One of the fundamental implications which the Riemann Hypothesis has for Mathematics relates to the nature of proof.
As I have suggested, when the true nature of such proof is properly recognised i.e. in dynamic experiential terms, then it is necessarily subject to the Uncertainty Principle.
However the full implications of this are very revealing for what this actually implies that every mathematical proof has in fact two distinct aspects (that are both quantitative and qualitative in relation to each other). So the Riemann Hypothesis establishes the axiomatic condition for consistency as between these two aspects. Thus the immediate corollary that follows is that every proposition with an (established) quantitative meaning equally can be given a coherent qualitative interpretation. This also implies that every theorem with an established proof in conventional terms equally can be given a coherent qualitative proof in holistic mathematical terms.
Indeed from a comprehensive mathematical perspective this would then entail that a theorem has not been properly proved until both aspects of proof have been fully established.
And this immediately points to the nature of the Uncertainty Principle that is involved.
So continual focus on just one aspect of proof (quantitative) thereby blots out recognition of the equally important alternative aspect (qualitative). This in fact explains the current nature of mathematics whereby total focus on the mere quantitative aspect of proof has completely blotted out recognition of the equally important alternative aspect.
Using an analogy from quantum physics it is as if we have concentrated so much on the mere particle existence of light, that we have no conception that light equally has a valid existence as a waveform!
The implications of what is being stated here are extremely far reaching in scope for once again it implies that strictly speaking the notion that Mathematics can have a mere abstract meaning (with no direct relevance to reality) is quite untenable.
So the very reason why this so often appears the case is precisely because of the lack of any developed - or even undeveloped - qualitative aspect to current mathematical thinking.
However when the qualitative aspect is properly integrated, then every mathematical proposition has a potential relevance to physical reality (in both quantitative and qualitative terms). And because physical and psychological aspects are complementary, this implies that every mathematical proposition has an equal potential relevance for psychological reality also!
Now we are still - to coin a phrase - a million light years from having any true conception of this comprehensive nature of Mathematics. However I can say with confidence, that as this is in fact the case, it will eventually be recognised.
It has to be stated however that the two aspects of proof (that I have outlined) are of a different nature (requiring uniquely distinct types of understanding). The quantitative aspect conforms to linear logic in the sequential establishing of unambiguous type rational connections between variables; the qualitative aspect by contrast conforms to circular logic in the simultaneous holistic recognition of complementary intuitive type connections (that are indirectly given a rational form).
One fascinating possibility is that a certain common pattern necessarily applies to both aspects. In other words with the appropriate type of proof for a proposition, one would be able to suggest the necessary structure of the qualitative aspect (from corresponding knowledge of the quantitative). Alternatively, one would be able to equally suggest the necessary structure of the quantitative aspect of proof (from corresponding knowledge of the qualitative).
So for example an established quantitative proof that did not lend itself readily to its qualitative partner would be deemed in some sense inefficient with a better proof still to be established.
In my own work I have given some - though necessarily limited - consideration to this issue. For example one of earlier "successes" was to resolve - what I call - the Pythagorean Dilemma, by providing the corresponding qualitative aspect of proof as to why the square root of 2 is irrational!
Now the Pythagoreans would have already established a quantitative proof as to why this root is irrational. However they implicitly recognised that it also required a qualitative aspect of proof (which they could not provide).
So a future stage of my own investigations will now relate to searching for this common pattern as between the two aspects of proof.
In other words the quantitative aspect really shows how the square root of 2 is irrational; the qualitative aspect then explains the deeper philosophical reason of why this root is irrational.
However there has to be a common structure to both aspects of an appropriate proof so that same symbols can be equally read in accordance with both linear establishing the quantitative and circular logic establishing the quantitative aspect of proof respectively.
Once again this is implied by the Riemann Hypothesis.
Friday, December 17, 2010
Thursday, December 9, 2010
Addendum
Yesterday I had a vivid insight into the nature of the Riemann Hypothesis showing me clearly why from the conventional mathematical perspective it can seem as if about to yield up its secrets while always remaining tantalisingly out of reach.
And from the redefinition of the Riemann Hypothesis that I have suggested as an intimate relationship as between quantitative and qualitative interpretation this is exactly what one would expect.
The very difficulty that the Riemann Hypothesis raises, points directly to a central unresolved problem with the nature of mathematical proof.
Once again the proof of a general proposition (such as the Pythagorean Theorem) strictly is of a qualitative nature that potentially applies to an infinite (unspecified) number of cases; however the quantitative application of such a proof is of a different nature applying in actual terms to a finite number of cases (that can be specified).
Now because of the reduced rational bias of conventional mathematical interpretation the qualitative distinct nature of the general proof is thereby reduced to mere quantitative interpretation leading to the characteristic - unjustifiable - absolute nature of conventional truth.
However the Riemann Hypothesis is altogether more subtle and points to the necessary condition for proper reconciliation of both infinite (general) and finite (specific) notions.
As we have seen this condition (on which subsequent conventional mathematical appreciation properly depends) both predates and postdates as it were all phenomenal (quantitative) manifestations and corresponding (qualitative) interpretations of such reality. So the Riemann Hypothesis - which establishes this mysterious fundamental correspondence as between quantitative and qualitative reality - is already implicit in the very axioms that are used in Conventional Mathematics while ultimately transcending any (phenomenal) attempt to understand its very nature.
So quite clearly - once we appreciate its true nature - the Riemann Hypothesis cannot be proven (or disproven) from within conventional mathematical axioms!
And we can see how this problem of attempted proof is manifesting itself. From one perspective at the general level theorists have seemingly been closing on the ultimate target of proof e.g. by demonstrating that an infinite no. of non-trivial zeros exist on the critical line (with real part .5), and also by slowly showing that a higher and higher percentage of possible zeros must lie on this line. However even if 99.9999...% of possible zeros could be demonstrated to lie on the critical line this would not constitute an acceptable proof!
Meanwhile from the quantitative empirical perspective all valid zeros (now exceeding countless billions) have been found to exist on the critical line. However once again now matter how much further we go in this direction (with no exceptions showing) this will never establish a proof of the Riemann Hypothesis.
And this is the very point as the Riemann Hypothesis indicates clearly that there is is fact no phenomenal identity as between the qualitative area of general proof and the corresponding quantitative area of specific examples!
So the very notion of mathematical proof - though still immensely valuable - needs to be redefined dynamically in the light of the implications of the Riemann Hypothesis, whereby it is understood to be of a merely relative nature and necessarily subject to uncertainty.
And from the redefinition of the Riemann Hypothesis that I have suggested as an intimate relationship as between quantitative and qualitative interpretation this is exactly what one would expect.
The very difficulty that the Riemann Hypothesis raises, points directly to a central unresolved problem with the nature of mathematical proof.
Once again the proof of a general proposition (such as the Pythagorean Theorem) strictly is of a qualitative nature that potentially applies to an infinite (unspecified) number of cases; however the quantitative application of such a proof is of a different nature applying in actual terms to a finite number of cases (that can be specified).
Now because of the reduced rational bias of conventional mathematical interpretation the qualitative distinct nature of the general proof is thereby reduced to mere quantitative interpretation leading to the characteristic - unjustifiable - absolute nature of conventional truth.
However the Riemann Hypothesis is altogether more subtle and points to the necessary condition for proper reconciliation of both infinite (general) and finite (specific) notions.
As we have seen this condition (on which subsequent conventional mathematical appreciation properly depends) both predates and postdates as it were all phenomenal (quantitative) manifestations and corresponding (qualitative) interpretations of such reality. So the Riemann Hypothesis - which establishes this mysterious fundamental correspondence as between quantitative and qualitative reality - is already implicit in the very axioms that are used in Conventional Mathematics while ultimately transcending any (phenomenal) attempt to understand its very nature.
So quite clearly - once we appreciate its true nature - the Riemann Hypothesis cannot be proven (or disproven) from within conventional mathematical axioms!
And we can see how this problem of attempted proof is manifesting itself. From one perspective at the general level theorists have seemingly been closing on the ultimate target of proof e.g. by demonstrating that an infinite no. of non-trivial zeros exist on the critical line (with real part .5), and also by slowly showing that a higher and higher percentage of possible zeros must lie on this line. However even if 99.9999...% of possible zeros could be demonstrated to lie on the critical line this would not constitute an acceptable proof!
Meanwhile from the quantitative empirical perspective all valid zeros (now exceeding countless billions) have been found to exist on the critical line. However once again now matter how much further we go in this direction (with no exceptions showing) this will never establish a proof of the Riemann Hypothesis.
And this is the very point as the Riemann Hypothesis indicates clearly that there is is fact no phenomenal identity as between the qualitative area of general proof and the corresponding quantitative area of specific examples!
So the very notion of mathematical proof - though still immensely valuable - needs to be redefined dynamically in the light of the implications of the Riemann Hypothesis, whereby it is understood to be of a merely relative nature and necessarily subject to uncertainty.
Sunday, December 5, 2010
True Significance of Riemann Hypothesis (5)
The Riemann transformation formula establishes an important link as between values of the zeta function for s > o on the RHS and corresponding values for 1 - s on the LHS of the equation. Now crucially from the conventional linear perspective, the zeta function will only converge for finite values of s > 1.
Therefore no adequate explanation can be given in linear terms for values of the zeta function where s < 1.
So the Riemann Zeta Function can be neatly subdivided for all real values of s as follows:
(i) as representing the standard reduced linear interpretation (directly in accordance with the dimension 1) for values of s > 1.
ii) as representing the alternative circular interpretation (directly in accordance with the dimensional number in question) in corresponding qualitative terms for s < 0.
Interestingly then the values in the critical range for 0 < s < 1 represent a hybrid mix of both quantitative and qualitative aspects. The condition therefore for equality with respect to both sides (with reference to the non-trivial zeros where both sides of the equation = o) is that s = 1 - s = .5.
So this condition representing the famous Riemann Hypothesis where for all non-trivial zeros, the value of the real part = .5, is required so as to obtain an exact correspondence as between both quantitative and qualitative type interpretation.
We can say therefore that when this condition is realised that there is no longer any gap as between the quantitative nature of prime number reality and our corresponding qualitative interpretations of such reality. This would equally imply that the quantitative nature of individual prime numbers can then be perfectly reconciled with the overall qualitative holistic nature of their distribution (among the natural numbers).
And once again because the Riemann Hypothesis - when correctly interpreted - points to this essential requirement for reconciliation of both quantitative and qualitative aspects with respect to prime number behaviour (and thereby by extension all number behaviour), it cannot be proved (or disproved) with respect to just one aspect i.e. the axioms pertaining to the conventional quantitative linear approach.
What the Riemann Hypothesis is directly implying in fact is that there are two equally important aspects to mathematical understanding i.e. quantitative and qualitative. However in the present mathematical approach (which has dominated understanding now for several milennia) only one of these aspects is formally recognised i.e. the quantitative.
There is I believe however a very simple way of expressing the relevance of the Riemann Hypothesis.
If we draw a circle and insert its line diameter, the point at the centre of the circle is equally the point at the centre of the line diameter. So in this sense both the line and the circle are identical at this mid-way point. So if we identify the line diameter as 1, the midpoint occurs at .5.
In similar qualitative terms the reconciliation of both linear (quantitative) and circular (qualitative) interpretation occurs at the same point. So the midpoint in this context represents the situation where opposite polarities of experience are perfectly balanced.
Thus maintainence of the most refined interaction possible as between (linear) reason and (circular) intuition, requires that these opposite polarities of understanding (which necessarily underline all phenomenal understanding of reality) be kept in perfect balance. In this way, one can temporarily separate and differentiate opposites (e.g. external objective phenomena and internal mental constructs) with respect to either pole in a refined rational manner while immediately seeing from an integral holistic perspective that both poles are complementary (and ultimately identical). In this way quantitative rational interpretation and qualitative intuitive appreciation can dynamically approximate the situation where they can then serve each other perfectly. However while approaching such an approximation a continual correction mechanism is required whereby unconscious i.e. imaginary projections (representing temporary imbalance as between the discrete nature of reason and continuous nature of intuition) are constantly emitted.
And the key to rapid adjustment here, is that like virtual particles in physics, these temporary imaginary projections should occur in pairs (whereby they can quickly cancel each other out).
This equally implies to the very nature of prime numbers, entailing an identical similar process where both the individual quantitative nature of each prime number can be kept in perfect balance with the qualitative holistic nature of the distribution of primes (among the natural numbers). And once again temporary imbalances as between the discrete nature of individual primes and the continuous nature of their general distribution are represented through appropriate imaginary dimensional numbers added to the real part of s (that occur in pairs). So prime number behaviour - properly understood - represents the interaction of two logical processes (linear and circular) that are kept perfectly in balance through the constant adjustments brought about through these imaginary dimensional number pairs.
So once again the quantitative nature of prime numbers cannot be ultimately distinguished from the corresponding qualitative nature (by which they are interpreted).
And this is the key message of the Riemann Hypothesis!
Therefore no adequate explanation can be given in linear terms for values of the zeta function where s < 1.
So the Riemann Zeta Function can be neatly subdivided for all real values of s as follows:
(i) as representing the standard reduced linear interpretation (directly in accordance with the dimension 1) for values of s > 1.
ii) as representing the alternative circular interpretation (directly in accordance with the dimensional number in question) in corresponding qualitative terms for s < 0.
Interestingly then the values in the critical range for 0 < s < 1 represent a hybrid mix of both quantitative and qualitative aspects. The condition therefore for equality with respect to both sides (with reference to the non-trivial zeros where both sides of the equation = o) is that s = 1 - s = .5.
So this condition representing the famous Riemann Hypothesis where for all non-trivial zeros, the value of the real part = .5, is required so as to obtain an exact correspondence as between both quantitative and qualitative type interpretation.
We can say therefore that when this condition is realised that there is no longer any gap as between the quantitative nature of prime number reality and our corresponding qualitative interpretations of such reality. This would equally imply that the quantitative nature of individual prime numbers can then be perfectly reconciled with the overall qualitative holistic nature of their distribution (among the natural numbers).
And once again because the Riemann Hypothesis - when correctly interpreted - points to this essential requirement for reconciliation of both quantitative and qualitative aspects with respect to prime number behaviour (and thereby by extension all number behaviour), it cannot be proved (or disproved) with respect to just one aspect i.e. the axioms pertaining to the conventional quantitative linear approach.
What the Riemann Hypothesis is directly implying in fact is that there are two equally important aspects to mathematical understanding i.e. quantitative and qualitative. However in the present mathematical approach (which has dominated understanding now for several milennia) only one of these aspects is formally recognised i.e. the quantitative.
There is I believe however a very simple way of expressing the relevance of the Riemann Hypothesis.
If we draw a circle and insert its line diameter, the point at the centre of the circle is equally the point at the centre of the line diameter. So in this sense both the line and the circle are identical at this mid-way point. So if we identify the line diameter as 1, the midpoint occurs at .5.
In similar qualitative terms the reconciliation of both linear (quantitative) and circular (qualitative) interpretation occurs at the same point. So the midpoint in this context represents the situation where opposite polarities of experience are perfectly balanced.
Thus maintainence of the most refined interaction possible as between (linear) reason and (circular) intuition, requires that these opposite polarities of understanding (which necessarily underline all phenomenal understanding of reality) be kept in perfect balance. In this way, one can temporarily separate and differentiate opposites (e.g. external objective phenomena and internal mental constructs) with respect to either pole in a refined rational manner while immediately seeing from an integral holistic perspective that both poles are complementary (and ultimately identical). In this way quantitative rational interpretation and qualitative intuitive appreciation can dynamically approximate the situation where they can then serve each other perfectly. However while approaching such an approximation a continual correction mechanism is required whereby unconscious i.e. imaginary projections (representing temporary imbalance as between the discrete nature of reason and continuous nature of intuition) are constantly emitted.
And the key to rapid adjustment here, is that like virtual particles in physics, these temporary imaginary projections should occur in pairs (whereby they can quickly cancel each other out).
This equally implies to the very nature of prime numbers, entailing an identical similar process where both the individual quantitative nature of each prime number can be kept in perfect balance with the qualitative holistic nature of the distribution of primes (among the natural numbers). And once again temporary imbalances as between the discrete nature of individual primes and the continuous nature of their general distribution are represented through appropriate imaginary dimensional numbers added to the real part of s (that occur in pairs). So prime number behaviour - properly understood - represents the interaction of two logical processes (linear and circular) that are kept perfectly in balance through the constant adjustments brought about through these imaginary dimensional number pairs.
So once again the quantitative nature of prime numbers cannot be ultimately distinguished from the corresponding qualitative nature (by which they are interpreted).
And this is the key message of the Riemann Hypothesis!
Saturday, December 4, 2010
True Significance of Riemann Hypothesis (4)
I have been discussing the true significance of the Riemann Hypothesis as establishing an intimate correspondence as between the quantitative and qualitative interpretation of mathematical symbols.
The question then arises as to why this should be especially relevant in the context of prime numbers!
Once again we identified the quantitative (analytic) aspect of interpretation with the linear use of logic (pertaining directly to reason); then we defined the qualitative (holistic) aspect of interpretation indirectly with the circular use of logic (pertaining directly to intuition).
Now prime numbers are especially relevant in this context as they combine extremes with reference to both systems. So once we can establish a correspondence as between quantitative and qualitative interpretation in such circumstances (with respect to prime numbers) we can then easily extend this correspondence to all other numbers.
The very definition of a prime number is that it has no factors (other than itself and 1). In this way prime numbers are the most linear (and independent) of numbers. Not surprisingly from this perspective, prime numbers are thereby seen as the basic building blocks for the entire natural number system.
However what is not properly realised is that prime numbers, when used as a qualitative means of interpretation, are also the most uniquely circular of all numbers (with a structural configuration that cannot be derived from other combinations).
So if we are to establish this unique correspondence as between the quantitative (analytic) and (holistic) qualitative use of mathematical symbols, we must first establish its application with respect to the prime numbers.
As I have stated the linear quantitative interpretation of prime numbers is well established in Conventional Mathematics, where once again they are viewed as the atoms or building blocks of the natural numbers.
However the circular holistic aspect relates to their opposite characteristics (en bloc) as being intimately dependent on the natural numbers (for their precise location).
Therefore though no clear pattern is evident with respect to the individual sequence of prime numbers, an amazing regularity of behaviour characterises their general distribution with respect to the natural numbers.
So there are two opposite tendencies at work (in extreme fashion) with respect to (linear) independence of individual prime numbers and (circular) interdependence with respect to the collective behaviour of primes.
Once again though Conventional Mathematics investigates both of these aspects in considerable detail, because of its linear bias it can only do so within a quantitative approach to interpretation.
However as the very key to appreciation of primes entails maintaining correspondence as between both quantitative and qualitative aspects (pertaining to two distinct logical systems) once again their true nature is overlooked.
We can actually learn a great deal about what is involved here by looking at the dynamic nature of prime (i.e. primitive) instincts with respect to human behaviour.
In earliest infant behaviour both conscious and unconscious remain strongly embedded with each other. Indeed one can say that human life begins from the point where they are totally confused with each other. So using psychological language, neither conscious (linear) nor unconscious (circular) activity can yet be distinguished. So here we have the perfect correspondence (in undifferentiated confusion) of both the quantitative and qualitative aspects of prime behaviour.
The very essence of primitive instincts is that holistic meaning (qualitatively pertaining to the unconscious) is identified with specific phenomena (quantitatively pertaining to conscious understanding). So in this sense primitive behaviour represents the confused correspondence of quantitative and qualitative meaning.
Now from a human development problem the ultimate solution to such behaviour requires the differentiation of conscious from unconscious paving the way for integral union in spiritual terms. Thus in this mature state we have the perfect correspondence of quantitative and qualitative meaning (equally entailing the perfect correspondence of reason and intuition) .
Thus the mystery of the primes relates to an initial state (where both the quantitative and qualitative aspects of understanding exist in perfect correspondence as mere immanent potential for existence combined with their pure actualisation in existence as realized transcendent experience of this correspondence.
This means that the secret governing the behaviour of the prime numbers (in time and space) is already encoded as a perfect correspondence as between two logical systems (prior to their experiential manifestation). However equally the full realisation of this secret entails the spiritual transcendence of all lesser phenomenal understanding in space and time (where this perfect correspondence is broken).
So Hilbert was right! When properly understood Riemann's Hypothesis is pointing to a truth that is not only central to the nature of mathematics but to life itself.
In the phenomenal realm of experience, we can never fully reconcile the quantitative with the qualitative, the discrete with the continuous, order with chaos, reason with intuition etc.
It is only in pure spiritual realisation of our destiny that we can approximate a state where these opposites can at last be truly bridged.
Understanding the nature of prime numbers entails exactly the same issues. Both in the beginning and in the end a perfect correspondence exists as between the quantitative nature of prime numbers and their qualitative nature (which is inseparable from psychological development). So resolving the dynamic issues in relation to the former is inseparable from resolving the development issues in relation to the latter. So only at last when the prime problem in qualitative terms is resolved through conscious (quantitative) and unconscious (qualitative) aspects of interpretation now operating as one, can the discrete and continuous aspects with respect to the quantitative behaviour of the primes be also fully merged (transcending phenomenal experience).
In the final contribution in this series I will deal a little more with the precise significance of the Riemann Hypothesis (with respect all non-trivial zeros lying on the line with real part = 1/2)
The question then arises as to why this should be especially relevant in the context of prime numbers!
Once again we identified the quantitative (analytic) aspect of interpretation with the linear use of logic (pertaining directly to reason); then we defined the qualitative (holistic) aspect of interpretation indirectly with the circular use of logic (pertaining directly to intuition).
Now prime numbers are especially relevant in this context as they combine extremes with reference to both systems. So once we can establish a correspondence as between quantitative and qualitative interpretation in such circumstances (with respect to prime numbers) we can then easily extend this correspondence to all other numbers.
The very definition of a prime number is that it has no factors (other than itself and 1). In this way prime numbers are the most linear (and independent) of numbers. Not surprisingly from this perspective, prime numbers are thereby seen as the basic building blocks for the entire natural number system.
However what is not properly realised is that prime numbers, when used as a qualitative means of interpretation, are also the most uniquely circular of all numbers (with a structural configuration that cannot be derived from other combinations).
So if we are to establish this unique correspondence as between the quantitative (analytic) and (holistic) qualitative use of mathematical symbols, we must first establish its application with respect to the prime numbers.
As I have stated the linear quantitative interpretation of prime numbers is well established in Conventional Mathematics, where once again they are viewed as the atoms or building blocks of the natural numbers.
However the circular holistic aspect relates to their opposite characteristics (en bloc) as being intimately dependent on the natural numbers (for their precise location).
Therefore though no clear pattern is evident with respect to the individual sequence of prime numbers, an amazing regularity of behaviour characterises their general distribution with respect to the natural numbers.
So there are two opposite tendencies at work (in extreme fashion) with respect to (linear) independence of individual prime numbers and (circular) interdependence with respect to the collective behaviour of primes.
Once again though Conventional Mathematics investigates both of these aspects in considerable detail, because of its linear bias it can only do so within a quantitative approach to interpretation.
However as the very key to appreciation of primes entails maintaining correspondence as between both quantitative and qualitative aspects (pertaining to two distinct logical systems) once again their true nature is overlooked.
We can actually learn a great deal about what is involved here by looking at the dynamic nature of prime (i.e. primitive) instincts with respect to human behaviour.
In earliest infant behaviour both conscious and unconscious remain strongly embedded with each other. Indeed one can say that human life begins from the point where they are totally confused with each other. So using psychological language, neither conscious (linear) nor unconscious (circular) activity can yet be distinguished. So here we have the perfect correspondence (in undifferentiated confusion) of both the quantitative and qualitative aspects of prime behaviour.
The very essence of primitive instincts is that holistic meaning (qualitatively pertaining to the unconscious) is identified with specific phenomena (quantitatively pertaining to conscious understanding). So in this sense primitive behaviour represents the confused correspondence of quantitative and qualitative meaning.
Now from a human development problem the ultimate solution to such behaviour requires the differentiation of conscious from unconscious paving the way for integral union in spiritual terms. Thus in this mature state we have the perfect correspondence of quantitative and qualitative meaning (equally entailing the perfect correspondence of reason and intuition) .
Thus the mystery of the primes relates to an initial state (where both the quantitative and qualitative aspects of understanding exist in perfect correspondence as mere immanent potential for existence combined with their pure actualisation in existence as realized transcendent experience of this correspondence.
This means that the secret governing the behaviour of the prime numbers (in time and space) is already encoded as a perfect correspondence as between two logical systems (prior to their experiential manifestation). However equally the full realisation of this secret entails the spiritual transcendence of all lesser phenomenal understanding in space and time (where this perfect correspondence is broken).
So Hilbert was right! When properly understood Riemann's Hypothesis is pointing to a truth that is not only central to the nature of mathematics but to life itself.
In the phenomenal realm of experience, we can never fully reconcile the quantitative with the qualitative, the discrete with the continuous, order with chaos, reason with intuition etc.
It is only in pure spiritual realisation of our destiny that we can approximate a state where these opposites can at last be truly bridged.
Understanding the nature of prime numbers entails exactly the same issues. Both in the beginning and in the end a perfect correspondence exists as between the quantitative nature of prime numbers and their qualitative nature (which is inseparable from psychological development). So resolving the dynamic issues in relation to the former is inseparable from resolving the development issues in relation to the latter. So only at last when the prime problem in qualitative terms is resolved through conscious (quantitative) and unconscious (qualitative) aspects of interpretation now operating as one, can the discrete and continuous aspects with respect to the quantitative behaviour of the primes be also fully merged (transcending phenomenal experience).
In the final contribution in this series I will deal a little more with the precise significance of the Riemann Hypothesis (with respect all non-trivial zeros lying on the line with real part = 1/2)
Friday, December 3, 2010
True Significance of Riemann Hypothesis (3)
Though again not properly realised in conventional mathematical interpretation, the nature of the roots of a polynomial equation indirectly points to the true multidimensional nature of mathematical interpretation.
Indeed an enormous amount can be gleaned from consideration of the simplest polynomial equation of degree 2 i.e. where x^2 = 1.
It is the very nature of original mathematical discovery to see a fundamental problem with an explanation (which every one else accepts without apparent question).
As I have so often stated Conventional Mathematics adopts a linear logical approach that is literally 1-dimensional from a qualitative perspective. Now one of the key characteristics of this approach is that it unambiguous in nature.
So crucially for example if one proves for example that a theorem is true, then this rules out the possibility of any other alternative (especially the polar opposite case of being false).
Thus if we designate the outcome that a proposition is true as positive, then the opposite case of it being false is thereby negative. In this sense therefore a proposition cannot be given both a positive and negative truth value.
Now there is a direct correspondence here as between qualitative and quantitative for in the equation where the value of x is raised to 1, its answer is entirely unambiguous.
So, as we see for example in the simplest case where x = 1, the equation has only one correct value.
So this represents the one-dimensional case where x ^1 = 1.
Once again, here the qualitative interpretation of the result (as 1-dimensional and unambiguous) corresponds directly with the quantitative nature of the result that is likewise 1-dimensional and unambiguous).
However in the case where the dimensional number (power or exponent) is now 2, a fundamental problem arises.
We will demonstrate this first by looking carefully at the quantitative aspect. Now again in conventional mathematical interpretation, the value of x in this equation (representing the two roots of 1) can be given as either + 1 or - 1.
So what is remarkable is that two answers - which are polar opposites of each other - are deemed as a correct solution.
Now this state of affairs points to an inherent - unresolved - ambiguity with respect to mathematical interpretation.
So let us probe more deeply where in fact Conventional Mathematics falls short in this regard.
Now the key point that I am making is that the use of a number as dimension directly relates to a unique mode of qualitative interpretation of mathematical symbols.
However because the qualitative mode of Conventional Mathematics is linear (i.e. 1-dimensional) when we raise 1 to a dimension (other than 1) the resulting quantitative result is given in reduced terms as 1 (i.e. 1^1).
So therefore from this perspective (i.e. in reduced quantitative terms) 1^2 is indistinguishable from 1^1! So here the dimensional number 2 is given a reduced qualitative interpretation as 1.
So the key to unlocking the apparent ambiguity attached to the two roots of 1 is to recognise that 1^1 and 1^2 are actually distinct from each other (in qualitative terms).
Thus, from this newly defined perspective, there is only one unambiguous root of 1 (i.e. - 1) which when squared = 1^1. The other (supposed) root (+ 1) when squared gives the result 1^2.
So 1^1 (i.e. + 1) is not properly the square of 1 but rather (1^2).
However, having redefined this relationship in quantitative terms, we must now complete the more difficult task of redefining it correctly also in corresponding qualitative terms.
And it as this stage that Holistic Mathematics properly starts.
So once again in qualitative terms 1-dimensional interpretation corresponds with the unambiguous linear logical approach (based on either/or distinctions).
The key to the qualitative interpretation of a dimensional number is the recognition that it is structurally identical with its corresponding root.
And as the second root of 1 is, as we have established, - 1 this entails that 2-dimensional understanding literally entails the qualitative negation therefore of linear (rational) type interpretation.
The very word "unconscious" that we use in psychological terms implies the negation of "conscious". So the clue to the nature of 2-dimensional understanding is that it is of a direct unconscious nature arising from the dynamic negation of what is considered positive (and thereby true) at a rational linear level.
Such understanding is holistic and intuitive . However intuitive understanding can be indirectly expressed in a rational manner in circular both/and logical terms (as the complementarity of opposite polarities).
Therefore, the significance of recognition of both 1-dimensional and 2-dimensional interpretation in qualitative terms is that intuition must be formally included with reason in mathematical understanding.
Briefly all "higher" dimensional interpretations can be expressed structurally in terms of their corresponding quantitative roots.
So for example if we wished to understand the true nature of 4-dimensional interpretation, in qualitative terms, we look at the structural nature of the 4th root of 1) which is i.
Therefore in qualitative holistic mathematical terms this requires explaining the precise philosophical meaning of what is meant by "imaginary" interpretation.
I have spent most of my adult life elaborating exactly such issues. However it is sufficient to state here that what we refer to as "imaginary", in qualitative terms, relates to the rational means through which circular both/and logic can be indirectly represented in a rational linear manner.
So just as in quantitative terms the comprehensive number system is complex (with real and imaginary components), likewise a comprehensive interpretation of mathematical reality includes both real and imaginary aspects (relating to two different logical systems).
And as all other roots of 1 entail both real and imaginary parts in quantitative terms, corresponding multidimensional interpretation of these roots entails unique configurations of both real and imaginary interpretation (ultimately relating to a precise relationship of reason to intuition in understanding).
And the truly wonderful - yet totally mysterious - correspondence as between quantitative and qualitative interpretations, that enables all this meaning to unfold, once again is directly implied by the Riemann Hypothesis.
So 1^(1/x) in quantitative terms corresponds directly with 1^x in a qualitative structural manner. (Again this key relationship as between quantitative values and corresponding qualitative interpretation - to which the Riemann Hypothesis relates - is completely missed from a conventional mathematical perspective. For when x = 1, both sides are identical so that qualitative interpretation is thereby reduced in merely quantitative terms)!
However in remains to be shown why such a correspondence is especially relevant to prime numbers (which we will do in the next contribution).
Indeed an enormous amount can be gleaned from consideration of the simplest polynomial equation of degree 2 i.e. where x^2 = 1.
It is the very nature of original mathematical discovery to see a fundamental problem with an explanation (which every one else accepts without apparent question).
As I have so often stated Conventional Mathematics adopts a linear logical approach that is literally 1-dimensional from a qualitative perspective. Now one of the key characteristics of this approach is that it unambiguous in nature.
So crucially for example if one proves for example that a theorem is true, then this rules out the possibility of any other alternative (especially the polar opposite case of being false).
Thus if we designate the outcome that a proposition is true as positive, then the opposite case of it being false is thereby negative. In this sense therefore a proposition cannot be given both a positive and negative truth value.
Now there is a direct correspondence here as between qualitative and quantitative for in the equation where the value of x is raised to 1, its answer is entirely unambiguous.
So, as we see for example in the simplest case where x = 1, the equation has only one correct value.
So this represents the one-dimensional case where x ^1 = 1.
Once again, here the qualitative interpretation of the result (as 1-dimensional and unambiguous) corresponds directly with the quantitative nature of the result that is likewise 1-dimensional and unambiguous).
However in the case where the dimensional number (power or exponent) is now 2, a fundamental problem arises.
We will demonstrate this first by looking carefully at the quantitative aspect. Now again in conventional mathematical interpretation, the value of x in this equation (representing the two roots of 1) can be given as either + 1 or - 1.
