Paraprosdokian

"I want to die like my father, quietly, in his sleep—not screaming and terrified like his passengers." Bob Monkhouse

Dase's algorithm

I just saw a TED talk about Mathemagics by Arthur Benjamin, and googled a few related things. With the notable exception of Alexander Aitken, it seems that very few calculator prodigies are able to explain how they actually calculate so quickly. The fuzziness of the explanations suggests some less-rational procedure that seems closer to intuition than to any algorithm.

One such calculator prodigy was Zacharias Dase (1824-2861). He is shortly mentioned in Hofstadter’s book, Gödel-Escher-Bach. His calculation skills were such that Gauss himself was impressed and he had him hired by the Hamburg Academy of Sciences to compile some mathematical tables. Dase had other amazing abilities, like counting sheep in a flock in a single glance.

In the MacTutor History of Mathematics archive, you can read the following sentence about Dase: "He multiplied two 20 digit numbers in 6 minutes; two 40 digit numbers in 40 minutes; two 100 digit numbers in 8 hours 45 minutes."

When you put these figures on a logarithmic plot it seems that Dase’s unknown algorithm was slightly slower than quadratic (solid line). It is likely that he was using the same algorithm as anyone of us: multiplying each digit of the first number by the second number, and taking the sum. He was very good at it, and he obviously had some tricks, but that was probably all of it.

Had Zacharias Dase lived lived today, he might have tried to use the FFT-like algorithm by Schönhage and Strassen (dotted line), which is faster for multiplying large numbers.

http://www.ted.com/talks/lang/eng/arthur_benjamin_does_mathemagic.html.

What is the difference between an ape and the number pi?

My message in French of December 19th had to do with tomorrow's newspaper being hidden somewhere in the number pi. Express pi in base 26 and write it down with our alphabet. You obtain an infinite series of random letters, in which any given finite sequence of letters (including e.g. Hamlet, on any text yet to be written) appears with a probability equal to 1.

A related paradox is that of an ape rewriting Hamlet by just typing randomly on a keyboard. The difference though is that it will take the ape a time longer than the age of the universe before scene 1 is finished. In pi, it seems to be there from the beginning. Just say "the ratio of a circle's perimeter to its diameter" and here is Hamlet!

The ape's paradox was invented by Borel, who solved it in a quick-and-dirty way by arguing that small probabilities ought to be considered as impossibilities, and large probabilities as certainties [1]. Such a pragmatic approach is so surprising under the pen of a mathematician, that I cannot but understand it as a real upset from Borel's part. Apparently, he stopped working on infinity in the twenties [2].

The difference between the ape and the number pi has to do with the difference between actual and potential infinity. The ape may potentially write Hamlet if he keeps typing for some time, while Hamlet is actually in pi. The possibility of actual infinities drove mathematicians nuts, at the end of the nineteenth and early twentieth century. Quite literally for a few of them [2] who - unlike Borel - kept working on these problems.

Another reading of mine sheds other light on that issue [3]: actuality/potentiality is a dichotomy, just like empiricism/speculation, creation/evolution, etc. We love opposing things because it helps us understand; a dichotomy is a simplified model that puts order in our mind, not in our world. We are constantly shrinking the complexity of the world to make it fit in the much smaller space of what we can understand. We most often shrink a hyper-dimensional reality to a 1-dimensional segment, and eventually we ignore the segment and consider only its two ends. Potentiality and actuality are two such ends, just as finite and infinite as a matter of fact.

Choosing one of the two options in a dichotomy cannot be right or wrong. Dichotomies are fruitful because they pave the way towards a better understanding of our fundamentally non-dichotomous world. When dichotomies are no longer taken too seriously, this generally means that the phenomenon they describe is becoming better understood: solid/liquid, rational/intuitive, good/bad, wave/particle, etc. Other dichotomies are quite lively but they will no doubt end up in the same way. I would love to be there when this happens for deterministic/random, past/future, matter/spirit and dead/alive.

