From 73715cd175f544df7b4af148f43863beee5903b7 Mon Sep 17 00:00:00 2001 From: oskar Date: Sat, 25 Apr 2026 08:07:07 +0200 Subject: [PATCH] MacBookPro.lan 2026-4-25:8:7:7 --- ...tre XII et ses cercles non-concentriques.md | 33 ++++++++++++++++-- .../fiche_lecture.pdf | Bin 271784 -> 279272 bytes .../figures/demi_cercles_ppc.tikz | 32 +++++++++++++++++ .../figures/mouvement_tube.tikz | 3 +- 4 files changed, 64 insertions(+), 4 deletions(-) create mode 100644 sources/0 - cours/LOGOS S2/le savoir en mathématiques/figures/demi_cercles_ppc.tikz diff --git a/sources/0 - cours/LOGOS S2/le savoir en mathématiques/Fiche de lecture - La lettre XII et ses cercles non-concentriques.md b/sources/0 - cours/LOGOS S2/le savoir en mathématiques/Fiche de lecture - La lettre XII et ses cercles non-concentriques.md index 10bc4e0d..d945d173 100644 --- a/sources/0 - cours/LOGOS S2/le savoir en mathématiques/Fiche de lecture - La lettre XII et ses cercles non-concentriques.md +++ b/sources/0 - cours/LOGOS S2/le savoir en mathématiques/Fiche de lecture - La lettre XII et ses cercles non-concentriques.md @@ -175,7 +175,6 @@ Ainsi, Camerini obtient la traduction suivante : [^barbaras-incommensurabilite]: \say{Ce premier argument porte donc sur l’incommensurabilité entre des \emph{choses} qui, n’ayant pas d’unité de mesure commune, ne peuvent être exprimées l’une en fonction de l’autre par \emph{aucun} nombre. On peut montrer – par exemple, par un raisonnement par l’absurde comme le font les mathématiciens anciens – l’impossibilité de trouver \emph{un} nombre par lequel pourrait s’exprimer le rapport entre diverses parties de la même figure. Si ces choses doivent néanmoins être considérées comme des grandeurs, leur grandeur ne peut être mesurée par aucun nombre.} [@barbarasSpinozaScienceMathematique2007 chap.6, Paragraphe 19] [^camerini-trad-spatii]: [@camerini-lettre12 p.9] [^ppc-cercles]: [Principes de la philosophie de Descartes, @spinoza-pleiade p.141 proposition 9] -[^correspondance-rovere]: [@spinoza-correspondance-rovere] [^ppc-parties]: \say{par partie de la matière il [Descartes] entend tout ce qui est transporté ensemble, même si cela peut être à son tour constitué de multiples parties} [Principes de la philosohpie de Descartes, Partie II, Définition VIII @spinoza-pleiade p.226] @@ -183,15 +182,43 @@ $$*\,*\,*$$ Après avoir analysé philosophiquement le texte pour en produire une traduction la plus fidèle possible (travail qu'il appelle exégétique), Camerini passe à une approche plus historique, qui va inscrire la figure des cercles non-concentrique dans une lignée de figures et de réflexions au sujet de l'infini actuel. -La figure des deux cercles est déjà présente dans les *Principia Philosophiae* de Descartes de 1644. Elle s'inscrit dans le cadre de la physique de Descartes, dont il est utile de présenter quelques principes. Pour Descartes, la matière n'est pas composée d'atomes ; le vide n'existe pas ; le mouvement est pensé à travers la notion de *mouvement local*, c'est-à-dire que l'on considère une portion de matière qui se transporte ensemble, à une même vitesse[^ppc-parties] ; la matière est potentiellement infiniment divisible, mais n'est *actuellement divisée* qu'en tant qu'elle se concoit en parties ayant des mouvement distincts, autrement dit des vitesses distinctes. C'est alors qu'intervient la figure des cercles : on imagine que ces cercles représentent la coupe d'un tube de diamètre variable, à l'intérieur duquel se trouve un fluide parfait. Dans le cas de cercles concentriques (c'est-à-dire pour un diamètre constant) on peut diviser la matière en parties finies et expliquer ainsi un mouvement circulaire du fluide (voir \figref{fig:mvt-tube}, à gauche). La vitesse de chaque partie de matière sera la même, l'espace étant homogène, et chaque partie suivra un déplacement le long du tube (représenté par les flèches de la \figref{fig:mvt-tube}) (la matière agissant comme un tout de vitesse uniforme, il n'est en fait pas nécessaire de la diviser pour expliquer son mouvement. Mais dans le cas de deux cercles non-concentriques (si le diamètre varie) (voir \figref{fig:mvt-tube}) il faudra que, en même temps qu'une certaine quantité de fluide passe par $AB$, une même quantité passe par $CD$. Les diamètres $AB$ et $CD$ étant inégaux, il faudra que la matière aille plus vite lorsqu'elle passe par $CD$ que lorsqu'elle passe en $AB$ : l'espace non-homogène entraine des vitesse non-homogènes. Plus encore, la matière située entre $AB$ et $CD$ dans le tube (par exemple, située en $E$) aura une vitesse distincte de celle située en $AB$ et de celle située en $CD$. +La figure des deux cercles est déjà présente dans les *Principia Philosophiae* de Descartes de 1644. Elle s'inscrit dans le cadre de la physique de Descartes, dont il est utile de présenter quelques principes. Pour Descartes, la matière n'est pas composée d'atomes ; le vide n'existe pas ; le mouvement est pensé à travers la notion de *mouvement local*, c'est-à-dire que l'on considère une portion de matière qui se transporte ensemble, à une même vitesse[^ppc-parties] ; la matière est potentiellement infiniment divisible, mais n'est *actuellement divisée* qu'en tant qu'elle se concoit en parties ayant des mouvement distincts, autrement dit des vitesses distinctes. Autrement dit \say{Une matière qui est diversement mue a au moins autant de parties divisées en acte qu'on observe simultanément en elle différents degrés de vitesse.}[^ppc-division-vitesse]. +C'est alors qu'intervient la figure des cercles : on imagine que ces cercles représentent la coupe d'un tube de diamètre variable, à l'intérieur duquel se trouve un fluide parfait. +Dans le cas de cercles concentriques (c'est-à-dire pour un diamètre constant) on peut diviser la matière en parties finies et expliquer ainsi un mouvement circulaire du fluide (voir \figref{fig:mvt-tube}, à gauche). La vitesse de chaque partie de matière sera la même, l'espace étant homogène, et chaque partie suivra un déplacement le long du tube (représenté par les flèches de la \figref{fig:mvt-tube}) (la matière agissant comme un tout de vitesse uniforme, il ne serait en fait pas nécessaire de la diviser pour expliquer son mouvement). +Mais dans le cas de deux cercles non-concentriques (si le diamètre varie) (voir \figref{fig:mvt-tube}) il faudra que, en même temps qu'une certaine quantité de fluide passe par $AB$, une même quantité passe par $CD$. Les diamètres $AB$ et $CD$ étant inégaux, il faudra que la matière aille plus vite lorsqu'elle passe par $CD$ que lorsqu'elle passe en $AB$ : l'espace non-homogène entraine des vitesse non-homogènes. Plus encore, la matière située entre $AB$ et $CD$ dans le tube (par exemple, située en $E$) aura une vitesse distincte de celle située en $AB$ et de celle située en $CD$. En observant la division finie des cercles non-concentriques, à droite de la \figref{fig:mvt-tube}, il apparaît que cette division ne peut pas expliquer le mouvement du fluide ; par exemple, la partie $E$ ne pourra pas se déplacer "d'un bloc" à l'emplacement de la partie $F$. On comprends que, dans ce cas, les variations de diamètre étant infinies (bien que comprises entre $AB$ au maximum et $CD$ au minimum), il faudra que la matière soit divisée infiniment (en cela, la partie droite de la \figref{fig:mvt-tube} est "fausse" puisqu'elle ne présente qu'une division finie). Ainsi, dans cet espace non-homogène, pour expliquer le mouvement du fluide, il faut le concevoir comme *actuellement infiniment divisé*. On voit aussi en quoi cette division surpasse tout nombre : si on divise l'espace en un certain nombre de parties, on constatera que cette division n'est pas capable d'expliquer le mouvement. \begin{figure}[h] \center - \resizebox{!}{4cm}{\input{figures/mouvement_tube.tikz}} + \resizebox{!}{5cm}{\input{figures/mouvement_tube.tikz}} \caption{mouvement des parties de la matière} \label{fig:mvt-tube} \end{figure} +Dans les *Principes de la philosophie de Descartes*, Spinoza commence par exposer l'exemple physique repris de Descartes (le tube circulaire de diamètre changeant), puis il en vient à une situation géométrique (dont la figure est reproduite par la \figref{fig:demi-cercles}) qui lui permet d'expliquer en quoi l'espace sera partout inégal dans le cas de cercles non-concentriques. + +\begin{figure}[h] + \center + \resizebox{!