Counting and Cognition...

A bit ago, someone on Twitter linked to this year-old post that I found interesting, having to do with counting and mental images of number lines (revealing how individuals’ cognitions differ):

https://www.cambridgemaths.org/blogs/say-what-you-see/

…also interesting, this note left in the comments section:

“In note of Richard Feynman’s memoirs he mentions an experience with a room-mate. One of them found he could keep counting while he was reading, and the other could keep count while he was talking, but neither could do both. They discovered that for one of them perceiving his numbers meant hearing them (so talking blocked his counting), while the other was seeing them (and so couldn’t simultaneously read).”

Flourishing With Francis Su

When I was part-way through Francis Su’s new book I tweeted out, “Am about halfway through Francis Su's "Mathematics For Human Flourishing"... and astounded at what a beautiful read it is (& suitable for such a wide swathe of readers)!” 

Since finishing it I’ll just say that 100x; certainly in the running for my ‘popular math book of the year,’ and, it’s only January (…of course that assumes that that rascal Ben Orlin doesn’t enter the fray this year with an entry). Some call Su’s book “inspiring,” though I think “meditative” and “reflective” may be more apt terms. It is certainly not at all the ‘typical’ popular math book (though there are a few fun math problems included). Indeed, words used in the blurbs for the book on the back cover include, "compelling," "galvanizing," "sublime," "lyrical," "all-embracing"... words not often associated with math volumes.

The book has already received so many accolades from so many quarters that I feel no compulsion to offer a review here. As most likely know, Su’s book is an outgrowth of a farewell address he gave to the MAA upon departing his term as President of the organization. That address pretty much went viral through the math cybersphere, if not beyond. If you are familiar with it than you know the flavor and themes of this book and Su's vision. 

Often when speeches, essays, short pieces and the like are turned into book-length projects they come off as artificially stretched or composed of a lot of redundant ‘filler.’ Not so with Su’s volume; it maintains a rich, creative feel throughout most of the text. And as my original tweet indicates I believe it can be easily enjoyed by math professors and middle school students alike (and everyone in-between), as Su combines two words, “mathematics” and “flourishing” that rarely mingle in the public mind.

Oddly, in a way, this very current book reminds me of some words from the Introduction to a Laurie Buxton book, "Mathematics For Everyone" over 35 years ago, that I'll end with (from the Introduction):

"Mathematics is a living and growing organism; within it are intricate and delicate structures of strong aesthetic appeal.  It offers opportunities for surprise as unexpected vistas open the mind to new lines of thought. There is a moment when, after struggling with an intractable problem, the mechanism is suddenly clear, and the problem resolves with a deep sense of rightness. In this it offers the most intense sense  of intellectual pleasures. 

"Mathematics was created by all manner of people. There were religious bigots and atheists, political reactionaries and wild revolutionaries, snobs and egalitarians; some were people of great charm, some odious. If there is any common denominator, it is a driving curiosity, a desire to understand, a need to build, even if the structures be abstract. Admirably suited though mathematics is to modelling the real world, it can be developed totally without dependence on anything outside itself. Parts of it are simply mind creations, owing nothing to the physical world. It is a playground for the mind... 

"Regrettably, many of us have never been allowed to see what mathematics is. It has been obscured by pointless emphasis on routines rather than ideas. This failure to distinguish what is important has led many people to see mathematics as a collection of totally arbitrary rules which have to be learnt by rote, and performed with the exactness and precision of a religious rite. Ask a person if there is much to be remembered in mathematics; if they speak of an overwhelming mass of material, their education in this area has been counter-productive; not merely neutral. Mathematics, properly seen, is a connected web; grasp at one piece and all the surrounding region comes to mind."

ADDENDUM:  Since I'm not giving a detailed review of the volume here I'll offer this link to a fuller review:
https://harvardmagazine.com/2020/01/francis-su-math

Our Fragile Nation

American democracy is fragile, yet resilient. We'll soon see if Republicans turn the impeachment trial of demagogue Donald Trump into a farcical sham (as they'll no doubt attempt to do). How many Republican Senators will refuse to cave to money, power, and autocracy, and actually seriously consider the Constitution, rule-of-law, and founding principles?....

Today (because it seems timely) I am just re-posting, with minor changes, an entry from close to a year ago:

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Most readers here likely know the famous story of Kurt Gödel’s 1947 visit to an examiner’s office to apply for U.S. citizenship — it’s been briefly told many places, as I did back HERE.

Gödel, logician that he was, thought he’d found a 'logical flaw' or 'contradiction' in the U.S. Constitution that would allow a dictator to take power in the U.S., not unlike what Europe had witnessed. Friends, Oskar Morgenstern and Albert Einstein, talked Gödel out of bringing this up at his examination, believing, according to some accounts, that his worry was 'far-fetched and outlandish.'  No one knows for sure what his qualms centered upon, but the most widespread guess is that he was concerned about Article V of the Constitution allowing for amendments to the Constitution… even amendments that might weaken/eliminate various checks-and-balances and hand more authority to a despotic leader. We could theoretically amend ourselves right into a dictatorship.

The appeal of this explanation is that it reflects Gödel's well-established interest in self-reference: i.e., the Constitution could be amended, the amendments could be amended, the amendments to the amendments could be amended, etc.

While certainly possible, I’ve never been fully comfortable with that ‘guess’ of Gödel’s thought process, because Kurt probably realized what a slow, arduous, unwieldy path amending the Constitution actually is… with ample opportunity along the way to redress or put the brakes on ill-founded changes.

I’ve begun to wonder if just perhaps what Gödel had in mind was, alternatively, something far simpler, more direct, and more mathematical:

He would’ve clearly understood the math of the Electoral College (as spelled out in Amendment XII of the Constitution), and recognized that a demagogic individual could become President with a minority of the citizens' vote (not even a plurality, let alone a majority) by simply concentrating on a handful of key states. Then, with backing of a subservient Party he might run roughshod over most checks-and-balances simply with the judicious use of Executive Orders, Executive Privilege, emergency measures, martial law, judicial appointments, and the President’s function as Commander-In-Chief of the military... may or may not be a better explanation than the Article V focus, and might or might not be viewed as a 'logical flaw.'

In the oft-quoted words of Sinclair Lewis (who was contemporaneous with Gödel):

 "When fascism comes to America it will be wrapped in the flag and carrying a cross."

In short, if enough of the electorate is naive and ignorant enough to elect one, isn’t every democracy at risk of putting a despot-to-be into power, and made even easier with our Electoral College system? The German experience may be more 'normal' than we care to contemplate...

Is THAT what Gödel realized 72+ years ago?

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More Book Lists…

Book lists always make good space-fillers! So here we go ;)

Awhile back I stumbled upon an 8-year-old post from prolific author Ian Stewart listing his  “Top 10” popular math books:

https://www.theguardian.com/books/2012/jan/18/ian-stewart-top-10-popular-mathematics

In 2012 that list was as follows (no particular order):

The Man Who Knew Infinity  — Robert Kanigel

Gödel, Escher, Bach  — Douglas Hofstadter

The Colossal Book of Mathematics  — Martin Gardner

Euclid and The Rainforest  — Joseph Mazur

Four Colors Suffice   — Robin Wilson

What Is Mathematics, Really?  — Reuben Hersh

Magical Mathematics    — Persi Diaconis and Ron Graham

Games of Life   — Karl Sigmund

Mathematical Tales of Mathematical Wonder   — ed. by Rudy Rucker

The Mathematical Principles of Natural Philosophy   — Isaac Newton

(don't know what Stewart might add to that list in the dozen years since it appeared)

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This made me start looking around the Web for other favorite popular math book lists. There were fewer than I expected.

This 13-member list comes from math teacher Ali Kayaspor:

• Zero: The Biography of a Dangerous Idea — Charles Seife.
• Prelude to Mathematics — W.W. Sawyer.
• Measurement   — Paul Lockhart
• The Joy of X — Steven Strogatz.
• An Imaginary Tale — Paul Nahin.
• Proofs From the Book  — Aigner and Ziegler
• Things to Make and Do in the Fourth Dimension    — Matt Parker
• What is Mathematics?   — Courant and Robbins
• A History of Pi    — Petr Beckmann
• e: The Story of a Number    -- Eli Maor
• Imagining Numbers   -- Barry Mazur
• Journey Through Genius   -- William Dunham
• Prime Obsession   -- John Derbyshire

Kayaspor also has a separate book-list "for math people and designers":

https://medium.com/however-mathematics/25-interesting-books-for-math-people-and-designers-1c7ffcebb66a

From the GoodReads website comes this list of a dozen math-related books:

Gödel, Escher, Bach — D. Hofstadter

Fermat’s Enigma  — Simon Singh

Flatland  — Edwin Abbott

The Code Book  — Simon Singh

Zero    — Charles Seife

The Man Who Loved Only Numbers  — Paul Hoffman

Journey Through Genius  — W. Dunham

A Beautiful Mind   — Sylvia Nasar

The Drunkard’s Walk   — Leonard Mlodinow

How to Lie with Statistics  — Darrell Huff

Euclid’s Elements — Euclid

What Is Mathematics   — Courant and Robbins

The FiveBooks site offers this (2018) list of 10 best math 'history' books:

Prime Obsession  — J. Derbyshire

Mathematics for the Nonmathematician   — Morris Kline

Zero  the biography of a dangerous idea   — Charles Seife

A Concise History of Mathematics   — Dirk Struik

Unknown Quantity  — J. Derbyshire

The Math Book   — Clifford Pickover

A History of Mathematics   — Merzbach and Boyer

God Created the Integers   — Stephen Hawking

Fermat’s Enigma   — Simon Singh

Journey Through Genius   — W. Dunham

The Fivebooks site has several other science/math related listings possibly worth perusing:

https://fivebooks.com/category/mathematics-and-science/maths-statistics/

——————————--——

This list comes from Peter Flom on Quora:

https://www.quora.com/What-are-your-favorite-math-books-1

A Mathematician's Lament — Paul Lockhart

Out of the Labyrinth — Robert and Ellen Kaplan

Conversations with a Mathematician — Gregory Chaitin

Proofs and Refutations  — Imre Lakatos. 

Mathematics: Coffee Time in Memphis — Bela Belobas

The Measure of Reality: Quantification and Western Science 1250-15400 — Alfred Crosby

Godel Escher Bach — Douglas Hofstadter

The Proof and Paradox of Kurt Godel — Rebecca Goldstein

Group Theory in the Bedroom and Other Mathematical Diversions — Brian Hayes
 Pretty much anything by Martin Gardner or Ian Stewart

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Here's one of several Reddit threads related to favorite math books:

https://www.reddit.com/r/math/comments/2efz50/best_math_book_youve_ever_read/

Simon Singh offers up a long list of recommendations here:

https://simonsingh.net/books/recommended-books/mathematics-books/

And finally, I’ll add my own tentative baker's-dozen list (in no special order, and subject to change on a different day-of-the-week) to the mix:

The Prime Number Conspiracy  — ed. by Thomas Lin

Single Digits   — Marc Chamberland

Math With Bad Drawings  — Ben Orlin

Things To Make and Do In the Fourth Dimension  — Matt Parker

How Not To Be Wrong   -- Jordan Ellenberg

The Language of Mathematics — Keith Devlin

The Colossal Book of Mathematics   — Martin Gardner

How Mathematicians Think  — William Byers

The Music of the Primes  — Marcus du Sautoy

Mathematics For Everyone   — Laurie Buxton

Grapes of Math   — Alex Bellos

Unknown Quantity   — John Derbyshire
The Penguin Book of Curious and Interesting Mathematics  -- David Wells

———————————————

3 additional books are among my all-time favorites, but their subject matter crosses so many boundaries that I don’t really think of them as ‘popular math books’:

The Outer Limits of Reason  — Noson Yanofsky

Gödel, Escher, Bach    — Douglas Hofstadter

When Einstein Walked With Gödel — Jim Holt
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By the way, for those whose taste runs to fiction Alex Kasman maintains a large site of recommended math-related fiction/novels:
http://kasmana.people.cofc.edu/MATHFICT/
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Honestly though, all of these barely scratch the surface of the wonderful math reading that is out there. Indeed, I think we're currently experiencing a kind of golden age for popular math!

ASMR Monday

OK, to start your week, the first ASMR post of 2020. One genre of ASMR I only stumbled upon a few months back (and quickly became a favorite type) is often called "fast and aggressive ASMR," generally involving rapidly produced and changing auditory stimuli. Because the focus is on the audile and not visual aspect it can simply be played in the background while doing other things online, or simply closing one's eyes and listening. Here is one example:

Lies, Damned Lies, and....

(via Wikipedia)

                                                    
One of the late chapters in David Spiegelhalter’s fine volume, “The Art of Statistics,” focuses on the problems and reproducibility crisis in psychology research.

This sentence gave me a bit of a startle: “In a 2012 survey of 2,155 US academic psychologists, just 2% admitted to falsifying data.” The part that gave me a gasp was the phrase, “just 2%” as if that was a small figure. I’m not surprised that 2% have falsified data; I’m surprised that that many would admit to it! Indeed, I feel sure (though am only guessing) that the majority who have done it would NOT admit to it, and that the true figure is therefore probably at least double the 2% given — that would be at least 4% who haven’t just made mistakes, or fudged a little, or spun their conclusions, but outright falsified data! 

I don’t know how many total academic psychologists there are, but depending where they are and how much they publish, 2 - 4+% represents a pretty serious problem in my mind.  And while psychology is probably especially vulnerable to such falsification this doesn’t even address how much may additionally go on in biological, medical, and physical science fields. Spiegelhalter goes on to note that 94% of those in the study admitted committing at least one of 10 "questionable research practices" looked at.  Of course there are many reasons for such a state of affairs, but ultimately none very defensible. The award-winning site "Retraction Watch" has been tracking, for years now, published scientific papers that are retracted due to various issues, including fraud -- and they never seem to run out of material! :(( (definitely a site worth following and supporting if you're not already).

I was a psychology major myself 45 years ago and complained, to deaf ears, about the sloppiness of the field, but sloppiness and fraud are almost separate issues. Still, glad to see it all attracting more attention these days.

The study Spiegelhalter cites is here (with a lot more details):

https://www.cmu.edu/dietrich/sds/docs/loewenstein/MeasPrevalQuestTruthTelling.pdf

Reuben and Martin...

     “The working mathematician is a Platonist on weekdays, a formalist on weekends.”  

                                                                                      — Reuben Hersh

Mathematician and popular prolific writer Reuben Hersh died on Jan. 3, apparently in Santa Fe, NM., but I haven’t seen any details about his demise or major tributes as yet on the Web (no doubt forthcoming).  Hersh started off with a B.A. in English literature in 1947, and only later returned to school to attain his Ph.D. in mathematics in 1962.

Hersh’s writing was almost always interesting and thought-provoking (advocating what he termed a “humanist” view of mathematics) whether one agreed with him or not. He famously feuded with recreational mathematician/writer Martin Gardner as exemplified in a well-known and critical 1997 review Gardner wrote about Hersh’s popular volume, “What Is Mathematics, Really?” Gardner was a vocal and sharp-tongued "Platonist;" Hersh was not. Definitely worth a read if you’ve never seen it (the review):

https://www.latimes.com/archives/la-xpm-1997-oct-12-bk-44915-story.html

Gardner leads off saying he has “such high respect for [Hersh] as a mathematician and such low respect for his philosophy of mathematics.” And it gets harsher from there.

Hersh did eventually reply to Gardner though I couldn’t find a direct, free link to it on the Web (if someone knows of one let us know). This page offers up the first page of Hersh’s response and means for seeing the rest if you so wish:

https://www.deepdyve.com/lp/springer-journals/reply-to-martin-gardner-1IRiQyGw5c

==> ADDENDUM: Thanks to .mau (in comments below) for finding this unformatted link to Hersh’s response:

http://139.6.160.6/litecat/litie/?param[]=litie_200&value[]=Hersh,%20R

Many of Hersh's other articles are here:

https://sites.google.com/site/reubenhersharticlesetc/home/articles

Martin Gardner, who like Hersh died in his 90's, wanted to believe there was an 'afterlife' following death. Perhaps if there is, he and Reuben are carrying on their fascinating debate somewhere in the great beyond... or, better yet, maybe they've found a final answer.
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ADDENDUM:  Here's one tribute to Reuben that came in a few weeks after his death:
https://www.santafe.edu/news-center/news/tribute-reuben-hersh

It's A Contest!

MoMath

                                                

I’m sure word of this is getting disseminated adequately without my little added promo for it, but if you haven’t heard yet the National Museum of Mathematics has announced the "Steven H. Strogatz Prize for Math Communication" for applicants between 15 and 18 years of age. How cool is that! Read all about it here:

https://momath.org/the-steven-h-strogatz-prize-for-math-communication/

The contest is worldwide, and I’ll bet there are going to be some sensational, maybe mind-blowing entries -- so much untapped creativity across the globe. But deadline for submissions is April 22, 2020, not all that far off. All sorts of forms of communication are encouraged, falling into various categories, and will be judged on the basis of “content, creativity, communication.” There will be cash prizes and of course posting of winning entries on the Web.

What a great idea to encourage young math minds early on along their path. Hope/assume this will be an annual event.

Twitter Resolve….

                                   

It's that ridiculous time of year again for well-intended (if improbably-kept) resolutions:

1)  Re-read more math and statistics books… and, peruse less politics on Twitter. 

2)  Lift more weights, do more back exercises… and, tweet less. 

3)  Play more pickleball/tennis… and, post fewer re-tweets. 

4)  (Pretend to) play my guitar more… while responding on Twitter less.  

5)  Eat fewer carbs, breakfast pastries, and treats… resist growling, grumbling over tweets.   

6)  Dwell more on the awesome and transcendent… and less on the terse and snarky.

…catch my drift; there's a theme here (now if someone would hold me to it).

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...With an impeachment trial soon underway (I presume) and state primaries little over a month away I may be too distracted by politics for awhile to blog much (though I do have plenty of ASMR videos saved to post sporadically!). So IF posts become sparse it just means that I'm busy with other things (...like, you know, saving democracy!), and not gone into hiding.