Wednesday, August 27, 2014

Introduction to Incompleteness


"Incompleteness is one of the most beautiful and profound proofs that I’ve ever seen. If you’re at all interested in mathematics, it’s something that’s worth taking the effort to understand."  -- Mark Chu-Carroll 

Mark Chu-Carroll (of "Good Math, Bad Math") is in the process of re-posting his own splendid discussion/explanation of Gödelian Incompleteness this week. If it's a subject that interests you, or you've always wanted a detailed introduction, his first three four posts (with more to come) are here:

http://www.goodmath.org/blog/2014/08/25/godel-reposts/

http://www.goodmath.org/blog/2014/08/26/godel-numbering/

http://www.goodmath.org/blog/2014/08/27/godel-part-3-arithmetic-and-logic/

[just added] http://www.goodmath.org/blog/2014/08/28/gdel-part-3-meta-logic-with-arithmetic/


Monday, August 25, 2014

Ahhh, Mickey Mantle...



Just for fun today, contemplating the wonderful number "7" with these two appreciations:





(image via SGT141/WikimediaCommons)


Sunday, August 24, 2014

Sunday Morning With Paul


"Mathematical reality is an infinite jungle full of enchanting mysteries, but the jungle does not give up its secrets easily. Be prepared to struggle, both intellectually and creatively. The truth is, I don't know of any human activity as demanding of one's imagination, intuition, and ingenuity. But I do it anyway. I do it because I love it and because I can't help it. Once you've been to the jungle, you can never really leave. It haunts your waking dreams….

"The solution to a math problem is not a number; it's an argument, a proof. We're trying to create these little poems of pure reason. Of course, like any other form of poetry, we want our work to be beautiful as well as meaningful. Mathematics is the art of explanation, and consequently, it is difficult, frustrating, and deeply satisfying."


-- Paul Lockhart from "Measurement"


[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one submitted by a reader, I'll cite the contributor.]


Friday, August 22, 2014

Epiphanies...


Love this newly-posted (by MAA) video of James Tanton answering the question, "What was the hardest thing you learned when studying math?" Especially timely to me since it ties in beautifully with the last two 'Sunday Reflections' I've posted here:

http://math-frolic.blogspot.com/2014/08/prime-synchrony.html
http://math-frolic.blogspot.com/2014/08/somethings-going-on-here.html

 

And, for more mathy stuff check out this Friday's link collection over at MathTango.

 

Thursday, August 21, 2014

And a Few More Puzzles


 If the prior puzzle was a bit too much for you, a few below that are more manageable...

Been reading "Mathematical Curiosities," new from Alfred Posamentier and Ingmar Lehmann. It is, as the subtitle suggests, "a treasure trove of unexpected entertainments" -- especially entertaining if you have a geometry bias.

In the middle of it come 90 "curious problems with curious solutions." Several of these are classics with which you'll be familiar, and others are a little fresher, all interesting. I'll pass along three to whet your appetite (these are paraphrased from the volume):


#1.  I feel like EVERYone should know this first one, so just passing it along for any readers not already familiar with it:
On a certain pond the water lilies double in number every single day. After the 50th day the pond is completely covered. How many days were required for the pond to be half-covered?

#2. Given the following four numbers:

8,932
12,668
85,423
9,165

What percentage of their sum, is their average?

#3. What time is it now if in 2 hours it will be one-half as long 'til noontime as in 1 hour from now?

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. answers below
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1)  49 days

2)  25%  (if you work this out the 'long' way, you may then see there's an easier, more general solution)
3)   9 am.



Tuesday, August 19, 2014

This Is a Puzzle You've Hated (or Loved) Before


A puzzle re-run today....
Two years ago I ran the below mind-numbing, self-referential puzzle that became one of the most frequent links back to this blog... I think primarily from computer programmers who enjoyed writing code to solve it. Anyway, if you missed it first go-around, here's another chance (answer posted further down):

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Given the following list of 12 statements which of the statements are true?

1.   This is a numbered list of twelve statements.
2.   Exactly 3 of the last 6 statements are true.
3.   Exactly 2 of the even-numbered statements are true.
4.   If statement 5 is true, then statements 6 and 7 are both true.
5.   The 3 preceding statements are all false.
6.   Exactly 4 of the odd-numbered statements are true.
7.   Either statement 2 or 3 is true, but not both.
8.   If statement 7 is true, then 5 and 6 are both true.
9.   Exactly 3 of the first 6 statements are true.
10.  The next two statements are both true.
11.  Exactly 1 of statements 7, 8 and 9 are true.
12.  Exactly 4 of the preceding statements are true.

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answer:  1, 3, 4, 6, 7, 11 are true

Monday, August 18, 2014

Dr. Yanofsky and Category Theory


Delighted to see Dr. Noson Yanofsky getting some further publicity in this piece from FQXi on category theory:

http://fqxi.org/community/articles/display/192

I interviewed Yanofsky last year after reviewing his popular work, "The Outer Limits of Reason," which I regard as the best, most important book I've read in a very long spell. I'll again reiterate that anyone interested in cross-disciplinary math-science-related fields ought devour this volume!




