Monday, March 30, 2015

Book Matters

Don't know if I'll get around to fully reviewing any of these volumes, but will make note of a few books today:

1)  First, happy to see that Matthew Watkins has a new publisher and new publicity for his delightfully wonderful trilogy on prime numbers (I loved the first volume, but haven't actually gotten around to the other two yet):

Volume 1 reviewed here:

2)  A h/t to Richard Elwes for pointing out this book review of Cédric Villani’s "Birth of a Theorem"... unlike almost any book review you've previously read:

...and a couple more reviews of the same volume below:

3)  Meanwhile, just became aware that one of my favorite writers, William Byers, was out with a new book at the end of last year, "Deep Thinking: What Mathematics Can Teach Us About the Mind." Here's one review:

Though a trained mathematician, Byers tends to be both philosophical and psychological, and if you enjoy exploring the cerebral foundations of mathematics, or thinking about mathematical thinking, I always recommend his work.

4)  Finally, the one book I am currently reading/enjoying is Ian Stewart's latest, "Professor Stewart's Incredible Numbers." With over 30 popular math books to his credit, any new offering falls prey to a certain amount of redundancy, but if by chance you don't have (m)any Stewart books this is a great one to start with. The Professor has pulled together a fantastic and wide range of topics to refresh the engaged mathematician or stir the mind of the younger person on their mathematical path. Both text and illustrations are excellent. While it certainly includes some recreational math (often a mainstay for Stewart), the book really is more of an instructive adjunct guide to a high school or basic college math text. And it's released in paperback so the price is right. My only small issue with it is that it lacks an index (is this a new trend, I've run across several books in the last year foregoing indexes???) which makes it impossible to find all instances of mention of a topic that one may be keenly interested in.

Anyway, it's not even quite April yet, and there are already so many good math books out there... 2015 appears to be another banner year for numbers fans.
I've already reviewed two books over at MathTango this year that will be among my year's favorites:

Sunday, March 29, 2015

Of Engineers and Mathematicians

Okay, a little different 'Sunday reflection' this morning.... ;-)

A mathematician and an engineer capsize a canoe and end up on a deserted island. They quickly notice that there are just two palm trees on the island and each bears one lone coconut at the top. The engineer doesn't hesitate to climb up one palm, grab the coconut, and eat it. Seeing this, the mathematician scurries up the other tree, grabs the single coconut, shimmies back down, and climbs up the first tree, replacing the now missing coconut. Then he climbs back down and, with a wide grin on his face, proudly announces to his engineer friend, "Now we've reduced the situation to a problem we already know how to solve."

-- old, old math joke

Wednesday, March 25, 2015

Of Jocks and Graph Laplacians

I first read the story via Jason Rosenhouse here:

...just a brief post about a journal article (that you may have been awaiting... or, NOT): "A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians" by mathematician John Urschel... who, just happens to play pro lineman for the Baltimore Ravens football team.
Who knew!?...

The story has received some significant coverage, including this audio piece at NPR:

also, from Huffington Post:

and at Bloomberg news:

Urschel's Erdös number, by the way, is now 4 (rather better, I dare say, than Peyton Manning's).

Monday, March 23, 2015

Flippin' Probabilities

I suspect we all get a kick from mathematical problems that lead us merrily down wrong intuitive paths; statistics is often a source for such misdirecting problems. Mike Lawler pointed to a good one over the weekend. The problem arises in an old Peter Donnelly TEDTalk (below), starting at about the 4-minute mark (but his entire 22-minute talk is definitely worth a listen). The problem is simply this:

When flipping a fair coin repeatedly (heads/tails), which 3-part sequence is more likely to appear first:  HTH or HTT?
i.e., In a running sequence, is HTH likely to show up before HTT, or HTT before HTH, or, over many trials, are the probabilities equal?

MANY people jump to the conclusion that the two possibilities are equally likely, no doubt thinking in terms of the true equal probability of each single flip being an 'H' or a 'T.'
BUT NO, the somewhat surprising answer is that the HTT sequence is more likely to occur first, and the explanation Donnelly gives (again starting at ~4-min. point), has to do with the 'clumping' of overlapping 3-part chunks:

It may be worth noting that one could identify 3-part sequences that DO have equal probability of occurrence, as well as additional sequences, which like the above two, have unequal probabilities.

