lovers Than You Do

"You spend your time tweeting, friending, liking, poking, and in the few minutes left, cultivating friends in the flesh. Yet sadly, despite all your efforts, you probably have fewer friends than most of your friends have. But don’t despair — the same is true for almost all of us. Our friends are typically more popular than we are."                                                                     -- Steven Strogatz

The above was Steven Strogatz's lead-in to his piece on the "friendship paradox" for the NY Times back in 2012:

http://opinionator.blogs.nytimes.com/2012/09/17/friends-you-can-count-on/?_r=0

Presh Talwalkar did a nice treatment of the paradox as well at his blog awhile back:

http://tinyurl.com/lba7qu7

The friendship paradox was first recorded in 1991 by sociologist Scott L. Feld, demonstrating that most people, on average, have fewer friends than their friends have!  It is essentially a result of sampling bias in social networks.

The basic notion can be used to show that, on average, most of your social media contacts as well (Facebook, Twitter, Google+, etc.) have more contacts/followers than you do (assuming your name isn't "Justin Bieber"); or even perhaps (if you care to think about it) that most of your sex partners have had more sex partners than you. In a more serious vein, the paradox has been used to study the course of epidemics, spread through human contact:

“We think this may have significant implications for public health,” said Christakis [Harvard researcher]. “Public health officials often track epidemics by following random samples of people or monitoring people after they get sick. But that approach only provides a snapshot of what’s currently happening. By simply asking members of the random group to name friends, and then tracking and comparing both groups, we can predict epidemics before they strike the population at large. This would allow an earlier, more vigorous, and more effective response.”

From HERE.

Math... Is That All There Is?

"So nature is clearly giving us hints that the universe is mathematical. I’ve taken it to the extreme by proposing that our entire physical reality isn’t just described by math, but that it is a mathematical structure, having no properties besides mathematical properties."  -- Max Tegmark (physicist)

 "There are still some unanswered questions. For example, would the Higgs boson exist if there wasn't the mathematics to describe it? Perhaps this is a question best solved after a few drinks."  -- Brian Butterworth (neuroscientist)

Interesting discussion (among 2 physicists and 2 neuroscientists) from the Kavli Foundation on the 'origins of math' (the word "Platonism" barely even arises, and yet it is once again largely a debate of two different views, Platonist versus non-Platonist). Is mathematics all-and-only in our heads, or is the Universe completely a mathematical structure, regardless of our human presence to recognize it?:

http://www.kavlifoundation.org/science-spotlights/kavli-origins-of-math

(edited transcript)

Tao Tackles Riemann

Honk if you can read and understand this:

http://terrytao.wordpress.com/2013/07/19/the-riemann-hypothesis-in-various-settings/

...I'm pretty much lost from the word "meromorphically" on… :-(

But seriously, Terry Tao has put up a verrrry long entry on the Riemann hypothesis, which no doubt, serious number theorists will be scrutinizing.
For me the most interesting bit comes at the end, with the first comment posted, invoking a dose of Andrew-Wiles-deja-vu! I quote it in its entirety:

"So Terence Tao has posted a blog regarding the Riemann Hypothesis. He notes that his blog is one that makes 'no new progress' on the hypothesis, but later refers to the entry as one in which he is 'simply arranging existing facts together.'
"And THAT, ladies and gentlemen, would be the perfect way for someone as amazing as Tao to provide a subtle, suave introduction to a long series of posts, the culmination of which could be an actual proof to Riemann’s actual hypothesis.
"Has the already-brilliant Terry Tao solved the Riemann Hypothesis? Stay tuned to his quadrant of internet space to find out…."

I can't imagine that the above wishful scenario would be fulfilled… but hey, it never hurts to stay tuned. (And if Dr. Tao schedules a series of 3 lectures at an upcoming conference, well, the room will be packed to the rafters.)