So what is remarkable is that two answers - which are polar opposites of each other - are deemed as a correct solution.
Now this state of affairs points to an inherent - unresolved - ambiguity with respect to mathematical interpretation.
So let us probe more deeply where in fact Conventional Mathematics falls short in this regard.
Now the key point that I am making is that the use of a number as dimension directly relates to a unique mode of qualitative interpretation of mathematical symbols.
However because the qualitative mode of Conventional Mathematics is linear (i.e. 1-dimensional) when we raise 1 to a dimension (other than 1) the resulting quantitative result is given in reduced terms as 1 (i.e. 1^1).
So therefore from this perspective (i.e. in reduced quantitative terms) 1^2 is indistinguishable from 1^1! So here the dimensional number 2 is given a reduced qualitative interpretation as 1.
So the key to unlocking the apparent ambiguity attached to the two roots of 1 is to recognise that 1^1 and 1^2 are actually distinct from each other (in qualitative terms).
Thus, from this newly defined perspective, there is only one unambiguous root of 1 (i.e. - 1) which when squared = 1^1. The other (supposed) root (+ 1) when squared gives the result 1^2.
So 1^1 (i.e. + 1) is not properly the square of 1 but rather (1^2).
However, having redefined this relationship in quantitative terms, we must now complete the more difficult task of redefining it correctly also in corresponding qualitative terms.
And it as this stage that Holistic Mathematics properly starts.
So once again in qualitative terms 1-dimensional interpretation corresponds with the unambiguous linear logical approach (based on either/or distinctions).
The key to the qualitative interpretation of a dimensional number is the recognition that it is structurally identical with its corresponding root.
And as the second root of 1 is, as we have established, - 1 this entails that 2-dimensional understanding literally entails the qualitative negation therefore of linear (rational) type interpretation.
The very word "unconscious" that we use in psychological terms implies the negation of "conscious". So the clue to the nature of 2-dimensional understanding is that it is of a direct unconscious nature arising from the dynamic negation of what is considered positive (and thereby true) at a rational linear level.
Such understanding is holistic and intuitive . However intuitive understanding can be indirectly expressed in a rational manner in circular both/and logical terms (as the complementarity of opposite polarities).
Therefore, the significance of recognition of both 1-dimensional and 2-dimensional interpretation in qualitative terms is that intuition must be formally included with reason in mathematical understanding.
Briefly all "higher" dimensional interpretations can be expressed structurally in terms of their corresponding quantitative roots.
So for example if we wished to understand the true nature of 4-dimensional interpretation, in qualitative terms, we look at the structural nature of the 4th root of 1) which is i.
Therefore in qualitative holistic mathematical terms this requires explaining the precise philosophical meaning of what is meant by "imaginary" interpretation.
I have spent most of my adult life elaborating exactly such issues. However it is sufficient to state here that what we refer to as "imaginary", in qualitative terms, relates to the rational means through which circular both/and logic can be indirectly represented in a rational linear manner.
So just as in quantitative terms the comprehensive number system is complex (with real and imaginary components), likewise a comprehensive interpretation of mathematical reality includes both real and imaginary aspects (relating to two different logical systems).
And as all other roots of 1 entail both real and imaginary parts in quantitative terms, corresponding multidimensional interpretation of these roots entails unique configurations of both real and imaginary interpretation (ultimately relating to a precise relationship of reason to intuition in understanding).
And the truly wonderful - yet totally mysterious - correspondence as between quantitative and qualitative interpretations, that enables all this meaning to unfold, once again is directly implied by the Riemann Hypothesis.
So 1^(1/x) in quantitative terms corresponds directly with 1^x in a qualitative structural manner. (Again this key relationship as between quantitative values and corresponding qualitative interpretation - to which the Riemann Hypothesis relates - is completely missed from a conventional mathematical perspective. For when x = 1, both sides are identical so that qualitative interpretation is thereby reduced in merely quantitative terms)!
However in remains to be shown why such a correspondence is especially relevant to prime numbers (which we will do in the next contribution).
Thursday, December 2, 2010
True Significance of Riemann Hypothesis (2)
We have already explained the Riemann Hypothesis in terms of that intimate connection enabling consistency as between reality as viewed objectively in quantitative terms and the corresponding psychological constructs necessary for overall qualitative interpretation of its nature.
An even deeper appreciation of this relationship entails the incorporation of both (holistic) intuitive and (analytic) rational type interpretation.
In a direct sense, (holistic) intuition relates to the empty spiritual aspect of reality (that is infinite in potential terms); by contrast (analytic) reason relates to the formal material aspect (that is finite in an actual manner). So the dynamic relationship of reason and intuition in experience pertains to the central relationship of finite to infinite.
And in a crucially important sense, the Riemann Hypothesis again can be expressed as the fundamental requirement for consistency with respect to this relationship.
For example in strict terms - though this is overlooked with respect to conventional mathematical interpretation - a general proof of a theorem applies to an infinite no. of potential cases. However any specific examples, illustrating the general proof, relate to a finite no. of actual cases.
So the proof of the Pythagorean Theorem for example applies potentially to all possible right hand triangles (in infinite terms). However the applicability of the theorem necessarily relates to a finite no. of actual right hand triangles.
Thus inherent in the belief that we can apply a general (potential) proof to specific (actual) examples is an implicit acceptance of the fundamental consistency as between finite and infinite realms.
And this is exactly the same consistency that is implied by the Riemann Hypothesis!
Unfortunately, as Conventional Mathematics is formally based merely on rational interpretation, the significance of the Riemann Hypothesis will always remain out of reach (when approached from this perspective).
Looked at in yet another equivalent manner, the fundamental axiom - to which the Riemann Hypothesis relates - enables consistent dynamic switching as between the opposite polarities of experience (such as external and internal).
I will explain here the basic nature of this dynamic interaction before explaining its precise mathematical significance in the next contribution.
All experience - indeed all development processes - entail twin processes of differentiation and integration respectively (based on two distinct logical systems).
Differentiation essentially entails linear logic whereby opposite polarities are clearly separated.
Integration by contrast entails circular logic whereby these same opposites are viewed as complementary (and ultimately identical).
In human experience differentiation and integration relate to the conscious and unconscious respectively. Reason is directly identified with conscious understanding and best suited to detailed analytic understanding of specific aspects of reality (using linear logic); intuition is directly identified with unconscious understanding and correspondingly suited to holistic appreciation of the overall nature of reality (using circular logic).
Conventional mathematical appreciation - as stated - is based on sole recognition of reason (using linear logic). And central to this appreciation that objective truth can be clearly separated from subjective (in absolute type fashion).
Now in order to switch from conscious recognition with respect to external (objective) reality to corresponding conscious recognition with respect to internal (psychological) constructs (and then back again), the unconscious is always implicitly involved. The unconscious essentially is based on intuitive recognition that external and internal (as opposite poles of experience) are dynamically complementary (and indeed ultimately identical). So in a sense to view objective reality in an absolute manner is very unbalanced (from this unconscious perspective). So in an attempt to establish balance, experience switches direction (to the internal psychological pole).
However when mathematical reality is viewed in a merely rational manner, the internal pole is also given a somewhat absolute identity. This leads to the view that there is thereby a direct correspondence (in absolute terms) as between the mental constructs that Mathematics uses for interpretation and the objective reality to which these relate.
This belief in turn leads to a somewhat rigid relationship between polarities whereby experience continues to confirm the same rigid assumptions (on which interpretation is based).
Indeed this rigidity has been so great that it has blotted out recognition altogether of the equally important holistic aspect of mathematical understanding. Again this is directly based on a special type of intuitive understanding (conveyed through coherent qualitative interpretation of mathematical symbols).
So true qualitative appreciation of reality - relative to quantitative - is of a holistic intuitive nature. So when we attempt to express such appreciation in merely rational linear terms, we simply reduce the qualitative aspect to the quantitative.
So not surprisingly the very nature of Holistic Mathematics cannot be understood from this perspective.
An even deeper appreciation of this relationship entails the incorporation of both (holistic) intuitive and (analytic) rational type interpretation.
In a direct sense, (holistic) intuition relates to the empty spiritual aspect of reality (that is infinite in potential terms); by contrast (analytic) reason relates to the formal material aspect (that is finite in an actual manner). So the dynamic relationship of reason and intuition in experience pertains to the central relationship of finite to infinite.
And in a crucially important sense, the Riemann Hypothesis again can be expressed as the fundamental requirement for consistency with respect to this relationship.
For example in strict terms - though this is overlooked with respect to conventional mathematical interpretation - a general proof of a theorem applies to an infinite no. of potential cases. However any specific examples, illustrating the general proof, relate to a finite no. of actual cases.
So the proof of the Pythagorean Theorem for example applies potentially to all possible right hand triangles (in infinite terms). However the applicability of the theorem necessarily relates to a finite no. of actual right hand triangles.
Thus inherent in the belief that we can apply a general (potential) proof to specific (actual) examples is an implicit acceptance of the fundamental consistency as between finite and infinite realms.
And this is exactly the same consistency that is implied by the Riemann Hypothesis!
Unfortunately, as Conventional Mathematics is formally based merely on rational interpretation, the significance of the Riemann Hypothesis will always remain out of reach (when approached from this perspective).
Looked at in yet another equivalent manner, the fundamental axiom - to which the Riemann Hypothesis relates - enables consistent dynamic switching as between the opposite polarities of experience (such as external and internal).
I will explain here the basic nature of this dynamic interaction before explaining its precise mathematical significance in the next contribution.
All experience - indeed all development processes - entail twin processes of differentiation and integration respectively (based on two distinct logical systems).
Differentiation essentially entails linear logic whereby opposite polarities are clearly separated.
Integration by contrast entails circular logic whereby these same opposites are viewed as complementary (and ultimately identical).
In human experience differentiation and integration relate to the conscious and unconscious respectively. Reason is directly identified with conscious understanding and best suited to detailed analytic understanding of specific aspects of reality (using linear logic); intuition is directly identified with unconscious understanding and correspondingly suited to holistic appreciation of the overall nature of reality (using circular logic).
Conventional mathematical appreciation - as stated - is based on sole recognition of reason (using linear logic). And central to this appreciation that objective truth can be clearly separated from subjective (in absolute type fashion).
Now in order to switch from conscious recognition with respect to external (objective) reality to corresponding conscious recognition with respect to internal (psychological) constructs (and then back again), the unconscious is always implicitly involved. The unconscious essentially is based on intuitive recognition that external and internal (as opposite poles of experience) are dynamically complementary (and indeed ultimately identical). So in a sense to view objective reality in an absolute manner is very unbalanced (from this unconscious perspective). So in an attempt to establish balance, experience switches direction (to the internal psychological pole).
However when mathematical reality is viewed in a merely rational manner, the internal pole is also given a somewhat absolute identity. This leads to the view that there is thereby a direct correspondence (in absolute terms) as between the mental constructs that Mathematics uses for interpretation and the objective reality to which these relate.
This belief in turn leads to a somewhat rigid relationship between polarities whereby experience continues to confirm the same rigid assumptions (on which interpretation is based).
Indeed this rigidity has been so great that it has blotted out recognition altogether of the equally important holistic aspect of mathematical understanding. Again this is directly based on a special type of intuitive understanding (conveyed through coherent qualitative interpretation of mathematical symbols).
So true qualitative appreciation of reality - relative to quantitative - is of a holistic intuitive nature. So when we attempt to express such appreciation in merely rational linear terms, we simply reduce the qualitative aspect to the quantitative.
So not surprisingly the very nature of Holistic Mathematics cannot be understood from this perspective.
Wednesday, December 1, 2010
True Significance of Riemann Hypothesis
I have already redefined the Riemann Hypothesis demonstrating how it relates to an intimate correspondence as between both the quantitative and qualitative interpretation of mathematical symbols.
Personally I find this connection to be of enormous significance. For many years now I have been developing an alternative type of Mathematics where each mathematical symbol is given a holistic as opposed to a strictly analytic interpretation as in Conventional Mathematics. So this has continually raised the question of this mysterious relationship as between quantitative and qualitative.
So like two blades of a scissors, properly understood we have two aspects of Mathematics that - in a more balanced appreciation - would be recognised as of equal importance i.e. standard and holistic.
Then in a comprehensive approach to Mathematics both of these aspects would increasingly interact in a dynamic manner that would be both immensely productive and highly creative (representing the full expression of both rational and intuitive type capacities).
However this very interaction already assumes an intimate correspondence as between both aspects. And this in fact is the very same axiom or assumption that underlines the truth of the Riemann Hypothesis.
So in strict terms we can never prove that the Riemann Hypothesis is true (or indeed false). The reason for this is that the axiom - to which the Hypothesis relates - is already implicit in the very axioms that are used in Conventional Mathematics.
Put another way if we wish to doubt the validity of the Riemann Hypothesis, then we must necessarily doubt the validity of all the axioms that we conventionally use, which would in turn undermine belief in the truth of any mathematical proposition.
Thus in the deepest sense the truth implied by the Riemann Hypothesis is not implied by reason but rather by faith! For without such faith the whole mathematical edifice constructed so painstakingly over the past few thousand years would be without any foundation.
However it is indeed possible to probe a little more into what the Riemann Hypothesis truly implies.
Central to this truth is that we can never divorce objective knowledge of reality from the psychological constructs that we must necessarily use to interpret this reality. In other words we cannot divorce quantitative type results from qualitative type interpretation.
Now the great appeal of Mathematics to so many is a belief in its pure objective nature. So for example when a proposition is proven as true the belief is that this proof possesses an "objective" validity that is absolute.
Strictly speaking however this absolute view of mathematical truth is not warranted. For the very "objective" truth that we demonstrate through a mathematical proof is itself but a reflection of the mental constructs that we deem appropriate in arriving at such a conclusion.
So expressed now in more refined terms, that directly relates to experience, the faith that we place in mathematical truth (such as a proof of a theorem) in dynamic interactive terms expresses the belief that an automatic correspondence exists as between what is objectively demonstrated to be true and the subjective mental constructs that we must necessarily use in reaching such a conclusion.
Now in static terms there are two equally valid ways that we could express such proof:
(i) as pertaining to objective reality (as independent of psychological interaction). In other words the proposition is thereby absolutely true in quantitative terms.
(ii) as pertaining to psychological mental reality (as independent of any physical interaction). So again the proposition is absolutely true in qualitative terms.
Now in effect - when we interpret in this static absolute manner as in Conventional Mathematics - both possible interpretations will directly correspond with each other. So it does not matter in effect which polar aspect of explanation we might give. In other words the qualitative can thereby be reduced to the quantitative aspect.
However once we view mathematical understanding in a dynamic interactive manner, the relationship as between both aspects is of a merely relative nature. So quite literally we now accept that both external (objective) and internal (psychological) aspects are (dynamically) related to each other in experience.
Though interpretation of a mathematical proposition cannot now be strictly of an absolute nature, a high level of trust can still be placed in proof (by accepting this intimate correspondence as between both aspects). And once again this intimate correspondence - enabling us to place such great value in what (objectively) corresponds to our (psychological) interpretations of reality - is the very same axiom to which the Riemann Hypothesis applies.
So the key significance of the Riemann Hypothesis is completely missed through conventional mathematical interpretation.
And it is this very fact which makes it so central. For once we understand what it really is implying, then we must accept that it is exposing the most fundamental weakness possible with respect to Conventional Mathematics. In other words - properly understood - there are two aspects of equal importance to Mathematics. Yet one of these aspects has been effectively ignored almost entirely throughout our mathematical history!
And the Riemann Hypothesis - when appropriately appreciated - is there as that essential axiom enabling the proper integration of both branches (on which is based - what I refer to as - Radial Mathematics).
Yes! the Riemann Hypothesis has indeed considerable implications for the true nature of prime numbers!
More importantly however it has even greater implications for the true nature of Mathematics.
Personally I find this connection to be of enormous significance. For many years now I have been developing an alternative type of Mathematics where each mathematical symbol is given a holistic as opposed to a strictly analytic interpretation as in Conventional Mathematics. So this has continually raised the question of this mysterious relationship as between quantitative and qualitative.
So like two blades of a scissors, properly understood we have two aspects of Mathematics that - in a more balanced appreciation - would be recognised as of equal importance i.e. standard and holistic.
Then in a comprehensive approach to Mathematics both of these aspects would increasingly interact in a dynamic manner that would be both immensely productive and highly creative (representing the full expression of both rational and intuitive type capacities).
However this very interaction already assumes an intimate correspondence as between both aspects. And this in fact is the very same axiom or assumption that underlines the truth of the Riemann Hypothesis.
So in strict terms we can never prove that the Riemann Hypothesis is true (or indeed false). The reason for this is that the axiom - to which the Hypothesis relates - is already implicit in the very axioms that are used in Conventional Mathematics.
Put another way if we wish to doubt the validity of the Riemann Hypothesis, then we must necessarily doubt the validity of all the axioms that we conventionally use, which would in turn undermine belief in the truth of any mathematical proposition.
Thus in the deepest sense the truth implied by the Riemann Hypothesis is not implied by reason but rather by faith! For without such faith the whole mathematical edifice constructed so painstakingly over the past few thousand years would be without any foundation.
However it is indeed possible to probe a little more into what the Riemann Hypothesis truly implies.
Central to this truth is that we can never divorce objective knowledge of reality from the psychological constructs that we must necessarily use to interpret this reality. In other words we cannot divorce quantitative type results from qualitative type interpretation.
Now the great appeal of Mathematics to so many is a belief in its pure objective nature. So for example when a proposition is proven as true the belief is that this proof possesses an "objective" validity that is absolute.
Strictly speaking however this absolute view of mathematical truth is not warranted. For the very "objective" truth that we demonstrate through a mathematical proof is itself but a reflection of the mental constructs that we deem appropriate in arriving at such a conclusion.
So expressed now in more refined terms, that directly relates to experience, the faith that we place in mathematical truth (such as a proof of a theorem) in dynamic interactive terms expresses the belief that an automatic correspondence exists as between what is objectively demonstrated to be true and the subjective mental constructs that we must necessarily use in reaching such a conclusion.
Now in static terms there are two equally valid ways that we could express such proof:
(i) as pertaining to objective reality (as independent of psychological interaction). In other words the proposition is thereby absolutely true in quantitative terms.
(ii) as pertaining to psychological mental reality (as independent of any physical interaction). So again the proposition is absolutely true in qualitative terms.
Now in effect - when we interpret in this static absolute manner as in Conventional Mathematics - both possible interpretations will directly correspond with each other. So it does not matter in effect which polar aspect of explanation we might give. In other words the qualitative can thereby be reduced to the quantitative aspect.
However once we view mathematical understanding in a dynamic interactive manner, the relationship as between both aspects is of a merely relative nature. So quite literally we now accept that both external (objective) and internal (psychological) aspects are (dynamically) related to each other in experience.
Though interpretation of a mathematical proposition cannot now be strictly of an absolute nature, a high level of trust can still be placed in proof (by accepting this intimate correspondence as between both aspects). And once again this intimate correspondence - enabling us to place such great value in what (objectively) corresponds to our (psychological) interpretations of reality - is the very same axiom to which the Riemann Hypothesis applies.
So the key significance of the Riemann Hypothesis is completely missed through conventional mathematical interpretation.
And it is this very fact which makes it so central. For once we understand what it really is implying, then we must accept that it is exposing the most fundamental weakness possible with respect to Conventional Mathematics. In other words - properly understood - there are two aspects of equal importance to Mathematics. Yet one of these aspects has been effectively ignored almost entirely throughout our mathematical history!
And the Riemann Hypothesis - when appropriately appreciated - is there as that essential axiom enabling the proper integration of both branches (on which is based - what I refer to as - Radial Mathematics).
Yes! the Riemann Hypothesis has indeed considerable implications for the true nature of prime numbers!
More importantly however it has even greater implications for the true nature of Mathematics.
Friday, November 19, 2010
Holistic Mathematical Connections (3)
We are still looking at the holistic mathematical significance of the Euler Zeta Functions for positive integer values.
We have seen that in qualitative terms, each dimensional value of s is associated with a unique structural manner of rational interpretation.
Once again from this perspective the great limitation of conventional mathematical appreciation is that in formal terms it is solely confined to linear type appreciation (i.e. where s = 1).
And the key defining characteristic of this mode of interpretation is the qualitative aspect of understanding is not recognised and in effect is thereby reduced to the quantitative!
However an infinite set of more refined alternative "higher" interpretations exist where both quantitative and qualitative aspects are given distinctive recognition.
This entails that Mathematics is no longer formally defined in mere conscious rational terms but rather in more subtle terms where both conscious and unconscious aspects interact.
In direct terms the unconscious aspect of such understanding relates to holistic intuition. However indirectly this holistic aspect can be indirectly translated in rational terms as imaginary (based on a circular manner of interpretation).
So just as in quantitative terms the structure of 1 (for 3 or more roots) always entails both real and imaginary parts, likewise in corresponding qualitative terms, the holistic structure of 1 (for 3 or more dimensions) always contains both real (conscious) and imaginary (unconscious) aspects of interpretation.
Though all my essential thinking on these matters had been long formulated before I became interested in the Riemann Hypothesis, this Hypothesis does indeed provide an extremely important application of such notions. And in reverse manner understanding the Hypothesis in this way has helped to considerably clarify my own understanding with respect to the higher structures of psychological understanding.
Using spiritual type language, the higher integral dimensions (positive) relate to the transcendent type structures of understanding that progressively unfold through the process of contemplative type development. So through increasing dynamic interaction of both rational and intuitive aspects of understanding, interpretation becomes ever more refined in rational terms.
Once again the even numbered structures relate to integral type understanding at such higher dimensions (where one attempts to define the appropriate manner through which overall holistic interdependent nature of reality can be rationally encoded).
Then the odd numbered dimensions relate to corresponding differentiated understanding at these higher dimensions (where one now tries to equally define the appropriate manner through the relatively independent nature of specific phenomena of form can be likewise encoded).
Now ultimately both the - relatively - continuous nature of the integral aspect and the discrete nature of the differentiated aspect become so refined as to be indistinguishable from each other which would relate to pure contemplative experience of reality. However this represents a limiting state that can only be approximated imperfectly in human terms.
Incidentally this brings us back to the holistic mathematical nature of e (which plays such crucial role with respect to the behaviour of prime numbers).
Just as in conventional mathematical terms both the integral and differential of e^x are indistinguishable from each other, in holistic mathematical terms it is quite similar. So here we are no longer able to distinguish the discrete (differentiated) aspect of phenomenal form from the corresponding continuous (integral) aspect of holistic emptiness.
And again we can identify such a state as one of pure contemplative awareness (in the proper balanced sense of fully harmonising both intuitive and rational aspects of understanding). So this ideal state - insofar as it can be humanly approximated - combines both profound intuitive depth with incredible rational clarity).
Indeed it is in this state that the mystery of the prime numbers is finally resolved (which is the same state any lingering problem with respect to involuntary primitive instincts) is also resolved. So once again the quantitative nature of prime number behaviour and the qualitative nature of primitive instinctive behaviour entail the relationship of both specific conscious and holistic unconscious aspects of understanding (corresponding to both the linear and circular aspects of understanding respectively).
We have seen that in qualitative terms, each dimensional value of s is associated with a unique structural manner of rational interpretation.
Once again from this perspective the great limitation of conventional mathematical appreciation is that in formal terms it is solely confined to linear type appreciation (i.e. where s = 1).
And the key defining characteristic of this mode of interpretation is the qualitative aspect of understanding is not recognised and in effect is thereby reduced to the quantitative!
However an infinite set of more refined alternative "higher" interpretations exist where both quantitative and qualitative aspects are given distinctive recognition.
This entails that Mathematics is no longer formally defined in mere conscious rational terms but rather in more subtle terms where both conscious and unconscious aspects interact.
In direct terms the unconscious aspect of such understanding relates to holistic intuition. However indirectly this holistic aspect can be indirectly translated in rational terms as imaginary (based on a circular manner of interpretation).
So just as in quantitative terms the structure of 1 (for 3 or more roots) always entails both real and imaginary parts, likewise in corresponding qualitative terms, the holistic structure of 1 (for 3 or more dimensions) always contains both real (conscious) and imaginary (unconscious) aspects of interpretation.
Though all my essential thinking on these matters had been long formulated before I became interested in the Riemann Hypothesis, this Hypothesis does indeed provide an extremely important application of such notions. And in reverse manner understanding the Hypothesis in this way has helped to considerably clarify my own understanding with respect to the higher structures of psychological understanding.
Using spiritual type language, the higher integral dimensions (positive) relate to the transcendent type structures of understanding that progressively unfold through the process of contemplative type development. So through increasing dynamic interaction of both rational and intuitive aspects of understanding, interpretation becomes ever more refined in rational terms.
Once again the even numbered structures relate to integral type understanding at such higher dimensions (where one attempts to define the appropriate manner through which overall holistic interdependent nature of reality can be rationally encoded).
Then the odd numbered dimensions relate to corresponding differentiated understanding at these higher dimensions (where one now tries to equally define the appropriate manner through the relatively independent nature of specific phenomena of form can be likewise encoded).
Now ultimately both the - relatively - continuous nature of the integral aspect and the discrete nature of the differentiated aspect become so refined as to be indistinguishable from each other which would relate to pure contemplative experience of reality. However this represents a limiting state that can only be approximated imperfectly in human terms.
Incidentally this brings us back to the holistic mathematical nature of e (which plays such crucial role with respect to the behaviour of prime numbers).
Just as in conventional mathematical terms both the integral and differential of e^x are indistinguishable from each other, in holistic mathematical terms it is quite similar. So here we are no longer able to distinguish the discrete (differentiated) aspect of phenomenal form from the corresponding continuous (integral) aspect of holistic emptiness.
And again we can identify such a state as one of pure contemplative awareness (in the proper balanced sense of fully harmonising both intuitive and rational aspects of understanding). So this ideal state - insofar as it can be humanly approximated - combines both profound intuitive depth with incredible rational clarity).
Indeed it is in this state that the mystery of the prime numbers is finally resolved (which is the same state any lingering problem with respect to involuntary primitive instincts) is also resolved. So once again the quantitative nature of prime number behaviour and the qualitative nature of primitive instinctive behaviour entail the relationship of both specific conscious and holistic unconscious aspects of understanding (corresponding to both the linear and circular aspects of understanding respectively).
Thursday, November 18, 2010
Holistic Mathematical Connections (2)
We have now looked at the Euler Zeta Function for positive even integers of s (2, 4, 6,...) to find that the resulting value can be always be expressed in terms of Pi.
I have also been at pains to indicate the qualitative significance of this fascinating numerical behaviour.
One might initially think therefore that a similar expresion would exist for corresponding odd integers of s (3, 5, 7,...) but, as is well known, this is not the case. No closed value expressions have yet been found (though several ingenious closely approximating formulae have been derived).
Once again this is where holistic mathematical understanding can prove very illuminating.
In all psychological behaviour we have the two related aspects of differentiation and integration respectively entailing both conscious and unconscious appreciation. Differentiation is (analytically) associated with the linear logical system thus enabling the separation of opposite polarities in experience (as independent).
By contrast integration is (holistically) associated with the circular logical system thus enabling the complementarity - and ultimate identity - of these same polarities (as interdependent).
Now when we look at the higher dimensions of understanding, whereby rational understanding becomes refined in an increasingly intuitive manner, both processes of differentiation and integration are at work.
We have already identified 1-dimensional appreciation as the representative of the linear and 2-dimensional as representative of the pure circular aspect (that exhibits full complementarity) respectively.
So 1-dimensional appreciation is properly geared for the differentiated interpretation of reality (in analytic terms); then 2-dimensional appreciation is properly geared for integrated interpretation (in holistic terms) .
In quantitative terms we see this reflected in the corresponding root structures (of 1).
The 1st root of unity = + 1 is fully independent of its negative and thereby exhibits no complementarity; by contrast the two roots of unity = + 1 and -1 exhibit perfect complementarity (as interdependent).
Now, the higher even numbered dimensions (that are positive) can best be seen as the movement towards ever more refined integral sttructures . So, as we have seen, when we obtain the roots of unity for any positive even integer, we are always able to match each of these roots with a corresponding negative. For example the four roots are + 1, - 1, + i and - i. So here each root (as positive) can be matched with a corresponding root (as negative).
In like manner, the holistic mathematical interpretation of such even dimensions relate to qualitative structures of higher integration (representing ever more refined rational appreciation of interdependence).
However the higher positive odd integral dimensions do not represent such perfect symmetry. In fact with all such root structures we will always have one root that is + 1 separated from the rest (with all other complex roots representing symmetry only with respect to their imaginary parts). Though the sum of the corresponding real parts will indeed sum up to - 1, this represants a form of broken symmetry (with respect to the real part). So - quite literally - in psychological terms, there is an inevitable delay with respect to the unconscious process of negation of phenomenal forms. In this way though experience is indeed of an increasingly refined nature, the very process of differentiating phenomena creates a degree of linear rigidity (by which they are given a degree of independence).
So the holistic mathematical interpretation of such odd dimensions (in qualitative terms) is perfectly reflected in the quantitative structure of its corresponding roots.
The clear implication therefore is that pure circular complementarity (requiring the perfect complementarity of opposites) cannot exist with the positive odd integral dimensions).
Because of a degree of breaking (of integral symmetry) in qualiative terms, such dimensions cannot represent the pure relationship of circular to linear understanding. In like manner the quantitative value of such zeta expressions cannot be conveyed through closed pi expressions.
Now a great deal of debate has continued as to whether such zeta values (for positive odd integer values of s > 1) are rational or irrational.
It has indeed been proved that - at least when s = 3 - the value is irrational.
Now there are good holistic mathematical reasons for believing that not alone is the value of such expressions always irrational (but necessarily of a transcendental nature).
If the value was in fact rational (for s > 3) this would imply a rational value in the prime product formula over the infinite range of terms. This would imply that each term bears a relationship to previous terms that can be expressed in a rational manner. However the very nature of prime numbers is that their individual quantitative nature is uniquely distinct from their overall holistic behaviour with respect to each other. Therefore no such rational formula could exist.
Therefore the convergent sum of terms inevitably entails a relationship involving both linear and circular notions (which directly implies that it is transcendental). And as we have seen what distinguishes the even from the odd dimesnions here, is that with the even we have a pure relationship of circular to linear (whereas with the odd this is not the case).
Once again the key assumption here (regarding the inherent nature of prime numbers) literally transcends conventional mathematical interpretation. Indeed it is the same assumption that ultimately establishes the axiomatic nature of the Riemann Hypothesis!
I have also been at pains to indicate the qualitative significance of this fascinating numerical behaviour.
One might initially think therefore that a similar expresion would exist for corresponding odd integers of s (3, 5, 7,...) but, as is well known, this is not the case. No closed value expressions have yet been found (though several ingenious closely approximating formulae have been derived).
Once again this is where holistic mathematical understanding can prove very illuminating.
In all psychological behaviour we have the two related aspects of differentiation and integration respectively entailing both conscious and unconscious appreciation. Differentiation is (analytically) associated with the linear logical system thus enabling the separation of opposite polarities in experience (as independent).
By contrast integration is (holistically) associated with the circular logical system thus enabling the complementarity - and ultimate identity - of these same polarities (as interdependent).
Now when we look at the higher dimensions of understanding, whereby rational understanding becomes refined in an increasingly intuitive manner, both processes of differentiation and integration are at work.
We have already identified 1-dimensional appreciation as the representative of the linear and 2-dimensional as representative of the pure circular aspect (that exhibits full complementarity) respectively.
So 1-dimensional appreciation is properly geared for the differentiated interpretation of reality (in analytic terms); then 2-dimensional appreciation is properly geared for integrated interpretation (in holistic terms) .
In quantitative terms we see this reflected in the corresponding root structures (of 1).
The 1st root of unity = + 1 is fully independent of its negative and thereby exhibits no complementarity; by contrast the two roots of unity = + 1 and -1 exhibit perfect complementarity (as interdependent).
Now, the higher even numbered dimensions (that are positive) can best be seen as the movement towards ever more refined integral sttructures . So, as we have seen, when we obtain the roots of unity for any positive even integer, we are always able to match each of these roots with a corresponding negative. For example the four roots are + 1, - 1, + i and - i. So here each root (as positive) can be matched with a corresponding root (as negative).
In like manner, the holistic mathematical interpretation of such even dimensions relate to qualitative structures of higher integration (representing ever more refined rational appreciation of interdependence).
However the higher positive odd integral dimensions do not represent such perfect symmetry. In fact with all such root structures we will always have one root that is + 1 separated from the rest (with all other complex roots representing symmetry only with respect to their imaginary parts). Though the sum of the corresponding real parts will indeed sum up to - 1, this represants a form of broken symmetry (with respect to the real part). So - quite literally - in psychological terms, there is an inevitable delay with respect to the unconscious process of negation of phenomenal forms. In this way though experience is indeed of an increasingly refined nature, the very process of differentiating phenomena creates a degree of linear rigidity (by which they are given a degree of independence).
So the holistic mathematical interpretation of such odd dimensions (in qualitative terms) is perfectly reflected in the quantitative structure of its corresponding roots.
The clear implication therefore is that pure circular complementarity (requiring the perfect complementarity of opposites) cannot exist with the positive odd integral dimensions).
Because of a degree of breaking (of integral symmetry) in qualiative terms, such dimensions cannot represent the pure relationship of circular to linear understanding. In like manner the quantitative value of such zeta expressions cannot be conveyed through closed pi expressions.
Now a great deal of debate has continued as to whether such zeta values (for positive odd integer values of s > 1) are rational or irrational.
It has indeed been proved that - at least when s = 3 - the value is irrational.
Now there are good holistic mathematical reasons for believing that not alone is the value of such expressions always irrational (but necessarily of a transcendental nature).
If the value was in fact rational (for s > 3) this would imply a rational value in the prime product formula over the infinite range of terms. This would imply that each term bears a relationship to previous terms that can be expressed in a rational manner. However the very nature of prime numbers is that their individual quantitative nature is uniquely distinct from their overall holistic behaviour with respect to each other. Therefore no such rational formula could exist.
Therefore the convergent sum of terms inevitably entails a relationship involving both linear and circular notions (which directly implies that it is transcendental). And as we have seen what distinguishes the even from the odd dimesnions here, is that with the even we have a pure relationship of circular to linear (whereas with the odd this is not the case).
Once again the key assumption here (regarding the inherent nature of prime numbers) literally transcends conventional mathematical interpretation. Indeed it is the same assumption that ultimately establishes the axiomatic nature of the Riemann Hypothesis!
Monday, November 15, 2010
Holistic Mathematical Connections
As we have seen when s is a positive even integer, the value of the Euler Zeta function can be expressed in terms of an expression comprising (Pi^s) * k (where k represents a rational number).
Now what I am concerned with here is to give a holistic mathematical interpretation of the significance of this result.
To do this it is easiest to concentrate initially on the simplest - and best known - case where s = 2.
As we have seen here the value of the Zeta function = (Pi^2)/6
As stated repeatedly, in holistic mathematical terms, standard conventional interpretation is qualitatively of a linear nature (i.e. 1-dimensional).
What this means in effect is that the polarities of experience are inherently separated giving just one positive (unambiguous) direction.
So for example all understanding - including of course mathematical - entails a dynamic interaction of both external (objective) and internal (subjective) aspects.
However in mathematical interpretation these are clearly separated. So mathematical truth is given just one direction i.e. dimension as objective (with which psychological mental constructs are thereby assumed to statically correspond).
So linear logic is decidedly of the unambiguous either/or variety!
However corresponding to all other dimensions is a unique qualitative mode of interpretation and the simplest of these is 2-dimensional.
So with 2-dimensional interpretation, two polar directions are always given (which dynamically interact in a paradoxical manner).
So for example interpretation is not identified with either the external or internal pole of understanding (as separate) but rather with their mutual interdependence. This then leads to an alternative circular mode of both/and logic that is based on the complementarity (and ultimate identity) of polar opposites.
So the qualitative nature of 2-dimensional interpretation corresponds with the structural nature of the two roots of unity.
However whereas the two roots of 1, i.e. + 1 and - 1 are interpreted using either/or (linear) logic in quantitative terms, qualitatively they are interpreted using circular both/and circular logic.
However the use of 2 initially always requires the corresponding use 1-dimensional interpretation. For if we are not able to identify polar opposites as initially separate in experience, then we cannot hope they see how they are complementary!
Thus a key issue that thereby arises is the reconciliation of both the linear and circular modes of understanding.
Now in quantitative terms the constant pi relates to the relationship as between the (circular) circumference and its (linear) diameter.
Likewise in qualitative terms, for correct 2-dimensional interpretation we must reconcile both linear (either/or) and circular (both/and) interpretation.
Thus the very form of the quantitative result for the Euler Zeta function (where s = 2) has its corresponding qualitative holistic mathematical explanation (i.e. as the relationship of linear and circular understanding).
Though in quantitative terms, the Euler function here does indeed have a direct quantitative numerical result, indirectly it also contains a qualitative significance in the form of the result (entailing pi).
Now we will see later, when s takes on negative integer values, that the numerical results arising have but an indirect quantitative significance. Rather in direct terms, the numerical values that result can only correctly be given a qualitative interpretation!
Briefly the holistic mathematical interpretation for all other even integer values of s operates along the same lines. Just as the 2-dimensional case represents the simplest version of the complementarity of opposites, all higher dimensions point to more intricate arrangements of the same complementarity. This corresponds in quantitative terms with the fact that the s roots of 1 can also be arranged in a complementary manner.
In my own work, I have mainly concentrated on interpretation that corresponds to 2, 4 and 8 dimensions respectively (which I believe are the most important). However in principle the basic requirement regarding matching positive and negative directions (now taken in the complex plane) can be easily extended to all even integer values of s.
And once again this constitutes the holistic mathematical explanation as to why the form of all such zeta results can be expressed in the form of pi.
Now what I am concerned with here is to give a holistic mathematical interpretation of the significance of this result.
To do this it is easiest to concentrate initially on the simplest - and best known - case where s = 2.
As we have seen here the value of the Zeta function = (Pi^2)/6
As stated repeatedly, in holistic mathematical terms, standard conventional interpretation is qualitatively of a linear nature (i.e. 1-dimensional).
What this means in effect is that the polarities of experience are inherently separated giving just one positive (unambiguous) direction.
So for example all understanding - including of course mathematical - entails a dynamic interaction of both external (objective) and internal (subjective) aspects.
However in mathematical interpretation these are clearly separated. So mathematical truth is given just one direction i.e. dimension as objective (with which psychological mental constructs are thereby assumed to statically correspond).
So linear logic is decidedly of the unambiguous either/or variety!
However corresponding to all other dimensions is a unique qualitative mode of interpretation and the simplest of these is 2-dimensional.
So with 2-dimensional interpretation, two polar directions are always given (which dynamically interact in a paradoxical manner).
So for example interpretation is not identified with either the external or internal pole of understanding (as separate) but rather with their mutual interdependence. This then leads to an alternative circular mode of both/and logic that is based on the complementarity (and ultimate identity) of polar opposites.
So the qualitative nature of 2-dimensional interpretation corresponds with the structural nature of the two roots of unity.
However whereas the two roots of 1, i.e. + 1 and - 1 are interpreted using either/or (linear) logic in quantitative terms, qualitatively they are interpreted using circular both/and circular logic.
However the use of 2 initially always requires the corresponding use 1-dimensional interpretation. For if we are not able to identify polar opposites as initially separate in experience, then we cannot hope they see how they are complementary!
Thus a key issue that thereby arises is the reconciliation of both the linear and circular modes of understanding.
Now in quantitative terms the constant pi relates to the relationship as between the (circular) circumference and its (linear) diameter.
Likewise in qualitative terms, for correct 2-dimensional interpretation we must reconcile both linear (either/or) and circular (both/and) interpretation.
Thus the very form of the quantitative result for the Euler Zeta function (where s = 2) has its corresponding qualitative holistic mathematical explanation (i.e. as the relationship of linear and circular understanding).
Though in quantitative terms, the Euler function here does indeed have a direct quantitative numerical result, indirectly it also contains a qualitative significance in the form of the result (entailing pi).
Now we will see later, when s takes on negative integer values, that the numerical results arising have but an indirect quantitative significance. Rather in direct terms, the numerical values that result can only correctly be given a qualitative interpretation!
Briefly the holistic mathematical interpretation for all other even integer values of s operates along the same lines. Just as the 2-dimensional case represents the simplest version of the complementarity of opposites, all higher dimensions point to more intricate arrangements of the same complementarity. This corresponds in quantitative terms with the fact that the s roots of 1 can also be arranged in a complementary manner.
In my own work, I have mainly concentrated on interpretation that corresponds to 2, 4 and 8 dimensions respectively (which I believe are the most important). However in principle the basic requirement regarding matching positive and negative directions (now taken in the complex plane) can be easily extended to all even integer values of s.
And once again this constitutes the holistic mathematical explanation as to why the form of all such zeta results can be expressed in the form of pi.
Sunday, November 14, 2010
Thr Pi Connection
Euler also demonstrated another remarkable connection as between his Zeta function and the value of pi.
So whenever (representing the dimension) in the function is a positive even integer, then the resulting value can be expressed as pi (to the power of s) * by a rational no.
So in the simplest - and best known - case when s = 2,
∑[1/n^2] = ∏{p^2/(p^2 – 1)} = (pi^2)/6
One of the interesting implications of this result is that it provides another means of proving the infinitude of primes (i.e. in the accepted reduced nature of the infinite).
For if the the no. of primes was finite then the product formula (involving the primes) would ensure a rational value.
However because the actual answer is irrational (involving pi), then this implies that the no. of primes must be infinite.
However the deeper qualitative implications of this result are not properly appreciated.
Again no matter how many finite terms are included in the product formula, a rational result will ensue. Therefore the fact that the result is irrational, implies that an additional qualitative aspect of understanding is required.
Now some infinite series in the limit of an infinite process - again accepting the reduced conventional appreciation of the infinite - do result in a rational finite answer.
For example if we sum the series 1 + 1/2 + 1/4 + 1/8 +......,
the actual sum for any finite no. of terms will be rational. However the limiting value for an infinite no. of terms will also be rational (i.e. 2).
However one reason why this infinite series does not lead to an irrational value is the fact that each successive term can itself be expressed as a rational fraction of the previous term. So just as rational number can be expressed in decimal form with a consistent repeating sequence of digits, likewise if successive terms in a series are related in a similar manner (through the application of a consistent rational operation) then a rational limit will result.
However clearly in the case of series entailing the prime numbers, this is not in fact the case. Rather, as I have stated before the actual location of each prime number intimately depends on the overall holistic relationship of the primes to the natural numbers.
And, again the key point is that this holistic relationship is qualitatively of a different nature as it relates to the potential infinite nature of these numbers whereas rational interpretation relates properly to actual finite type considerations.
So once again the key limitation of Conventional Mathematics is that it can only attempt to deal with potential infinite notions - properly relating to circular paradoxical type appreciation - in a grossly reduced manner. Here they are treated as an extension of the finite (so as to become amenable to an unambiguous linear type logic).
As we know the constant pi (which is irrational and transcendent in nature) pertains quantitatively to the relationship of the (circular) circumference to its (linear) diameter.
Likewise in holistic mathematical terms, qualitative understanding (that is irrational and transcendental) pertains to the corresponding relationship as between (pure) circular and linear type interpretation.
This thereby implies that - correctly understood in an appropriate qualitative manner - interpretation of the Zeta function (for even values of s) implies the pure relationship as between linear and qualitative type notions that is expressed in an indirect rational manner.
In other words - when appropriately understood - we then realise that the very nature of prime numbers entails a pure relationship as between actual finite notions (in the precise identity of specific prime numbers)and potential infinite notions (in the general distribution of the primes among the natural number system).
However there is another fascinating connection in these pi expressions and the nature of the primes.
If we confine ourselves to the rational nos. by which the pi expressions must be multiplied, then the denominators of such expressions bear a very important relationship to the primes.
So, when s = 2n (where n = 1, 2, 3, 4,...)
then, where 2n represents an integral power of 2, the denominator of the rational number part will represent a product of all prime numbers (in various combinations) up to 2n + 1 and only these primes.
In all other cases, the denominator of the rational part will represent the product of all prime numbers - in varied combinations - from 3 to 2n + 1(and only these primes).
For example when s = 2n = 4, this represents an integral power of 2. The denominator of the rational part = 90 = 2 * 3^2 * 5 which represents all primes from 2 to 2n + 1(i.e. 5).
Then when s = 2n = 6, this does not represent an integral power of 2.
The denominator of the rational part = 945 = 3^3 * 5 * 7 which represents all primes from 3 to 2n + 1 (i.e. 7).
I will just give one more example to illustrate
Wne s = 2n (i.e. n = 8) = 16, s can again be expressed as 2 (raised to a positive integral power) i.e. 2^4.
The denominator here of of pi^16 = 325641566250.
And 325641566250 = 2 * 3^7 * 5^4 * 7^2 * 11 * 13 * 17. So we can see here how all the prime numbers from 2 to 2n + 1 i.e. 17, are included as factors in the denominator (and only these primes).
So whenever (representing the dimension) in the function is a positive even integer, then the resulting value can be expressed as pi (to the power of s) * by a rational no.
So in the simplest - and best known - case when s = 2,
∑[1/n^2] = ∏{p^2/(p^2 – 1)} = (pi^2)/6
One of the interesting implications of this result is that it provides another means of proving the infinitude of primes (i.e. in the accepted reduced nature of the infinite).
For if the the no. of primes was finite then the product formula (involving the primes) would ensure a rational value.
However because the actual answer is irrational (involving pi), then this implies that the no. of primes must be infinite.
However the deeper qualitative implications of this result are not properly appreciated.
Again no matter how many finite terms are included in the product formula, a rational result will ensue. Therefore the fact that the result is irrational, implies that an additional qualitative aspect of understanding is required.
Now some infinite series in the limit of an infinite process - again accepting the reduced conventional appreciation of the infinite - do result in a rational finite answer.
For example if we sum the series 1 + 1/2 + 1/4 + 1/8 +......,
the actual sum for any finite no. of terms will be rational. However the limiting value for an infinite no. of terms will also be rational (i.e. 2).
However one reason why this infinite series does not lead to an irrational value is the fact that each successive term can itself be expressed as a rational fraction of the previous term. So just as rational number can be expressed in decimal form with a consistent repeating sequence of digits, likewise if successive terms in a series are related in a similar manner (through the application of a consistent rational operation) then a rational limit will result.
However clearly in the case of series entailing the prime numbers, this is not in fact the case. Rather, as I have stated before the actual location of each prime number intimately depends on the overall holistic relationship of the primes to the natural numbers.
And, again the key point is that this holistic relationship is qualitatively of a different nature as it relates to the potential infinite nature of these numbers whereas rational interpretation relates properly to actual finite type considerations.
So once again the key limitation of Conventional Mathematics is that it can only attempt to deal with potential infinite notions - properly relating to circular paradoxical type appreciation - in a grossly reduced manner. Here they are treated as an extension of the finite (so as to become amenable to an unambiguous linear type logic).
As we know the constant pi (which is irrational and transcendent in nature) pertains quantitatively to the relationship of the (circular) circumference to its (linear) diameter.
Likewise in holistic mathematical terms, qualitative understanding (that is irrational and transcendental) pertains to the corresponding relationship as between (pure) circular and linear type interpretation.
This thereby implies that - correctly understood in an appropriate qualitative manner - interpretation of the Zeta function (for even values of s) implies the pure relationship as between linear and qualitative type notions that is expressed in an indirect rational manner.
In other words - when appropriately understood - we then realise that the very nature of prime numbers entails a pure relationship as between actual finite notions (in the precise identity of specific prime numbers)and potential infinite notions (in the general distribution of the primes among the natural number system).
However there is another fascinating connection in these pi expressions and the nature of the primes.
If we confine ourselves to the rational nos. by which the pi expressions must be multiplied, then the denominators of such expressions bear a very important relationship to the primes.
So, when s = 2n (where n = 1, 2, 3, 4,...)
then, where 2n represents an integral power of 2, the denominator of the rational number part will represent a product of all prime numbers (in various combinations) up to 2n + 1 and only these primes.
In all other cases, the denominator of the rational part will represent the product of all prime numbers - in varied combinations - from 3 to 2n + 1(and only these primes).
For example when s = 2n = 4, this represents an integral power of 2. The denominator of the rational part = 90 = 2 * 3^2 * 5 which represents all primes from 2 to 2n + 1(i.e. 5).
Then when s = 2n = 6, this does not represent an integral power of 2.
The denominator of the rational part = 945 = 3^3 * 5 * 7 which represents all primes from 3 to 2n + 1 (i.e. 7).
