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Tuesday, February 28, 2012

Calling All Mathematical Musicians… or Musical Mathematicians


First, just a couple of recent links to the notion of mathematics as the study of (or 'science of') "patterns" -- there's plenty more on the Web about this simple notion, but here are a couple of starting points:

http://mathpalette.com/2012/02/mathematics-the-science-of-patterns/#.T0pXARsQezs.twitter

http://devlinsangle.blogspot.com/2012/01/patterns-what-patterns.html

The second piece is once again from Keith Devlin and toward the end of it he throws in some discussion of mathematics and music, an association I always find interesting.

I've known a number of individuals who were very skilled at both math and music (playing and/or writing music)… indeed, I've known several folks who were double majors in college, math and music. Often these two interests, one more-or-less from the humanities, and one more-or-less from the sciences, are perceived as opposites involving different mental realms, yet clearly there is an underlying thread that unites them.

Having no great musical talent myself, I am especially impressed with (envious of!) those who do, especially when they simultaneously exhibit the knack for mathematics -- those are the folks I'd like to hear from: Can you explain in your own words how math and music are associated in your minds; how the two intersect for you (assuming they do)? Which interest or talent did you experience first, and how quickly after one did the other follow? Do the two interests carry approximately equal weight for you, or is one more dominant than the other, or does one in some sense 'play off' the other? Are the two creative processes very similar in your mind or do they differ significantly?

Meanwhile, here are a few (of many) webpages that address the mathematics-music connection:

http://www.woodpecker.com/writing/essays/math+music.html

http://serendip.brynmawr.edu/exchange/node/1869

http://en.wikipedia.org/wiki/Music_and_mathematics

Sunday, February 26, 2012

I'm Not a Sudoku Fan, but….


... if YOU are, head right over to "Wild About Math" blog which is already up with it's third podcast… this time a longer one (49 min.) with Jason Rosenhouse and Laura Taalman, authors of "Taking Sudoku Seriously":

http://wildaboutmath.com/2012/02/24/jason-rosenhouse-and-laura-taalman-inspired-by-math-3/

I've not read the book, but based on Rosenhouse's great "The Monty Hall Problem" (the best treatment of that wonderful puzzle I know of), I feel no hesitation in recommending it sight-unseen. The podcast, by the way, does touch on more things than just Sudoku.

It's always been interesting to me that while I very much enjoy Ken-Ken puzzles, Sudoku has never 'grabbed' me… I don't know if that somehow relates to the fact that Ken-Ken at least requires some arithmetic thinking, whereas Sudoku, unlike first appearances, requires no arithmetic (it is simply the logical manipulation of 9 symbols; it relates to mathematical thinking, but not to "arithmetic" or computation). Any others have a similar experience? or care to comment on what makes Sudoku so addictive for some of you???

Finally, Keith Devlin's wonderful review of the Rosenhouse/Taalman volume is here:

http://online.wsj.com/article/SB10001424052970204301404577173022950738492.html

Here's one excerpt:
"The authors show vividly that mathematics is really about the power of abstraction, the push to explain as much as possible in the most compact form possible. Numbers and arithmetic are a part of that enterprise, but there is a lot more besides. "Taking Sudoku Seriously" is an excellent vehicle whereby devotees of the puzzle can come to understand the nature of mathematics."
But I love even more a passage from Dr. Devlin that follows the above:
"The puzzle never really interested me. Not because I did not recognize at once its mathematical character. Rather, it didn't offer me anything I did not get from my day job. To me it was mathematics in significantly diminished form. It reminded me of the younger me who loved rock climbing but never enjoyed indoor climbing on artificial walls. Yes, the individual moves were the same, but the climbing wall lacked the grandeur and sense of reality—and truth be told the heightened senses aroused by the slight chance of death—of a genuine rock face.
"In fact, I have never found puzzles satisfying; I always get a far greater thrill from mathematics, which has a rich and deep aesthetics, to say nothing of a huge importance to human life, that puzzles lack, though at a micro level the intellectual challenge is much the same."
Mathematics as 'rock-climbing'... gotta love it!
Anyway, I recommend you read the book, Devlin's review... and "Wild About Math" blog!

Friday, February 24, 2012

Friday Puzzle

(First, I'm still interested in further responses to the prior post if any care to contribute...)

But on to a Friday puzzle I've adapted from the same essential problem recently presented over at CTK Insights:

What is the final answer for this computation:

3/2 x 4/3 x 5/4 x 6/5 x…. x 1999/1998 x 2000/1999 = ______

Answer below:
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ANSWER: 1000

Peruse the first 3 possible results…

3/2 x 4/3 = 12/6 = 2/1 or 4/2

3/2 x 4/3 x 5/4 = 60/24 = 5/2

3/2 x 4/3 x 5/4 x 6/5 = 360/120 = 3/1 or 6/2

...it becomes clear that the answer always entails the final given numerator over the initial denominator or "2".
Thus, the answer for the problem as stated is 2000/2 or 1000.

Wednesday, February 22, 2012

What Brung Ya To Mathematics?...

M.I.B.T.! ....
(Math Iz A Beautiful Thang!)


If you're following Sol Lederman's new podcast series over at 'Wild About Math' (that I've been linking to) you know that the question he is using as a jumping off point is, "How is it that some people are inspired by math... and how can we bottle that?" This is a pretty fundamental question that I imagine interests many of us.
For purposes of this blog, I'm less interested in the "bottling" part, which has mostly to do with teaching math and introducing it to young people in particular -- very important subjects, but ones dealt with (and debated over) by a great many other Websites already, without me entering the fray. But I am interested in why so many people can easily recognize the beauty of a piece of music or art, yet only a small subset seem moved by the beauty (granted, it's less sensory, more abstract/cerebral) of mathematics. An old, well-worn clip from Richard Feynman touches on the matter, and most of you have probably seen it, but as I always say, you just can't watch Feynman too often, sooooo:



(As an aside, I've never read a bad biography of Feynman, but I just finished last year's volume from Lawrence Krauss, entitled "Quantum Man," and it is particularly good and recommended.)

Anyway, the question Sol is posing to some well-known/recognized mathematicians I think probably has interesting answers from all manner of folk who find mathematics inspiring. I don't have nearly the readership of Sol (so don't know that I'll draw much response!), and I don't mean to steal his thunder, but thought I'd just use this post to open up the question to all math fans who care to express themselves: what is it about math that you find beautiful/inspiring, or what initially attracted you to it, whether that was at the age of 5 or 10 or 20 or much later? (...Did it involve a specific teacher, parent, class, book, or problem, or just seem innate?) -- express your views in the comments. (I'm especially interested to see what elements are common to many people, versus to what degree everyone's experience is a little different?)....

Tuesday, February 21, 2012

For Your Reading Enjoyment...

In case you're looking for just a tad more math-reading on the Web….

Great listing of over 500(!) math-related blogs/sites broken into 15 categories here:

http://www.mathblogging.org/bytype

...and if you're on Twitter, a list of 50 pertinent Twitter feeds for math fans here:

http://www.bestcollegesonline.com/blog/2011/09/07/the-50-best-twitter-feeds-for-math-geeks/

Monday, February 20, 2012

"Does Math Really Exist"


"The bottom line is that human beings have brains capable of counting to high numbers and manipulating them, so we use mathematics as a useful tool to describe the world around us. But numbers and math themselves are no more real than the color blue – ‘blue’ is just what we tag a certain wavelength of light because of the way we perceive that wavelength. An alien intelligence that is blind has no use for the color blue. It might learn about light and the wavelengths of light and translate those concepts completely differently than we do."
The above comes from a Forbes magazine piece, of all places, that touches on (I think pretty weakly) the whole Platonist/Non-Platonist debate within mathematics... (does the human mind discover mathematics or create it?):

http://www.forbes.com/sites/alexknapp/2012/01/21/does-math-really-exist/

Long-term readers here know that I am fascinated with this very basic (philosophical) question of whether mathematics is a real part of the extant world, or merely a human cognitive construction. (see Web Platonism articles here or here.)

