Fibonacci Type Relationships as Fractals

The mystery of the Fibonacci ratio (phi = 1.618...) is often expressed geometrically in terms of a Nautilus Shell which is a good example of a self-repeating spiral pattern (found frequently in nature).

Therefore from this perspective the very structure of the number phi is inherently dynamic and of a fractal nature (that endlessly repeats a simple pattern).

Now the Fibonacci ratio itself can be obtained as the positive solution to the simple polynomial equation,

x^2 - x - 1 = 0.

We can use a fascinating way to approximate this solution - and indeed any polynomial equation with an algebraic solution - through an easy iterative procedure.

In general terms for the equation x^2 + bx + c = 0, we start with the two numbers 0, 1 and add 1 * (- b) + 0 * (- a). So for the Fibonacci equation, this gives (1 * 1) + (0 * 1) = 1.

So we now have in the sequence 0, 1, 1 .

Continuing on in the same fashion the next term = (1 * 1) + (1 * 1) = 2.

So we now have 0, 1, 1, 2,

In this manner the well known Fibonacci sequence can be derived

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597,...

Now, phi can be approximated as the ratio of a term and its preceding term (which approximation continually improves with higher terms).

So if we take the ratio of the last two terms we get 1587/977 = 1.61803444782..

This is already an extremely good approximation to the true value of phi = 1.6180339887.. .

However phi can equally be approximated in terms of a simple formula relating to the terms in the sequence that involve powers of 2.

So phi = 1/t1 + 1/t2 - 1/t4 - 1/t8 - 1/t16 - ......

Thus calculating up to t16 we get

phi = 1 + 1 - 1/3 - 1/21 - 1/987 = 1.61803444782..

So the value of this series up to t16 gives the same result for phi as t17/t16!


Just as the Golden Ratio (phi) can be represented in dynamic terms as a number fractal, in principle every (real) algebraic irrational number can be expressed in like manner as a fractal. This follows from the fact such an algebraic irrational must correspond to some polynomial equation with (real) integer coefficients. And as all such equations give rise - like the Fibonacci - to unique number sequences with recursive features, we can use these numbers to approximate (to any required degree of accuracy) the irrational numbers involved.


For example for the equation x^2 - 2x - 1 = 0, we derive the following sequence

0, 1, 2, 5, 12, 29, 70, 169, 408, 985, ...

The solution to this equation 2.414213562373... is approximated again by ratio of successive terms.

Thus using the last two terms we obtain 985/408 = 2.41421568... (which already is a pretty good approximation).

In exactly the same manner as with the Fibonacci, we can use this sequence of numbers to drive a simple expression to derive the square root of 2.

So square root of 2 = 1/t1 + 1/t2 - 1/t4 - 1/t8 - ...

= 1 + 1/2 - 1/12 - 1/408 -....

= 1.41421568... (i.e. 2.41421568... - 1).

Partition Numbers

I recently came across the interesting fact that a new finite method for calculating partitions has been discovered by Ken Ono with some collaborators at Emory University.

This is a fascinating development with respect to an area of number theory that seems deceptively simple yet proves to be fiendishly difficult.

Partitions simply relate to the number of ways that a particular number can be broken down. It may help to initially think of this in concrete terms.

So for example the various partitions of 4 could be likened to the manner in which we could break up arrangements of - say - four pebbles.

At one extreme we could take the four together (i.e 4).
Then we could divide the four into a group of three with one left over (i.e. 3 + 1).
We could also divide into two groups of two pebbles (i.e. 2 + 2).

Then we could split one group of 2 into two single pebbles while maintaining the other group intact (i.e. 2 + 1 + 1).

Finally we could break the 4 into 4 single pebbles (i.e. 1 + 1 + 1 + 1).

So the total number of partitions of 4 is thereby demonstrated to be 5.


However whereas it is relatively easy to work out the partitions in this manner for the lower numbers, it quickly becomes increasingly more difficult so that for example the number of partitions of 100 is 190569292!

In arriving at the partitions in this manner we are considering various combinations without rearrangement. So for example 3 and 1 and 1 and 3 in this interpretation represent the same partition.

Now when we allow for rearrangement, the calculation of the number of partitions is surprisingly simple.

Thus taking once again the number 4 we can include here as additional partitions 1 + 3, 1 + 1 + 2 and 1 + 2 + 1 giving eight partitions in all.

In fact the general formula for sum of partitions (with rearrangement) is 2^(n - 1).

So the answer of 8 represents the case where n = 4.

Now this result of the number of partitions (with rearrangement allowed) can be expressed as the sum 1 + 2^0 + 2^1 + 2^2 + ... + 2^(n - 2).

Interestingly - dating from Euler - the number of unrestricted partitions (without rearrangement) can be expressed as a generating function entailing the partition numbers.


One fruitful exercise would be the exploration of the relationship as between unrestricted partitions (without rearrangement) and restricted partitions (with rearrangement).

Clearly the number of unrestricted is considerably less than restricted for large n.

It struck me that Mersenne primes can be seen to represent a unique relationship with 2^(n - 1).
Thus all Mersenne primes therefore are related to (appropriate) restricted partition numbers (through the subtraction of 1).

So using the restricted formula for partition numbers i.e. 2^(n - 1) where once again rearrangement is allowed, the first Mersenne prime when n = 3 is 2^2 - 1 = 3.

The second Mersenne prime for n = 4 is 2^3 - 1 = 7. The third Mersenne prime for n = 6 is 2^5 - 1 = 7 and the fourth for n = 8 is 2^7 - 1 = 127.


It also struck me that perhaps a more general relationship involving the relationship of the prime to natural numbers also pertains to the relationship as between the unrestricted (without arrangement) and restricted partitions (with arrangement).

For example the 100th restricted partition number = 2^99 and the 100th unrestricted partition number (190569292) lies between 2^29 and 2^30. Now as the number of primes contained in the first 99 natural numbers = 25, it is tempting to believe that perhaps there is some link here with the general distribution of the prime numbers.


However for much higher values of n this apparent relationship breaks down.

In other words when we express an unrestricted partition number n, as a power of 2, the power of this number ultimately bears very little relationship with the frequency of primes up to n - 1!

However it is still tempting to surmise that - even if less apparent - that an important relationship relating to the distribution of primes (among the natural numbers) underlies the relationship of unrestricted to restricted partition numbers.

In this context, Ono and his team demonstrated a pronounced recurrence pattern to partition numbers whereby - ultimately - all terms could in principle be shown to recur at regular intervals (as multiples of the original term) in the sequence of partition numbers.

Ramanujan had already demonstrated this recurrence pattern for 5, 7 and 11. However though much less obvious this can be extended to the other partition numbers!

For example the first 30 terms of the (unrestricted) partition number sequence are

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718 and 4565.

Now if we look at 5 in this sequence we can see that a multiple of this number recurs with every 5th succeeding number. So 30 is clearly a multiple as are 135, 490, 1575 and 4565.

If we now look at 7 we can see that a multiple of this occurs with every 7th succeeding number. So 77, 490 and 2436 are all multiples of 7!

Now finally if we look at 11, we can see that a multiple of this number occurs with every 11th succeeding number. So 297 and 3718 (in this sequence) are multiples of 11!

Not surprisingly these recurrence patterns lead to the notion of the partition numbers as very interesting examples of fractals. So partition numbers in their inherent structure give rise to fractals.

From a related piece of work that I am investigating at present, I have come to the conclusion that all algebraic irrational numbers are inherently dynamic in their very structure exhibiting unique fractal patterns.

The deeper significance of this finding is that all such numbers entail the relationship as between discrete and continuous notions (that in qualitative terms are linear and circular with respect to each other).

So ultimately the very nature of partition numbers entails this same relationship!

Cardinal and Ordinal Numbers (2)

As we have seen every number can be given both a cardinal and ordinal meaning which are quantitative and qualitative with respect to each other.
And where the ordinal number is the last in a group of numbers, the ordinal can be represented as the reciprocal of the cardinal.

So if we write 1^4 this can be given a quantitative meaning - where 4 is a cardinal number representing the dimension (or power) in question whereby 1^1^1^1 = 1 (in reduced quantitative terms).

However 4 equally here has an ordinal meaning as the 4th dimension (where 4 dimensions overall are considered). So here the 4th dimension represents 1/4 (of all four dimensions)
Thus to express 4 with respect to the ordinal number 4 (as the the 4th dimension) we obtain the value of 1^(1/4) = i.

So i here has a qualitative interpretation as imaginary i.e. the indirect expression of holistic unconscious meaning in conscious terms.

Of course 1/4 here can be given a quantitative meaning so that 1^(1/4) = i represents a number on the circle of unit radius.

Now interestingly if we now consider the 4th dimension as one of 5 dimensions it no longer can be represented by 1/4!


Whenever a number ≠ 1, a complementary relationship exists between the dimensional power and its reciprocal (that are quantitative and qualitative with respect to each other).

Once again in the default case = 1, both this dimensional number and its reciprocal are identical (in Type 1 terms) so that qualitative is reduced to quantitative meaning.

Cardinal and Ordinal Numbers

I have frequently referred to the fact that all numbers have both quantitative and qualitative aspects which are dynamically interdependent in experience.

However Conventional (Type 1) Mathematics, in giving formal expression to the merely quantitative aspect, thereby distorts appreciation of the true nature of number.

One manifestation of the relationship of quantitative to qualitative aspects pertains to the relationship as between the cardinal and ordinal use of number.

The very nature of the conventional linear (1-dimensional) approach is that it reduces qualitative to quantitative type interpretation.

This is exemplified by the manner in which we look at natural numbers.

In cardinal terms we can speak of the natural numbers as 1, 2, 3, 4,... as marked along the standard number scale.

However equally in ordinal terms we can speak of the 1st, 2nd, 3rd, 4th .... numbers as marked along the same scale.

