Wednesday, September 17, 2014

The Mathematical MacArthurs


The 21 MacArthur Fellows for 2014 have been announced:

http://www.macfound.org/fellows/class/class-2014/ 

http://tinyurl.com/mdg4csr  (NPR coverage)

As usual quite a motley group, but including at least three with mathematical connections:

Danelle Bassett is a physicist, using mathematics to study the complex networking of the brain.

Yitang Zhang, the suddenly-famous mathematician who showed there were finite bounds to the prime gap problem.

and Jacob Lurie, a 36-year-old Harvard pure mathematician shown below:

 


Tuesday, September 16, 2014

A Threshold Reached...


Interesting little story of how "network theory" (via social media) underlied the solution to a 13-year-old mystery photograph from the 9/11 tragedy:

http://tinyurl.com/luhnov8

a blurb:

"...when information moves through a social network, it doesn’t move at random: The information pings from one 'node,' or person, to another, and the relationships between all those thousands of nodes create an intricate geography of influence and power"....
"...there’s a reason the photo went viral this year, when it didn’t go viral all the long years before. For the first time, it hit what information scientists call the epidemic threshold — the point at which a thing reaches enough nodes in the network that it can’t easily die out."

 


Monday, September 15, 2014

Calculus Going Viral?… Could It Be


Forbes Magazine reports on Ohio State professor Jim Fowler taking calculus to the Web via Coursera (and getting rave reviews):
http://tinyurl.com/lz2smt2

And he's on YouTube, if you don't want to sign up for Coursera without getting a sampling first:
https://www.youtube.com/user/kisonecat

here's his intro to the course:




Sunday, September 14, 2014

Sunday Reflection


From Philip Davis and Reuben Hersh's "The Mathematical Experience":
(I don't necessarily agree with this sentiment, but do find it interesting)
" We maintain that:

1)  All the standard philosophical viewpoints rely in an essential way on some notion of intuition.
2)  None of them even attempt to explain the nature and meaning of the intuition which they postulate.
3)  A consideration of intuition as it is actually experienced leads to a notion which is difficult and complex, but it is not inexplicable or unanalyzable. A realistic analysis of mathematical intuition is a reasonable goal, and should become one of the central features of an adequate philosophy of mathematics….

"Mathematics does have a subject matter, and its statements are meaningful. The meaning, however, is to be found in the shared understanding of human beings, not in an external nonhuman reality. In this respect, mathematics is similar to an ideology, a religion, or an art form; it deals with human meanings, and is intelligible only within the context of culture. In other words, mathematics is a humanistic study. It is one of the humanities."

[…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.] 


Monday, September 8, 2014

"Why Study Paradoxes?"


Given my fondness for paradoxes… and, for Raymond Smullyan... I couldn't help but love this weekend post from Ray T. Cook on paradoxes (and why to study them) at Oxford University Press blog:

http://tinyurl.com/ppvg4cp

After offering his own example that he once posed to the master logician Smullyan, Cook goes on to talk about the mathematical complexity of paradoxes before ending thusly:
"...that's why I work on paradoxes: their surprising mathematical complexity and mathematical beauty. Fortunately for me there is still a lot of work that remains to be done, and a lot of complexity and beauty remaining to be discovered."
Fortunately for me, there is still a lot of work that remains to be done, and a lot of complexity and beauty remaining to be discovered. - See more at: http://blog.oup.com/2014/09/why-study-paradoxes/?utm_source=feedblitz&utm_medium=FeedBlitzRss&utm_campaign=oupblogmathematics#sthash.VxyOQRJg.dpuf
that’s why I work on paradoxes: their surprising mathematical complexity and mathematical beauty. Fortunately for me, there is still a lot of work that remains to be done, and a lot of complexity and beauty remaining to be discovered. - See more at: http://blog.oup.com/2014/09/why-study-paradoxes/?utm_source=feedblitz&utm_medium=FeedBlitzRss&utm_campaign=oupblogmathematics#sthash.VxyOQRJg.dpuf
that’s why I work on paradoxes: their surprising mathematical complexity and mathematical beauty. Fortunately for me, there is still a lot of work that remains to be done, and a lot of complexity and beauty remaining to be discovered. - See more at: http://blog.oup.com/2014/09/why-study-paradoxes/?utm_source=feedblitz&utm_medium=FeedBlitzRss&utm_campaign=oupblogmathematics#sthash.VxyOQRJg.dpuf
And that’s why I work on paradoxes: their surprising mathematical complexity and mathematical beauty. Fortunately for me, there is still a lot of work that remains to be done, and a lot of complexity and beauty remaining to be discovered. - See more at: http://blog.oup.com/2014/09/why-study-paradoxes/?utm_source=feedblitz&utm_medium=FeedBlitzRss&utm_campaign=oupblogmathematics#sthash.VxyOQRJg.dpuf
And that’s why I work on paradoxes: their surprising mathematical complexity and mathematical beauty. Fortunately for me, there is still a lot of work that remains to be done, and a lot of complexity and beauty remaining to be discovered. - See more at: http://blog.oup.com/2014/09/why-study-paradoxes/?utm_source=feedblitz&utm_medium=FeedBlitzRss&utm_campaign=oupblogmathematics#sthash.VxyOQRJg.dpuf
that’s why I work on paradoxes: their surprising mathematical complexity and mathematical beauty. Fortunately for me, there is still a lot of work that remains to be done, and a lot of complexity and beauty remaining to be discovered. - See more at: http://blog.oup.com/2014/09/why-study-paradoxes/?utm_source=feedblitz&utm_medium=FeedBlitzRss&utm_campaign=oupblogmathematics#sthash.VxyOQRJg.dpuf
that’s why I work on paradoxes: their surprising mathematical complexity and mathematical beauty. Fortunately for me, there is still a lot of work that remains to be done, and a lot of complexity and beauty remaining to be discovered. - See more at: http://blog.oup.com/2014/09/why-study-paradoxes/?utm_source=feedblitz&utm_medium=FeedBlitzRss&utm_campaign=oupblogmathematics#sthash.VxyOQRJg.dpuf
that’s why I work on paradoxes: their surprising mathematical complexity and mathematical beauty. Fortunately for me, there is still a lot of work that remains to be done, and a lot of complexity and beauty remaining to be discovered. - See more at: http://blog.oup.com/2014/09/why-study-paradoxes/?utm_source=feedblitz&utm_medium=FeedBlitzRss&utm_campaign=oupblogmathematics#sthash.VxyOQRJg.dpuf
that’s why I work on paradoxes: their surprising mathematical complexity and mathematical beauty. Fortunately for me, there is still a lot of work that remains to be done, and a lot of complexity and beauty remaining to be discovered. - See more at: http://blog.oup.com/2014/09/why-study-paradoxes/?utm_source=feedblitz&utm_medium=FeedBlitzRss&utm_campaign=oupblogmathematics#sthash.VxyOQRJg.dpuf

Also, Ray has apparently written a short book entirely on Yablo's Paradox which I've mentioned here before (and which the above post is related to):
http://ukcatalogue.oup.com/product/9780199669608.do


Sunday, September 7, 2014

Complex Numbers (Sunday Reflection)


"…there is never a need to go beyond the complex numbers. No further numbers are needed. They suffice and so they bring to completion the very long effort at construction that over thousands of years yielded first the natural numbers, then the fractions, then zero and the negative numbers, and after that the real numbers. The complex numbers complete the arch.
"Beyond the theory of complex numbers, there is the much greater and grander theory of the functions of a complex variable, as when the complex plane is mapped to the complex plane, complex numbers linking themselves to other complex numbers. It is here that complex differentiation and integration are defined. Every mathematician in his education studies this theory and surrenders to it completely. The experience is like first love.
"I once mentioned the beauty of complex analysis to my great friend, the mathematician M.P. Schutzenberger. We were riding in a decrepit taxi, bouncing over the streets of Paris.
" 'Perhaps too beautiful,' he said at last.
"When I mentioned Schutzenberger's remarks to Rene' Thom, he shrugged his peasant shoulders sympathetically.
"This is one of the charms of the theory of complex numbers and their functions. It has broken men's hearts."


