Though a materialist, he fell in love with the beautiful logical structure of Euclidean geometry. In his book Leviathan (1651), Hobbes argued that, without a strong ruler, individuals were condemned to live in a “war of all against all” in which, as he famously said, life was “solitary, poor, nasty, brutish, and short.”īut Hobbes philosophized about more than politics. The English Civil War, which resulted in more than 100,000 dead out of a population of five million, provided a major impetus for Hobbes’ political philosophy. Battles continued off and on until 1688 when Parliament asked William of Orange and his wife Mary to rule the country. The political dispute over who should rule - an absolute king or a (somewhat) more representative Parliament - was linked to religious disputes between the established Church of England, Catholics, and dissenters like the Puritans. King Charles I and Parliament clashed, the Army purged Parliament of the king’s supporters, and Charles I was actually executed in 1649. So it is no wonder that Jesuit intellectuals opposed using indivisibles in geometry.Įven more surprising, the historical context for Hobbes’ philosophical objection to indivisibles was the English Civil War. Furthermore, Catholic theology owes much to Aristotle’s philosophy, and Aristotle, arguing for the potentially infinite divisibility of the continuum, had explicitly ruled out both indivisibles and the actual infinite. Bonaventura Cavalieri, who pioneered indivisible methods in geometry, was among Galileo’s followers. Besides opposing the Church about whether the earth went around the sun, Galileo treated matter as made of atoms, which are physical indivisibles. The doctrine of indivisibles was on the side of Galileo. Various rulers and nations lined up on one side or another of the religious divide one result was the Thirty Years War (1618–1648).Īll sorts of ideas were judged on the basis of which side they seemed to favor. The Peasants’ Revolt in the 1520s showed how attacks on one kind of authority could spill over into the political realm. The religious challenges became intertwined with political ones. The Catholic Church responded by firming up church doctrines and institutions. Why did the Church get involved in evaluating the “new math” of indivisibles, infinitesimals, and the infinite? Catholicism had dominated medieval Europe, but by the sixteenth century had been challenged religiously by Protestantism. Let’s first look at that context, and then evaluate his conclusions. But Hobbes thought that Wallis’s arguments weren’t good geometry, and that the social order itself depended on sticking to Euclidean rigor.Īlexander’s provocative new book begins with a lively account of these two disputes and their historical context. Wallis had used these ideas to find the areas under the graph of \(y=x^n\) for rational \(n\). In England a couple of decades later, the mathematician John Wallis (the first to use the symbol \(\infty\) for infinity) and the political philosopher Thomas Hobbes clashed over whether the infinite and the infinitely small were mathematically legitimate. My Solid Geometry teacher didn’t tell us that. What we now call Cavalieri’s Principle was thought to be dangerous to religion. Des exemples sont donnés pour un problème à petites et un problème à grandes déformations, ce qui illustre l'applicabilité en vue de son implémentation pratique pour la simulation de processus de formage.In the 1630s, when the Roman Catholic Church was confronting Galileo over the Copernican system, the Revisors General of the Jesuit order condemned the doctrine that the continuum is composed of indivisibles. Des kernels Lagrangiens et Eulériens sont comparés et leur influence sur la longueur intrinsèque est évaluée. Une attention particulière est donnée au type de non-localité utilisé pour le calcul de l'endommagement. Cette approche garande que le problème mathématique associé est bien posé, et élimine toute forme de sensitivité au maillage. Les deux cas font usage d'une variable d'endommagement isotrope, calculée d'un champs non-local. Fondée sur la réduction progressive de la contrainte d'écoulement par l'endommagement, la théorie est d'abord présentée dans un contexte de déplacements infinitésimals, après lequel une généralisation envers une formulation géométriquement non-linéaire en hyperélasto-plasticité suit. Cet article présente une approche phénoménologique pour l'incorporation de l'endommagement ductile dans un modèle élasto-plastique.
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