In it he links to this 28-min. talk he recently gave at a Swiss conference on the subject:
...and, for a bonus video today, here is Matt Parker with his Psychic Pets project, including Barry the psychic Labrador:
I've already requested that they ask Barry IF Trump will be impeached, but thus far I've heard nothing back. :( [...and currently over at MathTango, my commentary on the ongoing "Big Internet Math Off"]
As most have heard by now, Robert Langlands, at age 81 (and it’s always great to hear of an 81-year-old mathematician receiving an award! ;), is the winner of the 2018 Abel Prize in mathematics. Robert’s work, the Langlands Program, is fascinating even if all you grasp is the broad outline of what it attempts to do, without much understanding of its complex details. Here are 3 of the general audience pieces already out on this momentous occasion:
I suspect over the next week there will be additional excellent articles appearing on this subject (I may or may not add other links here as they come along.)
Added:
For those with the background, a longer, more technical piece from AMS here:
For any who've never read it, or are unfamiliar with it, Ed Frenkel's "Love and Math: The Heart of Hidden Reality" introduces readers, to the Langlands Program, Ed's specialty.
And rightly or wrongly, this whole unification of mathematics notion, reminds me of a favorite quote from Keith Devlin I’ve used multiple times before (from an interview he once did for the NPR program “On Being” — and, not meant to imply anything about his own specific knowledge of Langlands):
"...that's when I became a mathematician; that's what I stumbled on at age 15 or 16 when here I was learning all this mathematics because I needed it. I had a utilitarian view of mathematics. I was learning it because I needed to solve the equations because I was going to be solving them in physics. And then, at the age of about 16 or 17, it all fit because it all came together in my mind. It was no longer this disjointed collection of techniques you could use to solve problems. It all fell into place, into this wonderful landscape. It was as if I'd been stumbling around in a forest, and suddenly I've climbed to the top of a tree and looked out and thought, this is the most beautiful place in the world. You can't tell it when you're down in the trees, which I had been, but the moment you reach an elevation where it all falls into place and you can see the whole topographic display in front of you, then the beauty is incredible. And the moment I discovered it, I said, um, I want to study mathematics. And I've been studying it ever since."
(...not sure of the specific credit for creation of this fun map, that has been passed around a lot, or I'd give credit?)
Keith Devlin’s new volume, “Finding Fibonacci” is more historiography than math, but there is some simple fun math sprinkled along the way deriving from Fibonacci’s writings. One old problem (translated from Fibonacci’s “Liber abbaci”) that Keith quotes runs as follows [I’m re-wording it in updated English]:
A man buys 30 birds composed of partridges, pigeons, and sparrows, for 30 denari. A partridge costs 3 denari, one pigeon costs 2 denari, and 2 sparrows cost 1 denaro, or 1/2 denaro/each. How many birds of each kind has the man purchased?
Of course this looks like a classic multi-equation problem, except there are only 2 equations, yet 3 unknowns:
1) x + y + z = 30 (number of birds; x partridges, y pigeons, z sparrows)
2) 3x + 2y + z/2 = 30 (total price)
You can solve it, or look at the answer below…
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Keith notes, there is a third hidden piece of knowledge buried in the problem:
namely, that x, y, and z must be positive integers (because birds don’t come in fractions!)
Thus, the two equations above are easily reduced to:
5x + 3y = 30 (where both x and y must be whole positive numbers)
Keith notes the first and third terms are divisible by 5 and so the second term (3y) must also be divisible by 5. In turn, that means y must equal 5, 10, 15 etc… but 10 or more is too large to work in the equation, so only 5 can be the correct answer (and x = 3, and z = 22).
As I indicated previously, the book may be more appealing to math history buffs than for mathematicians themselves (I found the last few chapters, containing more math bits, the most interesting of the book), though, with warm weather approaching, I'm tempted to call it a good beach read for the nerdier among us. In it, Dr. Devlin makes a case that "Liber abbaci," from the under-appreciated Leonardo Bonacci ("Fibonacci") is "a book that changed the course of Western civilization," and seeing him build that case (including making an analogy between Fibonacci and Steve Jobs) is interesting in its own right.