So here there is a direct correspondence as between each successive number (as used in both a cardinal and ordinal sense).
Thus 1 corresponds with the 1st number, 2 with with the 2nd number, 3 with the 3rd number, 4 with the 4th number etc.

Indeed, without implicit recognition of the ordinal use of number, it would be strictly impossible to define the cardinal numbers.

However there is a subtle difference with respect to both types. The cardinal number 4 for example suggests a multiple of part units (that are literally 1).

So in concrete terms 4 could be used to denominate a collection of any four units (e.g. 4 stones).

However 4 in ordinal terms i.e. as the 4th unit inherently conveys the opposite relationship of part to whole. So the 4th unit specifically refers to just one unit which is seen as part of - what in cardinal terms - is a whole that comprises 4 units. So implicitly to give a ranking of 4 (as the 4th) necessarily implies recognition of 3 other units that are 1st, 2nd and 3rd. In this sense it represents the inverse (i.e. reciprocal) of the cardinal notion (i.e. 1/4).

Therefore when seen in this light, the relationship as between the cardinal and ordinal notion of number in experience represents the dynamic interaction as between whole and part (and part and whole).

Now, when written more fully 4 is represented as 4^1.
In other words the default quantitative interpretation of number (where powers or dimensions are considered redundant) is with respect to 1. And in qualitative terms this corresponds directly with linear (1-dimensional) understanding.

However the reciprocal 1/4 can be represented as 4^(- 1). So from a qualitative perspective the very manner through which we switch from the cardinal to the ordinal notion of number implies the negation of the 1st dimension.

As we have seen before, this entails switching from rational to intuitive type understanding.

At a deeper level this implies that the proper understanding of number requires an approach that combines both rational (specific) and intuitive (holistic) type interpretation (relating to quantitative and qualitative aspects respectively).

In Type 1 quantitative terms, the absolute value of a number arises when the sign is ignored. So in absolute terms - 1 is thereby indistinguishable from + 1.

Likewise in Type 2 qualitative terms, absolute interpretation arises when we ignore the sign (i.e. direction) of understanding. So linear understanding ( + 1) is thereby indistinguishable from its negation i.e. - 1.

Put another way with absolute type interpretation (as befits Type 1 Mathematics), quantitative cannot be clearly distinguished from qualitative meaning!

Wholes and Parts

An intimate mathematical relationship exists as between wholes and parts in both linear and circular logical terms respectively.

Now it is the very essence of the linear approach to give a merely reduced rational interpretation of wholes and parts (where the whole is viewed solely in quantitative terms as the sum of its constituent parts).
From a conventional (linear) perspective a cake for example might comprise four (equal) slices. Thus the whole cake in this context is interpreted as the sum of the four (4) part slices.

Then in reverse manner each individual slice is interpreted - again in quantitative terms - as a quarter (1/4) of the whole cake.


Now in (Type 1) mathematical terms the whole cake (containing the four slices) = 4^1.

Meanwhile the reverse relationship of each part to this whole = 4^(- 1) = 1/4.

Thus, fascinatingly, the very ability to switch in experience from whole to part recognition entails the corresponding ability to implicitly negate the 1st dimension. What this means is that understanding implicitly must switch from rational to intuitive mode before this important transition can be made. However when qualitative interpretation is absolute, this switch from positive to negative sign is ignored so that it formally remains at a merly rational level!

Thus it requires Type 2 mathematical understanding to provide this important insight of how intuition insight is necessarily involved in the understanding of the true relationship as between whole and part..

Therefore once again Type 1 interpretation - which is conducted in merely reduced rational terms - cannot explain the decisive qualitative means of switching from whole to part (and part to whole). In order words in confining itself to a merely (reduced) quantitative (Type 1) interpretation of its symbols, it ignores completely the equally important qualitative (Type 2) interpretation of the same symbols.

Thus the minus sign (-) has a well-defined meaning in quantitative (Type 1) terms. However the same sign has a largely unrecognised qualitative (Type 2) meaning as the negation of what is (consciously) posited in rational terms. In dynamic interactive terms, this implies the (unconscious) intuitive aspect of recognition. Thus from a Type 2 perspective, in transferring from whole to part recognition, linear (1-dimensional) interpretation of a rational kind must be (temporarily) negated in a qualitative form of holistic recognition. Thus though the part can then be likewise interpreted in a quantitative manner, the decisive switch from whole to part recognition requires Type 2 qualitative recognition.


However equally we have a circular number system that allows for a whole range of differing qualitative interpretations in each case entailing a unique configuration of both (conscious) rational and (unconscious) intuitive type appreciation. From this perspective the (whole) dimension given by the number 4, is qualitatively defined in Type 2 terms (as 1^4).

Then 1/4 i.e. 1 ^(1/4) represents the reverse quantitative relationship in part terms for this system.

In circular quantitative terms this is given as i (which lies on the unit circle in the complex plane).

Thus in reverse terms i refers to the qualitative meaning of 4 (as a whole dimension).

This clearly indicates again therefore that the very switch from whole to part recognition always entails a corresponding switch from qualitative to quantitative. However understanding here is much more refined allowing for imaginary (rather than real) appreciation.

So 4-dimensional interpretation from a qualitative mathematical perspective is of an imaginary (rather than real) rational nature, as the indirect expression of unconscious meaning.

One important physical application of this principle relates to quantum mechanics where the "higher" psychological interpretation of a qualitative nature is always necessary to properly interpret "lower" sub-atomic phenomena (of a physical nature). In other words if the manner of the "higher" intuitive recognition in qualitative terms is n-dimensional, the corresponding "lower" physical recognition with respect to physical "parts" in nature is 1/n. So once again their is a reciprocal qualitative to quantitative (and quantitative to qualitative) relationship as between whole and part.

Number as Dynamic Interaction

When properly appreciated all numbers represent a dynamic interactive process that necessarily entails both quantitative and qualitative aspects (which provide the very basis for such interaction).

A number perception representing a quantity is strictly speaking meaningless in the absence of its corresponding number concept (which relatively is of a qualitative nature).

So for example the number "2" representing a specific number perception has no meaning in the absence of the universal concept of number.

And this concept of number is strictly qualitative relating to what is potential and infinite. So in this qualitative sense the general concept of number applies potentially to all (as yet unspecified) numbers in an infinite manner.

However any specific number is necessarily of an actual finite nature. And this relationship of actual number perceptions (as quantities) and the potential number concept (as qualitative) provides the very dynamic for subsequent number interaction in experience.

Of course just as a left turn off a road becomes a right (and a right turn becomes a left) when one switches the direction of movement along the road, likewise through dynamic switching in experience number perceptions equally can attain a qualitative and the number concept - relatively - a quantitative aspect.

So from this latter perspective the number perception is seen to embody, as it were, the general property of number (that is qualitative in nature) while the concept is understood to apply to all actual numbers (in quantitative terms).

This thereby represents the reverse direction of the relationship as between quantitative and qualitative in number experience.

From a psychological perspective the dynamic interaction by which both the quantitative and qualitative aspects of number appreciation occur, requires in turn both rational (conscious) and intuitive (unconscious) understanding. In this interaction the intuitive aspect points to the holistic potential aspect of qualitative recognition (that switches as between concept and perception). The rational then points to corresponding actual quantitative recognition (that again switches - in relative fashion - as between perception and concept).

In the conventional (Type 1) approach, Mathematics is formally interpreted in a rational manner. This entails in turn that a merely reduced quantitative notion of number is given.

The starting point for a true interactive (Type 3) approach is the recognition that every specific number quantity implicitly implies a corresponding dimensional number concept that is - relatively - of a qualitative nature (and vice versa).

Thus for example "2", which conventionally in Type 1 terms is interpreted (solely) as a number quantity, equally has an implicit qualitative aspect (as representing a number dimension).

So in more complete terms, the natural number quantity "2" is defined with respect to a (default) number dimension 1 which, in experiential terms, is - relatively - of a qualitative nature.

So in this context 2 is properly 2^1.

Now, the very nature of 1, when used as a qualitative dimension is that qualitative is necessarily reduced to quantitative type meaning. Therefore though the number concept properly refers here to a potential - as opposed to actual - meaning, from a linear (1-dimensional) rational perspective, this is then interpreted in a merely reduced manner (as applying to all actual numbers).


However just as we can define the extreme quantitative (Type 1) approach in terms of a default dimensional number (i.e. 1), likewise we can define an extreme qualitative (Type 2) approach in terms of a default base quantity that is also "1".

So "2" in this approach representing a dimensional number is written as 1^2.
The significance of this latter approach is that each number represents a unique manner of qualitative interpretation of mathematical symbols.
Thus from this perspective we have a - potentially - infinite set of ways for the logical interpretation of mathematical relationships. And the qualitative structure of each logical system of interpretation exactly matches the corresponding root structure (in quantitative terms)! So just as the two roots of 1 - for example - in quantitative terms are + 1 and - 1, equally the qualitative logical corresponding to 2 (as dimensional number) is based on the complementarity of opposite poles (i.e. + 1 and - 1, taken as interdependent. So in quantitative terms we interpret the roots in a linear either/or logic (based on the independence of opposites) in corresponding qualitative terms we interpret the dimensional number in both/and terms (based on their interdependence).

Extending this realisation, every mathematical symbol that is given a specific quantitative interpretation in Type 1 Mathematics, can equally be given a holistic qualitative interpretation in Type 2 terms.

In Type 3 Mathematics we then attempt to interactively combine both Type 1 and Type 2 understanding.

Thus the simple mathematical expression 2^2, in Type 3 terms represents a dynamic interaction as between 2 as base number and 2 as dimensional number (with each number having both quantitative and qualitative aspects always in opposite relationship to each other).

Three Levels of Appreciation of Transcendental Numbers

I have mentioned on many occasions that three type of mathematics exist i.e Type 1, Type 2 and Type 3 respectively.

