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?
.
.answer below
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
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!

Ubiquitous Benford

Princeton University Press presents this offering on Benford's law:
http://tinyurl.com/qhfmccy

...and Benford also shines through in this recent video from Numberphile discussing the powers of 2:

A Monday Puzzle

To start the week, a puzzle I've adapted directly from a Quora thread:
https://www.quora.com/What-is-the-best-math-puzzle-ever

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.

M1:  Ahh, now I know!

Question: how old are the 3 daughters???

***********************************
.
.answer below
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
answer:
3, 3, and 8

"true, deep beauty... comes only with experience and familiarity" [Sunday Reflection]

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":

http://devlinsangle.blogspot.com/2015/10/letter-to-calculus-student-sequel.html

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/

Lastly, enjoy Dr. Feynman:

On Random and Deliberative Processes (College Admissions)

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':**

http://mathwithbaddrawings.com/2015/09/30/why-ive-stopped-doing-interviews-for-yale/

It's all about the "factory process" of today's college admissions, specifically at a place like Yale.

Not only a fun read, but either his cartooning has gotten better over time, or I've lowered my standards, 'cuz even his lovable drawings are a hoot.

Not much math involved, but just some life-experience most of us can relate to either from our own lives or via our children or friends.


**  apologies for not providing a trigger warning before proffering that evocative phrase...