-- David Berlinski from "Infinite Ascent"


[…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.]

Thursday, September 4, 2014

Books For a Desert Island... (meme?)


First off, 3 new mathy-related books I've been looking at recently are:

"Mathematics and the Real World" by Zvi Artstein (a historical look at mathematics over time)  
"Standard Deviations" by Gary Smith (another popular take on how statistics get used and misused) 
and
"Rock Breaks Scissors" by William Poundstone (another probability-meets-the-real-world sort of offering)

I like almost everything Poundstone writes so suspect I'll enjoy his latest work, and the Smith book looks good as well (though somewhat redundant in content to several other recent works), but the only volume I've actually started is the Artstein book... and so far not particularly enamored of it; it has plenty of information, just a more mundane or pedantic writing style (perhaps because it is a translation, from Hebrew) than several recent popular math books -- Jordan Ellenberg has spoiled me ;-)  I'm liking the second half of the volume more than the first half and will withhold judgment 'til finished, but Publishers Weekly (where I rarely see negative reviews) appears even more disenchanted with the volume than I am:

http://www.publishersweekly.com/978-1-61614-091-5

(Even at that it may still fill a certain niche on your math bookshelf, depending on what historical volumes you already have.)
Anyway, I may say more about any/all of these books later.

Now, departing from the mathy-track once again....

via Mr.TinDC flickr

Blog memes don't seem to go around much anymore as they did at one time, but I did notice a meme bubbling in some Facebook circles, asking for 10 books that most moved you or changed/affected your life. I probably couldn't name 10 volumes that affected me at that level, but thought about 10 books that were important to me, and that I'd grab to take to a deserted island, if need be.

Might be interesting to hear from other math communicators what would make it onto THEIR list for a desert island… if for no other reason than to show that math buffs are less nerdy and more diverse than some people might imagine!! (…although my list, admittedly, is somewhat nerdy, especially since I don't read fiction :-((

Anyway, with no annotations and in no particular order, here are my 10:

1)  "The Night Is Large"  -- Martin Gardner
2)  "Pilgrim At Tinker Creek"   -- Annie Dillard  
3)  "Metamagical Themas" (and "Gödel, Escher, Bach" as well)  -- Douglas Hofstadter
4)  "The Outer Limits of Reason"  -- Noson Yanofsky
5)  "Natural Prayers"    -- Chet Raymo
6)  "The Pleasure of Finding Things Out"  -- Richard Feynman
7)  "The Black Swan"   -- Nassim Taleb
8)  "Language In Thought and Action"    -- S.I. Hayakawa
9)  "Beyond the Hoax"   -- Alan Sokal
10)  "How Mathematicians Think"    -- William Byers

There are lots of other books I'd want along for sheer entertainment value (including, off to the side somewhere, book-compendiums of either "Dilbert," "New Yorker," or "Far Side" cartoons), but above are ones I'd want along to exercise my mind.

I, for one, would be interested to hear of other math bloggers'/writers' lists (post at your own blog or in the comments here).


Tuesday, September 2, 2014

Devlin Talks Geometry


I don't often see Keith Devlin focus on geometry in his blog posts… but today he did… and quite excellently!  In fact, from my standpoint the post, in some ways, makes for a nice counterpoint to the Sunday reflection I ran this weekend on Platonism (Dr. Devlin is a non-Platonist). Read Keith here:

http://devlinsangle.blogspot.com/2014/09/will-real-geometry-of-nature-please.html

Several snippets…:

"Mathematics provides various ways to model our perception and experience of reality. Different parts of mathematics provide different models, some better than others."

He goes on to talk about fractal geometry and cellular automata of Steven Wolfram as two geometric approaches to the world.

"Both approaches can be said to begin by looking at how nature works, but the moment you start to create a model, you leave nature and are into the realm of human theorizing."

"...make no mistake about it, we do begin with assumptions. Not arbitrary ones, to be sure—not even close to being arbitrary."

"...mathematics is not 'the true theory of the real world' (whatever that might mean). Rather, mathematical theories are mental frameworks we construct to help us make sense of the world."

