## Sunday, September 30, 2018

 [Any resemblance between this graphic & the author of this post is purely coincidental]

Suppose a man starts off with100,000 hairs on his head, but on Wednesday he loses 50 hairs. Is he now “bald.” Of course not. But say you don’t see him for a decade and when you next run into him he has only 50 hairs left on his head. Is he now effectively “bald.” Yes. But when, in time, did he “become” bald. This is just one of many ways of stating the ancient “Sorites” paradox, also known as the “heap paradox” or trying to describe how many grains of sand constitute a “heap” of sand.

This all came to my mind a bit ago when Mike Lawler tweeted out, “It is almost impossible to imagine -> 4:40 per mile pace for the entire marathon,” while referencing a newly-set record for a marathon race. Indeed, I do find it impossible to imagine… yet, it was accomplished:

I’m always amazed at how, over time, so many track-and-field records keep falling. There are of course improvements made in nutrition, training, equipment, etc. but still limits to human performance must, in the end, rule — there is never going to be a 5-second 100-yard dash, nor a 1-hour marathon (at least not as humans are currently constituted). Yet finding that ‘boundary’ which can be asymptotically-approached but not crossed seems a difficult task.
Famously, Roger Bannister barely broke the 4-minute barrier for the mile-run in 1954, following a century of efforts by others. In the 60+ years since, the record has gradually dropped to almost 3:43. How much lower can it go (can it break the 3:40 level)? Read the history/progression for this track event here:

Of course the longer the race, the more likely there is room for improvement: easier to imagine shaving a second off a marathon or 10K run (or even a mile) than the 100-yard dash. How about the pole vault, the long jump, the hammer throw, the shot put?... how easy to keep setting records there?
[One philosophical approach to Sorites is to argue that a definite boundary exists, but that it is unknowable. Perhaps a similar take exists for athletic activities: there are human (physical/physiological) limits, but it's unknowable exactly what they are...?]

...In logic, the law of the excluded middle, is both a staple, but also controversial. Claiming that a statement can’t be both true and false, the law seems simple and innocent… except that in normal discourse meaning and language are rarely so binary. Is it true that John is tall, or smart, or fast, or…. Obviously, it depends on how you define “tall…etc.”, but moreover no definition will likely neatly fit precisely all cases (especially since you also enter into issues over measurement, precision, and context). As applied in math and logic the law is somewhat more clean, but still controversial, and Sorites, with its boundary-ambiguity, gives some indication of its ongoing murkiness.
Loosely, this all also reminds me a bit of mathematical “surreal numbers” and “Dedekind cuts” where it is the ‘boundary' or middle ground that again becomes all-important. Like many ancient paradoxes, the Sorites paradox has a lot of depth.
"Fuzzy logic" and other multi-valued logics (which include 3 or more truth-values) are one alternative to the classical logic of two truth-values.
...In the meantime, there are enough variables at work in running a marathon that record-breaking can probably go on for quite awhile!

## Friday, September 28, 2018

### Chi-i-i-i-i-i-ll Friday.... *

WHAT a week! As Seinfeld's Frank Costanza would say (...or scream):  "Serenity Now... SERENITY NOW!" ;)

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

## Sunday, September 23, 2018

### Some Bits Crossing My Mind This Month

A miscellany today...:

1)  FIRST, in case you've been living under a rock... on the planet Zorka... in Galaxy 134-18B this last week and don't know, TOMORROW (Monday) Michael Atiyah is giving a 45-min. talk entitled simply, "The Riemann Hypothesis" at the Heidelberg Laureate Forum, claiming "a simple proof using a radically new approach." I have no idea how serious of a "proof" this is (other than Atiyah being a serious mathematician, but still hard to take this at face-value, given how many radically new, simple approaches have already been tried). Either way, math cyberspace should be abuzz tomorrow with commentary following the presentation:

I believe the talk will be live-streamed and recorded at the HLF YouTube channel here:
Also, a couple of the Aperiodical bloggers will be in attendance and reporting on the meeting (I assume they'll check in after the dust settles, but maybe they'll do some live-blogging or tweeting  as well? -- surely there will be some live-tweeting from #HLF18).

==> ADDENDUM 9pm. 9/23... the proof, by contradiction, has now been posted here (h/t to @sigfpe on Twitter):

ADDENDUM II 9/24:  one of the live-tweeted threads from Atiyah's talk (now over) is here:

...needless to say, a lot of skepticism being expressed across the Web by those who understand the math/logic; no doubt there will be a lot more commentary today, and even if negative, much food-for-thought may still emerge from this.
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2)  Speaking (loosely) of proofs... logician George Boolos would’ve been 78 this month… had he not died at the relatively young age of 55 in 1996. For any relative newbies, one of my favorite mathy pages on the Web is his famous, delightful single page explaining Gödel’s second incompleteness theorem “in words of one syllable” (worth reading for fun at least once every year!):

It can probably even make a nice introduction to Gödel for younger folks.
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3)  Curses, curses, curses to Jordan Ellenberg who has gotten me regularly reading Martin Shkreli’s prison-composed blog, ever since Jordan cited it in a tweet.  I first wrote about it back here:

And in his 9/6/18 entry, after seeking help with some math he was working on, Shkreli ended by writing:

Thank you to all the professors, postdocs and other math professionals who have reached out to help me. It has been great to communicate with you all. Bear with me as I order my thoughts and respond in the limited way I can.”

I don’t know if this is bluster, bluff, or actuality, but if it is for real, I’d sure be curious to hear about what substantive math any “math professionals” have taken up with Martin, if you’d care to share? Ought to be some sort of interesting backstory there.

[...Also, Martin regularly recommends Bio-Pharm stocks to buy or avoid (or short), and even though I don't dabble in bio-pharm stocks myself I'd be curious if anyone else has found his judgments useful/profitable.]
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4)  For those interested in cognition, science writer John Horgan has a new Web-accessible volume out on mind-body problems. I’ve enjoyed John’s writing in the past, but also tire a bit of this topic that seems forever shrouded in sound and fury, without much ever resolved. As John says, the book offers my subjective takes on my subjects’ subjective takes on subjectivity.” So I wasn’t expecting too much from his latest, but in fact enjoyed it immensely, partly because of the portraits it paints of specific diverse, fascinating thinkers; their foibles and makeup, in addition to their academic or cerebral selves, while delving into their thoughts on mind/body issues. You can read the whole volume here:

…or you can download it from the Web for a small price.
There are probably many Douglas Hofstadter fans out there, so as one sample chapter, I recommend Chapter Two which is with Dr. Hofstadter (p.s… one small side-note that I learned here, and didn’t even realize before, is that David Chalmers did his PhD. under Hofstadter):

Speaking of books, Scott Alexander (just a bit behind the times) offers a long review of Nassim Taleb’s “The Black Swan” here (followed by 250+ comments):
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5)  For those with the chops to follow it, Steve Strogatz recently passed along this history of the Langlands Program:

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6)  And I'll close out with this gem that surfaced on my Twitter feed yesterday:

## Friday, September 21, 2018

### Chi-i-i-i-i-i-ll Friday…

For some variety, awhile back Evelyn Lamb introduced a feature to her Twitter stream of sharing a musical entry once each week. I kinda like the idea (and hey, there's a lot of math within music!), so will experiment with posting a music video (usually instrumental), without commentary, here at the blog every Friday.
In my case it’s not for variety, nor for your edification, BUT, for my sanity! ;)))
By Friday, the current White House usually has me a tad bonkers, and a little meditative sojourn seems to be in order heading into the weekend. So we'll call it “Chi-i-i-i-i-i-ll Friday…” -- feel free to skip over these Friday interludes if not to your taste, just me here chillin’ out.
Have a pleasant weekend everyone:

## Monday, September 17, 2018

### The All-star Linkfest is Here...

Despite a few glitchy matters and some selections sent in that weren’t usable for various reasons, the “All-star linkfest” I sought is finally ready for prime-time. In retrospect, it seems like a near-impossible task, to pick out just 1 or 2 favorite math postings from a Web inundated with great mathematics, as the variety here will indicate:

Originally I kicked the project off citing a 2015 Lior Pachter piece on K-12 math education, with many great suggestions for math problems young people can work on:

…another long education piece I always think of in conjunction with Pachter’s piece (even though they are quite different) is this 2012 one from Fields Medalist Timothy Gowers on teaching math to non-mathematicians (with a couple hundred comments as well):

So much for me though, here are YOUR picks (in no special order):

Steven Strogatz surprised me a bit when he wrote that his first thought was this link to a mathematical fiction site (which is definitely useful for folks who like to link their love of fiction with their love of math — 100s of selections):

…but then as a more mathy choice (that he noted “everyone interested in math should read) Dr. Strogatz went with this classic from his former fellow Cornellian(?) William Thurston, “On Proof and Progress In Mathematics”:

No surprise that someone would pick Steven Strogatz himself for great postings, and Patrick Honner cited Dr. Strogatz's fantastic NY Times' series that started here (and led to an eventual book based on the series):
https://opinionator.blogs.nytimes.com/2010/01/31/from-fish-to-infinity/

I would’ve been shocked if no one had chosen something from Grant Sanderson’s incredible 3Blue1Brown YouTube site, and I wasn’t disappointed. Sol Lederman (who it was great to hear from), formerly proprietor of the immensely popular “Wild About Math” blog, picked out Grant’s video on Euler’s Formula & group theory:

And Benjamin Leis also opted for the fabulous 3Blue1Brown, singling out this one on Pythagorean triples:

Mathematician/computer-scientist/author Rudy Rucker sent in this highly graphic selection on the Mandelbulb, a 3-D version of the better known Mandelbrot Set:

Colin Beveridge went with a StackExchange discussion of a Gaussian proof:

[…this made me think that another possible interesting “linkfest” might be to have readers send in their all-time favorite math questions/discussions/debates from forums like StackExchange, MathOverflow, Quora, Reddit, etc. I don’t read any of these sites regularly myself, but know there have been some great postings on occasion there.]

p.s.... any mention of Gauss can't help but also make me think of this favorite old humor site on "Gauss Facts" (too funny):
http://www.gaussfacts.com

Of course none of us will forget the wonderful life-work Alexander Bogomolny left us with his Cut-the-knot site and Jim Wilder pointed to two problem-selections from there:

Jim Propp couldn’t contain himself and sent in the most links, six (including one from Evelyn Lamb and one from Ben Orlin), and because I like Jim so much I almost let him get away with it… am passing along 5 of his diverse picks here (the first four have a lot to say about mathematics, while the last one involves doing mathematics):

From Tim Chartier came this numerical math trick (requiring flash):

Meanwhile, leave it to Ben Orlin to send in cartoon work (not his own), on math myths, for readers to appreciate:

Statistician Adam Kucharski passed along this interesting one on random numbers and casinos:

Also related to randomness, Don McDonald sent in this recent Noson Yanofsky piece:

An entry I particularly liked came from great math popularizer Richard Elwes with this longish piece on math foundations:

One individual wished to remain anonymous (not sure why) and sent in this somewhat classic Terry Tao piece (that’s readable by a general audience) on rigor in mathematics:

Colm Mulcahy went with Tyler Vigen’s spoofy, always-good-for-a-chuckle ‘Spurious Correlations’ website, illustrating, believe-it-or-not, 'correlation is not causation' ;)

Another graphic site (Tumbler) came from Jo Morgan:

Meanwhile, James Tanton and Edmund Harriss sent along education-related websites:
James:

Edmund

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That's it! It's been fun for me to collect these and offer up some sense of the great variety of items that mathematicians enjoy/recommend within their own field.

## Sunday, September 9, 2018

### A Few Things Readers Have NOT Submitted Among ‘Greatest Hits’ Posts

When I asked folks to submit some of their favorite math postings of all time I had a definite idea what sort of things might show up.  I’ve been a li’l surprised by the sheer range of pieces people have suggested, as well as by some of the things that have been left out so far (…but please, please keep sending in your picks for now). [email: sheckyr at gmail..... ]

Anyway, will take this opportunity to mention a few varied bits that popped into my mind thinking about this, but that haven’t thus far been contributed by others:

Certainly Quanta Magazine can’t be ignored when it comes to memorable math postings (though no one has mentioned it yet). Just as one example from their great stable of writers, there's this 2016 Erica Klarreich piece on prime number digits:

David Mumford has written a number of great posts over the years at his blog. Here’s one cross-field example:

RadioLab,” one of the best long-running podcasts around, has done several episodes related to math. “Stochasticity” was a good one:

So much joyous math work from Numberphile over the years. Here was the incomparable Tadashi Tokieda with an early piece on “freaky dot patterns”:

As a Keith Devlin groupie, I can’t let him go missing. From his long-running “Devlin’s Angle” blog this is among my many favorites:

…and I also relished his appearance on Krista Tippett’s “On Being” broadcast several years back:

I’ll stop here… and almost hate even mentioning these, because there’s automatically so many great pieces I’m leaving out. No doubt if I searched more, I could find pieces by Barry Mazur, Doug Hofstadter, Jim Propp, Brian Hayes, Natalie Wolchover, Raymond Smullyan, maybe even L.E.J. Brouwer, and others I'd want to pass along, but those just represent some of my tastes/biases; yours will differ.
Anyway, in another week or so will hopefully have organized/formatted the picks readers did send in — but keep on contributing in the meantime. I never tire of seeing the math that others find interesting, inspiring, or just entertaining.

## Monday, September 3, 2018

### Two To Take Note Of

FIRST, great initial entries coming in to prior post. Please continue to send along your favorite postings for inclusion....
On to today's post:
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While scanning my bookshelf recently I noted that two authors I’ve especially enjoyed, are less referenced (in my experience) than several others, so thought I’d toss a little light in their direction:

1)  One is British mathematician (and retired teacher) David Wells, who I suspect is better known ‘across the pond’ than here in the U.S. I have three of his books and love them all!:

The Penguin Book of Curious and Interesting Mathematics
Prime Numbers: The Most Mysterious Figures in Math
Games and Mathematics: Subtle Connections

Highly recommend all of these, or, sight unseen, any of his other volumes:

He has a Martin Gardner-like knack for drawing attention to interesting mathematical content/ideas.
Unfortunately, I couldn’t find web links to many of his articles, but here is one that is often cited, having to do with beautiful equations (from “The Mathematical Intelligencer” — also, many of his books are accessible on Google Books):
[If anyone knows of free links to others of his popular math essays, please pass them along in the comments.]

Here also, a transcribed interview with him concerning undergraduate math education:

Anyway, if you enjoy popular math writing and aren’t familiar with Wells’ work I suggest looking him up!
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2)  The second person I want to cite here is Bart Kosko, a bit of a polymath with bachelors degrees in Philosophy and in Economics from the University of Southern California, a masters degree in Applied Mathematics from the University of California at San Diego, and Ph.D. degree in Electrical Engineering from the University of California at Irvine. He also has a J.D. degree from Concord Law School, and is a licensed California attorney.
And he’s been previously called “a celebrated maverick in the world of science.”
A partial list of his essays here (including many for John Brockman's "Edge" organization):

Kosko is especially well-known for his promotion of "fuzzy logic" as opposed to the conventional Aristotelian or binary logic we are accustomed to. His most well-known book is “Fuzzy Thinking: The New Science of Fuzzy Logic,” which you can read about here:
[You can also find the volume on Google Books.]
And he’s also author of “Nanotime,” “Heaven In a Chip: Fuzzy Visions of Society,” and “Noise.”

Short YouTube video of him here:

And finally here is audio of him on the late night talk radio show “Coast To Coast” talking about defense, AI, technology, and other matters:

Anyway, two very different folks and writers, both of whom I think deserving of attention.