I had been hoping for some time that someone would write a review for this long-awaited translation. Unfortunately none has appeared and until a more comprehensive and useful review is written, I hope these brief comments will help.
(A brief disclaimer. This review does not summarize or critique the arguments in this book--it would be unjust to attempt to do so in the space of a few paragraphs. I hope only to give some indication of the relevance of this work for those who are interested in Badiou's work and/or those who have heard the name "Badiou" and are trying to find a way in to what his work is all about. If my comments are elliptic or obscure because I use Badiou's terms without providing explication, this is only because I hope that I give enough indication of the direction of his ideas to promote the reading of the actual text.)
Unfortunately, I cannot comment on the quality of the translation, since I have not seen the French text. Feltham's familiarity with Badiou's work is unquestionable, however. He was, for example, one of the editors of the collection "Infinite Thought" (also published by Continuum). He has also contributed to a recent issue of `Polygraph' devoted to a discussion of Badiou's work (#17, 2005).
Until this translation, American readers were denied significant access to Badiou's philosophical method and concepts. The key sources were commentaries by people like Peter Hallward, Keith Ansell-Pearson, and Eric Alliez (and, of course, Slavoj Zizek). The closest one got to Badiou himself was the collection called "Theoretical Writings" (also published by Continuum). With the exception of "Deleuze: The Clamor of Being", it was difficult to know what Badiou's work was all about since just about all of his other translated works presuppose knowledge of the concepts and terms developed in "Being and Event".
Those who have read Badiou's "Deleuze" will have some idea of what occupies "Being and Event". The title recalls, of course, Heidegger's "Being and Time", and Badiou explicitly agrees with Heidegger that philosophy can only be done on the basis of the ontological question. In "Deleuze", Badiou argues that that great thinker was at bottom a thinker of the One and, as Keith Ansell-Pearson points out, the real quarrel between Badiou and Deleuze is over who can speak of being as pure multiplicity. For Deleuze, the concepts are those found in Bergson and the differential calculus; for Badiou one must look to post-Cantorian set theory. In both cases, one cannot approach ontology without a firm understanding of mathematics (anyone who does not have a working grasp of set theory will not be prepared for "Being and Event").
The ontological question cuts a diagonal through various trajectories. Although Badiou accepts the gauntlet Heidegger threw down to philosophy, like Deleuze he thinks that ontology has to be done post-phenomenologically. Badiou even rejects the later Heidegger's notion of "forgetting". Badiou's answer to the ontological question involves a second project in "Being and Event": the articulation of a post-Cartesian (and even a post-Lacanian) subject. If, Badiou says, mathematics is ontology (that is, only mathematics can write being as it is, even if there is no intra-mathematical sense to this writing), the question is no longer the Kantian "how is mathematics possible?" but, rather, if mathematics is the science of being, how is a *subject* possible? In accord with his notion that there are four (and only four) "truth procedures", there are only artistic, scientific, political, and amorous subjects. It is on this idea that Badiou's other works on ethics, politics, art ("inaesthetic"), and so forth, are predicated. In a sense, none of Badiou's other translated works make much sense without the doctrine of the subject laid out in "Being and Event".
(This project of a post-Cartesian subject is announced by the book itself in that it is written as a series of "meditations" that could not be more dissimilar in method to the meditations of either Descartes or Husserl. My own hunch is that any successful engagement and/or refutation of Badiou's work will have to be done on the question of method--viz., Badiou's axiomatic procedure.)
These theses on ontology and subjectivity cross the so-called analytic-continential divide in philosophy. Badiou offers readings of major thinkers throughout the history of philosophy and his readers are asked to have a similarly encyclopedic knowledge of both the post-Kantian analytic and continental traditions. This book is most certainly neither for laypersons, amateurs, or beginning students of philosophy. Throughout the introduction Badiou expresses consternation over the fact that his readers must not only be professional philosophers, but also well-trained in mathematics. One is usually well-trained in one or the other. Analytic philosophy tends to do better at this than Continental (indeed, one of Badiou's goals is to provide a way out of the aporias of the Vienna Circle), but Badiou equally draws from the continental tradition (by way of figures like Hegel, Heidegger, and Lacan) and continental readings of the history of philosophy. (And, until "Being and Event", one couldn't really find much after Quine on the philosophy of set theory except something like Mary Tiles' work from 1989.)
The ontological argument, premised on what Badiou has to say about the One and the presentation of multiplicity (i.e., the question that preoccupied the presocratics) hinges on this: "maintain the position that nothing is delivered by the law of the Ideas, but make this nothing be through the assumption of a proper name. In other words: verify, via the excedentary choice of a proper name, the unpresentable alone as existent; on its basis the Ideas will subsequently cause all admissible forms of presentation to proceed. ... It is because the one is not that the void is unique ... [which is equivalent] to saying that its mark is a proper name". This is how Badiou interprets the axiom of the null (or void) set and distills the question of the One and Many from Being and change (see, e.g., the history and development of the concepts of the calculus). The question is not simply "how does one think non-being?" but also (and Parmenides also recognized this) "how does one name non-being?" The proper name, as Badiou points out in a passage immediately following the above, is not the transcendent God or the promise of the One or presence but the "un-presentation and the un-being of the one" (cf. Derrida's comments on the possibility of a negative theology).
The payoff for working through Badiou's text is nothing less than a revitalization of philosophy (particularly for anyone who thinks philosophy in America has been boring since the waning of Rortyian pragmatism). The ontological debates surrounding Deleuze/Badiou have tended to be conducted in the margins of philosophical discourse in the US (with both thinkers more popular in circles of theory than philosophy and in the pages of journals on culture and politics than Nous or Mind), but the publication of "Being and Event" itself is precisely what Badiou means when he writes of an "event": something that disrupts the current situation. ("Event" and "situation" are, of course, technical terms for Badiou. The most succinct statement of these terms is probably "The Event as Trans-Being" in the Theoretical Writings.) Like his compatriot Ranciere (who too found his own voice after breaking with a youthful Marxism), Badiou is concerned with how it is possible that something new can be seen. "Being and Event" is compulsory for anyone who thinks ontology has been boring since Heidegger (even Millan-Puelles' ambitious "Theory of the Pure Object" fails to satisfy); and for those who weren't convinced by Deleuze that alternative ways to do ontology (viz. Bergson) were dead-ends, "Being and Event" the place to turn. (Whether one ultimately agrees with Deleuze or Badiou, however, is an open question. The basic difference is this: for Badiou, multiplicities are rigorously determined; Deleuze, obviously, denies this. In both cases being is pure multiplicity, nondenumerable, etc)
And for those who may be interested by Deleuze but are wedded to more traditionally analytic ways of writing: Badiou's writing is often praised for its clarity and in many ways it mimes the economy of analytic philosophy, avoiding the obscurity (while preserving the density) of many of his French contemporaries. Badiou has often been compared to Sartre (both being novelists and playwrights in addition to philosophers), but not only does Badiou in many ways stand apart from the French traditions of Sartre and Hyppolite, "Being and Event" is eminently more readable than "Being and Nothingness". Even if Badiou's writing lacks the brilliance of Derrida or Deleuze, this may be because he explicitly tells us that the poetic is subordinate; indeed, Badiou's writing itself is probably best described as "mathematical". While he is not immune to some amount of obscurity in some others of his writings, "Being and Event" certainly cannot be so faulted. At worst one might fault the author for demanding too much of his reader; but if this be a fault it is an admirable one to have, since it is a rare author indeed who can make such a demand.