I think that what is going on may be a little deeper. (01)
Stipulating for a moment the continuous function argument. In any
digital system, in any countable system actually, you can approximate
a real number's value to any desired level of precision. (02)
Now, there is some evidence that the universe is not truly continuous:
it may be granular both with respect to space and with respect to time. (03)
If your approximation of a continuous function is more accurate than
that granularity, then it can make no difference in the observed
results. I.e., digital simulations of fractal functions and chaotic
predictions can be as accurate as the universe will allow anyway. (04)
However, the weather man does not need to go to this level of accuracy
to get results that are best explained in terms of chaos theory. The
results of a prediction are unstable at any level of accuracy: the
higher the accuracy the longer the stable prediction; but the pattern
is the same. (05)
On Feb 2, 2008, at 10:34 PM, Rob Freeman wrote: (06)
> On Feb 3, 2008 12:11 PM, Randall R Schulz <rschulz@xxxxxxxxx> wrote:
>> On Saturday 02 February 2008 19:47, Rob Freeman wrote:
>>> Why can't we model chaotic behaviour on a digital computer Randall?
>> Because digital computers cannot represent or process real numbers.
> That's a good point.
> Here's Wikipedia to put it in context:
> "An early pioneer of the theory was Edward Lorenz whose interest in
> chaos came about accidentally through his work on weather prediction
> in 1961. Lorenz was using a simple digital computer, a Royal McBee
> LGP-30, to run his weather simulation. He wanted to see a sequence of
> data again and to save time he started the simulation in the middle of
> its course. He was able to do this by entering a printout of the data
> corresponding to conditions in the middle of his simulation which he
> had calculated last time.
> To his surprise the weather that the machine began to predict was
> completely different from the weather calculated before. Lorenz
> tracked this down to the computer printout. The computer worked with
> 6-digit precision, but the printout rounded variables off to a 3-digit
> number, so a value like 0.506127 was printed as 0.506. This difference
> is tiny and the consensus at the time would have been that it should
> have had practically no effect. However Lorenz had discovered that
> small changes in initial conditions produced large changes in the
> long-term outcome. Lorenz's discovery, which gave its name to Lorenz
> attractors, proved that meteorology could not reasonably predict
> weather beyond a weekly period (at most)."
> So this presents a problem.
> To draw the analogy, though, Lorenz didn't react to his discovery by
> abandoning computer modeling of the weather, or by assuming the
> weather was not really chaotic after all. He just accepted that tiny
> inaccuracies meant that any digital copy of the weather would diverge
> from the original over time.
> I'm guessing chaos is still assumed in all modern models of the
> weather, and further I'm assuming all those models are still digital.
> Digital models of the weather may be imperfect, but presumably to
> ignore chaos when you try to model the weather is not to model the
> weather at all.
> Note: there's an interesting corollary to this idea of sensitivity to
> arbitrarily small differences in initial conditions. It would mean
> that, assuming a chaotic model, any "copy" we might one day be able to
> make of an individual's thoughts would immediately start to diverge
> from the original. So while one day it may be possible to copy
> someone's thoughts, according to this model it would never be possible
> to predict them (beyond a week or so :-)
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