Another review of Number and Numbers:
http://ndpr.nd.edu/review.cfm?id=14345
Despite j.fisher's reassurance, I have been uneasy about this chap since he seems to be espousing a dressed up version of Quine's views (as the above review sort of suggests), the chief attraction of which (Quine's ideas) is their minimalism i.e., if you can get over the stuffy language.
But the review reveals to me that Badiou was one of Sokal's targets. That alone casts him in a better light to my eyes!
From the review:
> It's not that being ismathematical, but that mathematical discourse
> "pronounces" what is expressible of "being qua being." Theories of
> anything, but mostly the natural and social world as described using
> numeric methods, are re-presentations of this ontology. Consistent
> with the primacy of natural science, numbers and numeric structure
> have to be "immanent" (76, 101, 177), and especially, not
> "constructed" via syntax, grammar or other inductive procedures, of
> which Badiou is completely disdainful: "if it is true that
> mathematics, the highest expression of pure thought, in the final
> analysis consists of nothing but syntactical apparatuses, grammars
> of signs, then a fortiori all thought falls under the constitutive
> rule of language".
While I share his worship of mathematics, I find myself in no rush to claim that it *is* ontology to avoid reducing it to syntactical apparatuses, etc. Am I wrong?
j, you write that we should gain our impressions by reading the original material, but Number and Numbers is really tough material -- I have read Wittgenstein, Heidegger, philosophers like that, and its equally tough going, but mathematics is a space that has always defined clarity, so it’s a mental block as well to read someone talking about mathematics, and using mathematical concepts, but in some hitherto unseen ways! ;-) Well not really... Wittgenstein does the same sort of thing in his Cambridge lectures and I find myself reluctantly nodding in puzzlement with Turing (but not Sokal, I hope!).
Like take this part from the review that describe the first part of Badiou's book:
> The set theoretic universe just doesn't look much like the ordinary
> real number line, and this Badiou dislikes. In particular, Cantor's
> transfinite ordinals are discrete, unlike the real numbers. The
> symbol "ω" is for the first transfinite ordinal, and in set theory
> it is defined as the set {0, 1, 2, . . .}. You can add 1 to ω to get
> ω + 1, which is just the set {0, 1, 2, . . . , ω}, i.e. ω + 1 = ω
> ∩ {ω}. But there's no ordinal a between ω and ω + 1, just as
> there is no counting number nbetween 2 and 3, or 100 and 101, even
> though there are lots of such real numbers, like 2.002 or
> 100.101100111 . . . . There is some algebra for calculating with
> transfinite ordinals, but not like the algebra for rational and real
> numbers. For Badiou, Cantor's great insight, to generalize just the
> discrete counting numbers into the transfinite, is ultimately an
> ontological flaw.
This is a strange way to describe the history of the mathematical developments of the time, and even Cantor's project. Transfinite numbers were a natural consequence, AFAIK, of issues raised by consideration of countability and so on. However, their discrete nature is very much a source of worry -- it’s the very same Cantor who authored the continuum hypothesis, which Hilbert gave prominence in his list of 23. And it was 60+ years later that Cohen demonstrated that the continuum hypothesis remains a tricky customer and is unprovable from ZFC. In fact, Badiou seems to be following the same Platonist road as Gödel in being dissatisfied with the continuum hypothesis.
OTOH, it seems like Badiou is a big fan of forcing -- perhaps Mart can comment on this? Especially w.r.t whether Cohen's results had anything to say that presage the implications drawn by Badiou.
--ravi
-- Anyone who takes an effort to intellectually challenge the status quo and established habits is infinitely more venerable than hacks defending that status quo and established habits, regardless of the truth function of their propositions. -- W.Sokolowski