}{2.5cm}{\input{figures/demi_cercles_ppc.tikz}} + \caption{Lemme de la proposition IX, Principes de la philosophie de Descartes} + \label{fig:demi-cercles} +\end{figure} + +Si Spinoza passe déjà à la géométrie dans les *Principes de la philosohie de Descartes*, c'est seulement dans la *Lettre 12* qu'il se détache complètement de la physique. En même temps que ce passage de la physique à la géométrie, Spinoza opère un autre changement : \say{le passage de la­ considération des espaces inégaux aux inégalités de ­ l’espace (« omnibus inaequalibus spatiis » des Principes de la philosophie de Descartes à « omnes inaequalitates spatii » de la Lettre XII)}[^4]. De plus, par rapport à Descartes qui \say{était très réticent à affirmer la possibilité­ d’une division infinie ­ continue}[^5], Spinoza affirme un infini actuel de la division continue de l'espace, et affirme que l'intellect peut comprendre cet infini, même si l'imagination ne peut pas le faire sous le prisme du nombre. Il faut comprendre deux choses à propos de cette "impuissance du nombre" : + + - Ce n'est pas en tant qu'il est dû à l'imagination que le nombre ne peut pas déterminer cet infini. C'est bien plus à cause de son aspect discret que le nombre se trouve impuissant dans cet exemple où la division de l'espace est continue. Spinoza affirme d'ailleurs plus tôt dans la *Lettre 12* que nous pouvons imaginer certains infinis ; et la *mesure*, autre *auxilliaire de l'imagination*, semble pouvoir s'appliquer à ce cas, puisque Spinoza affirme qu'il existe un infini \say{[possédant] un maximum et un minimum sans pour autant [que nous soyons] en mesure d'en égaler et d'en expliquer les parties par aucun nombre}[^lettre-p1] mais surtout parce qu'il parle d'infinis \say{plus ou moins grands}. + - Ce n'est pas à cause d'une \say{excessive grandeur} des parties que les variations de la matière dépassent tout nombre ; cela est plutôt dû au fait qu'une détermination par un nombre de ces variations serait paradoxal : \say{L'espace compris entre des cercles de centre différents ne peut rien souffrit de tel. Si bien que, si quelqu'un veut déterminer [toutes ces inégalités de l'espace] par un nombre précis, il lui faudra faire en même temps qu'un cercle ne soit pas un cercle}[^lettre-num-impossible]. + + +[^ppc-division-vitesse]: [Principes de la philosohie de Descartes, Partie II, Axiome XVI, @spinoza-pleiade p.228] +[^4]: [@camerini-lettre12, p.16] +[^5]: [@camerini-lettre12, p.14] +[^lettre-num-impossible]: [Lettre 12, partie 2, @spinoza-pleiade p.1079] +[^lettre-p1]: [Lettre 12, partie 2, @spinoza-pleiade p.1074] + +$$*\,*\,*$$ + + + # Critique # Bibliographie diff --git a/sources/0 - cours/LOGOS S2/le savoir en mathématiques/fiche_lecture.pdf b/sources/0 - cours/LOGOS S2/le savoir en mathématiques/fiche_lecture.pdf index a729feac4c52947fe3613f515259d78d1ea24b1c..6637724f8d3cf3287db0d440e53372a9d33b0bbd 100644 GIT binary patch delta 114314 zcmZs>Q*hu-@GczNwr$(mIN8{?z40%}#>R#l+qP|EW81dA_kX@~E>6{%nv0pP?y2g& z=z03-E}cL*2|=yxFazUaO+>;3EG^kh*7TPS&jgPG!5)5)+e5`J))> 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left=45] (11.center) to (12.center); + \draw [bend left=45] (12.center) to (13.center); + \draw (8.center) to (10.center); + \end{pgfonlayer} +\end{tikzpicture} diff --git a/sources/0 - cours/LOGOS S2/le savoir en mathématiques/figures/mouvement_tube.tikz b/sources/0 - cours/LOGOS S2/le savoir en mathématiques/figures/mouvement_tube.tikz index 891af199..645a6b96 100644 --- a/sources/0 - cours/LOGOS S2/le savoir en mathématiques/figures/mouvement_tube.tikz +++ b/sources/0 - cours/LOGOS S2/le savoir en mathématiques/figures/mouvement_tube.tikz @@ -72,7 +72,8 @@ \node [style=none] (70) at (2.75, 0) {B}; \node [style=none] (71) at (2.75, -1.5) {C}; \node [style=none] (72) at (2.75, -2.5) {D}; - \node [style=none] (73) at (4.375, 0.625) {E}; + \node [style=none] (73) at (4.225, 0.575) {E}; + \node [style=none] (74) at (4.275, -0.975) {F}; \end{pgfonlayer} \begin{pgfonlayer}{edgelayer} \draw [bend right=45] (3.center) to (2.center);