Thursday, December 14, 2017

Need Some Things To Read...?

I'll have a Friday potpourri tomorrow morning at MathTango, but just to make sure you have enough to read/hear ;) below are some lists I've come across recently that may be of interest:

a)  Fine list of online math resources (including free online math-related magazines) that was tweeted out this week:

b)  Sean Carroll tweeted out this long list of 300+ philosophy interviews, at least a few of which may pertain to math or mathematical thinking:

c)  And a compendium here of Marcus du Sautoy’s many entertaining and varied podcasts (I don't think these were always readily available in U.S.):

Monday, December 11, 2017

There once was a...

A dozen, a gross, and a score
Plus three times the square root of four
Divided by seven
Plus five times eleven
Is nine squared and not a bit more. **

For this supposedly celebratory time of year it’s been a pretty depressing month of news and politics (…on top of the entire last year of infamy), so just turning today to some light-hearted links with math limericks (I’ve referenced these before, but there are some good ones worth re-visiting, and the one above is classic):

…and these from Ben Orlin, who would probably be the head-writer for Saturday Night Live of Math... if there was such a thing:

** This limerick comes in some variations and is attributed to various authors, so I'm not sure of it's specific origin or composer.

Sunday, December 10, 2017

Math and Contemplation

Sunday reflection:

"[Bertrand] Russell's analytic approach had its origins in numbers; mathematics was his first love. In his autobiography he recalled his miserable adolescence and a footpath down which he would wander on England's south coast. 'I used to go there alone to watch the sunset and contemplate suicide. I did not, however, commit suicide, because I wished to know more about mathematics.'"

-- From "Wittgenstein's Poker" by David Edmonds and John Eidenow

Friday, December 8, 2017

“…it took AlphaZero only four hours to ‘learn’ chess. Sorry humans, you had a good run”

Again this week there will be no “Potpourri” over at MathTango (hope to return to it NEXT Friday), but if you want a nice varied collection of readings, the newest "Math Teachers At Play" carnival is out HERE. And I will pass along just a couple of things from last two days...

This story certainly would have been included in my potpourri had there been one:
Following in the footsteps of AlphaGo, that mastered the complex board game “Go,” the folks at Google have now come up with AlphaZero which took 4 hours to teach itself the game of chess from scratch, and then “obliterated the highest-rated chess engine” in existence (Stockfish). In a 100-game match, it won 28 games (25 times when it was playing white), had 72 draws, and ZERO losses. Remarkable! Fascinating story:

A bit scary to contemplate our possible Google overlords… though far less scary than the current deranged overlord in the White House.

Meanwhile Andrew Gelman reports that people have lost confidence in so many of our institutions (I can't help but think this loss of confidence coincides with the ever-rising income gap that leaves large numbers feeling bitter and resentful):

Wednesday, December 6, 2017

General Semantics...

Bit of an offshoot today...
I recently mentioned “General Semantics” in a post over at MathTango, forgetting that a lot of folks have no familiarity with the movement that faded from its limited popularity decades ago. A few readers emailed me with either questions or comments about it, so I thought I’d try to find an introductory video that wasn’t too long or too old-and-grainy expounding on General Semantics. I haven’t found one that in a brief, entertaining way covers the contents of GS, but this 6-minute one does offer a non-detailed overview:

...and this 10-minute effort gets into a few more specifics, while relating some ideas of Alfred Korzybski (the founder of General Semantics) to mathematical thinking and language:

[...I will add though that, unlike the presenter above (Corey Anton), I DON'T recommend that people try to read Korzybski's difficult-to-digest old anti-Aristotelian tome "Science and Sanity," but instead go with a more popularized, but still old, version of GS (if you can even find one) from S.I. Hayakawa, Stuart Chase, Wendell Johnson, or Irving Lee.]

[...on a different side-note, for any readers who may contemplate emailing me Martin Gardner's old critical commentary on General Semantics, yes, I'm well familiar with it, and consider it one of the worst errors/judgments he made in his long writing/skeptical career (with that said, he was primarily critical of Korzybski, less so of the popularizers who followed). I once mentioned to a Gardner specialist my upset with Martin's criticism of GS, and he in turn noted that almost every fan (like myself) of Gardner, has some one thing on which they deeply disagree with Gardner... problem is, it's a different 'thing' for each one of those fans. ;) ]

Monday, December 4, 2017

Some Twitter Trivia... plus Futility Closet

I only follow 250 accounts on my Twitter feed… a relatively small, but manageable number: Dunbar’s number +100  ;) 
Recently, having too much free time on-hand, I noticed that of those 250 I follow, 122 of them follow Evelyn Lamb — Twitter gives statistics on how many of the people that you follow, also follow any other given account you follow. I was actually surprised that more of the 250 didn’t follow Dr. Lamb (might’ve guessed it would be closer to 200), so then decided to check out how other Twitterers stacked up — who among people I follow are most followed by others that I follow (...follow me? ;) — a tree or group diagram of all the myriad connections might actually be interesting, but then I don’t have THAT much free time on-hand!

Anyway, the top 10 individuals (I’ve removed groups, magazines, news outlets, etc.), with the numbers of accounts (out of the 250 I follow), that follow them are below; in parentheses I’ve added the TOTAL number of followers (for some highly-variable further context) each of these accounts had at the time I looked:

@StevenStrogatz       155          (36,100 total followers)
@EvelynJLamb          122            (7,433)
@DivByZero               121          (10,500)
(Dave Richeson)
@JohnAllenPaulos     118          (26,900)
@MrHonner                116          (11,100)
@AlexBellos               114          (24,100)
@RepublicOfMath      113          (69,400)
(Gary Davis)
@CutTheKnotMath     109          (15,900)
(Alexander Bogomolny)
@JamesTanton           109          (11,600)
@ProfKeithDevlin        106          (12,400)

All of which doesn't mean much; just idly interesting to me to see the interaction between people I follow — i.e., who do people I follow, follow. Among individuals, only Dr. Strogatz is followed by over half (62%) of those I follow. (On-the-other-hand, he’s followed, overall, by 5 times more accounts than Dr. Lamb who edges into second place in my group, despite having the lowest total number of followers.)

Anyway, other folks might find it interesting to keep an eye on their “followers you know” numbers.


...And for something more fascinating and different, this morning’s new Futility Closet podcast tells the incredible story (I’d never heard before) of Marvin Hewitt, a 20th century American imposter who taught physics, math, and engineering in various academic venues:

Sunday, December 3, 2017

Mathematical Technique

Sunday reflection from G.H. Hardy:
“Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.” 

Friday, December 1, 2017

A Staggering Concept...

Again, won't be doing a Friday potpourri (over at MathTango this week; not sure if I'll get back to them before Xmas), but will pass along this tweet from the week that I enjoyed, along with it's lengthy comment thread:

Wednesday, November 29, 2017

Andrew Wiles in Conversation

I haven't had a chance to view this myself, but figure a conversation with Andrew Wiles must be good enough to pass along immediately:

Meanwhile, over at MathTango I'm in a bit of a Bah Humbuggery mood today:

Sunday, November 26, 2017

To Toss or Not to Toss

The paradox known as Buridan’s Bridge:
Bridge Gatekeeper to the approaching passerby: “If the next sentence you utter is true I will permit you to cross. But if you speak falsely, I will throw you into the water."
Passerby: “You shall throw me into the water.”

Saturday, November 25, 2017

Sean Carroll Expounds...

Very nice, recent 1-hour summary of modern physics thought (at least one physicist's interpretation) from Sean Carroll (if you've not heard Sean cover this material before this is a great presentation):

Sunday, November 19, 2017

A Little Self-referential Humor

for Sunday reflection... just an old, classic joke:
“There are three kinds of people in the world: those who are good at mathematics and those who aren’t.”

[...also, a new post over at MathTango today in honor of Thanksgiving approaching.]

Thursday, November 16, 2017

Sunday, November 12, 2017

Playing Games

From the inimitable John Conway:
“You get surreal numbers by playing games. I used to feel guilty in Cambridge that I spent all day playing games, while I was supposed to be doing mathematics. Then, when I discovered surreal numbers, I realized that playing games IS mathematics.”

