### Science

# The Case of the Disappearing Mathematician

Starting in the 1930s, Nicolas Bourbaki published dozens of papers, becoming a famous mathematician. There was just one problem: He didn't exist. Author and scholar Amir Aczel talks about the genius mathematician who wasn't.

IRA FLATOW, host:

Up next, another sort of mystery about a mathematician. I want you to ask your nearest mathematician, of course the person sitting next to you, what he or she knows about Nicolas Bourbaki.

And you're in for a long answer because author Amir Aczel first encountered this legend as an undergraduate math major in the University of California at Berkeley. That was some time ago.

And it goes something like this. And I'm going to have him fill in the end. I'll just tell you the beginning. In the 1930s, a French mathematician by the name of Nicolas Bourbaki set out to unify and redefine mathematics. And he published many papers, many books. His works were considered some of the most widely influential mathematic text of the 20th century.

For example, do you remember the new math you learned as a kid? Well he's probably responsible for that.

There's only one little fly in this ointment, and that is Nicolas Bourbaki never existed. He was all a hoax. The math was real; the person wasn't. So who was he?

My next guest, Amir Aczel, has written extensively on science and mathematics, including the books, The Riddle of the Compass, Fermat's Last Theorem; his latest book is The Artist and the Mathematician, tells the story of how Nicolas Bourbaki came to be and how the longest running joke in mathematics has changed the world.

Dr. Aczel is a visiting scholar in the history of science at Harvard and a research fellow at Boston University. And he joins us today in our SCIENCE FRIDAY studios. Welcome back to the program.

Dr. AMIR ACZEL (Author, The Artist and the Mathematician): Thank you. It's a pleasure to be here.

FLATOW: Did mathematicians know this was a hoax all this time when you were an undergraduate and you heard about him or - and you just play along with this, or what's the story on this?

Dr. ACZEL: Well, when I was an undergraduate he was already well known in America.

FLATOW: Yeah.

Dr. ACZEL: In France, it was well known earlier. In one of the stories that - the funniest stories about Bourbaki is that they weren't sure whether Americans and people in other countries knew about them. And at some point Bourbaki - actually Andre Weil who was one of the main members, he pronounced it Vay(ph), one of the founders - they key founder wrote a letter to the American Mathematical Society in Providence, Rhode Island, saying I request membership in the American Mathematical Society.

(Soundbite of laughter)

FLATOW: Under that name.

Dr. ACZEL: Yes.

FLATOW: Why was that name chosen?

Dr. ACZEL: Well the name was chosen because there was a general, a German - I'm sorry, a Greek - of Greek origin, a French of Greek of origin by the name of Charles Bourbaki, who was a general and lost France, a major battle against oppression.

So he was an anti-hero. And he tried to commit suicide and couldn't succeed either. So he was a real loser. And these French mathematicians, who loved jokes and pranks, chose his name for their group.

I wanted to add, though, that Ralph Boas, who was the secretary of the American Mathematical Society, already knew who Bourbaki was.

FLATOW: Ah.

Dr. ACZEL: This was in the '50s.

FLATOW: Right.

Dr. ACZEL: So he wrote a letter back to France saying, I understand this is not an application from an individual. You'll have to pay the institutional rate, which is much higher.

(Soundbite of laughter)

FLATOW: Talking with Amir Aczel, the author of The Artist and the Mathematician: The Story of Nicolas Bourbaki, the Genius Mathematician Who Never Existed on TALK OF THE NATION: SCIENCE FRIDAY from NPR News.

Why create him? I mean, why - was this just a joke, really, of mathematicians who are because they're kind of funny people creating this fictitious person?

Dr. ACZEL: Yes. They loved pranks. But in addition to that there was a reason behind that. And the reason was that French mathematics was not doing very well in the beginning part of the 20th century.

There was a book by a person named Gorsa(ph), who was really not a very good textbook writer. And this was used in their calculus sequence, which is the most important basic math course.

FLATOW: Right.

Dr. ACZEL: And students were not doing well. It was poorly written. The examples were bad. It was just a very bad textbook.

So these six mathematicians by the name of Cartan, Chevalley, Delsarte, Dieudonne, Possel, and Weil, met together in a café in Paris. And the café is in the best place in Paris, of course.

FLATOW: Of course. But of course.

Dr. ACZEL: Yes. It's on the Boulevard Saint-Michel, right next to the, below the Pantheon in the center. Today it's a like a McDonald's, a French type of McDonald's.

