Author: lievenlb

NaNoWriMo (1)

Published November 5, 2016 by lievenlb

Some weeks ago I did register to be a participant of NaNoWriMo 2016. It’s a belated new-year’s resolution.

When PS (pseudonymous sister), always eager to fill a 10 second silence at family dinners, asked

(PS) And Lieven, what are your resolutions for 2016?

she didn’t really expect an answer (for decades my generic reply has been: “I’m not into that kinda nonsense”)

(Me) I want to write a bit …

stunned silence

(PS) … Oh … good … you mean for work, more papers perhaps?

(Me) Not really, I hope to write a book for a larger audience.

(PS) Really? … Ok … fine … (appropriate silence) … Now, POB (pseudonymous other brother), what are your plans for 2016?

nanowrimo_2016_webbanner_participant

If you don’t know what NaNoWriMo is all about: the idea is to write a “book” (more like a ‘shitty first draft’ of half a book) consisting of 50.000 words in November.

We’re 5 days into the challenge, and I haven’t written a single word…

Part of the problem is that I’m in the French mountains, and believe me, there always more urgent or fun things to do here than to find a place of my own and start writing.

A more fundamental problem is that I cannot choose between possible book-projects.

Here’s one I will definitely not pursue:

“The Grothendieck heist”

“A group of hackers uses a weapon of Math destruction to convince Parisian police that a terrorist attack is imminent in the 6th arrondissement. By a cunning strategy they are then able to enter the police station and get to the white building behind it to obtain some of Grothendieck’s writings.

A few weeks later three lengthy articles hit the arXiv, claiming to contain a proof of the Riemann hypothesis, by partially dismantling topos theory.

Bi-annual conferences are organised around the globe aimed at understanding this weird new theory, etc. etc. (you get the general idea).

The papers are believed to have resulted from the Grothendieck heist. But then, similar raids are carried out in Princeton and in Cambridge UK and a sinister plan emerges… “

Funny as it may be to (ab)use a story to comment on the current state of affairs in mathematics, I’m not known to be the world’s most entertaining story teller, so I’d better leave the subject of math-thrillers to others.

Here’s another book-idea:

“The Bourbaki travel guide”

The idea is to hunt down places in Paris and in the rest of France which were important to Bourbaki, from his birth in 1934 until his death in 1968.

This includes institutions (IHP, ENS, …), residences, cafes they frequented, venues of Bourbaki meetings, references in La Tribu notices, etc.

This should lead to some nice Parisian walks (in and around the fifth arrondissement) and a couple of car-journeys through la France profonde.

Of course, also some of the posts I wrote on possible solutions of the riddles contained in Bourbaki’s wedding announcement and the avis de deces will be included.

Here the advantage is that I have already a good part of the raw material. Of course it still has to be followed up by in-situ research, unless I want to turn it into a ‘virtual math traveler ’s guide’ so that anyone can check out the places on G-maps rather than having to go to France.

I’m still undecided about this project. Is there a potential readership for this? Is it worth the effort? Can’t it wait until I retire and will spend even more time in France?

Here’s yet another idea:

“Mr. Yoneda takes the Tokyo-subway”

This is just a working title, others are “the shape of prime numbers”, or “schemes for hipsters”, or “toposes for fashionistas”, or …

This should be a work-out of the sga4hipsters meme. Is it really possible to explain schemes, stacks, Grothendieck topologies and toposes to a ‘general’ public?

At the moment I’m re-reading Eugenia Cheng’s “Cakes, custard and category theory”. As much as I admire her fluent writing style it is difficult for me to believe that someone who didn’t knew the basics of it before would get an adequate understanding of category theory after reading it.

It is often frustrating how few of mathematics there is in most popular maths books. Can’t one do better? Or is it just inherent in the format? Can one write a Cheng-style book replacing the recipes by more mathematics?

The main problem here is to find good ‘real-life’ analogies for standard mathematical concepts such as topological spaces, categories, sheaves etc.

The tentative working title is based on a trial text I wrote trying to explain Yoneda’s lemma by taking a subway-network as an example of a category. I’m thinking along similar lines to explain topological spaces via urban-wide wifi-networks, and so on.

But al this is just the beginning. I’ll consider this a success only if I can get as far as explaining the analogy between prime numbers and knots via etale fundamental groups…

If doable, I have no doubt it will be time well invested. My main problem here is finding an appropriate ‘voice’.

At first I wanted to go along with the hipster-gimmick and even learned some the appropriate lingo (you know, deck, fin, liquid etc.) but I don’t think it will work for me, and besides it would restrict the potential readers.

