Wednesday, May 31, 2017

Number and Development (14)

There are in fact two complementary relationships as between the primes and the natural numbers.

From the standard well-known perspective, this relates to the quantitative (base) aspect of such numbers.

However from the little regarded alternative perspective it relates to qualitative (dimensional) aspect of such numbers i.e. as represented by factors.

So for example, from the former external perspective we might seek to calculate the frequency of primes up to a given natural number.

From the latter internal perspective, we might then seek to calculate the ratio of natural to prime  factors within this given number.

In both contexts, log n is of special importance.

In the former external case, it measures the average gap as between prime numbers. In the latter internal case, it measures the average frequency of natural number factors.

Thus again in the former case, n/log measure the average frequency of primes to n. In the latter case n/log n measures the average gap as between the natural number factors of n.

So one can see clearly, even in these simple illustrations, how two complementary measurements with respect to the primes and natural numbers, are at play which are - relatively - quantitative and qualitative with respect to each other.

So there is a (base) quantitative aspect which relates to the fundamental notion of such numbers as points on the (1-dimensional) line. However there is also a qualitative dimensional aspect aspect, which relates to the equally important aspect establishing the spatial - and indeed temporal - relationship as between such numbers.
And this is vitally important to emphasise because numbers (like physical matter) strictly can have no meaning in the absence of space and time dimensional characteristics.

It is therefore only the gross fallacy of the continual reduction of the qualitative (relational) aspects of number in quantitative terms that has led to the utterly mistaken  assumption that numbers can have an absolute abstract existence (independent of space and time).

So without a qualitative (relational) aspect, numbers can enjoy no meaningful existence as quantities. And without a quantitative (independent) aspect, no meaningful relationship can be established as between numbers.

However clearly both of these aspects can only be appropriately understood in a dynamic interactive manner, where both (quantitative) independence and (qualitative) interdependence
are understood in a truly relative fashion.

This is all very pertinent to interpretation of the true meaning of Riemann's Hypothesis.

The assumption here is that all the Zeta 1 (non-trivial) zeros lie on an imaginary line drawn through 1/2.

Now bear in mind what I have been repeatedly saying in these blog entries regarding the truly complementary nature of both quantitative and qualitative aspects of the number system!

And remember that I have also strongly maintained that the Riemann zeros in fact indirectly represent the hidden qualitative aspect of this system (in cardinal terms)!

However because of the reduced nature of Conventional Mathematics, where qualitative considerations are reduced in an absolute quantitative manner, the misleading rational assumption is made that all real number quantities already lie on the (1-dimensional) number line.

However when we view the number system appropriately in a truly dynamic interactive manner, then we are no longer entitled to make this assumption regarding the horizontal linear nature of the  quantitative aspect independent of the qualitative aspect (represented by the non-trivial zeros).

Equally we are not entitled to make the assumption regarding the vertical linear nature of the qualitative aspect (represented by the non-trivial zeros) independent of the quantitative aspect (represented by the real line).

In particular there is no way of proving the truth regarding the vertical nature (on an imaginary line) of the non-trivial zeros i.e. of proving Riemann's Hypothesis in conventional mathematical terms, as this very assumption is already implicit in the acceptance of the horizontal nature of the real numbers (on the real line).

In other words, in making the assumption that all real numbers can be consistently expressed as lying on a (1-dimensional) line, we already assume total consistency as between the independent aspect of number (in quantitative terms) and the interdependent aspect of number (in qualitative terms) whereby numbers can be consistently related with each other.

So once again the truth regarding the qualitative aspect of number, which the non-trivial zeros indirectly express, is already unwittingly assumed in the very assumptions regrading the real number line.

So the truly fundamental issue which the Riemann Hypothesis - when appropriately interpreted - raises, is the ultimate consistency with respect to both the quantitative (analytic) and qualitative (holistic) use of mathematical symbols.

In more psychological terms, it relates to the ultimate consistency of both the conscious (rational) and unconscious (intuitive) interpretation of these same symbols.

And once again this issue cannot remotely be solved in an absolute conventional mathematical manner, as it already blindly assumes that the qualitative aspect is consistent with the quantitative.

So properly understood, ultimate belief in such underlying consistency represents a giant act of faith in the entire subsequent mathematical enterprise.

What we can therefore say in a dynamic relative manner, is that if our quantitative assumptions regarding the number line are to be valid, then equally the assumption that all the non-trivial zeros lie on an imaginary line (through 1/2) must also be true.

Equally, if the assumption that all the non-trivial zeros must lie on an imaginary line (through 1/2) is to be valid then the assumption that all real numbers lie on the number line must also be true.

Monday, May 29, 2017

Number and Development (13)

I ended the last blog entry by attempting to succinctly explain the true significance of the famed Riemann (i.e. Zeta 1) zeros.

In fact what I said there requires just a little more clarification.
Remember the fruits of this understanding arise from a dynamic interactive manner of understanding number relationships (which always involves complementary opposite poles)!

So therefore when we start with the customary analytic view of the natural number system i.e. as independent cardinal numbers in quantitative terms, the Zeta 1 (Riemann) zeros then operate as the qualitative (holistic) counterpart of this system i.e. where the interdependence of these numbers, through their unique prime factor combinations, can be indirectly represented in a numerical fashion.

However because in dynamic terms reference frames continually switch, we can equally start with the (unrecognised) holistic view of the natural number system (where one is directly aware in an intuitive manner of the interdependence of prime factors). Then, from this perspective the Zeta 1 (Riemann) zeros operate as the quantitative (analytic) counterpart to this understanding, in providing an independent set of numbers on which the holistic understanding is necessarily grounded.

So in this interactive sense, both the natural numbers and Zeta 1 (Riemann) zeros can be seen to contain both quantitative (analytic) and qualitative (holistic) meanings, which necessarily keep switching with each other in the dynamics of experience.

And of course similar dynamics relating to continually switching reference frames likewise apply to the Zeta 2 zeros with respect to understanding of the true ordinal nature of the number system.

However  it is possible to now probe more closely the exact nature of the Zeta 1 (Riemann) zeros and the clue to this again lies in the appreciation of the meaning of complementary opposite relationships.

And in this important sense, these zeros directly complement - in dynamic interactive fashion - the primes!

So from one valid perspective, we have seen that the primes and natural numbers operate in a complementary opposite fashion.

Likewise the primes as numbers without constituent factors complement those composite numbers (with factors).

Therefore in looking for the complementary opposite of the primes we should be attempting to determine all of the factors (or divisors) of the natural numbers.

So it is in this way - though the interaction of such factors - that the qualitative interdependent nature of the primes is expressed.

Now as always a lot depends on how we precisely define factors.

In conventional terms even the primes have factors with 1 being a constituent factor and the prime number itself. Thus from this perspective, each prime has 2 factors (which represents the minimum that a number can contain).

However because 1 is necessarily a factor of all numbers, just as we treat 1 as a trivial root of the number 1, we likewise treat 1 as a trivial factor of every number.

Therefore from this perspective, each prime has just 1 factor, which directly concurs with its 1-dimensional nature.

So for example if we start with 2, the independent quantitative nature of 2 as a prime is expressed through the fact that this is the only factor of 2.

Likewise with 3, the independent quantitative nature of 3 as a prime is likewise expressed through the fact that this is the only factor of 3.

However with 4 a new situation arises in that 2 and 4 are now constituent factors..

Therefore 2 now acquires a new interdependent qualitative status as a constituent factor of 4. This can equally be expressed by the fact that 2 as a prime must be now combined with another prime 2 to generate 4.

So the interdependence here arises directly through the fact that the number 2 can be expressed as part of a multiplication operation (which directly implies a qualitative transformation).

However 4 in this single context - though a composite number - acquires a relatively independent quantitative status (i.e. as a number that can be placed on the number line).

However when 4 then subsequently exists as a sub-factor of a larger number e.g. 8, then it too now acquires a qualitative interdependent status.

So each new natural number - when initially uniquely generated by prime factor combinations - carries a relative independent quantitative status. However when this number then exists as a factor of a larger number, its qualitative interdependent status is revealed.

Of course ultimately all natural number factors can be expressed as combinations of primes.

So we can say for example that for the number 8, as well as the default 8 as a factor (in a relatively independent quantitative sense), 2 and 4 are also factors (in a relatively interdependent qualitative manner).

So again though 2 is indeed a prime number, as a factor of 8 it obtains a unique qualitative status. Likewise 4 also obtains a unique qualitative status in this context (as a sub-factor of 8).

However this can also be expressed by saying that the prime combination of 2 * 2 thereby obtains a unique qualitative status (as a sub-factor of 8).

Therefore what we are saying here is that the qualitative (holistic) nature of the number system - to which the Zeta 1 (Riemann) zeros are intimately associated - directly relate to the natural factors of each member of the number system.

In other words when a number exists as a factor of another number, this implies that it is directly connected to that number through a (non-trivial) multiplication operation, and because multiplication always - when appropriately understood - entails a qualitative type transformation, this essentially therefore is what defines the qualitative (holistic) nature of such factors.   

Now if we attempt to calculate the frequency of such factor combinations a surprising link exists to the harmonic series.

I have already mentioned that 2 is a factor of every 2nd number. Therefore 1/2 of all numbers contain 2 as a factor. Likewise 1/3 of all numbers contain 3 as a factor and 1/4 of all numbers contain 4 as a factor and so on.

Therefore in this way one might conclude that the average no of factors of the number n = 1/2 + 1/3 + 1/4 +...+ 1/n.

Seeing as we are leaving out 1 this would = log n – 1 + γ.

However there is a slight problem that arises with this logic in relation to discrete numbers. For example if we are counting to 10, this might suggest that 3 occurs 3 + 1/3 times, 4, 2 + 1/2 times, 6, 1 + 2/3 times, 7, 1 + 3/7 times, 8, 1 + 1/4 and 9, 1 + 1/9 times. However clearly each of these will occur just a whole number of times. Therefore to eliminate these fractions we need to make an adjustment by subtracting (1 – γ). See "Surprising Result". So this would give us log n  – 2 + 2γ (or in the case where 1 is included as a factor log n – 1 + 2γ). 

However if we ignore the Euler-Mascheroni constant (which arises in adjusting for discrete values) the simple formula for the average no. o factors in the number n = log n – 1. 

Then the total accumulated factors to n = n(log n – 1).

This then bears a remarkable similarity with the formula for calculating the frequency of Zeta (Riemann) non-trivial zeros to t which is given as t/2π.

Thus where n = t/2π, the two formulae are identical.

Now it must be remembered that qualitative (holistic) notions of number relating to dimensions i.e. factors properly relate to the Type 2 notion of number (based on the unit circle).

And as we have seen such circular notions of number can then be converted in an imaginary linear manner (i.e. as points on an imaginary axis).

However to convert from circular to linear units we divide by 2π.

Therefore if we want to approximate the accumulated sum of factors to n, we count the frequency of non-trivial zeros on the imaginary scale to n * 2π.

So for example the accumulated sum of natural number factors to 100 will match very closely the corresponding frequency of non-trivial zeros to 628.138 (approx). 

Saturday, May 27, 2017

Number and Development (12)

We are accustomed through conventional mathematical training to view the primes in quantitative terms as the independent "building blocks" of the natural number system.

However what is not all realised is that very nature of the primes changes when they exist - not individually - but rather as factor components of the unique product combinations that generate the composite numbers.

It is in this manner therefore that the qualitative nature of the primes arises i.e. through their interdependence with other prime factors.

So for example both 2 and 3 (as separate individual primes) can in a valid - though strictly relative - sense be viewed as independent "building blocks" in quantitative terms.

However when 2 and 3 are then combined through multiplication to uniquely generate the composite number 6, i.e. 2 * 3 = 6, both 2 and 3 now acquire a relative interdependent meaning in this context, which is thereby of a qualitative (holistic) nature.

And of course there is ultimately no limit to all the relative contexts in which each of the primes can be used (with respect to unique factor combinations with other primes). So 2 for example must necessarily exist as a factor with respect to every even composite number!

Just as we saw earlier that there is an inherent paradox in terms of the definition of each individual prime (with complementary quantitative and qualitative aspects), equally this is true with respect to the collective relationship of primes with respect to the natural number system.

Again from the conventional quantitative perspective, we are automatically trained to see the relationship between the primes and natural numbers unambiguously in a one-way manner.

So clearly from this perspective, the natural numbers appear to depend for their existence on the primes.
However implicit in this view is an unexpected problem which is rarely recognised, as our very understanding of the primes already requires the natural numbers for their proper comprehension.

In other words, the very ability to spatially separate in a meaningful fashion the primes from each other already implies a notion of order that applies to the natural numbers.

So the positioning of each prime already depends on the composite ordering of prime factors.

And if we cannot meaningfully assign a position to each prime, then equally we cannot meaningfully provide it with a definite numerical identity!

Thus again we have two complementary perspectives.

From the quantitative perspective, the natural numbers appear to depend on the independent primes as "building blocks".

However, from the qualitative perspective, the positioning of each prime appears to depend on the  interdependence of prime factors that uniquely generate the natural numbers.

Therefore to properly appreciate this paradox, we must once again move to a dynamic interactive appreciation of number behaviour, entailing both quantitative and qualitative aspects as equal partners.

This then leads inevitably to the realisation that - just as with micro  - the macro behaviour of the number system entails the synchronistic behaviour of both quantitative (analytic) and qualitative (holistic) aspects, which is ultimately ineffable.

And of course as micro behaviour (associated with the Zeta 2 function) and macro behaviour (associated with the Zeta 1 function) are themselves dynamically complementary, ultimately neither has a meaning independent of the other    

Thus properly understood, in dynamic interactive terms,  both the primes and natural numbers (and natural numbers and primes) mutually co-determine each other. both with respect to micro and macro aspects, in a synchronistic manner.

So again the key fallacy with respect to  conventional understanding is the attempt to view the number system in an absolute - merely quantitative - manner where the relationship as between primes and natural numbers is misleadingly viewed as one-way and unambiguous. 

We have already seen how the solutions to the Zeta 2 function provide an indirect quantitative means of expressing the qualitative nature of each individual natural number member of a prime number group (at the micro level).

Likewise - again in true complementary fashion - the solutions to the Zeta 1 (Riemann) function, provide an indirect quantitative means of expressing the qualitative nature of the collective interdependence of prime factors with respect to the natural number system (at a macro level).

So this is the key revelation that can now be made with respect to the Riemann zeros.

Remember how Hilbert, when once queried as to most important problem in Mathematics replied,
" the problem of the zeros of the zeta function, not only in Mathematics but absolutely the most important!"

And the true reason why these zeta zeros are indeed so important is that indirectly they express the hidden qualitative nature of the natural number system.

And in even more precise terms we can say that  the Zeta 2 zeros indirectly express the qualitative (holistic) nature of the ordinal natural number system, whereas the Zeta 1 (Riemann) zeros express the corresponding qualitative nature of the cardinal natural number system, And ultimately, both ordinal and cardinal aspects are completely interdependent  with each other.

Friday, May 26, 2017

Number and Development (11)

It may be useful again at this point to emphasise the key significance of what I have been articulating in these blog entries.

We are accustomed to thing of Mathematics - especially in its treatment of number - in an absolute unambiguous manner (where the meaning of symbols remains fixed).

However in truth an unlimited number of relative type interpretations can potentially apply, with the standard conventional approach representing just one special limiting case.

Put another way the standard interpretation is of a 1-dimensional nature, whereby qualitative type considerations with respect to mathematical symbols are reduced in a merely quantitative manner (within a rigidly fixed framework).

However associated with every mathematical symbol is a unique qualitative manner of interpretation.
Thus Analytic i.e. Conventional Mathematics relates to the quantitative interpretation of mathematical symbols whereas Holistic Mathematics relates to their qualitative inetrpretation.  And the integration of both approaches - in what I refer to - as Radial Mathematics, requires a dynamic interactive approach, whereby both quantitative (analytic) and qualitative (holistic) interpretations are inherently understood in a complementary manner.

So the huge unacknowledged problem is that just one special limited interpretation (where the qualitative meaning of symbols is reduced in a merely quantitative manner) has become synonymous with "all Mathematics", thereby completely blinding us to the infinite array of riches that would readily unfold through recognition of its hidden qualitative aspect as an equal partner.

So in psychological terms, the vitally important unconscious aspect of understanding has been completely blotted out in the misleading attempt to formally portray Mathematics as merely a rational (conscious) discipline.

However so far we have been looking at merely the "micro" aspect of number behaviour.

In other words we have started from the notion of an individual prime as a quantitative "building block" of each natural number to discover its hidden qualitative meaning as a number group whose individual ordinal members are uniquely defined in a natural number fashion.

So the inherent paradox of the nature of each prime is thereby clearly revealed. Thus from the quantitative perspective, the primes appear to unambiguously generate the natural numbers (in a cardinal manner); however from the (hidden) qualitative perspective each individual prime already is defined by an unbroken sequence of natural numbers (in an ordinal manner).

Thus from a dynamic interactive perspective - entailing both cardinal and ordinal aspects - it becomes clear that both the primes and natural numbers are ultimately co-determined in a synchronistic manner (which is ultimately ineffable).

However this realisation can have no meaning within the conventional mathematical perspective that misleadingly insists on viewing the nature of number in a merely quantitative manner!

However as well as the "micro" aspect of each individual prime, we likewise have the "macro" aspect of the collective behaviour of the primes with respect to overall natural number system.

And of course both individual "micro" and collective "macro" aspects of prime (and natural number) behaviour are themselves complementary in a dynamic interactive manner.

So strictly speaking we neither start with each individual prime (as somehow pre-defined) or the entire natural number system (as likewise somehow pre-defined).

Rather they both arise through mysterious dynamic interaction patterns that then subsequently enable their separate identities to be abstracted  in a fixed manner.  

In now looking at the "macro" behaviour of the number system, we make direct contact with Riemann's zeta function and the famed (non-trivial) zeros.

However, it was intense investigation over many decades relating to the less investigated ordinal nature of number (briefly outlined in the 10 previous blog entries) that eventually provided me with the appropriate framework to understand the true nature of these Riemann zeros.

However the starting point goes back to a classroom revelation when attending primary school in Ireland, regarding an unacknowledged problem regarding the nature of multiplication.

As we know in quantitative terms all natural numbers can be uniquely expressed as the product of primes.

So if we take the number "6" to illustrate this is uniquely expressed as 2 * 3.

Now the number "6" is represented as a point on the number line. In other words 6 is represented in a 1-dimensional manner as 61.

However if we represent 2 and 3 here in geometrical fashion - say as two sides of a rectangular tables (measured) in metres - then the area represented by 2 * 3 is given in square (i.e. 2-dimensional) terms. In other words, though each side relates to a measurement in 1-dimensional, the resulting area relates to a corresponding measurement in 2-dimensional units.

However with respect to the standard treatment of multiplication, this qualitative transformation in the nature of units is simply edited completely out of consideration.

Thus the result of  2 * 3 (and by extension every factor combination of primes) is given in a merely reduced (i.e. 1-dimensional) quantitative manner.

However though this early insight still remained fully valid, it was only at a later date that I realised an additional - perhaps even more fundamental - problem with the conventional nature of multiplication.

When we speak of the area of a table (as for example in my illustration) we are still operating at the analytic level of understanding dimensional numbers.

However as we have seen, all such numbers have a corresponding holistic interpretation and this too is intimately involved with the very nature of multiplication.

If for these purposes we have two rows with 3 coins in each row, each row would represent in analytic terms a dimension (e.g.. length) and each column (of 2 coins) another dimension (e.g. height).

Now without specific reference to rows or columns, we could attempt to treat each coin in an independent manner and obtain the result by adding up each of the (independent) units.

However the key point about multiplication that in order to use the operator of 2 we must recognise the mutual interdependence of the two rows (with 3 coins).
Thus crucially whereas with addition, we can proceed by recognising the independence of each individual item, for multiplication we must also recognise the mutual interdependence of rows and columns.

So when we recognise the two rows as interdependent we use 2 as operator to obtain 2 * 3.

Equally when we recognise the three columns as interdependent we use 3 as operator to obtain 3 * 2.

So conventional multiplication attempts to represent mutiplication  - very misleadingly - as a short-hand form of addition.

So with respect to addition when we add the two rows we get 3 + 3 = 6. And is each row is defined in a 1-dimensional manner the result through addition is likewise 1-dimensional in nature.

Thus 31 + 31  = 61.

Conventional multiplication then attempts to "speed up" this process. Because 3 is repeated 2 times with respect to addition, we now multiply 3 two times i.e. 2 * 3 to apparently get the same result.

Now this "speeding up"is of course not so evident when the operator is 2. However imagine if 3 was to be added to 3 one hundred times, then the expression of this through multiplication as 100 * 3 would indeed be much more efficient.

However the crucial unrecognised problem is that the very switch from addition to multiplication requires the recognition of the rows and columns (representing the two dimensions) not only as each containing independent items but also that these dimensions themselves as mutually interdependent (and thereby freely interchangeable with each other).

And this latter recognition requires the qualitative (holistic) interpretation of the nature of a dimension.

So I was already well primed - to  excuse the pun - to see that there was a distinct qualitative aspect to the nature of multiplication, which was completely unrecognised in conventional mathematical terms. And I was already confident that - when appropriately understood -  the Riemann (Zeta 1) zeros directly related to this hidden qualitative aspect of the cardinal number system.      

Thursday, May 25, 2017

Number and Development (10)

I will start this entry by contrasting the respective meanings in Type 1 and Type 2 terms of the fractions 1/3, 2/3 and 3/3 respectively.

In Type 1 terms these would be given as (1/3)1, (2/3)1 and (3/3)1 respectively.

So from the standard quantitative (analytic)  perspective (1/3)1 represents 1 of 3 (equal) parts i.e. one third;
(2/3)1 represents 2 of 3 (equal) parts i.e. two thirds.

(3/3)1 represents 3 of 3 (equal) parts i.e. 1 as a whole unit,  

However in Type 2 term, these fractions would be given as 11/3, 12/3 and 13/3 respectively.

Now in standard quantitative terms, these represent the 3 roots of 1 i.e. .5 + .866i, .5 .866i and 1 respectively.

However these also have an important qualitative (holistic) interpretation. And in dynamic interactive terms, when the Type 1 interpretation relates to the quantitative (analytic) aspect, then the corresponding Type 2 interpretation relates to the corresponding qualitative (holistic) aspect.

And the qualitative (holistic) interpretation of  1/3, 2/3 and 3/3 (i.e. 11/3, 12/3 and 13/3) would be expressed as 1st of 3 (related) dimensional units, 2nd of 3 (related) dimensional units and 3rd of 3 (related) dimensional units respectively.

Of course reference frames continually switch with respect to experience. So if we now start with the Type 2 aspect defined in the standard analytic manner (as the 3 quantitative roots of 1), then in Type 1 terms 1/3, 2/3 and 3/3 now take on a complementary holistic qualitative meaning as the 1st of 3 (related) base units, the 2nd of 3 (related) base units and the 3rd of 3 related base units respectively.

Now the key significance of a prime from the qualitative (holistic) perspective is that each of its ordinal members (with the exception of the last) is always uniquely defined.

In other words, when we omit consideration of the last unit (i.e. given in general terms as the nth of n units), the indirect numerical designation of all other ordinal positions for each prime, by definition, cannot be replicated with respect to any other prime.

So again from the quantitative (analytic) perspective, the primes collectively are unique as the fundamental factors or "building blocks" of the natural numbers.

However, from the qualitative (holistic) perspective, each individual prime group is unique in that all its natural number ordinal members (except the last) -indirectly represented in turn by the various roots of 1 (except the last) - are unique for that prime group.

So from this perspective, 5 (representing a prime group of individual members) is unique in that its 1st, 2nd, 3rd and 4th members - indirectly represented in turn in a qualitative manner by the 1st, 2nd 3rd and 4th roots cannot - by definition, be replicated with respect to the ordinal members of any other prime group.

It was at this time that I gave intense consideration as to the precise significance of the "trivial" last root i.e. 1 (which exists for every prime group).

Then it slowly dawned on me that it was the interpretation with respect to this root that naturally occurs in the standard ordinal interpretation of number (where ordinal notions are reduced in a cardinal manner).

So the number system starts with 1 (which or course is not prime). Then when we move on to consideration of 2, in standard analytic terms the 1st unit is unambiguously fixed as 1 with 2nd unit now in likewise manner fixed with the last remaining unit of 2.

Then when we move on to consideration of 3 the 1st and 2nd units have already been unambiguously fixed with the 1st two units so that the 3rd unit is now likewise unambiguously fixed with the last unit (of 3).

We started with the quantitative definition of 3 (as cardinal number) = 1 + 1 + 1.

If we now define this in ordinal terms, 3 = 1st + 2nd + 3rd units.

However because in conventional mathematical terms, each ordinal position is unambiguously fixed in an absolute manner with the last unit of its corresponding cardinal number group (= 1), from this perspective,  

1st + 2nd + 3rd = 1 + 1 + 1 = 3.

So there seems therefore from the quantitative (analytic) perspective that there exists a perfect correspondence as between cardinal and ordinal notions!

However from the corresponding qualitative (holistic) perspective, all ordinal positions are merely relative and can be interchanged with each other.

And as these are indirectly given in quantitative as the corresponding roots of 1, from this perspective,

1st + 2nd + 3rd =   .5 + .866i .5 .866i + 1= 0.

In others, strictly speaking from the holistic qualittaive perspective, the pure interdependence of all ordinal positions has no cardinal meaning!

So therefore the important point to grasp is that in standard quantitative (analytic) terms, each new ordinal unit is unambiguously fixed with the last unit of the number group in question. 

However the key point regarding the qualitative (holistic) interpretation of number is that ordinal positions can be undercharged with each other.

Thus in a group of 3 for example what is 1st from one relative perspective can equally be 2nd and 3rd from two other equally valid perspectives. Likewise what is 2nd from the first perspective, can be equally 1st and 3rd from the other perspectives, and finally what is 3rd from the first can equally be 1st and 2nd from the other perspectives.

So the holistic appreciation of ordinal positions implies the interdependence of each individual member with each other member of the respective group.

The analytic appreciation implies by contrast the independence of each individual member, whereby the ordinal position is unambiguously fixed with this member.

So as we have seen from this latter perspective, 1st is unambiguously identified with the last member (of a group of 1) , 2nd with the last member (of a group of 2), 3rd with the last member (of a group of 3) and so on.

It was at this stage that I suddenly saw how a striking complementarity in fact existed as between all this work on the holistic nature of ordinal numbers and the famed Riemann zeros. 

In fact I could see now that were in fact two sides as it were of the same coin.

This insight arose from the attempt to isolate the truly unique holistic solutions indirectly implied in general terms by the t roots of 1 (except the default root of 1).

So the t roots of 1 are obtained from the equation, 

xt – 1 = 0, or alternatively as better suits our purposes 1 xt = 0.

Thus to eliminate the default root (where x = 1), we divide by 1 – x, to obtain

1 + x1 + x2 + x3 + ... + xt – 1  = 0.

So I started to refer to this as the Zeta 2 in contrast to the complementary type Riemann zeta function (i.e Zeta 1) where

1– s + 2– s + 3– s + 4– s + ....  = 0.

So the famed non-trivial zeta zeros represent the solutions for s to this equation.

But note the complementarity as between both functions! 

Whereas the former (Zeta 2) is of of a finite nature that be extended without limit, the latter (Zeta 1) is infinite in nature.

Likewise whereas the sequence of natural numbers appear as dimensional powers (with respect to the Zeta 2), they appear as a sequence of base numbers (with respect to the Zeta 1).

Also, whereas the unknown to be solved from the equation is a base number (with respect to the Zeta 2), it is a dimensional power (with respect to the Zeta 2).

The realisation of the complementary nature of these two functions was then to prove invaluable in "unearthing" the true significance of the Riemann (Zeta 1) zeros.      

Wednesday, May 24, 2017

Number and Development (9)

As we have seen, there are two aspects with respect to the understanding of all numbers - and indeed by extension all mathematical relationships - that are quantitative (analytic) and qualitative (holistic) with respect to each other.

Both of these aspects can only be properly understood in a dynamic relative context, where each type of understanding  implies the other in a complementary manner.

Now, from a psychological perspective, the analytic aspect is directly related to rational type appreciation (of a conscious kind); by contrast the holistic aspect is directly related to intuitive type appreciation (of an unconscious nature).

Therefore, when understood appropriately, the role of intuition with respect to mathematical understanding is utterly distinct and cannot be confused with reason.

However, because in effect conventional mathematical understanding entails the reduction of holistic type meaning (in an analytic manner), equally this entails the corresponding reduction of intuitive  (directly related to the unconscious) with rational type appreciation (of a distinctive conscious nature).

So once again the truly central issue with respect to all mathematical interpretation is thereby missed.

From the external physical perspective, this relates to consistency with respect to both the quantitative and qualitative interpretation of its symbols; from the corresponding psychological perspective - which in dynamic terms complements the physical - this equally relates to consistency with respect to the rational and intuitive interpretation of these same symbols.

In fact as I have repeatedly stated in my blog entries, from the appropriate dynamic interactive perspective, the Riemann Hypothesis can be seen as a key statement with respect to this central issue.

And by the same token, because in conventional mathematical terms the qualitative aspect is not formally recognised (as distinct from the quantitative) this implies that such attempted "proofs" of the Riemann Hypothesis are rendered futile!

When one looks at the nature of the primes from this dynamic interactive perspective, appreciation of their very nature is thereby transformed.

Once again using "3" to illustrate this of course represents a prime number!

Now from the conventional analytic perspective in cardinal terms, this prime thereby represents a constituent "building block" of the natural number system.

However from the (unrecognised) holistic perspective, in ordinal terms, the position is reversed with each prime group representing a unique configuration of its constituent individual members.

So "3" for example, thereby represents a unique configuration with respect to its 1st, 2nd and 3rd members.

The next prime "5" would then represent a unique configuration with respect to its 1st, 2nd, 3rd, 4th and 5th members.

However this begs the significant question as to the derivation of the 4th member (which implies the number "4"). Therefore though from the cardinal perspective "5" is already viewed as an independent "building block", clearly from the ordinal perspective the composite natural number "4" is directly implied with respect to its 4th member!

In other words, from a dynamic interactive perspective it is quite untenable to maintain this absolute stance with respect to the primes as representing the independent "building blocks" of the natural number system!

Certainly from a relative perspective, the primes appear as the "building blocks" of the natural number system (in cardinal terms). However from an equally valid alternative relative perspective, each constituent prime appears as representing a unique configuration of its individual natural number members (in an ordinal manner).

So again in cardinal terms, the natural numbers appear to be determined by the primes; however from the ordinal perspective, each prime appears to be determined by its natural number members.

The key implication therefore is that from a dynamic interactive perspective - which represents the true nature of the number system - both the primes and natural numbers are co-determined in a synchronistic manner (that is ultimately ineffable).So the primes and natural numbers ultimately mirror each other in a perfect manner.
And through right understanding one can experientially approach, to an ever closer degree, true appreciation of this perfect mirroring.

So in terms of Band 5 development, a new appreciation of "dimensional" numbers started to open up, whereby they could now become fully grounded in the linear (1-dimensional) levels of Band 2.

So I now came to the clear realisation that the very means of "converting" the qualitative notions of 1st, 2nd, 3rd, ... entailed the holistic appreciation of the corresponding roots of 1.

So again for example in Type 2 terms, we can represent 3 as 13.

Thus is holistic terms 3 represents the notion of 3 dimensions as fully interdependent with each other (which is directly grasped in an intuitive manner).

Thus with each number is associated a distinctive "quality" of intuition. Thus associated with 2 is the quality of appreciating the interdependence of 2 related dimensions, with 3, the quality of appreciating the interdependence of 3 related dimensions, with 4 the quality of appreciating the interdependence of 4 related dimesnions and so on.

However we can indirectly convert such qualitative notions, in a quantitative (1-dimensional) rational manner, by taking the corresponding roots of 1 (associated with each number).

Thus for example the 2 roots of 1, i.e. + 1 and – 1 express - in a necessarily paradoxical "circular"  manner - the linear rational notion of the interdependence of two objects.

As I have repeated many times before this naturally arises in our appreciation of the paradoxical nature of turns at a crossroads.

So in approaching the crossroads from one direction one can unambiguously define left and right turns. So if "left" is designated as + 1 (as 1st), then "right" in this context is designated as – 1 (as 2nd).

However when the crossroads is approached approached from the opposite direction, what was formerly a left turn is now right and what was a right turn is now left. So what was + 1 (as 1st) is now – 1 (as 2nd), and what was  – 1 (as 2nd) is now + 1 (as 1st).

So in this context of mutual relative interdependence,  + 1 and – 1 can switch between each other (with each possessing a merely relative validity). In fact the interdependence of the two numbers is expressed through the requirement that their sum = 0.

Likewise the 3 roots of 1 i.e. + 1,.5 + .866i and .5 –.866i express (in an indirect linear rational manner) the interdependence of 3 numbers with respect to 1st, 2nd and 3rd positions (which can mutually switch in a relative manner as between each other).

And in more general terms the n roots of 1 likewise express (in an indirect linear rational manner) the interdependence of n numbers with respect to 1st, 2nd, 3rd,..., nth positions (which can all mutually switch in a relative manner as between each other)..

Monday, May 22, 2017

Number and Development (8)

There is an important (unappreciated) paradox with respect to the quantitative definition of any number.

For example, if we take the cardinal number "3"to illustrate, it can be defined in the conventional mathematical manner as,

3 = 1 + 1 + 1.

This represents - what I term - analytic interpretation, whereby the (whole) sum i.e. 3, is treated in an actual quantitative manner as the sum of its independent (part) units.

So here again, each of its three (sub) units is defined in an independent homogeneous manner i.e. without qualitative distinction.

Therefore, from an ordinal perspective, there is no way to distinguish (with respect to dimensions of space and time) which units are 1st, 2nd and 3rd respectively.

So in this ordinal context, each unit can potentially qualify as both 1st, 2nd and 3rd respectively.

In other words in - what I term - holistic interpretation, each (part) unit is treated in a qualitative manner as potentially representing the interdependence of all the ordinal elements of its corresponding group.

Thus from this holistic qualitative perspective, 1st, 2nd and 3rd (as ordinal positions) can equally be identified with each of the individual units of 3.

Therefore when one one appreciates this central paradox i.e. that a number that is defined from an extreme quantitative perspective (in analytic terms), gives rise to the opposite extreme qualitative perspective (in a corresponding holistic manner), then one must accept that the conventional attempt to define number in absolute fixed terms must itself be abandoned.

In other words, to resolve this paradox, one must move to a dynamic interactive interpretation of number, defined in a balanced relative manner, equally containing both quantitative (analytic) and qualitative (holistic) aspects.

So now from this new dynamic perspective, the quantitative aspect of number is viewed analytically in relatively independent manner; in complementary terms, the qualitative aspect is viewed holistically in a relatively interdependent fashion.

Therefore, again from this dynamic perspective - which concurs directly with the human experience of number - quantitative independence (in analytic terms) always implies qualitative interdependence (in a holistic manner); likewise qualitative independence necessarily implies quantitative independence, with both interpreted in a - necessarily - relative manner.

Thus with respect to the number "3" in our illustration, this number is now understood to entail the dynamic interaction of both the quantitative notion of 3 (understood in an analytic manner) and the qualitative notion of 3 i.e. as "threeness" (understood in a corresponding holistic fashion).

Now, when one properly appreciates what is stated here, then it should become apparent that the standard accepted interpretation of number is simply not fit for purpose. It reduces the distinctive qualitative aspect (which can only be properly understood in a holistic manner) in an absolute quantitative manner (that is interpreted in a merely analytic fashion).

I have explained on many occasions how I have sought to remedy this deficiency in the standard interpretation of number by employing a truly dynamic appreciation, which entails the complementary interaction of both Type 1 and Type 2 aspects.

Thus when we interpret "3" - now understood appropriately in a relative manner - with respect to its quantitative characteristics, in Type 1 terms, this is written as 31.

So the dimensional number (i.e. exponent) of 3 is defined with respect to its default status of 1, which implies that we can concentrate in this context on the relative quantitative nature of 3.

Then when we interpret "3" with respect to its corresponding qualitative characteristics, it is now written in Type 2 terms as 13 .

So 3 now directly represents its dimensional status defined with respect to the default base of 1, which implies that we can now concentrate in this alternative context, on the relative qualitative nature of 3.

Thus more simply expressed, when In Type 1 terms we are aware of the quantitative nature of 3 (in analytic terms), then - relatively - in Type 2 terms we are aware of the qualitative nature of 3 (in a holistic manner).

However reference frames continually switch in experience.

Therefore there is also a valid Type 1 sense in which 3 takes on a qualitative meaning (in a holistic manner), with 3 then - relatively - carrying a quantitative Type 2 meaning in analytic terms.

In fact this latter qualitative Type 1 aspect of number is central to proper interpretation of the nature of multiplication, whereas the quantitative Type 2 aspect arises for example in the geometrical appreciation of a cube (with 3 linear dimensions).

Therefore, properly understood, the number "3" - and by extension every number - keeps switching in experiential terms as between its quantitative (analytic) and qualitative (holistic) meanings with respect to both Type 1 and Type 2 aspects.

And once again, the standard interpretation of number (in absolute quantitative terms) distorts these key dynamics in a grossly reduced manner.

Friday, May 5, 2017

Number and Development (7)

In integral terms, the levels of Band 3 properly constitute both the new emerging "higher" stages of that band and the continually revisited "lower" stages of Band 1 (with which they are - in horizontal, vertical and diagonal terms, dynamically complementary).

Therefore from a true integral perspective, we do not have here individual stages (in a discrete separate manner) but rather the growing interpenetration of all the stages of Band 1 and Band 3.

However because integration is not yet fully balanced, typically more emphasis is placed initially on the differentiation of the new "higher" stages of Band 3 (without full consideration of the consequent need for integration of these with the corresponding complementary stages of Band 1).

So in this differentiated sense, it is still correct to give each new stage of Band 3 a relatively distinct independent identity.

In terms of my own journey through these stages (of Band 3), I was indeed well aware in one sense of the need for their two-way integration with those of Band 1.

I was convinced - even at this time - of the need for both top-down integration (where the "lower" stages of Band 1 would be integrated from the perspective of the "higher" stages of Band 3) and bottom-up integration (where the "higher" stages of Band 3 would be integrated from the revisited stages of Band 1).

However in practice, the "higher" levels, based predominantly on the cognitive mode (of reason) were in important respects still repressing the instinctive behaviour of the "lower" levels, based predominantly on the affective mode (of emotion).

And such imbalance as between cognitive and affective modes is in many ways inevitable until proper integration is eventually achieved (relating to the radials stages of Band 6 and 7 in my account).

And this is why I always emphasise the volitional mode as truly primary, as this needs to be used with ever-greater discernment to eventually bring both cognitive and affective into true harmony.

So one's ever more refined sense of something remaining "not quite right" with development, can only be addressed at the appropriate time, thus enabling eventual harmony to be achieved. And this balance - which always is of an approximate nature - is dictated by the volitional mode!

In this dynamic account of development, it is necessary to distinguish as between the default, diminished and enhanced experience of each stage.

At the very beginning of development the infant literally moves all over the spectrum in a confused manner (where neither the differentiation of distinct stages nor their integration with each other has yet taken place).

So from a discrete (differentiated) perspective, earliest development relates to the unfolding of the stages of Band 1. This is what I then refer to as the default understanding of these stages.

However, because all stages are necessarily to a degree still related to all other stages, this does enable a diminished perspective on "higher" stages.

Then when for example one moves to differentiation of the stages of Band 2, this then becomes the default understanding of those stages.

However one is now enabled to form both a diminished perspective on - still - "higher" stages, while, in revisiting the earlier stages from Band 2, form an enhanced view of their features.

So therefore we do not have just one experience i.e. default, of each level (of each band) on the spectrum but in fact a whole series of continually changing perspectives on other levels of both a diminished and enhanced nature.

Thus whereas, from a differentiated perspective, the earliest level of Band 1 is the first to unfold (and of the most primitive nature) from the opposite integral perspective, this likewise remains the last level to be properly integrated in terms of overall development.

So from this latter perspective (in both top-down and bottom-up terms) successful integration of this level (with all other levels) now represents the "highest" goal in development.

However because of the standard linear asymmetrical approach, far too-much attention is typically placed on the differentiated aspect of development (in the unfolding of distinct stage structures) which then runs directly counter to what is experientally required to achieve true integration of all stages.

So in the default understanding of the levels of Band 3, while I was aware of the need for bi-directional integration of "higher" levels with "lower" (and "lower" with "higher"), in practice attempted integration was still predominantly of the top down variety (where primitive instincts of the earliest levels were unwittingly repressed).

This was even evident in my holistic mathematical understanding of the nature of number at the time.

So with respect to the "higher" stages of  Band 3, there are in fact two directions.

One represents the "ascent", where one sees these stages as progressively transcending, in an intuitive contemplative manner, the "middle" dualistic stages of Band 2.

Thus my very understanding of the holistic nature of number as representing varying "higher" dimensions of experience fitted in very well with this notion of the "ascent".
In particular I associated the holistic notion of "2" with Level 1, the holistic notion of "4" with Level 2 and the holistic notion of "8" with Level 3.

And then the larger numbers were associated with even more refined contemplative development beyond these levels.

However I was only to later realise that this new holistic mathematical understanding needed to be properly grounded in the analytic levels of Band 2.

So in my account Band 5 (Level 1, Level 2 and Level 3) relates to the corresponding "descent" back to the middle levels, which then opened up marvellous new revelations regarding the ordinal nature of number.

The other direction at Band 2 is the subterranean "descent" from the middle levels of Band2, towards  the earliest primitive understanding of the levels of Band 1.

However because my psychological understanding of these levels still remained somewhat repressed (through the predominant influence of "higher" cognitive development) my corresponding holistic mathematical understanding did not readily emerge at this time.

I had indeed formed the conviction that somehow the prime numbers (given a new holistic interpretation) would be deeply relevant in terms of the structure of these levels. However I was not yet able to precisely see what such holistic understanding entailed. 

I was also aware that my "circular" understanding of number as dimension was also relevant, in the sense that the 3 "lower" levels represented in complementary terms the confused understanding of the corresponding 3 "higher" levels of Band 3.

So therefore the earliest (most primitive) level represented the confusion of all 8 polar directions (i.e. form with emptiness, wholes with parts and external with internal).

With the next level, form could be distinguished from emptiness, but the two other confusions still in large measure remained.

Then finally, as the "lower" levels approached the middle level, only the remaining confusion of external with internal directions remained.

However, I knew that a deeper holistic mathematical knowledge of the nature of the primes and how this ultimately could be grounded in accepted analytic understanding, was necessary.

Indeed this required nothing less than a radical new interpretation of the true dynamic nature of the number system, which was likewise to unfold during Band 5.  

Tuesday, May 2, 2017

Number and Development (6)

I have drawn attention to the great holistic mathematical significance of 2, 4, and 8 (representing dimensionl nos.) in terms of the developmental task of integration (with complementary applications in physical and psychological terms).

Once again, 2 directly relates to horizontal bi-directional integration (within a given level).
4, relates to horizontal and vertical bi-directional integration (within and between levels), where both aspects of integration are still pursued in a - relatively - separate manner.
8, then relates to diagonal bi-directional integration simultaneously within and between levels.

This is required to fully integrate the "higher" levels of Band 3 with the complementary "lower" levels of Band 1 (within and between levels).

However in dynamic experiential terms, the task of integration cannot be properly considered in the absence of the corresponding requirement for sufficient differentiation of the major levels of each band.

So we can have two extremes:

1. Where differentiation - especially with respect to the specialised development of the linear levels of Band 2 - takes precedence.  Here, limited exposure to the "higher" levels of Band 3 is likely to take place with integration significantly reduced and geared to the successful further spread of differentiated type experience, in what typically constitutes the "active" life..

2. Where integration - especially with respect to the specialised "higher" levels of Band 3 - takes precedence. However this often leads to the significant by-passing of the conventional everyday experience of the levels of Band 2. In former times this typically constituted the "passive" i.e. contemplative life, where spiritual aspirants left worldly concerns behind to live in confined monastic communities.

However the fullest development requires that an equal balance can be given to both differentiation and integration in a mixed approach. Then, when successfully attained, one becomes become fully grounded in the differentiated activities of everyday life, while maintaining in the midst of such activity, a deep contemplative vision (which then serves to properly integrate all things).

Thus the greatest exponents of such mixed development  - who often in the past were pioneering religious reformers - were thereby enabled to live amazingly productive lives, full of creative endeavour that served to dramatically transform the world in which they lived.

However this "full life" combining ever growing differentiation and integration always remains an ideal to which one can only ever roughly approximate.

However the lessons to be learnt from each - somewhat limited - attempt to achieve this exalted goal can possibly open up important new features with respect to development that have not been adequately recognised previously.

Again I have placed special emphasis on the holistic mathematical relevance of 2, 4 and 8 - which literally represents varying dimensions - for development. Though potentially all numbers possess a distinctive relevance, 2, 4 and 8 (especially 4) still maintain a special importance which can be illustrated as follows.

In quantitative terms, every number can be represented on the complex plane. This contains a real axis where a number can be positive or negative and an imaginary axis (where likewise a number can be positive or negative).

So all numbers can be shown to represent unique configurations with respect to these 4 aspects.

Likewise in qualitative (i.e. holistic) mathematical terms, all numbers can be shown to represent unique configurations with respect to these same four aspects (i.e. positive, negative, real and imaginary).

And the holistic mathematical appreciation of 4 opens up this latter realisation.

Therefore experience throughout all development entails unique configurations of wholes and parts with respect to their external and internal aspects.

Thus when one has mastered the 4-dimensional approach, one has the blueprint as it were to successfully map all experience.

Here, ever greater development at the "higher" levels then necessarily entails more refined ways in which one can experience the relationship between wholes and parts with respect to their internal and external directions.

However, I gradually came aware of a crucial distinction with respect to even numbered and odd numbered dimensions respectively.

As I have stated on many occasions, I was at the time very much influenced by the writings of St. John of the Cross.

However I slowly came to the view that he laid too much emphasis on "passive" rather than "active" nights.

Now the "passive" nights can be directly related to the deep task of unconscious spiritual integration.

However for healthy development - even during the intense purgation of the "dark nights"- passive must be balanced to a degree by active type development (where differentiated activity is necessarily involved).

So I saw the healthy unfolding of "higher" development as follows.

Each relatively more differentiated stage should then be followed by a subsequent stage where the primary emphasis would be on integration.

Thus in holistic mathematical terms, each odd numbered would be followed by an even numbered stage.
There is in fact a crucial mathematical distinction as between the even and the odd roots of 1. With the even roots a complementary relationship always exists as between the roots so that every positive root can be matched by a negative counterpart.
However with odd numbered roots this is not the case. One of these roots however will always be 1, with the other roots existing as complex conjugates of each other.

