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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reading one of the self-published books I created from an unpublished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 |