[FOM] Badiou review:Number & Numbers
joeshipman at aol.com
Mon Aug 31 18:17:37 EDT 2009
I have read the review linked below, and Badiou's book. I agree with
Kadvany that Badiou has a consistent philosophical vision and a correct
understanding of the relevant mathematics.
Kadvany's criticises Badiou's anti-constructivism but that does not
strike me as a serious problem -- Badiou is not trying to derive his
anti-constructivism from his mathematical development; you have to
accept his idealism as a presupposition not something he is claiming to
prove. Rather, he is concerned to show how rich a mathematical
ontology flows from simple idealistic presuppositions, and thereby to
establish mathematics (at least the "Number" part of it, comprising
number theory, real analysis, and the theory of ordinals and cardinals)
as part of philosophy.
Other philosophers have attempted to do this through logicism, but
according to some critics (see Burgess, "Fixing Frege") this project
doesn't get far enough. I have more hopes for logicism (as
traditionally practiced, starting from rudimentary arithmetic and
moving toward set existence axioms) than Burgess does, but Badiou comes
from the other direction, following Conway and Gonshor to get numerical
plenitude from pure being.
Badiou is impressed that Godel and Cohen have established the inability
of ZFC to settle the size of the continuum, but he is not disturbed at
such human limitations, and does not feel that this somehow
ontologically illegitimizes higher infinities. One may reject his
idealistic philosophy on pragmatic or other grounds, but his book is a
valuable contribution to the philosophy of mathematics.
From: John Kadvany <jkadvany at sbcglobal.net>
FOM readers may find my review of Alain Badiou's Number and Numbers of
interest, at Notre Dame Philosophical Reviews.
I focus on Badiou's use of Conway's surreal numbers, ZF, and Badiou's
account of logic and number in Peano, Frege et al. I cast Badiou as
something of an anti-constructivist. The review is sympathetic but
from that foundational perspective.
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