As we seen at Band 3, the movement is towards an increasingly holistic type understanding that expresses the interdependent nature of all reality.
And this applies equally to the number system.
So whereas formerly one understood the number system (at Band 2) in an absolute (1-dimensional) manner, now at Band 3 one realises that one can have an unlimited set of natural number dimensional understandings (with an increasing relative validity).
Thus in terms of psychological development, one moves - in the journey to "higher" super-conscious levels - from rigid rational notions to a refined intuitive contemplative worldview. In like manner, one moves from the static notion of number representing an absolute unchanging form to the opposite dynamic notion of number representing a pure energy state.
Then in the corresponding journey at Band 3 into the "lower" sub-conscious levels of personality, one equally moves from the absolute notion of prime numbers as fixed quantitative entities to the dynamic notion of primes as entailing both quantitative (conscious) and qualitative (unconscious) aspects.
And as one probes ever more deeply in psychological terms the primitive depths of personality (in the rapid but short-lived projection of instinctive phenomena), one likewise comes closer to appreciation of the truly relative nature of the primes.
However it is only at Band 5 that this emerging new mathematical understanding can start to reach maturity.
Thus again the super-conscious understanding of Band 3 represents the transcendent ascent away from the dualistic understanding of Band 2 towards a more rarefied intuitive contemplative understanding. However the danger here is that one can thereby lose significant touch with Band 2.
So Band 5 represents the corresponding immanent descent, in the reverse direction, in the attempt to properly integrate this newly developed holistic intuition with the former rational linear understanding (of Band 2).
And this in turn has dramatic implications for the understanding of number.
As we have seen, the default conventional understanding of number is 1-dimensional, where it is understood in an absolute quantitative manner.
However associated with all the other dimensional numbers are relative interpretations (corresponding to qualitative notions of interdependence).
Now one can indirectly express in a quantitative manner the structure of all these dimensions through obtaining the various roots of 1.
So for example the qualitative structure of 5-dimensional reality is indirectly expressed through the 5 roots of 1.
Now the key insight that then emerged is that these roots in fact express - again in an indirect quantitative manner - the ordinal nature of number.
Thus in the case of our example of the 5 roots of 1, these express - in indirect quantitative fashion - the qualitative ordinal notion of 1st, 2nd, 3rd, 4th and 5th respectively (in the context of 5 members of a number group).
These would be written as 11/5, 12/5, 13/5, 14/5 and 15/5 respectively
Now as I have explained before the last of these i.e. the 5th of 5 necessarily reduces to the accepted 1-dimensional notion of number (i.e. 15/5 = 11).
And this is how ordinal notions - which are inherently of a distinctive qualitative nature - are successfully reduced in a cardinal quantitative manner.
So each ordinal unit is fixed as the last of its corresponding group. So 1st is the last of 1 member, 2nd the last of 2 members, 3rd the last of 3 members and so on.
However properly understood each ordinal member can enjoy an unlimited number of relative identities. So 2nd in the context of 3, 2nd in the context of 4 and 2nd in the context of 5 members respectively all represent distinctive relative definitions of the ordinal notion of 2nd.
So the remarkable discovery here is that implicit in our everyday understanding of the ordinal notions of 1st, 2nd, 3rd, 4th and so on is corresponding higher dimensional interpretation (that is of an unconscious nature).
Thus again when we place 5 members of a group for example in ordinal relationship with each other, implicit in this understanding is a corresponding 5-dimensional notion (expressing the interdependence of these 5 dimensions). However though of necessity such understanding implicitly underlines interpretation of ordinal notions, for the most part it remains completely blind and undeveloped in conventional understanding.
So it is only with the intuitive type development corresponding to Band 3 that the holistic unconscious basis of number can be properly brought to light.
And then in the recognition that all these qualitative dimensions are intimately involved in the ordinal nature of number, the holistic (qualitative) then can become properly grounded with accepted analytic (quantitative) appreciation.
So in psychological terms, this represents the integration of both (conscious) rational and (unconscious) intuitive aspects. And this corresponds to Band 5 understanding.
A further refinement of this understanding was then to occur later, as I once more looked at the basic nature of multiplication.
I was considering the simple example of 2 * 3 and imagined a concrete example of two rows with 3 coins in each row.
Therefore in order to use 2 here as the initial operator, we must recognise that the 3 coins in one row match the 3 coins in the other. In other words we must recognise that the two rows are interdependent with each other.
So to recognise the 3 coins (in each row) we must treat them as - relatively - independent of each other. However to recognise that the 2 rows are similar (thereby enabling us to treat the relationship in a multiplicative fashion) we must then recognise the - relative - interdependence of each row.
So 2 in this multiplication example thereby applies to the qualitative notion of interdependence.
However we are using 2 here as a base - rather than dimensional - number.
Now once again when 2 is written in Type 1 terms as 21, 2 represents the base number (with 1 the default dimensional number).
However when 2 is written as 12, 2 represents the dimensional number (with 1 the default base number).
Therefore holistic interdependence can apply to both base and dimensional numbers (depending on relative context).
Thus whereas formerly, I had derived the ordinal notion as applying to base with respect to corresponding dimensional numbers, I realised that we have here the complementary opposite situation (where reference frames have switched). So here in reverse, the notion of ordinal numbers as applying to dimensions is derived with respect to their corresponding base numbers. So for example the notions of 1st and 2nd (in the context of two dimensions) is derived from the holistic interdependent notion of 2 (as applying to the base number).
So the key point in this example is that the number 2 - whether representing a base or dimensional number - can be given both quantitative (analytic) and qualitative (holistic) interpretations. And the quantitative corresponds with cardinal and the qualitative with ordinal meaning respectively.
And in the dynamics of experience, these keep switching as between cardinal and ordinal (and ordinal and cardinal) meaning respectively in a complementary manner.
And just as this applies to 2, it equally applies to every natural number.
However to properly understand, we must recognise that all numbers - indeed all mathematical symbols - have twin analytic and holistic meanings (which in the dynamics of appropriate understanding are fully complementary).
And this applies equally to the number system.
So whereas formerly one understood the number system (at Band 2) in an absolute (1-dimensional) manner, now at Band 3 one realises that one can have an unlimited set of natural number dimensional understandings (with an increasing relative validity).
Thus in terms of psychological development, one moves - in the journey to "higher" super-conscious levels - from rigid rational notions to a refined intuitive contemplative worldview. In like manner, one moves from the static notion of number representing an absolute unchanging form to the opposite dynamic notion of number representing a pure energy state.
Then in the corresponding journey at Band 3 into the "lower" sub-conscious levels of personality, one equally moves from the absolute notion of prime numbers as fixed quantitative entities to the dynamic notion of primes as entailing both quantitative (conscious) and qualitative (unconscious) aspects.
And as one probes ever more deeply in psychological terms the primitive depths of personality (in the rapid but short-lived projection of instinctive phenomena), one likewise comes closer to appreciation of the truly relative nature of the primes.
However it is only at Band 5 that this emerging new mathematical understanding can start to reach maturity.
Thus again the super-conscious understanding of Band 3 represents the transcendent ascent away from the dualistic understanding of Band 2 towards a more rarefied intuitive contemplative understanding. However the danger here is that one can thereby lose significant touch with Band 2.
So Band 5 represents the corresponding immanent descent, in the reverse direction, in the attempt to properly integrate this newly developed holistic intuition with the former rational linear understanding (of Band 2).
And this in turn has dramatic implications for the understanding of number.
As we have seen, the default conventional understanding of number is 1-dimensional, where it is understood in an absolute quantitative manner.
However associated with all the other dimensional numbers are relative interpretations (corresponding to qualitative notions of interdependence).
Now one can indirectly express in a quantitative manner the structure of all these dimensions through obtaining the various roots of 1.
So for example the qualitative structure of 5-dimensional reality is indirectly expressed through the 5 roots of 1.
Now the key insight that then emerged is that these roots in fact express - again in an indirect quantitative manner - the ordinal nature of number.
Thus in the case of our example of the 5 roots of 1, these express - in indirect quantitative fashion - the qualitative ordinal notion of 1st, 2nd, 3rd, 4th and 5th respectively (in the context of 5 members of a number group).
These would be written as 11/5, 12/5, 13/5, 14/5 and 15/5 respectively
Now as I have explained before the last of these i.e. the 5th of 5 necessarily reduces to the accepted 1-dimensional notion of number (i.e. 15/5 = 11).
And this is how ordinal notions - which are inherently of a distinctive qualitative nature - are successfully reduced in a cardinal quantitative manner.
So each ordinal unit is fixed as the last of its corresponding group. So 1st is the last of 1 member, 2nd the last of 2 members, 3rd the last of 3 members and so on.
However properly understood each ordinal member can enjoy an unlimited number of relative identities. So 2nd in the context of 3, 2nd in the context of 4 and 2nd in the context of 5 members respectively all represent distinctive relative definitions of the ordinal notion of 2nd.
So the remarkable discovery here is that implicit in our everyday understanding of the ordinal notions of 1st, 2nd, 3rd, 4th and so on is corresponding higher dimensional interpretation (that is of an unconscious nature).
Thus again when we place 5 members of a group for example in ordinal relationship with each other, implicit in this understanding is a corresponding 5-dimensional notion (expressing the interdependence of these 5 dimensions). However though of necessity such understanding implicitly underlines interpretation of ordinal notions, for the most part it remains completely blind and undeveloped in conventional understanding.
So it is only with the intuitive type development corresponding to Band 3 that the holistic unconscious basis of number can be properly brought to light.
And then in the recognition that all these qualitative dimensions are intimately involved in the ordinal nature of number, the holistic (qualitative) then can become properly grounded with accepted analytic (quantitative) appreciation.
So in psychological terms, this represents the integration of both (conscious) rational and (unconscious) intuitive aspects. And this corresponds to Band 5 understanding.
A further refinement of this understanding was then to occur later, as I once more looked at the basic nature of multiplication.
I was considering the simple example of 2 * 3 and imagined a concrete example of two rows with 3 coins in each row.
Therefore in order to use 2 here as the initial operator, we must recognise that the 3 coins in one row match the 3 coins in the other. In other words we must recognise that the two rows are interdependent with each other.
So to recognise the 3 coins (in each row) we must treat them as - relatively - independent of each other. However to recognise that the 2 rows are similar (thereby enabling us to treat the relationship in a multiplicative fashion) we must then recognise the - relative - interdependence of each row.
So 2 in this multiplication example thereby applies to the qualitative notion of interdependence.
However we are using 2 here as a base - rather than dimensional - number.
Now once again when 2 is written in Type 1 terms as 21, 2 represents the base number (with 1 the default dimensional number).
However when 2 is written as 12, 2 represents the dimensional number (with 1 the default base number).
Therefore holistic interdependence can apply to both base and dimensional numbers (depending on relative context).
Thus whereas formerly, I had derived the ordinal notion as applying to base with respect to corresponding dimensional numbers, I realised that we have here the complementary opposite situation (where reference frames have switched). So here in reverse, the notion of ordinal numbers as applying to dimensions is derived with respect to their corresponding base numbers. So for example the notions of 1st and 2nd (in the context of two dimensions) is derived from the holistic interdependent notion of 2 (as applying to the base number).
So the key point in this example is that the number 2 - whether representing a base or dimensional number - can be given both quantitative (analytic) and qualitative (holistic) interpretations. And the quantitative corresponds with cardinal and the qualitative with ordinal meaning respectively.
And in the dynamics of experience, these keep switching as between cardinal and ordinal (and ordinal and cardinal) meaning respectively in a complementary manner.
And just as this applies to 2, it equally applies to every natural number.
However to properly understand, we must recognise that all numbers - indeed all mathematical symbols - have twin analytic and holistic meanings (which in the dynamics of appropriate understanding are fully complementary).
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