I will now interpret the meaning of a transcendental number such as e according to the three types.

In conventional Type 1 appreciation, e represents e^1. So when one concentrates merely on the quantitative interpretation of a number, it is always defined with respect to the (default) 1st dimension. So here we have - literally - a linear rational interpretation of number.

Properly speaking as we will see, a number such as e transcends mere rational interpretation. In qualitative terms, rational (linear) is suited to the interpretation of rational discrete numbers that are finite in nature.

However the very essence of e - and indeed all transcendental numbers - is that they combine both finite and infinite aspects in their very nature. Thus though the quantitative value of e can be approximated to any required degree of accuracy, its true value always remains unknown (leading potentially to an unlimited number of terms in its decimal sequence).

Though there are many ingenious ways of representing e, as with all transcendental numbers it cannot be the solution to a polynomial equation.


In Type 2 appreciation, in a somewhat inverse fashion e represents the dimension to which the (default) no. 1 is raised.

So e in Type 2 terms is represented as 1^e.Therefore as a dimension, it now takes on a holistic qualitative significance (with respect to its default base quantity).


This is a crucial point that is not addressed in Type 1 interpretation. From Type 1 perspective, when e is used as a power it still represents a number quantity. However properly speaking the relationship between the base quantity and its dimensional power is always quantitative as to qualitative (and qualitative as to quantitative).

So e in Type 2 terms takes on an appropriate qualitative holistic meaning. Now a transcendental number in qualitative always expresses a relationship between rational (discrete) and intuitive (continuous) notions!
In this context e has a special significance as the number which uniquely reconciles both aspects within its own nature.
So in psychological (and corresponding physical terms) with e, both the processes of (discrete) differentiation and (continuous) integration are reconciled.
It serves therefore as an especially advanced qualitative symbol of such differentiation and integration (where both aspects are indistinguishable).


In Type 3 terms, understanding keeps switching as between both quantitative and qualitative interpretation with respect to e as both base and dimensional number respectively.

So we understand e as a (base) quantity. Then attention switches to understanding e in holistic terms as a dimensional quality; then attention switches again to now understanding e in holistic terms also as a base quality; and finally in this cycle attention switches to understanding e also as a dimensional quality.

So in this dynamic interactive manner, e whether representing a base or dimensional number, possesses both quantitative and qualitative aspects which keep alternating in experience.

This means that in Type 3 terms a proof always entails both quantitative and qualitative aspects and is subject to the Uncertainty Principle.

For example the Type 1 proof that any algebraic number raised to an algebraic power can in truth be given both a quantitative (Type 1) and qualitative (Type 2) interpretation. (In qualitative terms this amounts to a holistic mathematical interpretation of how development proceeds from the psychic/subtle to the causal level).

Then in Type 3 terms the Uncertainty Principle necessarily applies to the dynamic interpretation that combines both. In this manner, in comprehensive Type 3 terms, all mathematical proof is subject to the Uncertainty Principle.

Mathematical Dimensions and Psychological Development (2)

We have see in the last contribution that when the quantitative to qualitative relationship is maintained that raising a rational number to a rational fraction results in an irrational quantity.

So for example in the best known case, the square root of 2 i.e. 2^(1/2) = 1 .4142... is an irrational number.

In corresponding fashion when rational perceptions are dynamically related to rational concepts that are fuelled by appropriate intuitive appreciation of a qualitative nature (algebraic) irrational perception (and later conceptualisation) results. Expressed in more common language this entails the more refined appreciation of phenomena that are inherently paradoxical in nature.

The very nature of an irrational number is that it necessarily combines both finite and infinite aspects. Thus an irrational number can be approximated in rational terms to any required degree of accuracy. However equally it possesses an elusive infinite aspect in that its exact value can never be known.

Likewise in corresponding qualitative terms, irrational understanding with respect to perceptions and concepts (in what is sometimes referred to as the psychic/subtle realm) combines both finite and infinite aspects. Thus phenomena in experience still possess a distinct identity (of a dynamic relative nature). However equally they possess a numinous spiritual quality that is infinite in nature.

As always we can identify - though in truth considerable overlap may be involved - three stages at this level.

Firstly we have the unfolding mainly of the more superficial refined phenomena that are qualitatively irrational in nature. Now once again the very basis of rational understanding is that interpretation appears - especially in a mathematical context - unambiguous in nature. Thus for example the positing therefore of what is true, implies the corresponding negation of its opposite as false.

However in the very dynamics of understanding at this level, phenomena that are consciously posited in experience are quickly negated (in an unconscious manner), Thus propositions take on a merely relative i.e. paradoxical truth value.

The second stage then entails deeper conceptual structures of understanding that are also irrational (in qualitative terms).

Finally at the most advanced stage we have the growing interaction of both irrational perceptions and irrational concepts paving the way for a remarkable transformation of experience to a new level.

When the mathematician Hilbert detailed 23 unproven propositions at his famous address in 1900, one related to finding a proof that any rational (or irrational) number when raised to an irrational dimension (power) would result in a transcendental number quantity!

A transcendental number resembles an even more subtle form of irrational number.

For example the the square root of 2 and the well known constant pi are both irrational numbers. However whereas the former - and indeed any number of this type - can be expressed as the solution to a polynomial equation, the latter cannot be expressed in this manner.

There is a fascinating corollary in qualitative terms. Paradoxical (2-dimensional) understanding based on the complementarity of opposites is clearly paradoxical in terms of conventional linear (1-dimensional) reason that is unambiguous in nature.

However if we define reason in terms of the former (2-dimensional) variety then from this perspective it is now rational. Indeed Hegel did precisely this in his writings defining reason in terms of his dialectic while treating conventional logic as a "lower" form that he termed "understanding".
The trouble is that Hegel then effectively reduced the nature of such dialectical reason through his failure to emphasise the corresponding need for the necessary supporting intuition provided through authentic contemplation.

This in turn is a regular failing at the irrational (psychic/subtle) level where secondary rigid attachments to the paradoxical symbols in experience emerge.

The resolution of this problem requires the profound negation of such attachment. In this way one gradually develops the ability to preserve an increasingly harmonious balance as between (conscious) reason and (unconscious) intuition.

Put another way this implies maintaining an appropriate relationship (that is quantitative as to qualitative) as between both the paradoxical perceptions and concepts that typify the level.

It was eventually proven in 1934 that when a rational (or irrational) number is raised to an irrational power that a transcendental number quantity results. And remember this was one of Hilbert's 23 propositions!

Remarkably we can provide the qualitative corollary to this proposition by saying that when rational (or irrational) perceptions are appropriately related to irrational concepts that a transformation in understanding takes place whereby experience of a transcendental nature emerges. And this is the important transformation that enables successful transition from the psychic/subtle to the causal level.

Now, we can understand the true nature of a transcendental number with respect to the nature of pi, which represents the pure relationship of the circular circumference to its line diameter. In like manner transcendental understanding (which typifies the causal realm) represents the pure relationship between circular appreciation (that is paradoxical) and rational understanding (of a linear nature). In other words it points directly to the common relationship as between both.
Now the center of a circle equally represents the midpoint of its line diameter. In like manner it is through the still point of being (representing the naked will through pure volitional intent) that both circular and linear type appreciation are reconciled. In this way the transcendental structures properly evolve. Because this entails approximating ever closer to this still point of being (in both physical and psychological terms) I have always referred to the causal level in holistic mathematical terms as the point level!

In terms of development of such structures the most refined possible are of an imaginary - rather than real - nature.

A real transcendental perception (using holistic mathematical language in a precise manner) relates to a specific phenomenon as representing the refined interaction of both conscious and unconscious aspects of experience (with both operating in close harmony).
A real transcendental concept then represents corresponding general universal categories as again representing the refined harmonious interaction of both conscious and unconscious aspects of experience.

However an imaginary transcendental perception is even more elusive as representing the indirect recognition of a projection emanating from the unconscious where again both conscious and unconscious aspects of recognition with respect to its temporary phenomenal identity are maintained in close harmony. And then finally an imaginary transcendental concept would entail the corresponding recognition with respect to indirectly projected universal categories of experience. In other words when conscious and unconscious aspects of recognition become so closely related in experience so as to approximate simultaneous identity, then - by definition - remaining involuntary attachment to phenomena largely ceases.

Some 20 years ago when I wrote the "Number Paradigms" I recognised in holistic mathematical terms that the most refined conceptual structures possible in experience are - in holistic mathematical terms - of an imaginary transcendental nature and that these typify the most advanced stage of the causal level (approaching pure spiritual union).

It was only later that I was able to properly make the connection as between such understanding and the Euler Identity and realise its deeper significance.

Mathematical Dimensions and Psychological Development (1)

I have already pointed to the fact that - properly understood - every number expression represents a dynamic interaction as between a a base quantity and a dimensional number (that is relatively of a qualitative nature).

So what we might refer to in conventional (Type 1) terms as the number quantity 2, more accurately is expressed as 2^1 (where 2 is quantitative and 1 - relatively - of a qualitative holistic nature).

However because the very nature of linear (1-dimensional) understanding is to reduce qualitative to quantitative type meaning, from a Type 1 perspective, interpretation of numbers is invariably reduced in a mere quantitative manner.

However when correctly appreciated in holistic Type 2 terms, number expressions have an intimate bearing on the interpretation of all the main stages on the spectrum of psychological development.


As we know number is the best means we have for quantitative ordering in experience. In fact such ordering is synonymous with number appreciation. And seen from this Type 1 perspective we have various number types such as prime, natural, rational, irrational etc.

What is not recognised is that the same number types have a profound relevance for the qualitative interpretation of development.
In fact the holistic structure of each of the main levels of psychological development can be precisely matched to expressions entailing the main number types. Furthermore we can subdivide each "number" level into three coherent stages showing how the transformation into the next level on the spectrum takes place.

