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Friday, June 29, 2012

Friday Puzzle

We're back on the ol' island of liars and truth-tellers….

I've written a random number on a card and shown it to Mark, Nancy, Owen, and Patty, one of whom is a "liar" (who always lies, and the other 3 being truth-tellers who always tell the truth).

Mark remarks that "It is not 150"
Owen notes, "It has two digits"
Nancy says, "It divides evenly into 150"
Patty says, "It is divisible by 25"

Which islander is a liar:

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ANSWER:

Patty is the liar

Wednesday, June 27, 2012

Pi = 2

A well-circulated paradox with a circle inscribed in a square results in π being equal to 4, but what I hadn't previously noticed was this similar James Tanton offering making pi equal to 2 (obviously considering these two paradoxes, pi must, on average, be equal to 3 ;-))




Tuesday, June 26, 2012

Playing Devlin's Advocate


Just Devlin, Devlin... and more Keith Devlin today:

A few posts back I mentioned that Gregory Chaitin regards himself as "a Pythagorean" -- one who believes that mathematics is essentially what the Universe is made of… of course a great many mathematicians hold this view or something close to it -- BUT, not all… Keith Devlin, one of the most well-known popularizers around has a different take.
Devlin's neither a Pythagorean nor a Platonist, and not at all certain that the universality that so many see in mathematics is real, rather than merely a creation of the human brain. He's not at all dogmatic about it, nor does he have specific alternative candidates for reality, but simply thinks that humans may live "in a cave" and are only capable of seeing the walls of that cave, and the "shadows" thereupon. A different intelligent life-form from somewhere else in the Universe just might have evolved in a different (unimaginable to us) sort of cave, with a different reality, including a different mathematics. It's both a simple and a mind-blowing notion…

Devlin gave a talk at the SETI Institute expressing these views. His talk is about 45-minutes long, followed by a half hour question/answer session. The latter period indicates how difficult it is for many folks to accept or wrap their heads around the non-Platonist view -- Platonism is sort of the easy, or default, position here; Devlin's view takes more concerted effort to hold onto. I sort of wish Martin Gardner, outspoken Platonist, could've been in the audience to see what questions he would pose to Keith! (…or perhaps at some time, before Martin's death, these two had occasion to sit down over coffee and discuss the two viewpoints? Anyone know???)



This Edge essay from Devlin (which I've linked to previously), much more briefly summarizes the view he espouses above:

http://www.edge.org/q2008/q08_4.html

Here's another longish YouTube presentation from Keith, this time on math education, another favorite topic of his:



You ought also know, if you don't already, that Devlin is on Twitter here:

https://twitter.com/#!/profkeithdevlin

Finally, his many books (on Amazon) here:

http://www.amazon.com/Keith-Devlin/e/B000APRPC6

Monday, June 25, 2012

Turing 101

 "What do [people] say about the Turing machine? It's "the simplest computing device". It's the 'basis for modern computers'. It's 'the theoretical model for the microchips in your laptop'. It's the 'mathematical description of your computer'.  None of those things are true. And they all both over and under-state what Turing really did. 
In terms of modern computers, the Turing machine's contribution to the design of real computers is negligible if not non-existent. But his contribution to our understanding of what computers can do, and our understanding of how mathematics really works -- they're far, far more significant than the architecture of any single machine. 
"The Turing machine isn't a model of real computers. The computer that you're using to read this has absolutely nothing to do with the Turing machine. As a real device, the Turing machine is absolutely terrible. 
"The Turing machine is a mathematical model not of computers, but of computation. That's a really important distinction." -- Mark CC
At this time of tremendous media coverage of Alan Turing's work (100th anniversary of his birth), Mark Chu-Carroll offers an excellent primer on what Turing machines really are… and aren't (longread):

 http://scientopia.org/blogs/goodmath/2012/06/24/turing-machines-what-they-are-what-they-arent/

Sunday, June 24, 2012

For Your Viewing Pleasure...

A Sunday meditation (of sorts)... for the math-inclined, a lovely bit of short filmmaking here (hat tip to Dave Richeson):



Learn more about the film and the images used, here:

http://www.etereaestudios.com/docs_html/inspirations_htm/maths_index.htm

Friday, June 22, 2012

Birthday Puzzler

Reginald has a birthday today. After celebrating by blowing out all his cake candles, he noted that he'd blown out a total of 528 candles in his lifetime (assume he has blown out the correctly representative number for each year of life). How old is Reginald today?

