Tuesday, May 31, 2011

Odd Numbered Integers (7)

There are further fascinating aspects in qualitative terms associated with the rational fractions for negative odd integers of the Riemann Zeta Function.

Again - before any association with the Riemann Function - I had become aware of a certain important pattern associated with contemplative type development.

In my writings I distinguish carefully as between contemplative and radial develolopment.

Now the essence of the former type of development is that it represents an unfolding of intuitive type understanding. And I refer to this as Band 3 on the Spectrum of Development.

Just as the (differentiated) rational unfolds at the earlier Bands through a - literal - reduction in intuitive type type development, now in reverse this (integral) intuitive type awareness unfolds through a similar reduction in rational understanding.

So initially in development as the more contemplative type stages unfold, one learns to economise greatly on the use of associated rational faculties. So the task is to learn to give expression to this new intuitive understanding through temporarily allowing the rational faculties become largely dormant.

Now this qualitative notion of a reduction in rational type understanding is replicated by the magnituide of the rational fraction associated with the Riemann Zeta Function (for the negative odd integers). So it starts with - 1/12 for s = - 1 and then quickly drops (in absolute terms) so that for s = - 3 the result is 1/120.
So we could say therefore that by the 3rd dimension a great reduction (and corresponding) refinement has already taken place in the use of reason. This in turn is largely facilitated by a substantial degree of unconscious darkness in experience (so that one does not have the freedom to readily exercise the rational faculties).

For s = - 5, where the result is 1/252, the absolute value reaches its minimum. This is turn is associated with the darkest period of mystical development (in what St. John would refer to as the passive night of spirit). Indeed typically a crisis emerges here due to the considerable absence of associated intuitive light to support the rational structures. This in turn is usually due to an extreme transcendent focus to development at this time. So we have here the start of a rebalancing whereby immanence as well as transcendence is recognised in development. And associated with this growing immanence is a corresponding recovery in rational type development. So for s = - 7, the result is 1/240 (which is now slightly larger in absolute terms).

In my earlier writings I then associated the attainment of an extreme in pure contemplative type development with the 8th dimension.

Now gradually I came to realise that a further Band of development would now unfold whereby the specialisation of this intuitive type development could take place (just as the earlier specialisation of rational understanding took place with the second Band).

This in turn - as is emphasised in the mystical literature - is likewise associated with attaining proper balance as between emphasis on form and emptiness (i.e. immanence and transcendence).

This in fact requires the unfolding of a further 8 dimensions. So that radial development does not properly commence till the 16th dimension has been traversed.

Fascinatingly between s = - 15 and s = -17, the absolute value of the rational fraction for the Riemann Zeta function finally exceeds 1.

So this likewise signals in qualitative terms the full incorporation of form with emptiness.

I had been impressed in earlier years by the fact that the great exponents in my own Christian tradition of radial type development had so often been people of considerable action. They were in a sense superhuman with respect to the degree of their commitment to changing society, as they saw it, especially in religious terms. So this progressive ability to engage with phenomenal form is a key feature of ongoing development in the radial life (which in turn corresponds with the unfolding of dimensions of understanding > 16). Put another way because of a growing balanced immersion in spirit (as central) they maintained the progressing capacity to endure with equanimity the extremes of dealing with phenomenal type activities.

Seen in this light therefore the numerical results for the negative odd integers of the Riemann Zeta Function have a direct qualitative relevance in terms of the nature of rational development itself (at the higher dimensions). And again with the negative dimensions such rational development obtains its greatest refinement through the significant removal of associated intuitive support!

And once again because psychological and physical reality are complementary this means that the same rational values have a direct physical relevance for high energy physics.

What one could thereby postulate is at extremely high energy levels (that would still be way beyond present technological capacity to generate) progressively heavier phenomenal particles would be associated with energy. However the very detection of these particles would require that we can isolate them from their associated high-energy environment. Therefore because of the difficulty of doing this these heavier particles (of which there is no upward limit) would become increasing unstable.

So even here we can see the complementarity of physical and psychological. Whereas in physical terms these "heavy" particles are associated with instability, in corresponding psychological terms the ability to deal with increasing "heaviness", in the increasing demands of phenomenal activity, corresponds with an ever-growing stability in personality terms.

