Albert Lautman (1908-1944) was a French philosopher of mathematics whose work played a crucial role in the history of contemporary French philosophy. His ideas have had an enormous influence on key contemporary thinkers including Gilles Deleuze and Alain Badiou, for whom he is a major touchstone in the development of their own engagements with mathematics. Mathematics, Ideas and the Physical Real presents the first English translation of Lautman's published works between 1933 and his death in 1944. Rather (...) than being preoccupied with the relation of mathematics to logic or with the problems of foundation, which have dominated philosophical reflection on mathematics, Lautman undertakes to develop an understanding of the broader structure of mathematics and its evolution. The two powerful ideas that are constants throughout his work, and which have dominated subsequent developments in mathematics, are the concept of mathematical structure and the idea of the essential unity underlying the apparent multiplicity of mathematical disciplines. This collection of his major writings offers readers a much-needed insight into his influence on the development of mathematics and philosophy. (shrink)

The first English collection of the work of Albert Lautman, a major figure in philosophy of mathematics and a key influence on Badiou and Deleuze.

Lautman synthétise dans ce rapport quelques idées centrales qui seront par la suite développées dans ses Thèses . Il s’agit d’un manuscrit inédit, qui semble être le premier texte scientifique du jeune philosophe. Lautman étudie le local et le global suivant Galois, Riemann, Hilbert et Cartan, et propose une hypothèse sur les rapports structurels généraux du local et du global, qui préfigure l’essor de la théorie des faisceaux, laquelle apparaîtra une dizaine d’années après.Lautman synthetizes some of the (...) main ideas that will be developped in his Thesis . It is an unpublished manuscript, which may be the first scientific text of the young philosopher. Lautman studies the local/global dialectics, following Galois, Riemann, Hilbert and Cartan, and proposes an hypothesis on general structural relations between local and global, very close to the sheaf theory paradigm which will emerge ten years later. (shrink)

1. ptie. Les schémas de structure -- 2.ptie. Les schémas de genèse.

It is possible today to observe in hindsight the epistemological landscape of the twentieth century, and the work of Albert Lautman in mathematical philosophy appears as a profound turning point, opening to a true under- standing of creativity in mathematics and its relation with the real. Little understood in its time or even today, Lautman’s work explores the difficult but exciting intersection where modern mathematics, advanced mathe- matical invention, the structural or unitary relations of mathematical knowledge and, (...) finally, the metaphysical and dialectical tensions underly- ing mathematical activity converge. Well beyond other better-known names in philosophy of mathematics – who are focused above all on ques- tions concerning the logical problem of foundations, important but frag- mentary studies in the vast panorama of modern mathematics – Lautman broaches the emergence of inventiveness in the very broad spectrum of the development of the mathematical real. Group theory, differential geome- try, algebraic topology, differential equations, functional analysis, functions of complex variables and number fields are some of the domains of his preferred examples. He detects in them methods of construction, structu- ration and unification of modern mathematics that he connects to a precise Platonic interpretation in which powerful pairs of ideas serve to organize the edifice of effective mathematics. (shrink)

Nous posons l'oeuvre d'Albert Lautman comme une sorte d'opérateur de brisure de symétrie dans le cadre de l'opposition traditionnelle de la philosophie spéculative et des sciences physico-mathématiques. L'enjeu pour la philosophie en est, à de très rares exceptions près, encore très mal perçu. Sur ce plan, nous reprenons la question du « platonisme » supposé de Lautman, et nous le confrontons à sa lecture fondamentale de Martin Heidegger. Les conséquences de cette inscription dans le sillon heideggérien sont (...) fondamentales pour une juste restitution de cette pensée et de cette philosophie dans ce qu'elles conservent de plus actuel. Sur l'autre versant, celui des percées lautmaniennes dans le champ de la physique mathématique, nous insistons sur l'extrême alacrité prospectiviste de son regard, sa puissance philosophique venant renforcer cette singulière sensibilité à ce qui fit l'essentiel des mathématiques et de la physique théorique de son temps. En ce sens, sa position occupe bel et bien l'un des points nodaux du site « surrationaliste ». (shrink)

Albert Lautman. Mathematics, Ideas and the Physical Real. Simon B. Duffy, trans. London and New York: Continuum, 2011. 978-1-4411-2344-2 (pbk); 978-1-44114656-4 (hbk); 978-1-44114433-1 (pdf e-bk); 978-1-44114654-0 (epub e-bk). Pp. xlii + 310.

Dans cet article, j’explore dans un premier temps la conception que se fait Lautman de la dialectique en examinant ses références à Platon et Heidegger. Je compare ensuite les structures dialectiques identifiées par Lautman dans les mathématiques contemporaines avec celles qui émergent de ses sources philosophiques. Enfin, je soutiens que les structures qu’il a découvertes dans les mathématiques sont plus riches que le suggère son modèle platonicien, et que la distinction « ontologique » de Heidegger est moins utile (...) que semblait le penser Lautman. -/- In this paper, I first explore Lautman’s conception of dialectics by a consideration of his references to Plato and Heidegger. I then compare the dialectical structures that he found in contemporary mathematics with the model that emerges from his philosophical sources. Finally, I argue that the structures that he discovered in mathematics are richer than his Platonist model suggests, and that Heidegger’s “ontological” distinction is less useful than Lautman seemed to believe. (shrink)

In October 1984, Bruno Huisman stated with regards to Jean Cavaillès, ‘Let us be honest, or at least realistic: today, one can be a professor of philosophy without ever having read a single line of Cavaillès. Often invoked, sometimes quoted, the oeuvre of Cavaillès is little attended for itself’ (Huisman 1984). As for Albert Lautman, it would seem that the situation is even more extreme. In 1994, the publisher Hermann, under the impetus of Bruno Huisman and George Canguilhem, (...) collected almost the totality of the Jean Cavaillès papers in one volume (Oeuvres complètes de philosophie des sciences (Cavaillès 1994)). But, the Essai sur l’unité des mathématiques et divers écrits (Lautman 1977), published by the Union générale d’Éditions in 1977, had all but disappeared by the early 1980s and yet was never republished! This will remain one of the great indignities of French publishing, for as Jean Petitot rightly affirms: ‘Regarded as too speculative, in spite of his exceptional mathematical scholarship and his close connection with Hilbertian axiomatic structuralism, his mathematical philosophy has, until now, been devoid of any particular attention …. We would like to state clearly from the start, Albert Lautman represents, in our view, without exaggeration, one of the most inspired philosophers of this century’ (Petitot 1987, 79-80). (shrink)

In the chapter of Difference and Repetition entitled ‘Ideas and the synthesis of difference,’ Deleuze mobilizes mathematics to develop a ‘calculus of problems’ that is based on the mathematical philosophy of Albert Lautman. Deleuze explicates this process by referring to the operation of certain conceptual couples in the field of contemporary mathematics: most notably the continuous and the discontinuous, the infinite and the finite, and the global and the local. The two mathematical theories that Deleuze draws upon for (...) this purpose are the differential calculus and the theory of dynamical systems, and Galois’ theory of polynomial equations. For the purposes of this paper I will only treat the first of these, which is based on the idea that the singularities of vector fields determine the local trajectories of solution curves, or their ‘topological behaviour’. These singularities can be described in terms of the given mathematical problematic, that is for example, how to solve two divergent series in the same field, and in terms of the solutions, as the trajectories of the solution curves to the problem. What actually counts as a solution to a problem is determined by the specific characteristics of the problem itself, typically by the singularities of this problem and the way in which they are distributed in a system. Deleuze understands the differential calculus essentially as a ‘calculus of problems’, and the theory of dynamical systems as the qualitative and topological theory of problems, which, when connected together, are determinative of the complex logic of different/ciation. (DR 209). Deleuze develops the concept of a problematic idea from the differential calculus, and following Lautman considers the concept of genesis in mathematics to ‘play the role of model ... with respect to all other domains of incarnation’. While Lautman explicated the philosophical logic of the actualization of ideas within the framework of mathematics, Deleuze (along with Guattari) follows Lautman’s suggestion and explicates the operation of this logic within the framework of a multiplicity of domains, including for example philosophy, science and art in What is Philosophy?, and the variety of domains which characterise the plateaus in A Thousand Plateaus. While for Lautman, a mathematical problem is resolved by the development of a new mathematical theory, for Deleuze, it is the construction of a concept that offers a solution to a philosophical problem; even if this newly constructed concept is characteristic of, or modelled on the new mathematical theory. (shrink)

La question fondamentale d’Albert Lautman concerne la nature du réel et la capacité de l’esprit de l’appréhender. C’est pourquoi elle convoque les données de la physique, leur expression en concepts mathématiques et leur interprétation métaphysique qui doit préciser le rapport de la pensée humaine à la réalité du monde. L’examen des théories mathématiques les plus sophistiquées de son temps est destinée à montrer l’affinité de la genèse des concepts mathématiques avec une Dialectique supérieure, qui met en jeu les (...) Idées, comprises en un sens dérivé de Platon. Mon but est d’expliquer comment Lautman comprend les termes « Dialectique », « Idée », « genèse », simultanément sur le plan métaphysique et dans leur incarnation mathématique. Je montrerai que Lautman développe une conception très personnelle du platonisme, différente de celle de la version reçue par la tradition philosophique. Lautman rejette la séparation des Idées et du sensible, et adopte le style de pensée heideggerien pour montrer que les Idées sont mues par une Dialectique et qu’elles entretiennent une relation réciproque, dialectique, avec le sensible.Albert Lautman’s main questioning was about the nature of reality and the capacity of mind to grasp reality. The answer needs to go at the same time into the laws and tools of physics and the corresponding concepts of mathematics, and to search for the metaphysical interpretation that could give an account of the relation between scientific thought and reality. The examination of the most sophisticated mathematical theories was intended to show the affinity between the generation of mathematical concepts and the dialectical movement of Ideas. My aim is to explain how Lautman understood ‘Idea’, ‘Dialectic’, ‘genesis’ at once in a mathematical and in a metaphysical meaning. I will point out that Lautman had a very personal conception of Platonism, very different from Platonism of mathematicians of his time and very different from the traditional interpretation developed in philosophy. Lautman did not assume the separation between Ideas and the sensible world. By contrast, he adopted the Heideggerian style of thought to show the Dialectic between Ideas and the reciprocal relation between Ideas and the sensible. (shrink)

Albert Lautman,.philosophe, juif, fusillé par les Nazis pour faits de Résistance comme son ami et collègue Jean Cavaillès, a laissé inachevée une œuvre puissante et dont l'influence s'est fait sentir au-delà de son domaine. Lautman est une figure rare : un philosophe souverain, capable de mobiliser les grands noms de la tradition philosophique, de Platon à Heidegger, et suffisamment averti des mathématiques pour proposer une intelligence de leur développement acceptée des savants. Les membres du groupe Bourbaki ont (...) reconnu en Lautman un auteur exposant adéquatement les mathématiques auxquelles ils contribuaient ; Deleuze se réfère à lui de manière élogieuse ; Badiou, Petitot et Salanskis, notamment, lui font écho ou tentent de le prolonger. On examine ici cette œuvre dense, depuis Essai sur les notions de structure et d'existence en mathématiques et Essai sur l'unité des sciences mathématiques dans leur développement actuel aux textes posthumes, Symétrie et dissymétrie en mathématiques et en physique suivi du Problème du temps, après l'avoir située dans son contexte historique-scientifique. On espère ainsi montrer la profondeur et l'actualité d'une pensée singulière, celle d'un grand rationaliste qui était aussi un métaphysicien. (shrink)

Albert Lautman (b. 1908–1944) was a philosopher of mathematics whose views on mathematical reality and on the philosophy of mathematics parted with the dominant tendencies of mathematical epistemology of the time. Lautman considered the role of philosophy, and of the philosopher, in relation to mathematics to be quite specific. He writes that: ‘in the development of mathematics, a reality is asserted that mathematical philosophy has as a function to recognize and describe’ (Lautman 2011, 87). He goes (...) on to characterize this reality as an ‘ideal reality’ that ‘governs’ the development of mathematics. The relation between mathematical problems as they arise in the historical development of mathematics and the solutions that are provided to these problems by mathematicians, in the form of new mathematical theories, definitions or axioms, are governed by a what Lautman characterises as a dialectics of mathematics. The aim of this paper is to give an account of this Lautmanian dialectic and of how it can be understood to govern the development of solutions to mathematical problems. (shrink)