So, three main stages are involved with respect to unfolding of each level.

Firstly we have the unfolding of specific perceptions (characteristic of the level in question). In the second stage this gives way to unfolding of the more generalised conceptual understanding (again characterising the level). Now the relationship of such perceptions and concepts is as quantitative and qualitative with respect to each other!
Then in the third stage increasing dynamic interaction takes place as between perceptions and concepts causing a transformation to a new level (whose structure in turn corresponds with a new distinctive number type).


Development commences from a completely undifferentiated state where form is indistinguishable from emptiness. This relates in qualitative terms to - what I refer to as - the original numbers. So unity (1) is initially indistinguishable from nothingness (0). Then when development commences the first initial differentiation takes place leading to the birth of duality (2) in experience.

Though we still have great confusion, in a qualified sense we can identify three stages (1) where duality takes place with respect to incipient perception (2) when duality now takes place with respect to the incipient formation of concepts and finally where some level of interaction of both takes place. This can be identified with the archaic level.


All going well, the dynamic interaction of understanding that is quantitative and qualitative with respect to each other, leads to a transformation to the next level.

This level is then identified in terms of the holistic interpretation of the prime numbers that entails an intimate relationship of both conscious and unconscious aspects of experience. Indeed we can accurately use the word primitive to refer to such experience where holistic collective notions (pertaining to the unconscious) are continually confused with distinct specific notions (pertaining to the conscious). We can identify this with the magical level.

Again in a qualified manner we have the unfolding of three stages firstly with respect to specific perceptions of a primitive kind, then more general concepts (of a primitive nature) and finally the growing dynamic interaction of both perceptions and concepts.

When we raise a prime number to a prime number dimension in quantitative terms, a transformation takes place in that we derive a new (composite) natural number.

Likewise in holistic psychological terms when we relate specific perceptions of a primitive quantitative nature to their general concepts (that relatively are qualitative in nature) a transformation to a new level of understanding takes place which correlates with the holistic nature of natural numbers.

One of the key characteristics of natural numbers is that numbers are whole (and not yet divided into fractional components).
Likewise at this level (which relates to mythic development) wholes - especially with respect to concepts - cannot be properly broken into parts. Put another way, abstract ability is not yet sufficiently developed so that objects are still given a (whole) personality. Only later when the conscious aspect is further differentiated from the unconscious does true abstract ability of an impersonal nature properly unfold.

One fascinating feature of this level is that one learns to - temporarily - negate both perceptions and concepts (while holding them in memory). This is made possible through the greater level of phenomenal constancy characteristic of this level.
This literally means therefore that such dynamic negation of concepts in experience relates to negative rather than positive dimensions (in number terms).


Once again we can identify three stages 1) where natural perceptions (and their corresponding temporary negation) takes place 2) natural concepts (and their temporary negation occurs and finally (3) the growing interaction of both perceptions and concepts in both a positive and negative sense.
So during this level the natural numbers give way to the integers (where numbers can be both positive and negative) in holistic terms. And the scientific structure of the level is based on such understanding.

When we look at the simple number expression 4^ (- 1) we have an integer that is raised to the negative of 1 (as dimensional power). And this number expression in Type 1 terms leads to the generation of a fraction i.e. 1/4.

Likewise in development, when whole perceptions are related to concepts (that can be temporarily negated in experience) we generate analytic understanding of a fractional kind.


This leads to an important transformation in development whereby the mythic gives rise to the rational stages (that are so relevant in conventional scientific understanding).

What is remarkable here is that appropriate holistic mathematical interpretation implies that even the simplest rational task implies the ability to temporarily negate the linear (1-dimensional) mode of understanding.

Let us illustrate with respect to the task of cutting a cake into four slices.

Now, the cake obviously represents a whole and each slice another unique whole object. So the ability to recognise each slice also as a part requires the temporary negation of its whole status. This then enables the link of the slice to the whole status of the cake whereby it is now recognised as a part of that whole. So, initially each slice is posited as a whole perception and then negated (with respect to this status) enabling it to be thereby understood as a part of the greater whole of the cake.
So here we establish the link between whole and part perceptions which implicitly requires the ability to temporarily negate their respective identities while still holding them in memory. And the same ability also implicitly develops with respect to concepts. For example one can initially recognises a specific number such as 2 as a perception. However the recognition that this number belongs to the universal number class i.e concept of number requires the temporary ability to negate its specific status in recognition of its general identity. Then the reverse procedure of once again recognising 2 as a specific perception in turn requires the temporary negation of its number concept. Therefore - though still operating at a merely implicit level the continual dynamic positing and negating of both perceptions (as quantities) and concepts (in qualitative terms) takes place. Once again the (linear) rational level - which dominates conventional understanding of Mathematics and Science - has 3 stages.

The first corresponds to the rational appreciation of perceptions in the ability to break object perceptions into smaller parts and rearranging them again in composite wholes. This is generally referred to in Piagetian terms as conop.

The second stage corresponds to the ability to break general universal concepts into sub-categories and then synthesise them with respect to the original concepts. In Piagetian terms this is formop.

Finally the third stage entails the growing interaction of rational perceptions and concepts. When suitably refined - though this is not generally the case - this leads to a dynamic intuitive based form of rational understanding i.e. vision-logic.


Another remarkable transformation occurs in Type 1 quantitative terms when a rational number is raised to a fractional dimension (as power) in that an irrational number results.

For example in the well known case when 2 is raised to 1/2 we obtain the square root of 2 = 1.4142... which is an irrational number.


However, in Western culture there is very little evidence of the sustained growth of further more advanced stages of understanding (beyond the rational level).

One important reason for this is quite revealing. The source of the dynamic transformation that takes place in the emergence of each new psychological level results from the interaction of perceptions and concepts, that are implicitly experienced as quantitative and qualitative with respect to each other. This enables therefore a fruitful interaction of conscious and unconscious to emerge.

In formal terms the interpretations of Mathematics and Science directly reduce the qualitative in quantitative terms. As this leads to emphasis solely on the conscious aspect of experience, a considerable diminution in true holistic type qualitative understanding can result. For example Holistic (Type 2) Mathematics, which I am outlining in these contributions, is totally unrecognised at present by the mathematical profession!


In experiential terms therefore this can reduce the unconscious intuitive aspect of understanding (within which true qualitative appreciation is embodied) to such a significant degree that sustained progress beyond the rational level is not possible.
In other words the specialisation of rational understanding, which is so characteristic of Western culture can greatly reduce the role of the unconscious in experience with the consequence that dynamic transformation beyond the rational level is significantly impeded.

Multiplication and Addition

The key problem in reconciling addition with multiplication is that they represent mathematical processes that are quantitative and qualitative with respect to each other. And as Conventional (type 1) Mathematics is based on a merely reduced quantitative approach this creates enormous difficulties in properly appreciating the nature of the problem.


As we have seen properly we have two number systems that are quantitative and qualitative with respect to each other.

1) In the conventional (Type 1) system, the natural numbers 1, 2, 3, 4, 5,.... for example respect quantities are defined with respect to a (default) dimensional value of 1.

So written in full, this system is represented as:

1^1, 2^1, 3^1, 4^1, 5^1,........


2) In the unrecognised (Type 2) system, the same natural numbers 1, 2, 3, 4, 5,.... represent qualitative dimension that are defined with respect to a (default) base quantity of 1.

So written in full, this alternative system is represented as:

1^1, 1^2, 1^3, 1^4, 1^5,........


Now with respect to the first system when we add two numbers, say, 2 + 3,
this is fully represented as

(2^1) + (3^1) = 5^1


However when we add the same two numbers, 2 + 3, with respect to the second system, this is fully represented as

(1^2) * (1^3) = 1^5.


So whereas addition of these two numbers is involved with respect to the first (quantitative) system, multiplication of the same two numbers (now representing dimensions) is entailed with respect to the second (qualitative) system.

This clearly entails that whereas pure addition (i.e. with respect to numbers that are all defined with respect to 1 as default dimension) is of a direct quantitative nature. Pure multiplication (i.e. with respect to numbers that are all defined with respect to 1 as default base) is by contrast of a direct qualitative nature.

Therefore we cannot ultimately hope to reconcile addition and multiplication without equal recognition of both Type 1 (quantitative) and Type 2 (qualitative) numerical systems.

In practice therefore where non-unitary values are given to both (quantitative) base and (qualitative) dimensional numbers, Type 3 Mathematics (representing the coherent interaction of both Type 1 and Type 2 systems) must be used for comprehensive understanding.


A further problem relates to the reconciliation of multiplication and exponentiation.

If multiplication is now treated in a (Type 1) quantitative manner, then
2 * 3 for example is represented as (2^1) * (3^1) = 6^1.


However with respect to the second (Type 2) dimensional system 2 * 3 is represented as

(1^2)^3

So multiplication with respect to the two numbers in the Type 1 (quantitative) system represents exponentiation with respect to the same two numbers in the second.

And just as (2^1) * (3^1) = (3^1) * (2^1) (with respect to the first)

(1^2)^3 = (1^3)^2 (with respect to the second).

Nature of Number as Dimension

I have been thinking again in a deeper manner regarding the nature of number representing a dimension (or power).

Let us start with the convenient (default) case of 1.

Now clearly 1 can represent a unit quantity. So implicit therefore in the recognition of any specific object is the number 1 (as an actual finite quantity).

However when used to represent a dimension the number 1 takes on a distinctive holistic meaning (in a potential infinite manner).

So for example if we attempt to represent the number system on a straight line this automatically presumes a linear (1-dimensional) background that is - potentially - infinite.

Therefore though we can use the same symbol 1 to represent a base unit quantity or alternatively the linear dimension (within which such a number is expressed) clearly the meaning is very different in each case.

