In this paper is presented and discussed the notion of true finite by opposition to the notion of theoretical finite. Examples from mathematics and physics are given. Fermat’s infinite descent principle is challenged.

ABSTRACTIn this paper, I offer an account of propositional learning: namely, learning that p. I argue for what I call the “Three Transitions Thesis” or “TTT” according to which four states and three transitions between them characterize such learning. I later supplement the TTT to account for learning why p. In making my case, I discuss mathematical propositions such as Fermat's Last Theorem and the ABC Conjecture, and then generalize to other mathematical propositions and to non-mathematical propositions. I also (...) discuss some interesting applications of the TTT, and reply to some noteworthy objections. (shrink)

En partant du problème des partis, conçu comme un dialogue entre partenaires, nous ferons ressortir le tissage de sa structure avec des considérations religieuses et éthiques; puis nous reconsidérerons l’argument dit « du pari » en l’interprétant, mutatis mutandis, comme une préfiguration de la théorie des jeux plutôt que comme la résolution d’un problème de probabilité, ce qu’il est plus évidemment sous la plume de Fermat qui, à la limite, ne résout pas le même problème que Pascal, quoique les (...) deux épistoliers aient pu en avoir l’impression. (shrink)

This paper introduces a little-known episode in the history of physics, in which a mathematical proof by Pierre Fermat vindicated Galileo’s characterization of freefall. The first part of the paper reviews the historical context leading up to Fermat’s proof. The second part illustrates how a physical and a mathematical insight enabled Fermat’s result, and that a simple modification would satisfy any of Fermat’s critics. The result is an illustration of how a purely theoretical argument can (...) settle an apparently empirical debate. (shrink)

The first half of the 17th century was a time of intellectual ferment when wars of natural philosophy were echoes of religious wars, as we illustrate by a case study of an apparently innocuous mathematical technique called adequality pioneered by the honorable judge Pierre de Fermat, its relation to indivisibles, as well as to other hocus-pocus. André Weil noted that simple applications of adequality involving polynomials can be treated purely algebraically but more general problems like the cycloid curve (...) cannot be so treated and involve additional tools–leading the mathematician Fermat potentially into troubled waters. Breger attacks Tannery for tampering with Fermat’s manuscript but it is Breger who tampers with Fermat’s procedure by moving all terms to the left-hand side so as to accord better with Breger’s own interpretation emphasizing the double root idea. We provide modern proxies for Fermat’s procedures in terms of relations of infinite proximity as well as the standard part function. (shrink)

This is the diary of an observant mathematician who documented the drama of the resolution of Fermat's Last Theorem as it unfolded around him from 1993 to 1995. Pierre Fermat claimed around 1637, in the most famous marginalia in the history of mathematics, to have a proof of the theorem that xn + yn = zn has no whole number solutions for n greater than 2. The other principal figure is the British mathematician Andrew Wiles, who emigrated (...) to Princeton University in the 1980s and who in 1993, at what for mathematicians is the relatively advanced age of forty, announced a proof of FLT. In the course of Wiles's initial exposition to mathematicians, before a full‐fledged proof could be published, a gap was discovered. This was too late to forestall announcements in the world press of an imminent proof. What had been an almost secretive devotion to the task by Wiles now became a rather more public and pressurized effort over the course of nearly a year. Wiles wondered at times if success could be achieved, but with the help of other mathematicians the gap was filled in and the full formal proof published in 1995.The statement of the theorem is likely to be the only part of the 1990s story that a nonspecialist can immediately appreciate, and the author has not especially addressed his book to the nonspecialist. Rather, this account is analogous to a well‐illustrated exhibition guidebook that serves as a memento and record to those who visited in person but that is not intended fully to inform those who did not experience it. For a broader account of the historical and mathematical background a reader can consult the three principal books published previously on the same topic by Amir Aczel, Simon Singh, and Alfred J. van der Poorten. The author knew many of the main players, attended most of the key talks, took notes, made recordings, and photographed. A major part of the book is a selection of sixty‐two photographs from the total of three thousand made by the author. He asked questions of Wiles and others and recorded both formal and informal reactions that he observed around him during the many public and nonpublic presentations that were made. Also, in keeping with the journal nature of his work, he has recorded his own reactions. Appendixes include an excerpt from Wiles's 1995 Annals of Mathematics publication and a synopsis by Ram Murty of the key results by Gerhard Frey, Jean‐Pierre Serre, and Kenneth Ribet that formed a basis for Wiles's work.It may interest historians and archivists to learn of the extent to which the primary sources for this story include e‐mail, web pages, photographs, and newspaper and television interviews. Though we recognize this reality of contemporary science, a diary account probably brings the issue home more directly than would studies by professional historians, who might feel obligated to do more of their own interviews, for example, rather than rely on the media. Mathematics has rarely had such a public spotlight turned on it—the proof of the four‐color theorem in 1976 is the only comparable event that comes to mind. Consequently, it has opened an unusual window into how mathematicians do mathematics. (shrink)