Sunday, August 17, 2014

Prime Synchrony


Sunday reflection today, courtesy of Freeman Dyson and Hugh Montgomery (this reflection actually ties nicely into last week's Sunday offering as well)...:

"[Freeman] Dyson helped bring together the continuous and the discrete understandings of subatomic behavior. Similarly, by fusing his love of number theory with his expertise in creating the mathematical tools of physics, he would make the initial observation that would reinforce the connections between the discrete world of the integers and the continuous world of analysis, and thus galvanize research on the Riemann hypothesis.
"As Dyson recalls it, he and [Hugh] Montgomery [number theorist] had crossed paths from time to time at the [Princeton] Institute [for Advanced Study] nursery when picking up and dropping off their children. Nevertheless, they had not been formally introduced. In spite of Dyson's fame, Montgomery hadn't seen any purpose in meeting him. 'What will we talk about?' is what Montgomery purportedly said when brought to tea. Nevertheless, Montgomery relented and upon being introduced, the amiable physicist asked the young number theorist about his work. Montgomery began to explain his recent results on the pair correlation, and Dyson stopped him short -- 'Did you get this?' he asked, writing down a particular mathematical formula. Montgomery almost fell over in surprise: Dyson had written down the sinc-infused pair correlation function.
"Dyson had the right answer, but until that moment he had associated this formula with understanding a phenomenon that seemed completely unrelated to the primes and the Riemann hypothesis. In a flash he had drawn the analogy between the sinc-described structured repulsion of the zeta zeros and a similar tension seemingly exhibited by the different levels of energy displayed by atomic nuclei. Whereas Montgomery had traveled a number theorist's road to a 'prime picture' of the pair correlation, Dyson had arrived at this formula through the study of these energy levels in the mathematics of matrices. This connection is the source of most of the current excitement surrounding the Riemann hypothesis..."


-- from "Stalking the Riemann Hypothesis" by Dan Rockmore


[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one submitted by a reader, I'll cite the contributor.]

Friday, August 15, 2014

Of Love and Hate… sort of


"...journalism has rules about writing stories that don’t really work for math. When journalists are told to 'put a face on the story,' they end up with all face and no story."

'Mathbabe' (Cathy O'Neil) hits another home run, or should I say home rant, today with this piece on math and the Fields Medal... and "the incredible collaborative effort that is modern mathematics":

http://mathbabe.org/2014/08/15/i-love-math-and-i-hate-the-fields-medal/


...and for more mathy links this morning see the weekly MathTango potpourri:
http://mathtango.blogspot.com/2014/08/friday-bonanza.html

Wednesday, August 13, 2014

An Even Better Day Will Come…


I wrote a couple days back that I didn't plan to cover the Fields Medals here, since I believed they would receive good and widespread coverage elsewhere… little did I realize what an understatement that would be! Because of the first-ever female winner, Maryam Mirzakhani, the reportage has been even beyond what I anticipated, in both the popular press as well as math sites.
 I hope that everyone is right in thinking that this will be a huge boost for women in math -- that Maryam can be a role-model and inspiration to young female math enthusiasts everywhere. I almost fear that the continual, overriding emphasis on her gender plays into a perception that she has achieved some rarefied, super-human feat, no ordinary female can aspire to… but then, I probably worry too much. Still... better will be the day when there is no special hoopla surrounding a woman winning a major math prize… it will just be a common ordinary happenstance! Until then though, indeed, congratulations to Dr. Mirzahkani and her co-recipients, Manjul Bhargava, Artur Ávila, and Martin Hairer… I just wish I could understand anything that they wrote :-((

Anyway, here is a smidgen of the coverage that is out there (if you've been living under a rock and missed it somehow ;-):

Part of the original press release for both Fields and other prize winners:
http://www.mathunion.org/general/prizes/2014

Quanta Magazine's nice profiles of the winners, starting here:
http://www.simonsfoundation.org/quanta/20140812-a-brazilian-wunderkind-who-calms-chaos/

Lots more roundup of the Fields coverage from The Aperiodical:
http://aperiodical.com/2014/08/2014-fields-medals-awarded-coverage-round-up/

Also, Keith Devlin's quick take on the awards for NPR today (Keith and Maryam are both at Stanford):
http://hereandnow.wbur.org/2014/08/13/math-prize-mirzakhani

Meanwhile, on a side-note, the IMU has also created a 'Women in Mathematics' website:
http://www.mathunion.org/wim/
(not clear to me if this has been around for awhile, or was possibly created in anticipation of the first female Fields Medal winner being announced?)


Tuesday, August 12, 2014

Carpe Diem... no more

"Oh Captain! My Captain!"

