Sunday, March 9, 2014
Fabulous book news!:
Jason Rosenhouse has edited a new anthology devoted to Raymond Smullyan, "Four Lives: a celebration of Raymond Smullyan":
This is the sort of book I will heartily recommend, sight unseen (although I'll now be looking for it!).
I think my first of several posts involving Smullyan was this one from the first few months of the blog:
Besides his math and logic books I also very much enjoyed his book on Taoism entitled "The Tao Is Silent." If you can find it, and are into Eastern religion/philosophy, I recommend it.
(....For any who don't know, the "knights and knaves" of the post-title is a reference to a common set of Smullyan logic puzzles involving an island of knights and knaves, or truthtellers and liars.)
Thursday, March 6, 2014
just some fun today....
Last weekend I linked to this post from Keith Devlin talking about "breakthroughs" happening when disparate mental parts make certain connections (producing the so-called "Eureka" moment); in his case, he related it to mountain biking:
I thought back to Dr. Devlin's piece the other day, when I experienced my own simple "Aha" moment in a different context:
Fawn Nguyen tweeted a link to a really fun puzzle on the Web:
It's called the "Flash Mind Reader" and if you're not familiar with it, go play with it RIGHT NOW, so I don't spoil it for you with what follows.
I can usually figure out puzzles fairly quickly from having seen so many and being familiar with them… when I'm NOT familiar with a given puzzle, then I'm as big a sucker as anyone! And this puzzle initially stumped me (even though it was just a variation off some other tricks I've seen). In frustration I put it aside, did some other things, and then went to take a shower, the puzzle cast from my mind.
My shower-head spews forth a broad stream of droplets… somehow my mind suddenly flashed with the image of those droplets being the array of right-hand number choices given for the puzzle… in a moment (which I still can't well-explain) the thought occurred that if some subset of those droplets, which looked random, was actually always identical to one another, and, if my attention could somehow be consistently drawn to THAT subset, than this would solve the puzzle. Stepping out of the shower I needed only think about it for another 60 seconds to realize how the problem worked. (If you don't know how it works even after this hint I've given, you can always just google "flash mind reader" to find the solution somewhere; I won't spell it out here.)
Decades ago, Arthur Koestler promoted his notion of "bisociation" between two autonomous mental constructs as being the key to not only scientific and artistic creativity, but to humor as well. Some of the best jokes, in short, result from a comedian weaving together an analogy that the audience never saw coming! (as a complete sidelight, perhaps worth noting that Douglas Hofstadter's latest tome, Surfaces and Essences, also focuses on the central role of analogies in our thinking processes; not just for creativity but for all thought).
Interesting too, that the mathematician-writers of the animated Simpsons TV show, likewise discern a strong link between their senses of humor and their mathematical ability. As author Simon Singh noted of the jokes/math-puzzle linkage: "Both have carefully constructed setups, both rely on a surprise twist, and both effectively have punch lines. Indeed, the best puzzles and jokes make you think and smile at the moment of realization." So who says that underlying mathematics there isn't a lot of fun to be had... be it mountain biking, writing jokes for Homer Simpson, working out puzzles, or showering!
Wednesday, March 5, 2014
Nice piece in Nature about the rise of collaborative Web efforts in science, with special focus on Tim Gowers' ongoing Polymath Project (started as an experimental project in 2009):
"Gowers' online challenge was a radical suggestion for mathematics — a field that is often viewed as the domain of lonely, secretive figures who work for years in isolation. And it went against the grain of the wider academic culture, which tends to encourage researchers to share their ideas only by publishing them."The article points out that there are now some commercial or incentivized collaborative projects as a way to increase participation, with even the Government getting involved: https://challenge.gov/
Business, mass education, and social media (and porn!) may be among the Web's most dominant uses, but I've long thought that truly the most ideal and promising use of the Internet would come in the form of scientific collaboration… the hive-mind solving problems, one-after-another, in a fraction of the time formerly required, leading to an exponential growth of knowledge/progress previously inconceivable. We've barely just begun... kudos to Gowers, Terence Tao, and others at the forefront.
Tuesday, March 4, 2014
Below is yet another recent article espousing the "math is beautiful" theme, but a bit more original in that it focuses on the link between math beauty and current fMRI research studying "the neural basis of beauty."
