Re: chaos <=> paradox. Prove me wrong. A challenge.

From: Will Twentyman (wtwentyman_at_read.my.sig)
Date: 09/23/04 

Date: Thu, 23 Sep 2004 13:04:02 -0400

Lefty wrote:

>>>Conjecture:
>>>chaos <=> paradox.
>>
>>Question: what do you mean by chaos? There is chaos theory, which has
>>much less to do with randomness than you might think. There is pure
>>randomness, which can be described by probability/statistics if it is
>>"well behaved".
>>
>>Similarly, what do you mean by paradox? I usually interpret as a claim
>>that cannot be assigned a self-consistent truth value.
>
> My usage of the term chaos is as follows: Phenomena occuring in dynamical
> systems which exhibit order and randomness simultaneously. A mixture of
> order and randomness.
>
> There are lots of other things I'm neglecting to mention, but I'm narrowing
> the scope of the definition to indicate only this. A mixture of order and
> randomness.

Here's the problem: In what sense do you mean this? For example: set up
  Newton's Cradle (5 suspended balls just barely touching) and set it in
motion. At one level, there is order. The balls are exhibiting nearly
elastic collisions under damped oscillation. At another level, there is
randomness. The atoms are vibrating at random speeds/directions;
electrons are being transfered between atoms randomly as well.
Considering both order and randomness are present, do you really want to
call the click-clacking of Newton's Cradle chaos?

>>>Justification:
>>>We know that
>>>order + randomness <=> chaos.
>>
>>This suggests what you mean by chaos, but doesn't define it.
>
> A suitable definition of chaos would be much lengthier. I'm borrowing it for
> this restricted usage.

See the problem above.

>>>Assertion:
>>>logic + nonsense <=> paradox.
>>
>>This is patently false.
>
> Well, perhaps not false, but I certainly have a lot of proving to do.
>
> Look at any paradox you wish. You can see the logic to it. Yes ? Now, you
> get to the part where it starts to not make sense anymore and you have your
> nonsense component. It is a hybrid of logic and nonsense. It seems very
> clear to me.

Paradoxes that I encounter consistently make sense. There is a
(usually) single false/inappropriate assumption. Think of a paradox as
a contradiction that can be avoided by being more careful.

> Take Russel's Paradox. Everything makes sense for the most part, you have
> very logical sounding sentences regarding sets, set membership, and subsets.
> This is all very logical. But when you assemble those words and consider the
> global meaning of the sentences, it simply does not make any sense. Hence,
> it is nonsene. And I dont mean that in a bad way, only that nonsense is the
> complement of all things sensible (roughly I suppose).

No, it does make sense. What it tells you, however, is that those
definitions/axioms are inconsistent. With a few careful modifications
you can make the inconsistency go away and the paradox becomes useful as
an illustration of *why* the modifications are necessary. The paradox
simply does not exist when you are working in the corrected version.

>>>If logic = order, and randomness = nonsense, then chaos would be roughly
> the
>>>equivalent of paradox.
>>
>>Proof by analogy? This is an invalid style of argument.
>
> Actually, I'm not trying to set up an analogy, but merely show that the
> differences in these things is almost purely semantic.The definitions of
> these terms are nearly identical enough to consider that logic = order, and
> randomness = nonsense.

I suspect that the similarities will disappear as the definitions get
more precise.

> Obviously, there is much more to chaos theory than "randomness <plus> order
> <yields> chaos". But, this is the part which interests me in this context.
>
> Perhaps I'm producing nonsense instead of understanding it ? Dont worry, I
> wont keep it up for months on end.
>
> Let me say this. It is not reasonable that paradoxes should exist within the
> framework of mathematics. It does not seem that they belong, yet they do. I
> suspect that this has something to do with the properties of space/time in
> which we are embedded.

No, it has to do with the development of the subject and some of the
inconsistencies that were found in attempts to formalize certain
branches such as logic and set theory. Those inconsistencies were one
of the driving forces behind the revisions that took us to a consistent
form of those branches. They are historically significant to explain
*why* we use some of the odd definitions/constructions.

> Very possibly, but perhaps I've inspired some different thoughts. What
> good's a mind if you dont use it once in a while.

Thoughts are good. Exploration is good. The assumption that you are
right could lead to a paradox.

> So, if you had to answer the question, what does it yield when one mixes
> logic with nonsense ? Just more nonsense ? Diluted nonsense ? Dilute logic ?

I'd say nonsense.

> Please consider the following - knowing that it is intended to be nonsense.
> ------------------------------------------------
> Let a =/= a.
> Let b =/= b.
>
> Define a + b = c, such that d does not exist, and is not defined.
> ------------------------------------------------
>
> OK. Now, that was nonsense. I was not really a paradox, but perhaps akin to
> one.

If I choose to try and force some meaning out of it, I can start making
observations. 1) =/= is not the standard symbol for inequality, or
variables do not behave as they do in algebra. Perhaps the . is
significant, indicating that "a" =/= "a.". There are a number of things
that it could indicate. In logic, for example, the following is true:
"If 1=2 then 2=3". Why? Because (F -> F) = T.

The definition, at the very least, requires a larger context.

> So, I have something which is partially logical, afterall it is legible, and
> partially illogical because some of it makes no sense. It is a mixture of
> logic and nonsense and it is also not a paradox.
>
> OK, I have a couterexample, unfortunately it's working against me.
>
> This statement "logic + nonsense = paradox" is therefore definately not
> valid for all logic and all nonsense.
>
> In Russels paradox there is cerrtainly no nonsense until you consider the
> totality of it. Thats when the nonsense emerges, and you have paradox.

The totality gives a contradiction. The paradox is a result of assuming
  that you had defined a consistent system.

> OK, you've bloodied my nose and I'm on the ropes. Let me go to my corner for
> a respit before getting KO'd. Give me a minute.

Perhaps a more effective approach would be to give some context for how
your thinking and ask people what they think appropriate definitions for
your terms would be. 

-- 
Will Twentyman
email: wtwentyman at copper dot net

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