But there used to be a great café there. They met there. And they said, let's beat this guy, Gorsa. We'll write our own textbook. And since textbooks can't be written or they didn't think could be written by six people or they - and they didn't want to reveal their identities. They were the young Turks who were trying to take over French mathematics on the older generation.

So they decided to invent this person that didn't exist. Now there's a history behind it. Andre Weil was really into these jokes. He just loved these jokes.

And a few years earlier in 1923 a person by the name of Raoul Husson, H-u-s-s-o-n, played a prank on all the entering class in mathematics at the Ecole Normal Superieure, one of the main - the most prestigious French schools where mathematics was very important.

And he gathered all the freshman in a room and came dressed with a fake beard and strange outfit to look like the general, and wrote on the board, Theorem of Bourbaki, you are to prove the following.

And of course it was all nonsense. And the people were sitting there scratching their heads trying to figure out what it was. And they eventually knew it was a prank. He wasn't one of the people who were there. He had heard about it.

Somebody came to him and said, you know, I really did understand that theorem. So it was a joke that seemed to work on people.

FLATOW: Now what about all the mathematics that was behind it? Was it still - was the math a joke also or was it really good mathematics that they were coming out with?

Dr. ACZEL: What they were creating was excellent mathematics.

FLATOW: Right.

Dr. ACZEL: And it is still affecting us today.

FLATOW: Give us an idea of the kind of things.

Dr. ACZEL: Well mathematics was not done very rigorously and not with great abstraction or generality at the time of Bourbaki in the '30s. The meeting in that café was 1934.

At that time, for example, Poincare, Henri Poincare, who is a very famous now because of the Poincare conjecture, was the epitome - he epitomized mathematics that was imprecise. He had a great insight and could do mathematics very, very well. But he didn't care about details, Epsilon, Delta type things, as mathematicians would say. He didn't care about very rigorous proofs. And that frame of mind entered into instruction mathematics.

FLATOW: All right. We're going to have Amir take a breath, take a time out here. We're all going to come back and talk about this hoax of a mathematician. Stay with us. We'll be right back after this short break.

I'm Ira Flatow. This is TALK OF THE NATION: SCIENCE FRIDAY from NPR News.

(Soundbite of music)

FLATOW: This is TALK OF THE NATION: SCIENCE FRIDAY. I'm Ira Flatow.

A brief program note, coming up on Monday Neal Conan talks with comedienne Paula Poundstone about her new book, The Best Part of Her Arrest, and why she never wins on WAIT, WAIT, DON'T TELL ME. Plus, O.J. Simpson's latest collision with the media spotlight. That's on Monday's TALK OF THE NATION.

We're talking this hour on SCIENCE FRIDAY with Amir Aczel, author of The Artist and the Mathematician: The Story of Nicolas Bourbaki - got to get it right - The Genius Mathematician Who Never Existed, published this year by Thunder's Mouth Press.

Amir, tell us about the accomplishments they did make, including, if I read correctly, this new math that we were all taught as kids.

Dr. ACZEL: Right. So they were sitting in this café in the heart of the Left Bank in Paris. And they're talking about, these six mathematicians, - later, a seventh joins later, some other - so the number after the first meeting is never precise. Nobody knows.

FLATOW: They could keep this secret that this is a fictitious mathematician that they were imitating?

Dr. ACZEL: Right. They could keep the secrets for quite a while.

FLATOW: Yeah.

Dr. ACZEL: And so they start with a very limited goal, which is to rewrite the calculus textbook of Gorsa for the next 25 years. But once their work starts developing, and they meet in resort towns. These French mathematicians love to spice up their life.

FLATOW: Why not? Why not?

Dr. ACZEL: Yes. They go to resorts, beaches, country inns, places like that, skiing areas...

FLATOW: I'm living in the wrong era and the wrong business. Go ahead.

Dr. ACZEL: So they really have a great time. And they realize that their project is growing and growing and becoming more important. And what it really becomes is Elements of Mathematics, as the volumes that they produce together.

And that's named after Euclid's Elements. And instead of for the next 25 years, it's for the next 1,000 years. They're trying to rewrite mathematics for the next 1,000 years.

And what they do is they start with set theory, and that's where the new math comes in. Since they decided to build mathematics up from the foundations, from set theory as a foundation, that gave people the idea that mathematics could be taught and new math could be taught starting the sets operations rather and numbers and equations and things like that.

FLATOW: 1-800-989-8255 if you want to talk math with Amir Aczel, author of The Artist and the Mathematician.

I guess for a joke they sure took it pretty seriously, didn't they?

Dr. ACZEL: Yes. They took their jokes really seriously.

