Sadly, two years ago today, Sally Ride left us at the all-too-young age of 61. I wrote a brief bit about her one week ago. Her organization for young people, "Sally Ride Science" is here. Lynn Sherr's biography of her came out a little over a month ago. And here's a 1984 PBS Nova clip of Sally as the inspiring, professional, enthusiastic scientist she was:

Meanwhile, this week, Jim Al-Khalili's last seasonal BBC broadcast of "The Life Scientific" (which gets rave reviews) was with distinguished chemist Carol Robinson, about her struggles toward success and recognition as a female in science:

Boys are often encouraged, or even pushed, towards science careers,
while girls frequently have similar innate interests discouraged or even
blocked. A couple of viral ads from a month ago addressed that issue, and warrant repeated viewing... though I can only do so with my semi-cynical post from a month ago on the mixture of corporatism and message:

One quick side-note, speaking of daring, independent-minded women, tomorrow marks the birthday anniversary of trailblazing American aviator Amelia Earhart.

What attracts me to these women is not just their abilities, but their courage and determination… especially the courage of their convictions, regardless of societal bounds or expectations, seeking, in a sense, as Thoreau once exhorted, "to live deep and suck out all the marrow of life."
I'll end this by returning full circle to the U.S. Space Program... and John Denver's tribute to the seven astronauts who perished aboard Shuttle Challenger 28 years ago, including the first teacher in space, 37-year-old Christa McAuliffe, a role model for countless young people at the time (an additional female astronaut on the doomed crew was Judy Resnik):

Sally Ride served famously and admirably (alongside Richard Feynman) on the Presidential commission that investigated the Challenger disaster and found decision-making flaws in NASA's organization hierarchy.
I'm not a Ronald Reagan fan, but I'll close out with his famous words to a grieving nation 28½ years ago:

"We will never forget them, nor the last time we saw them, this morning, as they prepared for their journey and waved goodbye, and 'slipped the surly bonds of Earth' to 'touch the face of God.'"

1) An emailer asks me to lend support for a kickstarter project (documentary) of some independent NY filmmakers, focused on Erdös-Bacon numbers and networks.

Not sure how much general interest there is for such topics. But Simon Singh did cover them interestingly over a few pages in his popular book on The Simpsons last year. People have had fun with the 'six degrees of Kevin Bacon" as a parlour game for several decades, and of course Erdös is a fascinating character in his own right… so maybe there is a way to draw in a wider audience?
At any rate, see what you think, and pass it along if you know others who might be interested. Whatever brings more math to the public is a positive, and they have less than a month left to make their kickstarter goal.

2) And in a second emailed request, a survey is being conducted to study the philosophical intuitions of mathematicians "relating to the objects and methodology of mathematics." If you are a mathematician and can take time to participate please go here (they'd like as large a sample as possible, so feel free to pass on to others also):

...Lastly, this might be a nice time to re-link to one of my fave photos on all the Web -- a 10-year-old Terry Tao and Paul Erdös, almost mirroring each other, in contemplation of a math problem:

A nice old Martin Gardner puzzle to start the week, which he titled "Fifteen Finesse" in his slim "Aha!" volume (I've adapted it, though still using the same example he used):

There is a new carnival game in town called "Fifteen." It involves a board with the numbers 1 through 9 laid out in order (like above). You play against the carny; doesn't matter who goes first, but you will take turns back-and-forth. You each put coins down one-at-a-time on a single number and then "own" that number ('til someone wins or all 9 numbers are used up). The carny will be putting silver dollars down while you put nickels down. The object is to own any THREE numbers that add up to 15, before your opponent does (for example, 3, 5, and 7). Whoever does this gets all the money played, in cases of draws (no winner) you each take your money back.

So we'll take Gardner's example:
You go first putting a nickel on 7. Carny puts a dollar on 8. You put a nickel on 2. Carny puts a dollar on 6 (and blocks you), realizing that you will win the game if allowed to do so (7+2+6). Now you are forced to put a nickel on 1 to block Carny from winning the game on his next move (8+6+1). Carny puts his next dollar on 4. You again block his chance (6+4+5) at a win by putting a nickel on 5. But next Carny places a dollar on 3 and still wins, since 8 + 4 + 3 is 15. You lose your money. The question is, is there any strategy by which you could be assured a win?

Well, needless to say if Carny is giving you odds of a dollar to a nickel, there's no method for you winning a round… but there is a way for Carny to make sure he never loses a round (wins or draws every game). Can you see it?
.
.
.
.
.
.
.
.
What Gardner goes on to explain is that this game is "isomorphic" to classic tic-tac-toe (another game one can always win or draw, and never lose, if played strategically).
There are only eight possible 3-digit triplets that are winners for the Fifteen game:

1+5+9
1+6+8
2+4+9
2+5+8
2+6+7
3+4+8
3+5+7
4+5+6

and these can be arranged in a "magic" square formation:

2 9 4
7 5 3
6 1 8

such that EACH horizontal and vertical line and diagonal is one of the winning triplets.
The carny simply keeps a card illustrating this 'magic square' hidden from (your) sight and makes his number selections referring to it (trying to get 3 in a row, while preventing you from doing so), as if he were playing tic-tac-toe. Then, he can't lose (assuming he doesn't screw up ;-).

Most of you likely know of Steve Strogatz's book, "The Calculus of Friendship," about the touching lifelong relationship between Dr. Strogatz and one of his high school math teachers.
Today, for a Sunday reverie, I'll just link to this 2009 RadioLab episode (15 mins., entitled "Calculove") about the story… worth a 2nd listen if you've heard it before (or already read the book), and definitely worth a first-listen if you aren't familiar with the story:

The irrepressible Veritasium (Derek Muller) offers a quick lesson on randomness, information, entropy, quantum mechanics, and chaos (in 10 minutes no less!):

One good lay-friendly book that covers/expands on some of this same ground is Charles Seife's "Decoding the Universe."

Or, maybe I should've entitled this: "Sally Ride... R.I.P."

(Yet anotherbrief diversion today… maybe this will become a series… NOT)

When I posted the Richard Feynman piece last Sunday, an emailer wrote to say (scold?) that I hadn't mentioned a single female in my short list of admired people. So let me rectify that a little bit, with three who come to mind fairly quickly:

First is Jane Goodall, who I admire greatly, but more as a spokesperson and role model than as a scientist (nothing against her, I just don't hold high regard for anthropology!). In the political arena there is Elizabeth Warren who shows tremendous potential, but way too early to tell yet.
The only "scientist" though who really jumps to my mind is Sally Ride (of course there are plenty of female scientists I'm little-familiar with), who died two years ago this month. I largely lost track of her once she left NASA, though I knew she was involved in education (at different levels), and heard rumors of her gender preferences. When she died and the rumors of her personal life were confirmed, I was disappointed. NO, not because of the lesbianism, but because of her failure to publicly acknowledge it during her lifetime. I respect and fully understand that decision, but still her trepidations dropped her down a smidgen for me. It says something VERY sad about our society that someone with the courage to pursue the science/space career she had, nonetheless lacked the courage to simply be herself in public. Pretty clearly, what she feared was the judgment of others; the barbed, shrewish commentary and invasion of privacy that might ensue from culture police, made even worse in the day of the internet. Sally escaped unscathed, but what anguish did it bring to her life hiding constantly behind such a cloak of privacy...

I can almost imagine Sally and Dick Feynman off in another world now, sitting around a table somewhere, playing poker (...maybe even strip poker at Richard's request!), talking physics and differential equations... and laughing at what a screwed-up world they left behind.

Once again, departing from math today (sorry) to help keep this already-viral piece floating along the Web (ICYMI)... Corporate wretch #Comcast doing what comes naturally over the phone (being a corporate ass):

After yesterday's seriousness feel like lightening the mood a bit with this golden oldie I've posted before (there's even some potential math involved, but we'll ignore it):

[BTW, I tried to get back to each emailer earlier today, but some glitchy things were happening with email, so if you emailed me before ~1pm EDT but didn't receive a response yet try again.]