In Différence et répétition, Deleuze's ontology is structured by his theory of dialectical Ideas or problems, which draws features from Plato, Kant, and classical calculus. Deleuze unifies these features through a theory of Ideas/problems developed by the mathematician and philosopher Albert Lautman. Lautman worked to explain the nature of the problems or dialectical Ideas mathematics engages and the solutions or mathematical theories endeavouring to understand them. Lautman drew upon Heidegger to do this. This article clarifies Deleuze's (...) theory of dialectical Ideas/problems by analysing its debts to Lautman and Heidegger and demonstrates a Heideggerian influence on Deleuze via Lautman. (shrink)

Dans sa thèse complémentaire intitulée « Essai sur l’unité des sciences mathématiques dans leur développement actuel » Albert Lautman analysa la question de l’unité des mathématiques en considérant différentes paires antithétiques de concepts mathématiques, notamment le continu et le discret. Dans le cadre de sa refonte de la géométrie algébrique abstraite, le mathématicien français Alexandre Grothendieck considéra également l’opposition traditionnelle du continu et du discret selon un cadre conceptuel fort similaire à celui de Lautman. En comparaison, l’introduction (...) du concept de topos lui permit de donner une réponse strictement mathématique et parfaitement claire à cette question.Albert Lautman’s complementary thesis, entitled “Essai sur l’unité des sciences mathématiques dans leur développement actuel”, analysed the unity of mathematics through various antithetical pairs of mathematical concepts such as the continuous and the discrete. As part of his renewal of abstract algebraic geometry, French mathematician Alexandre Grothendieck also considered the traditional opposition between the continuous and the discrete and adopted a conceptual framework very similar to that of Lautman. However, the topos concept allowed Grothendieck to come up with a purely mathematical and, in comparison, much clearer solution. (shrink)

This book is about the ethical issues that clinicians encounter as they care for patients and is written to assist those who serve on hospital ethics committees as they deliberate about appropriate action in difficult ethical cases.

This book is an attempt to get to the bottom of an acute and perennial tension between our best scientific pictures of the fundamental physical structure of the world and our everyday empirical experience of it. The trouble is about the direction of time. The situation (very briefly) is that it is a consequence of almost every one of those fundamental scientific pictures--and that it is at the same time radically at odds with our common sense--that whatever can happen can (...) just as naturally happen backwards. Albert provides an unprecedentedly clear, lively, and systematic new account--in the context of a Newtonian-Mechanical picture of the world--of the ultimate origins of the statistical regularities we see around us, of the temporal irreversibility of the Second Law of Thermodynamics, of the asymmetries in our epistemic access to the past and the future, and of our conviction that by acting now we can affect the future but not the past. Then, in the final section of the book, he generalizes the Newtonian picture to the quantum-mechanical case and (most interestingly) suggests a very deep potential connection between the problem of the direction of time and the quantum-mechanical measurement problem. The book aims to be both an original contribution to the present scientific and philosophical understanding of these matters at the most advanced level, and something in the nature of an elementary textbook on the subject accessible to interested high-school students. Table of Contents: Preface 1. Time-Reversal Invariance 2. Thermodynamics 3. Statistical Mechanics 4. The Reversibility Objections and the Past-Hypothesis 5. The Scope of Thermodynamics 6. The Asymmetries of Knowledge and Intervention 7. Quantum Mechanics Appendix: Gedankenexperiments with Heat Engines Index Reviews of this book: The foundations of statistical mechanisms are often presented in physics textbooks in a rather obscure and confused way. By challenging common ways of thinking about this subject, Time and Chance can do quite a lot to improve this situation. --Jean Bricmont, Science Albert is perfecting a style of foundational analysis that is uniquely his own...It has a surgical precision...and it is ruthless with pretensions. The foundations of thermodynamics is a topic that has accumulated a good deal of dead wood; this is a fire that will burn and burn. --Simon W. Saunders, Oxford University As usual with Albert's work, the exposition is brisk and to the point, and exceptionally clear...The book will be an extremely valuable contribution to the literature on the subject of philosophical issues in thermodynamics and statistical mechanics, a literature which has been thin on the ground but is now growing as it deserves to. --Lawrence Sklar, University of Michigan. (shrink)