I will just give one more example to illustrate
Wne s = 2n (i.e. n = 8) = 16, s can again be expressed as 2 (raised to a positive integral power) i.e. 2^4.
The denominator here of of pi^16 = 325641566250.
And 325641566250 = 2 * 3^7 * 5^4 * 7^2 * 11 * 13 * 17. So we can see here how all the prime numbers from 2 to 2n + 1 i.e. 17, are included as factors in the denominator (and only these primes).
Wednesday, November 10, 2010
The Harmonic Series Again
Euler made great advances with respect to better understanding of the prime numbers.
In what must constitute one of the most truly memorable contributions to Mathematics he was able to demonstrate an intimate connection as between the natural numbers on the one hand and the prime numbers on the other.
Now once again the zeta function is defined as ∑1/n^s (for n = 1 → ∞)
The harmonic series results from setting s = 1
Thus ∑1/n = 1 + 1/2 + 1/3 + 1/4 +...... which is divergent.
Now the Euler zeta function is defined for values of s (> 1) with
∑[1/n^s] = ∏{p^s/(p^s – 1)} where again p ranging from 2 → ∞
Therefore when s = 2
1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 +... = {2^2/[(2^2) - 1]}*{3^2/[(3^2) - 1]}*{[5^2/([(5^2) - 1]}
So
1/1 + 1/4 + 1/9 + 1/16 +..... = 4/3 * 9/8 * 25/24 *.....
So we have here an intimate connection as between the natural numbers on the one hand (connected through addition)and the prime numbers (connected through multiplication).
Furthermore a unique dimensional connection exists between the two for all numbers where s > 1.
The question then arises as to whether a unique connection also exists in the vitally important case where s = 1.
Clearly in this case both the natural number sum series and the prime number product series will both diverge.
However if we confine ourselves to a limited finite number of terms, then an interesting connection does indeed exist.
So where the harmonic series is summed to a finite number of terms (n)
∑1/n → ∏{p + 1)/p}
So
1 + 1/2 + 1/3 + 1/4 + 1/5 +.... → 3/2 * 4/3 * 6/5 *.......
So as we obtain the sum of the first n terms of the harmonic series on one side, we approximate the corresponding product of all p terms up to n on the other side.
This relationship can also be written in another interesting way!
1 + 1/2 + 1/3 + 1/4 + 1/5 +.... → (1 + 1/2) * (1 + 1/3) * (1 + 1/5) *.....
Also when the no. of terms on the LHS = n, the corresponding no. of terms on the RHS approximates n/log n.
Thus on the LHS we have the sum of terms entailing the reciprocals of the natural nos (starting with 1); on the RHS we then have the product of terms entailing the reciprocals of the primes (in each case added to 1).
Now what this formulation clearly demonstrates is the intimate relation as between addition and multiplication in the connection of the primes to the natural number system.
However though - correctly - understood, multiplication involves both a quantitative and qualitative transformation with respect to number interpretation, Conventional Mathematics is based on the merely quantitative aspect. Therefore it inevitably reduces the qualitative aspect to the quantitative.
This quite simply then constitutes not alone the key barrier to solving the Riemann Hypothesis but in fact likewise the key barrier to its proper interpretation.
For ultimately the Riemann Hypothesis relates to the reconciliation of both the quantitative and qualitative aspects of Mathematics.
So again putting it bluntly, not alone can the Riemann Hypothesis not be solved in a conventional mathematical fashion, it is not even capable of being properly understood in this manner!
In what must constitute one of the most truly memorable contributions to Mathematics he was able to demonstrate an intimate connection as between the natural numbers on the one hand and the prime numbers on the other.
Now once again the zeta function is defined as ∑1/n^s (for n = 1 → ∞)
The harmonic series results from setting s = 1
Thus ∑1/n = 1 + 1/2 + 1/3 + 1/4 +...... which is divergent.
Now the Euler zeta function is defined for values of s (> 1) with
∑[1/n^s] = ∏{p^s/(p^s – 1)} where again p ranging from 2 → ∞
Therefore when s = 2
1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 +... = {2^2/[(2^2) - 1]}*{3^2/[(3^2) - 1]}*{[5^2/([(5^2) - 1]}
So
1/1 + 1/4 + 1/9 + 1/16 +..... = 4/3 * 9/8 * 25/24 *.....
So we have here an intimate connection as between the natural numbers on the one hand (connected through addition)and the prime numbers (connected through multiplication).
Furthermore a unique dimensional connection exists between the two for all numbers where s > 1.
The question then arises as to whether a unique connection also exists in the vitally important case where s = 1.
Clearly in this case both the natural number sum series and the prime number product series will both diverge.
However if we confine ourselves to a limited finite number of terms, then an interesting connection does indeed exist.
So where the harmonic series is summed to a finite number of terms (n)
∑1/n → ∏{p + 1)/p}
So
1 + 1/2 + 1/3 + 1/4 + 1/5 +.... → 3/2 * 4/3 * 6/5 *.......
So as we obtain the sum of the first n terms of the harmonic series on one side, we approximate the corresponding product of all p terms up to n on the other side.
This relationship can also be written in another interesting way!
1 + 1/2 + 1/3 + 1/4 + 1/5 +.... → (1 + 1/2) * (1 + 1/3) * (1 + 1/5) *.....
Also when the no. of terms on the LHS = n, the corresponding no. of terms on the RHS approximates n/log n.
Thus on the LHS we have the sum of terms entailing the reciprocals of the natural nos (starting with 1); on the RHS we then have the product of terms entailing the reciprocals of the primes (in each case added to 1).
Now what this formulation clearly demonstrates is the intimate relation as between addition and multiplication in the connection of the primes to the natural number system.
However though - correctly - understood, multiplication involves both a quantitative and qualitative transformation with respect to number interpretation, Conventional Mathematics is based on the merely quantitative aspect. Therefore it inevitably reduces the qualitative aspect to the quantitative.
This quite simply then constitutes not alone the key barrier to solving the Riemann Hypothesis but in fact likewise the key barrier to its proper interpretation.
For ultimately the Riemann Hypothesis relates to the reconciliation of both the quantitative and qualitative aspects of Mathematics.
So again putting it bluntly, not alone can the Riemann Hypothesis not be solved in a conventional mathematical fashion, it is not even capable of being properly understood in this manner!
Sunday, November 7, 2010
Zoning in on the Primes
We have seen how important the harmonic series is with respect to the nature of prime numbers.
Once again - as an initial approximation - the sum of the first n terms of the series provides an estimate of the gap as between primes (in the region of n).
Secondly the reciprocal of n provides a good estimate of the change in the gap as between primes in the region of n.
So as we have seen as we move for example from 1,000,000 to 1,000,001 the change in the gap would be close to 1/1,000,000.
Now the extent to which the harmonic series falls short in terms of predicting the actual gap between primes is explained by Euler's Constant (= .5772...).
More accurately the gap between primes (in the region of n) is given by log n.
And as the harmonic series approximates log n + λ (where λ = Euler's Constant) then using the harmonic series as a prediction will overestimate the gap by .5772.
However remarkably the value of Euler's Constant is itself related to all other positive integer values of the Zeta function.
So λ = ζ(2)/2 - ζ(3)/3 + ζ(4)/4 - ζ(5)/5 +.....
Therefore when we confine ourselves to finite values of n, the value of log n (representing the average gap between primes) can be expressed in terms of the positive integer values for s of the Zeta Function
i.e. ζ(1)/1 - ζ(2)/2 + ζ(3)/3 - ζ(4)/4 + ζ(5)/5 -......
We can likewise modify the initial estimate of the change in average spread between primes (in the region of n).
So now using all of these Zeta values a better approximation is given by
1/n - 1/{2*(n^2)} + 1/{3*(n^3)} - 1/{4*(n^4)} +.....
We could also express the above as the value of log (n + 1) - log n
However as n becomes very large the modifications to the initial estimate of the change in spread (i.e. 1/n) are so small as to be negligible. Therefore 1/n serves as a particularly good estimate!
We can also express log n in another way as the sum of Zeta function values for positive integer values of s > 1.
Therefore the average gap between primes is given as
ζ(2)/2 + ζ(3)/3 + ζ(4)/4 + ζ(5)/5 +.....
So n/log n representing the frequency of primes (among the first n natural numbers)
→ n/{ζ(1)/1 - ζ(2)/2 + ζ(3)/3 - ζ(4)/4 + ζ(5)/5 -…..}
or alternatively,
→ n/{ζ(2)/2 + ζ(3)/3 + ζ(4)/4 + ζ(5)/5 +…..}
As we have seen n in fact represents the value ζ(0) where the sum of terms is taken over a finite limited range.
Therefore
n/log n → ζ(0)/{ζ(1)/1 - ζ(2)/2 + ζ(3)/3 - ζ(4)/4 + ζ(5)/5 -…..}
or alternatively,
n/log n → ζ(0)/{ζ(2)/2 + ζ(3)/3 + ζ(4)/4 + ζ(5)/5 +…..}
Once again - as an initial approximation - the sum of the first n terms of the series provides an estimate of the gap as between primes (in the region of n).
Secondly the reciprocal of n provides a good estimate of the change in the gap as between primes in the region of n.
So as we have seen as we move for example from 1,000,000 to 1,000,001 the change in the gap would be close to 1/1,000,000.
Now the extent to which the harmonic series falls short in terms of predicting the actual gap between primes is explained by Euler's Constant (= .5772...).
More accurately the gap between primes (in the region of n) is given by log n.
And as the harmonic series approximates log n + λ (where λ = Euler's Constant) then using the harmonic series as a prediction will overestimate the gap by .5772.
However remarkably the value of Euler's Constant is itself related to all other positive integer values of the Zeta function.
So λ = ζ(2)/2 - ζ(3)/3 + ζ(4)/4 - ζ(5)/5 +.....
Therefore when we confine ourselves to finite values of n, the value of log n (representing the average gap between primes) can be expressed in terms of the positive integer values for s of the Zeta Function
i.e. ζ(1)/1 - ζ(2)/2 + ζ(3)/3 - ζ(4)/4 + ζ(5)/5 -......
We can likewise modify the initial estimate of the change in average spread between primes (in the region of n).
So now using all of these Zeta values a better approximation is given by
1/n - 1/{2*(n^2)} + 1/{3*(n^3)} - 1/{4*(n^4)} +.....
We could also express the above as the value of log (n + 1) - log n
However as n becomes very large the modifications to the initial estimate of the change in spread (i.e. 1/n) are so small as to be negligible. Therefore 1/n serves as a particularly good estimate!
We can also express log n in another way as the sum of Zeta function values for positive integer values of s > 1.
Therefore the average gap between primes is given as
ζ(2)/2 + ζ(3)/3 + ζ(4)/4 + ζ(5)/5 +.....
So n/log n representing the frequency of primes (among the first n natural numbers)
→ n/{ζ(1)/1 - ζ(2)/2 + ζ(3)/3 - ζ(4)/4 + ζ(5)/5 -…..}
or alternatively,
→ n/{ζ(2)/2 + ζ(3)/3 + ζ(4)/4 + ζ(5)/5 +…..}
As we have seen n in fact represents the value ζ(0) where the sum of terms is taken over a finite limited range.
Therefore
n/log n → ζ(0)/{ζ(1)/1 - ζ(2)/2 + ζ(3)/3 - ζ(4)/4 + ζ(5)/5 -…..}
or alternatively,
n/log n → ζ(0)/{ζ(2)/2 + ζ(3)/3 + ζ(4)/4 + ζ(5)/5 +…..}
Wednesday, October 27, 2010
More on Finite and Infinite
I have already stated that the conventional interpretation of the infinite leads to a reduced notion whereby it is viewed as an extension of the finite.
For example we still routinely refer to series as infinite e.g. the natural number series. Here the mistaken impression is given that with the terms getting progressively larger that "in the limit" they become infinite.
However strictly this is just nonsense. As terms get larger they always remain finite and at no stage pass over to a so-called infinite state.
It would be valid to say that we cannot set a limit to the size of possible terms in the series (while still accepting that they always remain finite). But this is quite different from saying that they become infinite!
A similar problem results in the treatment of "infinitesmals" which again is a meaningless notion.
As a quantity becomes progressively smaller, it still remains finite and at no stage becomes infinitesmal.
Though it may be quite appropriate to approximate such quantities as 0 when obtaining the numerical value of a calculation, an important qualitative distinction remains in that such calculations are always of a merely relative nature.
Put yet another way, the distinction as between infinite and finite concepts relates to the corresponding distinction as between potential and actual notions.
For example the number concept is of an infinite nature potentially relating to any number (not actually specified). However a specific number perception e.g. 2 is then actualised in a finite manner. Thus the dynamic interaction that necessarily exists in experience as between the number concept and related perceptions inevitably combines both finite and infinite notions.
Thus to avoid confusing the infinite with the finite we need to recognise that - again in dynamic terms - the finite determination of any number always implies that other finite numbers thereby remain indeterminate.
Thus what is loosely - and inaccurately - referred to as an infinite series relates to a situation where the progressive determination of certain terms implies that other terms remain actually indeterminate!
Therefore when correctly understood in dynamic experiential terms, the uncertainty principle applies to all mathematical interpretation.
Once again this relates to the fact that (discrete) rational and (continuous) intuitive aspects are always entailed with respect to understanding. Thus we can only make greater clarity with respect to one aspect through accepting corresponding fuzziness with respect to the other.
And as I have been demonstrating the apparent absolute nature of Conventional Mathematics stems from the - mistaken - view that mathematical interpretation is merely of a rational nature!
Once again when we properly accept the true interactive nature of mathematical possibility of "proof" in an absolute sense disappears.
The "proof" of a general hypothesis strictly relates to its merely potential nature as applying to all (non-specified) cases which is of an infinite continuous nature. However the application of the "proof" to any specific case entails actualisation in a finite discrete manner. So to maintain that the general "proof" logically applies to the specific case entails a basic confusion (whereby the infinite is reduced to finite notions).
This is not to suggest that mathematical "proof" is thereby of no value. Rather it is to suggest that it is subject to the uncertainty principle and thereby of a merely probable nature. Indeed momentary reflection on the issue will reveal that the very acceptance of mathematical "proof" entails a certain form of social consensus that can later transpire to have been mistaken. For example Andrew Wiles first proof of "Fermat's Last Theorem" was found to be in error after it had already passed through a rigorous process of verification. So the present status of the revised proof is strictly of a probable nature. In other words as time goes by with no further errors being found we can accept its probable truth with an ever greater degree of confidence!
So in dynamic experiential terms (which represents the true nature of mathematical understanding) all "proof" is subject to the uncertainty principle.
However even within the reduced assumptions of Conventional Mathematics there are certain problems that in principle can be shown to have no proof (or disproof).
And chief among these problems is the Riemann Hypothesis.
For example we still routinely refer to series as infinite e.g. the natural number series. Here the mistaken impression is given that with the terms getting progressively larger that "in the limit" they become infinite.
However strictly this is just nonsense. As terms get larger they always remain finite and at no stage pass over to a so-called infinite state.
It would be valid to say that we cannot set a limit to the size of possible terms in the series (while still accepting that they always remain finite). But this is quite different from saying that they become infinite!
A similar problem results in the treatment of "infinitesmals" which again is a meaningless notion.
As a quantity becomes progressively smaller, it still remains finite and at no stage becomes infinitesmal.
Though it may be quite appropriate to approximate such quantities as 0 when obtaining the numerical value of a calculation, an important qualitative distinction remains in that such calculations are always of a merely relative nature.
Put yet another way, the distinction as between infinite and finite concepts relates to the corresponding distinction as between potential and actual notions.
For example the number concept is of an infinite nature potentially relating to any number (not actually specified). However a specific number perception e.g. 2 is then actualised in a finite manner. Thus the dynamic interaction that necessarily exists in experience as between the number concept and related perceptions inevitably combines both finite and infinite notions.
Thus to avoid confusing the infinite with the finite we need to recognise that - again in dynamic terms - the finite determination of any number always implies that other finite numbers thereby remain indeterminate.
Thus what is loosely - and inaccurately - referred to as an infinite series relates to a situation where the progressive determination of certain terms implies that other terms remain actually indeterminate!
Therefore when correctly understood in dynamic experiential terms, the uncertainty principle applies to all mathematical interpretation.
Once again this relates to the fact that (discrete) rational and (continuous) intuitive aspects are always entailed with respect to understanding. Thus we can only make greater clarity with respect to one aspect through accepting corresponding fuzziness with respect to the other.
And as I have been demonstrating the apparent absolute nature of Conventional Mathematics stems from the - mistaken - view that mathematical interpretation is merely of a rational nature!
Once again when we properly accept the true interactive nature of mathematical possibility of "proof" in an absolute sense disappears.
The "proof" of a general hypothesis strictly relates to its merely potential nature as applying to all (non-specified) cases which is of an infinite continuous nature. However the application of the "proof" to any specific case entails actualisation in a finite discrete manner. So to maintain that the general "proof" logically applies to the specific case entails a basic confusion (whereby the infinite is reduced to finite notions).
This is not to suggest that mathematical "proof" is thereby of no value. Rather it is to suggest that it is subject to the uncertainty principle and thereby of a merely probable nature. Indeed momentary reflection on the issue will reveal that the very acceptance of mathematical "proof" entails a certain form of social consensus that can later transpire to have been mistaken. For example Andrew Wiles first proof of "Fermat's Last Theorem" was found to be in error after it had already passed through a rigorous process of verification. So the present status of the revised proof is strictly of a probable nature. In other words as time goes by with no further errors being found we can accept its probable truth with an ever greater degree of confidence!
So in dynamic experiential terms (which represents the true nature of mathematical understanding) all "proof" is subject to the uncertainty principle.
However even within the reduced assumptions of Conventional Mathematics there are certain problems that in principle can be shown to have no proof (or disproof).
And chief among these problems is the Riemann Hypothesis.
Tuesday, October 26, 2010
The Harmonic Series
The Pythgoreans have a crucial input into true appreciation of the Riemann Hypothesis in at least two important ways.
We have already discussed the first of these relating to the square root of 2 in "The Pythagorean Dilemma".
The Pythagoreans realised to their dismay that this number was irrational. It did not suffice for them to simply prove in quantitative terms that the number was indeed irrational. More importantly they were seeking a qualitative appreciation as to how such a number could arise.
The essence of an irrational number is that it contains both finite and infinite aspects. In the subsequent mathematical understanding however of irrational numbers a solely reduced quantitative interpretation is provided. Thus the real need, which the Pythagoreans clearly realised, to provide both quantitative an qualitative interpretation has thereby been avoided.
The square root of 2 also points to a circular appreciation of number where - paradoxically - both positive and negative interpretations can both be correct.
So correct to 4 decimal places the square root of 2 can be expressed as + 1.4142 or - 1.4142.
Now the relevance of all this to the Riemann Hypothesis is that we obtain the square root by raising 2 to the dimensional power of 1/2.
And of course the Riemann Hypothesis states that all the non-trivial zeros of the zeta function will relate to values of s i.e dimensional powers with real part = 1/2.
However the Pythagoreans became equally famous for the discovery of the significance of what has come to be known (in their memory) as the harmonic series.
This is the simple sequence comprising the reciprocals of the natural nos.
1 + 1/2 + 1/3 + 1/4 +.....
Now the Pythagoreans were able to connect this series with musical harmonics. They seemed to realise that if a vessel full of water was struck and then successively also struck when half full, a third full, a quarter full etc. that the musical notes generated would appear harmonious to the ear.
Indeed this provided for them striking confirmation of the overriding importance of the natural numbers in explaining nature's secrets. So the very terms "music of the spheres" derives from this discovery.
However the harmonic series provides the starting base for - what is known as - Riemann's Zeta Function where each of the natural number reciprocals can be raised not alone to the dimensional power of 1 but to any complex number power.
However the harmonic series in itself can be shown to have a striking relevance for understanding the behaviour of the primes.
The prime number theorem is often stated in its simplest form as where the ratio of n/log n to the true frequency of primes approaches 1 as n becomes progressively larger.
This implies that log n provides a very good measurement (especially for large values of n) of the average gap as between successive prime numbers.
However there is a close connection as between the harmonic series and log n.
In fact the harmonic series (where the denominator ranges over the natural numbers from 1 to n) = log n + k (where k is known as Euler's constant = .5772 approx).
Thus for very large values of n the sum of the harmonic series itself provides a very accurate estimate of the average gap between successive prime numbers.
So this provides just one striking example as to the close connection as between the primes and natural numbers!
It is also remarkable in another sense. Once again the sum of the harmonic series (where n is finite) = log n + k.
Thus when we differentiate with respect to n we get 1/n.
This therefore implies that the average gap as between primes increases by 1/n (as n increases by 1)
This would imply for example that as move from n = 1,000,000 to 1,000,001 that the average gap between primes itself increases by 1/1,000,000 (or perhaps more accurately 1/1,000,000.5).
This is truly remarkable in that it links an important aspect of prime number behaviour in an extremely simple manner to the reciprocals of the natural numbers!
It is also fascinating that the famous formula n/log n relating to the frequency of the primes can be expressed as the ratio of the two zeta functions (where n is taken over a finite range) for s = o and s = 1 respectively.
When s = o, the zeta function
= 1/1^0 + 1/2^0 + 1/3^0 + 1/4^0 + .....
= 1 + 1 + 1 + 1 +...... = n
When s = 1, the zeta function
= 1/1^1 + 1/2^1 + 1/3^1 + 1/4^1 + ..... (which approximates to log n when n is suitably large)
Therefore we can approximate n/log n as
zeta (0)/zeta (1) where the range of values for n is finite (and values of series calculated in the conventional linear manner).
So perhaps (though more inaccurate for lower values of n) this could provide the simplest formulation of the prime number theorem,
i.e. {zeta (0)/zeta (1)}/{actual occurrence of primes from 1 to n} approximates 1(when n is sufficiently large in magnitude).
We have already discussed the first of these relating to the square root of 2 in "The Pythagorean Dilemma".
The Pythagoreans realised to their dismay that this number was irrational. It did not suffice for them to simply prove in quantitative terms that the number was indeed irrational. More importantly they were seeking a qualitative appreciation as to how such a number could arise.
The essence of an irrational number is that it contains both finite and infinite aspects. In the subsequent mathematical understanding however of irrational numbers a solely reduced quantitative interpretation is provided. Thus the real need, which the Pythagoreans clearly realised, to provide both quantitative an qualitative interpretation has thereby been avoided.
The square root of 2 also points to a circular appreciation of number where - paradoxically - both positive and negative interpretations can both be correct.
So correct to 4 decimal places the square root of 2 can be expressed as + 1.4142 or - 1.4142.
Now the relevance of all this to the Riemann Hypothesis is that we obtain the square root by raising 2 to the dimensional power of 1/2.
And of course the Riemann Hypothesis states that all the non-trivial zeros of the zeta function will relate to values of s i.e dimensional powers with real part = 1/2.
However the Pythagoreans became equally famous for the discovery of the significance of what has come to be known (in their memory) as the harmonic series.
This is the simple sequence comprising the reciprocals of the natural nos.
1 + 1/2 + 1/3 + 1/4 +.....
Now the Pythagoreans were able to connect this series with musical harmonics. They seemed to realise that if a vessel full of water was struck and then successively also struck when half full, a third full, a quarter full etc. that the musical notes generated would appear harmonious to the ear.
Indeed this provided for them striking confirmation of the overriding importance of the natural numbers in explaining nature's secrets. So the very terms "music of the spheres" derives from this discovery.
However the harmonic series provides the starting base for - what is known as - Riemann's Zeta Function where each of the natural number reciprocals can be raised not alone to the dimensional power of 1 but to any complex number power.
However the harmonic series in itself can be shown to have a striking relevance for understanding the behaviour of the primes.
The prime number theorem is often stated in its simplest form as where the ratio of n/log n to the true frequency of primes approaches 1 as n becomes progressively larger.
This implies that log n provides a very good measurement (especially for large values of n) of the average gap as between successive prime numbers.
However there is a close connection as between the harmonic series and log n.
In fact the harmonic series (where the denominator ranges over the natural numbers from 1 to n) = log n + k (where k is known as Euler's constant = .5772 approx).
Thus for very large values of n the sum of the harmonic series itself provides a very accurate estimate of the average gap between successive prime numbers.
So this provides just one striking example as to the close connection as between the primes and natural numbers!
It is also remarkable in another sense. Once again the sum of the harmonic series (where n is finite) = log n + k.
Thus when we differentiate with respect to n we get 1/n.
This therefore implies that the average gap as between primes increases by 1/n (as n increases by 1)
This would imply for example that as move from n = 1,000,000 to 1,000,001 that the average gap between primes itself increases by 1/1,000,000 (or perhaps more accurately 1/1,000,000.5).
This is truly remarkable in that it links an important aspect of prime number behaviour in an extremely simple manner to the reciprocals of the natural numbers!
It is also fascinating that the famous formula n/log n relating to the frequency of the primes can be expressed as the ratio of the two zeta functions (where n is taken over a finite range) for s = o and s = 1 respectively.
When s = o, the zeta function
= 1/1^0 + 1/2^0 + 1/3^0 + 1/4^0 + .....
= 1 + 1 + 1 + 1 +...... = n
When s = 1, the zeta function
= 1/1^1 + 1/2^1 + 1/3^1 + 1/4^1 + ..... (which approximates to log n when n is suitably large)
Therefore we can approximate n/log n as
zeta (0)/zeta (1) where the range of values for n is finite (and values of series calculated in the conventional linear manner).
So perhaps (though more inaccurate for lower values of n) this could provide the simplest formulation of the prime number theorem,
i.e. {zeta (0)/zeta (1)}/{actual occurrence of primes from 1 to n} approximates 1(when n is sufficiently large in magnitude).
Sunday, October 24, 2010
Finite and Infinite
Ultimately the true nature of the Riemann Hypothesis pertains to the key relationship as between finite and infinite (not just in Mathematics but with respect to all living experience).
Indeed the various ways in which I have already expressed this relationship represent the finite/infinite connection.
For example we can formulate this as the essential relationship of quantitative and qualitative, linear and circular, discrete and continuous, classical and quantum mechanical, order and chaos etc.
However conventional mathematical understanding is properly geared solely for (rational) finite interpretation. Though infinite notions are of course recognised they are treated in a strictly reduced manner (i.e. that is amenable to rational type analysis).
In other words the true meaning of the infinite is thereby lost in Mathematics.
This remains the - largely unrecognised - elephant in the room as its key overriding central problem.
To understand what is involved here we have to once again recognise that actual mathematical understanding is never strictly rational (though explicitly in formal terms it is indeed represented as rational!)
Rather, such understanding always involves a dynamic interaction of both rational (conscious) and intuitive (unconscious) elements.
Put quite simply, whereas the rational element of this interaction pertains directly to appreciation of the finite, by contrast the intuitive aspect pertains directly to the infinite.
Though implicitly mathematicians may indeed recognise the importance of intuition - especially in the generation of creative insights - explicitly however Conventional Mathematics is interpreted in a merely reduced rational manner (where in effect the holistic notion of the infinite is reduced in a finite manner).
Thus if we are to properly incorporate the infinite with finite notions then we must include a new qualitative type of mathematical understanding (which I call Holistic Mathematics).
However it is important to recognise that Holistic Mathematics operates in terms of a distinctive logical system (which is circular in nature).
It is only through the appropriate circular use of logic that intuition - pertaining directly to appreciation of the infinite in experience - can be indirectly translated in rational terms.
Once again I will briefly express the key difference here as between the two logical systems.
With linear logic - which defines conventional mathematical experience - the key polarities of experience are treated as separate and independent.
For example internal and external - which necessarily underpin all phenomenal experience - represents one important example of such key polarities.
So in Mathematics the external (objective) aspect is treated as independent of the internal (subjective) aspect. This thereby creates the impression of a strict objective validity to mathematical truths (which ultimately is unwarranted).
Likewise in Mathematics, whole notions are treated as independent of parts so that we can study either aspect in isolation from each other. In fact perhaps the most characteristic feature of the linear approach is the manner in which wholes are reduced to parts. In this way both aspects can be treated in a merely quantitative manner (where the whole is literally seen as the sum of the parts).
Thus the key qualitative distinction as between wholes and parts is thereby lost through such interpretation!
With circular logic the key polarities are treated as interdependent (rather than separate). Ultimately of course this means that with full interdependence any notion of polarities as phenomenally separate disappears. We are then left in experience with pure formless appreciation (which is the very nature of intuitive awareness).
However we can characterise the nature of such awareness in an indirect rational manner through the complementarity of opposites. This leads to the recognition - like left and right turns on a road - that opposites have a merely arbitrary definition in any context (depending on the polar frame of reference). So for example what is considered as (quantitatively) whole in one context could be considered as part in another (and vice versa). Now the mysterious dynamic enabling this switching of reference poles is of a qualitative (intuitive) nature and not thereby confused with rational interpretation in isolated contexts.
Put another way - deliberately using holistic mathematical language - there are two key tasks in all understanding (which thereby includes Mathematics). First we must successfully differentiate phenomenal symbols (as independent); secondly we must successfully integrate those same symbols (as interdependent).
In actual mathematical experience, the first of these tasks (of differentiation) is properly achieved through (linear) reason using either/or logic; however the second equally important task (of integration) is properly achieved though intuition that is indirectly represented in a paradoxical manner through circular reason (using both/and logic).
Once again there is a huge unrecognised problem with conventional mathematical interpretation in that is based solely on (linear) reason. Therefore it can only deal with the qualitative task of integral interpretation in a reduced manner.
The relevance of all this for the Riemann Hypothesis is that - properly understood - it actually points to the fundamental condition for the successful harmonisation of both (analytic) quantitative and (holistic) qualitative type understanding.
Imagine in geometrical terms a straight line diameter = 1 unit circumscribed by its circular circumference. Now the midpoint which is common to both circle and line occurs at the midpoint of the line (i.e. at 1/2).
In a nutshell this is what the Riemann Hypothesis is all about i.e. the central condition that is necessary to reconcile both the quantitative (linear) and qualitative (circular) aspects of mathematical understanding.
Indeed the various ways in which I have already expressed this relationship represent the finite/infinite connection.
For example we can formulate this as the essential relationship of quantitative and qualitative, linear and circular, discrete and continuous, classical and quantum mechanical, order and chaos etc.
However conventional mathematical understanding is properly geared solely for (rational) finite interpretation. Though infinite notions are of course recognised they are treated in a strictly reduced manner (i.e. that is amenable to rational type analysis).
In other words the true meaning of the infinite is thereby lost in Mathematics.
This remains the - largely unrecognised - elephant in the room as its key overriding central problem.
To understand what is involved here we have to once again recognise that actual mathematical understanding is never strictly rational (though explicitly in formal terms it is indeed represented as rational!)
Rather, such understanding always involves a dynamic interaction of both rational (conscious) and intuitive (unconscious) elements.
Put quite simply, whereas the rational element of this interaction pertains directly to appreciation of the finite, by contrast the intuitive aspect pertains directly to the infinite.
Though implicitly mathematicians may indeed recognise the importance of intuition - especially in the generation of creative insights - explicitly however Conventional Mathematics is interpreted in a merely reduced rational manner (where in effect the holistic notion of the infinite is reduced in a finite manner).
Thus if we are to properly incorporate the infinite with finite notions then we must include a new qualitative type of mathematical understanding (which I call Holistic Mathematics).
However it is important to recognise that Holistic Mathematics operates in terms of a distinctive logical system (which is circular in nature).
It is only through the appropriate circular use of logic that intuition - pertaining directly to appreciation of the infinite in experience - can be indirectly translated in rational terms.
Once again I will briefly express the key difference here as between the two logical systems.
With linear logic - which defines conventional mathematical experience - the key polarities of experience are treated as separate and independent.
For example internal and external - which necessarily underpin all phenomenal experience - represents one important example of such key polarities.
So in Mathematics the external (objective) aspect is treated as independent of the internal (subjective) aspect. This thereby creates the impression of a strict objective validity to mathematical truths (which ultimately is unwarranted).
Likewise in Mathematics, whole notions are treated as independent of parts so that we can study either aspect in isolation from each other. In fact perhaps the most characteristic feature of the linear approach is the manner in which wholes are reduced to parts. In this way both aspects can be treated in a merely quantitative manner (where the whole is literally seen as the sum of the parts).
Thus the key qualitative distinction as between wholes and parts is thereby lost through such interpretation!
With circular logic the key polarities are treated as interdependent (rather than separate). Ultimately of course this means that with full interdependence any notion of polarities as phenomenally separate disappears. We are then left in experience with pure formless appreciation (which is the very nature of intuitive awareness).