Essay Review of “Les Arithmétiques de Diophante. Lecture historique et mathématique” by Roshdi Rashed and Christian Houzel, and Histoire de l’analyse diophantienne classique : d’Abū Kamil à Fermat by Roshdi Rashed.

Pierre de Fermat is known as the inventor of modern number theory. He invented–improved many methods useful in this discipline. Fermat often claimed to have proved his most difficult theorems thanks to a method of his own invention: the infinite descent. He wrote of numerous applications of this procedure. Unfortunately, he left only one almost complete demonstration and an outline of another demonstration. The outline concerns the theorem that every prime number of the form 4n + 1 (...) is the sum of two squares. In this paper, we analyse a recent proof of this theorem. It is interesting because: it follows all the elements of which Fermat wrote in his outline; it represents a good introduction to all logical nuances and mathematical variants concerning this method of which Fermat spoke. The assertions by Fermat will also be framed inside their historical context. Therefore, the aims of this paper are related to the history of mathematics and to the logic of proof-methods. (shrink)

Early in his mathematical career Leibniz discovered some important methods and results but had to recognize that his findings had been anticipated by other mathematicians such as Pierre de Fermat, James Gregory, Isaac Newton, François Regnauld, John Wallis, etc. This paper investigates the cases of Isaac Barrow and Pietro Mengoli who, earlier than Leibniz, had been familiar with the characteristic triangle, transmutations methods, the inverse connection between determining tangents and areas of curves or the sums of the reciprocal (...) figurate numbers, and the harmonic triangle. To what extent was Leibniz aware of the results and publications of his predecessors? How did he assess their methods and results? Why did Leibniz never acknowledge any influence of these two mathematicians on his own studies? After publication of Leibniz’s manuscripts concerning the prehistory and early history of the calculus in the Academy Edition these questions can be investigated on the solid foundation of original texts. (shrink)

ABSTRACT: What kinds of markets, market regulations, and business organizations are compatible with contemporary egalitarian theories of justice? This article argues that any thoughtful answer to this question will have to draw on recent developments in political philosophy that are concerned not only with the equality of the distribution of core goods but also with the requirements for equality of status, voice, and so on, in the relations between individuals and within organizations. The dominance of theories of distributive justice in (...) egalitarian political philosophy since Rawls may have contributed, on the one hand, to the oft-recognized gulf between these theories and their theorists and, on the other, to discussions of corporate governance and business ethics. The main purpose of this article is to introduce business ethicists to some of the less-familiar features of recent relational theories of justice and equality, and to suggest that some of these notions may help bridge the gap between business ethics and political philosophy more generally. (shrink)

We prove a couple of results on NTP2 theories. First, we prove an amalgamation statement and deduce from it that the Lascar distance over extension bases is bounded by 2. This improves previous work of Ben Yaacov and Chernikov. We propose a line of investigation of NTP2 theories based on S1 ideals with amalgamation and ask some questions. We then define and study a class of groups with generically simple generics, generalizing NIP groups with generically stable generics.

The main objective of this article is to capitalise on many years of research, and of practice, relating to the use of introspection in a research context, and thus to provide an initial outline description of introspection, while developing an introspection of introspection. After a description of the context of this research, I define the institutional conditions which would enable the renewal of introspection as a research methodology. Then I describe three aspects of introspective practice: 1) introspection as a process (...) of becoming aware, theorized through Husserl's model of consciousness modes; 2) introspection as recollection, through the model of retention and awakening in Husserl's theory of memory; 3) the use of universal descriptive categories for the description of all lived experiences, as a guide for skilled practice of introspection in research. Finally I examine the question of the validation of introspective data, suggesting a strong distinction between the ethical criterion and the epistemic criterion of truth. (shrink)

Un spectre hante l’Europe classique, l’antitrinitarisme – et sa variante la plus connue : le socinianisme. L’histoire commence par un double itinéraire européen, de l’Italie catholique et intolérante à la Suisse protestante et non moins intolérante, puis à la Pologne qui représente alors un havre de la liberté de conscience. Lelio Sozzini (1525-1562) s’était enfui de Sienne à Zurich en 1548, mais, traité d’hérétique par Calvin, il avait dû fuir de nouveau en Pologne. Dix ans plus tard, son ne..

