Sunday, May 20, 2018

Infallibility... Nope

Today's Sunday reflection:

“Mathematics has the completely false reputation of yielding infallible conclusions. Its infallibility is nothing but identity. Two times two is not four, but it is just two times two, and that is what we call four for short. But four is nothing new at all. And thus it goes on and on in its conclusions, except that in the higher formulas the identity fades out of sight.”
— Johann Wolfgang von Goethe

Thursday, May 17, 2018

New From Jim Holt on Infinity and More

One of my favorite past posts to write was a little tribute to David Foster Wallace for his volume on infinity, “Everything and More”:

Jim Holt now has a new book, When Einstein Walked With Gödel, excerpted here with this wonderful passage on Wallace and more:

Wednesday, May 16, 2018

9/11 Mortality

Here’s a “lateral thinking puzzle” I kind of liked from this week’s Futility Closet podcast, based upon a prior Guardian news piece (I’ve re-worded and adapted):

The official death toll from the 2001 September 11 hijacked-plane terrorist attacks stands at almost 3000 people, but a German professor specializing in risk, estimates that almost 1600 additional individuals died in the year following the 9/11 attacks due to that terrorism. These are NOT people who died of direct or physical exposure to the attack itself; i.e. first-responders/rescuers etc. exposed to debris/dust/air etc., but people who died from other choices or behavior in the year following. Can you guess the cause?
.answer below

.Answer:  the professor estimates that as a result of increased fears of airplane travel in the year following the 9/11 terrorism, air travel decreased by between 12 and 20%, while long distance automobile road trips increased significantly.  Statistically, car travel is far less safe than plane travel, and would result in a higher number of deaths than would be the case if those same individuals had opted instead to fly.

Monday, May 14, 2018

Stan the Man... and Mathematician

Stanislaw Ulam via Wikipedia

Yesterday marked the 34th anniversary of the death of Stanislaw Ulam. Most math fans know of him via his Ulam spiral and perhaps some other contributions, but given the wide range and scope of his efforts in mathematics I’m surprised he isn’t an even better known figure to many folks.
His almost 30-year-old autobiography,“Adventures of a Mathematician” is here:

…or you can also read sections of it online here:

And I highly encourage everyone to read this wonderful older piece by his friend and colleague Gian-Carlo Rota (lending a much richer profile of Ulam than does his Wikipedia piece):

To peak your interest it starts off thusly:

“One morning in 1946 in Los Angeles, Stan Ulam, a newly appointed professor at the University of Southern California, awoke to find himself unable to speak. A few hours later he underwent an emergency operation. His skull was sawed open and his brain tissue sprayed with newly discovered antibiotics. The diagnosis — encephalitis, an inflammation of the brain. After a short convalescence he managed to recover, apparently unscathed. 
In time, however, some changes in his personality became obvious to those who knew him. Paul Stein, one of his collaborators at Los Alamos, remarked that, while before his operation Stan had been a meticulous dresser, a dandy of sorts, afterwards he became visibly careless in the details of his attire, even though his clothing was still expensively chosen. 
When I met him, many years after the event, I could not help noticing that his trains of thought were unusual, even for a mathematician. In conversation he was livelier and wittier than anyone I had ever met, and his ideas, which he spouted out at odd intervals, were fascinating beyond anything I have witnessed before or since. However, he seemed to studiously avoid going into any details. He would dwell on a given subject no longer than a few minutes, then impatiently move on to something entirely unrelated.”

And elsewhere, Rota wrote of his friend, “Ulam's mind is a repository of thousands of stories, tales, jokes, epigrams, remarks, puzzles, tongue-twisters, footnotes, conclusions, slogans, formulas, diagrams, quotations, limericks, summaries, quips, epitaphs, and headlines. In the course of a normal conversation he simply pulls out of his mind the fifty-odd relevant items, and presents them in linear succession. A second-order memory prevents him from repeating himself too often before the same public.”

Just one more in the panoply of fascinating and brilliant mathematical characters....

Sunday, May 13, 2018

'devoid of factual, empirical content'

Sunday reflection from Carl Hempel:

“The nature of the peculiar certainty of mathematics is now clear: A mathematical theorem is certain relatively to the set of postulates from which it is derived; i.e., it is necessarily true if those postulates are true; and this is so because the theorem, if rigorously proved, simply re-asserts part of what has been stipulated in the postulates… A mathematical truth is irrefutably certain just because it is devoid of factual, or empirical content.

Sunday, May 6, 2018

The Teachers In Our Lives

A 'change in venue' today...  In place of the usual "Sunday Reflection" normally found here, I'll refer readers instead to a longer, new post now up at MathTango this morning. Enjoy...

Wednesday, May 2, 2018

Two For Wednesday

Just some mid-week offerings:

a)  If you like your math with a little physics and a little history, this recent piece is Evelyn Lamb's first for Symmetry Magazine:

...and Dr. Lamb's latest "TinyLetter" is newly-out, if you need to catch up with her doings for the month of April:

b)  A somewhat quirky blog from Mark Dominus that’s been around for quite awhile, but that I only recently stumbled upon (he touches on a number of subjects, but this link focuses on his math posts that many may find of interest):

Sunday, April 29, 2018


For today’s Sunday reflection, a little Rudy Rucker:
“…the ultimate success will never be ours [scientists]. Nowhere in the castle of science is there a final exit to absolute truth.
“This seems terribly depressing. But, paradoxically, to understand Gödel’s proof is to find a sort of liberation. For many logic students, the final breakthrough to full understanding of the Incompleteness Theorem is practically a conversion experience. This is partly a by-product of the potent mystique Gödel’s name carries. But, more profoundly, to understand the essentially labyrinthine nature of the castle is, somehow, to be free of it.”

Wednesday, April 25, 2018

Quiz for a Slow Week...

It’s a somewhat slow week at Math-Frolic (am too busy with other things), so will just run this revised quiz I put on Twitter not long ago:

Found the below photograph in a book recently. Is it a young:

a)  Warren Beatty
b)  Grigori Perelman
c)  John Travolta
c)  Marx brother
d)  Jordan Ellenberg

p.s…. Google, you’re drunk!! When I initially plopped the above picture into Google image search to see what would come up, it said the best guess was “afro” and took me to this page:

[this is a company making algorithms for driverless cars mind you; oy…]  ;)

Sunday, April 22, 2018

The Pleasure of Proof

For Sunday reflection, Bertrand Russell quoted in David Wells’ “The Penguin Book of Curious and Interesting Mathematics”:
“My friend G.H. Hardy, who was a professor of pure mathematics… told me once that if he could find proof that I was going to die in five minutes he would of course be sorry to lose me, but this sorrow would be quite outweighed by pleasure in the proof. I entirely sympathised with him and was not at all offended.”

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]