I Ching PLUS : MATHEMATICS AND THE I CHING

MATHEMATICS AND THE I CHING

INTRODUCTION.

Mathematics and the I Ching touches on such concepts as recursion, exponentiation, tetration, scalars and vectors, probabilities, randomness, binomial theorem (Pascal's Triangle), complexity/chaos (in the form of Sierpinski's gasket), and the universal category of number types, their qualitative nature, we use in Mathematics.

RECURSION/EXPONENTIATION.

In the I Ching's case exponentiation is expressed as powers of 2; tetration I will describe later. The method, exponentiation/tetration, seems to reflect how our brain derives details on ANY sensation, where the brain uses self-referencing in the form of recursion where we take X and apply it to itself.

In Western civilisation, the roots of the I Ching are reflected in the work of Aristotle and the notion of a 'Universe of Discourse' and the concept of A/NOT-A (a dichotomy) which reflects our analysis of details of a sensation and so a sense of EITHER/OR.

In the I Ching the 'Universe of Discourse' concept is expressed in the concept of "T'ai Chi'. Both concepts reflect the focusing on a 'whole' where the intent is to derive details of that whole, to reveal its parts, its relationships of parts to whole and the relationships of the whole to other wholes and as such what the whole IS as compared to what the whole IS NOT.

This process is done through the recursion applied to the concepts of 'yin' and 'yang'. Consider these as basic qualities with which we categorise things in reality; yin is negative to yang being positive; yin is female to yang being male; yin is 'what could be' to yang's 'what IS'; yin is 0, yang is 1.

It is 'obvious' that the world is more than just yin OR yang and here we move into recursion where we take the yin/yang dichotomy and apply it to itself; we apply yin/yang distinctions to yin and to yang. This fleshes out more details such that in yin there is yin but also elements that reflect a degree of 'yangness'. Thus the first level of derivation from yin/yang takes us from two qualities to four:

Pure yang (yang+yang)
Mixed yang (yang+ yin)
Mixed yin (yin+yang)
Pure yin (yin+yin)

Symbolically, we use an unbroken line to represent yang and a broken line to represent yin. Thus the four qualities expressed above become:

Pure yang:

Mixed yang:

Mixed yin:

Pure yin:

Note these symbols are created BOTTOM-UP. Thus when you add another line it goes on top of these lines. That other line can be either yin or yang and so the above four symbols develop into eight symbols - called trigrams in the I Ching - this reflects the exponential at work where each step in the recursion reflects developing categories at the rate of 2ⁿ where n is the level of recursion (such that 2³ gives us 8 categories/qualities we can use to describe qualitative differences within whatever we are analysing) - also see comments on Binomial Theorem below.

TETRATION.

We can keep going by adding yin and yang lines, reflecting the exponentiation process, but there is a 'quicker' way to reach the goal of 64 hexagrams and that is to take the eight trigrams and map to each trigram ALL of the others. This reflects the concept of 'tetration' which is 'faster' than exponentiation. Thus we take the 8 and apply them to each other by placing one trigram atop another, giving us 64 six-line symbols, these are called hexagrams. This step 'jumps' us from 2³ to 2⁶ directly rather than going through the slower steps of 2⁴ and 2⁵.

Our brain seems to use tetration where it (1) recruits the current set of 8 qualities and then (2) uses them as sources of analogy to describe differences WITHIN the existing 8 qualities, giving us 8 octets, sets of 8 hexagrams that can be used to describe differences WITHIN one quality. (3) the analogies are taken literally and the 8 octets now become 64 'independent' qualities used to describe anything (the brain LOVES to recruit and abstract as it tries to interpret things)

WHAT/WHERE.

Once we have a good set of qualities, 64 expressions in the form of hexagrams, they are then used as a 'filter' with which to derive meaning from reality; the I Ching is an 'ontology' - a general parts list of the qualities within 'reality' which we use to interpret reality.

From the brain perspective, ANYTHING is analysable using the set of 64 qualities, which if need be we can, using tetration, 'refine' into 4096 qualities (64 x 64) or even 16 million+ qualities (4096 x 4096). These extensions are often not required since what we are analysing may not reflect all of those qualitative differences! - we lose the resolving power needed to identify differences. However, in the I Ching we actually utilise the range of 4096 categories/qualities (64 x 64) - see MOVING LINES below.