[1] E. Borel, Probabilité et Certitude, Presses Universitaires de France (1950);
[2] L. Graham & J.-M. Kantor, Naming Infinity, A True Story of Religious Mysticism and Mathematical Creativity, Harvard University Press (2009);

[3] S.J. Gould, Time's Arrow, Time's Cycle: Myth and Metaphor in the Discovery of Geological Time, Harvard University Press (1987);

Why does the other lane seem to be more rapid?

The answer is: “because the other lane is actually more rapid”. And you can jump from one lane to the other without changing anything to it. Whatever you do, you will still be iost of the time in the slowest lane.

It is the high density of a lane that makes it slow, because anybody adjusts the speed of his car according to the distance to the front car. For the same reason, the traffic is the most rapid where it is least dense, because there the distance to the front car is the largest. If you take a snapshop of a traffic jam from a helicopter and count the cars, you will necessarily find more slow cars than rapid cars.

Suppose that the slow lanes contain 2/3 of the cars and the quick lane 1/3. Every driver is alternatively in a quick and in a slow lane, particularly if he keeps jumping from one to another. As the fraction of slow and rapid cars is 2/3 and 1/3, a driver spends 2/3 of the time in a slow lane and only 1/3 in a rapid lane.

When you look at the other lane, 2 times out of 3 it will be more rapid than yours.

Pi = C,CVEZCVBMLYZGEC...

Il y a une propriété du nombre pi qui me fait douter de son existence, ou plus modestement, qui me fait réaliser que je n’avais pas vraiment compris ce qu’est un nombre. Et cette propriété est : « pi est un nombre normal. » Cela veut dire que la suite des chiffres qui composent pi, et qui commence par 314159 à toutes les propriétés d’une séquence aléatoire infinie, où chaque chiffre apparaît avec la même fréquence 1/10.

Une conséquence de cette propriété est que toute suite de chiffres de longueur finie, comme 0123456789, est présente quelque part dans la suite des chiffres qui composent pi. Plus la séquence est longue plus sa fréquence est faible, mais comme pi est une suite infinie, on a la certitude que n’importe quelle suite finie y est présente une infinité de fois. Par exemple, la suite la 10 chiffres 0123456789 est présente en moyenne une fois tous les 10000000000 chiffres, tout comme 0000000000, ou 3141592653.

La même observation est plus choquante quand on l’exprime différemment. Quand on écrit pi = 3.1415, ce n’est qu’une manière concise d’écrire que pi est le nombre qu’on obtient en faisant le calcul 3*1+1*1/10+4*1/(10*10)+1*1/(10*10*10)+5*1/(10*10*10*10). C‘est ce qu’on appelle la base 10. Avec la numération en base 2 des ordinateurs on écrirait pi = 11,0010010… ce qui veut dire pi = 1*2+1+0*1/2+0*1/(2*2)+1*1/(2*2*2)+…Si on voulait utiliser une base plus grande que 10, il nous faut des symboles pour écrire tous les chiffres jusqu’à la valeur base. Pour écrire les nombres en base 27, par exemple, on peut convenir d’utiliser les lettres de notre alphabet et l’espace. On conviendrait 0 = «_», 1 = « A », 2 = « B», etc. jusqu’à 26 = « Z ». En base 27, et avec cette convention, les premières décimales de pi sont données dans le titre de ce post.

La normalité d’un nombre est indépendante de la base dans laquelle il est écrit, ce qui signifie que n’importe quelle suite finie de lettres se trouve quelque part dans pi. Par exemple, on s’attend à trouver le mot « PAPA » une fois toutes les 500000 décimales. Le texte du journal de demain figure aussi quelque part dans pi, de même que toute l’oeuvre de Voltaire. Tout cela y figure même plusieurs fois, une infinité de fois ! Pire, ce n’est pas propre à pi : la plupart des nombres sont normaux. Cela veut dire que la plupart des fois que vous faites un calcul, l’œuvre de Voltaire est cachée dans la réponse.

La situation est analogue à celle du singe imaginé par Emile Borel, qui reproduirait l’œuvre de Molière en tapant au hasard à la machine à écrire. Dans ce cas là, on peut se consoler de ce que cette situation n’est pas vraiment réelle, puisqu’il faudrait au singe un temps plus long que l’age de l’univers pour ne taper qu’un début de tirade. Ce que je trouve très choquant avec la normalité des nombres, c’est que tout y serait dès le début. De manière statique, et depuis toujours. Faut-il en conclure que la plupart des nombres ne sont pas vraiment réels ?