An entry completely off the beaten track...
those who don't care about or like Richard Feynman need not read. This
just goes out to his fans at a time he is drawing some heat on the
internet:
[...and I'm adding small bits to this post as time goes on, and different issues arise around the Web.]

I've never had many heroes in
life; don't even much like the term "heroes," as I tend to see people
with all their warts and weaknesses on display. But there are some
folks I hold in high esteem. In politics and public life, Ralph Nader,
Jerry Brown, Bill Bradley, Morris Dees, and historically, Abraham Lincoln and
Clarence Darrow, perhaps Bobby Kennedy if he had lived. I throw the
names out, not to name-drop, but just to establish my leanings [I offer a few female alternatives here]. In
science, no living heroes. I liked Martin Gardner of course, but
probably the closest to a hero, and in a league all his own, was Richard
Feynman. I fancied Feynman long before the internet came along (and
long before he died), for all the reasons that don't need spelling out
to his legion of admirers. As a scientist, a teacher, a thinker, an
inspirer, an inquirer, and just a character, he was perhaps unsurpassed
in the 20th century.

But, he had his human flaws… he
wrote about them; his biographers wrote about them; they were never
hidden, the way most of us keep our unsavory skeletons behind closet
doors. They were part of who he was, especially as a younger man. And
for some reason, now all these years later, they've become grist for
some on the internet; pertaining to what we view today as rampant
"sexism."
Feynman matured formatively in the 1930s/40s/50s largely
in an all-boys network (MIT, Caltech, the Manhattan Project). He was smart, attractive, clever. Did every
male in those networks have Feynman's sexist predilections… perhaps not,
but I suspect all those with Feynman's charm, intelligence, and hormones
did! (admittedly, just my guess)

I'll do a little of what some Webbites patronizingly call "mansplaining":
When
new mothers suffer post-partum depression and behave badly around their
new-borns, we don't call them lousy, incompetent, dangerous moms
(though we could) and recommend they give their babies up for adoption…
we try to treat the hormonal/biochemical imbalances they are falling
prey to. For that matter, we no longer castigate women suffering major PMS
mood disorders and send them off to asylums either, but again try to treat the
underlying conditions. Males too, especially young males, and especially
young males surrounded by other young males for lengthy periods, are
hugely subject to hormonal/biochemical drives in addition to peer
pressures. (Of course we are all subject to our biochemistry and neurology throughout our lives.) IF that was the case for Richard Feynman 60-70 years ago, and
he lacked certain impulse controls that are expected today, so be it;
it does little good to go back and rake over those ashes now. Move on. Nothing more or new to see here. This is all old, old news.

...More old news: My
father murdered people 70+ years ago. As a young, wide-eyed,
idealistic American he was shipped off to Europe in WWII and ordered to
kill Nazis (to 'save the world for democracy') -- don't argue with me about the definition of
"murder;" he deliberately killed people, under orders. He came home from
that experience a pacifist, having seen atrocities he couldn't fathom…
atrocities committed, not just by the Nazis, but repeatedly by American
soldiers. I can assure you if your dads fought in WWII or Korea or Viet
Nam they saw or participated in atrocities as well, even if they won't speak of it out loud... because, you know, 'War is Hell,' and
believe-it-or-not, well, Americans just aren't always choirboys and saints. But we
don't judge our fathers' lives by what they did in such a
context, no matter how wretched it might've been. We look over their
entire lives, motives, intents, behaviors, contributions, values, and
weigh it altogether.

If we were going to judge
people of the past by today's standards, then of course all the Founding
Fathers fail, just as most of us will fail the ethical standards in
place 100 years from now (hell, as far as I'm concerned we are already ethical failures, in addition to being hypocrites and
egotists -- a religious person would say we are all sinners, I simply
say we are all 'human,' occasionally doing the best we can, usually
not). How many of us would wish our entire personal lives opened up to public scrutiny and judgment. The "purists" out there want something that doesn't exist: a scientist with nothing objectionable in his past. I'm quickly judgmental myself of politicians and businessmen (they hold sway over my life and my country, moreso than do scientists), so I understand how easy being judgmental is... but I also know it comes off as sanctimonious and better avoided.

Attacking Feynman to improve the lot for women today, is like attacking George Wallace to improve race relations today; it serves little purpose or impact. Face today's culprits in the here-and-now and deal with them. Or, another analogy, from the corporate world: German transnational company Bayer AG, famous for cropsciences and healthcare, had Nazi associations back in the day (I believe Volkswagen did too???); should we dredge up that history of Bayer and judge them accordingly now? Or is it more appropriate to take companies to task today for what they are actually doing today? (Yo, Comcast...)

Anyway,
I'll stack up Richard Feynman's contributions to science, society,
education, and humanity, and even stack up his OVERALL values system,
against those of his new-born, petulant, self-absorbed internet critics,
anytime and have no doubt he'll leave them in his dust. It hurts to
see his name dragged through the mud at this late date, especially by
people who may never leave any similar lasting mark on the world.

It's
a shame that 26 years after his death I feel a need to say Richard
Feynman, R.I.P… Rest... In... Peace. But we live in a day when it is
hubris-driven sport to point fingers at the famous, the vaunted, the
iconic, and bring them down to size (and it's especially easy to impugn the
deceased). Except Feynman, more than most, already understood his size
in the Universe; I hope some other folks eventually understand theirs.

[I apologize to readers for this intrusion into the usual math fare of this blog, but found it necessary. For math fare see my interview with Jordan Ellenberg over at MathTango.]

ADDENDUM: Of course there are really too many to even choose from, but I'm adding one of my favorite clips of Feynman:

[I expect I may be adding more to this post as time goes on. In the meantime I recommend people read the above volume of his letters, edited by his daughter, and reviewed by Freeman Dyson here.]

A few general remarks about some of the stories (which I won't repeat) that have landed Feynman in hot water. First, I'd be cautious about anecdotes from Feynman's own two popular autobiographical volumes ("Surely, You're Joking Mr. Feynman" and "What Do You Care What Other People Think?"). These books were ghostwritten from Feynman narratives, and ghostwriters often embellish, or 'punch up,' stories for marketing purposes. Not all details there are corroborated. Also, Feynman lived through the 'sexual revolution' or 'sexual liberation' of the 60s/70s, a time when it wasn't uncommon for many to broadcast their sexual exploits and freedom… just possibly someone in a field viewed as nerdy as physics might feel compelled to burnish their reputation or prowess a bit with details that today seem appalling. NO, I'm not making excuses, but am acknowledging that I don't know the precise accuracy of all the tales told around and by Feynman… and, neither do you. For that matter most of us don't even know much about what went on behind the closed doors of our own parents' rooms… that we might find very perturbing in today's 'enlightened' times.

Several of the incidents focused on by critics relate to Feynman's second wife and divorce, a messy affair. The bitterness of the divorce alone mean the exact details as reported must be taken with caution. But moreover, in those days "incompatibility" was not a basis in most states for dissolving a marriage. If one couldn't show "abandonment" (having had no contact with the spouse for 1-2 years) the only alternative to opt for was some form of 'abuse' as grounds for divorce, which again meant that stories were embellished, or even concocted, just to get the legal process moving along.

None of this is to absolve Richard Feynman of his worst behaviors, but the thing is, I don't absolve any of us of our choices throughout the day (in food, clothing, material goods, transportation, entertainment, words, etc.), that send damaging ripples out into the world. All actions have consequences, and we render harm to other people, animals, and the planet, even if unintentionally. All of us (and our parents/grandparents) are vulnerable to character assassination by selective reporting of bits of our lives. I can't stand idly by and watch that happen to an individual who contributed SO much, and who, if alive today, would likely be one of the strongest proponents around for women and minorities in science.

"And yet the history of mathematics is a history of aggressive territorial expansion, as mathematical techniques get broader and richer, and mathematicians find ways to address questions previously thought as outside their domain. 'A mathematical theory of probability' sounds unexceptional now, but once it would have seemed a massive overreach; math was about the certain and the true, not the random and the maybe-so! All that changed when Pascal, Bernoulli, and others found mathematical laws that governed the workings of chance. A mathematical theory of infinity? Before the work of Georg Cantor in the nineteenth century, the study of the infinite was as much theology as science; now, we understand Cantor's theory of multiple infinities, each one infinitely larger than the last, well enough to teach it to first-year math majors. (To be fair, it does kind of blow their minds.)….

