samedi 29 mars 2008

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?

mardi 11 mars 2008

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.

vendredi 7 mars 2008