Wallace, Bolzano, Weierstrass et le paradoxe de Zénon

J’ai la chance d’aider - pour ce qui est de la terminologie et usages mathématiques français - le traducteur de “Everything & more, a compact history of ∞” de David Foster Wallace. Ça me permet la primeur d’un texte vraiment drôle et érudit qui donne des lettres de noblesse rock’n'roll et sexy à l’histoire du concept d’infini en mathématiques. C’est plein de passion et tout-à-fait accessible aux non-matheux, les parties techniques sont soigneusement balisées d’un “Si vous êtes intéressé”. Ceci étant, il arrive parfois que le soucis de vulgarisation desserve, à mon sens, la clarté du propos. C’est l’un de ces points que je voudrais reprendre ici, sans vouloir effrayer quiconque, quelque chose de très très résumé, juste pour mon bien-être spirituel.
Zénon d’Élée (Ve siècle av. J.-C) est un antique trouble-fête grec qui osa aborder de front la question de l’infini tandis que ses contemporains (et leurs descendants) avaient remisé l’idée au rayon néfaste pour la santé mentale.
Parmi les paradoxes de Zénon, prenons celui de la dichotomie : une pierre lancée sur un arbre doit parcourir, avant d’atteindre sa cible, la moitié du chemin qui les sépare, puis encore la moitié de la distance restante, puis la moitié de ce qui reste, etc. Elle doit donc occuper une infinité de positions avant d’atteindre l’arbre, chaque étape se faisant en un temps non nul. Puisqu’on peut toujours diviser le parcours restant en deux moitiés dont la première prend toujours un peu de temps à parcourir, avant de reconsidérer le problème et de recommencer le raisonnement précedent, la pierre ne peut jamais arriver jusqu’à l’arbre. On voit à merveille ici l’intrication de l’infini et du continu, je me souviens encore du délicieux frisson ressenti lorsqu’on m’a enseigné cela : la continuité offrait un espace à toutes les galipettes imaginables.
Entre lui et Cantor (le héraut de l’infini), 23 siècles d’histoire mathématique qui évitent plus ou moins d’aborder le concept, permettant malgré tout, de creuser, petit à petit, une voie à l’intuition. 2300 ans en quète de rigueur aussi, et c’est Bolzano puis Weierstrass, qui, dans l’élan mathématique de leur époque (le 19ème donc), ont donné des définitions sans biais des limites et de la continuité. Wallace utilise la continuité d’une manière pas forcément évidente pour aborder le paradoxe de Zénon, je trouve plus simple de rephraser cela en terme de limite d’une suite infinie, avec la définition rigoureuse (ce qu’il -DFW- n’a pas fait apparemment pour se démarquer des démonstrations approximatives) . La définition “ε, δ” qui est alors “ε, N” de la limite d’une suite infinie c’est la suivante : une suite un tend vers une limite l quand n tend vers l’infini, si pour tout écart de tolérance ε, il existe un rang fini N à partir duquel, pour tout n>N, un est proche de l à ε près.

Reprenons notre pierre et lançons-la contre un arbre. L’expérience nous montre qu’elle met un certain temps pour réaliser son trajet que pour simplifier nous prendrons égal à 1 et confrontons ceci au raisonnement de Zénon. Nous supposons alors qu’elle parcourt la moitié de la distance à l’arbre en un temps égal à 1/2 , le quart suivant en 1/4 puis le huitième ensuite en 1/8 etc. au bout de n itérations, il lui faut 1/2n supplémentaire pour effectuer son petit bout de chemin, son trajet a alors duré (1/2 + 1/4 + 1/8 ….+ 1/2n) sachant que la distance restant jusqu’à l’arbre correspond à un trajet de 1/2n exactement.
Pour prouver qu’on va bien parvenir à la cible malgré la dichotomie, il faut montrer qu’au final l’addition indéfinie de tous ces petits temps tend vers la valeur de 1. On considère la suite des sommes partielles Sn = 1/2 + 1/4 + 1/8 +…+ 1/2n (donc Sn+1 = Sn + 1/2n+1 ) et on veut montrer que lorsque n tend vers l’infini, la valeur de Sn tend vers 1. Telle qu’on a construit la suite Sn, on voit bien que Sn + 1/2n = 1 puisqu’on lui rajoute justement la dernière moitié que l’on s’apprêterait à couper en deux à l’itération suivante. Donc montrer que Sn tend vers 1 est équivalent à montrer que (1-1/2n) tend vers 1 aussi, soit simplement que 1/2n tend vers 0. Ce qui est immédiat avec un tout petit peu d’arithmétique : si l’on veut 1/2n < ε, ε étant voué à devenir aussi petit que nécéssaire, on a : 1/ε < 2n, on passe aux logarithmes on obtient une condition sur n : n > ln(1/ε) / ln(2). Par exemple avec ε = 0.0001, on a n > 13, c’est à dire que pour n > N = 13, Sn est proche au dix-millième de 1. On aura beau rajouter une infinité de petits termes 1/214 + 1/215 + … etc., tout ce que l’on fera c’est de se rapprocher encore et encore de 1 ; les sommes partielles en constituant une approximation que l’on peut toujours ajuster (via N) de manière arbitrairement précise (ε), la somme totale (série) — donc pour l’indice n décrivant l’ensemble (infini) des entiers naturels — valant 1 exactement.
C’est beau, non ?
Bon si vous n’avez rien suivi, pas de panique, il faut quand même à Wallace presque 200 pages pour arriver là.

145 Responses to “Wallace, Bolzano, Weierstrass et le paradoxe de Zénon”

  1. G Says:

    bah oui j’ai décroché vers la moitié :)

    mais avec Wallace aussi je décroche souvent (et pourtant….)

  2. Bérangère Says:

    Ah wiii que c’est beau ! Autrement t’as de beaux yeux tu sais ?

  3. Pascal Says:

    Hello,
    je suis confronté à une forme amusante (enfin pas toujours) du paradoxe de Zénon sur le plan cognitif dans ma vie de tous les jours.
    Mon cerveau ne semble pas équipé pour traiter les “finitions” (certains appellent ça être un “poor finisher”) et en particulier, s’il me faut un temps T pour amener un travail jusqu’à 90% d’avancement, il faudra un autre temps équivalent T pour l’amener à 99%, et encore une temps T pour l’amener à 99,9% etc.
    Lorque des gens me parlent d’un travail “achevé”, cela me procure un grand inconfort car j’ai l’impression qu’il faudrait à un “finisseur” une simple rallonge de deux heures pur “achever” le travail, alors que cela prendrait un temps en apparence infini pour moi. C’est juste que le fait de focaliser ma pensée sur une donnée précise me demande une énergie et une prouesse énorme.

    Par ailleurs, le travail avec une marge de flou ou d’incertitude me va très bien et j’arrive à le réaliser sans effort. Notamment, l’intuition des directions dans lesquelles une organisation pourrait évoluer, les lignes de forces d’une culture d’entreprise, les tendances qui influencent une société dans une situation donnée, la notion même de “situation”, ainsi que la perception de ce qui se joue sur le plan relationnel sont des modes de pensée qui consomment très peu d’énérgie et d’effort pour moi.

    La plupart des modèles de fonctionnement de l’esprit humain utilisés actuellement en “psychologie de la normalité” montrent pourquoi ce qui est facile pour l’un est difficile pour l’autre. Les modèles les plus modernes et les plus influencés par les neurosciences vont un peu plus dans le détail en expliquant le rôle de fonctions séparées comme par exemple le calcul exact (26 plus 17 égale 43) et le calcul approché (on en a pour environ six semaines à faire ce job) ou encore (plus sioux) la coordination et le visuo-spatial. Ces fonctions se trouvent localisées à différents endroits du cerveau, comme en attestent les observations réalisées sur des personnes ayant subi des lésions qui les rendent incapables d’effectuer certaines tâches alors qu’elles peuvent toujours effectuer d’autres tâches que nous ne dissocions pas habituellement. Ainsi tel pianiste classique sera toujours capable de jouer de mémoire après un accident mais ne saura plus “lire” la musique.

    Nous appartenons physiquement à la continuité du monde et de ses interactions jusque dans la façon dont notre pensée en recrée artificiellement une image…
    Les Tibétains ne disent pas autre chose.
    Penser est une activité physique.

    PJ

  4. cb Says:

    “poor finisher”, encore un mème dont je suis victime ! Damned !

  5. G Says:

    en fait en y revenant en pleine lecture du wallace, mon seul soucis dans ta démonstration, ce sont les logarithmes, qui sont des OMNI pour ceux qui n’ont pas fait de math. et du coup si ta démonstration me semble hum plus distanciée et surtout moins bordélique je ne sais pas si elle aide plus le non-matheux (mais foster wallace doit du coup faire trépigner tous les autres ?).

  6. cb Says:

    mais là les logarithmes ils ne servent à rien qu’à résoudre : trouver n tel que 2n >1/ε ; on pourrait s’en passer si on demande au lecteur de croire que par exemple avec ε = 0.0001, on trouve que pour n > 13 ça marche. À la limite, au moins pour ces valeurs précises, c’est vérifiable avec une calculatrice… maintenant pour vérifier l’assertion pour tout ε, on peut toujours affirmer qu’il est possible de trouver “empiriquement” une valeur de n qui correspondra… mais c’est plein de non-dits.
    Ou plutôt, si les logarithmes sont trop OMNI, est-ce que faire un joli petit graphe comme celui-là de la fonction 2x qui tend donc vers l’infini et montrer que quelle que soit la valeur choisie pour ε on peut toujours se placer au dessus de 1/ ε ? (en fait c’est exactement avec ce genre de petits dessins quand c’est approprié, que j’explique les choses à mes étudiants, avant de les démontrer plus formellement )
    sinon justement la rigueur -au moins dans un cadre didactique- repose dans le fait de ne pas chercher à “faire croire”, mais à prouver…
    et pis t’as qu’à relire mon post sur le logarithme baby.

  7. G Says:

    oui…. je vois bien qu’il me manque un truc. c joli le sdessins !

    moi je trouve quand même merveilleux qu’il ait fallu 2500 pour démontrer ce que les sens savent (et peut-être ce dont les sens ont très peur d’assumer): le monde est continu, indivisible. 2500 pour que l’outil de création/observation des lois du monde soit en accord avec la principale donnée : tout est irrationnel. on ne peut pas découper une tâche en plein de petites tâches. on converge tout l’temps. on est du potentiel pas du fini + fini, pas même une infinité de finis. enfin j’me comprends (je crois).

  8. G Says:

    (hmmmff j’ai pas compris avec le dessin, faudrait reprendre tout le post je crois)

  9. cb Says:

    oui je t’expliquerai le dessin + tard. no problem.

  10. cb Says:

    (sinon là où DFW fait trépigner, c’est quand il se la joue : “allez on met les mains dans le cambouis et on se paye une vraie démo” alors qu’il s’y vautre sérieusement)

  11. G Says:

    oui pour le trépignage,je suppose. mais en fait les enjeux métaphysiques (ainsi qu’il les appelle) sont les seuls qui m’importent. pour le moment en tout cas.

  12. cb Says:

    yes, c’est d’ailleurs l’intérêt de ce livre (en plus de rendre tout ça assez rigolo).

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