Monday, September 30, 2013

Books, Links, Calculus...

I'm busily putting final touches on a long review of Martin Gardner's autobiography [now up HERE], while simultaneously reading Edward Frenkel's new "Love and Math" and Raymond Smullyan's "The Gödelian Puzzle Book" -- don't know if I'll write reviews of either, but have no hesitation recommending both. The latter is typical 'Ray Smullyan,' and perhaps his best, most focused attempt yet to elucidate Gödel's theorems via paradoxical puzzles. The Frenkel book thus far looks wonderful (deep, yet accessible) and I hope will confer (to my naive brain) some sense of what the cutting edge Langlands Program is all about.
For now, I just have time to pass along to readers a few miscellaneous links I've enjoyed the last few days:

1) Just today, the same Edward Frenkel had an interesting piece in Slate on NSA and their 'backdoor' cryptography practices:

2) Another fun read from Simon Singh (on The Simpsons' comedy writers), promoting his new book:

3) And finally, recently Steven Strogatz enthusiastically tweeted a link to this intro for Robert Ghrist's 1st-year calculus course (from Coursera):

The reason I provide this link at all is because quite awhile back an emailer asked me to recommend a video site on the Web for learning first-year calculus, and even though I'm aware of many, I wasn't confident endorsing any particular one. When I asked readers if they had definite recommendations I got no response. But if Steven Strogatz is willing to give a thumbs-up to this one, I trust his judgment!
One thing that makes it look interesting, beyond the quality of graphics put into it, is Ghrist's unconventional use of Taylor series (which usually come at the end of 1st year calculus), near the beginning of the course.

Saturday, September 28, 2013

Weekend Reads

Just a couple of good reads for the weekend….

For any who missed it, this somewhat low-key, yet fascinating (and non-technical) interview with Yitang Zhang ('rags-to-riches/fame' worker on the twin-prime conjecture) last week in Nautilus:

Secondly, a Scientific American excerpt from Edward Frenkel's new book, "Love and Math: The Heart of Hidden Reality" (now showing up in bookstores):

Friday, September 27, 2013

Thinking Caps...

Couple puzzles for today (one easy, I think, one less-so):

1) Since I brought up Raymond Smullyan in the last post, the first comes from him, adapted from the "Mathematical People: Profiles and Interviews" book I've also  mentioned earlier. This seems surprisingly easy for a Smullyan puzzle, but perhaps it's one of those tricky ones that some people see through right away and others have more difficulty with:

100 politicians attended a certain convention. Each politician was either crooked or honest. The following two facts are known to be true:

a. At least one of the politicians was honest.
b. Given any two of the politicians, at least one of the two was crooked.

From these two facts can it be determined how many of the politicians total were honest and how many were crooked?

2) The 2nd problem I've adapted from a recent Matt Parker tweet:

It's possible to arrange all the numbers from 1 to 17 so that every adjacent pair adds up to a square number (...every number except the two end numbers will be part of 2 adjacent pairs, and each pair sums to a square no.).
.answers below
1st answer:  99 politicians must be crooked and 1 honest (to insure the two conditions are met).

2nd problem:  there may be more than one answer, but the first one I arrived at runs like this:
1, 15, 10, 6, 3, 13, 12, 4, 5, 11, 14, 2, 7, 9, 16, 8, 17
...see first comment below for a correct solution!

Thursday, September 26, 2013

Still Paradoxical After All These Years

When it rains it pours... Feel like I'm inundated with book news these days… probably feel that way, because I AM!!

Brand new from 94-year-old youngun' Raymond Smullyan "The Gödelian Puzzle Book: Puzzles, Paradoxes and Proofs" -- Gots to be good! Just don't know when I'll find time to work through it. The one reader-review thus far up at Amazon (no doubt written by his mother ;-)) reads as follows:
"Each time I think that Raymond Smullyan has reached his upper limit, he produces a book even more amazing, wondrous, and stupendous. He is a boundless source of creativity and ingenuity, and not even his advanced years deter him in the slightest. His latest "Gödelian Puzzle Book" is a true masterpiece - a mixture of humor and brilliance which entertainingly bares the very mind and soul of the eminent logician Kurt Gödel. I can not recommend this book enough!"
As I've written before, Ray Smullyan is another American gem on par with Martin Gardner... speaking of which, my first take (will say more later) on Martin's wonderful new autobiography is now up over at MathTango:

Wednesday, September 25, 2013

Mac Winners!

Twenty-four 2013 MacArthur 'genius grant' recipients have been announced, including one "statistician," one "computer scientist," and one "astrophysicist":

Winners get $625,000 (over 5 years) to do with what they wish in pursuit of their creative endeavors.