Monday, May 23, 2011

Odd Numbered Integers (6)

There are interesting features to the rational fractions generated by the Riemann Zeta Function (for negative odd integers of s) which have a fascinating qualitative significance.

When s = - 1, the result of the Function = - 1/12; then the value falls up to s = - 5 with the results for s = - 3 and s - 5, 1/120 and - 1/252 respectively. Then the absolute magnitude of the fraction starts to continually increase in an accelerating fashion.

Now it might appear that there is no discernible pattern to these values.

However a series of approximating formulaes can be generated that express the value of successive ratios of the Riemann Function (for the negative odd integers), and also the difference of successive ratios, and the difference of the differences of successive ratios. See Approximation Formulae for Negative Odd Integers of the Zeta Function. What is fascinating is that all these approximating formulae are based on s and pi and in the last case simply on pi.

In fact the difference of the difference of successive ratios can be approximated by the simple expression 2/(pi^2) based on absolute values for the Zeta fractions.

To see what this means we can illustrate with reference to the first four values of s = -1, -3, -5 and -7.

So {[Zeta (-7)/Zeta (-5)] - [Zeta (-5)/Zeta (-3)]} -

{[Zeta (-5)/Zeta (-3)] - [Zeta (-3)/Zeta (-1)]} is approximated by 2/(pi^2).

This approximation .20 is already correct to two significant figures.

Now in the region of Zeta values around 50, the approximation has already greatly improved to the extent that it is correct to 11 significant figures!

The significance of the pi connection here in qualitative terms can be easily explained.

For the positive even integers the results of the Riemann Zeta Function can be exactly expressed in terms of formulae involving pi. This in turn is due to the fact that these represent states of integration (entailing the full harmonisation of linear with circular type understanding).

For the negative odd integers the results can be approximated in terms of formulae entailing pi. And this approximation greatly improves as the absolute value of s increases!
What this entails from a psychological perspective is that, even though a degree of broken symmetry necessarily attaches to the negative odd values of s, that with higher numbered dimensions (in absolute terms) the differentiated element of understanding becomes so refined that it can be scarcely distinguished from (even) integral dimensions.

Thus the higher numbered dimensions of understanding are so refined and dynamic that differentiation (in explicit terms) becomes indistinguishable from integration.

The corresponding situation from a physical perspective is that interaction becomes so dynamic that material particles cannot be explicitly distinguished from (pure) energy.

Sunday, May 22, 2011

Odd Numbered Integers (5)

We now look at the qualitative significance of the precise numerical values for the Riemann Zeta Function (with respect to negative odd integers of s).

As we earlier have seen with respect to positive even integer values of s, a definite pattern applies to the denominators of the rational fractions associated with these values.
In this context the denominator for an even integer will always be exactly divisible by the largest prime number up to s + 1 (and by no primes higher than this value).

However the reverse does not necessarily hold. In other words if the denominator is exactly divisible by s + 1, this does not entail that s + 1 is a prime number.
For example where s = 8, the denominator is 945. And 945 is in turn exactly divisible by s + 1 = 9. However 9 is not of course a prime number.

A similar - though more compelling sort of pattern - applies to the denominators of the rational values associated with the results of the Riemann Zeta Function (for negative odd integers).

If where s is negative the denominator of the result is always exactly divisible by s - 2 where the absolute value of s - 2 is prime.

Fro example the result of the Zeta Function where s = - 9 is - 1/132. Therefore the denominator here 132 is exactly divisible by - 11. And 11 is here prime.
However in this case, we can perhaps go further by suggesting that the converse is also true.

In other words if the denominator is in fact exactly divisible by s - 2, then this does seem to imply (on the basis of examination of all results of the Function for s up to - 200) that the absolute value of s - 2 is thereby prime.

In this way the Zeta Function for negative odd integral values can be seen as directly related to the generation of successive prime numbers.

Though it could not be suggested as a practical way to generate prime numbers, it is conjectured here that the entire set of prime numbers could be generated with reference to the division of the denominators of the Function (again for negative odd integral values of s) by s - 2. So once again wherever the denominator is exactly divisible by s - 2, then the absolute value of s - 2 is thereby prime.

