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:

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).

"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

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):

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!

"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

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":

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):

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.

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/

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):

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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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."

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!" ;-)

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:

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 ;-)

“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."

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 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:

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:

"Physical concepts are free creations of the human mind, and are not, however it may seem, uniquely determined by the external world. In our endeavor to understand reality we are somewhat like a man trying to understand the mechanism of a closed watch. He sees the face and the moving hands, even hears it ticking, but he has no way of opening the case. If he is ingenious he may form some picture of the mechanism which could be responsible for all the things he observes, but he may never be quite sure his picture is the only one which could explain his observations. He will never be able to compare his picture with the real mechanism and he cannot even imagine the possibility of the meaning of such a comparison."

I'll soon be posting my own list of (math) book ideas from 2015 for the holiday season over at MathTango, but in case you were specifically interested in looking for some recreational math reading possibilities, worth checking out this older Quora thread:

I'd feel remiss if I failed to share with you these profound sentences ;-) from mathematician John Allen Paulos in his latest book/memoir, "A Numerate Life" (reviewed over at MathTango today):

"We tend to think we've arrived at our present station [in life] largely by dint of determination and hard work, but as my father used to say, we're all just farts in a windstorm. Less graphically put, we're all parts of various systems -- familial, professional, societal -- and these systems impact on us and direct our paths as if we were pinballs whirling through the quincunx of life. Nevertheless, we should heed the aforementioned title of Benjamin Franklin's essay, 'Fart Proudly.' That is, we should embrace our contingency even when it's unpleasant."

Riemann Zeta Function along critical line Re(s) = 1/2

Long-time readers know this blog is as interested as any in news of the Riemann
Hypothesis. I won't even dignify it with a link or any names, but there was a story
this week of the Riemann Hypothesis being proved by a Nigerian
mathematician... uhhh, yeah, sure.

The first problem was that I saw the
story a couple days after the fellow had apparently announced the proof
-- any genuine proof would've hit various legit math websites I follow
within 30 mins. of being announced; maybe 3 minutes! The two places I initially
saw the story were... well, let's just say, NOT the brightest bulbs in the
world of journalism (although some more legitimate sources
embarrassingly picked up the story-blurb later). And finally, call me
prejudiced, but my gut reaction at this point, to ANY odd story emanating from Nigeria is, "FA-A-A-AKE!" (don't blame me, Nigeria has allowed
it's own credibility to be trashed).

Anyway, plenty of others have voiced their skepticism, although admittedly, I've not yet seen the story specifically unmasked as a hoax or case of crackpottery from the get-go, or alternatively as someone
with actual math credentials sincerely making an over-the-top claim that doesn't pan out (if someone by now knows the full details or backstory feel free to elucidate in the comments).

For now at least, seems safe to say that
Riemann's 156-year-old
mystery still awaits a solution that will send legions of mathematicians into paroxysms of jubilation(!), and $1 million (Clay Millennium Prize) still awaits the person
who can do it.

ADDENDUM: A couple of folks have emailed me with
questions I'm not able to answer, but the following pieces from
George Dvorsky and a Quora thread will help make clear why the announcement is
given little credence:

What
remains unclear to me is whether the individual involved (claiming the
proof) is some sort of charlatan or a bonafide mathematician in error.
Mistaken and crackpot Riemann Hypothesis proofs have been common over
the decades and there's simply no basis for thinking this story is
anything other. But I'll certainly update if, incredibly, anything more
positive arises from the story.

"The Boy Who Loved Math: The Improbable Life of Paul ErdÃ¶s," is a children's picture book that has been out for a couple of years... oddly enough, about the life of Paul ErdÃ¶s ;-) ...no really, it is a bit odd that someone (Deborah Heiligman and LeUyen Pham) thought to make a children's book based on the eccentric life of a great mathematician.
Anyway, James Propp has a fabulous new and extended review of the volume (great job covering the book and some of the key ideas ErdÃ¶s worked on as well):

Not too early to be thinking of stocking stuffers for any math-inclined younguns on your Holiday list. And even if you don't have children or an interest in children's books, the above piece from Propp is a VERY worthwhile read for the included mathematics.