In the former case 1 represents a specific finite notion that is inherently quantitative in nature; in the latter case it represents a holistic - potentially - infinite notion that is of an inherently qualitative nature.


And as all numbers representing quantitative values must implicitly be expressed with respect to a corresponding number dimension (with the default value = 1), then every number expression - when properly appreciated - necessarily entails a relationship between two aspects which are quantitative and qualitative with respect to each other.


As we have seen, the default dimensional state of a number is 1. And as it is the very nature of linear (1-dimensional) understanding to reduce the qualitative aspect to the quantitative, this means in effect that the qualitative notion of number is effectively always ignored in Conventional (Type 1) Mathematics.


This also causes an important difficulty when dealing with dimensional values (other than 1) which are inevitably treated in a reduced linear manner.


For example 2-dimensional reality would relate to a potentially infinite plane (within which a 2-dimensional object can be placed). However because the qualitative nature of logical understanding remains 1-dimensional, in Type 1 Maths this entails considering the plane as (linearly) extended in two directions that are horizontal and vertical with respect to each other.


However once we depart from 1-dimensional qualitative interpretation, the true nature of dimension is revealed to be of a circular nature.

In fact - when again appropriately understood - this is actually demonstrated in Type 1 Mathematics through the notion of roots.

If we obtain the two roots of unity, they will lie as equidistant points on the circle of unit radius (in the complex plane). Now we can of course in quantitative terms recognise these as + 1 and - 1 respectively. However if we are to give an appropriate 2-dimensional interpretation in qualitative terms (as is appropriate) then we require a logical means of combining + 1 and - 1 as being both true. Now this is done through the paradoxical (both/and) logic of the complementary opposites where each pole like the left and right turns on a road has a merely relative validity.

So we can see here an important inverse relationship as between the 2 quantitative roots of the number 1 and the corresponding 2-dimensional qualitative interpretation (with which they are consistent).

Strictly speaking we do not have 2 roots of 1 i.e 1^1.

- 1 is indeed the (unique) square root of 1. + 1 is however the (unique) square root of 1^2. And 1^1 and 1^2 relate to distinct qualitative numbers (representing dimensions).



Now in a comprehensive appreciation, an even more subtle dynamic interactive understanding is required. Thus when we start with the base - say of 1^2 - here 1 is quantitative and 2 as dimension - relatively - qualitative. However equally the base number 1 can be given a quantitative meaning with the dimensional number 2, thereby in relative terms quantitative. For example all natural logs (representing numbers as powers) clearly have a quantitative meaning as do for example the values of s in the Riemann Zeta Function!

So in relative terms, where the base number has a qualitative meaning this implies that the conceptual nature of the number is highlighted in understanding.

Thus 1 for example can be seen in quantitative terms as a number perception (in relation to the qualitative concept of number). However in reverse terms it can be seen as the number concept (to which 1 relates). So in experiential terms both aspects necessarily interact with all numbers thereby possessing both quantitative (specific) and qualitative (holistic) aspects.

However this poses severe limitations on a mathematical approach that solely recognises the quantitative aspect. And this reduced interpretation is what we misleadingly refer to as Mathematics.

More properly it refers to Type 1 Mathematics. So enormous scope remains for the proper development of Type 2 Mathematics (focussing on the neglected qualitative aspect) and Type 3 Mathematics (where both aspects - quantitative and qualitative - are coherently related).

More on Euler Identity

I have used the Euler Identity in a holistic mathematical fashion (Type 2 Mathematics) to demonstrate three stages of specialised contemplative development (preceding the full radial unfolding of stages).

This can however be given a more precise expression.

So we now define e^(k*2*i*pi) = 1^x (where k = 1, – 1 or 0 and x = 1, 0 and both 1 and 0 respectively.

So when k = 1,

1) e^(k*2*i*pi)= e^(2*i* pi) = 1^1

In holistic terms this corresponds to the rational linear (i.e. 1-dimensional) appreciation of (mystical) union.

Now with the 1st stage (of specialised contemplative development) a slight imbalance remains whereby appreciation remains unduly transcendent in nature.
In other words in emphasising the transcendent nature of spiritual reality as inherently empty and thereby beyond all notions of form, one to a degree still represses the corresponding immanent nature of such reality as prior and thereby inherent in all form.
This in effect leads to a subtle lingering rational attachment (of a necessarily linear nature) to the notion of unity (as phenomenal form) thereby preventing full realisation of complementary intuitive recognition (that is literally non-dimensional as empty of all form)

2) So when k = –  1,

e^(k*2*i*pi)= e^(- 2*i*pi) = 1^0

This is so as e^(– 2*i*pi) = 1/{e^(2*i*pi)} = (1^1)/(1^1) = 1^(1 – 1) = 1^0.

With the 2nd stage of such specialised development, this remaining lingering attachment to the notion of union (as a phenomenal point) is gradually negated.
Therefore as remaining involuntary rational attachment is eroded, the complementary pure intuitive realisation of the nature of union can unfold. Strictly this does not mean that the rational aspect of understanding now ceases with union but rather that it can interact with intuition in a very refined - and thereby transparent - fashion (due to the erosion of an excess element of involuntary attachment).

So whereas in the 1st stage we emphasised the refined rational linear appreciation of form (with respect to pure spiritual awareness), here in complementary fashion we are emphasising the corresponding intuitive (non-dimensional i.e. 0-dimensional) aspect of such awareness.
Put another way the emphasis here is on the immanent - as opposed to the transcendent - aspect of spiritual awareness.

3) when k = 0

e^(k*2*i*pi)= e^0 = (1^1)*(1^0)

This is so as e^0 = e^(2*i*pi)* e^(- 2*i*pi) i.e. e^(2*i*pi - 2*i*pi).

and as we have seen,

e^(2*pi*i)* e^(– 2*pi*i) = (1^1)*(1^0)

Now in Type 1 Mathematics, we would simply add the powers 1 and 0. However in Type 2 Mathematics these remain of a qualitatively distinct nature.

So the significance of the third formulation is that it represents the most specialised stage of contemplative awareness, where form and emptiness are successfully united in experience. Put another way it represents the full integration of both the transcendent and immanent aspects of spiritual understanding (which in turn implies the most refined interaction possible as between both the rational and intuitive modes of understanding).

So here we have pure emptiness serving as the potential for the entire world of (actual) created phenomenal form.

In mathematical terms this represents the most complete appreciation of both the linear rational (1-dimensional) nature of dimension that serves as the basis for quantitative appreciation of Type 1 Mathematics and the purely circular intuitive (0-dimensional) appreciation that serves as the basis for corresponding qualitative appreciation of Type 2 Mathematics.

This in turn with the radial unfolding of stages allows for the growing interpenetration of both Type 1 and Type 2 (in what constitutes Type 3 Mathematics).


We can perhaps see here the truly remarkable nature of e.

In conventional Type 1 terms any number raised to the power of 0 = 1. This represents therefore a merely reduced linear interpretation of 0 (as dimension). and of course when e is raised to 0 (in this reduced quantitative sense) its value is likewise 0.

However what is unique about e is that when it raised to 0 in Type 2 terms (reflecting the circular nature of dimension) its value is likewise 0.

Now we can appreciate why this is so with reference to the fact that by its very nature e fully reconciles corresponding notions of both differentiation (linear) and integration (circular).

So in Type 1 terms both the differentiation and integration of e^x are identical.

Likewise in Type 2 terms at the the most developed level of contemplative awareness differentiated (discrete) and integrated (continuous) elements are seamless so that phenomena of form no longer appear to arise in experience.

So e uniquely combines in its inherent nature both quantitative (discrete) and qualitative (continuous) aspects.


Now I have long identified the inherent nature of a prime number in that it too combines both extreme quantitative and qualitative aspects in its nature. Thus from a discrete independent nature prime numbers seemingly display no pattern. However from a continuous holistic perspective they display (en bloc) a truly remarkable regularity.

Not surprisingly therefore e has a vital role with respect to understanding the nature of primes.

Quite simply if we want to find the average space between primes (in any region of the number system) we simply find the dimensional power to which e must be raised to attain that number.

So for example to obtain 1,000,000, one must raise e to 13.8155... (i.e. the natural log of 1,000,000).

That means in the region of 1,000,000 we would expect the average space between primes to be just less than 14!

Parallel Riemann Hypothesis!

We concluded the last contribution with the remarkable finding that

i^i = e^(- pi/2), which is a real number!

Now if we take natural logs of each side

then i(log i) = - pi/2,

therefore 1/i(log i) = - 2/pi.

So, - i/log i = - 2/pi

Thus i/log i = 2/pi.


As we know the prime number theorem relating to the general frequency of the primes among the natural numbers is most simply expressed as n/log n (with the proportionate frequency increasing as n becomes larger).

So by allowing n to become progressively larger we have the linear quantitative attempt to reach the infinite (in an actual manner).

Now properly understood i represents the corresponding holistic notion of the infinite where one attempts to appropriate it (in a potential manner).

We can see this in the common psychological appreciation of the imaginary as something that emanates from the holistic unconscious to be embodied in an actual (conscious) manner.

Therefore understood in this light i/log i represents the qualitative correspondent to the prime number theorem!

In quantitative terms, we attempt to understand prime numbers from a Type 1 perspective as base quantities i.e.

2^1, 3^1, 5^1, 7^1,.......

However the prime numbers have in Type 2 terms a corresponding qualitative interpretation as dimensions i.e.

1^2, 1^3, 1^5, 1^7,......

In an inverse quantitative manner we can obtain the circular structure of these dimensions through obtaining the reciprocal roots.

So therefore we can attempt to find the 2 roots, 3 roots, 5 roots, 7 roots of unity and so on for each of the prime numbers.

In this approach we consider all roots which will have both a real and imaginary component.

With respect to both parts we take the values in an absolute manner (ignoring negative signs). Then we sum up both parts (both real and imaginary taken separately) and then obtain the average.

We can demonstrate simply here for p = 3.

There are 3 corresponding roots of 1 involved i.e. 1, - .5 +.866i and - .5 - .866i

Ignoring negative signs the sum of the real part here = 1 + .5 + .5 = 2.

Therefore the mean average = 2/3 = .6666.. .

Then taking the magnitude of the imaginary part (ignoring the i) the sum = .866 + .866 = 1.732

Therefore the mean average = 1.7321/3 = .57735...

The remarkable finding here as the value of p increases is that the mean value of the absolute quantitative value for both the real and imaginary parts converges on 2/pi = .636619772...
We can readily find all these values through use of the Euler Identity,

e^(2*i^pi) = cos (2*pi) + i sin (2*pi).
So the 3 roots of 1 - where the dimensional numbers are 1/3, 2/3 and 3/3 respectively are calculated in this manner as

cos {(2/3)*pi} + i {sin(2/3)*pi},
cos {(4/3)*pi} + i {sin(42/3)*pi}, and
cos {(6/3)*pi} + i {sin(6/3)*pi}.

As p becomes ever larger the mean value (for both parts) approximates ever closer to i/log i.

So we seem here in fact to have a circular number equivalent to the prime number theorem (that is couched in a linear quantitative manner).


However it does not end here!

We can see from our example above that the mean value for the real part = .6666.. and the imaginary part = .57735... respectively.

Therefore the mean value for the real part exceeds 2/pi and the corresponding value for the imaginary part is less than 2/pi respectively.

In fact looking at the absolute differences the value is .6666... - .636619772..

= .03005 (approx)... for the real part

and .636619772... - .57735... = .05927(approx)

Now the ratio of this difference real/imaginary = .03005/.05927 = .507 (approx)

This already seems very close to .5 (sound familiar!)

In fact as the value of p increases the ratio of this difference does indeed tend ever more closely to .5!


So once again what we are stating is this!

As p becomes larger, the absolute mean value of both real and imaginary prime roots of 1 converges ever closer to 2/pi (i.e. i/log i).

Insofar as a difference remains the ratio of absolute deviation of real/imaginary value converges ever closer to .5.

Just as Riemann came up with improvements to prediction of the general frequency of the primes, I experimented with my own improvements.

Now let's say that we wish to calculate the deviation of the absolute mean value of the real part from 2/pi for a larger value of p (say 127).

What we do here is to multiply the deviation (for p = 3) by (p/p1)^2 where p = 3 and p1 = 127.

So this gives us .03005 * (3/127)^2 = .000016768 (approx)

Considering that we are using such an early prime number = 3, this compares extremely well with the true deviation = .000016232 (approx).

In fact this and any other calculation can be significantly improved by then dividing the result by (1 + d) where again in this case d = .03005.

So in in this case we can then approximate the true deviation as .000016277..

Thus our answer is already correct to 3 significant figures!

Predictions can be greatly improved through using the deviations of later prime numbers.

For example if we use the deviation associated with p = 61, we can calculate the corresponding deviation associated with p = 127 correct to 6 significant figures!

Variations of this approach can be used likewise to predict corresponding deviations associated with the imaginary part!

Now in principle just as the non-trivial zeros of the Riemann Zeta function can be used to correct the deviations from the actual with respect to the general distribution of the primes, a corresponding method should exist enabling - ultimately - an exact mean of absolute prime root values for both real and imaginary parts.

So just as the Riemann Hypothesis is used to accurately calculate the average number of primes (in a linear context), this latter approach is used to calculate the average value of these primes in a circular context.

And in each case .5 plays a key role. In fact the circular version provides a key indication of what the .5 actually represents.

As we have seen in the circular context .5 represents ratio of (real) cos to (imaginary) sin values which indicates in turn a quantitative (analytic) to qualitative (holistic) connection.

And this is really what the Riemann Hypothesis is all about i.e. in establishing the condition necessary for full reconciliation of both quantitative and qualitative aspects of interpretation!

A New Number System (3)

There are really two components to this new number system (where numbers are interpreted with respect to their pure dimensional (as opposed to their base quantitative) characteristics.

Once again the linear system - for base quantities - is defined in terms of a fixed dimensional number i.e. 1.

So the natural numbers 1, 2, 3, 4,... in this system are more fully represented as

1^1, 2^1, 3^1, 4^1,.....


However the corresponding circular system - for dimensional qualitative values - is defined in terms of a fixed base quantity 1.

So the natural numbers 1, 2, 3, 4,... in this system are more fully represented as

1^1, 1^2, 1^3, 1^4,.....

The circular nature of this latter system comes through raising 1 to the reciprocal of each dimension thus obtaining a quantitative value that lies on the circle of unit radius.

So when we raise 1 for example to the reciprocal of 4 i.e. 1/4 we obtain in quantitative terms i, which lies on the circle of unit radius!

Because there is a direct relationship as between each dimension (as quality) and its reciprocal (in quantitative terms), this means that the 4 as dimension is associated with the qualitative (i.e. holistic) interpretation of i.

Thus rather that just one valid interpretation of mathematical symbols, which in conventional terms is associated with the default value of 1, potentially an infinite set of possible interpretations exists for all all mathematical symbols, relationships etc.

So whereas Type 1 Mathematics is associated merely with the (reduced) quantitative aspects of mathematical symbols, Type 2 is associated with appropriate qualitative interpretation of these same symbols.

Thus i for example has not merely a quantitative, but also an important qualitative meaning. However this qualitative dimension is completely ignored in Type 1 conventional terms.

This makes no sense for ultimately the quantitative results that are derived for example in complex analysis are somewhat meaningless in the absence of appropriate qualitative interpretation!


Type 3 Mathematics - which is easily the most refined and demanding in nature, then involves consistently relating both quantitative (Type 1) and qualitative (Type 2) interpretation.


However just as the base quantitative system has an imaginary counterpart, likewise the dimensional qualitative counterpart has an imaginary counterpart.


So again the natural numbers in the first system would be

i^1, 2i^1, 3i^1, 4i^1,....,

whereas in the second system the corresponding imaginary version is

1^i, 1^2i, 1^3i, 1^4i,....


Now because in Type 1 Mathematics the second system is not formally recognised this entails with respect to the real part that

1^1 = 1^2 = 1^3 = 1^4 =.....= 1^n.

As we have seen this leads to the misleading conclusion that for example + 1 and - 1 are both the square root of 1.
(Through use of the Type 2 system we can see that - 1 is the square root of 1^1 and + 1 the square root of 1^2 (which are distinct in Type 2 terms).
In other words we cannot properly divorce here proper quantitative from proper qualitative interpretation!)


Also because in Type 1 Mathematics the second system is not recognised this entails with respect to the imaginary part that

1^i = 1^2i= 1^3i = 1^4i =.....= 1^n.

This comes from the corresponding assumption that

e^(2*i*pi) = e^(4*i*pi) = e^(6*i*pi) = e^(8*i*pi) =....= e^(2n*i*pi).

This then leads to the misleading conclusion that 1^i for example can have an unlimited number of possible quantitative solutions.

However because properly speaking in a Type 2 approach

1^i, 1^2i, 1^3i, 1^4i, =.....= 1^n are all distinct,

this means that 1^i has indeed just one unique quantitative value!


So once again - this type directly with respect to a quantitative result - we cannot properly divorce here quantitative from qualitative interpretation!

As we have seen 1^i = e^(- 2pi) = .00186744....

However as i = 1^(1/4), this means that i^i = 1^(i/4) = e^{(- pi)/2} = .2078795763...

It must be stressed that in accordance with Type 1 Mathematics than - as with 1^i - an infinite set of possible values exists for i^i.

So in Type 1 terms, 1^i = e^(2*i*pi)^i = e^(4*i*pi)^i = e^(6*i*pi)^i = e^(8*i*pi)^i =....

By this logic, for example 1^i = e^(2*i*pi)^i = e^(8*i*pi)^i

= e^(- 2*pi) = e^(- 8*pi)

So i^i = e^(- pi/2) = e^(- 2pi)

However this would therefore suggest that i^i = 1^i (which makes little sense).

Therefore Type 2 interpretation needs to be included to avoid such confusion!

Alain Connes

This time when reading Karl Sabbagh's "Dr. Riemann's Zeros" I came across an interesting quote from Alain Connes on P.205. In commenting on the relationship between Geometry and Algebra he states:

'It really is fantastic step' he said, to understand that the square of a number - which is just a geometrical square - and the cube, which is just a geometrical cube - can be added together, even though you would say, "But one has dimension the length squared and the other the length cubed" and you would never add things which have different dimensions. So algebra is an amazing achievement, and once you have formulated things in an algebraic terms then they take on a life of their own.'


Algebra of course is incredibly important. However the price that has been paid in terms of such rational abstraction is that a basic form of reductionism is involved. In other words we can see clearly in geometrical terms that 2-dimensional is qualitatively distinct from 3-dimensional reality. However in algebraic terms this qualitative distinction is quickly lost with variables interpreted with respect to their mere quantitative meaning.


Indeed it is rather ironic that Connes in speaking about the Riemann Hypothesis in an a later quote on p.208 states:

"It is probably the most basic problem in mathematics, in the sense that it is the intertwining of addition and multiplication' Connes said. It's a gaping hole in our understanding, because until we really understand it we cannot say that we understand the line. Even the line itself is extraordinarily mysterious.'


May I suggest once again that the very reason why the link between addition and multiplication seems so intractable is precisely because the qualitative nature of variable transformation - necessarily involved in all multiplication - is formally ignored in present Type 1 Mathematics.

So properly understood there are both Type 1 (quantitative) and Type 2 (qualitative) aspects to all mathematical interpretation.

