Sorry, but I get very annoyed when I receive text in this kind of format! Maybe it has something interesting to say, but I get blinded by the disjointedness of the thing and find it impossible to read.


>After his free-swinging attack on me, I asked Zackmann to send me more=20
>detailed criticisms of my defense of Deleuze against Sokal and Bricmont.  He=
>=20
>did so.  I asked him to forward it to SCIENCE AS CULTURE and he attempted to=
>,=20
>but it was returned as it was too long (10 pages).  It contains interesting=20
>stuff on chaos theory and Bohm and non-standard analysis, among other things=
>,=20
>even if some his criticisms really aren't of what I said, so I though it=20
>worth forwardsin
>
><<You asked for it, so here IT is: (I) Val Says: Obviously Deleuze is no=20
>mathematical virtuoso, but his treatment of the issues of the calculus is fa=
>r=20
>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?=
>=20
>But on with my  response. Now as to Deleuze whom you keep obtusely defending=
>=20
>when there is marvelous work on the philosophy of science out there to be=20
>assimilated, we get from Deleuze (and Guattari) the absolutely and thoroughl=
>y=20
>laughable claim that the  first difference between science and philosophy is=
>=20
>that philosophy deals with concepts [simpliciter??? I do ask.] while science=
>=20
>deals with functions [a manifest _conflation_ of science with mathematics.=20
>More Postmodernist rubbish.] Here are how our two Postmodernist Glitterati=20
>whom Val Dusek seems hell-bent to defend characterize this truly bizarre cla=
>im
>
>"[THE] first difference between science and philosophy is their respective=20
>attitude toward chaos. Chaos is defined not so much by its disorder as by th=
>e=20
>infinite speed with which every form taking shape in it vanishes." (from=20
>"What is Philosophy" by Deluze and Guattari, New York, Columbia University=20
>Press, 1994) - as an aside I would like to ask WHY such a prestigious=20
>university as Columbia would CON-descend to publish such utter drivel? Guys,=
>=20
>I REALLY want an answer to this one!!!
>
>DZ:  WRONG: Not only is this NOT a definition of the essence of mathematical=
>=20
>chaos, such an assertion does not even attempt to give us sufficient=20
>conditions for  the manifestation of chaos in natural and artificial=20
>phenomena -- in the technical sense of the word used in mathematics,=3D=20
>physics, and science generally.  Moreover, very many branches of mathematics=
>=20
>have NOTHING WHATSOEVER to do with fractals, chaos, or nonlinear dynamics.
>
>N.B.!: For all interested parties, and nontechnically put, the sufficient=20
>conditions for chaos are as follows: (i) nonlinearity, which can be the=20
>result of a large initial displacement away from equilibrium or feedback=20
>within the system, (ii) energy dissipation, and (iii) an external driving=20
>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 exhibit=
>s=20
>sensitive dependency on initial conditions (assuming continuum mathematical=20
>models in this context).
>
>(A) "Aperiodic long-term behavior" means that there are trajectories [of the=
>=20
>3 or more systems of ODEs] which do not settle down to fixed points, periodi=
>c=20
>orbits, or quasiperiodic orbits as t-->+Infinity. For pragmatic purposes, we=
>=20
>should require that such trajectories be not too rare. For instance, we coul=
>d=20
>insist that there be an open set of initial conditions leading to aperiodic=20
>trajectories, or perhaps that such trajectories should occur with P(E) =3D 0=
>,=20
>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=20
>nonlinearity, rather than from simply noisy driving forces.
>(C) "Sensitive dependence on initial conditions" means that nearby=20
>trajectories separate exponentially fast: That is, that the system has a=20
>positive Liapunov exponent.
>
>Clearly what I have been mostly describing above is Chaos which can only=20
>occur in systems of 3 or more nonlinear ODEs, while postponing discussing=20
>Chaos which can occur in nonlinear PDEs and iterated maps.  Moreover there i=
>s=20
>a myth that Chaos can occur in 2 dimensional Phase Space, but this contentio=
>n=20
>is refuted by The  Poincare=B4 - Bendixson theorem which states that if a=20
>trajectory is confined to a closed, bounded region, and there are NO fixed=20
>points in the region, then the region must eventually approach a closed=20
>orbit. In short, no Chaos in 2 dimensional Phase space under these condition=
>s.
>
>The Poincare-'Bendixson Theorem
>
>Suppose that
>
>(1) R is a closed bounded subset of the plane,
>
>(2) dX/dT =3D F(X) -- where X is a vector function of a vector variable, and=
> so=20
>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=20
>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=20
>t--->+Infinity. Consequently, R contains a closed orbit and we have NO chaos=
>=20
>in 2 dimensional phase space under these conditions.>>
>
>I thank Zackmann for this concise exposition of chaos and the lack of chaos=20
>in 2 dimensions.  However with respect to what I said
>
>1.) I granted that stuff Deleuze wrote with Guattari was pretty crazy and=20
>concentrated on works he wrote by himself.
>
>2.) The chaos Deleuze and Guattari are talking about in the book quoted abov=
>e=20
>is old-fashioned molecular chaos, ie. Brownian motion, not the post- 1970=20
>modern use of chaos in chaos theory.  As I recall, Gabriel Stolzenberg=20
>(probably on the STS list) wrote that he pointed this out to Sokal, who=20
>granted it, but later wrote in the article in Koertge's "House Built on Sand=
>"=20
>as if the chaos talked of was chaos theory, rather than old-fashioned=20
>molecular chaos.
>
>Val Dusek