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Category: stories

G-spots : Vendargues

Published March 16, 2013 by lieven

In a couple of days, on march 28th, Alexandre Grothendieck will turn 85.

To mark the occasion we’ll run a little series, tracking down places where he used to live, hoping to entice some of these villages in the south of France to update their Wikipedia-page by adding under ‘Personnalités liées à la commune’ the line

– Alexandre Grothendieck (né en 1928), mathématicien français ayant reçu la Médaille Fields.

as did the village of Le Chambon-sur-Lignon, where Grothendieck was kept safe from 1942-1945, separated from his mother who was send to an internment camp (his father was deported by the French authorities in august 1942 and killed by the Nazis in Auschwitz).

After the war, Alexandre was reunited with his mother and, according to Allyn Jackson’s As If Summoned from the Void: The Life of Alexandre Grothendieck, they “went to live in Maisargues, a village in the wine-growing region outside of Montpellier”.

Amir Aczel adds to this in his book The artist and the mathematician, the story of Nicolas Bourbaki: “From 1945 until 1948, mother and son lived in the small hamlet of Mairargues, virtually hidden among the vineyards, a dozen kilometers from Montpellier. They had a marvelous small garden: they never had to work at gardening and yet the earth was so fertile, and the rains so abundant, that the garden produced a plentiful harvest of figs, spinach, and tomatoes. Their garden was at the verge of splendid poppies. Grothendieck remembers his time there with his mother as “la belle vie”.”

But, there is no Maisargues nor Mairargues to be found in France.

There is the village of Caissargues, close to Nimes, about 50 kms from Montpellier, and, there is the village of Meyrargues, close to Pertuis, more than 170 kms from Montpellier.

So, where is the hamlet of “la belle vie”?

Jackson’s and Aczel’s info is based on a footnote in Grothendieck’s Recoltes et semailles (in fact, Aczel’s text is a mere translation of it):

“Entre 1945 et 1948, je vivais avec ma mère dans un petit hameau à une dizaine de kilomètres de Montpellier, Mairargues (par Vendargues), perdu au milieu des vignes. (Mon père avait disparu à Auschwitz, en 1942.) On vivait chichement sur ma maigre bourse d’étudiant. Pour arriver à joindre les deux bouts, je faisais les vendanges chaque année, et après les vendanges, du vin de grapillage, que j’arrivais à écouler tant bien que mal (en contravention, paraît-il, de la législation en vigueur. . . ) De plus il y avait un jardin qui, sans avoir à le travailler jamais, nous fournissait en abondance figues, épinards et même (vers la fin) des tomates, plantées par un voisin complaisant au beau milieu d’une mer de splendides pavots. C’était la belle vie.”

Although Grothendieck misspells Mayrargues, he points to the village of Vendargues which is situated 12 kms east of Montpellier and has a hamlet called Mayrargues (foto above). Via Google Maps you can visit “l’hameau de la belle vie” by yourself (it even has streetview).

If someone at the Mairie de Vendargues comes across this post, please consider adding to your list of famous (former) inhabitants:

– Marcelin Albert (1851-1921), séjourne au mazet de Montmaris, leader de la révolte viticole, est le parrain de Marcellin Guille né en 1907 et oncle d’Archiguille.
– Sabri Allouani (1978-), raseteur (Septuple Vainqueur du Championnat de France de la Course Camarguaise au As 2000-2007)
– Archiguille (Augustin François Guille, peintre contemporain “Transfigurations”) vivant en Suisse.
– Laurent Ballesta (1974-), Biologiste marin, plongeur, photographe, collaborateur de Nicolas Hulot)
– Le général Pierre Berthezène (1775-1847), baron d’Empire, pair de France (1775-1847)
– Jerôme Bonnisel (joueur de football professionnel)
– le baron Pierre Le Roy de Boiseaumarié, (1890-1967), fondateur des appellations d’origine contrôlées, vigneron à Châteauneuf-du-Pape.

this one:

– Alexandre Grothendieck (né en 1928), mathématicien français ayant reçu la Médaille Fields.

Thanks!

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16 ways to capture a lion (in 1938)

Published February 13, 2013 by lieven

A classic among mathematical jokes is the paper in the August/September 1938 issue of the American Mathematical Monthly “A contribution to the mathematical theory of big game hunting” by one Hector Petard of Princeton who would marry, one year later, Nicolas Bourbaki’s daughter Betti.

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There are two main sources of information on the story behind this paper. There are Frank Smithies’ “Reminiscences of Ralph Boas” in the book Lion Hunting & Other Mathematical Pursuits and the transcript of an interview with John Tukey and Albert Tucker at Princeton University on 11 April 1984, part of the oral-history project on the Princeton mathematics community in the 1930s.

Smithies recalls being part of a lively group of people in Princeton during the academic year 1937/38 including Arthur Brown, Ralph Traber, Lyman Spitzer, Hugh Dowker, John Olmsted, Henry Walman, George Barnard, John Tukey, Mort Kanner (a physicist), Dick Jameson (a linguist) and Ralph Boas. Smithies writes:

“At some time that winter we were told about the mathematical methods for lion-hunting that have been devised in Gottingen, and several of us came up with new ones; who invented which method is now lost to memory. Ralph (Boas) and I decided to write up all the methods known to us, with a view to publication, conforming as closely as we could to the usual style of a mathematical paper. We choose H. Petard as a pseudonym (“the engineer, hoist with his own petard”; Hamlet, Act III, Scene IV), and sent the paper to the Americal Mathematical Monthly, over the signature of E. S. Pondiczery.”