Saturday, May 21, 2011

Odd Numbered Integers (4)

We have explained in qualitative terms why the results of the Riemann Zeta Function for negative odd integer dimensional values of s are always of a rational nature.

This qualitative explanation also enables one to appreciate the complementary nature of (negative) even and odd results for s respectively.

Once again for even values degrees of pure intuitive awareness (to which the even values relate) always requires negation of associated rational elements.
In converse manner pure rational awareness (to which the odd values relate) always requires corresponding negation of associated intuitive elements.

The next step is to explain - again in qualitative terms - why these rational results (for negative odd integer values of s) - keep alternating as between negative and positive signs.

Once again the explanation is very revealing as regards the nature of how higher level contemplative development takes place.

Phenomenal recognitions takes place in relation to both external (physical) and internal (psychological) aspects. Typically for example an extrovert will be more engaged with the external and the introvert with the - relatively - internal aspects respectively.

So when dynamic negation with respect to phenomenal experience takes place it entails both external and internal aspects (which are - relatively - positive and negative with respect to each other).

However it is in the nature of experience that this does not take place in a balanced fashion. So typically more attention will initially be given for example to the external structures and then to the internal structures and so on. So for example in St. John.s approach the initial emphasis is on purgation i.e. dynamic negation of the senses (which correspond to the external structures). Then with the more deep-rooted purgation of spirit, the emphasis switches to the internal structures.

And in development until full stable equilibrium is obtained - which in human development can only ever be approximated - a relative switch is emphasis will keep taking place at each higher differentiated stage of development as between external and internal.
So if we denote rational understanding associated with the external physical environment as positive, then in corresponding fashion rational understanding associated with internal psychological development will be - relatively - negative.

Thus dynamic negation of the rational structures takes place through the erosion of all secondary intuitive support (which provides that customary light that greatly facilitates the use of reason).

And this erosion takes place both with respect for intuitive support for the external and internal structures respectively (which are positive and negative with respect to each other).

Indeed St. John in his writings deals well with the problem of scrupulosity which at certain times can become a major problem on the contemplative journey.
Now this problem actually relates to the dynamic negation of the customary intuitive supports that normally facilitates the taking of internal moral decisions.

So as one is left - literally - more and more in the dark as regards the correct decision - in any relevant context to take it becomes increasingly difficult to decide on what is appropriate.

What happens in fact is that reason is used in an ever more refined manner in balancing pros and cons as one waits for a very faint intuitive signal providing inner confirmation that one is making the right choice. However in extremes, such supporting intuitive light is taken away altogether. So without ant supporting intuition, even refined reason can no longer operate and one is left to operate purely by faith (which in this context actually represents the purest degree of reason).

So in experiential terms both reason and intuition are necessarily involved in all phenomenal understanding. Thus the two extremes are:

1) the pure intuitive light (when all secondary rational understanding of a phenomenal nature is removed.

2) the pure intuitive darkness i.e. pure faith (when all secondary intuitive understanding associated with phenomenal understanding is removed.
And such pure faith actually likewise represents the purest form of reason i.e. that is so refined that any remaining phenomenal aspect is impossible to detect in an explicit manner.

Once again a complementary explanation of these alternating rational results (for the Riemann Zeta Function) can be given with respect to physical reality.

Though we - wrongly accustomed to looking at physical reality in a merely external objective fashion, in truth all physical processes entail the dynamic interaction of complementary internal and external aspects that are - relatively - positive and negative with respect to each other.

One way of viewing this would be in terms of intimate sub-atomic reactions where matter and anti-matter aspects of particles are involved.

So in this context the existence of - relatively - independent matter particles requires the blotting out of associated energy reactions with respect to both matter and anti-matter manifestations. And such manifestations can only occur in an alternating fashion (where for example matter excludes its anti-matter equivalent).

Friday, May 20, 2011

Odd Numbered Integers (3)

We return again to the important issue of why in qualitative terms all results in the Riemann Zeta Function for negative odd number integers (- 1, - 3, - 5,...) result in rational values.

I have already referred to St. John of the Cross in the context of explaining the qualitative significance of the negative even integer values of s.

Now the key thing about the negative dimensions is that it psychological qualitative terms they literally imply dynamic negation with respect to phenomenal experience.