I have to believe that most of us with math inclinations grow up essentially as Platonists -- perceiving mathematics as real, whether or not any humans exist to explore it. But somewhere along the line, a lot of mathematicians step back to explore the question more openly, objectively, and casting biases aside, and end up swayed to the Non-Platonist side… I'm always impressed by some of these folks who not only see it that way, but seem to think it is rather obviously so (a mere human construction). While still leaning toward mathematical Platonism myself, each year I feel less certain of it. ...Circles, lines, prime numbers, etc. do not exist in the same sense that stars, hydrogen, or atoms do, but does that really make them less real than the latter??? -- they may not represent "real" things, but do they not represent real "relationships"?

One of the ideas that makes non-Platonism at least slightly more palatable is the notion, popular in many physics quarters these days, of a "Multiverse" -- the concept that the Universe we humans have long studied may be only one of many (or even an infinite number of) separate universes that exist. The "laws" and order we discover operating throughout 'our' universe, may simply not be operative in other unseen, separately-evolved universes (whose mathematics might therefore be hugely different). And then there is also (physicist) Max Tegmark's view that the entire Universe is nothing but mathematics ( “there is only mathematics; that is all that exists”).

The Platonist debate includes some purely semantic elements, and is less black-and-white than can be fleshed out here -- there are several possible non-Platonist stances, and the Platonist view itself comes in stronger and weaker forms -- certainly too complex an issue to ever resolve here... yet, a topic I'm continually drawn back to from time to time....

Lastly, a bit of an aside -- in the process of composing this post I chanced upon a couple of wonderful, past Martin Gardner (a well-known Platonist) book reviews bearing on the subject -- but you can read them for the sheer enjoyment of Gardner's prose apart from any philosophical content!:

http://www.newcriterion.com/articles.cfm/Larger-than-proof-2299

http://www.newcriterion.com/articles.cfm/Still-four-4349

Saturday, February 18, 2012

Connecting the Dots (or Cities)...

Via Wikimedia Commons

Sol Lederman, over at 'Wild About Math' is already up with his 2nd podcast interview, this time with William Cook, author of "In Pursuit of the Traveling Salesman." If this famous and fascinating math conundrum interests you, definitely tune in (…ohh, and p.s., it ought interest you! ;-) -- it is one of the Clay Institute's million-dollar problems):

http://wildaboutmath.com/2012/02/17/william-cook-inspired-by-math-2/

And here's a short review of Cook's book from elsewhere on the Web:

http://www.mathteacherctk.com/blog/2011/12/in-pursuit-of-the-traveling-salesman/

Friday, February 17, 2012

Friday Puzzle


For a pre-weekend puzzle I've adapted this algebraic riddle from a Math Mom's puzzle of a couple of months back:

Taylor Swift's traveling bus is coming down Melody Blvd. one day, and one of her fans wants to estimate the length of the vehicle while it is steadily, uniformly moving. He walks alongside the moving bus from back to front (same direction as bus is moving), at a consistent speed, and counts 30 steps. Then, immediately, he turns back and walks alongside the bus in the opposite direction, requiring only 15 steps to reach the back end (at the same speed). If every step he takes is consistently 2 feet, then what is the length of the bus?

I'll put Math Mom's approach to a solution in the comments (I computed it slightly differently, but arriving at same answer).

Thursday, February 16, 2012

The Impact of "Big Data"

"GOOD with numbers? Fascinated by data? The sound you hear is opportunity knocking."

So begins this recent NY Times article on the coming flood of analytical data:

http://www.nytimes.com/2012/02/12/sunday-review/big-datas-impact-in-the-world.html?pagewanted=1&hp

An excerpt:
"Most of the Big Data surge is data in the wild — unruly stuff like words, images and video on the Web and those streams of sensor data. It is called unstructured data and is not typically grist for traditional databases.
But the computer tools for gleaning knowledge and insights from the Internet era’s vast trove of unstructured data are fast gaining ground. At the forefront are the rapidly advancing techniques of artificial intelligence like natural-language processing, pattern recognition and machine learning."

Wednesday, February 15, 2012

Keith Devlin Is Wild About Math

"Wild About Math" brings us the always-interesting Keith Devlin in a podcast (30 mins. of great stuff):

http://wildaboutmath.com/2012/02/14/keith-devlin-inspired-by-math-1/

This is Sol's first go at a podcast format and if this is any indication we have some great material to look forward to in future podcasts (seriously, I'm not sure he left himself a lot of room for improvement!)...

Tuesday, February 14, 2012

MathLove


For Valentine's Day I was looking, naturally, for something to post connecting mathematics and love... a lot of possibilities, but in the end decided I could do no better than to re-run a link I posted just 3 months ago -- Jennifer Ouellette's fun ode ("Love Among the Equations") bearing on the subject here:


an excerpt:
"It turns out that the world is filled with hidden connections, recurring patterns, and intricate details that can only be seen through math-colored glasses. Those abstract symbols hold meaning.  How could I ever have thought it was irrelevant?
This is what I have learned from loving a physicist. Real math isn’t some cold, dead set of rules to be memorized and blindly followed. The act of devising a calculus problem from your observations of the world around you – and then solving it – is as much a creative endeavor as writing a novel or composing a symphony."

Monday, February 13, 2012

Ian Stewart on Black-Scholes

\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2} V}{\partial S^{2}}+rS\frac{\partial V}{\partial S}-rV=0.
             (Black-Scholes Equation)

Good piece (as usual) from Ian Stewart on some of the mathematics underlying the financial crash:

http://www.guardian.co.uk/science/2012/feb/12/black-scholes-equation-credit-crunch

From the article:
"Anyone who has followed the crisis will understand that the real economy of businesses and commodities is being upstaged by complicated financial instruments known as derivatives. These are not money or goods. They are investments in investments, bets about bets. Derivatives created a booming global economy, but they also led to turbulent markets, the credit crunch, the near collapse of the banking system and the economic slump. And it was the Black-Scholes equation that opened up the world of derivatives."

Saturday, February 11, 2012

Seife, Wallace, and Infinity, Oh My!…


If by any chance you're not familiar with science-writer Charles Seife's work, I highly recommend it (one of my favorite science expositors). More recently he put up a simple 'tweet' that caught my eye, noting that he was now in possession of "David Foster Wallace's annotations" to his (Seife's) book "Zero: the biography of a dangerous idea." He actually linked to an image (pdf) of one of the pages, which included the following passage, with Wallace's simple annotation, "Heavy." :-)
"How big are the rational numbers? They take up no space at all. It's a tough concept to swallow, but it's true.
Even though there are rational numbers everywhere on the number line, they take up no space at all. If we were to throw a dart at the number line, it would never hit a rational number. Never. And though the rationals are tiny, the irrationals aren't, since we can't make a seating chart and cover them one by one; there will always be uncovered irrationals left over. Kronecker hated the irrationals, but they take up all the space in the number line.
The infinity of the rationals is nothing more than a zero."
Fascinating stuff… and, 'heavy' indeed!; all the moreso, no doubt, for someone who was essentially an English major/novelist/writer, like Wallace, with a brilliant/eclectic mind (he did deeply study logic and mathematics in college though).

About 3 years after Seife's "Zero" came out, Wallace produced "Everything and More: A Compact History of Infinity" -- a remarkable take (300 pages that he calls "a booklet"), from a non-mathematician, on the whole history and thought surrounding mathematical infinity.  I happen to like this quirky volume a lot for its sheer uniqueness (I still can't imagine the mind that could write such a book, and if you are familiar with Wallace's innovative writing style, you know it is unique!), but I've seen professional mathematicians give the volume both negative and positive reviews. In any event, we may now have at least some inkling as to what moved Wallace to even tackle such subject matter. As Paul Harvey would say, perhaps from Wallace's Seife-annotations we will get a glimpse of "the rest of the story."
[-- Tragically, Wallace took his own life at age 46 in 2008, after lifelong bouts with depression.]

Friday, February 10, 2012

Friday Fun

Wow!, it's not April 1st yet, so I can only assume that someone spiked Vi Hart's coffee for her latest video escapade ;-):

http://tinyurl.com/7s4ggmf

And for a Friday puzzle, a simple, straightforward riddle (requires more thinking through than mathematics):

Sally notes that the day before yesterday she was 22 years-old, yet next year she will be 25. What day does her birthday fall on, and when did she voice the above statement?
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Answer: Her birthday is on Dec. 31, and her statement was made on New Year's Day, Jan. 1.

Thursday, February 9, 2012

Wednesday, February 8, 2012

Wolfram Education Portal

I grew up up at the wrong time (woe is me!). There are SO many cool digital applications for education these days; particularly math education. And the not-so-new kid on the block comes from venerable Wolfram Alpha with the "Wolfram Education Portal." If you're an educator (or, even NOT), check it out -- it's in beta and they want feedback:

http://education.wolfram.com/

Tuesday, February 7, 2012

Elsevier… Can You Hear Us NOWWWW!?

Wow!… back on January 22, I linked to what I called an "interesting, inspired post" from mathematician Tim Gowers recommending a boycott of scientific publisher Elsevier for various ill-publishing practices. Little did I know how much pent-up, anti-Elsevier feeling was out there, nor how quickly the boycott idea would snowball. Below, are some of the many blog entries that have joined in the effort:

http://gowers.wordpress.com/2012/01/21/elsevier-my-part-in-its-downfall/
http://freethoughtblogs.com/pharyngula/2012/01/16/elsevier-evil/
http://blogs.discovermagazine.com/cosmicvariance/2012/01/30/boycott-elsevier/
http://www.michaeleisen.org/blog/?p=890
http://thecostofknowledge.com/
http://www.scottaaronson.com/blog/?p=891
http://tylerneylon.com/b/archives/136
http://cscs.umich.edu/~crshalizi/weblog/864.html
http://terrytao.wordpress.com/2012/01/26/the-cost-of-knowledge/
http://www.newappsblog.com/2012/02/more-on-the-elsevier-boycott.html
http://michaelnielsen.org/blog/on-elsevier/

...and for a slightly broadened take on matters, the always-interesting Doron Zeilberger:

http://www.math.rutgers.edu/~zeilberg/Opinion120.html

Meanwhile, 4000+ scientists have signed a petition supporting the boycott idea.  This isn't simply a mathematics issue, since Elsevier publishes in several different subject areas -- but frankly, I'm kinda proud (even though I'm not even a professional mathematician myself) that this whole movement was raised to the forefront by a major mathematician. More generally, I'm proud of the pioneering work that mathematicians have done in utilizing the Web as a collaborative force to accomplish things.
The public often perceives mathematicians in a 'geeky,' loner sort of light, working quietly away in a corner, but in fact, when the spirit moves them ;-) they can be among the most powerful (and unselfish) movers of science and society! Give Tim Gowers a high-five!

ADDENDUM: I don't plan to do continuous "Addendums" on this issue, as related pertinent Web posts appear, but I do think the following update from Tim Gowers is worth noting:

http://gowers.wordpress.com/2012/02/08/a-more-formal-statement-about-mathematical-publishing/

Sunday, February 5, 2012

"The Math Book"… ho hum?…..


In recent years there have been a number of what I call math "nugget" books written for a mass audience -- books covering a range of interesting mathematical topics in a brief way, and introducing math to lay folks who might otherwise not give it much attention.  I've especially liked certain volumes from Tony Crilly and Richard Elwes that I've mentioned here previously. They do a good job of giving readers enough interesting material to chew on to possibly seduce them into further study, without overwhelming them.

Clifford Pickover is a well-known polymath (I would almost say poly-polymath) and math popularizer. He's authored several books I've enjoyed, and is just an all-around fascinating fellow! His "The Math Book," came out a couple of years back as a sort of "nugget"/coffee-table/math volume that met with wide acclaim and awards. I don't think I've read a single bad review of it! Still, I hesitated buying it because it just never much  enticed me despite frequent scans of it at my local bookstore. But, with a major discount coupon ;-), I did finally purchase it, and continue to have mixed feelings about it, including some difficulty recommending it to math fans, despite its wide acclaim.

But first let's cover the positives, since it is a book with so many fans. The style and format is beautiful; very glossy, very glitzy; very, as I say, coffee-tablish; unlike most math books. It also touches upon a very wide range of topics, or as the book says, "milestones" (250 by its own count), and does so in an organized, chronological way, that pulls the reader along from antiquity to modern times (although it need not be read from beginning to end, but can be read in a much more random order). The language and level of material is appropriate to a lay audience. Pickover's own love of math shines through on each page, and it seems clear that his goal for the book is not so much to educate people, as perhaps to imbue them with some of the same excitement about the field that he feels. Every page seems to say, 'Isn't this wondrous!'
All of that is obviously to the positive…

My problems with the volume however are these:

Starting with the practical: it is a thick, square, heavy book (especially the hard-back version), not conducive to reading in bed, on one's back, as I often like to do -- a small quibble to many, but for me a major drawback in what is essentially a 'pleasure' book, not a textbook. It's coffee-table quality almost makes it more a volume for show than for reading (which is unfortunate)... Stephen Hawking's "Brief History of Time" was called perhaps the most widely-selling, UNread book in publishing history -- I fear Pickover's book could suffer a similar fate (even though it is more readable than Hawking's). I realize Pickover is probably trying to reach out to a very broad audience, but for my taste it is simply TOO glossy and glitzy; almost gaudy, and almost a turn-off. Moreover, the short, bite-size text entries are simply too brief to really afford the reader much substantive information on the subjects at-hand. Whereas, Crilly and Elwes, give the reader enough, as I say, to chew on, Pickover often just scratches the surface here, instead of fleshing things out. Each page leaves me wanting more, and feeling shortchanged. Take away the artistic presentation and you are left largely with a pithy, somewhat pedantic regurgitation of quite interesting historical math tidbits, for which you need Google standing by if you wish to explore further… it's a somewhat "Cliff"Notes version of math history (...although I also realize Pickover likes to experiment with math writing styles and presentation -- still, this volume seems a bit of a 'puff-piece' compared to some of his other fine works, and it would be ashamed, I think, if he became better known for this book than his other contributions). Overall, I am still glad I bought the book, as a reference source... but am also glad I got it at a heavy discount ;-))

In short, I can recommend this volume to young people beginning their journey in mathematics, as a sort of starter book, an initiation into the field of math, but am more reluctant recommending it to readers who are further-along, for whom there may be many better selections... including fuller text and less glitz. Still, many may (and clearly DO) find its style, format, elegance, and sheer breadth of topics, captivating enough to want it for their bookshelf (...or coffee table) -- indeed, mine seems to be a minority view; those who are particularly enamored of this book, feel free to chime in, in the comments, about what makes it so laudable or useful from your standpoint…

(Also, if you DO like this offering from Pickover, it is worth noting that he followed it up with a similarly-styled volume called "The Physics Book.")

By the way, the books I cited above from Crilly and Elwes that I very much liked and do recommend are:

"The Big Questions: Mathematics" and  "50 Mathematical Ideas You Really Need to Know" -- both by Tony Crilly

and, "Mathematics Without the Boring Bits" -- by Richard Elwes
Also, Elwes did an even more encyclopedic/comprehensive volume of math topics than Pickover's book, called "Mathematics 1001," but minus the glossiness -- I prefer the Elwes offering, if you're looking for terse treatments, but understand it won't carry the popular appeal of Pickover's effort.

Finally, for anyone who especially likes Pickover's book for its 'historical' format, I might just mention that Pat Ballew does a wonderful job of regularly noting mathematics history in his "On this Day in Math" entries over at "Pat's Blog."

Friday, February 3, 2012

Clock Geometry

For a Friday puzzle, another one lifted from William Poundstone's recent "Are You Smart Enough To Work at Google":

At 3:15, what is the angle formed between the minute hand and hour hand on an analog (regular) clock?
(if the answer seems immediately obvious to you, you likely have the wrong answer)

I'll be giving the correct answer in the comments section below.

Thursday, February 2, 2012

Math Genius of the Female Persuasion

It seems like most stories of young math savants or geniuses involve males, but here's one of a German female, 21-year-old (at the time) Julia Ruscher, whose specialty is stochastic processes:



...and speaking of gender and math talent, consult this old take from xkcd:

http://xkcd.com/385/