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Ghost metro stations

Published April 1, 2021 by lieven

In the strange logic of subways I’ve used a small part of the Parisian metro-map to illustrate some of the bi-Heyting operations on directed graphs.



Little did I know that this metro-map gives only a partial picture of the underground network. The Parisian metro has several ghost stations, that is, stations that have been closed to the public and are no longer used in commercial service. One of these is the Haxo metro station.



Haxo metro station – Photo Credit

The station is situated on a line which was constructed in the 1920s between Porte des Lilas (line 3bis) and Pré-Saint-Gervais (line 7bis), see light and dark green on the map above . A single track was built linking Place des Fêtes to Porte des Lilas, known as la voie des Fêtes, with one intermediate station, Haxo.



For traffic in the other direction, another track was constructed linking Porte des Lilas to Pré Saint-Gervais, with no intermediate station, called la voie navette. Haxo would have been a single-direction station with only one platform.

But, it was never used, and no access to street level was ever constructed. Occasional special enthusiast trains call at Haxo for photography.



Apart from the Haxo ‘station morte’ (dead station), these maps show another surprise, a ‘quai mort’ (dead platform) known as Porte des Lilas – Cinema. You can hire this platform for a mere 200.000 Euro/per day for film shooting.

For example, Le fabuleux destin d’Amelie Poulin has a scene shot there. In the film the metro station is called ‘Abbesses’ (3.06 into the clip)

There is a project to re-open the ghost station Haxo for public transport. From a mathematical perspective, this may be dangerous.

Remember the subway singularity?

In the famous story A subway named Mobius by A. J. Deutsch, the Boylston shuttle on the Boiston subway went into service on March 3rd, tying together the seven principal lines, on four different levels. A day later, train 86 went missing on the Cambridge-Dorchester line…

The Harvard algebraist R. Tupelo suggested the train might have hit a node, a singularity. By adding the Boylston shuttle, the connectivity of the subway system had become infinite…

Now that we know of the strange logic of subways, an alternative explanation of this accident might be that by adding the Boylston shuttle, the logic of the Boston subway changed dramatically.

This can also happen in Paris.

I know, I’ve linked already to the movie ‘Moebius’ by Gustavo Mosquera, based on Deutsch’s story, set in Buenos Aires.

But, if you have an hour to spend, here it is again.

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Lockdown reading : Bacon

Published March 28, 2021 by lieven

In this series I’ll mention some books I found entertaining, stimulating or comforting during these Corona times. Read them at your own risk.



In an attempt to raise the level of this series, I tried to get through the latest hype in high-brow literature: The Death of Francis Bacon by Max Porter.

It’s an extremely thin book, just 43 pages long, hardly a novella. My Kindle said I should be able to read it in less than an hour.

Boy, did that turn out differently. I’m a week into this book, and still struggling.



Chapter 4(?) :Three Studies for a Self-Portrait, (Francis Bacon, 1979)

A few minutes into the book I realised I didn’t know the first thing about Bacon’s death, and that the book was not going to offer me that setting. Fortunately, there’s always Wikipedia:

While holidaying in Madrid in 1992, Bacon was admitted to the Handmaids of Maria, a private clinic, where he was cared for by Sister Mercedes. His chronic asthma, which had plagued him all his life, had developed into a more severe respiratory condition and he could not talk or breathe very well.

Fine, at least I now knew where “Darling mama, sister oh Dios, Mercedes” (p.7) came from, and why every chapter ended with “Intenta descansar” (try to rest).

While I’m somewhat familiar with Bacon’s paintings, I did know too little about his life to follow the clues sprinkled throughout the book. Fortunately, there’s this excellent documentary about his life: “Francis Bacon: A Brush with Violence” (2017)

Okay, now I could place many of the characters visiting Bacon, either physically sitting on the chair he offers at the start of each chapter (“Take a seat why don’t you”), or merely as memories playing around in his head. It’s a bit unclear to me.

Then, there’s the structure of the book. Each of the seven chapters has as title the dimensions of a painting:

  • One: Oil on canvas, 60 x 46 1/2 in.
  • Two: Oil on canvas, 65 1/2 x 56 in.
  • Three: Oil on canvas, 65 x 56 in.
  • Four: Oil on canvas, 14 x 12 in.
  • Five: Oil on canvas, 78 x 58 in.
  • Six: Oil on canvas, 37 x 29 in.
  • Seven: Oil on canvas, 77 x 52 in.

Being the person I am, I hoped that if I could track down the corresponding Bacon paintings, I might begin to understand the corresponding chapter. Fortunately, Wikipedia provides a List of paintings by Francis Bacon.