As most know, Hans Rosling passed away this week. As a Sunday reflection, a few timely sentences from Keith Devlin, in tribute to him, via a comment at his own blog:
“For all his engaging presentation skills, the numbers were at the heart of Rosling's talks. It was not his oratory that convinced us, in an instant, that our preconceptions of our world were wrong -- often violently so. It was the data -- the numbers displayed on the screen in front of us…“As it happens, Rosling's death comes at a moment in time when people in highly powerful positions are waging an assault on scientific facts, on numerical data, and indeed on truth in general…“An attack on truth is an attack on Society in general. Those of us whose lives revolve around discovering and communicating numerical and mathematical truth have a duty to speak up forcefully, in opposition. If our Society loses the respect for, and dependency on, truth, the loss of mathematics will be the least of our worries.”
Sunday reflection from Keith Devlin, on Euler's formula: "Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than skin deep, Euler's equation [eiπ + 1 = 0] reaches down into the very depths of existence."
“The use of a symbol such as a letter, a word, or a picture to denote an abstract entity goes hand in hand with the recognition of that entity as an entity. The use of the numeral ‘7’ to denote the number 7 requires that the number 7 be recognized as an entity; the use of the letter m to denote an arbitrary whole number requires that the concept of a whole number be recognized. Having the symbol makes it possible to think about and manipulate the concept.
“This linguistic aspect of mathematics is often overlooked, especially in our modern culture, with its emphasis on the procedural, computational aspects of mathematics. Indeed, one often hears the complaint that mathematics would be much easier if it weren’t for all that abstract notation, which is rather like saying that Shakespeare would be much easier to understand if it were written in simpler language.
"Sadly, the level of abstraction in mathematics, and the consequent need for notation that can cope with that abstraction, means that many, perhaps most, parts of mathematics will remain forever hidden from the nonmathematician; and even the more accessible parts — the parts described in books such as this one — may be at best dimly perceived, with much of their inner beauty locked away from view.”
Keith Devlin reviews and comments on a new documentary titled “Zero Days” about the “Stuxnet” computer virus hatched by the U.S. and Israel to destroy centrifuges involved in the Iranian nuclear production program. Devlin calls the film “arguably the most important movie of the present century” and titles his piece “Mathematics and the End of Days,” which may give you a hint as to its thrust (Variety calls it "a white knuckle thriller"). But in case you want stronger indication, here are some sentences near the end of Keith's piece:
“The weapon is, after all, just a mathematical structure; a piece of code. Designing it is a mathematical problem. Unlike a nuclear bomb, the mathematician does not have to hand over her results to a large, well-funded organization to build the weapon. She can create it herself at a keyboard. "That raw power has been the nature of mathematics since our ancestors first began to develop the subject several thousand years ago. Those of us in the mathematics profession have always known that. It seems we have now arrived at a point where that power has reached a new level, certainly no less awesome than nuclear weapons.”
I hadn’t even heard of this film prior to Dr. Devlin's piece, but now am certainly anxious to see it. An old Murphy-like aphorism says that ‘anything that can happen will eventually happen’ -- in this day of computer malware and cyber-warfare that’s never been a scarier thought.
Here is the official trailer for this must-see film:
A
beautiful, touching, scrumptious essay this week from Keith Devlin, on
the beauty of mathematics... a somewhat tiresome phrase that he breathes
life into here, focusing on calculus, or, as he quotes William Blake, "infinity in the palm of your hand":
It
deals with a student's recent response to a piece Keith had written almost 10
years earlier. I heartily commend it to all mathematicians, math
teachers, math majors, and students in general, and all those, who like
myself, simply love math from the sidelines. It almost has a fractal
quality, as a beautifully-crafted essay, about beautiful ideas, about
the beauty of beauty! ;-)
[p.s... Dr. Devlin suggests "if you are a math instructor at a college or university, maybe
print off this blog post and pin it somewhere on a corridor in the
department as a little seed waiting to germinate." I'll
second that suggestion, which derives, NOT from Keith's ego,
but from his infectious love of math teaching/learning.]
Actually,
half the post is simply a verbatim letter Dr. Devlin received from a
math student who had previously read another of Keith's essays, and now
was writing to say how much he finally appreciated that earlier
piece. Is there anything more rewarding to a teacher than to hear from a
student (and in this case not even Keith's own student) how much
something you said or did in the past has affected that student years
later!? Keith's earlier piece was about the deep, deep beauty of
calculus, or again from Blake,seeing "an infinite (and hence unending) process as a single, completed thing."
