[From Bruce Gregory (2003.12.01.1730)]
http://www.nature.com/nsu/031124/031124-4.html
"Everything that needs to be said has already been said. But since no
one was listening, everything must be said again."
Andre Gide
Hi, Bruce --
I probably saw this when it came out. I'd say (as the authors do) that the
irregularity of the hardness of the rocks leads to the irregularities of
the coast on a scale that damps the waves and slows further erosion, sort
of like random reorganization. Sounds like a good explanation.
The shape of the coast on a 100 Km scale has nothing to do with this
phenomenon, does it? Introducing the idea of fractals wasn't necessary and
added nothing to the explanations. And anyway, if you go out and look at
any coast, you'll find that the actual self-similarity at various scales
requires a lot of help from the imagination and a lot of ignoring of
inconvenient counterexamples (like tidal pools, cliffs, grains of sand,
smooth stretches of beach, and so on).
Bill
[From Bill Powers (2003.12.03.2149 MST)]
The article makes it clear that a fractal shape need not be a continuous
irregular line (though it can be), and "self-similar"does not mean identical.
You're right, of course. But why even mention fractals? The article was
about hard and soft granite being worn away into an irregular coastline on
the scale that would tend to damp wave action, thus limiting further
erosion. The "fractal" stuff was gratuitous window-dressing on an otherwise
interesting idea.
"Recursive" is a mathematical notion that unfortunately involves several
kinds of infinities. Any real implementation of a recursive process (as on
a computer) requires saving the state of the machine each time the
recursion is repeated, leading (except in a few special cases) to a
requirement for infinite storage capacity (known as stack overflow).This can occur only if recursion is unlimited. The structure of language
is recursive and therefore the set of sentences of a language is
technically infinite. That does not result in stack overflow nor require
infinite storage capacity in the human brain.
True, recursions can be limited. They don't have to cause stack overflow,
if the programmer carefully includes tests that stop the recursion after a
finite number of iterations.. I should have said what I really object to,
which is calling everything that looks like circular causality "recursion."
I can think of better ways to discuss that subject.
But it is equally true that many recursions are not infinite series.
I agree they are not. But if many are, generalizations about recursions
tend to lose their generality.
Best,
Bill P.
[From Bruce Gregory (2003.12.04.0523)]
Powers (2003.12.03.2149 MST)
The article makes it clear that a fractal shape need not be a
continuous
irregular line (though it can be), and "self-similar"does not mean
identical.You're right, of course. But why even mention fractals? The article was
about hard and soft granite being worn away into an irregular
coastline on
the scale that would tend to damp wave action, thus limiting further
erosion. The "fractal" stuff was gratuitous window-dressing on an
otherwise
interesting idea.
I'm not sure that identifying power laws is necessarily providing
"gratuitous window-dressing." Fractals are self-similar and
self-similarity leads to power laws. Is it gratuitous to point out that
earthquake magnitudes follow power-laws?
Bruce Gregory
"Everything that needs to be said has already been said. But since no
one was listening, everything must be said again."
Andre Gide
[From Bruce Nevin (2003.12.03 21:51 EST)]
Bruce Gregory (2003.12.01.1730)–
I take it this was in reply to (Bill Powers 2003.11.23.0914 MST):
Bruce Gregory (2003.11.23.0954)–
My other objection to such “complex
systems” ideas is that they are notvery good approximations to reality. Consider fractals. I’m sure you
haveread about the way fractals describe coastlines or mountain skylines,
whichare “self-similar” on all scales of magnification. Except that
they’re not.If you get close to a real example of either kind of “fractal”
landscape,you find that its character changes completely. Instead of a
continuousirregular line, you start seeing granules, cracks, and other
discontinuities that do not look like the large-scale picture at
all.Fractals fit imaginary landscapes, not real ones. Nothing in the real
worldactually looks like the Mandelbrot set.
The article makes it clear that a fractal shape need not be a continuous
irregular line (though it can be), and "self-similar"does not
mean identical. The Mandelbrot set is relatively simple to state:
“the set of all points that remain bounded for every iteration of z
= z*z + c on the complex plane, where the initial value of z is 0 and c
is a constant”. Here’s a selection of ‘zooms’ into it, beginning
with the view that has become iconic of fractals:
“Recursive” is a mathematical notion
that unfortunately involves severalkinds of infinities. Any real implementation of a recursive process (as
ona computer) requires saving the state of the machine each time the
recursion is repeated, leading (except in a few special cases) to a
requirement for infinite storage capacity (known as stack overflow).
This can occur only if recursion is unlimited. The structure of language
is recursive and therefore the set of sentences of a language is
technically infinite. That does not result in stack overflow nor require
infinite storage capacity in the human brain.
And of
course since many recursions are basically infinite series (like
sin(sin(sin…sin(x))), they require either infinite time to be
calculated,or infinite speed to be calculated in a finite time.
But it is equally true that many recursions are not infinite series.
This does not show that Bruce Gregory (in the “Re: Perceptions”
thread) was correct, only that this rebuttal is unconvincing.
/Bruce
Nevin
(Attachment a47d8a21.jpg is missing)
···
At 05:30 PM 12/1/2003 -0500, Bruce Gregory wrote: