2. Prime Beginnings

2. Prime Beginnings

Stages of Development

The dynamic holistic nature of number is best illustrated with respect to the (entire) spectrum of stages that potentially unfolds throughout development. Indeed quite remarkably, the structure of each major stage (or level) can be encoded in holistic number terms. So from this scientific qualitative perspective, we can precisely identify the structure of the stage through the (holistic) number type with which it directly corresponds.

So to place the nature of prime numbers in their proper holistic context, I will now briefly outline the fundamental nature of the major levels of development from this qualitative number perspective.

In my most recent (updated) version of the Stages of Development, I identify six principle Bands (with three levels in each Band).

The 1st Band - in discrete terms – is the Lower (L). Here development moves gradually from total confusion where both the (rational) conscious and (intuitive) unconscious aspects of personality remain totally entangled to a state where linear type logic tends to dominate especially with respect to conscious experience.

Then we have the 2nd Band which, I now refer to as the Lower Middle (LM).

This is associated with the specialisation of (rational) linear understanding and -where further spiritual development is destined to occur - implicit preparation for the alternative (intuitive) circular type.

The understanding of this Band is of special significance for conventional mathematical interpretation.

The 3^rd Band which is the Higher (H) then represents the explicit movement away from linear to circular type understanding i.e. rational to intuitive (which indirectly can be given a paradoxical rational expression).

The 4^th Band is the Upper Middle (UM) and represents the specialised development of circular (i.e.) intuitive type understanding which ultimately is of a formless nature. All going well i.e. where the full spectrum of stages is destined to substantially unfold, this Band also represents implicit preparation for unfolding of the radial stages.

The combination these two Bands (3^rd and 4^th) is of special significance for the understanding that is Holistic Mathematics.

The 5^th Band is the Radial (R) and represents - from one important perspective - the gradual movement away from specialisation of circular intuitive to a more balanced mix of both linear (rational) and circular (intuitive) type understanding.

Finally the 6^th Band is the Advanced Radial (AR) where the fullest expression of the mature interaction as between linear and circular understanding can take place.

The 5^th and 6^th Bands are of special significance for the understanding of Radial Mathematics.         

Though these major stages (i.e. Bands) are listed here in a somewhat discrete manner, complementary links are also in evidence. Thus the Lower Band is complementary with the Higher (in circular terms). Then because the linear and circular are themselves subject to a distinctive type of complementarity the Lower and Upper Middle levels are also related in this sense. Then finally the Radial and Specialised Radial are complementary with all other levels with respect to both linear and circular aspects. 

Because I am explicitly adopting a radial approach (inevitably of an extremely preliminary nature), each band (and level) is given a dynamic identity as both relatively discrete (independent) and continuous (interdependent) with respect to all other stages.

Basically in development discrete identity is related to linear type understanding which comes through successful differentiation in experience; corresponding continuous identity - which relates to (paradoxical) circular type understanding - comes through successful integration (of what has already been differentiated).

So all development entails two distinct types of appreciation, which relate in turn to two distinct logical systems, which are linear and circular with respect to each other.

Indeed in qualitative mathematical terms this represents an alternative holistic binary system with what is linear (in understanding) corresponding to 1 and what is circular to 0 respectively. Therefore the dynamic integration of 1 and 0 in this sense is deeply inherent in all development processes (relating to varying configurations of differentiation and integration respectively).

So just as the conventional quantitative (i.e. analytic) understanding of the binary digits (1 and 0) is so important in terms of the potential encoding of all information, likewise - though not yet properly recognised - the corresponding qualitative i.e. holistic understanding of the same digits (1 and 0) is equally important in terms of the potential encoding of all transformation processes (such as human development).

Before human development for the infant commences, (phenomenal) form cannot be distinguished from (spiritual) emptiness. So we start with the total confusion of form and emptiness. Thus in qualitative holistic number terms, 1 (as deeply inherent in the recognition of form) remains totally entangled with 0 (as equally inherent in the experience of emptiness). Because no meaningful development i.e. differentiation or integration has yet taken place, these two aspects remain as mere potential for existence (not yet actualised in experience).