Therefore, whereas the even roots can always be associated with integration (in the complementarity of opposites), the odd roots can be associated with new forms of differentiation (where linear understanding maintains a degree of independence).  

So the linear (1-dimensional) stage of highly differentiated dualistic activity is then followed by the 2-dimensional stage where nondual integration of external and internal polarities takes place.

However this should then be followed by the 3-dimensional stage, which represents a new "higher" attempt at engaging in differentiated activity, from the perspective of the spiritual integration that has already taken place.

However this leads inevitably to a new clash with respect to dual and nondual which then requires an even "higher" 4-dimensional stage to resolve.

So each new "higher" stage of differentiation (with an odd number) gives way to a corresponding new "higher"stage of integration (with an even number).

However, if development of the middle levels of Band 2 has not been especially prolonged (due to early intense exposure to contemplative experience), a limited basis may thereby exist for differentiation at the subsequent "higher" levels. So development will then be unduly intensive (in an unconscious manner) and insufficiently extensive (in a conscious fashion).

Furthermore, the continual need to bi-directionally integrate the 3 levels of Band 3 with the 3 corresponding levels of Band 1, will mean that experience becomes so dynamically interactive (in both directions) that one will then find it extremely difficult to take rest as it were in the more stable phenomena of Band 2.

So the levels of Band 2 are likely therefore to become significant by-passed at this time, with their successful integration with both Bands 1 and Band 3 requiring on-going development through further stages of understanding.      

Monday, May 1, 2017

Number and Development (5)

In my approach, 2-dimensional understanding - associated with Band 3 (Level 1) - is mainly geared towards the bi-directional  horizontal integration of external and internal polarities (within a given level). So from the external perspective, each stage represents a new understanding with respect to the physical world; then from the internal perspective each stage represents corresponding new understanding with respect to psychological reality.

In general with respect to human development, an unbalanced emphasis is placed on stages (solely with respect to their internal psychological characteristics).
What is not all properly realised however is that each new stage of psychological development is equally associated with a new stage of scientific - and of course mathematical - understanding.

So the present accepted scientific paradigm simply reflects the understanding associated with one limited band (i.e. Band 2) of the overall spectrum.

4-dimensional understanding - associated with Band 3 (Level 2) is mainly geared additionally towards the bi-directional vertical integration of whole and part polarities (between levels).

Thus from one perspective, each new "higher" stage represents a growth in "holism" (whereby earlier - somewhat fragmented - part features are integrated into a new collective whole); equally from the opposite perspective, each - relatively - "lower" revisited stage represents a growth in "partism" (whereby somewhat rigid whole features are properly integrated through becoming uniquely contained in each part). In this way balanced vertical integration must necessarily be of both a top-down and bottom-up nature.

Then a new more intricate 8-dimensional understanding is associated with Band 3 (Level 3). This is now geared towards the simultaneous bi-directional integration both horizontally (within levels) and vertically between levels), in what can be referred to as diagonal integration.

Typically with Level 1 (Band 3) undue attention is given merely to horizontal type integration, whereas at Level 2, undue attention is then given to vertical type integration. So the considerable task then remains of properly balancing (in a bi-directional fashion) both of these aspects of integration with each other.

So the holistic mathematical interpretation of the 8 dimensions (relating to - now - 8 relatively distinct directions of understanding), is given, in a reduced 1-dimensional manner, by the corresponding 8 roots of 1. So the 4 additional roots i.e. k(1 + i), k(1 – i), k(– 1 + i) and k(– 1 – i) where k = 1/ √2, represent the structure of the 4 diagonal poles.

Now, the striking feature of these diagonal lines is that they can be given two equivalent interpretations, which then in dynamic interactive terms illustrate a key feature of advanced contemplative development.

So from one perspective, we can say that each pole represents a balanced mix of both real and imaginary aspects (though the signs can vary).

What this entails is that successful diagonal development of the psyche - enabling two-way balanced integration within and between levels - requires that both cognitive and affective modes, which are relatively real and imaginary with respect to each other, operate in equal balance in all four quadrants (of the unit circle) in which they arise.

In other words, in order to freely relate to phenomena without rigid attachment arising, the fully balanced integration of both cognitive (control) and affective (response) is required within and between all levels, which is now directly enabled through the volitional mode.

However as is well known in Mathematics,  these diagonal lines equally have a remarkable feature in that that they can be portrayed as "null lines" with a magnitude = 0.

In fact these "null lines" explain the nature of light.

In terms of its own frame of reference, light "travels" in zero time. In other words, when travelling at "light speed", one remains continually in the present moment.

Put another way, one can equally maintain that at the speed of light, neither its real or imaginary components (i.e. its particle and wave-like features) can be distinguished.

However there is a fascinating psychological equivalent in that when dynamic interactivity within the psyche approaches "light speed" through pure spiritual contemplative intuition, time does not pass and one abides in the continual present moment.

So the experience of spiritual "emptiness" in pure contemplative union, coincides in dynamic terms with the highly dynamic interaction of form, that is so freely experienced, without attachment, that phenomena no longer even appear to arise.

Thus I have consistently referred to the 3rd stage of integration (associated with Level 3) as the integration in all 4 quadrants of the poles of (phenomenal) form and (spiritual) emptiness.

And with respect to balanced integration of the psyche, both aspects must be harmonised with each other in a dynamic experiential manner.

Now from the opposite perspective, these diagonal polarities also have considerable relevance in terms of earliest development with respect to purely instinctive psycho-physical behaviour, where cognitive and affective aspects still remain greatly confused with each other.

And these two extremes of behaviour always remain necessarily closely related with each other.

So one can never be fully free of involuntary instinctive reactions. Therefore the requirement for achieving further spiritual integration, which is always of an approximate nature, is the corresponding further unravelling of instinctive behaviour, where cognitive and affective aspects still remain confused with each other.

Thus to keep ascending "higher" into the purer spiritual realms, one must equally keep descending "lower" into the primitive depths of the psyche.
One does indeed have the capacity to be with the angels; however one always remains rooted in one's animal nature. And without constant reminders of this twin nature, one cannot aspire to true integration.

In fact it is the very nature of development that the attempt to solve one important problem, can then open up other serious problems that have not yet been apparent.

Thus in terms of my own development, I began to find myself subject to growing psychological stress as I pushed more into - what I identified as - Level 3.

This was supposed to lead to the full integration of the 3 levels of the "higher" Band 3 with the corresponding 3 levels of the "lower" Band 1 (both within and between levels).

However I gradually recognised in this desire for spiritual integration that the middle levels of Band 2 - on which everyday activities are largely based - were becoming significantly by-passed.

In other words I was failing to properly ground either the "higher" Band 3 or the "lower" Band 1 in the more linear levels of Band 2.

So I was getting psychologically stretched  so much in continually attempting to reconcile extremes that I felt as if I could scarcely breathe.

Therefore another decisive change in direction was now required.