The Riemann Hypothesis in fact is a key statement regarding the relationship as between these two aspects.

Likewise with the line, Type 1 Mathematics, due to the same lack of a qualitative dimension, one cannot properly distinguish finite (discrete) from infinite (continuous) notions. So once again though the infinite notion is qualitatively distinct from the finite, in Type 1 interpretation it is necessarily reduced quantitatively in a finite manner!

A New Number System (2)

In the last contribution, I showed the relationship as between the extended Euler Identity and the Type 2 Number System.

Thus once again,

where k = 1, e^(2*k*i*pi) = e^(2*i*pi) = 1^1

where k = 2, e^(2*k*i*pi) = e^(4*i*pi) = 1^2

where k = 3, e^(2*k*i*pi) = e^(6*i*pi) = 1^3

where k = 4, e^(2*k*i*pi) = e^(8*i*pi) = 1^4,

and so on.


As there is no recognition in Type 1 Conventional Mathematics of the qualitative dimensional aspect of interpretation, a reduced and - ultimately - faulty understanding is given of the Euler relationships.

So just as in Type 1 terms, 1^1 = 1^2 = 1^3 = 1^4 =......,

likewise in reduced quantitative terms,

e^(2*i*pi) = e^(4*i*pi) = e^(6*i*pi) = e^(8*i*pi) =.......

However this misleading interpretation can be shown to lead to a problem which is very revealing in its consequences.


Because e^(2*i*pi) = 1^1, then when we raise both sides to the power of i, we get

e^(2*i*pi)^i = 1^i

Therefore e^(- 2*pi) = 1^i

So 1/{e^(2*pi)} = 1^i

Therefore 1^i = .0018674427....


However according to Type 1 interpretation,

e^(2*i*pi) for example = e^(4*i*pi)

So therefore in Type 1 terms,

e^(4*i*pi)^i = 1^i

Thus e^(- 4*pi) = 1^i

And 1/{{e^(4*pi)} = 1^i

Thus 1^i = .00000348734...


And because in Type 1 terms,


e^(2*i*pi) = e^(4*i*pi) = e^(6*i*pi) = e^(8*i*pi) =....... ad infinitum,


this implies that we can have an infinite number of valid quantitative results for 1^i!

Now in Type 1 terms this myriad of embarrassing riches is handled in a merely pragmatic unconvincing fashion. Just as with the many possible (circular type) roots of a number the positive real numbered root is considered as the principle root (though strictly it does not represent the correct root), likewise in this situation the quantitative result pertaining to e^(2*i*pi) is taken as the principle value with the other possible values (of an infinite set) effectively ignored.


However by employing Type 2 interpretation we can easily resolve this problem


So from a Type 2 interpretation

e^(2*i*pi) ≠ e^(4*i*pi)≠ e^(6*i*pi) ≠ e^(8*i*pi)... and so on,

Rather e^(2*i*pi) = 1^1

e^(4*i*pi) = 1^2

e^(6*i*pi) = 1^3

e^(8*i*pi) = 1^4, and so on


Therefore for example whereas


e^(2*i*pi) = 1^i,

e^(4*i*pi) = 1^2i, and so on

Therefore when seen from this perspective 1^i does indeed have one unique answer.
The second answer that we calculated above i.e. .00000348734... does not correspond in fact with the value of 1^i but rather 1^2i!

What is remarkable here is that we have now used Type 2 interpretation - not alone to show a qualitative distinction - when 1 is raised to a real dimensional number, but now in reverse fashion to show that a quantitative distinction is likewise involved when 1 is raised to an imaginary dimensional number.

This also strongly hints at the true nature of the imaginary number i.e. as of a qualitative holistic nature (expressed indirectly in a real quantitative manner)!

So when we raise 1 to a real rational number (as dimension), the result will fall on the circle (of unit radius).

However when we raise 1 to an imaginary rational number (as dimension), the result will fall on the straight line!


Though I had for many years recognised that there was a qualitatively distinct approach to Mathematics (which I refer to as Type 2), For some time I considered that these two separate aspects could be conducted in relative independence of each other.
In other words I did not directly consider that Type 2 interpretation would have a direct relevance with respect to derivation of quantitative results!

However appreciation of the extended use of the Euler Identity has changed all this for its real message is that both quantitative and qualitative type interpretation are inextricably linked!

So ultimately we cannot have consistent type interpretation of quantitative results without corresponding consistency in qualitative terms.

Therefore in my own evolution appreciation of the true nature of the Euler Identity (from both a quantitative and qualitative perspective) was to prove a key landmark in eventually unravelling the true nature of the Riemann Hypothesis which is essentially the same message i.e. that both quantitative and qualitative type interpretation are inseparable!

However with respect to the Riemann Hypothesis, this poses insuperable problems as within Conventional Mathematics there is - as yet - no formal recognition of its equally important qualitative aspect!

A New Number System (1)

Let me first clarify the distinction I make as between linear and circular (with respect to number systems) on the one hand and quantitative and qualitative on the other.

Now in the conventional Type 1 mathematical approach - though the overall qualitative approach is decidedly linear - both linear and circular notions can be dealt with from a quantitative perspective. The real number system for example is viewed in a linear fashion (as points laid out in a line). Indeed the imaginary number system is viewed in a similar fashion (lying on a line vertical to those of the real). However when it comes to the roots of unity, these lie in the circle of unit radius (in the complex plane). Now the quantitative nature of these roots can be dealt with (without however their true significance being realised).

In the Type 2 mathematical approach both linear and circular notions of logical interpretation can also be brought to bear on these same number systems. Also one clear implication of this approach is that circular notions (in quantitative terms) cannot be properly interpreted in the absence of circular type interpretation (from the qualitative perspective).

Thus the full use of both linear and circular notions requires that they be given both Type 1 (quantitative) and Type 2 (qualitative) interpretations.


The key significance of the Euler Identity is that - though seemingly arising in the context of the Type 1 approach to Mathematics - it actually gives rise to the need for the Type 2 approach.

In a very true sense therefore it arises at the very intersection of Type 1 and Type 2 approaches (where both quantitative and qualitative mathematical notions are fully interdependent).

I have already defined the natural number system with respect to Type 1 and Type 2 interpretation.

The Type 1 approach is qualitatively linear in nature and is literally defined in 1-dimensional terms,

1^1, 2^1, 3^1, 4^1,.......

So here the base number quantity keeps changing while the default dimensional number quality remains fixed as 1.

The Type 2 approach is by contrast quantitatively linear in nature, where the dimensional number quality keeps changing,

1^1, 1^2, 1^3, 1^4,.......

The circular nature of this alternative number system can indirectly be shown
in quantitative terms through obtaining the corresponding root (i.e. the reciprocal of the dimension in question).
Therefore in the circular number system, there is an inverse relationship as between a qualitative dimensional interpretation and corresponding quantitative root.

So once again for example, to properly explain the square root of 1 i.e. 1^(1/2), we need the corresponding qualitative dimensional interpretation of 2 i.e. 1^2!

Now what is fascinating about the fundamental Euler Identity is that it leads directly to this Type 2 Number system

So e^(2*i*pi) = 1 i.e. 1^1

Now all other numbers for example in the Type 2 natural number system can be obtained from the expression

e^(2*k*i*pi) where k = 1, 2, 3, 4,....

So in fact e^(2*i*pi) is just the special case where k = 1.


Thus where k = 1, e^(2*k*i*pi) = e^(2*i*pi) = 1^1

where k = 2, e^(2*k*i*pi) = e^(4*i*pi) = 1^2

where k = 3, e^(2*k*i*pi) = e^(6*i*pi) = 1^3

where k = 4, e^(2*k*i*pi) = e^(8*i*pi) = 1^4,

and so on.


Now we have already dealt with the ambiguity in terms of conventional interpretation of roots where for example but + 1 and - 1 are both given as the square root of 1, i.e. 1^(1/2).

Indeed I have already argued using Type 2 interpretation that properly - 1 is (unambiguously) the square root of 1 i.e.1^(1/2).

Now this is born out directly through extension of the fundamental Euler Identity.

So when k = 1/2, then

e^(2*k*i*pi) = e^(i*pi) = - 1.

So one valid way of interpreting the Euler Identity (as it is generally presented) is that the square root of 1 i.e. 1^(1/2) = - 1.

However this conflicts directly with conventional reduced (Type 1) interpretation whereby the principle square root of 1 = + 1.
So for example if you take out any calculator, input 1 and then raise this to .5, you will be given the result of 1, which the Euler Identity tells us is erroneous!

One of teh great advantages of the extended Euler Identity is that it provides a ready means for calculating all roots of 1.


So e^(2*k*i*pi) = cos(2*k*pi) + i sin(2*k*pi)

Therefore for example to obtain the cube root of 1 i.e. 1^(1/3) we let k = 1/3

So e^(2*k*i*pi) = e^(2*i*pi)/3 = cos(2*pi)/3 + i sin(2*pi)/3

which represented in degrees (rather than radians) = cos 120 + i sin 120

= {- 1 + 3^(1/2)i}/2 = 0.5 + .8660i (correct to 4 decimal places)

Note once again that just one unambiguous answer corresponds with the cube root of 1 i.e 1^(1/3)!

Avoiding Contradiction

From the Type 1 mathematical perspective

1) e^(2*i*pi) = 1


2) e^(- 2*1*pi) = 1


3) e^(0) = 1.

One might thereby be attempted to conclude therefore that the dimensional expressions in each case must also be equal

i.e. that 2*i*pi = - 2*i*pi = 0!

This clearly is not permissible from a Type 1 (quantitative) perspective. However it requires inclusion of the Type 2 (qualitative) perspective to properly show why this is the case.

As we have seen in Type 1 terms 1^1 = 1^2 = 1^3 .... = 1^n. So from this merely quantitative perspective each of these terms = 1.

