Reconstructing Mythic Algebra
By Michael Griffin
Abstract
This is the tenth in a series of papers on an algebra, derived from mythology,
that can model symbolic processes. Previous papers used literary semantics,
mythology, semiotics, philosophy and mathematics. The main features of the
algebra are set-based elements and making association a new operation. While
such a system can be reductive, it need not be. It may reconstruct into a more
useful tool. Various implications for its foundations are considered.
6 Keywords:
algebra, association, mathematical modeling, numbers, reduction
1. Introduction
Mythic algebra began as a system to model
mythology, then expanded into narration, thence to cover a host of mental
processes using the operation of association (2003). Once the system was
completed, new applications were made into semiotics and mathematics (2008).
One may question just how far can a system go that is derived from mythology.
A guiding assumption was that the mental processes that manifested as myths
would also manifest as anything else, if only the underlying system was found.
The guiding principle was to use the features of
basic algebra, looking for group principles and two distinct operations. The
results became the following system represented in a formulaic lineup:
(p,q,x,y,s,t), M, R, M/R, R/M, ↆ, =, #, S, $,+, ౼, *, %
This lineup separates into levels as:
Sets (p,q,x,y,s,t) of elements people p and their actions q, things x and
their actions y, space s and time t.
Functions M, R, M/R, R/M, ↆ,
set states mythic M or real R that can map elements among them by / or just
alter elements or states by ↆ.
Relations =, #, S, $ of equality,
inequality, similarity, or dissimilarity.
Operations +, ౼, *, % of addition, subtraction, association or dissociation.
2. Discretion
Sets have discrete elements. By discrete, we mean
separate, distinct, and also countable if we cared to. The meaning of
countable will be explored in a later section, while here we look at the most
basic definition of elements, their separate, distinct quality. Whether
elements are in the same or different sets, they remain distinct from each
other. A set of (A,A,A) has three A’s just as (A),(A),(A) does or the listing
A,A,A does.
Concrete reality clearly has things we
perceive as distinct from each other: a rock is not a tree even if we have no
names for them. Nature builds itself by using discrete elements, from atoms up
to cells and beyond. Alfred North Whitehead (1967) in 1925 noted this fact of
reality and examples of the advance in human thought once we recognize it:
The influence of atomicity was not limited to chemistry. The living cell is to
biology what the electron and the proton are to physics. Apart from cells and
from aggregates of cells there are no biological phenomena. The cell theory
was introduced into biology contemporaneously with, and independently of,
Dalton’s atomic theory. The two theories are independent exemplifications of
the same idea of ‘atomism.’ (99-100)
Our languages have modeled or mimicked this
reality of discrete elements by using discrete words, regardless of the
particular syntax or semantics. Labeling with words requires otherness,
difference. Before Derrida wrote anything on deconstruction, the Zen
philosopher Alan Watts (1959) observed the troubles that arise when making
discrete distinctions that may be arbitrary:
For the function of these nonsense terms is to draw our attention to the fact
that logic and meaning, with its inherent duality, is a property of thought
and language but not of the actual world. The nonverbal, concrete world
contains no classes and no symbols which signify or mean anything other than
themselves. Consequently it contains no duality. For duality arises only when
we classify, only when we sort our experiences into mental boxes, since a box
is no box without an inside and an outside. (80)
This “inherent duality” goes at least as far back as Plato and Aristotle, whose ideas were expressed as three Laws of Thought: 1. The law of identity [A is A]. 2. The law of contradiction [either A or not- A] . 3. The law of exclusion, or excluded middle [only A or not- A] (Law of thought, 2019).
Not only languages, but thought itself may become biased into useless or
useful dichotomies. As the Taoists noted millennia ago (2007), once you make
a discrete thing or quality you then have to have something else that is not
that thing:
For is and is-not come together,
Hard and easy are complementary;
Long and short are relative;
High and low are comparative;
Pitch and sound make harmony;
Before and after are a sequence. (2)
Cut out windows and doors
In the house as you build;
But the use of the house
Will depend on the space
In the walls that is void.
So advantage is had
From whatever is there;
But usefulness rises
from whatever is not. (11)
(Blakney 60, 70)
Some basic dichotomies exist in mythic algebra, too. There is a
static-dynamic principle to divide elements: static p,x,s or dynamic q,y,t.
There is also a distinction of persons-things or neither, as p,q or x,y or
s,t. This has been critiqued, for what defines a person whereas a sentient
creature like a pet is a thing? If this criterion is a human body, then
sentient minds don’t count. One has to define the elements according to
need.
Then the states of sets divide into the basic dichotomy of mythic M or real
R, with mythic defined as not-real. Generalizing states into F or G still
means that state G is not-F. Any number of distinct states can then be
notated, all defined as not each other: A,B,C,D,E,F,G,H,I,J,K,L,N,O,U,V,W,Z,
etc.
What criteria make them not
equivalent? It depends on the context. For mythic stories, the M-R
distinction was based on an M(s,t) mythic spacetime which was not in our
everyday world. For an F-G distinction it could simply be two separated
sets, though seemingly equivalent elements: F(a,b,c), G(a,b,c). F is this
one, G is that one.
So we now have sets of six basic
elements which can vary in their definitions. The state functions of these
sets can also vary in definition, just as long as they differ from each
other: M-R, F-G, etc. The elements and states are discrete because they
differ in binary polarities. Elements p,x,s are similarly static, yet p,q
are similarly of people but x,s are not. The utility of a people-things
distinction fits realms in which people figure, such as the human mind or
cultures. In realms of physical nature or pure mathematics, this may not
matter.
These four different realms of use will be
examined also in a later section. For now we can conclude that elements and
states of sets are only defined by the quality of being discrete from each
other. Another meaning of discrete is unconnected, which brings us to
consider the next levels of mythic algebra.
3. Plus -- or Not
The basic operation that all mathematics is
founded upon is addition. Once we have discrete units we combine them.
Undoing addition is subtraction, extended addition is turned into the
shortcut of multiplication, or its undoing is division. Mythic algebra does
not consider multiplication-division operations, presuming they are implied
by addition-subtraction. But what else may elements do besides combine by
adding? They may add to stay distinct, such as: A+B = A+B, or they may add
to transform, such as: A+B C, like 1+2=3 as a new number.
These kinds of operations work well with
discrete elements, but elements can also have a continuous connection apart
from this, even if they maintain their discrete character. Such connections
may not fit addition, but rather addition may fit into such broader
connections. This continuous connection or linkage has been modeled as the
* operation of
association. Some sketches can show the inadequacy of addition to fully
convey all possible connections of discrete point-elements:
Points A,B,C,D are linked as A* B, A* C,
A*D and arbitrary
measures of the paths may be taken as the magnitudes of points B,C,D. In the
first example, A+B+C+D = 2+2+2 = 6, and in the second example 3+3+3 = 9 yet
in both examples the paths are the same lengths. Any scale of distance does
not really matter to A* B, etc.
Or consider defining point A by its unequal
links:
On the left, A= 1+1+2+3 =7
On the right, A= 1+1+1 =3
Yet A and B are the same path and actual distance in both cases. Even B in
these examples has magnitude 2 or just 1. Traditional number lines avoid
this confusion of miscounts by keeping connections in a single dimension to
measure, so any differences of scaling can be compared:
A__+__+__B is 3/3 which equals A___+____B or 2/2
But such additive lines are single components of possibly endless continuous
links:
whether on a two-dimensional page or any-dimensional space. As if out of
Flatland (1884), a line connecting two points may only convey part of
their total linkage:
A________B
And if we only define the point's magnitude by measure along that single
dimension, we miss all of the other magnitudes and path-connections notated
by the asterisk symbol
*.
4. Levels
Besides the extra dimensions of connectivity,
levels of organization may upscale or downsize in a fractal-like style of
structure, maintaining mythic algebra at any level just as arithmetic does.
The results of any use of mythic algebra may be subsumed into a new basic
set element, its opposite elements then noted, and the usual mythic algebra
operations and functions performed on these new elements.
By opposite, I mean pairs of static-dynamic
elements. Even the space-time pair (s,t) can be a basic static-dynamic pair,
if we define space as the field that allows differing elements to occur, and
time as the force that allows change to occur, such as motion or mapping or
any action.
This makes a kind of reduction, but not
reductive to the single operation of addition, as arithmetic does, and the
edifice of mathematics built upon that. The properties of one level may not
be predictable from the association links' principles at a different level,
so no perfect reductionism exists. A structure can still be sketched to show
how one group of associations may be arbitrarily defined as the next level's
new basic elements:
Circled collections would be new element-points.
With the newer elements we may have only a
sketch of:
And such a rescaling of perspective itself need not be on a uniform scale
measurable by arithmetic.
However, natural science does model reality using
levels of scaling of uniform numerical measure. We proceed from the units of
ecology – whole organisms, to units of biology – body parts, to units of
chemistry, to underlying physics, to theoretical mathematical models of the
basis of physics. Each level uses numerical addition to construct the next
level, but even here we get new properties unexpected and not predictable
from earlier levels.
Still, these new properties can usually
be modeled with the same arithmetic-based mathematics that also is used for
reduction between levels. The addition operation, using discrete numbers,
does well to model nature. It is simply a starting fact without prior
justification, such as is remarked upon mathematics by Wigner (1960).
If additive numbers are just one aspect of
a larger mythic algebra using
* connections, then
anything modeled by mathematics may have other real links missed by the
traditional mathematics. It remains to prove such links and avoid the
pitfalls of false analogies and wishful, magical thinking. And perhaps
nature is so only because it is limited to arithmetical number-mathematics.
But any emergent properties in nature may indicate the missing
* links of the
association operation.
5. New Math
I have always viewed askance any system of claims to knowledge that can’t be
turned into a visual diagram or also into a structural symbolic system such
as mathematics or logic, even though I consider all such systems merely
provisional, pragmatic knowledge. Nonetheless, any long tome I consider as
talking a subject to death without getting to any deep understanding. This
is undoubtedly a bias of my own brain, and an urge to simplify and unify.
Since words to me are mere labels, I look for symbols that convey real
force, more reality than labels. Our scientific civilization has a few such
systems, mainly of mathematics, and perhaps this has been its appeal to me,
a feeling that it is truer than mere word labels. Yet such truths seem so
limited, that one can believe that a better, truer mathematics is possible.
Perhaps this has been an urge upon me, too.
References
Abbott, Edwin A. Flatland, A Romance of Many Dimensions, 1884 ed.
www.gutenberg.org/ebooks/201
Blakney, R.B. Tao Te Ching. New York: Signet Classics, 2007.
Griffin, Michael. ‘Looking Behind the Symbol: Mythic Algebra, Numbers, and
the Illusion of Linear Sequence.’ Semiotica 171, 2008, 1-13.
www.mythicalgebra.blogspot.com/2015_01_01_archive.html
-- ‘More Features of the Mythic Spacetime Algebra.’
Journal of Literary Semantics 32, 2003, 49-72.
www.mythicalgebra.blogspot.com/2014_11_01_archive.html
Law of thought, From Wikipedia, the free encyclopedia, last edited on 29 August 2019,
https://en.wikipedia.org/wiki/Law_of_thought#The_three_traditional_laws
Watts, Alan W. The Way of Zen. New York: Mentor Books, 1959.
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Science and the Modern World: Lowell Lectures, 1925. New York: The
Free Press, 1967.
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