Last year, mathematician Maria Chudnovsky won one.

Congrats to all!

Tuesday, September 24, 2013


Nothing like Christmas in September!!
I realize that for some of you, shiny objects, or 72-inch TVs, or items that house 8-cylinder engines are examples of great presents, but for me nothing beats finding a delightful hitherto-unfamiliar math book in a thrift store for a couple of bucks (…ok, so maybe I'm kinda different). Such was my luck this past weekend when I stumbled upon a 1985 MAA volume, "Mathematical People: Profiles and Interviews," covering 25 mathematicians, including some I'd not heard of, in addition to ones I already revere. The interviews and profiles are fantastic… I could probably find enough quotes in this volume to inspire a year's-worth of blog posts.
Of course I read the interview with Martin Gardner first, and will note this fanciful passage that has nothing to do with mathematics, but does indicate Gardner's quirky, self-deprecating side (as so much of the interview does)...
The questioner has asked Gardner about one of his lesser known works, "The Annotated Casey At the Bat," and Martin replies as follows:
"Oh, I have always been interested in the fact that there are poems that are not great poetry but seem to outlast the entire poetic output of poets who were very famous in their day. I guess the best term for them is popular verse. They don't pretend to be great poems and yet a single poem written by an individual like Thayer, who wrote 'Casey At the Bat,' can go on and on and on, and everybody knows about it. It will probably be remembered after everybody's forgotten every poem ever written by, say, Ezra Pound. This has always struck me as a very curious phenomenon. So I did an article on the history of "Casey At the Bat" that I sold to "Sports Illustrated." That was how it started. After it appeared, it occurred to me that I might put together an anthology of sequels to Casey. That's what the book is, a collection of the original poem with sequels by various other people. Then I annotated all the poems with sort of fake annotations to tie them all together in a connected story. The book is done as kind of a joke. It didn't sell too well either."
Wonderful! ...and "a very curious phenomenon" indeed. Of course the entire 8-page interview is wonderful and covers a lot of ground.

As best I can tell, this volume was just the first of 2 or 3 that were promoted by MAA over the years, interviewing mathematicians of note.

Anyway, as to the synchronicity... I read the Gardner chapter and a couple of others on the weekend, and then yesterday (Monday) afternoon turned to the chapter on Persi Diaconis, who is likely familiar to many of you. I was aware of Persi but had little knowledge of his background... his story/interview almost reads like a novel it is so entrancing... I won't even try to tell it here, I couldn't do it justice, but great stuff! (and you're in luck, the whole volume is online as a pdf here: with Diaconis's interview beginning on page 58; Gardner's begins on pg. 86, but every profile worth reading).
Blown away from reading Diaconis's life-path, I put the book down to go retrieve my daily mail, and there in the mailbox was a plain brown manila package... it might as well have been crimson-wrapped with green ribbon and sparkles! -- it was a review copy of Martin Gardner's brand new autobiography... with Foreword written by... taa daaa!... Diaconis. The volume is a short, delicious 200 pages -- I'll no doubt have a review up at some point over at MathTango (though it's almost redundant and trite for me to write one given my bias for all things Gardner!).

I'm not sure that Martin would put much weight in "synchronicity" (at least of the Jungian sort)... but as for myself, well, I swear I'm feelin' some v-v-vvvibes here! :-)

(image via License)

Monday, September 23, 2013


If you're a Simpsons fan (and you darn well oughta be!) then the below post is for you, whether you relate most to Homer, Marge, Bart, Lisa, or none of the above. The secret nerdiness of one of the most popular, long-running shows in the history of TV is 'outed' by Simon Singh in his new mathematical expose' ("The Simpsons and Their Mathematical Secrets") of that all-American family. Wonderful longish post below hints at how delightful the book is (…really, one of the most fun articles I've read in awhile):

Here's a bit therefrom:
"My favourite freeze-frame gag appears in "The Wizard of Evergreen Terrace" (1998), in which Homer tries to become an inventor. In one scene, we see him busily scribbling equations on a blackboard. One of the equations relates to the mass of the Higgs boson, another concerns cosmology and the bottom line explores the geometry of doughnuts, but the most interesting equation is the second one, which appears to be a counterexample to Fermat's last theorem.

"Although it was only on screen for a moment, this equation immediately caught my eye, because I have written a book on Fermat's last theorem. Homer's scribble sent a shiver down my spine. I was so shocked that I almost snapped my slide rule…
[Singh goes on here to explain Fermat's Last Theorem to the reader before continuing]:

"...Homer's blackboard proves the opposite!