FLATOW: Did they give Bourbaki a fake history, the whole bit? Did they have to defend who he was?

Dr. ACZEL: Yes. They created a baptismal certificate for him...

FLATOW: No kidding.

Dr. ACZEL: ...and a godmother whose name was Evalin de Possel(ph). She was the godmother. And he sprang into life already as an adult because he had a daughter who was getting married. Bourbaki's daughter is Betty, and she has wedding invitations sent, you know, in her name.

FLATOW: Wow. Tell us about how Bourbaki nearly cost Andre Weil his life.

Dr. ACZEL: Andre Weil was so taken with this joke. And there are other jokes that he liked. At one point there was somebody playing - Boulevard Mount Parnassus in Paris is called after Mount Parnassus which is - there was a pile of garbage there at the bottom of the center of Paris.

And at some point there was somebody playing a prank on a passerby. He was standing behind a podium on the little stage there asking for money for the nation of Pauldavia(ph). He's the Prime Minister of Pauldavia whose people are so poor they have no money for pants.

And then he steps away from the podium and you see he's wearing no pants. He's in his underwear. So Weil loved these jokes. And at some point, you know, he also didn't want to serve in the French army. So escaped to Finland and he's caught by the - in November 1939, as the Russians start bombing Helsinki his - the police suspect him. They arrest him and they find a fake identity that he has in the name of Nicolas Bourbaki and wedding invitations for Betty Bourbaki and calling cards in the name of Bourbaki.

So they think obviously he's a spy. In addition, they find letters in Russian inviting him to give talks, mathematical talks in Russia.

FLATOW: No kidding.

Dr. ACZEL: But mathematician, he was no mathematician in (unintelligible). So they're sure he's a spy and they want to execute him. There's no trial here, nothing.

But Weil's life was just like a fable. It's very strange, or a fairly tale. They're going to execute him as a spy. There's absolutely no doubt in anybody's mind that this is a Russian spy.

In addition, he has been on the frontier area with Russia with his wife looking at things and writing things down. That was his papers. He was writing his mathematical papers.

So they go to this official named Nevanlinna, Rolf Nevanlinna. And they say, tomorrow we're going to execute this guy but he says he knows you. Nevanlinna was related to - did some mathematics.

And he says, what's his name? And he says Andre Weil. And he says, yes, I do know him. Do you really have to execute him? Why don't you just deport him to Sweden?

And the head of police says, oh, that's something I hadn't thought of. So they deport him to Sweden instead of executing him. So it almost cost him his life, this idea of the prank of Bourbaki.

FLATOW: Wow. That's very interesting.

1-800-989-8255. You know, I think if you ask anybody what they think of the stereotypical Hollywood mathematician or what happens when a group of mathematicians get together, they think it's going to be a lot of quiet little scribbling on a notepad. But that's not what happened when these guys got together, was it?

Dr. ACZEL: No. They had wild parties everywhere. They had a good time, and in fact Bourbaki championed doing mathematics in nature. So they'd take a blackboard and put it outside in the park somewhere and do mathematics in the open. They really not only redid mathematics starting at the foundations of set theory and really introducing rigor into mathematics.

Proofs had to proofs, not just hand waving, as mathematicians would say. So the mathematical proofs had to be correct and have some generality and abstractions so they are very valuable to mathematics today. So in a sense, what Bourbaki did - and this is being confronted by American and other mathematicians too, not just the French. Because of course the French would tell you Bourbaki was everything.

But other mathematicians say too that in fact Bourbaki did introduce into 20th century mathematics this rigor and abstraction which we have today. So the reason that we have mathematical proofs done correctly and elegantly is due to Bourbaki, or started with the work of Bourbaki.

FLATOW: When did the word get out? When was the cover blown?

Dr. ACZEL: Well, the cover was blown during the war because the group dispersed. And Andre Weil came to this country. He was in New York in 1942, '43, then he was a - I'm sorry - he went to the University of Chicago and he died at Princeton in 1998. And once they spread around the world because of the Second World War, the word, you know, leaked out about Bourbaki.

FLATOW: Are there still any groups that get together now?

Dr. ACZEL: Yes.

FLATOW: Any remnants of them?

Dr. ACZEL: Yes. The Bourbaki group still exists in paper, but former members swear that Bourbaki is dead because they left. They had to leave at 50. Everybody who reaches age 50 is no longer a member of Bourbaki. And they make a strong case that the members of Bourbaki today, none of them are among the top French mathematician, top 40 French mathematicians, and therefore the group no longer exists, they say, and Bourbaki is dead.