Donnelly doesn't really expound too fully on the explanation for the outcome, so you may wish to follow along Mike Lawler's videos (linked to above), or read up on "Penney's Game" (named after creator Walter Penney) which relates to all this:

The always-entertaining "Scam School" once did a 15-min. episode on Penney's Game here:

Sunday, March 22, 2015

Computers and Math

"Can computers be used to prove mathematical theorems beyond being a computational aid? Doron Zeilberger of Rutgers University in New Jersey claims that the answer is yes. Moreover, he claims, the computer can reveal mathematical facts outside human reach... Computer programs that operate  on symbolic expressions have existed for many years. Zeilberger used these programs to prove important identities in algebra and used a computer to reveal new identities. He valued the computer's contribution so highly that he added the computer as a coauthor of some of his scientific papers. The computer is named Shalosh B. Ekhad... (At the time of writing these lines, there are twenty-three papers listed in Shalosh B. Ekhad's list of publications, and it has cooperated with thirteen authors.) Beyond the healthy humor, I think there is something basic in this approach. Zeilberger claims, and that claim cannot be ignored, that the day will come when computers will reveal mathematical theorems that will be difficult for humans to understand."

-- Zvi Artstein (from "Mathematics and the Real World")

p.s... for a different style quote to start your day check out Futility Closet's lovely offering this morning from Lewis Carroll:

 p.p.s... also, please note a new interview up over at MathTango this morning.

Thursday, March 19, 2015

Puzzle Redux

I don't mind re-running posts or puzzles that I've posted before that are personal or reader favorites, so below is one from almost 3 years ago that still gets hits each month. (It's the sort of puzzle that Rick Kurshen just might love. ;-))
I'd already seen it multiple times on the Web when I first ran it (though I don't know where it originated???), so I realize many of you have seen it already, but if not, give your brain a whack at it, if you dare!

[The general consensus from those who have coded the problem and run it, is that there is one lone correct answer, which I'll eventually insert in comments if no one else does.]


Given the following 12 statements which of the statements below are true?

1.  This is a numbered list of twelve statements.
2.  Exactly 3 of the last 6 statements are true.
3.  Exactly 2 of the even-numbered statements are true.
4.  If statement 5 is true, then statements 6 and 7 are both true.
5.  The 3 preceding statements are all false.
6.  Exactly 4 of the odd-numbered statements are true.
7.  Either statement 2 or 3 is true, but not both.
8.  If statement 7 is true, then 5 and 6 are both true.
9.  Exactly 3 of the first 6 statements are true.
10.  The next two statements are both true.
11.  Exactly 1 of statements 7, 8 and 9 are true.
12.  Exactly 4 of the preceding statements are true.


Tuesday, March 17, 2015

Encryption and Uncertainty...

Two tidbits I encountered in the last 24 hours that I found interesting, and so am passing along (perhaps, in some odd way, there's even a tiny thread of linkage between the two!??):

1)  RSA encryption researchers make "an astonishing find" (essentially that encryption keys aren't nearly as random as one might expect):


2)  Ed Frenkel advocating the importance of human uncertainty in science (I couldn't agree more) -- even scientists who acknowledge uncertainty, nonetheless often fall into the trap of assuming certainty about specific matters. The epistemological nature of uncertainty is difficult to internalize (I think) because we must operate in our daily lives as if we are certain of a great many things... but this ought not detract from the importance of recognizing uncertainty in a scientific perspective... anyway, read Ed and the Facebook discussion that follows (be sure to click on "See More" to expand Ed's full initial response):

p.s.  -- the Lawrence Krauss New Yorker piece that is referenced (and inspired Ed's response) is here: 

Monday, March 16, 2015

Badass Mathematicians... and Blogs

Here's another fun question from the Quora mathematics site -- "Who is the most badass mathematician ever?":

Fun reading... All the 'nominations' are pretty much who you would expect, but still interesting to see how people make a specific case for their individual choices -- little mini-profiles of the greats of mathematics... EXCEPT, to my utter dismay (unless I missed it), haven't seen anyone make the case for Bernhard Riemann yet! C'mon folks, the man founded perhaps THE most important, far-reaching, unsolved, incredible conjecture in all of mathematics, still being pursued after 150+ years, and nary a mention... show a little respect for your elders!! ;-)

Meanwhile "Grey Matters" just celebrated their 10th Blogiversary (a major achievement, and on Pi Day, no less) and, in celebration, posted links to many of their best posts over the years. Check it out:


Sunday, March 15, 2015

God, Is She a Geometer?...

