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:
https://plus.google.com/+johncbaez999/posts/P7AN48F9pC7

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, truth-warping, money-worshipping, law-disrespecting, ignorant, imperious, corrupt, coarse, Kafka-esque, foul, faux-Christian, scam-loving, dysfunctional, deplorable, delusional, democracy-dismantling, ill-principled, nepotistic, police-state-leaning, patronizing, power-grasping, plutocratic, pompous, prissy, prevaricating, pathological, piggish, petty, Putin-obeying, phony, pathetic, prickish, press-bashing, cerebrally-challenged, pseudo-American, scared little, self-absorbed, simple-minded, non-stable, tweet-obsessed, whistling-in-the-dark, roguish, Russian-colluding, Alpha-malevolent, amateurish (and impeachable?), Aryan-embracing, routinely-ridiculed-as-clueless, tin-pot dictatorial, 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:






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:




ADDENDUM:
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

Statuesque


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?
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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.]


Sunday, August 20, 2017

Math Was Never the Same Again


Sandro Contenta provides this Sunday reflection from a profile of Canada’s Robert Langlands:

“In 1966, [Robert] Langlands almost abandoned mathematics. Deep mysteries in number theory discouraged him. He decided on a change of scenery and applied for a job in Turkey.
‘The decision itself freed me and I began to amuse myself with mathematics without any grand hopes or serious intentions,’ he said in written answers to a 2010 UBC interview.
Inspiration struck during the Christmas break, in an empty, grand old building on the Princeton campus, as Langlands gazed at a garden through leaded windows.
He described his revelation in a Jan. 16, 1967 letter to Andre Weil, a giant in the field of number theory: ‘If you are willing to read as pure speculation,’ he wrote Weil, I would appreciate that; if not — I am sure you have a waste basket handy.’
"Three years later, after he’d returned from Turkey, Langlands published his two theories, called functoriality and reciprocity, under the title ‘Problems in the Theory of Automorphic Forms.’ Math would never be the same again.”



Thursday, August 17, 2017

It Was a Wiles Wednesday


Two pieces on Andrew Wiles showed up yesterday… with little overlap ;)
Ben Orlin and his round-faced friends here for your light read:

…and Peter Cameron delving into the Langlands Program here with some heavy going:

And even if you've seen it before, always worth watching again:




Wednesday, August 16, 2017

If I Had A Hammer... I'd Hammer Out a Warning


Sorry, I’m in a mood (someone put me there), so just a bit more music for the moment (‘cuz as bad as the 60’s were, they seem glorious compared to today):




Sunday, August 13, 2017

American Tune...


“And I don't know a soul who's not been battered
I don't have a friend who feels at ease
I don't know a dream that's not been shattered or driven to its knees
But it's alright, it's alright, for we live so well, so long
Still, when I think of the road we're traveling on
I wonder what's gone wrong, I can't help it I wonder what's gone wrong”



Category Theory via Eugenia Cheng


For Sunday reflection, Eugenia Cheng describing 'category theory':
"This is how category theory arose, as a new piece of math to study math. In a way, category theory is an ultimate abstraction. To study the world abstractly you use science; to study math abstractly you use category theory. Each step is a further level of abstraction. But to study category theory abstractly you use category theory."

Thursday, August 10, 2017

"the psychology of unspeakable truths"


I hope you've already seen it, but in case not, Scott Aaronson's latest post is both a thoughtful tribute to A.N. Kolmogorov and a somewhat stoic commentary about the world we  find ourselves in:

...an important read, though not for any math.