The method of self-referencing, through exponentiation and tetration (the 'other' forms of self-referencing are slower in development - they are the use of addition and multiplication) is a method our brain seems to use. The brain works by recursion of the dichotomy of WHAT/WHERE where the term WHAT is refinable to give us the distinctions of WHO and WHICH, and the term WHERE is refinable to give us the distinctions of WHEN and HOW. We can abstract these concepts to that of OBJECTS and RELATIONSHIPS.

The recursion of the WHAT/WHERE dichotomy gives us a set of qualities which we can use to derive 'meaning'. The problem is that this is a SPECIES-wide set of qualities, it is too GENERAL in that all it gives us is a notion of 'wholeness' but we cannot communicate the differences of wholes, we cannot indicate one whole from the other, other than by physically pointing to them and saying "THAT whole is different from THIS whole".

Our evolution has 'solved' this problem where we take the set of general qualities and link them to a specific context and from that linkage develop a set of words, a lexicon, specific to the qualities-context relationship.

The I Ching is an example of this linkage where the specific context was the Chinese perspectives on categorising reality through the use of the concepts of 'yin' and 'yang'. (in mathematics this is like taking digits and creating numbers.)

Thus in the I Ching we have TWO hexagrams that deal with "the species concept of 'pure' wholeness" (which is different from DERIVED wholeness) in more refined terms based on the concept of 'pure yin' and the concept of 'pure yang'

SO FAR: we have taken the set of GENERAL qualities of the species and tied them to a PARTICULAR perspective of reality - the yin/yang concepts of the I Ching. What many don't realise is that the WHOLE SET OF QUALITIES IS APPLICABLE AT ANY ONE MOMENT OF TIME. For any moment what we seek is a bias, a particular hexagram that maps to the current moment - we seek 1:1 relationship,a 'best fit' of the set of hexagrams and as such an ORDERED LIST of hexagrams with the 'head' of the list being the hexagram that 'best fits' the situation under analysis. This ordering emerges out of the property of the I Ching as an interpreter of reality in that the I Ching reflects 'all there is' and the set of qualities expressed in the symbols of the I Ching are all applicable to any moment, sorted from 'most applicable' to 'least applicable'.

RANDOMNESS and PROBABILITIES.

In ancient times, where they though the I Ching as having some 'spiritual' element, the method of finding the particular hexagram, the 'best fit', was left to the 'gods' in that a method based on randomisation was used to create the particular hexagram, the 'best fit', and the thinking was that there was no 'randomness', the 'random' process in fact allowed one to interact with 'god' or the 'spirits of the ancestors' etc etc who determined the result of the random process.

In more recent times the random process is in the random generation of the six lines of the hexagram by using three coins. In more ancient times the process involved a complex sorting and re-sorting of 50 yarrow sticks (yarrow is a plant, so it is collecting and getting the stalks to use in the process).

The coin system allocates the value of '3' to the heads and the value of '2' to the tails. In tossing these coins, each of the six trials (each trial creates one line, bottom-to-top) will give you a number in the range 6 to 9. The specific line types are:

6 - changing yin line (three tails)
7 - fixed yang line (two tails, one head)
8 - fixed yin line (two heads, one tail)
9 - changing yang line (three heads)

(the probability is 50:50 for each coin toss - I will leave it to you the reader to estimate the probability for the above numbers)

The changing emphasis is that the line is changing into its opposite; yang changes into yin, yin changes into yang. This 'change' emphasis is strong in the I Ching since its English title is "The Book or Changes"; the changes are caused by the context. Note how, in the fixed line values (7,8) the yin or yang nature is determined by the odd man out, thus TWO TAILS, ONE HEAD means the line is a YANG line.

So - if I get a set of tosses, bottom-to-top of 988876 my hexagram will be:

--- X --- (the X means a changing yin line)
---------
---   ---
---   ---
---   ---
----0---- (the 0 means changing yang line)

This hexagram is changing into:

---------
---------
---   ---
---   ---
---   ---
---   ---

And as such the current situation, as reflected in the first hexagram, is supposed to be changing into that quality described by the second hexagram. (thus hexagram 03 is changing into hexagram 20. Note that the traditional use of yarrow stalks was NOT a 50:50 chance process, the method in fact has a BIAS to more yin lines than yang lines, supposedly reflecting the 'natural' bias to conserve energy (a yin concept) rather than expend energy (a yang concept), and as such this method is considered more 'precise' in mapping reality than the tossing of coins etc. Times have changed where our culture is strongly into expending energy and people are happy to use coins - some software systems allow for the use of either system, or some alternative.