As we have seen with negative integer dimensions (that are even), dynamic negation with respect to secondary rational elements of form is required, leading to a purely holistic intuitive awareness (which is - literally - 0 in phenomenal terms).

It is this form of negation (i.e. psychological purgation) that St. John of the Cross addresses when referring to the passive nights of sense and spirit.

However an issue that is not perhaps fully addressed by St. John is the need for ongoing active (as well as passive) purgations at the higher (dimensional) stages of development.

The problem with merely emphasis on intuitive type development is that experience can become unduly passive whereby a person loses the ability to properly engage with the world of form. So properly understood each new higher passive stage (culminating in a purer type of contemplative awareness) should be followed in turn by a more refined active stage. Now whereas the even dimensions refer to more passive intuitive type development with reality, the odd dimensions refer to more active involvement relating to engagement with phenomena of form.

Thus the negative odd integer dimensions relate to the purgation or dynamic negation of such active phenomenal understanding.

Fascinatingly, just as the more passive intuitive purgations take place through the negation of associated secondary rational elements, the more active rational purgations take place - in reverse manner - through the negation of secondary associated intuitive elements.

The ease with which - for example - one can deal with the normal familiar tasks of life, owes a great deal to the customary light or supporting intuition that is associated with such activity. However unfortunately it is likewise this same customary ease that provides an unwitting opportunity for all sorts of unrecognised attachments to develop.

So the manner in which one is thereby purged of such attachments is through the gradual diminishment of the light so that in extremes one is left totally in psychological darkness with no visible support of an unconscious nature.

Put another way, in such circumstances one is led - while carrying out phenomenal activity - to operate purely in faith.

This has the beneficial effect of forcing a person to greatly economise on conscious effort only tending to what is immediate and essential with respect to a task.

So just as pure intuition (which is qualitatively 0 in phenomenal terms) is developed through negation of associated rational support in personality, pure reason in converse manner is developed through negation of associated intuitive support.

And the negative odd integer dimensional values refer precisely to such pure reason.

Not surprisingly just as all the negative even dimensions give rise to 0, the negative odd dimensions give rise to rational values (which once again in this context have a direct qualitative significance).

As before, because of the complementarity of psychological and physical domains, these rational values also necessarily have a qualitative significance for the physical behaviour of matter.

One way perhaps of appreciating such a connection is the realisation that the phenomenal independent manifestation of particles requires in a sense blotting out the dynamic energy interactions with which they are necessarily associated.

Likewise as we have seen in psychological terms when dynamic interaction greatly increases as a result of authentic contemplative development, to keep one's feet on the ground as it were, one must be able to blot out such intuitive interaction to concentrate on specific tasks of a mundane nature.

Thursday, May 19, 2011

Odd Numbered Integers (2)

We now come to the qualitative interpretation of the Riemann Zeta Function for corresponding negative odd integer values i.e. where s = - 1, - 3, - 5,...

Now we can initially relate several interesting facts with respect to the numerical nature of these results.

1) Though indirectly these numerical values have indeed a discernible quantitative form, in direct terms they are qualitative in nature. Once again the Riemann Zeta Function for negative odd integer values of s, diverges to infinity.
So these alternative results that are obtained (which are finite) relate directly to the alternative qualitative - as opposed to quantitative - interpretation of number.

2) The numerical values (i.e. for the negative odd integers of s) are always rational in nature e.g. - 1/12, 1/120, - 1/252,...etc.

3) These numerical values consistently alternate as between their positive and negative expressions.

4) Though initially the magnitude of these rational values decreases (in absolute terms), the value then increases and indeed steadily accelerates with respect to the magnitude involved. These values in fact can be approximated very closely with simple formulae that are very revealing as regards their nature.

We will first deal here with the qualitative nature of these numerical results attempting to explain what is entailed.

Though the same principle is indeed involved for negative odd integer values, it may be easier to appreciate what is involved for s = 0.

Some time ago I set about trying to resolve this for myself.

Though the Zeta Function for s = 0 in linear terms leads to 1 + 1 + 1 + 1 + ...
(which in standard terms diverges to infinity) we can make progress by considering the alternative Eta Function where the terms alternate so that we obtain
1 - 1 + 1 - 1,...