Then, I thought of writing it as a series of children’s stories. It might be fun to try to explain SGA4 to a (as yet virtual) grandchild. A bit like David Spivak’s short but funny text “Presheaf the cobbler”.

Once again, all suggestions or advice are welcome, either as a comment here or privately over email.

Perhaps, I’ll keep you informed while stumbling along NaNoWriMo.

At least I wrote 1000 words today…

Where’s Bourbaki’s tomb?

Published October 23, 2016 by lievenlb

In according to Groth IV.22 we tried to solve one of the riddles contained in Roubaud’s announcement of Bourbaki’s death.

Today, we’ll try our hands on the next one: where was Bourbaki buried?

The death announcement gives this fairly opaque clue:

“The burial will take place in the cemetery for Random Functions (metro stations Markov and Gödel) on Saturday, November 23, 1968 at 3 o’clock in the afternoon.”

What happened on November 23rd 1968?

Bourbaki died on November 11th, 1968 (exactly 50 years after the end of WW1). Perhaps an allusion to the mandatory retirement age for members of Bourbaki, as suggested by the Canulars Bourbaki.

Be that as it may, I believe this date was chosen because it is conveniently close to the intended time of the burial.

But then, what’s so special about November 23rd, 1968?

Well, is there a more suitable moment to burry Bourbaki than during a Seminaire Bourbaki? And, yes, in the fall of 1968 the seminar was organised from saturday 23rd till monday 25th of november:

So, where would all of Bourbaki’s close family be at 3 o’clock on that particular saturday? Right, at l’Institut Henri Poincare.

But, it’s hard to view the IHP as a cemetery. Besides, it’s nowhere close to two metro stations as a quick look on the map shows. The closest one is the RER-station at the Luxembourg gardens, but the RER-line didn’t exist in 1968.

(True Parisians may object that the Gare du Luxembourg was at the time the terminus of the Ligne de Sceaux which has a fascinating history, but let’s try to remain on track…)

If the first clue is the Institut Henri Poincare, then if we are looking for a cemetery, we might ask:

Where’s Poincare’s tomb?

Jules Henri Poincare is burried in the family tomb at the Montparnasse cemetery

He’s not the only mathematician buried there. Évariste Galois, Jean Victor Poncelet, Joseph Liouville, Charles Hermite, and Gaston Darboux also found their last resting place in Montparnasse.

In fact, there are at least 104 mathematicians buried at Montparnasse.

This is hardly surprising as the Montparnasse cemetery is close to the IHP, the Collège de France, the Sorbonne, the “rue d’Ulm” aka the ENS, l’Observatoire and until 1976 l’École polytechnique.

Here’s a map with pointers to some of these tombs:

So, the Montparnasse cemetery appears to be a plausible place to host Bourbaki’s tomb.

But, what about the other “clues”?

“Cemetery of random functions (metro stations Markov and Gödel)”

There are several references lo logic, set theory and applied mathematics in Bourbaki’s death announcement. Why?

Roubaud (and many with him) feel that the Bourbaki enterprise failed miserably in these areas.

He writes on page 49 of his book Mathematics, a novel:

“But Bourbaki, that ‘collective mathematician”, as Raymond Queneau put it, also had a good knowledge of the current state of mathematics at the time when his Treatise was being composed; with, of course, a few “gaps”:

for example, probability, which was considered to be just an “applied” brand of measure theory”; and logic, especially logic, which was made almost a pariah because of (so it was rumored) the premature death of Herbrand, who, in the generation of founders, Normaliens to a man, had studied under Hilbert, and thus had been associated with his meteoric rise; in sum, logic had died in a climbing accident along with Herbrand.”

This might explain the cemetery of “random functions” and the metro stations named after the logicians and set theorists Kurt Gödel and A.A. Markov or the father of stochastic processes Andrey Markov.

Is there more into these references?

Probably not, but just to continue with our silly game, the two metro stations closest to the Montparnasse cemetery are Raspail and Edgar Quinet.

Now, François-Vincent Raspail was a French chemist, naturalist, physician, physiologist, attorney, and socialist politician.

More relevant to our quest is that the Centre d’analyse et de mathématique sociales (CAMS) was based at 54, boulevard Raspail. The mission statement on their website tells that this institute is clearly devoted to all applications of mathematics. That is, “Raspail” may be another pointer to applied mathematics and random functions.

As for the other metro station, Edgar Quinet was a French historian and intellectual. Is there a connection to logic or set theory? Well, sort of. The Encyclopedia Britannica has this to say about Edgar Quinet:

“His rhetorical power was altogether superior to his logical power, and the natural consequence is that his work is full of contradictions.”