"Will there be a mathematical theory of consciousness? Of society? Of aesthetics? People are trying, that's for sure, with only limited success so far. You should distrust all such claims on instinct. But you should also keep in mind that they might end up getting some important things right."

A few months back Evelyn Lamb introduced me to a humor site I love, called "Gauss Facts" -- a sort of over-the-top jocular fan-site for one of the greatest mathematicians of all time. Worth checking out every few months, or whenever you feel the need to inject a little humor into your day (...and, you're a mathematician):

"Gauss can recite all of pi -- backwards."
"Gauss shaves both himself and Bertrand Russell."
"Erdos believed God had a book of all perfect mathematical proofs. God believes Gauss has such a book."

I recently discovered a math misconception I was unaware of… and will pass along for any others not aware of it. "Pasch's axiom" is a little-discussed "axiom," discovered by Moritz Pasch in 1882, that is needed for, but missing from, Euclid's axioms of plane geometry. I'd heard of it previously but now learn that what I'd always considered Pasch's axiom is actually Pasch's "theorem," and that his actual "axiom" is slightly different.

Here is what Davis/Hersh write of Pasch's "axiom" in their classic, "The Mathematical Experience," where I think I may have first learned of it (you can use the line depiction below for visual reference):

____a__________b________________c_____d_________

"Just what constitutes the 'straightness' of the straight line? There is undoubtedly more in this notion than we know and more than we can state in words or formulas. Here is an instance of this 'more.' Suppose a, b, c, d are four points on a line. Suppose b is between a and c, and c is between b and d. Then what can we conclude about a, b, and d? It will not take you long to conclude that b must lie between a and d. This fact, surprisingly, cannot be proved from Euclid's axioms: it has to be added as an additional axiom in geometry. This omission of Euclid was first noticed 2000 years after Euclid, by M. Pasch in 1882! Moreover, there are important theorems in Euclid whose complete proof requires Pasch's axiom; without it, the proofs are not valid."

But it turns out that this almost trivial-sounding statement of Pasch's is actually his "theorem," while his "axiom," is stated rather more interestingly and differently:

"In a plane, if a line intersects one side of a triangle internally then it intersects precisely one other side internally and the third side externally, if it does not pass through a vertex of the triangle." (from Wikipedia)

This is interesting in part, because the "theorem" sounds so much simpler or more basic than the "axiom"… reminiscent of Euclid's 5th postulate sounding so much more involved than his other axioms (also interesting that it took until 1882 to discover this missing component of Euclid's axioms). In fact, bopping around the Web, the two Pasch outcomes are often confused, and it is not even clear to me what makes one an "axiom" and the other a "theorem" (and so the confusion is understandable, and not limited to Davis and Hersh).

Like many, I'm a big fan of NPR's "This American Life," often listening to it's 24-hr./day archival radio station, where I recently heard a replay of their "testosterone" episode from 2002. The whole show is pretty interesting, but I especially had to re-chuckle at a short part I've reported on before. I'll just re-run that post from about a year ago below:

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A dash of the offbeat today….

Some weeks back, NPR's "This American Life" ran its 500th show, playing bits from dozens of favorite episodes over the years. Early on, was a 2002 episode piece with an individual who was born female, but always felt more male, and took testosterone shots later in life to finally make the transition in gender. The end of this little segment still rattles around in my mind… After relating a lot of already interesting stories to host Ira Glass about how the change in gender affected him, the interviewee is asked by Ira if there are any other alterations due to testosterone he thinks worth mentioning. The individual responds that after taking testosterone he "became interested in science; I was never interested in science before." To this Ira can't help but chuckle and respond, "NO WAY!" adding that such a response is "setting us back 100 years." The individual goes on to insist that testosterone resulted in "understanding physics in a way I never did before."
You can hear the whole episode here (but the specific exchange occurs around the 22:30 point):

Anyway, what an amazing thought that a few shots of testosterone could, waaah-laaah!, boost science aptitude! I assume math aptitude would be susceptible as well, so I googled "testosterone +math" which led to a 2007 piece [from a blog which no longer exists!] showing some connection.