Now this series has apparently two values. When we take an even number of terms the series sums to zero. However if we take an odd number of terms it sums to 1.

So one way of resolving the issue to get a single unambiguous answer is to obtain the mean of the two results = 1/2.

Then by a simple equation whereby the Zeta Function can be expressed in terms of the Eta, we can obtain an unambiguous result for the Zeta where for s = 0, the value is -1/2.

Now the qualitative interpretation of the Eta series (from which the final Zeta result is obtained) is very revealing.

When we combine two terms i.e. 1 - 1, we are in fact combining complementary opposites. However with odd terms we always have a single linear term left over.

So the Eta Function corresponds in qualitative terms with the consistent movement from (circular) complementarity to (linear) separation.

So in qualitative terms, the significance of the numerical value 1/2 that is obtained from this series relates to the fact that we are obtaining an even balance as between the two logical systems (linear and circular).

Now once again, the Zeta Series for s = 0 properly diverges (when interpreted in standard linear rational terms).

Quite literally therefore this convergent result of 1/2 thereby refers to interpretation that is - literally - only half rational (with an equal emphasis on circular intuitive understanding - that indirectly corresponds to the complementarity of opposites).

Therefore we can refer to the numerical value of the Eta Function as representing a certain degree of rationality in interpretation. It is likewise similar with respect to the derived Zeta Function value (which properly represents a derived degree of rationality with respect to interpretation).

The important point to grasp here is that in the conventional linear approach - where only quantitative interpretation is formally allowed - a solely rational logic is used (that is linear in nature).

However once we admit the relevance of both circular - as well as linear - logic, then interpretation of numerical values has a qualitative (as well as quantitative) significance.

So in this case the appropriate way of interpreting the Zeta result is as representing the degree of rationality required for its qualitative interpretation.

And as always, psychological and physical reality are complementary. Therefore such numerical results must thereby have a relevance for the physical world. One obvious connection is the rational nature of quantum mechanical behaviour that involves integral or half integral values.

Thus with respect to the subatomic physical world we have the dynamic interaction of both material particles (that are relatively independent) and energy states (representing their interdependence). This likewise represents the interaction of linear type (independent) and circular type (interdependent) behaviour. So the numerical values here in a physical context would then relate to degrees of phenomenal identity (which would be lower where dynamic interactivity is especially strong).

Thursday, May 5, 2011

Odd Numbered Integers (1)

As Euler demonstrated it is possible to represent the result of the Riemann Zeta Function for all even values of s (2, 4, 6,...) in terms of an expression involving pi, i.e. (pi^s)/k where k is a rational number.

However - as is well known - a similar sort of expression cannot be provided for the corresponding odd values of s (3, 5, 7,....)

So, from my perspective the first task is to explain in qualitative terms why this situation arises!

From the psychological perspective each value of s represents a "higher" dimension of understanding (which traditionally has been associated with contemplative type development).

In this context a crucial distinction can be made as between the odd and even numbers which is very revealing.

Whereas the odd numbers represent a new level of (rational) differentiation, the even numbers represent, by contrast, new equilibrium states of (intuitive) integration.

Now once again, the conventional (1-dimensional) level is viewed as a means of (rational) differentiation of meaning. Though informally the importance of supporting intuition may be admitted, this is not formally included in interpretation. And as we have seen this leads to the inevitable reduction of qualitative to quantitative type analysis.

In general terms differentiation is based on the separation of poles in experience, whereas integration is based on their complementarity (and ultimate identity).

From another perspective there always remains a certain linear character to the differentiated aspect of understanding (even at higher levels) while the integral aspect is always characterised by complementarity.

As stated previously the qualitative structure of a dimension is inversely related to its corresponding root structure. So for example the two roots of unity are + 1 and - 1 respectively; in corresponding inverse qualitative fashion, 2-dimensional appreciation is characterised by the complementarity of opposite poles of form. So with the quantitative interpretation of roots we employ a (linear) either/or logic; with the corresponding qualitative interpretation of dimensions we employ a (circular) both/and logic.

Now when we look at the root structure for all even numbered roots, they are characterised by the complementarity of opposites where half of the roots can be balanced by the other half (that represents the negative of all roots in the first half).

