Friday, April 20, 2018

Math and Understanding

Physicists often remark that no one actually understands quantum mechanics (and those who say they do are lying), but they use it because it consistently works.
Similarly, polymath John von Neumann once famously said that , “…in mathematics you don’t understand things, you just get used to them.”
And in a similar vein David Wells quotes applied mathematician, Oliver Heaviside, thusly:

The prevalent idea of mathematical works is that you must understand the reason why first, before you proceed to practise.  That is fudge and fiddlesticks. I know mathematical processes that I have used with success for a very long time, of which neither I nor anyone else understands the scholastic logic. I have grown into them, and so understood them that way.

It seems to me that a lot of the emphasis these days from professional mathematicians, as well as in Common Core’s approach, is for students to develop a much deeper understanding of mathematical logic and connections first (and foremost), and for rote processes to follow thereafter. A change in perspective or outlook perhaps??? (or maybe math education has simply always been a mixture of both, in a sort of chicken-and-egg fashion).

Anyway, I’ve spent this week offering up a few snippets (with one more coming Sunday) from Wells’ wonderful 20-year-old volume “The Penguin Book of Curious and Interesting Mathematics.” It is one of the most delightful math reads I’ve stumbled upon in quite awhile, with no particular order (that I can detect) to its succinct, highly-varied contents. Some of the best bits in it are stories/narratives/anecdotes, too long to quote verbatim, about specific famous mathematicians (I especially found the background on Stanislav Ulam fascinating, for example). If you can find a copy I highly recommend it.

Thursday, April 19, 2018

Marilyn vos Savant Steps In It…

Marilyn vos Savant is widely-known for writing a column and giving answers to puzzles of all sorts... and also famous for sometimes having her answers challenged, only to have the critics often embarrassed — famously so in the instance of the Monty Hall problem, but occasionally with other problems as well (she’s also had to correct or refine given answers on various occasions).
I didn’t recall however her controversy over Andrew Wiles’ proof of Fermat’s Last Theorem ’til I read the below brief passage in David Wells' “The Penguin Book of Curious and Interesting Mathematics”:
“Less than five months after Wiles’s lecture, she published a book, The World’s Most Famous Math Problem (The Proof of Fermat’s Last Theorem and other Mathematical Mysteries), in which she claimed that non-Euclidean geometry is unsound, and so therefore is Wiles’s proof, because he uses non-Euclidean geometry. She also claimed that his proof depends on developments in mathematics that are relatively recent and poorly understood, and she encouraged her readers to try to ‘demolish Einstein’s theories of relativity’ by proving the parallel postulate.”
Wow, pretty harsh… I’ve never read her short (80-page) book, but as best I can tell, the consensus of professional mathematicians is more uniformly critical of it than some of her other writings; one mathematician calling the book “pure drivel” (and this is despite Martin Gardner writing the Foreward, initially calling the volume "a delightful, informative, and accurate book”).
IF I understand correctly, Marilyn did retract much of her criticism at a later date?
I recommend a very interesting read, by the way, about the whole Gardner/vos Savant relationship here:  (link corrected, I think?)

Anyway, if anyone out there is more directly familiar with this particular controversy (and book), and cares to add anything further about it, I’d be curious to hear.

Wednesday, April 18, 2018

George Polya... Stand-up Comedian

Again, from David Wells' "The Penguin Book of Curious and Interesting Mathematics" (as was Monday's post); this is one of the more humorous passages I've read in a math book (quoting verbatim):

"It was George Polya who admitted to studying mathematics at college because physics was too hard and philosophy was too easy. This is his view of the traditional mathematics professor. 
'The traditional mathematics professor of the popular legend is absentminded. He usually appears in public with a lost umbrella in each hand.
‘He prefers to face the blackboard and to turn his back on the class.' 
'He writes a, he says b, he means c; but it should be d.’
‘Some of his sayings are handed down from generation to generation.’
’In order to solve this differential equation you look at it till a solution occurs to you.'
‘This principle is so perfectly general that no particular application of it is possible.’
‘Geometry is the art of correct reasoning on incorrect figures.’
‘My method to overcome a difficulty is to go round it.'
'What is the difference between method and device? A method is a device which you use twice.'

After all, you can learn something from this traditional mathematics professor. Let us hope that the mathematics teacher from whom you cannot learn anything will not become traditional."

Tuesday, April 17, 2018

If Gluten-free Flour Is Your Thing....

H/T to Nalini Joshi yesterday 2 days ago, for tweeting the above bag of Anthony’s gluten-free, almond flour. You get the feeling that Anthony, at heart, is a frustrated mathematician-wannabe… the lower left corner of the bag reads as follows:
“For a free bag on us, and a personal high five from Anthony, prove that the real part of every nontrivial zero of the function below is equal to 1/2”

[no mention that with the $1 million in prize money you'd receive for proving the Riemann hypothesis you could purchase several dozen bags of Anthony’s fine flour AND have money left over for coffee (even expresso).]

Further, in the comments to Nalini’s tweet someone mentions that a new attempt to prove the RH was recently submitted to arXiv:

So seriously let us know what if any status that submission has.

Finally, drawing again from the David Wells' book that I mentioned yesterday ("The Penguin Book of Curious and Interesting Mathematics") a couple of old quotes from David Hilbert:

Upon being asked "What technological achievement would be the most important?" Hilbert replied, "To catch a fly on the moon. Because the auxiliary technical problems which would have to be solved for such a result to be achieved imply the solution of almost all the material difficulties of mankind.
Then asked what mathematical problem was the most important, he responded, "The problem of the zeros of the zeta function, not only in mathematics, but absolutely most important!"

