It may be helpful to keep looking at this issue from a variety of related perspectives.
So in stressing the need for the both quantitative and qualitative aspects of interpretation (in science and mathematics) the role of the unconscious needs to be equally incorporated with conscious type interpretation.
Though the unconscious is now accepted as an integral aspect of psychology, its counterpart complementary equivalent is not however yet recognised (especially in mathematical terms).
In physical terms this counterpart equivalent of the unconscious would be the recognition of an unmanifest hidden (potential) dimension that continually interacts with actual manifest phenomena in nature. We could perhaps fruitfully refer to the unmanifest aspect as the holistic ground (or holistic dimensional ground) of reality.
So whereas in direct terms, traditional conscious rational type interpretation would directly relate to the manifest features, unconscious intuitive type appreciation would characterise the unmanifest holistic aspect of reality.
And from a scientific perspective, such intuitive understanding can be indirectly represented through a circular logical approach (with various degrees of increasingly refined expressions available).
Admittedly there is already some recognition of the holistic nature of physical reality at the highly interactive dynamic level of sub-atomic particles. However an equal recognition of the need for explicit incorporation of an unconscious aspect with respect to an appropriate paradigm has not yet been admitted.
This problem is of a more fundamental nature with respect to Mathematics where the unconscious aspect of understanding likewise urgently needs to be explicitly incorporated in an altogether more comprehensive approach.
Putting it bluntly the somewhat absolute nature of mathematical appreciation is ultimately grounded in a paradigm that fundamentally misrepresents the true dynamic nature of mathematical experience and thereby is greatly limited in scope.
So once we accept the need to (explicitly) incorporate the role of the unconscious (with conscious interpretation) then we must likewise accept that underlying all manifest mathematical phenomena is an unmanifest or hidden holistic universal ground with which these phenomena continually interact. Now this might seem like a million light years away from the cold abstract view of static absolute mathematical forms that currently exists but this really indicates how rigid - and indeed unrepresentative - is the current mathematical paradigm (based solely on mere conscious rational interpretation).
Now we have already seen with respect to one key area of investigation i.e. prime numbers what the dynamic nature of this activity might entail!
However though mathematicians are indeed coming to accept that prime number behaviour (at a deep level) is not what they expected, there is absolutely no recognition yet of the central issue i.e. that the existing method of approach ultimately is not fit for purpose.
So if we are to talk of the prime numbers in this new light we are led to accept that their behaviour - necessarily - is of a merely relative nature, reflecting the continual interaction of its manifest features with a universal (unmanifest) holistic ground. Seen in this light we cannot divorce the (actual) finite nature of the prime numbers from their (potential) infinite nature so that in a sense prime numbers - from a dynamic interactive perspective - are in a continual state of becoming (exhibiting at any time a merely approximate identity).
So as we have seen the primes and the non-trivial zeros are mutually encoded in each other. Though from an isolated perspective when we freeze this interaction they both show quantitative features, their mutual interdependence depends on the equal incorporation of their hidden holistic features (which reflects qualitative understanding).
Putting it more simply we can say that the prime numbers dynamically vibrate (with this vibration necessarily reflecting both quantitative and qualitative aspects of interpretation).
When we look carefully at our actual experience of number, it necessarily entails both quantitative and qualitative aspects pertaining directly to (rational) conscious and (intuitive) unconscious aspects of appreciation.
The infinite nature of the number concept (as potentially applying to all numbers) is strictly holistic (pertaining to unconscious intuitive type appreciation).
We then give the number concept a reduced finite nature as applying to all actual numbers (pertaining to conscious rational interpretation).
Likewise we have actual individual number perceptions (again pertaining to conscious rational interpretation).
And finally we have the sense of such perceptions in some way embodying the potential number concept (i.e. as members of the number class that is potentially infinite). So this points once more to the unconscious aspect of understanding.
So in actual experience we have the dynamic interaction of both finite and infinite notions of meaning (pertaining to conscious and unconscious appreciation of both number concepts and perceptions respectively).
And yet in interpreting numbers, mathematicians attempt to view them in merely conscious rational terms. And the key casualty in this is a highly reduced notion of the infinite, which effectively is considered as a linear extension of finite notions.
Thursday, February 9, 2012
Wednesday, February 8, 2012
Quantum Mechanics
We have seen how both prime numbers and physical particles can be understood to share common characteristics of a dynamic nature at a deep level of investigation.
Thus at the complex analytic level, prime numbers have embodied in their very nature an intricate harmonic structure (associated with waveforms that relate to the non-trivial zeros of the Zeta Function). And a two-way interdependence exists as between the individual primes and the non-trivial zeros (and in reverse the non-trivial zeros and the primes). So at this level neither the individual primes nor the holistic nature of their general distribution can be properly understood in isolation from each other.
In like manner at the sub-atomic level, physical particles possess a complementary aspect as waves, with once again a two-way interdependence existing as between particles and waves (and in reverse fashion waves and particles).
So once more we cannot hope to understand either the particle or the wave nature of matter in isolation from each other.
The deeper implications of these findings - which requires nothing less that the most radical shift in the scientific paradigm ever undertaken - have not yet been addressed by either the mathematics or physics communities.
So in both cases we have both individual and holistic features of behaviour which are interdependent. And just like two turns at a crossroads must necessarily be left and right (or right and left) with respect to each other depending on the direction from which they are approached, likewise both Mathematics and Physics require a new dual paradigm with complementary quantitative and qualitative aspects of interpretation.
And as we have seen the great limitation of the present paradigm is that in formal terms it is still defined by merely the quantitative aspect!
At least at one level physicists have been forced to recognise - largely though experimental evidence - the true dynamic relative nature of physical behaviour at the sub-atomic level.
However, in practice they still cling on to the old paradigm. Because this lacks a holistic qualitative dimension, it cannot provide a philosophically coherent explanation of the nature of the quantum world.
In other words the findings of Quantum Mechanics seem highly paradoxical and non-intuitive when filtered through conventional linear rational methods of interpretation.
And of course this clearly implies - though not apparently recognised - that newer qualitative holistic modes are required through which these findings can indeed achieve coherence from a philosophical standpoint.
This is where the contemplative vision, that formerly was strongly associated with the pure religious quest for meaning, becomes so relevant.
Unfortunately in the past however, precious little attention was given, by those successfully traversing the various levels of spiritual transformation, to the mathematical and physical implications of their new understanding.
But this is precisely what is now required, for a complementary relationship exists in vertical terms as between psychospiritual and physical - and indeed ultimately mathematical - understanding.
In other words as we journey deeper into the physical and mathematical underworld, its structure changes from forms, which appear unambiguous and absolute at the conventional level of rational understanding, to highly dynamic interactive relationships with no fixed identity.
Likewise when one one journeys into the higher spiritual world, it is accompanied by a gradual - and sometimes dramatic - breakdown in all the dualistic assumptions that govern normal everyday life.
So one adjusts to this new reality through experiencing reality in a new highly interactive manner, where reason becomes increasingly imbued with refined forms of intuition, so that quantitative and qualitative aspects become inseparable.
So the very structures that govern these higher dimensions (in psycho spiritual terms) have a mirror complementary aspect in the manner in which the behaviour of the physical world at a deep intimate level manifests itself.
So once again the very reason why the findings of Quantum Mechanics seem so paradoxical is that we are still attempting to apply qualitative interpretations (that apply at the normal macro level) to an entirely different level (where such interpretation is inappropriate).
And of course the very nature of scientific interpretation at the macro level is that it reduces qualitative to mere quantitative type interpretation!
So the true lesson that needs to be learnt from Quantum Mechanics is this! Just as matter at this dynamic interactive level possesses twin aspects that are complementary, likewise the appropriate paradigm to study such reality likewise requires twin aspects that are complementary. In other words it should combine both quantitative and qualitative aspects (which are not reduced in terms of each other).
In an even more fundamental manner, this requirement is also true of Mathematics.
So for example, the proper appreciation of prime number behaviour, at a deep complex analytic level of interpretation, requires twin complementary aspects that are also quantitative and qualitative with respect to each other.
We could indeed truthfully say that such behaviour requires the marriage of reason with the spiritual contemplative vision (however with the significant proviso that we are talking here about a vision that is properly trained in a holistic mathematical manner)!
And as I have repeatedly said - when appropriately understood - the Riemann Hypothesis (in the context of Riemann's Functional Equation) represents the two-way balancing of quantitative with qualitative (and qualitative with quantitative) meaning.
Thus at the complex analytic level, prime numbers have embodied in their very nature an intricate harmonic structure (associated with waveforms that relate to the non-trivial zeros of the Zeta Function). And a two-way interdependence exists as between the individual primes and the non-trivial zeros (and in reverse the non-trivial zeros and the primes). So at this level neither the individual primes nor the holistic nature of their general distribution can be properly understood in isolation from each other.
In like manner at the sub-atomic level, physical particles possess a complementary aspect as waves, with once again a two-way interdependence existing as between particles and waves (and in reverse fashion waves and particles).
So once more we cannot hope to understand either the particle or the wave nature of matter in isolation from each other.
The deeper implications of these findings - which requires nothing less that the most radical shift in the scientific paradigm ever undertaken - have not yet been addressed by either the mathematics or physics communities.
So in both cases we have both individual and holistic features of behaviour which are interdependent. And just like two turns at a crossroads must necessarily be left and right (or right and left) with respect to each other depending on the direction from which they are approached, likewise both Mathematics and Physics require a new dual paradigm with complementary quantitative and qualitative aspects of interpretation.
And as we have seen the great limitation of the present paradigm is that in formal terms it is still defined by merely the quantitative aspect!
At least at one level physicists have been forced to recognise - largely though experimental evidence - the true dynamic relative nature of physical behaviour at the sub-atomic level.
However, in practice they still cling on to the old paradigm. Because this lacks a holistic qualitative dimension, it cannot provide a philosophically coherent explanation of the nature of the quantum world.
In other words the findings of Quantum Mechanics seem highly paradoxical and non-intuitive when filtered through conventional linear rational methods of interpretation.
And of course this clearly implies - though not apparently recognised - that newer qualitative holistic modes are required through which these findings can indeed achieve coherence from a philosophical standpoint.
This is where the contemplative vision, that formerly was strongly associated with the pure religious quest for meaning, becomes so relevant.
Unfortunately in the past however, precious little attention was given, by those successfully traversing the various levels of spiritual transformation, to the mathematical and physical implications of their new understanding.
But this is precisely what is now required, for a complementary relationship exists in vertical terms as between psychospiritual and physical - and indeed ultimately mathematical - understanding.
In other words as we journey deeper into the physical and mathematical underworld, its structure changes from forms, which appear unambiguous and absolute at the conventional level of rational understanding, to highly dynamic interactive relationships with no fixed identity.
Likewise when one one journeys into the higher spiritual world, it is accompanied by a gradual - and sometimes dramatic - breakdown in all the dualistic assumptions that govern normal everyday life.
So one adjusts to this new reality through experiencing reality in a new highly interactive manner, where reason becomes increasingly imbued with refined forms of intuition, so that quantitative and qualitative aspects become inseparable.
So the very structures that govern these higher dimensions (in psycho spiritual terms) have a mirror complementary aspect in the manner in which the behaviour of the physical world at a deep intimate level manifests itself.
So once again the very reason why the findings of Quantum Mechanics seem so paradoxical is that we are still attempting to apply qualitative interpretations (that apply at the normal macro level) to an entirely different level (where such interpretation is inappropriate).
And of course the very nature of scientific interpretation at the macro level is that it reduces qualitative to mere quantitative type interpretation!
So the true lesson that needs to be learnt from Quantum Mechanics is this! Just as matter at this dynamic interactive level possesses twin aspects that are complementary, likewise the appropriate paradigm to study such reality likewise requires twin aspects that are complementary. In other words it should combine both quantitative and qualitative aspects (which are not reduced in terms of each other).
In an even more fundamental manner, this requirement is also true of Mathematics.
So for example, the proper appreciation of prime number behaviour, at a deep complex analytic level of interpretation, requires twin complementary aspects that are also quantitative and qualitative with respect to each other.
We could indeed truthfully say that such behaviour requires the marriage of reason with the spiritual contemplative vision (however with the significant proviso that we are talking here about a vision that is properly trained in a holistic mathematical manner)!
And as I have repeatedly said - when appropriately understood - the Riemann Hypothesis (in the context of Riemann's Functional Equation) represents the two-way balancing of quantitative with qualitative (and qualitative with quantitative) meaning.
Tuesday, February 7, 2012
Clarifying the Riemann Hypothesis
As stated in the last blog all mathematical symbols possess both quantitative and qualitative aspects.
Though in isolation it is indeed possible to seek to interpret numbers with respect merely to their quantitative properties, when using interdependent frames (implying a dynamic interactive approach), numbers are necessarily quantitative and qualitative (and qualitative and quantitative) with respect to each other.
So applying this again to prime numbers, one can indeed attempt to interpret the individual primes and their general distribution (in isolation) using a merely quantitative frame of reference.
However when we seek to understand the interdependence of specific primes and their general distribution, we must incorporate both quantitative and qualitative aspects of appreciation. And this poses insuperable difficulties for the conventional (Type 1) mathematical approach, which in formal terms is solely based on mere quantitative interpretation.
And the interdependence of primes, with respect to their individual identity and overall distribution, is clearly manifested in the relation of the trivial non-zeros on the one hand to the general prime number distribution (and the corresponding relationship as between the individual primes and the general distribution of the non-trivial zeros).
The non-trivial zeros themselves represent the unlimited possible solutions for s (where s represents a complex dimensional number of the form a + it). And as discussed in the last blog, the relationship of (base) numbers to their dimensional powers likewise is as quantitative to qualitative (and qualitative to quantitative).
Though this understanding is absolutely central to true appreciation of the nature of the Riemann Hypothesis, it completely eludes conventional (Type 1) analysis, which once again totally lacks, in formal terms, any distinctive qualitative aspect of interpretation.
As we have already seen the non-trivial zeros can be used in an ingenious manner (after a couple of other small adjustments) to gradually correct any remaining deviations arising from Riemann's general function for the prediction of prime number frequency. So in principle through using this approach we should be gradually able to zone in on the precise location of each individual prime, while ultimately correctly prediction the overall frequency of primes (up to a given number).
Now this is based on acceptance of the Riemann Hypothesis.
It is often stated in this manner that given the truth of the Riemann Hypothesis (i.e. that the real part of all these zeros of s lies on the line = 1/2, then in principle we can exactly predict the prime numbers (from their general frequency).
However much greater subtlety is required in this statement, which indeed is required to reveal the true nature of the Riemann Hypothesis.
As befits the proper distinction as between quantitative and qualitative, we need likewise to carefully distinguish as between (actual) finite and (potential) infinite meaning. By its very nature Conventional Mathematics inevitably reduces in any context (potential) infinite to (actual) finite notions of meaning!
So it is true as we progressively add in the corrections based on the non-trivial zeros that we move ever closer to the integer values of the primes.
However in actual terms this process can never be completed for no matter how many non-trivial zeros we seek to consider, an unlimited set of non-trivial zeros will remain. So there is an inherent uncertainty attached to this process whereby the successive approximation through determination of non-trivial zeros of the location of the primes is always based on an unlimited set of non-trivial zeros, which must remain indeterminate.
So in actual finite terms (which is the proper domain of quantitative interpretation)we can never exactly pinpoint the location of the primes, with an uncertainty thereby necessarily attached to their precise values.
However when we switch to a potential infinite context (which is the proper domain of qualitative meaning) we can indeed say that in potential terms, if the infinite set of non-zeros is included, that then we would indeed exactly obtain the discrete integer values of the primes. And in doing this our reconciliation of the primes with the trivial non-zeros would be complete.
However a purely potential state equally implies that no phenomenal identity can remain to the primes (in actual terms).
In other words the full reconciliation of the primes with the non-trivial zeros (both of which are mutually encoded in each other) points to an ineffable state with no phenomenal existence.
And as this process is based on the assumption that the Riemann Hypothesis is true, it thereby is pointing to this ineffable state.
So the Riemann Hypothesis is directly concerned with the ultimate reconciliation of quantitative and qualitative meaning (where finite and infinite can at last become identical).
So in this respect Hilbert was indeed correct. The implications of the zeros of the Zeta function in the context of Riemann's Hypothesis could not be more important.
The famous Buddhist heart sutra states this identity of finite and infinite in the following manner:
"Form is not other than Void;
Void is not other than Form"
The Riemann Hypothesis in fact is simply a restatement of this sutra related to the ultimate nature of mathematical meaning:
"The Quantitative is not other than the Qualitative;
The Qualitative is not other than the Quantitative"
The deeper implications of the true nature of the primes are awe inspiring.
There are two capacities that reveal themselves in nature, one for independence and the other for interdependence i.e quantitative and qualitative aspects (which ultimately relates to the nature of the prime numbers). In an original state - as mere potential for physical phenomenal existence - these two capacities are identical.
Then when operating through the veils of phenomenal reality, they become separated with full understanding of their identical nature again ultimately taking place in an ineffable spiritual manner.
So the task of understanding the mathematical (objective) nature of the primes cannot be ultimately divorced from the psychological nature of their (subjective) interpretation.
The mystery of the primes can therefore be validly seen as embracing the entire course of created evolution.
