Wednesday, October 29, 2014

Moravec's Paradox

This isn't exactly math, but it's artificial intelligence (AI), and that's close enough... especially since a few posts back I wrote about IBM's "Deep Blue" and its 1997 defeat of chess grandmaster Gary Kasparov (at the time, a long-held goal of AI). Well, Moravec's paradox is the interesting idea that advanced or high-level reasoning and logic is much more easily mimicked by a computer system than are low-level sensori-motor skills that have evolved over millions of years... it's easier for a computer to learn to play chess, than to recognize human faces. This is one of those things that is fairly obvious when you stop to think about it... but, we often don't stop to think about it!
Here's what Steven Pinker wrote in "The Language Instinct":
“The main lesson of thirty-five years of AI research is that the hard problems are easy and the easy problems are hard. The mental abilities of a four-year-old that we take for granted – recognizing a face, lifting a pencil, walking across a room, answering a question – in fact solve some of the hardest engineering problems ever conceived…. As the new generation of intelligent devices appears, it will be the stock analysts and petrochemical engineers and parole board members who are in danger of being replaced by machines. The gardeners, receptionists, and cooks are secure in their jobs for decades to come.”   
A more recent blog piece applies the paradox to Google's self-driving cars, a creation I've certainly had trouble comprehending, given the countless issues/variables involved:

[p.s. -- actually, where are the dang flying jetpacks I grew up believing we would all have by now... forget the cars Google, I want my personal commuting jetpack!]

anyway, below, another somewhat provocative post applying Moravec's paradox to brain processing:

Tuesday, October 28, 2014

Just Passing This Along

Colin Hegarty, who runs Hegartymaths has been bestowed a "Gold" tech-teaching award for his free math-tutorial site in Britain:

His videos are here:

I've not actually experienced the site or videos, so not directly endorsing it, but just recognizing that others attest to its value. It sounds a lot like (and was indeed inspired by) Khan Academy, which remains controversial in various quarters.

Anyway, check it out if you're looking for adjunct math tools.  Also, Colin tweets here:  @hegartymaths

Sunday, October 26, 2014

Taleb on Randomness

Today, a number of bits from an older Nassim Taleb volume, "Fooled By Randomness":

"Probability is not a mere computation of odds on the dice or more complicated variants; it is the acceptance of the lack of certainty in our knowledge and the development of methods for dealing with our ignorance. Outside of textbooks and casinos, probability almost never presents itself as a mathematical problem or a brain teaser. Mother Nature does not tell you how many holes there are on the roulette table, nor does she deliver problems in a textbook way (in the real world one has to guess the problem more than the solution)."

"This book is about luck disguised and perceived as nonluck (that is skills) and, more generally, randomness disguised and perceived as non-randomness (that is, determinism). It manifests itself in the shape of the lucky fool, defined as a person who benefited from a disproportionate share of luck but attributes his success to some other, generally very precise, reason."

"We are still very close to our ancestors who roamed the savannah. The formation of our beliefs is fraught with superstitions -- even today (I might say especially today). Just as one day some primitive tribesman scratched his nose, saw rain falling, and developed an elaborate method of scratching his nose to bring on the much-needed rain, we link economic prosperity to some rate cut by the Federal Reserve Board, or the success of a company with the appointment of a new president 'at the helm.'"

"Disturbingly, science has only recently been able to handle randomness (the growth in available information has been exceeded only by the expansion of noise). Probability theory is a young arrival in mathematics; probability applied to practice is almost nonexistent as a discipline"

"Indeed, probability is an introspective field of inquiry, as it affects more than one science, particularly the mother of all sciences: that of knowledge. It is impossible to assess the quality of the knowledge we are gathering without allowing a share of randomness in the manner it is obtained and cleaning the argument from the chance coincidence that could have seeped into its construction. In science, probability and information are treated in exactly the same manner. Literally every great thinker has dabbled with it, most of them obsessively.

[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know ( If I use one submitted by a reader, I'll cite the contributor.]