I did go to a seminar Bourbaki - a Bourbaki seminar in Paris and these were - in the heydays the rooms were full and there was excitement. A lot of mathematics - important mathematics was done there. When I came to this room in Ecole Normal Superieure where all these pranks took place - actually Institute on Refrancorei(ph), which is nearby.

There's a little room there and probably 15 mathematicians sitting, half of them asleep, somebody writing on the board a theorem, and then they all left the room. So the excitement of the Bourbaki group is no longer there.

FLATOW: Let me ask you this question, and just peripherally, because I watch this program called Numbers. Have you ever seen Numbers on television?

Dr. ACZEL: No.

FLATOW: This is a program that actually uses mathematics to solve crimes. You know, there are always crime-solving programs, but they use numbers and mathematics. And it's kind of interesting because they do generate an excitement of the kind that you are talking about, at least amongst the little group. Because you don't find very much excitement, you know, amongst students these days, you know, to study mathematics or to get them to study mathematics.

Dr. ACZEL: Well, that's the problem with mathematics. What is mathematics? Mathematics is a very - it can be a very abstract structure up in the air here that has no real connection directly with the real world. Although, to argue that is to miss how mathematics developed.

Of course, the ancient Greeks thought of it as a very abstract discipline. They called it geometry, and much of it was geometry. And they worked on theorems. They had no applications in the real world. But of course the calculus was developed by Newton and at the same time by Leibniz in Germany - he was actually living in Paris at that time - as a way of solving problems of the real world. And that's the beauty of the calculus. And there are other areas in mathematics that have very strong applications. So when you find amazing applications, that makes mathematics very exciting.

And in fact Bourbaki did have some connections with the real world, despite the fact that all said their pure mathematicians with no interest in the world around him.

FLATOW: 1-800-989-8255. Let's get a phone call or two. Mike in Kansas City, Kansas. Hi, Mike.

MIKE (Caller): Hi. Hey, I've got and survived college calculus - and actually even use it every now and then - but I don't really remember how I learned the basic math and I don't quite understand what you mean by talking about formulas versus sets. And how do people learn from the beginning?

FLATOW: When you talk about the - when you talked about they brought rigorousness to mathematics what do you mean by that?

MIKE: Right.

FLATOW: Until the late (unintelligible)

Dr. ACZEL: Well, what I mean is there is a substructure behind mathematics that tells you how to do a proof, a mathematical proof. Now let me try to answer the question. When you're talking about calculus, you're talking about a certain function. For example, you're taking the derivative or constructing the integral of that function. For example, x-squared, the integral would be x-cubed over three, the indefinite integral.

So that's what you would be doing in calculus. There are no sets here. We're talking about a function.

MIKE: Right.

Dr. ACZEL: And you're finding the integral or the derivatives, which is the opposite of finding an integral. So when you do that there's rarely any idea of a set behind it. But when you start doing mathematics, you're talking about open sets and closed sets.

FLATOW: Sets means a group of numbers or range.

Dr. ACZEL: Yes. Exactly. Now in the case of calculus, usually it would be talking about an interval of numbers as the sets. It could be an open interval from zero to five, not including zero and five.

MIKE: Right.

Dr. ACZEL: Or it could be the closed interval, which means including the endpoints. So here you have the set as a set of numbers on which you are operating. You are trying to find the integral, the derivative of a certain range of numbers. That range of numbers is the set. So these sets sort of underlie the calculus that's above you.

FLATOW: I understand now my calc 101 class 35 years ago. It was 8 o'clock in the morning at Buffalo. It's not easy to study calculus. That first day when (unintelligible) professor wrote a big thing of sets down - I couldn't understand what it had to do with calculus. Now, it's forty years later I figured - I'm glad you came today.

Dr. ACZEL: Okay.

FLATOW: 1-800-989-8255 is our number. Let's go to the phones. Let's go Joe in Oxford, Ohio. Hi, Joe.

JOE (Caller): Hello. I have a comment about Bourbaki that you might be interested in hearing. I had Zorn for a text as a course in the late '50s. Zorn is the man from him Zorn's...

FLATOW: Zorn's lemma, right.

JOE: Well, anyway, Zorn had us use for a text Bourbaki's set theory.

FLATOW: Wow.

JOE: It was written in French. Zorn was a - is a German and he spoke English - or German with English words in it, and it was hard to understand what was going on. And he was just as abstract as he could possibly be. He would never tell you anything. He would write stuff on the board that didn't make any sense and you had to scramble to try to understand it. It was a beautiful course.

I don't think I ever had a course that I enjoyed any more. Just struggling all the time to try to make some sense of it and in the - as a result you learned a great deal of mathematics. It was wonderful.

FLATOW: Let me just remind everybody that this is TALK OF THE NATION: SCIENCE FRIDAY from NPR News. I'm Ira Flatow talking with Amir Ace, author of the Artist and the Mathematician. Amir, what do you react to...

Dr. ACZEL: I'm not surprised that it was abstract because Zorn's lemma is a very abstract lemma in mathematics...

JOE: Yes.

Dr. ACZEL: ...in the foundations of mathematics.

JOE: Yes. This is a - so difficult course.

FLATOW: Did you have - Joe, did you have to teach yourself this basically, then?

JOE: Well, yeah. A lot of it, yeah. It was very - you just scrambled all the time trying to guess what was going on. But that was - the best part of it was trying to make some sense out of the thing.

FLATOW: Yeah.

JOE: And as a result, you learned a great deal about yourself and about the material.

FLATOW: Thanks for calling. Have a good weekend.

JOE: Thank you.

FLATOW: You said that there were some practical things that came out of this...

Dr. ACZEL: Yes.

FLATOW: ...as Bourbaki. Can you give us - well, I assume...

Dr. ACZEL: Oh, yes. I have a wonderful story that's my favorite story in the whole book.

FLATOW: Well, we've got to hear your favorite story.

Dr. ACZEL: This happened in New York in 1943. Andre Weil came here, by the way, after avoiding being executed. He had to be deported back to France from Sweden and Britain.

FLATOW: Wow.

Dr. ACZEL: And he - they put him - he was supposed to serve in the army, which is why he was in trouble, as an officer but he became a private to avoid greater punishment than that. He sort of escaped to Britain with the rest of the troops, was repatriated to France and ended up in New York, like many Jewish refugees during the war.

So he was here in New York. And Claude Levi-Strauss, the famous anthropologist was here too; also being Jewish, also playing the Nazis who lived here in New York and worked at the school for - the new school.

FLATOW: Right.

Dr. ACZEL: For social research. And he was working on a very interesting problem about Australian aborigines, and I actually went to Australia in part to research that story. The tribes of aborigines - they live under amazing laws that go back perhaps 50,000 years because Australian aborigines supposedly came 50,000 years ago to Australia.

And their societies haven't really changed. They are contiguous, they remain there and they're descendants of descendants and so on. And the rules are very strange. You must marry your father's sister's daughters if you're a guy, if such a person exists. And you are not allowed to marry your mother's brother's daughter. So these are cross cousins, one is taboo and the other one is a must-marry.

And I actually interviewed the woman who - a white woman who lived among these tribes for a while, and she said people who are taboo, you're not even allowed to look at them. And everybody in the tribe knows who is taboo and who is must-marry. So are - and people - other people are sort of neutral.

FLATOW: Right.

Dr. ACZEL: So Claude Levi-Strauss was working on the very beginnings of structural anthropology and trying to solve the mystery of these marriages. Why? Why do you have these rules?

FLATOW: Right.

Dr. ACZEL: And what do they tell you about the society that has these such rules. Is it one society or is it really several groups living together who never intermarry. And he couldn't solve it in any way and he realized that mathematics could give him the answer.

He came to a person named Jacques Hadamard, which is a very famous French mathematician. He was very - he was rather old at the time, and also Jewish, was also escaping Europe for New York. And he came to him, and Hadamard looks at him in a very typically French way. He says mathematics has four operations. He meant addition, subtraction, multiplication, division. of course there's exponentiation, too

FLATOW: I got 45 seconds, Amir...

Dr. ACZEL: Sure. And then he says mathematic - I'm sorry - marriage is not one of these operations. But Weil solved his problem using abstract algebra. He used group theory.

FLATOW: Wow.

Dr. ACZEL: A very abstract area to solve practice problem of Australian aboriginal on marriage laws.

FLATOW: That's a great - and it's in your book.

Dr. ACZEL: Thank you. Yes.

FLATOW: If you want to read the book, I highly recommend it. It's the Artist and the Mathematician: The Story of Nicholas Bourbaki, the Genius Mathematician Who Never Existed. It's been my pleasure to have Amir Aczel back here with us on SCIENCE FRIDAY. Good luck to you.

Dr. ACZEL: Oh, thanks.

FLATOW: Thank you for coming on and being with us today.

Dr. ACZEL: Sure, I do.

FLATOW: Have a great weekend. We'll see you next week. I'm Ira Flatow in New York.

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