Once again it is all about the reconciliation of quantitative and qualitative notions of meaning.
And Mathematics would make enormous strides in simply grasping this key fact!
Though in isolation it is indeed possible to seek to interpret numbers with respect merely to their quantitative properties, when using interdependent frames (implying a dynamic interactive approach), numbers are necessarily quantitative and qualitative (and qualitative and quantitative) with respect to each other.
So applying this again to prime numbers, one can indeed attempt to interpret the individual primes and their general distribution (in isolation) using a merely quantitative frame of reference.
However when we seek to understand the interdependence of specific primes and their general distribution, we must incorporate both quantitative and qualitative aspects of appreciation. And this poses insuperable difficulties for the conventional (Type 1) mathematical approach, which in formal terms is solely based on mere quantitative interpretation.
And the interdependence of primes, with respect to their individual identity and overall distribution, is clearly manifested in the relation of the trivial non-zeros on the one hand to the general prime number distribution (and the corresponding relationship as between the individual primes and the general distribution of the non-trivial zeros).
The non-trivial zeros themselves represent the unlimited possible solutions for s (where s represents a complex dimensional number of the form a + it). And as discussed in the last blog, the relationship of (base) numbers to their dimensional powers likewise is as quantitative to qualitative (and qualitative to quantitative).
Though this understanding is absolutely central to true appreciation of the nature of the Riemann Hypothesis, it completely eludes conventional (Type 1) analysis, which once again totally lacks, in formal terms, any distinctive qualitative aspect of interpretation.
As we have already seen the non-trivial zeros can be used in an ingenious manner (after a couple of other small adjustments) to gradually correct any remaining deviations arising from Riemann's general function for the prediction of prime number frequency. So in principle through using this approach we should be gradually able to zone in on the precise location of each individual prime, while ultimately correctly prediction the overall frequency of primes (up to a given number).
Now this is based on acceptance of the Riemann Hypothesis.
It is often stated in this manner that given the truth of the Riemann Hypothesis (i.e. that the real part of all these zeros of s lies on the line = 1/2, then in principle we can exactly predict the prime numbers (from their general frequency).
However much greater subtlety is required in this statement, which indeed is required to reveal the true nature of the Riemann Hypothesis.
As befits the proper distinction as between quantitative and qualitative, we need likewise to carefully distinguish as between (actual) finite and (potential) infinite meaning. By its very nature Conventional Mathematics inevitably reduces in any context (potential) infinite to (actual) finite notions of meaning!
So it is true as we progressively add in the corrections based on the non-trivial zeros that we move ever closer to the integer values of the primes.
However in actual terms this process can never be completed for no matter how many non-trivial zeros we seek to consider, an unlimited set of non-trivial zeros will remain. So there is an inherent uncertainty attached to this process whereby the successive approximation through determination of non-trivial zeros of the location of the primes is always based on an unlimited set of non-trivial zeros, which must remain indeterminate.
So in actual finite terms (which is the proper domain of quantitative interpretation)we can never exactly pinpoint the location of the primes, with an uncertainty thereby necessarily attached to their precise values.
However when we switch to a potential infinite context (which is the proper domain of qualitative meaning) we can indeed say that in potential terms, if the infinite set of non-zeros is included, that then we would indeed exactly obtain the discrete integer values of the primes. And in doing this our reconciliation of the primes with the trivial non-zeros would be complete.
However a purely potential state equally implies that no phenomenal identity can remain to the primes (in actual terms).
In other words the full reconciliation of the primes with the non-trivial zeros (both of which are mutually encoded in each other) points to an ineffable state with no phenomenal existence.
And as this process is based on the assumption that the Riemann Hypothesis is true, it thereby is pointing to this ineffable state.
So the Riemann Hypothesis is directly concerned with the ultimate reconciliation of quantitative and qualitative meaning (where finite and infinite can at last become identical).
So in this respect Hilbert was indeed correct. The implications of the zeros of the Zeta function in the context of Riemann's Hypothesis could not be more important.
The famous Buddhist heart sutra states this identity of finite and infinite in the following manner:
"Form is not other than Void;
Void is not other than Form"
The Riemann Hypothesis in fact is simply a restatement of this sutra related to the ultimate nature of mathematical meaning:
"The Quantitative is not other than the Qualitative;
The Qualitative is not other than the Quantitative"
The deeper implications of the true nature of the primes are awe inspiring.
There are two capacities that reveal themselves in nature, one for independence and the other for interdependence i.e quantitative and qualitative aspects (which ultimately relates to the nature of the prime numbers). In an original state - as mere potential for physical phenomenal existence - these two capacities are identical.
Then when operating through the veils of phenomenal reality, they become separated with full understanding of their identical nature again ultimately taking place in an ineffable spiritual manner.
So the task of understanding the mathematical (objective) nature of the primes cannot be ultimately divorced from the psychological nature of their (subjective) interpretation.
The mystery of the primes can therefore be validly seen as embracing the entire course of created evolution.
Once again it is all about the reconciliation of quantitative and qualitative notions of meaning.
And Mathematics would make enormous strides in simply grasping this key fact!
Monday, February 6, 2012
Momentous Change
We are perhaps already at the beginning of a truly momentous shift in mathematical - and by extension - all scientific understanding where the prevailing paradigm will change in a radical and unprecedented fashion.
We saw in the past two blogs how our very understanding of prime numbers needs to be altered in the light of Riemann's ground breaking discoveries.
So - again using the analogy from Quantum Mechanics - prime numbers can no longer be understood as static fixed entities (with specific integer values). Rather they need to be considered in relative terms as dynamic interactive entities (possessing complementary wave and particle aspects).
Putting it another way they possess both specific and holistic aspects (or - in the terms that I customarily use - quantitative and qualitative).
As I say, when appropriately interpreted this is already inherent in Riemann's discoveries with respect to his Zeta Function where both knowledge of specific prime numbers and their corresponding holistic aspect through the non-trivial zeros (relating to the unlimited solutions of the complex Zeta function) can be seen to be mutually encoded in each other.
Now why the obvious connection here with Quantum Mechanics is not clearly seen owes a great deal I believe to the limited - merely quantitative - approach that is employed in Conventional (Type 1) Mathematics.
To see the problem involved we will digress once again to my favourite example of road directions.
If I am travelling up a straight (vertical) road and come to a crossroads I can unambiguously identify - for example - a left turn.
Now if I later travel down the same road from the opposite direction, I can again unambiguously identify a left turn.
However we will now have designated both turns unambiguously as left.
And this results from using isolated independent frames of reference.
So in the former case the frame of reference was unambiguously "up" while in the latter it was unambiguously "down".
Now clearly of course when we use simultaneous frames of reference the two turns must necessarily be left and right (in relation to each other).
The relevance of this for the conventional mathematical approach is striking (as it solely recognises the quantitative approach).
Therefore when for example we approach the primes from one direction with respect to their specific attributes their quantitative nature seems apparent. Likewise when we then approach them from the opposite direction with respect to their general attributes, again their quantitative nature seems readily apparent.
So in effect the quantitative approach is used with respect to both specific and holistic appreciation of the primes. And admittedly when using isolated frames of reference in studying their aspects independently, both indeed exhibit quantitative features.
However when we bring both aspects together (simultaneously, as it were) to understand how both are interrelated, then the specific and holistic nature of the primes are quantitative and qualitative with respect to each other. Thus we cannot hope to properly understand - in this context - the relationship between quantitative and qualitative aspects while using a merely quantitative type approach!
This is why I have consistently maintained that the Riemann Hypothesis cannot be satisfactorily understood - not alone resolved - in the absence of the complementary qualitative aspect of mathematical understanding.
And this problem is not just isolated to the Riemann Hypothesis but is in fact endemic in the most common of mathematical processes.
Even as a child of 10, I had marked difficulties with the mathematical treatment of squares and the corresponding treatment of square roots.
I could see then that whenever a number is squared that a qualitative - as well as quantitative - transformation takes place. So when for example we square 3 (3^2) in quantitative terms a transformation takes place (to 9) and in qualitative terms from 1-dimensional to 2-dimensional (i.e. square) units. So the quantitative aspect is tied up with the base unit (3) and the qualitative aspect with the dimensional aspect of number (in this case 2).
So we have here the same old problem. When for example when we raise a number to a certain power e.g. a^b, both the base number a and the dimensional power b do indeed have a quantitative aspect (when considered in isolation from each other). However when we combine both aspects, then a and b are actually quantitative and qualitative with respect to each other.
So as I would see it, the very way that powers and roots are handled in Conventional (Type 1) Mathematics is grossly reductionist.
Now, remedying this problem requires that an alternative (Type 2) Mathematics be put in place to deal with the qualitative aspect (initially in isolation). Then when both are combined we need to move to Type 3 Mathematics for a comprehensive understanding of what is involved through interaction of both types. However, there is still precious little recognition of the truly fundamental nature of this problem in Mathematics.
Thus my own attempts for 40 years have been largely concerned with formulating the bones of this Type 2 Mathematics (which I call Holistic Mathematics). It is only in recent times that I have attempted to combine Type 1 and Type 2 in a very preliminary manner with reference to the most beautiful formula in Mathematics (Euler's Identity) and then the greatest unsolved problem in Mathematics (the Riemann Hypothesis). So what I have been attempting - again in a necessarily preliminary manner - serves as an introduction to Type 3 Mathematics.
Thus, through approaching the problem from the background of immersion in an entirely distinctive qualitative type approach, it quickly became apparent to me that the Riemann Hypothesis is not in fact a problem that can be solved in conventional (Type 1) terms. It properly relates to the basic condition necessary for reconciling both the quantitative (specific) and holistic (qualitative) aspects of the primes thus making it in effect a central axiom for Type 3 Mathematics.
When we look at Riemann's Zeta Function, we can see that this relationship as between base natural numbers and varying dimensional powers (s) is central (with the added complication that the dimensional powers are formulated in a complex manner).
It should also be obvious that the Riemann Zeta Function throws up a whole series
of values for s (with real part < 1) that have no strict meaning from a standard linear rational perspective. For example 1 + 2 + 3 + ..... diverges to infinity (in standard terms) yet acquires a value of - 1/12 (in the context of Riemann's Zeta function). The question then arises as to how we can reconcile both quantitative interpretations (which I have never seen satisfactorily answered).
And now further we see - in the light of the Function - that the primes exhibit two complementary aspects (of a specific and holistic nature) respectively.
However though both of these aspects can be investigated in isolation from a merely quantitative perspective, relative to each other they are quantitative and qualitative (and qualitative and quantitative) respectively.
The deeper meaning of all this is that the prime numbers as quantities have a mirror side as a qualitative means of interpreting reality. So we cannot solve the physical mystery of the primes without equally solving in mirror terms the psychological mystery of their appropriate interpretation (a task that greatly transcends the methods of Conventional Mathematics).
In psychological terms we could say, from a Jungian perspective, that they possess both conscious aspects (as distinct entities) and unconscious aspects (as archetypes of a universal reality) with both ultimately identical. In complementary physical terms they possess specific attributes and holistic communication abilities (reflecting their quantitative and qualitative nature).
These were originally identical as mere potential for existence and yet have now considerably evolved to yield the wonderful universe we inhabit.
We saw in the past two blogs how our very understanding of prime numbers needs to be altered in the light of Riemann's ground breaking discoveries.
So - again using the analogy from Quantum Mechanics - prime numbers can no longer be understood as static fixed entities (with specific integer values). Rather they need to be considered in relative terms as dynamic interactive entities (possessing complementary wave and particle aspects).
Putting it another way they possess both specific and holistic aspects (or - in the terms that I customarily use - quantitative and qualitative).
As I say, when appropriately interpreted this is already inherent in Riemann's discoveries with respect to his Zeta Function where both knowledge of specific prime numbers and their corresponding holistic aspect through the non-trivial zeros (relating to the unlimited solutions of the complex Zeta function) can be seen to be mutually encoded in each other.
Now why the obvious connection here with Quantum Mechanics is not clearly seen owes a great deal I believe to the limited - merely quantitative - approach that is employed in Conventional (Type 1) Mathematics.
To see the problem involved we will digress once again to my favourite example of road directions.
If I am travelling up a straight (vertical) road and come to a crossroads I can unambiguously identify - for example - a left turn.
Now if I later travel down the same road from the opposite direction, I can again unambiguously identify a left turn.
However we will now have designated both turns unambiguously as left.
And this results from using isolated independent frames of reference.
So in the former case the frame of reference was unambiguously "up" while in the latter it was unambiguously "down".
Now clearly of course when we use simultaneous frames of reference the two turns must necessarily be left and right (in relation to each other).
The relevance of this for the conventional mathematical approach is striking (as it solely recognises the quantitative approach).
Therefore when for example we approach the primes from one direction with respect to their specific attributes their quantitative nature seems apparent. Likewise when we then approach them from the opposite direction with respect to their general attributes, again their quantitative nature seems readily apparent.
So in effect the quantitative approach is used with respect to both specific and holistic appreciation of the primes. And admittedly when using isolated frames of reference in studying their aspects independently, both indeed exhibit quantitative features.
However when we bring both aspects together (simultaneously, as it were) to understand how both are interrelated, then the specific and holistic nature of the primes are quantitative and qualitative with respect to each other. Thus we cannot hope to properly understand - in this context - the relationship between quantitative and qualitative aspects while using a merely quantitative type approach!
This is why I have consistently maintained that the Riemann Hypothesis cannot be satisfactorily understood - not alone resolved - in the absence of the complementary qualitative aspect of mathematical understanding.
And this problem is not just isolated to the Riemann Hypothesis but is in fact endemic in the most common of mathematical processes.
Even as a child of 10, I had marked difficulties with the mathematical treatment of squares and the corresponding treatment of square roots.
I could see then that whenever a number is squared that a qualitative - as well as quantitative - transformation takes place. So when for example we square 3 (3^2) in quantitative terms a transformation takes place (to 9) and in qualitative terms from 1-dimensional to 2-dimensional (i.e. square) units. So the quantitative aspect is tied up with the base unit (3) and the qualitative aspect with the dimensional aspect of number (in this case 2).
So we have here the same old problem. When for example when we raise a number to a certain power e.g. a^b, both the base number a and the dimensional power b do indeed have a quantitative aspect (when considered in isolation from each other). However when we combine both aspects, then a and b are actually quantitative and qualitative with respect to each other.
So as I would see it, the very way that powers and roots are handled in Conventional (Type 1) Mathematics is grossly reductionist.
Now, remedying this problem requires that an alternative (Type 2) Mathematics be put in place to deal with the qualitative aspect (initially in isolation). Then when both are combined we need to move to Type 3 Mathematics for a comprehensive understanding of what is involved through interaction of both types. However, there is still precious little recognition of the truly fundamental nature of this problem in Mathematics.
Thus my own attempts for 40 years have been largely concerned with formulating the bones of this Type 2 Mathematics (which I call Holistic Mathematics). It is only in recent times that I have attempted to combine Type 1 and Type 2 in a very preliminary manner with reference to the most beautiful formula in Mathematics (Euler's Identity) and then the greatest unsolved problem in Mathematics (the Riemann Hypothesis). So what I have been attempting - again in a necessarily preliminary manner - serves as an introduction to Type 3 Mathematics.
Thus, through approaching the problem from the background of immersion in an entirely distinctive qualitative type approach, it quickly became apparent to me that the Riemann Hypothesis is not in fact a problem that can be solved in conventional (Type 1) terms. It properly relates to the basic condition necessary for reconciling both the quantitative (specific) and holistic (qualitative) aspects of the primes thus making it in effect a central axiom for Type 3 Mathematics.
When we look at Riemann's Zeta Function, we can see that this relationship as between base natural numbers and varying dimensional powers (s) is central (with the added complication that the dimensional powers are formulated in a complex manner).
It should also be obvious that the Riemann Zeta Function throws up a whole series
of values for s (with real part < 1) that have no strict meaning from a standard linear rational perspective. For example 1 + 2 + 3 + ..... diverges to infinity (in standard terms) yet acquires a value of - 1/12 (in the context of Riemann's Zeta function). The question then arises as to how we can reconcile both quantitative interpretations (which I have never seen satisfactorily answered).
And now further we see - in the light of the Function - that the primes exhibit two complementary aspects (of a specific and holistic nature) respectively.
However though both of these aspects can be investigated in isolation from a merely quantitative perspective, relative to each other they are quantitative and qualitative (and qualitative and quantitative) respectively.
The deeper meaning of all this is that the prime numbers as quantities have a mirror side as a qualitative means of interpreting reality. So we cannot solve the physical mystery of the primes without equally solving in mirror terms the psychological mystery of their appropriate interpretation (a task that greatly transcends the methods of Conventional Mathematics).
In psychological terms we could say, from a Jungian perspective, that they possess both conscious aspects (as distinct entities) and unconscious aspects (as archetypes of a universal reality) with both ultimately identical. In complementary physical terms they possess specific attributes and holistic communication abilities (reflecting their quantitative and qualitative nature).
These were originally identical as mere potential for existence and yet have now considerably evolved to yield the wonderful universe we inhabit.
Sunday, February 5, 2012
Approximate Nature of Prime Numbers
I stated in my last blog that the Uncertainty Principle lies at the heart of the number system (and my extension all mathematical procedures). So my intention here is to clarify in a little more detail what this situation actually entails.
