End-of-Year Retrospective

Some sort of year-end listing of favorite blog posts from the prior 12 months is a tad traditional (...and makes for a nice space-filler ;-) so I'll list these for any readers who may have missed them:

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1)  In Feb. I re-ran what was actually one of my very favorite posts from prior years, on David Foster Wallace and his volume, "Everything and More":
http://math-frolic.blogspot.com/2015/02/infinity-and-angst-david-foster-wallace.html

2)  In Mar. a post related to "Penney's Game" (and probabilities) was fun:
http://math-frolic.blogspot.com/2015/03/flippin-probabilities.html

3)  In July I recounted a quirky paradox from Futility Closet (one of the greatest purveyors of fun math out there!):
http://math-frolic.blogspot.com/2015/07/gladly-paying-110-for-dollar-bill.html

4)  In Aug. there was this quickie half-fun, half serious post:
http://math-frolic.blogspot.com/2015/08/have-you.html

5)  Not very mathy, but also from Aug. my personal listing of some favorite blogs/sites for following science on the Web:
http://math-frolic.blogspot.com/2015/08/following-science-on-web.html

6)  In Sept. I linked to a great Lior Pachter post regarding math education (this was actually one of my favorite links from the whole year!):
http://math-frolic.blogspot.com/2015/09/not-so-common-common-core-remarkable.html

7)  Also in Sept. just a fun, little oddball post honoring Pierre de Fermat:
http://math-frolic.blogspot.com/2015/09/get-life.html

8)  This Oct. post touched on math heroes:
http://math-frolic.blogspot.com/2015/10/pursuing-alexander.html

9)  Every year Keith Devlin inspires me with one or more of his essays, as he did this year in Oct.:
http://math-frolic.blogspot.com/2015/10/true-deep-beauty-comes-only-with.html

10)  A brief November post/link referenced a study connecting math and music:
http://math-frolic.blogspot.com/2015/11/good-vibrations.html

And finally, from MathTango I'll just re-mention my Nov. review of the year in math books here:
http://mathtango.blogspot.com/2015/11/the-year-in-books.html

Enjoy.... and Happy/Safe New Year to all, in the event I don't post again until next year!
(...I do plan to have a Friday potpourri back up this week at MathTango).

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Mathematics: "A Veritable Fairyland"

"Mathematics is often erroneously referred to as the science of common sense. Actually, it may transcend common sense and go beyond either imagination or intuition. It has become a very strange and perhaps frightening subject from the ordinary point of view, but anyone who penetrates into it will find a veritable fairyland, a fairyland which is strange, but makes sense, if not common sense. From the ordinary point of view mathematics deals with strange things. We shall show you that occasionally it does deal with strange things, but mostly it deals with familiar things in a strange way."

-- from "Mathematics and the Imagination" by Edward Kasner and James R. Newman

T'was The NIght Before...

via Redbubble

Good tidings to all, and to all a good night.....

Whoooa!

Taking off from an earlier post by Mike Lawler on mathy things that make us go "whoa!," including Cantor's diagonalization proof,  Evelyn Lamb posts about some of her own "mathematical wonders" (with several good further links):

http://blogs.ams.org/blogonmathblogs/2015/12/21/mind-blowing-math-reminiscence/

Lamb writes at one point that as "a late mathematical bloomer"... "Not a lot of math really blew my mind in college because my attitude at the time tended towards the utilitarian. Diagonalization notwithstanding, I didn’t often appreciate the beauty of what I was learning or even know that I should be surprised by it. As time passes, I gain more and more respect for many ideas in math, even ones I’ve been familiar with for years."

Somehow, I find that a fascinating confession, since I imagine (maybe incorrectly?) most professional mathematicians arriving at their destination specifically because of an early captivation with the wonders/beauty of math and (in Wigner's terms) its "unreasonable effectiveness," versus duller, mere utilitarian application. But the detour-ridden roads to our final destinations are often long and winding, and mathematics, with its many possible footpaths, side-tracks, byways, may be no different than any other.

Anyway, there are too many 'whoa'-inducing ideas in math to pick a favorite, but I will link once again to one of my own mind-blowing faves, the Cantor Set:
http://platonicrealms.com/encyclopedia/Cantor-set

Honestly, it's not hard for me to imagine how Cantor was driven from sanity, given the matters he persistently tackled and wrestled with. If you stare at the sun you risk going blind, and if you stare at the heart of mathematics, as Cantor did, perhaps there are risks as well.

Rebecca Goldstein wrote a couple of decades back, "Mathematics and music are God's languages. When you speak them...you're speaking directly to God." 

I like that metaphor; whether it be God, Creation, the center of the Universe, or some other essence-of-being, when you "speak" mathematics or music (or, I would add certain forms of prayer/meditation), you reach a place, outside the narrow human realm, unattainable by any other means. WHOOOA indeed!

A Fractal Universe?

Today's 'Sunday reflection':

"I believe that scientific knowledge has fractal properties, that no matter how much we learn, whatever is left, however small it may seem, is just as infinitely complex as the whole was to start with. That, I think, is the secret of the Universe."   -- Isaac Asimov

Don't Mess With Popper ;-)

 Natalie Wolchover, ran a piece in Quanta recently with a title I love, "A Fight For the Soul of Science," covering some of the dissing of Popper falsification, in favor of more shoddy (IMO) induction-focused approaches (turning parts of modern-day physics into glorified metaphysics, by some accounts), leading to "a crisis" in which "the wildly speculative nature of modern physics theories... reflects a dangerous departure from the scientific method":

https://www.quantamagazine.org/20151216-physicists-and-philosophers-debate-the-boundaries-of-science/

As the article notes, "Theory has detached itself from experiment. The objects of theoretical speculation are now too far away, too small, too energetic or too far in the past to reach or rule out with our earthly instruments." That's a nice excuse for the science playground that has resulted, but in some form it could probably have been said at any point in the history of scientific method.
The discussion leads into Bayesianism (and specifically, "Bayesian confirmation theory"), and as always, Wolchover does a great job attempting to present different sides of a sticky topic. And I have no problem with (indeed I enjoy) speculative theorizing... I'm just unwilling to label it 'good science' (at best, it is good speculation, and that's often different).

Anyway, Andrew Gelman balanced some of the discussion with a more nuanced assessment, including lots of comments (and the debate goes on elsewhere, as well; see also an earlier Deborah Mayo take on Popperianism HERE):

http://andrewgelman.com/2015/12/17/gathering-of-philosophers-and-physicists-unaware-of-modern-reconciliation-of-bayes-and-popper/

In actuality, "the soul of science" has ALWAYS been threatened by different philosophical outlooks, but it ought be understood by all, that in general, "induction" (while necessary because it is unavoidable) is always a WEAK mode of empiricism, and it's no wonder a lot of folks are losing patience with the loosey-gooseyness in some areas of theoretical physics; a looseness that has long been present in biomedicine, psychology, economics, and some other areas, and in a kind of mission-creep (driven perhaps by academic/publication/career pressures), is now, to our detriment, expanding outward.

Celebrating the Season With a Tribute and Some Geometry

Pat Ballew 'celebrates the season' this morning with some "beautiful geometry" from "a little known mathematical dilettante":

http://pballew.blogspot.com/2015/12/the-geometric-beauty-of-troubled-mind.html

Pat writes that George Odom Jr. "found five different simple geometrical approaches to the golden ratio using equilateral triangles, and platonic solids" that "are too beautiful to be so unknown." A nice tribute to someone likely unknown to most of us.

Also, a wonderful, 2007 piece by Siobhan Roberts (...you may have heard of her) on Odom, and his connection to John Conway, here:
http://thewalrus.ca/2007-04-field-notes-2/

Making the Incomprehensible a Little More Comprehensible

Wow! Seems like everyone has been writing for awhile now about how incomprehensible Shinichi Mochizuki's "proof" of the ABC conjecture is... leave it to Mathbabe to find someone, Brian Conrad, willing to take a stab at making it a little MORE comprehensible! Long, informative (but still technical) post (certainly the best effort I've seen to address the topic... IF you can set some time aside):

http://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-workshop-by-brian-conrad/

Marilyn's Marbles...

Always easy when I can kickstart the week with a puzzle from Marilyn vos Savant's column in the Sunday Parade magazine, ICYMI. And once again it's a probability teaser that I'll re-phrase below:

In a gameshow, contestants Donald, Ted, and Marco, and the gameshow host, each have a bag holding 3 colored marbles in front of them. In each bag there is one red, one white, and one blue marble. The host randomly pulls one marble from his bag. Then Donald randomly draws one, then Ted, and then Marco, in that order (each from their own bag). The winner is the FIRST contestant to draw out a marble that matches the color of a previously-drawn marble (by anyone).
WHO has the best chance of winning?
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answer:  Ted  (if you need to see the simple math involved you can visit the problem here:
http://parade.com/442184/marilynvossavant/a-puzzle-of-percentage/ )

6 billion years from now...

Straying from mathematics this Sunday to offer a reflection from cosmologist Martin Rees:

"Most educated people are aware that we are the outcome of nearly 4 billion years of Darwinian selection, but many tend to think that humans are somehow the culmination. Our sun, however, is less than halfway through its life span. It will not be humans who watch the sun's demise, 6 billion years from now. Any creatures that then exist will be as different from us as we are from bacteria or amoebae."

 

Laysplaining, Mathsplaining, and Weeds...

When I wrote my Master's thesis a few eons ago, for fun I slipped in a few casual, informal bits... which my adviser saw and asked, "You weren't planning to leave that in the final draft were you?" To which I responded, "Well, actually, yes; you know, just trying for a little levity and less stodginess." And he said, "You can't do that." Needless to say, the final version reverted to academese.

I was reminded of that long-ago episode after Jordan Ellenberg tweeted out a link this week to the below math thesis which describes itself as "a fascinating tale of mayhem, mystery, and mathematics." It's been buzzing around the intertubes ever since, and may just become THE most viewed math dissertation in history!:
 https://twitter.com/JSEllenberg/status/674245895580426241

It hails from Princeton graduate Piper Harron, and the original (more academic) version of the material was posted on arXiv a couple years back:
http://arxiv.org/abs/1309.2025

There's already been a lot of commentary about the dissertation on the Web. Among my favorite remarks was this:

"I don't know enough about higher math to evaluate her work, but I can tell she's absolutely brilliant. Because you have to be brilliant to get away with that amount of sheer attitude."

Indeed, I've also seen some quite negative commentary... emanating from folks I suspect are lacking in appreciation for humor, creativity, and certain attitude! (there's no real reason that math, even pure math, can't include those).

The actual mathematics involved may weight you down, so try to stay focused on the larger storyline/ideas Piper is conveying. A few lines from the "Prologue" to get you started:

"Respected research math is dominated by men of a certain attitude. Even allowing for individual variation, there is still a tendency towards an oppressive atmosphere, which is carefully maintained and even championed by those who find it conducive to success... My thesis is, in many ways, not very serious, sometimes sarcastic, brutally honest, and very me. It is my art. It is myself. It is also as mathematically complete as I could honestly make it...
"It is not my place to make the system comfortable with itself. This may be challenging for happy mathematicians to read through; my only hope is that the challenge is accepted."

...and perhaps then too, keep in mind the old saying, "Attitude is everything!" ;-)


ADDENDUM:  the inimitable Mathbabe (Cathy O'Neil) now has a guest post up from Piper herself further explaining her "thesis grenade":
http://mathbabe.org/2015/12/11/piper-harron-discusses-her-artistic-and-wonderful-math-ph-d-thesis/

Amir Aczel, Popularizer 1950 - 2015

Back on Nov. 26, science/math writer Amir Aczel died at the age of 65, yet I could find almost no information about it on the Web... even 4 days later! (a couple of Twitterers, in-the-know, mentioned it, and his Wikipedia page was updated). A bit odd for an author of several popular books. At any rate, this week, the NY Times finally did publish an obit of his death (still not many details, though cancer is mentioned as the cause), and further oddly initially mis-stated Andrew Wiles' name as "Peter Wiles" (since, corrected) -- I tried to imagine what possible name mix-up might cause such an error, but couldn't come up with any candidates??? Just a small compendium of oddities.
 Aczel died in France; perhaps that country's current overwhelming focus on terrorism since mid-Nov. has something to do with the paucity of news about his passing -- I really have no idea why there has not been more coverage and obituaries for this loss, at a somewhat young-ish age, of an author of close to 20 books?
In any event, from the NY Times:
http://www.nytimes.com/2015/12/08/us/amir-aczel-author-of-scientific-cliffhanger-dies-at-65.html

Aczel's books were not heavy reads, but they were nice little introductions to each topic he addressed, and I enjoyed several. Some of his more math-related volumes were:

"Fermat's Last Theorem"
"Finding Zero"
"The Mystery of the Aleph"
"Chance"
"The Artist and the Mathematician"
"A Strange Wilderness: The Lives of the Great Mathematicians"

Below is an interesting talk (~1 hr.) Aczel gave at Google on his book "Finding Zero":

A beats B, B beats C, C beats A...

                                      A                            B                            C


I've referenced "non-transitive dice" here before, but Mike Lawler recently posted about them... AND it's gift-giving time... so perhaps worth reminding readers of them:
http://singingbanana.com/dice/article.htm

purchaseable here:
http://mathsgear.co.uk/products/non-transitive-grime-dice

Non-transitivity is one general category of paradoxes, often exemplified using voting patterns, but these dice are a great, striking introduction to the notion for young people.... and p.s., at heart, we're all young people ;-)

Think or Swim...?

Sunday reflections:

“The question of whether a computer can think is no more interesting than whether a submarine can swim.”
-- Edsgar Dijkstra

"...in a broader sense, the term thinking machine is a misnomer. No machine has ever thought about the eternal questions: Where did I come from? Why am I here? Where am I going? Machines don't think about their future, their ultimate demise, or their legacy. To ponder such questions requires consciousness and a sense of self. Thinking machines don't have these attributes, and given the current state of our knowledge they're unlikely to attain them in the foreseeable future."

-- Leo Chalupa  (in John Brockman's  "What To Think About Machines That Think")

Looking Forward and Backward (at books)

Since listing my favorite math books of 2015, I was recently reminded that the new Barry Mazur/William Stein volume on the Riemann Hypothesis is due out at the end of January 2016:
http://amzn.to/1lZuxxb

Too late for Christmas, but what a great start to the new year. David Mumford calls it "a soaring ride." I suspect once out, this short volume will be THE book (out of many available) to introduce folks to possibly the most important unresolved, far-reaching conjecture in all of mathematics. (...Perhaps I already know my favorite book of 2016!)

Meanwhile, I just obtained a couple of fine prior books on paradoxes, and feel safe recommending both well-ahead of finishing them. Roy Cook's 2013 "Paradoxes" is a good, fairly standard treatment of what I believe is one of the most important topics in all of math/philosophy, for bright high-school-level-and-above students.

Stanley Farlow's 2014 "Paradoxes In Mathematics" looks to be an especially wonderful introduction to several of the classics for middle-to-high-school students particularly, in breezy but broad-covering fashion. I was previously unaware of this succinct little volume from Dover, and am delighted to have stumbled upon it. Again a great stocking-stuffer for that distinctively math-inclined youngun on your list.

Brian and Dan (and Ramsey and Desmos)

Two great pieces you ought not miss from the last 48 hours:

1) The always wonderful Brian Hayes with a delightful post on Ramsey Theory:
http://bit-player.org/2015/ramsey-theory-in-the-dining-room

Brian works/writes over at American Scientist in addition to his personal blog above (and is also a Scientific American alum). He's such a clever, insightful writer I can't help but think he could've been a fine successor to Martin Gardner over at SA (where he did briefly do a similar computer science column). Anyway, much more of his writing linked to at this page:

http://www.americanscientist.org/authors/detail/brian-hayes

2) Secondly, a fairly glowing (and well-deserved) New Republic piece on Dan Meyer and his approach to teaching mathematics. Dan (and his work with Desmos) will need no introduction to any secondary math teacher in America who is active on the Web, but whether you do or don't know of him read up:

https://newrepublic.com/article/124750/man-will-save-math