Archive for the ‘Math’ Category

Grigory Perelman

April 12, 2010 Leave a comment

Henri Poincaré

In my last post, on a Wall Street Journal article about Martin Gardner, I noted that it’s “a rare day when any major newspaper has an article with mathematical content.” I didn’t want to let the moment pass, so I wrote about the article. Now the latest New York Review of Books has arrived, with yet more mathematical content, a review by John Allen Paulos of Masha Gessen’s biography of Gregory Perelman, the Russian mathematician who proved the Poincaré Conjecture in 2002.

The conjecture, made by the great French mathematician Henri Poincaré in a 1904 paper, was one of the great unsolved problems in mathematics. Every mathematics graduate student learned about it when studying topology. It’s so easily stated, yet it resisted the efforts of mathematicians for a century, even as analogues were proven in dimensions 5 and higher by Stephen Smale, John Stallings, and E.C. Zeeman in the 1960s and in dimension 4 by Michael Freedman in 1982. Smale and Freedman were awarded Fields Medals for their work.

Perelman posted three papers in 2002 purporting to prove the conjecture, but the papers did not provide complete proofs. This led to a complex sequence of events that I won’t try to recount, with other mathematicians studying the approach he laid out, filling in details, and publishing their own papers verifying that he had indeed proved the conjecture. A 2006 New Yorker article by Sylvia Nasar and David Gruber that received a lot of attention gives an account of some of the controversy that ensued. Perelman was himself awarded a Fields Medal in 2006, but declined it. He was in the news again just two weeks ago when he declined the million-dollar prize awarded him on March 18 by the Clay Mathematics Institute for solving one of their seven Millennium Prize Problems. (Learn more by reading the Clay Institute’s short press release and full press release.

You can learn more about Perelman from the New Yorker article of 2006, Paulos’s review of the Gessen biography, or Jascha Hoffman’s NYT review of the biography last December. Hoffman notes in closing that Gessen “has written something rare: an accessible book about an unreachable man.”

Categories: Math

Gathering For Gardner

April 9, 2010 Leave a comment

I wrote two days ago about buying an iPad, mentioning in passing that one of the apps I had downloaded for it was the WSJ app. Yesterday I explored how it works. It’s really good. What it does is download and keep on the iPad the last seven days of the paper. The next day — if you bring up the app the next day — it deletes the oldest of the seven days and downloads the current day. You select the day you want, choose the section of the paper you want, and then start reading. In one mode, all the articles of that section are listed in a column on the right. When you tap on one of the articles, it comes up, with the column still there on the right so that you can go straight to any other article you wish. To continue reading a multi-page article, or to go back a page, you do the standard horizontal swipe.

Of course, this isn’t free. I don’t know what it costs to subscribe anew. As a print/online subscriber, I get iPad access, for now, at no additional cost. Apparently the WSJ will soon charge print subscribers.

Anyway, since we were back in New York a week ago, we didn’t get last Friday’s paper. As I explored the iPad edition of the WSJ yesterday, I realized I could look at Friday’s missed paper with just a tap. So I did, heading straight to the Weekend Journal, where I happily discovered an article on Martin Gardner that I would otherwise have missed. It’s a rare day when any major newspaper has an article with mathematical content. I’m glad I found this one.

Though not himself a mathematician, Gardner is one of history’s great popularizers of mathematics, through his long-running “Mathematical Games” column in Scientific American. He is as well one of the great debunkers of pseudo-science. The WSJ article describes the 9th annual conference in honor of Gardner, held two weeks ago in Atlanta. From the article:

. . . a four-day conference in honor of Martin Gardner, 95, a public intellectual whose most famous pulpit was “Mathematical Games,” written for Scientific American between 1956 and 1981. Mr. Gardner’s column illuminated the beauty of math and logic in discussions of fractals, origami, optical illusions, puzzles and pseudoscience. It challenged readers to discover how finely math and logic are interwoven through the world.

. . .

Puzzles are instructive, Mr. Gardner found, for they teach us to appreciate hidden structures of the world that are not owned by any particular discipline and are potentially useful to all. He saw the world as resembling not a magazine, where the subject of each section bears little relation to that of the next, but a well-written novel, where ideas introduced in one chapter are apt to reappear—transformed, modulated and extended—in others. He taught his readers to see the world in the same way, inculcating in them an openness and alertness to the often surprising possibilities of the world, and the desire to seek them out.

Categories: Math, Newspapers, Technology

2010 Abel Prize

March 24, 2010 Leave a comment

John Tate

Just by chance, I went to the website of the American Mathematical Society tonight and thereby stumbled on the news (which I would have learned soon enough) announced earlier today that John Tate is the 2010 recipient of the Abel Prize. The prize, established in 2001 by the Norwegian government, has been awarded by the Norwegian Academy of Science and Letters each year since 2003 to one or two outstanding mathematicians. It is named in honor of the great, early-nineteenth-century Norwegian mathematician Niels Abel and may be regarded as the mathematical counterpart of Nobel Prizes.

I wrote about the Abel Prize last June, in the wake of an article that week in the NYT reporting on the news that three of the recipients were NYU faculty members. As I noted at the time, according to the history of the Abel Prize given at their site, the idea for a math prize that would parallel the Nobel Prizes and be named after Abel goes back to 1899, when it was championed by that other great Norwegian mathematician Sophus Lie.

John Tate spent much of his career at Harvard, later moving to the University of Texas. The award is being given to him “for his vast and lasting impact on the theory of numbers.” You can read more about today’s announcement here and here.

I didn’t take a course from Tate when I was an undergraduate, but I wasn’t so foolish as to miss out on the opportunity altogether. During my third year in graduate school, I made it a point to attend his graduate number theory course, walking down to the Harvard Science Center from my apartment three mornings a week before heading over to MIT. Great course.

Categories: Math

Calculus Cartoon

December 15, 2009 1 comment

Ted Rall’s latest cartoon (above) is the rare political cartoon with an embedded calculus lesson. We might well wonder, as Rall does, why we are so excited by the news that unemployment continues to increase, only at a slower rate.

Rates of change are exactly what calculus allows us to discuss precisely. Rather than being some forbidden subject, calculus is simply the language to quantify and discuss such matters. But even without studying (or remembering) calculus, we all understand the basic issues, at least qualitatively.

Given a quantity we wish to measure, like the number of people unemployed, or how far we have traveled from home, the derivative tells us how quickly this quantity is changing (increase or decrease in unemployment figures per month; increase or decrease in distance from home per hour, otherwise known as velocity). And the second derivative, which is what has been in the news lately with regard to unemployment figures, measures how quickly that first rate of change is changing. This is indeed a subtle notion, but one we talk about all the time. In the unemployment example, it is the rate at which the change in unemployment is going up or down. Thus, unemployment may still be increasing this month, but perhaps it is increasing more slowly than it has increased in recent months. That slower rate of increase is measured by the second derivative, and its slowing means the second derivative is negative. In the example of leaving home, we might be driving away, but the velocity at which we are driving is decreasing — perhaps we braked but haven’t yet come to a stop. This means the second derivative of the distance from home is negative. And we have a familiar name for that second derivative. Acceleration. So acceleration is negative when the velocity is going down, even though the velocity may still be large and we may still be moving rapidly away from home. Just less rapidly.

So that’s that. Simple enough ideas, but fundamental.

By the way, let’s say I desperately wanted to get home for dinner, but I had taken a wrong turn and found myself entering the freeway in the wrong direction, taking me farther away from home. I had no choice but to drive to the next exit, another five miles away. Alas, a couple of miles down the road, traffic slowed because of a car stuck on the side of the road. I would have to brake gently and begin to drive 20 mph rather than 55 mph. Thus, the rate at which my distance from home is increasing would have begun to decrease. I would continue to get farther from home, that’s for sure, but at a slower speed. Is this good news?

Categories: Cartoons, Economy, Math

Logicomix Again

October 7, 2009 1 comment

Two Saturdays ago, I wrote a short post about the new graphic novel Logicomix: An Epic Search for Truth, right after reading Jim Holt’s review of it in the next day’s New York Times. Now that I’ve read it, I’ll say a little more.

As noted in my earlier post, the novel tells the story of Bertrand Russell‘s failed effort to build a logical foundation for mathematics. Why math? Well, there’s the obvious reason that it’s more interesting than anything else. But more to the point, one can imagine that if there’s any hope of building a foundation for some subject– a foundation allowing us to know the truth of its statements with certainty — then the subject most likely to yield to such a construction project is mathematics. The Russell depicted in the novel (and it is a novel, based on the real Russell and his compatriots, but not a genuine biography or history) is excited in his youth by the beauty of Euclidean geometry, thrilled that logic and reason can yield truths, but disturbed that there was something missing in the foundations of the subject. At Cambridge, his disquiet grows. He observes, while courting his future wife, that “At Cambridge, no one talks about the real issues of mathematics. Like what is the nature of mathematical truth?” He adds, “If only you knew how much depends on these questions. How crucial they are!”

And she married him! How about that? I had similar interests as an undergraduate. And I wasn’t as smart as Russell. But I did know that talking about the nature of mathematical truth wasn’t a promising approach to dating. (Then again, I didn’t exactly have a lot of success with other approaches. Maybe I should have tried it.)

The novel isn’t just about logic and math. Irrationality, madness, pacifism, the limits of reason, the Vienna Circle and Nazis all play major roles. Plus, of course, it’s a graphic novel, so there are all the drawings, which I didn’t give sufficient attention the first time around, since I was so eager to follow the story. I will need to re-read it with a closer look at the artwork. Along the way, the authors get to poke a little fun at those annoyingly logical people who make normal conversation difficult. For instance, there is the imagined visit Russell and his wife pay to the great logician Gottlieb Frege in Germany some time in the 1890s. They arrive at a home and ask the fellow who is seen in the yard, trimming the hedge, “Is this Professor Frege’s house?” “No,” he replies. “This is his garden. His house is in there.” Russell asks if the professor is at home. “No, he is in the garden.” Maddening. Which gets back to the recurring theme of the interplay between logic and madness.

An important character throughout the novel, inevitably, is Alfred North Whitehead, the co-author with Russell of Principia Mathematica, the three-volume work in which they lay out their logical foundations for mathematics. But the one who steals the show — for me — is Ludwig Wittgenstein, who appears about three-fourths of the way through the book (page 223) when he arrives at Russell’s door in his Cambridge University rooms, having been sent from Germany by Frege to learn logic from Russell. Seemingly an admirer, Wittgenstein soon becomes Russell’s most powerful critic. World War I intervenes, dramatically altering both Russell and Wittgenstein. Then, as the book nears its conclusion, Kurt Gödel inevitably arrives, demolishing the dream that a proper logical foundation for mathematics can assure the existence of a proof for every true mathematical statement. One of the amusing conceits of the novel is that Gödel, who laid waste to Russell’s program, may have been the only person who ever bothered to read the Principia Mathematica in full. Yet, perhaps only by building on the Russell and Whitehead’s development of logical foundations could Gödel have developed the methods that showed the limitations of logic as a foundation for mathematics. The book can only touch on this, one of the great intellectual discoveries of the twentieth century.

The authors and artists themselves appear throughout the novel, along with a pet dog, in interludes in which they discuss the book’s issues while working or walking in Athens. All the ideas come together when they attend a local production of Aeschylus’s Oresteia Trilogy.

Math, logic, war, peace, theater, dogs. I haven’t even mentioned the failed marriages, crazed experiments in education, and messed-up children. Something for everyone.

A final note: I just noticed a link at the book’s website to a trailer, which is the youtube video I have inserted at the top of the post. Have a look. It includes an appearance by Barry Mazur, a fabulous mathematician from whom I learned algebra in my sophomore and junior years. He then became my senior thesis advisor.

Categories: Biography, Books, History, Logic, Math


September 26, 2009 Leave a comment


In looking ahead, online, at tomorrow’s NYT Sunday Book Review, I came upon Jim Holt’s review of the graphic novel Logicomix: An Epic Search for Truth, written by Apostolos Doxiadis and Christos H. Papadimitriou and illustrated by Alecos Papadatos and Annie Di Donna. I read the review, looked at some sample pages available from the review’s webpage, went to the book’s website, realized that there will be a book reading here in Seattle in two weeks, and ordered it. I have a bit of a book backlog, as usual, but I knew I would want to read this eventually, so why not just get it?

As best I can tell, the book tells the story of Bertrand Russell’s decades-long failed effort to find a logical foundation for mathematics. Along the way, other famous logicians and mathematicians appear, including Gottlieb Frege, Georg Cantor, David Hilbert, Russell’s co-author Alfred North Whitehead, and, of course, Kurt Gödel, the greatest of all logicians, hero to us all. Plus, various world events intervene. How could I resist?

Perhaps I’ll have more to say after I read it, or after I attend the book reading. If you’re skeptical that such a book might be interesting, I suggest that you read the NYT review and visit the book’s website. See too, among many choices, Rebecca Goldstein’s recent book Incompleteness: The Proof and Paradox of Kurt Gödel.

Categories: Biography, Books, History, Logic, Math

Why Educate?

August 27, 2009 Leave a comment


When the September issue of Harper’s arrived at the house the week before last, I immediately read Mark Slouka’s article Dehumanized: When Math and Science Rule the School. I was going to write about it at the time, but the website still had the August up. September is online now (though you may need an account to read the full article). If you can get access to the article, I recommend it. Slouka makes a good case for the dangers of de-emphasizing the arts and humanities in favor of math, science, and preparation to participate in the market economy. I think he mis-represents the nature of mathematics at times. Whether he does so out of ignorance or in service to his argument I have no way to tell. But any errors in this direction shouldn’t distract from his larger warning about an imbalance in US education, with which I largely agree.

It is difficult, indeed unwise, for a university administrator to resist the temptation to build strength in disciplines that have the potential to bring in external research funding (at a major research university anyway). But at least when one makes such decisions, one should be aware of the issues Slouka raises. After the jump, I’ll quote some passages from the article to give an idea of his argument.

I am reminded of my son Joel’s initial first grade homework assignments years ago. On the first evening, he was to establish a location in the house where he would put his completed homework, so that he would be able to remember on a consistent basis to bring it to school each day. There was a similar assignment the next night, maybe involving setting up a regular work location. I had the sinking feeling that the underlying goal was to train him for the workforce rather than educate him. A year later, at our parent-teacher conference to review his work, I was struck even more forcefully by the realization that that teacher’s concern was his success at developing proper work habits, as opposed to his giving free rein to his curiosity.

This is an old tension in education, workforce development and socialization versus creativity and imagination. Many have written far more eloquently about it than I can, Slouka in particular. So I won’t say more. Except to note that science and math are not on one side of this. They are very much a haven for creativity and imagination. The problem that arises is how to respond when business and legislative leaders argue that math and science, as the areas most likely to lead to new business opportunities and most in demand by highly desirable businesses, should be given extra funding so that a university can train more students to prepare for careers in these fields. This is a good problem. Yet, it can open the door to mis-understanding about what a research university’s mission is, what the larger benefits of math and science education to all citizens can be, and how important arts and humanities are as well for an educated citizen.

Let me leave it at that. Here are representative excerpts from Slouka’s article:
Read more…

Categories: Culture, Education, Math, Science

Erdős Number 1?

July 17, 2009 Leave a comment


A hat tip to Arnold Zwicky at Language Log for pointing me to the recent xkcd cartoon above, in which mathematicians the world over are given hope that they might yet be able to acquire an Erdős number of 1.

I described Erdős number in a post last Christmas Eve, explaining that it is how mathematicians measure “their level of connectedness to the late, prolific Hungarian mathematician Paul Erdős. Erdős has an Erdős number of 0. If you wrote a joint paper with Erdős, your Erdős number is 1. If you didn’t, but you wrote a joint paper with someone who wrote a paper with Erdős, your Erdős number is 2. And so on.” (See here for more information.) I also noted the more familiar but essentially identical method of measuring an actor’s degree of separation from Kevin Bacon. The purpose of the post was to measure my Erdős-like distance from mathematician and Iraqi politician Ahmed Chalabi.

A week later I had a post on my Duke Ellington number. At the time, I did some research for a post on my Ty Cobb number, but I never wrote it. Perhaps I’ll do so soon. (I have several two-step connections to baseball players. Fairly serious connections, not just that I saw them at a game. Rather, I know someone well who is a friend or relative of a major league baseball player.)

My own Erdős number is 4. I hadn’t expected it to get any lower, but the cartoon gives me hope.

Categories: Cartoons, Math

Average Rider

July 12, 2009 Leave a comment


Light rail comes to Seattle next Saturday. We’ve been watching the line get built for years now. It’s hard to miss on trips to the airport, between the elevated tracks and the rise of the giant station. (And for those who live or work along its right of way, its presence has been especially noticeable.) It will run from downtown to just north of the airport, and in half a year it will reach the airport.

Given the locations of our house, work, downtown, and shopping, we are not likely to be frequent users of the line. Construction recently started on the next phase, which will run from downtown to the university, with the expectation, once more funding is secured, that it will continue on northwards from the university to the Northgate area, the densest part of north Seattle. This will make the line more convenient to us, but who knows where we’ll be living by then. We’re talking many years down the road. Or down the rail.

Anyway, however often we are destined to use the line, I plan to be on it on Saturday. I can’t wait. And I’ve been a close reader of the articles the Seattle Times has been carrying on it in the buildup to opening day. Each station has had its own feature article. Today there’s an overview of the line, along with a discussion of the public art at the stations, an interactive map, and a graphic showing the light-rail train.

Which brings me to the point of this post. In the graphic on the train, the accompanying text tells us, “To the average Seattle transit rider, a standard 190-foot, two-car Link train will look huge — about three times as long as an articulated bus.” I’ve long tired of lazy or inaccurate uses of the word ‘average’. For instance, whenever there’s a ballot measure to approve some new property tax, we learn that the average homeowner will pay an additional $200/year, or whatever.

I would prefer that we speak about taxes on a house with the average assessed value, if we are to use such formulations, rather than the tax for an average homeowner, whatever that is. But this seemed to take the lazy use (I might call it the wrong use, but I don’t want to be so judgmental) of ‘average’ to a new level. The train will look huge to the average Seattle transit rider. What could this possibly mean? I’m above average in height and well above average in weight. Will it look less huge to me? Is that what they’re getting at? Or has the average Seattle transit rider not gotten around much, so the rider doesn’t know that there are vehicles larger than buses. Like, um, trains. Commuter trains, like the ones I grew up riding in New York. Or Amtrak trains, like the ones that pull out of Seattle daily. Or, my gosh, freight trains. They’re galactic.

Am I missing something?

Categories: Language, Math, Transportation

Saucepan Reasoning

June 14, 2009 Leave a comment


Two weeks ago, I wrote again post about the edible idiom feature at Clotilde Dusoulier’s blog Chocolate & Zucchini, in which she discusses a French idiom related in some way to food or cooking. Once more I can’t resist writing about her latest, which this week is “Raisonner comme une casserole.” Clotilde offers the translation “reasoning like a saucepan” and the explanation that “it means demonstrating poor logic, formulating arguments that are evidently flawed. It is a colloquial expression that should only be used in informal conversation.” She goes on to reveal the underlying pun, which becomes merely a near-pun in English:

It’s not hard to imagine that debating philosophical matters with a saucepan would lead you nowhere, but there is actually a little more to this idiom than that: it is in fact a pun that plays upon two homophonous verbs, raisonner, which means to reason, and résonner, which means to resound. So when you say, “il raisonne comme une casserole,” it is really a double entendre, meaning that the person has as much sense as a saucepan, but also implying that if you banged him on the head, it would likely echo.

I should explain that I may have been particularly charmed by this expression because I graded the last homework assignment and the final exams for my spring quarter course in the two days before Clotilde’s post appeared. The course is named Introduction to Mathematical Reasoning. It is intended to prepare students who have taken our standard lower-level math courses (calculus, linear algebra, differential equations) for the more rigorous courses that lie ahead. I had something to do with the department’s decision to introduce this course a decade ago, but by the time we started offering it, I had begun my multi-year teaching hiatus. Now that I’m back in the classroom, teaching it seemed like a good idea.

I would prefer to adhere to my general policy of not discussing my teaching experiences here at ronsview. I’ll restrict myself to two points. First, I’ve been humbled by the discovery (or, really, re-discovery) of how hard it is to teach reasoning. Second, I’ve had the opportunity to hear a lot of saucepan reasoning. My ears are still resounding.

Categories: Language, Math, Teaching