Marcus du Sautoy is back on BBC with a three part series where he argues that mathematics is the secret code behind nature's secrets. Though Maths is apparently abstract, yet the behaviour of nature is wonderfully written in mathematical language.
Though du Sautoy confines himself to the conventional quantitative aspect of Mathematics, of even greater wonder to me at present is the realisation that nature (and indeed all life) is likewise written mathematically in a qualitative code which we do not yet even properly realise.
Indeed recently I have come to the view that the true nature of reality is indeed mathematical in both quantitative and qualitative terms.
One implication of this is that nature's ultimate physical secrets cannot be discovered merely through phenomenal investigation of reality, for these very phenomena already embody dynamic mathematical configurations with respect to both its quantitative and qualitative aspects.
One of the reasons why the qualitative aspect of Mathematics is missed is because it does not initially conform to rational investigation of the standard logical kind. Rather it conforms to an appreciation of interdependence (rather than independence) which is then indirectly conveyed through paradoxical interpretation of a circular kind.
So every mathematical symbol can be given a valid interpretation according to two logical systems that are linear and circular with respect to each other. Whereas the former corresponds with quantitative appreciation, the latter relates to qualitative appreciation (indirectly conveyed through mathematical symbols).
Put another way, whereas we now realise that numbers in quantitative terms can be both real and imaginary, the corresponding corollary in qualitative terms is that mathematical logical interpretation can likewise be both real and imaginary. So once again real in this context corresponds with linear type rational appreciation, whereas imaginary relates to circular or paradoxical type rational awareness.
One remarkable implication arising from this perspective is that qualitative type appreciation is directly of an affective kind, that indirectly can then be given a valid mathematical interpretation.
This would imply that ultimately a comprehensive mathematical appreciation implies substantial balance being maintained as between artistic (affective) and scientific (cognitive) type awareness, though the language of mathematics will be be rightly couched in cognitive terms.
In one valid sense, qualitative mathematical appreciation relates to the subtle appreciation of the the true dimensional nature of reality that - apart from some developments in string theory - is not currently recognised.
This would again imply that affective experience provides the direct means of appreciating such dimensions, which embody all phenomena with their unique qualitative characteristics.
Finally it struck me forcibly today that all this gives a new meaning to my interpretation of the Riemann Hypothesis as a statement regarding the ultimate reconciliation of both the quantitative and qualitative aspects of mathematical experience.
So before phenomena can even come into being, a crucial condition regarding the nature of prime numbers must be fulfilled guaranteeing their consistency according to two sets of logic that must necessarily diverge somewhat with respect to the phenomenal world. Therefore the subsequent manifestation of phenomena in nature is already based on deep mathematical principles that precede and ultimately also transcend their very existence in actual form. Therefore the Riemann Hypothesis can have no proof in conventional terms, for the very truth to which it relates already precedes any partial logical investigation either in standard (linear) or unrecognised (circular) terms.
Thus when we probe nature to its very limits, we must eventually leave the world of the merely physical to embrace what is truly mathematical. Indeed the very rigidity that defines phenomenal objects already implies a degree of reduction in the - ultimately ineffable - mathematical principles governing their nature. And this applies most readily to the qualitative aspect (the mathematical nature of which is not yet even recognised).
So the true nature of Mathematics - with respect to its quantitative and (unrecognised) qualitative aspects - lies at the very bridge that serves to connect the phenomenal world of physical form with the ineffable world of spiritual emptiness.
Thursday, July 28, 2011
Thursday, July 21, 2011
Odd Numbered Integers (9)
Unexpectedly this morning, while trying out an insight that struck me yesterday, I seemed to have detected a very interesting pattern that governs the denominators of values of the Riemann Zeta Function for negative odd integer values.
This pattern relates to divisibility of the denominator by the first two perfect numbers 6 and 28 and can be stated succinctly as follows.
(i) The denominator of such values is always divisible by 6.
(ii) in every 3rd case the denominator is divisible by both 6 and 28 (and only in such a case).
(iii) The denominator does not appear to be divisible by any other perfect numbers.
For example the 1st zeta result where s = - 1 is - 1/12 and the denominator is clearly divisible by 6.
The 2nd zeta result where s = - 3 is 1/120 and again the denominator is divisible by 6.
The 3rd zeta result where s = - 5 is - 1/252 and the denominator here is divisible by both 6 and 28.
And this trend continues. So the denominators (240 and 132) for both s = - 7 and s = - 9 are divisible by 6, whereas the denominator for s = - 11 (32760) which is the 3rd in the sequence, is divisible by both 6 and 28. Indeed in this case it is divisble by 6 * 28. However that is not generally the case!
Using zeta results compiled in Mc Gill University this trend can be verified for the first relevant 100 zeta values (i.e. up to s = -199).
It should be also stated that this numerical behavioural characteristic does not extend to denominators of the zeta function for positive even integer values of s!
However with respect to any qualitative interpretation of the meaning of such results it is perhaps too early to speculate.
Indeed even more dramatic numerical patterns exist with respect to the denominator of these zeta values (for negative odd integers).
As we have seen each successive value is divisible by 6 (i.e. 2 * 3).
Then every 2nd successive denominator value is divisible by 5; every 3rd succesive denominator nvalue is divisible by 7; every 5th successive denominator value is divisible by by 11; every 6th successive denominator value is divisible by 13; every 8th successive denominator value is divisible by 17; every 9th successive denominator value by 19; every 11th successive denominator value is divisible by 23; every 14th successive denominator value is divisible by 29 and so on.
In other words where the absolute value of s is prime, every {|s - 1|/2)th denominator value in the zeta sequence for negative odd integral values is thereby divisible by |s|.
Put another way if therefore every {|s - 1|/2)th denominator value is divisible by |s|, then |s| is prime.
For example if |- 31| is prime then every 15th value in the zeta sequence should be divisible by 31
Now the absolute value of the denominator is the first of these cases (15th value in sequence) is 85932 which is divisible 31.
Now for every further 15th value in sequence the absolute value of denominator will be divisible by 31.
For example the absolute value for the denominator of 30th value is 3407203800 which once again is divisible by 31.
Therefore we could conclude from this that 31 is a prime number!
Not alone does this pattern appear to hold unbiversally but equally for all absolute prime values of s > 3, the only time when the demominator is divisible by |s| is for the {|s - 1|/2)th denominator value in the sequence.
Therefore we could safely conclude for this value of |s| where = 31, that 31 is indeed a prime number.
And of course this would hold for all other values of |s| where the same principle applies!
Put finally yet another way if s = 2, 4 ,6, 8,....
then for zeta (1 - s), where the value of (s + 1) is prime, the absolute value of denominator will be divisible by all factors of s where with the addition of 1 are prime (and only these factors).
So one again for example when s = 36, zeta (1 - s) is zeta (- 35).
In this case s + 1 = 37 which is prime. Now all the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.
Now with addition of 1 in each case we get 2, 3, 4, 5, 7, 10, 13, 19 and 37.
However since 4 and 10 are not prime we can exclude these numbers.
Theerfore the denominator of zeta (-35) i.e. 69090840 is divisible by 2, 3, 5, 7, 13, 19 and 37 (and only these prime numbers).
And what is remarkable is that when the denominator is divided by the product of all these prime factors the result is s.
Therfore 69090840/(2 * 3 * 5 * 7 * 13 * 19 * 37) = 36.
Though this final result does not universally hold it does so in some cases i.e. when the denominator is divided by product of all prime factors the result is s.
This pattern relates to divisibility of the denominator by the first two perfect numbers 6 and 28 and can be stated succinctly as follows.
(i) The denominator of such values is always divisible by 6.
(ii) in every 3rd case the denominator is divisible by both 6 and 28 (and only in such a case).
(iii) The denominator does not appear to be divisible by any other perfect numbers.
For example the 1st zeta result where s = - 1 is - 1/12 and the denominator is clearly divisible by 6.
The 2nd zeta result where s = - 3 is 1/120 and again the denominator is divisible by 6.
The 3rd zeta result where s = - 5 is - 1/252 and the denominator here is divisible by both 6 and 28.
And this trend continues. So the denominators (240 and 132) for both s = - 7 and s = - 9 are divisible by 6, whereas the denominator for s = - 11 (32760) which is the 3rd in the sequence, is divisible by both 6 and 28. Indeed in this case it is divisble by 6 * 28. However that is not generally the case!
Using zeta results compiled in Mc Gill University this trend can be verified for the first relevant 100 zeta values (i.e. up to s = -199).
It should be also stated that this numerical behavioural characteristic does not extend to denominators of the zeta function for positive even integer values of s!
However with respect to any qualitative interpretation of the meaning of such results it is perhaps too early to speculate.
Indeed even more dramatic numerical patterns exist with respect to the denominator of these zeta values (for negative odd integers).
As we have seen each successive value is divisible by 6 (i.e. 2 * 3).
Then every 2nd successive denominator value is divisible by 5; every 3rd succesive denominator nvalue is divisible by 7; every 5th successive denominator value is divisible by by 11; every 6th successive denominator value is divisible by 13; every 8th successive denominator value is divisible by 17; every 9th successive denominator value by 19; every 11th successive denominator value is divisible by 23; every 14th successive denominator value is divisible by 29 and so on.
In other words where the absolute value of s is prime, every {|s - 1|/2)th denominator value in the zeta sequence for negative odd integral values is thereby divisible by |s|.
Put another way if therefore every {|s - 1|/2)th denominator value is divisible by |s|, then |s| is prime.
For example if |- 31| is prime then every 15th value in the zeta sequence should be divisible by 31
Now the absolute value of the denominator is the first of these cases (15th value in sequence) is 85932 which is divisible 31.
Now for every further 15th value in sequence the absolute value of denominator will be divisible by 31.
For example the absolute value for the denominator of 30th value is 3407203800 which once again is divisible by 31.
Therefore we could conclude from this that 31 is a prime number!
Not alone does this pattern appear to hold unbiversally but equally for all absolute prime values of s > 3, the only time when the demominator is divisible by |s| is for the {|s - 1|/2)th denominator value in the sequence.
Therefore we could safely conclude for this value of |s| where = 31, that 31 is indeed a prime number.
And of course this would hold for all other values of |s| where the same principle applies!
Put finally yet another way if s = 2, 4 ,6, 8,....
then for zeta (1 - s), where the value of (s + 1) is prime, the absolute value of denominator will be divisible by all factors of s where with the addition of 1 are prime (and only these factors).
So one again for example when s = 36, zeta (1 - s) is zeta (- 35).
In this case s + 1 = 37 which is prime. Now all the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.
Now with addition of 1 in each case we get 2, 3, 4, 5, 7, 10, 13, 19 and 37.
However since 4 and 10 are not prime we can exclude these numbers.
Theerfore the denominator of zeta (-35) i.e. 69090840 is divisible by 2, 3, 5, 7, 13, 19 and 37 (and only these prime numbers).
And what is remarkable is that when the denominator is divided by the product of all these prime factors the result is s.
Therfore 69090840/(2 * 3 * 5 * 7 * 13 * 19 * 37) = 36.
Though this final result does not universally hold it does so in some cases i.e. when the denominator is divided by product of all prime factors the result is s.
Friday, June 3, 2011
Odd Numbered Integers (8)
It might help to summarise the rationale of what has been involved in making these qualitative connections with the Riemann Zeta Function (for negative odd integer values).
Once again it is vital to appreciate that standard (unambiguous) quantitative type interpretation of numbers is associated with a default dimensional value of 1. So for example when numbers are raised to a power (other than 1), the attempt is made to obtain a reduced numerical result (in terms of 1).
Thus in this context 2^2 = 4 (i.e. 4^1). Therefore, though a qualitative change in the nature of the number takes place (through raising to the power of 2) the result is expressed in a reduced merely quantitative manner.
However the Riemann Zeta Function diverges for negative odd integer values of the dimensional power s.
So when s = - 1, we obtain in this context the series 1 + 2 + 3 +..... which clearly in standard terms sums to infinity.
However through the process of analytic continuation the Riemann Zeta Function can be given an alternative finite interpretation for all negative odd integers of s as a rational number.
Now the fascinating explanation for this alternative behaviour is that the finite numerical result now obtained relates to a circular rather than linear interpretation.
For simplicity, I have referred to this result as qualitative (rather than quantitative). However strictly speaking this is not properly correct.
So in a more complete fashion we have distinguished two distinct types of explanations for numerical results of the Riemann Zeta Function.
For values of s > 1, they can be given the standard linear (absolute) type interpretation. This results from the assumed absolute separation here of qualitative interpretation from the objective numerical result obtained.
However for values of s < 0, they can only be given an alternative circular (relative) type interpretation inherently in terms of the actual value for s utilised.
Here we operate according to the different assumption that both qualitative interpretation with resulting numerical values are always in dynamic interaction with each other. Thus the rational values that result from the Riemann Zeta Function thereby reflect the circular interrelationship of aspects of understanding that are always qualitative and quantitative with respect to each other. Furthermore this relationship tends towards complementarity (whereby both aspects are perfect mirrors of each other).
Now strictly this perfect complementarity properly only applies for negative even integer values of s. Thus the resulting value of 0 reflects that we can no longer separate any phenomenal result (in merely quantitative terms). So we have here pure intuitive realisation (in psychological terms) which perfectly mirrors appreciation of the empty holistic ground (underlying all phenomenal awareness of physical type reality)
However in a provisional limited sense for the negative odd integers - even though interdependence again characterises the relationship between qualitative interpretation and quantitative type results - a certain relative degree of separation can take place.
Thus from the psychological perspective the numerical result of the Function has a qualitative interpretation (as a certain mode of rationality).
Meanwhile from the physical perspective the same numerical result can be given a quantitative interpretation (i.e. representing a certain quantum relationship).
However as the numerical magnitude of dimensions increases a progressively higher level of pure energy characterises all relationships.
Thus from the psychological perspective, rationality becomes so refined that it ultimately cannot be separated from the spiritual intuitive energy with which it interacts.
Likewise from the corresponding physical perspective, quantitative phenomena become so unstable and short-lived that they can no longer be distinguished from the pure physical energy with which they are associated.
And this pure energy itself ultimately becomes inseparable from an empty holistic ground of nature (as the source of all physical reality) that is complemented by a pure empty spiritual experience (as the goal or realisation of all reality).
I will draw attention to another crucial distinction.
From the standard linear perspective, all numerical values are given an abstract identity (as essentially independent of all physical and psychological behaviour).
However from the corresponding holistic circular perspective, all numerical values necessarily express fundamental phenomenal relationships (with both physical and psychological aspects) that ultimately tend to full complementarity.
This suggests therefore that the results of the Riemann Zeta Function - as I have been explaining - have a direct relevance to both physical and psychological reality.
Likewise this is true of the famed non-trivial zeros which now have dual interpretations (both in the standard abstract sense and the new holistic sense as intimately related to both physical and psychological reality).
Whereas there is now some recognition that these zeros may indeed have a direct physical interpretation, there is no recognition as yet of their corresponding psychological relevance (as representing various refined states of interpretation).
And ultimately - as I have repeatedly stated - the very message of the Riemann Hypothesis is that physical and psychological aspects (through both quantitative and qualitative type interpretation) are ultimately inseparable.
Once again it is vital to appreciate that standard (unambiguous) quantitative type interpretation of numbers is associated with a default dimensional value of 1. So for example when numbers are raised to a power (other than 1), the attempt is made to obtain a reduced numerical result (in terms of 1).
Thus in this context 2^2 = 4 (i.e. 4^1). Therefore, though a qualitative change in the nature of the number takes place (through raising to the power of 2) the result is expressed in a reduced merely quantitative manner.
However the Riemann Zeta Function diverges for negative odd integer values of the dimensional power s.
So when s = - 1, we obtain in this context the series 1 + 2 + 3 +..... which clearly in standard terms sums to infinity.
However through the process of analytic continuation the Riemann Zeta Function can be given an alternative finite interpretation for all negative odd integers of s as a rational number.
Now the fascinating explanation for this alternative behaviour is that the finite numerical result now obtained relates to a circular rather than linear interpretation.
For simplicity, I have referred to this result as qualitative (rather than quantitative). However strictly speaking this is not properly correct.
So in a more complete fashion we have distinguished two distinct types of explanations for numerical results of the Riemann Zeta Function.
For values of s > 1, they can be given the standard linear (absolute) type interpretation. This results from the assumed absolute separation here of qualitative interpretation from the objective numerical result obtained.
However for values of s < 0, they can only be given an alternative circular (relative) type interpretation inherently in terms of the actual value for s utilised.
Here we operate according to the different assumption that both qualitative interpretation with resulting numerical values are always in dynamic interaction with each other. Thus the rational values that result from the Riemann Zeta Function thereby reflect the circular interrelationship of aspects of understanding that are always qualitative and quantitative with respect to each other. Furthermore this relationship tends towards complementarity (whereby both aspects are perfect mirrors of each other).
Now strictly this perfect complementarity properly only applies for negative even integer values of s. Thus the resulting value of 0 reflects that we can no longer separate any phenomenal result (in merely quantitative terms). So we have here pure intuitive realisation (in psychological terms) which perfectly mirrors appreciation of the empty holistic ground (underlying all phenomenal awareness of physical type reality)
However in a provisional limited sense for the negative odd integers - even though interdependence again characterises the relationship between qualitative interpretation and quantitative type results - a certain relative degree of separation can take place.
Thus from the psychological perspective the numerical result of the Function has a qualitative interpretation (as a certain mode of rationality).
Meanwhile from the physical perspective the same numerical result can be given a quantitative interpretation (i.e. representing a certain quantum relationship).
However as the numerical magnitude of dimensions increases a progressively higher level of pure energy characterises all relationships.
Thus from the psychological perspective, rationality becomes so refined that it ultimately cannot be separated from the spiritual intuitive energy with which it interacts.
Likewise from the corresponding physical perspective, quantitative phenomena become so unstable and short-lived that they can no longer be distinguished from the pure physical energy with which they are associated.
And this pure energy itself ultimately becomes inseparable from an empty holistic ground of nature (as the source of all physical reality) that is complemented by a pure empty spiritual experience (as the goal or realisation of all reality).
I will draw attention to another crucial distinction.
From the standard linear perspective, all numerical values are given an abstract identity (as essentially independent of all physical and psychological behaviour).
However from the corresponding holistic circular perspective, all numerical values necessarily express fundamental phenomenal relationships (with both physical and psychological aspects) that ultimately tend to full complementarity.
This suggests therefore that the results of the Riemann Zeta Function - as I have been explaining - have a direct relevance to both physical and psychological reality.
Likewise this is true of the famed non-trivial zeros which now have dual interpretations (both in the standard abstract sense and the new holistic sense as intimately related to both physical and psychological reality).
Whereas there is now some recognition that these zeros may indeed have a direct physical interpretation, there is no recognition as yet of their corresponding psychological relevance (as representing various refined states of interpretation).
And ultimately - as I have repeatedly stated - the very message of the Riemann Hypothesis is that physical and psychological aspects (through both quantitative and qualitative type interpretation) are ultimately inseparable.
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.
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.
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.
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).
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).
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