This is probably the most heavy, dense mathematical piece (on the Riemann Hypothesis, quasicrystals, Freeman Dyson, and more) I've ever linked to, and I don't comprehend it… but, all things Riemannian fascinate me, and even without following the mathematics here I think this very long piece from John Baez communicates the beauty and profundity of methodical, thoughtful, mathematical analysis. Simply put, it is, I think, a beautiful piece of mathematical exposition (not at all unusual for Baez)….
"Freeman Dyson is a famous physicist who has also dabbled in number theory quite productively. If some random dude said the Riemann Hypothesis was connected to quasicrystals, I’d probably dismiss him as a crank. But when Dyson says this, it’s a lot more interesting. So I’ve been trying to understand his remarks on this. And it’s been productive, in that I’ve learned some interesting things, and I now feel closer to seeing why the Riemann Hypothesis is a natural and important conjecture."
...On a side-note, I also find it interesting that Freeman Dyson
advanced this original quasicrystal notion, which has generated a lot of
interest (like so many things Dyson has advanced over the years), at the ripe young age
of 85! Who says mathematics must be a young man's game!
It's supposed to measure an individual's ability to think through a problem more reflectively or deliberatively, instead of jumping to an initial intuitive judgment.
The simple three questions involved (fairly familiar to many mathematics fans) are:
1)A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost?
2)If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?
3)In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?
People of a certain impulsive cognitive bent will tend to jump to the wrong answers on most of these questions, while those with a more patient cognitive style will take more time, arriving at correct answers. And these cognitive styles of course have further consequences.
The test has all the outward appearance of pop psychology, and it's hard to imagine a 3-question test as an accurate measure of anything (even if it correlates highly with something), but its creator is a former MIT professor, Shane Frederick (who has, BTW, worked with the esteemed Daniel Kahneman), and he explains the rationale for the test here:
A lot of follow-up research on the test seems to lend it some credence, although I haven't researched it enough to know how much criticism/skepticism of the test might exist.
...In any event, interesting to see someone attempt to get so much psychological mileage out of just three mathy questions!
Sunday (June 23rd) was the 20th anniversary of Andrew Wiles' 3rd talk at Cambridge's Isaac Newton Institute wherein he ended with "I think I'll stop here," to wild applause, having just finished a proof of Fermat's Last Theorem (to be later modified/patched). Plus.maths.org were among those on the Web noting the anniversary:
I've previously linked to this beautiful, moving 50-minute documentary/tribute to Wiles' accomplishment by BBC's Horizon show... if you've never seen it before make time for it... THAT'S an order! ;-) (...coincidentally, this documentary was done by Simon Singh, subject of yesterday's post.)
Are you a Simpsons fan? (...if you have a pulse you should be!)
"The Simpsons" is one of the longest running shows in the history of television, and D'OH! this looks grand: turns out best-selling science-writer Simon Singh's next volume is entitled, "The Simpsons and Their Mathematical Secrets" (due out in October). You may or may not know that several PhD. mathematicians work as writers for the show (who says mathematicians aren't funny!), but if you're a loyal viewer you know that mathematical references slip into the show with some regularity. A post I did earlier this year touches on it:
About the upcoming book, Singh says, "I think it is a fairly lighthearted and hopefully entertaining read, but it also contains some serious mathematics." There's a combination I look forward to!
Read a li'l more from Singh here:
or, just watch him talk about it in this interview excerpt:
And on a side-note, going from the brand new to the decades-old... I saw William Poundstone's classic "The Recursive Universe" in Barnes & Noble this weekend, apparently it's been newly released (by Dover) having originally appeared in 1985. Anything Poundstone writes is usually good, and this was one of his earliest works, a delightful exposition centered around John Conway's 'Game of Life.' In another bit of synchronicity, a little over a week ago I posted about Conway, only to now stumble upon this almost 30-year-old volume which, despite it's age, is a wonderful interweaving of Conway's work with cosmology, information theory, game theory, and complexity. If you're not already familiar with it, I recommend it, as do some other folks.
My blogging, or at least my attention, has been hijacked by Keith Devlin again for a bit...
First, he's written an appropriately longish response, taking to task the authors of a NY Times piece ("The Faulty Logic of the Math Wars") about math education; definitely worth a read if you're in math education yourself (LOTS of morsels to chew on):
On a side-note, Seb Schmoller has written a lengthy review of Keith's own MOOC course now that it is completed (Seb also wrote an interim review at about the half-way point):
I imagine Dr. Devlin will have his own further summary of how the 2nd-go of his MOOC course went, from his perspective, on his own MOOC blog soon, and will no doubt take into account the feedback from people like Seb.
AND, over at MathTango I now have up another interview with Keith… unlike any interview I've ever done in the Math-Frolic series (let's just say he has some things on his mind):
Leave it to "Futility Closet" to introduce me to yet another self-referential paradox I don't recall seeing before (although I've seen similar ones). This one called, "The Parity Paradox," is both simple yet a tad mind-numbing… and hey, that's how I like my paradoxes!:
2) And looking ahead... I've just received answers back from Dr. Keith Devlin for my second interview with him, and Keith's fans won't want to miss it... even though there is NO discussion of math. It will take me a couple of days to format and prepare it for posting, but I can't thank him enough for taking time to respond to my inquiries on some non-math timely matters. So do check back for that, also at MathTango; if current events interest you, you won't want to miss Dr. Devlin's take.
Beginning with reference to a classic 1959 C.P. Snow lecture, Stewart argues that mathematics, unlike almost any other field of study, "spans the divide" between the arts and sciences, and makes many interesting points along the way.
Some excerpts to poke your interest…:
"...the mathematical subculture is gravely misunderstood by most members of the public, many prominent scientists, nearly all artists, the vast majority of politicians and almost every bureaucrat on the planet." "For mathematics to be useful in today’s society, it has to be invisible. If anyone using a mobile phone had been required first to take a PhD in mathematics, the device would never have appealed to a large enough market to make it worth anyone’s while to manufacture. However, without a significant number of engineers who know some very advanced and apparently esoteric mathematics, mobile phones wouldn’t work." "If 10 per cent of the $9bn spent on Cern’s Large Hadron Collider had been allocated to research in the mathematical sciences instead, the benefits to society would have been far greater and would have occurred more rapidly. The development of the next generation of supercomputers, a project estimated to cost about $1bn, is struggling to find funding. The result would be a machine with applications throughout the sciences – for example, in climate change and in the design of new materials and alternative energy sources – and it would dominate future computer technology." "However, the nature and importance of mathematics do not rest solely on its practical uses or its artistic merit. It has an intrinsic intellectual interest and an idiosyncratic beauty. What makes it so hard to grasp the subject’s nature is that it combines many disparate elements in one ever-growing, ever-changing body of knowledge. It is practical, arcane, precise, obscure, abstruse, rooted in nature, rooted in the human imagination; its history spans 4,000 years, and what was discovered long ago is often just as valid and important today. It is absolutely huge, increasing by a conservative one million pages a year, and it is all one thing. Its internal connections link entirely different sub-disciplines in an intricate, dynamically changing web, so that yesterday’s dead end may suddenly become today’s essential technique."
Set aside some time to read... and think about... the whole piece.
I'm not even sure what that phrase means ("a mathematician's mathematician"), but it just sounds right for John Conway!....
John Horton Conway
Over at companion blog MathTango I now have up a review of "The New York Times Book of Mathematics," an anthology of superb past NY Times columns that relate to mathematics. I love the volume, and recommend it. And because it's a compendium of past newspaper columns, many can be found online, so the book is a rich source for future posts/links here as well!
I'll start with this wonderful 1993 profile by Gina Kolata of Princeton's fascinating John Conway (renowned for his math creativity, and inventing the "Game of Life") which comes near the end of the anthology:
A few endearing lines about Dr. Conway from the piece:
"But Dr. Conway has so many peculiar interests and quirks, and he can so
easily be made to sound eccentric, that his deep love of mathematics and
the natural world can be lost in the fluff. In a way, he is oblivious
to the routines and customs of ordinary life... "Yet though indifferent to fashion or fads, Dr. Conway is intensely aware
of nature, and wants to know it deeply and intimately. He is infinitely
curious and observant, seeing nature not only in a spider web or the
details of a daffodil but in mathematics."
There are probably some more recent, up-to-date profiles of Conway available, but likely none any more splendidly composed. Give it a read; you'll be glad you did!
Most folks are familiar with the Traveling Salesman Problem, one of the most famous dilemmas in all of math -- finding the shortest possible route between a given set of points (particularly ubiquitous in discussions of P vs. NP).
One real life example of TSP is route-scheduling for UPS delivery drivers who, every single workday, make on average, 120 deliveries. How most efficiently to drive that delivery route? There are a lot of consequences.
Apparently UPS has been field-testing a system, designated ORION ("On-Road Integrated Optimization and Navigation") which is their best algorithm for approximating a solution to TSP. So far they estimate it has saved them 35 million driving miles. Read more about it in the articles below (although they don't really give much information about the actual math behind ORION).
“ 'Advanced analytics should be one of the top priorities for CIOs,' says
Levis [UPS Director], who can talk of math in near-koans: 'Beyond knowledge is wisdom,
and beyond that is clairvoyance.' Math simply can solve problems that
humans can’t."
The recent Edward Snowden revelations have brought the NSA into the limelight more than usual (and probably more than they'd like). According to many, NSA is the largest employer of mathematicians in the world. For a little comic relief recall this scene from the movie "Good Will Hunting," with Matt Damon as a young math prodigy:
Here is NSA's own brief "career path" page for mathematicians:
The timing of the Snowden leaks was a bit of synchronicity for me, as I am currently reading "The NY Times Book of Mathematics" (an anthology, that I highly recommend, to lay readers especially) and had just reached chapter 5, which focuses on cryptography, where almost every reading includes mention of the NSA... interestingly, there was a time when a lot of friction existed between NSA and academic mathematicians who were being forced to work under various constraints by the governmental agency).
Anyway, for more info on the agency, there is of course a Wikipedia page for NSA here:
(BTW, if PRISM interests you, you may want to be sure you're familiar with ECHELON and Carnivore as well.)
Regular readers here know that I'm a huge fan of Keith Devlin, and for more than his mathematical writings. Coincidentally, he briefly worked for the NSA at one time, and on Twitter, has expressed major concerns since the NSA disclosures -- indeed, of over 200 math/science persons I follow on Twitter I've not seen anyone more vocal than Keith in his upset with the Government (and support for Snowden) in this matter.
He wrote a recent piece for HuffPost below on the subject, though it just scratches the surface:
He exhorts a kind of statistical (or "sense of risk") argument before ending thusly:
"Are we really, as a nation, going to give up personal freedoms that are the envy of the world -- a beacon to humanity -- because of a collective numerical stupidity which could be eradicated in a single generation by a small change in K-12 education?"
The unnerving part is, that I suspect at least 50% of the citizenry will answer "yes" to that....
Fear is a powerful stimulus (and political tool)… and we, or at least our leaders, have been living in fear since 9/11/2001.
(Little wonder that sales of George Orwell's 60+ year-old novel "1984" have skyrocketed since the Snowden story broke.)
As someone who has diddly artistic talent I often think that working artists literally perceive the world differently than those of us who don't do art… I suspect a painter or drawer may actually see his/her subject differently than the way my eyes see it.
And perhaps the same is true for those of us who love mathematics… we perceive the world, or at least think about it, differently than those who fear or dislike math… the 'beauty' of math (as I've noted often before) is obvious to some and oblivious to others.
Evelyn Lamb has a transcribed interview over at her Scientific American blog with two female mathematicians (in the field of dynamical systems), Laura DeMarco and Amie Wilkinson. So many great responses, especially if you're interested in the experience of female mathematicians. Please give it a read. But here's a small excerpt that touches on the 'beauty' aspect (as related to calculus) that always catches my attention:
Lamb: Are there any math topics that are particularly appealing or beautiful for you? Wilkinson: I like calculus a lot, probably because I learned it when I was young, and I learned it well. To me, it’s always comforting to use calculus to do something. The invention of calculus was certainly revolutionary. DeMarco: A conceptual breakthrough.
Wilkinson: It’s funny, because it’s like we just toss it out there to high school students, and I think a lot of them have no idea of the beauty.
DeMarco: What the ideas really were.
Wilkinson: Certainly some of the most beautiful mathematics I’ve learned is just calculus.
DeMarco: It’s funny you mention calculus. I don’t think I really appreciated it until I taught it as a graduate student. I was lecturing to these first-year students. I was just wowed by this subject. I had this moment of, holy cow, this is really beautiful! I remember my grandmother asking me what I was thinking about these days. I said, “Well, I’m teaching calculus right now, and you know what, calculus is really beautiful.” She said, “OK, Laura, what is calculus? Can you just tell me in 20 minutes, what is calculus?” And it was just the greatest thing to have this opportunity to just sit down with my grandmother, of all people, and tell her.
This section happened to strike a chord for me, but if you're a female and interested in mathematics there are so many other passages that will be of special interest, so do check out the whole interview (...and guys can read it too; I did ;-):
Right from the start he writes: "... let me just say that I am never going back to the Stand and Deliver
model that so many teachers are reluctant to alter in any way."
2) Haven't had a puzzle for awhile, so here's a simple-looking one from Ben Vitale that might keep you busy for a wee bit. A 3-digit number and a 2-digit number are multiplied... what are all the numbers involved?:
ALLE's represent even integers, and ALLO's are odd integers (but not the same integer):
O E E
x E E ___________ E O E E + E O E _____________ O O E E
In response to recent news, mathematician John Allen Paulos has tweeted a link to this "old, still relevant piece of mine on privacy, terrorists, and Bayes"…:
"....the fact remains that since almost all people are
innocent, the overwhelming majority of the people rounded up using any
set of reasonable criteria will be innocent."
