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Tuesday, April 25, 2017

"Gauss can recite all of pi -- backwards"

Blogging may continue to be a bit slow while I'm catching up on a number of things (and waiting very patiently for impeachment hearings ;), so may just re-run some old posts in the meantime. Anyway, will start by referencing this favorite old "Gauss Facts" site that's always good for a chuckle:


Sunday, April 23, 2017

Truth, Certainty, Explanation... and Mathematics

Physicist David Deutsch reflecting on mathematics (from his "The Fabric of Reality"):
[There is] "...an ancient and widespread confusion between the methods of mathematics and its subject-matter. Let me explain. Unlike the relationships between physical entities, relationships between abstract entities are independent of any contingent facts and any laws of physics. They are determined absolutely and objectively by the autonomous properties of the abstract entities themselves. Mathematics, the study of these relationships and properties, is therefore the study of absolutely necessary truths. In other words, the truths that mathematics studies are absolutely certain. But that does not mean that our knowledge of those necessary truths is itself certain, nor does it mean that the methods of mathematics confer necessary truth on their conclusions. After all, mathematics also studies falsehoods and paradoxes. And that does not mean that the  conclusions of such study are necessarily false or paradoxical.
"Necessary truth is merely the subject-matter of mathematics, not the reward we get for doing  mathematics. The objective of mathematics is not, and cannot be, mathematical certainty. It is not even mathematical truth, certain or otherwise. It is, and must be mathematical explanation."

Sunday, April 16, 2017

"Whence Certainty?"

Sunday reflection... from Rebecca Goldstein in "Incompleteness: the proof and paradox of Kurt Gödel":
"So the question is: Whence certainty? What is our source for mathematical certainty? The bedrock of empirical knowledge consists of sense perceptions: what I am directly given to know -- or at least to think -- of the external world through my senses of sight and hearing and touch and smell. Sense perception allows us to make contact with what's out there in physical reality. What is the bedrock of mathematical knowledge? Is there something like sense perception in mathematics? Do mathematical intuitions constitute this bedrock? Is our faculty for intuition the means for making contact with what's out there in mathematical reality? Or is there just no 'there'?"

Tuesday, April 11, 2017

For Your Funnybone…

 At a recent large used book sale I headed to the math section and picked up a few older volumes, including John Allen Paulos’ “Once Upon a Number” from 1998. Of course mathematics is timeless, but I was pleasantly surprised to see how much of the less-mathematical content of this volume is still relevant today (heck, maybe even more so since Nov. 8, 2016); much of it concerning logic, reasoning, meaning, information, clear/critical thinking and the like.
Anyway, I’ll put all that aside to only pass along this non-math joke Paulos tosses in at one point (the book is sprinkled with his typical humor):

A young man is on vacation and calls home to speak to his brother.

 “How’s Oscar the cat?”

 “The cat’s dead, died this morning.”

 “That’s terrible. You know how attached I was to him. Couldn’t you have broken the news more gently?”


“You could’ve said that he’s on the roof. Then the next time I called you could have said that you haven’t been able to get him down, and gradually like this you could’ve broken the news."

‘Okay, I see. Sorry.”

 “Anyway, how’s Mom?”

 “She’s on the roof.”

Sunday, April 9, 2017

Science As Uncertainty Reduction

Sunday reflection:

“It feels like there are two opposite things that the public thinks about science: that it’s a magic wand that turns everything it touches to truth, or that it’s all bullshit because what we used to think has changed… The truth is in between. Science is a process of uncertainty reduction. If you don’t show that uncertainty is part of the process, you allow doubt-makers to take genuine uncertainty and use it to undermine things…
“And it’s absolutely crucial that we continue to call out bad science. If this environment forces scientists to be more rigorous, that’s not a bad thing.”

— Christie Aschwanden (of FiveThirtyEight )

Wednesday, April 5, 2017

Riemann In the News...

Lots of interesting mathy stuff out there this week, but hey, you can’t go wrong with the crown jewel of number theory, so I’ll direct you to two pieces on the Riemann Hypothesis, if you’ve not seen them:
First, a brief interview with Barry Mazur and William Stein, authors of “Prime Numbers and the Riemann Hypothesis” (one of my favorite 2016 books):
…and then the incomparable Natalie Wolchover summarizing the latest intriguing approach from physicists to Riemann’s 150+ year-old, million-dollar conundrum:
The actual (physics) work was published last year but is just now being widely disseminated on popular media:
There are a great many other introductions to RH on the internet, including some video ones such as these:
From Numberphile:

…and from 3 Blue1Brown:

Sunday, April 2, 2017

Beyond the Boundary of Logic

For a beautiful Sunday reflection, the ending words from Eugenia Cheng in "Beyond Infinity":
"The most beautiful things to me are the things just beyond that boundary of logic. It's the things we can get quite a long way towrd explaining, but then in the end they just elude us. I can get quite a long way toward explaining why a certain piece of music makes me cry, but after a certain point there's something my analysis can't explain. The same goes for why looking at the sea makes me so ecstatic. Or why love is so glorious. Or why infinity is so fascinating. There are things we can't even get close to explaining, in the realm far from the logical center of our universe of ideas. But for me all the beauty is right there on that boundary. As we move more and more things into the realm of logic, the sphere of logic grows, and so its surface grows. That interface between the inside and the outside grows, and so we actually have access to more and more beauty. That, for me, is what this is all about.
"In life and in mathematics there is often a trade-off between beauty and practicality, along with a contrast between dreams and reality, between the explicable and the inexplicable. Infinity is a beautiful dream, inside the beautiful dream that is mathematics."

Thursday, March 30, 2017

For Mathematics Lovers

A great read for anyone who wants to learn what math really is, no prerequisites required. And those of us in the field are reminded of what first drew us to it."    Maria Chudnovsky, Princeton University and 2012 MacArthur Fellow
An elegant sampler of many beautiful and interesting mathematical topics. This could become one of the best books available for a popular audience interested in what mathematics really is.”  —Jayadev Athreya, University of Washington

Apologies for my slightly redundant book blurbs lately, but I again want to re-mention a new volume I already praised. The above publicity blurbs are for “The Mathematics Lover’s Companion” by Ed Scheinerman, and it is so far, my favorite book of this young year. I’m a sucker for volumes that offer up a buffet of interesting math topics without lingering too long on any one. You can view the Table of Contents (and some of the content) here:
I especially like Part 3 of the volume on “Uncertainty” (though it is the shortest section), but the other two parts, covering a nice selection of algebraic and geometry topics, are very good as well.
The reason for even bothering to note this book again, is my disappointment at how little “buzz” I see it getting in cyberspace (I do see it regularly in my local Barnes & Noble outlets, so I know it’s being well-distributed). My only guess is that the publisher, Yale University Press, just doesn’t put as much energy into promotion as some of its counterparts. This is a fabulous book, especially for a lay audience, but also for folks farther along their mathematics journey, so I'd hate to see it ignored... especially if you already love mathematics!

Wednesday, March 29, 2017

Fun From Fibonacci (via Keith Devlin)

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…
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.

Monday, March 27, 2017

Re-playing Testosterone

To start the week, a blast from the past….

This weekend’s “This American Life” episode was a replay of a crowd-favorite dating back 15 years to August 2002, on the subject of testosterone. The whole episode is wonderfully entertaining, but I always especially enjoyed “Act 2” with a transgender man, born as a female, but reporting on the results of undergoing years of testosterone treatments:

I’ve written about it here before because of one brief section that is fascinating (in a non-PC sort of way). I’ll simply re-quote from that earlier posting:
….After relating a lot of already interesting stories to host Ira Glass about how the change in gender affected him, the interviewee is asked by Ira if there are any other alterations due to testosterone he thinks worth mentioning. The individual responds that after taking testosterone he "became interested in science; I was never interested in science before." To this Ira can't help but chuckle and respond, "NO WAY!" adding that such a response is "setting us back 100 years." The individual goes on to insist that testosterone resulted in "understanding physics in a way I never did before."  (…the specific exchange occurs around the 22:30 point of the whole episode)
It is concerning, but also funny, to think of our abilities/skills being so subservient to our biochemistry, even if this is just one lone anecdotal case. Anyway, the entire segment is fascinating and worth a listen (assuming you enjoy the style of “This American Life;” however the part I’m noting above is the only bit that actually relates back to math or science; the rest is mostly social/psychological in nature).

In a quite long interview over at the Edge site a few years ago, Simon Baron-Cohen, a major researcher in this area, had this to say following a question from Marcus du Sautoy about the relationship between tendencies toward math, and human biology/testosterone:
“…you mentioned mathematicians, and I think you're right, that there are these areas of human activity, math is one of them, where we do see very disproportionate sex ratios. My understanding is that in mathematics, at university level, it's about 14 males for every one female sitting in the audience in those lectures. That's a very big difference. And there are other sciences, as we know, which used to be like that but which have changed dramatically. Medicine is a very good example. It used to be male dominated and it's now certainly 50-50, or if anything, it's gone beyond and there are now more female applicants and thankfully, successful applicants. If you look at the audience in medical lectures, the sexes are, if not equally represented, maybe even more women than men. But there remains this puzzle why mathematics, physics, computer science, engineering, the so-called STEM subjects, why they still remain very male biased. I’m the first to be open to anything we can do to change the selection processes at university, or change the way we teach science and technology at school level, high school level, to make it more friendly to females, to encourage more women to go into these fields. But there remains a puzzle as to why some sciences are attracting women at very healthy levels, and other sciences, including mathematics, remain much more biased towards males. Whether that's reflecting more than just environmental factors, and something about our biology, is something that I think we need to investigate.”

Controversy continues....

[...much of Baron-Cohen's research, by the way, studies the possible link between high pre-natal testosterone exposure and autism]

Sunday, March 26, 2017

Mathematical Values

This week's Sunday reflection, courtesy of Roger Penrose:
"How, in fact, does one decide which things in mathematics are important and which are not? Ultimately, the criteria have to be aesthetic ones. There are other values in mathematics, such as depth, generality, and utility. But these are not so much ends in themselves. Their significance would seem to rest on the values of the other things to which they relate. The ultimate values seem simply to be aesthetic; that is, artistic values such as one has in music or painting or any other art form."

Thursday, March 23, 2017

About Infinity...

I blurbed a little bit earlier about Eugenia Cheng’s new book “Beyond Infinity.” Very much enjoying it, now that I’m farther in (…but do realize it’s an entire book about infinity — so you need a significant interest in the topic to enjoy it; the typical popular math book might only have a chapter or two on infinity, touching a few highlights; this volume goes deeper).
For now just wanted to mention one small matter that came up:
Quite awhile back on Twitter I asked if there was any sort of “proof” that aleph-null must in fact be the ‘smallest’ infinity; i.e. infinity is full of so many counterintuitive outcomes, and the whole question of whether aleph1 really is the second infinity is so complicated, that I wondered how we could even be sure that the natural numbers, for example, represent the lowest degree of infinity.
The few replies I got implied that the minimalness of aleph-null was axiomatic or established by definition. BUT Dr. Cheng does offer a short form of something like a proof in her volume. Her basic argument is simply to indicate that there is no subset of the natural numbers that can be put into one-to-one correspondence with the natural numbers and have anything leftover (sort of a reversed diagonalization argument). Or as she concludes, “This means that every subset of natural numbers is either finite or has the same cardinality as the natural numbers. There is no infinity in between. So we have found the smallest possible infinity: it’s the size of the natural numbers.”
I don’t know if I’m quite fully convinced (that there is much more than tautology or definition at work here), but I was glad to at least see an argument put forth. Dr. Cheng herself admits “This is not quite a proof, but is the idea of a proof…” It’s at least better than saying that the natural numbers are the lowest infinity by edict ;)
A lot of the difficulty in wrapping one’s brain around infinity lies in our deep-seated entrenchment in one view of what “numbers” are. As Cheng writes at one point, “Infinity isn’t a natural number, an integer, a rational number, or a real number. Infinity is a cardinal number and an ordinal number. Cardinal and ordinal numbers do not have to obey all the rules that earlier types of number obey.” 
I still have several chapters to go, and they look like they will be quite good. As with her earlier work ("How To Bake Pi") Dr. Cheng writes in an off-hand, almost conversational style meant to draw readers in to sometimes difficult or abstract ideas. I don't think she is always successful, but admire her making the effort. And her own passion for her subject-matter is clear.

Sunday, March 19, 2017

The Universe as a Mathematically-designed Machine

"To the divine understanding, all phenomena are coexisting and are comprehended in one mathematical structure. The senses, however, recognize events one by one and regard some as the causes of others. We can understand now, said Descartes, why mathematical prediction of the future is possible; it is because the mathematical relationships are preexisting. The mathematical relationship is the clearest physical explanation of a relationship. In brief, the real world is the totality of mathematically expressible motions of objects in space and time, and the entire universe is a great, harmonious, and mathematically designed machine. Moreover, many philosophers, including Descartes, insisted that these mathematical laws are fixed because God had so designed the universe and God's will is invariable. Whether or not humans could decipher God's will or penetrate God's design, the world functioned according to law, and lawfulness was undeniable, at least until the 1800s."

-- Morris Kline (in "Mathematics and the Search For Knowledge")

Tuesday, March 14, 2017

Just For Fun

Just for fun today, some simple arithmetic….

Math Tricks” blog put up the somewhat classic '30-cows-in-a-field' riddle yesterday, and I’ll post another video of it here today for any not familiar with it:

(what I love about this is that it is simple, appropriate for all ages, and nicely demonstrates language/speech ambiguity)

Sunday, March 12, 2017

Atoms and Primes

 Beginning of Edward Scheinerman's new book, "The Mathematics Lover's Companion":
"The physicist Richard Feynman believed that if humanity were to be faced with the loss of all scientific knowledge but was able to pass on just one sentence about science to this postapocalyptic world, that sentence should describe how matter is composed of atoms. In that spirit, if we could pass on only one bit of mathematics to the next generation, it should be the solution to the problem: How many prime numbers are there?"
[...he goes on to describe some of the proofs for the infinity of primes, before embarking on a wide array of other topics throughout the book.] 

Wednesday, March 8, 2017

"Experimental Math"... never-ending explorations

ICYMI, John Horgan interviewed Stephen Wolfram recently at his blog:

That was followed up shortly by a long, interesting post from Wolfram himself on “experimental mathematics,” iteration, cellular automata, Mathematica, etc. (h/t to Mike Lawler):

...and then Mike Lawler followed that up incorporating some of Stephen's inventive ideas into his own "Family Math" series:

Sunday, March 5, 2017

Sunday With Hermann

Sunday reflection from a 2014 paper on mathematical neural correlates:
"Hermann Weyl is recorded as having said, 'My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful.' Relevant here is the story of Weyl's mathematical formulations, which tried to reconcile electromagnetism with relativity. Rejected at first (by Einstein) because it was thought to conflict with experimental evidence, it came subsequently to be accepted but only after the advent of quantum mechanics, which led to a new interpretation of Weyl's equations. Hence the perceived beauty of his mathematical formulations ultimately predicted truths even before the full facts were known."

Wednesday, March 1, 2017

Evelyn Lamb Serves Up Her Mathiness From 'Hard-hitting' February

The 2nd edition of Evelyn Lamb's new newsletter is out... GREAT place to keep up with Dr. Lamb's writings (since she shows up in multiple outlets), as well as other things on her mind:


If you're not already a subscriber I encourage you to become one (so you don't have to rely on me pointing out each new issue):  https://tinyletter.com/evelynjlamb

Sunday, February 26, 2017

"replication is central to science"

For a Sunday reflection, this from Andrew Gelman in "The Best Writing on Mathematics 2016":
"To resolve the replication crisis in science, we may need to consider each individual study in the context of an implicit meta-analysis. And we need to move away from a simplistic, deterministic model of science with its paradigm of testing and sharp decisions: accept/reject the null hypothesis and do/don't publish the paper. To say that a claim should be replicated is not to criticize the original study; rather, a replication is central to science, and statistical methods should recognize this. We should not get stuck in the mode in which a 'data set' is analyzed in isolation, without consideration of other studies or relevant scientific knowledge. We must embrace variation and accept uncertainty."

Saturday, February 25, 2017

Our Bizarro World

This isn’t much about math, but just idle commentary… Hardly a week passes anymore without a bizarre story from some corner of the world appearing in the news. This week it seemed to be the hands-on assassination of Kim Jong-un’s half-brother in a public airport. I’ve only read snippets of the story, so perhaps everything I’m about to say has already been well-covered and I’ve just missed it.

Two women apparently simply walked up to the half-brother with cloths in their hands that they applied briefly to his face, before running off. Within a short span of time he was dead of what was found to be VX nerve agent exposure.
VX is one of the deadliest chemical weapons known to exist — potent in very small quantities. As I understand it though, VX can be produced in a “binary” form where two separate components, that are not particularly dangerous apart, only become effective when combined. I suppose it's possible there were TWO attackers in this bizarre crime so that, in the event one chickened out, the other might still succeed… BUT far more likely it seems the reason for TWO attackers would be having each bearing a different component, relatively safe for themselves, but fatal when combined on the face of the target. The women could carry their separate cloths, and run off to wash hands afterwards in a rest room, probably with safety to themselves and those around the victim, while still accomplishing the task. 
With several arrests in the crime I suppose we’ll get answers to some of this soon. But my point is simply to say how scary it is to think that such an agent can perhaps thusly be employed in a highly-trafficked public environment to pinpoint a single victim, without harm to others. (One of the problems with anthrax, as I recall from older events, is that it is very difficult for someone to both produce it and deliver it to a single victim and maintain safety to themselves and others.)

Having said all that, it sounds fine in theory, but surely the North Korean regime (pretty clearly behind this) would not send these women on a one-time, never-before-tried mission, without having thoroughly tested it first. They must (one would think) have practiced this technique on victims, perhaps political prisoners, in their own closed society, to test for any pitfalls in the procedure, before making such a brazen effort in a public international airport. So one wonders how many others have died unreported and unknown in N. Korea over the years from VX poisoning in tests (and what other similar experiments are ongoing now)?

Sunday, February 19, 2017

Mathematical Thinking, Not Rule-Following

Sunday reflection:

"What books are to reading, play is to mathematics... I believe we have the power to make mathematical thinking flourish everywhere. We can't afford to misuse math to create passive rule followers."

-- Dan Finkel (TEDTalk)

Wednesday, February 15, 2017

What I'm Reading...

In the middle of reading several books simultaneously, three of which I'll mention as likely recommendations (possible reviews or longer blurbs later):

The Best Writing on Mathematics 2016” ed. by Mircea Pitici 
Newly out, Pitici’s latest (7th) volume of this anthology. I’ve barely started it and it already looks fantastic; quite possibly the best yet in the series. Will review at a later date.

Know This— another of John Brockman’s Edge/essay compendiums with a great set of science thinkers on a wide variety of topics bearing on our future. Very short nuggets of thought. Once-in-awhile his volumes disappoint me, but so far not this one.

Superforecasting” by Philip Tetlock & Dan Gardner 
A volume I’ve seen nothing but positive reviews for. Since Trump’s November election I’ve especially been touting books that pertain to critical thinking and this falls in that genre.

These are all available in paperback and I think worth your consideration.

Finally, in honor of Raymond Smullyan I'm very tempted to order one of his later books, "Reflections," which appears to be very autobiographical, and reminiscent of Martin Gardner's own rambling autobiography. I always think of Gardner and Smullyan together and they were close friends (also, just recently realized that Smullyan, like Gardner, had attended the University of Chicago and studied with Rudolf Carnap).

Monday, February 13, 2017

Canary In the Press?

(image via pixabay)

We seem to be living, as noted by many, in an anti-intellectual, anti-expert, anti-science time-frame. I’ve been tempted to write a commentary on the relatively tepid response from the STEM community to the Trump presidency, and earlier voiced dismay at how few scientists spoke out loudly and often during the election campaign (THANK YOU to those who did)… but for now, I'll refrain adding my puny voice here to the growing numbers finally speaking up, almost in a sudden panic (…now that he’s in office slowly dismantling democracy).

BUT… last week Raymond Smullyan died. I’ve been taken aback at the paucity of press for Smullyan’s passing. The NY Times finally ran an obituary 5 days after his death. Where are the articles though from the Washington Post, the LA Times, the Boston Globe, Chicago Trib, USA Today...? Obviously too, I might expect something more expansive soon from MAA, AMS, philosophy associations, and others. 
What does it say about our times (where a demagogue can not only run for president, but win) that a major proponent/author of rationality, logic, and clear thinking, passes away and is accorded so little attention. Losing Smullyan, at the age of 97, is not particularly unexpected, but the lack of coverage of this loss is discouraging. The silence is like a canary within the press dying, and indicating something awry with our values and focus. What two-bit celebrity will die next month and receive multi-columns of notice? Have we, after 200+ years, lost our way? In the word of our (so-called) President, it is “sad.”

Sunday, February 12, 2017

Assault on Truth

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.”

Thursday, February 9, 2017

Raymond Smullyan, A Knight Among Men... +ADDENDA

"Recently, someone asked me if I believed in astrology. He seemed somewhat puzzled when I explained that the reason I don't is that I'm a Gemini." -- R. Smullyan

At a time when we need his likes more than ever, brilliant polymath Raymond Smullyan has died at the age of 97. One of the undersung thinkers of our times — with a name far less well-known to the public than several other mathematicians.
As word gets out, I’m sure there will be many wonderful tributes to follow, but for now, I’ll just leave here a few of the Tweets I quickly found popping up:
from @shanewag1 :
Very sad to say goodbye to one of the world's great polymaths. Rest In Peace, Smullyan. There was something a little wrong with dualism. 

from @mathematicus :
I don't really do heroes but if I did Smullyan would be one of mine

from @J_Lanier :
The brilliant and playful Raymond Smullyan has passed away. I am grateful for the many happy hours I've spent reading and sharing his books.

from @bphopkins :
RIP the great Raymond Smullyan, many of whose books I shall someday gleefully subject my children to. So goes the Dao.

from @BradleyPallen :
The two sentences in this tweet are false.
Raymond Smullyan will never die.

I'll mention again that Jason Rosenhouse edited a nice tribute volume, "Four Lives," to Smullyan some years back:

Smullyan is best known for his logic works (both recreational and academic), but he also wrote several volumes on "spirituality." The best known was probably "The Tao Is Silent," but my own favorite is perhaps "A Spiritual Journey."

And lastly, I'll end with one more quote from Raymond:
"A joke is told that Epimenides got interested in eastern philosophy and made a pilgrimage to meet Buddha. He said to Buddha: 'I have come to ask you what is the best question that can be asked and what is the best answer that can be given.' Buddha replied: 'The best question that can be asked is the question you are asking and the best answer that can be given is the answer I am giving.'"

==> [It’s now 2:30pm EST and I had expected by now to see some more official notice or more-detailed obituary for Dr. Smullyan than the Facebook posting that started the news. I've seen at least a couple of people pass the news along who I don’t believe would have done so if they were not certain of its validity, but once some more official press links are available I will add them here.]

==> Apologies, that I may have multiple updates to this post as warranted…
For those who haven’t seen it, the initial news on Raymond was broken by a Facebook post from a personal friend HERE
I only just now noticed the post’s date is Feb. 7, saying Dr. Smullyan died “yesterday” (so, I assume Feb. 6), making it even more surprising that there are by now (3 days later) no formal press releases (though the poster does say she expects the NY Times to have a “big tribute” to him soon. Who knows what clever final instructions the iconoclastic Smullyan may have left for any announcements of his death… or alternatively, perhaps his lesser name recognition (compared to say his dear friend Martin Gardner) is causing a delay in more details getting out.

In any event, stay tuned… Raymond straddled a world between mathematicians, logicians, philosophers, recreationalists, cognitive scientists, academics, and layfolk… and musicians and magicians… and punsters ;) and he deserves the highest recognition.

==> 2/10/17  Perhaps Raymond has left us, as he lived, giving us one more puzzle to ponder. I awoke at 5 this morning and immediately searched Web for official news of his demise, and still it awaits. People have repeatedly tried to edit his Wikipedia page only to be rebuffed by editors who are also waiting for official confirmation. Maybe Feb. 11, being a prime number, will be the day of notification (…and I’m only half-joking). 
For those repeatedly asking, no, I think it clear this is not any sort of hoax or prank, but for whatever reason, and despite his worldwide fan base, official news just hasn’t come yet. I did glean from all my searching that Dr. Smullyan apparently died “peacefully in his sleep” from “complications of a stroke,” I believe the evening of Feb.6.
And I have to admit there is something almost delicious, that even in death, Dr. Smullyan continues to puzzle us from the great beyond.

For any readers who don’t know much about Smullyan or wonder why I'm spending this much time on him, until longer tributes appear, you can get a feel for him and his impact from the messages flowing in on this Facebook page:

==> Wikipedia page finally updated based in part on this account:

Probably much more to follow in next 24 hrs.

2/11/17 : The NY Times has weighed in with their obituary:

Tuesday, February 7, 2017

"Now" for a Book Blurb

In my 2016 end-of-year book wrap-up I briefly mentioned that Richard Muller’s book, “Now” (focusing on the nature of time and entropy) was one popular physics book I was looking forward to reading. I’ve now read it, and generally do recommend it (finding it less inscrutable and more satisfying than most popular physics books)… but with a caveat. While the volume has had mostly positive reviews there have been a few negative ones that are often put off by a couple of chapters near the end of Dr. Muller’s book. Muller is known as a rather independent thinker with a gadfly streak, and toward the end includes significant discussion that some will find too metaphysical (almost supernatural). I actually enjoyed seeing a physicist’s take on such matters, but some won’t. He is especially skeptical of “physicalism,” the approach most physicists take to scientific study; i.e. that everything is ultimately explainable in terms of “physical” elements that we do or can eventually understand. 
One of the examples he uses over and over of something we simply don’t understand is what it means to “see” the color “blue,” nor do we know if other people see (inside their brains/minds) blue the same way we see it. This is one of those profound questions that many children ask (and never receive an adequate answer), and you either get what he means by the query or you don’t. Philosophers have long discussed it at length, with no resolution. He also dives into a long discussion of free will, and why he views it as incompatible with "determinism" (some don't), again a topic too philosophical for many. 
One can get even more abstract by asking what does it mean to feel “wonder,” or “awe,” or “love,” and if these are nothing more than neurons firing in certain patterns (as many would say), then can we construct robots that experience these “feelings.” Also, long amazing to me, as someone with an interest in psycholinguistics, is our lack of real understanding of how everyday speech is either produced or processed, even though all normal humans do it effortlessly. Anyway, I’m going far astray from the discussion Muller has, just as a way of saying I’m not put off by seeing a physicist talk about things he finds inexplicable within a “scientific” framework. But yes, the chapters do stick out a bit awkwardly in an otherwise empirical look at some of the deepest questions faced by physics theorists today.
For an actual review of the volume see here:
…and here, an excerpt from the book:

Sunday, February 5, 2017

Number Sense

Sunday reflection:
 "[Alain] Connes thinks that expert mathematicians are endowed with a clairvoyance, a flair, a special instinct comparable to the musician's fine-tuned ear or to the wine taster's experienced palate that enables them to directly perceive mathematical objects: 'The evolution of our perception of mathematical reality causes a new sense to develop, which gives us access to a reality that is neither visual, nor auditory, but something else altogether'."  
-- from "The Number Sense" by Stanislas Dehaene

Wednesday, February 1, 2017

Who Doesn't Want More Evelyn Lamb!

For all of you who were just saying to yourselves, 'Ya know I need to subscribe to just one more email newsletter'... have I got good news. Evelyn Lamb announces she will begin putting out such a newsletter, which she describes in a tweet as “Math, math, math (And other stuff)," and the first issue will even be today!
Go here to subscribe (…that’s an order!):

Sunday, January 29, 2017

the depths of existence...

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
[e + 1 = 0] reaches down into the very depths of existence."

Thursday, January 26, 2017

Update on Poker-playing Robot

A couple of weeks ago I mentioned that AI programs were now taking on poker, and a nice followups to that storyline appear this week:
In an ongoing Texas hold’em tournament a poker-playing robot named “Libratus” is so far up by almost $800,000 against its human competitors.
From the the first article: 
“Poker requires reasoning and intelligence that has proven difficult for machines to imitate. It is fundamentally different from checkers, chess, or Go, because an opponent’s hand remains hidden from view during play. In games of ‘imperfect information,’ it is enormously complicated to figure out the ideal strategy given every possible approach your opponent may be taking.”Further it is noted that “...an AI player has to randomize its actions so as to make opponents uncertain when it is bluffing.”
And from the 2nd article:
"One of the things Libratus does well is bluff..."Mastering the art of the bluff requires AI that can calculate risk and reward in real time without having perfect information about what its opponent can do in return. It implies the system does more than simply play a perfectly safe game where it only grinds out wins when it has the stronger hand."
Libratus is specifically programmed (as I understand it) to be skilled at just one specific poker game, so even if it wins this tournament (and it looks like it will, as it seems to be getting stronger over time), it doesn’t mean that bots are on the verge of taking over all professional poker… or, at least not yet. Of course the real congratulations go, not to the bot, but to the clever humans (in this case from Carnegie-Mellon) programming it.

Monday, January 23, 2017

Winning At Blackjack... and more

To start the week, this delightful (19-min.) Planet Money podcast on Ed Thorp, starting with his blackjack-beating escapades (h/t to Francis Su for this one):


Try to find time for it (all the more so if you don't already know Thorp's story.)

via WikimediaCommons

Sunday, January 22, 2017

Seen and Unseen...

"We have never seen any curve or solid corresponding to my square root of minus one.  The horrifying part of the situation is that there exist such curves or solids. Unseen by us they do exist, they must, inevitably; for in mathematics, as on a screen, strange, sharp shadows appear before us. One must remember that mathematics, like death, never makes mistakes. If we are unable to see those irrational curves or solids, it means only that they inevitably possess a whole immense world somewhere beneath the surface of our life."

-- Yevgenii Zamyatin, quoted in Michael Harris's "Mathematics Without Apologies"

Friday, January 20, 2017

A Pig Is A Pig...

Given the events in Wash. DC. today, seemed only fitting to end the week with a joke... so, a little recursive humor I came across in Thomas Cathcart’s and Daniel Klein’s volume “Plato and a Platypus Walk Into a Bar.” Quoting verbatim:
“A woman sues a man for defamation of character, charging that he called her a pig. The man is found guilty and made to pay damages.  After the trial, he asks the judge, ‘Does this mean that I can no longer call Ms. Harding a pig?’ The judge says, ‘That is correct.’ ‘And does it mean that I can’t call a pig Ms. Harding?’ ‘No,’ says the judge, ‘you are free to call a pig Ms. Harding. There is no crime in that.’ The man looks Ms. Harding in the eye and says, ‘Good afternoon, Ms. Harding.’”
(Yeah, I was tempted to re-write the joke somehow so as to change “Ms. Harding” to “Mr. Trump,” but I restrained myself.)

Have as happy a weekend as you are able, under the circumstances....

Wednesday, January 18, 2017

"Ideology In Math Education"

At 70 pages (pdf), the latest “Archimedes Mathematics Education Newsletter” from David Wells (h/t A. Bogomolny) is almost more a small booklet than a newsletter, and is chockfull of rich, interesting discussion on math education:
It includes reviews, philosophy, quotations, history, arguments, discussion of pure vs. applied mathematics, an interview with Douglas Hofstadter... in short, something for anyone involved with math education to enjoy or tussle with. Don't expect to read it all in one sitting.
...prior issues of Wells' newsletter are listed here, by the way: http://amendavidwells.blogspot.com 

Monday, January 16, 2017

Staying In The Middle...

Been some discussion around math-Web this week about the “Median Game.” It got started with this Gil Kalai posting:

A quick, simple game (requires exactly 3 people to play, and just a pad-and-pencil), intriguing because of its recursive nature, and the resultant strategizing required.
The game actually shares origins with games called “Hruska” or “Mediocrity,” created by ever-inventive Doug Hofstadter, and described in Chapter 28 of his fantastic volume, “Metamagical Themas.”
Mike Lawler took the plunge and played Median with his boys recently and they quickly picked up on some of the nuances of the game:

As has been done with other games, would be interesting to have AI people write programs to compete at Median and hold an all-computer tournament to see which program (strategy) works best (…or perhaps this has already been done?)
Anyway, check it out at Gil’s site, and if you can, get 3 folks together to wile away some time playing it.

But fair warning, it can quickly play havoc with your brain! ;)