Chaos is not a "theory"

P8475423 at vmsuser.acsu.unsw.EDU.AU P8475423 at vmsuser.acsu.unsw.EDU.AU
Wed Sep 20 22:27:34 MDT 1995


Jerry's reply to Lisa, which began:
Do not mathematicians have established ways of proving or disproving
axioms? Are there not established ways in all of the natural sciences
...
seems to me to be based on a lack of appreciation for the
incompleteness of maths. For example, anyone can "prove" the
solution to a quadratic equation (even those who hate maths
should be able to vaguely recall the quadratic formula taught at
elementary school). There's also a formula for cubics, quartics
(powers of 4) and quintics (x^5 etc.). But it has been proven
(using group theory) that no such formula exists for powers
above 5.

Likewise, Lisa mentioned differential equations as a tool of
dynamic analysis, and that the exploration of these using computers
has shown chaotic properties for some systems. The vast majority
of such equations cannot be solved analytically: you simply have
to run the numbers through a computer to see what happens.

Mathematicians have developed various tools to try to work out
the general properties of an analytically insoluble system (things
like Lyapunov coefficients, linearisation about an equilibrium,
etc.), but these don't change the fact that these systems have
no analytic solution.

In udder woids, there's no simple "grab-bag" of techniques that
can be applied to a dynamic system, unlike the situation for
solving a quadratic where we can simply trot out the old
x=(-b +/- (b^2-4ac)^1/2)/2a. No such beast exists for dynamic
systems in general--though some do exist for very limited but
important instances of them, as in soliton theory, harmonics,
etc.

So Jerry's request for a comprehensive taxonomy of dynamical
systems, which can then be correlated against economically
meaningful systems, simply can't be fulfilled.

Or if anyone out there CAN do it, a Nobel Prize for Mathematics
awaits you...

Cheers,
Steve K


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