Monday, April 16, 2018

A Book of Chocolates

I’ve enjoyed every David Wells’ math book I’ve ever encountered so didn’t hesitate, for $1, snapping up a 20+ year-old volume by him I saw at a recent used book sale, “The Penguin Book of Curious and Interesting Mathematics” —  260+ pages of fun, entertaining mathematical anecdotes, factoids, quotations, curiosities/tidbits.
I’d heartily recommend this volume to any math-lovers. It’s a veritable big box-of-chocolates for the math fan!

Here’s one simple morsel (I've re-written) from early in the book, just an old Paul Erdös puzzle:

Prove that if you have n + 1 positive integers all of which are less than or equal to 2n, then at least one pair of them are relatively prime.
.answer below

. there are several ways to prove it, but one logical way is simply to realize that having n+1 integers less than or equal to 2n necessarily entails at least two of the integers being consecutive, and therefore relatively prime (...for n integers less than or equal to 2n this would no longer hold for ANY set of even integers).

[I'll probably offer a few more bits from the volume through this week.]

Sunday, April 15, 2018

Experiencing Pleasure... Music and Math

Sunday reflection via Gottfried Leibniz:
"Music is the pleasure the human mind experiences from counting without being aware that it is counting."

Wednesday, April 11, 2018

Birthday Boys

Today is Andrew Wiles 65th birthday, making him old enough to be a long lost son of Tom Lehrer!
For all the fans of mathematician Lehrer, there have been several tributes to him since his 90th birthday on Monday, including:

…for any younguns saying “Tom whoooo?” you can visit YouTube videos on him here:

or here:

A sample:

...ohhh, and in honor of birthday folks out there anywhere this week:

Sunday, April 8, 2018

Physics and Common Sense

Succinct from Bertrand Russell:

"Common sense implies physics and physics refutes common sense."

[...meanwhile, a new book blurb over at MathTango today]

Wednesday, April 4, 2018

Two For Wednesday

Just passing along a couple of posts from today that I enjoyed:

a)  Longish final installment in a series of posts from Keith Devlin on heuristics, mathematical thinking, problem-solving, education... (a rich read):

b)  Jim Henle with an interesting post playing with numeration… and, food (a quirky read that may yield ideas for the classroom):

Sunday, April 1, 2018

"When things get tough..."

Sunday reflection from physicist Victor Weisskopf:
When things get tough, there are two things that make life worth living: Mozart, and quantum mechanics.

Sunday, March 25, 2018

Improvising... Music and Math

For Sunday reflection, from Stephon Alexander’s “The Jazz of Physics”:
“Humans are the only creatures that can discover advanced mathematics, and the only creatures that can create and formalize music. If the beauty and physics of the universe, and the beauty and physics of music are linked, the links exist uniquely in human brains… What makes us uniquely able to do what nonhuman brains cannot: appreciate music and understand mathematics? And to create new things under the sun: compose, improvise, discover new mathematical facts about the universe?   
 “A few musicians, like Coltrane, have an uncanny ability to improvise, to find the hidden patterns and regularities underlying harmonic forms and to use those insights to generate brand-new kinds of melodic sequences. And a few scientists, like Einstein, can find regularities that have eluded even other great scientists — such as taking the Maxwell equations and reducing them into a single unifying formulation.”

Tuesday, March 20, 2018

Taking In the Forest From Above...

(via Wikipedia)
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:

Alex Bellos in The Guardian:

Kevin Hartnett at Quanta:

Davide Castelvecchi for Nature:

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.)

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?)

Sunday, March 18, 2018

Bernie's Self-control...

Sunday reflection:
"[Walter] Mischel has priceless videos from some of the early experiments that demonstrate the difficulty kids had in exerting self-control. There is one kid I am particularly curious about. He was in the toughest setup, in which the bigger prize, three delicious Oreo cookies, was sitting right in front of him. After a brief wait, he could not stand it anymore. But rather than ring the bell, he carefully opened each cookie, licked out the yummy white filling, and then put the cookie back together, arranging the three cookies as best he could to avoid detection. In my imagination, this kid grows up to be Bernie Madoff."
                                                             -- Richard Thaler in "Misbehaving"

Friday, March 16, 2018

Wednesday, March 14, 2018

A Sad Pi Day

In honor of Stephen Hawking today I’ll just link to perhaps my favorite Hawking story (granted it’s probably more of a Sean Carroll story). Most science buffs likely already know it, but in case you’ve missed it over the years:

Additionally, here’s an older lecture Dr. Hawking published in Plus Magazine about his work:

Tuesday, March 13, 2018

Math Story-Collider

H/T to Jim Propp for pointing out this current 'Story Collider' edition offering up two quite different narratives (with important messages) from Ken Ono and Piper Harron:

Transcripts of the talks are also presented, but give the under-15-min. talks a listen if you have the time.

Sunday, March 11, 2018

"the distilled essence of who we are"

Paul Lockhart expounds:

“And I'll go even further and say that mathematics, this art of abstract pattern-making — even more than storytelling, painting, or music -- is our most quintessentially human art form. This is what our brains do, whether we like it or not. We are biochemical pattern-recognition machines and mathematics is nothing less than the distilled essence of who we are.” 

Thursday, March 8, 2018

In Honor Of...

In honor of International Women's Day a few links I’ve posted here before, but seem especially appropriate for today:

…and Evelyn Lamb maintains this list of female math tweeters:

Lastly, a little nostalgia: