Authentic Math...

Short and simple; for Sunday reflection, this recent tweet from Eric R. Weinstein:

"Unless you majored in math you have no idea whether you are good at math. That thing you weren't good at? Yeah, so, that wasn't really math."

Mellifluous...

Yeah, sure, James Grime, Vi Hart, Matt Parker, are good, buuuuuuut...

is there any more mesmerizing voice for presenting math videos than Tadashi Tokieda!? The Perry Como of math perhaps... ;-) 
His latest for Numberphile here, on non-transitive dice:

And his other great vids here:

https://www.youtube.com/playlist?list=PLt5AfwLFPxWI9eDSJREzp1wvOJsjt23H_

(Non-transitive dice, by the way, come in different forms, some of which are available here:
https://mathsgear.co.uk/collections/dice )

Sir Roger Penrose Expounds

A blurb today about a new book I’m less than 100 pages into…

“Fashion, Faith, and Fantasy in the new physics of the Universe” is the latest from renowned Sir Roger Penrose. This is another beautifully produced, beautifully laid-out and diagrammed book from Princeton University Press. Given the breadth of these 400+ pages and Roger’s age I can’t help but wonder if this may be the last major work from him for a general audience. For all those reasons (and hey, simply because it IS from Roger Penrose) I’m sure it is worth recommending it to all readers interested in modern physics/cosmology. Peter Woit wrote a favorable review here (which I’ll point you to, since I won’t write a review):

http://www.math.columbia.edu/~woit/wordpress/?p=8759

I imagine the catchy main title (“Fashion, Faith, and Fantasy”) is especially intended to attract a wide general audience. Make no mistake about it though, it is heavy reading for anyone lacking acquaintance with theoretical physics, focused on the standard theory, inflation, quantum mechanics, string theory, and Penrose’s own twistor theory. There is a fair amount of math in the volume as well, and in fact, for math fans, the 70-page “Mathematical Appendix” may even be the best part of the book.

I won’t attempt a full review though because so much of the physics is simply beyond my grasp. Despite having read many popular cosmology books in the last two decades, the whole field is less, not more, comprehensible to me over time. I well understand the frustration (and rise) of the many skeptics of modern physics in recent years (Penrose himself is skeptical of many aspects). I don’t know of any other field where the more I’ve read, the fuzzier and less-comprehensible it all seems (not to mention being vociferously-debated-over by those who DO understand it!).

But with that said, I am looking forward to slowly slogging my way through this volume and gleaning from it what I can; there is possibly no better or more original expositor than Penrose to draw from. If modern physics theory is of interest to you you certainly won’t want to ignore this book either, and the more you already understand, the brighter read it will be.

I do have one curiosity about the book that maybe some knowledgeable reader can comment on in a few sentences:

I no more comprehend the specifics of the Langland’s Program in mathematics than the advanced cosmological theories in physics, but am still interested to hear about progress in cutting-edge Langlands work that hopes to unify mathematics (and in so doing may offer some unification to physics as well). Given that Penrose is a mathematical physicist I hoped there might be an update on Langlands somewhere in these 400+ pages, but it appears absent (not included in the Index to the volume). Am I wrong to think Langlands should tie in to some of this very theoretical discussion, or is there some reason one would NOT expect to see it in such a volume — is it too early on in Langlands work to be tying it back to "fashionable" ;-) physics, or perhaps the opposite problem, and it is simply too advanced for inclusion in a general audience volume? Just curious, if someone can enlighten me; I was looking forward to seeing Penrose's take on it all (or maybe it's just an area he has not much dabbled in?).

Anyway, here's how Woit ends his review of Sir Roger's volume:

"The range of non-crackpot speculative ideas about fundamental physics that normally get much attention is unfortunately quite narrow. In this environment Penrose is a breath of fresh air, providing here a different point of view on several topics, backed by serious and detailed argument. In some ways this is a popular book, but in others it is something else, deserving the attention of experts in the subject. I can’t recommend it too highly to anyone with a serious interest in fundamental questions about physics."

p.s.... Penrose will be at the Museum of Math in NY this coming Wed. evening promoting his book.

"...a fundamentally creative act"

Sunday reflection:

"Early encounters with math can be misleading. The subject seems to be about learning rules -- how and when to apply ancient tricks to arrive at an answer. Four cookies remain in the cookie jar; the ball moves at 12.5 feet per second. Really, though, to be a mathematician is to experiment. Mathematical research is a fundamentally creative act. Lore has it that when David Hilbert, arguably the most influential mathematician of fin de siecle Europe, heard that a colleague had left to pursue fiction, he quipped: 'He did not have enough imagination for mathematics."

-- Gareth Cook in NY Times (2015)

20 Years Ago...

Last weekend I noted that this year marked the 20th anniversary of one of the greatest events in academic publishing.

...and now serendipitously, yesterday, Keith Devlin notes this is also the 20th anniversary of his column "Devlin's Angle":

http://devlinsangle.blogspot.ca/2016/09/then-and-now-devlins-angle-turned.html

A lot to celebrate! ;-)

I briefly searched to see what other notable math-related events took place in 1996. Here are a few:

a.  Paul Erdös died.

b.  IBM's "Deep Blue" became the 1st computer to win a chess game against a World Champion (Garry Kasparov).

c.  Harvard-educated mathematician Ted Kaczynski was finally arrested as the "Unabomber," one of the most sought-after criminals of all time!

Count 'em

Today, just passing along, verbatim, one of Alex Bellos’ Guardian brainteasers from this week, if you haven’t seen it:

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“F is the first and the seventh letter of this sentence."

Using the sentence above as a model, fill in the gap in the following sentence to make it correct:

C is the first and the [….] letter of this sentence.

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What I like about this problem is not only that it’s easy-to-state and is recursive, but even though it sounds harder, it solves fairly quickly just working through it logically. However, in the end I’m not sure there’s any algorithmic or logical system for reaching the final answer other than brute-force trial-and-error with the ultimate solution candidates. Bellos simply states the answer with no further explication, and I give it below:

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Answer:

C is the first and the forty-sixth letter of this sentence.

Now In the Bookstores

Timely…: 

“The math-powered applications powering the data economy were based on choices made by fallible human beings. Some of these choices were no doubt made with the best of intentions. Nevertheless, many of these models encoded human prejudice, misunderstanding, and bias into the software systems that increasingly managed our lives. Like gods, these mathematical models were opaque, their workings invisible to all but the highest priests in their domain: mathematicians and computer scientists. Their verdicts, even when wrong or harmful, were beyond dispute or appeal. And they tended to punish the poor and the oppressed in our society, while making the rich richer…

“Big Data has plenty of evangelists, but I’m not one of them. This book will focus sharply in the other direction, on the damage inflicted by WMD [weapons of math destruction] and the injustice they perpetuate. We will explore harmful examples that affect people at critical life moments: going to college, borrowing money, getting sentenced to prison, or finding and holding a job. All of these life domains are increasingly controlled by secret models wielding arbitrary punishments.

“Welcome to the dark side of Big Data.”

— Cathy O’Neil, from the Introduction to “Weapons of Math Destruction”

"Given These Premises"....Inference

Am copying this verbatim from a recent Futility Closet posting about a hand of cards:

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Given these premises, what can you infer?

1. If there is a king in the hand then there is an ace, or if there isn’t a king in the hand then there is an ace, but not both.
2. There is a king in the hand.

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What is your answer???

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Now, go read the Futility Closet post:

https://www.futilitycloset.com/2016/09/07/a-cognitive-illusion/

As you will see, the post claims that “almost no one sees” the correct answer, and "practically everyone" infers (wrongly) instead that “there is an ace in the hand.”  The correct answer seems fairly obvious to me, but the post implies that most all fall for this “cognitive illusion.” Unfortunately there’s no way for me to know how many readers here immediately see the proper answer, but I’m wondering if math fans, perhaps more grounded in logic than the general populace, don’t answer this correctly at a much higher rate than other groups of people... IF that were indeed the case, it would be another indication of why training in mathematical thinking ought be encouraged.

The article says it is "unclear" why people mess up on this particular problem, though I think it's just one more example of how verbal cues are often very ambiguous or misleading for people... language is rarely as precise as individuals tend to assume. It all even reminds me a bit of a very old classic math conundrum that throws most people off (most of you will be familiar with it), which in one version (from Wikipedia) runs like this:

"Three people check into a hotel room. The clerk says the bill is $30, so each guest pays $10. Later the clerk realizes the bill should only be $25. To rectify this, he gives the bellhop $5 to return to the guests. On the way to the room, the bellhop realizes that he cannot divide the money equally. As the guests didn't know the total of the revised bill, the bellhop decides to just give each guest $1 and keep $2 as a tip for himself. Each guest got $1 back, so now each guest only paid $9, bringing the total paid to $27. The bellhop has $2. And $27 + $2 = $29 so, if the guests originally handed over $30, what happened to the remaining $1?"

OR, alternatively, here's a more recent example from the Web that many of you will recall, where the answer is actually fairly simple, yet many people, once again, are misdirected by the wording**:

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Jack is looking at Anne, but Anne is looking at George.
Jack is married, but George is not.
Is a married person looking at an unmarried person?

A) Yes       B)  No       C)  Cannot be determined

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...I s'pose the ability of language to hinder or interfere with rational thought has never been better demonstrated than by the current American presidential election :-( 

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**  the correct answer is "A"

"Authentic Maths" (...and proofs)

A sort of fun read from the week, "The Time I 'Nearly' Solved the Twin Prime Conjecture":

https://medium.com/@fjmubeen/the-time-i-nearly-solved-the-twin-prime-conjecture-8f033030fe90#.itcm4ftbq

With some important points toward the end, including:

a)  recognizing "failure" as "an inevitable part of the problem-solving process," and experiencing math "in a manner that engenders the learner’s sense of identity as a problem-solver."

and

b) "Authentic maths relies on having students develop and confront their intuitions as they straddle the lines between truth and uncertainty."

To Be(lieve) or Not to Be(lieve)

For Sunday reflection:

A story has been told of a visitor once to the office of Nobel-prize-winning physicist Niels Bohr who noticed a horseshoe hanging over the scientist's desk. The visitor asks, "Surely you don’t believe that horseshoe will bring you good luck, Professor Bohr?”
To which the brilliant Bohr chuckled and replied, “I believe no such thing, my good friend. Not at all. I am scarcely likely to believe such foolish nonsense. However, I am told that a horseshoe will bring one good luck whether you believe it or not.”

Curious...? Any Ideas?

Just a bit of light housekeeping today… see if anyone in the hivemind out there can explain to this Luddite what’s going on here….

I’ve used an adblocker in the past with Firefox and never noticed this, but recently added an adblocker to Safari and have this happening:

Normally while visiting sites the blocker shows anywhere from 1 to ~15 ads being blocked on any given page, but sometimes when I go to Yahoo Finance suddenly it starts rapidly scrolling into the 1000s — when I leave the page it’s been as high as 6000+ and still moving higher... can’t believe that many ads are really being blocked at once. Further, I can open the very same (YahooFinance) page twice (2 tabs, side-by-side essentially), and have one page show 10 blocks while the other one is showing over 6000??? (Doesn't occur on any other Yahoo page, just Finance.) Anybody know what’s likely happening here (and anything to be concerned about)?

I’ve looked at the Mac “Activity Monitor” and not noticed anything out of the ordinary (but I may not know what to look for). Also, I did recently have one of those ubiquitous MacCleaner-type malwarish elements infiltrate my laptop, but thought I’d managed to get rid of it — could some bit of it still be lingering somewhere on the machine causing this behavior, or completely unrelated?

Any ideas appreciated? I don't recall an adblocker ever registering blocks in the thousands. Not overly-concerned about it, just curious.

"Bits of DNA" from Lior Pachter

We live in a scientific culture populated simultaneously, by increasing numbers of specialists who, even in the same general field, can sometimes barely communicate with one another, and, on-the-other-hand, increasing numbers of “polymaths” who cross boundaries and work in multiple fields at once. This latter category has long fascinated me, especially since the most important future knowledge/findings may well come from them (though specialists too have much to contribute).

Lior Pachter is one such multi-disciplined individual. He calls himself “a computational biologist” though his PhD. from MIT is in mathematics, while his work focuses on genomics and crosses the boundaries of statistics, biology, and computer science. One of his longish blog posts is a favorite of mine, relating to the cross-cultures of biology and math (a really rich read):

http://tinyurl.com/j8dz3yn

[...Another of his posts that is a long-time favorite, by the way, is related to Common Core, education, and math problems (again, very rich with ideas):

http://tinyurl.com/nq56f5p  ]

I often can’t follow the depth and detail of Lior’s biology blogposts, but that doesn’t prevent me from appreciating the range and manner of his thinking.

While he has been at UC Berkeley for many years, he moves to Caltech (one of his alma maters) next year; a great addition to their talented faculty... and what better place to be working on cutting edge science.

The general field of “mathematical biology” has grown substantially in the last decade (not sure it even existed way back when I was in college), but it’s the sort of field requiring great care — like “Big Data” and other algorithmic areas it can be fraught with misunderstanding or misuse… Pachter seems to approach it with the necessary care and critical eye… and, as a bonus, engages in public communication about it (many scientists don't bother).  He is also a strong proponent of 'open access,' and doesn't shy away from controversy either (sometimes called a 'gadfly'), though I won't delve into that.

I’ve written previously of my delight in Jim Propp’s rich math blogposts for general readers; and broadening out, have cited Brian Hayes' blogposts, which, even when not on math, are fun and thought-provoking for math enthusiasts. And then there is Scott Aaronson's "Shtetl-Optimized" I've noted previously as perhaps my favorite math-related (but wider-ranging) blog (and I never know what to expect from Scott!).  So I’ll add Pachter’s blog ("Bits of DNA") to this list (broadening out still further to more biology than math), of blogs that really shouldn’t be missed, even if not your particular field:

https://liorpachter.wordpress.com
(he's also on Twitter: @LPachter )

[p.s…. I have no personal connection to Dr. Pachter, beyond an admiration for his online presence.]

Ford Circles

(via WikimediaCommons)

Awhile back I mentioned Alfred Posamentier’s latest volume “The Circle,” and around now it should be showing up in bookstores -- another great little geometry offering from Dr. Posamentier (and Robert Geretschlager). One of so many interesting tidbits in it is about “Ford circles”:

Imagine you have two tangent circles sitting atop a number line, one tangent to that line at “0” and the other tangent at “1.” Now in the space between these circles draw another circle tangent to both the “parent” circles and to the number line as well — it will touch the number line at the 1/2 position. You can keep iteratively drawing such circles (to infinity) in the space created with each new (smaller) circle. Now, quoting from the book:

“Of course, the circles get very small very quickly. As it turns out points of tangency of all these infinitely many circles with the number line have a quite unexpected property. The points of tangency are precisely the rational numbers in the interval between 0 and 1. No circle created by this process touches the number line at an irrational point, and every rational number is the point of tangency for some circle created in this manner.”

Pretty amazing, and a nice demonstration of one area of mathematics, plane geometry, connecting to other areas of infinity and number theory. Further, these circles relate back to Farey sequences.

Here’s one of several treatments of Ford circles on the Web:

https://nrich.maths.org/6594

Of Dogs and Financial Crashes

Recently finished (and very much enjoyed) Charles Wheelan's latest book, "Naked Money," and, though not very mathematical, will draw this Sunday reflection from it:

"...my dog was offered a preapproved Visa card with a five-digit credit limit sometime around 2005. (I subscribed to The New Yorker in his name, W. Buster Wheelan, and some credit card issuer obviously bought the list.) This would suggest that the credit markets were out of control. The subsequent real estate bust would not have been nearly as catastrophic if financial institutions had not been lending to dogs, literally and figuratively."

[...makes me think, that since you as a consumer are given a FICO score, perhaps financial institutions ought be assigned FIDO scores ;-) ]

18 and Counting

Not sure how he finds the time, but John Cook puts out 18 math-related Twitter accounts for our delectation (in addition to his blog):

http://www.johndcook.com/blog/twitter_page/

If you’re on Twitter and not following at least a couple of John’s feeds you probably want to check them out.  Most of these accounts tweet about once-a-day, always with some pithy bit of interesting information (or link), ranging over topics from general science to more technical math or computer-science niches.

While I don’t doubt that there are scamsters/trolls with more than 18 Twitter accounts, I’m guessing there may be no serious mathematician/scientist in John’s league of multiplicity. Or am I wrong? Anyone know of someone who surpasses Dr. Cook's prolific accounts-output?

Of course by the time I post this John may have added another 1 or 2 accounts… ;-) 

Also, just realized, that an old (now defunct?) account I enjoyed, @TautologyFacts, almost has the look/feel (in a parody sort-of-way) of a Cook account, though I don't believe it had any connection to John?:

https://twitter.com/TautologyFacts

Erdös Trivia...

Just a line I recently found surprising in the Wikipedia entry for Paul Erdös....

Though Erdös collaborated with over 500 different individuals in his lifetime, publishing more than 1500 mathematical articles, he not only never won the Fields Medal himself, "nor did he coauthor a paper with anyone who did, a pattern that extends to other prizes." That struck me as remarkable (and I assume it's accurate, unless someone knows a counter-example?). I'm not quite certain what the last phrase ("a pattern that extends to...") even means -- does it mean he never collaborated with the winner of any prize he had not himself won??? Anyway, apparently there have never been any Fields Medalists with "Erdös numbers" of "1" -- wouldn't have guessed that!

Life of Gardner...

Another YouTube video today, this time from Martin Gardner's official biographer, Dana Richards... over an hour long, but worth it if you're a Martin Gardner fan; lots of good stuff and interesting tidbits in addition to fairly well-known information about Martin:



(p.s... yesterday's 'Sunday reflection' came from Gardner)

Dissolving Into Pure Mathematics

Sunday reflection, via Martin Gardner:

"In light of today's physics the entire universe has dissolved into pure mathematics. The cosmos is made of molecules, in turn made of atoms, in turn made of particles which in turn may be made of superstrings. On the pre-atomic level the basic particles and fields are not made of anything. They can be described only as pure mathematical structures. If a photon or quark or superstring isn't made of mathematics, pray tell me what it is made of?"