"...we should not lose track of the fact that mathematics is not the truth.
 "Rather, it provides us with useful models of the world. As a result, it is a powerful and useful way of making sense of the world, and doing things in the world.
"

He ends with his vocal support again for Common Core (while admitting more focus is needed on "how to properly implement the Standards").

Read the entire piece, or like me, read it 3-4 times to squeeze out as much food for thought (and I dare say food for controversy as well!) as you can from it.


Monday, September 1, 2014

Is Savant Itchin' For a Fight?


Switch or don't switch... does that sound familiar?

Marilyn vos Savant is famous for (among other things) posing the original "Monty Hall puzzle" to a national audience, and baffling many, including experienced mathematicians. By now, almost anyone having interest in the puzzle no doubt knows the correct answer and why.

So it seemed a bit curious that in yesterday's Sunday "Parade" magazine column Marilyn deals with a similar-sounding puzzle that arrives at a different answer (the answer, 50/50, many had sought for the original Monty Hall). It's almost as if she were itchin' fer a fight, because I imagined that some folks, thinking back to Monty Hall, would reflexively argue she is wrong here. She is, of course, right, because the conditions or set-up are different from the Monty Hall example, but because she doesn't offer any lengthy explanation, it's predictable that she would churn up some naysayers who think she's inconsistent, and try to take her to task (...it has already begun in the comments).
See the column here:

http://parade.condenast.com/333629/marilynvossavant/should-you-swap-or-not/

By the way, a great book covering the Monty Hall puzzle in all its variations is, "The Monty Hall Problem" by Jason Rosenhouse from 2009.

Meanwhile, please be sure to also check out the very different, completely NON-mathy post over at MathTango today.


Sunday, August 31, 2014

Pickover on Platonism…


Some extended discourse via Cliff Pickover today from his volume, "A Passion For Mathematics" (one of my favorite Pickover offerings):

"I think that mathematics is a process of discovery. Mathematicians are like archaeologists. The physicist Roger Penrose felt the same way about fractal geometry. In his book The Emperor's New Mind, he says that fractals (for example, intricate patterns such as the Julia set or the Mandelbrot set) are out there waiting to be found:

'It would seem that the Mandelbrot set is not just part of our minds, but it has a reality of its own… The computer is being used essentially the same way that an experimental physicist uses a piece of experimental apparatus to explore the structure of the physical world. The Mandelbrot set is not an invention of the human mind: it was a discovery. Like Mount Everest, the Mandelbrot set is just there.'

I think we are uncovering truths and ideas independently of the computer or mathematical tools we've invented. Penrose went a step further about fractals in The Emperor's New Mind: 'When one sees a mathematical truth, one's consciousness breaks through into this world of ideas… One may take the view that in such cases the mathematicians have stumbled upon works of God.'

Anthony Tromba, the coauthor of Vector Calculus, said in a July 2003 University of California press release, 'When you discover mathematical structures that you believe correspond to the world around you, you feel you are seeing something mystical, something profound. You are communicating with the universe, seeing beautiful and deep structures and patterns that no one without your training can see. The mathematics  is there, it's leading you, and you are discovering it.'

"Other mathematicians disagree with my philosophy and believe that mathematics is a marvelous invention of the human mind. One reviewer of my book The Zen of Magic Squares used poetry as an analogy when 'objecting' to my philosophy. He wrote,

'Did Shakespeare 'discover' his sonnets? Surely all finite sequences of English words 'exist,' and Shakespeare simply chose a few that he liked. I think most people would find the argument incorrect and hold Shakespeare created his sonnets. In the same way, mathematicians create their concepts, theorems, and proofs. Just as not all grammatical sentences are theorems. But theorems are human creations no less than sonnets.'

Similarly, the molecular neurobiologist Jean-Pierre Changeux believes that mathematics is invented: 'For me [mathematical axioms] are expressions of cognitive facilities, which themselves are a function of certain facilities connected with human language.'
"


[…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.]


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?

.
. answers below
.
.
.
.
.
.
.
.
.
.
.
.
.
.

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.