Triletter DéS

What recursions may come

Hofstadter's Law: It always takes longer than you expect, even when you take Hofstadter's Law into account.

(Douglas Hoftstadter: Gödel, Escher, Bach: an eternal golden braid)

Use, misuse, and abuse of science

Cold Spring Harbor Laboratory (of which James DNA Watson was president) hosted the Eugenics Record Office during the first half of the twentieth century. A very interesting website about eugenics can be found at

http://www.eugenicsarchive.org/eugenics/.

Anything than can go wrong with science is found in the history of eugenics.

Devinette

What is this?

Try to guess.

Marcello Truzzi

"In science, the burden of proof falls upon the claimant; and the more extraordinary a claim, the heavier is the burden of proof demanded. The true skeptic takes an agnostic position, one that says the claim is not proved rather than disproved. He asserts that the claimant has not borne the burden of proof and that science must continue to build its cognitive map of reality without incorporating the extraordinary claim as a new "fact." Since the true skeptic does not assert a claim, he has no burden to prove anything. He just goes on using the established theories of "conventional science" as usual. But if a critic asserts that there is evidence for disproof, that he has a negative hypothesis—saying, for instance, that a seeming psi result was actually due to an artifact—he is making a claim and therefore also has to bear a burden of proof. "

(Marcello Truzzi, Zetetic Scholar, 12-13, 1987. )http://www.anomalist.com/commentaries/pseudo.html

Marcello Truzzi was an interesting character. He was born in 1935 in Copenhagen, in a Russian family of circus performers. The family moved to the US in the 1940, and Marcello became professor of sociology at Eastern Michigan University. He is one of the founders of the Society for Scientific Exploration (http://www.scientificexploration.org/), which aims at "the rigorous study of unusual phenomena that may be ignored or inadequately studied within mainstream science". He died in 2003.

Combien dure une partie de bataille?

Ce post est un exemple de mathématiques appliquées au bonheur familial. Ma fille Perrine a découvert l'interminable (?) jeu de bataille, et elle veut que je joue avec elle. Je me suis amusé à quelques simulations, histoire de savoir à quoi je m'engage exactement en lui disant oui.

Ci-dessus la distribution de la longueur des parties, pour 2000 donnes. Il y a deux versions qui diffèrent par la manière dont l'ex-aequo (la "bataille") est traité. En bleu: on rejoue et celui qui gagne empoche les 4 cartes; en rouge: on ajoute d'abord chacun une carte retournée avant de rejouer, et celui qui gagne empoche les 6 cartes. C'est assez surprenant de voir comme le fait d'ajouter des cartes retournées raccourcit sensiblement les parties (bien que je compte la mise des cartes retournées comme un coup). J'avais a priori imaginé le contraire, parce que les cartes retournées auraient pu etre une manière de filer des bonnes cartes à un adversaire qui aurait eu un mauvais jeu et n'auraitpas eu la possibilité de les gagner à la régulière. Ca aurait pu équilibrer les forces, mais apparemment c'est autre chose.

Quoi qu'il en soit, c'est quand meme tres long. Une partie sur deux dure plus de 270 coups dans le premier cas, et 200 coups dans le second. On peut raccourcir le jeu en n'utilisant pas tout le jeu. Encore une fois, la "bataille" a un drole d'effet. Ci-dessous les distributions de longueur des parties pour un jeu de quatre couleurs de 6 cartes (bleu) ou de deux couleurs de 12 cartes (rouge).

Ces deux jeux different par la fréquence d'occurrence des ex-aequo (qui est plus grande pour le bleu). Les parties les plus courtes sont obtenues avec le jeu qui rend les "batailles" les plus probables. Pourquoi?

Dialogue d'un chien avec son maître sur la nécessité de mordre ses amis

J'adore ce titre!

http://www.jeanmariepiemme.be/dialogue-chien.htm

And another two

Some other

Why would it be natural to oppose science and religion?

At first sight, religion and science have little in common, of course. One is about faith and the other is about doubts. In their nature, however, they are similar. Both are a quest of truth.

