[lit-ideas] Re: How long is the coast of Britain?
- From: John McCreery <john.mccreery@xxxxxxxxx>
- To: lit-ideas@xxxxxxxxxxxxx
- Date: Thu, 21 Oct 2010 13:55:34 +0900
But Richard, approximate is all we ever are, unless, of course, we restrict
ourselves to axiomatic language games. Let's put that aside for a moment.
To say that, with these rare exceptions, we always speak in metaphor is now
so common sense that it verges on cliché. The only serious question is "How
good a fit, for what?" While skeptics will always scoff that there is a
ding-an-sich out there beyond our understanding, the people who understand
the math will keep on changing our world in ways that our ancestors never
envisioned. Keeps life interesting.
John
On Thu, Oct 21, 2010 at 1:31 PM, Richard Henninge <
RichardHenninge@xxxxxxxxxxx> wrote:
> From Wikipedia:
>
> For many shapes that are often considered in mathematics, physics and other
> disciplines, the Hausdorff dimension is an integer. However, sets with
> non-integer Hausdorff dimension are important and prevalent. Benoît
> Mandelbrot, a popularizer of fractals, advocates that most shapes found in
> nature are fractals with non-integer dimension, explaining that "[c]louds
> are not spheres, mountains are not cones, coastlines are not circles, and
> bark is not smooth, nor does lightning travel in a straight line." [1]
>
> Hold that thought: Mandelbrot (as a heuristic, note that his name is
> literally almond-bread and that he gave himself a supernumerary middle
> initial B. that stands for nothing--or should that be B without the
> vestigial period, and not initial, since initial of nothing?), who is a
> "popularizer" (like few mathematicians) of fractals, his word, "advocates,"
> like the sophists in Ancient Athens with their beautiful set speeches, paid
> to be suasive (ask yourself why the Wiki-writer couldn't bring herself [a
> nod to Walter's pronominally feminine world in which everybody is she] to
> have him "argue" like everybody else, "that most shapes found--"Eureka," I
> found it, in Greece!--in nature are fractals, while "explaining" that
> "clouds are not spheres," mountains cones, coastlines circles, bark smooth,
> nor lightning straight-lined. Well, d'oh. Wouldn't it be pretty to think so,
> though, like the things dreamt of in our philosophy, our ideal geometrical
> forms? But neither-neither are they obedient to any other quadratically
> expressed function discovered, or better, shaped out of airy nothing by
> giving it a name--just a more complexly chaotic-seeming name. In every chunk
> of the real "separable" world (the ideal mathematical equivalent of which,
> the topologist's "separable metric space," can be conceived of as a sort of
> finer, uniform tofu) one could carve out, so to speak, scrimshaw of
> intricate fractal forms, but our delight in such forms is that we (with our
> lightning robotic assistant, or personal extension, the semiconductor
> processor) can generate forms that--dare we say so--beggar nature's own
> forms in their complexity, but a complexity that we can explain--flatten
> out, unpleat--into its every nook and cranny--we can even write out the code
> of it, give you a copy, you could even do it yourself, by the numbers, as
> many times as you want. See if your vaunted natural world can do that, with
> such ease and grace! Henry Ford meets fields of daffodils and begins pumping
> them out into the new animated world in vast arrays of numberless smileys.
>
> An early paper by Benoit Mandelbrot entitled How Long Is the Coast of
> Britain? Statistical Self-Similarity and Fractional Dimension and subsequent
> work by other authors have claimed that the Hausdorff dimension of many
> coastlines can be estimated. Their results have varied from 1.02 for the
> coastline of South Africa to 1.25 for the west coast of Great Britain.
> However, 'fractal dimensions' of coastlines and many other natural phenomena
> are largely heuristic
>
> make that totally heuristic, essentially, quintessentially heuristic
>
> and cannot be regarded rigorously as a Hausdorff dimension. It is based on
> scaling properties of coastlines at a large range of scales, but which does
> not however include all arbitrarily small scales, where measurements would
> depend on atomic and sub-atomic structures, and are not well defined.
>
> The best prophylaxis to brake these hubristic heuristic tendencies is
> Wittgenstein's checker: The world is everything that is the case, and that
> of which we cannot speak, of that we must remain silent. There will be no
> end to the coast of Britain if we start "finding" programmable patterns in
> it. Fractals may alter the length of that coastline, as people change
> people-based impressions, definitions, programs, sentences, but they are not
> about to make a new
> coastline in the world about which we can say, lo, it is the case that this
> coastline is fractal, without being accusable of making a metaphor, giving
> an illustration, speaking in parables, and being approximate, and human, all
> too human.
>
> Richard Henninge
> University of Mainz
>
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--
John McCreery
The Word Works, Ltd., Yokohama, JAPAN
Tel. +81-45-314-9324
jlm@xxxxxxxxxxxx
http://www.wordworks.jp/
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