Toward the end it concludes:

"The important thing to get out of this is that there is an intra-sex correlation between sex hormones and math scores, and this is in the direction one would expect. More prenatal exposure to the male sex hormone testosterone correlates with higher math scores."

before adding this cautionary, anecdotal note:

"However, I also believe that this finding, by itself, does not fully explain the connection between math ability and sex hormones. The stereotype is that the kids who are good at math are the nerdy non-athletic kids, the opposite of the high-testosterone football playing kids. So although there is a scientifically validated correlation between prenatal testosterone and math, there is an anecdotal opposite correlation for teenage boys. Less testosterone in the teenage years seems to predict higher math performance."

This is just one study of course and there are other studies of the relationship between hormones and science/math ability... and plenty of complicating, intervening variables, as well. Still, it is fascinating that at least one individual, albeit on a pop radio show, feels that the administration of testosterone, like a magic potion, brought on an interest in science that previously didn't exist.

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p.s.... for other lovers of "This American Life," if you don't already know this, the 24/7 radio-stream of old episodes can be accessed here: http://www.thisamericanlife.org/listen/stream

If you missed last weekend's "On Being" episode (hosted by Krista Tippett on NPR) with mathematician Jim Bradley and philosopher of science Michael Ruse in discussion, you may enjoy it:

In case you're not familiar with "On Being" here is part of how the show bills itself:

"On Being is a Peabody Award-winning public radio conversation
and podcast, a Webby Award-winning website and online exploration, a
publisher and public event convener. On Being opens up the
animating questions at the center of human life: What does it mean to be
human, and how do we want to live? We explore these questions in their
richness and complexity in 21st-century lives and endeavors. We pursue
wisdom and moral imagination as much as knowledge; we esteem nuance and
poetry as much as fact."

"Why, sometimes I've believed as many as six impossible things before breakfast."
-- Lewis Carroll (Alice In Wonderland)

I love it when disparate mathy things I stumble across over a couple of days start bouncing off one-another in my head....

First, let me point readers to this per-usual great response today from Keith Devlin to a NY Times piece I had referenced here a couple days back (about Common Core): http://devlinsangle.blogspot.com/2014/07/the-power-of-dots.html
(DO NOT miss this post... whatever side of the issues you're on!)

...Next, a h/t to Vi Hart for introducing me to the YouTube channel of "Looking Glass Universe":

Vi mentioned it for its excellent short vids on physics topics, but there are also some more strictly math videos as well, including the below, wonderful treatment of one of my favorite, mind-boggling, math puzzles/conundrums (...Keith Devlin writes above that "understanding structure... requires human insight. It is not trivial and it is
difficult" -- I think this video possibly captures that in 3 minutes):

And finally, the above video, in turn, led me to recall a brief posting from a couple days back on the exquisite "Painter's Paradox" of Gabriel's Horn (another lovely mind-bender):

The sudden inter-meshing (cross-fertilization, if-you-will, in my mind) of Gabriel's Horn, the diagonal paradox, and Dr. Devlin's emphasis on structure, pattern, and insight/creativity (...which can make the seemingly impossible, possible), was for me, a beautiful thing!...

I'm a number-luvin' primate; hope you are too! ... "Shecky Riemann" is the fanciful pseudonym of a former psychology major/lab-tech (genetics), now cheerleading for mathematics! A product of the 60's who remains proud of his first Presidential vote for George McGovern ;-) ...Cats, parrots, and shelties adore him.
Li'l more bio here.

Herein, links & posts to prod, illuminate, amuse, and entertain the cerebrum, 'cuz... M.I.B.T.! (Math Iz a Beautiful Thang!)

............................... --In partial remembrance of Martin Gardner (1914-2010) who, in the words of mathematician Ronald Graham, “...turned 1000s of children into mathematicians, and 1000s of mathematicians into children.” :-) ...............................