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ANSWER:

32 yrs-old

[ x(x+1)/2 = 528 (solve for x) ]

Thursday, June 21, 2012

A Nice Find

"Regrettably, many of us have never been allowed to see what mathematics is. It has been obscured by pointless emphasis on routines rather than ideas. This failure to distinguish what is important has led many people to see mathematics as a collection of totally arbitrary rules which have to be learnt by rote, and performed with the exactness and precision of a religious rite." --Laurie Buxton

One of the perks of living in a university town is the number of used bookstores around, and used book garage sales through the year. I often find older math book gems at such places, and picked up several at a recent sale. One (I'd never seen before) quickly impressed me as yet another fantastic introduction to math for a young person (middle/high school): "Mathematics For Everyone" by Laurie Buxton, originally published in 1984! I'd never heard of either the book or the author, and only after reading several pages into the volume did I realize it was once again a British writer and publication -- I apologize to my British readers who likely already know of the book and author. It turns out Mr. Buxton died about a decade ago (here's an obituary from the time: http://www.guardian.co.uk/news/2002/jan/15/guardianobituaries.highereducation ).

I've commented previously how many wonderful British explicators of math there are and am convinced this is not an illusion on my part, nor a sampling error -- for their relative population, the Brits just seem to produce more excellent math writers than America! Don't know if this has something to do with the way math is taught in Britain (where there seems to be almost as much math education criticism as in America), or perhaps has more to do with the way writing is taught!?? At any rate, Mr. Buxton's book was a delightful find for $1!

If you have any thoughts on British vs. American math education or writing, or further thoughts on Mr. Buxton, or perhaps just other old unheralded book 'gems' you're fond of, please feel free to comment.


Wednesday, June 20, 2012

Gowers On Math

A recent LOOOONG, rich post from the ever-thoughtful/insightful Tim Gowers on math education here:

http://tinyurl.com/6q6dfln

It's chockfull of LOTS of examples and ideas to chew on and think about (...and debate over). Also, LOTS of comments... don't expect to read it all in one sitting!

Tuesday, June 19, 2012

Math Fun & Games


From "Futility Closet," interesting visual proof to a simple geometry question here:

http://tinyurl.com/7mbebmn

Meanwhile, the latest (51st!) Math Teachers At Play blog carnival for your delectation here:

http://mathmamawrites.blogspot.com/2012/06/math-teachers-at-play-51.html

Monday, June 18, 2012

A Mathematician Tells Her Story

Any math lover ought enjoy this... Lovely video of a mathematician explaining why she does what she does:

http://vimeo.com/33615260

I especially like the last minute-or-so of her comments.

Sunday, June 17, 2012

"Proving Darwin"… Readin' Chaitin



I've already briefly referred to Gregory Chaitin's new compact volume, "Proving Darwin: Making Biology Mathematical," in a couple of posts, but time for a longer blurb…

Chaitin is known for his emphasis on uncertainty and 'unknowability' in mathematics, a topic I find of great interest (even though what most of us love about math is its precision!). His previous popular book, "Meta Math," was a volume that some might say starts where Gödel left off.

His new book (barely 100 pages), is a compendium of university course lectures -- as such, it is a bit choppy, terse, repetitive at points, and informal, but still, I think, a wonderful read from this stimulating, creative thinker. According to Chaitin the book was, interestingly, provoked in part by David Berlinski's "The Devil's Delusion."

In the volume, Chaitin proposes a new field of study he calls "metabiology," or the study of "the random evolution of artificial software (computer programs) instead of natural software (DNA)." He regards DNA as a computer program and his "mad dream" is to develop a "fundamental mathematical theory for biology." His "purpose" is "to lay bare the deep inner mathematical structure of biology, to show life's hidden mathematical core"… or, as he also puts it, "If Darwin's theory is as fundamental as biologists think, then there ought to be a general, abstract mathematical theory of evolution that captures the essence of Darwin's theory and develops it mathematically."

Chaitin presents what he terms a "toy model of evolution" and tries to demonstrate among other things that mathematically, the evolutionary diversity we witness today, can be explained in terms of mathematical algorithms of random mutation over the given immense time scale. I don't know how many will feel he succeeds in his goals, but it is fascinating to see him try [mind you, I don't myself grasp the technical aspects of Chaitin's presentation here. Also, it seems unfortunate that he calls the model of mutation he ends up favoring, "intelligent design," even though it doesn't necessarily imply the more widespread use of that term entailing a deity at work.]
And Chaitin himself knows he may not yet be convincing: at the conclusion of the book he remarks, "Even if almost everything in this book is wrong, I still hope that "Proving Darwin" will stimulate work on mathematical theories of evolution and biological creativity. The time is ripe for creating such a theory." "Creativity" (both mathematical and biological) is what Chaitin is all about, and the 'algorithmic information' that underlies it, and in this case predicts the rate of evolution. (Possibly worth noting that this book might be more apt for computer science and bioinformatics folks than for either straight mathematicians or biologists.)

Chaitin's views are hard to capsulize. He calls himself "a Pythagorean" in the sense that he believes "the world is built out of mathematics, that the ultimate ontological basis of the universe is mathematical, which is the hardest, sharpest, most definite possible substance there is, static, eternal, perfect." Yet he is not a strict reductionist… as I've said, his overriding emphasis is on the "creative" nature of humans and of biology more generally. At one point he notes, "In biology nothing is static, everything is dynamic. Viruses, bacteria and parasites are constantly mutating, constantly probing, constantly running through all the combinatorial possibilities. Biology is ceaseless creativity, not stability, not at all." I'm not sure how professional biologists themselves will feel about Chaitin's ideas/assertions here. References to biologists aren't all that many in the volume (some to Haldane, Maynard Smith, Leigh Van Valen, and Ernst Haeckel), and I'm not sure many current biologists won't feel Chaitin has bypassed a lot in trying to make his case -- he accuses biologists of talking too much "about stability and fixed points" and not enough "about creativity" (they might take exception to that). It's just that the algorithms for biology are incredibly complex (far moreso than in physics) and thus difficult to readily discern. We'll be working on them for decades (centuries?) to come.
Chaitin notes that "every cell is run by software, 3- to 4-billion-year-old software" and "Our bodies are full of software, extremely ancient software. We have subroutines from sponges, subroutines from amphibia, subroutines from fish." It will require "postmodern math" (since Gödel and Turing, or what he calls "open, non-reductionist kind of math") to comprehend biology. This will be achieved via the computer: "The computer is not just a tremendously useful technology, it is a revolutionary new kind of mathematics with profound philosophical consequences. It reveals a new world."
Among those referenced frequently and favorably in the book are Leibniz, Gödel, Turing, Stephen Wolfram, and von Neumann (to whom the book is dedicated); even philosopher Paul Feyerabend gets a favorable nod.

Chaitin's book seems very much a work-in-progress, and it will be interesting to see what comes of his ideas over forthcoming years. If you're interested in both math and biology, or if you've read and enjoyed Chaitin's work in the past you'll want to peruse this small volume… or you can simply watch videos of his lectures on the subject via YouTube:

http://tinyurl.com/6mutwov

also, an older hour-long podcast interview with Chaitin from "Math Factor" here:

http://mathfactor.uark.edu/2010/05/gy-chaitin-on-the-ubiquity-of-undecidability/  



...Another book side-note: One other new volume recently arriving in my local bookstore catching my eye (though not strictly a math book), is John Casti's "X-Events: the Collapse of Everything." Casti is a mathematician and complexity theorist who is always interesting (though this volume looks a tad foreboding).

http://www.amazon.com/X-Events-Collapse-Everything-John-Casti/dp/0062088289



Friday, June 15, 2012

Friday Puzzle

For the Friday puzzle-minded, this 'milk and soda' offering from James Tanton:





Thursday, June 14, 2012

Language, Hard Work, E. Asians, & Math Ability


"Singapore, Japan, South Korea, Taiwan and Hong Kong usually top international surveys of mathematics and science skills in the world. Countries we should expect to do as well, such as the United States, United Kingdom, Germany, score lower in the Trends in International Math and Science Study (TIMSS)…."

Why do East Asians do so well in math? Article has some thoughts on the matter:

http://tinyurl.com/7lacnhk


Wednesday, June 13, 2012

Turing... in Film

LOTS of Alan Turing stuff showing up this year in honor of the 100th anniversary of his birth, including this free full-length film thriller on the internet, "The Turing Enigma":

http://theturingenigma.com/

Turing books here:

http://tinyurl.com/74qnjla


Tuesday, June 12, 2012

No Nobel In Mathematics


Here, a brief post on why, perhaps, there is no Nobel Prize in mathematics:

http://mathandmultimedia.com/2012/06/10/nobel-prize-mathematics/

The Fields Medal, Abel Prize, and Wolf Prize are three awards that somewhat substitute for a Nobel in math. (The Fields, presented every 4 years, will next be awarded in 2014, while the Abel and Wolf Prizes are generally awarded annually.)


Monday, June 11, 2012

Measurement

Paul Lockhart on his new upcoming book, "Measurement":



In case you're not familiar with Lockhart, check out these older commentaries from Keith Devlin on Lockhart's earlier, famous piece, "A Mathematician's Lament":

http://www.maa.org/devlin/devlin_03_08.html

http://www.maa.org/devlin/devlin_05_08.html

Friday, June 8, 2012

Friday Logic Puzzle

Given a list of all the prime numbers under 2000, multiply them together by one another to arrive at a single result. What digit will be in the ones position of that final product?
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final digit will be "0" (2 and 5 are primes with a product of 10; anything multiplied by 10 will end in 0)



Thursday, June 7, 2012

Mathematical Truth...

