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Thursday, April 24, 2014

Truthiness From Yablo


This isn't everyone's cup-o-tea, but I've mentioned Yablo's Paradox before, and, since I love it, will do so again! As Sam Alexander states, Yablo's Paradox is "a cute version of the Liar’s Paradox" that manages "to achieve paradox without any direct self-reference." The simply-stated paradox involves a countably infinite number of sentences, each of which refer only to sentences that come after it:
  • Sentence 1:  Sentence n is false for every n > 1
  • Sentence 2:  Sentence n is false for every n > 2
  • Sentence 3:  Sentence n is false for every n > 3
  • Etc....
Read Alexander's post here:  http://www.xamuel.com/dangerous-graphs/  (he discusses it in terms of "graphs").
And here is (Stephen) Yablo's amazingly short, original (1993) piece introducing the paradox:

http://www.mit.edu/~yablo/pwsr.pdf

Worth noting, that while Yablo claims his paradox involves neither self-reference nor circularity (because all steps along the way reference sentences that are yet to come), others disagree with this contention. If you're logically-inclined, see Graham Priest here:

http://www.accionfilosofica.com/misc/1183297103crs.pdf

and JC Beall here: http://ferenc.andrasek.hu/papersybprx/jcbeal_is_yablo_non_circular.pdf

(This is another good example of how things become muddled when infinity is involved; in the case of Yablo's Paradox, an infinite number of sentences.)


Wednesday, April 23, 2014

Teaching, Conceptualizing, Common Core


As much as I try to avoid the primary/secondary education debates, I'd feel remiss if I didn't pass along two of the pieces from this morning's Web:

1) California teacher Fawn Nguyen interviewed by Cathy O'Neil on Common Core:
http://mathbabe.org/2014/04/23/interview-with-a-high-school-math-teacher-on-the-common-core/

2) Grant Wiggins on "conceptual understanding" (longread +comments):
http://grantwiggins.wordpress.com/2014/04/23/conceptual-understanding-in-mathematics/

Fascinating (to me) how something (Common Core) intended to bring unification/standards, is instead stirring up vastly more debate and discussion than likely ever anticipated, with resolution seemingly a long way off. One thing I'm not clear on (as a non-educator), maybe someone can quickly clarify, is Common Core being sharply debated in realms OTHER than math (i.e., English/verbal content), or is the debate mostly a math thing?

Tuesday, April 22, 2014

Sociopaths, PhDs, Mathematicians, and more


Jordan Ellenberg offers an observation on why mathematicians (generally) are NOT sociopaths (unlike, perhaps, investment bankers ;-):

http://quomodocumque.wordpress.com/2014/04/22/why-arent-math-professors-sociopaths/

Meanwhile, possibly on the opposite end of the spectrum, is Jason Padgett's story, which I've reported here previously -- an individual with "acquired" savant syndrome, which appeared later in life following significant brain trauma, as recounted in Padgett's new volume "Struck by Genius: How a Brain Injury Made Me a Mathematical Marvel."  Interesting stuff...
Another review here: http://tinyurl.com/nqk8g8t
Or, you can learn a lot more about Jason by googling him:  http://tinyurl.com/kbozcnm


Monday, April 21, 2014

Family Math


Just filling some time today passing along this li'l algebraic Friday puzzler that Richard Wiseman offered last week (I've re-stated it slightly):

In a certain family with 2 or more children, each daughter has the SAME number of brothers as she has sisters, while each son has TWICE as many sisters as brothers. How many sons and daughters are in this family?

answer below:
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answer:  4 daughters & 3 sons

Thursday, April 17, 2014

Thursday Meditation From Mathodist Pastor Keith Devlin ;-)


Just a mathematical meditation for today...:

About 10 days ago in a longish post over at MathTango I included some links to Keith Devlin materials -- one of my favorites was a podcast he did for NPR's "On Being," an always-wonderful radio-hour.  In that conversation he made the following observation, which I fancy, regarding a transition on the road to becoming a mathematician [bold added]:
"...that's when I became a mathematician; that's what I stumbled on at age 15 or 16 when here I was learning all this mathematics because I needed  it. I had a utilitarian view of mathematics. I was learning it because I needed to solve the equations because I was going to be solving them in physics. And then, at the age of about 16 or 17, it all fit because it all came together in my mind. It was no longer this disjointed collection of techniques you could use to solve problems. It all fell into place, into this wonderful landscape. It was as if I'd been stumbling around in a forest, and suddenly I've climbed to the top of a tree and looked out and thought, this is the most beautiful place in the world. You can't tell it when you're down in the trees, which I had been, but the moment you reach an elevation where it all falls into place and you can see the whole topographic display in front of you, then the beauty is incredible. And the moment I discovered it, I said, um, I want to study mathematics. And I've been studying it ever since."
I love this imagery of most of us exploring down amongst the trees and forest floor, versus looking out above the canopies over the whole forest landscape that is mathematics. This is what Ed Frenkel essentially talks about, especially in working on the Langlands Program which brings together disparate aspects of mathematics... the separate sections of the forest linking together when we zoom out far enough.  The same idea comes up in discussions of symmetry, group theory, number theory, and other areas of math, unlike the more disjointed way math must often, perhaps of some necessity, be taught in the classroom. The individual trees and groves can of course be beautiful and stately all by themselves… but, ohhh, to take in that majestic view from above the treetops!!


Monday, April 14, 2014

Numerosities


For the philosophically, or foundationally, inclined, the below post argues for something called "numerosities" that give "part-whole" set relationships priority over "one-to-one" relationships, and in so doing, counter the usual Cantorian orthodoxy, which permits a partial set (say odd numbers), to be deemed equal in size to an entire set (all integers):

http://www.newappsblog.com/2014/03/counting-infinities.html

Excerpt: "The philosophical implications of the theory of numerosities for the philosophy of mathematics are far-reaching... Philosophically, the mere fact that there is a coherent, theoretically robust alternative to Cantorian orthodoxy raises all kinds of questions pertaining to our ability to ascertain what numbers ‘really’ are (that is, if there are such things indeed)."



Sunday, April 13, 2014

This Sunday's Math Sermon…


"...every decent maths teacher knows that teaching maths is about understanding how different people think. Some students need to see the bigger picture first – they need context, they need a reason for doing things, they need to know what the end result will be. Others are happy to discover things for themselves and they enjoy the process as much as the outcome. Some are comfortable thinking in an abstract way; others need a more concrete approach. Many students impose their own rules which don’t quite work and we need to unpick what they are doing and figure out what their underlying thinking is.  I like this challenge of figuring out how people think. Studying history gave me a good grounding in understanding that not everyone sees the world in the same way."
The above comes from a li'l musing (dare I say rant?) from a British secondary math teacher who has a history degree where one might expect a math background. I LOVE what she has to say here:

http://mathematicalmagpie.blogspot.co.uk/2014/04/you-mean-you-have-arts-degree-pulls.html

Toward the end she writes, "I’d like to finish by arguing that, rather than being afraid of people who cross the invisible arts/science divide, more of us should come out and celebrate it."

...Amen

And lastly she vows: "All I can say is, if anyone tries to put me in a box, I’ll soon be fighting my way out."

As a bird lover, I even love this blogger's Internet handle: "the mathematical magpie"... go read the whole lovely post.


Thursday, April 10, 2014

Stats and Primes...


Today, a couple of statistics-related readings from the latest John Brockman/Edge volume, "What Should We Be Worried About?"

1) Bart Kosko on what he deems five "lamplight probabilities" that help explain the world we observe, but also limit our analysis, and 'especially restrict modern Bayesian inference':

http://edge.org/response-detail/23856

2) And Nassim Taleb talking about one of his favorite subjects, "fat tails," and why "having skin in the game" is a necessary component for honest, accurate measurement of risk:

http://edge.org/response-detail/23839

...In other matters, an update on the twin-prime conjecture, now down to an upper bound of 252 (from the original approach based on Zhang's work):

http://tinyurl.com/m3pt98s


Wednesday, April 9, 2014

From Seinfeld to Alex Bellos










Apparently George Costanza was onto something…:



"1" is perhaps the loneliest number, and "110," according to Alex Bellos, the least loved, but his research pinpoints "7" as the public's favorite number:

http://tinyurl.com/n24nqjr

ADDENDUM (I'll just pass along a few more points from the article):

Bellos admits it wasn't necessarily a rigorous or fair statistical sampling, but nonetheless over 30,000 respondents took part. A few interesting tidbits:

8 of the top 10 favorite numbers are single digits, the other two slots taken by 11 and 13 (but all numbers from 1 to 100 were chosen at least once). I was surprised that 13 came out as high as sixth place, but perhaps that has something to do with Taylor Swift's choosing it as her well-publicized favorite number.  Numbers ending in either 0 or 5 (except for 5 itself) were particularly avoided as favorites. Bellos notes that 7 is unique in many ways and that probably accounts for its popularity.

This material all comes from Alex's new book, "Alex Through the Looking-Glass," (out in the UK, but not available until June in the US, and then under the title "The Grapes of Math").




Tuesday, April 8, 2014

Good Listenin' and Readin'


Sol Lederman's latest inspiring interview is with Tim Chartier, newly out with his slim, but rich, book, "Math Bytes" (also famous for his basketball "bracketology" research):

http://wildaboutmath.com/2014/04/06/tim-chartier-inspired-by-math-34/

As usual, Sol covers a lot of interesting ground here, and we learn a lot about Dr. Chartier.

And another new book, "Infinitesimal," by Amir Alexander is reviewed in NY Times and sounds good. A lot of folks who take calculus somewhere along the way, don't ever learn about the checkered history of it, and the original controversy surrounding the concept of an "infinitesimal." So read up:

http://tinyurl.com/mnj9wzy

It's going to be another fertile year for popular math books!

If you're in a hurry for more good and varied readin' then the 109th Carnival of Mathematics is newly-posted and just a click away:

http://tonysmaths.blogspot.co.uk/2014/04/109th-carnival-of-mathematics.html

Sunday, April 6, 2014

Four From Friday


Friday brought several interesting, varied reads across my computer screen:

1) Scientific American got us started with a little math history lesson, centered around "infinitesimals" coming in and out of favor, and the re-casting of calculus:

http://tinyurl.com/k5ym5qd

2) On the light side, The Guardian offered an excerpt about our response to numbers, from Alex Bellos' newest book, "Alex Through the Looking Glass":

http://www.theguardian.com/books/2014/apr/04/why-all-love-numbers-mathematics

It starts off telling about a man named Jerry Newport, a retired taxi driver with Asperger's Syndrome who has "an extraordinary talent for mental arithmetic." Also, turns out that Jerry's living room includes "a cockatoo, a dove, three parakeets and two cockatiels"… THIS is a man I can relate to! ;-) -- I've had a similar living room in the past!… though I lack Jerry's number talents. Anyway, many of Jerry's unexplained skills interestingly center around prime numbers. (BTW, I touched on the subject of linkage between Asperger's and math ability a bit ago.)
The rest of Bellos' piece deals with various subjective (and seemingly inexplicable) oddities about integers, and our relationships to them.
An interesting, fun read, touching on language and psychology in addition to math.  If the rest of the book is this entertaining, jolly good!

3) A bit more advanced, Adam Kucharski writes a piece for Nautilus on the startling Weierstrass Function, which pre-saged "fractals" -- a fascinating function that is continuous, yet lacks a derivative at any given point (is "smooth" NOwhere). This was contrary to all prior math thought, and "With one bizarre equation, Weierstrass had demonstrated that physical intuition was not a reliable foundation on which to build mathematical theories":

http://nautil.us/issue/11/light/maths-beautiful-monsters

(this interesting extract comes from a longer article Nautilus subscribers can access)

4) Finally, in a fascinating bit of logical legerdemain, Arkady Bolotin has linked the P vs. NP Millennium Problem to quantum mechanics, and in so doing reached conclusions about both. A longstanding puzzle in quantum theory is how to apply equations that work so well at the quantum level to the world we actually live in and experience. Bolotin argues that while Schrodinger's equation has relatively simple solutions at the atomic-level, at the macro-level it becomes NP-hard (essentially unsolvable). Essentially he's killing two birds with one stone: claiming that P ≠ NP (which is what most assume, but have yet to prove) and that the quantum inscrutability of our world is the result of Schrodinger's equation being essentially unsolvable (within reasonable time) at the macro level:

https://medium.com/the-physics-arxiv-blog/7ef5eea6fd7a

And if you have any energy left after reading all these you can go catch the latest lengthy blather (re: math education) I've put up on MathTango this morning.