In medical ethics, business ethics, and some branches of political philosophy (multi-culturalism, issues of just allocation, and equitable distribution) the literature increasingly combines insights from ethics and the social sciences. Some authors in medical ethics even speak of a new phase in the history of ethics, hailing "empirical ethics" as a logical next step in the development of practical ethics after the turn to "applied ethics." The name empirical ethics is ill-chosen because of its associations with "descriptive ethics." Unlike descriptive (...) ethics, however, empirical ethics aims to be both descriptive and normative. The first question on which I focus is what kind of empirical research is used by empirical ethics and for which purposes. I argue that the ultimate aim of all empirical ethics is to improve the context-sensitivity of ethics. The second question is whether empirical ethics is essentially connected with specific positions in meta-ethics. I show that in some kinds of meta-ethical theories, which I categorize as broad contextualist theories, there is an intrinsic need for connecting normative ethics with empirical social research. But context-sensitivity is a goal that can be aimed for from any meta-ethical position. (shrink)

_Time_'s 'Man of the Century', Albert Einstein is the unquestioned founder of modern physics. His theory of relativity is the most important scientific idea of the modern era. In this short book Einstein explains, using the minimum of mathematical terms, the basic ideas and principles of the theory which has shaped the world we live in today. Unsurpassed by any subsequent books on relativity, this remains the most popular and useful exposition of Einstein's immense contribution to human knowledge.

From the author of Quantum Mechanics and Experience, a hugely influential book that challenged key assertions by Niels Bohr and other founders of quantum mechanics, A Guess at the Riddle provides a major metaphysical overhaul of one of physics' most intractable problems-the quest to bridge quantum and classical physics in order to understand the nature of reality.

This book is about the ethical issues that clinicians encounter as they care for patients and is written to assist those who serve on hospital ethics committees as they deliberate about appropriate action in difficult ethical cases.

The political theory of the property-owning democracy can be seen as a way of overcoming the ideological conflict between individualism and collectivism. Rawls offers the contemporary reference-point for this theory. Rawls contrasted the ideal-type of the property-owning democracy with the ideal-type of a capitalist welfare state. However, the terms of that contrast are not well drawn and raise a number of questions, in particular regarding Rawls’s a priori specification of the welfare state. An inductively derived specification of ideal-typical welfare states (...) suggests that horizontal redistribution, in line with the principle of social savings, is more important than vertical redistribution. Rawls’s preference for a social dividend or negative income tax scheme can be contrasted with the use of social insurance, but the latter has a claim to instantiate Rawlsian ideals better than a social dividend. There is a potential problem with the pre-emption of private savings in the welfare state, but this turns out not to be troublesome empirically or conceptually. The irony of the discussion is that those who have interpreted Rawlsian theory as justifying the welfare state have the better of the argument, despite Rawls’s own views. (shrink)

By turns historical, sociological and autobiographical, this book investigates racism as social pathology - a cultural disease that prevails because it allows one segment of society to empower itself at the expense of another.

Das Buch gibt einen Überblick über die Arbeiten von Hans Albert und behandelt ausgewählte Konsequenzen seines Werks für die Philosophie des Kritischen Rationalismus und ihre Anwendungen in Gebieten der Soziologie, Wirtschaftswissenschaften, Rechtswissenschaften und weiteren Wissenschaften.

Throughout his writings, Hume extols the benefits of an attribute he calls “Strength of Mind,” which he defines as the “prevalence of the calm passions over the violent” (T 2.3.3.10). But there is some question as to how he thought a person could attain this important trait. Contemporary scholars have committed Hume to the view that only indirect and social methods, such as state punishment or sympathetic pressure, could effectively cultivate it. Yet a closer examination of Hume’s corpus reveals a (...) more direct approach applicable at the individual level. Though rarely achieved and difficult to execute, self-government of the passions is possible according to Hume. The key to success lies in harnessing the power of habit. In its most sophisticated form, the process centrally involves cultivating a certain transformative connoisseurship, the ability to appreciate regularities in one’s own mental activity and thereby alter it. (shrink)