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.
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