Evidently there’s a newish biography out about Grigori Perelman, the man primarily responsible for solving the Poincaré Conjecture. Masha Gessen, a Russian journalist and author, has released “Perfect Rigor: A Genius + the Mathematical Breakthrough of the Century”, a work about the life of the curious mathematician who has vanished from the professional math community. She explains her work in an interview with failuremag.com, in which she describes the rationale behind the work, and gives some insights into both the Poincaré Conjecture and the life of Perelman.

There’s no doubt that Perelman’s response to solving one of math’s longest standing problems is part of what is so intriguing to the lay reader. He turned down a Field’s medal and withdrew from professional mathematics. Pieces on him tend to accentuate his eccentricities. Part of what makes the story so very interesting is Perelman’s response, as well as the other cast of characters involved in the solving of the problem. Though it came out in mid-2006, the New Yorker has a fantastic article about the solving of the Poincaré Conjecture and the controversy surrounding it. If you’d like to understand to a greater extent Perelman’s response to his solution, read this article. It’s long, but extremely informative and complete.

There have been a few posts I’ve seen recently about math related comics. First, Boing Boing clued me into The Thrilling Adventures of Lovelace and Babbage, which is pretty much exactly what is sounds like. This webcomic gives some great info (both truth and pseudo-truth) about both Charles and Ada. Sydney Padua, the author of the comic, explains:

It’s a punchline.  As in “Wouldn’t it be hee-larious if there was a comic about Ada Lovelace and Charles Babbage fighting crime? Thanks, I’ll be here all week!”  The imaginary comic was the punchline to Lovelace: The Origin, drawn because my good buddy Suw Charman got me drunk started a Noble Enterprise called Ada Lovelace Day.

Secondly, Jim Holt wrote a book review last Sunday that introduced me to Logicomix, which the official website describes as follows:

Covering a span of sixty years, the graphic novel Logicomix was inspired by the epic story of the quest for the Foundations of Mathematics.

This was a heroic intellectual adventure most of whose protagonists paid the price of knowledge with extreme personal suffering and even insanity. The book tells its tale in an engaging way, at the same time complex and accessible. It grounds the philosophical struggles on the undercurrent of personal emotional turmoil, as well as the momentous historical events and ideological battles which gave rise to them.

The role of narrator is given to the most eloquent and spirited of the story’s protagonists, the great logician, philosopher and pacifist Bertrand Russell. It is through his eyes that the plights of such great thinkers as Frege, Hilbert, Poincaré, Wittgenstein and Gödel come to life, and through his own passionate involvement in the quest that the various narrative strands come together.

VERY cool. I’ll definitely be picking this up before too long. The art looks fantastic, and I’m sure the story matches up! Check out the review linked to above for a fuller introduction to the story.

I’m rather late on the draw in regard to this note, but I just wanted to say that I was profoundly glad to hear that the Prime Minister of England released a letter of apology on September 10 recognizing Alan Turing for his monumental “contribution to humankind”, including work in computation, mathematics, and code-breaking.

Despite his contributions, Turing was convicted under English homophobic laws and sentenced to chemical castration. As PM Brown writes, “In 1952, he was convicted of ‘gross indecency’ – in effect, tried for being gay.” Given the choice between going to prison and undergoing hormone treatments, he chose the latter. Less than three years later Turing was found dead, apparently the result of a cyanide overdose. His death was judged to be a suicide.

While the note obviously cannot change the past, I’m glad to see the British government deliver the note. And I’m extremely proud of those individuals who lobbied the government to acknowledge this terrible mistake. Again, as PM Brown wrote, “So on behalf of the British government, and all those who live freely thanks to Alan’s work I am very proud to say: we’re sorry, you deserved so much better.” So very true.

I came across a great little tip on David Bau’s blog today about using Wolfram Alpha to create quick GIF images of mathematical formulae to be included in a blog. For example, by typing “integrate sin x dx from x=0 to pi” into the Wolfram Alpha search box, you’re presented with both of graph of the curve as well as a typeset version:

I’ve written a few times about using LaTeX in WordPress, but this is certainly an easy way to include typeset equations in a blog. I’m not sure why this hadn’t occurred to me when I first started playing with Wolfram Alpha, but I suppose that’s why people like David write great tips like this!