MOVING LINES.

The concept of moving lines is in fact a 'sneaky' way of representing not 64 possible qualities but 4096! What has happened is we have converted the use of 12-line symbols, the number of lines required to reflect 4096 qualities based on 2¹², to 6-line symbols where each line position can now have FOUR values:

yin,
yang,
yang changing into yin,
yin changing into yang.

In exponential form this is expressed as 4⁶ which = 4096, thus we have converted a STATIC 12-line system to a DYNAMIC 6 line system and as such retained the hexagram format to describe now 4096 possible qualities rather than just 64. BUT the traditional I Ching still retains the emphasis on the 64 hexagrams, in other words we interpret a hexagram as the basic quality of the hexagram being 'manipulated' by the context expressed in the form of the changing lines; the traditional system does not explicitly identify 4096 independent qualities (which it CAN do, the originators didn't take it all that far) and so we refer to 'hexagram 45 with changing lines 1,2, and 5' rather than giving a unique number to cover this combination within the range 0 to 4095.

BINOMIAL THEOREM and PASCAL'S TRIANGLE

Our 'playing' with the powers of 2 and 4, and so exponentials, introduces us to the binomial theorem where the general notation is (a + b)ⁿ. Since we are dealing with, originally, just yin and yang so a = yang = 1, b = yin = 1, and n = level of recursion. If the level is 6 then we get 64 hexagrams and the binomial theorem refines the general statement of '64 hexagrams' by giving us the details on how many hexagrams are made-up of all yin lines or all yang lines (1 of each), how many hexagrams have 1 yin, 5 yang lines, how many hexagrams have 1 yang, 5 yin lines etc etc etc These are expressed in the coefficients in each row of PASCAL's TRIANGLE (here we are at row 6) which stems from the expansion of the binomial theorem. ( I leave it to the reader as an exercise to map this ordering out).

PASCAL's TRIANGLE, COMPLEXITY/CHAOS, AND THE ZIERPINSKI GASKET.

If I take a diagram of Pascal's Triangle, say to 64 levels, and colour-in all of the ODD coefficients what will emerge is a pattern; a pattern reflecting self-referencing. This pattern has ITS roots in randomness, or more so in the encapsulation of 'randomness' where a container of, for example, triadic/tetradic form (such as a triangle (2D) or tetrahedron (3D)) if made into the 'capturer' of 'a moment' of reality (and so a 'random' state where all that is received is not yet categorised other than GENERAL distinctions (e.g. the visual opponent processing in the eye etc, our sense of smell, hearing etc etc)) will, over time, lead to the emergence of a pattern within the container - the Zierpinski gasket.

Out of the refinement of that gasket comes Pascal's triangle and all of the properties that come with that - and that includes recursion etc and so the (a+b)ⁿ and so ..... the I Ching.

REVERSIBILITY and REPEATABILITY.

Lets focus on the trigram level of the I Ching. At this level we have recursively derived 8 qualities. These qualities are ORDERED from (using yin = 0, yang = 1) 000 to 111, in full:

000 001 010 011 100 101 110 111 (reflecting the ordering of yin/yang lines bottom to top)

I can use this as a SCALE, a scale to reflect such concepts as temperature changes, thermostat controls, and so a process that is REVERSIBLE - e.g. 100 degrees to 102 degrees down to 101, up to 103 etc. This reflects the measurement of EXPRESSIONS of energy and so of MAGNITUDE, aka a SCALAR form of representation. This scalar form is usable to describe magnitude but will fail if I want to use it to measure something that is NOT reversible - e.g. TIME/DIRECTIONS; to measure time we move to CYCLIC patterns, patterns that are REPEATABLE but not REVERSIBLE. These distinctions get us into a consequence of self-referencing where given the distinction of BEGIN-END (as reflected in the thermodynamic nature of time, and so its relation to ENTROPY etc). This sense of magnitude + direction (as in time) means we shift from SCALAR representations to VECTOR representations, where the quality expressed by a vector is the combination of energy (magnitude) and time (direction). (note that in the above scale I can convert it into a WAVE format of four 'yang' dominant elements followed by four 'yin' dominant elements - as such I have a 'sine' wave format emerging)

A REPEATABLE sequence in the I Ching is derived from zooming-in to EACH of the trigrams used above to express magnitude and deriving a binary sequence WITHIN that particular trigram. For example, in the above trigram of 000 we zoom-in an apply a binary process to derive a TIME sequence that is REPEATABLE but is NOT REVERSIBLE:

Start with 000.

change first bit gives 100.
change second bit gives 010
change first TWO bits gives 110
change third bit gives 001
change first and third bits gives 101
change second and third bits gives 011
change all bits gives 111

The full sequence is 000 100 010 110 001 101 011 111.

This sequence reflects a path through 000 that is repeatable but when you analyse the MEANINGS of the trigrams the path is not reversible - this reflects the thermodynamic nature, the sense of 'begin-end' that is in the VECTOR perspective that is not so strong in the SCALAR perspective. At the full scale level of the I Ching these begin-end sequences are 64, or even 4096, hexagrams long and as such describe cycles of a quality, you cannot go directly 'back' to the step just prior to the last step, something the scalar system allows you to do, you have to go around the FULL set of qualities first to get back to the step just prior to the last step!

The point here is that the concepts of SCALARS and VECTORS seem to be encoded in the I Ching or can be 'seen' to be 'within' the I Ching. This is not so. The common element is the use in both the I Ching and Mathematics of recursion in the brain to derive basic qualities with which to 'map' reality; the commonality of I Ching with Mathematics etc stems from BOTH of these forms of 'interpreter of reality' being derived from the ONE GENERAL form used by the brain which gets specialised when linked to a context (in these cases where the context becomes a discipline which comes with its own language, its own lexicon)

Why get into reversible/repeatable? Because it touches on the concepts of countable, not countable, and the whole realm of set theory and the concepts of CARDINALITY and ORDINALITY, where Cardinality is biased more to magnitude and Ordinality is more biased to include direction.

QUALITIES of TYPES of NUMBERS.

Note that the formal method used by Peano to derive the set of integers was based on the recursion of the 'empty' set, and so we tie the source of the concept of an integer to Pascal's Triangle and the Zierpinski gasket - the same sources of the set of qualities in the I Ching.

Is there more to this? Lets go back to the brain level and the concept of what/where, objects/relationships and so the concepts of wholes, parts etc. We can map these GENERAL terms more fully to:

OBJECTS:
wholes
parts ( a whole in a relationship to a greater whole)

RELATIONSHIPS:
static
dynamic

We can then make an interesting 'discovery', namely that the general types of numbers we use in Mathematics are derived from these basics thus:

to describe:
WHOLES - we use whole numbers
PARTS - we use rational numbers (include the previous distinction of wholes in the form of X/1)
STATICS - we use irrational numbers (the harmonic series reflects the rationals. Irrationals use that series to bring-out invariant relationships and so we find such 'numbers' as PI and e, etc etc in this category)
DYNAMICS - we use imaginary numbers (these are symbols to reflect oscillations & cyclic/morphic processes, reversible/repeatable etc)

The set of REAL numbers is made up of the first three types, wholes, rationals, irrationals. The set of COMPLEX numbers is made up of combining REALS and IMAGINARIES.

Note we are still at the level of SCALAR representations. When we introduce VECTORS we move into mapping 'reality' where the HAMILTONIANS of QM etc are vectors.

The point here is that these distinctions of wholes, parts, statics, dynamics, are reflected QUALITATIVELY, in the I Ching - same patterns in seemingly 'different' disciplines. (of note - the concept of OBJECT:RELATIONSHIP is 'encoded' in all of the types - e.g. in WHOLE numbers we find the object:relationship pattern in the form of PRIMES:COMPOSITES. This all stems from our use of Self-Referencing where the moment you focus on A/NOT-A and apply it recursively you 'encode' that dichotomy IN ALL LEVELS and as such this reflects the encoding of the WHOLE in all PARTS, allowing for a PART to elicit the WHOLE (gets into the concept of metonymy)).

For additional work on LOGIC and the I Ching see XORing and the I Ching.

I hope the above has given the reader a good overview of the shared qualities in the I Ching and Mathematics as well as a good overview of the Mathematics of the I Ching. The shared qualities reflect the common ground, how our brain creates general sets of qualities that we can particularised into refined qualitative as well as quantitative descriptions of reality.

People 'see' Mathematics in the I Ching (and the I Ching in Mathematics) due to BOTH disciplines being derived from dichotomies and within the METHOD of dichotomisation is the set of general MEANINGS that come with the method - all meaning is determined by the method used to derive it.

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