Hmmm… Who knows what the significance of that number is?
It is (as of May 2013) the largest known first member of a prime triplet" [a grouping of 3 successive prime numbers with the smallest possible gaps between them: of the form (p, p + 2, p + 6) or (p, p + 4, p + 6)]
I thought with all the recent talk of twin primes, from Yitang Zhang's recent findings, it might be worth noting that mathematicians recognize other prime successions as well.
This leads, quite naturally, to yet another conjecture… that there are an infinite number of prime triplets. By the way, there are also prime quadruplets, and all of these sorts of categorizations are known as "prime constellations."
Yitang Zhang's recent proof regarding twin primes keeps generating press pieces… including some good recent ones for lay readers, first from the Wall Street Journal:
(I think all 3 of these are worth a gander as they touch on different aspects of the twin prime/Zhang story.)
Things are moving fast… Zhang's work was quickly taken up by a bevy of individual mathematicians, and now also by the Polymath Project group (a Web collaboration of many mathematicians). Zhang's computed upper limit of ~70 million has, in just a few brief weeks, been reduced to under 400,000! (the latest figure I've seen, but it will keep changing). Phenomenal!
The Polymath Project (largely inspired by math icons Tim Gowers and Terry Tao) is, to my mind, an example of the premier use of the internet, to utilize the 'hive mind,' as never before in history, to solve or work on significant problems (in this case math, but it can be applied to other subject areas). In the past, dozens or perhaps 100s of individuals could be brought together to work somewhat collaboratively on a problem, but today 100s of 1000s can be easily drawn from. Books have already been written about the power of the "hive mind" (some quite critical). Solutions to many problems will come at a warp-speed previously unattainable. Can the 'hive mind' be cluttered with junk science… pseudoscience… quackery… tomfoolery… Yes, of course, GOBS of it; But the beauty of the hive is how rapidly and efficiently it can sort through masses of information and ignorance to arrive at the productive core. "Open access," "crowdsourcing," collaboration, and the hive, are the wave of the future.
My current post over at MathTango is a doff-of-the-cap to those leading the way in the current evolution (revolution???) of math education, with a focus, by the end, again on how rapid/broad Web collaboration is currently leading the way to shape the 'flipped classrooms' and MOOCs of the near future… and, obviously, so much more.
Beal's Conjecture popped up in the news today. It's another one of those somewhat simple sounding, but very difficult-to-prove notions [a sort of take-off of Fermat's Last Theorem, but much more recently (1993) formulated], which now has a new $1 million bounty on its head:
The simple statement of it runs as follows:
If A^x + B^y = C^z where A, B, C, x, y, z are all positive integers with x, y, z > 2 then A, B, and C have a common factor.
It made the news because the originator, banker and dabbler in number theory, Andrew Beal, has just upped the ante to the $1 million mark for anyone who can prove the conjecture (bringing it in line, monetarily, with the Clay Institute Millennium Problems), and hopefully encouraging more takers.
You'll likely want to employ some programming skills to work on it, and on a sidenote, the always worth-reading John McGowan has some interesting things to say below about the push to teach coding/programming to all young people:
Keith Devlin's latest "Devlin's Angle" blogpost was inspired (in part) by his moderation of the World Science Festival panel (shown in my May 31 post):
He muses about whether the "study of infinity – in particular the hierarchy of larger infinities that Cantor bequeathed to us – would ever have any practical applications." (...and he thinks likely, not).
Part of what makes Cantor's work so eternally interesting is the huge divergence of opinion about it from his very own time. As Devlin writes, "Reactions to Cantor’s revolutionary new ideas ranged from outraged condemnation to fulsome praise." And these completely opposed viewpoints came from individuals equally-well-established and respected in the field. One of the paradoxical aspects of infinity is the equal ease with which it may be discussed in either direction: i.e., the infinitely large or the infinitesimally small. In any event, it was David Hilbert who eventually coined the term "Cantor's Paradise" for the new Cantorian thought that did slowly take hold.
Even so, still today, "infinity" can be such a difficult concept to grasp, that Cantor's ideas remain a frequent target of attacks by "crackpots" of the sort Mark Chu-Carroll often hears from:
In a world of depressing news, just "four silly stories" today (who knew the number four is "considered deeply unlucky in Chinese culture") passed along by "The Aperiodical":
As the poster admits, "None of these stories merits being reported on here on its own" but the four of them together (and completely unrelated) make for some entertaining reading.
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