Friday, April 4, 2014

Visit MathTango


Nothing new here, but linkfest up at MathTango now, and on Sunday will be a longish post there related to math education.

Tuesday, April 1, 2014

The Joy of Number Theory ;-)


Number theory is both one of the most interesting and arcane areas of all mathematics… arcane in that oftentimes findings or proofs within the field appear to have no practical application. Such was true of a finding from four decades ago which had no known application until this very week. As reported back in 1975:
"…when the transcendental number e is raised to the power of π times √163, the result is an integer. The Indian mathematician Srinivasa Ramanujan had conjectured that e to the power of π√163 is integral in a note in the Quarterly Journal of Pure and Applied Mathematics (vol. 45, 1913-1914, p. 350). Working by hand, he found the value to be  262,537,412,640,768,743.999,999,999,999,…. The calculations were tedious, and he was unable to verify the next decimal digit. Modern computers extended the 9's much farther; indeed, a French program of 1972 went as far as two million 9's. Unfortunately, no one was able to prove that the sequence of 9's continues forever (which, of course, would make the number integral) or whether the number is irrational or an integral fraction.
"In May 1974 John Brillo of the University of Arizona found an ingenious way of applying Euler's constant to the calculation and managed to prove that the number exactly equals 262,537,412,640,768,744. How the prime number 163 manages to convert the expression to an integer is not yet fully understood."
Only recently was it realized that this arcane mathematical number-theory result, which is now much better understood, could be put to practical use… by bloggers wishing to entertain on April 1st, 2014! ;-))

Yes, the above, for any who don't immediately recognize it, was an April Fool's fabrication from prankster Martin Gardner for his classic April 1975 column in Scientific American. While the reference to Ramanujan is true (except that Ramanujan knew the number involved was transcendental), and the computed number, as given, is accurate as far as it goes, the rest of the passage was a hoax that fooled many at the time ("John Brillo" was a play on the name of another number theorist). The next digit following the string of 9's that Martin listed, is actually a "2".
You can read a bit more about the interesting number from this old journal article (Pi Mu Epsilon Journal, Vol. 5, Fall 1972, No. 7, pgs. 314-15; "What Is the Most Amazing Approximate Integer in the Universe?" by I.J. Good):

http://www.pme-math.org/journal/issues/PMEJ.Vol.5.No.7.pdf

My quotation above comes from chapter 50 of Martin Gardner's "The Colossal Book of Mathematics." Of course, after all these years he still entertains us.

Ohhh, and by the way, your shoelaces are untied!....


Saturday, March 29, 2014

Make It An Ed Frenkel Weekend


Ed Frenkel

NOT Ed Frenkel
'Some truths are out there...'  -- Edward Frenkel

The hottest ticket in mathematics these days surely must be Ed Frenkel!

Yesterday I offered plenty of mathy links for the weekend over at MathTango (including a podcast with Ed)… but when you're done with those, carve aside another hour+ for the below video of Dr. Frenkel giving a public presentation (for about 20 minutes) at the LA Public Library before Chris Carter of X-Files fame joins him on stage (and later they take questions from the audience) -- I absolutely LOVE this whole talk which covers a lot of ground, but nothing too techincal! (my favorite video yet involving Dr. Frenkel). If you don't find it inspiring, well, you may need to go have your pulse checked ;-)

Edward Frenkel and Chris Carter: Love, Mathematics and The X-Files from ALOUDla on Vimeo.