Thursday, November 9, 2017

Happy Birthday Carl

It’s Carl Sagan’s birthday today; an apt time to re-read some of his words and view classic video:
    “If you wish to make an apple pie from scratch, you must first invent the universe.”
“Extinction is the rule. Survival is the exception.”
    “The nitrogen in our DNA, the calcium in our teeth, the iron in our blood, the carbon in     
    our apple pies were made in the interiors of collapsing stars. We are made of starstuff.” 
“We live in a society exquisitely dependent on science and technology, in which hardly
anyone knows anything about science and technology.”
    “The dangers of not thinking clearly are much greater now than ever before. It's not  
    that there's something new in our way of thinking -- it's that credulous and confused  
    thinking can be much more lethal in ways it was never before.”
Science is more than a body of knowledge; it is a way of thinking. I have a foreboding of an America in my children's or grandchildren's time -- when the United States is a service and information economy; when nearly all the manufacturing industries have slipped away to other countries; when awesome technological powers are in the hands of a very few, and no one representing the public interest can even grasp the issues; when the people have lost the ability to set their own agendas or knowledgeably question those in authority; when, clutching our crystals and nervously consulting our horoscopes, our critical faculties in decline, unable to distinguish between what feels good and what's true, we slide, almost without noticing, back into superstition and darkness. The dumbing down of America is most evident in the slow decay of substantive content in the enormously influential media, the 30-second sound bites (now down to 10 seconds or less), the lowest common denominator programming, credulous presentations on pseudoscience and superstition, but especially a kind of celebration of ignorance.
In memory...

Wednesday, November 8, 2017

Perception of Streaks

H/T to Dan Goldstein (on Twitter) for pointing out an Interesting plotting of (likely and VERY likely) “streak” probabilities:

The author looks mostly at "coin-tossing"-like scenarios, but notes such analysis potentially relates back to other theoretical discussions (such as "hot hand" observations). One can imagine a lot of other ways to play with similar data (and the author presents R code for doing such).

Tuesday, November 7, 2017

Twitter Quirkiness

Just a quirky episode from the other day... below is a verbatim copy of a tweet that someone posted (I've taken their name off) which I read quickly, thought was kind of fun, and re-tweeted to my followers, giving no more consideration.  If you've not already seen it, take a quick look, and then scroll further below for the little follow-up:


the non-commutativity of custard, according to . i'm going to have to try this. so weirddd  

A smidgen interesting I thought, but did you catch the error? (and no, I don't mean the spelling of the word "weird" which was clearly deliberate)... because it slid right by me, despite being rather obvious! And it interests me because of the way language and words function cognitively (psycholinguistics, including speech processing, was my primary focus decades ago).
So, in case, like me, you didn't notice it, the graphic example shown above is one of associativity, and is clearly labelled as such. Yet the poster's description calls it an example of "non-commutativity" (which of course it is NOT).
I simply found interesting the tendency (on my part anyway) to be so drawn to the graphic as to read right through the clear, posted words and not have them accurately register -- you think you know what they say, but the precise meaning may be translated in one's mind quite otherwise, overridden by expectation or the simple speed of processing (part of it too I think is just the mental 'appeal to authority,' resulting from invoking Eugenia Cheng's name, even though she is NOT the one who makes the error). It's almost like an optical illusion that you "see" one way, but is really another -- here, a word takes on an illusory effect.
Anyway, this may be of interest to no one else; I'm just eternally intrigued/concerned by how language plays with our minds (and often in far more deleterious ways!).
p.s.: Brian Hayes was the one who first pointed out the error above to me, and the author of the post apparently heard from several other folks as well; so to many, the mistake no doubt jumped right out.

In any event, I'm in the mood for some custard. :)

Monday, November 6, 2017

Sunday, November 5, 2017

The Dismal Science

Sunday reflection:

To put it bluntly, the discipline of economics has yet to get over its childish passion for mathematics and for purely theoretical and often highly ideological speculation, at the expense of historical research and collaboration with the other social sciences. 
...This obsession with mathematics is an easy way of acquiring the appearance of scientificity without having to answer the far more complex questions posed by the world we live in

-- Thomas Piketty

Sunday, October 29, 2017

Keeping It Short

One of the shortest Sunday reflections ever, courtesy of Paul Erdös:

"If numbers aren't beautiful, I don't know what is."

Wednesday, October 25, 2017

The Universe... Even or Odd

I should be writing a blurb about the various 2017 mathy books that have passed my way the last few months, but instead the volume I just finished reading is an older classic, Roy Sorensen’s 2003 A Brief History of the Paradox. Toward the end comes a ‘paradox’ (perhaps known by some as 'the odd universe' paradox) I was unfamiliar with and frankly don’t quite understand, though it doesn't appear too difficult. Am passing it along because some of you may find it interesting (…or be able to explain it better to me!).
Verbatim from the book (I’ve bolded a few bits that I especially have difficulty following):
“Meanwhile, Nelson Goodman kept sharpening the knife of nominalism. In 1951 he published The Structure of Appearances. This book contains a logic of parts and wholes. Goodman denies that there are sets. Instead, there are fusions built up from smaller things. Unlike a set, a fusion has a position in space and time. You can touch a fusion. I’m a fusion. So are you. Goodman’s ‘calculus of individuals’ says that there are only finitely many atomic individuals and that any combination of atoms is an individual. Objects do not need to have all their parts connected, for instance, Alaska and Hawaii are parts of the United States of America. Goodman does not let human intuition dictate what counts as an object; he also thinks that there is the fusion of his ear and the moonIn a seminar Goodman taught at the University of Pennsylvania around 1965, John Robison pointed out that The Structure of Appearances implies an answer to ‘Is the number of individuals in the universe odd or even?’ Since there are only finitely many atoms and each individual is identical to a combination of atoms, there are exactly as many individuals as there are combinations of atoms. If there are n atoms, there are 2n - 1 combinations of individuals. No matter which number we choose for n, 2n - 1 is an odd number. Therefore, the number of individuals in the universe is odd! The exclamation point is not for the oddness per se. Aside from those who think the universe is infinite, people agree that the universe contains either an odd number of individuals or an even number of individuals. What they find absurd is that there could be a proof that the number of individuals is odd. ‘Is the number of individuals in the universe odd or even?’ illustrates the possibility of one good answer being too many. Our expectation is that this question is unanswerable. The lone good answer confounds beliefs about what arguments can accomplish.”
Anyway, seems like an interesting thought exercise to play with.
(If you can explain it any more lucidly in the comments feel free to give it a go. The primary part I'm unclear about is, in the 2nd part that I've bolded, why does the 2nd sentence necessarily follow from the prior sentence?)

Monday, October 23, 2017

Gauss... the Rodney Dangerfield of Mathematics?

A John Golden tweet this weekend reminded me that I should check in on GaussFacts every now-and-then (…like when Trumpsky makes me want to slit my wrists, or, even more assuredly, his) for a few guffaws.
It's one of my favorite ongoing math-humor bits, but truly Gauss gets a lot more respect than Rodney ever did:

Sunday, October 22, 2017

The Minds of Brilliant Mathematicians

A Sunday reflection from Kaja Perina in a piece on Alexander Grothendieck:
“The minds of brilliant mathematicians are of perennial fascination. But in the onrushing era of synthetic neurobiology and genomic reconfiguration, the possibility that genius and mental illness are intertwined takes on monumental significance. If scientists are eventually able to alter living brains or edit human embryos with an eye to mitigating conditions such as autism and schizophrenia, do we risk excising brilliant outliers from the gene pool? Isaac Newton, John Nash, and Alexander Grothendieck are low-frequency, high-impact minds; they advanced civilization in the domain on which they trained their high beams. It is worth turning the high beams of scientific inquiry on those same unusual minds.”

Friday, October 20, 2017

Foundations, Randomness, Free Will, the Aaronson Oracle

To end the week, another wonderful new episode from PBS's Infinite Series, this time on the foundations of mathematics:

Also, sort of cool… in the commentary after the episode the show host, Kelsey Houston-Edwards, briefly mentions the Aaronson Oracle, which I was unfamiliar with, and which interactively demonstrates the difficulty of 'randomness.' It's a program from Scott that predicts a choice (generally succeeding well-over half the time, with two possible choices) that you will make in attempting to randomly press two computer keys:
Read a little about it here:
...and then try it out here:

Thursday, October 19, 2017

Crazy... Like A (Cunning, if Unstable, Mentally-Ill) Fox

I suspect one reason Dotard Trump was so willing to let Steve Bannon, Seb Gorka, Reince Priebus (and others) depart his Administration is because of the amount they were leaking, for their own benefit, to the press.
The crazy stories that eke out of this White House, may of course indeed reflect craziness within the West Wing, but more and more they look orchestrated and planted selectively just to see which ones end up reaching manipulated media outlets, thus signifying who is doing the ongoing leaking (which is NOT to say that there isn’t still much real craziness within the Oval Office)… all of which was hinted at by this puzzle post I did just a couple of months back:

If, alternatively, there is no method to the madness of this White House, then we are left with just pure unstable, narcissistic sociopathy in the midst of enablers. Oy.

Tuesday, October 17, 2017

Fawnzie Nguyen…

Yesterday afternoon I noticed my Twitter feed popping up with accolades for Fawn Nguyen’s keynote address to the Northwest Mathematics Conference. Unfortunately, it wasn’t recorded so those of us not in attendance have missed out.
I don’t have anything special in the works for posts this week, so it seems like a good time to refer any readers who have never read it to my 2014 interview with Fawn, which has always been one of my favorite interviews here (especially since at the time I knew relatively little about her). The same insightful, funny, inspiring spirit she exhibits on stage (and in writing and in the classroom and on Twitter) comes through I think in her answers here:

Also, in the interview I asked her about her favorite own postings of all time and she referenced just one (from 2012), which if you’ve not read before, you must:

Worth noting too that Ms. Nguyen has a book on teaching math coming out in the future.

p.s.… Twitter posters yesterday kept referring to the “last line” of Fawn’s keynote (apparently very memorable and powerful!), but I don’t know what it was??? :-(
So hey, can someone tell us what that line was with maybe enough context to get a full sense of it (or will it not carry as much weight without hearing the talk preceding?). Or, maybe Fawn or someone else can post a transcript of her keynote. Puhhh-leeeeze!

Sunday, October 15, 2017

The Darkness of Axioms

A little Sunday reflection from Bernhard Riemann:
“It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible. From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor the philosophers who have laboured upon it.” 

Friday, October 13, 2017


End of another crappy week for America, democracy being dismantled day-by-day; will just re-reference a previous post from 5+ months ago…:

Sunday, October 8, 2017

The Art of Mathematics

Sunday thought:
“Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness.” —E. T. Bell

Wednesday, October 4, 2017

American Tune...

Had no idea that Eva Cassidy had ever recorded Paul Simon's "American Tune"... until today:

Sunday, October 1, 2017

Of Birds and Frogs

A well-known passage from Freeman Dyson today:
"Some mathematicians are birds, others are frogs. Birds fly high in the air and survey broad vistas of mathematics out to the far horizon. They delight in concepts that unify our thinking... Frogs live in the mud below and see only the flowers that grow nearby. They delight in the details of particular objects, and they solve problems one at a time. I happen to be a frog, but many of my best friends are birds."

Wednesday, September 27, 2017

Fuzzy Thinking

Lofti Zadeh, the “father of fuzzy logic,” died earlier this month -- yes, "fuzzy logic" had a more technical meaning long before the current White House place-holder took office ;)
One of Zadeh's students was Bart Kosko, a scientist/engineer/author whose writings I’ve enjoyed previously (if you’re not familiar with ‘fuzzy logic,’ his older book, "Fuzzy Thinking: The New Science of Fuzzy Logic" is an easy introduction). 
I recommended to folks on Twitter a couple days ago to listen to him on late-night “Coast To Coast” talk radio where he was appearing (a show I don’t often recommend!). Then, I myself missed most of that program, but to recompense I looked him up on YouTube to see what might be available, and found this 10-minute piece easily suitable for a lay audience:

Sunday, September 24, 2017

Vintage Erdös

In 1953, Paul Erdös was invited to spend a year teaching at the University of Notre Dame. In his volume, “My Brain Is Open” Bruce Schechter relays the following story:
“Erdös was an avowed atheist, and his friends at Notre Dame enjoyed teasing him about his working at a Roman Catholic university. ‘He said in all seriousness that he liked being there very much,’ Melvin Henriksen, a colleague from those days, recalled, ‘and especially enjoyed discussions with the [priests].’ Only one thing bothered him. ‘There were too many plus signs,’ he irreverently remarked."

Thursday, September 21, 2017

Intrepid Math

Anthony Bonato’s “The Intrepid Mathematician” blog has caught my attention several times this week:

1) Interesting post on some neuroscience of math versus language:

2)  He’s  posted two interviews this week with wonderful mathematicians:
Maria Chudnovsky HERE and
Ken Ono HERE

3)  And today, this news on Ramsey Theory:

4)  I’m not much of a film buff myself, but if you are, you may want to additionally read his post on math and science in the movies here:

Read up!