Today, in remembrance of Robin Williams, am just re-running material from a post I did one year ago:

Came across this quirky little posting that linked together math, teaching, and one of my favorite Robin Williams' movies, "Dead Poets Society":

http://casualcalculations.wordpress.com/2013/08/21/dead-mathematicians-society/
(the post is probably even more pertinent today with all the debate over math reform, than it was a year ago)

Watch the scene in the above post and then, if you've seen the movie, re-live the ending, that still tugs at me (not specific to math and perhaps only meaningful if you've seen the film):



...and apparently I'm not the only one moved by the above scene; check out the Twitter feeds started last night for "stands on desk" and "standing on desk":
https://twitter.com/search?f=realtime&q=stands%20on%20desk&src=typd

https://twitter.com/search?f=realtime&q=standing%20on%20desk&src=typd

R.I.P.  Mr. Keating. . . .

ADDENDUM:  [There are lots of wonderful tributes to Williams pouring in today, but the best one I've read thus far comes from Russell Brand in The Guardian:  http://tinyurl.com/ltr2qo9
(VERY worth reading; H/T to N. Ghoussoub for pointing me to it)]


Monday, August 11, 2014

Of Medals and Cocktail Party Chatter


Evelyn Lamb ran a timely piece at her Roots of Unity blog today on the Fields Medal, to be awarded to four people this coming Wednesday (I hadn't previously seen the number of recipients listed this year, so I'll assume she's right; it's always 2-4 individuals):

http://blogs.scientificamerican.com/roots-of-unity/2014/08/11/how-to-talk-about-the-fields-medal-at-your-next-cocktail-party/

She touches on the 'age-ist' nature of the Fields, before discussing some of math's other prestigious prizes. Most interesting part to me was learning of the "Chern Medal" for lifelong achievement in math, which I'd not heard of, and which Evelyn calls "The hipster candidate for the 'Nobel Prize of mathematics,'” first awarded in 2010 (and only every four years thereafter). She urges we keep an eye on it now, well before it becomes "cool." :-)

Anyway, the excitement is building for Wednesday's announcement, so stay tuned.
[...since it'll get widespread coverage on the InterTubes, I'll likely tweet about the Fields awards, but not blog about it here, other than to include as part of the Friday potpourri on MathTango]


Sunday, August 10, 2014

'Something's Going On Here'


Today's Sunday reflection, a passage about the role of intuition in mathematics (from "How Mathematicians Think" by William Byers):

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"For the mathematician, the idea is everything.  Profound ideas are hard to come by, and when they surface they are milked for every possible consequence that one can squeeze out of them. Those who describe mathematics as an exercise in pure logic are blind to the living core of mathematics -- the mathematical idea -- that one could call the fundamental principle of mathematics. Everything else, logical structure included, is secondary.
"The mathematical idea is an answer to the question, 'What is going on here?' Now the mathematician can sense the presence of an idea even when the idea has not yet emerged. This happens mainly in a research situation, but it can also happen in a learning environment. It occurs when you are looking at a certain mathematical situation and it occurs to you that 'something is going on here.' The data that you are observing are not random, there is some coherence, some pattern, and some reason for the pattern. Something systematic is going on, but at the time you are not aware of what it might be…
"The feeling that 'something is going on here' can even be brought on by a single fact, a single number. A case in point happened in 1978, when my colleague John McKay noticed that 196884 = 196883+1. What, one might ask, is so important about the fact that some specific integer is one larger than its predecessor? The answer is that these are not just any two numbers. They are significant mathematical constants that are found in two different areas of mathematics. The first arises in the context of the mathematical theory of modular forms. The second arises in the context of the irreducible representations of a finite simple group called the Monster. McKay intuitively realized that the relationship between these two constants could not be a coincidence, and his observation started a line of mathematical inquiry that led to a series of conjectures that go by the name 'monstrous moonshine.' The main conjecture in this theory was finally proved by Fields Medalist winner Richard E. Borcherds. Thus the initial observation plus recognition that such an unusual coincidence must have some deep mathematical significance led to the development of a whole area of significant mathematical research….

"…But still it is possible to say 'we do not really understand what is going on.' Understanding what is going on is an ongoing process -- the very heart of mathematics."


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I think, in essence, this is very much what Eugene Wigner's famous notion of "the unreasonable effectiveness of mathematics" revolves around… how is it that the human brain is capable of such seemingly successful intuitions about the world around us….

[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one you submit, I'll cite the contributor.]





Friday, August 8, 2014

The Nobel and Noble In Math


Today's NY Times carries an interesting historical piece on the Fields Medal (to be awarded to unannounced winners next Wednesday). The interesting part is the story of Stephen Smale, a University of California at Berkeley winner back in 1966, and the political shenanigans that shadowed him because of his political activism. Politics and mathematics, to the surprise of many, do sometimes intertwine. With many mathematicians today taking stands against NSA work or certain other Government activities… and, with the Fields Medals being given out next week... it's a timely piece:

http://www.nytimes.com/2014/08/10/opinion/sunday/how-math-got-its-nobel-.html?_r=2