I rarely post to issues of the Carnival of Mathematics from my blog, and I really don’t have a good reason for it. The 54th edition was released yesterday on Todd and Vishal’s blog here, and there are some interesting posts. First, I enjoyed the post and the comments on the Gödel’s ontological proof, which Wikipedia tells us “is a formalization of Saint Anselm’s ontological argument for God’s existence.” Secondly, there is a link to The Lebombo Bone, which may be the oldest mathematical artifact in the world. Good stuff. Check out the carnival itself for links to these posts.

I must admit that the contents of these carnivals I enjoy the most are ones that take an interesting piece of the math world and make it accessible to a wide reaching audience. Which, I suppose, is generally what I try to do on this blog. The hardcore math is also good, but I like the immediate mystery present in the more accessible posts. I suppose this is because the mystery drew me to math in the first place. Alas.

The following TED video, given by mathemagician and professor Arthur Benjamin (about whom I’ve previously blogged about here), embodies the best idea I’ve heard about math education in a LONG time. Perhaps ever. Just as I recently posted about how games like backgammon embody the 21st century in replacement of games like chess for the 20th, statistics is the central branch of mathematics for the 21st century rather than the calculus centric view of the 20th century. If you’re into math and math education, this will probably be the best 3 minutes you’ll spend today.

In 1997, chess champion Gary Kasparov was beaten in a 6-point match against a computer. It was the first time this had ever happened. The computer, named Deep Blue, was developed by IBM after some Carnegie Mellon University graduates joined the company. Here’s what Wikipedia has to say about the hardware computing power of Deep Blue:

The system derived its playing strength mainly out of brute force computing power. It was a massively parallel, RS/6000 SP Thin P2SC-based system with 30-nodes, with each node containing a 120 MHz P2SC microprocessor for a total of 30, enhanced with 480 special purpose VLSI chess chips. Its chess playing program was written in C and ran under the AIX operating system. It was capable of evaluating 200 million positions per second…In June 1997, Deep Blue was the 259th most powerful supercomputer according to the TOP500 list, achieving 11.38 GFLOPS on the High-Performance LINPACK benchmark.

Brute force. That’s how the computer got the job done. Of course, it’s never that simple. But there is one thing that can be said for certain: If you lose a game of chess, it is because you were outplayed. Plain and simple. And I think it’s for this reason that chess became an apt metaphor for modernist notions of intelligence. Stereotypically speaking, if you ask a person the question of what game smart people play, I would guess that chess would be the most common answer in the western world (perhaps Go in the eastern world). The fate of this game is in the hand of the players entirely. There is no chance involved, with the one exception of which player plays first.

As a child, I had a hard time enjoying games that involved a substantial amount of probability. “What’s the point,” I thought, “of playing a game skillfully if it’s possible for me to lose at the last possible moment due to a bad roll of the dice or a badly dealt card?” But as I’ve grown older, I’ve come to enjoy games like this MORE on average than straightforward skill games like chess. Enter backgammon.

For those of you who don’t know backgammon, I suggest checking out the Wikipedia page here. Backgammon has been played for 5,000 years, and has evolved substantially over that time. For example, of the additions to the game, the doubling cube, drastically changed play and was introduced less than 100 years ago. Backgammon is not like chess. In a single game of backgammon, it’s quite possible for a novice to beat a master due to elements of chance. Said another way, it’s possible to play the best possible game of backgammon you can based on your dice rolls and still lose. And this is the aspect of the game that makes it an apt metaphor for the 21st century. While the 20th century dealt with certainty, the 21st will deal with probability.

And this is not to say that games like backgammon are somehow more subjective than games like chess. There are some amazing machine learning techniques used to study the game (e.g. TD-Gammon), and there are quite a few computer programs, such as GNU Backgammon, that use these techniques to outplay human opponents. Poker games like Texas Hold’em also involve an element of probability, and have grown wildly popular over the last many years. And those of you who know poker know that there are rules that govern “right” playing. Though the cards dictate play, there are strategies that maximize gain and minimize risk. The same is true of backgammon. And with the game popping up in popular culture a bit more, like in the television show Lost, I can only see backgammon growing in popularity.

Happy Pi Day, everyone. It seems that the U.S. Congress has actually declared it this year. That’s funny. It’s also funny to surf around to various websites to see how much merchandise is available to commemorate this day. I guess everybody needs a holiday, huh? Well, if you celebrate Pi Day in any way, I suggest not getting caught up in all the glitz, but instead think back on the various ways that Pi has influenced your life; e.g. your trig class when you were 15. Check out the first link in this post to see the official Pi Day website, including some fun ideas of how to celebrate the day!

