Monthly Archives: November 2006

Nicolas Bourbaki

Near the middle of the 20th century, Nicolas Bourbaki published several mathematics texts in areas such as Set Theory, Algebra, Topology, Functions of a Real Variable, etc. These texts had a profound impact on the mathematical landscape of the day and their affect is felt to this day. Many of the readers of this blog may remember the introduction of the “New Math” where the fundamental concepts of mathematics were built from the basic concepts in set theory. So what’s so significant about Bourbaki? How about the fact that he never existed. He is a fictional character invented by a group of French mathematicians who adopted the name Bourbaki from a French general of Greek origin, who lost a significant battle to the Prussians. That’s right, publications, major mathematical contributions, all made by a group hiding behind the allonym, Nicolas Bourbaki. What’s even more interesting is that for some time, the fact that Bourbaki was fictional was not known to the general mathematical community.

The first Bourbaki congress, July 1935. From left to right, back row: Henri Cartan, René de Possel, Jean Dieudonné, André Weil, university lab technician, seated: Mirlès, Claude Chevalley, Szolem Mandelbrojt.

One of the primary founders was André Weil who, along with the rest of the founders, was quite fond of the practical joke. Originally this group was founded in order to author an improved Calculus/Analysis text. Eventually, their goal expanded to basically re-develop the whole of mathematical theory around the premise of greater rigor.

For a time, the group was very secretive. Once the secret got out, it spread fairly slowly. Bourbaki was already making waves in the mathematical community. Even when most of the French were aware of the fictional nature of the Bourbaki character, they wondered if the Americans were aware. At one point, Weil sent an application to the American Mathematical Society requesting membership. The group had gone so far as to counterfeit official documents and create an imaginary life for Nicolas Bourbaki. Nevertheless, the president of the AMS at the time had become aware of the Bourbaki “group” and sent a “tongue-in-cheek” reply saying that he understood that the application was not from an individual and that they would have to pay the substantially higher institutional member dues.

For more information on Bourbaki, check out some of the following links:

Do not sign an email petition

I imagine everyone’s received an email similar to the one that I got today: Prayer Petition, DR. DOBSON’S PLEA FOR ACTION. You can look here for more details on this email “urban legend”:

Anyway, the part that bugs me the most, and what I’ve seen on several such emails, is the request at the end to sign a petition. It really is absurd to try to form a petition via email. Here are a couple of reasons that come to mind.

First, there is no formality to the medium of email. Anyone could place anyone’s name on the email. I’m reminded of a local politician who likely cost himself an election by placing names on a published list of his supporters. It turned out there were several of those individuals who were not supporters. They took out ads in the paper to make sure everyone in the community knew it. Nevertheless, many online petitions, whether email or through some web application, make little effort to avoid this problem.

Second, there is no final or comprehensive list. As the email is circulated, one person signs their name and passes it on to some set of individuals, of which only a fraction will sign and pass along. At this point there are as many versions of this list as there are people that signed it (plus one, the list that died with those who did not respond). This propagates every time it is passed along. The list I received has 2,071 names and, in all likelihood, this email has passed through at least 10 times that number of inboxes.

If you want to participate in an electronic petition, there are much better ways to do so. For example, I was recently notified of a petition being compiled online to help repeal the No Child Left Behind Act. The NCLB is coming up for renewal in 2007 and significant number of individuals are in favor of it being rewritten or completely abandoned. I, personally, will not be signing that petition. However, it illustrates a “more” appropriate way to collect petition signatures online. (Here’s the petition, if you’re interested.)

So, if you don’t mind, you can take me off your email list when you are sending out an email petition, and I recommend ignoring them altogether.

Klein Four Group

The Klein Four

I have been a fan of this musical group since I learned about them some 2 years ago. I’ve seen a flurry of content regarding them come out over the web in the last month or so, so I must jump on the bandwagon.

The first song I listened to was the Finite Simple Group of Order 2, a love song for mathematicians. It is definitely worth your time even if you don’t understand what they are talking about. The real clincher for me was their performance of “Where in the World is Carmen Sandiego?” For any of you that don’t remember, “Where in the Wolrd is Carmen Sandiego?” was a computer game from the late 80s. It happened to be one of the games that hooked me on computers and technology as a kid. After the success of the video game, there was a short lived game show for kids with the same name and there was an a capella group that sang the theme song. Klein Four is not that group, even though they perform the song quite well.

You might want to check out their site and hear a few tunes for yourself. Math nerds will be hooked immediately. Check out the Altoid Challenge while you’re there, it’s equally impressive.


I just went to their site which has change significantly since I went there over a year ago. (how dare they!) You can find the links to Performances and the Altoid Challenge under Quick Links.


Oh, and by the way for more information on the “real” Klein four group, check out Wikipedia.

Mathematical Explosions

explosion.jpgWhen teaching limits of sequences, I often use the phrase that some part of the general term “blows up”. I typically use it to indicate that it is going to infinity. For example, in the statement that [tex]displaystyle frac{1}{n} longrightarrow 0[/tex], I’d say that the denominator of the fraction blows up and causes the entire fraction to go to zero. I have heard it so often and use it so much that it just sounds natural. I realized that it is slightly unnatural to the less mathematically experienced. On a recent Calculus III Exam, a student was using the Alternating Series Test for Convergence on the expression [tex]displaystyle sum_{n=1}^{infty} frac{(-1)^n n}{2^n}[/tex], and he stated, “The denominator is going to explode. . .” He’s right, it does.

Exploding denominators, run for cover!

Where’s the other dollar?

647617_pizza_calabreza.jpgThree guys in a hotel call room service, place an order for two large pizzas. The delivery boy brings them up with a bill for exactly $30.00. Each guy gives him a $10.00 bill, and he leaves. That’s fact! When he hands the $30.00 to the cashier, he is told a mistake was made. The bill was only $25.00, not $30.00. The cashier gives the delivery boy five $1.00 bills and tells him to take it back to the 3 guys who ordered the pizza. That’s fact! On the way to their room, the delivery boy has a thought…these guys did not give him a tip. He figures that since there is no way to split $5.00 evenly three ways anyhow, he wil keep two dollars for a tip, and just give them back three dollars. So far so good! He knocks on the door and one fellow answers. He explains there was a mix up in the bill, and hands the guy the three dollars, then departs with his two-dollar tip in his pocket. Now the fun begins! Remember! $30-$25=$5. Right? $5-$3=$2. Right? All is well, right?

Not quite. Answer this: Each of the three guys originally gave $10.00 each. They each got back $1.00 in change. That means they paid $9.00 each, which multiplied by three is $27.00. The delivery boy kept $2.00 for a tip. $27.00 plus $2.00 equals $29.00.

Where is the other dollar?

Continue reading Where’s the other dollar?