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).

----------------------------------------------

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/
----------------------------------------------

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/
----------------------------------------------

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).

Expectations and Discoveries... and a Passing

This Sunday's reflection from Freeman Dyson (in "Dreams of Earth and Sky"):

"A typical proton-proton collision in the LHC will produce a large spray of secondary particles, and the collisions are occurring at a rate of millions per second. The machine must automatically discard the vast majority of the collisions, so that the small minority that might be scientifically important can be precisely recorded and analyzed, The criteria for discarding events must be written into the software program that controls the handling of information. The software program tells the detectors which collisions to ignore. There is a danger that the LHC can discover only things that the programmers of the software expected. The most important discoveries may be things that nobody expected. The most important discoveries may be missed."

                                                                      __  __  __

....on a side note, Oliver Sacks died this morning; I don't consider that a sad note -- his life and contributions were so glorious, and his approach to death so thoughtful and intrepid, that death is more like a punctuation mark at the end of another inspiring sentence from his rich, productive life and writings.
There will no doubt be profuse tributes/obits in coming days. This from the NY Times:

http://tinyurl.com/pxg3ak7

 

What Did You Do On Summer Vacation?

Siobhan Roberts probably didn't really get a summer vacation, being busier-than-ever with a book tour, but she reports on the mixture of "research and play" that filled the summers of some others:

http://www.newyorker.com/tech/elements/cogito-ergo-summer-the-beauty-of-summer-science

Drawing Straws with Chris and Marilyn

via WikimediaCommons

Today, Chris Harrow called attention to a Marilyn vos Savant probability problem from earlier in the month (an inquiry from a reader at Parade Magazine) that ran as follows:

"Say four people are drawing straws for a prize. My friends and I agree that the first person to draw has a 1 in 4 chance of getting the short straw. However, if he or she does get it, the second person’s chances drop to zero. And if the first person doesn’t get the short straw, the second person’s chances increase to 1 in 3, and so on. Our disagreement: Is it better to draw first, last, or does it make any difference?"

To which Marilyn simply responded:

"The order makes no difference. Envision all four people drawing straws, but instead, not looking at them yet. At this point, each person has a straw. Does it help to be the first or last one to look? No."

But Chris nicely fleshed it out a bit more as a teaching moment to make the distinction between frequentist and conditional probability; i.e. there are potentially two different probability sets here: 1) if all 4 straws are drawn and only THEN looked at, versus 2) if each straw is looked at as it is drawn:

https://casmusings.wordpress.com/2015/08/26/marilyn-vos-savant-and-conditional-probability/

And at the end of his post Chris makes an analogy from the straw-drawing to physics' quantum theory (or perhaps really Schrodinger's cat); I'm not sure if a physicist would be pleased with the analogy, but it's a nice thought exercise.

50-Cent Puzzles....

It's always a good weekend when I find a Raymond Smullyan book on sale for 50¢ in a local thrift store. This was a good weekend.
So we'll ease into the week with three simple quickies from the master logician (the first two I've adapted from the book, and the last one is verbatim from Smullyan):

1)  $5000 is stolen from a local store.  The robber or robbers make their get-away in a car. Soon thereafter 3 well-known criminals, A, B, and C, are picked up by law enforcement for questioning, and 3 facts are established:

  a)  NO ONE other than A, B, and/or C was possibly involved in the crime.
  b)  C NEVER pulls a robbery without A (and possibly others) also being involved.
  c)   B does NOT know how to drive.

Is A innocent or guilty?

2)  A train leaves from Boston heading to New York at 2 pm., and one hour later another train leaves New York heading for Boston on a parallel track, moving the same speed as the first train. Which of the two trains will be closer to Boston when the two meet?

3)  And lastly, this riddle from Smullyan:

"Those of you who know anything about Catholicism, do you happen to know if the Catholic Church allows a man to marry his widow's sister?"

answers below:
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1)  yes A is guilty (B couldn't do it by himself, because he couldn't drive the get away car, and if he did it with C, than, by #2 A must also be a guilty party. C couldn't have acted on his own without A.
A could've acted on his own, or with B or C, or both, but in some form had to partake.

2)  of course at the time they meet the two trains will be the same distance from Boston (...though a technical sort might argue that if by "meeting" one means the front of the locomotives meeting, than the tail/caboose end of the Boston train would still be "closer" to Boston).

3)  a dead man (widow's spouse) can't marry anyone!