Platonism about mathematics (or mathematical platonism) isthe metaphysical view that there are abstract mathematical objectswhose existence is independent of us and our language, thought, andpractices. Just as electrons and planets exist independently of us, sodo numbers and sets. And just as statements about electrons and planetsare made true or false by the objects with which they are concerned andthese objects' perfectly objective properties, so are statements aboutnumbers and sets. Mathematical truths are therefore discovered, notinvented., Existence. There are mathematical objects.

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)

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)

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)

Gilles Deleuze’s engagements with mathematics, replete in his work, rely upon the construction of alternative lineages in the history of mathematics, which challenge some of the self imposed limits that regulate the canonical concepts of the discipline. For Deleuze, these challenges provide an opportunity to reconfigure particular philosophical problems – for example, the problem of individuation – and to develop new concepts in response to them. The highly original research presented in this book explores the mathematical construction of Deleuze’s philosophy, (...) as well as addressing the undervalued and often neglected question of the mathematical thinkers who influenced his work. -/- In the wake of Alain Badiou’s recent and seemingly devastating attack on the way the relation between mathematics and philosophy is configured in Deleuze’s work, Simon Duffy offers a robust defence of the structure of Deleuze’s philosophy and, in particular, the adequacy of the mathematical problems used in its construction. By reconciling Badiou and Deleuze’s seeming incompatible engagements with mathematics, Duffy succeeds in presenting a solid foundation for Deleuze’s philosophy, rebuffing the recent challenges against it. -/- Simon B. Duffy is a Senior Lecturer in Philosophy at Yale-NUS College, Singapore, and Honorary Research Associate in the Department of Philosophy at the University of Sydney, Australia. He is the author of The Logic of Expression: Quality, Quantity, and Intensity in Spinoza, Hegel and Deleuze (2006). (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)

Ce volume reproduit sous un titre nouveau l'ensemble des textes de philosophie mathematique d'Albert Lautman, ecrits entre 1933 et sa mort en 1944. Il fut introduit a la beaute des mathematiques par son genial ami Jacques Herbrand, ce qui l'a preserve de la focalisation, alors frequente, sur le probleme des fondements. Comme Jean Cavailles, qui fut aussi son ami, il s'inscrit dans la poursuite mais aussi la contestation de la voie ouverte par leur maitre commun Leon Brunschvicg dans Les etapes (...) de la philosophie mathematique. Trois themes sont dominants. Les creations mathematiques peuvent et doivent etre lues comme des approfondissements d'oppositions dialectiques au sens platonicien entre des contraires tels que le Continu et le Discontinu, le Meme et l'Autre, l'Essence et l'Existence. Les theories mathematiques sont une matiere et la tache du philosophe est concue comme effort pour degager la realite ideale a laquelle elles participent. Ainsi apparaissent les lignes d'une theorie des rapports entre essence et existence ou la structure d'un Etre s'interprete en termes d'existence pour d'autres etres. Le deuxieme theme est l'unite profonde des mathematiques, invisible au temps ou l'Analyse dominait, visible avec la revolution introduite par Riemann, Hilbert et l'Algebre abstraite. Le troisieme est l'harmonie profonde entre theories mathematiques et developpements de la physique quantique comme de la Relativite. (shrink)

One of the most important philosophers of the twentieth century, Martin Heidegger was primarily concerned with the 'question of Being.' However, recent scholarship has tended to marginalize the importance of the name of Being in his thought. Through a focused reading of Heidegger's texts, and especially his late and often overlooked Four Seminars, Richard Capobianco counters this trend by redirecting attention to the centrality of the name of Being in Heidegger's lifetime of thought. Capobianco gives special attention to Heidegger's resonant (...) terms Ereignis and Lichtung and reads them as saying and showing the very same fundamental phenomenon named 'Being itself '. Written in a clear and approachable manner, the essays in Engaging Heidegger examine Heidegger's thought in view of ancient Greek, medieval, and Eastern thinking, and they draw out the deeply humane character of his 'meditative thinking.'. (shrink)

Martin Heidegger's Contributions to Philosophy reflects his famous philosophical "turning." In this work, Heidegger returns to the question of being from its inception in Being and Time to a new questioning of being as event.