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.

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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???

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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...

Women In Mathematics

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":

Let's Hear It For 7...

It was a slow math weekend, so here's all I got for you:

Having a child anytime soon... have you considered the name "Seven"?  Mona Chalabi reports finding 1584 people in the U.S. with that very appellation, more than any other integer between 1 and 20:
http://fivethirtyeight.com/datalab/there-are-1584-people-in-america-named-seven/

As you may recall, in a survey less than 2 years ago, "Seven" was also found to be the world's "favorite number." Soooo, it's a beautiful name.

...as George Costanza was thrilled to inform you:



And if you don't want to name your kid in honor of Mickey Mantle, well, fine, name him/her "Yogi" instead.

The Essence of Mathematics

Sunday's Reflection:

"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.'"

-- Daniel Tammet, from "Thinking In Numbers"

 

Overflow Links

Had so many links to use for the potpourri over at MathTango this Friday, decided to move a few over to here for this week:

Latest (126th) "Carnival of Mathematics" from last Friday:
https://cavmaths.wordpress.com/2015/09/05/carnival-of-mathematics/

New "Math Teachers At Play" blog carnival posted, as well:
https://lifethroughamathematicianseyes.wordpress.com/2015/09/21/math-teachers-at-play-90/

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!

...and as always, Mike's Math Page covered a lot of things this week:
https://mikesmathpage.wordpress.com/

Groucho Marx: Philosopher/Logician

Greg Williams caricature via WikimediaCommons

"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!).
 ;-)

Riemann...

Yesterday, Peter Woit passed along some interesting Riemann Hypothesis links here:
http://www.math.columbia.edu/~woit/wordpress/?p=7996 

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.

Not-so-common Common Core... (Remarkable Post)

A super post from biologist Lior Pachter addressing Common Core from a different angle, employing unsolved problems (LOT of potential food for thought here):

https://liorpachter.wordpress.com/2015/09/20/unsolved-problems-with-the-common-core/

As Pachter puts it, he believes there is a major "shortcoming in the almost universal perspective on education that the common core embodies:
The emphasis on what K–12 students ought to learn about what is known has sidelined an important discussion about what they should learn about what is not known."

Pachter proposes several unsolved problems that can be introduced to young people at different levels.  While admitting that K-12 students aren't likely to find solutions to such problems he argues that the problems "provide many teachable moments and context for the mathematics that does constitute the common core, and (at least in my opinion) are fun to explore (for kids and adults alike). Perhaps most importantly, the unsolved problems and conjectures reveal that the mathematics taught in K–12 is right at the edge of our knowledge: we are always walking right along the precipice of mystery. This is true for other subjects taught in K–12 as well, and in my view this reality is one of the important lessons children can and should learn in school."

Just a remarkable post I commend to all educators! (some of the perspective Pachter is proposing I think may already be inherent to the goals of Common Core, but not in the precise way he outlines).

Ahhh, The Calculus

Sunday reflection....

"The calculus is humanity's great meditation on the theme of continuity, its first and most audacious attempt to represent the world, or to create it, by means of symbolic forms that in their power go beyond the usual hopelessly limited descriptions that we habitually employ. There is more to the calculus than the fundamental theorem and more to mathematics than the calculus. And yet the calculus has a singular power to command the attention of educated men and women. It carries with it the innocence of an abstract pursuit successfully accomplished. It is a great and powerful theory arising at the very moment human beings contemplated the infinite for the first time: sequences without end, infinite additions, limits flickering in the far distance. There is nothing in our experience that suggests that mathematics such as this should work, so that the successes of the calculus in unifying aspects of experience are tantalizing but incomplete evidence that of the doors of perception, some at least may open and some at least may lead to someplace beyond."

-- David Berlinski (from "A Tour of The Calculus")


[p.s., over at MathTango this morning I recommend two recent books.]

A Friday Puzzle

To end the week, a problem very similar to the famous "two envelope paradox," except that while the two envelope version continues to be a source of contentious debate, the "three envelope problem" has a definite answer.
I've adapted this from Thomas Povey's "Professor Povey's Perplexing Problems," a volume I'll say more about in a Sunday posting over at MathTango:

You're handed an envelope, which upon opening, has X number of dollars in it. The presenter now places (out of your view) 2X dollars into another envelope and X/2 dollars in a third indistinguishable envelope (i.e. the values could be 100, 200, and 50). Now you are asked if you wish to hold onto your current envelope with X dollars or swap for either of the other two envelopes. Should you swap???

This sounds very similar to the two-envelope situation but is subtly different. In the two-envelope case, the X value used must simultaneously or ambiguously be viewed as potentially the largest or the smallest value when computing the various probable outcomes. In the three envelope case we have 3 distinctive and fixed values, X/2, X, 2X. As a result it turns out that the computed "expected value" of switching is more definitively 5X/4, and thus worth doing (i.e., 5X/4 is greater than X).