Many of Bacon’s paintings are triptychs, and the dimensions refer to those of a single panel. So, even if I found the correct triptych I still had to figure out which of the three panels corresponds to the chapter.

And often, there are several possible candidates. The 14 x 12 in. panel-format Bacon often used for studies for larger works. So, chapter 4 might as well refer to his studies for a self portrait (see above), or to the three studies for a portrait of Henrietta Moraes:



Chapter 4(?) : Three studies for portrait of Henrietta Moraes (1963)

Here are some of my best guesses:



Chapter 3(?): Portrait of Henrietta Moraes (1963)



Chapter 6(?): Three Studies for Figures at the Base of a Crucifixion (1944)



Chapter 5(?): Triptych Inspired by the Oresteia of Aeschylus (1981)

No doubt, I’m just on a wild goose chase here. Probably, Max Porter is merely using existing dimensions of Bacon paintings for blank canvases to smear his words on, as explained in this erudite ArtReview What Does It Mean To Write a Painting?.

Here’s the writer Max Porter himself, explaining his book.

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Boolean and Heyting islands

Published March 26, 2021 by lieven

Raymond Smullyan‘s logic puzzles frequently involve Knights (who always tell the truth) and Knaves (who always lie).

In his book Logical Labyrinths (really a first course in propositional logic) he introduced islands where the lying or truth-telling habits can vary from day to day—that is, an inhabitant might lie on some days and tell the truth on other days, but on any given day, he or she lies the entire day or tells the truth the entire day.

An island is said to be Boolean if is satisfies the following conditions:

  • $\mathbf{N}$ : For any inhabitant $A$ there is an inhabitant who tells the truth on all and only those days on which $A$ lies.
  • $\mathbf{M}$ : For any inhabitants $A$ and $B$ there is an inhabitant $C$ who tells the truth on all and only those days on which $A$ and $B$ both tell the truth.
  • $\mathbf{J}$ : For any inhabitants $A$ and $B$ there is an inhabitant $C$ who tells the truth on all and only those days on which either $A$ tells the truth or $B$ tells the truth (or both). (In other words, $C$ lies on those and only those days on which $A$ and $B$ both lie.)

On any given day there are only Knights and Knaves on the island, but these two populations may vary from one day to the other. The subsets (of all days) for which there is an inhabitant who is a Knight then and a Knave on all other days form a Boolean algebra with operations $\wedge = \cap$ ($\mathbf{M}$eet), $\vee= \cup$ ($\mathbf{J}$oin) and $\neg=$ set-complement ($\mathbf{N}$egation).

Here’s a nice puzzle from Smullyan’s book:

Solomon’s Island also turned out to be quite interesting. When Craig arrived on it, he had the following conversation with the resident sociologist:

Craig : Is this island a Boolean island?
Sociologist : No.
Craig : Can you tell me something about the lying and truth-telling habits of the residents here?
Sociologist : For any inhabitants $A$ and $B$, there is an inhabitant $C$ who tells the truth on all and only those days on which either $A$ lies or $B$ lies (or both).

Show that the sociologist didn’t go native, and that his research is lousy.
(My wording, not Smullyan’s)

Smullyan’s version: This interview puzzled inspector Craig; he felt that something was wrong. After a while he realized for sure that something was wrong, the sociologist was either lying or mistaken!

Extending Smullyan’s idea, we can say that an island is Heyting if, in addition to $\mathbf{M}$ and $\mathbf{J}$ is satisfies the following rules

  • $\mathbf{T}$ : at least one inhabitant tells the truth on all days.
  • $\mathbf{F}$ : at least one inhabitant lies on all days.
  • $\mathbf{I}$ : For any inhabitants $A$ and $B$ there is an inhabitant $C$ sharing Knight-days with $A$ only when $B$ tells the truth, and there are no inhabitants doing this while telling the truth on more days than $C$.

Let’s give an example of an Heyting island which is not Boolean.

On Three-island there are only three kinds of people: Knights, Knaves and Alternates, who can neither lie nor tell the truth two days in a row. All Alternates tell the truth on the same days.

Here’s a riddle:

You meet John, who is a Knight, James, an Alternate, and William, a Knave. You don’t know who is who. You can only ask one question containing at most four words, giving you a Yes or No answer, to just one of the three. The answer must tell you whether that person is James or not.

You may like to watch Smullyan on the Carson show for a hint.

Or, you might just watch it reminiscing long forgotten times, when talkshow-hosts still listened to their guests, and could think for themselves…

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