Copyright © 2001 Chris Lofting
(This is made-up of two sections. The first one deals with derivations of Logic
etc from the more concrete sensory systems and the second with the implied structure of the analytic mind including
the derivation of the Sierpinski gasket from boundary/colour processing leading to Pascal's Triangle etc being
'hard coded' into the species. This latter also touches on Charles Peirce's categories. See another page covering analytical + dialectical logic
)
Section 1 - A and NOT A : Logic from the Senses and the Source of Semiotics
Section 2 - Charles Peirce's Categories, Self-Reference,
and the Structure of the Analytic Mind
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SECTION 1:
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With the current amount of data available describing neurocognitive/affective processes within the species I believe
we can clearly identify the source of our abstract analytical skills as being in our senses and in particular the
senses of vision and audition in that these senses allow for processing both near-by and far away data when compared
to the olfactory, gustatory, and kinesthetic senses and as such have taken-over our brain or more so become the
core elements in the formation of a hybrid sense - the Mind.
From the Brain's processing of data we can identify:
(a) The concept of the excluded middle as having its source in the development of the visual system to identify
clearly, to establish EITHER/OR determinations.
(b) The source of communicating meaning as well as the cause of arguementation at the Mind level reflected in the
abstraction of the Brain's way of dealing with sensory paradox through the method of oscillations. (e.g. A/NOT
A sharing the same space as in Necker Cubes etc)
(c) A developmental pathway in identifications from a focus on attributes to focus on relationships and so a logic
of attributes vs a logic of relations. (there is an implicit development here of 'dot' precision FROM relational processes ('field' precision) that
we only become consciously aware of through encapsulation. Thus an explicit perception of attributes-to-relationships
(particulars to general) hides the implicit development of encapsulation from general-to-particular)
(c) follows from (a) in that (a) reflects a sense of identity-establishment that is then refined through (c) with
attributes (properties) and relationships (methods) analysis. In 'lower' lifeforms identifications seem to be made
more by attribute identifications, for example, a chick, if having had a negative experience with an object coloured
red, will bypass all red objects regardless of differences in object structure. This avoidence reflects primitive
logic being applied at the level of propositional calculus with its emphasis on ALL as compared to the more refined
predicate calculus that includes the notion of SOME. (Note that this sensitivity to attributes is also common in
humans who are autistic, any expertise is in ONE theme that is highly precise, all aspects identified - as we witness
with Savants).
At the Brain level we identify two perceptual biases - one of which favours a sense of the particular with an emphasis
on clear, precise, identifications and includes details analysis through the use of self-reference. This perspective
is an analytical perspective and focuses on staying 'in the box' to acquire details on whatever is under analysis.
Overall the emphasis in the analytic is a Logic of Attributes bias where the focus is WITHIN something.
Thus the analytical perspective works by encapsulation that enables a precise identification but is a little too
mechanistic at times in that the notion of thermodynamics is ignored - time is treated as something 'eternal' when
compared to 'realtime' that is definitly begin-end oriented and as such reflects entropy at work. Note how this
distortion of time favours a sense of ALL, the 'eternal', vs that of SOME.
The other mode of interpretation is best described as being dialectical in that it is sensitive to dynamics, to
cyclic & morphic processes and so a bias to transformations (the analytic perspective seems to be more biased
to transcendence). Note that Kant's notion of synthetic truths, where we step out of the box to recruit other boxes
to identify a truth, stems from a dialectical perspective, whereas Kant's notion of analytic truths, where we stay
in the box to establish truth, stems from an analytic perspective (Hegel is also here with the emphasis on only
knowing something by knowing its opposite and so a more 'dialectic' perspective)
The dialectical perspective focuses on patterns of begin-end and in general the joining of coordinates to identify
something. This is a geometric perspective in that joining the dots, forming a constellation, IMPLIES a meaning,
as we find in basic Logic of Relations with its focus on BETWEEN many things; the analytical perspective on the
other hand is 'clear' and so EXPLICIT about a meaning but lacks in-depth relational identifications; the analytical
is more 'literal-minded' when compared to the more 'metaphor-minded' of the dialectical.
Since we can identify structures within the Brain that process data analytically or dialectically, so we can identify
Logics of Attributes and Relations, where Logic serves to intergrate, as being 'hard-coded' into our Brains and
as such recruited by our Minds to transcend the Brain.
Note that the Brain works more on making 1:many distinctions in dealing with the everyday - this is dialectical.
To zoom-in for details reduces the many to allow for a 1:1 perspective - this is analytical. The success of this
'zooming-in' has allowed for the emergence of a perspective of the everyday that is analytical
in form and at time 'delusional' as compared to the dialectical that
allows for illusions. Thus in 'formal' Logic which lacks understanding of well-defined dynamics concepts, the concept
of Obsessive-Compulsive Disorder is 'illogical' and rejected whereas the disorder is very real. Thus any 'logical'
mapping of reality needs extention to identify these sorts of disorders etc.
The Logic of Attributes favours a Stimulus/Response (S/R) bias where the emphasis is on patterns based more on
syllogistic processes (Aristotle's Logic) as in:
A is Y
ALL Y are M
therefore A is M.
The 'Y' is an attribute of A as M is an attribute of Y. This logic, although clear and precise, is 'sterile' due
to its 'purity' in that one thing the universe is is diverse - mixed. (The purity in the analytical is in the 256
algebraically-determined forms of syllogisms. These are then reduced to about 19 'meaningful' forms)
The Logic of Relations (Frege, Peirce, Russell, Tarski etc etc) takes on a more Mediations/Representations bias
which goes beyond S/R in that the relational focus means an emphasis not on just the stimulus and the response
but the space in-between; a realm of possibles. The purpose here is the formation of a representation of the whole
S/R event in the form of a symbol and/or in the form of a habit.
The process of mediation is an oscillatory process that acts to BIND the elements of the S/R into a 'whole' that
becomes the symbol/habit. This oscillatory process seems to be rooted in the Brain's method in dealing with paradox,
the notions of A and NOT A sharing the same space which leads to an oscillation as the brain tries to 'best fit'
the perception by firstly presenting A and then NOT A and then A etc etc.
In the realm of S/R so the R is 'NOT S' and deriving meaning requires oscillations that can lead to an encoding
of the whole event, a 'transcendence' in the form of something (a word etc) that compacts the whole S/R into a
single 'space' at a 'higher' level of understanding and that compaction then becomes communicatable where, once
transmitted, other Brains treat it as a complex state and 'de-compact' the A/~A elements into 'meaning' (or misunderstanding
where the complexity appears to be non-reducable due to a lack of understanding due to limits on the pool of categories
used to de-cipher data etc) The token that encapsulates the S/R can work both as a stimulus (symbol) and as a response
(habit).
Thus the mediation process acts to achieve this 'compactness' where success is in the form of a Symbol of the whole
S/R event and as such serves as a source of stimulus itself, as well as in the formation of a habit - an instinctive
response to the WHOLE S/R event (or more so a variation of response to the orginal - a refinement in response).
Overall the formation of a symbol/habit expresses the concept of BONDING where the representation is not 'the thing'
but as close as possible to 'the thing'. (the BOND is the sharing of the same space by two+ elements intent on
keeping their separate identity and as such the symbol is a SIGN)
Thus symbols and habits reflect complex encodings of discrete objects and their relationship(s) and as such reflect
the recruitment by the Mind of a 'concrete' process of the Brain not just to decode complex states but to create
complex representations in that the created symbol/habit are abstractions of the original S/R event. These abstractions
can in their own right operate at the level of abstraction as if concrete and as such a level of abstraction can
become a 'base level' perspective seemingly 'free' of any underlying processes - this reflects some notions regarding
Mind and its relationship to Brain (e.g. see http://pages.prodigy.net/lofting/light.html )
The combination of the visual system's 'need' for clear identifications together with the processing of pardox
by oscillations seems to have led to the emergence of the abstract notion of Logic and Language in general. The
Audition system, with its wave emphasis and allowance for the superpositions of harmonics to make chords but also
retain the ability to 'pick-out' each harmonic, reflects a 'higher' degree of precision than vision (where the
excluded middle takes visual harmonics - red, green, blue colours - and combines them into a unique colour NOT
'pickable' with aids of some sort.). As such the spoken word has replaced indications where we no longer need the
context in which to work, to point at, we fold-up the context in words and take it with us to tell stories far
away.
High precision introduces probability concepts where a 'dot' has a range of expressions depending on the context
rather than the eternal 'IS' vs 'IS-NOT' distinctions of the excluded middle concept and as such we need to develop
such concepts as 'fuzzy' logic to deal with superpositions etc.
Working within the concepts of analytic/dialectic and working with the notions of oscillations and the excluded
middle, as well as including the notions of transformations and transcendence, notions linkable to basic neurological
processes, we can identify the following:
(1) The 'fundamental' nature of Logic and Arguement rooted in the Brain's analytical/dialectical methods of dealing
with reality. In particular, the oscillations of A/NOT A of the concrete Brain seem to be have been recruited and
generalised as the foundations for solving all 'paradox' where my assertion of A and your assertion of NOT A is
perceptually an expression of paradox such that for BOTH of us instinctively our Brains drop into 'paradox' resolution
mode, expressed at the Mind level as arguement.
(2) The failure in explicitly linking/reducing Mathematics/Mind to Logic/Brain in that the former reflects transcendence
and as such contains elements that are irreducable to Logic/Brain other than the identifications of 'oscillations'
in Logic/Brain that IMPLY Mathematics/Mind. Furthermore the transcendence process allows for abstractions to take
on a sense of being explicitly irreducable such that attempts to reduce can lead to only implicit identifications
in the form of the oscillations that led to the transcendence in the first place. This perceived irreducability
allows for an abstract level to become the 'base' level, to be treated as if 'concrete' and so abstractable even
further.
(3) Due to the oscillations in the Brain across analytically and dialectially biased work areas (as on left/right
hemispheres etc), Mind is expressed as a hybrid form of analytical/dialectical processing that can be used as feedback.
For example the concept of Logic is an encapsulation and as such will reflect more 'purist' concepts however the
feedback process acts to include the perception within an encapsulation such that the relational bias of the dialectical
perspective become observable as that which was encapsulated is 'fleshed-out' to a degree where we need to step
out of the box to identify.
This 'stepping-out' indicates that the nature of reality for us is always implied no matter how precise we get; the analytical is a methodology that allows the refinement,
the modification, of the 'everyday' that is the Universe but despite the explicitness of the analytic the Universe
can contain complex patterns irreducable to us since these patterns express sensory data outside the realm of the
senses of the Species; all we can ever 'see' are oscillations. (or else we mutate ;-))
(4) The step-up/step-down process we see in communications, where A/~A gets converted to a complex pattern to be
later 'de-coded', is a Brain concept recruited and abstracted by the Mind to enable precise communications with
a limited cost in energy where the 'natural' paradox-processing of the Senses/Brain has been recruited and now
works to 'reveal' meaning for the Mind. (this is highly energy-conserving and reflects the recruitment process
that favours the 'dont rewrite the bible' perspective)
Recent work has shown that the oscillations of the Brain lead to the emergence of Mind states and as such we can
identify Mind states as oscillations but not identify the location of Mind in Brain other than by these oscillations.
The variance in these oscillations, the accumulated timing differences, become reflected in behaviour where the
underlying neurocognitive biases for parts of the Brain emerge to dominate thinking in general and so demonstrating
Mind is tied to Brain. (Some favour Mind as independent and so taken literally, others favour Mind as a metaphor
for describing the Brain in a 'higher' gear. Transcendence allows for one or more Minds to emerge within the Brain).
(5) The process of Sign-making stems from the mediation/representation process (Transform/Transcend) which is at
a 'higher' level of function than the stimulus/response process where the latter is mindless and the former mindful
(but the former also reflects mindless mediations in the form of genetics).
(6) The formation of Signs reflects the more dialectial emphasis in nature where we link 'dots' to give us an IMPLICIT
identification. This linkage is refined in the A/~A oscillations where out of the oscillations comes a 'transcendence'
in the form of labels - words, images etc. These labels, although indicative of 'something', are not 'the thing'
and as such take-on the structure of Icons (representation by reflection), Indices (representation by indication),
and Symbols (arbitrary representation) etc. However, due to the transcendence emphasis these symbols can be treated
'as if' real - IOW literally rather than as Signs. Here we see the 'map is not the territory' issue.
(7) Logic as a whole must reflect all harmonics WITHIN an object as well as BETWEEN objects in that the object
distinction is a fundamental property of the species (forced by the A/NOT A of the excluded middle). As such, all
properties of vision/audition reflect properties abstractable to Logic which must include A/~A oscillations processing
- we see the beginnings of this in Spencer-Brown's calculus of indication (as well as Varela and Kaufman's refinements).
(8) One of the major distinctions in the Brain, regardless of sense, is that of WHAT from WHERE. This reflects
the object/analytic emphasis (WHAT) as compared to the relationships/dialectic emphasis (WHERE). Since the latter
is dependent on relativity it lacks the 'dot' precision (which is self-referencing) of the WHAT but maintains the
'field' precision of the WHERE (others-referencing and so a strong sense of the IMPLICIT).
(9) From (8) we can derive a 'root' set of categories based on applying self-reference to the WHAT/WHERE dichotomy.
This initially appears as an analytical bias BUT if we remove the historic emphasis in the dialectical there is
no perception of 'difference' between the two perspectives. IOW a 'root' perspective, as in the mind of a new born
child lacks all awareness of history other than that which is encoded in the genes as a form of species-history
and as such these are ignored. What is implied here is that since we have no sense of history/time so the analytical
perspective will 'rule' the infant, or any 'newborn' for that matter - and this seems to be the case in that the
Logic of Attributes will be preferred (as in the chicks and the colour Red, autistics and their single contexts
etc. I also emphasise that collectives can form with analytical perspectives that reflect 'child-mindedness' in
behaviour even though 'adult'. Since transcendence is strongly tied to the analytical so any transcendence will
be manifest as an intitial period of 'child-mindedness' as we deal with the 'new'. It is noteworthy that fundamentalist
collectives - secular or religious, often reflect 'child-mindedness').
(10) From (9) it follows that the set of categories identified will serve for BOTH perspectives - the analytical(statics)
and the dialectical (dynamics) but serve more as a source for metaphor creation since the general 'root' set of
categories will be too general to serve as communicatable forms; IOW we need A/~A oscillations within a specific
context to achieve transcendence/transformation and so something communicatable as 'meaningful' since the Brain
DEMANDS abstraction; just sending 'what/where' distinctions means we are sending irreducables (since they are the
BASE representations), we need to send the abstractions that include the link of the base representations to a
context.) - this step-up leads to the required step-down to A/~A oscillations and out pops 'meaning'.
(11) Since the focus here is on self-reference so what emerges from self-reference is a binary tree format. Going
through three levels of reference:
[1]A (self)
[2]A :: NOT A (self/not or 'negative' etc)
[3]A : NOT A in A :: A in NOT A : NOT A
We can see from these the roots of logic where level [2] reflects propositional calculus (ALL) and level [3] reflects
predicate calculus (ALL + SOME). There is a strong sense of WHOLE:PARTS in this derivation that does not get involved
in rich dynamics until level [4] where the focus is on:
Objects:
Wholes, Parts
Relationships:
Static, Dynamic
This level has EIGHT states - an octet, 2^3 - where to the above four distinctions is associated a variable mapping
a qualifier dichotomy - e.g. positive/negative, expansive/contractive, text/context etc etc).
These eight states reflect the minimum level of analysis required to clearly differentiate objects and relationships.
Otherwise, if you try to work with objects/relationships in more 'compact' spaces you will come across 'oscillations'
where the one space contains TWO categories - one static, one dynamic and so uncertainty in identification. (This
is reflected in Charles Peirce's notion of thirdness where we find an oscillation between mediation (dynamic) and
representation (static).) In these compact spaces it is CONTEXT that determines the actual value to be used and
as such shows us the lack in precision in that the context dependency forces us to think outside of the box and
so fail to identify 'clearly' or 'perfectly'; a failure to maintain 'purity'. (for a fuller description of a proposed
template of meaning derived from self-reference of WHAT/WHERE - see http://pages.prodigy.net/lofting/plate.html
)
(12) The recruitment process in the Brain/Mind, focused on self-reference, will take the eight states mentioned
above and apply them to themselves where the whole set of categories, being categories, are applicable to fleshing-out
the particulars of EACH category. This shifts us from eight categories to a set of sixty-four and we can do this
ad infinitum based on the set of mathematical examples of self-reference - tetration, exponentation, multiplication,
and addition. What stops the process either the loss of resolution needed to identify difference or the lack of
enough difference such that whatever is being analysed runs out of difference.
These examples of recruitment reflect the recruitment of different dimensions to 'flesh-out' details, thus addition
recruits one dimension at a time to gain details where as full recruitment is more like tetration where all dimensions
in the current are applied to the current. In an octet we would see a sequence of 1-8-64-4096-16Million+ reflecting
the intensity of the recruitment process; this is obviously more 'intense' than exponentiation which is more intense
than multiplication which is more intense than addition.
(13) To return to the visual system and the source of 'meaning' derivable from self-reference - the eye is responsible
for colour processing and in particular the gathering of information on the THREE harmonics that aid to construct
light - Red, Green, and Blue (complemented by their destructive form, Cyan, Magenta, Yellow). [A non-colour trichotomy
is also derivable from edge detection of bright-dark-boundary]
The three harmonics are linked to specific receptor cells in the eye such that the 'whole' image is defined by
working within the confines set by the THREE harmonics. What this light processing emphasises is the capturing
of 'out there' and as such a 'dynamic' and seemingly 'random' set of data and the 'bouncing around' within the
bounds set by the RGB.
What is notable in this 'bouncing around' is that once the data is 'in' the RGB box, which is triangular, and it
stays within the box but the self-reference used in detailed analysis of data by the Brain elicits feedback processes
that work like reduction and the R, G, and B are like lens that reduce an image to only R or G or B and then intergrates
them into a whole.
The self-reference ensures a reduction of 50% (1/2) of the previous 'something' for each level of analysis such
that after a few iterations of the self-reduction process WITHIN the 'triangle' of RGB a pattern starts to emerge.
This pattern is called the Sierpinski gasket. What is so special about the gasket? (for more on Sierpinski and
complexity/chaos etc see:
Peitgen, Jurgens, & Saupe (1992) "Chaos and Fractals : New Frontiers
of Science" Springer-Verlag)
SECTION 2:
---------
One of the more interesting discoveries in the realm of complexity/chaos theories is the consequences of encapsulating
a 'random' process. The discovery relates to the fact that once encapsulated the process can 'bounce' around within
the borders of the encapsulation and in doing so start to 'settle' in dynamics and form a pattern.
One particular pattern is called the Sierpinski gasket, named after the Polish discoverer of the pattern (see http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Sierpinski.html as well as http://math.rice.edu/~lanius/fractals/
)
This pattern reflects the concept of self-reference (and so self-similarity) where the act of encapsulation forces
all reference to be contained 'within the box' as as such be self-referencing.
Can we 'see' this in the human brain, where the taking-in of sensory data is akin to capturing the 'random' and
the enclosure is the brain/mind operating within the skull? Yes we can.
A fundamental tool in Mathematics is the binomial theorem ((1+x)^n) and its manifestation in the form of Pascal's
Triangle. (see http://ptri1.tripod.com/
for full details, but we can generalise where any emphasis on exponentiation
will reflect patterns of self-reference) When we make the distinctions of odd numbers from even numbers and colour
them (or just blacken the odd) out pops the Sierpinski gasket.
The process of deriving Pascal's triangle is rooted in self-reference in the form of applying the function (1 +
x)^n where n = 0,1,2,3,... and as such we identify a pattern in self-reference in that the act of self-reference
imposes order and enables precise identification of 'details'. (as emphasised earlier, exponentiation manifests
self-reference at work)
This order, due to the self-reference, will have the property of 'seeing' the whole in all derived parts and so
a 'fractal' emphasis will emerge, implicitly detected in Pascal's Triangle by the odd/even number colourings to
bring-out the Sierpinski gasket.
We can here generalise to say that the encapsulation of 'out there' will at least implicitly lead to the identification
of the binomial theorem as a 'fundamental' element of that encapsulation. This MUST include patterns emergent from
Brain/Mind activities where the encapulation of any set of data will format to patterns WITHIN the S. gasket.
We can identify some of these patterns in stable developments where there seems to be a range of text-to-context
relationships in that patterns in development emerge based on considerations of past context as influencing present
development. Thus reflections on the previous two contextual frames - regardless of scale - will lead to the emergence
of patterns we associate with the "Golden Mean", a ratio of 1:1.618 and allocated the name of PHI. As
the above links show, the Fibonacci sequence is encoded within the S.gasket. Also encoded in the S.gasket is the
other end of the development scale, where consideration of ALL previous contexts reflects a ratio of 1:2 and the
emergence of the Binary sequence (the sum of the coefficents of each ROW of Pascal's Triangle equal a member of
the Binary sequence based on 2^n).
The sequence that reflects no consideration of context is in the form of 1,1,1,1,1 where each moment is 'unique'
and this is expressed in the initial and terminating coefficients in each row of Pascal's Triangle.
The identification of a binary tree format within Pascal's Triangle, and so implicitly encoded within the S.gasket
(the function is (1+x)^n where b = 1) reflects the ease in which we can detect binary sequences within enclosed,
self-referencing, spaces.
With all of the above in mind, lets go off on a tangent for a moment.
The American philosopher Charles Peirce (http://pages.prodigy.net/lofting/peirce.html ) developed a perspective of reality based on the notion of trichotomies rather than
the more 'traditional' notion of dichotomies, and the trichotomy notion was fundamental to his derivation of a
set of categories he applied to mapping reality and in particular to his development of Semiotics - the 'science'
of Signs.
Peirce's categories are notions of:
Firstness
Secondness
Thirdness
Generally speaking, these notions reflect Stimulus(firstness), Response(secondness), Mediations/Representations
(thirdness).
Of importance here is the dual occupancy of the notion of thirdness by the concepts of mediation and representation.
In Peirce's writings there is a strong contextual sensitivity on EITHER mediation OR representation and this oscillation
is only stopped when we move to dealing with OCTETS (2^3) where a full set of categories emerges that is clear
enough to avoid the oscillations. (Peirce intuitively picked-up on this oscillation but lacked the knowledge we
have today to be able to explicitly map things out and at the same time had an attraction to threes that would
have 'skewed' his thinking ;-))
Earlier I have suggested that oscillations in thinking are rooted in the mindless, concrete processes of the brain
and its senses. In particular the root seems to be in (a) our neurology developing from the processing of visual
information - light - and (b) vision's enforcement of the law of the excluded middle.
What is important to recognise is that the derivation of Peirce's categories stemmed from self-reference in that
the assertion of the categories works as an encapsulation.
Given this encapsulation by Peirce, how is it that rather than identify a binary tree pattern he (a) focused on
trichotomies and (b) did not notice any 'problems' with the methodology until it was too late? By this I mean that
the assumed successes in the original work but when attempting to derive larger numbers of signs categorisations
seemed to dissapear into ga-ga land?
The clue to this is in the Sierpinski gasket and more so Pascal's Triangle. Given the fact that encapsulation of
'reality' will lead to the emergence of patterns associated with the Triangle lets focus on the first four sequences
of coefficents:
1
11
121
1331
Firstly note the sum of each row reflects a binary sequence development, 1-2-4-8. Secondly note that at the level
of 'mindless' processing there is only the notions of self (1) at each level such that we have the notion of the
previous level and its continuation in the current level together with a finer distinction:
[a]1 (self)
[b]11 (orginal self + refinement)
[c]1111 (each of [b] + refinement)
[d]11111111 (each of [c] + refinement)
An aside here in that the Brain of the species has developed skillsets one of which is the encapsulation of data
to enable precise analysis (identified
earlier as 'analytical' thinking). Although sourced at the level of the neuron, this skillset is abstracted to
the level of the hemispheres of the neocortex where biases emerge with a more explicitly precise, more 'dot', more object, oriented left hemisphere
(in most) complements the more implicitly precise, more 'field', more relationships, oriented right hemisphere.
These left/right distinctions reflect not so much opposition as more the extracting of something from the field,
the left encapsulates, and so precisely names 'X' which can then be re-inserted back into the 'river' of life.
The encapsulation process MUST reflect the Sierpinki gasket in that the encapsulation process leads to self-reference
as the 'core' of identifying all components of that which has been encapsulated - AND THAT INCLUDES THE NOTION OF RANDOMNESS.
Now lets return to the first three rows of Pascal's Triangle. A fine level analysis will only focus on the self-reference
from a binary perspective - all 1s such that the focus is on the pattern of:
[a]1 (self)
[b]11 (orginal self + refinement)
[c]1111 (each of [b] + refinement)
[d]11111111 (each of [c] + refinement)
A LESS refined analysis will focus more on the formal derivation process of (1+x)^n which gives:
[a]1
[b]1 + 1x
[c]1 + 2x + 1x^2
[d]1 + 3x + 3x^2 + 1x^3
Now the emphasis here is that the process of encapsulation and self-reference becomes reflected in the method use
by the Mind to 'go for details' and the number of elements in the set of categories derived by this 'going for
details' is listed in the coefficients of x given in each row of Pascal's Triangle. Thus a dichotomous perspective
[row b] will give you TWO categories (as in A and NOT A). A trichotomous perspective [row c] will give you FOUR
categories but TWO are compressed into ONE giving you the perception of THREE categories.
The point here is that there is no differentiation on the ORDER of distinctions within the coefficients. What I
mean is that in a trichotomous system you have FOUR symbols:
00, 10, 01, 11
The P.triangle identifies the uniqueness of these symbols by summing the contents thus there is 1 symbol of all
0s, 2 of mixing 1s and 0s, and 1 of all 1s. In other words the mathematical methodology hides
dependencies (which are dialectical concepts) such that 01 and 10 are
'the same'. I propose that this 'hiding' of dependencies led to Peirce letting 01 and 10 share the same space in
the form of the distinctions of mediation/representation rather than flesh them out to be distinct properties;
the 'problem' with the P.triangle is this 'encapsulation' process that focuses on sameness by ignoring orderings
and this reflects a common theme in analytical thinking as compared to dialectical thinking in that in the realm
of the analytical Time is considered 'eternal' and even reversible and so there is no 'difference' in 10 or 01
within the context of 'pairs'.
However when we zoom-in to the binary tree format which lists all combinations we see differences, we experience
differences, where, for example, the notion of relationships is clearly 'cut' into the distinctions of static relationships
and dynamic relationships. (see the template diagram http://pages.prodigy.net/lofting/plate.html
)
Furthermore we need to drop to at least the level of octets to fully map 'reality' with some degree of confidence.
The P.triangle just gives us eight elements grouped into:
000
001
010
011
100
101
100
111
Where (ignoring x) the relational elements are grouped into two lots of 3s:
1x + 3x + 3x + 1x. When fleshed-out *semantically* we find eight distinctions:
passive whole
passive static relationships
passive parts
passive dynamic relationships
active dynamic relationships
active parts
active static relationships
active whole
The reduction in the P.triangle does not make fine levels of distinctions, all 'non-whole' concepts are folded
into the general '3x' grouping since they reflect 'mixing' of 0s and 1s rather than the PURE concepts of 000 and
111. But the triangle, gasket are mapped by self-reference which is rooted in binary tree formation and out of
which we can acquire 'finer' distinctions that includes some awareness of dialectical distinctions.
Conclusion.
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I think you can 'see' from the above sections how something like the Sierpinski gasket, Pascal's triangle, binary
trees, notions of 'purity' and 'mixing', analytical and dialectical, and the different calculi of Logic have emerged
from our brain's 'mindless' adaptions to the immediate context - a world dominated by light and sound - and how
the Mind can 'transcend' the Brain and recruit all of these 'wonders' at a more abstract level to aid in mapping
reality and create our collective and individual worlds of abstractions - which can be 'delusions' if we are not
careful ;-)