All
of us who've taken calculus will probably freely admit that, no matter
what our grade or ability in a first-year course, we lacked any deep
grasp of the subject at that point. To a lesser degree maybe that even
holds for algebra, geometry, trig… the student can't fully appreciate
these subjects 'til s/he has taken in much more mathematics for context,
depth, nuance. The "inner beauty" of math requires persistence and
commitment to fully access.
Dr. Devlin's post reminded
me slightly of the well-known Richard Feynman blurb that I've placed
below (and am sure most of you have already seen), wherein he speaks of
the "beauty of a flower," and how,
despite what an artist friend thinks, he as a physicist also has access
to seeing that beauty; perhaps even perceiving it at a deeper level than
does the artist.
I WISH I could see the
beauty of math the way Keith, and Ed Frenkel, and Steven Strogatz, and
others see it (seeing it, as Keith has previously written, from a
treetop overlooking the vast but inter-connected forest below). But
alas, as a rank-amateur, my vision is far more limited, far more myopic
than theirs. Yet even from my lowly vantage point mathematics resounds
in beauty, in "excitement, mystery, and awe" as Feynman refers to.
Some
of course call mathematics the language of science, or even the
language of God. But at base, I think its beauty lies in being a pure,
grand, and almost inexplicable creation (or discovery) of the human
mind... the pinnacle of that which our brains are capable. In a day
when our lives, politics, and society, seem inundated with violence,
intolerance, and irrationality, mathematical thinking stands out as a
beacon for the future, if we as a species are to have a future.
Growing
up, I watched my grandfather (and other seniors) become increasingly
cynical about the world as they aged, and swore to myself I would never
be like that. But I do now find myself saddened each day when I turn on
the news… cynicism is hard to repress. My hope today though, is that
every teacher out there, at least once in your lives, receives a letter
like the one Dr. Devlin has shared, or if you're not a teacher, that you
hear from some young person, when you're not expecting it, what a
difference you made in their lives.
The oddball Count
(and father of General Semantics), Alfred Korzybski wrote that we humans
are a "time-binding" species (different from all other species that
only "space-bind") because of the way we routinely transfer our
increasing knowledge across generations. That, in part, is what I see
going on in Dr. Devlin's piece, "time-binding" with a younger
generation... and, as always, the younger generation is our real hope
for the future... and, our shield against cynicism!
Finally, as I was completing this post a new blogpost from Megan Schmidt
crossed my webfeed. If you need a reminder that teachers impact young
lives (or even if you don't) I hope you will read it as well, (be
sure to click on and read the student exposition she provides): http://mathybeagle.com/2015/10/03/where-do-we-go-from-here/
"For reasons that I don’t fully understand, our mathematical culture encourages us to define our mathematical ability by what we don’t know, what we aren’t able to do, rather than by what we do know and have learned how to do. The power of culture is strong, with deep roots…"
A thoughtful post (including the above) from AMS blogs, that I suspect everyone can think about:
It's about our self-perception, as math students, of our own abilities, and how that aids or hinders us.
Then, interestingly, the latest post from Keith Devlin reviewing a new math-oriented movie, "A Brilliant Young Mind," that somewhat dovetails the above AMS post (especially see Devlin's "postscript"):
This Sunday re-running a quote from Keith Devlin that I first posted last year and that derives originally from his Sept. 2013 appearance on NPR's wonderful "On Being" podcast :
"...that's when I became a
mathematician; that's what I stumbled on at age 15 or 16 when here I was
learning all this mathematics because I needed it. I had a utilitarian
view of mathematics. I was learning it because I needed to solve the
equations because I was going to be solving them in physics. And then,
at the age of about 16 or 17, it all fit because it all came together in
my mind. It was no longer this disjointed collection of techniques you
could use to solve problems. It all fell into place, into this wonderful
landscape. It was as if I'd been stumbling around in a forest, and
suddenly I've climbed to the top of a tree and looked out and thought,
this is the most beautiful place in the world. You can't tell it when
you're down in the trees, which I had been, but the moment you reach an
elevation where it all falls into place and you can see the whole
topographic display in front of you, then the beauty is incredible. And the moment I discovered it, I said, um, I want to study mathematics. And I've been studying it ever since."
[p.s. -- just yesterday, Keith was in a bad biking accident, when some of his equipment failed... I believe the bicycle suffered more damage than Keith (at least his tweeting hand seemed to be working :-), but in any event, I'm sure we all wish him well in any recovery needed.]
Fitting I s'pose that I should start a new year with Keith Devlin, who has probably graced my posts more often than any other individual. His blog-year started off (yesterday) with another piece about the educational system in his ongoing attempt to persuade Americans of the need for Common Core. I'm afraid by now he's preaching to the choir -- views on this topic are so hardened. Either you believe in Common Core (perhaps with some reservations, but nonetheless support it), or you think it the product of overly-liberal educators and incompetent, intrusive government... and not a lot of wiggle-room in-between.
Keith writes succinctly at one point:
"The fact is, any parent who opposes adoption of the CCSS is, in effect, saying, 'I do not want my child prepared for life in the Twenty-First Century.' They really are. Not out of lack of concern for their children, to be sure. Quite the contrary. Rather, what leads them astray is that they are not truly aware of how the huge shifts that have taken place in society over the last thirty years have impacted educational needs."
Common Core opponents focus on the past, which somehow they think was just fine, while Keith is focused on the future, especially in terms of needed job skills for rising generations.
He also links to some great TEDTalks (Sugata Mitra and Ken Robinson) that I suspect most readers here have seen... but if you haven't, by all means, watch.
A central part of his post is a chart showing the "skills" most sought-after in new workers by Fortune 500 companies back in 1970 versus 1999 (is there not an even more up-to-date list?)
"Problem-solving," which didn't appear in the top 10 in 1970, is #2 in 1999 (and certainly one of the impetuses for education change). In fact, the top 3 sought-after skills had completely changed by 1999, and I have to give some cynical, non-mathematical commentary about the other two:
1)#3 on the list is "interpersonal skills," non-existent in the top 10 from 1970! I can't help but think that this is partly the result of today's world becoming a courser, less-civil place than the world of 1970. Perhaps in 1970 it was presumed that if you made it to adulthood and were seeking employment, then you had sufficient interpersonal skills for a workplace -- today employers must seek out people with such interpersonal skills! Also, on an even more cynical note, I suspect there are more jobs today (sales, marketing, public relations, management etc.) that require a person to be persuasive and controlling of others, than in 1970, when "honesty" or "sincerity" may have been greater virtues than the ability to manipulate and sway people.
2) Meanwhile, #1 on the 1999 list is "Teamwork" (it was #10 in 1970) -- this troubles me a bit! One of the most common questions prospective employees hear these days is along the lines of "Are you a team player?" I've felt for some time now that this is often code for, "will you do whatever the company asks you to, regardless of laws and ethics?" Employers don't want workers with consciences or personal ethics (who might object to something or be whistleblowers), so much as company drones who will 'look the other way' when needed and march to the company anthem. That's a broad generalization, but from observation of big business behavior over the last 3 decades (yes, there are some ethical businesses out there; their numbers just seem in steady decline).
I usually agree with Dr. Devlin's views, but was once troubled by a response he gave me to a question I posed to him last year. He wrote:
"...the only possible answer to the provision of good education in this country is by private enterprise. The state system is a century out of date and broken beyond repair."
Wow!, "broken beyond repair"... As a proponent of public education and critic of private industry, that gloomy reply stung! The thought of private companies in charge of our education system (and it's already happening in some places) disturbs me (I continue to believe it possible for public education, and for that matter, big government, to work successfully). Keith is part of a small entrepreneurial company (BrainQuake, Inc.) and I suppose his optimism stems in part from that experience. I'm not concerned with enterprises of that size... but am worried by the companies that will, if Keith's company is successful, gobble up his enterprise and strictly subordinate whatever its good works and intentions are, to a bottom-line (...and then burp, ignominiously).
No doubt in my mind that Dr. Devlin is right and the educational system badly needs reforming... but it's so much broader than that... the Fortune 500 itself is also in desperate need of reform! And, unfortunately, I'm not convinced either will happen anytime soon.
Keith and I are close to the same age, and he once again closed out his post with some classic 60's music... so once again he inspires me to do the same ;-):
I don't often see Keith Devlin focus on geometry in his blog posts… but today he did… and quite excellently! In fact, from my standpoint the post, in some ways, makes for a nice counterpoint to the Sunday reflection I ran this weekend on Platonism (Dr. Devlin is a non-Platonist). Read Keith here:
"Mathematics provides various ways to model our perception and experience of reality. Different parts of mathematics provide different models, some better than others."
He goes on to talk about fractal geometry and cellular automata of Steven Wolfram as two geometric approaches to the world.
"Both approaches can be said to begin by looking at how nature works, but the moment you start to create a model, you leave nature and are into the realm of human theorizing."
"...make no mistake about it, we do begin with assumptions. Not arbitrary ones, to be sure—not even close to being arbitrary."
"...mathematics is not 'the true theory of the real world' (whatever that might mean). Rather, mathematical theories are mental frameworks we construct to help us make sense of the world."
"...we should not lose track of the fact that mathematics is not the truth. "Rather, it provides us with useful models of the world. As a result, it is a powerful and useful way of making sense of the world, and doing things in the world."
He ends with his vocal support again for Common Core (while admitting more focus is needed on "how to properly implement the Standards").
Read the entire piece, or like me, read it 3-4 times to squeeze out as much food for thought (and I dare say food for controversy as well!) as you can from it.
And in a similar vein, still more from Jordan Ellenberg on mathematical thinking (including a new podcast from "Inquiring Minds" with Chris Mooney): http://tinyurl.com/ojtsykp
"The CCSS were created to ensure that all students who graduate from an
American high school do so with the skills and knowledge necessary to
succeed in college, career, and life in the Twenty-First Century,
regardless of where they live." -- K. Devlin
Dr. Keith Devlin addresses the math Common Core debate in his latest posting for Huffington Post (he supports CC):
...just from the first two comments to his piece, I can imagine this is going to generate some lively, effusive discussion!
In all honesty, it's hard for me to see how this whole debate ever ends well, both sides, pro and con, having quite hardened, opposing views. Even if Common Core works well, it might be years, or even a generation, before we are able to fully recognize that. Meanwhile, in the short-term (which seems to be all people focus on these days) there are bound to be difficulties with the implementation of this education overhaul, and both sides are poised to blame each other for whatever travails result.
[None of that, by the way, is meant as a criticism of Keith's piece, which I think is great, but just an acknowledgment, that I don't believe it will be persuasive to the audience it's aimed at: generations of adult parents who have only ever known one way of learning math; or, as Keith writes, "...they were only ever exposed to the algorithmic-skills math instruction
developed for earlier times -- a form of teaching that is hopelessly
inadequate for life in today's world."] I hope I'm wrong, but, from my standpoint, this whole squabble isn't looking pretty heading into the future... :-(
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? "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?"
All of which leads to some interesting cognitive/neuroscience speculation and praise for the amygdala's role in problem-solving. Enjoy the whole piece... and then, go take a bike ride!
In his latest Devlin's Angle post, Keith Devlin looks back over 23 years of writing math columns (his "mathemaliterary journey" as he calls it) that divide into 3 main themes: 1) "What is multiplication?" ...the theme that created a "firestorm," 2) "Mathematical Thinking," and 3) MOOCs:
1) First, Keith Devlin continues his fascinating series on his MOOC experience (now 6 recent posts to catch up on in his latest series, if you haven't been following along):
2) And if you missed Keith delightfully talking on NPR this weekend about the excitement generated by Yitang Zhang's attack on the Twin-prime conjecture earlier this year, give that a listen here:
3) Meanwhile, Peter Woit writes about the incredible difficulty involved in verifying last year's proof from Shinichi Mochizuki of the ABC conjecture (perhaps unparalleled in the history of math):
5) Perhaps mathematicians are too shy or uncomfortable talking about their (or their colleagues') politics, but I'm still interested to hear from you, if you aren't:
b) Just this morning, Presh Talwalkar posted a nice triangle geometry problem with both a traditional algebraic solution and a 'quickie' simpler solution available:
