When we look at the neurology of the human brain we seem to find a process of feedback where the fundamental dichotomy of WHAT (objects)/WHERE(relationships) is fed back onto itself (recursion at work). This process, taken over a few steps leads to a layout as shown in figure 1:
+---------------------------------------------------------------+
| | | | | | | | |
| what |where | what | where | what | where | what | where |
| | | | | | | | |T3
+---------------------------------------------------------------+
| | | | |
| what | where | what | where |T2
+---------------------------------------------------------------+
| | |
| WHAT | WHERE |T1
+---------------------------------------------------------------+
Fig 1. WHAT/WHERE applied recursively bottom-up.
An analysis of the feelings associated with the different types of numbers used within mathematics suggests the presence of four fundamental types which, when including the concept of negative and positive can be mapped to the above chart. This is shown in a bottom-up fashion in fig 2 but with a change in axis such that we see the mirror effect of positive/negative:
+---------------------------------------------------------------+
| (what)|(where)|(what) |(where)|(where)|(what) |(where)|(what) |
| whole |irratio|ration |imagine|imagine|ration |irratio| whole |
| |nal |al | | |al |nal | |T3
+---------------------------------------------------------------+
| | | | |
| whole |rational (part)| rational | whole |T2
+---------------------------------------------------------------+
| | |
| +ve whole numbers | -ve whole numbers |T1
+---------------------------------------------------------------+
| Numbers (Whole) |T0
+---------------------------------------------------------------+
Fig 2. Types of numbers and their 'realms' of operation.
(Note: If we move to a level beyond T3 so the cells linked to wholes' at level T3 would breakdown into (a) a cell for prime numbers and (b) a cell for expression of the relationship of prime numbers, namely composite numbers. At level T2 we see rational numbers where the emphasis is on an object in the form of a relationship aka a part. In general, here we have the generation of all the number types we need to 'map' reality. More complex types emerge from the combination of the basic types. Mathematics is thus tracable to neurology and does not require any metaphysical explanations).
In the context of the template, whole numbers have two forms, prime numbers and composite numbers. Prime numbers are uncutable and so manifest the concept of objects. The application of a relationship to primes leads to the expression of that relationship in the form of composite numbers. In passing note that the feeling of oneness, of BLENDING , is a feeling applicable to the categorisation of 'wholeness'.
Composite numbers are the 'first' level of quantifying relationships. If we look at this hierarchically, starting with 1, then 1+1 is a relationship symbolised by the symbol 2. The symbol 3 is a complex symbol in that it is at a 'new' level of '1' and so included the previous with it ((1+1)+1). The symbol 4 captures the inclusion of relational aspects at the 3 level and so ((1+1)+1+1). At the next level so we have the symbol 5 which is ((1+1)+1+1)+1 and so on. What this leads to is the concept that even numbers express dependencies whereas odd numbers express independencies. All primes are odd with the exception of 2 which is the 'root' of all even numbers; all even numbers are variations on the 'evenness' theme. The same could be said for all odd numbers except that prime numbers have added characteristics that make them standout -- they assert a degree of 'independence'. The question is 'how'?
If we consider the symbol 1 as capturing 'a whole' in the form of a particular, a 'this' from 'that', so the symbol 2 captures two wholes but also includes a semantic relationship; a tie linking the two wholes, and so the symbol 'points' to the 'space' inbetween rather than the wholes themselves. If we view this as the 'base' level for all numeric representations then there are only two 'numbers' and they are 1 and 2. All other numbers are symbols for making things 'easier'. In this sense so the symbol 2 introduces a relational marking, an 'invisible' connection, a qualitative element, as well as the quantitative element.
When we go 'beyond' 1 so we deal with relational considerations and this also happens when we go 'in' 1 in that we apply the same methods of analysis to 1, we cut it. This process of cutting the whole introduces us the the first 'aspects' of 'the whole' and that is harmonics. known here has 'rational' numbers.
These are simply all the parts that a whole can be cut into. They make-up all the elements of the harmonic series:
1/1,1/2,1/3,1/4,..2/3,..50/51....
(Older forms expressed this sequence as always of the form 1/X so 50/51 is transposable to this format)
The ability to make parts is the ability to seperate, to bound. What is of interest is that to develop the full list it appears as if I must include jumps. For example the sequence 2/1,2/2,2/3,2/4.. has within it an element that seems to be excluded from the harmonic sequence, 2/1. Any number of the form n/1 is supposidly either a prime number or a composite and is thus a whole. However, context plays a part. 2 is whole number and is not the same as 2/1 which is a harmonic. Using the harmonic concept we find that as we develop the harmonic sequence so each level is the equivalent of the n/1 number and all of the parts within its domain:
1/1 1/1 1/2 1/3...
2/1 2/2 2/3 2/4 2/5 2/6...
3/1 3/2 3/3 3/4 3/5 3/6....
4/1 4/2 4/3 4/4 4/5 4/6 4/7 4/8....
Fig 3 The emergent continuum of number
Notice that as we develop downwards so we head towards a refined continuum, and also notice that certain parts when decimalized have repeating decimals thus manifesting the presence of 'uncutability' within the context of parts but of an implicit nature. This raises the question as to the nature of decimalization. Is it a process of numbering or is it a process showing relational aspects (and thus the presence of infinity - see next section)? Since decimalization is the conversion of 'parts' to a symbolism based on powers of ten, the presence of infinity is not surprising as any numeric symbol derived within the context of a base is in fact representing an aspect of that base (a 'whole'), and thus relational properties rather than 'unique' identity.
In this context, 2.0 is NOT the same as 2/1 nor is it the same as the whole number '2' which is a base-free symbol of 'twoness'. The other symbols are twoness within other contexts (decimal and harmonic) and thus have more refined relational aspects.
These are symbols for static relationships of the whole rather than being 'wholes' or 'parts' themselves. Symbols like Pi, e, phi, sqrt(2), etc all fit into this category. The bias for the so-called transcendentals (PI, e) to having infinite decimal forms is as if the trancendentals are the 'primes' of the set of irrational numbers - the manifestation of 'pure' wholeness within the context of irrationality (with other irrationals being just repeating decimals). This would imply that the pure/composite dichotomy is valid in all numeric 'types' and supporting the conjecture that decimalization is a representation of the relational aspects of using bases rather than a 'pure' numeric symbol.
Irrational numbers can be shown to be values derivable from infinite series, which are in fact the grouping of parts extracted from the harmonic series. Thus they show static relationships created by the association of various parts. (they can also be combinations of other irrational numbers with these parts)
Like the irrationals, these too represent relational aspects, but deal with dynamic relationships. Since parts are seperated from wholes these numbers find their use in describing "influences" rather than attachements, especially the concepts of transformation (manifest in morphic behaviour) and transition (manifest in cyclic behaviour). Any form of change/oscillation is best maped using these types of numbers. In mixing terms we have the concept of BINDING in that we see apparently 'independent' forms (e.g. Sun and Earth) that over time are seen to be in some sort of a relationship.
The concept of 'infinity' plays an explicit part in both irrational and complex numbers. It is a symbol that demonstrates the relational characteristics of these numbers, and thus the inability to seperate. Therefore the number of elements in the set of all numbers is infinite. But this set is intended as a relational set and thus the presence of infinity. Infinity thus takes on contextual aspects. Cantor demonstrated (see next section) that there were different 'levels' of infinity (hierarchy) as well as types; just as there are different levels of oneness (e.g. fig 3) - we call them numbers. But this is just playing with the large numbers that we can characterize as being of an 'infinite' nature. The other aspect is infinitys relational function and thus the use of the term infinity as both a whole/parts concept and as a relational concept. It is in fact often used as a meta-symbol and has it's roots in the concept of 'a lot', which is in fact a relational term and not a numeric term.
