Mathematics of the Transcendental: Onto-logy and being-there

by Alain Badiou

In Mathematics of the Transcendental, Alain Badiou painstakingly works through the pertinent aspects of Category Theory, demonstrating their internal logic and veracity, their derivation and distinction from Set Theory, and the 'thinking of being'. In doing so he sets out the basic onto-logical requirements of his greater and transcendental logics as articulated in his magnum opus, Logics of Worlds. This important book combines both his elaboration of the disjunctive synthesis between ontology and onto-logy (the discourses of being as such and being-appearing) from the perspective of Category Theory and the categorial basis of his philosophical conception of 'being there'.

Hitherto unpublished in either French or English, Mathematics of the Transcendental provides Badiou's readers with a much-needed complete elaboration of his understanding and use of Category Theory. The book is an essential aid to understanding the mathematical and logical basis of his theory of appearing as elaborated in Logics of Worlds and other works and is essential reading for his many followers.

Genres: Philosophy, Mathematics

296 pages, Hardcover

First published November 2, 2013

About the author

Alain Badiou, Ph.D., born in Rabat, Morocco in 1937, holds the Rene Descartes Chair at the European Graduate School EGS. Alain Badiou was a student at the École Normale Supérieure in the 1950s. He taught at the University of Paris VIII (Vincennes-Saint Denis) from 1969 until 1999, when he returned to ENS as the Chair of the philosophy department. He continues to teach a popular seminar at the Collège International de Philosophie, on topics ranging from the great 'antiphilosophers' (Saint-Paul, Nietzsche, Wittgenstein, Lacan) to the major conceptual innovations of the twentieth century. Much of Badiou's life has been shaped by his dedication to the consequences of the May 1968 revolt in Paris. Long a leading member of Union des jeunesses communistes de France (marxistes-léninistes), he remains with Sylvain Lazarus and Natacha Michel at the center of L'Organisation Politique, a post-party organization concerned with direct popular intervention in a wide range of issues (including immigration, labor, and housing). He is the author of several successful novels and plays as well as more than a dozen philosophical works.

Trained as a mathematician, Alain Badiou is one of the most original French philosophers today. Influenced by Plato, Georg Wilhelm Friedrich Hegel, Jacques Lacan and Gilles Deleuze, he is an outspoken critic of both the analytic as well as the postmodern schools of thoughts. His philosophy seeks to expose and make sense of the potential of radical innovation (revolution, invention, transfiguration) in every situation.

Reviews

[Benjamin · Rating 4 out of 5]

A dense, difficult and possibly very beautiful piece of philosophy. No politics, no art, no lengthy discussions of poetry; just pure maths.

I can only speak from my own experience. I found the first book to be incredibly difficult to follow, despite his typical claim that anyone with basic mathematical skills can navigate his work (French schools have a rigour that English ones don't). I picked up very little from it and found it hard to understand. However the second book was clear and, though not simple, certainly manageable for anyone with any training in logic. I found the exercises impossible to complete but could follow their explanations and the whole conceptual framework gradually fell together.

Probably one for die-hard Badiou fans or those who study his work. Not for those who fancy dipping in.

[J. · Rating 5 out of 5]

Badiou as educator: MotT is a beautiful - and probably thankless - work of elucidation through metaphorical transfer.

[Himanshu Damle · Rating 5 out of 5]

Alexandar Grothendieck, possibly the greatest post-WW2 mathematician died a hermit at the age of 86 in 2014 in France. He emancipated mathematics from the Euclidean tyranny of statics in pushing the discipline into the fluidity of relativity and probability and imaging a synthetic unification of the branches of algebra, geometry and topology. Spending a significant part of his latter life in near-total desolation and delirium, his memoirs 'Harvests and Sowings' published eight years posthumously in 2022 has rekindled an interest in his Topos Theory, considered by many as a cardinal platform for the NextGen AI. Grothendieck introduced the eponymous category as a kind of Abelian Category that helped homological algebra in unifying the modules and sheaves into algebraic geometry, the very language of complex mathematical structures.

The Grothendieckian Category is employed by the Colombian Mathematician and Philosopher, Fernando Zalamea in promulgating transitory ontology, almost congruent with the French mathematician moving from statics into the flux by way of morphisms or cartographies. This movement is a movement away from the traditional set-theoretic (Zermelo-Frankael Set Theory) construct towards redefining the philosophical notions of being and reality by focusing on transformations and relationships. Zalamea has clinically explored Alain Badiou's 'Being and Event', 'Logics of World ', and the 'Immanence of Truths' by an elementary Peircean perspective and categoric-theoretic reading of the Trilogy specifying the dialectics along the way.

Badiou's Category Theory is covered in Bartlett and Ling's putting into place battered photocopies of Saturday seminars in the 1990s and published under the title, 'Mathematics of the Transcendental'. This is a technically demanding work that eventually puts into parallel the logics of Topos Theory with Badiou's earlier Mathematics is Ontology claim drawn from set theory's assertion of pure multiplicity. The book is a reflection of softening of Badiou's adamantine mistrust in Category Theory to a realisation of its potential in providing a structure capable of accounting for the varied intensities of appearing, or in other words, the capacity to formalize appearing itself. This in no way is suggestive of two exclusive phases in Badiou's trajectory, but rather are complementary, in that, the later phase takes the philosophical journey from the intrinsic to the extrinsic, or from Aristotlean 'being qua being' as pure multiplicity to being-there, coextensive with being itself as the localisation of a being, or of its appearance in the world.

Spread over two parts, viz. Topos; and Being There, the lectures are heavy in diagrams that show the shape of definition spatially (Category is defined that way, ain't it?) while transferring the weight heavy-handedly to Being's preventative localisation, whose space is the situation to which it belongs, or Da-sein. The latter part of the book is discursive on the seizure in a situated relational network, such that this being is more or less different from another multiple that belongs to the same situation. The book, of which the first half is relatively easier, is still not for the uninitiated. A grounding in Aristotle, CS Peirce, Grothendieck, and Zermelo-Frankael (as well as Cantor) is necessitated to an extent. Mathematics of the Transcendental is transitory from the ontology of Being and Event, and therefore indispensable.

[Jeff · Rating 3 out of 5]

Badiou effectively conveys what the philosophical stakes are in the mathematical areas he's working in (category theory, topology), and in doing so provides an entrance point into grappling with some of the ideas and problems of higher level math (beyond set theory). For instance: how do you account for the existence of transcendental logical structures (truth-evaluations) within an immanent system of objects and operations? How do you translate the logical coherence of sets (which depends on an overarching framework of global consistency) into other spaces that are structured by more 'local' (immanent/emergent) structures?

That said, the actual proofs and accounts of the underlying mathematical reasoning were often far too terse and under-developed for readers without higher level math. (The implications of) crucial definitions or premises were tossed off in the course of long, complex proofs in a way that made them extremely difficult to turn back to later. In some cases, I was only able to make sense of certain proofs by concluding that lines I didn't follow must contain typos, or that I'd needed to assume something I hadn't been instructed to.

(Having tried to read actual works of higher math before too, I gather this is typical of how mathematics is written, not a specific limitation of Badiou's. I wish it weren't. Of course, I realize that math can involve extreme degrees of abstraction that cannot be expressed exhaustively in ordinary language, but the field's apparently obsessive focus on precision, elegance, and above all concision at the expense of making itself understood to larger audiences is still alienating and disappointing.)