Badiou's Gauntlet, the challenge that Badiou issues to any kind of philosophy, is that the categories are three. No more than three, but also no less than three. Badiou's Gauntlet is that there are bodies and languages, but also truths.
The challenge permeates all of Badiou's work. One particularly clear expression of it comes at the start of Logics of Worlds, where Badiou differentiates cleanly between the two-category thinkers and the three-category thinkers. The two-category thinkers claim that there are only bodies and languages, only two basic categories of existence. These “bodies and languages” may appear under any number of alternate names. Some philosophers talk about objects and relations, others about extension and thought. Some contemplate matter and mind, others wrestle with things and information. The names are not as important as the commitment they entail, that there are only two basic things in the world, only real entities and the languages through which they interrelate.
As a three-category thinker, Badiou claims that there are three co-original categories, a trio of aspects not simply a duality. Hence not just bodies and languages but also truths. Not just objects and relations but also change. Not just matter and mind but also becoming. Not just things and information but also process.
(It's telling that philosophers have disciplinary labels for the first two -- ontology and epistemology -- but no good names for the third. Praxis? Revolution? Paradigm shift? Theory of the subject? Philosophers typically account for the third thing inside ontology or epistemology. Thus it's not uncommon to say that being and becoming are both ontological questions. Or to say that process is a form of relation, and hence an epistemological question. But a canonical science of the event, this Badiou has to fight for.)
Using somewhat infelicitous terminology, Badiou labels the first camp “democratic materialism” and the second camp “dialectical materialism.” Although later in Logics of Worlds he uses the concept of atone (“atonal” or “flat”) and tendu (“tense” or “taut”) to mean something similar. In Kuhnian language the two modes would be called “normal science” and “revolutionary science.” Or we might use terms like neutral/neutralizing for the former, and transformational for the latter.
"There are only bodies and languages, except that there are truths" (en4, fr12). To be clear, the three categories are not equivalent, and the conjunction indicates exception rather than series or similarity. To say “being and event” really means “there is being but there are also events” or “events are exceptions in being.” And, given time, much more could be said about the rhetoric of exception that exists in this section and throughout the book (on pages 4, 6, 32, 45, 360, and passim).
In sum, Badiou's Gauntlet means you are shortsighted if you assert only two categories. If all you say is “bodies and languages” and no more, then you've failed. Any real philosophy must account for bodies and languages, but also truths/events. (Corollary: you are also misguided if you assert only one category: everything is an object, everything is a mind, everything is made of cheese, etc. Thus, while another portion of Badiou's work hinges on the assertion that everything is a set, he also wants to deny that this constitutes a form of ontological monism.)
The last thing I'll add has to do with the concept. I'm an avid reader of Badiou and think his work is endlessly fascinating and important. And for anyone interested in the political, Badiou is certainly the most significant thinker to come along in some time -- I say political, since the ethical is another question altogether. At the same time there are a few fundamental issues that I have a hard time accepting. Chief among them are his use of mathematics and his affection for idealism, two characteristics that are closely related. I've often thought that Marx's basic assessment of Hegel also applies to Badiou, that Badiou is standing on his head and needs to be put back on his feet again. In other words, we need a fully materialist interpretation of Badiou today, just as when Marx put forth a fully materialist interpretation of Hegel in the mid Nineteenth Century. Such an interpretation of Badiou would suppress his yen for highly abstract mathematical formalism, bordering on a kind of mathematical divination, along with Badiou's self-described “Platonism of the multiple,” in favor of an evental materialism rooted in the real.
That said, I do think it somewhat irresponsible to veto Badiou on grounds of idealism, in a vain attempt to wash one's hands of the matter. Thus I find it most useful to consider Badiou's idealism in terms of the concept. Idealism is, at root, the science of subjectivity. And Badiou's science of subjectivity revolves around the concept. But what is a concept? Here we might differentiate between two recent definitions of the concept, the first by Deleuze & Guattari and the second by Badiou.
In What Is Philosophy? Deleuze & Guattari wrote that philosophy is the creation of concepts. Philosophy is not contemplation, reflection, or communication. It's not the “love of wisdom.” Rather, philosophy is the act of formulating new concepts, where “concept” means any kind of interesting structure furnished to thought. A concept might be “the body without organs,” or it might be “the affection-image.” Not merely synonymous with abstraction or generality as such, the concept is a multiplicity, a kind of assemblage or space of mixture where bodies flow and exchange. In this sense, “democracy” probably wouldn't be a concept for Deleuze & Guattari, who might rather frame the notion as “an assemblage of bodies and powers,” that is, as a concept.
If for Deleuze & Guattari the concept is an opening, for Badiou the concept is a condition. For him “concept” means a law, a promise, or a hope. Concepts are the things that persist, the things to which subjects are bound. And, in binding themselves to concepts, subjects graduate into full existence. If Deleuze's concept is about creativity and expression, Badiou's concept is fundamentally a question of law, limitation, and faithfulness.
Badiou is an “idealist,” yes, but only in the way that activists are idealists. They both use concepts to bind subjects to truths. Here “concept” means less “idea” than “ideal,” that is, a desired state of affairs. The activist is an idealist because he or she adheres to an ideal -- climate justice or the eradication of poverty -- and subordinates his or her action to it. Badiou's theory of the subject is identical. The subject adheres to an event and becomes a subject to truth. Thus the concept is the key to understanding Badiou's idealism.
Yes, this means that Badiou is a moralist of the highest order. But then again the political and the moral are, quite literally, the same thing.