...a companion blog to "Math-Frolic," specifically for interviews, book reviews, weekly-linkfests, and longer posts or commentary than usually found at the Math-Frolic site.

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." ---Bertrand Russell (1907) Rob Gluck

"I have come to believe, though very reluctantly, that it [mathematics] consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-legged animal is an animal." ---Bertrand Russell (1957)

******************************************************************** Rob Gluck

Friday, June 21, 2013

Keith Devlin… as you may not have heard him before….

Math-Frolic Interview #15 (...not the usual fare)

"As a naturalized American, I have an immigrant's reverence for those words of our National Anthem, 'Land of the free, home of the brave.' For many of my fellow citizens born here, I fear these are just words they learned to recite in elementary school. For the fact that 56% of Americans declare that they would give away fundamental freedoms to reduce the risk of terrorist attack indicates that we may become the 'land of the enslaved, home of the scared.' "  -- Dr. Keith Devlin in Huffington Post

I'm not sure there's anyone more generous with their time and energy than Keith Devlin….
Since Edward Snowden's NSA revelations (and subsequent controversy), I noticed Dr. Devlin expressing himself on the topic (as much as one can in 140 characters!) on Twitter (@ProfKeithDevlin) more than any other mathematician/scientist I follow. The passion of his opinions intrigued me and I asked if he'd do another Math-Frolic interview, but this time just on his views about this NSA controversy -- NO MATH (my prior math interview with Keith is HERE) -- I thought he deserved more time or space than available on Twitter or even Huffington Post, where he has a piece. So read up, Keith doesn't much mince his opinions!

1) The recent NSA revelations have generated a broad range of opinion across the spectrum (from outrage, to 'ho-hum, nothing new here'). Of the many math/science persons I follow on Twitter you've been among the most harshly outspoken. Can you explain a little more deeply where that sense of betrayal stems from… as a 'naturalized' citizen do you perhaps appreciate American democratic ideals even more than a lifelong born-citizen who just takes them for granted… and how much does your British background (another country with a strong democratic history) come into play? Or, is it mostly just a straightforward legal/Constitutional issue for you, unrelated to background?
Appreciating the iconic ideals of US Democracy as enshrined in the Constitution is part of my outrage at the way the US (the nation, not just the government) has allowed those ideals to slide. I certainly am under no illusions as to the many deficiencies in the US, for instance, its third-world levels of poverty and infant mortality and its medieval prison system. But those words of the Founding Fathers are one of Humanity's greatest achievements. It's surely worth remembering that the US owes its existence to the fact that those Founding Fathers were traitors. So far, everything I have seen of Edward Snowden puts him into the same camp as the Nation's founders. Certainly, his public statements and actions so far qualify him as a greater American than the current President who complicitly -- and secretly -- allowed the slide away from the founding ideals to continue.

The personal twist in my case is that, twice in my life, I've found myself as the "suspected outcast". I'll describe the first. As a young assistant professor in Germany in the 1970s, when Germany was struggling with massive student unrest and genuine internal terrorism (Bader-Meinhoff, etc.), security forces surveillance of some of my students brought me into their radar, and for a few weeks I was followed around and my mail was regularly intercepted. The fact that I was very aware of this indicated that the intention was probably to scare me as much as to find any incriminating evidence against me, and my wife and I actually regarded the whole affair with amusement. I was clearly low level, peripheral fish in their surveillance sweep, and after a few weeks they were (as far as I know) out of my life. Still, it made me realize how easy it is for a totally innocent individual to find him or herself on a government security list, simply by virtue of the people they interact with. (To this day I have no idea if any of my students were active in the political unrest of the time, or indeed if any engaged in illegal activities.)

The second time was in the UK, and contributed to my leaving my homeland for the US, but the one example I have described should be enough to indicate why I simply don't buy the frequently touted idea that "If you have done nothing wrong, you have nothing to fear." In the age of big data, just as we can easily find ourselves with a wrecked credit rating that can take years to sort out, so too we can find ourselves on a government "person of suspicion" list. In my case, I had the psychological strength to shrug it off -- albeit I did emigrate from the UK to the USA. Had I a different psychology, the ending could have been tragically different, as it was for Aaron Swartz, who was unable to sustain the inhuman persecution by US attorney Carmen Ortiz and Massachusetts assistant US attorney Stephen Heymann, who clearly viewed him as a mere pawn to advance their careers.
2) Related to the above, you actually worked for the NSA at one time in your life (contractor???). I suspect you can't say a lot about that work, but can you say, in a more general way, if that specific experience with NSA, contributes to your strong feelings? And are there any details from your own NSA experience you can tell us about which are pertinent to this ongoing story?
In the early post-9/11 era, I did work on a large, non-classified (albeit not publicized) project to improve the quality of the actionable intelligence that could be obtained from massive amounts of data. I was glad to play my small role, though when that project came to an end, I held the same view I did at the start: absent a significant HUMINT lead (human intelligence), trawling through massive amounts of data is a waste of time. There is no chance you will be able to prevent a terrorist attack. I know that a number of intelligence leaders have made statements of late that claim otherwise, but all I can say is that after working hard on the problem for five years, I reached a very different conclusion. To be sure, I do not know the computing capabilities the NSA has, but based on my understanding of the problem, without a good HUMINT lead, it won't be enough. Mathematically, the problem is known as combinatorial explosion. (With a HUMINT lead, on the other hand, you don't need to trawl the data, you just have to search for confirming evidence, starting from that one lead.)
3) Author Kurt Eichenwald wrote a book a few years ago, "500 Days," apparently divulging info similar to what Ed Snowden has revealed. Have you by any chance read this volume? He claims on Twitter (as do others) that there is nothing new in the current revelations, and that in fact some of the details, as covered by the press, are simply wrong or misleading. To those who would say, there's no real news here, and moreover, private companies (Google, Facebook etc.) snoop on individual lives FAR MORE than the Gov't., what would you say?
I've written elsewhere (Huffington Post) that the Snowden revelations were akin to Lance Armstrong's appearance on Oprah. In both cases, we learned nothing we did not already strongly suspect. But making it public knowledge, as opposed to widely believed suspicion, changes the debate. In the Armstrong case, within days, he had lost all his multi-million dollar sponsorship deals. After Snowden, the intelligence chefs could not respond to questions by saying there was no wrong doing, they had to provide actual details, in at least one case revealing a clear-cut case of perjury before Congress. Maybe heads will roll -- they should -- but maybe not. (In fairness to those involved, the nature of intelligence does put people in a difficult position with regard to being truthful. Few of us have the courage of Edward Snowden.) In any event, even ordinary citizens had a pretty good idea of what the NSA was doing, so for sure our enemies did. Statements that the Snowden revelations damaged national security are clearly absurd. The security lies in the data, not the knowledge that is exists. The only damage from the Snowden revelations is the embarrassment of people in power. (It surely cannot be international relations, except on the surface, since all the other countries harbored the same suspicions as we all did, and for sure the many countries with the technological capabilities knew for sure what we were up to!)
4) Personally, while the massive net for "metadata" concerns me, what troubles me even more (and doesn't get much coverage) is the potential for NSA individuals to target specific politicians/leaders for scrutiny and use that info for strictly partisan purposes… possessing knowledge about the personal lives of political opponents is an even greater danger to democracy than knowledge of the general citizenry. Any thoughts?
This is the real worry. Right now, we have President Obama saying "Trust me, this immense security apparatus is being used for your safety." As it happens, I am inclined to give him that trust, though in so doing I am making a leap based on no first-hand knowledge. But that's not the point. Who knows who will hold the reins in the future? It was not long ago that J Edgar Hoover was in charge of the FBI. We've had despots in positions of power before, it can happen again. When I was living and working in West Germany, I traveled occasionally to East Berlin to consult with university colleagues, and learned enough about the STASI to never want to live in a state with such a powerful and intrusive security apparatus.
5) Some people view Snowden (thus far) as a highly-intelligent, sincere, courageous, deeply-patriotic  individual, and others label him narcissistic, self-aggrandizing, delusional (some have even said, why can't he be ALL of the above!). Care to say, how you would characterize him?
I already did. I think history will portray him as a twenty-first century "Founding Father", who initiated a return to the principles by which the country was founded. Assuming, that is, that we do indeed step back from the abyss. The current attempts to discredit him are as predictable as they are transparent. His personal character actually makes little difference. He did the US a great service (that's the part history will remember) by performing a heroic act, clearly at high risk to himself. Exactly the same can be said of the Founding Fathers. Acts can endure, personalities are replaced by stories.
6) One of the interesting major disagreements is between those who say that the sort of massive "dragnet" surveillance that is going on is outright illegal and not authorized by Patriot Act measures, versus those who say there is NO "surveillance" but only the collection of large-scale metadata (which does not constitute surveillance), and only when a 'pattern' of interest is found in the data can the Gov't. then seek a court order to do further actual surveillance. I know you are interested in the uses of language and meaning, and clearly that is what we have here… Any comments?
It is clearly illegal, being against the Constitution. It's also immoral. Period.
7) Do you feel very disillusioned (as some do) by the Obama presidency over the various issues of transparency/secrecy that have arisen, or are your issues more with the intelligence community than with the White House?
We live in a democratic republic. The intelligence community do their job, and implicit in that is to collect as much information as they can. The elected government are the ones setting the limits and calling the shots. If there has been a breakdown in that line of command, it is the government that has the responsibility to put things right. If ever we were at a juncture where a president should offer real leadership, now is that time. I understand Obama would like to go down in history as another Lincoln. Now is his chance. I wonder if he has it in him.
8) Supposedly Glenn Greenwald/Guardian have several more disclosures to make from the information Snowden provided. Care to make any predictions (and I know you think that predictions, especially about the future, are difficult ;-) about what may happen over the course of say the next year? …Will Snowden be extradited and prosecuted here in the U.S.? Will the Patriot Act be revisited and revised by Congress? Will the stand politicians' take on this affair (with or against Snowden/NSA) have a major effect on the 2014 mid-term elections?….
Since I don't know what information Snowden has, I don't see how anyone can make predictions. Whatever he has clearly already exists in multiple copies, held by different people, so it will likely eventually come out. So in practical terms, the best option for the US is to simply leave Snowden alone in Hong Kong. Public interest being as it is, "the Snowden story" will soon go away -- though I hope that real reforms result. Trying to have him extradited to the US, in contrast, will not only keep the story on the front pages for months and more likely years, but if the attempt succeeds, we will have a martyr on our hands. And martyrs are dangerous. Do we want to turn Snowden into another Nelson Mandella? How do we respond if, for instance, an imprisoned Edward Snowden is awarded the Nobel Peace Prize? (Those Scandinavians have a strong sense of social justice and are not easily pressured, so that could very well happen!) Better not to go that route. There is a slew of downsides, but the only "upside" is revenge, and there is no way the US could come out with dignity and respect if we throw our immense power going after one of our own citizens so it would prove to be a hollow upside.
9) And one last crystal ball inquiry… many have contended for a long while now that in the future there simply will be NO privacy… some think current young generations have ALREADY forfeited any significant concern over privacy. I truly wonder if, a century from now, "privacy" won't be just a quaint little term in historical footnotes. You and I might not wish to live in that world, but is not the near-complete loss of privacy inevitably coming? :-(
I think that here in the US we have a choice. In 1789, a bunch of traitors to the ruling authority formulated the First and Fourth Amendments as they set the new nation on its course.  Like him or hate him, Edward Snowden has put the questions of public information and personal privacy on the table once again. As a result, we have an opportunity to correct our course. Because of the Founding Fathers, we are currently able to debate this issue freely and openly. If we don't live up to those two-hundred-years ideals now, that great episode of human society (great for all its flaws, which lie in the execution, not the ideals) will have come to an end. We will be the "Land of the enslaved, home of the scared."

....I don't completely agree with everything Keith says here, but I surely love the man's passion... as he demonstrates in everything he takes an interest in or speaks about. And further, as someone who has experienced unwarranted governmental suspicion/surveillance elsewhere -- albeit by his admission short term and low level -- his views deserve close attention. THANKS again for taking the time to respond Dr. Devlin.

I'll close out (...for some comic relief) with this "Good Will Hunting" scene that I've already used over at Math-Frolic, and most of you have likely seen:

Tuesday, June 18, 2013

Waxing Platonic…

The Platonic divide in math....

The book I've highlighted recently, "The New York Times Book of Mathematics," ends with a chapter of readings on various notable mathematicians… Erdos, Ramanujan, Conway, Gödel, Wiles, etc. I suspect most (if not all) of the brilliant figures profiled were/are Platonists (mathematics is discovered, not merely created). Yet many other recent math figures (Reuben Hersh, William Byers, Keith Devlin, Jim Holt, and more) have forcefully argued that mathematics is indeed a mental creation that might even differ considerably in a different Universe than ours -- indeed some almost seem to find the notion of mathematical Platonism so wrong-headed as to be silly (while Martin Gardner found the non-Platonist view almost silly). And occasionally such writers cause me to sway toward their non-Platonist stance though I always seem to float back toward Platonism.

One thing that so many of the greatest, most productive mathematicians seem to share is an uncanny, almost inexplicable ability to tap into a realm of intuition or mental landscape not readily accessible to most of us. Ramanujan is certainly the unparalleled, most inexplicable, example of this; producing amazing mathematical results that are still today being explored and proven. Reading James Gleick's portrait of Ramanujan in the Times volume it really hit me… was Ramanujan, who routinely produced such results/theorems without ever showing the steps that led to the outcome, in direct access of the "Platonic realm?" He himself claimed his insights came in dreams and trances directly from the Indian Goddess Namagiri... Who are we to argue (and where did she reside)?!!

In many ways, Ramanujan's extraordinary talents are reminiscent of the incredible abilities of various mathematical savants and prodigies who usually can't explain how they do what they do. Their brains seem clearly to operate, or even be wired, differently from those of 'ordinary' people.

My point in all this is simply that such rare, yet nonetheless real, individuals DO give an appearance of tapping into a realm… call it perhaps the Platonist realm… that the rest of us lack ready access to, where numbers and math really DO exist apart from our day-to-day world.  Naysaying non-Platonists will simply argue that however Ramanujan and the rest gain their special knowledge, it ultimately still arises via the firing of neurons within a physical human brain situated between two ears… i.e. it is still a human creation. I can't prove that reductive view wrong, but the notion that there are worlds out there that only some of us can easily tap into, and only some of the time, through means we don't even comprehend… is so much more appealing! As Shakespeare put it long long ago, “There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.

I think Martin Gardner might well relate to this idea too… For all his empirical skepticism, Gardner also described himself as a "Mysterian," a philosophical view which holds that ultimately consciousness cannot be explained by any human brain. In the famous words of computer scientist Emerson Pugh, "If the human brain were so simple that we could understand it, than we would be so simple that we couldn't."  Is it possible that humans are able to draw upon a Platonic world, and can recognize 'consciousness,' yet perhaps never, with our limited minds, fully grasp either? Does the 'Platonic world' exist, but like the Continuum Hypothesis, fall into a nether land of things that simply can't be proved true or false by human logic?
Speaking of certain mathematical proofs, Paul Erdos would famously say, "This one is from The Book!" I'm not so sure he was speaking in metaphor... perhaps The Book, in some (Platonic) manifestation, exists. Is the alluring beauty of math only in our heads, or is it an integral part of all creation? MIT physicist Max Tegmark has argued for some time now that the entire physical universe, as we perceive it, is nothing more than mathematics, or a mathematical structure (called the MUH, or "mathematical universe hypothesis").

Anyway, read Gleick's beautiful 1987 portrait of Ramanujan and just imagine the Indian mystic-mathematician dreaming and dipping into a realm where numbers are as 'real' as rocks and chairs are to most of us:


a couple of brief lines from therein:
" 'When he [Ramanujan] pulled extraordinary objects out of the air, they weren't just curiosities but they were the right things,' said Jonathan M. Borwein of Dalhousie University in Halifax, Nova Scotia...
" 'He seems to have functioned in a way unlike anybody else we know of,' Dr. Borwein said. 'He had such a feel for things that they just flowed out of his brain. Perhaps he didn't see them in any way that's translatable.' "
It's probably also worth noting that the very first entry in the entire NY Times anthology is a 1998 George Johnson piece also addressing the subject of Platonism:

"Useful Invention or Absolute Truth: What Is Math?" by George Johnson

At the end of the piece, Johnson cites a 1995 book, "Conversations on Mind, Matter and Mathematics" that covered a debate between French mathematician Alain Connes and French neurobiologist Jean-Pierre Changeux over the subject of math Platonism. An interesting and rich review of that book here (even makes brief reference to Ramanujan):

Connes and Changeux didn't resolve the debate... and we won't here... but still, nourishing food-for-thought.

