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Wow. Absolutely god dmaned right. But Badiou my dear, never get out of the boat, or you might get eaten by tigers...
Here is the problem with the deep platonism of this idea. Crudely put it was destroyed by Goedel's dual theorems on consistancy and completeness. This in effect killed the most simple sort of ideal statements about number and being as the great castle of eternal and rational truth. Remember we have had a conceptual unity about number and being since at least Pythagorus.
On a different level ignoring these details, understanding some of the conceptual forms within mathematics helps a great deal to understand the nature of our techno-scientific ontological premises, and the nature of the kinds of minds we inhabit.
``His major work, Being and Event, kicks off with this stark assertion, and proceeds to derive a series of bold conclusions – the wresting of ontology from Heidegger’s embrace, the construction of a rigorous and rationalist metaphysics..''
I'll have to read it, but this conflict was part of the famous Davos dust-up between Cassirer, Heidegger, and Carnap. I'll have to go over that again too. Roughly to sum that up: a neo-Kantian analysis of symbolic forms v. existantial analysis of being in the world, a logio-mathematical analysis of thought and language.
One of the more important spin-offs of these discussions is found in the strange problem of the mind-language seen in some of the current survey of work in biological-linguistic studies Kenneally does. I still haven't finished it either.
Here's the basic problem. You have to have theory of mind in order to do empirical work on the nature of mind. And most of the researchers in biological sciences and linguistics do not have a theory of mind. The way I imagine this goes, is you have to find room for both mathematics and arts, science and animal behavior. Your concepts have to be foundationally comprehensive enough to not be exclusionary. That's a pretty tall order.
Okay, I just read the article, which is a review of Badiou's two books. I am familiar with all these subjects in my own junior level.
If you want to make the claim that mathematics is ontology, then the best place to start is not with number, but with mathematic's other representation as space. (This was Cassirer's brillant move.) This leads to point-set topology and its analog of combinatorial topology, downward into geometries and current physical theory. This in turn leads to all the most wild of current cosmological theory, where the unity between philosophy, mathematics, and the physical cosmos join up with pre-socratic concepts of the Cosmos, all that is. (BTW, I have another book I haven't gotten more than fifty pages, Roger Penrose, The Road to Reality. Penrose also starts off with space.)
So then getting back to space. You can see that space is a concept that can be shared with a great variety of animals, since we all have to navigate our environmental spaces, and we have evolutionary adaptations of great variety of anatomical systems to do that. The consequence is that space-nagivation is part of the foundation of a theory of mind.
If you want to develop an anti-capitalist theory of the political economy, then space comes in remarkably handy, since the very first act of a capitalist is to expropriate land through money lending, Land is the material embodiment of space, the space we all must live in to reproduce our society.
CG