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?

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:

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.”

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) ;)

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]

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.]

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.”

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:

https://mathwithbaddrawings.com/2017/08/16/three-rules-for-tackling-a-world-famous-math-problem/

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

https://cameroncounts.wordpress.com/2017/08/16/wiles-and-langlands/

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

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):

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."

"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:

http://www.scottaaronson.com/blog/?p=3376

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

In Case I’m Banished to a Gulag

People love… and... hate, lists… at least they’re a fun time-and-space-filler, so I've been thinking about which books I’d grab off the shelf if Donald Trump, in his wisdom (spelled “p-a-t-h-o-l-o-g-y”) decided to banish me to a remote Gulag, only letting me take along 10 of my math-related books; which ones might I grab quickly for sustenance and entertainment? In no particular order, here’s what I chose (some aren’t particularly mathy though):

a) The Colossal Book of Mathematics  — Martin Gardner (...so much fun and games and puzzlement!)

b) How Mathematicians Think — William Byers  (...a long time favorite of mine about ideas permeating and underlying mathematics)

c) The Outer Limits of Reason — Noson Yanovsky  (...my favorite volume from the last few years, weaving together so many important subjects)

d) How Not To Be Wrong — Jordan Ellenberg  (...popular best-selling treatment of mathematical thinking)

e) Things To Make and Do In the Fourth Dimension — Matt Parker  (...jaunty, wise, diverse, instructive topics)

f)  Love and Math — Ed Frenkel  (...fascinating bio and intro to the Langlands Program)

g) Math In 100 Key Breakthroughs — Richard Elwes  (...succinct overview of key math topics)

h) The Music of the Primes — Marcus du Sautoy  (...'cuz I gotta have one volume devoted to the Riemann Hypothesis)

i)  Metamagical Themas — Douglas Hofstadter  (...some of the best stuff from Hofstadter's fertile mind)

j)  Beyond the Hoax — Alan Sokal  (...not math, but rich overview of critical thinking and much more)

Oddly two of my favorite math expositors, Keith Devlin and Ian Stewart, didn’t quite make the cut, though I’ve happily read more of their books than any of the above authors. Nor does it include the single volume I still most frequently recommend to lay people: Strogatz’s “The Joy of X.”  And a lot of other wonderful picks, including some older classics, go unmentioned as well.

Admittedly, an eclectic list, framed to my interests, that wouldn’t satisfy many of the math-folks likely beside me at the Gulag. Oh well, at the very least I suspect I'd have the company of Devlin, Ed Frenkel, and John Allen Paulos along to help entertain me! ;) (...and probably many more of you as well; hey, maybe even Andy Borowitz would be there to keep us all in good humor).

Let the banishments begin… or, not.

Of Math and Physics

From Numberphile once again, this time on physics and math, for a general audience:

Math Methods Versus Math Tricks

This week's Sunday reflection comes from Jim Propp in a recent Web piece:

"Mathematicians are people who like solving problems, and have the persistence to work on problems that take time to solve, and have collected a mental tool-kit consisting of methods that have helped them solve problems in the past. Some mathematicians distinguish between methods and tricks. A method is a tool that solves more than one problem, while a trick is a tool that applies to only one. Under this definition, I’d say that there are no tricks in math, and part of the discipline of getting good at math is to study every trick you encounter until you see the method hiding inside it."

Derangement

James Grime is a bit deranged in this recent Numberphile episode (or maybe he's a bit deranged in any Numberphile episode… and I mean that in a good way!):