Tag: bourbaki

The dangerous bend symbol

Published May 31, 2025 by lievenlb

With the upcoming Breach album by 21 pilots I’ve noticed an influx of clikkies’ clicks on this blog looking for info about the Bourbaki group.

Perhaps this is a good opportunity to add some new posts in which I’m looking for potential connections between the ‘Dema/Trench lore’ surrounding 21 pilots’ albums Blurryface, Trench, Scaled and Icy, and Clancy, and historical facts about Nicolas Bourbaki.

Here’s a list of all lore-related Bourbaki posts so far:

The Clancy album was rather disappointing lore-wise, even though it started out promising with a clear reference to Bourbaki in its first song Overcompensate.

I said, I fly by the dangerous bend symbol (wait, what? Wait, what?)
Mm, don’t hesitate to maybe overcompensate
And then by the time I catch in my peripheral (wait, what? Wait, what?)
Mm, don’t hesitate to maybe overcompensate

First, let us clear up the confusion between the dangerous bend symbol, invented and used by the Bourbaki group, and the dangerous bend (road) sign which features on the back of the Clancy-album.

The dangerous bend road sign predates Bourbaki by at least a decade, see the Wikipedia commons on Historic road signs in Germany.

Already in 1907 it’s used on a road sign of the ‘Kaiserlicher Automobil Club’ (on the left) and from 1927 on as we know it today (on the right).

It’s unclear to me why they put the road sign on their album (perhaps for design reasons) rather than the bend symbol (the curly Z inside) which was used by the Bourbakistas (several of them studied in Germany in the 1920s) to indicate difficult paragraphs in their texts.

But even Bourbaki used the bend symbol only in their later works. In the first versions of their ‘Theorie des Ensembles’ they still used the ‘Danger de Mort’ or ‘Skull and Bones’ sign as precursor:

Now, what is the use of the dangerous bend symbol in 21pilots’ lore?

As far as I know, there is no further mention of this symbol but for one interview in which they relate it to the road sign. In the music videos of the Clancy-songs the dangerous bend symbol (not the sign) is clearly visible on the garbage can in front of the shop in the Backslide-video.

Here’s something strange.

If you look at the clip frame-by-frame from 0:18 till 0:21 you’ll see that twice some frames are cut out, first when he walks to his bike and then when he leaves with it. Both times in the vicinity of the trash can, or better, of the dangerous bend symbol.

It is as if the dangerous bend symbol is an indication of a time warp, action speeds up in its vicinity.

Or perhaps, it indicates a region where memory lapses occur?

Anyway, this could be a coincidence and merely an editing quirk.

However, in the remainder of the clip no further such quirks appear, until the very end (from 3:00 till 3:02) when he returns to the shop.

Then again, frames are missing twice. First when he arrives with the bike, and then after he dropped the sack in the garbage can.

To me this looks like something deliberate, and connected to the dangerous bend symbol.

What exactly is anybody’s guess, but no doubt all will become clear when Breach comes out in September.

This gives me just a couple of months to come up with more wild theories…

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Clancy and Nancago

Published May 3, 2024 by lievenlb

Later this month, 21 pilots‘ next album, “Clancy”, will be released, promising to give definite answers to all remaining open questions in Dema-lore.

By then we will have been told why Andre Weil and the Bourbaki group show up in the Trench/Dema tale.

This leaves me a couple of weeks to pursue this series of posts (see links below) in which I try to find the best match possible between the factual history of the Bourbaki group and elements from the Dema-storyline.

Two well-known Bourbaki-photographs seem important to the pilots. The first one is from the september 1938 Dieulefit/Beauvallon Bourbaki congress:

At the time, Bourbaki still had to publish their first text, they were rebelling against the powers that be in French mathematics, and were just kicked out of the Julia seminar.

In clikkies parlance: at that moment the Bourbakistas are Banditos, operating in Trench.

The second photograph, below on the left, is part of a famous picture of Andre Weil, supposedly taken in the summer of 1956.

At that time, Bourbaki was at its peak of influence over French mathematics, suffocating enthusiastic math-students with their dry doctrinal courses, and forcing other math-subjects (group theory, logic, applied math, etc.) to a virtual standstill.

In clique-speech: at that moment the Bourbakistas are Bishops, ruling Dema.

Let me recall the story of one word, associated to the Bourbaki=Bishops era which lasted roughly twenty years, from the early 50ties till Bourbaki’s ‘death’ in 1968 : Nancago.

From the 50ties, Nicolas Bourbaki signed the prefaces of ‘his’ books from the University of Nancago.

Between 1951 and 1975, Weil and Diedonne directed a series of texts, published by Hermann, under the heading “Publications de l’Institut mathematique de l’Universite de Nancago”.

Bourbaki’s death announcement mentioned that he “piously passed away on November 11, 1968 at his home in Nancago”.

Nancago was the name of a villa, owned by Dieudonne, near Nice. Etc. etc.

But then, what is Nancago?

Well, NANCAGO is a tale of two cities: NANcy and ChiCAGO.

The French city of Nancy because from the very first Bourbaki meetings, the secretarial headquarters of Bourbaki, led by Jean Delsarte, was housed in the mathematical Institute in Nancy.

Chicago because that’s where Andre Weil was based after WW2 until 1958 when he moved to Princeton.

