Re: Infinite series, Zeno, simulations...

[John M <***>]

Judith, you are amazing.

Cauchy brings back memories of my youth when I studied
his polynoms (anybody for Lagrange?) - long forgotten.

Am I right that C-y was monotonous, while L-e overshot
the previous member once up, once down - converging
luckily to a final (unattainable) value? 
These considerations assume our precise omniscience
that there is NOTHING beyon the conditions we started
with. So we can trust your final 'sum' value of 2. 
Not so in a natural system, with unfathomable variety.

Now let me transpose this into 'modeling'. IMO the
formulation of models is NOT like a math series, it is
simply cutting off the tails and including what easily
comes to mind (knowledge-base). When one continues the
model ad infinitum - making it a maximujm model, (the
'thing itself'), it enters a different system - the
natural one vs the scientific (reductionist) views. So
your interim conclusions are applicable beyond math. 
And this still did not take into account that the
omitted 'beyond boundary' incorrectness may contain
quite 'out-of-order' domains with effects totally
scewing up the correct math results. This is IMO what
RR expressed in his "Turing Non Computable" criterion.

I could not muster all that endurance to pay full
attention to all you wrote (it is your list, so I
cannot ask you to truncate the sequences of topics
within one post), but please, have mercy on us, poor
readers! Your flow of thoughts is natural for you, the
reader has to 'follow' and after several changing of
directions I for one get exhausted. I would be happy
to read and follow them all (can you imagine the
length of replies?) but once a text is put aside,
there is always another one coming up, so
comprehensive recapitulations are on;y feasible in
consecutive lives. Not too much hope for me: my sins
will degrade me in the next incarnation into a
seaweed. Not much of an intellect there.

Thanks for the process you wrote about, to revamp RR's
unpublished notes for our enjoyment.

John Mikes

--- Judith Rosen <***> wrote:

> 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. 
> 
=== message (Thalidomide) truncated ===