The focus here is on how analysis of an esoteric discipline has brought about the discovery of a property of
recursion that is essential to understand for those dealing with self-referencing systems,
and that includes the brain.
The I Ching or "Book of Changes" is an ancient text having its roots in divination but then developing
into a fundamental source of Chinese Philosophy and influencing Confucian and Taoist thinking.
The book is made-up of symbols, hexagrams, derived from the recursion of the yin/yang dichotomy and so the whole
system is grounded in self-referencing. These symbols are formed from building, bottom-up, an image based on broken
(yin) and unbroken (yang) lines. For example here is hexagram 51:
If we assign 0 to yin and 1 to yang, we can demonstrate the recursion using bit (Binary DigIT) patterns for each
level of recursion but now expressed left-to-right rather than bottom-to-top. For demonstration purposes we will
limit ourselves to three-bit patterns:
0 / 1 (first level of distinction)
00, 01 / 10, 11 (second level)
000, 001, 010, 011 / 100, 101, 110, 111 (third level)
In the I Ching these eight three-bit patterns are 'trigrams' and come with unique qualities for each. Hexagrams
are often interpreted as two trigrams, one atop the other, and so from eight trigrams come all of their bottom/top
combinations to give us 64 hexagrams (e.g. 001-100 is hexagram '62'. The trigram of 'mountain' (001) is at the
bottom, the trigram for 'thunder', 100, at the top. In the above figure we have hexagram 51, 100-100, the trigram
of thunder in both bottom and top positions..Note that the numbering of hexagrams does not match their binary forms
but the numbers have become 'traditional' and so are retained in most texts despite their inefficiency in use)
A cognitive analysis of the I Ching hexagrams has shown that the generic qualities represented by these symbols
are isomorphic to qualities derivable from the recursion of the WHAT/WHERE dichotomy, the core dichotomy manifest
in the brain as it processes information (this dichotomy is specialist in name and can be generalised to a dichotomy
of differentiating/integrating).
We can map-out the qualities derived from recursing differentiate/integrate three times thus (using the bit notation of above):
111 - Wholeness categorised by differentiation
000 - Wholeness categorised by integration
101 - Partness categorised by differentiation
010 - Partness categorised by integration
110 - Static relatedness (invariance, sharing of space with another/others) categorised by differentiation
001 - Static relatedness categorised by integration
100 - Dynamic relatedness (sharing of time with another/other; representations of cyclic/morphic change) categorised by differentiation
011 - Dynamic relatedness categorised by integration
If we focus on the PAIRS in the above we find them to be isomorphic to the types of numbers we use in Mathematics:
WHOLE - whole numbers
PART - rational numbers
STATIC relatedness - irrational numbers
DYNAMIC relatedness - imaginary numbers
Thus from consideration of 'mindless' dynamics of the brain recursing WHAT/WHERE (aka differentiate/integrate) we have come across qualities fundamental to our mathematical representations of reality.
The analysis of the I Ching maps the generic qualities of the trigrams to the above eight categories of WHAT/WHERE:
111 - heaven (devotion to self, issues of self-wholeness by expanding, pushing-out)
000 - earth (devotion to another, issues of self-wholeness by contracting, drawing-in)
101 - fire
010 - water
110 - lake
001 - mountain
100 - thunder (issues of the 'new', sudden change)
011 - wind (issues of cultivation-based change)
Consideration of the above properties allows us to come up with terms more associated with feelings:
Blending - issues of wholeness
Bonding - issues of static relatedness (sharing space with another/others)
Bounding - issues of partness (boundaries, cuts of 'them' vs 'us' etc)
Binding - issues of dynamic relatedness (sharing of time - cyclic/morphic change)
When we add a qualifying dichotomy so we have the eight categories (expansive blending, contractive blending etc. This material is covered in more detail in the IDM pages. )
In other words the methodology in deriving meaning in the I Ching appears to be the same as that used by the brain
in general where the focusing of attention and the oscillations across the WHAT/WHERE dichotomy manifest 'passive'
recursion of the WHAT/WHERE dichotomy and so the elicitation of qualities mixing the elements of the dichotomy
just as in the I Ching we have the mixing of yin/yang.
Further analysis of the meanings of the hexagrams in the I Ching showed an interesting property, one that
appears to be a property of the methodology, I.e. recursion of a dichotomy.
This property is where the WHOLE is encoded in all PARTS (as in our full genetic code in each cell). What this
means is that for each hexagram we find that all hexagrams contribute to the description of the properties and
methods of that hexagram with LOCAL context distorting the contributions.
This initial discovery stemmed from a casual observation some years ago that if one 'flipped' the top and bottom
lines of a hexagram (using bit notation, convert 1 to 0 and 0 to 1) we would get a hexagram that seemed to describe
the 'generic' qualities of the hexagram in question. For example, the qualities of hexagram 01, 111111, are all
'yang', the focus is all 'male', all 'light' etc. If we flip the first and last bits we get 011110. This is hexagram
28 and it covers the concept of Excess - applied to 01, a focus on 'too much yang'; a good description of the infrastructure,
the skeletal form, of hexagram 01.
Further investigation showed that this process applied to all hexagrams, flip the top and bottom lines and out
popped a hexagram describing the 'raw' or 'skeletal' form of that hexagram. This was obviously a form of analogy
where, due to the limit in symbols (64, or 4096 when we include line patterns), combined with the I Ching supposed
to describe 'all there is', so symbols represented MANY contexts and as such, in most cases, were sources of analogy
in describing reality. What we see here is a self-referencing system being able to describe itself through the
use of 'bit' representations and the XOR logic operator.
In the original work, of noticing the line 1 and 6 relationship, the discovery was worked upon and interpretations
made as to how it came about but they were not comforting in nature - they rang more of rationalisations.
My background is in the IT industry and so always aware of programming needs and so familiar with logic operators
such as AND and XOR (Exclusive OR).
My interests are in the derivation of meaning by the brain and so I am heavily involved in analysis of brain functions,
one of which is in the processing of sensory paradox (for details see my web page on paradox
processing )
An observation made whilst reflecting on paradox was the extraction process of 'details' from a complex form; e.g.
the cubes of the Necker cube paradox from a 'complex line drawing'. This same form of dynamic is not limited to
vision, it also occurs in audition and as such indicated a neurological 'issue' rather than something unique to
a sense.
The complex line drawing is an integrated whole and so representative of the AND or IOR logic operator. The Necker
cubes represent the extraction from that data of a particular pattern and as such an XOR function in that what
is extracted excludes all else. Furthermore, the exclusion elicits the paradox in that we 'see' two different cubes
sharing the one space and we deal with that by oscillating between the two cubes - we present one, then the other,
and so an EXCLUSIVE OR dynamic.
The XOR operator works where, given x and y we create a third representation based on rules derived from this 'truth'
table:
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X |
Y |
X XOR Y |
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0 |
0 |
0 |
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0 |
1 |
1 |
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1 |
0 |
1 |
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1 |
1 |
0 |
Thus, XOR-ing hexagram 27 with hexagram 01 gives us hexagram 28:
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Hexagram 27 |
Hexagram 01 |
27 XOR 01 = Result - Hexagram 28 |
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(line 6) 1 |
1 |
0 |
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(line 5) 0 |
1 |
1 |
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(line 4) 0 |
1 |
1 |
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(line 3) 0 |
1 |
1 |
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(line 2) 0 |
1 |
1 |
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(line 1) 1 |
1 |
0 |
Put ANY hexagram in the first column and the XOR process will describe, by reference to some other hexagram, the expression of that first hexagram 'through' the middle hexagram - and so an expression of a part. - and so the 27-ness of 01 is described by analogy to the generic qualities of hexagram 28. This information is given in the seperate pages for each hexagram linked off the below table (under the heading "line positions"):
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Bottom/Top |
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Thus what has followed has been an extensive analysis of the XOR-ing process involving all hexagrams and through
that it can be reasonably said that we have discovered a feature of recursion not clearly identified in the literature
on recursion. For further example:
Lets consider hexagram 48 (011010) and its expression through hexagram 57 (011011) - we find that, using XOR, the
48-ness of 57 is described by analogy to 23 (000001):
011010 (48)
011011 (57)
-------- (XOR)
000001 (23)
57 is about cultivation, becoming influencial etc and so a focus on anticipation. This focus gets its nourishment
(represented by the qualities of 48, "The Well"), what keeps it 'going' from qualities represented in
23 - a sense of pruning, of housekeeping. I think you can see how this focus on pruning/housekeeping sets the foundations
for the development of full blown 57 with its focus on cultivation/anticipation.