Perhaps worth noting also that there are two wonderful, older bios of ErdÃ¶s for the adults on your shopping list as well (no one would believe ErdÃ¶s' life if someone wrote him up as a character in a work of fiction... YET he was REAL!):

It's all above my pay-grade ;-), but I am wondering if this in any way relates back to previous interesting work (Freeman Dyson and Hugh Montgomery) finding linkage between quasi-crystals, prime numbers, the Riemann Zeta function, and sub-atomic structure (here and here)? No clear connection is made in the above article, but in both cases concepts from pure mathematics appear unexpectedly in a quantum mechanics context, so just wondering?
Anytime that pure Platonic-like math raises its head in an area as fundamental as atomic structure it gives one pause to ponder....

Interesting short reading (pdf download) a few days back, "On Things That Do Not Average or the Mean Field Problem," from irascible Nassim Taleb in what I presume is an excerpt (preliminary draft) from his next book:

In it, he rebukes "psychology, 'evolutionary theory,' game theory, behavioral economics, neuroscience and similar fields not subjected to proper logical (and mathematical) rigor" (...can't believe he left out epidemiology ;-) for their inadequacy in dealing with nonlinearity.

Toward the end he writes:

"Much of the local research in experimental biology, in spite of its
seemingly 'scientific' and evidentiary attributes fail a simple test
of mathematical rigor.
"This means we need to be careful of what conclusions we can and cannot
make about what we see, no matter how locally robust it seems. It is
impossible, because of the curse of dimensionality, to produce information
about a complex system from the reduction of conventional experimental
methods in science. Impossible."

On a side-note, a guest post in October at Cathy O'Neil's blog drew LOTS of comments pro-and-con about the likelihood that computer scientists will ever truly simulate the human brain (with huge MONEY being poured into such projects).
Taleb makes it clear here that he's in the camp arguing we will "never" understand the workings of the brain based on an understanding its parts, and not because it is too difficult, but because it is mathematically "impossible."

ADDENDUM:yesterday, Taleb followed up the above paper with this far more technical version (again pdf) on the subject:

We'll kickstart the week with an "Ask Marilyn" (Marilyn vos Savant) puzzle column, from yesterday's Parade Magazine. It's another of those easy-to-understand, but tricky, probability brainteasers:

A writer asks (and the wording is important), "Among parents with four children, what is the most common distribution of boys and girls? My friends think it’s two of each sex." . .answer below . . . . . . . . . . . . . . . . . . . .

Most would probably give the answer of 50/50, two boys and two girls. But Marilyn contends the most likely distribution is in fact three children of one sex and one of the
other. She goes on to list ALL (16) of the possible birth outcomes:

Then she notes that families with 3 children of one sex occur 8 different ways (or 50% of the time), while 2 of each sex occur in only 6 ways (or 37.5%).

She'll no doubt get pushback on this though (not uncommon for her) since the term "distribution," and the wording of the question, can be interpreted in crucially different ways:

Marilyn is only looking at distribution of "same" or "different" sexes, but if you look at distribution in terms of specific sexes then you have 2-boys/2-girls occurring in six cases, 3-boys/1-girl in four cases, and 3-girls/1-boy also in four cases... thus, the 50/50 boy/girl case IS indeed the most common.

"It's time for science to retire the fiction of statistical independence. "The world is massively interconnected through causal chains. Gravity alone causally connects all objects with mass. The world is even more massively correlated with itself. It is a truism that statistical correlation doesn't imply causality. But it is a mathematical fact that statistical independence implies no correlation at all. None. Yet events routinely correlate with one another. The whole focus of most Big Data algorithms is to uncover just such correlations in ever larger data sets.... "A revealing problem is that there are few tests for statistical independence. Most tests tell at most whether two variables (not the data itself) are independent. And most scientists would be hard pressed to name even them. So the overwhelming common practice is simply to assume that sampled events are independent. Just assume that the data are white. Just assume that the data are not only from the same probability distribution but also statistically independent. An easy justification for this is that almost everyone else does it and it's in the textbooks. This assumption has to be one of the most widespread instances of groupthink in all of science."

Gives an overview of what is famously-designated "the hardest logic puzzle ever" (created by Smullyan and solved by Boolos).

Gallagher ends the piece noting the puzzle demonstrates "how essential one of the supposed
fundamental laws of logic -- the law of excluded middle -- seems to be" (which assumes that "every statement is either true or false -- there is
no middle ground"), or in Boolos' words, “Our
ability to reason about alternative possibilities, even
in everyday life, would be almost completely paralyzed were we to be
denied the use of the law of excluded middle.”

A practical problem of course is that the law of the excluded middle only operates within narrow, well-defined contexts, and NOT in most of day-to-day life... language and life are far more characterized by ambiguity, continuity, and gray areas, than the discrete black-and-whiteness implied by a simplistic excluded-middle law. Thus, my own increased recent interest in so-called "fuzzy logic" (mentioned awhile back) over classic Aristotelian logic... but still, for puzzle and logic purposes, a great article.

And her responses in the accompanying interview are fascinating as well (more-so than one might expect within a statistics context!), as she touches upon her lack of an academic background in statistics, the use of narrative in math writing, mathematics difficulty as a "spiritual" wound, Florence Nightingale as her statistics "hero," and her own medical experience with 'uncertainty.'

An interesting piece last week in the Washington Post, about the connection between mathematics and the music that makes us feel good: http://tinyurl.com/pgjkyrt

Fast tempo, major chords, and positive lyrics are among the elements that tend to associate with music that is mood-uplifting.

The article includes a list of the Top 10 feel-good inducing songs (below), based on a formula neuroscience researchers have worked out (wouldn't quite jive with my own Top 10 list, but so be it):

1. Don’t Stop Me Now (Queen)
2. Dancing Queen (Abba)
3. Good Vibrations (The Beach Boys)
4. Uptown Girl (Billie Joel)
5. Eye of the Tiger (Survivor)
6. I’m a Believer (The Monkeys)
7. Girls Just Wanna Have Fun (Cyndi Lauper)
8. Livin’ on a Prayer (Jon Bon Jovi)
9. I Will Survive (Gloria Gaynor)
10. Walking on Sunshine (Katrina & The Waves) In a related note, NPR's RadioLab re-ran an episode this week, with less math, but relating music to language: https://www.wnyc.org/radio/#/ondemand/542333

Samantha Oestreicher writes today about her experience with mathematics in the "Ivory Tower" world and in "Industry," and as a result, how she has "radically changed the way[she] views the world." She writes that the "core of [her] mathematical faith rests on those building blocks of real analysis, probability theory and dynamical systems" sometimes ignored by the needs of industry... but, "It’s not okay to ignore the building blocks of my field."

If you've had your math degree for awhile now and used it in a corporate job, read her piece and see if you can relate to her experience. I imagine she'd enjoy hearing from both those who've had similar or different experiences in applying Ivory Tower math to the real world out there (as Industry sees it):

"Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. 'Immortality' may be a silly word, but probably a mathematician has the best chance of whatever it may mean." -- G.H. Hardy

...and for comic relief, Woody Allen:

"'I don't want to achieve immortality through my work; I want to achieve immortality through not dying."

2)Less scary, but more mind-racking perhaps than differentiation, is the 'Sleeping Beauty Problem/Paradox,' which I haven't mentioned for awhile, but do now (...at least one version of it):

He posted the answer as 20% and then, at the coaxing of some commenters, lowered it to 15%.
Then...
later that evening, he raised the probability to 1... because of course
Murphy (of the New York Mets) did just that, hit a home run in the 8th
inning (playing against the Chicago Cubs, surely a major factor ;-)

And so, in a matter of hours the "probability" of something went from 20% to 15% to 100%... a nice demonstration of why, given human complexity, "probability" is often a near-meaningless
concept when it comes to individual behavior and events.

Not much math here, but another fabulous post from Scott Aaronson, this time (in general) on the social sciences (...the comments, as usual, are fascinating as well):

(I'm dang near wanting to declare Aaronson a national treasure for the thoughts and discussion he generates! ...seriously, anyone know if Scott has ever been nominated for a MacArthur Award? hint, hint...)

