After his free-swinging attack on me, I asked Zackmann to send me more 
detailed criticisms of my defense of Deleuze against Sokal and Bricmont.  He 
did so.  I asked him to forward it to SCIENCE AS CULTURE and he attempted to, 
but it was returned as it was too long (10 pages).  It contains interesting 
stuff on chaos theory and Bohm and non-standard analysis, among other things, 
even if some his criticisms really aren't of what I said, so I though it 
worth forwardsin

<<You asked for it, so here IT is: (I) Val Says: Obviously Deleuze is no 
mathematical virtuoso, but his treatment of the issues of the calculus is far 
more detailed, informed and serious than Sokal and  Bricmont let on.

DZ: Ah yes, how Zen-like of you -- the old mystical Superstitions again -no? 
But on with my  response. Now as to Deleuze whom you keep obtusely defending 
when there is marvelous work on the philosophy of science out there to be 
assimilated, we get from Deleuze (and Guattari) the absolutely and thoroughly 
laughable claim that the  first difference between science and philosophy is 
that philosophy deals with concepts [simpliciter??? I do ask.] while science 
deals with functions [a manifest _conflation_ of science with mathematics. 
More Postmodernist rubbish.] Here are how our two Postmodernist Glitterati 
whom Val Dusek seems hell-bent to defend characterize this truly bizarre claim

"[THE] first difference between science and philosophy is their respective 
attitude toward chaos. Chaos is defined not so much by its disorder as by the 
infinite speed with which every form taking shape in it vanishes." (from 
"What is Philosophy" by Deluze and Guattari, New York, Columbia University 
Press, 1994) - as an aside I would like to ask WHY such a prestigious 
university as Columbia would CON-descend to publish such utter drivel? Guys, 
I REALLY want an answer to this one!!!

DZ:  WRONG: Not only is this NOT a definition of the essence of mathematical 
chaos, such an assertion does not even attempt to give us sufficient 
conditions for  the manifestation of chaos in natural and artificial 
phenomena -- in the technical sense of the word used in mathematics,= 
physics, and science generally.  Moreover, very many branches of mathematics 
have NOTHING WHATSOEVER to do with fractals, chaos, or nonlinear dynamics.

N.B.!: For all interested parties, and nontechnically put, the sufficient 
conditions for chaos are as follows: (i) nonlinearity, which can be the 
result of a large initial displacement away from equilibrium or feedback 
within the system, (ii) energy dissipation, and (iii) an external driving 
force.

REALITY CHECK:  Back to Rigorous Mathematics

To be even more precise, Chaos may be more rigorously defined as follows:

Chaos is APERIODIC long-term behavior in a deterministic system that exhibits 
sensitive dependency on initial conditions (assuming continuum mathematical 
models in this context).

(A) "Aperiodic long-term behavior" means that there are trajectories [of the 
3 or more systems of ODEs] which do not settle down to fixed points, periodic 
orbits, or quasiperiodic orbits as t-->+Infinity. For pragmatic purposes, we 
should require that such trajectories be not too rare. For instance, we could 
insist that there be an open set of initial conditions leading to aperiodic 
trajectories, or perhaps that such trajectories should occur with P(E) = 0, 
given random initial conditions.
(B) "Deterministic" means that the system has no random or noisy inputs or
parameters. The irregular behavior arises from the system's intrinsic 
nonlinearity, rather than from simply noisy driving forces.
(C) "Sensitive dependence on initial conditions" means that nearby 
trajectories separate exponentially fast: That is, that the system has a 
positive Liapunov exponent.

Clearly what I have been mostly describing above is Chaos which can only 
occur in systems of 3 or more nonlinear ODEs, while postponing discussing 
Chaos which can occur in nonlinear PDEs and iterated maps.  Moreover there is 
a myth that Chaos can occur in 2 dimensional Phase Space, but this contention 
is refuted by The  Poincare´ - Bendixson theorem which states that if a 
trajectory is confined to a closed, bounded region, and there are NO fixed 
points in the region, then the region must eventually approach a closed 
orbit. In short, no Chaos in 2 dimensional Phase space under these conditions.

The Poincare-'Bendixson Theorem

Suppose that

(1) R is a closed bounded subset of the plane,

(2) dX/dT = F(X) -- where X is a vector function of a vector variable, and so 
is F -- is  a C^(1) vector field on an open set containing R,

(3) R does not contain any fixed points and,

(4) There exists a trajectory C that is confined in R in the sense that it 
starts in R and stays in R for all future time.

THEN, either C is a closed orbit, or it spirals toward a closed orbit as 
t--->+Infinity. Consequently, R contains a closed orbit and we have NO chaos 
in 2 dimensional phase space under these conditions.>>

I thank Zackmann for this concise exposition of chaos and the lack of chaos 
in 2 dimensions.  However with respect to what I said

1.) I granted that stuff Deleuze wrote with Guattari was pretty crazy and 
concentrated on works he wrote by himself.

2.) The chaos Deleuze and Guattari are talking about in the book quoted above 
is old-fashioned molecular chaos, ie. Brownian motion, not the post- 1970 
modern use of chaos in chaos theory.  As I recall, Gabriel Stolzenberg 
(probably on the STS list) wrote that he pointed this out to Sokal, who 
granted it, but later wrote in the article in Koertge's "House Built on Sand" 
as if the chaos talked of was chaos theory, rather than old-fashioned 
molecular chaos.

Val Dusek