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is this all about triangles?

Until now, 21 pilots didn’t give us much to go on in order to continue this series of posts on the Bourbaki/Trench story (links to previous posts below).

They gave us the impression that Bourbaki might still be important in the new era from the I am Clancy video, and the inclusion of the dangerous bend symbol in almost all clips so far released, and more importantly, on the back of the cover of Clancy album (image leaked a week ago)

Interestingly, they do not include the Bourbaki dangerous bend symbol (the ‘Z’ like figure), but rather the variant of the triangular road sign, alerting for a dangerous situation ahead. Keep that thought.

So we have to fall back on previous unsolved hints, and to me, the most important of them is the identity of the bearded man in the photoshopped image on Tyler’s desktop, during the finishing phase of the Trench album.

For months I’ve been looking for his identity. I even had a shortcut for bringing up all Google-image of any person I encountered in my readings of Bourbaki and the Weil-twins.

With the imminent release or leak on the Clancy-album I resurrected my Twitter account (yeah,yeah, unfollow me on Mathstodon) and could get in contact with some longtime Clique-members via DM.

One of them @New_Era_News told me this man was ‘half-faced’ on purpose in Tyler’s image because “the man is in the past (Dema), the present for him (with the Bourbaki Group), and in the future (now with us in our present time) – caught in the doorway portal – between all of the areas this story takes place in.”

For me, this was a true eye-opener. His true identity might be less important than the coded message he is emitting.

So, what are the key characteristics of this man? Well, he’s balding and has a beard, both features not very helpful for our story.

What else, well he seems to be wearing a gilet-suit. So what?

Well, in 1937 Andre Weil married Eveline Gillet (pronounced ‘gilet’) who was a bit shorter than him, consistent with the observation I made in the Bourbakis=Bishops or Banditos-post that bearded man is substantially shorter than Andre, so could hardly be Nico.

Here’s a thought: what if bearded man is a coded image for Eveline Gillet, in order to embed the complicated triangle of relationships between Simone Weil, her brother Andre, and his wife Eveline into this photograph?

Andre’s daughter Sylvie recounts this relationship in her book Chez les Weil quite openly. On page 112 she writes: “Eveline et Simone, les deux femmes d’Andre”.

For me it is hard to shake off this triangle relationship when listening to The craving.

I wonder whether anyone going to the ‘listening parties’ tomorrow will see the lyrics of Clancy any other way.

Previously in this series:

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What about Simone Weil?

While we wait for Clancy to be released/leaked, it is laudable that some people develop their own theory about Dema and its lore.

In recent weeks, a theory that Simone Weil is the key to it all is getting a lot of traction.

Image credit

In two words, this theory is based on the assumption that Vialism=Weilism and on textual similarities between the writings of Simone Weil and the Clancy letters.

The Keons YouTube channel explains all of this in great detail. Here’s their latest summary:

I may explore this further in an upcoming post, but today I want to recall what my own take on what Simone Weil’s role was until now.

I thought that Andre Weil was crucial to the story, and that Simone’s role was to have a boy/girl or brother/sister archetypical situation.

There’s this iconic photograph of them from 1922, taken weeks before Andre entered the ENS:

The same setting, boy on the left, girl to the right was used in the Nico and the niners-video, when they are in Dema

and, when they are a quite a bit older and in Trench, at the end of the Outside-video.

These scenes may support my theory that Dema was the ENS (both Andre and Simone studied there) as is explained in the post Where’s Bourbaki’s Dema?, and when they were both a bit older, and at the Bourbaki meetings in Chancay and Dieulefit, that they were banditos operating in Trench, as explained in the post Bourbaki = Bishops or Banditos.

There are two excellent books to read if you want to know more about the complex relationship between Andre and Simone Weil.

The first one is The Weil Conjectures: On Math and the Pursuit of the Unknown by Karen Olsson.

From it we get the impression that, at times, Simone felt intellectually inferior to Andre, who was three years older. She often asked him to explain what he was working on. Famous is his letter to her written in 1940 when he was jailed. Here’s a nice Quanta-article on it, A Rosetta stone for mathematics. This was also the reason why she wanted to attend some Bourbaki-meetings in order to get a better understanding of what mathematics was all about and how mathematicians think.