Just want to quickly pass along this new fun "n-Category Cafe" post which includes links back to two other rich reads (that I haven't fully digested yet), one being from David Mumford. It all has to do once again with mathematicians and the experience of beauty (from a neuroscience perspective):

A departure from the norm for this Sunday's reflection... instead of a quotation, I'll just refer you to this entire month-old post from Michael Harris:

Often, people find the most oddball, neurotic, reclusive mathematicians to be the most fascinating, even heroic, ones (I touch slightly upon math eccentricity in the prior post at MathTango), but Evelyn Lamb points to a woman who actually
approached and met (before he died) one of those unorthodox mathematical geniuses, Alexander Grothendieck:

The protagonist here, Katrina Honigs, writes early on of her 2012 encounter: "...I am driven to demystify -- it is part of what motivates me to be a mathematician -- and when we tell ourselves and others that our heroes are inhuman and on a pedestal that is not just high but unattainable, we are actually pushing ourselves down rather than climbing." And so she actually trespasses and carries baked goods along to meet the object of her fascination. There's no great drum-roll or clash of cymbals to her story, just the brief, unlikely encounter of two different individuals. She sums it up simply as "a story worth telling: a bit odd, a bit funny, and, at least to me, a bit meaningful."

I wouldn't go so far in such pursuit as Katrina does, but her story did make me wonder what living math-giants I might feel driven to meet if I could simply wave a magic wand and be plopped into their presence. Three names that came to mind quickly were Raymond Smullyan, Ed Witten, and Freeman Dyson, though I'm sure there are others... but what I would possibly say to any of those three, were I to meet them, I barely have a clue! :-(
Who might you most like to chat with over coffee and scones, given a magic wand?

"Classical logic is like a person who
comes to a play dressed in a black suit, a white, starched shirt, a
black tie, shiny shoes, and so forth. And fuzzy logic is a little bit
like a person dressed informally, in jeans, tee shirt, and sneakers. In
the past, this informal dress wouldn't have been acceptable. Today, it's
the other way around."
-- Lofti Zadeh (1984)

Though it's been around for a good while I only recently began dabbling in "fuzzy logic,"
and now enjoying it as an approach that makes a lot of sense (reminds me also of the non-Aristotelian approach of General Semantics, and getting rid of the "law of the excluded middle"). I've enjoyed various essays by Bart Kosko in the past, but only recently learned of his connection to fuzzy logic (which drew me to the subject). Kosko's 1993 read, "Fuzzy Thinking" is a great introductory volume.
Another popular old-read (also 1993) on the topic is "Fuzzy Logic" by McNeill and Freiberger, but I didn't find it nearly as satisfying as Kosko's volume.

There
are also many web videos available on fuzzy logic, but the few I've
looked at didn't seem all that helpful or effective. I'd still like to find a good visual presentation. So if someone cares to
recommend a good video, feel free to (and save me some time ;-) Or feel
free to recommend other books and websites for the interested layperson.

You receive a letter on a Friday that is either a rejection letter or an acceptance letter to medical
school. You have a wonderful weekend planned and don't want bad news interfering with it. Can you devise a way to learn the contents of the letter BUT ONLY if it is good news?
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Have a friend open the letter.

Instruct them that IF it is good news they are to flip a coin and tell you the news ONLY if it comes up heads, otherwise tell you nothing.

AND, if it's bad news, tell you nothing.

This way you will either be told good news, OR STILL have 33% hope for good news, if they tell you nothing.

Not precisely mathematics, but this week's Sunday reflection by physicist Max Tegmark on why we need to be careful when it comes to programming artificial intelligence:

"If you're walking on the sidewalk and there's an ant there, would you actively go and stomp on it just for kicks? (Me: 'No.')
"Now, suppose you're in charge of this big hydroelectric plant that's
gonna bring green energy to a large region of the U.S. And just before
you turn the water on, you discover there's an anthill right in the
middle of the flood zone. What are you gonna do? It's too bad for the
ants, right? It's not that you hate ants. It's not that you're an evil
ant-killer. It's just that your goals weren't aligned with the goals of
the ants, and you were more powerful than the ants. Tough luck for the
ants. We want to design AI in the future so that we don't end up being
those ants."

Ben Orlin tapped my funny bone again this week... and brings out the toddler in all of us... with this offering on the role of rote repetition/practice in learning and mastery:

The format will be familiar to many of you.
I've given the answer farther below, but without explanation, so if you need that, you can go to the link, find the problem, and check the responses there.

***********************************

Two math grads run into each other at the shopping mall, having not seen each other in 20 years. Their conversation proceeds like this:

M1: How have you been?

M2: Great! I got married and now have 3 daughters.

M1: Wonderful... how old are they?

M2: Well, the product of their ages is 72, and the sum of their ages is the same as the number on that building over there.

M1: Sure, ok... er wait... Hmmm, I still don’t know their ages.

M2: Ohh sorry, the oldest one just started piano lessons.

A
beautiful, touching, scrumptious essay this week from Keith Devlin, on
the beauty of mathematics... a somewhat tiresome phrase that he breathes
life into here, focusing on calculus, or, as he quotes William Blake, "infinity in the palm of your hand":

It
deals with a student's recent response to a piece Keith had written almost 10
years earlier. I heartily commend it to all mathematicians, math
teachers, math majors, and students in general, and all those, who like
myself, simply love math from the sidelines. It almost has a fractal
quality, as a beautifully-crafted essay, about beautiful ideas, about
the beauty of beauty! ;-)
[p.s... Dr. Devlin suggests "if you are a math instructor at a college or university, maybe
print off this blog post and pin it somewhere on a corridor in the
department as a little seed waiting to germinate." I'll
second that suggestion, which derives, NOT from Keith's ego,
but from his infectious love of math teaching/learning.]

Actually,
half the post is simply a verbatim letter Dr. Devlin received from a
math student who had previously read another of Keith's essays, and now
was writing to say how much he finally appreciated that earlier
piece. Is there anything more rewarding to a teacher than to hear from a
student (and in this case not even Keith's own student) how much
something you said or did in the past has affected that student years
later!? Keith's earlier piece was about the deep, deep beauty of
calculus, or again from Blake,seeing "an infinite (and hence unending) process as a single, completed thing."
All
of us who've taken calculus will probably freely admit that, no matter
what our grade or ability in a first-year course, we lacked any deep
grasp of the subject at that point. To a lesser degree maybe that even
holds for algebra, geometry, trig… the student can't fully appreciate
these subjects 'til s/he has taken in much more mathematics for context,
depth, nuance. The "inner beauty" of math requires persistence and
commitment to fully access.

Dr. Devlin's post reminded
me slightly of the well-known Richard Feynman blurb that I've placed
below (and am sure most of you have already seen), wherein he speaks of
the "beauty of a flower," and how,
despite what an artist friend thinks, he as a physicist also has access
to seeing that beauty; perhaps even perceiving it at a deeper level than
does the artist.

I WISH I could see the
beauty of math the way Keith, and Ed Frenkel, and Steven Strogatz, and
others see it (seeing it, as Keith has previously written, from a
treetop overlooking the vast but inter-connected forest below). But
alas, as a rank-amateur, my vision is far more limited, far more myopic
than theirs. Yet even from my lowly vantage point mathematics resounds
in beauty, in "excitement, mystery, and awe" as Feynman refers to.

Some
of course call mathematics the language of science, or even the
language of God. But at base, I think its beauty lies in being a pure,
grand, and almost inexplicable creation (or discovery) of the human
mind... the pinnacle of that which our brains are capable. In a day
when our lives, politics, and society, seem inundated with violence,
intolerance, and irrationality, mathematical thinking stands out as a
beacon for the future, if we as a species are to have a future.

Growing
up, I watched my grandfather (and other seniors) become increasingly
cynical about the world as they aged, and swore to myself I would never
be like that. But I do now find myself saddened each day when I turn on
the news… cynicism is hard to repress. My hope today though, is that
every teacher out there, at least once in your lives, receives a letter
like the one Dr. Devlin has shared, or if you're not a teacher, that you
hear from some young person, when you're not expecting it, what a
difference you made in their lives.

The oddball Count
(and father of General Semantics), Alfred Korzybski wrote that we humans
are a "time-binding" species (different from all other species that
only "space-bind") because of the way we routinely transfer our
increasing knowledge across generations. That, in part, is what I see
going on in Dr. Devlin's piece, "time-binding" with a younger
generation... and, as always, the younger generation is our real hope
for the future... and, our shield against cynicism!

Finally, as I was completing this post a new blogpost from Megan Schmidt
crossed my webfeed. If you need a reminder that teachers impact young
lives (or even if you don't) I hope you will read it as well, (be
sure to click on and read the student exposition she provides): http://mathybeagle.com/2015/10/03/where-do-we-go-from-here/

Woodbridge Hall/Yale U. via Nick Allen/WikimediaCommons

Well, Ben Orlin leaves me ROFLOL once again as he explains
why... if you can believe it... he purposefully avoids things that 'feel like spiders
crawling out of his eyeballs':**

h/t to Julie Rehmeyer for pointing to some short (~4-5 min.) video clips relating the issue of gender in mathematics, as touched upon by the play entitled, "One Girl's Romp Through M.I.T.'s Male Math Maze":

"Mathematics and contemporary art may seem to make an odd pair. Many people think of mathematics as something akin to pure logic, cold reckoning, soulless computation. But as the mathematician and educator Paul Lockhart has put it, 'There is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics.' The chilly analogies win out, Lockhart argues, because mathematics is misrepresented in our schools, with curricula that often favor dry, technical and repetitive tasks over any emphasis on the 'private, personal experience of being a struggling artist'…

"…During his four minutes, Alain Connes, a professor at the Institut des Hautes Etudes Scientifiques, described reality as being far more 'subtle' than materialism would suggest. To understand our world we require analogy -- the quintessentially human ability to make connections ('reflections' he called them, or 'correspondences') between disparate things. The mathematician takes into another hoping that they will take, and not be rejected by the recipient domain. The creator of 'noncommutative geometry', Connes himself has applied geometrical ideas to quantum mechanics. Metaphors, he argued, are the essence of mathematical thought. "Sir Michael Atiyah, a former director of the Isaac Newton Institute for Mathematical Sciences in Cambridge, used his four minutes to speak about mathematical ideas 'like visions, pictures before the eyes.' As if painting a picture or dreaming up a scene in a novel, the mathematician creates and explores these visions using intuition and imagination. Atiyah's voice, soft and earnest, made attentive listeners of everyone in the room. Not a single cough or whisper intervened. Truth, he continued, is a goal of mathematics, though it can only ever be grasped partially, whereas beauty is immediate and personal and certain. 'Beauty puts us on the right path.'"

I'll remind folks that Presh Talwalkar also does a weekly wrap up of math picks later on Fridays at his "Mind Your Decisions" blog (usually quite different from my MathTango selections): http://mindyourdecisions.com/blog/

...and Crystal Kirch has been doing Sunday linkfests for teachers at her "Flipping With Kirch" blog: http://flippingwithkirch.blogspot.com/ (check 'em out on Sun.)

If
there are other regular weekly math linkfests you think worth knowing
about, feel free to send them along (via comments or email). I'm happy
to publicize other sites that are spreading the math wealth!

"I don't want to belong to any club that would accept me as a member."

Hmmm, after using this quote for decades, I just suddenly realized what a deep-thinking set-theorist
Groucho Marx was (...and, a whole LOT funnier than Bertrand Russell too!).
;-)

Recommended to everyone is the freely downloadable book (pdf) on RH by Barry Mazur and William
Stein. Get it! ==> UGHH, looks like link for download no longer works, so consider yourself lucky if you already got it; otherwise look forward to the book when eventually published. I understand the publisher not wishing free downloads to be available; on-the-other-hand I suspect most of those downloading will eventually want a hard copy of the final version anyway.