[FOM] Badiou review:Number & Numbers

joeshipman@aol.com 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.

-- JS

-----Original Message-----
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.

  John Kadvany


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