She was then critical about mathematics because all that thinking about illusory objects had no immediate relevance for real life. Well Simone, that’s the difference between mathematics and philosophy.

The second one is Chez les Weil, Andre et Simone written by Andre’s eldest daughter Sylvie.

From it we get another impression namely that Andre may have been burdened by the fact that, after Simone’s death, his parents life centered exclusively around the preservation of her legacy, ignorant of the fact that their other child was one of the best mathematicians of his generation.

Poor Andre, even on their family apartment in the Rue Auguste-Comte, which Andre used until late in his life while in Paris, is now this commemorative plaque

Well Andre, that’s the difference between a mathematician and a philosopher.

But, already at the end of the TØP PhotoShop mysteries post I hinted at the possibility that Simone’s role might be more important than I thought until now.

For, if you superimpose the two pictures Tyler used in his photoshop-trick, both appearances of Simone almost perfectly mirror each other.

Let’s dive deep into this new theory.

Previously in this series:


Dema-lore for the rest of us

If you are a longtime follower of this blog, you probably got lost in recent posts about a band, 21 pilots, and the stories waved around their albums Blurryface, Trench, Scaled and Icy, and soon to be released: Clancy.

Suffice it to say that this story, ‘Dema-lore’ as they call it, involves riddles about the Bourbaki group and Andre Weil, and you know I can’t resist any of those. That’s why.

But then, perhaps this story is all too recognisable for far too many early career mathematicians. So, in this post I’ll try to pitch it that way.

To get everyone on the same page at the start of the Clancy-era, the pilots released a summary of Dema-lore. It’s just 4 minutes long so if you’re new to all that ‘Dema-lore’, may I suggest to watch it first?

Twenty years ago Peter Woit and Lee Smolin addressed the problem of Groupthink in Physics, and in particular in String Theory, in their books ‘Not Even Wrong’ and ‘The Trouble with Physics’:

But, the modus operandi of groupthink is not restricted to Physics. I’ve seen it at work in almost every niche-topic in mathematics I encountered.

Right, now keep this one word, groupthink, in mind and watch the clip again. This time, block out the visuals and concentrate on the story.

I am trapped
Stuck in a cycle I have never been able to break
I want to believe this is the last time, but
I don’t know for sure

I’ll start with what I do know
I am a citizen of an old city
Well, they say it’s old, but there’s just no proof
I can feel my friends rolling their eyes
I’ll keep it simple

I am a citizen of Dema
a circular cement city in the lower portion
of an otherwise wild and green continent, Trench
We aren’t allowed to go out there
Most haven’t seen it. But I have

You’re a starting Ph.D. student in mathematics. Your advisor has suggested you’d work in the niche-topic (s)he works in.

(S)he tells you it is a very important branch of mathematics, but you’ve seen enough of mathematics to question this.

You try to broaden your horizon, but feel you’re trapped.

I am an escapee
Getting better at it with every attempt
But they always find me
Well, he does
Or, Blurryface is what he calls himself
He’s the leader of the nine Bishops
who govern the city

Their authority comes from two things
a miraculous power and a hijacked religion
One feeds the other
A cycle
It’s called Vialism
And all you really need to know
is that it teaches that self-destruction
is the only way to paradise
It also conveniently allows you
to become an available vessel for the Bishops to use

And that’s where the miracle comes in
We call it seizing
The rules are
that you can only seize, or control, a dead body
and only for a short while
Also, they, the Bishops
are the only ones who can do it

This niche-topic has its key figures, your advisor may be one of them (which would be nice), but probably (s)he is only aspiring to become one of them, one day.

These people get their power from managing the journals in which you tend to publish, and because they’re asked to write letters of reference, judging your contributions to the topic.

If you try to enter another topic, they’ll get you back, because now you experience that it is much harder to get your results published or get enough support. These key leaders have hijacked part of mathematics, they want a large group around them to self-convince them of their importance.

