This paper attempts to show that mathematical knowledge does not grow by a simple process of accumulation and that it is possible to provide a quasi-empirical (in Lakatos's sense) account of mathematical theories. Arguments supporting the first thesis are based on the study of the changes occurred within Eudidean geometry from the time of Euclid to that of Hilbert; whereas those in favour of the second arise from reflections on the criteria for refutation of mathematical theories.
The present article aims at showing that it is possible to construct a realist philosophy of mathematics which commits one neither to dream the dreams of Platonism nor to reduce the word ''realism'' to mere noise.It is argued that mathematics is a science of patterns, where patterns are not objects (or properties of objects), but aspects, or aspects of aspects, etc. of objects. (The notion of aspect originates from ideas sketched by Wittgenstein in the Philosophical Investigations.).
Bertrand Russell's contributions to last century's philosophy and, in particular, to the philosophy of mathematics cannot be overestimated.Russell, besides being, with Frege and G.E. Moore, one of the founding fathers of analytical philosophy, played a major rôle in the development of logicism, one of the oldest and most resilient1 programmes in the foundations of mathematics.Among his many achievements, we need to mention the discovery of the paradox that bears his name and the identification of its logical nature; the generalization to (...) the whole of mathematics of Frege's idea that it is not possible to draw a demarcation line between logic and arithmetic; the programme, carried out with Whitehead, of derivation of mathematics from the logical system of Principia Mathematica ; and the ramified theory of types, devised by Russell to protect the system of PM from the known paradoxes.Although there is an ample literature on these topics, it is quite important to reconsider Russell's contributions to the foundations of mathematics at a time when, as a consequence of the crisis of the classical programmes in the foundations of mathematics, new trends are beginning to develop within the philosophy of mathematics. These are trends which move in a very different direction from that of logicism, intuitionism, and Hilbert's programme.To see this we need to consider that, in spite of profound disagreements on the nature of mathematical activity, on the relationship existing between logic and mathematics, on the causes of and therapies for the paradoxes, etc., logicism, intuitionism, and Hilbert's programme share an important metaphor: the idea that mathematics is an edifice built on unshakable foundations,2 an edifice which makes possible only a cumulative growth of mathematical knowledge.Such a metaphor—which, together with more specific theses belonging to these schools of thought, remained unsupported …. (shrink)
The present paper aims at showing that there are times when set theoretical knowledge increases in a non-cumulative way. In other words, what we call ‘set theory’ is not one theory which grows by simple addition of a theorem after the other, but a finite sequence of theories T1, ..., Tn in which Ti+1, for 1 ≤ i < n, supersedes Ti. This thesis has a great philosophical significance because it implies that there is a sense in which mathematical theories, (...) like the theories belonging to the empirical sciences, are fallible and that, consequently, mathematical knowledge has a quasi-empirical nature. The way I have chosen to provide evidence in favour of the correctness of the main thesis of this article consists in arguing that Cantor–Zermelo set theory is a Lakatosian Mathematical Research Programme (MRP). (shrink)
L’ouverture en juin 2010 des paris sportifs, hippiques et de poker en ligne constituait une véritable révolution culturelle auprès des joueurs et des professionnels du secteur. Il s’agissait essentiellement d’encadrer et de réguler en France les jeux d’argent et de hasard sur Internet, notamment par la création de l’Arjel , lutter contre les sites illégaux basés à l’étranger et protéger les joueurs du risque de dépendance. De très fortes retombées financières étaient également attendues par l’État. Au-delà de ces attentes économiques, (...) pas toujours tenues, un an et demi plus tard, le gouvernement semble hésiter sur sa politique de gestion de la cyberdépendance. D’un côté, en légalisant les jeux d’argent en ligne, l’État français incite implicitement les joueurs-parieurs à se connecter et à miser sur Internet massivement. Dans le même temps, les responsables politiques créent des organismes de sensibilisation face au danger que représente la dépendance informatique, fortement relayée par des médias, volontiers alarmistes. Ce paradoxe communicationnel s’inscrit alors dans l’analyse d’un double discours fortement controversé, incitant et diabolisant simultanément auprès des joueurs la pratique des jeux d’argent en ligne.The launch of online poker and betting on sports and horse-racing in June 2010 was a real cultural revolution for punters and professionals in the gambling sector. The aim was essentially to control and regulate online betting and gambling, especially with the introduction of ARJEL , to take action against illegal sites based abroad and to protect gamblers from the risk of addiction. The State also anticipated very substantial tax revenues. But over and above these financial expectations, not all of which had been realised a year and a half later, the government seems to be hesitating on its policy for managing cyber-addiction. On the one hand, by legalising online gambling, the French State is implicitly encouraging gamblers to log on and place bets on a massive scale. But at the same time, the authorities are setting up agencies to raise awareness on the dangers of Internet addiction, to a chorus of often alarmist media attention. This communicational paradox is therefore directly relevant to the analysis of a dual and highly controversial discourse that encourages online gambling while simultaneously demonising the practice. (shrink)
