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I've been doing a bit more research on the whole situation of sums
and products in infinite series and I am struck by just how much information
there is on the internet about all this stuff. Sheesh! My father had a knack for
finding the hornet's nests, didn't he?!
Mainly, though, I have been forced to conclude that it's a waste of
time to try and get inside the thought process involved... my entire problem
with all mathematical logic can be easily illustrated by one example
from what I've learned about convergence and infinite series: It's the
convergent series: 1 + 1/2 + 1/4 + 1/8 + 1/16 +.......... =
2. The mathematical "logic" involved in order for the
field of mathematics to officially decide that arbitrarily close
to 2 means that the series converges to 2; 2 is the
limit of the series-- That's the kind of thing that my mind balks at.
As if getting close to 2 is "good enough", "might as well be 2"
or "is the same thing as 2"... when mathematics ironically prides itself on
exactness..... My conclusion is that what is referred to as formal
logic is inconsistent, even within the realm of
mathematics-- and the reason it is is because human minds are
trying too hard to retain a limited notion of consistency in the face of
complexity. That was pretty much Robert Rosen's diagnosis, as well,
and while I admit to the overwhelming influence of a Rosennean point
of view, both genetically and because of my life experience... my father was
able to speak, understand, and navigate the language of mathematics as
fluently as any pure mathematician. The fact that our two points of view wind up
in the same place after we each do our analysis is what I have meant in the past
when I said that Robert Rosen's scientific work passes my "common sense
test".
For any who are interested, here's my (non-formal) logical
assessment of the whole "infinite series" sums and products, thing:
If they're infinite, they're infinite. Period. If you make them
into something finite, you're not working with the same system, anymore. Worse;
the information gained by the exercise is not necessarily applicable-- so it is
dangerous to use such information for
answering questions which pertain to the infinite
realm. Wouldn't it be more useful to study the
various mathematical entailment relations of the finite and the
infinite realms, via the behaviors of mathematical systems and
their interactions in each, rather than attempting to make the infinite
realm behave as if it's finite? This was partly where my little
mathematical insight, posted last week, came from. We need to do the same exact
thing in science.
The way I see what Cauchy was looking at, there will
never be convergence. It is not possible. An infinite series
is, among other things, an infinite process-- of
either addition or multiplication-- and it's the process that makes the series
infinite, not the physical numbers themselves. Mathematics as a language is
a complex system, in and of itself, yet mathematicians keep focusing on little
pieces of it-- in this case, the numbers (as if the process or any other
aspect doesn't really matter). In a finite world, that may make sense
(commutativity, associativity, etc...) and when it does make sense,
then the strategy may be useful. But this particular
situation is about processes taking place in the infinite
realm. Humanity, it seems to me, has a
tendency for always trying to apply finite rules to "the problems
of dealing with" an infinite world-- even after we recognize that it's infinite
(as is the case under discussion here). One of the truths about
mathematics as a whole that comes out of my father's work in complex systems
theory is that any particular field of mathematics will have
active relational information from the larger system of
mathematics "encoded" into it. In this particular case, it's actually
visible: Even though we are supposed to be dealing strictly with sums
and products, both addition and multiplication are specified by a larger
system of entailment which includes the opposites of both (adding fractions
together is a case of adding processes of division; multiplying fractions is a
case of multiplying processes of division; adding positive and negative numbers
together is a process of adding processes of subtraction)...
The paradoxes arise precisely because a single realm of mathematics
is treated as if it's not connected to the rest and does not carry the
entailments of the larger system within it. People
have labeled such relational aspects "semantics,"
defined semantics as an impediment, and tried to get rid of the
relational aspects by replacing them with different entailment
(syntactic/formal entailment). They oversimplified, which
is practically guaranteed to cause side effects, and the paradoxes are the
result. Paradoxes can be defined as a symptom of an imbalance somewhere in
the inferential entailments of a formal system. They are a side effect of lost,
or otherwise missing, information. An analogy to this
is the past experience in medical science of thalidomide damage
to babies borne of mothers who were prescribed thalidomide at a critical
point in their pregnancies. This was a classic case where a side
effect was caused by tinkering in a complex system
using medical answers based on information derived
from simplistic models.
[Incidentally, the most important aspect those models failed
to take into account in their experiments, in my opinion, were the relations
of time. There are specific time relations involved in pregnancy that
are related to but distinct from the time relations in "human female"
organization, another distinct set of different time relations involved in human
embryonic development, and then there are yet other time relations
governing the interaction between the other two sets. They are all related,
they all affect one another, but they are distinct from one
another. When the researchers did their drug
safety experiments for thalidomide, they assumed that all phases of
embryonic development were equally vulnerable to any exposure of drug. This
is not the case. The drug's characteristic side effect; a baby born with some
variation on hands growing from shoulders and feet growing from hips, was
generated because thalidomide only interfered with embryonic development
during the phase of long-bone formation. This particular
embryonic phase happens to occur during a
pregnancy's first-trimester phase, so it coincides with the worst
nausea for the pregnant woman. Thalidomide was being prescribed as a therapy for
nausea.]
I think the main problem of the infinite series "solutions" in
mathematics is also partly the "forgetting" or formal
elimination of the relations of time that are an integral aspect,
built into the process. Any infinite process can only be infinite if time is
part of the organization. Taking time out of the series and dealing with the
result as a finite process is analogous to trying to scientifically analyze
a complex system via "states".
Unfortunately, the finite system that convergence relations create
(no offense to Cauchy) to deal with the symptoms (paradoxes) is based on the
same effort and the same thinking that caused the symptoms in the first place.
As often happens in such a situation: the therapy seems
to eliminate the symptom... but it does nothing about the
generator of it; the symptom was merely an observable consequence of
entailments. Rather than addressing whatever created an imbalance in
the system entailments, this approach gets rid of a symptom
by applying more of the same therapy that caused the imbalance in the
first place. The result is guaranteed to generate MORE SYMPTOMS; mixed up
together because we now have two generators. In a relational world, we can
pretty much expect that the two generators will begin interacting with each
other and with the rest of the system, over time. What is the
next solution? To fractionate further and create a whole family
of convergence relations, each designed to deal with specific
categories of symptoms.... and in effect, they have created a new infinite
process of their own; what RR called "an infinite regress". There are currently
many different sets of convergence relations for addressing all manner
of different kinds of infinite series-- it's just nuts.
The properties of finite mathematics which were so starkly
inapplicable in infinite series of addition or
multiplication are commutativity and associativity. Commutativity [ 5 + 6 =
6 + 5 ] and associativity [ (1 + 2) + 3 = 1 + (2 + 3) ]
are properties that hold in the (finite) process
of multiplication as well. Therefore, these are considered to be
mathematical laws in multiplication and addition and the fact that "you
can depend on this" has been built into a whole bunch of other operational
rules and mathematical procedures. I guess that's why the realization that they
don't apply in infinite cases caused such distress. But there's a lot more
than just those examples in the set of things you can do in a finite
world that don't transfer to the infinite. For example, in a finite
world all of the following are absolutely equal to each other and
to the final answer, which is exactly 1: [6-5], [-5+6], [-5- -6], and so
on. Furthermore, in a finite world, the following are
all equivalent to each other, because their end results are all
equal: [3+3-5], [6-6+1], [1006-1005], etc. Furthermore because the first group
and the second group all have the same final answer, that
means the first and second group are equivalent to each other. My
immediate reaction to such statements during my research was: Just
because the final answer is the same doesn't mean that these examples are the
same-- in fact they are all different. It's just that the ways they are
different from one another don't impact the conclusion: the final result of
each of these different processes is the same. On the other hand, that is only
because they are happening in a finite realm, where time stands still
and where relational aspects are minimized.
There is another analogy here between mathematics and
science: to reduce an entire set of mathematical processes down to a
single, numerical result and define each process in the
set entirely on the basis of the result is just like calling a
simulation "the same as" the system it simulates. That's pure nonsense
if the entailment underneath is entirely different. However, mathematics
and science have both been really lucky: When dealing with simple systems, the
negative impact is "equivalent" to zero!!! (sorry... couldn't resist!) In a
finite realm, if we create a simulation of a finite system, like a car-- a
new system that has all the same outward appearances and all observable
properties and behaviors of a car, it will perhaps actually BE A CAR.
So, if we simulate it accurately enough, we may actually
create another of the system being simulated. But even if there is no
negative impact from doing so, it's a bad habit to get into. The rules
that appear to work in a finite realm are not necessarily applicable at
all; it's just not showing up as negative impact. This is exactly the
situation with the machine metaphor in physics.
In the course of this past week or so, I researched the three
most famous of the "Zeno paradoxes" (one of which is "Achilles and the
Tortoise"). I did so because the Zeno paradoxes were often invoked in any
description of the ideas surrounding convergence and infinite sums, etc.
After reading these paradoxes and the stories of the trouble they
caused in the history of both science and mathematics, I confess to a
strong feeling of bemusement. Frankly, I think the Zeno
paradoxes are just as illogical as the "problem" and "the
solutions" when dealing with infinite sums and products in a mathematical
series. In a strange way, they are sort of inverses of each other. The
Zeno paradoxes were created by trying to make the finite,
infinite. In contrast, the mathematical convergence rules
(regardless of which "type" of convergence it is) all try to make the infinite
finite. In both cases, the system entailments are completely
warped after being processed in that way and therefore, the results
are less than useful.
I find it a little mind-blowing that those paradoxes drove
people nuts for centuries. Even more mind-blowing is the fact that they never
came up with the analysis that it is something they were doing to the
system prior to the generation of the paradox that needs to be
looked into-- rather than concentrating solely on the paradox
itself and labeling IT as the problem to be solved.
Judith
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