Content:
- Introduction
- The WHAT
- The WHERE
- Precision
- Objects & Relationships
- Cardinality & Ordinality
- Recursion and Emerging Numeracy
- Proto-Numeracy
- Neurocognitive Roots of the Concept of Prime Numbers
- Blending, Bonding, Bounding, & Binding
- Points and Fields; Particulars and Generals
- Mathematics
Introduction
A question often asked by many is “where does Mathematics come from?”. Recent work in the
context of Cognitive Science has attempted to answer this question (Lakoff, G., & Nunez, R., (2000) “Where
Does Mathematics Come From?” Basic Books) however, although extremely useful in discussing the roots of Mathematics,
the book seems to have failed in its intent. The following aims to present a model of Mathematics that may aid
in ‘refining’ the G. Lakoff and R. Nunez text and so aid in answering the “where does…” question. This is achieved
by going a little deeper into the neurocognitive/affective elements that inevitably lead to the development of
Mathematics.
We start by recognising that the neurocognitive/affective activity in the human brain has
developed to the degree where our species ‘instinctively’ makes the distinctions of WHAT and WHERE
when processing ANY information - these terms being derived from the methods of differentiation and integration,
methods identified as forming the roots of meaning - see the IDM material on the derivation
of meaning in general.
The WHAT
The emphasis on WHAT is an emphasis on an object, a bounded ‘thing’ that can be tangible
(as in a ball) or intangible (as in a marriage). Note how the intangible reflects what we call nominalisation where
a process (and so a relationship, ’.. getting married’) has been converted into a noun, a thing (‘this marriage…’).
Although the term ‘what’ has a general nature about it, it still has a ‘point’ or ‘dot’ emphasis
and we can refine this emphasis further by introducing additional terms such as WHO and WHICH. These
terms act to particularise the general in that the ‘what’ realm is strongly ‘dot’ oriented and as such favours
clear, precise, identifications and so a more LOCAL, discrete perspective.
This emphasis on ‘dot’ precision forces a degree of focus that can distort all considerations
of the context in which the dot exists in that the precision requires a dependence on a universal context to support
it.
The WHERE
The emphasis on WHERE is an emphasis on a relationship, there is a coordinates bias
‘relative’ to something else. There is a more intangible element
here in that a set of relationships can go towards identifying an object by implications; there is an intuitive
emphasis where a pattern based on linking a set of coordinates is ‘suddenly’ recognised as implying ‘something’;
in other words there is a ‘constellations’ emphasis where objects are linked together to form a pattern that is
then itself objectified; for example there is a strong emphasis here to geometric forms –e. g. ‘triangles’, ‘cubes’
etc. which in basic mathematics come out of joining coordinates.
This emphasis on constellation formation means that, when compared to the realm of the ‘what’,
the ‘where’ reflects a LACK in precision where (!) the identification of something is made by identifying a pattern
of landmarks ‘around’ the something. There is thus a strong context-sensitivity in the ‘where’ analysis when compared
to the more precise, almost context-free (or local context-ignored) emphasis in the ‘what’ analysis. Thus the transference
of a ‘where’ to a ‘what’ through the process of nominalisation acts to de-contextualise or more so encapsulate
the context with the text. (See figure 1).
In general the term ‘where’ is as general as the term ‘what’ and as such we can introduce
additional terms such as WHEN and HOW to aid in particularising the general. When compared to the
distinctions of WHO and WHICH, the WHEN and HOW terms are highly dependent on coordinates (space and/or time),
on establishing specific ‘begin-end’ positions rather than emphasis on a point free of any extensions.
Precision
Reflection on the WHAT/WHERE dichotomy discussed above, or more so the dynamics involved,
suggests that the elements of the dichotomy are not ‘independent’ but in fact serve as windows onto a dimension
where the windows reflect aspects of the dimension that are different enough to warrant different terms.
Figure 1 may help in understanding where the left-facing and right-facing triangles reflect
degrees of precision, moving from the gross (triangle base) to the refined (triangle apex):
Figure 1 Dot (Text) vs Field (Context) Sources of Precision
From the figure 1 we can identify patterns attributed to the hemispheres of the brain. For
example:
Focus on one element (either text/dot/foreground or context/field/background) can cause a
distortion of the other element. Thus focusing on high precision text forces a dependency on a universal and so
non-local context for support (universal constants etc). This will distort local/median contexts.
The processing of negation is more ‘right’. This seems to stem from (a) the negation is NOT
the dot and (b) the negation is part of the context. Note that processing is different to expressing. ‘Right’ expression
is limited to implications etc when compared to the at times extremely explicit ‘Left’ expression.
The ‘left’/’right’ distinctions combined with the positive/negative distinctions shows the
general emergence of the A/~A (NOT A) dichotomy in the form of a 1:many emphasis (where ‘many’ is variable and
so reducible to 1) that reflects the brain’s primary system used for identifying/re-identifying reality.
Overall, the emphasis on expression favours a more ‘left’ bias in that expression is a particular
and so more ‘dot’ biased. However as I pointed out, there is a play here on the explicit/implicit dichotomy where
a ‘right’ expression is possible but also more ‘hidden’, requiring some degree of interpretation when compared
to the explicitness of the ‘left’.
The ‘middle’ position is the normal position reflecting a balance between text/context distinctions
implying that the far left and far right of the diagrams reflect extreme positions when seen in the context of
the ‘everyday’ but then the ‘everyday’ is more of a balanced state, full of potentials, habits/intuitions that
can elicit journeys to the extremes to resolve an ‘out of balance’ situation. In other words oscillations across
the brain.
‘Dot’ precision from the right-side perspective favours meaning being spread-out, as in a
constellation or waypoint-type map where local landmarks are recruited to identify a pattern (there is a geometric
bias present). This is reflected in esoteric disciplines such as Tarot or Astrology etc where a ‘spread’ or constellation
based on planetary positions etc is used to symbolise a meaning of some form. This symbolisation is qualitative
and so rich in possible interpretations. (Pythagoras’ school showed tendencies rooted in this area where the geometry
was linked to spirituality etc. This area of diffuseness favours a sense of ‘what could be’ when compared to the
‘dot’ precision of ‘what is’)
Refinements in maps, and the contents and methods within disciplines etc favours a shift ‘left’
with an increase in words (or numbers and so a more algebraic emphasis) to replace the esoteric symbols. This does
not remove the esoteric disciplines from use since they can still contain meaning as they remain as valid metaphors
for object(what)/relationship(where) distinctions.
Each shift left favours the increasing ‘dot’ precision as the current source of interpretations,
replacing all that have been before (as the more esoteric disciplines have been replaced by Science which is now
occupying the ‘high resolution slice’ position in figure 2 – This shift is also noticeable in the discretisation
of mathematics where we witness shifts from dimensions and the geometric to the dimensionless and the algebraic.)
Despite the shift left, ‘Old’ systems, such as the esoteric disciplines, can remain and develop
‘sideways’ in precision. In other words the lexicon etc remain but go through a process of refinement that still
keeps them at their ‘less precise’ station (e.g. low resolution slice
in figure 2) but allows for the content of that level of precision to be developed and applied further and so still
maintain a sense of meaning without the need to delve into Science etc.
Figure 2 Dot Precision Resolution Slices
Each slice, when seen in its context appears to be ‘precise’ until compared to others. Thus
the dot precision of Science, combined with its need on a universal context to support it, is reflected in the
low resolution slice where stars and planets are used but are not at the same level of precision as in Science
and are more metaphors than to be taken literally. (Science is strongly
associated with a universal context as in universal constants, forces, algorithms and formulas. These are all taken
literally as were the influences of the stars etc in less precise times prior to the dissociation of species from
planetary influences)
In the development of infants there seems to be an initial bias to a more medium to low-resolution
level. In other words the high-resolution develops over time and becomes strong with the development of the spoken
word, mathematics etc. This development reflects an increase in a more ordinal perspective when compared to the
early development stages that reflect a more cardinal emphasis (amplitudes of sensations etc).
We start to see here a development process from general (more right) to particular (more left)
and then from particular to general (as in the use of induction/abduction etc to make maps (hypotheses)). Furthermore
the development process does not require a movement all the way to the left, it can develop from low to medium
where the local context determines the medium to be the ‘best’ context to work with and so development works within
that framework.
Note that the ‘low’ resolution is so in regarding details, i.e. dot precision but is more
‘high’ resolution when dealing with fields (context). Thus the low resolution ‘dot’ contains constellations in
the form of instincts, habits, and intuitions and as such has more of a sense of ‘all is [potentially] linked together’
and ‘all is [potentially] meaningful’. The process of discretisation can compress, can concentrate, the constellation
into a ‘dot’ or it can cut up the constellation and so remove connections; the ‘cant see the forest for the trees’
syndrome.
In dealing with the concept of negation note that negation deals with the unknown and so a
more ‘unresolved’ sense when compared to the positiveness of the known. In humans (and in other species using the
neuron as an information processor) the ‘left’ seems to be more attuned to processing the known and the ‘right’
is more attuned to processing the unknown. This bias reflects the above distinctions on precision where the more
context-sensitive right is recruited to analyse local context in detail to achieve some sense of general identification
that can then be focused upon and eventually identified/named.
In general we need to note that the left-right emphasis for the hemispheres is repeated at
lower scales WITHIN each hemisphere. E.g. relationship of temporal lobe to parietal lobe is like that of left-right
(categories vs coordinates). This is in EITHER hemisphere BUT the distinction of left=particular, right=general
seems to be maintained. Thus KNOWN faces elicit a left-led response, UNKNOWN faces a more right-led response.
Furthermore the same patterns are reflected within particular lobes (anterior temporal vs
posterior temporal) and in the general anterior-posterior relationships of the brain with the most particular areas,
the association areas of the anterior brain (frontal lobes etc), areas used for planning etc, being the last to
develop in humans, completing initial development in early 20s.
Finally, note that, from the perspective of the ‘dot’ precision, the more ‘right’ emphasis
is more ‘as is’, more linked to the reality of sensory events. The ‘dot’ precision acts to zoom-in on these events
to analyse them and so interpret them and as such is more precise than ‘reality’.
Objects & Relationships
With these basic what/where distinctions established (or more so the dimension onto which
what/where serve as windows) let us go a little further. The distinctions of what and where have been associated
with the more abstract concepts of objects and relationships however these rigid distinctions are not so rigid
in that neuro-cognitive research has shown that in the new-born (and just pre-birth) there is a definite bias to
WHERE over WHAT. However closer examination of sensory development in infants suggests that what we identify as
the realm of the ‘where’ is more so a realm of ‘as is’ processing combined with a sense of ‘what COULD BE’ when
compared to the realm of the ‘what’ emphasis or ‘what IS’.
In other words the realm of ‘where’ reflects a mindset that lacks precision when compared
to the developing ‘what’ bias but retains a qualitative precision that the ‘what’ can sometimes loose touch with.
Thus the bias to ‘relationships’ and so WHERE is more so an expression of GROSS object identification
or more so the failure to differentiate objects and so a failure to particularise, a failure in ‘precise’ precision;
we have to identify through ‘landmarks’ or ‘waypoints’ since we cannot determine with precision the ‘exact’ thing
or path etc.
Just prior and post birth the infant has as yet not clearly differentiated sensory processing
such that there is a response to particular data only in a general or holistic way; an auditory stimulus will elicit
full sensory system responses, auditory, visual, kinaesthetic, gustatory, olfactory respond ‘as one’.
This ‘total’ response reflects the constellation emphasis where different systems are linked together, recruited,
to assimilate data.
Only after exposure to social processes will these responses be less total, more specific
such that an auditory stimulus will elicit an auditory response (as in spoken language), in other words the sensory
systems have become differentiated enough to allow for discernment in responses, we see precision start to emerge.
(In some cases the early ‘entanglements’ continue as we find in cases of synesthesia).
The WHERE bias that we seem to detect in infants favours a bias to responding to a dynamic
context with the focus on the activities BETWEEN objects and so a bias to relational space. From a constellations
perspective this favours an ‘all is linked together’/’all is [potentially] meaningful’ emphasis where the objects
serve more as markers, like pins on a map around which we weave ribbons to show patterns etc. This has a geometric/visual
bias and is like a topological perspective where our sensory cortex acts like a ‘rubber’ sheet, bending and twisting
as the infant attempts to ‘fit in’ with the ‘new’ context it finds itself. Thus all sensations ‘resonate’ across
the whole sheet rather than be localised as we find after further development.
Cardinality & Ordinality
The emphasis in sensations is their cardinality, a sensation’s intensity, its size. The cardinal
lacks numeracy until the development of ordinality, the concept of sequence and so discreteness, and as such the
cardinal is initially measured qualitatively, thus a sensation is measured intuitively, it is compared to ‘some
other like sensation’; a process that can act to generate a general concept of ‘sameness’ or ‘difference’ without
a precise measurement being made.
The development of ordinality as a derivation from sensory experiences acts to refine cardinality
by supplying discrete units of measurement such that the ordinal and cardinal interact to improve our ‘need’ for
precision; the discreteness enables us to count the cardinal rather than be restricted to ‘feeling’ the cardinal.
As such this discretisation process allows for the communication of a sensation relatively ‘free’ of subjective
assessments. (e.g. ‘the temperature was 98’ rather than ‘it was hot!’). The added bonus of the development of ordinality
is the precision gained through analysis/categorisation of sequence and so a developing emphasis on algorithm/formula
identifications to further satisfy our need for precision. (Elsewhere I suggest the emergence of ordinality from
the spectral analysis of expressions).
Before we can get to the particulars of numeracy we need to identify a source for numeracy
and we seem to be able to do that through the use of recursion applied to the original WHAT/WHERE dichotomy associated
with neuro-cognitive processing of information.
Recursion and Emerging Numeracy
The recursion process is where an element is applied to itself, thus the identification of
an object causes us to zoom-in on that object for details. This process leads to the recognition of such concepts
as an object’s negation that at the general level relates to the entire universe exclusive of the object, and at
the particular level the objects direct opposite (e.g. positive/negative, earth/sky etc).
Analysis of the patterns that emerge from applying the what/where dichotomy to itself leads
to the identification of four fundamental distinctions which we can tie to feelings and so tie to pre-linguistic
understandings of reality. These distinctions are:
Objects:
Wholes
Parts
Relationships:
Static
Dynamic
Note that a ‘part’ is the term we use for the combination of (a) an object and (b) a relationship
to a greater object and it is the word ‘part’ that reflects what we can call the superposition of two distinctions
– the distinction of ‘wholeness’ combined with the distinction of ‘relatedness’.
Proto-Numeracy
Once we have identified the basic feelings associated with object/relationship experience
it soon becomes obvious that their nature reflects the fundamental feelings associated with the development of
number types. In other words from the basic neurocognitive distinctions of objects/relationships, applied recursively,
has emerged feelings that set a context for the development of numeracy. These associations are:
Wholes (blend) – whole numbers. The object/relationship pattern is present in the form of
the distinctions of prime numbers (what we can call ‘pure’ objects) and the distinction of composite numbers, reflecting
relational processes where primes are summed to give composites. There is an overall ‘dot’ emphasis with whole
numbers.
Parts (bound) – rational numbers. Here we list all of the ‘cuts’ we can apply to the whole
and at the same time include rational representations of wholes in the form of X/1 etc. Thus we have included the
distinction of whole numbers within the set of rational numbers. Rational numbers make up the elements of the harmonic
series.
Static relationships (bond) – irrational numbers. These are not numbers that we can count
with but more so markers of invariance. E.g. PI or e etc Irrational numbers are more often derived by summing elements
taken from the harmonic series and so reflect not the 1:1 mappings of rational to whole (as in ½ or 1/6
etc) but more 1:many mappings where a GROUP of harmonics bring out a specific aspect of the whole.
Rational numbers combined with irrational numbers form what we call Real numbers.
Dynamic relationships (bind) – imaginary numbers. Like the irrationals these are not numbers
you can count with, they are more representatives of change – of cyclic or morphic change – transitions and transformations
and are therefore fundamental in dealing with reality.
Combining real numbers with imaginary numbers gives us complex numbers and it is these numbers,
in the disguise of terms such as ‘Hamiltonians’ etc that are used to describe reality; thus the wave equation used
in quantum mechanics is a complex number.
Note that as we shift up in number-type dimensions so we lose 'laws':
Complex - lose law of commutation [ a + b != a'] {in other words you cannot count with complex
numbers}
Quaternions - lose law of association [ (ab)c != a(bc) ]
Octonions - lose law of distribution [ab + ac != a(b+c)]
Also note that with complex numbers we use conjugation to convert to reals:
(a + ib)(a - ib) = a2 - iab + iab + b2 = a2 + b2
|
NUMBER TYPES
|
REAL REPRESENTATIONS
|
| REAL NUMBERS (R) - 1 dimension (20) |
(a) [also interpretable as a PAIR of (a,1)] |
| COMPLEX NUMBERS (C) - 2 dimensions (21) |
a+ib expressable as (a,b) [pair of reals] |
| QUATERNIONS (H) - 4 dimensions (22) |
((a,b),(a,b)) [pair of complex = pair of a pair of reals] |
| OCTONIONS (O) - 8 dimensions (23) |
(((a,b),(a,b)),((a,b),(a,b))) [pair of quaternions] |
Overall, The processing of information seems to be managed by the neuron with the aid of hormones (aka neurotransmitters/neuromodulators).
The process includes the 'drive' to recruit and abstract to aid in analysis where the recruitment process reflects
a need to increase bandwidth to process the data. The primary focus at the level of the neuron is on Amplitude
Modulation (AM) and Frequency Modulation (FM) processing where the AM is expressed in the waveforms that come from
the summing of sensory data + feedback and the filtering of that data from (a) the instincts/habits of the species
encoded in the dendrites - reflects integration - and (b) the recruitment processes that to function require synchronisations
and as such are tied to the inhibit/excite activity at the soma of the neuron. The FM is in the output of the processes
of (a) and (b).
This recruitment and abstraction process allows for a network of neurons to function as if a single neuron; for
a lobe to function as a single neuron, for a hemisphere to function as a single neuron, for an individual to function
as a single neuron ... all the way up to the species level. The differences are in what is being processed, the
choices, in that it all runs in parrallel, a focus on the local all the way up to the universal.
The Neurocognitive Roots of the Concept of Prime Numbers
From the analysis of the make-up of the brain we can safely assert that the major functions
involved in information processing are those of differentiation and integration; in the neuroscience texts these
functions are expressed more as biases to processing WHAT (differentiation) and WHERE (integration, requires coordinates
and so linkage of relationships).
Furthermore, focusing on the differentiation/integration processes indicates that the 'left hemisphere/right hemisphere'
distinctions, often interpreted as 'opposites', reflect cooperative systems whereby the more differentiating hemisphere
is recruited by the more integrating hemisphere to flesh-out details on a sensation; variations in degree of differentiation
allow for a dimension of precision to emerge and so differences in the clarity of perceptions such that the 'purest'
precision, the 'dot' level precision requires high energy focus that allows for a very EITHER/OR, IS/IS-NOT perspective;
a very binary perspective. Reductions in precision elicit different patterns of meaning (e.g. patterns reflecting
the fibonacci sequence) and as such a more qualitative level of assessment, as compared to the quantitative assessments
possible at the 'dot' level of precision.
These general processes of differentiation/integration force the expression of the more differentiating hemisphere
to be more 'dot' sensitive, to focus on the WHAT, the WHO, the WHICH, when compared to the more integrating hemisphere
that is more 'field' sensitive and focuses more on the WHERE, the WHEN, the HOW; as such the differentiation bias
is on the expression, the integrating bias is on additional support distinctions for the expression as well as
reflecting the universe 'AS IS'; the dynamics of the everyday as detected by our sensory systems.
The BIASES of hemisphere focus does not mean there is no integration function in the differentiating hemisphere,
there is but it is focused on WITHIN what has been differentiated. On the other hand, the differentiating element
in the integrating hemisphere is more general, more focused on differentiation WITHIN integration where the integration
focus dominates and so a bias in perspective to relationships BETWEEN what has been differentiated.
The dimension of precision that we seem to see operating in the brain reflects our interactions with the everydayness
of the Universe in that attempts to map the everyday, and so 'improve' it from the perspective of the species,
requires we differentiate particulars from the generality of the everyday; as such we focus on apparent DIFFERENCES
(the task of differentiation) and so reject SAMENESS (our senses are in fact over sensitive to differences and
quickly habituate to sameness. This over sensitivity seems to be tied to the instincts of 'all sensations are [potentially]
meaningful' and 'all sensations are [potentially] linked together'. These instincts are survival tools, useful
in a competitive environment)
The neurocognitive processes of our brain allow us to extract a 'sensation', to focus upon it, to isolate it, so
we can analyse it and derive a parts list of it. Once done we can create a representation of the sensation in the
form of a symbol as well as create a habit that is used to respond to the same or like sensations in the future;
in other words we generalise the particular sensation into a symbol as well as generalise our response to the sensation
in the form of a habit. The degree of symbolisation is relative with an iconic form of symbol being the closest
to reflecting the sensation (see the work of Charles Peirce and his Semiotics for more on signs etc - summarised
at http://pages.prodigy.net/lofting/peirce.html )
This process of focusing upon a sensation, analysing it, and forming representations of it, reflect a path from
general-to-particular and THEN from particular-to-general (the latter is the generalisation that creates a symbol
and/or habit for later use if need be)
It seems that the path from general-to-particular is often ignored or not even experienced in that only when the
sensation is at the level of the particular is it noticed as such in that to identify something precisely we need
to clearly differentiate it; this process of general-to-particular reflects the path from unconscious-to-conscious
where we become 'aware' of something.
As a result of this 'missing' of the general-to-particular processing (often unconscious), we will more often put
the notion of the 'beginning' of things at the level of the particular and generalise from there. This means we
in fact start 'half way' in our analysis of the FULL set of properties and methods of a sensation where the realm
of the everyday is at the level of the GENERAL (and so more 'unconscious', more stimulus/response) and our species
has developed to being able to change the everyday to some degree through the development of a higher degree of
precision than is shown by the universe! (no straight lines 'out there' other than those made by us). As such we
often confuse the 'precise', 'aware' levels as being the 'everyday'; they are not.
This tendency to start at the particular is reflected in Mathematics where we more often start with the concept
of '1' and the assertion of that concept, together with the forming of a number line etc from that concept, reflects
we are at a high level of differentiation and already starting to generalise (mathematical induction processes
etc)
The 'truth' of the matter is that the process of 'strong' differentiation acts to identify DIFFERENCES and as such
will focus on the realm of the everyday to which it will apply filtering processes as it attempts to extract and
identify 'pure' differences and in doing so reject all sameness; reject all variations on themes since the intent
is to identify the 'pure' themes. ('weak' differentiation would be something like identifying all of the variations
on the theme)
The process of differentiation will thus (a) extract a difference and then (b) remove all apparent differences
that are in fact reflections of the original difference; this process reflects the 'drive' to differentiate 'clearly'
and 'precisely', we seek *pure* differences, absolutes rather than relatives.
This sense of the absolute comes from the process of focusing attention upon a particular within a context such
that increasing focus, and so energy, on the precision of the particular can distort the local context into becoming
a universal context. This is due to the everyday being highly dynamic with a definite sense of time (thermodynamic
links dominate and so a sense of direction that is not reversible) and the focus of energy actually bringing-out
a property of the physiology in that an increase in energy has a reciprocal relationship with subjective time experience;
the more we focus the more time seems to 'slow' or even 'stop' - from this can emerge a sense of the eternal which
is what we need to clearly identify a specific, we need to convert the dynamic of the everyday, the material, into
a static and thus form an ideal.
The increase in precision means we increase our need for support systems to ensure the 'eternal' nature of the
precision; once we identify 'X' we want that identification to stay around for a while, if not 'forever'.
This requirement of 'forever', for the sense of the 'eternal', means that we have to recruit universal constants,
found in the CONTEXT, to support the precision we seek since those universals reflect 'eternal' values that will
exist as long as the universe exists (or more so the general context in which whatever it is we are being 'precise'
about resides. Thus this context can be local or as big as the Universe but the push for 'dot' precision will force
the interpretations of the context to be of a universal type.)
The increase in precision will thus force the making of links to the constants in the universal context and the
more precision required so the more links are made such that the 'purest' degree of precision of 'something' is
where we have links to EVERY ELEMENT within the context that that 'something' resides in; as such the 'something'
becomes a node in a network of relations, the sum of which is 'the whole'. Of course since elements within this
'whole', this 'universe of discourse' have dynamic properties so the increase in precision can force the introduction
of probabilities due to the inability to 'stop' these dynamics operating as they support the clear, precise identification
of the 'something'. (that said, the identification of 'quality' links can aid in reducing the excessive number
of links seemingly required for 'dot' precision - pragmatism is required)
With all of this in mind, let us focus on the concept of Prime numbers. Where do they come from? How do they exist?
Are they 'real' or concepts that emerge from our methodology in categorising numeric concepts?
To summarise: Analysis of the neurocognitive processes dominating hemisphere activity in the brain, and in particular
the exaggeration process used to differentiate a particular from the general, leads to the realisation that in
dichotomisation, where the discrete element (reflecting a focus on objects) is derived from, is an exaggeration
of, the continuous element (reflecting a focus on relationships), the mathematics concept of primality is not a
'root' concept, but a derived concept and reflects the differentiation of an as yet unidentified integration process
(as such the SET of natural numbers is the whole and differentiation using filters will remove numbers from the
set, leaving behind, in principle, the prime number sequence)
Thus the dichotomy of prime/composite maps to the standard dichotomy format of differentiate-bias/integrate-bias,
(for a fuller description of these dichotomies, see the chart in http://pages.prodigy.net/lofting/hemis.html )
except that rather than the composites reflecting the summing of, the integration of primes, the primes reflect
the differentiation of composites sourced at a more general level of analysis; the realm of the 'everyday'; the
realm where 'all is linked together', the realm dominated by integration.
The task of differentiation is the identification of difference and so the filtering-out of sameness. In the identification
of primes, the sequence of primes cannot be initially identified 'as a whole' in that the application of filters
derived from the earlier numbers in the natural number sequence are required to bring-out a prime. E.g. From the
set of natural numbers, the filtering of all numbers made-up of a number added to itself (as in 1 + 1 generalised
to x + x) removes all even numbers other than the original value of (1 + 1) aka 2. Thus the identification of the
prime nature of 5 requires the filter of 2 to remove the 4 and the 6 where the 4 and 6 fit the rule "remove
all x + x" (more on this below). The application of the rule leaves the 5 'outstanding' and, since the only
previous filter also applicable is derived from 3s, "remove all x + x + x", which does not apply to the
5, so the 5 joins the set of 'primes' and in doing so imposes the "remove all x + x + x + x + x" rule
on all remaining numbers that follow since all of these numbers reflect 'sameness' once the 'root' number type,
the 'prime', has been identified.
This stepping function means a prime is identified by its presence remaining after all of the rules, the filters,
for the *previous* numbers have been applied (reflections upon this over the centuries have found easier methods
such as to only consider values up to the square root of the number under consideration etc.)
As the step function moves through the sequence of natural numbers the number of rules increase and so the number
of possible numbers remaining to be processed is, in principle, reduced (for example, the "remove all x +
x" rule removes all even numbers 'for ever', the *concept* of these types of numbers is removed OTHER THAN
the 'root' concept, the 'pure form', here expressed as '2') as such, if we retain the sequence of natural numbers,
the 'whole', analysis will show a diminishing in the frequency of primes as we move 'up' the sequence. This reflects
a context problem in that you cannot retain the whole sequence since its content 'changes' with the application
of each filtering rule. (there are 'issues' here re infinite sets etc but we do not need to consider them at this
level of analysis)
Primacy is therefore dependent on the filtering processes used to identify difference. The set of primes represent
the symbolisms of the filters which reflect the basic form of self-reference, the use of addition of x to x (self-reference
concepts move from addition through multiplication and exponentiation to tetration; differentiation seeks the pure
self-referencing patterns and then removes all derivatives since they are not the 'pure' of the 'pure')
NOTE:[there is an indication here that the sequence of prime numbers reflects properties of optimisation as shown
in the fibonacci sequence where the structures derived from the fibonacci sequence reflect the 'balance' of differentiation
and integration, ensuring the best possible differentiation WITHIN an integrated whole and as such maintaining
an even distribution of the concepts of differentiation/integration. The increase in the values in the sequence
reflect the tightening of the integration, e.g. the higher the number of buds in a plant, the tighter the spiral
patterns in that plant.
The primes as such reflect 'pure' differentiation, a pattern of values that are 'stand alone', remaining from applying
filters to the set of all natural numbers. In principle, once the set of all primes has been created, the "remove
all" rule can be applied to all primes and as such leaves us with just two numbers, 1 and 2. Note also that,
in principle, since we are demonstrating the results of neurocognitive processes we can generalise this process
to ANY 'universe of discourse' and so extract 'pure' differences; prime forms etc]
Since primes are, by definition, 'irreducible', and since elsewhere (http://pages.prodigy.net/lofting/paradox.html
) we have identified the irreducible as being in fact reducible but to some form of oscillation process we need
to identify the make-up of the oscillation.
The lowest prime number is 2 which can be differentiated from, or more so abstracted from, the integration of 1
and itself (1 + 1 = 2) and as such the labelling of an 'oscillation' of 1 + 1 that at a more abstract level of
differentiation is labelled as '2'. With this in mind, if we use the output of x + 1 to become the x value for
the next iteration we have from 1 to 49 (where x = 1 to start; and P = prime number):
x + 1
----------------------------------
01 + 1 = 02 P ...........1 + 1.
02 + 1 = 03 P ...........1 + 1 + 1.
03 + 1 = 04, but so does 2 + 2.
04 + 1 = 05 P .......... 1 + 1 + 1 + 1 + 1.
05 + 1 = 06, but do does 3 + 3.
06 + 1 = 07 P ...........1 + 1 + 1 + 1 + 1 + 1.
07 + 1 = 08, but so does 4 + 4.
08 + 1 = 09, but so goes ..3 + 3 + 3
09 + 1 = 10, but so does 5 + 5
10 + 1 = 11 P
11 + 1 = 12, but so does 6 + 6
12 + 1 = 13 P
13 + 1 = 14, but so does 7 + 7
14 + 1 = 15, but so does ..5 + 5 + 5
15 + 1 = 16, but so does 8 + 8
16 + 1 = 17 P
17 + 1 = 18, but so does 9 + 9
18 + 1 = 19 P
19 + 1 = 20, but so does 10 + 10
20 + 1 = 21, but so does ..7 + 7 + 7
21 + 1 = 22, but so does 11 + 11
22 + 1 = 23 P
23 + 1 = 24, but so does 12 + 12
24 + 1 = 25, but so does .. 5 + 5 + 5 + 5 + 5
25 + 1 = 26, but so does 13 + 13
26 + 1 = 27, but so does ..9 + 9 + 9
27 + 1 = 28, but so does 14 + 14
28 + 1 = 29 P
29 + 1 = 30, but so does 15 + 15
30 + 1 = 31 P
31 + 1 = 32, but so does 16 + 16
32 + 1 = 33, but so does ..11 + 11 + 11
33 + 1 = 34, but so does 17 + 17
34 + 1 = 35, but so does .. 7 + 7 + 7 + 7 + 7
35 + 1 = 36, but so does 18 + 18
36 + 1 = 37 P
37 + 1 = 38, but so does 19 + 19
38 + 1 = 39, but so does ..13 + 13 + 13
39 + 1 = 40, but so does 20 + 20
40 + 1 = 41 P
41 + 1 = 42, but so does 21 + 21
42 + 1 = 43 P
43 + 1 = 44, but so does 22 + 22
44 + 1 = 45, but so does.. 9 + 9 + 9 + 9 + 9
45 + 1 = 46, but so does 23 + 23
46 + 1 = 47 P
47 + 1 = 48, but so does 24 + 24
48 + 1 = 49, but so does...7 + 7 + 7 + 7 + 7 + 7 + 7
....
....
The first pattern of (x + x) reflects the failure of all even numbers to be primes other than the first one (1
+ 1). As such all of the form (x + x) are repetitions of the root form (1 + 1) that gives us the root prime of
2. As such, for x = 1 to infinity, all pairs of (x + x) can be removed.
There is nothing to stop us using even filters (1 + 1 + 1 + 1) but there will be no values due to them being filtered
out by the initial application of the (1 + 1) filter and so all of the form 1 + 1 remove all even values since
ANY number added to itself gives us an even number, thus there is ALWAYS ONLY ONE EVEN NUMBER AS A PRIME NUMBER.
All of the remaining numbers are applied the 1 + 1 + 1 filter and the samenesses removed.
All of the remaining numbers are applied the 1 + 1 + 1 + 1 + 1 filter and the samenesses removed.
and so on. (I maintain this laborious level of description to focus upon as pure a concept as possible, avoiding
the abstractions, the generation of formulas/algorithms so 'instinctive' to us)
With the above method, all that are left are values that map to the single filter of 1 (single identity, a unique
form), in that the possible filters used are:
1 {x} - if you used this filter to start with you would have no numbers since ALL numbers would be removed other
than 1.
We use these filters to identify a 'pure' difference and then remove all samenesses -
1 + 1 {x + x}
1 + 1 + 1 {x + x + x}
1 + 1 + 1 + 1 + 1 {x + x + x + x + x}
etc
.... for x = 1 to infinity.
Recall we could use 1 + 1 + 1 + 1 (x + x + x + x) but the filter of (1 + 1) would remove this filter and all others
that reflect even numbers.
The result, in principle, of this differentiation and removal process, is a sequence of 'pure' differences - the
prime numbers.
Thus the list of 'prime' numbers stems from the process of starting with (1 + 1), generalised to (x + x) and so
on, and so filtering out all numbers, leaving those that map only to the {x} filter. The application of the filter
algorithm guarantees that all resulting numbers are primes.
However, recall the comment made earlier regarding what would happen if, once we had derived the prime number sequence,
we could again apply the 'remove' function in that once we have identified the 'pure' prime number, we can remove
all others since they are 'variations' of the prime number theme. This leaves us with just two numbers, 1 and 2.
To keep the notion of 'prime' we cannot apply the reduction rule of (x) since that would remove the 2, only leaving
us the 1 which would be a 'lie' in the context of primacy and the identification of 'pure' differences.
What is implied here is that the 'infinite' nature of the prime number sequence etc is in fact an illusion, or
more so an exaggeration, in that it is a property resulting from the characteristics of just two numbers - 1 and
2; the 'one' and the 'many', the WHAT and the WHERE, the differentiated and the integrated such that the purest
differentiation of difference, within the context of differentiation, is 1 whereas the purest differentiation of
difference, WITHIN THE CONTEXT OF INTEGRATION, is the number 2. ALL else is useful exaggerations, useful variations
on these basic themes, themes seemingly hard-coded into our brains.
There is a relationship of the properties we see in the prime number sequence with the properties
we see in the fibonacci sequence and all LIKE sequences. See the diagram at the bottom of http://pages.prodigy.net/lofting/dicho2.html
These sequences represent points on the dimension that stretches from 1 to 2. Each 'dot' on that dimension serves
as the ground for a perspective on reality where it is the consideration of past contexts that guide the sense
of value of patterns seen in the immediate context (the 'true' pattern is at the top, '1,1,1,1,1...' and reflects
the lack of feedback) Thus a perspective that is grounded in the fibonacci sequence (maps to the point numerically
expressed as 1.618 - it reflects a ratio) derives all 'high value' meaning from the identification of patterns
reflecting the fibonacci sequence; the next sequence is the tribonacci sequence all the way up to the highest sequence,
where ALL previous context is considered; this sequence is the binary sequence.
All of these patterns are scaleless - they apply at all levels of analysis.
Note that in the prime number sequences discussed above the structure of the filters, as in (1+1), (1+1+1)... and
so on reflect the SAME pattern in the fibonacci sequence etc but now in the form of how many past context frames
do we consider as influencing the current frame. Thus the fibonacci sequence is derived from the focus upon past
two contexts (1+1), the trib on past three (1+1+1) and so on (note how the prime number focus is initially on difference
that is then formed into a sequence of sameness. This is reflected in all of the sequences referred-to in the above
link).
The context considerations reflect energy issues, the fib is slow, energy conserving, with a balance of integration/differentiation.
As you demand more energy to focus on higher precision so you move towards the high energy, high differentiation,
of the binary sequence. This leads to highly 'dot' precise identifications, pure EITHER/OR, IS/IS-NOT but can lose
touch with the thermodynamics of the universe - gets too idealist. At the same time, putting more energy into this
identification process, and so moving 'beyond' the binary, leads to emergence - complexity/chaos concepts develop
where the feedback comes from memories etc and as such takes any moment beyond what it 'truely' is as far as the
universe is concerned.
BUT there is also a process, tetration, that allows for any perspective to achieve the same degree of precision
as the binary, just in an at times too 'confusing' or too 'esoteric' form. This process is where the original qualities
used in each perspective, to ensure you stay in that perspective, to ensure you stay 'in the box', are recruited
as sources of analogy to describe the differences of expression of a particular quality in a particular context.
In turn these analogies are abstracted to give a full set of 'autonomous' qualities and as such 'refine' the precision
of the perspective even though it is not 'binary' (although tetration is the 'highest' form of self-referencing).
We can do this 'ad infinitum' but the lack of a more quantitative focus (that dominates the binary perspective
in the form of the sequence of integers - which are formally defined from the recursion of the empty set) means
the overall focus remains qualitative.
Thus the elimination of 'uncertainty' has different perspectives, all of which can eliminate the uncertainty BUT
are grounded in a context, a 'universe of discourse' that develops from historic developments of a perspective
due to the influence of context on species/collective/individual.
Blending, Bonding, Bounding, & Binding
In object identification there is a sense of ‘completeness’ and of ‘discreteness’ and the
feelings we can best associate with the whole/parts distinctions are:
Wholes – a feeling of blending, of becoming ‘one’.
Parts – a feeling of bounding, of partition, of cutting, separation, boundary distinctions.
In relationships identification there is a sense of ‘between-ness’ with an emphasis on the
space ‘inbetween’ objects. The distinctions of static and dynamic are best associated with feelings thus:
Static relationships – a feeling of bonding where objects share the same space ‘forever’
but also retain a sense of individual identity.
Dynamic relationships – a feeling of binding where relationships between objects is
identifiable over time, we notice a seemingly ‘invisible’ tie (e.g. the binding of Sun and Earth through gravity
or the binding of individuals by contract of some sort).
We can map these basic distinctions to the triangle diagrams dealing with dot/field precision
(below is for dot. Reverse the diagram for field) What this emphasises is an increasing emphasis on ‘dot’ precision
(blend – whole) will favour a universal bind context (dynamic relationships):
Figure 3 Mapping of Feelings to Dot Precision
Points and Fields; Particulars and Generals
What is noticeable about the derivation process of these feeling types is that we move from
a point emphasis that is context-free (as in the ‘pure’ nature of prime numbers) to an increasing context-sensitivity
in the form of Hamiltonians where each quantum event is described by a Hamiltonian that contains all of the fundamental
elements in the local and non-local context that act to identify the object under description.
In other words there is a change in emphasis from dot reference to field reference (the field
surrounding the dot ‘identifies’ the dot). What this emphasises is the increasing dependency on the precision of
the dot to a universal context. (Note that the general A/~A distinction is still present in that the wave equation
of quantum mechanics, Psi (A), requires its conjugate (~A) to enable ‘clear’, precise, identification in the form
of a real number).
Elsewhere we have identified a vector form of the qualities derived from basic recursion.
This form is derived by rotating the scalar qualities to become vector qualities.
This rotation process reflects a property and method that comes with the general method of
recursion as used to derive sets of meanings. Of special note is that this gets into the nature of integers etc
in that they are formally derived by recursion of the empty set (Peano). This gives you a scalar format. How do
you turn integers on their head? unless 1 becomes -1... and that gets us into complex numbers perhaps? ...possible
in that the requirement to convert a complex number into a real is a form of 'turning it on its head' [use of the
conjugate] and so the vector is converted to a scalar (real number).
Mathematics
When we enter the realm of Mathematics proper we discover two fundamental distinctions, those
of Geometry and Arithmetic, the latter of which includes algebraic processes as well as the more familiar addition,
subtraction, multiplication, division.
What is noticeable in these distinctions is that Arithmetic is more precise than Geometry
in that it allows for the identification of relationships often not obvious from a geometric perspective. In other
words the dichotomy of Arithmetic/Geometry is analogous to that of Particular/General and so precise/approximate.
What is allowed however, is for as discipline to develop ‘laterally’ such that within the realm of Geometry we
can develop seemingly very precise data BUT when compared to what is possible algebraically this data will ‘lack’
precision.
Conclusion
Numeracy is sourced in our neurology, within the fundamental distinctions of objects and relationships.
The object/relationship distinctions have been derived through adaptation of our species to our environment with
the development of precision; we have internalised information concepts ‘out there’ into ‘in here’ such that we
can model any external event mathematically and achieve ‘good’ measurement – a form of ‘resonance’. At the same
time the development of mathematics through the development of a refined sense of ordinality has allowed us to
create ‘what if’ scenarios of events rarely occurring on this planet if at all.
The blending, bonding, bounding, and binding distinctions reflect the development of a FEEL
for mathematics but these distinctions are in fact applicable to ALL information processing other than just mathematical
representations. In other words our language is built on the fundamental distinctions of objects (nouns) and relationships
(verbs) and out of these fundamentals develop more complex expressions that in their own right can create far more
complex contexts that in turn support even more complex expressions. (And the diagrams show how nouns/verbs can
interact, one transforming into another).
Esoteric disciplines, rooted in a more iconic age, encoded the SAME what/where distinctions
into icons which over time became symbols and as such continue to retain feelings of ‘value’ simply because they
are representative of blending, bonding, bounding, binding, and/or their more complex forms.
In the model of information processing we have the formula:
f(a,b) = c
Where:
a = stimulus
b = response
f = mediation function
c = representation
c contains two forms of representation - symbol creation where the original stimulus is generalised into a symbol
that can elicit the same response as the original stimulus - and habit creation where the original response is
generalised into a habit usable to deal with other 'like' stimuli.
Note that the symbol and habit of c can be fed-back into the mediation arguments, symbol to a, habit to b, and
so more refinements are possible as is 'emergence'.
For references/further reading see http://www.ozemail.com.au/~ddiamond/brefs.html