Much more on the history of Nancago can be found in the newspaper article by Bourbaki scholar par excellance Liliane Beaulieu: Quand Nancy s’appelait Nancago (When Nancy was called Nancago).

Right, but then, if Nancago is the codeword of the Bourbaki=Bishops era, what would be the corresponding codeword for the Bourbaki=Banditos era?

As mentioned above, from 1935 till 1968 Bourbaki’s headquarters was based in Nancy, so even in 1938 Nancy should be one of the two cities mentioned. But what is the other one?

In 1938, Bourbaki’s founding members were scattered over several places, Jean Delsarte and Jean Dieudonne in Nancy, Szolem Mandelbrojt and Rene de Possel in Clermont-Ferrand, and Andre Weil and Henri Cartan in Strasbourg. Claude Chevalley was on a research stay in Princeton.

Remember the Bourbaki photograph at the Beauvallon meeting above? Well, it was taken in september 1938 when the Munich Agreement was reached.

Why is this relevent? Well, because Strasbourg was too close to the German border, right after the Munich agreement the Strasbourg Institute was ordered to withdraw to the University of Clermont-Ferrand.

Clermont-Ferrand lies a bit south of Vichy and remained in WW2 in the ‘free zone’ of France, whereas Strasbourg was immediately annexed by Germany.

For more on the importance of Clermont-Ferrand for Bourbaki during 1940-1942 see the article by Christophe Eckes and Gatien Ricotier Les congrès de Clermont-Ferrand de 1940, 1941 et 1942.

That is, all Bourbaki members where then either affiliated to Nancy or to Clermont-Ferrand.

A catchy codeword for the Bourbaki=Banditos era, similar to Nancago as the tale of two cities, might then be:

CLermont-Ferrand + nANCY = CLANCY.

[For clikkies: rest assured, I’m well aware of the consensus opinion on the origins of Clancy’s name. But in this series of posts I’m not going for the consensus or even intended meanings, but rather for a joyful interplay between historical facts about the Bourbaki group and elements from Dema-lore.]

In this series:

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Bourbaki and Dema, two remarks

Published February 26, 2024 by lievenlb

While this blog is still online, I might as well correct, and add to, previous posts.

Later this week new Twenty One Pilots material is expected, so this might be a good time to add some remarks to a series of posts I ran last summer, trying to find a connection between Dema-lore and the actual history of the Bourbaki group. Here are links to these posts:

In the post “9 Bourbaki founding members, really?” I questioned Wikipedia’s assertion that there were exactly nine founding members of Nicolas Bourbaki:

I still stand by the arguments given in that post, but my opinion on this is completely irrelevant. What matters is who the Bourbaki-gang themself deemed worthy to attach their names to their first publication ‘Theorie des Ensembles’ (1939).

But wait, wasn’t the whole point of choosing the name Nicolas Bourbaki for their collective that the actual authors of the books should remain anonymous?

Right, but then I found this strange document in the Bourbaki Archives : awms_001, a preliminary version of the first two chapters of ‘Theorie des Ensembles’ written by Andre Weil and annotated by Rene de Possel. Here’s the title page:

Next to N. Bourbaki we see nine capital letters: M.D.D.D.E.C.C.C.W corresponding to nine AW-approved founding members of Bourbaki: Mandelbrojt, Delsarte, De Possel, Dieudonne, Ehresmann, Chevalley, Coulomb, Cartan and Weil!

What may freak out the Clique is the similarity between the diagram to the left of the title, and the canonical depiction of the nine Bishops of Dema (at the center of the map of Dema) or the cover of the Blurryface album:

In the Photoshop mysteries post I explained why Mandelbrojt and Weil might have been drawn in opposition to each other, but I am unaware of a similar conflict between either of the three C’s (Cartan, Coulomb and Chevalley) and the three D’s (Delsarte, De Possel and Dieudonne).

So, I’ll have to leave the identification of the nine Bourbaki founding members with the nine Dema Bishops as a riddle for another post.

The second remark concerns the post Where’s Bourbaki’s Dema?.

In that post I briefly suggested that DEMA might stand for DEutscher MAthematiker (German Mathematicians), and hinted at the group of people around David Hilbert, Emil Artin and Emmy Noether, but discarded this as “one can hardly argue that there was a self-destructive attitude (like Vialism) present among that group, quite the opposite”.

At the time, I didn’t know about Deutsche Mathematik, a mathematics journal founded in 1936 by Ludwig Bieberbach and Theodor Vahlen.

Deutsche Mathematik is also the name of a movement closely associated with the journal whose aim was to promote “German mathematics” and eliminate “Jewish influence” in mathematics. More about Deutsche Mathematik can be found on this page, where these eight mathematicians are mentioned in connection with it:

Perhaps one can add to this list:

Whether DEutsche MAthematik stands for DEMA, and which of these German mathematicians were its nine bishops might be the topic of another post. First I’ll have to read through Sanford Segal’s Mathematicians under the Nazis.

Added February 29th:

The long awaited new song has now surfaced:

I’ve only watched it once, but couldn’t miss the line “I fly by the dangerous bend symbol“.

Didn’t we all fly by them in our first readings of Bourbaki…

(Fortunately the clique already spotted that reference).

No intention to freak out clikkies any further, but in the aforementioned Weil draft of ‘Theorie des Ensembles’ they still used this precursor to the dangerous bend symbol

Skeletons anyone?