And if you’ve never seen the following visual representation of Pi, enjoy!

So I certainly won’t be covering any new ground by bringing up the debate of whether mathematics is created or discovered. There are plenty of resources online covering this topic. Some of the better ones I’ve read are here and here. I’ve been thinking about this question over the past day given the Radiolab program I mention in my last post. Suffice is it to say that Plato certainly believed that mathematics was discovered, as evidenced through his theory of forms. While thinking over the issue, I remembered an interview that I had read earlier this year in Discover Magazine with Max Tegmark, an Associate Professor of Physics at MIT. Though most of his works centers around conventional cosmology, he has some interesting theories about the universe, and how it is physically composed of mathematics. As Max puts it in the interview, “Mathematical things actually exist, and they are actually physical reality”. In this sense, he doesn’t align himself with Plato’s theory of forms given that mathematics IS reality, rather than mathematical forms existing in some ideal way outside of reality. The interview is informative from a cosmological perspective as well, particuarly in regard to the four levels of multiverse he describes.

The latest edition of WNYC’s Radiolab is named “Yellow Fluff and Other Curious Encounters“, and brings up a few interesting questions. Namely, it asks whether scientific knowledge is discovered, or whether it is created. With its typical casually educational tone, this issue brings us stories from individuals such as Steve Strogatz, who sets the theme of the hour long program with a story about parabolas. Steve is currently a professor in Applied Mathematics with Cornell, and in the opening of the program he talks about being a young student, and how a simple experiment in school led him to a eureka moment. Though Radiolab is an audio program (which can be downloaded or listened to on their site), they’ve bundled this episode with a video interpretation of Steve’s story, which can be watched below.

Kurt Gödel died 31 years ago today. From the little I’ve read of his life, and from the even smaller amount that I truly grasp from his work, I believe that only in reality could such a fantastic and somewhat lamentable figure come to be. He was included in the infamous Vienna Circle, but was himself a Platonist. He was shy, reclusive, and prone to illnesses both physical and mental. He was a friend to Albert Einstein. And he shook the world of mathematics in a way that destroyed the Hilbert program. In simple terms, he showed that the mechanization of mathematics could not be fully automated, or that mathematics was not something that could be neatly placed in a box and tied up with a bow.

John W Dawson Jr. explains the first of Gödel’s Incompleteness Theorems by saying, “In his 1931 paper Gödel showed that, no matter how you formulate the axioms for number theory, there will always be some statement that is true of the natural numbers, but that can’t be proved. (That is, objects that obey the axioms of number theory but fail to behave like the natural numbers in some other respects do exist.)”

John Von Neumann, certainly one of the greatest mathematicians of the 20th century, had the following to say in a letter shortly after the publication of the Incompleteness theorems:

Thus today I am of the opinion that 1. Gödel has shown the unrealizability of Hilbert’s program. 2. There is no more reason to reject intuitionism (if one disregards the aesthetic issue, which in practice will also for me be the decisive factor). Therefore I consider the state of the foundational discussion in Königsberg to be outdated, for Gödel’s fundamental discoveries have brought the question to a completely different level.

Another way of summing this up is to say, “this work has changed the way we must view mathematics.” I have to imagine that the fame of the majority of famous people peaks in the prime of life, only to wane with time and death. Only the smallest number of people see their influence grow with time, as reflection shows their achievements to be truly monumental. Gödel, I believe, sits comfortably in the latter group.

Obviously, I have a bit of a crush.

Welcome to the 37th edition of the Carnival of Mathematics!

In preparation for this edition, I actually managed to secure an exclusive interview with the number 37, and have included a small portion of our conversation below:

Logic Nest (LN): So 37, what have you been up to lately?

37: Oh, not much. I’ve always had a fairly good life given that I’m not only a prime number, but a lucky, irregular, AND unique prime. It’s summertime where I live, so mostly I hang out by the pool with my good friends 16, 21, and 28. We’re in a band together called the Padovan Sequence.

LN: Wow. That’s very interesting. I’ve heard that some people think you’re unlucky though. What do you say about that?

37: That’s totally a fabrication. Just because I’m the number 666 divided by its digits added together [37=666/(6+6+6)] doesn’t mean a thing.

LN: Understandable. I can see the confusion. I’ve heard that there’s a website out there that’s all about you, is that true?

37: Yes, and I must admit that I’m a bit embarrassed about it. Just because I pop up in all sorts of scientific, cultural, and historical situations doesn’t mean that I should have a fansite. I mean, come on now, people…

And it went on like that for a while…

Speaking of prime numbers, let’s kick the carnival off with this article submitted by Jeffrey Shallit from Recursivity about a Rutgers graduate student named Eric Rowland who has proved a new prime-generating formula that’s quite simple. There are some great comments on this post that include various programming implementations of the formula.

Over at Walking Randomly, Mike Croucher has posted his second Integral of the Week involving an exponential function and the square root of pi. The twist on this problem is that he gives you the evaluation and asks you to prove it. In addition, he’s asking readers to exclude the common evaluation method of converting the integral to polar coordinates. He’s taking solutions via the comments on the site. There are already a few proposed solutions, but take some time to think it over before jumping straight to the comments!

“A” presents an editorial on Being Bad at Math posted at It’s the Thought that Counts. This post is about the popular idea that it’s acceptable to confess a total lack of math ability, even though equivalent statements about difficulty in something like one’s native tongue would be seen as embarrassing. This post explores a cultural brushing off of mathematics, and how this idea should no longer be tolerated in the twenty-first century.

Another great lesson in math and culture comes from Barry Leiba, who points out a personal pet peeve of mine in his article That’s a mean median posted at Staring At Empty Pages, namely that people often incorrectly equate “median” with “average”, even at the New York Times. This one should get the blood of you stats people out there boiling!

Given the impending American presidential election, Barry Wright, III presents an educational post entitled Plurality Winner, Condorcet Loser? at fashionablemathematician – mathematics. The contents of the article explores various ideas that Barry is exploring from Donald Saari’s Basic Geometry of Voting, which is a text he is using “both for research purposes and to prepare to TA a class on the mathematics involved in Democracy, voting systems, and the like”. By definition, “a Condorcet winner is one which is ranked higher than every other alternative in a majority of decisions” while a “plurality winner is an alternative which receives more first-place votes than any other alternatives”. As the title implies, there is an interesting case when one can be both a plurality winner and a Condorcet loser. Good stuff.

In The Universe of Discourse : Period Three and Chaos posted at The Universe of Discourse, Mark Dominus gives us some information about Möbius functions, which are of major importance in complex analysis, where they correspond to certain transformations of the Riemann sphere. In particular, he looks at Möbius functions with real coefficients. In this post he talks about functions with a periodic point of order 3 (where f(f(f(x))) = x for some x) in connection to the Sharkovskii’s theorem. Both of these concepts are explained more fully at the link above.

Denise presents Math History on the Internet posted at Let’s play math!. She presents links for some WONDERFUL historical resources available on the web. As she says, “the story of mathematics is the story of interesting people. What a shame it is that our children see only the dry remains of these people’s passion. By learning math history, our students will see how men and women wrestled with concepts, made mistakes, argued with each other, and gradually developed the knowledge we today take for granted.” There’s some really great stuff available at this link for anyone interested in picking up some mathematics history!

In his post Playing with Permutations at Reasonable Deviations, Rod Carvalho proposes a 2-player game. The goal is to find out whether a necessary condition is also sufficient. This game blends Combinatorics with Algebra, and even Algebraic Geometry. It’s an interesting game to consider and builds on a few other posts that Rod has written since January 2008.

Ron Cook from The Endeavour gives us an explanation of Random Inequalities in this three part series. Random inequalities are often used in Bayesian clinical trial methods, and should interest all the stats people who are reading. The first part introduces the reader to the concept of random inequalities, the second part shows how they are analytically evaluated, and the third shows how they are numerically evaluated when analytical evaluation is not possible.

Lastly, Heather wants us to think about Burnt Pancakes and Godzilla at her article 4, 6, 8, 10, 12, 14,?.What comes next? posted at 360. As she explains, “the Burnt Pancake problem involves pancakes of different sizes, each with one burnt side, piled up on top of one another.” It’s great content explained in an entertaining manner. FYI, Godzilla evidently wears a chef’s hat when cooking pancakes.

Here are a few more submissions that have come in since I initially published last night:

Alvaro Fernandez presents Top 10 Brain Training Future Trends posted at Sharp Brains.This article discusses the concept of “brain training”, or how we keep our brains fit. This is particularly interesting given that mathematics is commonly perceived as a game for the young, as evidenced by this XKCD comic. Take care of your brains, people!

Are you aware that there is an Encyclopedia of Triangle Centers? David Eppstein is, and he describes another kind of triangle center, different from the ones at the Encyclopedia, here.

Catsynth asks the question, “What do you get when you mix a cat and a Fourier Transform?” in this post. Education and entertainment ensue! The lesson to be learned is simple: be careful of what mathematical transforms you perform on your pets. Obviously.

That’s all for this edition! If you’d like to post any additional articles to this edition of the Carnival, please contact me. I’ll be taking submissions through Sunday evening. Otherwise, stay tuned for the next edition which will be hosted by CatSynth.

I came across this article today on Science Daily that talks about the Pirahã, which, according to Wikipedia, are “an indigenous hunter-gatherer tribe of Amazon natives, who mainly live on the banks of the Maici River in Brazil”. The Science Daily article introduced me to the fact that this tribe has no concept of precise numbers. While they do use indefinite numerical terms such as “some” and “more”, this group does not seem to have any representation for concepts such as “one” or “two”. As MIT professor Edward Gibson states, “here is a group that does not count. They could learn, but it’s not useful in their culture, so they’ve never picked it up.” Absolutely fascinating. You should certainly check out the two links above, especially the portion in the Science Daily article that describes some of the experiments carried out by Gibson and his MIT team that have further illuminated this portion of the Pirahã culture.

This article intrigued me so much that I dug a bit deeper, and found that Daniel L. Everett, the Chair of Languages, Literatures, and Cultures from Illinois State University, has spent a good portion of his career working with the Pirahã people. He has collaborated in the past with Gibson on various projects in the past. Some info can be found here. There’s a great New Yorker story that was published in April 2007 on Dan here that’s certainly worth a look. Here’s a teaser from this article:

The Pirahã, Everett wrote, have no numbers, no fixed color terms, no perfect tense, no deep memory, no tradition of art or drawing, and no words for “all,” “each,” “every,” “most,” or “few”— terms of quantification believed by some linguists to be among the common building blocks of human cognition.

It’s a very long article, but it paints a beautiful picture of linguistics, cognition, faith, and personal relationships. It’s packed full of great questions. There’s a LOT that’s in these writings I’ve linked to that I haven’t even brought up (including the idea of recursion in linguistics), so I urge you all to read more! There are also some great links for further reading in the Wikipedia article linked to above, including several scholarly papers.

A friend let me know quite a while ago about this story presented on NPR’s site entitled “Mathematicians Explain Tape’s Tendency to Tear”. It’s an explanation of a recent Pedro Reis article in the journal Nature Materials describing the annoying tendancy of tape to narrow while unpeeling it from the roll. As the article explains, Reis’ work “could help engineers test thin films for strength and reliability” The audio of the story is also available on the NPR site.

I love this story because I can imagine Pedro first thinking about this problem while unpeeling a roll of tape. I don’t know if the inspiration actually came this way, but its a great mental image that conveys the idea that some of the most interesting problems to solve are right under our noses.

Here’s the abstract of the paper from Dr. Reis’s website:

Thin adhesive films have become increasingly important in applications involving packaging, coating or for advertising. Once a film is adhered to a substrate, flaps can be detached by tearing and peeling, but they narrow and collapse in pointy shapes. Similar geometries are observed when peeling ultrathin films grown or deposited on a solid substrate, or skinning the natural protective cover of a ripe fruit. In this work, we have shown that the detached flaps have perfect triangular shapes with a well-defined vertex angle; this is a signature of the conversion of bending energy into surface energy of fracture and adhesion. In particular, this triangular shape of the tear encodes the mechanical parameters related to these three forms of energy and could form the basis of a quantitative assay for the mechanical characterization of thin adhesive films, nanofilms deposited on substrates or fruit skin.

Let’s be honest…there are certain subjects that a math-ish kind of blog must mention at some point. One of these obligatory topics happens to be the “0.9999999… = 1″ proof. It’s one of those facts that delights the mathematically inclined. It’s sort of like the joke that Grandpa always tells when the family gets together: you know it’s coming, and you know how much pleasure he gets out of relaying the joke, but for goodness sake, this is the 99th time you’ve heard the punchline. At any rate, there is a set of about 15 math facts that people love to talk about simply because they’re all totaling mind-blowing or sound totally nonsensical. I tend to think that the “0.9999999… = 1″ proof belongs in the latter category.

The previous digression leads me to mention the Things of Interest blog, and their absolutely fantastic post on various forms of the “0.9999999… = 1″ proof. You can find that post here. In case one proof doesn’t do it for you, this site offers several, each of which occurs at a various level of mathematical rigor. There will definitely be a proof for you here that you’ll understand.