Infinity play a strong active part in complex number processing, in that it is often used to counter its own emergence within the complex equations developed (known as renormalization), this is found in such areas as quantum electro-dynamics (QED).
Cantor made an intense study on the concepts of infinite and in doing so discovered the transfinites where the
concepts of transfinite cardinality (TC) and transfinite ordinality(TO) become 'untangled' from their finite mathematics
format to show that the processing, the arithmetic, of TC and TO are very different and reflect underlying conceptual
differences.
The major differences are sharp emphasis that ordinality is sequenced bias whereas cardinality is size biased;
thus the alephs used by Cantor in cardinality distinctions reflect set theory based mappings where the members
in the sets are without sequence. Cantor's thinking in this reflects a bias to linkage BETWEEN whereas the ordinal
bias reflects linkage WITHIN.
Thus the set of X, in the context of precision WITHIN the set lacks detain in the form of ordering; the content
is randomly distributed.
We can relate these 'basic' distinctions directly to the *general* asymmetries we find in the brain. Elsewhere
I emphasis these are BIASES and in fact can be linked to general characteristics of neurons that get refined with
grouping and out if which can emerge 'novel' behaviours unsupportable in less refined contexts. The whole brain
at birth is full of potentials with immediate context causing variations in the differentiation of senses etc This
also includes the ability, in children, of one hemisphere taking-over the management of tasks usually performed
by the other if the other is removed etc.
Consider the following list of some of the characteristics biases I have identified reflected in this asymmetry,
I use the distinctions of 'thread' to emphasise the weaving nature within these distinctions in that they apply
at all scales from hemispheres to lobes within hemispheres to neural nets within lobes to the neuron itself:
|
Left Thread |
Right Thread |
| Particular | General |
| Local | Non-Local |
| Objects | Relationships |
| The ONE | The MANY |
| What (Who,Which) | Where (How, When) |
| Tonic | Harmonic |
| Internal Linkage (within) | External Linkage (Between) |
| Syntax | Semantics |
| Single Context | Multi Context |
| As Is | Exagerate, Distort |
| Known | Unknown |
| Ordered | Disordered |
| Non-Change | Change |
| Ordinality | Cardinality |
| What IS | What IS NOT |
| delusion | illusion |
| repression | supression |
| What WAS | What COULD HAVE BEEN |
| What WILL BE | What COULD BE |
| +1/-1 | zero/infinity |
| Text | Context |
| Foreground | Background |
| Quantitative | Qualitative |
| Expression | Behind Expression |
| Self | Others |
| The Dot | The Field |
| Particle Interpretations | Wave Interpretations |
| Metonymy (part-for-whole) | Metaphor (whole-for-whole) |
| Axon-like (pulse, FM = SEQUENCE, Ordinality) | Dendrite-like (wave amplitude, AM = SIZE, Cardinality) |
| neuron | synaptic 'soup' |
| dopamine biased (internal linkage emphasis, internal integrity) | serotonin biased (external linkage emphasis, social integrity) |
| psychosis, schizophrenia | neurosis, depression |
| identify | re-identify |
| blend, bound (feeling terms for whole, parts) | bond, bind (feeling terms for statics, dynamics) |
There is a suggestion that the concept of 'truth' is tied to our animal cousins and their mapping of territory; the abstract concept of truth is rooted in the simple, local distinctions of identifying 'my' territory from that of someoneelses. The method in which this is done, through what is called waypoint mapping, when generalised and so abstracted is transformed into the feeling of 'right' from 'wrong' and that is tied in to syntax processing. (This feeling has been sourced in the left hemisphere of humans, note how EITHER/OR this feeling is. This either/or emphasis reflects the distinction of square-wave from fourier transform where the latter sums waves to approach the 'pure' square-wave; in other words fourier tranforms are like summing aspects, harmonics, of a whole but despite the level of precision possible you can never absolutely assert 'the whole' since the list of harmonics is infinite.)