What I found even more interesting though comes at the end of the article when it cites the above formula (from Ramanujan) as the equation a consensus of mathematicians deemed the 'most ugly'! (can't say as I blame them... but it still remains beautiful that a human mind could even come up with it!):
Monday, March 3, 2014
Ed Frenkel's latest piece for the LA Times, with his message of bringing math appreciation (and ability) to a wider circle of students, has already been cited quite a bit around the Web, but in case you missed it:
In it he notes that, "...abstraction is all around us — and math is the language of abstraction...
"For the next generation to operate effectively, they must gain proficiency with abstraction, and that means mathematical knowledge plus conceptual thinking times logical reasoning — all things that a wider view of math would bring to the math classes at our schools."
Others frequently talk about this same focus on abstraction in terms of 'pattern recognition,' and in turn, many now use that focus on patterns to stress engaging children in math via their natural, playful interest in patterns.
The Atlantic had a great piece with Maria Droujkova that nicely dovetails, and fleshes out, Frenkel's article (somewhat interestingly, both Frenkel and Droujkova are immigrants from the old USSR):
Droujkova notes that the complexity of ideas and the difficulty of doing them "are separate, independent dimensions,” and children are capable of much more (math) learning than people think. She notes a progression from more informal ideas to more abstract ones and maintains that a "playful aspect" can be "retained along the entire journey": “This is what mathematicians do—they play with abstract ideas, but they still play.”
Both Droujkova and Frenkel are combating the contagion of so many young people being turned off to math at an early age.
I would urge all parents and educators of youngsters, not already familiar with Droujkova's work, to read the article, and followup with visits to her "natural math" website and related online materials:
I've said it before, and will say it again, we are so lucky to be living in a time when not only does the Web present a vast array of resources at one's fingertips for learning math, but a time when so many passionate people actively take the message of math farther and wider than ever before.
ADDENDUM: on a tangential subject, this morning's Diane Rehm Show (NPR) spent the last hour in a wide-ranging discussion of the controversial Common Core standards (as wide-ranging as could be squeezed into 1 hour):
Saturday, March 1, 2014
Two people who have pretty obviously been cloned, since they keep showing up everywhere(!) evangelizing for mathematics literacy, are Ed Frenkel and Keith Devlin... (no doubt, in some sort of brilliant disinformation strategy, they've been cloned by the very governmental agencies they keep warning us about ;-)
As for Frenkel... science sites, sure; math sites, well of course; NY Times, why not; and needless-to-mention Twitter... now this week Ed (or one of those clones) appears in Mother Jones online edition exhorting the importance of mathematics and his frequent message that it is not just for the gifted few, but IF taught right, for the masses.
Includes a podcast interview with Frenkel, whose Russian accent is almost as enticing as Keith Devlin's British one!
Lots of good points made by the man who dared to title a book, "Love and Math."
excerpt (from Mother Jones):
"...Frenkel views math as an 'archipelago of knowledge' that's universally available to all of us, and he's been everywhere of late spreading the word. In particular, Frenkel is intent on warning us about how people are constantly using (or misusing) math to get our personal data, to hack our emails, to game our stock markets. 'The powers that be sort of exploit our ignorance, and manipulate us more when we are less aware of mathematics,' said Frenkel."...
"To him, math—not religion—is the one shared body of firm, unchanging knowledge that we all possess and that nobody can ever take away from us… 'It's a great equalizer,' Frenkel says"....
"Forget the idea that [math is] alienating and hard. According to Frenkel, life is hard without it."
And Keith Devlin continues his math promotion, equating solving math problems with mountain biking (another of his passions), while hailing the lowly amygdala!:
His discussion of math's "Eureka" moments is especially interesting:
"How does the human mind make a breakthrough? How are we able to do something that we have not only never done before, but failed many times in attempts to do so? And why does the breakthrough always seem to occur when we are not consciously trying to solve the problem?All of which leads to some interesting cognitive/neuroscience speculation and praise for the amygdala's role in problem-solving.
"The first thing to note is that we never experience the process of making that breakthrough. Rather, what we experience, i.e., what we are conscious of, is having just made the breakthrough!
"The sensation we have is a combined one of both elation and surprise. Followed almost immediately by a feeling that it wasn’t so difficult after all!
"What are we to make of this strange process?"
Enjoy the whole piece... and then, go take a bike ride!
