Local axioms in disguise: Hilbert on Minkowski diagrams

While claiming that diagrams can only be admitted as a method of strict proof if the underlying axioms are precisely known and explicitly spelled out, Hilbert praised Minkowski??s Geometry of Numbers and his diagram-based reasoning as a specimen of an arithmetical theory operating ??rigorously?? with geometrical concepts and signs. In this connection, in the first phase of his foundational views on the axiomatic method, Hilbert also held that diagrams are to be thought of as ??drawn formulas??, and formulas as ??written diagrams??, thus suggesting that the former encapsulate propositional information which can be extracted and translated into formulas. In the case of Minkowski diagrams, local geometrical axioms were actually being produced, starting with the diagrams, by a process that was both constrained and fostered by the requirement, brought about by the axiomatic method itself, that geometry ought to be made independent of analysis. This paper aims at making a twofold point. On the one hand, it shows that Minkowski??s diagrammatic methods in number theory prompted Hilbert??s axiomatic investigations into the notion of a straight line as the shortest distance between two points, which start from his earlier work focused on the role of the triangle inequality property in the foundations of geometry, and lead up to his formulation of the 1900 Fourth Problem. On the other hand, it purports to make clear how Hilbert??s assessment of Minkowski??s diagram-based reasoning in number theory both raises and illuminates conceptual compatibility concerns that were crucial to his philosophy of mathematics.     

Conceptual framework from the Paris Psychosomatic School: A clinical psychoanalytic approach to oncology1

This article presents further clinical material from the Paris Psychosomatic School ( Aisenstein, 2006 ). The Freudian foundations of psychosomatics are detailed and post-Freudian developments focusing on the contribution of the Paris Psychosomatic School are outlined, in particular, the somatizing process as a result of regression and the somatizing process as a result of drive unbinding. The authors argue that the latter possibly gives rise to progressive and serious illness leading to death. The relationship of classical psychoanalysis to psychotherapeutic treatment from the angle of the Paris school is commented on. The authors then turn to two clinical presentations of women suffering from breast cancer. The method of evaluating the patients’ capacities for undergoing psychotherapeutic treatment and their mental capacity for healing is discussed. The face-to-face psychoanalytic treatment undertaken with the second case is discussed. Finally, the authors recall Freud’s insistence after 1920 on the opposition of the life drives and the death drives, which placed self-destruction at the centre of psychic functioning. They conclude that current research in biology and medicine, notably research concerning programmed cell death, will converge with psychoanalytic psychosomatics to illuminate somatizing processes and demonstrate the relevance of psychoanalytic treatment to patients who are capable of mental reorganization in the course of their illness and medical treatment.     

The Œdipus complex, crystallizer of the debate between psychoanalysis and anthropology

The way that anthropologists understand the Oedipus complex, in particular, is a good example of how they understand psychoanalysis in general. Indeed, it has crystallized a set of reactions marked by ignorance, misunderstanding, distortion and screening out and at the same time has provoked suspicion among anthropologists as to psychoanalysis, according to the preconceptions of the various schools of thought and authors implied, and this from the very first contacts up to nowadays. In what way did the psychoanalysts contribute to this and what representation did they, in turn, elaborate of anthropology? The purpose of this paper is to expose the epistemological and historical conditions of the emergence of this debate, and then to develop it by following chronology up to the 1950s and 1960s, while differentiating three major cultural areas, Great Britain, the USA and France, in order to get a clearer picture. From that point on, we will try to diversify our inquiry and to formulate some interpretative hypotheses. In particular, we think that a traumatic event may have inaugurated and organized the history of the relationship between the two disciplines, producing a situation of acculturation with multiple impacts, if we identify them with two cultures coming into contact: what is at stake here is Totem and Taboo in which Freud carries through the first major psychoanalytical approach of the interpretation of ethnographic facts, that leads him to transplant the universality of the Oedipus complex to the very root of the first social institutions and to pinpoint the presence of unconscious processes in their genesis.     

How Discrete Patterns Emerge from Algorithmic Fine-Tuning: A Visual Plea for Kroneckerian Finitism

This paper sets out to adduce visual evidence for Kroneckerian finitism by making perspicuous some of the insights that buttress Kronecker’s conception of arithmetization as a process aiming at disclosing the arithmetical essence enshrined in analytical formulas, by spotting discrete patterns through algorithmic fine-tuning. In the light of a fairly tractable case study, it is argued that Kronecker’s main tenet in philosophy of mathematics is not so much an ontological as a methodological one, inasmuch as highly demanding requirements regarding mathematical understanding prevail over mere preemptive reductionism to whole numbers.