Truths come in degrees, personal, cultural, universal, and this reflects the hierarchic format we find operating in brain function. Consider the following table:
| Objects | Relationships |
| Whole | Parts & other Relationships (Statics, Dynamics) |
| Parts (wholes in a relationship to a greater whole) | Relationships (Statics, Dynamics) |
| Static | Dynamic |
Moving from top to bottom, there is a zig-zag pattern in this table reflecting analysis using dichotomisations. At the second level (whole- parts & other relationships) there is a GENERAL distinction that we then refine by objectifying the concept of parts, we identify them as wholes in their own right. We next zoom-in again on the relationships side and change scales to make the distinctions of static vs dynamic where a static emphasis, an invariance emphasis, implicitly ties to an 'object' concept.
This zig-zap, or oscillations, pattern is reflected in the neurology where when processing information we oscillate
across the hemispheres of the neocortex. This process by implication flows down and through all of the subsections
of the brain . For example the WHAT/WHERE dichotomy is repeated WITHIN EACH hemisphere; the overall hemisphere
biasing distinction being that of the PARTICULAR (left) / GENERAL (right). This distinction favours a more general
object bias to the left (and so a more WHAT bias with an emphasis on the KNOWN, the previously identified .. note
the emphasis on PREVIOUS which immediately ties us to the concept of SEQUENCE, what came BEFORE. This sequence
is strongly ORDINAL in form and also single-context setting; it acts to GROUND, can emphasise a 'start' but not
necessarily an end.)
The WHERE emphasis reflects a coordinates emphasis which is more context sensitive and includes the general identification
of the UNKNOWN (abstracted to the concept of negation). In doing so the emphasis is more on EXTERNAL linkage, foreground-to-background
etc (1-to-many). The WHERE thus usually acts as a support system to the WHAT in that the WHAT associated with SELF
and the known compared to the WHERE that is more associated with unknown and so OTHERS; the WHAT thus has a degree
of self-containement about it, a 'WITHIN-ness'.
The WHAT is thus more associated with the 'dot', precision and the eternal (a 'start' with no explicit end). There
is a structural emphasis, archetypal. The WHERE is more associated with the space in-between the dots, a dynamic
realm, strong in pattern matching and mixing and so a typal emphasis. The archetypal asserts and retains identity,
the typal acts to re-identify which includes the 'refinement' of an identity and so an emphasis on transformations.
To qualify these statements, the same general patterns detailed above are present at each scale of analysis IOW
we use the SAME method of distinction-making. For example the left/right distinctions applied BETWEEN left brain/right
brain is also found WITHIN lobes of either hemisphere e.g. temporal (object bias) vs parietal (relationships bias).
These biases can then be identified WITHIN each of these lobes and so on down to the 'basic' functions of a single
neuron, AM biased dendrites and an FM biased axon where refined EXPRESSION is predominately, proactively, axonic.
Dendrite (AM) 'leakage' can influence surrounding dendrites of local neurons demonstrating a BETWEEN emphasis as
compared to the more sequence-oriented axon that in turn influences MANY dendrites of MANY neurons, and so on.
Add feedback loops and you get the entanglement of 'left/right' processes that can both filter 'up' and 'down'
the neurology.
We can relate all of the above to the context of transfinite numbers, where what is noticable is that using the
common general emphasis on LINKAGE so the transfinite ORDINALS 'fit' the left whereas transfinite CARDINALS 'fit'
the right.
The left has a bias to SEQUENCE (as shown in its bias to syntax processing), to the CORRECT internal linkage that
asserts 'the ONE'. The right is not so quantitively 'precise', it has a qualitative emphasis based on one-to-many
(many is variable and so reducable to one) associations that are more 'size' related and the right is strongly
associated with exageration, with expanding or contracting things to emphasise; IOW a relational bias (bigger than/less
than etc); in the right the bias to external linkage (all members of set A to all members of set B) has no rigid
sequence involved (if at all).
Combining the sides gives us mind where there are BIASES presented. Thus focusing on a particular left bias will
reveal right biased elements at a different scale of analysis; the sequencing in ordinality reflects a particular
type of *relationship* where GENERAL relational concepts are compressed to 'fit' a single context.
Across the species the distinctions made above re 'left brain' and 'right brain' are not always physically linked;
typal processes, both in nature and nurture, allow for variations but at the general structural level in expressions
of thought you will find strong expression of the above distinctions.
Our brain continously oscillates between the left/right and out of this comes mental states expressing left/right
characteristics ('pure' left-handers are supposed to have their hemispheres 'switched' in characteristics but that
does not totally change expressions other than allowing for a degree of dyslexia in a right-handed world!)
Brain oscillation is not uniform with, for any distinction, the accumlated time spent in one hemisphere acting
to assert its general influence on specific thinking. Cantor's development of the distinctions, the characteristics
of TO and TC are shown to be isomorphic to general neurological processes applied to all particular identifications
and so points to the identification of mathematics in general as being a product of neurological processes based
on 1:many distinction making and in particular the recursion of the what(one - objects)/where(many - relationships)
dichotomy.
From this summary of the number system we find a correlation with the mixing template derived from dichotomy, and the fact that mathematical development is rooted in dichotomous thought makes this not surprising. However, the explicit association leads to a finer level of understanding of the roots of mathematical symbolisms. Furthermore, we can feedback these types of numbers into each other, creating concepts like whole numbers within the context of complex numbers. These would emerge at deeper levels of the dichotomy-tree.
The use of a cartesian-like coordinate system (e.g. the Argand diagram for complex numbers) implicity introduces number as a form of dichotomy. For example, if given the dichotomy of positive/negative, one intuitively associates these states with extremes and if asked to measure degrees of positive/negative, one will usually use a graphic system based on, perhaps, cartesian line coordinates thus:
negative --------------------------+------------------------ positive
-10 neutral (0) 10
What is noticable is that the insertion of an origin and of a numeric scale either side is the equivalent of a dichotomy of the form -10/10 where 10 is the number of points either side of the origin. We thus combine ordinality with cardinality.
This is the same as using 10 units of measurement of the form -1/1; or simply put, positive/negative. In other words, we have added ten dichotomies of this type (pos/neg). We distinguish each one using number. As stated earlier, the more dichotomies we use the more refined becomes our data, and that is precisely what mathematics allows us to do; we replace a repeating single word-based dichotomy with numbers and thus achieve a level of measurable refinement which is not confusing (as would be found in an expression like 'positive of positive of pos...')
negative / positive
^
we insert numbers here to get a range.
negative...0....positive
this is the same as adding alternative word-based
dichotomies to display the increasing levels of positive
or negative. (negative 1, negative 2, ...)
Numbers therefore replace a list of possible words that could describe the states, and this system 'hides' the use of the explicit use of dichotomy in mathematical analysis. What happens when we analize is that we take the dichotomy we are using and feed it back into our analysis and apply it to itself; it serves as the context for the next level of distinction:
A negative(neg)/positive(pos)
B analysis of A using A gives: neg., not_so_neg., not_so_pos pos.
C analysis of B using B. We then replace the words with numbers
on a number line:
neg . . . 0 . . . pos
3 2 1 0 1 2 3
We then assign the +/- symbols to develop a scale:
neg . . 0 . . pos
-3 -2 -1 0 1 2 3
Numbers symbolize degrees of dichotomous analysis and emerge from the development of the middle or 'middle space' of the intitial dichotomy. (In formal logic the middle space is excluded and thus the very 1:1 bias in logic-based dichotomies (e.g. law of the excluded middle). Modern times have led to the emergence of 'Fuzzy' logic which is the opening-up of middle space leading us into the 1:many type of dichotomy).
Pure mathematics is when we just use the symbols, removing them as adjectives and treating them as nouns but that does not remove their dichotomous nature in that the fundamental operations on these numbers is dichotomous (add/subtract, multiply/divide).
All dichotomies are expressions of an axis and can be grouped to form increasing degrees of dimensionality as long as all axis are orthoganal to each other.
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