We have already shown that properly distinguishing - 1 (as the root of 1) from + 1 requires a Type 2 explanation. So in qualitative terms 1^1 is recognised as distinct from 1^2. So strictly speaking therefore - 1 is the square root of 1^1, whereas + 1 is the square root of 1^2!

Appropriately distinguishing 2*i*pi, - 2*i*pi and 0 from each other (as dimensional expressions) requires a similar qualitative interpretation!


1) e^(2*i*pi) = 1 (i.e. 1^1)


2) e^(- 2*1*pi) = 1 (i.e. 1^0)

This is easily seen by the fact that 2) is the inverse of 1) so that

e^(- 2*1*pi) = 1/e^(2*i*pi) = 1^1/(1^1) = 1^(1 - 1) = 1^0.


3) e^(0) = 1) * 2) = e^(2*i*pi) * e^(- 2*1*pi) = e^(2*i*pi - 2*i*pi) = e^0.

So to conclude e^0 = 1^1 * 1^0.
Now of course from a merely Type 1 perspective 1^1 * 1^0 = 1^1. However the very point in this context is that we are using a Type 2 (qualitative) interpretation.

Thus in a qualitative context, the three different dimensional expressions 2*i*pi, - 2*i*pi and 0 have a subtly distinct interpretation.

So 2*i*pi = 1^1 corresponds to the linear rational component of non-dimensional understanding (insofar as we can give the the non-dimensional notion a finite actual meaning)!

- 2*i*pi = 1^0 corresponds to the circular intuitive component of non-dimensional understanding (where we are relate to a formless non-phenomenal experience).

0 = 2*i*pi - 2*i*pi combines both rational and intuitive comprehension. So in dynamic experiential terms, our notion of 0 as in the expression e^0, necessarily entails both quantitative and qualitative aspects that are interdependent. However in terms of the the two number systems (quantitative and qualitative) these are necessarily split up.


To conclude the notion of 0 as a dimension literally relates to the concept of a point (which is non-dimensional). This in turn implies the identity of both linear and circular interpretations (in both quantitative and qualitative terms).
Clearly as Type 1 Mathematics is devoid of a circular qualitative dimension, it lacks the means to adequately interpret the Euler Identity. It can indeed provide the quantitative demonstration of its validity, but then lacks the means to convey its deeper significance (which is of a qualitative nature).


The real mystery of the Euler Identity is that it beautifully combines both the quantitative and qualitative meanings of its symbols (in a manner where they become indivisible). However this requires that both Type 1 and Type 2 mathematical interpretations be coherently combined!

When 0 is not Nothing!

The significance of the dimensional expression 2*i*pi is highly elusive. In fact there are direct connections to the mystical notion of "bindu" or point centredness or even more recently the physical notion of a singularity.

The way I look at it is as the centre point of the "imaginary" circle.

We can approach this initially with reference to the "real" circle.

Now the centre of this circle is equally the centre point of its line diameter. So in this sense it is at this point that both linear and circular notions are reconciled as identical. Put another way we could say that the quantitative and qualitative notions of interpretation (characterising Type 1 and Type 2 Mathematics respectively) are reconciled.

However the problem with the real circle is that it cannot be conceived in the absence of linear notions (relating to the size its radius). So though with the central line diameter we can depict the middle point as 0 with the right hand side in positive units and the corresponding left hand side in negative units (with both equal) in actual fact we represent both positively geometrically as lines (with equal extension). Now if we could imagine the circle in more dynamic terms where positive and negative aspects cancel each other out the circle would shrink to a point (with no linear extension). Likewise of course the line diameter would shrink to the same point. So at this point the line and the circle would be identical. Now inherent in the qualitative notion of the imaginary is this complementarity of positive and negative polarities. So the imaginary unit circle which is represented by 2*i*pi is therefore identical with this point (where in a sense the circle and line have collapsed to a point where they are indistinguishable from each other).

Now quantitative interpretation is of a linear rational nature based on finite notions of form; qualitative interpretation is directly of a circular intuitive nature based on infinite notions of emptiness.

So in fact we can have 3 different expressions for this non-dimensional point.

First we can have the rational interpretation where it is approached from the perspective of actual form i.e. as being represented by a quantity - rather like the infinitesimal notion - that is in the process of becoming nothing i.e. 0.

Secondly we can have the intuitive interpretation where it approached from the perspective of emptiness (through negation of the quantitative aspect). So from this perspective when we have surrendered any rational notion of nothingness we are intuitively left with the experiential awareness of its true qualitative nature.

Thirdly we can have the combined appreciation of both quantitative (rational) and qualitative (intuitive) notions.


As I say this area is in fact dealt with at length in the mystical contemplative literature where the three notions I have mentioned represent three stages in the specialised growth of pure spiritual understanding.


The first is often referred to a the arrival at a pure state of concentrated transcendent awareness i.e. as spirit as the centre of one's being. Here spirit is understood as beyond all phenomenal form.

However because initially there is lack of corresponding immanent awareness i.e. where spirit is seen as equally inherent in all form, this leads to a lingering phenomenal attachment to the very notion of this point. Put another way though maintaining the absolute (separate) nature of ineffable spirit one thereby still understands the notion in an unduly phenomenal manner.

So the second stage requires the negation of this lingering - merely - phenomenal understanding a spiritual point. This - when successful - leads in turn to corresponding free intuitive awareness of its qualitative nature.

However because both rational and intuitive are themselves dynamically complementary in experience, ultimately both refined rational and intuitive interpretation must be combined in the realisation of creation as both form and emptiness. This thereby entails both the immanent aspect (where spirit is inherent in matter) and the transcendent aspect (where spirit is beyond all form).
And by this third stage both the quantitative and qualitative aspects of understanding are coherently related.


We will see how these notions are related to the mathematical treatment of 2*i*pi in the next contribution.

Spiritual Illumination

Again with reference to Marcus du Sautoy's "The Music of the Primes", I found an interesting quote on P.297 attributed to Andre Weil.

"Every mathematician worthy of the name has experienced ... the state of lucid exaltation in which one thought succeeds another as if miraculously...this feeling may last for hours at a time, even for days. Once you have experienced it you are eager to repeat it but unable to do it at will, unless perhaps by dogged work..."

What Weil is desribing here is in fact - what spiritual writers refer to as - "illumination" where for a while one has a peak experience of holistic intuitive insight. It is in such moments the truly great mathematical insights are obtained and those decisive creative breakthroughs where for a brief moment one is able to "see" certain important relationships - perhaps for the first time - in an enhanced manner.

In fact quite clearly such moments relate directly to the qualitative - rather than quantitative - aspect of mathematical appreciation. Though the intuitive insights obtained may indeed be later expressed in a (reduced) rational manner that wins the acceptance of the mathematical community, the initial intuitive realisation properly remains of a qualitative nature.

Remarkably however, the qualitative aspect of Mathematics is given no formal recognition at all in conventional terms.

In other words though every mathematical symbol, relationship, hypothesis etc. has a distinctive (holistic) qualitative as well as (analytical) quantitative interpretation in Type 1 Mathematics only the the latter is recognised.

Thus in a comprehesive treatment we should have both Type 1 (quantitative) and Type 2 (qualitative) aspects that initially are developed in relative independence from each other.

Then when both of these aspects have achieved appropriate degrees of specialisation they can be fruitfully combined with each other in the most advanced form of Mathematics (i.e. Type 3).

Interesting Comments

I was just re-reading Du Sautoy's "Music of the Primes" in the Chapter where he links up the Riemann Zeros with findings from quantum mechanics. In one paragraph he makes statements that link up perfectly with the qualitative approach that I have adopted.

For example he sees on P.267. "For as long as the quantum world remains unobserved it exists only in the world of imaginary numbers".

Now I have long maintained that actual observed and potential unobserved reality are properly interpreted according to two logical systems that are linear and circular with respect to each other. And in the simplest version we use both 1-dimensional (linear) and 2-dimensional (circular) interpretation!

Also I have maintained that the imaginary concept - as used in Mathematics - represents an indirect means of converting 2-dimensional to 1-dimensional format.
So not surprisingly therefore the 2-dimensional world of potential (unobserved) reality would be represented through imaginary numbers.

He the says later in the same paragraph

"When we observe an event in the quantum world, it is as though we are not seeing the event itself in its natural domain, but a shadow of the event projected into our "real" world of natural numbers"

This statement in fact perfectly equates with the physical complement of Jungian notions of the unconscious.

So when experience is directly of a holistic unconscious nature (which cannot be phenomenally appropriated in its own domain) it becomes projected into the conscious world (where it is generally confused with the specific phenomena involved). And of course in Jungian terms this is the very means by which the "shadow" (unconscious) personality expresses itself.

Now the direct implication of this for the world of quantum reality is that - properly understood - it entails the physical aspect as complementary to the notion of the unconscious in psychological terms.
We can refer to this physical complement as the holistic ground or - perhaps - holistic dimensional ground of reality. However the implication is clear!

Just as psychological experience entails the continual dynamic interaction of two aspects that are conscious and unconscious with respect to each other, likewise physical reality - especially at the quantum level - entails the continual dynamic interaction of manifest (observed) reality and a hidden (unobserved) holistic potential from which the observed phenomena emerge.

Now the problem in physics is that because no recognition is given to the intuitive mode of the unconscious in formal interpretation, it has great difficulties in accurately portraying the true nature of quantum reality (from a qualitative perspective).

It is now recognised that the non-trivial zeros of the Riemann Hypothesis bear a close relationship to certain quantum chaotic energy vibrations in the physical world.

The clear implication therefore that adequate interpretation of the Riemann Hypothesis entails a qualitative (Type 2) as well as quantitative (Type 1) approach.

And indeed once again the key significance of the Hypothesis - as I have repeatedly stated - is that it expresses the condition necessary for the consistent reconciliation of both aspects.