In correct holistic mathematical terms, the goal of experience is to both differentiate these two aspects and then meaningfully fully integrate them, so that oneness (as form) is again inseparable from nothingness (as emptiness). This is directly expressed in the famous Buddhist sutra:

“Form is not other than Void, Void is not other than Form,"

Now to illustrate this process in greater detail showing in turn how the number types - with special reference to the primes - emerge we will look at the various levels, which potentially unfold throughout development. 

Lower Band

The Lower Band entails the three levels Lower 3 (L3), Lower 2 (L2) and Lower 1 (L1) respectively. These can be roughly equated with - what can be referred to as - the Archaic, Magical and Mythic stages respectively. Now it might seem strange that the first stage (to unfold in discretely terms) is labelled L3. However the lowest is most complementary in continuous terms with the highest stage (i.e. H3). Therefore to preserve both the discrete and continuous aspects of stages a distinctive labelling of stages is required!

L3 (The Original Holistic Numbers 1 and 0)

We have seen that before development meaningfully unfolds, Oneness (1) as form is indistinguishable from Nothingness (0) as emptiness.

This can be expressed by the holistic identity (+) 1 – 1 = 0.

So initially the opposite polarities e.g. internal and external - which condition all phenomenal experience - cannot be meaningfully separated. So the positive direction in this sense is thereby equal (in an undifferentiated manner) to the negative.

Now the very word “prime” is derived from the Latin word “primus” which means the first. And because the first implies 1, I would see 1 along with 0 as the first (i.e. original) prime numbers.

Thus to distinguish them from what - are commonly referred to as the primes - I refer to them as the original prime numbers. So from this perspective the (conventional) primes are secondary as the derived prime numbers all related to 2 and ultimately rooted in 1 and 0.

Likewise in earliest infantile experience, we have the expression of the most primitive instincts where form is distinguishable in only the most fleeting manner from emptiness.

However with the first successful differentiation of a distinct bodyself in experience, not only can form (as holistic oneness) be distinguished to some degree from emptiness (as holistic nothingness), but likewise we have the birth in experience of duality (as the holistic experience of 2).

This can likewise be explained with some clarity in a qualitative mathematical manner. The very differentiation of the positive from the negative polarity (whereby the infant can experience a world as opposed to self) leads the corresponding ability to posit both polarities separately in experience.

So in holistic terms (+) 1 + 1 = 2.

From a dimensional perspective, this enables the infant to obtain a first appreciation of the existence of space and time. Also, coinciding with this development, the commencement of phenomenal object constancy takes place (that can be sustained through the corresponding dimensional appreciation).

However in the beginning both objects and dimensions lack any true perspective remaining tightly embedded with each other.

However this distinction as between (quantitative) object phenomena and the related (qualitative) dimensional background has a corresponding important distinction in number terms.

So, as illustrated in the first part of this series of articles, we can define a horizontal (quantitative) number system (with respect to a default dimension of 1).

Thus the natural numbers in this conventional approach would be written,

1¹, 2¹, 3¹, 4,¹,….. 

Now this exemplifies the linear approach where - quite literally - numbers can be represented by successive points marked out on a straight line.

The vertical (qualitative) system is then defined with respect to a default number quantity of 1.

So the natural numbers - now referring directly to dimensions - would be written in this alternative system

1¹, 1², 1³, 1,⁴,….. 

Finally the diagonal (both quantitative and qualitative) system is defined with respect to both the number and power (≠ 1).

So 2² for example is an expression in the diagonal number system!

Now a key point relates to the fact that whereas numbers in the horizontal (quantitative) system correspond to a linear (either/or) logic, properly understood numbers as dimensions (i.e. in the vertical (qualitative) system correspond directly to the alternative circular (both/and) logical system.

This circular aspect - in quantitative terms - can be appreciated by obtaining the reciprocal of the dimensional power and then expressing 1 with respect to this fractional number.

So in the simplest case i.e. 1², we convert the dimension (to which 1 is raised) to its reciprocal i.e. 1^1/2. We then obtain the two roots + 1 and – 1, which can be represented as two equidistant points on the circle of unit radius (in the complex plane).

Now the corresponding holistic (qualitative) interpretation of these roots relates to the simplest version of circular logic based on the complementarity of opposite poles.

Thus when we apply two-dimensional - as opposed to one-dimensional - qualitative interpretation to mathematical symbols an entirely distinctive interpretation ensues.

