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.
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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.
(Photo by Jeephead)
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!
I’ve decided this year to read or reread as much of Bertrand Russell’s work as possible. I’ve been reading The History of Western Philosophy and The Problems of Philosophy (the latter of which is now public domain), and have been trying to digest On Denoting by reading it every few weeks. With this in mind, I was excited to recently find a set of several articles written by Douglas Anele for Vanguard Online on Russell. They serve as a good biographical introduction, and also provide a bit more information than average on Russell’s view of sexual ethics and religion. Perhaps most interesting is that the essays are written from a Nigerian perspective. They’re certainly worth reading.
Part 1: The Power of Reason: A Celebration of Bertrand Russell
Part 2: The Power of Reason: A Celebration of Bertrand Russell
Part 3: The Power of Reason: A Celebration of Bertrand Russell
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, Ξ_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.
I came across this wonderful introduction to several famous paradoxes quite a while ago, but haven’t taken the time to inform you all about it. Daniel Haggard presents a non-technical explanation of five age-old paradoxes that have both delighted and confused humanity. It’s a very accessible read and I recommend it for everyone interested in the strange logical conundrums that surround us. I particularly enjoy his section on Newcomb’s Paradox, which boggles my brain every time I think about it. Honestly, I’m glad that paradoxes exist. I mean, nothing I say is true, right?
I’m usually quite interested in the attempts of individuals to apply math to the Bible, and the Church Hopping blog has a fun little article about some people that are actually using interesting mathematical principles on the text of the Bible. Check it out.
UPDATE: Daniel from the Logos Blog has contacted me about a great post over on that site. As he said, “Thought you might be interested in today’s Logos Blog post looking at The Top 50 People in the Bible and using the IBM Many Eyes visualization. Cool thing is that anyone can play around with the charts and data…” Check it out here.
Wolfram, the makers of the software Mathematica, are offering a $25,000 prize to the first person who can prove whether the above 2, 3 Turing machine is universal. From the website (here):
“A universal Turing machine is powerful enough to emulate any standard computer.
The question is: how simple can the rules for a universal Turing machine be?
Since the 1960s it has been known that there is a universal 7,4 machine. In A New Kind of Science, Stephen Wolfram found a universal 2,5 machine, and suggested that the particular 2,3 machine that is the subject of this prize might be universal.
The prize is for determining whether or not the 2,3 machine is in fact universal.”
What a great idea! I’m really curious to see how long it take for someone to claim the prize. If you’re interested in understanding more about what a Turing machine is, please check out the above links.
Here’s yet another reason why you should make sure to learn basic math. I suppose that this is one way to lose your job…
There was an announcement yesterday that a collaboration of mathematicians from the United States and Europe have mapped the structure of E8, which is a 248-dimensional Lie group. It’s actually even more rich than that, but I think the concept of a Lie group is intense enough for one post. What interests me most about this particular problem is that there was some SERIOUS computer horsepower that went into the solution. As the Yahoo! news story (link) indicates, “While the human genome, which contains all the genetic information of a cell, is less than a gigabyte in size, the result of the E8 calculation, which contains all the information about E8, is 60 gigabytes in size.” Yikes. Amongst other practical applications this result will provide some good information for physicists who study string theory. The reason for this is that structure of E8 is both symmetrical and extremely complex. Please check out the American Institute of Mathematics page on the E8 project here for more information. There’s a lot of great information on their site. So what does the structure of E8 look like? Here’s the picture:
I’ve decided to try to do a weekly feature called “Proof of the Week,” where I’ll explain a mathematical proof that I find particularly illuminating or intriguing. Part of the reason that I write so many math posts on this blog is that I feel that much of the beauty of math is an acquired taste. So my desire is to help serve as a “waiter” who introduces people to some of the fascinating tidbits of the subject. I know a lot of people who run (or roll their eyes) when they hear the word “math.” It brings back terrifying memories of grade school multiplication tests and what not. I don’t blame you. My fourth grade math teacher used to slam a book shut at the end of every minute long mad-dash times test. It scared the bejeus out of me every time. Even so, I still love math.