Thursday, February 27, 2014
1) "The central change in real-world maths of the last 50 or so years is that we automated the hell out of calculating." That's Conrad Wolfram in the course of espousing his view of digital reform for British math education (and singing the praises of Estonia for adopting a more computer-based math education system):
(...not surprisingly, he gets some push-back in the comments)
2) Several science bloggers do regular end-of-week linkfests to articles they found interesting over the prior week. I've been doing little "potpourri" posts haphazardly here at Math-Frolic from time to time to direct readers to pieces that I don't care to write a whole post on. I'll now try (as an experiment) to do a once-per-week "potpourri" offering over at MathTango (probably on Fri. or Sat.) of all the extra stories I want to take note of... a sort of weekly mini-math carnival solely of my own picks-of-interest.
Check MathTango tomorrow for the first one! (And then on Sun. or Mon. the next Math-Frolic interview will be up over at MathTango as well.)
3) Speaking of interviews... Frederick at White Group Mathematics recently interviewed yours truly for his blog (...wherein you learn that I'm waiting for a call from Taylor Swift ;-):
Tuesday, February 25, 2014
Monday, February 24, 2014
Monday, February 17, 2014
(Now with Addenda, at bottom....)
Not for the first time ;-), a tweet by Steven Strogatz caught my eye today.
But before I get to that tweet let me say that Strogatz was actually responding to another tweet from Jordan Ellenberg linking to a recent piece Strogatz did about the need for "empathy" in effective math communication:
A wonderful read, especially delightful for its interesting discussion of three of Strogatz's science communication "heroes": Richard Feynman, Stephen Jay Gould, and Lewis Thomas.
So DO read that piece. The tweet, however, from Dr. Strogatz that caught my eye ran as follows:
"A lot of us in math are on the Asperger's-autism spectrum, which can make the empathy issue even more challenging."
I thought that was a rather interesting remark, that 140 characters couldn't do justice to, so I googled around to see what I might find about Aspergers relation to math. Essentially, from what I saw, it seems safe to say that Aspergers individuals exhibit no significantly better mathematical (or other analytical) skill than the general population; indeed, many struggle greatly with math, the Asperger's spectrum being quite wide. But this isn't really what Strogatz is hinting at anyway… he's coming from the other side of the equation and implying that the population of (professional) mathematicians may have a higher number of Asperger's individuals within it than the population as a whole (i.e. the population of mathematicians might tend toward high Aspergers scores, even if Aspergers individuals, as a whole, don't tend toward mathematical aptitude) -- I really didn't find much in my brief search empirically addressing that question.
So am curious if anyone knows of any studies that have looked at say PhD.-level mathematicians or just working mathematicians, to see what percentage of them may score high for Aspergers, and is it greater than the general population? It would be an easy study to conduct, since there are simple verbal tests to indicate (not diagnose, but nonetheless, indicate) one's potential position on an Aspergers scale, and by administering such a test to a large enough random sample of working mathematicians, one might get an initial indication of mathematicians' standing relative to the overall population.
ADDENDUM: Thanks to all who sent along references/links to studies of this question. Possibly the best, easily-referenced source is this 2001 study from Simon Baron-Cohen and colleagues:
Baron-Cohen is one of the main proponents of the notion that mathematicians/scientists do indeed have increased predilection for the high end of Asperger's spectrum. Not everyone agrees with that, and the issues/variables are very complex, but here's part of the conclusion from the above study which utilized the AQ test as a measure of tendency to high-functioning autism:
"Finally, scientists score higher than non-scientists, and within the sciences, mathematics, physical scientists, computer scientists, and engineers score higher than the more human or life-centred sciences of medicine (including veterinary science) and biology. This latter finding replicates our earlier studies finding a link between autism spectrum conditions and occupations/skills in maths, physics, and engineering."Of course all this plays into the stereotypical view people often hold of nerdy, geeky mathematicians... I don't really have too great a problem with that, except to caution that all generalizations are mushy, and "mathematician," like any other category includes a wide range of individuals and personalities.
ADDENDUM II: Now someone sends along this link to an abstract further indicating a relationship of high-functioning autism with mathematicians:
A point I'd want to emphasize (given my cautionary statement above) is that while the mathematicians here exhibit significantly higher diagnoses than a control group, even among the mathematicians the rate of autism remains low at 1.85%. (For what it's worth, might also note that the subjects in this study, were all undergraduates at a single university, not randomly-selected professional or working mathematicians).