Wednesday, September 20, 2017

A Sphere In Any Other Dimension Is Still A Sphere

Can spheres be spiky? According to Matt Parker yes they can, once you escape your puny 3-dimensional world:

Sunday, September 17, 2017

Healthy Mathematics

This Sunday reflection from Ian Stewart in the 2nd edition (1992) of “The Problems of Mathematics”:
“Some observers have professed to detect, in the variety and freedom of today’s mathematics, symptoms of decadence and decline. They tell us that mathematics has fragmented into unrelated specialties, has lost its sense of unity, and has no idea where it is going. They speak of a ‘crisis’ in mathematics, as if the whole subject has collectively taken a wrong turning. There is no crisis. Today’s mathematics is healthy, vigorous, unified, and as relevant to the rest of human culture as it ever was… If there appears to be a crisis, it is because the subject has become too large for any single person to grasp… today’s mathematics is not some outlandish aberration: it is a natural continuation of the mathematical mainstream. It is abstract and general, and rigorously logical, not out of perversity, but because this appears to be the only way to get the job done properly. It contains numerous specialties, like most sciences nowadays, because it has flourished and grown. Today’s mathematics has succeeded in solving problems that baffled the greatest minds of past centuries. Its most abstract theories are currently finding new applications to fundamental questions in physics, chemistry, biology, computing, and engineering. Is this decadence and decline? I doubt it.”

Wednesday, September 13, 2017

Measuring Infinities

Fantastic article from Quanta Magazine (Kevin Hartnett) about new findings/proof of the equivalency of two variant infinities — actually findings published a year ago; am amazed it’s just now reaching the wider press (at least I’d not heard about this ’til now!):

Part of what makes the proof interesting (IF I understand matters correctly) is that it didn't require any re-statement of fundamental set theory, but only a bringing together of disparate math models that had not been linked up before. Even if you (like me) don't understand the details of the finding, just recognizing that a 50+ year problem has been resolved is very exciting. The solvers, Maryanthe Malliaris and Saharon Shelah, received the Hausdorff Medal for their work earlier this year.

…In a bit of irony, the above article got tweeted out yesterday on the very anniversary of the death of David Foster Wallace whose book on infinity, “Everything and More,” I've discussed earlier here:

Tuesday, September 12, 2017

Quantitative Literacy

According to a recent study, 36 percent of college students don’t significantly improve in critical thinking during their four-year tenure. 'These students had trouble distinguishing fact from opinion, and cause from correlation,' Goldin explained.
The above words from mathematician/statistician Rebecca Goldin come near the beginning of this new piece in Quanta Magazine:

The title of the piece is “Why Math Is the Best Way To Make Sense of the World.” I fear the title may be the very sort that turns people away from it, or at least many of those who most need to read it — just mention 'math' in some sort of positive light and a lot of the ‘I-was-never-any-good-at-math’ folks will turn away out of disinterest :(
And if college-bound students aren’t gaining critical thinking skills over their 4-year sojourn, what can we expect of the non-college crowd who may have even less opportunity to be exposed to critical-thinking skills?
But critical thinking shouldn’t even begin with college; it should begin back in elementary school with language skills, which are themselves fundamentally entwined in critical thinking. Nonetheless, the above article (and interview with Goldin) is excellent and focused on the societal value of math and science at the university level -- there are several lines in it I’d love to quote, but just go read it for yourself and take to heart this central message: “…if we don’t have the ability to process quantitative information, we can often make decisions that are more based on our beliefs and our fears than based on reality.
Interestingly, this article appears at a time that topics like critical thinking, quantitative reasoning, innumeracy and the like are getting a fair amount of discussion in society, though I’m not confident that we’re even close to dispensing such skills to the population-at-large, nor to upcoming generations. In fact I fear quite the opposite; it may be too little too late in a digital world of speed, simplification, and reality-manipulation... hope I'm wrong, but the Machiavellians who plotted the path of our current Oval Office interloper knew all-too-well that appeals to base instincts could overcome appeals to critical thought. :(

Wednesday, September 6, 2017

ABC... A Baez Commentary

ICYMI, the more hardcore among you may want to see John Baez's recent commentary (and the comments that follow) on Mochizuki's "proof" of the ABC conjecture:

Mathematician Go Yamashita has written a 294-page "summary" of Mochizuki's 500-page inscrutable(?) proof... if that's any encouragement to you ;)

Here's a few lines of the summary as quoted by Baez:
"By combining a relative anabelian result (relative Grothendieck Conjecture over sub-p-adic felds (Theorem B.1)) and "hidden endomorphism" diagram (EllCusp) (resp. "hidden endomorphism" diagram (BelyiCusp)), we show absolute anabelian results: the elliptic cuspidalisation (Theorem 3.7) (resp. Belyi cuspidalisation (Theorem 3.8)). By using Belyi cuspidalisations, we obtain an absolute mono-anabelian reconstruction of the NF-portion of the base field and the function field (resp. the base field) of hyperbolic curves of strictly Belyi type over sub-p-adic fields (Theorem 3.17) (resp. over mixed characteristic local fields (Corollary 3.19))."
...Have at it!

Sunday, September 3, 2017

Springtime For...

We’ve ended another wrenching week with this current unfit, anti-science, authoritarian, truth-warping, money-worshipping, manipulative, law-disrespecting, ignorant, imperious, elitist, hedonistic, corrupt, coarse, Kafka-esque, foul, faux-Christian, thin-skinned, dysfunctional, despotic, defaming, deplorable, deceit-prone, delusional, demagogic, democracy-dismantling, disingenuous, despicable, demented, ill-principled, nepotistic, police-state-leaning, patronizing, propagandistic, power-grasping, plutocratic, pompous, petulant, prissy, prevaricating, pathological, piggish, petty, Putin-obeying, phony, pathetic, prickish, press-bashing, bullying, BS-ing, bribing, bluster-driven, golddigger-harboring, shameful, pseudo-American, self-absorbed, simple-minded, scam-loving, swamp-infested, cerebrally-challenged, self-serving, snowflakey, non-stable, slacker, sociopathic, slimeball, thuggish, tweet-obsessed, tax-evading, whiny, whistling-in-the-dark, roguish, reckless, Russian-colluding, Alpha-malevolent, NRA-owned, amateurish (and impeachable?), Aryan-embracing, odious, Emperor-without-clothes, routinely-ridiculed-as-clueless, treasonous, tin-pot dictatorial, jerkwad, bed-wetting, Making-America-Gag-Again, 4th-Reichian, pussy-grabbing Regime (...but you didn't hear any of that from me**), and somehow I feel compelled to again run this classic Jacob Bronowski clip:

...BUT, so as not to end on too sad a note, we'll close out sliding from Bronowski to Brooks:

**  ...or, if you did, it was locker room talk.

Tuesday, August 29, 2017

Collatz… So what’s the history of it???

I see the always-intriguing Collatz conjecture going around a bit again on Twitter (as it seems to every few months), but just started wondering what the history/background of it is, which I’ve never seen much about, other than that it originated with Lothar Collatz maybe in the 1930s(?).
The simple statement of it, is that you take any positive integer and apply the following 2 rules iteratively:
  • If the number is even, divide it by two, or
  • If the number is odd, triple it and add one. (Then repeat.)
Doing so successively you will always conclude with a sequence of integers ending at 4, 2, 1 (...or so goes the conjecture).
People write a lot about the conjecture and continue to work on it, but what I’m wondering now is how did Collatz stumble upon those two specific iterative rules to begin with out of essentially an infinite number that might be imagined (even if many would pretty obviously not lead to anything interesting)? Or, you could even come up with 3 iterative rules! Or, or, or… Did he try LOTS of others… have other people since tried LOTS of others? Is there something unique about his two rules, as opposed to ANY others that might be concocted and have some interesting result?
Anyone know, or can point to some informative links?

...And for anyone who's missed it, here's a nice Numberphile introduction to the Collatz conjecture:

In the comments below Brian Hayes responds with this link to an old piece he wrote for Scientific American on the subject. Like other pieces, it’s largely analysis of the conjecture, written in Brian’s always-superb exposition, but there is a bit of history on page 12. He also references a piece by Lothar himself, but what I found most interesting in tracking it down, was seeing a number of folks say that though Lothar explored many iterative functions, he never actually claimed specific credit for the so-called 3N+1 problem that took on his own name!

And with all that said, what I’m still not clear about is whether the two conjecture rules involved in 3N+1 were arrived at primarily by sheer trial-and-error, or was there a more methodological/quantitative approach to hitting upon them?

Monday, August 28, 2017

Too Good Not To Pass Along

By now we've probably all seen plenty of Richard Feynman videos. But h/t to Paul Halpern for tweeting out this old clip (that I don't recall viewing previously) of Feynman and Fred Hoyle in brief conversation (3+ mins.) about scientific revelation:

Sunday, August 27, 2017

Logical Consistency/Objectivity

This week's Sunday reflection taken from Michael Guillen's “Bridges To Infinity” (1983):
“…the world of today’s mathematician is one not only in which truth is not synonymous with logical proof but also in which merely trusting in the validity of a logical proof is itself a matter of faith. This is because Gödel not only showed that any logical system is unable to prove all the mathematical statements that are actually true, but also that any system of logic is unable to prove its own logical consistency. Believing in logic, in other words, is no less subjective a frame of mind than believing in, say, a secular or mystical principle of faith, because even logic itself cannot be verified logically or objectively.”

Wednesday, August 23, 2017


Recently on Twitter @mathematicsprof asked for suggestions on who ought be commemorated if a statue of a native-born American (U.S.) mathematician was to be erected in Washington D.C. Of course famous mathematics names (including several that were mentioned) have a tendency to be British or otherwise European, so it’s not surprising that many different names arose to the tweet, without any one standing out above all others. Among those getting at least one mention were the following (in no particular order):
David Mumford
Paul Cohen
Ken Ribet
John Milnor
Ed Witten
John Nash
Julia Robinson
Martin Gardner
Katherine Johnson
Raymond Smullyan
Claude Shannon
William Thurston
John Tate
Jim Simons
Stephen Smale
Ronald Graham
Donald Knuth
Persi Diaconis
Alonzo Church
Definitely a tough choice! I very slightly lean toward Thurston, but good arguments can certainly be made supporting many of these choices (Nash, Witten, Shannon, Milnor were among those with multiple votes). And I'd throw Barry Mazur into the mix as well. Also, was a little surprised that several popular math writers didn’t seem to get a mention: Reuben Hersh, Morris Kline, Philip Davis, James Newman, Ed Kasner, Paul Lockhart.  A bit odd too, that despite responders citing a great many non-U.S. born mathematicians (mostly European) I don't recall Grothendieck or Perelman coming up -- political bias or mathematicians just not wanting to be represented by social outliers? (or perhaps repliers simply knew the latter two were foreigners, while unaware that many others named, some of whom were naturalized Americans, were born elsewhere.)
Anyway, interesting to think about... (ya know, in case any of you were hoping to replace some Robert E. Lee monument with a mathematician) ;)

Tuesday, August 22, 2017

A Total Parody of Bonnie Tyler

Long-time readers here may recall my affinity for self-reference and recursion, so in that vein (and just for fun), this outlandish rendition of Bonnie Tyler's classic hit, "Total Eclipse of the Heart" which many folks tweeted yesterday in honor of the celestial show:

[p.s.: apparently her original 1983 hit became the #1 popular streaming song this week as eclipse-viewers re-fell in love with it]

Monday, August 21, 2017

Probably Not Just Coincidence...

I’ve adapted this little puzzle from one of the recent "Riddler" postings over at FiveThirtyEight blog:
Say (you know just for the sake of imagination), that you’re the President of the U.S. and your panties are in a wad because there are too many leaks coming from your Administration. So of course you wish to catch the sniveling culprit and axe them from your staff. Thus, you hatch a plan: You will give, at different times, different stories out to each of your 100 staffers and watch to see what bits end up in the press — we’ll assume there is just one leaker and they always leak what they know to the media. How many different concocted stories, minimum, do you need to feed to your staff of 100, in what manner, over time to be able to identify the leaker?
.answer below
7 stories are required IF you release them sequentially as follows:
The first story is told to half your staff (50 people) and withheld from the other 50 staffers. If it is leaked, you immediately know the leaker is among the first 50, or, if it doesn’t leak, the leaker is in the other half. Whichever 50 staffers are still suspects, give 25 of them a new story, and withhold same from the other 25. Repeat this process and you get a sequence like this: 100, 50, 25, 13, 7, 4, 2, 1, such that within 7 steps you’ve narrowed the search down to one culprit.
[p.s…: any resemblance between this process and our current Administration is probably not just coincidence.]