Potpourri

Just another potpourri of bits catching my attention lately:

1) An odd, quirky, but I thought interesting, little blog post here:

http://unimodular.net/blog/?p=453

2) Kiki Prottsman (founder of Thinkersmith ), with a special interest in steering more girls toward computer science, is a great listen in the latest podcast from "Inspired By Math":

http://wildaboutmath.com/2013/07/14/kiki-prottsman-inspired-by-math-31/

3) Will note that Hilda Bastian is moving to the Scientific American blog corral with a blog entitled "Absolutely Maybe" on statistics, epidemiology, and various clinical issues. I like that title… it reflects how I feel about pretty much most "science" these days!:

http://blogs.scientificamerican.com/absolutely-maybe/

4) Off on a tangent, I suspect most math fans either are, or were at some time, interested in chess, in which case you'll find this blog post drawing attention to three of the greatest chess games ever played (from 1851, 1956, and 1999) of interest:

http://mathwithbaddrawings.com/2013/07/17/historys-greatest-chess-matches/

5) And finally, for entertainment (and since I've been seeing a lot of posts pertaining to 'game theory' lately), I'll repeat this clip that Steven Strogatz called "a stunning display of game theory," which I played here just a few months ago (...and which leads inevitably to the deep, pressing cultural question: is the game title of "Golden Balls" as snicker-inducing to a British crowd as it would be to an American audience???):

Happy Tao Day

 Just a quick Happy Birthday to Dr. Terence Tao today:

In honor of his special day, I'll link again to this wonderful old photo he once shared:

https://plus.google.com/u/0/114134834346472219368/posts/fiZbgKv4Yew

And below a couple of interviews I've also previously linked to with Fields Medalist Tao:

http://www.austms.org.au/Publ/Gazette/2009/Nov09/MathMinds.pdf

http://blogs.ams.org/mathgradblog/2009/10/04/an-interview-with-terence-tao/

The Future Will Be Crowd-sourced

Still more from weekend NPR....

I've long touted the value of large-scale collaboration, crowd sourcing and the like that the internet has turned into a reality (100,000 used to be a huge number for collaboration, but in the digital-age millions can take part at once). This week's TED Radio Hour from NPR covered the subject well -- TED Radio Hour is a great offering from NPR, but doesn't yet have the distribution of long-time favorites like This American Life or RadioLab,
so if you missed it the entire hour-long show is here:

http://www.npr.org/programs/ted-radio-hour/

 It includes segments with Jimmy Wales from Wikipedia and author/commentator Clay Shirky, but my favorite segment, which was both fascinating and eye-opening, was from Luis Von Ahn on massive crowd sourcing that you're not even aware you're participating in (you'll learn what's actually going on in the background of those annoying "captcha" spammer-filtering challenges!):

http://www.npr.org/2013/07/12/191620023/can-you-crowdsource-without-even-knowing-it

Give the whole show a listen. The material isn't altogether mathematical, but is so vitally important, so timely, and related enough to things that mathematicians have a hand in, or are behind, that all math buffs ought find it interesting.

Sunday With Steven...

This weekend NPR's "RadioLab" replayed an older piece in which Steven Strogatz joins the regulars to elucidate the well-known game theory conundrum, the Prisoner's Dilemma. If you missed it, listen to Dr. Strogatz's as-always clear narrative focused on Robert Axlerod's solution to the puzzle (from around the 7 min. to the 15 min. point, but enjoy the whole 25 min. piece):

http://www.radiolab.org/2010/dec/14/one-good-deed-deserves-another/

And, what-the-heck might as well make it a Strogatz-Sunday… here's another of my favorite Strogatz-included RadioLab episodes, all about his lifelong relationship/friendship with a high school math teacher named Don Joffray; very touching, and the subject of his earlier book, "The Calculus of Friendship." This is the sort of story, that even if you've heard it before, is still enjoyable (and moving) on the second or third hearing (~16 mins.):

http://www.radiolab.org/2009/nov/30/calculove/

If you're a Strogatz fan you can look up several RadioLab episodes that have included him over the last several years, here:
 
http://www.radiolab.org/search/?q=strogatz#q=strogatz

ADDENDUM: Dr. Strogatz tweets this page for further Weblinks to him:

http://www.stevenstrogatz.com/media.html

Some Weekend Reading

No full reviews, but will just mention/recommend two books I'm reading now… one old, one newer.

I've always enjoyed anything I've read from math-writer David Wells and science-popularizer K.C. Cole, and that goes for these offerings as well. Am halfway through Wells' 2012 book, "Games and Mathematics: Subtle Connections," and that's enough for me to already feel comfortable recommending it to anyone interested in those topics; as one reviewer calls it, "a very approachable yet erudite book." (I will caution that it is not so much a Martin Gardner-style mathematical games book, as a more academic or scholastic approach.)

K.C. Cole usually writes lyrically about physics or general science, so I was pleasantly surprised to stumble across her 1998 volume, "The Universe and the Teacup," and discover it's a volume of essays focused around mathematical notions. A bit dated, and not at all technical, but an enjoyable, accessible, almost lilting compendium of math-related essays for the layperson.
I'll end with a few passages therefrom:

"Curiously, the human brain is a product of exponential amplification. According to Anthony Smith in his book "The Mind," the human brain contains 10 to 15 billion nerve cells -- three times as many as there are humans on the planet. If you add in the number of connections between nerve cells, the sum is more than the number of humans who ever lived. Fifteen billion is also more or less the number of stars in the galaxy.
"The brain built up all this marvelous gray matter by doubling. To get to 15 billion nerve cells takes only thirty-three doublings of the first cell; to get to half that number takes just thirty-two doublings -- which is about the size of the number of cells in the brain of an ape. Our brains are only one doubling away from our simian relatives."

"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."

"In fact, probability permeates just about any attempt to pin down a scientific 'fact.' The fact itself might wiggle away from any precise attempt at measurement, or the measurement (or measurer) might be wobbly or overwhelmed with background static and interference.
"Take a straighforward 'fact' such as my height. Recently asked how tall I was in a doctor's office, I answered five feet five inches. Then I added: 'At least in the morning.'
"The sad truth is, by evening, I've shrunk at least an inch. And so, dear reader, have you. There's no mistaking that gravity gets us down -- everybody equally. Give it a full day to pull us toward the center of the Earth, and we compress like an accordion."

"Math has its own inherent logic, it's own internal truth. Its beauty lies in its ability to distill the essence of truth without the messy interference of the real world. It's clean, neat, above it all. It lives in an ideal universe built on the geometer's perfect circles and polygons, the number theorist's perfect sets. It matters not that these objects don't exist in the real world. They are articles of faith."

70,000,000 to 7000? Impressive!

Articles just keep coming along about the twin-primes work started by Yitang Zhang… this time a good piece from the always layman-accessible Amir Aczel (who it turns out actually worked with Zhang in a university math department a few years back):

http://tinyurl.com/ml6ula2

Mathematicians on the Web taking a collaborative approach to improve Zhang's initial 70,000,000 figure for an upper boundary twin-primes gap, have, in just a couple of months, reduced the figure to barely over 12,000; possibly even 7000 (awaiting some technical verifications)… and at this rate, only a short matter of time likely before even those figures may be improved upon... an amazing and fascinating accomplishment over such a brief time period, on a conjecture that had evaded great progress for decades.
Terry Tao's "progress report" on all of this (for the technically-inclined) was posted less than two weeks ago:

http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/

Jason and Jacob... Unfathomable Minds

"If the human brain were so simple that we could understand it, we would be so simple that we couldn't." -- Emerson M. Pugh

I've previously mentioned the case of Jason Padgett here, but Cliff Pickover recently tweeted links to these 2 stories about the savant/artist, who only acquired his distinctive talents after a severe mugging (it's such a fascinating story, worth re-posting about):

http://www.huffingtonpost.com/2012/04/30/college-dropout-jason-pad_n_1464835.html

http://tinyurl.com/866msww

Examples of his fractal art here:

http://fineartamerica.com/profiles/jason-padgett.html

In at least a slightly related matter, this weekend I skimmed through "The Spark" by Kristine Barnett, a fantastic account of raising her autistic savant son, Jacob ("Jake"), full of touching, powerful, fascinating moments. Anyone interested in autism, savantism, learning, or heck, just raising kids, should read the volume. I've reported on Jake (currently a Masters student in quantum physics, at age 14) here before as well, but just Google him to find lots more information, including videos.

And here's a fuller review of Barnett's wonderful book:

http://articles.washingtonpost.com/2013-05-17/opinions/39327938_1_book-club-alphabet-jake-s

Finally, a bit of recent BBC video (May, 2013) of Jake ...stunning, just stunning!:



Some Kurzweillian thinkers these days believe it is only a matter of time before we will fully understand and even artificially duplicate the workings of the human brain... Jason and Jacob are among the examples that make me, like Emerson Pugh above, doubt we ever will.

Math and Montana

This weekend, NPR's "RadioLab," one of the best programs on all of radio,  re-ran a story on their "Oops" episode ("stories of unintended consequences"), having to do with psychology, but with an odd indirect connection to mathematics at the end. If you didn't hear it, I won't say anymore so as not to spoil the finish, but you can listen at the below URL. The story runs from about the 4 min. point to the 13 min. point, but the entire hour-long RadioLab episode is fascinating as always if you wish to listen in its entirety:

http://www.radiolab.org/popup_player/#

Sunday Recreation (Okidoku)

A piece on recreational grid math puzzles (the Sudoku/Ken-Ken variety):

http://www.edutopia.org/blog/recreational-educational-value-math-puzzles-deepak-kulkarni

...introduces the reader to a relatively new one called OkiDoku (similar, but perhaps more challenging than Ken-Ken).

Here are a few more websites on OkiDoku:

http://anglepointgraphics.org/?page_id=15
http://www.matholympiad.info/Pages/OkiDoku.aspx
http://www.okidoku.org

Done Too Soon...

First, to put you in the right mood (some old Neil Diamond)…



   A short while back, Guillermo Bautista compiled an annotated list of 10 notable mathematicians who died too young (between the ages of 20 and 40):

http://tinyurl.com/mabghvl

The World As We Think Maybe Perhaps Possibly We Know It

Two great science communicators….:

I mini-reviewed Jim Holt's book "Why Does the World Exist?" awhile back at MathTango HERE, and he and physicist Sean Carroll recently sat down for an interesting discussion of the topic below:


Jim Holt and Sean Carroll from ALOUDla on Vimeo.


This is more physics and philosophy than mathematics, but still of potential interest to many math buffs. The conversation is close to an hour long, followed by audience questions, so you'll need to make some time for it, or play it in the background as you do other Webby things.

Unfortunately, they weren't able to settle the question once-and-for-all in the allotted time... ;-)

Wax On, Wax Off...

Another great post (with many good links) from Keith Devlin, on the use of video games for math education, with specific reference to the game "DragonBox" and 'algebraic learning':

http://devlinsangle.blogspot.com/2013/07/it-only-takes-about-42-minutes-to-learn.html

(I feel like he's over-reacting a bit at the start to a magazine headline -- such looseness of language is pretty darn common in the press, generally, and headlines in particular -- but once you get past that part, a lot of interesting food for thought.)

Moreover, I have to love any post that works the original "Karate Kid" (one of my favorite movies), into a discussion of math education!! (via the topic of 'transfer learning').

Tic-Tac-Toe… on Steroids

Just an interesting post on a simple way to turn tic-tac-toe into a more challenging, strategic game:

http://mathwithbaddrawings.com/2013/06/16/ultimate-tic-tac-toe/