Most of the proofs I’ll be talking about from week to week won’t be overly intense. I’m sure that many of them will require some general knowledge background, but nothing too academic. My hope is that by explaining some interesting results that you too might see a little bit more of the grandeur contained in this subject. I remember when I took my first proof-based math class during my sophomore year of college. I knew that a lot of rigorous math had to do with proofs, but it wasn’t until my 20th year of life on this planet that I learned what they were really all about. And here’s one of the many revelations I came to rather quickly:
Math is nowhere near as objective as I thought it was growing up. In other words, I always thought that there was a unique answer to every problem. Because of this, I think that many people regard math as some sort of rigid 60 year old person wearing starched clothing who eats the exact same three meals a day and whose house is painted a single shade of grey. To use another image, many people view math problems as some sort of assembly line. You insert a problem at the beginning of the line, perform a bunch of robotic methods, and the answer plops out at the end of the line. If this is your view of math, no wonder you think it’s boring! There’s no art in these images. There’s no movement or color in these pictures.
Math is nowhere near as simple as an assembly line. At least not at its heart. But since most of us grow up learning rote methods to solve problems many of us find the subject to be too tedious or mundane. And I don’t blame you for thinking that. What I WOULD like for you to consider is that you’ve been misled. Like any other academic discipline, math is a growing organism. Hopefully in these “Proofs of the Week” I’ll be able to illuminate some of the beauty that is contained in math. The first of the series will be up in a day or two. Stay tuned!
Fractals are beautiful things. If you don’t know what a fractal is, you should read this for a general overview. The most famous fractal (and one of the most mathematically simple) is the Mandelbrot Set, which is named after its discoverer Benoît Mandelbrot. For awhile I’ve wanted to include some sort of video of the Mandelbrot Set “in action”. The following video shows what happens when you “zoom in” on a portion of this fractal. It’s quite interesting. Suffice is it to say that if I ever fall into a bottomless pit, I hope that bottomless pit is like falling into the Mandelbrot Set. At least that way there would be good stuff to look at. There are several other videos out there on the web that show other perspectives of zooming into this particular fractal, so if you like what you see here head over to YouTube or what not and search for some more! The math rock song in the video was written by Jonathan Coulton. If you listen to the lyrics they actually explain a little bit about how to graph this particular fractal. Check out his website here. [Warning: For those of you with sensitive ears, the song that accompanies the video has a few curse words scattered throughout!]
Voila! Turing machine muffins! What a delicious idea. If I had used this method while learning about these universal machines I probably would’ve been much happier. What’s a Turing Machine, you ask? Read about them here and here. Check out other pictures of muffin madness here and here. Thanks to Boing Boing for this info!
Yes, it seems that Edwin A. Abbott’s wonderfully original novel about the travels of the square named A. Square through one, two, and three dimensional space will soon be brought to video. The website for Flatland: The Movie can be found here. The trailer for the movie is available on the website or on YouTube here. Here’s the synopsis of the movie:
Flatland: The Movie is an animated film inspired by Edwin A. Abbott’s classic novel, Flatland. Set in a world of only two dimensions inhabited by sentient geometrical shapes, the story follows Arthur Square and his ever-curious granddaughter Hex. When a mysterious visitor arrives from Spaceland, Arthur and Hex must come to terms with the truth of the third dimension, risking dire consequences from the evil Circles that have ruled Flatland for a thousand years.
Well, it sounds like there has definitely been some license given to modify the orginal plot of the novel. But I have to say that the plot modifications were immediately forgiven once I found out that Martin Sheen was going to do the voice of A. Square. Who can argue with that? Also, Tony Hale, of the late TV show Arrested Development, will be playing the King of Pointland. At any rate, as the website explains, “The movie will be part of an educational DVD, which will include the original text from Abbott’s book.” Also, it looks like it will be coming out in spring 2007, which isn’t too far away! If you’re dying to get a copy you can sign up on the website for priority access to the DVD. While you’re anxiously awaiting its release, I suggest reading Abbott’s original work. It’s a really quick read and is imaginative and original.