A great, older interview with Rebecca Goldstein on Gödel, Platonism, and everything in-between, from 2005:

http://edge.org/conversation/godel-and-the-nature-of-mathematical-truth

an excerpt:
 "Gödel mistrusted our ability to communicate. Natural language, he thought, was imprecise, and we usually don't understand each other. Gödel wanted to prove a mathematical theorem that would have all the precision of mathematics—the only language with any claims to precision—but with the sweep of philosophy. He wanted a mathematical theorem that would speak to the issues of meta-mathematics. And two extraordinary things happened. One is that he actually did produce such a theorem. The other is that it was interpreted by the jazzier parts of the intellectual culture as saying, philosophically exactly the opposite of what he had been intending to say with it. Gödel had intended to show that our knowledge of mathematics exceeds our formal proofs. He hadn't meant to subvert the notion that we have objective mathematical knowledge or claim that there is no mathematical proof—quite the contrary. He believed that we do have access to an independent mathematical reality. Our formal systems are incomplete because there's more to mathematical reality than can be contained in any of our formal systems. More precisely, what he showed is that all of our formal systems strong enough for arithmetic are either inconsistent or incomplete. Now an inconsistent system is completely worthless since inconsistent systems allow you to derive contradictions. And once you have a contradiction then you can prove anything at all."


Wednesday, June 6, 2012

Crankery


Vortex THIS!....

Ya know sometimes I just wish Mark over at Good Math, Bad Math would actually speak his mind and tell us what he really thinks ;-) :

http://scientopia.org/blogs/goodmath/2012/06/03/numeric-pareidolia-and-vortex-math/


Tuesday, June 5, 2012

Books, Books, Books… Old & New

But first, for sheer entertainment, just a cool (geometry-impinged) illusion:




on to the books:

I'm forever amazed at the number of wonderful math-related books available for the non-professional, like myself! ...Have been looking through (…or looking forward to) several recently, and give all the following a thumbs-up (…apologies that some of these are older works you may well be long-ago familiar with):

"To Infinity and Beyond" -- Eli Maor's work from 1987 may be the best intro to infinity I've yet come across (how I missed it all these years I don't know!); a rich, varied, and comprehensive introduction to an endlessly fascinating topic. A review of it here:

http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=1029

 "The Universal Computer" -- from Martin Davis, 2000; perhaps the best and most readable summary of the historical persons and events that brought us computers as we know them today. A great read and chronology. A review here:

http://www.publishersweekly.com/978-0-393-04785-1

"A Tour of the Calculus"  -- David Berlinski's 1995 volume which I started when it came out almost 2 decades ago, but never got through… having recently read and enjoyed his "One, Two, Three" I decided to give "Tour" another try, and I must've grown accustomed to his quirky, unconventional writing style, because I'm finding it much more delightful (although still ponderous at points) this go-around; certainly the most unique, outside-the-box volume on calculus out there (even for a non-textbook).

Interestingly, I've now read Gregory Chaitin's short new volume on math and biology, "Proving Darwin," and it begins by mentioning that it was inspired in part by yet another "delightfully polemical book" from Berlinski, "The Devil's Delusion." Chaitin's book is a bit choppy but still, to me, a wonderful read that I may say more about after a second reading (although if you haven't cared for Chaitin's writings/ideas in the past, you might not care for this pithy volume).

Finally, tangential to math, I'm a long-time Douglas Hofstadter fan, though didn't much care for his last book, "I Am a Strange Loop" (see Martin Gardner's review here: http://tinyurl.com/7skxw7g), so have been waiting to see what he would do next. Apparently, he has a new volume coming in the fall entitled "Surfaces and Essences," relating the importance of analogy to human thinking.

http://tinyurl.com/6qgpftp

Also, a reminder that Steven Strogatz's book "The Joy of x: A Guided Tour of Math From One to Infinity" is also due out in the fall.

So much good stuff coming our way!! (...and I'm still awaiting the paperback version of Daniel Kahneman's "Thinking Fast and Slow"!)

Lastly, speaking of math books, 'Wild About Math' blog did its latest podcast with Vicki Kearn, an editor at Princeton University Press, which consistently puts forth some of the best popular math books out there (give it a listen):

http://wildaboutmath.com/2012/06/03/vickie-kearn-inspired-by-math-8/


Monday, June 4, 2012

Friday, June 1, 2012

Friday Puzzle

A 10-digit number is represented below, but with the digits replaced by letters of the alphabet (each letter always representing the same digit). Given the clues following the "number" figure out what the whole 10-digit number may be (there are 2 possible answers):

ABDCDDDCDD

1) "A" is the number of 0's in the full number.
2) "B" is the number of 1's in the full number.
3) "C" is the number of 2's in the full number.
4) "D" is the number of 3's in the full number.

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6,201,000,100 or 6,102,000,200