Amara Graps wrote:
>From: Lee Corbin <lcorbin@ricochet.net>
>
>> If you ask anyone who is reading this what a good mathematical
>> model of the conscious development in real time of an organism
>> should be, they'll sooner or later point you in the direction
>> of non-linear dynamics (chaos theory), because of the way that
>> our lives, as well as so many other physical phenomena, exhibit
>> features not found in differential equations.
>
>This makes no sense to me. Differential equations are one of the main
>mathematical handles for linear and nonlinear physics, as well as for
>every other physical approximation that I'm familiar with. Even if
>your differential equations computes into a coupled system of a
>million-by-million eigenvalue matrix, the mathematical formulation of
>your differential equations is still valid.
We have since the days of the Bernoullis been formulating differential
equations to describe a lot of physical phenomena. Whether it's heat
flow in solids (Fourier's big breakthrough) or cables hanging in
catenaries, using the language of mathematics to express infinitesimal
constraints has been extremely productive. But it is only in this
arena, important as it is, that differential equations are of any use.
>Are you maybe mixing "continuous" and "discrete" ??
No. Discrete mathematics is less amenable to this sort of "global"
technique. Sure, there are difference equations in finite mathematics
that resemble differential equations, and we surely use algebra to
specify constraints. But no one has ever seriously proposed, for
example, that solutions to chess problems might be facilitated by
equations. Nor do quantitative equations illuminate axiomatic
developments. And I hope that no one has tried to frame
"intelligence" in terms of differential equations.
In my original post of Thursday May 10, from which the above
quote was taken, I tried to explain more fully why computation,
which, since the days of Turing we've come more and more to see
as fundamentally discrete, can't be straitjacketed by differential
equations. Its very nature, wonderfully and visually epitomized
by Conway's Life and other cellular autonoma, seems significantly
different. The closest that I myself have come to capturing the
essential difference is what I mentioned yesterday in this thread,
namely that from a computational perspective, the myriads of intermediate
results necessary in the numerical solution of differential equations
are merely approximations to the mathematical truth. But in iterative
calculations of this other kind, e.g., Life, chess, Turing machines,
and the disjointed details of our lives on Earth, the necessary
intermediate results **are** the "mathematical" truths themselves.
Lee
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