Though I’m a little bit late on this, Science Magazine recently published a great article on the scientific breakthroughs of 2006. Topping the list was the proof of the Poincare Conjecture, which I’ve posted about several times on this blog. You can read their synopsis of the breakthrough proof here. It turns out that from the media’s perspective the drama behind the proof is almost greater than the mathematical result. Basically there was a lot of name calling among some members of the mathematical community concerning who made certain contributions toward the eventual proof. Sad. Apart from the soap opera, the author explains the Poincaré Conjecture in a very accessible way, which should be understandable by anyone who’s interested in reading it. This proof will be a huge deal for mathematics over the coming decades, and should help mathematicians better understand topics such as the “Navier-Stokes equation [of fluid dynamics] and the Einstein equation [of general relativity].”
Perhaps the most interesting thing to note is that the article focuses not only on the result of the problem (the proof itself), but also the methods used to solve the problem. This is an hugely understated part of the mathematical process. I’m of the opinion that when the general populace thinks about math that they are fixated on two things: the problem and the answer. What people tend to overlook is the process of problem solving. In math, there are not always clear-cut methods that explain how to get from point A to point B. A lot of thought is sometimes necessary to figure out how to traverse the path. The Poincaré Conjecture is a monumental achievement not only because of the end result, but also because of the original steps the solvers of the problem (especially Grigori Perelman) took to get there. These steps will be used in other problems; they are not exclusively tied to this one specific problem. Once again, congratulations to Perelman and the other mathematicians who had a hand in making this historic achievement!
I love when people intentionally mix together mathematics and art, and one of the best examples of this merger that I’ve seen for awhile can be found here. As the site itself says, “this experiment attempts to convert the first 10,000 digits of pi into a musical sequence.” You have the ability to choose several preset music scales, or can choose 10 notes either manually or randomly. It takes a few minutes to play through the sequence, and the sounds are quite transfixing. Even though this meshing of pi and music is somewhat artificial, the result is wonderful. It’s worth checking out.
My friends Tim and Megan over at The Franktuary recently sent me a math puzzle and asked me why it worked. I thought it would be a fun little exercise to explain on the site. So here’s the puzzle:
1. First pick the number of times a week that you would like to go out to eat (more than once but less than 10).
2. Multiply this number by two.
3. Add five.
4. Multiply by fifty.
5. If you have already had your birthday this year add 1757. If you haven’t, add 1756.
6. Subtract the four digit year you were born.
You should have a three digit number.
The first digit of this was your original number (i.e., how many times you want to go out to eat in a week).
The next two numbers are your age.
It’s said that this is the only year (2007) that this will work.
Here’s why this works:
First, you should notice that this puzzle has nothing to do with the number of times that you’d like to eat out every week. You could ask a person to randomly pick a number between 1 and 9 (including 1 and 9) and the puzzle would work out the same way.
If we go through step by step here’s why this works:
1. Let’s call the number that you initially pick x. It’s important to the puzzle that this number be between 1 and 9, including 1 and 9. You cannot pick 10. After we go through the explanation of the problem you should quiz yourself and ask why you cannot pick 10.
2. Multiply your number by 2. Now we have 2x.
3. Add 5 to this number. Now we have 2x + 5.
4. Multiply this number by 50. Now we have (50)(2x + 5). If we multiply this out we have 100x + 250.
5. Let’s assume that you’ve already had your birthday this year. According to the puzzle we should next add 1757 to our number. So we have (100x + 250) + 1757. Again, if we simplify we get 100x + 2007. Notice that the second half of this equation (after the plus sign) is the four digit year. Once we’re done with the puzzle you should ask yourself why we only add 1756 if we haven’t had our birthday yet this year.
6. At this point we’re supposed to subtract our four digit year of birth from our number. Since I was born in 1981 I’ll subtract this number. So our number is now (100x + 2007) – 1981. Simplifying, we get 100x + 26. This final equation is split into two parts. The second part (after the plus sign) will be your age. I am 26 years old. This is correct. The first part of the equation will be a multiple of 100, and will always be a three digit number. In other words, if you consider the three individual digits of this number, it will always be digit x followed by two digits of 0. So the number represented by the first part of this equation will basically be the number x00, if that makes sense. If you add 26 (or whatever your age is) to x00, you’ll always get x26.
The first digit will always be the number of times you’d like to eat out every week and the last two digits will be your age. And there’s the answer!
Here’s another two quiz questions for you:
1. Will this puzzle work for people of ALL ages, or just for specific ages?
2. How could you modify this puzzle so that it works in 2008 (or any other year for that matter)?
Well, hopefully that explains the puzzle. Let me know if you have any questions!
Yet another reason why I’m a total dork. If you don’t understand what’s going on just laugh when the people in the video laugh. Trust me, it’s funny. Also, this is actually what you do in graduate school. 
Watch the music video here.
Well good grief, how can I possibly improve on this headline? In the spirit of Halloween, check out this article over at Live Science, which attempts to deny the existence of vampires through some simple exponential reasoning.
I have always enjoyed Halloween. It conjures up images of mischievousness in my mind, rather than images of gore or fear. Then again, I’m a sucker for the mysterious, so Halloween fits that category quite nicely. Also, we’re coming up on All-Saints day as well, which you can read about here. At any rate, Emily and I will be passing out candy at our house this Halloween, so stop by! Also, thanks to my mom for this article, who recently celebrated her birthday on one of the most mysterious days of the year: Friday, October 13!
I was meandering through the Slashdot archives this evening and came across this discussion about philosophy’s role in computer science. I think the conversation is illuminating on several levels. There are blatant IT professionals coming from one angle versus hard-core philosophers coming from another. While there’s a lot of overlap in perspective each person tends to accentuate a particular part of the (dis?)connection. I think the threads are worth reading both from a professional and an academic standpoint.
There are obvious links between the disciplines, notably the fact that concepts such as computability were born from the work of folks like A.M. Turing, but I often wonder if computer scientists think about this with any sort of regularity.
I’m sure most of you have heard by now, but on Tuesday Grigori Perelman refused to accept the Fields Medal for his work on the Poincaré Conjecture. I’m sure there are people with all sorts of opinions about this, but the First Post has an interesting take on the situation. Also, if you’re looking for another explanation of why any of this matters anyway, I suggest reading Jordan Ellenberg’s Slate article here.
If you have no idea at all what I’m talking about, then you should read my first and second posts on the subject, which should supply a bit of mathematical background information and redirect you to more extensive resources.
I started thinking about the RSA Factoring challenge the other day when I received my RSA SecureID® fob to log into the UPMC network offsite. According to the RSA website, “The RSA Factoring challenge is an effort, sponsored by RSA Laboratories, to learn about the actual difficulty of factoring large numbers of the type used in RSA keys. A set of eight challenge numbers, ranging in size from 576 bits to 2048 bits is posted here. Each number is the product of two large primes, similar to the modulus of an RSA key pair.” So if you feel like trying to make yourself an easy $200,000, try to factor the following number as the product of two primes:
25195908475657893494027183240048398571429282126204
03202777713783604366202070759555626401852588078440
69182906412495150821892985591491761845028084891200
72844992687392807287776735971418347270261896375014
97182469116507761337985909570009733045974880842840
17974291006424586918171951187461215151726546322822
16869987549182422433637259085141865462043576798423
38718477444792073993423658482382428119816381501067
48104516603773060562016196762561338441436038339044
14952634432190114657544454178424020924616515723350
77870774981712577246796292638635637328991215483143
81678998850404453640235273819513786365643912120103
97122822120720357
If you’re interested in learning about the history of RSA, which is an algorithm for public key encryption that helps to make internet security tick, you should read this Wikipedia article or check out the RSA Laboratories website.
It looks like the New York Times has a story today about the proof of the Poincaré Conjecture. You can read the article here. I just love how the media always plays up the “insatiably crazed introvert” image of mathematicians, in this case overly dramatizing Grigory Perelman, the Russian mathematician responsible for much of the proof methodology. I blogged about this a while back, but this article definitely gives a much more in-depth explanation of the history of the conjecture. Happy reading everyone!
I’ve noticed a trend…
My high school calculus teacher used a calculator that looked like it had been built in 1985. My undergraduate algebra professor used a calculator that looked like it had been built in 1990 (perhaps earlier).
And though I hate to admit it in some strange sense, my calculator of choice is my TI-85, which was introduced to the world in 1992.
Maybe it’s nostalgia that I always reach for my TI-85 when sitting down to do personal finances. Maybe it’s the fact that I must’ve spent YEARS of my life playing Tetris on that sucker. But it’s probably because I’ve learned how to use it with incredible efficiency over the years. For some reason the calculator seems to violate my desire for extreme technological innovation. I constantly desire the cutting edge in consumer electronics. So what is it exactly that keeps me from updating to a TI-89? I may never know.
So does anyone out there have a specific piece of technology (calculator or otherwise) that has followed you through the years? Leave a comment if you have something. And if you’re looking to have some Texas Instruments nostalgia of your own you should check out ticalc.org. They’ve got some good stuff over there.
First off, I want to welcome everyone from StumbleUpon. Hopefully you’ll enjoy your stay while you’re here. Also, I’m going to be shifting the focus of this blog away from the theological for the indefinite future. I’ve had my fill of adding to the vast landscape which is the emergent conversation. I’ve reached saturation point, so to speak.
So into the future I’m going to try to post more on the topics of math, logic, and the philosophical underpinnings thereof. Check out the Logic/Math category on this blog to see what I’ve written so far on these topics. I’m also going to install a LaTeX plugin in the near future so I’ll be able to post some more symbol intensive stuff. Also, if you have any recommendations of topics you’d like to read about or related articles/blogs that would be good for me to link to, please let me know by leaving a comment. Take it easy everyone.
Check out this article over at Science News Online that explores some of the wonderful mathematics references that have found their place in episodes of The Simpsons. For instance, in one episode “Kwik-E-Mart proprietor Apu brags that he can recite pi to 40,000 decimal places. “The last digit is 1,” he announces. To get that detail right, the Simpsons writing team faxed a query to NASA, where mathematician David Bailey obliged with the digit in question.”
It’s good to know that great comedy doesn’t always have to be brainless.
There’s a great article over at Seed Magazine about the seventh annual “Gathering for Gardner” conference, which in short celebrates the literal magic of mathematics. As the write-up explains, the Gathering for Gardner conference is “a bi-annual pilgrimage honoring Martin Gardner, who, from 1957 to 1981, enraptured mathematicians and scientists, hobbyists and professionals, magicians and puzzlists and skeptics alike with his “Mathematical Games” column in Scientific American.”
I highly recommend reading their account of the conference, which is highly entertaining. Ah, mystery and magic!
264 years ago today Christian Goldbach wrote a letter to fellow mathematician Leonhard Euler in which he conjectured a very simple idea.
This conjecture, though hauntingly simple, has resisted proof for over two and a half centuries. Though I have no evidence to support it, I would say that the so-called Goldbach conjecture has led to the insanity of more than one mathematician.
Simply stated in its modern form, here is the Goldbach Conjecture: Every even number greater than four can be expressed as the sum of two prime numbers. The number 2 is excluded since it is itself prime. Sounds simple, doesn’t it? Well, one can only wish that every easily stated problem had an easily stated answer.
In fact, sometimes the simplest sounding problems are the most resilient. This conjecture would be a prime example. Although millions of even numbers have been computer tested in confirmation to the conjecture, there is always that faint possibility that there’s an untested even number out there that doesn’t fit in (does anyone remember the Fermat Primes?). So while the Goldbach Conjecture seems experimentally obvious, its proof is still hidden.
I love mystery. I suppose that’s at least partially why I love math and logic. There are century old mysteries that exist in these disciplines. These mysteries provide a portion of the shared history of mathematicians and logicians. Also, I believe these mysteries provide hope. They give hope that even though math and logic have historically come a long way that there are still beautiful discoveries to be made. In other words, the mathematical landscape is lush.
It’s a shame to me that the popular perception of math views the discipline as total “objective” fact. When seen in this way the actual organic nature of the subject is skewed or forgotten. Math is not as “objective” as you’d think. At least not in the way you were taught as a child. To borrow a famous example, when first learning about the structure of an atom, a physicist may be asked to imagine a model similar to our solar system, with the planets revolving gracefully around the sun. It’s only later when the true nature of the atom is revealed as a much more complex reality governed by probability. A similar type of metaphor may be given for the mathematical sciences.
Aim to think deeper. Aim to revel in mystery. Thank you Christian Goldbach.
The Xinhua News Agency website is reporting that two Chinese mathematicians, Zhu Xiping and Cao Huaidong, have proved the Poincaré Conjecture. Or rather, I should say, that the proof is currently being scrutinized by a panel from the Asian Journal of Mathematics, where the proposed proof will be published upon consensus.
According to its MathWorld entry, the Poincaré Conjecture states “that the three-sphere is the only type of bounded three-dimensional space possible that contains no holes”, where “a three-sphere is simply a generalization of the usual sphere to one dimension higher”.
The Clay Mathematics Institute also had listed the conjecture as one of its Millenium Problems, each of which carried a one million dollar reward for a proof. Their explanation of the problem puts it this way:
If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is “simply connected,” but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since.
As the article makes clear, the movement toward this proof spanned both decades and nationality. American, Russian, and Chinese mathematicians were instrumental in progression of ideas which finally cracked the conjecture. I hope that this example illustrates the global and community effort of current mathematicians in their work.
Congratulations to Zhu Xiping and Cao Huaidong!
According to its Wikipedia entry, a numbers station is “a shortwave radio station of uncertain origins.” According to both direct and indirect evidence a numbers station is a radio station used by a government to communicate secretive information to its spies. A stereotypical numbers station broadcast is basically comprised of a seemingly nonsensical sequence of numbers or letters read by a human voice. The idea is that the spy listens to the broadcast and then is able to decipher the sequence using a previously received decoder. In theory, if the spy has kept the decoder secretive, the code is almost literally unbreakable. In other words, this is a fairly secure way to pass sensitive information.
You can listen to a good example of a numbers station broadcast here. I suggest not listening to this alone at night, as it sounds a bit creepy. This recording is one of many included in the Conet Project, which according to Archive.org is “the first comprehensive collection of Numbers Stations recordings released to the public”.
Given the news of various NSA activities in America over the last several months I thought it would be interesting to write something about “spy” activity. There’s a lot of great in-depth information (including many audio examples of numbers station broadcasts) located in both the Wikipedia article and on Archive.org (both previously linked), so I suggest checking these out if you’re interested at all. Who knows, the next time you turn on a shortwave radio maybe all you’ll hear is numbers.
So this isn’t new at all, but when I saw it about a month ago for the first time I thought it was hilarious. Yeah yeah I know it should be “as x approaches 8 from above”, but come on now, let’s just let ourselves laugh a bit. Let me know if you have any funny math jokes/quotes!
For those who don’t know, April happens to be Mathematics Awareness Month (MAM). According to the MAM website, “Mathematics Awareness Month is held each year in April. Its goal is to increase public understanding of and appreciation for mathematics.” The organization which sponsors this month is the Joint Policy Board for Mathematics, which “is a collaborative effort of the American Mathematical Society, the American Statistical Association, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics.”
Each year has an appropriately labeled theme, and this year is no different. 2006 is the year of Mathematics and Internet Security. There are a few links to articles related to math and internet security on the website, including topics such as internet voting, public key cryptography, internet password security, computer viruses, and secure data storage. So if you fall into the category of people who doesn’t understand what math does in the real world, read some of this stuff. There is no escape from math, MWAHAHAHAHA!
The website for MAM can be found here.
Happy Mathematics Awareness Month Everyone!
It happens to be the case that there are two different verses in the Old Testament which provide for an approximation of π. In the NIV translation both I Kings 7:23 and II Chronicles 4:2 give the following measurements for a tank which would be enclosed in the “First” Temple:
He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it.
Taking these measurements along with the ratio for π given as circumference over diameter we have a coarse approximation for π of 3. While not astounding in accuracy, I’m always excited to see how subtle tidbits of mathematics invade even religious scripture. In other words, sometimes math is beautifully inescapable.
In you’re interested in learning more about these two verses I suggest reading an article called “On the Rabbinical Exegesis of an Enhanced Biblical Value of π” written by Shlomo Edward G. Belaga. An online version of the article can be found here. The article surmises that the Biblical narrative lends itself to a much more accurate approximation of π. It’s worth a look if you’re intrigued by such ideas.
I’m reading a book on the history of zero which I’m sure I’ll talk more about later, but for the time being I want to reiterate a proof as to why division by zero is a mathematical no no. This isn’t a complete answer at all but it’s a quick little proof by contradiction that is easily grasped. Assume for the time being that division by zero is okay.
Take, as an assumption, that 3 does not equal 11.
We know that both 3 * 0 = 0 and 11 * 0 = 0.
So 3 * 0 = 11 * 0.
Then (3 * 0) / 0 = (11 * 0) / 0.
On both the left and right sides of the equal sign cancel the zeroes in the fraction.
So 3 = 11.
But we know from our initial assumption that 3 does not equal 11.
Contradiction.
The only other assumption we made was that division by zero was possible. Since this assumption led to a contradiction it is a false assumption. So division by zero is a questionable practice. I used 3 and 11 as example numbers, but the same argument holds for arbitrary numbers x and y, where x does not equal y.
I wanted to take a moment to let everyone out there know about the summer school in logic and formal epistemology that will be offered through Carnegie Mellon Philosophy this summer. Here’s a short excerpt from the webpage:
There is a long tradition of fruitful interaction between philosophy and the sciences. Logic and statistics emerged, historically, from combined philosophical and scientific inquiry into the nature of mathematical and scientific inference; and the modern conceptions of psychology, linguistics, and computer science are the results of sustained reflection on the nature of mind, language, and computation. In today’s climate of disciplinary specialization, however, foundational reflection is becoming increasingly rare. As a result, developments in the sciences are often conceptually ill-founded, and philosophical debates lack scientific substance.
In 2006, the Department of Philosophy at Carnegie Mellon University will launch a three-week summer school in logic and formal epistemology for promising undergraduates in philosophy, mathematics, computer science, linguistics, and other sciences. The goals are to:
* introduce promising students to cross-disciplinary fields of research at an early stage in their career; and
* forge lasting interdisciplinary links between the various disciplines.
Other points to note are:
1. The program is open to all undergraduates or those who will have just received their undergraduate degrees.
2. The costs of housing and tuition are $0. In other words, this opportunity won’t break the bank. Some costs will apply, such as food, but all in all the price is quite inexpensive.
3. The dates of the program run from Monday, June 12 to Friday, June 30, 2006
For more information please reference the web page for the summer school here. There is also a PDF flyer for the program located here.
If you have a stray hour here or there I highly suggest checking out my multimedia link for “The Elegant Universe”. It’s a 3 hour (broken into smaller segments) PBS adaptation of Brian Greene’s book of the same name. It surveys 20th century physics into the present, showing how the developments and subsequent mismatches of relativity and quantum mechanics are potentially bridged by string theory. The program is extremely accessible and informative. Even if you don’t consider yourself scientific I would recommend this one. I’ve said for a long time that if you wanted to follow the zeitgeist of the 20th century that all you had to do was follow the theories of the physicists (my hypothesis for the 21st is that we’ll be following the theories of the biologists). In other words, watching the program will bring you more than just a physics tutorial. As least I hope it does.
In this vein…
Once upon a time a man was branded as a heretic by the church for claiming that the earth was not the center of the universe. You may have heard of this man at some point in your life travels. It turns out he was right. Well, he was least slightly more right than the presently accepted truth. He posited that the sun was the center of the universe. Now we know that this is in fact not the case. Not only is the sun moving along with every other star in our galaxy, but our galaxy is also moving. Every galaxy is moving. To and fro the heavens move. What I’m getting at is that we’re never finished with discovering the beautiful and complex facts about existence. We shouldn’t be surprised when the popular understanding of reality is shaken beyond recognition. Nor should we be worried.
Even as I write this post scholars at the forefront of physics question whether our universe is unique or whether it is just one of an infinite number of universes. Does that possibility bother you? Think about it. An infinite number of universes. Well, I don’t think it should bother you. In fact, my belief is that it should comfort you. As we discover the deeper complexities of our universe I see more and more only the hands of God.
Similarly, I feel the same way about the so-called emerging worldviews. I embrace postmodernity. I embrace relativism. Not without question, but I do embrace them. New perspectives bring both greater clarity and greater confusion. It’s the job of those who care to help work out which is which. Will postmodernity or relativism destroy Christian or any other religious belief? Of course not. And if you believe that it will then you are naïve.
Look at relativity and quantum mechanics. They are two branches of physics that smart people have been trying to unite for the past half century. Alone, each tells us unique facts. But when these facts try to mesh together in the current spheres of thought there are inconsistencies. Does this mean that the two oppose one another? I don’t think it does. On one levels there is definite disagreement. But I believe with no doubt that there is a larger canvas through which the two fit in harmony. It is beyond right and wrong.
Why can’t we get past right and wrong? Why can’t we let go of trying to control everything we do? It is so utterly binding. There is always, I am convinced, one step farther down the line that explains the currently unanswerable. And at the end of this line I do believe that God stands with a smile.
I’m not sure what to do with this website ultimately. I feel tripolar about it. First, I have all these random “state-of-the-union in-Ian’s-life” stuff, comprised of dreams and musings and incoherent nonsense. Some of the posts in this category are probably better off somewhere else, but I know that there’s probably someone out there that likes to read them.
Next I have the logic and math stuff, which I’m currently pursuing educationally. I love this stuff, but when I post about it I realize that just about no one is going to read it or understand it (unless they expressly came here to check out what I’m doing at Carnegie Mellon, which is a possibility). I do recommend that even if you don’t have any desire to learn anything about math/logic that you try to read one or two papers/posts anyway. Some of the ideas behind the papers are easily grasped and philosophically extensive. It’s good stuff, and wherever I end up in life I’ll always respect these disciplines.
Lastly, I have all this theology stuff everywhere, which is something I’m passionate about. But the people who are coming here to read about logic/math are probably not at all interested in reading about Christian concepts of absolute truth in the postmodern age. Alas, for now they’ll just have to sift through the muck, although from a philosophical and cultural point of view some of the theology writings and website links are extremely informative even for thos who hold no faith.
So there it is. My tripolar nature. And I don’t apologize about it. It’s just not externally consistent. I hesitate to split this up into three separate blogs, but I may have to do it. Any suggestions from anyone out there?
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