Relation of poetic language and mathematical notations

(post to JCS-online @ 2010-06-22)

Relation of poetic language and mathematical notations
(this might have a very strict relation to consciousness - or indeed
does -, but it's necessity to understand most of what I write to make
these distinctions and associations)

I try to put into English, what is a part of my try to research the
(equivalent) paradigms - a poetic language in relation to mathematical
symbolism.

Jung has defined "abstraction" - abstraction is when you connect a
paradigm (coordinate system) with symbol (space). Abstraction as
activity (monad) is definition and appliance of such coordinate system
whereas abstraction as a thing (aggregate) is a specific sign,
division or symbol of such paradigm. Paradigm makes up a coordinate
system - the lines inside a whole.

Jung tells us that we can abstract thoughts, feelings or experiences.
Using whatever coordinate system with whatever symbol (I have used
symbol "om" or "omn" as the total reality, starting point of every
abstraction - symbols are first-level abstractions whereas
abstractions are last-level abstractions and they are linear as they
can be connected to graph, a formal system) we can find abstractions
with same essence or form. Those abstractions have rules of acausal
connections (dividing one symbol is actually a generation act of
acausal connections - everything abstracted out from one symbol is
acausally connected as the symbol is one thing; logic is based on
acausal connections of such kind as is math); such rules can be used
everywhere we can match the abstractions. We can find two symbols,
which have coordinate systems such that the whole abstractions are the
same, but symbols are different - this is because symbol, in relation
to abstraction, is a continuum.

There are paths of thought, which are purely deductive, which are not
results of logic. If there is a continuum of allowed deductive
movements inside symbol space, which all relate to shifting the
coordinate system on some "axis" (this is convenient simplification
that I talk about axes in this context, it comes from my visual
thinking). On that continuum, we can take an abstraction or a symbol
and start using the paths of this continuum to transform it as whole
in such ways that the result is as true as the starting point. On that
continuum, logic is a small subset of points.

We know that throughout the history - especially at all kinds of
eastern, Jewish or Muslim cultures, also Hellenistic world and other
places, maybe most except the late west -, scientists have written
poetry. We tend to call this kind of poetry a "metaphoric way" of
speaking of reality, whereas math is a direct way. Anyway, poetic
language in it's pure essence is an abstraction. We make this
discrimination based on difference of feeling - poetry activates in us
the kind of feelings, which are activated only by pure kind of math at
early childhood; formal math or habitual math does not have that
feeling - this is because we are not fully conscious about what we are
doing, when we solve some kind of math problems at school. In fact, we
have not read the proofs nor see the underlying reality and that is
why we do not have the poetic feeling.

Poetic language is an abstraction, because it carefully finds
sentences, which would give only partial meaning. Take the symbol
"yin". It is told to be yielding in relation to firm, Earth in
relation to Sun or woman in relation to man. There are countless
number of such examples. Yin is not a pure abstraction as you can not
form a graph - it does not have those "slots", where you can draw the
connections, but it's a continuum. Math carefully tries to put
non-discrete things into discrete, abstract form. Yin is not an
abstraction, but a symbol - thus it has a whole continuum of those
"slots", which might not be points, but patterns or fields of that
continuum. Fortunately, eastern philosophy tells us that after
counting a number of yin-yang pairs, you can catch the meaning of a
symbol. In reality, this meaning is symbol - as a meaning of any
acausal connection is that connection itself, seeing the "meaning" of
a word we have caught the reality behind. Mostly, indeed, that reality
behind a word is an experience related to something - "the world is my
idea" is how Scopenhauer calls it in beginning of "The world as will
and idea" (I quote that one quite much, because I really think that
everyone should grasp the ideas in that book).

Thus, as a poetic language is a math of symbols. Math itself tries to
catch the symbols based only on understanding of graphable
abstractions - those having the slots. Symbols can, by coordinate
systems, usually (or always?) converted to many different such
abstract systems, which in turn is basis on non-discrete math. Because
of the limitations of paper and pencil, math looses it's sharpness
quality, when we try to work directly on symbols. Thus, we try to make
non-linear and non-discrete things into discrete abstractions before
starting to work on them. This would be OK if our minds had the same
kind of limitations but this is not OK to stay on that level as they
are not. It is, anyway, reasonable to base a lot of communication -
always when we can - on those abstractions.

Poetry uses different kinds of abstractions not covered by Aristotle's
logic. Aristotle's logic itself, on other hand, would have been
indifferent to poetic language at ancient world - we would have
perceived it as some kind of poetry.

How poetry works:
* Abstract out some important abstractions of a set of things, which
might or might not Aristotle's abstractions.
* Take a number of real-life cases of such abstractions (things, where
it applies) and show each case - many poems do this.
* Show some implication rules, which might be horse-jumps of logic or
uncovered by logic (I usually take "too complex" and "impossible" as
synonyms as I am not seeing Turing machines around me, but only a
normal finite-state-machines, where even halting problem won't apply
as you can always run the program until it finishes or the values of
all variables equal to some historical state; I think that when
talking about human thinking, the differentiation between "too
complex" and "impossible" is only important when you are talking
specifically about this difference; so I don't ask a question if logic
is eventually able to describe all those cases).
* Better make the result practical - that the implication really works.

Or, let's look at the classics:

Tyger! Tyger! burning bright,
In the forests of the night,
What immortal hand or eye
Could frame thy fearful symmetry?

In what distant deeps or skies
Burnt the fire in thine eyes?
On what wings dare he aspire?
What the hand dare seize the fire?

And what shoulder, and what art?
Could twist the sinews of thy heart?
And when thy heart began to beat,
What dread hand, and what dread feet?

What the hammer? What the chain?
In what furnace was thy brain?
What the anvil? What dread grasp
Dare its deadly terrors clasp?

When the stars threw down their spears,
And watered heaven with their tears,
Did he smile his work to see?
Did he who made the Lamb, make thee?

Tyger! Tyger! burning bright,
In the forests of the night,
What immortal hand or eye
Dare frame thy fearful symmetry?

First - tiger can not burn. By such obvious "contradiction" we jump
right into abstraction (and look like non-mathematical to someone, who
thinks that math and contradictions don't go hand-by-hand). We take
two symbols - "tiger" and "burning" and say that those two have some
common abstraction. Thus we define an abstraction, which is the
solution to this contradiction. We could as well show two birds, two
fishes and two stones and ask - what they all have in common? Twoness.
If we want to show twoness, we surely show three different pairs and
state them being the same on some abstraction level (counting).

Next - we declare that there is a thing "deeps or skies" (what is
common between zero and infinity?) in which something burnt the fire
into eyes of our Tiger (and we have defined before that this tiger
*is* her eyes, because we specifically stated that the whole tiger is
burning). Thus, in an infinity-zero is something burning fire in
something, what has the properties of ability to burn and ability to
see (the latter is common abstraction for eye and tiger). Tiger is
also strong, thus we might suggest that strength and fire have a
connection here - tiger is red and strong, fire is red and strong.

In following lines, the poem actually goes into enormous complexity
and then goes back to beginning - showing, not, not the question, but
an answer, because we have put together a number of facts already.

And here we actually see, how poetry has done the following:
* Made the purified abstraction - and I doubt if it's possible or
reasonable to express the same abstraction as math, but I am sure that
the abstraction is worth expressing and not less exact than math.
* Connected this abstraction with a set of symbols - we can see it's
relation to parts, to whole and to many experiences of us all; this is
math, because it applies to everything.

This, now, shows, how an I Ching, Taodejing or other such stuff might
be the same kind of endeavours as logic of Aristotle. We also see,
very clearly, the limitations of that logic. We will understand more
deeply, why we need different paradigms. And last, but not least - we
can understand that many things we have opposed to logical directness
as metaphorical obfuscations might be actually the same thing as
logic; they might be (and often are) designed to remove the
complexities and obfuscations and to conjecture a set of symbols and
rules, which would apply on all cases - wherever you match the symbols
or relation (like yin-yang), you can use all implications.

And this is, actually, how our mind works - it abstracts things out,
symbolizes things, connects abstractions and symbols. And it is also
able to grasp the wholes; at least we could assume that it can grasp
the om symbol - but not without growing. Without growing in size, mind
will grasp an abstraction of om symbol - moving our mind in such
direction, we would drop on land completely different from ours, but
as small detail of whole as ours. When we would cover the whole
Universe, but retain the size of our mind, we would simply be in
different point - we would live in a kind of generalization and see
big, big universe around us. We would be as abstractions, as symbols,
as parts and as small as we are now. Just the world around us would be
different.