[p.s. -- as I watched this I suddenly couldn't help but think how much Dr. Frenkel resembles David Duchovny! ;-) -- above pics, Frenkel from his Google+ page and Duchovny from Wikipedia]

Secondly, in a 15 minute NPR "Snap Judgment" segment Frenkel describes his incredible, inspiring math upbringing in the then Soviet Union (even if you've read or heard of his Russian background before, again, I recommend listening to this wonderful narrative):

http://tinyurl.com/oxdpym7


Thursday, March 27, 2014

Freethinker Dyson...


via Wikipedia

Fascinating article and interview with mathematician/thinker/puzzlemeister/physicist (and "rebel") Freeman Dyson over at Quanta Magazine:

https://www.simonsfoundation.org/quanta/20140326-a-rebel-without-a-ph-d/

Somehow I never realized that he DOESN'T have a PhD. and at 90-years-old sounds as feisty, insightful, brilliant as ever! I'd quote some lines from the article but it's SO good (especially the interview portion) it's impossible to pick, so go enjoy the whole thing (also includes a 5-minute video).

A couple years back I did a short post on "Dyson numbers" which included a famous anecdote about Freeman here:

http://math-frolic.blogspot.com/2012/10/next-up-dyson-numbers.html

I think some evening I'd like to have Dyson and Raymond Smullyan (94 years-old) over for pizza (maybe throw in 76-yr.-old whippersnapper John Horton Conway, as well) and see if, bouncing ideas off one another, they couldn't prove the Riemann Hypothesis before David Letterman comes on ;-)

[side-note: tomorrow I should have some weekly potpourri picks posted at MathTango.]


Wednesday, March 26, 2014

In Honor of a Non-mathematician...


Dave Richeson, just back from Atlanta and the 11th Gathering For Gardner (which got rave reviews!) reports on the interesting "gift exchange" therefrom:

http://divisbyzero.com/2014/03/25/gabriels-paper-horn/


The theme for the conference was "John Horton Conway," though I dare guess presentations and activities strayed pretty far afield, covering Martin Gardner's wide-ranging interests in math, magic, science, and philosophy.

I suspect there will be more blog reports (from some of the 350 attendees) in the next week on the always-unique conference … anyone who attended and writes a blog-post, please feel free to add a link to your post in the comments.

Also, note the new wonderful Martin Gardner website in honor of this, his Centennial year:

http://www.martin-gardner.org


One can imagine Martin, wherever he is, almost embarrassed by all the attention (while he plays nimbly with a hexaflexagon) :-)


I'll end with some of the words from Doug Hofstadter's eulogy to him that I've quoted previously:
"He is totally unreproducible -- he was sui generis -- and what's so strange is that so few people today are really aware of what a giant he was in so many fields -- to name some of them, the propagation of truly deep and beautiful mathematical ideas (not just 'mathematical games,' far from it!), the intense battling of pseudoscience and related ideas, the invention of superb magic tricks, the love for beautiful poetry, the fascination with profound philosophical ideas (Newcomb's paradox, free will, etc. etc.), the elusive border between nonsense and sense... Martin Gardner was so profoundly influential on so many top-notch thinkers in so many disciplines -- just a remarkable human being -- and at the same time he was so unbelievably modest and unassuming. Totally. So it is a very sad day to think that such a person is gone, and that so many of us owe him so much, and that so few people -- even extremely intelligent, well-informed people -- realize who he was or have even ever heard of him. Very strange. But I guess that when you are a total non-self-trumpeter like Martin, that's what you want and that's what you get." 
 Thanks to all who keep spreading his name and legacy to a wider audience and future generations.


Sunday, March 23, 2014

"Streetfighting Math"


I've not seen this book myself, but if Steven Strogatz endorses it I figure it must be worth passing along:

http://streetfightingmath.com/

Here's a promotional trailer for it:



It's also a course over at EdX:

https://www.edx.org/course/mitx/mitx-6-sfmx-street-fighting-math-1501

Finally, there is an interesting review of the book in AMS Notices here:

http://www.ams.org/notices/201107/rtx110700960p.pdf

It starts off somewhat negatively, and than wavers back-and-forth between positive and negative points, before concluding as follows:
"…'Street-Fighting Mathematics' is an engaging, well-written, insightful book. I do know that the book will provide any reader with new tools for making quick estimates and will introduce new ways of viewing problem solving. And I do know that I will read this book again. And again. And again. And maybe one day I will take the leap and actually use these methods to solve a problem."