Now the very nature of such complementary understanding is that it is perfectly circular in nature. However because customary mathematical understanding is based on linear type distinctions we can only attempt to convey the nature of such circular understanding (which ultimately is of a purely intuitive nature) in a reduced linear fashion.

So from a rational perspective the nature or 2-dimensional understanding entails the relationship as between what is circular and linear. In corresponding quantitative fashion the very nature of pi involves the relationship as between circle and line (i.e. as the ratio of circular circumference to line diameter).

Thus the qualitative nature of all even dimensions (as representing the refined rational linear manner of conveying circular meaning) is perfectly replicated in quantitative terms in the numerical expressions of the Zeta Function for even dimensional values that are positive.

When we consider the structure of odd number roots, perfect quantitative complementarity does not exist. This is evidenced by the fact that the odd numbered roots cannot be arranged in a fully complementary manner! In corresponding fashion, qualitative complementarity likewise does not exist. Therefore in both cases, they represent a situation of broken symmetry.
In fact with odd numbered roots, + 1 is always one root obtained that in a sense remains separated from all other roots where a complementary pattern does exist with respect to the imaginary parts of these roots. So the broken symmetry relates here to a distinction as between the behaviour of real and imaginary parts. Likewise in qualitative terms, at the higher odd numbered dimensions a degree of linear understanding is maintained thereby enabling consious differentiation of phenomena that is ultimately inconsistent with the (unconscious) holistic understanding of these dimensions. Thus the imbalance of conscious and unconscious remaining requires progression to the next (even) higher dimension where a new level of integration can be obtained.

So from a psychological perspective, again it is easy to suggest why such broken symmetry might exist.
In development each new dimensional stage of integration (where a temporary equilibrium is reached) follows a corresponding previous dimensional stage of differentiation (where such equilibrium is temporarily broken).

Therefore the (positive) odd-numbered dimensions represent a situation where a new configuration of both (refined) rational and intuitive understanding go hand in hand. In other words experience is literally somewhat uneven whereby the relationship as between both linear and circular type interpretation entails a degree of confusion. When such confusion is then unravelled one moves on to the next integral stage (represented by an even numbered dimension). And then the deepening of such integral awareness requires the further progression through ever "higher" alternating odd and even-numbered dimensions.

Now ultimately with very high numbered dimensions the differentiated rational element of understanding becomes so refined that it is no longer (explicitly) separable from the intuitive. So here both integral and differentiated understanding closely approximate (where the discrete becomes inseparable from the continuous).

Remarkably here we have the qualitative correspondent of e. As is well known both the differentiated and integrated expressions of e^x remain the same.
The same e also plays an enormous role with respect to the distribution of prime numbers. In corresponding qualitative fashion e represents the dynamic state where (integral) intuition cannot be distinguished from (differentiated) reason. This plays an enormous role with respect to the holistic mastery of primitive desires (so that their discrete involuntary nature can be controlled). And it is this same experience that characterises the qualitative appreciation of the holistic nature of prime numbers!

2-Dimensional Proof

I have already pointed to the remarkable fact that the square root of 1 yields - according to conventional interpretation - two equally valid answers i.e. either + 1 and - 1 that are diametrically opposite to each other.

The (unrecognised) qualitative corollary of this is that mathematical proof at the (inverse) 2-dimensional level of understanding yields a totally paradoxical appreciation whereby a proposition that is both positive (+) and thereby true is equally negative (-) and thereby false.

Put another way a merely relative - rather than absolute - truth value applies at this level of understanding.

To see more clearly what is involved here, we must remember that in dynamic experiential terms, all mathematical interpretation involves both external (objective) and internal (subjective) poles. Whereas in linear (1-dimensional) terms these are clearly separated in corresponding circular (2-dimensional) terms these are complementary and ultimately identical.

So in conventional mathematical terms a mathematical proof may be viewed in two distinct ways:

(i) as what is considered true in an unambiguous objective sense (that is not altered through psychological interaction for its confirmation). In this sense for example one would maintain that the Pythagorean Theorem is objectively true (irrespective of the nature of one's psychological appreciation);

(ii) as what is considered true in an unambiguous internal mental sense (that is not altered through interaction with the objective referents of such mental constructs). So from this alternative perspective the truth of the proof is identified directly with the (internal) mental interpretation involved.

Now it might be readily admitted by mathematicians that external and internal aspects are necessarily positive (+) and negative (-) with respect to each other.

However in conventional terms this distinction is ignored resulting - literally - in a qualitatively absolute interpretation.

In other words the crucial assumption is made that a direct correspondence necessarily exists as between the (internal) mental constructs used in interpretation and the external reality (to which they relate).

However once we accept that these opposite poles in experience (external and internal) necessarily interact then a different form of interpretation results.

In other words from this new dynamic perspective external and internal do not enjoy an (absolutely) independent existence but only obtain meaning through relationship with each other.

Therefore if we attempt to identify this interaction (of external and internal) with either pole as independent then this excludes the equal truth of its opposite pole.

So in dynamic interactive terms if we maintain the proof of a mathematical proposition in a merely (external) objective sense, this thereby excludes the equal validity of its (internal) mental aspect; in corresponding fashion if we then maintain the the proof in a merely (internal) mental fashion this thereby excludes the equal validity of its (external) objective aspect.

Therefore from a balanced interactive perspective, a proof of any proposition must necessarily be both true (+) and not true (-).

In fact this merely exemplifies in the context of experience the simple example of road directions that I have so frequently given.

When we use an independent frame of reference (e.g. where movement is either up or down a road) a turn off the road will have an unambiguous meaning (as either left or right). However when we then attempt to combine both frames simultaneously (as interdependent), then a turn has a merely paradoxical meaning (as both left and right).

So in the context of mathematical interpretation, conventional (1-dimensional) appreciation is based on independent reference frames (where external and internal aspects are considered as separate).

However where - in qualitative terms - a circular 2-dimensional approach is employed interdependent reference frames are required (where external and internal are now considered as complementary).

And just as a turn off a road has a merely paradoxical interpretation in this context, likewise the truth of any mathematical proposition likewise has a merely paradoxical interpretation.

Therefore from a 2-dimensional qualitative perspective all mathematical interpretation (which of course includes all mathematical proof) is of a paradoxical nature.

What this implies is that meaning at this level is understood in a dynamic interactive manner that entails continual transformation (in both the external and internal aspects) of experience.

Put another, way truth is now of a merely relative nature or to use an analogy from quantum physics, all mathematical truth is now subject to the uncertainty principle.

So the true meaning of a dimension (which is inversely related to its corresponding root number structure) implies direction.

With linear (1-dimensional) interpretation only one direction is admitted leading to unambiguous interpretation. So crucially for example from this perspective a proposition is either true or false. In other words in has just one truth value (which is interpreted in an absolute fashion). Also in physical terms, this is identified with the conventional assumption that movement in both time and space takes place in a forward (positive) direction.

With circular - which in its simples manifestation implies 2-dimensional - interpretation, two directions are always involves (that are polar opposites of each other). So from this perspective for example a mathematical proposition is always both true and false (when communicated in reduced linear terms). Likewise in physical terms movements in space and time now necessarily have both positive (forward) and negative (backward) directions.

Of course an unlimited number of such "higher" dimensional interpretations potentially exist.

For example at the extremely important 4-dimensional level, all mathematical propositions would be given four directions.

So from this perspective a proposition would have a positive and negative truth value in both real and imaginary terms. The imaginary in this qualitative context arises from the true relationship as between wholes and parts (which in dynamic interactive terms are "real" and "imaginary" with respect to each other). In psychological terms this relates to the distinction as between perceptions and concepts. Now when one reflects on it one cannot deal with any perception (such as specific number) without implying the corresponding concept of number. However crucially in qualitative terms the relationship of "perception" to "concept" is as of "real" to "imaginary".

And at the 4-dimensional level of interpretation, understanding is now qualitatively so refined that this distinction can be made.

Though potentially as I have stated an unlimited number of qualitative dimensional inetrpretations exist, the most important would be confined to the earlier numbers (esp. 1 - 8).

In general terms it is much easier here to explain the even numbered dimensions (especially those that are powers of 2 such as 2, 4 and 8).

In turn this all has a fascinating connection with the Riemann Zeta Function.

So we will now turn to consideration of the odd integer values for s (both positive and negative) with respect to this function.