...Science fiction, savantism, mushy Common Core, MOOCs, take your pick:
1) Both math and science fiction geeks should find Sol Lederman's latest wide-ranging podcast with Chuck Adler, physicist and recent author of "Wizards, Aliens, and Starships," interesting; lots of ideas tossed around:
2) I sometimes take note of prodigies and savants here, and the Jason Padgett story is one of the most interesting (Jason attained his mathematical artistic talents only after having been mugged and receiving a severe head injury). A new book out, "Struck By Genius," chronicles his story:
3) and then there's this:
“If you were to graph the creative flexibility afforded our highly educated and maximally qualified teachers over time with common core, you would find that both the first derivative of the function and, most alarmingly, the second derivative of the function, are negative. There is no point of inflection as the function approaches infinity (i.e., increasingly decreasing teacher autonomy, with no turnaround in sight.)”If you haven't a clue what that's all about, read the rest of an engineer's commentary on Common Core here:
4) Finally, MOOCs are full of good, bad, and uncertainty, and continue receiving lots of criticism from outside observers -- sure, there are various numbers/statistics that give rise to such negative views, BUT I for one continue to think we're still very early in the game of a revolutionary development. Perhaps no one has thought about (and worked on) MOOCs any more than Keith Devlin, and so another quick take from him defending their future:
Thursday, February 13, 2014
1) I'm delighted to learn that Noson Yanofsky's "The Outer Limits of Reason" has won a PROSE Award in the "Popular Science and Popular Mathematics" category for 2013.
"The PROSE Awards annually recognize the very best in professional and scholarly publishing by bringing attention to distinguished books, journals, and electronic content in over 40 categories. Judged by peer publishers, librarians, and medical professionals since 1976, the PROSE Awards are extraordinary for their breadth and depth."
I LOVED this volume, calling it a "phenomenal book" in my review last November (indeed it is my FAVORITE book of the last couple decades, and I'm glad to see it get further acknowledgement!):
Congratulations to Dr. Yanofsky!
2) And in the 'suddenly-it-came-to-me' category, another nice story on the continuing saga of Yitang Zhang and his work on the twin-prime saga conjecture:
" 'There's nothing wrong with working at a Subway, but normally these proofs, these breakthroughs, are achieved by those that are working at Princeton, Harvard, these kind of really elite places,' Tony Padilla, a physics professor at the UK's University of Nottingham, says... 'And now we've got somebody who's literally come out of nowhere, that no one expected to produce this kind of results, and has done something impressive that many great minds were unable to do'...3) Last week I was writing about the age-old topic of math and beauty over at MathTango and now a new study points to a neuroscience substrate linking the two:
"Zhang himself, a self-described 'shy person,' said in a UNH statement that the proof came to him during a vacation in Colorado, when he was feeling particularly relaxed. 'I didn't bring any notes, any books, any paper,' he said. 'And suddenly it came to me.' "
4) Finally, h/t to Derek Smith of AMS blogs for pointing out a nice listing of interesting math-related documentaries available from this MathOverflow page:
Wednesday, February 12, 2014
I'm delighted today to direct readers over to MathTango for an interview with Cathy O'Neil, "Mathbabe" of the blogosphere:
If you don't already follow her blog regularly you should! -- I consider it must-reading. Cathy is a former Wall Street 'math quant' who deals with a variety of topics and issues, including some not generally found on math blogs (and also some pure humor), and it is one of the best-written math blogs out there. At the end of the interview I link to another excellent video interview she did with PBS's Frontline series 2 years ago, and you ought try to find 40 mins. free to watch that as well (gooood stuff!).
...Also, in interview mode, Patrick Honner conversed extensively with Steven Strogatz in the current (Feb.) issue of Math Horizons and a nice excerpt is available on the Web here:
I love that this particular section goes into some detail about the writing of Strogatz's 2009 book "The Calculus of Friendship" -- a volume he calls "emotional" and "raw, yet understated." Just a few days ago I tweeted that every math-lover should read this book (and it has more math in it than you might expect, even though it is primarily an intimate, moving life story), and so the timing of this excerpt is excellent.
What better way to spend a snowed-in wintry day than with Cathy and Steve!
Sunday, February 9, 2014
My last two posts (Feb. 4 & 9) at MathTango touch on the frequent topic of "beauty" in mathematics (including Frank Wilczek's take); if you missed them, easy reading for a lay-back Sunday morning:
…and if you missed this engaging Mike Lawler rendition of a Fawn Nguyen problem for her middle-schoolers, also worth a Sunday gander:
Friday, February 7, 2014
Je l'aime quand vous parlez français....*
Perhaps I've just been too long outside the loop of higher academia, but here's something I was completely unaware of:
"...a little-known fact: to get a PhD in maths from Harvard, you need to be a language buff. The university points out that 'almost all important work' is published in French, German, English or Russian, and so 'every student is advised to acquire an ability to read mathematics in French, German and Russian'. This makes it sound optional, but if you want the PhD you have to pass a two-hour written exam in two of these three languages."That's the beginning of a new piece in the New Statesman which is actually about the recent claim of a Kazakh mathematician to have proven the generality of Navier-Stokes equations, one of the Millennium Prize problems. Because of the complexity of the proof, and the fact that it is written in Russian, it will take significant time to confirm, and so we face what the author calls "a mathematical pile-up at the language barrier."
The article is interesting for its brief discussion of Navier-Stokes, but I was more struck by the "little known fact." Assuming the author is correct, and a knowledge of two languages outside English is indeed a requirement (not merely highly-advised) for the Harvard math PhD., I'm curious if this is now the norm at most top-flight mathematics graduate schools, or does it vary considerably from PhD. program to PhD. program? Also, does the "ability to read mathematics in French, German and Russian" perhaps entail significantly less proficiency than would be required for fluency or conversational ability in the languages?
(* "I love it when you speak French" ;-)
Thursday, February 6, 2014
Returning from their excursion into the addition of infinite series, to just plain ol' normal levels of incredibleness ;-) Numberphile is back with the AKS primality test (initials from the names of its three originators), which amazingly, is an algorithm discovered only in 2002, for testing whether or not any integer is a prime. There are some other similar algorithms, but as Wikipedia states, "AKS is the first primality-proving algorithm to be simultaneously general, polynomial, deterministic, and unconditional." And from Wolfram MathWorld this Paul Leyland quotation regarding the finding: "One reason for the excitement within the mathematical community is not only does this algorithm settle a long-standing problem, it also does so in a brilliantly simple manner. Everyone is now wondering what else has been similarly overlooked."
Anyway, watch and enjoy as James Grime explains further:
ADDENDUM: just discovered that Grey Matters blog has also done a nice explanatory post on the Numberphile video here:
Monday, February 3, 2014
|(via Rachel CALMUSA/WikimediaCommons)|
In chapter 11 ("Is Time an Illusion") of Max Tegmark's new book "The Mathematical Universe" the author, while discussing the nature of time and human consciousness, touches upon the "Sleeping Beauty" puzzle/paradox, which I mentioned here almost two years ago:
This is one of the most interesting and delicious (perhaps even complicated, in some ways) puzzles around, as people argue vociferously for either of two different answers (1/2 or 1/3), because of the conditional probabilities involved.
[Here is one statement of the puzzle: Sleeping Beauty undergoes the following experiment, being told all these details ahead of time. On Sunday she will be put to sleep. A fair coin will then be tossed to determine which experimental procedure is undertaken. If the coin comes up heads, Beauty is awakened and interviewed on Monday, and the experiment ends. If the coin comes up tails, she is awakened and interviewed on Monday AND Tuesday. But when she is put to sleep again on Monday, she is given an amnesia-causing drug which ensures she cannot remember the prior awakening. In this case, the experiment ends after she is interviewed on Tuesday. Whenever Sleeping Beauty is awakened and interviewed, she is asked, "What do you believe is the probability that the tossed coin landed on heads?" -- What is her answer?]In re-visiting the links I provided in my original blog post I discovered that the "Tanya Khovanova" link has since added further lo-o-ong discussion of the issues by two commenters back-and-forth, which is probably worth checking out if you are especially interested in probability in general, or this problem in particular (if these areas don't interest you, don't visit it, lest you fall into a deep, deep coma, or alternatively, your head explode ;-)
Sunday, February 2, 2014
Haven't had much time for posting lately, but, in case you've missed them, here are several recent, fun puzzles from the ever-entertaining Futility Closet to tide you over:
"Futility Closet," the book, by the way, is available here:
Tuesday, January 28, 2014
i.e... Think! (this comes from math teacher Rick White)
And, in a different form, more "advice to a young mathematician" here from the Princeton U. Press's "Princeton Companion to Mathematics":