Friday, June 14, 2013

Math via The New York Times

Non-technical math anthologies are rare critters… when one comes along my instinct is to pounce on it. "The New York Times Book of Mathematics," edited by Gina Kolata, was worth the pounce!

This volume covers a wide and interesting array of topics. Here are the 7 chapter headings (though they don't fully hint at the range of material touched on):

1) What Is Mathematics?

2) Statistics, Coincidences and Surprising Facts

3) Famous Problems, Solved and As Yet Unsolved

4) Chaos, Catastrophe and Randomness

5) Cryptography and the Emergence of Truly Unbreakable Codes

6) Computers Enter the World of Mathematics

7) Mathematicians and Their World

That should give you a sense of the breadth of topics on display here. The pieces are vibrant, terse treatments (no doubt only intended to fit within a certain column length). The writing is so good that the pithiness leaves one reaching the end of most pieces wanting more... just one more page pl-e-e-ease.

I think Gina Kolata sets the tone and 'feel' of this engaging volume very aptly when she writes in her Introduction:
"A mathematician once dismissed the very idea that people outside his circle could ever understand the true essence of the field. Mathematics is an art form, like music or painting. Translating math into the English language, he said, is harder than translating Chinese poetry. The beauty is lost, the elegance, and a proof that is a thing of ineffable iridescence becomes reduced to a baffling or mundane-sounding bottom line....
"But even if the rest of us cannot appreciate mathematics as an art form, are we really shut out? Articles in the New York Times may not give the details of proofs, but they reveal a rich world that can be exciting, surprising, and can even tug at the heartstrings."
Yet several reviews I've seen of the volume are rather ho-hum about it, but these are usually from professional mathematicians -- for the working mathematician there may not be that much here to excite -- although I think any math lover will find at least a few pieces that strike a chord. But for lay folks with an interest in math (my core readership!!) this may be the BEST anthology I've ever come across! There is no technical material or equations to weigh down your enjoyment, nor slow your consumption. It is all about math and mathematicians... without doing math.

One downside is that because this is limited to NY Times' writers, many excellent popular math writers are absent.  Indeed, I'd normally be skeptical of an "anthology" that was restricted to the number of writers this one is -- it is very heavy on pieces from Gina Kolata and James Gleick -- but these writers are SO good at their craft that skepticism quickly fades away. While Kolata and Gleick's pieces are perhaps the best, there are numerous fine contributions as well from George Johnson, John Markoff, Dennis Overbye, and others. Oddly, there are no entries from Steven Strogatz here (author of some of the most popular math pieces the Times has carried in recent years), but perhaps his offerings simply didn't make the 2010 cutoff for the volume. The one other thing that may be missing from the collection, it seems to me, are more articles which relate math to the other sciences, particularly physics and biology (I believe the Times has run several such pieces).

Most of the entries come from the last 3 decades or so, but some go back as far as the late 1800s. I wasn't particularly enamored of several of the older entries that were probably included more for the sense of history or progression they illustrate than for the math covered. Still, overall the mix is appealing.
Chapter 5, focusing on cryptography, would have been interesting in its own right, but became even more-so, in light of current events, as almost every article makes mention of the NSA and its relationship with mathematicians (by most accounts, by the way, NSA is the largest employer of mathematicians in the world). But there isn't a bad chapter in the volume.

In short, I love this compendium, even more than I expected to. If it wasn't such a thick, heavy volume I would almost recommend it, at this time of year, as a 'beach-read'… for the mathematically-inclined. In the distressed world of print journalism, the NY Times has been cutting back on science journalism, so it is wonderful to have this hard-copy of delicious math-related essays to keep on one's shelf as a permanent source of popular math writing stretching across decades. Hats off to Ms. Kolata on a job well-done!

Wednesday, June 5, 2013

Flipped Classrooms, MOOCs, and Having a Blast

Timing is everything....

Well, this was great… I was planning to write a post musing a bit more about math education in regards to both "flipped classrooms" and MOOCs… but then discovered Keith Devlin has just put up a new (longish) post on his MOOC blog saying most of what I wanted to say, and with more authority than I could say it. So please read it:


Do note that I think his title may be a bit misleading so follow carefully all he has to say. I was afraid his long lapse in blogposts might mean that the 2nd rendition of his 'mathematical thinking' MOOC hadn't proceeded well (though his insanely busy schedule could also account for it), and luckily it doesn't sound like that was the case… though he does still write with caution about MOOCs, and will have more to say in the future about this last go-around.

Here are a few of the most trenchant comments he makes (I've added some emphasis):
"the vast majority of people under twenty now interact far more using social media than in person.
We could, of course, spend (I would say “waste”) our time debating whether or not this transition from physical space to cyberspace is a good thing. Personally, however, I think it is more productive to take steps to make sure it is – or at least ends up – a good thing. That means we need to take good education online, and we need to do so for the same reason that it’s important to embed good learning into video games…
"The media of any age are the ones through which we must pass on our culture and our cumulative learning."

"Something else that digital technologies and the Web make possible is rapid iteration guided by huge amounts of user feedback data – data obtained with great ease in almost real time."
The one place where I think Keith sounds a little too negative is when he writes:
"Experimentation and rapid prototyping are fine in their place, but only when we all have more experience with them and have hard evidence of their efficacy (assuming they have such), should we start to think about giving them any critical significance in an educational system which (when executed properly) has served humankind well for several hundred years. Anyone who claims otherwise is probably trying to sell you something."
Actually, I think "experimentation and rapid prototyping" may now be an integral part of our quickly evolving world and education system… more than ever before change can happen with such speed that we may try 4 failed experiments and still succeed at #5 in an acceptable/practical amount of time (even before the "hard evidence of efficacy" is fully in or agreed upon. Just the speed with which the MOOC movement has grown is a testament to that, and as Keith implies, the time is ripe for us to "make sure" they [MOOCs] work in some form.

So much for MOOCs…
What actually got me thinking again about education was a recent Twitter tweet that led me to this blog I was previously unfamiliar with:


Despite the uk URL appendage it's from a California high school math teacher (Crystal Kirch) focused on the "flipped classroom" concept. Just scanning over it, it looks interesting and impressive to me, but as someone not in the loop of secondary education I don't want to assume too much. What definitely caught my attention though (and those of you in secondary education likely already knew this) was the sheer number of other blogs with a similar focus on flipped instruction (as well as a network of teachers with this interest) that Mrs. Kirch links to. The "flipped classroom" has been around long enough that LOTS of teachers are trying it, tweaking it, playing/experimenting with it, blogging about it, and just generally sharing their experiences (good and bad) with their peers. What a great collaborative endeavor!!… and not brought on by some agency-directed-commissioned group-on-high, but by the spontaneous interest of those who share similar goals. Again, before the internet this sort of rapid cross-communication effort wasn't possible.

The term "flipped classroom" came about, so far as I'm aware, from early uses of Khan Academy videos (and Khan Academy still has many vocal critics), but of course there are now MANY internet resources available to choose from, and Khan itself constantly evolves. (Some have noted that the 'idea' of the flipped classroom, though not the term itself, actually long precedes Khan Academy.)

It is fascinating to me how both "flipped classrooms" and MOOCs, which in some ways share little in common, and operate on different levels of education, have simultaneously sprouted up like mushrooms in the cyber landscape, both controversial and rapidly-evolving, yet giving tremendous promise.

As Keith writes so aptly at the end:
"Those of us in education are fortunate to be living in a time where there is so much potential for change. The last time anything happened on this scale in the world of education was the invention of the printing press in the Fifteenth Century. As you can probably tell, I am having a blast."
And some of us are just having a blast... watching those of you who are in the trenches having a blast.
To Keith, and Mrs. Kirch, and all others doing the nitty-gritty work that will shape the education of future generations... THANK YOU!