Wednesday, March 4, 2015

Primes, Zeta, Riemann, Oh My (from Marcus du Sautoy)

Love this podcast (~28 mins.) with Marcus du Sautoy on prime numbers, the zeta function, Riemann Hypothesis, etc.:

Monday, March 2, 2015

Geometry Fun

Just a few past Futility Closet geometry puzzles to get your week rolling:

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.]


Thursday, February 26, 2015

...Or, You Can Major in Art History (just sayin')

OK, next time, one of your students asks, "When am I ever gonna use this stuff?" perhaps this tidbit will help inspire them....

Recently, on the Quora website someone asked "Who is the wealthiest math major on Earth?".
The top answer (last I looked), drawn from a Forbes billionaires listing, cited these four gents:

Sergey Brin -- $29 billion,  BA in mathematics and computer science from the Univ. of Maryland
Steve Ballmer -- $21.6 billion, BA in applied mathematics and economics from Harvard
James H. Simons -- $14 billion, BA in mathematics from MIT and PhD from Berkeley
Andrew Beal -- $11.7 billion, studied math at Michigan State but dropped out before graduating

...just something to shoot for.

Wednesday, February 25, 2015

A Beach Read for the Middle-of-Winter

Like many folks, "big data" at one time interested me, but I became bored with it a year or more ago as it seemed waaaay over-hyped, imprecise, and not living up to promises/claims.  I've avoided most of the popular books out on "big data" waiting for the field to catch up a bit to its vaunted notoriety. Still, having seen a lot of positive reviews of Christian Rudder's "Dataclysm," I pulled it off a library shelf not long ago, and scanned the first 20+ pages... and, he hooked me!
I won't really review the volume, but here are some longer reviews for those wanting such:  (by Jordan Ellenberg)  (a more critical review from Cathy O'Neil)

Christian Rudder was a co-founder of the dating site OKCupid, from which much of the info/data for this book derives (is that a "representative" sample of humanity?). But he's also worn many other hats, and additionally has a mathematics degree from a place called "Harvard" i.e., he's an earnest mathematician/data-analyst. I like that he doesn't seem to take either himself or his subject matter too seriously here, and admits he stayed away from a lot of the deeper possible math to write an engaging book for a general audience.
I view this as more of a recreational book than a terribly serious math, or even human behavior, treatise, and probably am more hesitant than Christian to reach some of the conclusions that are speculatively drawn. But still, plenty of interesting tidbits in a very fun, entertaining package, with a lot of grist for hearty discussions/debates among friends... I can definitely recommend it, if only as, what Steven Strogatz calls it on the back-cover, "a guilty pleasure." Human behavior, is after-all, inherently curious and fascinating, and Rudder does have a wealth of data to draw from, lending us a fun, ultimately hopeful (perhaps Pollyannishly-so?) treatise on the future of "big data."

If it were the right time of year, I would call this book a beach-read for math geeks, but since it's the middle of winter I'll just call it a great weekend read as long as you don't take the conclusions too seriously or uncritically. For now, 'big data' is still in its infancy... perhaps even its gawky, 'terrible twos' stage so-to-speak, but it is advancing rapidly and certainly we need to keep a close eye on it.

Monday, February 23, 2015

From 6th Grade to Advanced Physics...

Just a couple of links that hint at the breadth of mathematics, to start the week with:

a)  Brian Hayes, who is not a 6th-grader ;-) is off-and-running with an analysis of an arithmetic problem Fawn Nguyen previously posed to her 6th graders (...both delightful and interesting):

b)  Despite David Hilbert famously chastising the detractors of Cantor's "infinity," that "no one shall drive us from the paradise which Cantor has created for us," physicist Max Tegmark argues for finding "the true laws of physics" by finally getting rid of infinity:

Sunday, February 22, 2015

The Means and the Ends of Math

This Sunday's reflection comes from Ben Orlin of the "Math With Bad Drawings" blog. He penned these lines after attending the 2014 Joint Math Meetings:
"There’s a thin fault-line running through all our conversations about math. Is mathematics a means, or is it an ends?
On the one hand, math is a warehouse of applications, the world’s favorite toolkit. It enables the technologies and discoveries that have carried our species from caves to houses to rocket ships. In that sense, math is a means.

Math is also a self-contained realm of pure ideas. This doesn’t mean math is insulated from human activity; it means that math is a quintessentially human activity. Math is the pursuit of patterns, not necessarily for the sake of faster computers, but for the sake of the patterns themselves, for the sake of their elegance and beauty. In that sense, math is an ends unto itself.

The two camps sound irreconcilable. But really, these are just the happy and inherent contradictions of a pursuit that transcends even our most focused efforts to describe it.

Math is a means and an ends. It’s a world-changing toolkit and a beautiful world in its own right. Math belongs not just to mathematicians, but to scientists, engineers, financiers, actuaries, artists, even television writers. It belongs to teachers and students and infants learning to count. It belongs to the 6,000 humans who gathered in Baltimore, and to the 7 billion who didn’t."

...And this morning over at MathTango, Ben answers my questions for Math-Frolic Interview #27.
You'll enjoy it.

Friday, February 20, 2015

Infinity and Angst (David Foster Wallace)

Tomorrow would have been American writer David Foster Wallace's 53rd birthday.  Over 7 months ago I posted this as a "Sunday Reflection" over at MathTango. I re-post it again today, here at Math-Frolic, in his honor and for his many fans:

"Here is a quotation from G.K. Chesterton: 'Poets do not go mad but chess players do. Mathematicians go mad, and cashiers; but creative artists very seldom. I am not attacking logic: I only say that this danger does lie in logic, not in imagination.' Here also is a snippet from the flap copy for a recent pop bio of Cantor: 'In the late nineteenth century, an extraordinary mathematician languished in an asylum… The closer he came to the answers he sought, the further away they seemed. Eventually it drove him mad, as it had mathematicians before him.'
"The cases of great mathematicians with mental illness have enormous resonance for modern pop writers and filmmakers. This has to do mostly with the writers'/directors' own prejudices and receptivities, which in turn are functions of what you could call our era's particular archetypal template….

"Chesterton above is wrong in one respect. Or at least imprecise. The danger he's trying to name is not logic. Logic is just a method, and methods can't unhinge people. What Chesterton's really trying to talk about is one of logic's main characteristics -- and mathematics'.  Abstractness.  Abstraction."

-- From "Everything and More" by David Foster Wallace

 I've lately been re-reading parts of David Foster Wallace's "Everything and More: A Compact History of Infinity."
I don't read fiction, but that doesn't stop me from sometimes being intrigued by fiction writers, or in this instance one who additionally wrote non-fiction. And I have other reasons to be fascinated by Wallace:
He spent much of his childhood in Champaign/Urbana, Illinois, very near my own hometown, and died tragically in 2008 while teaching at my alma mater, Pomona College, in Claremont, California. In-between, his brain seemed to gallop effortlessly all over the place.
Of course it's not his widely-acclaimed, award-winning fiction that interests me; it's his, slightly-lesser-known, non-fiction. That a person of the humanities with an English degree, who poured himself into long, involved, complex novels and wordplay, was capable of also writing about deep mathematics is fascinating. It's a little less strange given that Wallace did deeply study philosophy, logic, and mathematics at the college level... but still amazing to me that "Everything and More" could be born in the same mind that authored "Infinite Jest"(…interesting that "infinite" makes its way into this title as well).

Wallace called his 300+ page volume ("Everything and More") on infinity a "booklet," and he no doubt genuinely considered it but an introduction to the whole subject; scratching the surface of a topic that so often baffles undergraduates, even leading to incredulity or heated arguments, amongst young math majors. Yet the book is a meticulous parsing of the subject as virtually never found in a popular work. In fact, I suspect it falls into that category of 'widely-bought, least-read books ever,' with a large percentage of buyers never completing it; purchasing it solely based on the author's reputation, and then abandoning it after the first 30-50 pages.

The book received a number of favorable reviews upon release, but several professional mathematicians also harshly critiqued it, finding it peppered with technical errors… I s'pose that I, as a non-professional, tend to be more forgiving, spellbound as I am by Wallace's ability to even approach these strenuous subjects innovatively… not that that justifies inaccuracies, but just that my joy with the volume stems not simply from the mathematics/philosophy entailed, so much as from the sheer audacity of a renowned novelist crossing boundaries to tackle such matters. I can't even imagine who this book was intended for… surely not the same audience who loved Wallace's fiction; but nor for professional mathematicians who would find faults in it. And not just for me, an audience of one ;-) Somewhere out there must be other "mees," I guess, who stand almost in awe of what Wallace accomplished: the mix of language and math, of thought and meta-thought, of narrative and cerebral-wrestling, while attempting to communicate it all to a mass(?) audience. As I wrote once before, this volume is "written in an informal and conversational tone about ideas that are utterly UN-informal and UN-conversational (...and the multitudinous footnotes are virtually as fascinating as the main text)."
And Wallace's prose isn't just deep, but in some ways, prescient... anticipating something we now commonly hear about high school math education, here's what he said in 2003 about college-level math:
"The trouble with college math classes -- which classes consist almost entirely in the rhythmic ingestion and regurgitation of abstract information, and are paced in such a way as to maximize this reciprocal data-flow -- is that their sheer surface-level difficulty can fool us into thinking we really know something when all we really 'know' is abstract formulas and rules for their deployment. Rarely do math classes ever tell us whether a certain formula is truly significant, or why, or where it came from, or what was at stake. There's clearly a difference between being able to use a formula correctly and really knowing how to solve a problem, knowing why a problem is an actual mathematical problem and not just an exercise."
This remains one of the quirkiest, both convoluted and semi-profound, volumes on my math bookshelf, from one of the quirkiest, most imaginative minds America has produced. Possibly there is some irony, some stinging irony, that Wallace, a long-time sufferer of depression, died tragically at his own hands, via hanging at the young, fertile age of 46... possibly even suffering mental demons not altogether dissimilar from Georg Cantor, a century earlier; died perhaps an example of the same stereotype or "archetypal template" he points to in the opening passage above ("the closer he came to the answers he sought, the further away they seemed").

One of the endorsement blurbs on the back of my copy of the volume says, "...David Foster Wallace is the perfect parachute buddy for a free fall into the mathematical and metaphysical abyss that is infinity." I think "abyss" may be too strong a word, but I do like the imagery of 'free-falling' into infinity... with an English major no less!
This volume won't suit a lot of people's taste, but reading it more as a treatise on human thought/genius and psychology, than a math treatise, I return to it... in wonderment and reflection... each year.


Thursday, February 19, 2015

A Congruency of Triangles

Futility Closet reports succinctly on a recent geometric theorem from the ever-interesting Lee Sallows:

Seems odd that this simple finding on triangles hasn't been reported previously elsewhere(???), but that's part of the beauty of mathematics, that such elementary conclusions can be sitting out there just waiting to be discovered.
Now, does this one have any particular applications...?

Tuesday, February 17, 2015

Quantum Physics and Pure Math Creeping Toward One Another

Seed Magazine has just re-run a fascinating 2006 piece from Marcus du Sautoy on a possible deep connection between the mystery of prime numbers and the quantum physics of atomic structure:

It relates back to one of my favorite 'Sunday reflections' here involving a famous encounter between Freeman Dyson and Hugh Montgomery.

Here is a more recent longish piece (2013) on the same subject from Princeton's Institute for Advanced Study that brings quasi-crystals into the discussion:

And a whole lot more from this 2013 John Baez posting over at n-Category Cafe blog:
(again, technical stuff!)

Matthew Watkins' trilogy of books on prime numbers, for a more general audience, also relates to this discussion.

One gets the feeling there is something very fundamental going on here... verrrrry fundamental... but, can the human brain unravel it!?

Sunday, February 15, 2015

Rocky Mountain Mathematical High

"When his friend Ronald Graham bet Erdös $500 he couldn't quit amphetamines for a month, Erdös won the bet but complained, 'I didn't get any work done... I'd have no ideas, like an ordinary person. You've set mathematics back a month.' Erdös went back to taking pills and writing papers.
"Hamilton and Erdös indulged their drug habits, it's said, to maintain their stamina. California chaos theorist Ralph Abraham is the only mathematician I know who has claimed drugs actually affected the contents of mathematical research, and for the better. In a 1991 interview with the style magazine
GQ, Abraham claimed, 'In the 1960s a lot of people on the frontiers of math experimented with psychedelic substances. There was a brief and extremely creative kiss between the community of hippies and top mathematicians. I know this because I was a purveyor of psychedelics to the mathematical community.' The interview is maddeningly short on specifics. Molecular biologist Kary Mullis 'seriously doubt[s]' that he would have invented the PCR technique for which he won the Nobel Prize if he hadn't taken LSD. Timothy Leary wrote in 1977 that he expected 'the new wave of turned on young mathematicians, physicists, and astronomers... to use their energized nervous systems... to provide new correlations between psychology and science.'...
"Pharmaceutical enhancement may be redundant if mathematics is itself the drug. Mathematicians stress the addiction even more than the buzz:

"Marcus du Sautoy: 'Doing mathematics is like taking a drug. Once you have experienced the buzz of cracking an unsolved problem or discovering a new mathematical concept, you spend your life trying to repeat that feeling.'"

-- From "Mathematics Without Apologies" by Michael Harris [my review of this new book now up at MathTango]

Thursday, February 12, 2015

Plethora of Podcasts

If you're a member-in-good-standing of the Cathy O'Neil Fan Club ("mathbabe") you'll want to give this recent 46-minute Samuel Hansen interview with multi-talented Cathy a listen:

If you wish to catch up on older Hansen interviews go here:

And finally, in a guest post at Cathy's blog, Sam has listed many other math podcasts that may interest you (this ought keep you occupied until Season 2 of "Serial" returns ;-):