# Re: Is the Mandelbrot Set Real?

Hara Ra (harara@shamanics.com)
Wed, 03 Sep 1997 22:21:14 -0700

John K Clark wrote:

> 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.
>
Yes, Pi is an Areal value, and the definition provided above is an Areal
algorithm...... As for computing the trillionth digit, there are more
complex derivations which will yield the trillionth digit with enough
calculation. (I believe the last reported value for Pi has some 52
>
> >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.

Please provide a summary of the various current interpetations of
"complexity". For the moment I am going with the definition mentioned
below. However, I'd like to know more here...
>
> >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.
>
Misplaced precision here. Yes, finding n, the minimum length of string S
within a propositional system L is similar to the halting problem and in
most cases impossible may be true (and proven as you say). However,
finding a string S1 of 2n or less in length may be easily findable. 2n
is good enough for this line of argument.

> Three questions.
>
> 1) Exactly what is this property you call "physical" and how can I test to
> see if an object has it?
>
Each element of the object can be one for one mappped into the integers,
and the number of integers required is less than the number of integers
required to count the number of particles in the Universe. The Universe
is presumed to contain a finite number of particles, appproximately
10^80.

> 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?
>
The dragons are not, but the idea of the dragons is Mreal.

> 3) If the integers or the Mandelbrot Set are not part of Mreal then how do
> we know anything about them?
>
the idea of the integers and the idea of the Mandelbrot set, and the
various theorems, lemmas, etc, are all statable as Mreal finite length
strings.

> >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.
>
Peano Postulates = Mreal set of strings, finitely statable.
Integers = Areal, not finitely listable.
Application of Peano Postulates results in the Integers. Note the term
"Integers" is Mreal, but the set it names is Areal.

It's a tradeoff. The Mreal strings are much more manipulatable, as in
provability, than the sets themselves, which due to their infinite size,
cannot be manipulated. Mreal we can work with, Areal is not. We can in a
Mreal way discuss the Areal, which is what this thread is doing.

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| Hara Ra <harara@shamanics.com> |
| Box 8334 Santa Cruz, CA 95061 |
O----------------------------------O
so many stupid people, so few comets