Pondiczery was Princeton’s answer to Nicolas Bourbaki, and in the interview John Tukey recalls from (sometimes failing) memory:

“Well, the hope was that at some point Ersatz Stanislaus Pondiczery at the Royal Institute of Poldavia was going to be able to sign something ESP RIP. Then there’s the wedding invitation done by the Bourbakis. It was for the marriage of Betti Bourbaki and Pondiczery. It was a formal wedding invitation with a long Latin sentence, most of which was mathematical jokes, three quarters of which you could probably decipher. Pondiczery even wrote a paper under a pseudonym, namely “The Mathematical Theory of Big Game Hunting” by H. Petard, which appeared in the Monthly. There were also a few other papers by Pondiczery.”

Andrew Tucker then tells the story of the paper’s acceptance:

“Moulton, the editor of the Monthly at that time, wrote to me saying that he had this paper and the envelope was postmarked Princeton and he assumed that it was done by some people in math at Princeton. He said he would very much like to publish the paper, but there was a firm policy against publishing anything anonymous. He asked if I, or somebody else that he knew and could depend on, would tell him that the authorship would be revealed if for any reason it became legally necessary. I did not know precisely who they were, but I knew that John [Tukey] was one of them. He seemed to be in the thick of such things. John agreed that I could accept Moulton’s terms. I sent a letter with this assurance to Moulton and he went ahead and published it.”

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The empty set according to bourbaki

Published February 9, 2013 by lieven

The footnote on page E. II.6 in Bourbaki’s 1970 edition of “Theorie des ensembles” reads




If this is completely obvious to you, stop reading now and start getting a life. For the rest of us, it took me quite some time before i was able to parse this formula, and when i finally did, it only added to my initial confusion.

Though the Bourbakis had a very preliminary version of their set-theory already out in 1939 (Fascicule des Resultats), the version as we know it now was published, chapter-wise, in the fifties: Chapters I and II in 1954, Chapter III in 1956 and finally Chapter IV in 1957.


In the first chapter they develop their version of logic, using ‘assemblages’ (assemblies) which are words of signs and letters, the signs being $\tau, \square, \vee, \neg, =, \in$ and $\supset$.

Of these, we have the familiar signs $\vee$ (or), $\neg$ (not), $=$ (equal to) and $\in$ (element of) and, three more exotic ones: $\tau$ (their symbol for the Hilbert operator $\varepsilon$), $\square$ a sort of wildcard variable bound by an occurrence of $\tau$ (the ‘links’ in the above scan) and $\supset$ for an ordered couple.

The connectives are written in front of the symbols they connect rather than between them, avoiding brackets, so far example $(x \in y) \vee (x=x)$ becomes $\vee \epsilon x y = x x$.

If $R$ is some assembly and $x$ a letter occurring in $R$, then the intende meaning of the *Hilbert-operator* $\tau_x(R)$ is ‘some $x$ for which $R$ is true if such a thing exists’. $\tau_x(R)$ is again an assembly constructed in three steps: (a) form the assembly $\tau R$, (b) link the starting $\tau$ to all occurrences of $x$ in $R$ and (c) replace those occurrences of $x$ by an occurrence of $\square$.

For MathJax reasons we will not try to draw links but rather give a linked $\tau$ and $\square$ the same subscript. So, for example, the claimed assembly for $\emptyset$ above reads

$\tau_y \neg \neg \neg \in \tau_x \neg \neg \in \square_x \square_y \square_y$

If $A$ and $B$ are assemblies and $x$ a letter occurring in $B$ then we denote by $(A | x)B$ the assembly obtained by replacing each occurrence of $x$ in $B$ by the assembly $A$. The upshot of this is that we can now write quantifiers as assemblies:

$(\exists x) R$ is the assembly $(\tau_x(R) | x)R$ and as $(\forall x) R$ is $\neg (\exists x) \neg R$ it becomes $\neg (\tau_x(\neg R) | x) \neg R$

Okay, let’s try to convert Bourbaki’s definition of the emptyset $\emptyset$ as ‘something that contains no element’, or formally $\tau_y((\forall x)(x \notin y))$, into an assembly.

– by definition of $\forall$ it becomes $\tau_y(\neg (\exists x)(\neg (x \notin y)))$
– write $\neg ( x \notin y)$ as the assembly $R= \neg \neg \in x \square_y$
– then by definition of $\exists$ we have to assemble $\tau_y \neg (\tau_x(R) | x) R$
– by construction $\tau_x(R) = \tau_x \neg \neg \in \square_x \square_y$
– using the description of $(A|x)B$ we finally indeed obtain $\tau_y \neg \neg \neg \in \tau_x \neg \neg \in \square_x \square_y \square_y$

But, can someone please explain what’s wrong with $\tau_y \neg \in \tau_x \in \square_x \square_y \square_y$ which is the assembly corresponding to $\tau_y(\neg (\exists x) (x \in y))$ which could equally well have been taken as defining the empty set and has a shorter assembly (length 8 and 3 links, compared to the one given of length 12 with 3 links).

Hair-splitting as this is, it will have dramatic implications when we will try to assemble Bourbaki’s definition of “1” another time.

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