3987^12 + 4365^12 = 4472^12.

Check it for yourself on your phone calculator and you will find that the equation balances!"

[The problem famously lies in the error factor of the digital output for a handheld calculator; i.e. Homer does NOT disprove Fermat.]

Sunday, September 22, 2013

Talk, Talk, Talk

On the heels of my transcribed interview with Dr. Colm Mulcahy, Sol Lederman now has a nicely-detailed podcast with Colm up for "Inspired By Math" covering a lot of ground: 

And simultaneously, fresh over at MathTango is my new interview with... well..., drop in and see for yourself:

Finally, below, a little Sunday meditation, this time from David Wells' volume, "Games and Mathematics: Subtle Connections":
"The vast landscape of modern mathematics guarantees that most mathematicians will only be at home in a small corner, and must lack the deeper understanding needed to make very delicate aesthetic judgements about regions of which they know little.
" 'Which Is the Most Beautiful?' was a questionnaire for readers of the 'Mathematical Intelligencer'  designed to test the plausible proposition that mathematicians no more agree on their judgements of beautiful mathematics than art lovers agree on favourite paintings or music lovers agree over Mozart and Beethoven. The items were chosen to be elementary, so that all respondents would be more-or-less familiar with them...
"It worked well! The introduction quoted several classic comments such as John von Neumann's that, 'I think it is correct to say that [the mathematician's] criteria of selection, and also those of success, are mainly aesthetical,' and Poincare's comment on this phenomenon: 'It is true aesthetic feeling which all mathematicians recognise. The useful combinations are precisely the most beautiful.'  In other words, the beauty of mathematics is not an add-on, it's not a bonus, and attention to beauty is not an option but an essential feature of mathematical creativity."

Friday, September 20, 2013

Listening To Devlin

Yet another beautiful explication of what mathematics is by math 'evangelist' Keith Devlin, this time on the NPR show "On Being" with Krista Tippett (podcast). Keith has covered this ground before, but, as his view is so different from views frequently held regarding mathematics, very much worth taking in again (50+ mins long). Includes talk of "mathematical thinking," the connection between math and music, Euler's identity, the Pythagorean theorem, MOOCs, some history, and more.

With his British accent I could probably listen to Keith read the New York City telephone directory and be entertained for an hour ;-) ....50 minutes of math discussion just flies by:

Thursday, September 19, 2013

"The Heart of Hidden Reality"

Two great math memoirs in one month!... Martin Gardner's autobiography should be showing up in bookstores shortly, and already University of Calif./Berkeley mathematician Edward Frenkel's book "Love and Math" is popping up as well. The title of this post, is the subtitle of the book. Peter Woit has a positive review here:

Frenkel is a leader among those studying the "Langlands program" and mathematical physics.

From Peter's review:
"What’s really wonderful though is his dedication to the cause of the opposite of obscurantism, that of doing the hard work of trying to explain mathematical insights to as wide an audience as possible. His book is packed with mathematics and physics, full of enlightening explanations of difficult topics at all different levels of mathematical sophistication.
"Perhaps the most remarkable part of the book though is the way it makes a serious attempt to tackle the problem of explaining one of the deepest sets of ideas in mathematics, those which go under the name of the “Langlands program”….
"I heartily recommend this book to all with an interest in mathematics or its relation to physics. If the 'Love' of the title has you hoping for a tale of romance between two people, you’re going to be disappointed, but you will find something much more unusual, a memoir of the romance of mathematics and its relation to the physical world."
...and if that's not enough to get you interested, read the scintillating reviewer blurbs at the Amazon page linked to above (which affords you about 20 pages of free reading, plus ~40 pages of endnotes!).

Wednesday, September 18, 2013

"Computer Based Math"

A new education effort calling itself "Computer Based Math" appears to be taking a foothold. It is initially backed by Conrad Wolfram (and Wolfram Research) who has argued for some time now for a sort of revolutionary computer-based approach to math education, wherein calculating is downplayed in favor of learning coding and information technology applications. It is to be an international effort, and they are joining forces with UNICEF as announced here:

main page for the organization:
with an "about" page here:

They state that what is needed is "...a fundamental change to the school subject we call math. It needs to be clearly articulated and decisively acted upon. That's why Conrad Wolfram has founded He and many others see a growing chasm between math in education and math outside, between the increasingly irrelevant school math curriculum that contrasts with the critical and growing importance of math and its uses in the real world. They've observed how many of those involved in school math fail to appreciate the total transformation and fundamental change that computers have brought to this ancient subject in recent decades."

The website appears young at this point, rather thin on details and speaking in generalities, but probably worth following as things progress.

They have a Twtter feed (@ComputerMath) here:

(image from Open Clip Art Library)

Tuesday, September 17, 2013

Happy Birthday Bernhard!

It's Bernhard Riemann's birthday today... or would be if he were around to celebrate his 187th.
Died at just 39, in his "prime," and never having a chance to prove the hypothesis that bears his name... and earn a million dollars! ;-)

(image via "Dark Meadow"/Wikimedia)

Sunday, September 15, 2013

Luck o' the Irish...

If you know Colm Mulcahy don't dilly-dally but hop on over to MathTango pronto for a wonderful lengthy interview with him... and, if you don't know of Colm, all the more reason to click on over and learn about this mathemagical professor:

Thursday, September 12, 2013

A Couple of Popularizers

First, a quick note that Brit Richard Elwes, one of my favorite popular math writers, has a new volume out "Maths In 100 Key Breakthroughs":

I'll note that Elwes' books, published in Britain, unfortunately are not always readily available in the U.S. right away, and sometimes show up at a later date, under a different title! (but worth keeping an eye out for)

Speaking of favorite math popularizers, I've had occasion to think about Martin Gardner lately, and so will re-run one of his classic puzzles that I used here a couple years back -- am quoting it directly from his "The Jinn From Hyperspace" volume:
"Now for a final paradox. There is a certain event that I guarantee will or will not take place during the next ten minutes. You are absolutely  incapable of predicting correctly whether it will or won't occur. I don't mean that it's unlikely you can predict it. I mean it is logically impossible to predict it!

"You don't believe it? Then do the following. If you think the event will occur write 'Yes' inside the blank rectangle below. If you think it won't happen, write 'No' inside the rectangle.
"If you predicted correctly, I'll send you a million dollars.

"The event is: You will write 'No' inside the rectangle."  

A wonderful paradox/conundrum entangled with self-reference, causation/prediction, and human language and logic.
I should have a bit more to pass along about Martin in an upcoming post at MathTango, but for now you can read the current post  up there about "skepticism." 

Tuesday, September 10, 2013

P vs. NP vs. NSA

Since the press leaks regarding NSA subversion of online encryption through various means, dicey articles (and forum discussion) have begun speculating as to whether or not the agency may be working with a quantum computer, or even might have already proven P = NP (but for obvious reasons would never disclose such a proof -- cue in "Travelling Salesman" movie). Not much to back up such speculation… but always fun guessing, I s'pose. Here's one 'popular' piece (focused on RSA encryption, no actual mention of P vs. NP):

And for those up to more of a challenge, here's a recent take from the always interesting, excellent Scott Aaronson:

Monday, September 9, 2013


I've reported here on young genius Jacob Barnett before... here again is his incredible story and current status at Santa Fe's Perimeter Institute:

Sunday, September 8, 2013

"Genius Recognizing Genius"

83 Years ago yesterday...:
"…the steps leading up to Gödel's startling conclusions are both logically tricky and intricately intertwined…
"An indicator of the degree to which Gödel's results were unexpected can be found in the reaction to his original announcement of the theorem at a philosophy-of-science symposium in Königsberg, Germany, on September 7, 1930. Ironically, Königsberg happened to be Hilbert's hometown, which perhaps partially accounts for the lukewarm reception given to Gödel's presentation of his results. In fact, the transcript of the discussions at the meeting gives no indication whatsoever of Gödel's remarks, and there is no mention of Gödel at all in an article published later summarizing the papers given at the meeting! So like many belief-shattering ideas, Gödel's appears to have been so unexpected and revolutionary that even the professionals didn't at first understand what he had accomplished.  But one participant who did see immediately the implications of the work was John von Neumann, who cornered Gödel after his talk and pressed him for more details -- a case of genius recognizing genius, I suppose."
          -- from John L. Casti's "Searching For Certainty"

and the rest, as they say, is math-logic-philosophy history....

Saturday, September 7, 2013

Math Enlightenment...

This morning, Steven Strogatz tweeted the following:

"Huh? Marilyn vos Savant, of Monty Hall problem fame, bashes math … Say it ain't so!"

which struck me as a little odd, since I'd read the same piece and thought it a fairly innocent (even if simplistic) commentary on math's place within most people's fields of study. She was responding to someone's sincere question as to why they (like a lot of people) suffer from so much math anxiety. And in fact I find Marilyn's suggestion that we start math education earlier and stop it earlier (except for those who will go on to use it professionally), interesting.
But some other Twitter responders were critical of her take, the most button-pushing lines possibly coming when she wrote: "Math doesn’t enlighten us the way literature, social studies, or art appreciation do. Instead, it’s an extremely valuable tool that many of us simply don’t need to use much." This is unfortunately ambiguous, since some may be reading it to mean that 'math doesn't enlighten us, while other fields do,' when I think she more likely simply meant that math often doesn't enlighten people in the same manner that some other areas do (certainly math enlightens most people at some level, at some time; physics also doesn't enlighten people in the same way, say that music does -- there are different forms of 'enlightenment').

Anyway, probably the more important take-away notion from the piece is her belief that math phobia is not an inherent trait suffered by some individuals so much as a by-product of the educational system… and that should be fixable. She wasn't so much 'bashing' math as bashing the pedagogical system that renders math to very young minds (...and these days who doesn't bash that system!).

But enough verbiage, read the piece yourself (it's not long), if you haven't already, and see what you think...

-- ADDENDUM: It occurs to me this might be a good place to bring in two links I'd saved but not yet used here, for Keith Devlin's ( new "Wuzzit Trouble" game:

and a review from "The Aperiodical":

Dr. Devlin, through InnerTube Games, has been arduously working on the conviction that young minds can learn mathematical concepts and thinking through games… essentially learning math without even realizing you're learning (or being taught) math.

I suspect Marilyn would approve….


Friday, September 6, 2013

Timing Is Everything...

What if (against almost everyone's expectations) P actually equals NP???....

With fresh reports of the NSA cracking/hacking of internet encryption in the news, probably a good time for the award-winning, mathematical thriller* "Travelling Salesman" movie (about the P vs. NP problem) to be making its worldwide release... and... lo-and-behold it is!:

reviewed a bit here:

and interview with the director here:

* yes Virginia, there is such a thing ;-)

Thursday, September 5, 2013

Prime Entertainment

A few days back The Aperiodical nicely summarized some of the major prime number conjectures (including Wieferich primes which I'd not heard of):

And if you haven't seen it, enjoy passionate Aussie Adam Spencer's TEDTalk ode to BIG prime numbers (gets better and better as it goes along):

Wednesday, September 4, 2013


Feels like Christmas in September!....

Yesterday, at 7:40 pm EST ;-)) Princeton University Press tweeted the link to Martin Gardner's autobiography with a note that the first chapter ("Earliest Memories") was available (pdf) for reading over the Web (gloriosky!!!):

It's delicious Gardner with a few vintage memories from childhood... and 20 more chapters to look forward to… (the volume should be showing up in bookstores later this month).

ADDENDUM: has just put up a review here:

Tuesday, September 3, 2013

Form and Balance...

 Sometimes I have to post something for sheer enjoyment and reflection... so, moving from yesterday's cerebral, philosophical realm to the less-mathy simple visual below (I dare say there is some math intrinsically involved, as math underlies everything!):
 Several examples of the art of "Sanddornbalance" have appeared on the Web (it was actually created back in 1996), but for any who've missed this beautiful, breath-holding act, enjoy:


(feather image via Hariadhi at WikimediaCommons)

Monday, September 2, 2013

Laboring Over Formalism

Happy Labor Day to US readers... and here's a little reading you can labor over:

A piece (not sure how old it is???) from non-Platonist Timothy Gowers on philosophy and mathematics, focusing on Platonism, logicism and formalism (good stuff, BUT ONLY if you're already inclined toward philosophical underpinnings):

It's brimming with interesting ideas, including (in the "#6 Truth and Provability" section) the notion that somewhere in the decimal expansion of pi there ought surely be a string of a million 7's, on the basis of it being a "normal number."

Here's a little bit of his wrap-up to the longread:
"...the point remains that if A is a mathematician who believes that mathematical objects exist in a Platonic sense, his outward behaviour will be no different from that of his colleague B who believes that they are fictitious entities, and hers in turn will be just like that of C who believes that the very question of whether they exist is meaningless...
"So why should a mathematician bother to think about philosophy? Here I would like to advance a rather cheeky thesis: that modern mathematicians are formalists, even if they profess otherwise, and that it is good that they are...

"When mathematicians discuss unsolved problems, what they are doing is not so much trying to uncover the truth as trying to find proofs….

"I also believe that the formalist way of looking at mathematics has beneficial pedagogical consequences. If you are too much of a Platonist or logicist, you may well be tempted by the idea that an ordered pair is really a funny kind of set -- the idea I criticized earlier. And if you teach that to undergraduates, you will confuse them unnecessarily. The same goes for many artificial definitions."