I rest my case.

from chocolate bars to constructivism

Published October 19, 2016 by lievenlb

A fun way to teach first year students the different methods of proof is to play a game with chocolate bars, Chomp.

The players take turns to choose one chocolate block and “eat it”, together with all other blocks that are below it and to its right. There is a catch: the top left block contains poison, so the first player forced to eat it dies, that is, looses the game.

Let’s prove some results about Chomp illustrating the strengths and weaknesses of different methods of proof.

[section_title text=”Direct proof”]

By far the most satisfying method is the ‘direct proof’. Once you understand it, the truth of the statement is unescapable.

Theorem 1: The first player to move has a winning strategy for a $2 \times n$ chocolate bar.

Here’s the winning move: you take the lower right block!

What can the other player do? She can bring the bar again in rectangular shape of strictly smaller size (in which case you eat again the lower right block), or she can eat more blocks from the lower row (but then, after her move, you eat as many blocks from the top row bringing the bar again in a shape in which the top row has exactly one more block than the lower one. You’re bound to win!

Sadly, it is not always possible to prove what you want in such a direct way. That’s why we invented another tactic:

[section_title text=”Proof by induction”]

A proof by induction is less satisfactory because it involves a lot more work, but still everyone considers it as a ‘fair’ method: it shows how to arrive at the solution, if only you had enough time and energy…

Theorem 2: Given any Chomp position, either the first player to move has a winning strategy or the second player has one.

A proof by induction relies on simplifying the situation at hand. Here, each move reduces the number of blocks in the chocolate bar, so we can apply induction on the number of blocks.

If there is just one block, it must contain poison and so the first player has to eat it and looses the game. That is, here the second player has a winning strategy and we indicate this by labelling the position with $0$. This one block position is the ‘basis’ for our induction.

Now, take a position $P$ for which you want to prove the claim. Look at all possible positions $X$ you get after one move. All these positions have strictly less blocks, so ‘by induction’ we may assume that the result is true for each one of them. So we can label each of these positions with $0$ if it is a second player win, or with $\ast$ if it is a first player win.

How to label $P$? Well, if all positions in $X$ are labeled $\ast$ we label $P$ with $0$ because it is a second player win. The first player has to move to a starred position (which is a first player win) and so the second player has a winning strategy from it (by moving to a $0$-position). If there is at least one position in $X$ labeled $0$ then we label $P$ with $\ast$. Indeed, the first player can move to the $0$-position and then has a winning strategy, playing second.

Here’s the chart of the first batch of positions:

But then, it may take your lifetime to work through the strategy for a complicated starting position…

What to do if neither a direct proof is found, nor a reduction argument allowing a proof by induction? Virtually all professional mathematicians have no objection whatsoever to resort to a very drastic tactic: proof by contradiction.

[section_title text=”Proof by contradiction”]

The law of the excluded third (or ‘tertium non datur’) says that for any proposition $P$ either $P$ or not $P$ is true. In 1927 David Hilbert stated that not much of mathematics would remain without the use of this rule:

[quote name=”David Hilbert”]To prohibit existence statements and the principle of excluded middle is tantamount to relinquishing the science of mathematics altogether.[/quote]

Theorem 3: The first player has a winning strategy for a rectangular starting position.

We will prove this by contradiction. That is, we will assume that it is false, that is, the first player has not a winning strategy, and derive a contradiction from it. Therefore $\neg \neg P$ is true which is equivalent to $P$ by the law of the excluded
middle.

If the first player does not have a winning strategy for that rectangular chocolate bar, the second player must have one by Theorem2. That is, for every possible first move the second player has a winning response.

Assume the first player eats the lower right block.

The second player must have some winning response to this. Whatever her move, the resulting position could have been obtained by the first player already after the first move.

So, the second player cannot have a winning strategy.

This strategy stealing argument is a cheap cheat. We have not the slightest idea of what the winning strategy for the first player is or how to find it.

Still, one shouldn’t stress this fact too much to first year students. They’ll have to work through plenty of proofs by contradiction in the years ahead…

[section_title text=”Accepting constructive mathematics?”]

Sooner or later in their career they will hear arguments in favour of ‘constructive mathematics’, which does not accept the law of the excluded third.

Andrej Bauer has described was happens next. They will go through the five stages people need to come to terms with life’s traumatising events: denial, anger, bargaining, depression, and finally acceptance.

His paper is just out in the Bulletin of the AMS Five stages of accepting constructive mathematics.

This paper is an extended version of a talk he gave at the IAS.

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