"Our minds may indeed be just swirls of electrons in nerve cells; but those cells are part of the universe, they evolved within it, and they have been molded by Nature's deep love affair with symmetry. The swirls of electrons in our heads are not random, not arbitrary, and not -- even in a godless universe, if that is what it is -- an accident. They are patterns that have survived millions of years of Darwinian selection for congruence with reality....
"Perhaps we have created a geometer God in our own image, but we have done it by exploiting the basic simplicities that nature supplied when our brains were evolving. Only a mathematical universe can develop brains that do mathematics. Only a geometer God can create a mind that has the capacity to delude itself that a geometer God exists.
"In that sense, God is a mathematician; and She's a lot better at it than we are. Every so often, She lets us peek over her shoulder.

-- Ian Stewart in "Letters To a Young Mathematician"

[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know ( If I use one submitted by a reader, I'll cite the contributor.]   

Saturday, March 14, 2015

Bah Humbug...

For we Pi Day curmudgeons, Vi Hart comes through as usual:

(ohh, but, hey,  Happy Birthday to Albert Einstein... now that's something I can celebrate... with, a piece of pie perhaps!)

Friday, March 13, 2015

Mathematicians Drunk on Moonshine... so-to-speak

 “Math is all about building bridges where on one side you see more clearly than on the other. But this bridge was so unexpectedly powerful that before you see the proof it’s kind of crazy.” -- John Duncan (Case Western Reserve)

Linking number theory and physics... Fantastic Erica Klarreich piece for Quanta Magazine on the Monster group, j-function, Monstrous Moonshine, Kac-Moody algebra, string theory, serendipity in math, and more... these are the sort of connections that are almost spine-tingling, they're so spooky (in a positive way):

Wednesday, March 11, 2015

Answer Me This...

Just a couple of interesting queries on Quora recently:

1)  "What is the largest non-integer that naturally occurs in mathematics?"

You can see the top answer ("Ramanujan's constant"), and others, here:

and, even more broadly,  
2) "What is the most intriguing mathematical concept you have ever encountered?"

Of course many good selections for this one, though I was a bit surprised to not see one of my own favorites, "Cantor's dust" or "Cantor Set," mentioned in the responses when I last leafed through them:

Sunday, March 8, 2015

Doing Mathematics (What It's Like)

I originally posted this last year when MAA first uploaded it to YouTube, but will revisit it now as a nice Sunday reflection via James Tanton, talking about the experience of doing mathematics:

Wednesday, March 4, 2015

Sunday, March 1, 2015

Black Swans…

Today's Sunday reflection comes from the "Prologue" to Nassim Taleb's "The Black Swan":

"Before the discovery of Australia, people in the Old World were convinced that all swans were white, an unassailable belief as it seemed completely confirmed by empirical evidence. The sighting of the first black swan might have been an interesting surprise for a few ornithologists… but that is not where the significance of the story lies. It illustrates a severe limitation to our learning from observations or experience and the fragility of our knowledge. One single observation or experience can invalidate a general statement derived from millennia of confirmatory sightings of millions of white swans….
"I push one step beyond this philosophical-logical question into an empirical reality, and one that has obsessed me since childhood. What we call here a Black Swan is an event with the following three attributes.
"First, it is an outlier, as it lies outside the realm of regular expectations, because nothing in the past can convincingly point to its possibility. Second, it carries an extreme impact. Third, in spite of its outlier status, human nature makes us concoct explanations for its occurrence after the fact, making it explainable and predictable.
"I stop and summarize the triplet: rarity, extreme impact, and retrospective (though not prospective) predictability. A small number of Black Swans explain almost everything in our world, from the success of ideas and religions, to the dynamics of historical events, to elements of our own personal lives. Ever since we left the Pleistocene, some ten millennia ago, the effect of these Black Swans has been increasing. It started accelerating during the industrial revolution, as the world started getting more complicated, while ordinary events, the ones we study and discuss and try to predict from reading the newspapers, have become increasingly inconsequential"…..

"…I stick my neck out and make a claim, against many of our habits of thought, that our world is dominated by the extreme, the unknown, and the very improbable (improbable according to our current knowledge) -- and all the while we spend our time engaged in small talk, focusing on the known, and the repeated. This implies the need to use the extreme event as a starting point and not treat it as an exception to be pushed under the rug."

[Meanwhile, over at MathTango this morning a new interview with one of my favorite current math writers, Richard Elwes.]