...a companion blog to "Math-Frolic," specifically for interviews, book reviews, weekly-linkfests, and longer posts or commentary than usually found at the Math-Frolic site.

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." ---Bertrand Russell (1907) Rob Gluck

"I have come to believe, though very reluctantly, that it [mathematics] consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-legged animal is an animal." ---Bertrand Russell (1957)

******************************************************************** Rob Gluck

Friday, December 30, 2016

Close-out Potpourri of 2016

Good-bye (riddance) to 2016, but not before pointing to a few more math reads:

1)  Ben Orlin’s round-faced friends took a closer look at the (not-so) boring number line this week:

2)  “Flowing Data” called attention to this video introduction to Bayesian thinking:

3)  This Nikola Tesla bit of history almost reads like a Martin Gardner prank, but is apparently real:

4) “Ethics and Statistics” (…or is that an oxymoron? ;-) via Andrew Gelman:

5)  God-only-knows what scientific data will be destroyed once Donald Trump and his minions take reins in the White House, so John Baez continues his campaign to save climate data for posterity (…you know, just in case the Earth is worth saving):

6)  A relatively new blog for educators that I’d not seen before and David Wees pointed out yesterday:

7)  Mike Lawler posted a year-end review citing some of his favorite 2016 math projects with his sons; so many great ideas and projects:

8) And couple days back, I felt compelled to write about the inmates running the asylum (sorry, but this truly is the most depressing New Years of my entire lifetime):

Potpourri BONUS! (extra NON-mathematical links of interest):

End the year with just some relaxing music once again:

Friday, December 23, 2016

Unwrapping a Few Math Reads From the Week

Math just keeps coming:

1)  Jim Propp’s latest monthly offering is on David Kelly’s “Hampshire College Summer Studies in Mathematics” program.  If, like me, you’ve never heard of it (and perhaps even moreso if you have) fascinating stuff, with lots of good links:

2)  Dylan Kane on using instructional visual patterns:

3)  Andrew Gelman revisits the “hot hand” discussion yet one more time:
(…wonder if ‘hot hands’ apply to bloggers, ‘cuz Andrew has been on a tear lately putting up blog posts ;)

4)  Just in time for Christmas, new from James Grime & Numberphile on Euler’s number ‘e’:

5)  “Solve My Maths” keeps putting up great geometry problems week after week:

…and I only recently realized (“DOH!”) the site also has a separate blog with interesting posts:

6)  The “Christmaths edition” of the “Math Teachers At Play” blog carnival is up here:

9)  It's the giving time of year, so I'll just mention that Greg Ross who runs "Futility Closet," a favorite eclectic site for 11 years (with many science/math bits) is always in need of monetary support:

10)  Welcome to your new algorithm-overlords... Cathy O'Neil points to this Wall St. Journal piece hinting at our superlative future:

11)  ICYMI, my final interview of the year, last weekend, was with Grant Sanderson of the amazing 3Blue1Brown website:

12)  And one last piece I'm throwing in just because I found it oddball and difficult/confusing to follow — don’t know if that’s just me, or if it really is poorly composed — it’s Stephen Senn on placebos in research, in a guest post at Deborah Mayo’s site, where I usually find more clearly written statistics posts:
(p.s… I’m not necessarily commenting on the merits of what he’s written, only on the confusion of it)

Potpourri BONUS! (extra NON-mathematical links of interest):

1)  Last weekend’s “This American Life” was a rerun of one of their classics about how children perceive the world. Even if you’ve heard it before worth hearing again at this season of children’s dreams:

2)  Speaking of children, and sorry to be a downer, but hope everyone keeps the children of every God-forsaken war-torn place on this planet in your thoughts at this time of year… will close out with some (somber) Mary Chapin Carpenter holiday music:

Sunday, December 18, 2016

Grant Sanderson…. An Eye for Math Video Instruction

Math-Frolic Interview #40

"Hi Grant, Thank you for making math videos. When I watched the topology video, I was hanging on the edge of my seat in suspense as if watching Game of Thrones, while enjoying the beauty of the problem, the solution, and simply the graphics and animations."
-- a commenter at Patreon

With strong interests in both math and computer science, Grant Sanderson now produces some of the best, most cutting-edge, entertaining and instructional math videos out there on his 3Blue1Brown YouTube site. When you see the beauty and quality of his videos you'll understand how he has turned this into full-time work.  I believe Steven Strogatz was the first to bring Grant’s work to my attention, and you know when Steve recommends something it’s going to be good. 
Grant describes his effort this way:

3Blue1Brown is some combination of math and entertainment, depending on your disposition. The goal is for explanations to be driven by animations and for difficult problems to be made simple with changes in perspective.

He also has a Patreon account here:

And now a little more about him:


1)  You run an amazing math video site on YouTube, “3Blue1Brown.” How many hours per week or month, approximately, do you spend working on that?  And how do you decide what to cover with each new video you do?

It's hard to speculate on specific hour counts.  For one thing, it's only relatively recently that I started doing this full-time, so the honest answer is that I don't have enough data to answer yet.  Also, the question seems more applicable to occupations with less of a work-life blur than I have.  For example, when I read math, does that count as work towards 3blue1brown?  What if some of it eventually makes its way into a video?  When I work on improving the animation tool, but not for a particular video, is that work or just a side-project?  In general, I'd say I work quite a lot, but the word "work" doesn't really do justice to what a playful process it is.  

As far as deciding on what to cover next, there's just a long list of things that I think could make good videos, and every time I have a thought or come across something I think is worthy it goes on the list.  The more interesting question is how to sort the list, and for that I try to prioritize ideas that I don't think are commonly expressed elsewhere on the internet, and which strike the right balance between approachable yet deep.

2)  We’ll go back to some basics:
What is your academic background and your current professional/career role? 

I studied math and CS as an undergrad at Stanford.  The original plan was to continue on the PhD track, and my later years at Stanford were spent increasingly in graduate classes in preparation for that.  But I personally wanted to spend a few years out of academia before doing so, even though for whatever reason that is a less common thing to do.  After graduating, I worked as a content creator for Khan Academy, making things related to multivariable calculus.  At the time, I spent my nights/weekends working on 3blue1brown as an unassociated side-project.  Thanks to the huge support people showed for 3blue1brown, this is actually what I do full time now.  With things going how they are now, and given that I've always tended more towards the teaching/outreach side of math than the research, it's looking less and less like I'll return to the PhD path.  Maybe that's why more people don't take those years in-between.

3)  You say at one point on your site that you’ve “loved math for as long as I can remember.” Do you recall what initiated your interest in math, and when did you know you wished to pursue it professionally?

My dad, definitely.  Even though he would tell you that his own math education ended earlier than he would have liked, sometime shortly after calculus, he has a great appreciation for beautiful problem solving.  That appreciation was matched by his eagerness to pass it on to me, as embodied by the countless times he exposed me to neat puzzles and patterns when I was young.  I remember a particular game when I was very young where he’d stack sugar cubes in some interesting geometric way, and if I correctly gave the number of cubes in the configuration, he'd give me one as a reward.

Beyond that, I had the good fortune of some caring and encouraging math teachers to bump up my interest as I grew up.  I am particularly thankful to Phil Sakashita, my calculus teacher, who did more than almost anyone else to open my eyes to what math could be.

As far as going into it professionally, it was probably sometime in high school that I thought becoming a mathematician would be incredibly cool.  But somewhere in college, I actually toggled to the CS side of things, and had you interviewed me then I would have been quite certain that my future lay somewhere in software or data science.  But at the end of each tech internship that I did, I found myself lamenting the fact that I wasn't doing more math, so I decided to switch gears and point myself towards a math career.

4)  Any idea how long you’ll be putting new material on your site, or what’s ahead otherwise in your life, mathwise? Any special goals for the future you’re actively striving toward?

Right now, the focus is just on getting into a flow of regular content creation.  I'm working on an Essence of Calculus series behind the scenes, and I'll always be working on more "Essence of" series of some sort, so look out for those.

5)  You explain in a FAQ that the odd name for your site is a reference to an eye condition you have, “sectoral heterochromia,” and you say the name puts “a genetic signature on my work.” I’m not quite sure what that means or why it was important for you; can you flesh that out a little more?

I'll be the first to admit that making this my logo is somewhat strange, but my right eye is 3/4 blue and 1/4 brown.  When I say it puts a "genetic signature" on my work, I mean that just as other people put their names on their work, I chose to put a little piece of who I am in a different sense.  The more important point, though, is that the channel is about seeing things, so an eye is somewhat fitting.

6)  Increasingly, there’s more and more competition online for math videos and instruction. Do you have some favorite other sites that you don’t mind recommending to folks? 

Mathologer is, of course, great.  And for classrooms, I really like the work that Desmos is doing.  The Art of Problem Solving site/books/courses were influential for me growing up.  Also, "How to fold a Julia Fractal" is a must-see for anyone not yet familiar with it.

7)  And how about books?… I’m always interested to hear what ‘popular’ math books were especially inspiring to a math enthusiast, that you’d recommend to others?

Again, I quite liked The Art of Problem Solving books growing up.  Vladimir Arnold's book on ODEs is fantastic.  John and Barbara Hubbard's "Vector Calculus, Linear Algebra and Differential Forms" is great for any early undergrad hoping to get a deeper feel for what they're learning.  Munkres' "Topology" (of course).  Cox's, "Primes for the form x^2 + ny^2".  There are also these three little books on number theory by Kato, Kurokawa and Saito which are a delight, though probably best read with a supplement.  "Proofs from the Book" by Aigner and Ziegler is just filled with gems of cleverness.  Larson's "Problem solving through problems".  Many of Terence Tao's books are great, especially the one on measure theory.  I'm sure I'm neglecting some great ones, but those are what come to mind right now.

8)  When you’re not engaged in mathematical pursuits, what are some of your other interests/hobbies/activities?

I enjoy playing violin and mandolin, and pretending like I know how to play guitar, bass and piano.  Hiking and running are also solid default activities.  I do some private teaching on the side, which I count as a hobby because I do it for my own pleasure.


Thanks Grant, hope any readers not already familiar with your site will check it out soon. It's like having a mini-college math education at your fingertips... and the price is right!

Friday, December 16, 2016

Some Reads From the Week

1)  “Visualizing the Riemann zeta function” from Grant Sanderson (video):

2)  Couple of problems from DataGenetics this week:

3)  Keith Devlin contemplates the secret of changing a bicycle tire and doing mathematics:

4)  Andrew Gelman once again on Bayesian statistics:

…and in another post Andrew discusses skepticism in science, and the viewpoint of science-writer John Horgan:

5)  Love the game of Monopoly?… Matt  Parker & Hannah Fry report on some math behind winning Monopoly strategies:

6)  I’ll drop a quick plug for a site I just heard from this week (but don't know that much about); an educational videos site covering lots of subjects (the breadth of videos and few clips I looked at seemed impressive). The math selections are here:

7)  h/t to Steven Strogatz passing along this long NY Times piece on AI:

8)  ICYMI, a few jottings I’d made over time, about Martin Gardner, General Semantics, and dysgenics, all came crashing together after the Presidential election:

...and in a post with some similar underlying concerns, Brian Hayes wonders how we deal with fakery and truth in a world where so many people seem unable (or unwilling) to recognize the difference between the two:

On Sunday, by the way, I’ll be squeezing in one more interview for 2016, right here at MathTango, so check back then.

Potpourri BONUS!:

Natalie Wolchover looks at where Grand Unification theories in physics stand today:

Tuesday, December 13, 2016

Martin Gardner Helped Wreck My Country…

Martin Gardner
Alfred Korzybski

[Bit of a long ramble ahead through some things on my mind for awhile.]

We’ll start with a news story (…because we live in this wonderful time when you get to make up any damn thing you want and pass it along as "breaking news"):

==> According to top-classified PRIME Security documents uncovered by TMD special investigator Alexus Jones, the Donald Trump we see is in fact an electronic Slovenian-built, fully-programmable cyborg under direct control of long-time Moscow operative Melania Kvorninski and her Russian handlers (no one has ever seen him actually use a rest room for example). 25 years in the hatching, this sinister-plot is scheduled to reach fruition in mid-2017, with complete dissolution of the American military and Congress.

Que sera sera…. (p.s… by the way, NOBODY can prove the above false; that's the beauty of fake news, where more layers of lies can simply be added on).

Moving on….
There were many factors of course for Hillary Clinton's election defeat, but the simplest is that James Comey’s late intervention torpedoed her campaign, halting her momentum and moving many undecideds (cringing at the possibility of hearing about emails for the next 4 years), over to the Trump side. Without that single news event all indications are HRC would've won, and of course she DID win the popular vote handily. But with that said, at least the Comey announcement was an actual news event. 
Another factor getting a lot of attention lately is all the “fake news” reports that got passed along (and believed) as real by the unaware (or dare I say, the none-too-bright)… some of it so ridiculous on-its-face as to bode ill for Democracy’s future! Which leads me to the two topics I want to raise here. Because... what-the-hell is wrong with people's basic critical thinking skills these days!?

American demographics are changing; specifically the percentage of older people, say 70 and over, is far greater than in the past relative to the number of young people, say 30 and under. Medical/health advancements and lower birth rates mean age proportions are becoming more skewed all the time.
But ANOTHER potential skewing is FAR more controversial...
It has been called "dysgenics" or "retrogressive evolution" (the opposite of eugenics). Unfortunately, controversial Nobel-Prize winner William Shockley brought the idea to public attention decades ago, though it well-precedes him in the biology community. If you remove notions of race and IQ from the discussion, the more general concept is worth reviewing as a partial explanation for the anti-intellectualism, anti-science, anti-establishment, and fringe views prominently on display these days. 

Shockley spoke out at a time that the "Zero Population Growth" movement, family-planning, and widespread access to birth control were spreading across America. In brief, his idea was that the most educated, most successful, most fit individuals were primary practitioners of these new trends, while the least educated, least successful, and least fit, were largely unaware of, or uninterested in, such trends. In short, the most fit, capable couples were often deliberately limiting families to 2-3 children at most, while their less fit (less capable) counterparts were more likely to have 4 or more offspring, who in turn grew up in less ideal conditions, received less education, were generally less fit or successful, and continued the cycle — self-perpetuated skewing (of ability and education). 

70 years since the world collectively said "never again," the rise of xenophobia, anti-semitism, isolationism/nationalism, anti-elitism, anti-science, and simplistic-thinking across the globe is unmistakeable, and may coincide with factors Shockley noted decades ago. 
[An alternative explanation for recent political events specifically in the U.S. is that the right-wing have always been a part of the American populace, but were not very active politically in prior generations. Beginning in the 1960’s the Republican Party deliberately set out to contact, mobilize, and activate conservative rural and fundamentalist elements (the "Silent Majority" as it were) through targeted mass mailings and talk radio... and now also with digital social networking. They were successful -- millions who were never previously active in politics now are.]

In any event, if the numbers of less-educated, less critically-thinking people have risen in recent decades (partly perhaps because dysgenics, mathematically-speaking, may be baked into the demographics), then widespread quality education and social services may be the only antidote to all this. Which leads me finally to Martin Gardner….

The greatest skeptical mistake Gardner ever made, in my opinion, was his derision of non-Aristotelian "General Semantics."  While some elements of G.S. (and its founder, Alfred Korzybski) are easily critiqued, the broader intent and purposes of G.S. were admirable. G.S. fundamentally trains people to be aware of the many ways, words and language manipulate thinking, perception, decision-making. In short, it assists in 'critical thinking,' in dismally short supply these days. 
Gardner, of all people, should have been a fierce adherent of G.S.; instead his scorn helped make it a fringe field, doing for G.S. what Noam Chomsky did for “behaviorism” in psychology. In fact I don't know of any way to teach people critical (and skeptical) thinking without employing tenets from General Semantics. They are simple and basic, yet often not internalized by people without some instruction. G.S. inculcates a wariness of language in advertising/marketing, politics, relationships, religion, news-coverage, and even science, and a recognition of the subtle dogmatism that infuses so much communication. On a sidenote, here are a couple of popular 'taxonomies of logical fallacies' which actually relate back (unknowingly) to a number of G.S. basic principles:

(You may be able to embiggen them on your screen for better readability, or download them -- they ought be somewhere in every high school, IMO!)

Seriously, it amazes me that secondary students are forced to read Shakespeare or Dickens or Hardy, but never taught rudiments of how language operates on our cognition, a far more important faculty in today's world.

I won’t try to summarize G.S. principles here, and I haven’t kept up with G.S. volumes over the years, but two of the early popularizations (there are many) that most influenced me were:
Language In Thought and Action  by S.I. Hayakawa
The Tyranny of Words  by Stuart Chase

In its own way, Alan Sokal’s “Beyond the Hoax,” which I’ve referenced here before, also touches on many of these issues. I was recently a bit miffed to see Ben Goldacre, in a tweet, imply (if I interpreted it correctly) that Sokal’s hoax may have actually contributed to the large-scale mistrust people now exhibit towards science and academics. I don’t think that’s true at all; on the contrary Sokal’s target was specifically 'postmodernism,' and some of the ’softer’ disciplines that brush on a veneer of science, but not science or true expertise itself. We need more Alan Sokals, not fewer, to combat the growing anti-elitism, anti-science sentiments thriving today.
Anyway, at the bottom of the page I’ve also tacked on a couple of weblinks that further detail the whole rift between Gardner and G.S. Some people even viewed G.S. as a “cult,” in part probably because L. Ron Hubbard was said to have incorporated parts of it for his early “Dianetics” program, but many of G.S.'s principles were broad enough that they could be employed in any number of programs.

In fairness to Gardner I will note that he was less critical of the “popularizers” of G.S. than of its foundations and some of its technical notions.

In his autobiography Gardner uses an example of how "E-prime," a G.S. idea that never caught on (which advocated avoiding all forms of "to be"), would alter some simple doggerel... E-prime version on left, standard in parentheses on the right:

Roses look red  (Roses are red)
Violets look blue  (Violets are blue)
Honey tastes sweet   (Honey is sweet)
As sweet as you

Now, I suppose to many (like Gardner) these may seem trivial, innocuous changes, but buried in language/words are deep meanings/mindsets and effects, and the effects of these two versions ARE different (the first being more accurate, less dogmatic; the second being assertive, but unprovable and potentially inaccurate). Emotions, prejudices, ambiguity are intrinsically buried and maintained in our language use. I’m a bit flummoxed that a writer as clear and incisive as Gardner didn’t show a deeper grasp of how words (dangerously) manipulate people.

Or take a different simple sentence: “Mary had a little lamb.” 
Sounds simple; you likely think you understand it. But in fact you CAN’T really understand it without more context because it has too many possible meanings. Just emphasizing different words shifts the meaning:
MARY had a little lamb.
Mary HAD a little lamb.
Mary had a LITTLE lamb.
Mary had a little LAMB.

And putting these varying sentences into different extended contexts can further significantly alter what is being said. The point is that routine language (that we take for granted) is very imprecise and ambiguous, yet people react to it as if it is clear and explicit. I’m using minor examples here, but there are far more nefarious ones out there in the world (especially in this day of bountiful conspiracy theories and lies).

There is some irony that a spurned figure like Shockley was perhaps prescient in foreseeing where the U.S. was headed, while Gardner’s much-touted skepticism led him to rebuke one program that could have prevented this electoral outcome; prevented the very gullibility and irrationality he spent his adult life battling.

Properly taught at young ages, General Semantics, would be an antidote to the nationalism, anti-elitism, anti-intellectualism we now face. Though it is still around, G.S. basically flopped back in the 50s/60s, shortly before the time that Shockley began worrying aloud about dysgenic factors in this country. 50 or 60 years later a demagogue gets elected President of the U.S. Well, no shit Sherlock!! ;-)
As a self-described “democratic-Socialist,” Gardner would be spinning in his grave (or wherever), at this year’s election outcome. But I’m here to say, even if a bit facetiously, that he was unknowingly partially responsible. 

Meanwhile, between now and mid-2017, beware of thin-skinned, orange cyborgs….

p.s… an aside: I once briefly mentioned to Jim Propp my disappointment in Gardner’s failure to take General Semantics seriously, and he countered that every fan of Martin has some one beef with Gardner; some particular thing they think he got very wrong. So rest-assured, in that context, I still consider myself a typical, inveterate admirer of Martin.

For anyone wanting to know more about the Gardner/G.S. clash here are a couple of pages that go into further detail:

ADDENDUM:  Found it a bit serendipitous that the latest post from Ben Orlin deals with “the catchy nonsense of ‘two negatives make a positive'” — this is at least somewhat similar to some of the issues General Semantics deals with in pointing out language’s weaknesses. In turn, this brings to my mind also, Keith Devlin’s longstanding crusade against teaching “multiplication as repeated addition” (it is NOT). And I’ve pointed out here before, math puzzles where the actual math or logic is simple, yet the language used easily leads people astray to wrong answers. In short, even mathematics (or at least math education) is not immune to the imprecise, misleading nature of words.

Friday, December 9, 2016

Some Reads From the week

ICYM any of these:

1)  A fun new Numberphile:

2)  Sort of a cool (unexpected?) factoid that Colin Wright tweeted out this week:
Except in exactly one case, the digit sum of the product of twin primes is always 8.
[The exception, btw, is 3 and 5.]

3)  Andrew Gelman and Deborah Mayo, frequentism and Bayesianism:

4)  New book on Frank Ramsey, one of the great mathematicians of the 20th century:

5)  Nice tribute to Richard Guy, on his 100th birthday! (h/t John Allen Paulos):

6)  Latest issue of “Math Munch”:

7)  More of Cathy O’Neil and “Weapons of Math Destruction”:

…and here, Cathy on Facebook and democracy:

8)  It sometimes bothers me when bloggers beat around the bush with their opinion… THIS isn’t one of those occasions (Gelman ranting on a statistics text and publisher):

9)  Evelyn Lamb has suggestions for the "Breathrough Prize" awards (concurring in part with Peter Woit):

...and here she writes about the "Pseudocontext" notion of Dan Meyer:
10)  A new “Carnival of Mathematics”:

Potpourri BONUS! (extra NON-mathematical links of interest):

1)  The issue of right-wing manipulation of fake news and Facebook:

2)  If you missed this powerful viral video from the week, you ought see it, especially if you’re a teacher, parent, or student (…but warning, while it’s an important message, it’s NOT a joyous Holiday one):

Friday, December 2, 2016

Weekend Reads, if you missed them...

A short list of picks from the week:

1)  A few frequently asked questions/answers from John Cook:

2)  Alex Bellos presents one of the all-time greatest, most contentious paradoxes (Newcomb’s):

3)  A newly-discovered prime number:

4)  Interesting piece on (mathematician) Jim Simon’s incredible Medallion Fund (h/t Steven Strogatz):

5)  Evelyn Lamb suggests some possible videos for generating math discussion at the secondary level:

…and in one of her more complex/weird “favorite spaces,” Dr. Lamb explains “lexicographic ordering on the unit square”:

Potpourri BONUS! (extra NON-mathematical links of interest):

1)  My own interests in philosophy are fairly narrow, but the five (2016) books recommended in this long post look interesting (h/t Graham Farmelo):

2)  A paper called (by it’s AUTHOR) a steaming pile of dung from start to meaningless finish” is retracted from a journal:

(folks, you can’t make this stuff up… or, maybe in a sense, you can!)