A more realistic statement would be that the purpose of science is to produce useful models of reality [1]. One can of course argue on the exact meaning of words, but this definition is apt to create a consensus. As for religion, being religious can be considered equivalent to believing in the existence of supernatural entities (referred to hereafter as gods) that rule or influence the natural world.

One can only speculate on why people have first come to believe in the existence of gods [2, 3]. It seems to me that gods contribute to make the world more understandable, if not predictable. A catastrophe occurs, say, heavy rains followed by a flood. Attributing this to the anger of some god, rather than to mere chance, is a way to preserve the comfortable idea that everything happens for a reason. Moreover, this explanation is also enticing because it provides people with a means of controlling Nature (do not irritate the gods). Therefore, humans might well have created gods because their world ought to be understandable and controllable; in other words, the inventors of gods believed in causality and in determinism.

From that perspective, gods are simply a mental representation of the principles that control Nature. So to speak, gods are a model of reality. And that early model was definitely useful in helping people accept their faith and keep living, which is sometimes described as an evolutionary asset [2]. According to the definition given above, the creation of gods can be though of as a scientific process. Another obvious usefulness of religions is that they helped people believe in causality and determinism, which beliefs eventually happened to be so fruitful in the development of our technological world during the last three centuries or so.

However complex they may seem, nowadays models - like general relativity, or DNA - are far simpler than the psychology of an irascible god. Simple models are useful only because their domains of applicability are very limited. The laws of physics explain only a very small fraction of the physical world because they are all conditional [4]. No prediction can be made unless the initial state of a system is known, and most human questionings about Nature concern the initial conditions that physics says nothing about. Even more fundamentally, the way physics tries to grasp the notion of time is via evolution laws, which are somehow opposed to the truly creative evolution that is central to anybody’s life [5].

Science accumulated spectacular achievements by restricting its models to more and more limited domains of applicability. The evolution of religions is quite opposite: they evolved towards more and more general principles, such as Good and Evil, which generalization culminates in monotheism. The achievements of religions, and notably of monotheistic religions, are unquestionable given the civilizations they contributed to develop and the way they still shape the human mentalities all over the world. The appeal of their generalizing and unifying ideals to the human nature makes no doubt.

Not so surprisingly is now science moving towards more general models and paradigms than in the past. The hypothetical grand unification theory between nuclear interactions and electromagnetism is just an example. The appeal of multi- or trans-disciplinary researches is also very similar. Trans-disciplinary researches, by attempting to enlarge the initial domains of applicability of models, have already resulted in shaking some of the founding ideas of modern science. Reductionism – which is a basis of the experimental method – is now known to be of limited applicability as some systems are subject to emergence [6].

In summary, the self-confidence of the positivists of the 19th century is not defensible at the beginning of the 21st century. Moderate religion and lucid science are just two different aspects of the same questioning of mankind, torn between the need of simple models for making accurate predictions and the appeal of general principles that would give life a meaning. The endless lines about creationism –just to mention one- that are superficially presented as an opposition between religion and science are merely a clash between fanaticism and intellectual honesty.

[1] This nice definition was in a previous version of the entry about Science in Wikipedia. It seems that the article in question was recently victim of much vandalism.
[2] R. M. Henig, Darwin’s God, New York Times Magazine, March 4 2007.
[3] U. Eco, God isn't big enough for some people,

http://www.umbertoeco.com/id-49/Umberto_Eco_About_God.html.
[4] E. Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, Communications in Pure and Applied Mathematics 13 (1960);
http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
[5] I. Prigogine, From Being To Becoming, Freeman: 1980.
[6] P.W. Anderson, More Is Different, Science 177, (1972) pp. 393-396.

Another one

La vache du voisin est malade...

C'est pas grand chose mais ça fait toujours plaisir.
(Sagesse populaire)

Bohmian mechanics and the organized skepticism

A few days ago, I read the paper about Bohmian mechanics on the website of the Stanford Encyclopaedia of Philosophy (SEP) [1]. The last time I came across the name of David Bohm [2] was during my stay in Princeton in 2005. I was subletting a house of Princeton University and people in the neighborhood had the habit of leaving on a shelf at the laundry any book they didn’t want to carry along when they were moving out. In one of these, there was a chapter about scientists in Princeton who had been the victims of McCarthy. David Bohm was one of them.