[ 5X/4, by the way, is one of the solutions to the two-envelope paradox as well, obviously arguing for swapping; the problem is that alternative calculations are logically possible that lead to a don't-swap conclusion -- and the back-and-forth arguments, based on small nuances, could give you a migraine! ;-)]

Vi Hart Climbs Infinite Trees

The latest from the remarkable Vi Hart (and you'll be pleased to know it involves a notebook, not a microwave oven ;-):

Mathematical Discovery... Deconstructed

This morning's Sunday reflection from Stanislas Dehaene's "Consciousness and the Brain":

"[Jacques] Hadamard deconstructed the process of mathematical discovery into four successive stages: initiation, incubation, illumination, and verification. Initiation covers all the preparatory work, the deliberate conscious exploration of a problem. This frontal attack, unfortunately, often remains fruitless -- but all may not be lost, for it launches the unconscious mind on a quest. The incubation phase -- an invisible brewing period during which the mind remains vaguely preoccupied with the problem but shows no conscious sign of working hard on it -- can start. Incubation would remain undetected, were it not for its effects. Suddenly, after a good night's sleep or a relaxing walk, illumination occurs: the solution appears in all its glory and invades the mathematician's conscious mind. More often than not, it is correct. However, a slow and effortful process of conscious verification is nevertheless required to nail all the details down."

Get A Life!

In his latest book, "Numbers: Their Tales, Types, and Treasures," Alfred Posamentier mentions what he labels, "Pythagorean Curiosity #4":

It seems that in the mid-1600s the ever-inquisitive Pierre de Fermat sought a Pythagorean triple wherein the SUM of the two smaller values (a + b) was a square integer, AND the largest triple (c) was also a square integer.
Well, he found one such triple:

(a) 4,565,486,027,761
(b) 1,061,652,293,520 and
(c) 4,687,298,610,289

where a + b = 5,627,138,321,281 or 2,372,159²  and c = 2,165,017² 

Mind you, of course, no computers in those days!

MOREOVER, Fermat proved that this was the smallest such Pythagorean triple! (I don't know if any more such triples have been found in the almost four centuries since?)

All of which leads me to imagine being alive in 1643 (when Fermat concocted the problem) and sayin', "YO Pierre, uhhh, GET A LIFE!"  ;-)

Some Fave Math Sites

A bit ago, in reaction to a list compiled at Wired Magazine, I shared a list of my favorite science communication sites. Well, I'm involved with the group over at mathblogging.org site (which, if you're not already familiar with, you should get to know) and they are requesting that some of us designate our favorite math sites, so figure I might as well share my current math faves here as well:

Will start with this baker's dozen (in alphabetical order):

Bit-player
http://bit-player.org/

CAS Musings
https://casmusings.wordpress.com/

CTK Insights
http://www.mathteacherctk.com/blog/

Devlin's Angle
http://devlinsangle.blogspot.com/

Error Statistics Philosophy
http://errorstatistics.com/

Gödel's Last Letter and P=NP
https://rjlipton.wordpress.com/

Math blog
http://math-blog.com/

Math With Bad Drawings
http://mathwithbaddrawings.com/

Mike's Math Page
https://mikesmathpage.wordpress.com/

Mind Your Decisions
http://mindyourdecisions.com/blog/

Numberphile
https://www.youtube.com/channel/UCoxcjq-8xIDTYp3uz647V5A

Roots of Unity
http://blogs.scientificamerican.com/roots-of-unity/

Tanya Khovanova's Math Blog
http://blog.tanyakhovanova.com/

There are LOTS of primary and secondary math education blogs that are excellent as well, but I don't follow many of them closely enough to include in a personal faves list (but they're out there).

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Below are five additional sites whose mathematical content I GREATLY enjoy, BUT hesitate to list as 'favorite math sites' since a high percentage of their postings are not specific to math:

Evolution Blog 
http://scienceblogs.com/evolutionblog/
 
Futility Closet
http://www.futilitycloset.com/

Mathbabe
http://mathbabe.org/

Quanta Magazine
https://www.quantamagazine.org/

Shtetl-Optimized 
http://www.scottaaronson.com/blog/
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Finally, to round up to 20 picks, I'll throw in two other sites that aren't blogs, but are full of good stuff:

James Tanton
http://www.jamestanton.com/

Plus maths
https://plus.maths.org/content/
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Though a bit short of education blogs and highly-technical sites, the above 20 picks should offer a wide-ranging, varied mix of content for the popular math reader (and at mathblogging.org there are 100s more to sample).

Analogies...

Quick and short Sunday reflection today:

"Good mathematicians see analogies between theorems or theories, but the very best ones see analogies between analogies."
-- Stefan Banach

Two Good Mid-week Reads

2 longish reads to pass along today....

"For reasons that I don’t fully understand, our mathematical culture encourages us to define our mathematical ability by what we don’t know, what we aren’t able to do, rather than by what we do know and have learned how to do.  The power of culture is strong, with deep roots…"

A thoughtful post (including the above) from AMS blogs, that I suspect everyone can think about:

http://tinyurl.com/o7vnwdf

It's about our self-perception, as math students, of our own abilities, and how that aids or hinders us.

Then, interestingly, the latest post from Keith Devlin reviewing a new math-oriented movie, "A Brilliant Young Mind," that somewhat dovetails the above AMS post (especially see Devlin's "postscript"):

http://devlinsangle.blogspot.com/2015/09/a-brilliant-young-mind-imo-goes-to.html

Good reads, and food for thought, if you can set aside a little time....
(also, both posts contain several additional interesting links)

Parsing the Failure To Replicate

This study in Science, involving attempts to replicate 100 prior research studies, has been making the rounds (I've linked to the story before) for finding only about 1/3 of psychology studies statistically replicable; in some places setting off a bit of a firestorm.
Virginia Mayo takes an extended, more nuanced look at the issue here:

http://tinyurl.com/nkqd7bd

Worthwhile reading if you've been following the controversy (one dispute is whether the statistical methods employed are inherently weak/poor, even in need of being discarded, or are the methods fine, but simply abused/misused often in their application).