La cyberdépendance est un phénomène dont l’étude relève, encore aujourd’hui, largement de la psychologie. En abordant cette problématique sous l’angle pluridisciplinaire des sciences de l’information et de la communication, ce nouvel objet peut permettre la mise à distance de discours convenus sur la technophilie, mais également, sur une certaine technophobie. Ainsi, en abordant l’univers des jeux vidéo et d’Internet, il est désormais possible de prendre la mesure de la croissance multiforme des mondes virtuels, mais aussi et surtout, des risques inhérents (...) de dépendance, liés à ces pratiques et, plus largement, aux usages des techniques de l’information et de la communication. L’intérêt de notre démarche serait donc de relativiser, d’une part, un discours particulièrement élogieux à l’égard des nouvelles technologies, mais dans le même temps, montrer que la technique, aussi puissante soit-elle, reste tributaire de variables culturelles fortes. Les résultats d’une enquête empirique permettront alors de mieux comprendre comment s’articulent les relations entre l’individu, la technique et la culture, en montrant comment cette dernière parvient finalement à canaliser la technique.Studies on Internet addiction are still largely confined to the field of psychology. However, addressing this new object from the multidisciplinary angle of the information and communication sciences can help to step back from conventional discourses on technophilia, but also on some instances of technophobia. Investigating the world of video and Internet games makes it possible to take stock of the many-faceted growth of virtual worlds, but also, and especially, of the inherent addictiveness of game playing and the use of information and communication technologies in general. The value of our approach lies therefore in taking an objective view of the consistently eulogistic discourse on new technologies while showing at the same time that technology, for all its power, will always be dependent on strongly marked cultural variables. The results of our empirical investigation shed light on the interaction of relationships between people, technology and culture, to show that, ultimately, it is culture that channels technology and not the reverse. (shrink)
Available from UMI in association with The British Library. Requires signed TDF. ;The central position defended in this work is that Analytical philosophy is the best realization so far of Descartes' concern for the construction of a method of scientific investigation in philosophy. ;In Frege, the father of Analytical philosophy, such concern was embodied in his Conceptual Notation, which should have provided philosophy with a rigorous and powerful deductive system, and in the 'linguistic turn', i.e. the study of the structure (...) of thought conducted by means of an analysis of the structure of language. This 'linguistic turn' should have produced, in Frege's view, definitive conclusions about the limits of what is thinkable not by setting up arbitrary systems of Pure Reason, but through the controllable investigation of the logical structure of an entity--language--of which we have conceptual experience. ;Wittgenstein's later philosophy is seen here as an attack against the aims and tenets of Analytical philosophy. In fact, what Wittgenstein shows in Philosophical Investigations is, among other things, that by means of an investigation of language it is impossible to justify the assumption that language is an expression of thought. But, if this were true, it would follow that we were not justified in assuming that there exists an isomorphism between language and thought. That there is such an isomorphism is the presupposition which lies at the basis of Frege's 'linguistic turn'. ;In conclusion I attempt to prove that there is no solution to Wittgenstein's objections unless the original exclusively linguistic Analytical philosophy model is modified by a 'pragmatic turn'. The key element of the 'pragmatic turn' is that the truth of certain assumptions--and therefore the justification for their introduction--can be confirmed by the impact that the system based on them has on reality in controlling and transforming it successfully through techniques which are logically dependent on the system. (shrink)
1.1 ContextIn the period following the demise of logicism, formalism, and intuitionism, contributors to the philosophy of mathematics have been divided. On the one hand, there are those who tend to focus on such issues as: Do mathematical entities exist? If so, what type of entities are they and how do we know about them? If not, how can we account for the role that mathematics plays in our everyday and scientific lives? Contributors to this school—let us call it the (...) analytic school—do not, on the whole, concern themselves with careful analyses of important historical developments in mathematics. On the other hand, there are those who contribute to an historical school in the philosophy of mathematics. Contributors to this school tend to concern themselves almost exclusively with detailed historical analyses of important developments in mathematics. They are typically interested in answering questions concerning the growth of mathematical knowledge.In recent years, interest in the historical school has been growing, as has its influence on the analytic school. This book marks another stride in this direction. Oliveri aims to employ tools developed for use within the historical school to address one of the major issues investigated by the analytic school, i.e., whether we should be realists or anti-realists about mathematics. This is a laudable objective, since even a successful partial integration of these two schools would be valuable to contributors within both.1.2 Noteworthy ContributionsIn attempting a partial integration of these two schools, Oliveri makes several noteworthy contributions. The most significant is his development of a new type of argument for structural realism about mathematics. This argument exploits tools for theorizing about mathematics that were developed by Imre Lakatos  in his work on scientific research programs . It focuses attention on the progressive mathematical research program that started with …. (shrink)
"While ancient Greek thought is widely acknowledged as the major source of political ideals such as freedom and equality, ancient Greek practices including slavery, the subordination of women, and imperialism have been condemned as undemocratic and immoral. So is ancient Greek political thought still relevant today? In this wide-ranging history, Ryan Balot shows what ancient Greek political texts might mean to citizens of the twenty-first century."--BOOK JACKET.
Let us imagine for a moment that God is a most accomplished cithara player who nevertheless is not playing because he does not have a cithara; in other words, he is someone who has all the skills to act in the most masterly manner, but refrains from acting due to a lack of material implements. As no bodily counterpart can match his active power, he finds himself in the awkward situation of not being able to express himself. This is the (...) conundrum that the early modern philosopher Giordano Bruno brings to the attention of his readers in De l'infinito, universo e mondi, one of the... (shrink)