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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