You are only important to them in as much as you give them plenty of references, and spread their message at conferences.

I am a citizen
I am an escapee
And I am an exception to the rule

Okay this is what happened recently
I tricked Nico into taking me outside the walls
I created a fiery diversion
I escaped
I wandered
grew weak
and was tracked down
But this drag path was different
I saw them. They watched me. The Banditos
Legends, only stories of a group that lived out here

Shortly after being back inside the walls
my new people got me out
They needed me for something
They brought me in, taught me their colours
But the cycle was too strong
I was recaptured. Back inside

Now, let’s assume you’re exceptionally good at maths, not just their subject but more.

They give you the opportunity to speak at a prestigious conference, hoping you’ll talk about their subject, but you decide to give the lecture about your other findings.

People in that other subject notice you but their initial support is too weak, so you’re drawn back to your original niche-subject.

After a while, you get invitations to workshops and conferences of the other subject, but still your regular contributors remain in the niche and it’s too hard for you to make new connections in the other subject, so again you’re recaptured.

I guess word got around, I became known in Dema
The Bishops didn’t like this
but decided to use it to their advantage

They made me entertain the people
Lie to them
They made me perform for them

Okay, but more people in the niche become aware of your growing reputation outside.

The key figures decide to give you a bit more power, you’ll be an editor to one of their journals and you may be one of the organisers of the next annual meet-up of your topic.

They make it perfectly clear to you that all of this will be withdrawn if you’d pull another stunt at some outside conference. When you’re invited, you’ll have to deliver their story.

Then Nico was betrayed
And I escaped
This time
I found myself at a new place, washed up on an island

And there, I was given a gift
thought to be extinct
I now had the same exact miraculous power
they wielded from their towers

Secretly, you’ve been working on the other subject, opening up new approaches to several open problems.

You’re setting up your own new topic in maths, lonely at first, but quite quickly people will join you because they see the potential.

You may not like it, but you’ve become the key figure in your own topic.

I am a citizen
I am an escapee
I am an exception
I am returning to Trench
I am Clancy

I don’t wish this on you, but perhaps this new status gives you a purpose of revenge.

You will overthrow the powers that be in your niche topic, and guide all their followers in a better direction. You’ve become a bishop yourself.

[End of this pitch of Dema-lore]

Clearly, this story is not limited to physics or mathematics. I guess it’s recognisable for more people working in academia, and even to everyone who has to survive in a toxic work-environment.

Now comes a tricky point for clikkies (the pilots’ fanbase): what is 21 pilots’ Dema?

Can it be the clique itself?

They’ve said over and over again that they change the lore according to input from the clique. Maybe this has become too much of a burden? In this new album ‘Clancy’ they may just play the songs they want to play.

The heck whether they fit into the lore, or whether they explain that lore, let alone that dammed Bourbaki/Weil-angle.

Until yesterday I would have been disappointed by this, but then I was completely blown away by the little ballad Tyler performed for the first time at their small event in London.

But hey Tyler, if you ever come across this post, and have a minute: please let me know in the comments who that guy is you photoshopped in the doorway of that picture of the 1938 Bourbaki meeting in Dieulefit? Thanks!

Previously in this series:

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Clancy and Nancago

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.]

Previously in this series:

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Bourbaki and Dema, two remarks

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?

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Exactly 20 years ago I wrote my first blogpost, ‘a blogging 2004’. I wasn’t using WordPress yet (but something called pMachine), and this blog was not called ‘neverendingbooks’, but ‘’ (the URL of the mac still running this blog).

At the time I wanted to find out whether blogging was something for me. “I’m just starting out. Give me a couple of weeks/months to develop my own style and topics and I’ll change the layout accordingly.”

Well, after 20 years I know what I can, and more important, what I cannot do within this framework. Time to move on.

There are other reasons why this might be the right time to pull the plug.

– I’m on retirement since October 1st and soon I’ll have to vacate my office, containing the webserver on which NeB runs.

– My days are filled with more activities now, and I don’t think you want to read here for example about my struggles with chestnut-farming.

– I like to explore other channels to talk about mathematics. This may happen on Mathstodon, MathOverflow or YouTube. Or it might be through teaching or writing a book, perhaps even a children’s book.

NeB will remain reachable until mid 2024. I’ll check out options to preserve its content after that (suggestions are welcome).

I wish you a better 2024.



Grothendieck’s gribouillis (6)

After the death of Grothendieck in November 2014, about 30.000 pages of his writings were found in Lasserre.

Since then I’ve been trying to follow what happened to them:

So, what’s new?

Well, finally we have closure!

Last Friday, Grothendieck’s children donated the 30.000 Laserre pages to the Bibliotheque Nationale de France.

Via Des manuscrits inédits du génie des maths Grothendieck entrent à la BnF (and Google-translate):

“The singularity of these manuscripts is that they “cover many areas at the same time” to form “a whole, a + cathedral work +, with undeniable literary qualities”, analyzes Jocelyn Monchamp, curator in the manuscripts department of the BnF.

More than in “Récoltes et semailles”, very autobiographical, the author is “in a metaphysical retreat”, explains the curator, who has been going through the texts with passion for a month. A long-term task as the writing, in fountain pen, is dense and difficult to decipher. “I got used to it… And the advantage for us was that the author had methodically paginated and dated the texts.” One of the parts, entitled “Structures of the psyche”, a book of enigmatic diagrams translating psychology into algebraic language. In another, “The Problem of Evil”, he unfolds over 15,000 pages metaphysical meditations and thoughts on Satan. We sense a man “caught up by the ghosts of his past”, with an adolescence marked by the Shoah, underlines Johanna Grothendieck whose grandfather, a Russian Jew who fled Germany during the war, died at Auschwitz.

The deciphering work will take a long time to understand everything this genius wanted to say.

On Friday, the collection joined the manuscripts department of the Richelieu site, the historic cradle of the BnF, alongside the writings of Pierre and Marie Curie and Louis Pasteur. It will only be viewable by researchers.“This is a unique testimony in the history of science in the 20th century, of major importance for research,” believes Jocelyn Monchamp.

During the ceremony, one of the volumes was placed in a glass case next to a manuscript by the ancient Greek mathematician Euclid.”

Probably, the recent publication of Récoltes et Semailles clinched the deal.

Also, it is unclear at this moment whether the Istituto Grothendieck, which harbours The centre for Grothendieck studies coordinated by Mateo Carmona (see this post) played a role in the decision making, nor what role the Centre will play in the further studies of Grothendieck’s gribouillis.

For other coverage on this, see Hermit ‘scribblings’ of eccentric French math genius unveiled.


A question of loyalty

On the island of two truths, statements are either false (truth-value $0$), Q-true (value $Q$) or K-true (value $K$).

The King and Queen of the island have an opinion on all statements which may differ from their actual truth-value. We say that the Queen believes a statement $p$ is she assigns value $Q$ to it, and that she knows $p$ is she believes $p$ and the actual truth-value of $p$ is indeed $Q$. Similarly for the King, replacing $Q$’s by $K$’s.

All other inhabitants of the island are loyal to the Queen, or to the King, or to both. This means that they agree with the Queen (or King, or both) on all statements they have an opinion on. Two inhabitants are said to be loyal to each other if they agree on all statements they both have an opinion of.

Last time we saw that Queen and King agree on all statements one of them believes to be false, as well as the negation of such statements. This raised the question:

Are the King and Queen loyal to each other? That is, do Queen and King agree on all statements?

We cannot resolve this issue without the information Oscar was able to extract from Pointex in Karin Cvetko-Vah‘s post Pointex:

“Oscar was determined to get some more information. “Could you at least tell me whether the queen and the king know that they’re loyal to themselves?” he asked.
“Well, of course they know that!” replied Pointex.
“You said that a proposition can be Q-TRUE, K-TRUE or FALSE,” Oscar said.
“Yes, of course. What else!” replied Pointex as he threw the cap high up.
“Well, you also said that each native was loyal either to the queen or to the king. I was just wondering … Assume that A is loyal to the queen. Then what is the truth value of the statement: A is loyal to the queen?”
“Q, of course,” answered Pointex as he threw the cap up again.
“And what if A is not loyal to the queen? What is then the truth value of the statement: A is loyal to the queen?”
He barely finished his question as something fell over his face and covered his eyes. It was the funny cap.
“Thanx,” said Pointex as Oscar handed him the cap. “The value is 0, of course.”
“Can the truth value of the statement: ‘A is loyal to the queen’ be K in any case?”
“Well, what do you think? Of course not! What a ridiculous thing to ask!” replied Pointex.”

Puzzle : Show that Queen and King are not loyal to each other, that is, there are statements on which they do not agree.

Solution : ‘The King is loyal to the Queen’ must have actual truth-value $0$ or $Q$, and the sentence ‘The Queen is loyal to the King’ must have actual truth-value $0$ or $K$. But both these sentences are the same as the sentence ‘The Queen and King are loyal to each other’ and as this sentence can have only one truth-value, it must have value $0$ so the statement is false.

Note that we didn’t produce a specific statement on which the Queen and King disagree. Can you find one?

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the strange island of two truths

Last time we had a brief encounter with the island of two truths, invented by Karin Cvetko-Vah. See her posts:

On this island, false statements have truth-value $0$ (as usual), but non-false statements are not necessarily true, but can be given either truth-value $Q$ (statements which the Queen on the island prefers) or $K$ (preferred by the King).

Think of the island as Trump’s paradise where nobody is ever able to say: “Look, alternative truths are not truths. They’re falsehoods.”

Even the presence of just one ‘alternative truth’ has dramatic consequences on the rationality of your reasoning. If we know the truth-values of specific sentences, we can determine the truth-value of more complex sentences in which we use logical connectives such as $\vee$ (or), $\wedge$ (and), $\neg$ (not), and $\implies$ (then) via these truth tables:

\downarrow~\bf{\wedge}~\rightarrow & 0 & Q & K \\
0 & 0 & 0 & 0 \\
Q & 0 & Q & Q \\
K & 0 & K & K
\end{array} \quad
\downarrow~\vee~\rightarrow & 0 & Q & K \\
0 & 0 & Q & K \\
Q & Q & Q & K \\
K & K & Q & K
\end{array} \]
\downarrow~\implies~\rightarrow & 0 & Q & K \\
0 & Q & Q & K \\
Q & 0 & Q & K \\
K & 0 & Q & K
\end{array} \quad
\downarrow & \neg~\downarrow \\
0 & Q \\
Q & 0 \\
K & 0

Note that the truth-values $Q$ and $K$ are not completely on equal footing as we have to make a choice which one of them will stand for $\neg 0$.

Common tautologies are no longer valid on this island. The best we can have are $Q$-tautologies (giving value $Q$ whatever the values of the components) or $K$-tautologies.

Here’s one $Q$-tautology (check!) : $(\neg p) \vee (\neg \neg p)$. Verify that $p \vee (\neg p)$ is neither a $Q$- nor a $K$-tautology.

Can you find any $K$-tautology at all?

Already this makes it incredibly difficult to adapt Smullyan-like Knights and Knaves puzzles to this skewed island. Last time I gave one easy example.

Puzzle : On an island of two truths all inhabitants are either Knaves (saying only false statements), Q-Knights (saying only $Q$-valued statements) or K-Knights (who only say $K$-valued statements).

The King came across three inhabitants, whom we will call $A$, $B$ and $C$. He asked $A$: “Are you one of my Knights?” $A$ answered, but so indistinctly that the King could not understand what he said.

He then asked $B$: “What did he say?” $B$ replies: “He said that he is a Knave.” At this point, $C$ piped up and said: “That’s not true!”

Was $C$ a Knave, a Q-Knight or a K-Knight?

Solution : Q- and K-Knights can never claim to be a Knave. Neither can Knaves because they can only say false statements. So, no inhabitant on the island can ever claim to be a Knave. So, $B$ lies and is a Knave, so his stament has truth-value $0$. $C$ claims the negation of what $B$ says so the truth-value of his statement is $\neg 0 = Q$. $C$ must be a Q-Knight.

As if this were not difficult enough, Karin likes to complicate things by letting the Queen and King assign their own truth-values to all sentences, which may coincide with their actual truth-value or not.

Clearly, these two truth-assignments follow the logic of the island of two truths for composed sentences, and we impose one additional rule: if the Queen assigns value $0$ to a statement, then so does the King, and vice versa.

I guess she wanted to set the stage for variations to the island of two truths of epistemic modal logical puzzles as in Smullyan’s book Forever Undecided (for a quick summary, have a look at Smullyan’s paper Logicians who reason about themselves).

A possible interpretation of the Queen’s truth-assignment is that she assigns value $Q$ to all statements she believes to be true, value $0$ to all statements she believes to be false, and value $K$ to all statements she has no fixed opinion on (she neither believes them to be true nor false). The King assigns value $K$ to all statements he believes to be true, $0$ to those he believes to be false, and $Q$ to those he has no fixed opinion on.

For example, if the Queen has no fixed opinion on $p$ (so she assigns value $K$ to it), then the King can either believe $p$ (if he also assigns value $K$ to it) or can have no fixed opinion on $p$ (if he assigns value $Q$ to it), but he can never believe $p$ to be false.

Puzzle : We say that Queen and King ‘agree’ on a statement $p$ if they both assign the same value to it. So, they agree on all statements one of them (and hence both) believe to be false. But there’s more:

  • Show that Queen and King agree on the negation of all statements one of them believes to be false.
  • Show that the King never believes the negation of whatever statement.
  • Show that the Queen believes all negations of statements the King believes to be false.

Solution : If one of them believes $p$ to be false (s)he will assign value $0$ to $p$ (and so does the other), but then they both have to assign value $Q$ to $\neg p$, so they agree on this.

The value of $\neg p$ can never be $K$, so the King does not believe $\neg p$.

If the King believes $p$ to be false he assigns value $0$ to it, and so does the Queen, but then the value of $\neg p$ is $Q$ and so the Queen believes $\neg p$.

We see that the Queen and King agree on a lot of statements, they agree on all statements one of them believes to be false, and they agree on the negation of such statements!

Can you find any statement at all on which they do not agree?

Well, that may be a little bit premature. We didn’t say which sentences about the island are allowed, and what the connection (if any) is between the Queen and King’s value-assignments and the actual truth values.

For example, the Queen and King may agree on a classical ($0$ or $1$) truth-assignments to the atomic sentences for the island, and replace all $1$’s with $Q$. This will give a consistent assignment of truth-values, compatible with the island’s strange logic. (We cannot do the same trick replacing $1$’s by $K$ because $\neg 0 = Q$).

Clearly, such a system may have no relation at all with the intended meaning of these sentences on the island (the actual truth-values).

That’s why Karin Cvetko-Vah introduced the notions of ‘loyalty’ and ‘sanity’ for inhabitants of the island. That’s for next time, and perhaps then you’ll be able to answer the question whether Queen and King agree on all statements.

(all images in this post are from Smullyan’s book Alice in Puzzle-Land)

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some skew Smullyan stumpers

Raymond Smullyan‘s logic puzzles are legendary. Among his best known are his Knights (who always tell the truth) and Knaves (who always lie) puzzles. Here’s a classic example.

“On the day of his arrival, the anthropologist Edgar Abercrombie came across three inhabitants, whom we will call $A$, $B$ and $C$. He asked $A$: “Are you a Knight or a Knave?” $A$ answered, but so indistinctly that Abercrombie could not understand what he said.

He then asked $B$: “What did he say?” $B$ replies: “He said that he is a knave.” At this point, $C$ piped up and said: “Don’t believe that; it’s a lie!”

Was $C$ a Knight or a Knave?”

If you are stumped by this, try to figure out what kind of inhabitant can say “I am a Knave”.

Some years ago, my friend and co-author Karin Cvetko-Vah wrote about a much stranger island, the island of two truths.

“The island was ruled by a queen and a king. It is important to stress that the queen was neither inferior nor superior to the king. Rather than as a married couple one should think of the queen and the king as two parallel powers, somewhat like the Queen of the Night and the King Sarastro in Mozart’s famous opera The Magic Flute. The queen and the king had their own castle each, each of them had their own court, their own advisers and servants, and most importantly each of them even had their own truth value.

On the island, a proposition p is either FALSE, Q-TRUE or K-TRUE; in each of the cases we say that p has value 0, Q or K, respectively. The queen finds the truth value Q to be superior, while the king values the most the value K. The queen and the king have their opinions on all issues, while other residents typically have their opinions on some issues but not all.”

The logic of the island of two truths is the easiest example of what Karin and I called a non-commutative frame or skew Heyting algebra (see here), a notion we then used, jointly with Jens Hemelaer, to define the notion of a non-commutative topos.

If you take our general definitions, and take Q as the distinguished top-element, then the truth tables for the island of two truths are these ones (value of first term on the left, that of the second on top):

\wedge & 0 & Q & K \\
0 & 0 & 0 & 0 \\
Q & 0 & Q & Q \\
K & 0 & K & K
\end{array} \quad
\vee & 0 & Q & K \\
0 & 0 & Q & K \\
Q & Q & Q & K \\
K & K & Q & K
\end{array} \quad
\rightarrow & 0 & Q & K \\
0 & Q & Q & K \\
Q & 0 & Q & K \\
K & 0 & Q & K
\end{array} \quad
& \neg \\
0 & Q \\
Q & 0 \\
K & 0

Note that on this island the order of statements is important! That is, the truth value of $p \wedge q$ may differ from that of $q \wedge p$ (and similarly for $\vee$).

Let’s reconsider Smullyan’s puzzle at the beginning of this post, but now on an island of two truths, where every inhabitant is either of Knave, or a Q-Knight (uttering only Q-valued statements), or a K-Knight (saying only K-valued statements).

Again, can you determine what type $C$ is?

Well, if you forget about the distinction between Q- and K-valued sentences, then we’re back to classical logic (or more generally, if you divide out Green’s equivalence relation from any skew Heyting algebra you obtain an ordinary Heyting algebra), and we have seen that then $B$ must be a Knave and $C$ a Knight, so in our new setting we know that $C$ is either a Q-Knight or a K-Knight, but which of the two?

Now, $C$ claims the negation of what $B$ said, so the truth value is $\neg 0 = Q$, and therefore $C$ must be a Q-Knight.

Recall that in Karin Cvetko-Vah‘s island of two truths all sentences have a unique value which can be either $0$ (false) or one of the non-false values Q or K, and the value of combined statements is given by the truth tables above. The Queen and King both have an opinion on all statements, which may or may not coincide with the actual value of that statement. However, if the Queen assigns value $0$ to a statement, then so does the King, and conversely.

Other inhabitants of the island have only their opinion about a subset of all statements (which may be empty). Two inhabitants agree on a statement if they both have an opinion on it and assign the same value to it.

Now, each inhabitant is either loyal to the Queen or to the King (or both), meaning that they agree with the Queen (resp. King) on all statements they have an opinion of. An inhabitant loyal to the Queen is said to believe a sentence when she assigns value $Q$ to it (and symmetric for those loyal to the King), and knows the statement if she believes it and that value coincides with the actual value of that statement.

Further, if A is loyal to the Queen, then the value of the statement ‘A is loyal to the Queen’ is Q, and if A is not loyal to the Queen, then the value of the sentence ‘A is loyal to the Queen’ is $0$ (and similarly for statements about loyalty to the King).

These notions are enough for the first batch of ten puzzles in Karin’s posts

Just one example:

Show that if anybody on the island knows that A is not loyal to the Queen, then everybody that has an opinion about the sentence ‘A is loyal to the Queen’ knows that.

After these two posts, Karin decided that it was more fun to blog about the use of non-commutative frames in data analysis.

But, she once gave me a text containing many more puzzles (as well as all the answers), so perhaps I’ll share these in a follow-up post.

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