Re: Judith's mathematical insight...

[James N Rose <***>]

You have hit upon the crucial factor, Judith, which, though
in and of itself is difficult to specific - since we live 
inside it and can't readily pare it down (through standard
reductionist logic as your father did in your example here) -
the harder challenge comes next.

The step which you just depicted essentially corresponds to this:

- within a natural environment there are multiple simultaneous
frames of refence (habitats), which, though they use the same 
mathematics, ordinate them differently and therefore coordinate 
in different functional ways ... both independently -and- inter-
involvedly.  They are functionaly independent -and- bound at the
same time.

The challenge is specifying their coordinated behaviors, when
one uses base=0 and the other uses base=1.

:-)

Jamie 


 

> Judith Rosen wrote:
> 
> I actually had a mathematical insight over the weekend  while reading one of the
> self-published books I created from an unpublished partial manuscript of my
> father's ("The Limits of the Limits of Science"). I thought I might as well post
> it for the list:
> 
> In this particular passage, Robert Rosen described the nature of the problem in
> mathematics in dealing with infinite sums and infinite products. The problem
> is that judicious placing of the parentheses can create entirely different
> results, in spite of the fact that the numbers haven't changed at all. In the
> example he used: 1+ (-1) + 1+ (-1) +1 +(-1)...=? If the parentheses are placed
> like this: (1 + (-1)) + (1 + (-1)) +..... the answer will be "zero". If the
> parentheses are placed like this, however:
> 1 + ((-1) + 1) + ((-1) + 1) +..... the answer will be "one"
> 
> He than goes on to talk about Cauchy, a mathematician, who introduced the notion
> of "convergence", specifically to solve this problem. Cauchy's solution
> formulated a set of criteria for discriminating between cases where the "value"
> (limit) of an infinite sum or product was dependent on how it was evaluated and
> when it was INdependent of that. RR's conclusion: Cauchy showed that the
> generalization of the behavior of mathematics-- and therefore the
> assumed applicability of approach-- from finite to infinite (at least with
> regards to the behaviors and properties of sums and products)-- wasn't
> appropriate: To preserve the independence of a value or limit from how it is
> evaluated, an independence which is automatic in the finite realm,  we need to
> cut the "infinite" world back down again-- by imposing convergence conditions.
> 
> Here's my little insight from the weekend: What's being illustrated by this
> whole situation is the impact of the relational effect. The relational aspect of
> causality, which is represented here by the parentheses, is magnified in
> infinite processes, whereas in finite processes it can be safely ignored. The
> reason has to do with the nature of complexity: The parentheses represent the
> temporal aspects of sequencing... It's common knowledge in non-mathematical
> realms that the timing of "when" something happens is sometimes more important
> than the fact that it happens at all.
> 
> 
> Furthermore: the transition from the way mathematics predictably behaves in the
> process of generating sums and products in the FINITE realm to the behavior of
> mathematics engaged in the exact same processes-- but in the INfinite realm-- is
> a transition that creates the potential for relational aspects to have a far
> greater influence than they have in the finite realm. Therefore, it is
> inappropriate to generalize from finite to infinite. This exactly mirrors the
> case between the impact of relations in systems with simple organizations and
> systems with complex organizations. As such, it's not a paradox at all: it's
> telling us something about the potential entailments of relations on
> causality in an infinite realm. It's also an object lesson for science on what
> not to do in making a similar transition from studying simple systems to
> studying complex systems. Generalizing from finite to infinite is a dangerous
> thing to do.
> 
> Judith Rosen
>