Though initially it might seem very hard to accept, the implications of Riemann's discoveries in relation to the primes is that at a complex level of investigation, our knowledge is necessarily of a merely approximate nature.
Now it might be helpful in this regard to keep relating to the analogous situation in physics. Here at the "real" everyday level of existence, objects appear to possess an objective unambiguous identity. However at the deeper subatomic levels of reality, revealed through Quantum Mechanics, objects have a merely relative approximate existence. From one perspective at this level, we can no longer clearly divorce the (subjective) observer from what is (objectively) observed. So "objects" are now necessarily seen in a dynamic interactive manner (where both polarities - external and internal - are involved).
Likewise "objects" now possess both wave and particle aspects with again both necessarily interacting in a dynamic relative manner.
So in an analogous manner at the customary "real" level of mathematical experience, numbers - such as the primes - appear to possess a clear unambiguous objective identity (in discrete terms).
However at the deeper complex level of analytical investigation this clear picture irretrievably breaks down (with the full implications of this yet to be addressed by the mathematical community).
It is customarily portrayed that Riemann somehow has managed to support the traditional "real" view of the primes (as having a discrete integer identity).
His famous formula for eliminating deviations - so we are told - opens up the way in principle to exactly calculate both the general frequency of the primes (up to any required natural number) while also precisely providing the location of each prime number.
However this is merely valid in an approximate manner and it is in the clear realisation of this point that we can only begin to appreciate the true nature of the primes at this deeper level - literally - as approximate values.
Riemann's method - which is indeed truly ingenious - provides a means from moving from the general distribution of the primes (which initially is represented by a smooth continuous curve) to gradually approximate - principally through adjustments based on the non-trivial zeros - to the original step function nature of the actual primes.
So having catered for the other small adjustments, seemingly all that is required so as to exactly pinpoint the location of the primes is to progressively add in each contribution that is made by the non-trivial zeros and through doing this zone in ever more closely on their precise location.
However the problem with this procedure is that - no matter how many non-trivial zeros are allowed to make their contribution - we will never arrive at the discrete integer values (that we are so familiar with from our "real" everyday understanding).
Now it might be said that if in principle we were to add in the contributions of the "infinite" number of non-trivial zeros that exist, then we would indeed finally obtain the integer values of the primes.
However with respect to a finite procedure we cannot - by definition - ever add in the contributions of all the non-trivial zeros, for no matter how many are included an unlimited amount must necessarily always remain excluded! In other words we can only determine any set of non-trivial primes by leaving an unlimited set of other non-trivial primes indeterminate!
So once again it can be seen here that the very belief that we can calculate integer prime values from Riemann's method represents the old fallacy of confusing infinite with finite notions (i.e. qualitative with quantitative). And as I have said all along ultimately this is what the Riemann Hypothesis is all about i.e. the need to reconcile finite with infinite notions (which properly belong to two distinctive domains of understanding).
So to be strictly accurate, the relationship which Riemann establishes as between primes and non-trivial zeros, is of a dynamic relative nature.
Thus from one perspective, starting with Riemann's general continuous function for prime distribution - after allowing for two other small adjustments - we can use the non-trivial zeros to move ever more closely to the actual location of the individual primes (and the overall frequency of such primes).
However strictly this is always of an approximate relative nature (and subject to uncertainty). So the prime numbers can never obtain a discrete integer identity through this process!
Likewise from the other perspective we can move from the knowledge of these approximate values obtained through the process to more precisely measure the frequency and precise location of the non-trivial zeros.
And of course the more we concentrate on fixing one aspect e.g. the location of individual primes, the more fuzzy our knowledge of non-trivial zeros will become. If on the other hand we try to concentrate on fixing the precise location of non-trivial zeros, the more fuzzy our knowledge of the precise location of the primes.
Part of the problem that we have in appreciating this point is that we are always starting from a pre-conceived knowledge of individual integer primes on the one hand (and a preconceived knowledge of the location of non-trivial zeros on the other).
Thus in testing Riemann's procedures we already know the answers we are seeking to obtain from the procedure.
So for example if we use his methods to predict the location of all primes up to 1,000,000 we already know the location of all these primes. So when the approximations get close to these - already known - prime integers we automatically round off the approximate values to conclude that we have exactly located the primes!
However this is essentially to miss the crucial point that the two levels of primes are of a distinct nature.
The first (i.e. of already known discrete integer values) reflects the absolute type notions of number that characterises the conventional linear approach.
Once again this is directly analogous to our identification of objects in unambiguous terms at the everyday macro level of existence (in accordance with Newtonian type physics).
The second (i.e. of merely approximate values) reflects the relative type notions that - properly understood - characterises the complex analytic view of the primes.
And of course here the analogy with the quantum level of behaviour of sub-atomic particles is especially relevant.
And as I have demonstrated, uncertainty naturally reigns here as we must always compromise to a degree as between our "particle" knowledge of the location of individual prime numbers and our "wave" knowledge of the "momentum" (or wavelength) of a non-trivial zero!
Physicists found the findings of Quantum Mechanics extremely difficult to accept (and still do). However largely on experimental grounds they were eventually led to accept its validity.
I would strongly imagine that mathematicians will find the merely approximate nature of prime numbers (at the corresponding "sub-atomic" level of mathematical investigation) even more difficult to accept. For one thing the - ultimately mistaken - belief in the absolute validity of mathematical procedures has led to a mind set that will create enormous barriers to such new acceptance.
Furthermore the same kind of experimental evidence will not be available as in physics.
So ultimately I believe a slow philosophical process will be involved where it will become gradually apparent that much mathematical interpretation in many crucial respects leaves a great deal to be desired. And in all of this the need to to recognise the unacceptable manner in which the infinite notion is persistently reduced in finite terms will be key.
And really, as I see it, this is the message that lies at the heart of the Riemann Hypothesis.
Though initially it might seem very hard to accept, the implications of Riemann's discoveries in relation to the primes is that at a complex level of investigation, our knowledge is necessarily of a merely approximate nature.
Now it might be helpful in this regard to keep relating to the analogous situation in physics. Here at the "real" everyday level of existence, objects appear to possess an objective unambiguous identity. However at the deeper subatomic levels of reality, revealed through Quantum Mechanics, objects have a merely relative approximate existence. From one perspective at this level, we can no longer clearly divorce the (subjective) observer from what is (objectively) observed. So "objects" are now necessarily seen in a dynamic interactive manner (where both polarities - external and internal - are involved).
Likewise "objects" now possess both wave and particle aspects with again both necessarily interacting in a dynamic relative manner.
So in an analogous manner at the customary "real" level of mathematical experience, numbers - such as the primes - appear to possess a clear unambiguous objective identity (in discrete terms).
However at the deeper complex level of analytical investigation this clear picture irretrievably breaks down (with the full implications of this yet to be addressed by the mathematical community).
It is customarily portrayed that Riemann somehow has managed to support the traditional "real" view of the primes (as having a discrete integer identity).
His famous formula for eliminating deviations - so we are told - opens up the way in principle to exactly calculate both the general frequency of the primes (up to any required natural number) while also precisely providing the location of each prime number.
However this is merely valid in an approximate manner and it is in the clear realisation of this point that we can only begin to appreciate the true nature of the primes at this deeper level - literally - as approximate values.
Riemann's method - which is indeed truly ingenious - provides a means from moving from the general distribution of the primes (which initially is represented by a smooth continuous curve) to gradually approximate - principally through adjustments based on the non-trivial zeros - to the original step function nature of the actual primes.
So having catered for the other small adjustments, seemingly all that is required so as to exactly pinpoint the location of the primes is to progressively add in each contribution that is made by the non-trivial zeros and through doing this zone in ever more closely on their precise location.
However the problem with this procedure is that - no matter how many non-trivial zeros are allowed to make their contribution - we will never arrive at the discrete integer values (that we are so familiar with from our "real" everyday understanding).
Now it might be said that if in principle we were to add in the contributions of the "infinite" number of non-trivial zeros that exist, then we would indeed finally obtain the integer values of the primes.
However with respect to a finite procedure we cannot - by definition - ever add in the contributions of all the non-trivial zeros, for no matter how many are included an unlimited amount must necessarily always remain excluded! In other words we can only determine any set of non-trivial primes by leaving an unlimited set of other non-trivial primes indeterminate!
So once again it can be seen here that the very belief that we can calculate integer prime values from Riemann's method represents the old fallacy of confusing infinite with finite notions (i.e. qualitative with quantitative). And as I have said all along ultimately this is what the Riemann Hypothesis is all about i.e. the need to reconcile finite with infinite notions (which properly belong to two distinctive domains of understanding).
So to be strictly accurate, the relationship which Riemann establishes as between primes and non-trivial zeros, is of a dynamic relative nature.
Thus from one perspective, starting with Riemann's general continuous function for prime distribution - after allowing for two other small adjustments - we can use the non-trivial zeros to move ever more closely to the actual location of the individual primes (and the overall frequency of such primes).
However strictly this is always of an approximate relative nature (and subject to uncertainty). So the prime numbers can never obtain a discrete integer identity through this process!
Likewise from the other perspective we can move from the knowledge of these approximate values obtained through the process to more precisely measure the frequency and precise location of the non-trivial zeros.
And of course the more we concentrate on fixing one aspect e.g. the location of individual primes, the more fuzzy our knowledge of non-trivial zeros will become. If on the other hand we try to concentrate on fixing the precise location of non-trivial zeros, the more fuzzy our knowledge of the precise location of the primes.
Part of the problem that we have in appreciating this point is that we are always starting from a pre-conceived knowledge of individual integer primes on the one hand (and a preconceived knowledge of the location of non-trivial zeros on the other).
Thus in testing Riemann's procedures we already know the answers we are seeking to obtain from the procedure.
So for example if we use his methods to predict the location of all primes up to 1,000,000 we already know the location of all these primes. So when the approximations get close to these - already known - prime integers we automatically round off the approximate values to conclude that we have exactly located the primes!
However this is essentially to miss the crucial point that the two levels of primes are of a distinct nature.
The first (i.e. of already known discrete integer values) reflects the absolute type notions of number that characterises the conventional linear approach.
Once again this is directly analogous to our identification of objects in unambiguous terms at the everyday macro level of existence (in accordance with Newtonian type physics).
The second (i.e. of merely approximate values) reflects the relative type notions that - properly understood - characterises the complex analytic view of the primes.
And of course here the analogy with the quantum level of behaviour of sub-atomic particles is especially relevant.
And as I have demonstrated, uncertainty naturally reigns here as we must always compromise to a degree as between our "particle" knowledge of the location of individual prime numbers and our "wave" knowledge of the "momentum" (or wavelength) of a non-trivial zero!
Physicists found the findings of Quantum Mechanics extremely difficult to accept (and still do). However largely on experimental grounds they were eventually led to accept its validity.
I would strongly imagine that mathematicians will find the merely approximate nature of prime numbers (at the corresponding "sub-atomic" level of mathematical investigation) even more difficult to accept. For one thing the - ultimately mistaken - belief in the absolute validity of mathematical procedures has led to a mind set that will create enormous barriers to such new acceptance.
Furthermore the same kind of experimental evidence will not be available as in physics.
So ultimately I believe a slow philosophical process will be involved where it will become gradually apparent that much mathematical interpretation in many crucial respects leaves a great deal to be desired. And in all of this the need to to recognise the unacceptable manner in which the infinite notion is persistently reduced in finite terms will be key.
And really, as I see it, this is the message that lies at the heart of the Riemann Hypothesis.
Saturday, February 4, 2012
Fundamental Implications of the Nature of Primes
At first glance the specific prime numbers 2, 3, 5, 7,... appear to display no obvious pattern.
And till the beginning of the 19th century, this situation remained. Then largely due to Gauss this situation changed when he discovered a striking collective pattern to the primes based on the natural log function.
So from an individual perspective the primes numbers formerly seemed the most independent of numbers (exhibiting no obvious pattern), Yet from the new general perspective it could now be seen that they displayed an amazing regularity.
It was the genius of Riemann to try and connect these two aspects. He made amazing strides in analysing the complex nature of the primes so that he was able to suggest a formula that ultimately could eliminate deviations arising from the general estimate of prime number frequency.
He made the wonderful discovery that associated with the primes are a - potentially - unlimited number of solutions to the Riemann Zeta Function (defined in the complex plane). These solutions are commonly referred to as the non-trivial zeros (of the Zeta Function) or more simply Riemann's zeros. And associated with these zeros are elaborate wave forms which can - magically as it were - be used to correct any remaining deviations arising from his general estimate of prime number frequency.
So underlying the prime number system, seemingly is composed of discrete individual members, is a subtle wave form pattern. And it is in the relationship as between the two aspects that the mystery of the prime numbers resides.
Thus from one perspective we can use the waveforms associated with the non-trivial zeros in a systematic pattern to eliminate deviations with respect to the general estimate of prime number frequency (up to a given natural number).
In corresponding reverse manner we can also use knowledge of the individual prime numbers to eliminate any remaining deviations in the general estimate of the frequency of the non-trivial zeros (up to a given number on the imaginary number scale).
So this displays admirably the two-way interdependence as between the individual primes on the one hand and the corresponding wave patterns associated with these primes.
Then coming from either direction, mediated through the natural numbers, is a means of moving from individual primes to their collective wave patterns on the one hand and likewise in reverse manner from these wave patterns to the individual primes.
Right away this new newly discovered nature of the primes (by Riemann) suggests a direct correspondence with quantum physics.
Again it is truly remarkable in this regard that Riemann can be directly linked with the two great developments in physics of the 20th century. His developments in Geometry (Riemannian Geometry) was to prove invaluable to Einstein in formulating his Theory of General Relativity. However equally - though not yet sufficiently recognised - his work on the prime numbers provides a direct mathematical link with Quantum Mechanics.
As is now commonly known at the sub-atomic level of matter, particles possess a dual complementary existence exhibiting - depending on the context of observation - both wave and particle effects. Indeed even at the everyday macro level, objects exhibit both particle and wave aspects (though in practical terms the wave aspect can be effectively ignored)!
We can now say exactly the same thing in relation to prime numbers. At a deep complex level of investigation, prime numbers likewise exhibit both particle and wave aspects. However once again at the macro level of "real" interpretation these wave aspects do not reveal themselves and are effectively ignored. So once again it requires the complex analytic techniques first used by Riemann to reveal - as it were - the subatomic structure of the primes.
It is revealing that the primes are often referred to as the "atoms" of the number system which then serve as the "building blocks" for the natural numbers. However though indeed valid from one partial perspective, this is unduly reductionist for the primes - as we now know - contain a remarkable wave pattern system. And because of the two-way interdependence as between the particle and wave aspects, we can no longer hope to understand primes as mere "building blocks".
So as with subatomic particles we now recognise that the particle aspect of the primes also contain a wave aspect and the wave likewise a particle aspect.
However now we get to the crucial point! Just as it is now understood that the behaviour of particles at the sub-atomic level is governed by the Uncertainty Principle, equally we can assert that the prime numbers - at a deeper level of investigation - are governed by a corresponding mathematical Uncertainty Principle.
In physics if we try to precisely fix the position of a particle we blot out recognition of its corresponding momentum. And in turn, if we then try to precisely measure momentum it tends to block out any precise knowledge of its position.
It is similar in relation to the primes. When we try to precisely track the individual location of the primes, it tends to block out corresponding knowledge of their general distribution. And mathematical history testifies to this point! For though the primes were known and studied for well over 2000 years, it was not until the 19th century that this customary stance with respect to prime number analysis was abandoned.
And in turn, when we then concentrate on the general distribution of the primes this tends to block out recognition of the precise location of each individual prime!
So let me put it bluntly so that the point cannot be easily ignored. Uncertainty - by its very nature - lies at the heart of the number system and by extension at the heart of all mathematical relationships!
Furthermore the Uncertainty Principle that is now accepted with respect to the physical world - though not unfortunately its many implications - has its deeper roots in the nature of the prime numbers!
In other words the Uncertainty Principle that governs the physical world is ultimated rooted in the uncertainty that lies at the heart of mathematics.
The wave forms associated with prime numbers were discovered by Riemann long before the breakthroughs in Quantum Physics. However just as Einstein's General Theory of Relativity has its mathematical roots in Riemannian Geometry, I am confident it will be accepted in the future that Quantum Mechanics - especially in relation to the Uncertainty Principle - has its roots in Riemann's amazing discoveries with respect to the complex Zeta Function.
Then, even deeper implications of what I am saying here, have not yet been addressed by either the physics or mathematical community.
Indeed it is again ironic that the Uncertainty Principle itself can be effectively used in a related manner to highlight the key dilemma!
In truth all mathematical and scientific understanding entails two aspects of understanding which are interrelated in experience.
One the one hand we have reason - pertaining directly to the conscious - which in the current mathematical and scientific means simply linear reason. Such reason properly relates to differentiated interpretation with respect to specific actual phenomena (that are necessarily of a finite nature).
However equally we have intuition, pertaining directly to the unconscious. Though the importance of such intuition might well be informally recognised especially for creative work, in formal terms it simply is reduced through the conventional paradigm to reason. And in contrast to reason, intuition relates to integral holistic interpretation with respect to general phenomena of a universal potential nature (that are necessarily infinite in origin).
So a great price that has been paid for the admittedly enormous advances using the present paradigm. Thus we have learnt to automatically reduce - in any context - what is truly qualitative and distinct to mere quantitative interpretation.
Whereas reason tends to be linear, intuition is inherently of a circular nature (when indirectly interpreted in rational fashion). In other words linear reason is based on the clear unambiguous separation of opposite polarities (such as external and internal) whereas intuition is based on their complementary - and apparently paradoxical - identity.
So the key implication for mathematics and science is that a more comprehensive paradigm requires the incorporation of both linear and circular modes of logical interpretation. And remember these circular modes serve as the indirect expression of ever more refined forms of reason! Put another way, in qualitative terms we need a paradigm that contains both real and imaginary aspects!
We have seen that an attempt to precisely fix the position of a particle blocks out recognition of its complementary (unrecognised) aspect.
Thus the obsession with a paradigm that is geared to mere quantitative interpretation has all but blotted out recognition of its corresponding qualitative aspect.
And this problem cannot be solved through ever more sophisticated attempts to adapt scientific interpretation of reality to the conventional linear rational approach!
Since my own "conversion" over 40 years ago, I have consistently proposed that a more comprehensive Mathematics entails at least three aspects.
The first is what I now term Type 1 Mathematics which relates to the standard conventional approach based on mere quantitative interpretation. This does not mean of course that qualitative considerations can play no part, but rather that they inevitably become reduced, due to the limited nature of an approach adopted, to quantitative interpretation!
The second is what I now term Type 2 Mathematics (which formerly I referred to as Holistic Mathematics). This is directly based on training in ever more refined states of intuition (that often are associated with the spiritual contemplative vision). In more formal terms, it relates to ever more intricate circular logical systems which have a precise mathematical basis (in holistic terms). Putting it in a Type 2 manner, every number for example has a direct qualitative significance as representative of a dimensional meaning.
So In Type 2 terms, conventional mathematical interpretation is based on a linear (1-dimensional) approach. However potentially an unlimited number of possible other interpretations exist, according to numbers qualitatively used as alternative dimensions.
When seen from this perspective, what is currently known as Mathematics simply refers to 1-dimensional interpretation. However this leaves a potentially unlimited set of alternative dimensions of interpretation (that have yet to be explored). The general nature of all these alternative interpretations is that each dimensional number relates to a unique means of configuring the relationship between reason and intuition (that now is understood in merely relative terms). Put another way, in Type 2 terms all mathematical symbols have both particle and wave aspects (with the wave aspect corresponding to a unique type of holistic appreciation that depends on the quality of intuition that inspires it).
My own mathematical journey has largely consisted of developing the implications of this type of appreciation which - even from my necessarily limited perspective - I can see will be truly breath-taking in scope.
The third aspect of Mathematics, which promises to be by far the most comprehensive, is what I now refer to as Type 3 (formerly Radial) Mathematics. It is designed to combine both Type 1 and Type Mathematics in ways that can be both extraordinarily productive and creative.
Properly understood all Mathematics is implicitly of a Type 3 nature. However - as I have stated - because one aspect (i.e. the holistic qualitative) has been entirely blotted out from formal interpretation, in effect it is reduced to Type 1.
The insights offered here are highly unlikely to arise from within the conventional mathematical community as existing practitioners are still strongly wedded to conventional interpretation. So I offer them as an outsider who from an early stage could see deep cracks in the mathematical edifice.
So coming back to Riemann and the famous legacy of The Riemann Hypothesis, it can now be restated in an especially cogent form as relating to the central condition necessary for the reconciliation of both Type 1 and Type 2 Mathematics.
When seen in this light, not alone can it not be proved (or disproved) in Type 1 terms, it cannot even be properly appreciated in this manner.
In other words there is a quantitative and qualitative meaning that can be given to every mathematical symbol which cannot be successfully reduced in terms of each other.
So the wonderful mystery then arises as to how - despite these distinctive meanings - a remarkable overall coherence is preserved. And this coherence lies deep in the nature of prime numbers themselves that preserve a unique quantitative identity yet display in collective terms remarkable holistic qualitative characteristics.
It is as if prime numbers are able to communicate with each other so effectively, that while contributing distinct individual notes, no discordance arises with respect to their collective symphony. Indeed this is precisely what is happening as - properly understood - in dynamic interactive terms, the primes are living entities (that are necessarily mediated though phenomena of nature). And once again, here we have the connection with subatomic particles which can communicate with each other (even at a great distance).
And this mystery cannot be interpreted in mere quantitative terms! Rather it relates to a prior mystery that ultimately governs the entire pattern of evolution (in both a quantitative and qualitative manner).
So properly understood, the implications of the Riemann Hypothesis are already contained in the mathematical axioms we use (in Type 1 Mathematics). Its acceptance thereby represents a massive act of faith in the subsequent consistency of the whole mathematical enterprise.
And till the beginning of the 19th century, this situation remained. Then largely due to Gauss this situation changed when he discovered a striking collective pattern to the primes based on the natural log function.
So from an individual perspective the primes numbers formerly seemed the most independent of numbers (exhibiting no obvious pattern), Yet from the new general perspective it could now be seen that they displayed an amazing regularity.
It was the genius of Riemann to try and connect these two aspects. He made amazing strides in analysing the complex nature of the primes so that he was able to suggest a formula that ultimately could eliminate deviations arising from the general estimate of prime number frequency.
He made the wonderful discovery that associated with the primes are a - potentially - unlimited number of solutions to the Riemann Zeta Function (defined in the complex plane). These solutions are commonly referred to as the non-trivial zeros (of the Zeta Function) or more simply Riemann's zeros. And associated with these zeros are elaborate wave forms which can - magically as it were - be used to correct any remaining deviations arising from his general estimate of prime number frequency.
So underlying the prime number system, seemingly is composed of discrete individual members, is a subtle wave form pattern. And it is in the relationship as between the two aspects that the mystery of the prime numbers resides.
Thus from one perspective we can use the waveforms associated with the non-trivial zeros in a systematic pattern to eliminate deviations with respect to the general estimate of prime number frequency (up to a given natural number).
In corresponding reverse manner we can also use knowledge of the individual prime numbers to eliminate any remaining deviations in the general estimate of the frequency of the non-trivial zeros (up to a given number on the imaginary number scale).
So this displays admirably the two-way interdependence as between the individual primes on the one hand and the corresponding wave patterns associated with these primes.
Then coming from either direction, mediated through the natural numbers, is a means of moving from individual primes to their collective wave patterns on the one hand and likewise in reverse manner from these wave patterns to the individual primes.
Right away this new newly discovered nature of the primes (by Riemann) suggests a direct correspondence with quantum physics.
Again it is truly remarkable in this regard that Riemann can be directly linked with the two great developments in physics of the 20th century. His developments in Geometry (Riemannian Geometry) was to prove invaluable to Einstein in formulating his Theory of General Relativity. However equally - though not yet sufficiently recognised - his work on the prime numbers provides a direct mathematical link with Quantum Mechanics.
As is now commonly known at the sub-atomic level of matter, particles possess a dual complementary existence exhibiting - depending on the context of observation - both wave and particle effects. Indeed even at the everyday macro level, objects exhibit both particle and wave aspects (though in practical terms the wave aspect can be effectively ignored)!
We can now say exactly the same thing in relation to prime numbers. At a deep complex level of investigation, prime numbers likewise exhibit both particle and wave aspects. However once again at the macro level of "real" interpretation these wave aspects do not reveal themselves and are effectively ignored. So once again it requires the complex analytic techniques first used by Riemann to reveal - as it were - the subatomic structure of the primes.
It is revealing that the primes are often referred to as the "atoms" of the number system which then serve as the "building blocks" for the natural numbers. However though indeed valid from one partial perspective, this is unduly reductionist for the primes - as we now know - contain a remarkable wave pattern system. And because of the two-way interdependence as between the particle and wave aspects, we can no longer hope to understand primes as mere "building blocks".
So as with subatomic particles we now recognise that the particle aspect of the primes also contain a wave aspect and the wave likewise a particle aspect.
However now we get to the crucial point! Just as it is now understood that the behaviour of particles at the sub-atomic level is governed by the Uncertainty Principle, equally we can assert that the prime numbers - at a deeper level of investigation - are governed by a corresponding mathematical Uncertainty Principle.
In physics if we try to precisely fix the position of a particle we blot out recognition of its corresponding momentum. And in turn, if we then try to precisely measure momentum it tends to block out any precise knowledge of its position.
It is similar in relation to the primes. When we try to precisely track the individual location of the primes, it tends to block out corresponding knowledge of their general distribution. And mathematical history testifies to this point! For though the primes were known and studied for well over 2000 years, it was not until the 19th century that this customary stance with respect to prime number analysis was abandoned.
And in turn, when we then concentrate on the general distribution of the primes this tends to block out recognition of the precise location of each individual prime!
So let me put it bluntly so that the point cannot be easily ignored. Uncertainty - by its very nature - lies at the heart of the number system and by extension at the heart of all mathematical relationships!
Furthermore the Uncertainty Principle that is now accepted with respect to the physical world - though not unfortunately its many implications - has its deeper roots in the nature of the prime numbers!
In other words the Uncertainty Principle that governs the physical world is ultimated rooted in the uncertainty that lies at the heart of mathematics.
The wave forms associated with prime numbers were discovered by Riemann long before the breakthroughs in Quantum Physics. However just as Einstein's General Theory of Relativity has its mathematical roots in Riemannian Geometry, I am confident it will be accepted in the future that Quantum Mechanics - especially in relation to the Uncertainty Principle - has its roots in Riemann's amazing discoveries with respect to the complex Zeta Function.
Then, even deeper implications of what I am saying here, have not yet been addressed by either the physics or mathematical community.
Indeed it is again ironic that the Uncertainty Principle itself can be effectively used in a related manner to highlight the key dilemma!
In truth all mathematical and scientific understanding entails two aspects of understanding which are interrelated in experience.
One the one hand we have reason - pertaining directly to the conscious - which in the current mathematical and scientific means simply linear reason. Such reason properly relates to differentiated interpretation with respect to specific actual phenomena (that are necessarily of a finite nature).
However equally we have intuition, pertaining directly to the unconscious. Though the importance of such intuition might well be informally recognised especially for creative work, in formal terms it simply is reduced through the conventional paradigm to reason. And in contrast to reason, intuition relates to integral holistic interpretation with respect to general phenomena of a universal potential nature (that are necessarily infinite in origin).
So a great price that has been paid for the admittedly enormous advances using the present paradigm. Thus we have learnt to automatically reduce - in any context - what is truly qualitative and distinct to mere quantitative interpretation.
Whereas reason tends to be linear, intuition is inherently of a circular nature (when indirectly interpreted in rational fashion). In other words linear reason is based on the clear unambiguous separation of opposite polarities (such as external and internal) whereas intuition is based on their complementary - and apparently paradoxical - identity.
So the key implication for mathematics and science is that a more comprehensive paradigm requires the incorporation of both linear and circular modes of logical interpretation. And remember these circular modes serve as the indirect expression of ever more refined forms of reason! Put another way, in qualitative terms we need a paradigm that contains both real and imaginary aspects!
We have seen that an attempt to precisely fix the position of a particle blocks out recognition of its complementary (unrecognised) aspect.
Thus the obsession with a paradigm that is geared to mere quantitative interpretation has all but blotted out recognition of its corresponding qualitative aspect.
And this problem cannot be solved through ever more sophisticated attempts to adapt scientific interpretation of reality to the conventional linear rational approach!
Since my own "conversion" over 40 years ago, I have consistently proposed that a more comprehensive Mathematics entails at least three aspects.
The first is what I now term Type 1 Mathematics which relates to the standard conventional approach based on mere quantitative interpretation. This does not mean of course that qualitative considerations can play no part, but rather that they inevitably become reduced, due to the limited nature of an approach adopted, to quantitative interpretation!
The second is what I now term Type 2 Mathematics (which formerly I referred to as Holistic Mathematics). This is directly based on training in ever more refined states of intuition (that often are associated with the spiritual contemplative vision). In more formal terms, it relates to ever more intricate circular logical systems which have a precise mathematical basis (in holistic terms). Putting it in a Type 2 manner, every number for example has a direct qualitative significance as representative of a dimensional meaning.
So In Type 2 terms, conventional mathematical interpretation is based on a linear (1-dimensional) approach. However potentially an unlimited number of possible other interpretations exist, according to numbers qualitatively used as alternative dimensions.
When seen from this perspective, what is currently known as Mathematics simply refers to 1-dimensional interpretation. However this leaves a potentially unlimited set of alternative dimensions of interpretation (that have yet to be explored). The general nature of all these alternative interpretations is that each dimensional number relates to a unique means of configuring the relationship between reason and intuition (that now is understood in merely relative terms). Put another way, in Type 2 terms all mathematical symbols have both particle and wave aspects (with the wave aspect corresponding to a unique type of holistic appreciation that depends on the quality of intuition that inspires it).
My own mathematical journey has largely consisted of developing the implications of this type of appreciation which - even from my necessarily limited perspective - I can see will be truly breath-taking in scope.
The third aspect of Mathematics, which promises to be by far the most comprehensive, is what I now refer to as Type 3 (formerly Radial) Mathematics. It is designed to combine both Type 1 and Type Mathematics in ways that can be both extraordinarily productive and creative.
Properly understood all Mathematics is implicitly of a Type 3 nature. However - as I have stated - because one aspect (i.e. the holistic qualitative) has been entirely blotted out from formal interpretation, in effect it is reduced to Type 1.
The insights offered here are highly unlikely to arise from within the conventional mathematical community as existing practitioners are still strongly wedded to conventional interpretation. So I offer them as an outsider who from an early stage could see deep cracks in the mathematical edifice.
So coming back to Riemann and the famous legacy of The Riemann Hypothesis, it can now be restated in an especially cogent form as relating to the central condition necessary for the reconciliation of both Type 1 and Type 2 Mathematics.
When seen in this light, not alone can it not be proved (or disproved) in Type 1 terms, it cannot even be properly appreciated in this manner.
In other words there is a quantitative and qualitative meaning that can be given to every mathematical symbol which cannot be successfully reduced in terms of each other.
So the wonderful mystery then arises as to how - despite these distinctive meanings - a remarkable overall coherence is preserved. And this coherence lies deep in the nature of prime numbers themselves that preserve a unique quantitative identity yet display in collective terms remarkable holistic qualitative characteristics.
It is as if prime numbers are able to communicate with each other so effectively, that while contributing distinct individual notes, no discordance arises with respect to their collective symphony. Indeed this is precisely what is happening as - properly understood - in dynamic interactive terms, the primes are living entities (that are necessarily mediated though phenomena of nature). And once again, here we have the connection with subatomic particles which can communicate with each other (even at a great distance).
And this mystery cannot be interpreted in mere quantitative terms! Rather it relates to a prior mystery that ultimately governs the entire pattern of evolution (in both a quantitative and qualitative manner).
So properly understood, the implications of the Riemann Hypothesis are already contained in the mathematical axioms we use (in Type 1 Mathematics). Its acceptance thereby represents a massive act of faith in the subsequent consistency of the whole mathematical enterprise.
Wednesday, February 1, 2012
Approximation Formulae for Zeta Function (Negative Odd Integers)
These are some approximation formulae that I devised for my own use.
Let s = negative odd integer
Now to simplify the following expression a little I write k = - (s + 1)/2
So for example when s = -3, k = - (- 3 + 1)/2 = 1.
ζ(s-2)/ζ(s) → .1 + 11k/{[2(pi^2)]} + k(k–3)/(pi^2)
Thus if we already know the value for ζ(s), we can use it to approximate the corresponding value for ζ(s-2).
We can also give an approximating formula for difference of ratios
ζ(s-4)/ζ(s-2) - ζ(s-2)/ζ(s) → 11/{2[(pi^2)]} + 2(k-1)/(pi^2)
Also we can provide an even simpler formula for approximating the difference of difference as between successive ratios!
{ζ(s-6)/ζ(s-4) – ζ(s-4)/ζ(s-2)} – {ζ(s-4)/ζ(s-2) – ζ(s-2)/ζ(s)} → 2/(pi^2).
Here is one other formula that involves a link with even values.
So if we take s now as an even integer, then
│{ζ(1-s)}^(1/s)/{ζ1-s-k)^[1/(s-k)]}│ → s/(s-k)
Let s = negative odd integer
Now to simplify the following expression a little I write k = - (s + 1)/2
So for example when s = -3, k = - (- 3 + 1)/2 = 1.
ζ(s-2)/ζ(s) → .1 + 11k/{[2(pi^2)]} + k(k–3)/(pi^2)
Thus if we already know the value for ζ(s), we can use it to approximate the corresponding value for ζ(s-2).
We can also give an approximating formula for difference of ratios
ζ(s-4)/ζ(s-2) - ζ(s-2)/ζ(s) → 11/{2[(pi^2)]} + 2(k-1)/(pi^2)
Also we can provide an even simpler formula for approximating the difference of difference as between successive ratios!
{ζ(s-6)/ζ(s-4) – ζ(s-4)/ζ(s-2)} – {ζ(s-4)/ζ(s-2) – ζ(s-2)/ζ(s)} → 2/(pi^2).
Here is one other formula that involves a link with even values.
So if we take s now as an even integer, then
│{ζ(1-s)}^(1/s)/{ζ1-s-k)^[1/(s-k)]}│ → s/(s-k)
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