On the other hand, the 27-ness of 57, its skeletal form prior to adding the 'furniture' etc (the notion represented
in 27), is described by analogy to:
100001 (27)
011011 (57)
------ XOR
111010 (05)
Hexagram 05 represents waiting for the 'right' moment to come along and to then take the opportunity. Thus the 27-ness of 57 is described by analogy where the basic sense of cultivation/anticipation is in waiting for the 'right' moment. That is the skeletal form that develops into full blown 57.
The feature of recursion is that for each row of qualities derived in the process, the qualities are all linked
together and the use of bit representations allows us to bring out this linkage using the XOR operator where we
can extract the expression of quality A through quality B in the form of description by analogy to quality C. (as
mentioned previously, this favours a 'wave' model and a focus on topological dynamics/holographic processes where
a unique frequency allows for the extraction of some detail from a complex wave form. The unconscious element at
work here re constructs etc is discussed in the page covering our processing of foreground/background
information)
For the I Ching this dynamic indicates the acquiring of a hexagram's "spectrum" where its properties
and methods are described by reference to all of the other hexagrams that make-up the row of categories when recursing
yin/yang to six levels (2^6 = 64). (Of note is that the quality of the information is poor at levels of 4 or less;
there is not enough differentiation to allow for clear perspectives.)
This XOR feature is NOT a property of the I Ching, the same dynamics have been found in categories of personality
types etc (MBTI) and in categorisation of human emotions,
it is a property of recursion and as such will apply to anything derived from this method. That includes, for example,
the derivation of DNA codons from the recursion of the purine/pyrimidine dichotomy.
What is implicit here is the generation of meaning 'codes' from recursion of dichotomies where the recursion 'down'
the page is, in principle, infinite, but in this process form rows of categories across the page where each row
is finite (due to the nature of the original dichotomy, these categories come in pairs).
Each row thus serves as a source of meaning in the form of categories usable to encode/decode and so represent
information but in the form of analogies/metaphors where the finite nature of the set of representations means
each category is used many times to represent something where the only difference is the local context. IOW we
are witnessing here the roots of all languages be they molecular, emotional, or the written/spoken word.
This focus on association of 'universals' and a local context takes us into the realm of specialisations where
the ONE set of categories is used as many times as necessary to represent 'all there is' with the use of re-labelling
allowing for the link of universal with local context. In this process each specialisation will create their own
lexicon with which to describe their unique context (as has the I Ching, Mathematics, the MBTI etc etc where all
have their roots in self-referencing - different labels, same qualities). This customisation process is a property
of 'small world networks' where the set of POTENTIAL expressions is exposed to a 'random'
environment and the potentials become actualised through the dynamic. This dynamic elicits 'smaller
world networks'.
Despite all of the apparent differences of each specialisation, they all share the SAME set of core categories
and it is this sameness that allows us to use one specialisation to flesh-out details of another - e.g. Mathematics
applied to Physics or to Sociology. IOW analogies/metaphors are fundamentals in our derivation and communication
of meaning.
The advantage of some specialisations over others is in the representations used (E.g. Mathematics or yin/yang)
and in these we can 'see' all others as in all others we can 'see' these.
What is of fundamental interest here is that the XOR dynamic, being a property of self-referencing, applies to
all self-referencing models of reality and so showing us 'hidden' meanings encoded into non-linear systems; IOW
there is a lot more going on than we have previously imagined.