David Bohm had worked on the Manhattan project, he was assistant professor at Princeton University in 1949. After being suspected of being a communist, he was fired from Princeton and he could not find any position in any other American university. He worked in several countries -notably in Brazil- and he eventually settled in London, where he died in 1992.

The exile of David Bohm was actually twofold: not only was he persona non grata in his own country, he was also left out of the international clique of physicists who mattered. Not surprisingly is Bohmian mechanics unorthodox; it is the work of an outcast.

What I realized with the SEP article is that the Copenhagen interpretation of quantum mechanics (QM) is not all there is about QM. David Bohm proposed a causal (deterministic) interpretation of quantum mechanics, the predictions of which are identical with the orthodox QM [3]. Philosophically, however, the two theories are at odds. In Bohmian mechanics the particles have a well defined position and momentum, and the wave function acts as a potential (a so-called pilot wave) that guides the particles. Such an approach was claimed to be impossible, and proved to be so by a famous theorem due to John Steward Bell [4]. Interestingly enough, here is what Bell himself wrote in 1987 about Bohmian mechanics [1]:

But why then had Born not told me of this ‘pilot wave’? If only to point out what was wrong with it? Why did von Neumann not consider it? More extraordinarily, why did people go on producing ‘‘impossibility’’ proofs, after 1952, and as recently as 1978? ... Why is the pilot wave picture ignored in text books? Should it not be taught, not as the only way, but as an antidote to the prevailing complacency? To show us that vagueness, subjectivity, and indeterminism, are not forced on us by experimental facts, but by deliberate theoretical choice?

This is the story of David Bohm and of his unorthodox theory. Yet another sad example of organized skepticism turned de facto into dogmatic denial.

[1] http://www.science.uva.nl/~seop/entries/qm-bohm/
[2] http://en.wikipedia.org/wiki/David_Bohm
[3] D. Bohm, Phys. Rev. 85 (1952) 166-180.
[4] http://en.wikipedia.org/wiki/Bell%27s_Theorem

Triletters

A few days ago was HMB's birthday. Happy birthday, H!

Nicolas Caritat, ci-devant marquis de Condorcet

Il y a une statue de Condorcet sur le quai Conti à Paris, à coté de l'hôtel des monnaies, dont il a été contrôleur général. Je connaissais un peu Condorcet pour avoir lu sa biographie écrite par les Badinter. J'ai lu, il n'y a pas très longtemps sa "Lettre d'un père proscrit à sa fille" (lien ci-dessous).

Mon sentiment après l'avoir lue a été: "Merde alors, le pauvre vieux!" Ce n'est pas que je plaignais particulièrement sa situation quand il a écrit cette lettre, toute précaire qu'elle eut été (il se terrait dans un grenier, d’où il n’est sorti que pour mourir emprisonné quelques jours plus tard). C'est plutôt ce que cette lettre laisse transparaître de la succession de souffrances intimes qu'a été sa vie.

A mes yeux, cette lettre est une énumération en 10 pages de toutes les recettes qu'une personne hypersensible a mises au point tout au long de sa vie pour ne pas souffrir, notamment de ses rapports avec les autres. Et il transmet ces recettes à sa fille comme un savoir très précieux.

En vrac. A propos du travail: "Rien n'est donc plus nécessaire a ton bonheur que de t'assurer des moyens dependans de toi seule pour remplir le vide du tems, écarter l'ennui, calmer les inquiétudes, te distraire d'un sentiment pénible". Ou encore: "On peut n'être pas maître de ne pas écouter son coeur, mais on l'est toujours de ne pas l'exciter; et c'est le seul conseil que la raison puisse donner à la sensibilité" Un autre: "N'attends, n'exige des autres, qu'un peu au-dessous de ce que tu ferais pour eux". Etc., etc., etc.

(http://books.google.com/books?id=cis0AAAAMAAJ&pg=PA603&dq=Avis+d%27un+proscrit+%C3%A0+sa+fille).