Is the Mandelbrot Set Real?

John K Clark (
Tue, 19 Aug 1997 21:27:32 -0700 (PDT)


Hara Ra <> Wrote:

>A Real object such as the Mandelbrot set's shape cannot be directly

Agreed, at least I can't.

>Its abstract definition is easily stated in a few hundred bytes

At the most.

>In fact, the Mandelbrot set is a rather simple object in terms of
>complexity theory since its description is very much smaller than
>its resulting representation!

Pi = 4 - 4/3 +4/5 -4/7 +4/9 - 4/11 + ...

Please tell me what the trillionth digit of Pi is. The short line above is
all the information you need to figure it out, so you should be able to do it
in a snap. It's simple.

>Ah, serious confusion here. Indeed, to our perception, the Mandelbrot
>Set is infinitely complex, in its overwhelming amount of detail. In
>terms of complexity theory, the Mandelbrot Set is a simple object.

In terms of which complexity theory? You act as if complexity is a clear cut
obvious idea universally agreed upon. It's not.

>The mathematical definition of complexity is basically the length of
>the string describing the algorithm used to create the object.

Obviously we must talk about the minimum string or it is not unique, but as
I've already mentioned, it's been proven that we can never know if a smaller
string that does the same thing can be found, so if this is the definition of
complexity it's not a very useful concept because we can never know waht it
is for anything.

>Let me make a slight modification to my terminology:
>Areal - Abstractly Real, ie, Platonic reality
>Preal - Physically Real, representable in the physical universe
>Mreal - Mentally Real, representable in the mind
>Mreal is a subset of Preal. Preal is a subset of Areal.
>The set of integers, and the Mandelbrot Set are Areal objects. Said
>sets cannot be represented in the Preal or Mreal.

Three questions.

1) Exactly what is this property you call "physical" and how can I test to
see if an object has it?

2) If Mreal is a subset of Preal and I have the idea of a fire breathing
dragon in my mind, does that mean that dragons are physically real?

3) If the integers or the Mandelbrot Set are not part of Mreal then how do
we know anything about them?

>The definitions or algorithms for these objects are Mreal, and
>therefore Preal. The definitions are not complex, and the sets
>themselves are.

The definition is real but the set is not? Can you give an example of
something the definition has but the set does not.

John K Clark

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