Wednesday, October 22, 2014

"The Man vs. The Machine"

(via MichaelMaggs/Wikimedia)

Math fans usually like chess, so I'll refer readers to FiveThirtyEight's first mini-documentary film (17 mins.), on the historic 1997 match between then-World-Champion Garry Kasparov and IBM's "Deep Blue" (actually it's the RE-match that Kasparov LOST). Some interesting history... and following its victory and acclaim, Deep Blue "retired":

ADDENDUM:  I've now discovered, for the more-thoroughly chess-ensconced (who have 90 minutes to devote to the Kasparov/Deep Blue battle), this older film on the same topic:

Tuesday, October 21, 2014

He'd Be Embarrassed By All the Attention...

Wasn't planning to do a post on Martin Gardner's Centennial today (...I did my little reflection post on him this past Sunday), since I've covered him plenty in the past, and knew many others would be paying tribute this week. But so many good posts have gone up, I don't want to ignore them, and thus offer a small sampling below.
It's impossible to overdose on Martin Gardner, incredible thinker/writer that he was (who hardly took a math course beyond high school!), so enjoy...  (possibly I'll add additional links in next 24 hrs., but really there are too many to choose from!):

Also, my year-old review of Gardner's autobiography here:


One suspects Martin is now somewhere off demonstrating the joy of hexaflexagons to a whole new audience of enthralled angels... or, just maybe, he and Paul Erdös are sitting together, sipping coffee, and excitedly reading each other passages from "The Book." ;-)

Monday, October 20, 2014

Some Monday Stuff

So many interesting, varied things passing by my computer screen the last 48 hrs.; have to pass a few along rather than hold onto until the Friday "potpourri" collection:

First, this wonderful video on P vs. NP... about as good as any quick (11-min.) intro I've ever seen on this important subject:

On the education front, fans of Robert Talbert should read his "Medium" piece outlining the future of his "Casting Out Nines" blog.

And Grant Wiggins has an update on the education post that made the rounds at his blog last week, and turns out to have been written by his daughter! More soon to come from her:

And finally HERE, Tracy Zager re-visits the below Robert Kaplinsky video, covering an interesting problem/issue that actually goes back to at least 1986:

Sunday, October 19, 2014

A Martin Gardner Sunday

This coming Tuesday marks the 100th anniversary of Martin Gardner's birth, so for a Sunday reflection, some quotes about the man:
[several of these are taken from the Martin Gardner "testimonial" page: ]

Douglas Hofstadter, in tribute to Martin, upon his death in 2010:
"This is really a sad day…  sad because his  [Gardner's] spirit was so important to so many of us, and because he had such a profound influence on so many of us. He is totally unreproducible -- he was sui generis -- and what's so strange is that so few people today are really aware of what a giant he was in so many fields -- to name some of them, the propagation of truly deep and beautiful mathematical ideas (not just "mathematical games", far from it!), the intense battling of pseudoscience and related ideas, the invention of superb magic tricks, the love for beautiful poetry, the fascination with profound philosophical ideas (Newcomb's paradox, free will, etc. etc.), the elusive border between nonsense and sense, the idea of intellectual hoaxes done in order to make serious points... and on and on and on and on. Martin Gardner was so profoundly influential on so many top-notch thinkers in so many disciplines -- just a remarkable human being -- and at the same time he was so unbelievably modest and unassuming. Totally. So it is a very sad day to think that such a person is gone, and that so many of us owe him so much, and that so few people -- even extremely intelligent, well-informed people -- realize who he was or have even ever heard of him. Very strange. But I guess that when you are a total non-self-trumpeter like Martin, that's what you want and that's what you get."
"Several decades passed by before I rediscovered the elegance, simplicity, and depth of his writing, and most importantly, the validity of his approach to mathematics. Then, in due course, I had the great pleasure of meeting him in his old age. He was nothing like the stern-looking man on all those book covers: in reality he was a sweet-natured, kind, wise and modest to a fault, with a twinkle in his eye, and a total joy to be with. While I can't say that Martin's columns or books steered the early course of my life, his extraordinarily diverse written legacy, his devotion to learning, his generous sharing of his toys, and his sheer decency, all conspired to reset my course in midlife.
He was also extremely egalitarian and generous with his time: he didn’t care if you were a prince or a pauper, if you had an interesting idea then he wanted to know about it, and he’d encourage you to get it in front of others. In a sense he was the original (mathematical) community organizer, at a time when it was neither profitable nor popular."
-- Colm Mulcahy  
"Martin Gardner was an artist of mathematical writing. His work stands and will continue to stand the test of time. It is a springboard for others. I can gaze and contemplate his work over and over and see new things and create new ideas."
-- Tim Chartier
 "All of us who dare to aim our writing at 'the general reader' follow as best we can in Martin's footsteps. He is the Archimedes of mathematical writing."
        — Keith Devlin
and this:
"We glibly talk of nature's laws
but do things have a natural cause?
Black earth turned into yellow crocus
is undiluted hocus-pocus."

-- Piet Hein
used as the frontispiece to Martin's autobiography, "Undiluted Hocus Pocus"

To conclude, this wonderful, older and rare (14-min.-edited) interview with Martin was uploaded this week to YouTube... delightful:

Finally, has this tribute page running this month in honor of the Gardner Centennial:

Friday, October 17, 2014

"A New Universal Law"

Natalie Wolchover never fails to enthrall. Her latest piece at Quanta is on a "curiously pervasive statistical law" that connects math, physics and biology.  It's known as the Tracy-Widom distribution, after the founders who discovered that "Systems of many interacting components — be they species, integers or subatomic particles — kept producing the same statistical curve." In other words, similar to the bell curve, the Tracy-Widom distribution seems to operate universally, describing many complexly-interacting systems. Unlike the bell curve though, the tails are asymmetric in some manner relating to the universal nature of phase transitions, and phase transitions, the article notes, "are for statistical physicists 'almost like a religion.'" The article also notes that where physicists are often satisfied with "a preponderance of evidence," mathematicians want more rigorous "proof" of a relationship. Read more here:

Thursday, October 16, 2014

A Mathematical Parable... and Ebola

I've shortened and simplified this post considerably. It wasn't even intended for the regular math-literate readership here who almost certainly know the classic story of grains of wheat accumulating on a 64-square chessboard; but rather for their possible math-phobic friends who need a more vivid understanding of the potential exponential nature of numbers-growth, in lieu of the Ebola story unfolding.

Ever since the virus spread (completely unlike prior decades) beyond the villages it was usually confined to, and especially since its spread beyond the shores of West Africa, some of us have had a more cautionary, skeptical view than the CDC's confident stance, because of the simple mathematics of the situation (combined with the fact that NO amount of medical protocols/regulations realistically offers 100% prevention of spread, given that humans who must carry out such protocols are imperfect, suffer lapses, make mistakes, are forgetful or tired or ill-trained, or in a hurry, etc.etc. (And that's no fault of theirs, that's just being normal humans, instead of machines). While the 70+ contacts of the Dallas Liberian victim might seem a manageable number, 300, 500, or 1000 potential contacts/exposures will not be easily manageable. (The fact that the virus doesn't spread through the air is lucky for us, but by no means precludes widespread infection.). Enough said:

[p.s., in a recent release the World Health Organization warned that before the end of this year there could be as many as 10,000 new cases of Ebola in Western Africa alone every week -- I'd be a bit surprised if that happened... but that IS the point of the above video, it could happen that fast.] Somewhere between calm and panic there is an appropriate state of alarm and alertness that the American public needs to find, to be prepared for the major disruption this epidemic, and consequent public health measures, could cause society. "Be prepared" is often a more trenchant maxim than "stay calm." Or, to put it a different way, the "precautionary principle" again takes hold (better to be overly precautious, than not precautious enough).

As an aside, in the short term, I'll say that my own confidence lies, not in our ability to necessarily control the spread of this disease, but rather in our ability to attain early diagnosis and more effective treatment for it, cutting the current 70% fatality rate significantly (but that too certainly isn't assured).

Wednesday, October 15, 2014

Grant Wiggins Presents "A Veteran Teacher Turned Coach"

 Chances are, if you read this blog, you've already seen this... because at last check well over ~300,000 500,000 had... a super piece from Grant Wiggins -- actually a posting he's taken from another educator/writer ("a veteran teacher turned coach") -- about a teacher becoming a student for a day (because we all forget too quickly what it's like!).
If somehow you've missed it, take a gander; it makes so much common sense, of the sort we often look right passed, with lots of suggestions (and lots of interesting comments as well):

This morning, I noticed someone on Twitter responding to the post by mentioning that they knew a couple of classrooms where regular school seats were replaced with ball chairs -- I thought that was a fascinating idea, even if not always practical -- just an example of the simple (and perhaps healthier?) outside-the-box thinking the article encourages.
By the way, Grant promises that the writer will be doing a follow-up to the piece.

ADDENDUM: Michel Reed, the teacher who mentioned the above ball chairs example, later tweeted this photo of such a classroom :-):

Monday, October 13, 2014

A Puzzle to Kickstart the Week

A sweet, simple puzzle to kickstart your week, taken directly from a recent Brian Brushwood "Scam School" episode; and it's one of those grand facepalm-type puzzles, you'll kick yourself for, IF you don't solve it:

Multiply together a long sequence as follows:

(a-x) X (b-x) X (c-x) X (d-x)...... (y-x) X (z-x)  i.e., utilizing ALL the letters of the alphabet once

What will be the end product of this sequence multiplied out???

["Scam School" has been around a long time, but if you've missed it by any chance, you can check out it's many entertaining videos HERE.]
.Answer below
Answer = 0 ...just before the final two sequence entries listed (but not shown), would be (x-x)

Sunday, October 12, 2014

The Tao of Tao ;-)

Terry Tao from his blog, "What's New":

"The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture. Without one or the other, you will spend a lot of time blundering around in the dark (which can be instructive, but is highly inefficient). So once you are fully comfortable with rigorous mathematical thinking, you should revisit your intuitions on the subject and use your new thinking skills to test and refine these intuitions rather than discard them. One way to do this is to ask yourself dumb questions; another is to relearn your field."

[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know ( If I use one submitted by a reader, I'll cite the contributor.]

Friday, October 10, 2014

"Right Now the Math Still Favors the Virus"

This started out as a link, but turned into a bit of commentary...

Society often walks a fine line between panic and laxity over any potential crisis. Ebola is no different, and I understand why the medical community took a somewhat pollyannish view toward it in pronouncements to the public. But the numbers involved, our highly mobile society, and the fact that complex containment protocols on paper will not be completely carried out (because they never are) calls for a more sober assessment. This Washington Post article, "The Ominous Math of the Ebola Epidemic," offers a somewhat more realistic view of the numbers and potential exponential growth involved:

I'm not terribly confident of success in "containing" Ebola in the short-term, but I do suspect we will find effective treatments for managing it in those diagnosed early, and thus significantly cutting the fatality rate (the successes we've already had are quite encouraging). In trying to avoid panic the CDC and others bent over too far in the direction of preaching calm and confidence. And the problem with that scenario is the backlash it may lead to.
This country already has a perturbingy large, vocal anti-science component in it. If the medical community misplays the Ebola epidemic it will add ammunition to their arsenal: 'see, we can't trust scientists; they don't really know what they're doin'. The anti-vaccers, anti-evolutionists, climate-denialists etc. will have a field day, long-term, if, after all the calls for calm/trust, the epidemic spreads widely. I'm almost as concerned over that as I am over the medical crisis itself.
In so many ways of course we have a wonderful medical community in this country, especially when it comes to medical emergencies. One just hopes they're not already in over their head in this case. We are probably already in the stage of swinging from calm to panic (there is so little middle-ground):

I don't know if the medical community could've done any better in their public communications -- they were caught between a rock and hard place... walking a tightrope... over a mass public that little understands how real science operates.
Anyway, to those on the front lines, where so much courage, care, commitment, and selflessness are now required, I sit in awe of you.

ADDENDUM:  highly-respected Laurie Garrett has now posted a piece that I think pretty well nails the proper cautionary stance/tone needed in this circumstance, while addressing "five myths about Ebola" (glad to see her do it!):

Thursday, October 9, 2014

"The Upside-Down World Paradox"

Given my fondness for paradox, just linking today to this quirky, fun little (non-mathematical, but logical) post about the 'upside-down world game' (an offshoot of 'the liar paradox'):

The blogger's seven-year-old daughter enjoys the game in the post, so if you have young kids maybe they will as well.

Wednesday, October 8, 2014

Of Friends, Face-to-Face and Virtual

Always love to see math making it into the popular or mainstream press, so nice to see this Maria Konnikova article on Dunbar's Number and social networks (and the new ramifications of digital social media) in the New Yorker:

As the article states, " one really knows how relevant the Dunbar number will remain in a world increasingly dominated by virtual interactions," or as Dunbar himself is quoted, “We haven’t yet seen an entire generation that’s grown up with things like Facebook go through adulthood yet.

There are potential neuroscience, and in turn social, implications to all this reliance on virtual interaction. It is, for now, a sort of grand, ongoing experiment, outcome unknown.

Tuesday, October 7, 2014

Happy Birthday Neil Sloane and OEIS!

Wonderful Alex Bellos piece in The Guardian today, on Neil Sloane and the OEIS (Online Encyclopedia of Integer Sequences) he founded. Fascinating reading:

I was happy to learn of the Kolakoski sequence which combines a number sequence with self-reference (one of my favorite topics), and of which Bellos writes, "Mathematicians drool over this sequence."
 The OEIS has been around as a go-to resource for mathematicians of all stripes for 50 years now, and today includes some 250,000 number sequences, while still growing, according to the article, at a rate of about 40 new sequences each day! Some sequences can be quite creative of course, and open up interesting, difficult-to-solve questions.
Rutgers' Doron Zeilberger goes so far as to say that the OEIS has made Sloane the world's most influential mathematician!
Lots more in the article, including a video.

Sunday, October 5, 2014

Couple of Physicists' Views

Sunday thoughts on science...:

"Science in its everyday practice is much closer to art than to philosophy. When I look at Gödel's proof of his undecidability theorem, I do not see a philosophical argument. The proof is a soaring piece of architecture, as unique and as lovely as a Chartres cathedral. The proof destroyed Hilbert's dream of reducing all mathematics to a few equations, and replaced it with a greater dream of mathematics as an endlessly growing realm of ideas."
-- Freeman Dyson in "Nature's Imagination"

...and from another physicist, a related view:

"…to sum up, science is not about data; it's not about the empirical content, about our vision of the world. It's about overcoming our own ideas and continually going beyond common sense. Science is a continual challenging of common sense, and the core of science is not certainty, it's continual uncertainty -- I would even say, the joy of being aware that in everything we think, there are probably still an enormous amount of prejudices and mistakes, and trying to learn to look a little bit beyond, knowing that there's always a larger point of view to be expected in the future."
-- physicist Carlo Rovelli in "The Universe" edited by John Brockman

or, just perhaps, Richard Feynman summed it up succinctly ;-):

"Physics is like sex: Sure, it may give some practical results but that’s not why we do it.”

[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know ( If I use one submitted by a reader, I'll cite the contributor.]

Thursday, October 2, 2014

How Would American Kids Do?

An interesting pair of successive tweets from Alexander Bogomolny this morning, showing a Hong Kong 1st-grade "admissions" test question :

Wednesday, October 1, 2014


Seeing an awful lot written on Bayesian ideas in the last year (and week!).
Jason Rosenhouse uses the Monty Hall problem and a NY Times article as a launching point for a discussion of the subtlety of Bayes here:

Rosenhouse takes the Times' article to task, and ends simply with:
"Applying statistics correctly is hard, even for people with professional training in the subject. But the problems are found in the complexity of real-life situations, and not in the underlying philosophical approaches to probability and statistics."
(Rosenhouse's 2009 book on the The Monty Hall problem is great, by the way, if you've never read it -- yes, an entire volume on that one problem)

Meanwhile, The Guardian also presented Bayesian statistics this week:

And in other statistics briefs, Jeff Leek tries to come to the defense of much-maligned p-values here: