Chi-i-i-i-i-i-ll Friday *

[ *  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]

When Losing Is Winning

Jim Propp’s fertile posts or tweets often get me thinking about tangential things… 

Yesterday in a tweet he whimsically mentioned wanting to lose quickly at Monopoly when he plays against his kid.

Which got me immediately thinking about game variants where the object is to lose (you WIN by losing!). The only thing I could find, quickly googling around, was this somewhat technical piece on a checkers variant, sometimes called “suicidal checkers” with the object to lose: 

https://arxiv.org/pdf/1806.05657.pdf

Seems to me over the years I’ve read other such game variations, but a quick search didn’t turn much up (there are plenty of common game variants, just not where losing becomes the goal).

I did find this year+ old Scam School video showing a similar fun variation for Tic-Tac-Toe (the main description beginning ~2:16 mark) -- this is essentially a version of what's been called "misere tic-tac-toe," reverse or anti tic-tac-toe, or even "eot-cat-cit," where whoever gets 3 in-a-row first loses:

Presh Talwalkar did a nice, more expansive analysis of this game a couple years back at his Mind Your Decisions blog:
https://mindyourdecisions.com/blog/2016/11/01/the-best-first-move-in-misere-tic-tac-toe-3-in-a-row-is-losing-game-theory-tuesdays/

If anyone can point to other such win-by-losing variations of well-known games let us know.

Lipogrammatic Fun and Gams ;)

A few days back Jim Propp tweeted out the following:

“I’m thinking of an irrational quantity important in calculus (it’s hard to discuss natural logarithms without it). What constant am I thinking of, and why am I talking about it in this odd roundabout way?”

It seemed fairly clear that Jim was referencing “e,” but I completely missed (’til it was pointed out, DOH!) that he had composed a "lipogram" — a sentence deliberately leaving out a specific letter or letters, in this case, “e”. "E" is frequently used because it is the most common letter in the English alphabet, and thus more challenging.

Another mathematician, A. Ross Eckler also dabbled in lipograms, some of which are presented here (along with other fun wordplay):
https://digitalcommons.butler.edu/cgi/viewcontent.cgi?referer=https://www.google.com/&httpsredir=1&article=4164&context=wordways

In fact, I always find it intriguing how many mathematicians seem additionally innately interested in language play and in music. Music is the easier to understand since it clearly involves many mathematical aspects and patterns, and indeed several books address such. I suspect that language, and particularly the prosodic elements thereof (stress, pauses, intonation, rhythm, etc.), likewise may be governed by many mathematical rules that we have yet to fully appreciate or understand. Music, language, science, all very math-driven perhaps.

Anyway, returning to lipograms, several years ago NPR ran a contest asking listeners to create lipograms without the letter “i” and they got some great ones:

https://www.npr.org/2012/06/29/155364882/contestant-lipograms-the-best-of-the-best

And Douglas Hofstadter once composed a lengthy autobiographic profile, again leaving out "e":

http://prelectur.stanford.edu/lecturers/hofstadter/autolipography.html

More famously, Ernest Vincent Wright wrote a 50,000 word novel, Gadsby, remarkably without a single "e"... certainly not something ever accomplished by that poet Cummings. ;) 

By the way, lipograms are just one of several categories under the heading of what's deemed "constrained writing."

Chi-i-i-i-i-i-ll Friday *

[ *  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]

Age... Modes, Medians, Musings

“Death be not proud, though some have called thee 

Mighty and dreadful, for, thou art not so…”

   — John Donne

“We are stardust

Billion year old carbon

We are golden

Caught in the devil's bargain

And we've got to get ourselves

back to the garden”

   — Joni Mitchell

George H.W. Bush’s passing has me musing a bit about aging…

I’ve never honestly understood the widespread desire of folks to live into their 80s and 90s. Quality of life, and not length, has always been my stoic main concern. What Steve Jobs accomplished in 56 years blows me away; I wish we’d had him around another half-dozen years, but only if his quality of life had been maintained… and it wouldn’t have been.

George H.W. Bush died at 94 years of age, Jimmy Carter’s current age. Gerald Ford and Ronald Reagan both passed at 93. Those are long lives by current standards  (and yet I hear some scientists talking, I think ridiculously and grievously, about humans living routinely to 120+). The average longevity for a male in America is currently around 76, and for a female about 82. Of course ex-presidents get the best of care and opportunities, so it’s not surprising they may outlive the averages.

Anyway, those averages are what we always hear about, but I started wondering about the median and mode of longevity, and was at least slightly surprised by what I found. The median expected longevity age (at birth) for a male is ~80, and for a female is ~85, while the modal male age is ~86 and ~89 for females (I was viewing 2014 stats, but assume they haven’t changed much).

Of course it’s to be expected that the median ages would be higher than the average longevity ages since plenty of people die at say 15 or 20 (and younger). For males to average 76 years of longevity it means a male who dies at say 15 must either be ‘averaged out’ by a male dying at 137 (which ain’t gonna happen), or several males must die past 76. Still I find it rather amazing that essentially half or more of the population is living past ~80 (…what, and dying bankrupt to the medical system???… sorry). Seriously, the proportional skewing of populations (as never before seen) to an older cohort has worrying ramifications for the future of society, and perhaps for the well-being of younger generations, if more-and-more of society’s money and resources must be siphoned off to an expanding older population), but that’s fodder for a different discussion.

The mode is more interesting: for men ~86, for women ~89. That’s the one that really surprises me (partly only because I scan local obituaries fairly often and would guess most deaths I see are between mid-70s and low 80s). I wonder (but haven’t looked up) what the second and third closest modal numbers are, and what (if anything) accounts for those particular numbers (or is it sheer happenstance more than anything else)?

My peer group (and I haven’t even hit 70 yet), with our aches and pains, rickety joints, hips, and knees, high blood pressure, cholesterol, or acid reflux, etc. etc. often joke to each other that we were sold a bill-of-goods when young about how wonderful retirement and old-age would be. Yeah, age has its privileges and freedoms (woo-hooo, discount coffee at McDonalds ;)… but also its frustrations, like incessantly watching the world take 3 steps forward and 4 steps back; seeing problems/issues we thought were resolved return over-and-over again. And hearing, eyesight, mobility etc. all decline; virtually nothing of our physicality improves with age; we merely adapt to the gradual infirmities. Living vicariously through the lives of children/grandchildren is rewarding; I’m less certain that living in the day-to-day real world is! (but maybe that’s just my brain living under Donald Trump speaking). It’s famously said that “youth is wasted on the young” — that has more meaning for me now than it did even 15 years ago; as Kierkegaard put it, "Life can only be understood backwards; but it must be lived forwards." Then there’s the David Mamet adage, ”Old age and treachery will always beat youth and exuberance" — that’s a fun one (I employ it in pickleball whenever I can). But truthfully, it is only young people, with each new generation, who are left to fix the world their parents… and old Presidents… screw up royally… time and time again. If only young people would never grow up! ;)

It Was a Very Good Year... In Books

via ClipartSuggest

Time to Holiday shop for the math bibliophiles on your list….

This year was the hardest choice I’ve ever had picking a ‘book-of-the-year’ due to two very different books I relished so much (though, if you read my post of October 14th maybe you’ve already guessed my choice). In April Jim Holt’s fantastic essay compendium, When Einstein Walked With Gödel appeared and I couldn’t imagine any book surpassing its rich, thought-provoking content, crossing the boundaries of math, physics, philosophy, and culture.

Then on May 23rd, Ben Orlin announced he had written a book… and, well, the rest is (delightful) history. I loved his September volume, Math With Bad Drawings, more-and-more the further I got into it. Completely different of course than the Holt volume, but in the end had to go with the one that contained more actual math (though the subjects of Holt’s essays fascinate me), fresh, original content (Holt’s brilliant essays are fabulous but are previously-published material), and simply possessed a creative flair I’ve rarely-if-ever seen in a math volume. So it’s Ben Orlin's by a sliver as my book-of-the-year. And to look at him, Ben appears to be fresh out of middle-school… so no telling how many more great volumes he has to give us in the future!

By the slimmest of margins after these two, comes another fabulous compendium, The Prime Number Conspiracy (from those fine folks at Quanta Magazine). A collection of the great pieces they've been handing us for free for years now, so go ahead and pay up to read them again. With it they've released a companion volume of Quanta pieces on the sciences (especially physics) entitled, Alice and Bob Meet the Wall of Fire. (These are both paperbacks, and so very reasonably-priced, btw.)

A couple of other books I enjoyed this year were (like Holt’s volume) a bit tangential to math. Technically, Exact Thinking In Demented Times  by Karl Sigmund, shouldn’t be on my list. It was released in the original German in 2016, and was first published in English (translated by none-other-than Douglas Hofstadter) in late 2017. But I didn’t get it ’til early 2018, and loved it, so am including it here, though it is only for those who find the history and personages of analytical philosophy (specifically, the “Vienna Circle”) interesting.  Just a great account of a rich, potent time and place in academic history.

Nassim Taleb’s Skin In the Game was one of the most fun, entertaining volumes of the entire year. There is mention of probability of course (and also a 10-page mathy “technical Appendix”), but otherwise it’s not really a popular math book, so much as a pop-psychology or pop self-help (or even social anthropology) volume. Taken as an actionable financial or life guide the book could ultimately disappoint, but if taken as simply a fun, regaling read, with irascible, pontificating Nassim continuing to cultivate his burnished, blustery public persona, it’s readily recommended, even if not as substantive as his prior Antifragile.
[If you do want more of Nassim's serious mathematical work you can find him on YouTube.]

As long as I’m veering away from math with some recommendations, will venture further off the rails with physicist Alan Lightman’s Searching For Stars on an Island in Maine, a wonderful volume of meditative essays with more philosophy, metaphysics, or simply speculation and musings, than either physics or math. The sort of volume I think of as a beach-read for the more cerebrally-inclined.

And finally, also departing from math, Freeman Dyson was out with his autobiographical book of letters Maker of Patterns, which I haven’t read but suspect his many fans will enjoy. It’s likely reminiscent of Richard Feynman’s older book of letters (collected by his daughter), Perfectly Reasonable Deviations From The Beaten Track — most of those letters were quite mundane, but here and there are the ones that exhibited the brilliant, iconoclastic, playful, quirky image he took on publicly late in life (showing, I think, that that image wasn’t wholly hyperbolic, but very much a part of him). I'm guessing that Maker of Patterns is similarly a mix of the mundane with the insightful.

Anyway, returning now to actual mathematics, if you want a book you can sink your math chops into more, Vicky Neale’s short volume Closing the Gap, on one of the most fascinating mathematical narratives of recent years, is wonderful: all about Yitang Zhang’s contribution to the Twin-Prime conjecture, and tangential topics; short, but including a lot of ideas you need to slow down to contemplate.  Her writing is terse and straightforward (at times, a little more explication or illustration might've been helpful) with interesting detours from the main topic. A bit pricey for a 150-page book (but normal coming from Oxford University Press). Not necessarily for a general audience, but certainly timely and of interest to most mathematical types who hold any fascination with prime numbers or number theory.

Physician/statistician Hans Rosling’s book, Factfulness was published after he died in 2017 and doesn’t directly contain a lot of math, but is about related topics: data, knowledge, misinformation, patterns, critical thinking, and in a day of so much gloom-and-doom it carries an optimistic message of hope despite the ignorance prevalent around the globe. A very good, worthwhile, and acclaimed book.

Hannah Fry’s well-received Hello World is yet another offering in this burgeoning genre of volumes on big data and algorithms that are increasingly running the world and our lives. Very entertaining from beginning to end; a bit similar (I think) to the Rosling book above, and an excellent selection for all interested in this area.

But if you want a volume that offers much more of the math surrounding models, big data, and probability, than the above two books do, The Model Thinker by Scott Page may well be for you. More math and technicality, suited to a more academic crowd, but a solidly and surprisingly good effort.

Mircea Pitici’s latest Best Writing On Mathematics 2018 is another wonderful, highly diverse collection of math offerings from 2017. Something for every math-lover in this volume, as well as mention of the many popular math writings that didn’t make the cut for the edition, but may be worth checking out. Although his slant or themes change slightly from year to year, if you've read Pitici's prior iterations of this volume you know what to expect from this great series he's created.

A couple of other noteworthy books I enjoyed from the year that didn’t make my top tier, against such stiff competition:

The Art of Logic In an Illogical World is the third offering from ever-popular Eugenia Cheng, this time taking on the need for critical thinking in an increasingly polarized world. More and more of these volumes focused on critical thinking seem to be appearing... and they can't come too soon! ;)

The Calculus Story by David Acheson; a surprisingly nice, short intro to calculus, for anyone who is approaching the subject or wants to re-introduce themselves to it.

There were a slew of popular books I never got around to reading, but based on reviews or other buzz, here are a few I feel worth mentioning:

Lost In Math  — Sabine Hossenfelder’s contrarian take on modern physics cosmology, critical of the modern obsession with “beauty” in current-day physics theory; creating a lot of buzz, perhaps even polarization, among physicists.

Deborah Mayo’s  Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars is probably more for the professional statistician than a lay audience, taking on current day issues in the so-called “statistics wars” (Mayo is a philosopher, who runs an active blog on the issues covered in the book).

Alfred Posamentier et.al. — The Mathematics of Everyday Life, another typical Posamentier volume (always interesting, well-written) for a general audience.

Oliver Roeder  — The Riddler a compendium of great puzzles from the puzzle writer for the FiveThirtyEight blog.

Millions, Billions, Zillions: Defending Yourself in a World of Too Many Numbers — Brian Kernighan

...a small, stocking-stuffer-sized book aimed at bestowing basic, much-needed numeracy to readers.

From two well-established authors: John Stillwell was out with Reverse Mathematics and Eli Maor with Music By the Numbers.

Anyway, those are just volumes that favorably caught my eye in the prior 12 months. As usual there are plenty more where these came from! Hope some of them make their way into your Holiday festivities.

--------------------------------------

Meanwhile, some years I've done an expanded end-of-year list of posts I most enjoyed writing in the prior 12 months; this year I’ll only pass along four, none particularly mathy. In case you missed them:

Teachers in our lives…:

https://mathtango.blogspot.com/2018/05/teachers-in-our-lives.html

A bit about humor…:

https://math-frolic.blogspot.com/2018/06/of-cartoons-and-rorschach-tests-and.html

Language etc…:

https://math-frolic.blogspot.com/2018/06/her-name-is-not-kurt-of-language.html

Just another ramble…:

https://math-frolic.blogspot.com/2018/11/sunday-night-ramble.html

And that's a wrap! Now get to your local bookstore.

Chi-i-i-i-i-i-ll Friday *

[ *  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]

What A Life!

Many readers here will have enjoyed various videos done by Stanford mathematician Tadashi Tokieda for the Numberphile site or elsewhere, but probably, like me, didn’t know much about his background. Erica Klarreich has now given us a wonderful profile of the man behind the mellifluous voice and sparkling “toy” videos (and his "unusual path into mathematics"). I encourage everyone to read one of the most fascinating brief portraits I’ve ever encountered:

https://www.quantamagazine.org/tadashi-tokieda-collects-math-and-physics-surprises-20181127/

Here’s a summary of some of the high points, but do read the entire interview to fill in all the details:

1)  He doesn’t find puzzles, games, and the like particularly interesting, because they are man-made with artificial man-made rules and set-ups. The “toys” he finds intriguing are simply a part of nature or life that become interesting when closely examined.

2)  When young he thought he would become a painter/artist — something he was very good at. And even now acknowledges that “In some sense, drawing and pictures are still what I care about most.”

3)  Later, living in Japan, he had “a real epiphany” about language, eventually leading to a decision to learn various languages and become a philologist.

4)  By chance, needing something to read on a train trip, he picked up a biography of Russian physicist Lev Landau. It opened his eyes to science, math, and specifically calculus, which he then decided he must learn.

5)  Landau suggested learning math, not with classes or lectures, but by finding “a book with the largest number of solved exercises and go through them all.” The book Tokieda found was in Russian which he didn’t know, but as a philologist was willing to learn.

6)  Eventually (within months actually) he says he found he “was fairly good at this kind of silly manipulative exercise,” and then proceeded to enroll at Oxford in a two-year undergraduate program in mathematics. He didn’t know English, by the way, but what the heck, just another language to quickly learn!

7)  Soon he realized mathematics was what he wanted to do for a living, and was off to Princeton for a Ph.D. program.

…And, as they say, the rest is history. ;)

(He's currently a professor of mathematics at Stanford.)

I’ll stop here, but the last several paragraphs of the Quanta piece are also great reading (Tokieda’s take on math and his own videos... including making children happy), so be sure to read the whole piece.

…If I had read of Tokieda’s life, as Klarreich reports it here, in a novel, I would’ve thought, ‘what a pile of non-believable fiction this is… there could never be such a character in real life.’ …And yet apparently there is!

One last note: Quanta is recently out with two compendiums of their many superb articles on math, and on the sciences, particularly physics... this Klarreich profile is one more of many they've now done on individual scientists/mathematicians. I suspect that somewhere in the future there may be a collection of these profiles available as well.

[Addendum: I've now noticed that several of these profiles are already included in the math volume, The Prime Number Conspiracy.]

A Li'l Physics, and more, With Eric Weinstein

Polymath (and member of the so-called “Intellectual Dark Web”) Eric Weinstein is a bit of an odd-duck — I often agree with his take on various matters and usually find him interesting (whether in agreement or not), but also occasionally find him infuriating. Anyway, he recently did a long (almost 4 hr.!) stint with Joe Rogan where he attempts at one point (beginning ~41:20 mark) to give Joe and listening audience a primer on fundamental physics that lasts about an hour:

A couple other shorter segments I enjoyed:

A bit of discussion of the ‘craft’ of comedy versus that of physics (starting ~1:36:10):

Followed by a short lesson from Eric on harmonica music (starting ~1:40:20):

There are many other interesting bits in the whole podcast (which touches on a LOT of diverse topics), though I thought the first half more interesting overall than the 2nd half.

Eric, an economist by trade and mathematical physicist by academic degree (who, in his spare time, is working on his own 'theory of everything'), has appeared in many other podcasts/videos as well.

[...by the way, he should NOT be confused with "Eric Weisstein" who runs "Wolfram MathWorld" -- 2 completely different people.]

Chi-i-i-i-i-i-ll Friday *

[*  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]

Book Mentions....

First off, John Golden recently tweeted out this video on math and analogies from Kalid  Azad over at “Better Explained,” that I thought worth passing along (once again it touches on the interplay of math and language as I was musing about in my recent post, and as Jim Propp also broaches in his latest offering:

————————————

(via: http://www.clipartsuggest.com )

Moving on, it’s been another great year in mathy popular books, with a mini-flurry of volumes showing up in the final three months of the year.  Of the books I’ve already read, I recently tweeted out my 5 faves, one of which I’ll soon cite as my 2018 ’book-of-the-year,’ but for now will hold ya all in suspense ;)

When Einstein Met Gödel -- Jim Holt

Math With Bad Drawings -- Ben Orlin

Exact Thinking In Demented Times -- Karl Sigmund

Closing the Gap -- Vicky Neale

Hello World -- Hannah Fry

Meanwhile, Thomas Lin (as editor) is newly-out with “The Prime Number Conspiracy,” a collection of pieces from that outstanding stable of writers at Quanta Magazine.  I haven’t seen it yet, but no doubt it would easily break into my select group above if I had.

And since citing the above, Mircea Pitici’s latest “Best Writing On Mathematics” for 2018 has appeared; delighted to see it materialize in my mailbox... knowing that Mircea has had some topsy-turvy changes in his life this year, I wasn’t sure he’d have the time/inclination to do another edition — but he has and it contains his typical variety of diverse selections (something for everyone). I received the volume just a couple of days after my Nov. 11 posting and so was heartened to see several picks in it dealing with creativity, paradox/puzzles, and beauty in math, as well as Wigner’s “unreasonable effectiveness” notion (all things I’d been thinking about lately).

From Brian Kernighan (and Princeton University Press) comes Millions, Billions, Zillions: Defending Yourself in a World of Too Many Numbers a small, stocking-stuffer-sized book aimed at bestowing basic numeracy to readers.

And am currently reading/enjoying “The Model Thinker” by Scott Page (from Basic Books) — a book for which I'd seen no prior buzz or publicity before its arrival. It’s another in the string of volumes covering big data, modeling, algorithms, probabilities and the like, for a mass audience, but with more textbook-like intros to an array of data/statistical subjects than most previous volumes have offered. Lots of interesting topics. If you’ve been looking for a ‘meatier,’ more mathematical, take in this genre this may be it.

Enough said for now… in another week or two I’ll have my final book wrap-up of the year (including a few additional book mentions) for anyone with math bibliophile friends-or-family on their Holiday list… or, for themselves!

Chi-i-i-i-i-i-ll Friday *

[ *  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]

Sunday Night Ramble…

"If my mental processes are determined wholly by the motion of atoms in my brain, I have no reason to believe that my beliefs are true... and hence I have no reason for supposing my brain to be composed of atoms."--- J.B.S. Haldane, "Possible Worlds" (1927)

“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.”— Eugene V. Wigner

(longish ramble ahead....)

Last Friday was one of those oddly serendipitous days in some ways — and that’s despite the fact that I was stewing over missing Ben Orlin’s night-before presentation at our local University. ARRRRRRRGH!! — had planned for weeks to attend, but for a whole series of reasons didn’t make it. Luckily, someone had posted his hour-talk from a previous stop (which I assume was the same as here), so I went online later and viewed that.

A couple days prior, a mathematician/blogger had sent along something to read for any comments, and parts of it reminded me of a favorite quote I’ve used here previously from provocateur David Berlinski (for those who’ve seen me employ it multiple previous times I beg forgiveness, and indeed apologies for all the long quotes coming below):

"Like any other mathematician, Euclid took a good deal for granted that he never noticed.  In order to say anything at all, we must suppose the world stable enough so that some things stay the same, even as other things change. This idea of general stability is self-referential. In order to express what it says, one must assume what it means. Euclid expressed himself in Greek; I am writing in English. Neither Euclid's Greek nor my English says of itself that it is Greek or English. It is hardly helpful to be told that a book is written in English if one must also be told that written in English is written in English. Whatever the language, its identification is a part of the background. This particular background must necessarily remain in the back, any effort to move it forward leading to an infinite regress, assurances requiring assurances in turn. These examples suggest what is at work in any attempt to describe once and for all the beliefs 'on which all men base their proofs.' It suggests something about the ever-receding landscape of demonstration and so ratifies the fact that even the most impeccable of proofs is an artifact."-- D. Berlinski (from "The King of Infinite Space")

The interplay of language, meaning, abstraction, perception… and, mathematics/science is an ongoing interest of mine. Another quote I’ve used elsewhere is from venerable Bertrand Russell toward the end of a frustrating career (1957) trying to formalize all of mathematics:

"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."  

Again, I think this reflects on the complex interplay of language and human thought — hmmm, all of mathematics as tautologies?… maybe all of meaning is tautological, just substituting one set of human scratchings or sounds for another… is all of knowledge just one gigantic Thesaurus? ;)) No, I don’t believe that, but a lot of “knowledge” does seem illusory or mirage-like and certainly changing. Often meanings, metaphors, analogies and the like simply refer back on themselves within an enclosed bubble.

Later on Friday afternoon I stumbled across this new essay at the Scientific American site, “Proofs and Guarantees” (actually reprinted from “The Mathematical Intelligencer”), which seemed to hit some of the same notes, questioning assumptions. It ends accepting the “fallibility of the initial axioms or other first principles” of math, and thus acknowledging an evolutionary nature to mathematics, as something that is not necessarily fixed over time (just as science is not static and self-corrects over time):

https://www.scientificamerican.com/article/proofs-and-guarantees/

In arguments between math Platonists and non-Platonists it’s often contended that even (philosophical) non-Platonists ARE Platonists-at-heart when it comes to their livelihood… i.e., a pure mathematician must presume math exists out there to be discovered, in order to carry on their daily work. That may overstate the case, but on-the-other-hand one of my favorite volumes is by non-Platonist, retired mathematician William Byers, “How Mathematicians Think.” The subheading to the book title is: “Using Ambiguity, Contradiction, and Paradox to Create Mathematics” and that succinctly sums up what he argues in the volume, that mathematics is “created” out of the very things that most people presume run counter to it.

Interestingly, a second major influence in Byers’ life (besides mathematics) is Zen Buddhism, known for its mystical focus on contradictions. Here is one passage where he touches upon it:

“The second strand in my life was and is a strenuous practice of Zen Buddhism. Zen helped me confront aspects of my life that went beyond the logical and the mathematical. Zen has the reputation for being antilogical, but that is not my experience. My experience is that Zen is not confined to logic; it does not see logic as having the final word. Zen demonstrates that there is a way to work with situations of conflict, situations that are problematic from a normal, rational point of view. The rational, for Zen, is just another point of view. Paradox, in Zen, is used constructively as a way to direct the mind to subverbal levels out of which acts of creativity arise.”

Later in the volume he writes:

“…every human being lives in a bubble. This bubble contains all their perceptions and cognitions. What exists outside the bubble is not knowable. Radical constructivists 'do not make claims about what exists in itself, that is, without an observer or experiencer.' This is a point that I also made earlier when I claimed that there exists no mathematical knowledge that is completely objective. Mathematical knowledge and truth must be considered as a package with both objective and subjective aspects. The belief in ‘objective mathematical knowledge,’ that is, knowledge that is independent of the beings who know it, is itself a belief and therefore nonobjective. There is no knowledge that is independent of knowing. There is no absolute, objective truth.”

And finally, here’s Byers, at length, in another book, “The Blind Spot,” hitting the same theme:

"It is certainly conceivable that the clarity we perceive in the world is something we bring to the world, not something that is there independent of us. The clarity of the natural world is a metaphysical belief that we unconsciously impose on the situation. We consider it to be obvious that the natural world is something exterior of us and independent of our thoughts and sense impressions; we believe in a mind-independent reality. Paradoxically, we do not recognize that the belief in a mind-independent reality is itself mind-dependent. Logically, we cannot work our way free of the bubble we live in, which consists of all of our sense impression and thoughts. The pristine world of clarity, the natural world independent of the observer, is merely a hypothesis that cannot, in principle, ever be verified. To say that the natural world is ambiguous is to highlight this assumption. It is to emphasize that the feeling that there is a natural world 'out there' that is the same for all people at all times, is an assumption that is not self-evident. This is not to embrace a kind of solipsism and to deny the reality of the world. It is to emphasize that the natural world is intimately intertwined with the world of the mind. In consequence, the natural world is a flow just like the inner world. By stabilizing the inner world through language, logic, mathematics, and science, we simultaneously stabilize the outer world. The result of all this is the recognition that the clarity we assume to be a basic feature of the natural world merely masks a deeper ambiguity. One of the functions of mathematics and science is precisely to deny this ambiguity. This is really the motivation behind the science of certainty." 

Anyway, finally coming back around to Dr. Orlin’s talk, which is all about the relationship between mathematics and science, Ben concludes that they share a symbiotic relationship — two quite DIFFERENT activities feeding off one another (as opposed to the more common conception of math being foundational to science). I couldn’t help but think that perhaps that viewpoint might be broadened out to describe the relationship/interplay between language, thought, and math… entities that are separate but very much feed off one another (though many mathematical aspects of language may not even yet be understood/appreciated).

Give Ben's entertaining, thoughtful talk a watch if you’ve not seen it:

David Chalmers famously talks about consciousness as the “hard problem” of philosophy and psychology, left untouched by resolving the other “easy” or soft problems. How do subjectively-felt  experiences arise out of the conglomeration of matter that is our physical brain? Or, in Thomas Nagel’s famous take, what does it feel like to be a bat?

In recent years there has been a lot discussion and competing theories over “consciousness.” Certainly some of Doug Hofstadter’s past writings touch on these matters, as does Joselle sometimes over at her Mathrising blog… and many many more [including, if you haven't seen it, John Horgan's latest 'free' volume on consciousness, where he speaks to several major thinkers on the topic, HERE].

But then another favorite quote of mine (from Emerson Pugh) is, “If the human brain were so simple that we could understand it, we would be so simple that we couldn’t,” implying that  we will never be able to turn the brain on itself to reveal its own deepest secrets. That’s a sort of “Mysterian” viewpoint (and I’m in the mysterian camp with Colin McGinn, Martin Gardner, Roger Penrose, and others, but plenty of folks oppose it, believing the brain can be fully understood, even duplicated).

Anyway, similarly, I think mathematics has a ‘hard’ problem (philosophically-speaking). It is the one made famous by physicist Eugene Wigner (quoted above). How does one account for the exquisite fit of abstract mathematics with the physical world as we interpret it? How indeed! Max Tegmark’s tempting answer is that fundamentally, mathematics is all there is… mathematics IS the core foundational structure/component of the Universe, or of reality; an intriguing notion, but difficult to flesh out, and not one I see a lot of others gravitating toward. Even if you're a full-out Platonist (like Gödel) and believe mathematics exists in the world, independent of humans, the question remains where did it come from and how are we humans able to access it so successfully? Or has some alien civilization, a million years more advanced than us, recognized mathematics as a truly tautological illusion, and moved on to something else more fundamental by now?
Finally, there's been a lot of emphasis in recent years on "beauty" in mathematics, but now even that view is being drawn into question, with a lot of buzz in particular around Sabine Hossenfelder's recent volume, "Lost In Math" (on modern physics), where she argues the myopic focus on beauty simply leads us astray. Is nothing sacred anymore!... first taking away certainty and truth, and now even our wistful love of beauty. ;)
More and more, I find pieces I'm reading connect somehow back to these interests in language, cognition, consciousness, recursion... maybe some day our grandchildren's grandchildren... or, Ben Orlin... will actually make sense of it all! 

Well, I've let the atoms in my brain (assuming they exist) bounce around a bit too much tonight... time to put them to bed, and perhaps go flying -- something I do quite splendidly in my dreams, but, frustratingly, can't seem to do upon awakening (...though, as Chuang Tzu might ponder, perhaps my flying is real and it's this blog-writing that is just a dream....)

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==> Enough of my stream-of-consciousness, IF you want to hear from a real mathematician, I listened to Numberphile's maiden podcast earlier today with Grant Sanderson (of 3Blue1Brown), and it's quite good:
https://www.numberphile.com/podcast/3blue1brown

Chi-i-i-i-i-i-ll Friday *

[ *  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]

More Collaboration and Less Collusion!

via Redbooth.com

January marks the 10-year anniversary of the collaborative “Polymath Project” begun by Fields Medalist Tim Gowers. Let’s celebrate early!

Here is the main page for the Project:

https://polymathprojects.org

…and here is the Tim Gowers posting that originally proposed such a crowdsourcing effort be attempted:

https://gowers.wordpress.com/2009/01/27/is-massively-collaborative-mathematics-possible/

I find it interesting how little widespread digital collaboration has made a dent in other sciences, but been productive in mathematics — one suspects there’s some sort of whole sociological dissertation that could be written here! (although perhaps the pen-and-paper, and computer-assisted nature of mathematics makes it more suited to such endeavors than other more applied sciences)

Anyway, Vicky Neale in her fabulous volume “Closing the Gap,” writes a lot about Polymath’s contributions to the ongoing work on the Twin Primes conjecture. She includes many wonderful quotes from Gowers along the way. Here he is when first proposing a digital collaboration on significant unsolved math problems:

“The ideal solution would be a solution of the problem with no single individual having to think all that hard. The hard thought would be done by a sort of super-mathematician whose brain is distributed amongst bits of the brais of lots of interlinked people. So try to resist the temptation to go away and think about something and come back with carefully polished thoughts: just give quick reactions to what you read and hope that the conversation will develop in good directions.”

Neale then writes, “Gowers was very keen to encourage participants to share their immediate thoughts, rather than working on ideas independently in private, with an emphasis on expressing their immediate thoughts as clearly as possible so that others could build on them.” 

And then she quotes Gowers again:

“When you do research, you are more likely to succeed if you try out lots of stupid ideas. Similarly stupid comments are welcome here. (In the sense to which I am using ’stupid,’ it means something completely different from ‘unintelligent’. It just means not fully thought through.)”

Later on he writes, “…even ‘frivolous’ observations can (and should) be posted on this thread, if there is even a small chance that some other participant may be able to find it helpful for solving the problem.

“Similarly, ‘failed’ attempts at a solution are also worth posting: another participant may be able to salvage the argument, or else the failure can be used as a data point to eliminate some approaches to the problem, and to isolate more promising ones.”

Gowers also foresaw the problem of assigning 'credit’ for such widespread collaborations, and so addressed that ahead of time as well:

“Suppose the experiment actually results in something publishable. Even if only a very small number of people contribute the lion’s share of the ideas, the paper will still be submitted under a collective pseudonym with a link to the entire online discussion.”

Later in the book Neale references how Terry Tao deferred to recent-grad James Maynard to insure he would get the most credit for advances to the Twin Prime conjecture that both had made independently — Tao, absolutely established in his field with no need for acclaim, knowing that credit to Maynard might boost his career greatly — I don’t know that this type of gesture/collegiality would occur very often in any field outside of mathematics!

At this point the Polymath Project has met much success (with fits and starts) and is working on its 16th project. Congratulations to all participants involved, and to Tim Gowers' foresight. As the post-title above suggests, here's to more collaboration in life, and less collusion! ;)

For entertainment purposes only, I’ll end with a completely different take on collaboration in math, in the form of a wonderful short piece of film fiction (on the Goldbach conjecture) that I haven’t re-posted for awhile — for any who have never seen it, “The Calculus of Love”:

Chi-i-i-i-i-i-ll Friday *

[ *  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]

Thesis Psalm

Some time ago while re-reading an old John Barrow volume, “Pi In the Sky” I ran across this delightful bit of parody at the end of a section essentially on the foundational math conflict between formalism and intuitionism:

The Lord is my thesis adviser; I shall not err.

He arranges for me to be published in the

respectable journals; he teaches me how

to use the reductio argument.

He enshores my validity; he leads me by

the classical logic, for the truth's sake.

Yea, though I walk through the valley of

the existence proofs, I will fear no contradiction;

for he edits my work.

The Axiom of Choice and Zorn's Lemma, they comfort me.

He invites a colloquium on classical analysis,

for my participation, in the absence of the constructivists;

He frequently and approvingly abstracts me

in Mathematical Reviews;

my reputation flourishes internationally.

Surely, honors and grants shall follow me

all the days of my career,

and I shall rise in the ranks of the

Department,

to Emeritus.

Amen.

[the verse was not attributed to anyone, so I’m not clear if Barrow penned it himself — a quick internet search didn’t help, but if someone knows the author is other than Barrow please let us know]