Pour comprendre la correspondance de Spinoza, il faut s’appuyer sur les règles généralement en usage au XVIIe siècle, ainsi que sur les caractères spécifiques de l’écriture spinozienne : ses lettres sont toujours des réponses, elles s’adaptent à des questions extérieures et, loin d’être autonomes, elles renvoient à l’ensemble du système.

Georg Misch, auteur d'une monumentale histoire de l'autobiographie, met en oeuvre les principes de Dilthey: le récit de la vie est aussi vieux que l'histoire, mais il se modifie selon les époques, et les détenninations qu'il reçoit à la Renaissance et à I' Âge classique l'orientent vers la description des lois de développement de l'individu.

Is a policy of economic growth compatible with the neutrality of the State? Some liberals (Rawls, Dworkin, Ackerman, Larmore, Kymlicka) think so. I do not. I begin by explaining and discussing the different meanings of the neutrality thesis, then I show that, whatever meaning we give to the idea of neutrality, it is very difficult to argue convincingly that a policy of economic growth does not favour some conceptions of the good.

Nothing has yet been done, and, here, in the middle of the twentieth century, it is fast becoming too late to draw up a suitable catalogue of the works of human wisdom. We are forced to project for the future the complete realization of our desires. This future will no doubt discover a conscious and effective organization of thought and action—a constant good fortune in the pursuit of legitimate satisfactions through a total mastery of natural forces—in a word, a perfect (...) and reciprocal adaptation between man and that part of the universe in which he lives. All this, or most of it, still remains to be done; but much has already been said in the five thousand years that men have been writing. An intelligent observer arriving from Sirius would doubtless be especially struck by the extraordinary gap separating here on earth what is so well said from what is so badly carried out. Or rather he would wonder how it is that beings as highly gifted as men, judging from the knowledge and wisdom they have accumulated, still make such ineffective and even harmful use of the means at their disposal. The most indulgent hypothesis would be to see in this the result of a temporary misunderstanding owing to the great speed with which human thought evolves. Our observer would resume his journey then, resolved to return after a hundred years or two, sure that he would then find that man had bridged the gap and become as great in what he does as in what he knows. (shrink)

"This volume assembles twelve texts published between 1892 and 1915.... The editors allow one to see the genesis of the ideas of Duhem, philosopher and historian, of the variety of his styles, and sometimes also the limits of his work.... A useful index, probably unique in the field of Duhemian studies, completes the book.... The English-language public may be assured an exemplary translation and a reliable critical apparatus." --Jean Gayon, _Revue d'Histoire des Sciences_.

The article is an instructive exposition of the central problem of scholastic metaphysics, namely: the problem of analogia entis. The author sees the origin of this idea in a historical development, which began with late neoplatonism, continued with Arabic philosophy and ended with medieval authors, especially S. Thomas Aquinas.

As stressed by Perruchet & Vinter, the SOC model echoes Johnson-Laird's mental model theory. Indeed, the latter rejects rule-based processing and assumes that reasoning is achieved through the manipulation of conscious representations. However, the mental model theory as well as its modified versions resorts to the abstraction of complex schemas and some form of implicit logic that seems incompatible with the SOC approach.

Ce livre révisionniste (sans connotation péjorative) s’inscrit dans la littérature historienne nous parlant des guerres qui ont ensanglanté le Viêt-nam pendant trente ans (1945-1975). L’intention première de son auteur est de rompre avec une historiographie dominante, héroïsante et partisane qui donne le beau rôle au Nord Viêt-nam qui faisait une « guerre juste » contre l’impérialisme étranger « et ses valets sud vietnamiens », pour accomplir un noble dessein, celui de la libération puis celu...

Femmes voilées : les Grecs aussi. À propos d’un livre de Lloyd Llewellyn-Jones. 2003 marque une date importante dans l’historiographie de l’Antiquité grecque : le livre de Lloyd Llewellyn-Jones « ose » comparer de façon systématique le port du voile par les Grecques à d’autres usages, à d’autres référents culturels comme ceux du monde araboislamique. Ce court hommage a pour objet d’insister sur l’originalité et l’importance de cette contribution majeure à l’histoire des femmes grecques ; il s’agit aussi de dénoncer (...) le silence et l’aveuglement des érudits. (shrink)

v. 1. Raison et conscience de soi.--v. 2. Raison et histoire.

Le texte de la leçon inaugurale prononcée au Collège de France, le 23 avril 1982.