Bourbaki: Mathematics and Truth

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“Mathematicians have always been convinced that what they prove is ‘true’. It is clear that such a conviction can be only of a sentimental or metaphysical order, and cannot be justified, or even ascribed a meaning which is not tautological, within the domain of mathematics. The history of the concept of truth in mathematics therefore belongs to the history of philosophy and not of mathematics; but the evolution of this concept has had an undeniable influence on the development of mathematics, and for this reason we cannot pass over it in silence.” (p. 306f.) #Bourbaki #mathematics #truth

Bourbaki, Nicolas, Elements of Mathematics: Theory of Sets. Berlin, Heidelberg, New York: Springer 2004.

Brunschvicq: Temps et espace

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“De là on ne peut pas conclure que l’arithmétique soit la science du temps comme la géométrie est la science de l’espace. Le temps n’est pas un objet; il est une condition de l’arithmétique, ou plus exactement de la mathématique en général.” (p. 269) #Brunschvicq #mathématique #arithmétique #temps #géométrie #espace

Brunschvicg, Léon, Les étapes de la philosophie mathématique. Paris: Alcan 1912.

Thom: Descartes’ vs. Newton’s Physics

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“History gives another reason for the physicist’s attitude toward the qualitative. The controversy between the followers of the physics of Descartes and of Newton was at its height at the end of the seventeenth century. Descartes, with his vortices, his hooked atoms, and the like, explained everything and calculated nothing; Newton, with the inverse square law of gravitation, calculated everything and explained nothing. History has endorsed Newton and relegated the Cartesian constructions to the domain of curious speculation. The Newtonian point of view has certainly fully justified itself from the point of view of its efficiency and its ability to predict, and therefore to act upon phenomena.” (p. 5) #Thom #Descartes #Newton #physics

Thom, René, Structural Stability and Morphogenesis: An Outline of a General Theory of Models. Boca Raton, FL: CRC Press 2018.

Bourbaki: Greek Mathematics

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“The originality of the Greeks consists precisely in a conscious effort of arranging mathematical proofs in a sequence so that the passage from one step to the next leaves nothing in doubt and compels universal assent. Of course, the Greek mathematicians, just like their present-day successors, made use of ‘heuristic’ rather than rigorous arguments in the course of their researches”. (p. 297) #Bourbaki #Greek #mathematics #proof #passage

Bourbaki, Nicolas, Elements of Mathematics: Theory of Sets. Berlin, Heidelberg, New York: Springer 2004.

Ruelle: La source d’information

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“La source d’information est censée produire une suite au hasard de messages permis (ou un message infiniment long avec certaines propriétés statistiques). On ne demande pas que les messages soient utiles ou logiquement cohérents, ni qu’ils aient aucun sens. Dire qu’un message contient une grande quantité d’information revient à dire qu’il est choisi dans une grande classe de messages permis, ou que beaucoup de hasard est présent. Ce hasard peut correspondre en partie à de l’information utile, en partie à du bruit sans intérêt.” (p. 176) #Ruelle #information #hasard #message #bruit

Ruelle, David, Hasard et Chaos. Paris: Éditions Odile Jacob 1991.