Once again however conventional interpretation remains rooted in the linear (one–dimensional) format where qualitative is necessarily reduced to the mere quantitative interpretation of symbols.

In other words though both number (as quantity) and number (as dimension) relate to two distinctive logical interpretations - that are linear and circular with respect to each other - in standard interpretation the circular (as qualitative) is simply reduced to the linear interpretation (as quantitative).

This means in effect that in place of the three number systems, that I have identified, horizontal, vertical and diagonal, only one is really considered i.e. the horizontal.

Therefore because the reduced quantitative value of all numbers in the vertical system i.e. 1¹, 1², 1³, 1,⁴,…..   = 1, this system is considered trivial (and of little relevance).

Likewise number expressions with respect to the diagonal system e.g. 2² , are reduced to mere quantitative interpretation as numbers in the horizontal system.

So 2² = 4 (i.e. 4¹).

Thus though an important qualitative change is involved whenever a number is raised to a dimension (≠ 1) this is not reflected in the (reduced) quantitative interpretation of the result in conventional mathematical terms.

Though it may seem initially surprising, the three-fold interpretation of number in horizontal (quantitative), vertical (qualitative) and diagonal (both quantitative and qualitative) terms, has major implications in terms of the dynamic manner in which both the sub-levels of each level and ultimately all the levels of development unfold.

So we can profitably investigate the evolution through each level in terms of three corresponding sub-levels.

Though in practice these will overlap in varying degrees, the first sub-level (of each level) i.e. SL1 relates primarily to horizontal development i.e. of phenomenal perception, within that level.

The second sub-level i.e. SL2 relates to vertical development i.e. of the more general conceptual - or dimensional - framework appropriate to that level.

The third sub-level i.e. SL3 relates primarily to diagonal development, where greater interaction can now take place as between both the (quantitative) perceptions and (qualitative) conceptual dimensions of that level.

This dynamic interactive experience can then lead to a distinctive type of transformation culminating with the experience of a new level (which likewise always has corresponding equivalents in number terms).

Now we will attempt to trace development of the first level through the various stages.

As we have seen the first successful differentiation in experience leads to the emergence of duality i.e. the holistic number “2” (where opposite poles, which formerly remained totally entangled with each other, can now be gradually separated).

In perceptual terms - relating to SL1 - this enables a certain degree of object constancy to be obtained. where the infant is dimly aware of phenomena (as objects) distinct from the self (as subject).

Then in conceptual terms - relating to SL2 - the infant gradually becomes aware - in general terms of an overall environment, that is somewhat distinct from that of the self, leading to the growth of existential separation.

Now these two sub-levels are characterised by deeply primitive experience (relating holistically to the original numbers 1 and 0).

In other words the infant starts from the state (before meaningful differentiation or integration) where form is indistinguishable from emptiness i.e. 1 – 1 = 0.

However after the birth of duality the infant still blindly seeks this original state of emptiness (relating to spiritual peace). However instead of the original undifferentiated form (of 1) we now have the dualistic form of 2.

So instead of obtaining 1 – 1 = 0, the infant in seeking that original emptiness holistically obtains 2 – 1 = 1.

In other words deeply implicit in the most primitive experience, is the direct misidentification of form (1) with emptiness (0). So the infant in seeking (original) emptiness directly confuses it with form.

However when experience reaches the third sub-level, with a degree of interaction now taking place as between experience that is now dualistic in both quantitative (perceptual) and qualitative (conceptual) terms, a remarkable transformation can take place (which again has an equivalent in holistic number terms).

So in the transition from L3 to L2, we have a dramatic evolution - in holistic terms - from the original numbers 0 and 1 together with the number 2 (already formed) to the proliferation of the secondary primes (which are all ultimately rooted in these earlier numbers).

L2 (Introducing the Primes)  

In order to clarify the situation, we will start with some definitions.

I refer to the numbers 0 and 1 as the original prime numbers. Then 2, which we have already dealt with in holistic terms, is the first of what are conventionally referred to as the prime numbers. However for good reasons, I refer to them as the secondary prime numbers. So, of the secondary primes, 2 is the starting and most important member. Indeed because all other (secondary) prime numbers can be traced back to 2, I refer to it as the root prime number. 2 is also unique among these numbers as the only even member! 

We have seen, in holistic mathematical terms, how the duality (2) arises during the first level and how the infant in trying to restore that original experience of emptiness through negation of form (1) directly confuses emptiness with form. This in turn represents the most primitive instinct, where in the drive to unify form with emptiness the infant directly confuses form with emptiness.

So originally in an undifferentiated holistic manner 1 – 1 = 0; however now the infant confuses 2 – 1 = 1 with 1 – 1 = 0 in both perceptual (quantitative) and conceptual (qualitative) terms.

However when both perceptions and concepts are related to each other we now get the diagonal expression 2².

Then when the same drive to return to that original emptiness manifests itself, a new primitive expression arises i.e. 2² - 1  = 3.

So instead of the original prime number of 1 resulting, we now have the secondary prime number of 3. In this manner the infant enters a new level of primitive experience that is matched - in holistic terms -  by an equivalent new type of prime number.

When appreciated in experiential dynamic terms, prime number behaviour operates in a very distinctive manner (where such numbers are very closely related to the natural).

Indeed in the very derivation of the prime number 3, the composite natural number 4 is necessarily likewise involved.

We could explain this in developmental terms by saying that the infant’s first experience of the natural world (where perceptions and concepts are necessarily related to each other) remains very unstable. This is due to the fact that the infant still largely acts out of the original undifferentiated experience where form and emptiness coincide. Therefore as soon as a phenomenon is posited in natural (composite) terms, it is immediately reduced in a primitive instinctual experience.

Thus the infant’s experience of the natural world still remains dominated by primitive experience (though now of a secondary kind that allows for the natural emergence of phenomena).

So in line with our three distinctive number systems that I have already mentioned we also should have three distinctive approaches to the primes.

So corresponding to the horizontal system we have numbers representing (secondary) prime quantities i.e. 2¹, 3¹ , 5¹, 7¹ ,….

This corresponds to the linear approach to prime numbers which - when alone considered -  leads to the somewhat limited view that the prime numbers are the building blocks of the natural number system.

However corresponding to the vertical system we also have numbers representing (secondary) prime qualities (i.e. dimensions) i.e. 1², 1³ , 1⁵, 1⁷ ,….

Though the direct meaning of these dimensional numbers is qualitative (referring to a distinctive expression of circular type logic with which to study mathematical symbols), indirectly their circular nature can be demonstrated in quantitative terms through taking corresponding roots of unity.  So for example the three roots of unity will appear as three equidistant points on the circle of unit radius (drawn in the complex plane).

However when we look at prime numbers from this circular perspective, we must accept that they are already intimately related (by their very nature) to the natural numbers.

The ultimate paradox regarding prime numbers is then revealed when we view them in their diagonal context.

This entails - in this context - raising a prime number (to a prime number dimension).  And because 2 is the root prime number, the most productive type of diagonal prime number expression arises from raising 2 to another prime number.

So diagonal numbers are in fact composite numbers (with a prime number of factors).

However a close relationship between diagonal numbers based on raising 2 to a prime dimension and horizontal primes exists in that the diagonal often converts to a horizontal prime through subtraction of 1 (resulting in the well-known Mersenne primes).

In this way prime numbers (with no factors) are in fact most closely related with their opposite i.e. composite numbers with the maximum number of factors possible for their size.

For example 128 = 2⁷ has 7 factors (which is the maximum number of factors possible for a number of this size).

However 2⁷ – 1 = 127 is prime (with no distinct factors).

Interestingly then, such composite number and least composite (i.e. prime) numbers can always be related to each other leading to perfect numbers.

So   {(2⁷ – 1) * 2⁷}/2 is a perfect number.

In psychological terms we can look on this relationship holistically where the perfect personality represents a mean - as it were - as between characteristics ranging between extreme independence (i.e. what is prime) and extreme interdependence (what is most composite) respectively. Alternatively this can be expressed as the perfect balancing of masculine (independent) and feminine (interdependent) characteristics.

So from the linear (independent) perspective we can say that the primes are the building blocks of the natural numbers i.e. every natural number can be expressed as the product of one or more prime factors.

However from the complementary circular perspective we can equally say that every prime is uniquely obtained from a (composite) natural number through subtraction of 1.  So for example 127 (which is prime) = 2⁷ – 1.

This complementary relationship as between linear (independent) and circular (interdependent) characteristics can also be demonstrated in another interesting manner.

Once again we can look at the prime number quantity (i.e. in the horizontal system) as representing the most linear (independent) of numbers.

So 7¹ in this sense, is the most independent of numbers.

However when we negate this unitary dimension and obtain 7^-1 = 1/7 we now obtain the most circular (interdependent) type of number.

In other words the digits in the decimal expansion i.e. 142857 keep recurring and contain many amazing circular properties.

And the most circular of all the numbers in this important sense are the reciprocals of certain primes esp. where a full cycle is obtained (where the unique sequence in the decimal expansion is one less than the prime number involved).

So in seeking a better understanding of the primes, a distinctive perspective is required where both linear and circular understanding is closely associated.

Thus to sum up, a horizontal prime is geared to the quantitative appreciation of numbers conforming to linear type logic.

This type of prime is always defined with respect to a default dimensional value of 1.

A vertical prime is geared to the qualitative appreciation of numbers conforming to circular type logic.

This type of prime is defined with respect to a default quantitative value of 1.

Finally a diagonal number - obtained from a prime number raised to a prime dimension - is geared to the combined quantitative and qualitative appreciation of numbers conforming to both logical systems.

So a diagonal number (which in fact is composite) has both a prime quantitative and a qualitative aspect with a prime number of factors.

In experiential terms the stable development of phenomenal recognition requires a diagonal prime process where perceptions become gradually rooted in a coherent dimensional framework.

Primitive confusion arises however when the infant attempts prematurely to identify this emerging experience with the original undifferentiated experience of emptiness (through subtraction of 1).

So we have seen how the simplest type of diagonal holistic number experience

2² becomes thereby reduced through subtraction of 1 leading to a new (secondary) prime in quantitative terms.

i.e.  2²  - 1 = 3.

It is important to stress that primitive confusion in experience essentially derives from a failure to distinguish (conscious) linear from (unconscious) circular type understanding.

Once again we have directly identified the linear aspect of experience with the stable conscious identification of the object perception and the circular with the corresponding stable (indirect) unconscious recognition of the dimensional conceptual background that contains the corresponding object.

However when experience is primitive the dimension (as qualitative concept) becomes directly confused with the object (as quantitative perception). So the dimensional perspective necessary to stably locate the object collapses - as it were - with only a momentary fleeting experience of both being possible. And the more primitive the dynamics of the experience the less stable the phenomenal identity can be.

A complementary type of experience exists at the sub-atomic levels of matter.

Here because of the dynamic interaction of particles with their corresponding dimensions, stable identification of either aspect is not possible. So at the level of string reality, the “particles” thereby contain their corresponding “dimensions”. 

Going back to our number relationship we can now demonstrate a fascinating recurring sequence of primes through continually inserting the quantitative prime derived by the diagonal number relationship as the qualitative prime (i.e. power or dimension) in a new diagonal number expression.

So  2²  - 1 = 3.

Now we insert 3 (as the derived quantitative prime) as a corresponding qualitative prime dimension of 2 (in the diagonal number expression) again subtracting 1 from the result.

So we now have

2³  - 1 = 7 (which is prime)

Once again inserting the newly derived quantitative prime as dimension of 2 (and subtracting 1) we obtain

2⁷  - 1 = 127 (which is prime)

Again continuing to switch we now obtain

2¹²⁷  - 1 = 170141183460469231731687303715884105727 (which is prime)

Now it seems quite remarkable that this Mersenne prime - which is so easily generated from this recursive approach (that dynamically underlines prime number behaviour) - held the World record for nearly 100 years as the largest prime number known.

It begs the question as to whether in continuing this process

2^{170141183460469231731687303715884105727} – 1 is prime.

If this could be verified (which is well beyond the reach of present day computation) it would be incomparably larger than the largest prime presently known.

Though it will remain an open question, I would however be confident that because of the sound “prime” rationale underlying the generation process, that its inclination to be prime would be very strong (thus ensuring very few factors for a number so large).

Though it may be accidental, I have found a very interesting parallel with another prime generating sequence (which once again demonstrates the two-way inherent relationship as between primes and the natural numbers).

Though we can identify prime number quantities (in cardinal terms) their very ranking (or ordering) implicitly entails the natural numbers. 

So 2, 3, 5 and 7 represent the 1st, 2^nd, 3^rd and 4^th (secondary) primes respectively. 

So starting with the 2nd prime 3, one could maintain a continual switching approach so as to ensure that both cardinal number and ordinal ranking remains prime.

So 2^nd prime is 3;

the 3^rd prime is 5;

the 5^th prime is 11;

the 11the prime is 31;

the 31^st prime is 127;and so on indefinitely.

Now it seems very interesting that 127 once again appears as a prime which is generated through both approaches (that recognise both linear and circular type approaches to prime number behaviour).

Once it is recognised that both the natural numbers and primes are intimately associated with each other it is not perhaps surprising that prime numbers eventually largely give way to the natural.

In other words as experience becomes more differentiated, the infant is implicitly enabled to better separate the perceptual (quantitative) and conceptual (qualitative) nature of experience.

So the higher dimensional organisation of objects stably posited in experience, now becomes increasingly prevalent (where the infant no longer actively seeks original emptiness through negation of form).

So the holistic number equivalent of this is where prime numbers are multiplied together to achieve greater natural wholes.

e.g. 2 * 3 * 5 = 30 (a higher whole number)

Though prime numbers still can result through subtraction of 1, the more coherent and organised the experience (which entails higher and higher levels of dimensionality) the less frequent are prime occurrences.

So even though for example  2 * 3 * 5 - 1 = 29 (which is prime), such occurrences recede as the number of factors generating whole numbers increases.

Thus though primitive understanding never completely dies out of experience, assuming development goes according to plan, stable understanding of the natural world (entailing both perceptions and concepts) gradually begins to dominate.

Putting it another way, the very reason why it is impossible to precisely locate all of the prime numbers is that they are in fact interdependent with the natural numbers. The primes are as much derived from natural numbers therefore as the natural numbers are derived from the primes.

However it requires a new way of looking at the issue - where linear (quantitative) and circular (qualitative) understanding are coherently combined - to properly appreciate this basic fact.  

Finally at this stage, I will briefly summarise the sub-levels of L2.

At SL1, the primitive diagonal relationship where conscious and unconscious – though gradual separation occurs – remains entangled. Here perception is confused with conceptual (dimensional) experience so that objects thereby possess magical qualities.

At SL2, the focus switches more to conceptual experience which however remains confused to a degree with perception.

Finally at SL3, a better interaction can now take place leading to the stable growth of natural phenomena. Here experience entails a coherent organisation of both quantitative and qualitative aspects with objects (as phenomena placed within a more stable dimensional framework of space and time).

In holistic mathematical terms this is expressed by the fact that all natural numbers result from the interaction of one or more prime factors (representing holistically in experience the higher order integration of both quantitative and qualitative aspects of experience).

L1 (The Natural Numbers)

As we have seen a distinct evolution is in evidence through the lower stages with coherent parallels in holistic number terms.

Initially form and emptiness are themselves identical (in a completely undifferentiated and non-integrated manner) So 1 in holistic terms cannot be meaningfully separated from 0).

Then with first separation of diagonal form and emptiness duality arises (as the holistic expression of 2).

However for some time a close - though still confused – relationship exists in diagonal terms in experience as between perceptions and concepts. Again in holistic terms this relates to number (as quantity) raised to number (as power or dimension) which in the simplest case entails 2 raised to the power of 2.

Then with the attempted negation of form (in the continuing quest to achieve the original experience of emptiness) a reductionism in experience takes place with emptiness confused with this new primitive experience.

In the most basic holistic number sense this entails 2² - 1 = 3 (as the next prime number).

However in the very dynamics of experience the (composite) natural number 4 is very closely associated with the new derived prime of 3.

Further prime numbers can then arise through the switching of the derived prime (as quantity) so that it becomes the new dimension (as quality) of 2. Then in repeating the process e.g. 2³ - 1 = 7, we derive a new prime quantity. This replicates in holistic number terms the pattern in experience whereby conscious and unconscious are continually confused with each other leading to the generation of new primitive experience.

However it is important to recognize that such instinctive behaviour coincides with the gradual unfolding of (composite) natural phenomena (where perceptions are more stably placed within a dimensional environment of space and time).

Nest with the growing separation of the vertical phenomena (whole concepts and part phenomena) primitive is more successfully “converted” into natural experience though once again more refined primitive experience remains (though receding in strength).

In holistic terms this can be expressed by saying that all natural numbers are the product of various primes with however new primes frequently still arising (though at a diminishing rate) through subtraction of 1 from such composite numbers.

However when experience reaches the next level, the focus is now on the separation of the remaining horizontal polarities of internal and external. In psychological terms this leads to the successive stabilisation of natural phenomenal perceptions (within a given conceptual background). Because of the lack therefore of a specific (unconscious) qualitative emphasis, objects can gradually be understood in a more abstract manner where they lose any residual magical properties. However myth in various forms - which in essence represents a more refined form of magical thinking - still remains.

In holistic mathematical terms, L1 is associated in turn with a switch from emphasis on multiplication where (quantitative) number objects and (qualitative) number dimensions are combined to addition (where all quantitative transformations take place within the same default dimension of 1).  

So as conscious differentiation strengthens in development, we have the gradual reductionism of the circular (which is inherently of unconscious origin) to the linear (which is directly conscious). Put another way, from a rational perspective, the qualitative aspects of experience become reduced in terms of mere quantitative interpretation.

Likewise in mathematical terms (unconscious) intuitive insight - though implicitly still necessary - becomes formally reduced in terms of mere rational interpretation.

Thus understanding gradually becomes more linear - where quite literally - all interpretation takes place with respect to the first dimension.

And once again it is the logical interpretations of this dimension that characterises conventional mathematical understanding.

So just as in this approach, the natural numbers are represented in linear terms (as points on a straight line of one dimension). Likewise in holistic number terms, L1 can be associated with the natural numbers (where interpretation is linear corresponding to mere quantitative meaning).

Once again it will be very instructive to demonstrate the holistic mathematical nature of this development with respect to the three sub-levels of L1.

With SL1 the emphasis is mainly on natural perception in the ability to abstract repeated perceptions of the same class (from an unchanging conceptual background). Indeed it is this very process that brings increasing stability to perceptions (and when carried to excess increasing rigidity). 

In holistic terms this relates to increased ability to operate on numbers (within the general number class where they maintain a merely reduced quantitative identity).

With SL2 there is now a greater emphasis on the general concepts (to which varied perceptions are related). This turn leads to a growing ability to abstract concepts (from perceptions) though in truth still in a somewhat limited manner.   

In holistic terms this leads to a growing ability to operate on the general concept of number (as for example in algebraic formulations) to which specific number perceptions are related.

However though inherently of a qualitative nature, conceptual understanding is now greatly reduced interpreted merely in terms of more generalized quantitative relationships.

Also at this level of understanding there is a certain coincidence of conventional and holistic mathematical understanding with the holistic aspect providing the wider philosophical context with which to interpret the conventional (analytic) type understanding of these levels.

However it is again important to note that at other levels e.g. higher holistic interpretation leads to a very distinctive type of mathematical understanding (that by its very nature is more qualitative than quantitative).

With SL3 we now have the growing interaction of perceptions and concepts (that are both understand in a relatively independent fashion).

This leads to another significant development in understanding with an important holistic mathematical explanation.

Thus to stably identify either perceptions (or concepts) in experience one must be able to posit these as external objects (in contrast to the understanding self as subject).

However with growing interaction of both perceptions and concepts it then likewise becomes necessary to equally negate with respect to both aspects. So to switch from a perception (as posited) to the corresponding concept (as posited) one must implicitly negate the perception. Then in turn when switching back from the (posited) concept to the (posited) perception, one must likewise negate the concept.

However because the identity of both these aspects is now sufficiently stable, neither is permanently lost - as it were - through negation and remain sufficiently in memory to be posited once again when the need arises.

Now in holistic (and analytic terms) terms, this entails that we can give the natural numbers a negative as well as positive direction.

In this way we move from the natural numbers to the integers. (This also includes the reduced quantitative interpretation of 0). 

However with the ability to both posit (explicitly) and negate (implicitly) now established in experience we move on to the Middle Band (where the specialization of linear type understanding now unfolds).