The Man Who Almost Defies Comprehension

Ramanujan is much in the news lately with release of the new biopic of his life, "The Man Who Knew Infinity":



Colm Mulcahy newly-reviews the film here:
http://blogs.scientificamerican.com/guest-blog/a-math-genius-like-no-other-comes-to-the-big-screen/

...and Stephen Wolfram presents this riveting account of Ramanujan (h/t Steven Strogatz):
http://blog.stephenwolfram.com/2016/04/who-was-ramanujan/

And finally here, Ken Ono speaks briefly about his involvement with the film as a consultant:



If it comes to a theater near you, don't miss it!

Good Science?

via WikiMediaCommons

Early in my career I did ~6 years of animal research... but couldn't stick with it as I saw too much poor/sloppy science being done. The variables are so many, so complex, so ill-defined and easily overlooked, as to make it almost surprising how often progress is actually made (or at least perceived).
Here's just one more problem:
https://www.statnews.com/2016/04/19/lab-mice-temperature/

IF temperature affects the results of experiments with lab mice, what about sounds, sights, lighting, diet, human touch, air circulation, altitude, and on and on and on... of course no one truly knows all the effects on physiology and brain chemistry of sensory inputs (anymore than anyone knows what weather events may be affected halfway around the globe eight months after a butterfly flaps its wings). Cause-and-effect, when it comes to living beings, is nothing if not chaotic.

That's not to be too harsh about such work, but simply to prompt a skeptical stance, especially toward initial, and unverified-or-unreplicated results (let alone the hype of headlines).

I've previously voiced dismay here with those who proclaim themselves "skeptics," yet who largely grant a free pass to weak science and methods published routinely in major journals (luckily, now, decades since I experienced my cynicism, such skepticism is creeping into more mainstream outlets).
There is pseudoscience, speculative science, and real or good science... and the lines blur far more than admitted. Even theoretical physics, revered in my youth, today stands accused from many quarters, of bordering on metaphysics or philosophy, and not true empiricism... I'm not judging it one way or the other, except to say that even such an accusation, from bright people, is telling.

No one said good science should be easy... or common... indeed, it is difficult and rare. Mediocre science is the norm. And wrong-headed science is not uncommon... but is correctable. The (scary) anti-science attitudes/backlash of so many Americans today is a direct result of being sold a naive bill-of-goods and never understanding the true tentative, uncertain nature of science, its strengths and too-often-unacknowledged weaknesses. Still, it remains the best, by far, we've got... and its cornerstone, by the way, is mathematics.

On Twitter I've often used the below graphic (sorry I don't know its origination), but with the suggestion that you can replace the word "success" with the word "science" and it remains true:

Bottom line, good science is incredibly complex at a time when many increasingly gravitate toward simple answers (ala the absurd rise of Donald Trump). Science, misunderstood and misused, can destroy us... yet it is also probably the ONLY thing that can save us... from ourselves.

Bongard Problems

Good Alex Bellos column to start the week in The Guardian today on "Bongard problems" (visual comparison puzzles):
http://tinyurl.com/jeyb7y8

Further interesting reading on philosophical, cognitive, and AI aspects of Bongard problems:

http://app.ebape.fgv.br/comum/arq/Linhares2.pdf

http://www.foundalis.com/res/diss_research.html

 

Numbers are Very Real

Sunday reflection: 

"It’s quite astonishing and I still don’t understand it, having been a mathematician all my life. How can things be there without actually being there? There’s no doubt that 2 is there or 3 or the square root of omega. They’re very real things. I still don’t know the sense in which mathematical objects exist, but they do. Of course, it’s hard to say in what sense a cat is there, too, but we know it is, very definitely. Cats have a stubborn reality but maybe numbers are stubborner still. You can’t push a cat in a direction it doesn’t want to go. You can’t do it with a number either. I’m only using the word number because you’ll have a vague idea in your head as to what I mean. The objects that a mathematician studies are more abstract than numbers but very real.
"I often think of cats. I think of trees. I think of dogs occasionally but I don’t think of them all that much because dogs are agreeable. They do what you want them to do to some extent. Some people believe that mathematics is what we think it is and it’s created by our thoughts. I don’t. I’m a Platonist at heart, although I know there are very great difficulties in that view."

           -- John Conway

Three for Wednesday

1)  This morning, Futility Closet introduced many of us to "Rayo's Number," "the largest finite number ever written on an ordinary-sized chalkboard":
http://www.futilitycloset.com/2016/04/20/the-outer-limits/

2)  Ben Orlin delights again with his comically insightful characters trying to define math, because it's "crazy hard to define"':
https://mathwithbaddrawings.com/2016/04/20/math-is-crazy-hard-to-define/

3)  and h/t to Julie Rehmeyer for pointing out this more general piece on some of the ills of current science, "Big Science Is Broken." One line toward the end reads as follows: "Science, at heart an enterprise for mavericks, has become an enterprise for careerists":

http://theweek.com/articles/618141/big-science-broken

Pale Blue Dot... Indeed

For Sunday reflection today just some classics that warrant regularly re-visiting:

Beware of Dating Logicians

            
                       YES    NO   ????


Got a hot date this weekend?
For the men out there (...though I s'pose women could turn the tables and use this as well), two questions to pose to your next date after that romantic dinner:

1)  Will you answer THIS question the very SAME way you answer the NEXT question?

and

2)  Will you come back to my place and make mad, passionate love to me tonight?


(I adapted this specifically from a fine little volume, "Paradoxes: Adventures In the Impossible" by Gary Hayden and Michael Picard, but have run across versions of it in other places.)

John Baez Ruminates

John Baez is one of those fellows virtually incapable of writing a dull post!
A couple of recent challenging ones from him; not easy reads, but interesting:

1)  This one on "surprises in logic" focuses on "a complexity barrier built into the very laws of logic," with some Chaitin, some Gödel, some Kolmogorov, some Kritchman-Raz all rolled in together, before closing out with Joel Hamkins and the "computability of incomputable functions" (also includes lots of good links):
http://math.ucr.edu/home/baez/surprises.html
(this is mainly for logicians, but generalists can find bits of interest as well)

2)  and this one, completely different, from his personal blog ("Azimuth") on "Diamonds and Triamonds" which leads to mention of one of his "favorite entities," the E₈ lattice, and closes out with 5 puzzles:
https://johncarlosbaez.wordpress.com/2016/04/11/diamonds-and-triamonds/
(again a challenging read)

Searching For Simplicity

"Simplify, simplify, simplify."  -- Thoreau  ;-)


I'll start with an old puzzle many or most will know:

25 basketball teams compete in a single-elimination tournament -- as soon as you lose one game you're out of the tournament, and with 25 teams one team will get a bye in the first round. How many games total will be played by the end in order to crown the champion?

Clearly you can reach the correct answer by patiently working out all possible pairings or games and counting up their number; that will take some time... BUT there's a much easier way that gives the answer literally in seconds. It requires a simple insight that cuts through the process and in 'Aha!' fashion let's you see the answer in a moment.  For any who don't know it, I give the answer at the bottom of this post.**

I mention this puzzle as an intro to the post from "Gödel's Last Letter..." that I'm linking to, to launch the week. It starts with chess, then moves to the Reimann Hypothesis and then P vs. NP, while generally wondering out loud about the "easiest possible kinds" of proof or solutions:
 https://rjlipton.wordpress.com/2016/04/09/missing-mate-in-ten/ 

An interesting read. The gist of it is pondering whether there could be simpler, overlooked solutions to some of the exceedingly difficult problems that appear in mathematics... it's like wondering, as folks did for many years, if Fermat could have actually had a simple proof for his "Last Theorem." In most cases, the answer is likely 'no,' but somewhere out there, who knows? What might our human logic be overlooking, or our limited, biased minds be completely missing? The post includes links to a couple of recent "proofs" offered for the Riemann Hypothesis, one of which is relatively simple (or short) -- though a further commenter, "Gentzen," IDs it is a "fake proof," for reasons that exceed my pay grade :-(

 And as long as we're talking about the difficulty of proofs, I'll also point you to this nice, succinct exposition on the Continuum Hypothesis, re-published in Nautilus this weekend (a case where, to this point at least, the only "proof" has been to show that there are no proofs, under current set theory):
http://nautil.us/blog/how-a-hypothesis-can-be-neither-true-nor-false

 ------------------------------------

** the answer is 24, because in a single elimination tournament of 25 teams there must, by the end, be 24 LOSERS (and one champion) -- and if there are 24 losers, there must be 24 games played. 

Of Mathematics and Diamonds

"No diamond can compete with the clarity of a mathematical sentence. When you successfully prove a new sentence, you hold in your hands a clear fragment of truth. It is an exceedingly rare privilege to devote your adult life to the search for truth. I reflect on this day and night."

-- mathematician Dr. Nathan Linial (from his 2016 Rothschild Prize acceptance speech)

Talking Politics...

Off-topic today, for another detour into political La-La Land... because, why not!
I don't usually have much luck with political prognostication, but fairly satisfied with my efforts this year thus far....

I wrote here last June that only 3 people had a real chance at the Republican presidential nomination: Cruz, Rand Paul, and Paul Ryan.  Then, after 3 debates it was clear that R. Paul didn't have the charisma I'd imagined, and I substituted Kasich, as the main alternative to Cruz, since Ryan hadn't announced... the large contingent I'd expected Rand Paul to draw from was instead going to Donald Trump, who I never took seriously, AND STILL DON'T -- between convention rules, the RNC & establishment Republicans, the IRS, CIA, operatives from Russia, China, Europe (seriously), and a 65+% disapproval rating, I've never granted Trump any real chance at the nomination (more likely he'd be indicted for something than nominated)... nor of completing a campaign if he did somehow get nominated (in fact, he can't hardly afford the pay cut of being Presdent!) -- if he did end up running, he'd lose, and finally if somehow he won the popular vote, the Electoral College simply won't select him... that final arbiter of our democracy the Founding Fathers instituted, now suddenly seeming rather useful and insightful! ...And God help us if I'm wrong!!
 
So, still betting on Cruz, Kasich, or Ryan (who I suspect has been running all along and laughing diabolically at how well his strategy has played out); probably on a 3rd or 4th ballot (though even five wouldn't be a surprise, followed by gunfights in the breezeways). In any event, this has been the most interesting, bizarre primary season of my lifetime... and, I hope I never EVER see another like it.

"The Heart of Fermat's Last Theorem"

 Another video today... this time a wonderful Numberphile effort to pass along a sense of the proof at "The Heart of Fermat's Last Theorem":

The Math Behind Beethoven

But first, let me re-try a question for Mac users I posted on Twitter and only got a couple of responses to:
Does anyone have a favorite, freely-downloadable anti-virus/malware program (there are several) for Macs (actually for an older MacBook Pro still on Lion OS X)? How about Mac "tune-up"/cleanup software... favorites?.

...and now to the main feature:

H/T to "3 Quarks Daily" blog which passed along this 4-min. YouTube video on music and math (watch it for the math, or for the Beethoven!):

3 To Launch the Week With

Might as well kickstart the week with a couple of pieces from folks I recognized yesterday at MathTango (NOT that they need any recognition from ME):

1)  Scott Aaronson has a nice essay (and less technical than much of his stuff) over at Aeon on "the great mystery of mathematics":
https://aeon.co/opinions/the-great-mystery-of-mathematics-is-its-lack-of-mystery
(this was posted a few days ago, though I believe it's a re-post of an earlier essay)

2) ...And from Keith Devlin this new piece on the history of algebra and the nature of mathematics (in part a response to a certain book and author I shall not mention):
http://devlinsangle.blogspot.com/2016/04/algebraic-roots-part-1.html

3)  Finally, not math, but a quite fun 13-min. talk from physicist Sean Carroll on his career and how Stephen Hawking figures into it:
https://soundcloud.com/the-story-collider/sean-carroll-what-would-stephen-hawking-do
(again, I think this is much older, but I only came across it today)


Enjoy....

Interconnected Phenomena

The Sunday reflection:

"Science loves to know the direct links between causes and effects, but it does not require us to know that there are such links. Scientists may suspect a correlation between two complex phenomena. The real problem is that humans naturally tend to make connections where there are none, and also tend to ignore connections that are too complex to predict. We see coincidences as events that are mysteriously fated by some deeply significant design. That might be true, and it might not be. In a highly complex world of interconnected phenomena, some connections are so subtly coupled through long chains of indirect links that we can never envision the effect of one on another."

-- Joseph Mazur in "Fluke"

[...and by a stroke of coincidence, according to Google blogspot stats, 2 milestones today: the above marks the 1500th posting in the history of Math-Frolic, while over at MathTango the 200th entry has just been posted this morning!]

Public Service Announcement...

Just a public service warning today: TOMORROW IS APRIL 1st... otherwise known as APRIL FOOL'S DAY...

With that said, my recommendation is to NOT take too seriously any blog entries published tomorrow with the following phrases in the subject line:

1)  "the Riemann Hypothesis"
2)  "P vs. NP"
3)  "squaring the circle"
4)  "Goldbach Conjecture" 
5)  "finally solved"
6)   "Elvis Presley"
7)  "jet contrails"
8)  "A Nigerian Prince"
9)  "Your lottery winnings"
10)  "Donald Trump"


Otherwise, have a good weekend.
...now I have to get back to writing out an exception I've discovered to the Collatz Conjecture; perhaps I can get it done and posted by tomorrow morning....

Of Marrieds and Singles...

In case you missed it, there was an Alex Bellos puzzle in The Guardian recently that almost 70% of responding readers missed (and supposedly 80+% of the general public usually miss) -- I was astounded by those numbers, since it doesn't appear terribly difficult, given just a modicum of thought. It ran as follows:

*************************
Jack is looking at Anne, but Anne is looking at George.
Jack is married, but George is not.
Is a married person looking at an unmarried person?

A)  Yes     B)  No     C)  Cannot be determined

*************************

James Grime covers it well here (and he also throws in a second, completely unrelated, problem worth seeing):



After this puzzle began making Twitter rounds, David Wees pointed out the very similar problem I've placed below (I advise NOT looking at it 'til you work the above puzzle), which interestingly seems much easier than the Bellos problem -- a strong indication of how words/language can get in the way of clearly understanding relationships or structure, or simply what is being asked.
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"A coin is flipped 3 times. 1st flip comes up heads and 3rd flip is tails. Is heads ever followed directly by tails?"

 

Here's To Amateurs... and Professionals at Play

Astronomy is often considered the best science for amateurs because over the centuries amateur or backyard astronomers have contributed so many important findings to the field. The heavens are so expansive that they offer many niches for even backyard astronomers to make significant discoveries, or be involved in "citizen" science.

Mathematics... not so much.  With its highly-specialized, technical and abstract content, math is usually not seen as a playground for non-professionals. Indeed many who try (and there ARE many) end up classified as "crackpots," their ideas or approaches so off-base and unworthy of attention.
In recent times there is the famous case of "homemaker" Marjorie Rice who, playing around with tessellations, made important contributions to the geometry of tilings. And there are a few others... but the number is small. In the last couple decades, how many "proofs" have come along for the Riemann Hypothesis or other "Millennium" problems from people dabbling way outside their competency, largely wasting time and energy.

I mention all this because of my ongoing fascination with the recent findings regarding the "pattern" of consecutive prime last-digits. Robert Lemke Oliver and Kannan Soundararajan who discovered it are professional mathematicians, but it is the sort of thing that could have been discovered by amateurs just 'playing around' with prime numbers (as people often do).  Indeed, a common response to their finding has been, 'HOW did this go UNnoticed so long!?" In this day of Mathematica and similar programs it seems the door is wide open to all manner of analyses of prime number digits/succession/position that amateurs could imagine doing -- it might well take a trained number theorist or other specialist to explain a given outcome, but just generating that outcome might be do-able by brute-force amateurs.