Friday, March 21, 2014

T.S. Eliot Teaches Evelyn Lamb Some Math


Well not exactly, but I do love the way Evelyn brings Eliot's words into a mathematical context for this wonderful meditation at her "Roots of Unity" blog:

http://tinyurl.com/lwo6cbz

Patrick Honner similarly notes in a tweet that "reinventing the wheel" is in a sense what a teacher needs to do in order "to learn to teach something well."

It is a common complaint that too often students learn mathematics procedures or methods, without really comprehending the deeper process behind those methods.
Evelyn's mention of calculus, in this regard, caused me to recall my own dismal experience with college calculus:
[Evelyn's words]: "Calculus is far from a basic topic in math, but it is one that is learned very early, before we understand much about just how intricate it is. We do a lot of exploring before we come back and truly understand calculus for the first time." 
My experience: most days our prof would spend 45 mins. with his back to us as he wrote on the blackboard some long drawn-out proof of whatever we were covering that day, and then with 3 minutes of time left on the clock would turn around and ask, "any questions?" Those of us who hadn't fallen asleep didn't have the nerve to explain that we were lost from step 4 on… and then the bell rang anyway. On the one hand it was a horrible way to teach, but in retrospect I also understand now that he believed to really grasp the 'intricacy' of calculus you needed to be shown how each piece of it was derived from the ground up.

I mention all this because most of my younger life I had little interest in (or patience for) proofs or explanations, but just wanted to learn the 'facts' of math and how to apply them. Only later in life did I come to appreciate that it is by fully understanding such proofs or processes that one acquires a deep grasp of mathematics and mathematical thinking. My concentration on doing the methods and working the numbers got me through high school in fine form, but probably contributed to my math downfall in college.
Moreover, in the logic, reasoning, and proofs of mathematics, once comprehended, lie more of the beauty that is so easily missed if you treat math as just a bag of manipulative symbolic tricks, as many may perceive it.
As Evelyn implies, math is more a constant "exploration" than a rote process or recitation, and that remains so whether you are a student… or teacher.


Thursday, March 20, 2014

The Bigness of Math and Other Things


Unification... it's not just for physics anymore. A "theory of everything" (or TOE), unifying our knowledge of the Universe's laws), has been a goal of physicists for quite awhile now. In a nice, short piece, Peter Lynch points out that unification is similarly an ongoing objective within mathematics:

http://thatsmaths.com/2014/03/20/the-unity-of-mathematics/

He writes that there is a "tendency for mathematics to fracture into many disparate areas," but "From time to time, sweeping simplifications arise [in mathematics] when seemingly unrelated areas are embraced in a single unifying framework."

He points out a few historical examples before noting that there is "a marked distinction between discrete and continuous mathematics," and then citing the Langlands Program, which is also "intimately related to modern physics," as the current attempt to unify all of mathematics (it was given widespread publicity with Ed Frenkel's 2013 book, "Love and Math.")

Anyway, a quick, non-technical read. 

In another straightforward, non-technical... and timely... read, Mark Chu-Carroll explains why it could take soooo long to find Malaysia flight 370 -- i.e., why our intuition for big numbers is not very good. A Boeing 777 is a big, BIG, BIG plane... but even a patch of ocean is so many times BIGGER!:

http://www.goodmath.org/blog/2014/03/19/big-numbers-and-the-lost-plane/

[...As I type these words the latest news rumor coming in is that debris spotted off of Australia MIGHT be from the lost plane...]

 

Monday, March 17, 2014

New Interview...


If you can pull yourself away from today's BICEP-2 cosmology news ;-), the latest interview over at MathTango is with author/math professor John Allen Paulos!:

http://mathtango.blogspot.com/2014/03/john-allen-paulos-spreading-math.html