Digital is discrete and discontinuous. Analogue
is continuous (more or less). Real numbers and
rational numbers are continuous. Whole numbers
are not. They are discrete and discontinuous. Of
course, much of this is a matter of perceptual levels.
Things look continuous at one level (perception of
a movie) when they are discontinuous or discrete at
another level (the successive images in the film). Is
one looking at the trees or at the forest?
At the ultimate level we do not know whether reality
is discrete or continuous. Quantum mechanics suggests
discreteness, but string theory suggests a deeper level
of continuity. But then, we do'nt know whether string theory
is true or even how to test it, at least not now.
Barkley Rosser
-----Original Message-----
From: Charles Brown <[EMAIL PROTECTED]>
To: [EMAIL PROTECTED] <[EMAIL PROTECTED]>
Date: Tuesday, May 30, 2000 4:02 PM
Subject: [PEN-L:19761] Re: Re: Re: Dialectical materialism and ecology
>
>
>>>> "J. Barkley Rosser, Jr." <[EMAIL PROTECTED]> 05/25/00 11:27AM >>>
> For those who are curious, I have a recently published
>paper on these issues.
>"Aspects of dialectics and non-linear dynamics," _Cambridge
>Journal of Economics_, May 2000, vol. 24, no. 3, pp. 311-324.
> It is also available on my website without the figures at
>http://cob.jmu.edu/rosserjb.
>Barkley Rosser
>
>(((((((((((((((
>
>Barkley,
>
>Related to mathematically modelling qualitiative change, how about the
integers , counting numbers or whole numbers ? In 1,2,3,4,5... aren't there
discontinuities, leaps, on the real number line ?
>
>
>Charles
>
>))))))))))))))))))
>
>III. Catastrophe Theory and Dialectics
> The key idea for analyzing discontinuities in nonlinear dynamical
systems is bifurcation, and was discovered by Poincaré (1880-1890) who
developed the qualitative theory of differential equations to explain
more-than-two-body celestial mechanics. Consider a general family of n
differential equations whose behaviour is determined by a k-dimensional
control parameter m, such that
>
> dx/dt = fm(x); x 0 Rn, m 0 Rk, (1)
>
>with equilibrium solutions given by
>
> fm(x) = 0. (2)
>
> Bifurcations will occur at singularities where the first derivative of
fm(x) is zero and the second derivative is also zero, meaning that the
function is not at an extremum, but is rather at a degeneracy. At such
points structural change can occur as an equilibrium can bifurcate into two
stable and one unstable equilibria.
> Catastrophe theory involves examining the stable singularities of a
potential function of (1), assuming that there is a gradient. Thom (1975A)
and Trotman and Zeeman (1976) determined the set of such stable
singularities for various dimensionalities of control and state variables.
Arnol?d, Gusein-Zade, and Varchenko (1985) generalized this analysis to
higher orders of dimensionalities. These singularities can be viewed as
points at which equilibria lose their stability with the possibility of a
discontinuous change in a state variable(s) arising from a c
>
>