I've been following Mike Lawler, inspired by Oliver/Soundararajan's work, play with prime triplets for the last week with results that, while beyond my comprehension, could hold significance for others (Mike is not an "amateur," as he has a math PhD., but he is not a number theorist or prime specialist, and what he is doing could be done by a non-math PhD.). Current posts for his work are here:

https://mikesmathpage.wordpress.com/2016/03/27/my-fun-interaction-with-prime-numbers-this-week/

https://mikesmathpage.wordpress.com/2016/03/27/prime-triples-and-the-sieve-of-eratosthenes/

And he has been recording his results (looking at prime-last-digit-triples in billion increments) on an ongoing Google spreadsheet here:

https://docs.google.com/spreadsheets/d/1fZ0wkYrei3CR1XtUuqWKVubAbuM0EzFZ-SGOkKnapd0/edit#gid=0

This isn't for everyone, but for those mesmerized by the mysterious way prime numbers weave their way through our integer system, tantalizing us with their secrets and their non-random randomness(!), it can almost be addictive (beware). So here's to amateurs and professionals alike playing in that heady world of pure math, never quite knowing what they might find, where it might lead, or what it might mean.


...ADDENDUM:  after posting this, Mike put up another entry summarizing somewhat his experience thus far:
https://mikesmathpage.wordpress.com/2016/03/28/what-ive-learned-playing-around-with-primes/

The Contradiction of Primes

Primes been on my mind lately, so, for Sunday reflection:

"There are two facts about the distribution of prime numbers which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts.
"The first is that despite their simple definition and role as the building blocks of the natural numbers, the prime numbers... grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout.
"The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behaviour, and that they obey these laws with almost military precision."

-- D. Zagier  in "The First 50 Million Prime Numbers" The Mathematical Intelligencer  (1977)

...and:

"One of the remarkable aspects of the distribution of prime numbers is their tendency to exhibit global regularity and local irregularity. The prime numbers behave like the 'ideal gases' which physicists are so fond of. Considered from an external point of view, the distribution is -- in broad terms -- deterministic, but as soon as we try to describe the situation at a given point, statistical fluctuations occur as in a game of chance where it is known that on average the heads will match the tail but where, at any one moment, the next throw cannot be predicted. Prime numbers try to occupy all the room available (meaning that they behave as randomly as possible), given that they need to be compatible with the drastic constraint imposed on them, namely to generate the ultra-regular sequence of integers."

-- G. Tenenbaum and M. Mendèfs France, in "The Prime Numbers and Their Distribution" (AMS, 2000)


[These come from a great collection of quotes about prime numbers from a page at Matthew Watkins' website:
http://empslocal.ex.ac.uk/people/staff/mrwatkin//isoc/quotes.htm ]

Calling All Number Theorists....

Those persnickety primes... Earlier today Mike Lawler posted some results from Mathematica for the patterns in last digits of prime-triples, essentially in intervals of a billion primes (not sure if I'm stating that very clearly, but read his post):

https://mikesmathpage.wordpress.com/2016/03/24/weird-clustering-with-last-digits-of-3-consecutive-primes/

His columns are a bit hard to read, but you should be able to spot the "clustering" (or I would call it "coupling") he refers to which seems to largely hold for all 3 columns of the post. He notes in a comment that the "average" expected value for each entry would be about 15.6 million, so you can see how widely the values diverge from that, as well as see how they tend to pair up.
He is in the process of adding more columns at the below easier-to-read spreadsheet -- I assume he'll be going out to 10 billion primes, to restore data he originally had, but lost -- as I write this, columns for 4 billion primes are listed, and it appears to me (merely eyeballing it), that the numbers for the paired triplets are getting even closer(???): 

https://docs.google.com/spreadsheets/d/1fZ0wkYrei3CR1XtUuqWKVubAbuM0EzFZ-SGOkKnapd0/edit?usp=sharing

 As each set of a billion primes is a somewhat independent and random-like group of integers, this pattern of the same ordered-triplet of last digits re-occurring in associated pairs seems, on the surface at least, rather odd and striking!?  What (if anything) is it about those pairs? Perhaps a number theorist can see through to a simple explanation for it (if so, I'm sure Mike would love to hear it). Or does this finding piggy-back in any way on to the peculiar result from a week prior of prime number last digits tending to avoid repetition in consecutive primes?
WHAT is going on here....?

ADDENDUM:  I should have included in this post that Mike has already recognized that the paired triplets involved are consistently of the form (a, b, c) and (-c, -b, -a) in mod 10. Now THAT surely must mean something! (Again, perhaps something obvious to a number theorist, but WHAT?)