However we can characterise the nature of such awareness in an indirect rational manner through the complementarity of opposites. This leads to the recognition - like left and right turns on a road - that opposites have a merely arbitrary definition in any context (depending on the polar frame of reference). So for example what is considered as (quantitatively) whole in one context could be considered as part in another (and vice versa). Now the mysterious dynamic enabling this switching of reference poles is of a qualitative (intuitive) nature and not thereby confused with rational interpretation in isolated contexts.
Put another way - deliberately using holistic mathematical language - there are two key tasks in all understanding (which thereby includes Mathematics). First we must successfully differentiate phenomenal symbols (as independent); secondly we must successfully integrate those same symbols (as interdependent).
In actual mathematical experience, the first of these tasks (of differentiation) is properly achieved through (linear) reason using either/or logic; however the second equally important task (of integration) is properly achieved though intuition that is indirectly represented in a paradoxical manner through circular reason (using both/and logic).
Once again there is a huge unrecognised problem with conventional mathematical interpretation in that is based solely on (linear) reason. Therefore it can only deal with the qualitative task of integral interpretation in a reduced manner.
The relevance of all this for the Riemann Hypothesis is that - properly understood - it actually points to the fundamental condition for the successful harmonisation of both (analytic) quantitative and (holistic) qualitative type understanding.
Imagine in geometrical terms a straight line diameter = 1 unit circumscribed by its circular circumference. Now the midpoint which is common to both circle and line occurs at the midpoint of the line (i.e. at 1/2).
In a nutshell this is what the Riemann Hypothesis is all about i.e. the central condition that is necessary to reconcile both the quantitative (linear) and qualitative (circular) aspects of mathematical understanding.
Thursday, October 14, 2010
Riemann Hypothesis and Physical Connections
When I first read about connections as between the Riemann non-trivial zeros and certain physical energy states in quantum physics, I was not at all surprised.
For I had long believed that important physical applications of the Riemann zeros would necessarily exist.
Indeed I would go considerably further. Not alone do the zeros have implications for physics but equally they have a deep relevance for psychological understanding of various spiritual energy states. Furthermore in the holistic mathematical understanding of what is involved the two forms of understanding (physical and psychological) are fully complementary.
Though I have always recognised - in principle - the potential physical relevance of the zeros (and of course the associated Riemann Hypothesis) my main focus has been on the psychological relevance of the Riemann zeros.
And here there is an interesting twist! For in attempting to provide a coherent interpretation of what the Hypothesis actually entails (which then leads to a simple resolution of the Hypothesis), I have found the trivial zeros to be of greater significance.
Indeed my initial interest in the Riemann Hypothesis was sparked by the desire to provide a coherent explanation for the trivial zeros.
For example if the take the first of these for the zeta function (where s = -2) this would result in the series
1 + 4 + 9 + 16 +.....
Now clearly from a conventional linear quantitative perspective, this series diverges so that it has no finite sum.
However according to Riemann's Zeta Function, the sum of this infinite series = 0.
So it was through attempting to explain the nature of this unexpected value that I realised that it referred directly - not to a quantitative but rather - to a qualitative interpretation of number.
Basically in qualitative terms, 2-dimensional understanding refers to the complementarity of (real) polar opposites in experience such as internal and external. Though in linear (1-dimensional) terms these are clearly separated in experience enabling for example the unambiguous objective interpretation of mathematical calculations, in circular (2-dimensional) terms these are seen as complementary and interdependent (with ultimately no division possible between them).
Strictly 2-dimensional understanding (as positive) relates to the rational paradoxical interpretation of this circular type relationship.
However 2-dimensional understanding (as negative) relates to the purely intuitive appreciation of the same relationship (where secondary rational distinctions are negated).
Put another way when s = - 2, correct qualitative understanding relates to a pure contemplative i.e. spiritual energy state. (It would certainly have resonated with the Pythagoreans before the unfortunate subsequent split in mathematical understanding!) Though it is correctly represented as zero, it relates directly in this instance to a qualitative - rather than quantitative - meaning.
Therefore we give a coherent numerical explanation to the first of these trivial zeros through interpreting the relationship directly in qualitative terms. Equally all the other trivial zeros can be explained as "higher" intuitive contemplative states of understanding (that are nothing in phenomenal terms).
Riemann also provided a fascinating transformation formula enabling one to calculate from conventional values for s > 1, corresponding non-conventional values for s < 0. So the clear implication here is that the transformation formula - when correctly understood - provides a clear (indissoluble) relationship as between numerical results with a direct quantitative value (for s > 1) on the right hand side of his equation and corresponding results with a direct qualitative holistic value (for s < 0) on the left hand side.
We can also continue this procedure in the critical strip (0 < s < 1) where now values for the zeta series represent a hybrid mix of both quantitative and qualitative aspects. The exact matching of both left and right hand side values requires that s = 1/2 where - by definition - quantitative and qualitative aspects exactly match with each other.
And this in short is what the Riemann Hypothesis is all about i.e. the crucial condition enabling the consistency of both quantitative and qualitative aspects of mathematical interpretation.
Unfortunately in a Mathematics that only formally recognises the quantitative aspect of understanding the true nature of the Riemann Hypothesis will always prove elusive.
Incidentally attempts to "prove" the Riemann Hypothesis with respect to establishing an exact correspondence with certain physical energy states seems to me somewhat confused.
Though it would indeed be exciting and important to conclusively establish such a link this would only demonstrate an especially strong correlation as between corresponding mathematical and physical systems.
However perfect correlation - even if demonstrated to exist - does not constitute proof!
For I had long believed that important physical applications of the Riemann zeros would necessarily exist.
Indeed I would go considerably further. Not alone do the zeros have implications for physics but equally they have a deep relevance for psychological understanding of various spiritual energy states. Furthermore in the holistic mathematical understanding of what is involved the two forms of understanding (physical and psychological) are fully complementary.
Though I have always recognised - in principle - the potential physical relevance of the zeros (and of course the associated Riemann Hypothesis) my main focus has been on the psychological relevance of the Riemann zeros.
And here there is an interesting twist! For in attempting to provide a coherent interpretation of what the Hypothesis actually entails (which then leads to a simple resolution of the Hypothesis), I have found the trivial zeros to be of greater significance.
Indeed my initial interest in the Riemann Hypothesis was sparked by the desire to provide a coherent explanation for the trivial zeros.
For example if the take the first of these for the zeta function (where s = -2) this would result in the series
1 + 4 + 9 + 16 +.....
Now clearly from a conventional linear quantitative perspective, this series diverges so that it has no finite sum.
However according to Riemann's Zeta Function, the sum of this infinite series = 0.
So it was through attempting to explain the nature of this unexpected value that I realised that it referred directly - not to a quantitative but rather - to a qualitative interpretation of number.
Basically in qualitative terms, 2-dimensional understanding refers to the complementarity of (real) polar opposites in experience such as internal and external. Though in linear (1-dimensional) terms these are clearly separated in experience enabling for example the unambiguous objective interpretation of mathematical calculations, in circular (2-dimensional) terms these are seen as complementary and interdependent (with ultimately no division possible between them).
Strictly 2-dimensional understanding (as positive) relates to the rational paradoxical interpretation of this circular type relationship.
However 2-dimensional understanding (as negative) relates to the purely intuitive appreciation of the same relationship (where secondary rational distinctions are negated).
Put another way when s = - 2, correct qualitative understanding relates to a pure contemplative i.e. spiritual energy state. (It would certainly have resonated with the Pythagoreans before the unfortunate subsequent split in mathematical understanding!) Though it is correctly represented as zero, it relates directly in this instance to a qualitative - rather than quantitative - meaning.
Therefore we give a coherent numerical explanation to the first of these trivial zeros through interpreting the relationship directly in qualitative terms. Equally all the other trivial zeros can be explained as "higher" intuitive contemplative states of understanding (that are nothing in phenomenal terms).
Riemann also provided a fascinating transformation formula enabling one to calculate from conventional values for s > 1, corresponding non-conventional values for s < 0. So the clear implication here is that the transformation formula - when correctly understood - provides a clear (indissoluble) relationship as between numerical results with a direct quantitative value (for s > 1) on the right hand side of his equation and corresponding results with a direct qualitative holistic value (for s < 0) on the left hand side.
We can also continue this procedure in the critical strip (0 < s < 1) where now values for the zeta series represent a hybrid mix of both quantitative and qualitative aspects. The exact matching of both left and right hand side values requires that s = 1/2 where - by definition - quantitative and qualitative aspects exactly match with each other.
And this in short is what the Riemann Hypothesis is all about i.e. the crucial condition enabling the consistency of both quantitative and qualitative aspects of mathematical interpretation.
Unfortunately in a Mathematics that only formally recognises the quantitative aspect of understanding the true nature of the Riemann Hypothesis will always prove elusive.
Incidentally attempts to "prove" the Riemann Hypothesis with respect to establishing an exact correspondence with certain physical energy states seems to me somewhat confused.
Though it would indeed be exciting and important to conclusively establish such a link this would only demonstrate an especially strong correlation as between corresponding mathematical and physical systems.
However perfect correlation - even if demonstrated to exist - does not constitute proof!
Wednesday, October 13, 2010
The Invisible Gap
In the most fundamental sense, the Riemann Hypothesis relates to the invisible gap, as it were, that divides quantitative (linear) and qualitative (circular) type interpretation of reality. In psychological terms this relates to the inevitable interaction that necessarily takes place with respect to all understanding (including of course mathematical) between rational (analytic) and intuitive (holistic) type processes of understanding.
Unfortunately the problem for Conventional Mathematics is that it in formal terms it recognises solely the role of rational interpretation. Therefore it can only deal with this interaction in a reduced manner i.e. by attempting to explain - what properly relates to - the rational and intuitive, in merely rational terms.
Alternatively, it can only attempt to deal with the relationship of both quantitative and qualitative type mathematical understanding in a reduced quantitative manner.
And this is in a nutshell is the very reason why a satisfactory "proof" of the Riemann Hypothesis has proven so elusive.
Quite simply - when the nature of the problem is properly appreciated - the Riemann Hypothesis can have no solution in conventional mathematical terms.
This same problem can be expressed in other ways that directly impinge on the understanding of the true nature of the Hypothesis.
For example the Riemann Hypothesis lies in that invisible gap where the discrete and continuous interpretation of number is united. And this is central to appreciation of the true nature of prime numbers in the attempt to successfully unite their discete individual identities with the continuous nature of their overall frequency among the natural numbers.
There is a key problem here which again is not properly recognised. The study of individual primes and their overall general frequency have both quantitative and qualitative aspects that are of a (conscious) analytic and (unconscious) holistic nature with respect to each other. Just we can choose in isolation to investigate an atomic particle with respect to either its particle or wave aspect in quantitative terms, likewise we can attempt to study in isolation both individual primes and their general distribution with respect to their quantitative characteristics.
However this approach will inevitably break down in the simultaneous integration of both aspects (which are now complementary). So here we must incorporate both quantitative and qualitative type appreciation. Conventional Mathematics by its very nature is not geared for this task. Not alone can it not resolve the Hypothesis, it is not even capable of providing a coherent explanation of its true nature.
Another way of expressing the same problem is that all classical systems have counterpart systems that are quantum mechanical in nature. Once again we can indeed attempt to study both systems in a separate manner with respect to their - mere - quantitative aspects. However if we wish to properly relate both types once again we have to broaden appreciation to include both quantitative and qualitative modes of interpretation.
Finally, as is now evident from particle physics, the Riemann zeros can be given a coherent explanation in terms of the energy states of certain chaotic quantum mechanical systems. This again clearly points in my mind to an inevitable interaction as between two distinct modes of behaviour with properties that are analytic and holistic with respect to each other.
However through all this the elephant in the room is ignored.
There is an entirely distinctive holistic interpretation that can be given to every mathematical symbol (in what I term Holistic Mathematics) that constitutes the missing qualitative aspect of mathematical understanding.
It is in the very relationship of the two aspects of Mathematics (quantitative and qualitative) that the simple resolution of the Riemann Hypothesis is found not as a proof but rather as a fundamental axiom that underlies all the lesser axioms on which conventional mathematical interpretation is based.
When asked once what was the most important problem in Mathematics, the great mathematician Hilbert - as claimed in Constance Reid's book "Hilbert" - replied.
"The problem of the zeros of the zeta function. Not only in mathematics. But absolutely most important"
And in fact Hilbert was right!
There is a famous sutra in Buddhism which ultimately is equivalent to the Riemann Hypothesis
"Form is not other than Void; Void is not other than Form".
One could equally say with respect to Mathematics.
"The quantitative aspect is not other than the qualitative; the qualitative is not other than the quantitative."
So it is in this mysterious intersection of quantitative and qualitative aspects that the Riemann Hypothesis resides.
So the end the resolution of the Hypothesis is attained not through reason but in that ineffable spiritual experience where quantitative and qualitative distinctions are no longer necessary (nor even can be made).
Unfortunately the problem for Conventional Mathematics is that it in formal terms it recognises solely the role of rational interpretation. Therefore it can only deal with this interaction in a reduced manner i.e. by attempting to explain - what properly relates to - the rational and intuitive, in merely rational terms.
Alternatively, it can only attempt to deal with the relationship of both quantitative and qualitative type mathematical understanding in a reduced quantitative manner.
And this is in a nutshell is the very reason why a satisfactory "proof" of the Riemann Hypothesis has proven so elusive.
Quite simply - when the nature of the problem is properly appreciated - the Riemann Hypothesis can have no solution in conventional mathematical terms.
This same problem can be expressed in other ways that directly impinge on the understanding of the true nature of the Hypothesis.
For example the Riemann Hypothesis lies in that invisible gap where the discrete and continuous interpretation of number is united. And this is central to appreciation of the true nature of prime numbers in the attempt to successfully unite their discete individual identities with the continuous nature of their overall frequency among the natural numbers.
There is a key problem here which again is not properly recognised. The study of individual primes and their overall general frequency have both quantitative and qualitative aspects that are of a (conscious) analytic and (unconscious) holistic nature with respect to each other. Just we can choose in isolation to investigate an atomic particle with respect to either its particle or wave aspect in quantitative terms, likewise we can attempt to study in isolation both individual primes and their general distribution with respect to their quantitative characteristics.
However this approach will inevitably break down in the simultaneous integration of both aspects (which are now complementary). So here we must incorporate both quantitative and qualitative type appreciation. Conventional Mathematics by its very nature is not geared for this task. Not alone can it not resolve the Hypothesis, it is not even capable of providing a coherent explanation of its true nature.
Another way of expressing the same problem is that all classical systems have counterpart systems that are quantum mechanical in nature. Once again we can indeed attempt to study both systems in a separate manner with respect to their - mere - quantitative aspects. However if we wish to properly relate both types once again we have to broaden appreciation to include both quantitative and qualitative modes of interpretation.
Finally, as is now evident from particle physics, the Riemann zeros can be given a coherent explanation in terms of the energy states of certain chaotic quantum mechanical systems. This again clearly points in my mind to an inevitable interaction as between two distinct modes of behaviour with properties that are analytic and holistic with respect to each other.
However through all this the elephant in the room is ignored.
There is an entirely distinctive holistic interpretation that can be given to every mathematical symbol (in what I term Holistic Mathematics) that constitutes the missing qualitative aspect of mathematical understanding.
It is in the very relationship of the two aspects of Mathematics (quantitative and qualitative) that the simple resolution of the Riemann Hypothesis is found not as a proof but rather as a fundamental axiom that underlies all the lesser axioms on which conventional mathematical interpretation is based.
When asked once what was the most important problem in Mathematics, the great mathematician Hilbert - as claimed in Constance Reid's book "Hilbert" - replied.
"The problem of the zeros of the zeta function. Not only in mathematics. But absolutely most important"
And in fact Hilbert was right!
There is a famous sutra in Buddhism which ultimately is equivalent to the Riemann Hypothesis
"Form is not other than Void; Void is not other than Form".
One could equally say with respect to Mathematics.
"The quantitative aspect is not other than the qualitative; the qualitative is not other than the quantitative."
So it is in this mysterious intersection of quantitative and qualitative aspects that the Riemann Hypothesis resides.
So the end the resolution of the Hypothesis is attained not through reason but in that ineffable spiritual experience where quantitative and qualitative distinctions are no longer necessary (nor even can be made).
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