Bye Bye Statistical Independence

Sunday reflection:

"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."

-- Bart Kosko, in John Brockman's "This Idea Must Die"

Put On Your Thinking Caps

Wonderful new Brian Gallagher article in Nautilus yesterday covers some classic Ray Smullyan/George Boolos logic conundrums:

http://nautil.us/issue/30/identity/how-to-solve-the-hardest-logic-puzzle-ever?

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.

Julie Rehmeyer Succeeds...

Congratulations to Julie Rehmeyer for winning another writing award, specifically for statistics writing:

http://linkis.com/magazine.amstat.org/hP40c

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.'

Good Vibrations...

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

...and just to end on a feel-good note ;-):

The Ivory Tower and Beyond...

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):

http://socialmathematics.net/2015/11/02/outside-the-ivory-tower/

Immortality...

Another short, simple Sunday reflection today:

"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."

Revisiting Seven

"The Magical Number Seven, Plus or Minus Two" was a famous, influential paper (1956) by cognitive psychologist George A. Miller introducing the idea of cognitive 'chunking' of items.

Related to it, in a post yesterday, Nathan Kraft passed along an interactive number memory test that some may find fun/interesting:
http://www.humanbenchmark.com/tests/number-memory

Give it a try, or like Kraft, you may even find it a useful game to explore in a classroom situation.

Be Afraid, Be Very Very Afraid

Two for the price of one today:

1)  First, in time for Halloween, DO NOT miss the frightful tale of Differentiation... as only Ben Orlin can tell it (bwaaahaaaahaaaaa):
http://mathwithbaddrawings.com/2015/10/28/the-differentiation-a-survivors-tale/


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):



The correct answer is: 1/2, 1/3?; 1/2, 1/3?; 1/2 or 1/3???.... two different logical answers, splitting the mind in two, with no final resolution. Spine-tingling stuff! ;-)
My original post on it was back in 2012 with a number of additional links:
http://math-frolic.blogspot.com/2012/03/sleeping-beautynot-your-childhood-fairy.html

Also, Tanya Khovanova had lengthy previous discussion of it on her blog here:
http://blog.tanyakhovanova.com/2011/08/the-sleeping-beauty-problem/

And even physicist Sean Carroll covered it a year ago, drawing 240+ comments:
http://www.preposterousuniverse.com/blog/2014/07/28/quantum-sleeping-beauty-and-the-multiverse/

Pick your side... you'll find some good arguments (and thinkers) backing you up either way.  Spooky indeed!

The Topology of DNA

New from Numberphile; amazing stuff (of life):

The Illusion of Pure Mathematics

This Sunday, a quickie!:

"There are two kinds of mathematics: applied mathematics and mathematics that is not yet applied."

-- Jim Callahan (quoted in Jim Henle's "The Proof and the Pudding")

The Improbability of Knowing Probability

via Gerald G/WikimediaCommons

On Oct. 21 Andrew Gelman asked on his blog, "What's the probability that Daniel Murphy hits a home run tonight?" (in a record-setting 6th straight playoff game):
http://andrewgelman.com/2015/10/21/whats-the-probability-that-daniel-murphy-hits-a-home-run-tonight/

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.

Ben and Natalie Deliver

1)  Of math tests, fair and otherwise...
Another post from Ben Orlin that had me alternating between chuckling and thinking (...and trying to figure out how he gets SO MUCH expression into stick figures!):
http://mathwithbaddrawings.com/2015/10/21/the-best-way-to-kill-a-math-test-try-to-make-it-fair/

And YO, Ben, get your tail back over HERE... we can't be sharing your level of talent with Scotland, when you're needed badly in the U.S.! ;-)

2)  Now if you're wanting a little more serious, mind-stretching math, DO NOT MISS Natalie Wolchover's latest exquisite piece on "graph coloring" for Quanta Magazine:
https://www.quantamagazine.org/20151020-perfect-graph-coloring/

Aaronson Optimized...

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):

http://www.scottaaronson.com/blog/?p=2494

(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...)

Of Math Tribes, Brains, and Beauty...

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):

https://golem.ph.utexas.edu/category/2015/10/four_tribes_of_mathematicians.html

Mechanization, Mathematics, Mencken...

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:

https://mathematicswithoutapologies.wordpress.com/2015/09/08/does-mathematics-work-like-this/

To whet your appetite though, it starts off by referencing H.L. Mencken:

"When H.L. Mencken, an avowed atheist, was asked if he believed in baptism, he replied 'Believe in it? I’ve seen it done!'"

Pursuing Alexander...

A. Grothendieck via Wikipedia

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:

http://blogs.scientificamerican.com/roots-of-unity/a-close-encounter-with-grothendieck-and-his-fence/

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?

Still Fuzzy on Fuzzy Logic

"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.

Another Monday Puzzler

Another little brainteaser I've adapted from a recent Quora.com mathematics thread:

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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answer:

1. Have a friend open the letter.
2. 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.
3. 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.

Robots, Humans, Ants

 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."

Calling All Publishers...

via Stilfehler/WikimediaCommons

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:

http://mathwithbaddrawings.com/2015/10/07/practice-math-like-a-baby/

p.s... It will be a missed opportunity (and a loss to present and future generations), if some publisher out there doesn't eventually put out a compendium of Ben's work!