Re: Judith's mathematical insight...

[James N Rose <***>]

Hi Judith,

First, the alternate results that come from having
the group partitioned differently, is an amazing thing 
to identify.  Interpretations of why its that way
are several .. including your dad's and then your
reasonings, and mine, which look at even other aspects.

My first question is whether this is a known phenomenon
or not to mathematicians (earlier and better than we).

Before going into discussion directions mentioned from the
start of this thread, more basic questions arise, like:
the commutative ans associative laws of essential math
can no longer be taken so casually or cavalierly.

>From other arithmetic relations I've toyed with over
the years a different reality arose: that current math
neglects, avoids and is otherwise blind to the situation
that all manipulations and equationings really involve
both commutative and non-commutative results with any and
every math-act that's done or modeled.

That's essentially like acknowledging that - by default of
operations, and all possible relations - multiple simultaneous
panoramas of mathematical conclusions branch off from even the 
simplest math-act, even simple addition; but numeric-manipulators
opt or bias for tracking small specific subsets that are convenient,
easy, locally consistent, etc. just because it avoids clutter
of too much extra-information (like the real natural world is prone
to, but that linear logic can't readily handle).

Limited models are pre-selected even in the presence of some
vast 'more' of information.   

But I'd be quick to not lay blame.  Every living creature needs
to practice simple behaviors before achieving competence to
handle clusters of 'the simple' and then 'complexness'. The same
goes for thinking.   We have to practice and master 'addition'
-alot- before we casually toss around 'alternative meanings of
sum-series'.  :-)  Whether in an individual, or over generations.


But let me get back to your thoughts Judith.  I need to point out that
divergence and convergence have added aspects .. they can be closed
-or- open ; i.e., there can be specified limits for div and conv
functions, and there can be conv and div functions which continue
without -any- definite limit, and ones that are asymptotic.

Next, when you write "timing is critical", this translates into
the mathematical notion that -everything- is non-commutative.
(take a step forward and turn left, is different from, turn 
left and take a step forward ; where as casual arith says
5 + 6 is always and exactly 6 + 5).  Natural reality is both 
while certain pragmatism lets us avoid 'all natural reality'
a lot.

So RR was fighting a massive battle, trying to convince people
to accept and juggle all the added messiness of all these
other co-realities, when efficiency and habit was and had driven
them to -avoid- extranities and all that wonderful messiness.
Even the most complex and advanced thinkers.  They were trained
too well inhabituated thinking.  Most people are, unfortunately.

Anyway, I have to rush out to work again this morning.  I want to
continue writing about this before getting anything back, would you mind?
Will try to get it in this evening or tomorrow AM earlier.

+1/0/-1 are possibles.  but then again, so are fractal aspects.
But, I think the first is to explore dimensions, recursions
and waves ... that come from the alternative partitionings.

this is really important, what you've started discussion about.

Jamie







>Hey Jamie,
> 
>I've got a few more thoughts on this whole situation;
>lemme know what you think...
> 
>I wonder if it would be accurate to say that an infinite
>series is convergent as long as it only has a certain 
>similarity of process in it, like; only addition and only
>positive numbers, for example? In other words-- if it is
>always a process of progressing from smaller to larger, 
>with no reversals in the process...
> 
>Because it seems to me that the reversals in the process are
>what create the divergency-- adding then subtracting (or 
>adding negative numbers to positive numbers which is the 
>same thing)... one step forward, one step back. A rhythm is
>created, with the time signature specified by the parentheses.
>As soon as time is involved, "when" those reversals happen is
>capable of having a major impact on the whole thing.
> 
>I see this as a bit of yin and yang action: if it's all yin
>or all yang, it's convergent, but as soon as there's a mix of
>both, then the way the two are mixed is going to matter and
>timing/sequencing becomes a critical issue. The fact that the
>universe is a balance of yin and yang means that there are
>not likely to be many convergent situations, it seems to me.
>Would there even be ANY?
> 
>Judith
>PS: A friend pointed out a third result from this exercise:
>it could equal -1. Antimatter! We'd have to shuffle the
>order of the numbers as well as the parentheses, though: I'm
>not sure what that means, mathematically. Any feedback on that?