Yesterday, my attention was drawn to a headline in a newspaper relating to a story regarding “the circle of the 60 most influential people in Ireland”.
This set me thinking once again about the fundamental nature of the number system and how in fact it is substantially misrepresented in conventional mathematical terms.
We are accustomed through training – especially in a formal mathematical context – to think of number with respect merely to its quantitative aspect. Indeed the identification of people nowadays by a number (rather than a personal name) has become synonymous with the impersonal nature of modern society.
Now this quantitative impersonal treatment of number correlates well with the linear interpretation that characterises Conventional (Type 1) Mathematics.
At its deepest level the linear approach is characterised by the use of isolated uni-polar reference frames for mathematical interpretation.
All experience of reality (including of course mathematical) is necessarily conditioned by polar opposites which dynamically interact in a relative manner.
So strictly an external object such as a number has no meaning in the absence of a corresponding mental perception that - relatively – is of an internal nature.
Likewise the actual identification of a specific number in quantitative terms has no meaning in the absence of a holistic number concept (potentially applying to “all” numbers) that is - relatively - of a qualitative nature.
However when linear (one-dimensional) polar reference frames are used, numbers are misleadingly given an absolute identity (independent of subjective interpretation); likewise they are given a merely quantitative identity (independent of holistic qualitative considerations).
In other words, though the true nature of number is inherently of a dynamic interactive nature, it is misleadingly portrayed through Conventional Mathematics in a somewhat static absolute manner.
Thus, despite its so-called rigour, the most fundamental issues of mathematical interpretation are glossed over leading to reduced interpretation at every turn.
For if numbers are defined in absolute terms as independent entities, then this excludes – except in a reduced manner – corresponding consideration of their interdependence with other numbers. Thus the very notion of interdependence – which inherently is of a qualitative nature – is thereby inevitably confused in conventional mathematical terms with the quantitative aspect.
And as I have pointed out this for example is the key problem in the failure to recognise the true relationship of the primes to the natural numbers (and the natural numbers to the primes).
So the starting point for the more accurate understanding of number – and indeed all mathematical relationships – is the recognition that they must be defined in a merely relative manner (befitting the dynamic interaction entailed by complementary opposite poles).
Now it is interesting how informally the use of number with respect to its relational aspect of interdependence is characterised by a circular rather than linear reference. So a grouping of a number of friends for example (thereby entailing close knit relationships) will be referred to as a “circle of friends”.
And unfortunately we often have in society “golden circles” where the relationships of power and influence among a small group of people can be too strong and exclusive.
However this informal use of circles to suggest relationship provides a direct clue as to the nature of the qualitative aspect of mathematical understanding.
Rather than being based on the line, numbers are now considered in a circular manner. Now the roots of 1 comprise such a circular system; however we cannot possibly attempt to understand the inherently qualitative nature of this system while still using linear logic!
So the qualitative nature of the circular system of numbers is only revealed when appropriately viewed through a corresponding circular logical approach (based on the dynamic relationship between opposite poles of understanding).
Thus in this new understanding of numbers both the linear and circular aspects of interpretation are closely integrated in a dynamic interactive manner.
Therefore the linear (cardinal) aspect - Type 1 - is initially defined with respect to the relatively independent aspect of number behaviour; the circular (ordinal) aspect - Type 2 - is then defined with respect to the relatively interdependent aspect. However just as in physics, particles have wave aspects and waves particle aspects respectively, likewise it is with number.
So we will see that corresponding to this dynamic understanding, the number system possesses both a cardinal (particle) aspect that is quantitative and a corresponding ordinal (wave) aspect that is qualitative in relative terms.
However the cardinal aspect as quantitative equally has a corresponding aspect as qualitative; likewise the ordinal aspect as qualitative equally has an ordinal aspect as quantitative. And this recognition of the pure interdependence of quantitative and qualitative aspects gives rise - to what I term - Type 3 Mathematics.
Now until Riemann’s pioneering work only the cardinal quantitative aspect of number was properly realised. What Riemann has shown is that underlying this particle notion of number is a harmonic wave system (which is however viewed solely in quantitative terms).
What is still totally missing from this picture is any clear recognition of the corresponding (mirror) qualitative interpretation of number (which itself has both particle and wave aspects). And without this qualitative aspect the quantitative aspect itself cannot be properly appreciated.
So to put it simply, Riemann’s findings regarding an underlying wave harmonic structure to the number system points to its inherent dynamic nature.
However we cannot properly appreciate this dynamic nature while still trying to understand number relationships in absolute terms (with respect to their merely quantitative aspect).
The very dynamism that can now be seen to exist arises directly from the fact that number has both quantitative and qualitative aspects (in continual interaction with each other).
And with this realisation the mystery of the primes can at last be resolved where it is now seen from this dual perspective that both the natural and prime numbers (and prime numbers and natural) are perfect reflections of each other in an ultimate identity that is ineffable!
However once again, we will never appreciate this mutual identity while trying to understand numbers in a merely quantitative manner!
Monday, April 30, 2012
Thursday, April 5, 2012
Another Perspective on the Riemann Hypothesis
I have identified a central problem with Conventional Mathematics in that within its own terms of reference it has no way of satisfactorily distinguishing the cardinal and ordinal aspects of number (which relate to the quantitative and qualitative aspects of number respectively).
Once again if we attempt to give numbers an absolute quantitative identity (as independent entities) which is the rationale of Conventional Mathematics, then this leaves - by definition - no role for a corresponding qualitative identity (whereby numbers can be appropriately viewed within a relational context of interdependence with each other).
So, Conventional Mathematics can only proceed through gross reductionism (whereby the qualitative relational aspect of number is effectively reduced to the quantitative)! And this is inevitably associated - as I have been continually pointing out - by a litany of confused thinking (especially in relation to treatment of the infinite notion).
Thus, the starting point for a more comprehensive mathematical vision is that mathematical entities (including of course numbers) have both independent (quantitative) and interdependent (qualitative) aspects that are relative.
This of course implies that the cardinal and ordinal aspects of number (when properly interpreted) are necessarily of a dynamic relative nature.
One important consequence of this new relative definition (comprising complementary aspects) is that it quickly leads to corresponding recognition that numbers have both particle and wave aspects (that again are of a relative nature).
Now it might be helpful initially to identify the cardinal aspect with the (independent) particle nature of number and the ordinal aspect with its (relational) wave aspect. However ultimately - in what I refer to as Type 3 mathematical understanding - the particle is clearly seen (from a complementary perspective) to be wave like, and equally the wave aspect (again from a complementary perspective) as particle like!
However it must be stressed once again that the relationship between both of these aspects cannot be properly interpreted from a merely quantitative perspective, as these distinctions relate to the twin quantitative and qualitative aspects, (the very interaction of which properly constitutes the true nature of number)!
So, as I have repeatedly stated on this blog since the very first entry, the clear message is that its long neglected qualitative aspect now urgently needs to be incorporated within Mathematics.
Indeed it is exactly the same message for Physics as the deeper understanding of wave/particle duality in Quantum Mechanics points equally to the fact that physical reality itself cannot be properly interpreted with respect to merely its quantitative aspect!
To conclude this entry, in light of our recent discussions we can provide a new way of stating the Riemann Hypothesis as the fundamental condition necessary for maintaining consistency as between both the cardinal and ordinal aspects of number.
However once again as this relates to the ultimate reconciliation of quantitative with qualitative meaning, it is thereby futile attempting to seek a proof in - mere - quantitative terms! Indeed as I have repeatedly stated, the essential nature of the Riemann Hypothesis cannot be properly grasped in conventional mathematical terms!
Once again if we attempt to give numbers an absolute quantitative identity (as independent entities) which is the rationale of Conventional Mathematics, then this leaves - by definition - no role for a corresponding qualitative identity (whereby numbers can be appropriately viewed within a relational context of interdependence with each other).
So, Conventional Mathematics can only proceed through gross reductionism (whereby the qualitative relational aspect of number is effectively reduced to the quantitative)! And this is inevitably associated - as I have been continually pointing out - by a litany of confused thinking (especially in relation to treatment of the infinite notion).
Thus, the starting point for a more comprehensive mathematical vision is that mathematical entities (including of course numbers) have both independent (quantitative) and interdependent (qualitative) aspects that are relative.
This of course implies that the cardinal and ordinal aspects of number (when properly interpreted) are necessarily of a dynamic relative nature.
One important consequence of this new relative definition (comprising complementary aspects) is that it quickly leads to corresponding recognition that numbers have both particle and wave aspects (that again are of a relative nature).
Now it might be helpful initially to identify the cardinal aspect with the (independent) particle nature of number and the ordinal aspect with its (relational) wave aspect. However ultimately - in what I refer to as Type 3 mathematical understanding - the particle is clearly seen (from a complementary perspective) to be wave like, and equally the wave aspect (again from a complementary perspective) as particle like!
However it must be stressed once again that the relationship between both of these aspects cannot be properly interpreted from a merely quantitative perspective, as these distinctions relate to the twin quantitative and qualitative aspects, (the very interaction of which properly constitutes the true nature of number)!
So, as I have repeatedly stated on this blog since the very first entry, the clear message is that its long neglected qualitative aspect now urgently needs to be incorporated within Mathematics.
Indeed it is exactly the same message for Physics as the deeper understanding of wave/particle duality in Quantum Mechanics points equally to the fact that physical reality itself cannot be properly interpreted with respect to merely its quantitative aspect!
To conclude this entry, in light of our recent discussions we can provide a new way of stating the Riemann Hypothesis as the fundamental condition necessary for maintaining consistency as between both the cardinal and ordinal aspects of number.
However once again as this relates to the ultimate reconciliation of quantitative with qualitative meaning, it is thereby futile attempting to seek a proof in - mere - quantitative terms! Indeed as I have repeatedly stated, the essential nature of the Riemann Hypothesis cannot be properly grasped in conventional mathematical terms!
Wednesday, April 4, 2012
Riemann Zeta Function and Odd Integer Values (1)
It has been well known since the time of Euler that the value of the Zeta Function for odd integers of s (i.e. s = 3, 5, 7, 9,...etc.) behaves in a very different manner than for corresponding even values. Euler, as we know, was able to prove that for any even integer s,
ζ(s) = k*(π^s) where k is a rational fraction.
However no such relationship characterises values of the Zeta Function for odd integers of s!
Indeed it took some time to prove that the first of these values for s = 3, is irrational (Apéry's constant) though it is not known if it is transcendental.
Also though it is known that other values of the Zeta function for odd integers values must likewise be irrational, little clarity can as yet be provided as to how this applies in specific cases.
Though of course there is a validity to the the quantitative attempt of attempting to provide a (Type 1) proof as to the precise status (as number type) that characterises Zeta Function results for the odd integers, inherently such an issue relates more to qualitative - rather than quantitative - considerations.
Now the first key indication that behaviour of the Zeta Function for odd integral values is quite distinct from corresponding behaviour with respect to the even integers is given through examination of the root structure for odd integer roots which reflects in quantitative terms corresponding interpretation of these same integers (as dimensional numbers).
Whereas for even numbered integers, roots can always be arranged in a complementary manner, this is never the case for odd numbered integers.
For example if we look at the 3 roots of 1 we have,
1, - 1/2 + {[(3^(1/2)]* i}/2 and - 1/2 - {[(3^(1/2)]* i}/2, which cannot be arranged in a complementary manner.
Thus the perfect matching of independent terms with interdependence through the direct complementary relationship as between roots, characterising the pure relation of linear to circular meaning (that defines the qualitative nature of π) is thereby missing with respect to odd integer roots and corresponding dimensional numbers.
When one looks more closely at the odd numbered roots one can see that + 1 is always one of these roots that stands in a sense alone. With respect to the other roots they always comprise conjugate pairs where the imaginary part is complementary with its partner (in the pairing).
There is also a significant clue as to what this entails in qualitative terms provided through the nature of higher psychospiritual contemplative development which involves the process of traversing these higher dimensions.
Basically the pattern keeps switching from differentiation (which always implies a degree of linear type understanding) to integration (where a new harmonious contemplative state is established).
So each even numbered stage (as the qualitative development with respect to such stage) represents the restoration of a new temporary equilibrium (characterised by the attainment of a contemplative state appropriate to such development).
However each odd numbered stage by contrast represents the breaking up of that (temporary) equilibrium is a new more refined experience of linear phenomena (which always however entails a degree of dislocation and asymmetry with respect to the harmony of the previous stage).
Then eventually with sufficient evolution in development, one again moves to the next even numbered stage and the restoration of a higher state of contemplative attainment.
Put more precisely in a qualitative mathematical manner, the dislocation and asymmetry associated with the odd numbered stages arises from a degree of inevitable confusion as between rational understanding (geared to linear interpretation) and irrational understanding (reflecting the corresponding attempt to apply the higher states of intuitive awareness already attained to such understanding).
In other words one tries to combine both rational and true intuitive appreciation - which is paradoxical to reason - in a coherent manner. However before a new integral state can be attained this is always necessarily associated with a degree of confusion as between both aspects (rational and irrational).
Now we have already mentioned how complementary type interpretation characterises the relationship as between RHS and LHS of the Riemann Zeta Function.
Interestingly when we look at the values of the Riemann Zeta Function for odd integers of s (negative) on the LHS (where s < 0), they are always represented as rational values.
This would therefore imply that the corresponding values of the Zeta Function for these odd integers (positive) would be thereby represented (in complementary fashion) by irrational values.
And these irrational values would be algebraic (rather than transcendental) in nature and necessarily apply in the case of all the odd integers.
Now the relationship of these values to π is still important.
In other words as the odd dimensional number increases an ever closer degree of complementarity characterises roots (belonging to the set of conjugate pairings).
What this means in psychospiritual terms is that the linear understanding characterising very high odd integer stages become so refined as to be almost transparent (indicating that it can now interpenetrate with intuition with a remarkable lack of confusion remaining).
Interestingly once again in complementary fashion, when this happens the rational values for associated negative integer values increase dramatically. What this implies in qualitative terms is that with independence and interdependence both successfully combined to a very significant degree in experience, that a very high level of (refined) rational activity becomes compatible with an advanced contemplative state.
So just as the Riemann Zeta Function has important physical implications, equally it has extremely important psychospiritual implications (which have not yet been addressed). And once again this unfortunate blockage in understanding is due to the mistaken emphasis of Conventional Mathematics on merely quantitative meaning!
ζ(s) = k*(π^s) where k is a rational fraction.
However no such relationship characterises values of the Zeta Function for odd integers of s!
Indeed it took some time to prove that the first of these values for s = 3, is irrational (Apéry's constant) though it is not known if it is transcendental.
Also though it is known that other values of the Zeta function for odd integers values must likewise be irrational, little clarity can as yet be provided as to how this applies in specific cases.
Though of course there is a validity to the the quantitative attempt of attempting to provide a (Type 1) proof as to the precise status (as number type) that characterises Zeta Function results for the odd integers, inherently such an issue relates more to qualitative - rather than quantitative - considerations.
Now the first key indication that behaviour of the Zeta Function for odd integral values is quite distinct from corresponding behaviour with respect to the even integers is given through examination of the root structure for odd integer roots which reflects in quantitative terms corresponding interpretation of these same integers (as dimensional numbers).
Whereas for even numbered integers, roots can always be arranged in a complementary manner, this is never the case for odd numbered integers.
For example if we look at the 3 roots of 1 we have,
1, - 1/2 + {[(3^(1/2)]* i}/2 and - 1/2 - {[(3^(1/2)]* i}/2, which cannot be arranged in a complementary manner.
Thus the perfect matching of independent terms with interdependence through the direct complementary relationship as between roots, characterising the pure relation of linear to circular meaning (that defines the qualitative nature of π) is thereby missing with respect to odd integer roots and corresponding dimensional numbers.
When one looks more closely at the odd numbered roots one can see that + 1 is always one of these roots that stands in a sense alone. With respect to the other roots they always comprise conjugate pairs where the imaginary part is complementary with its partner (in the pairing).
There is also a significant clue as to what this entails in qualitative terms provided through the nature of higher psychospiritual contemplative development which involves the process of traversing these higher dimensions.
Basically the pattern keeps switching from differentiation (which always implies a degree of linear type understanding) to integration (where a new harmonious contemplative state is established).
So each even numbered stage (as the qualitative development with respect to such stage) represents the restoration of a new temporary equilibrium (characterised by the attainment of a contemplative state appropriate to such development).
However each odd numbered stage by contrast represents the breaking up of that (temporary) equilibrium is a new more refined experience of linear phenomena (which always however entails a degree of dislocation and asymmetry with respect to the harmony of the previous stage).
Then eventually with sufficient evolution in development, one again moves to the next even numbered stage and the restoration of a higher state of contemplative attainment.
Put more precisely in a qualitative mathematical manner, the dislocation and asymmetry associated with the odd numbered stages arises from a degree of inevitable confusion as between rational understanding (geared to linear interpretation) and irrational understanding (reflecting the corresponding attempt to apply the higher states of intuitive awareness already attained to such understanding).
In other words one tries to combine both rational and true intuitive appreciation - which is paradoxical to reason - in a coherent manner. However before a new integral state can be attained this is always necessarily associated with a degree of confusion as between both aspects (rational and irrational).
Now we have already mentioned how complementary type interpretation characterises the relationship as between RHS and LHS of the Riemann Zeta Function.
Interestingly when we look at the values of the Riemann Zeta Function for odd integers of s (negative) on the LHS (where s < 0), they are always represented as rational values.
This would therefore imply that the corresponding values of the Zeta Function for these odd integers (positive) would be thereby represented (in complementary fashion) by irrational values.
And these irrational values would be algebraic (rather than transcendental) in nature and necessarily apply in the case of all the odd integers.
Now the relationship of these values to π is still important.
In other words as the odd dimensional number increases an ever closer degree of complementarity characterises roots (belonging to the set of conjugate pairings).
What this means in psychospiritual terms is that the linear understanding characterising very high odd integer stages become so refined as to be almost transparent (indicating that it can now interpenetrate with intuition with a remarkable lack of confusion remaining).
Interestingly once again in complementary fashion, when this happens the rational values for associated negative integer values increase dramatically. What this implies in qualitative terms is that with independence and interdependence both successfully combined to a very significant degree in experience, that a very high level of (refined) rational activity becomes compatible with an advanced contemplative state.
So just as the Riemann Zeta Function has important physical implications, equally it has extremely important psychospiritual implications (which have not yet been addressed). And once again this unfortunate blockage in understanding is due to the mistaken emphasis of Conventional Mathematics on merely quantitative meaning!
Tuesday, April 3, 2012
Why the Trivial Zeros are not so Trivial!
We are returning again to interpretation of the - so called - trivial zeros of the Riemann Zeta Function, which occur for the negative even integers of s, i.e. s = - 2 , - 4, - 6, - 8, ....etc.
The first hint as to their nature is given through consideration of the values of the Function for corresponding even integer values of s, i.e. s = 2, 4, 6, 8,... etc.
Now for each of these values, the Function can indeed be given a finite quantitative value (in accordance with accepted linear notions of interpretation that define Type 1 understanding). In other words, in each case the sum of terms for the Function can be seen to converge to a limiting value.
Furthermore since Euler's pioneering contributions, all of these values (with s as an even integer) can be expressed in the form k*(π^s) where k is a rational fraction.
So for example in the best known case where s = 2,
ζ(2) = (π^2)/6.
So in direct terms on the RHS for s > 1, the Riemann Zeta Function (i.e. the Zeta 1 Function) can be given a quantitative interpretation in accordance with Conventional (Type 1) Mathematics (that is defined qualitatively in default 1-dimensional terms). And once again, this always implies that qualitative type considerations are reduced in a quantitative manner.
However when one allows for a uniquely distinctive (Type 2) aspect to Mathematics, this implicitly implies a (hidden) alternative Zeta 2 Function that indirectly can provide a distinctive qualitative interpretation (for Type 1 quantitative results).
So just as π in quantitative terms implies the pure relationship of the circular circumference to its line diameter, in corresponding qualitative (Type 2) terms, π equally implies the pure relationship of circular and linear type meaning.
What this in turn implies - as I have illustrated so many times before with respect to the crossroads analogy - is the ability to give opposite poles in understanding both an independent meaning (as separate) and then a truly interdependent meaning (as complementary and - indeed - ultimately - identical).
And with respect to the simplest example (where s = 2) this implies in Zeta 2 terms, qualitative appreciation of 2 as a dimensional number.
This then quantitatively corresponds with the 2 roots unity which can can be geometrically portrayed as the circle of unit radius, drawn in the complex plane (with a single line diameter that is positive to the right of centre and negative to the left).
Now it has to be understood that when s is positive, understanding is conducted in a rational linear manner. This therefore leads to paradox in terms of the qualitative interpretation of all dimensions (other than 1).
Thus, in attempting to convey the very nature of interdependence in rational terms, one must to a degree keep both poles separate. So therefore the best one can do from a rational perspective is to directly imply that both poles (positive and negative) are indeed ultimately identical (when properly understood as complementary).
Though it is simplest to understand the nature of such circular complementarity (in paradoxical rational terms) for the case where s = 2, the same basic principle applies for all even integer values of s.
What this implies from a quantitative perspective is that roots of 1, for all such even values can be arranged in a complementary manner. So for example if s = 6, this means that 3 roots can be chosen that can then be identically matched in complementary fashion (i.e. by multiplying by - 1) with the 3 remaining roots.
Therefore when we understood the Zeta Function properly in qualitative terms (i.e. through recognition of the alternative Zeta 2 aspect that qualitatively interprets Zeta 1 results) we can then recognise - in a necessarily indirect rational manner - that for all even integer values of s, a perfect complementarity can ultimately be achieved with respect to both independence and interdependence.
In other words, polarised interpretations that arise through initially interpreting in a relatively independent manner (with respect to one isolated frame of reference) are ultimately understood to be fully interdependent with each other when these frames are related.
So again if we take the simplest example (for s = 2) of the two turns at a crossroads, one can initially understand using - relatively - isolated reference frames with respect to one fixed direction of movement, that both left and right turns have an unambiguous meaning.
However when we then relate frames of reference simultaneously (as interdependent) one now appreciate that left and right turns have a purely relative meaning, depending on context.
Thus, what is a left turn travelling up the road will be right when travelling down in the opposite direction; likewise what is right when travelling up the road will be left when approached in the opposite (down) direction.
Now the deeper understanding of such a relationship implies that whereas - in direct terms - independent type distinctions are provided through reason, interdependent appreciation - in direct terms - comes from holistic type intuition.
So though one can indeed indirectly come to a very refined rational appreciation of the nature of interdependence in a circular paradoxical manner, the full realisation of such interdependence can only come directly through (holistic) intuition.
Therefore, to put it another way, though the very nature of Conventional (Type 1) Mathematics - defined as its is by the linear (1-dimensional) approach - is to attempt to understand number with respect to its mere cardinal (quantitative) nature, in truth both cardinal and ordinal aspects always interact (reflecting both the quantitative and qualitative aspects of number respectively).
And though the qualitative (ordinal) aspect can be indirectly conveyed in a refined (circular) rational manner, its direct understanding requires holistic intuitive appreciation!
The shocking indictment therefore that must be made against Conventional (Type 1 Mathematics) is that it cannot possibly deal with both the cardinal and ordinal aspects of number in a consistent manner (as it formally defined solely with respect to the cardinal i.e. quantitative aspect)!
However in the more comprehensive Type 3 approach (which I am attempting to demonstrate here) even when explicitly recognising the quantitative numerical validity of mathematical expressions from the Type 1 perspective (as with the Riemann Function for values of s > 1), one implicitly still recognises the - indirect - need for a corresponding Type 2 interpretation (of a distinctive qualitative nature).
This provides the appropriate perspective to realise that when one then switches to the LHS of the Function, that it is the qualitative aspect that now is given a direct expression (with corresponding numerical values arising, having a merely indirect quantitative interpretation)!
Stating it in an equivalent alternative fashion, though on the RHS of the Function (for s > 1) the interaction of both cardinal and ordinal aspects of number does lend itself explicitly to quantitative type interpretation (and implicitly qualitative), with respect to the LHS of the Function (for s < 0) the position is reversed.
Here the interaction of both cardinal and ordinal aspects of number now lends itself explicitly to qualitative type interpretation of an intuitive kind (with the rational numbers arsising implicitly possessing merely an indirect quantitative meaning).
Appreciation of this very point is truly vital! For, when properly grasped, this entails that Riemann's famous Functional Equation - when properly interpreted - always establishes a complementary type connection as between quantitative and qualitative (and - in reverse - qualitative and quantitative) type interpretation with respect to RHS and LHS values of the Function!
And this in turn means that we cannot possibly interpret the Function in a comprehensive manner, without recognition of its two vital aspects i.e. the Zeta 1 Function and the Zeta 2 Function respectively!
So again on the RHS (for s > 1) the Zeta 1 Function is explicitly used for interpretation in quantitative terms (with the Zeta 2 Function implicitly used from a qualitative perspective).
Then on the LHS (for s < 0), it is the Zeta 2 that is now explicitly used to interpret numerical values from a qualitative perspective (with the Zeta 1 implicitly used in a quantitative manner).
Now it should be patently obvious that from the conventional perspective, quantitative values of the Riemann Zeta Function for s < 0 make no sense!
For example when s = - 2,
ζ(- 2) = 1 + 4 + 9 + 16 + ....., which clearly diverges from a Type 1 (linear) mathematical perspective. Yet according to the Riemann Functional Equation,
ζ(- 2) = 0.
So much as modern mathematical interpretation attempts to cover over this important issue through technical abstract explanations with respect to analytic continuation, domains of definition, holomorphic functions etc., it cannot properly explain within its own terms such an ambiguous result.
And the true reason for this is that proper interpretation requires recognition of the qualitative - as well as the quantitative - aspect of mathematical interpretation! So quite simply, the result for ζ(- 2) seems counter-intuitive when interpreted in conventional linear terms (where by definition the qualitative nature of such intuition is reduced in a quantitative rational manner).
Indeed the first real indication as to the truly qualitative (ordinal) nature of the Riemann Zeta Function (where s < 0) is provided through appropriate dimensional interpretation of the negative sign for values of s.
I have explained before how dynamic negation (of rational interpretation) is the very means by which holistic intuitive appreciation arises in experience.
So if we wish to truly appreciate the interdependent nature of qualitative (ordinal) type number relationships, then we must switch directly to this intuitive mode (which only indirectly can be given rational expression).
So the important point to grasp is that the trivial zeros of the Riemann Zeta Function directly relate to the qualitative - rather than the quantitative - appreciation of number.
So what the trivial zeros thereby demonstrate is the nature of perfect ordinal interdependence (where number relationships are involved).
Expressed more precisely, the trivial zeros represent a direct qualitative (ordinal) appreciation of the interaction of both the cardinal and ordinal features of number (where perfect integration of both is achieved).
In other words when one appreciates number relationships in such a manner, one does so - not with respect to the initial independence of the numbers involved - but rather from the resulting interdependence arising in the relationship.
So, let us explain again this carefully with respect to the first of the trivial zeros i.e. s = - 2.
Now we already have dealt with the rational counterpart (for s = 2) where the relationship is given its geometric expression (in complementary quantitative terms) with respect to the 2 roots of 1. So we have a circle and a line diameter with the radius portion to the right of the centre positive (and the radius to the left negative). Now of course in dynamic interactive terms, what is positive or negative here is merely of an arbitrary relative nature!
So once again one can - literally - posit a left or right turn at a crossroads in unambiguous linear terms using relatively independent frames of reference i.e. when the direction of approach to the crossroads is fixed in one-way.
So when one moves up the road and approaches the crossroads from this direction, what is left and right (in this linear context) will have an unambiguous meaning!
Then, when having gone through the crossroads one switches direction and heads back down the road, one can again fix left and right turns in an unambiguous manner (from this alternative independent direction).
However when one now simultaneously relates both turns as interdependent, deep paradox arises (from a rational linear perspective). For what is left from one direction is also right from the alternative direction; likewise what is right from one direction is also left from the alternative:
So rational appreciation of interdependence is fully paradoxical (in linear terms). And this is the best one can manage in terms of indirect RHS appreciation of the qualitative nature of the 2-dimensional result.
However when one now switches to the LHS and considers the same dimensional number qualitatively in a negative fashion, this implies direct holistic intuitive realisation of the true nature of such interdependence.
Alternatively it represents the direct appreciation of the pure ordinal nature of number (which implies complete relational interdependence)!
So the startling implication is that the trivial zeros for the Riemann Zeta Function do not - directly - represent a quantitative but rather a qualitative interpretation of number.
And just as the quantitative (cardinal) interpretation of number relates directly to rational appreciation (of a linear kind), the corresponding qualitative interpretation relates directly to holistic intuitive appreciation (which appears as circular when indirectly expressed in rational terms).
The true qualitative nature of results for the non-trivial zeros is even symbolically indicated trough the very use of 0 as the symbol for zero (which directly indicates its circular nature)!
Now this might indeed seem strange (and out of bounds of what normally is considered Mathematics).
But this is precisely my point! For what is normally considered as Mathematics can be seen on close examination to be of an extremely reduced nature and quite frankly not fit for purpose.
And Conventional (Type 1) Mathematics is certainly not suited for proper interpretation of the Riemann Zeta Function.
In fact I now realise that for many years I have been paying Conventional Mathematics a deference which is not in truth warranted. Thus, while recognising the distinctive qualitative aspect for several decades now, I believed that Conventional Mathematics could still preserve a valid place (within its own terms of reference). However I now clearly realise that this position is not in fact tenable!
As it is ultimately impossible to properly understand the quantitative aspect of Mathematics (in abstraction from the qualitative) without confusion upon confusion being heaped on each other, I can now see that whole mathematical enterprise is in urgent need of rebuilding from the ground up (which is exactly what I am now, in my own way, attempting to do).
Of course this does not mean that the very considerable achievements of Conventional Mathematics have all been in vain. Rather it means that they will have to undergo radical reinterpretation, so that they can then be placed in their proper context (as one aspect of a much more comprehensive vision of Mathematics).
Indeed in the very term used for these zeros of the Zeta Function, the customary bias of Conventional Mathematics is clearly demonstrated.
They are referred to as the trivial zeros (implying that - unlike the non-trivial -they have no important role from a quantitative perspective (with respect to clarification of the distribution of primes).
However if one was to obtain a clearer understanding of the true nature of even the first of the trivial zeros (for s = - 2), then one's appreciation of both prime numbers and the Riemann Zeta Function would be changed forever!
Indeed it was precisely this insight, obtained many years ago, that gave me the initial confidence to pursue the Riemann Hypothesis with a view to unlocking its true nature.
From my earlier explorations in Holistic Mathematics - where I was seeking a satisfactory qualitative mathematical manner of mapping out the nature of contemplative development - I had already reached an understanding that the first of these "trivial zeros" related properly to a contemplative state of understanding (rather than rational interpretation).
In this regard I was deeply influenced by the writings of the Spanish mystic St. John of the Cross (which I have recounted in other blog entries). Now St. John is the most profound spiritual writer on the nature of the dynamic negations leading to contemplative development (of a pure intuitive kind). And he literally refers to the nature of such negation as nada i.e. nothing! In other words when one directly obtains a pure realisation of the nature of interdependence, then nothing - by definition - remains in phenomenal terms!
And of course this insight equally applies to the pure nature of numerical interdependence within Mathematics (which is - literally - nothing from an independent quantitative perspective)!
So here we are at the other extreme of the pure ordinal appreciation of number (which is nothing from an independent cardinal perspective)!
Once again the treatment of number in Conventional Mathematics is in terms of its - merely - independent (cardinal) identity, which is misleadingly interpreted in absolute terms.
However if numbers have an absolutely fixed quantitative identity, then there is no way without gross reductionism of giving such numbers a corresponding ordinal (i.e. relational) identity!
So to avoid such confused reductionism, in truth numbers have to be redefined with both independent (cardinal) and interdependent (ordinal) aspects which are now understood in relative terms.
Thus at one extreme we have the quantitative (cardinal) appreciation of numbers (in rational terms) where their - relatively - independent nature is highlighted.
However at the other extreme, we have the qualitative (ordinal) appreciation of numbers (in intuitive terms) where their - relatively - interdependent relational nature is highlighted.
Thus in the context of the Riemann Zeta Function, the trivial zeros represent a state of pure relational number interdependence.
So again if we take the simplest case where s = - 2, this would imply in the context of our crossroads illustration, a pure intuitive realisation of the interdependence of left and right directions (when opposite frames of reference are simultaneously considered).
Though the nature of such interdependence does indeed become more complex for the other negative even integer values of s, in principle the basic position remains the same.
Once again for any even integer dimensional value, we can through the Type 2 Zeta Function, provide a complementary matching set of roots (in quantitative terms) so that they always - literally - cancel out in this manner.
Thus the corresponding negative values for these integers relate to the pure intuitive (ordinal) interpretation of such interdependence (that is directly of a qualitative nature).
So far from being trivial, the true nature of the "trivial zeros" provides a fundamental key for proper comprehension of both the Riemann Zeta Function and its associated Riemann Hypothesis.
For the clear realisation that the trivial zeros relate to a pure qualitative (ordinal) rather than cardinal (quantitative) appreciation of number, immediately suggests the complementary nature of quantitative and qualitative type interpretation that exists with respect to RHS and LHS of the Function.
And the Riemann Hypothesis is then simply seen in this context as the central condition required for preserving the harmonious distribution of the primes (i.e. where quantitative and qualitative interpretations are identical).
The first hint as to their nature is given through consideration of the values of the Function for corresponding even integer values of s, i.e. s = 2, 4, 6, 8,... etc.
Now for each of these values, the Function can indeed be given a finite quantitative value (in accordance with accepted linear notions of interpretation that define Type 1 understanding). In other words, in each case the sum of terms for the Function can be seen to converge to a limiting value.
Furthermore since Euler's pioneering contributions, all of these values (with s as an even integer) can be expressed in the form k*(π^s) where k is a rational fraction.
So for example in the best known case where s = 2,
ζ(2) = (π^2)/6.
So in direct terms on the RHS for s > 1, the Riemann Zeta Function (i.e. the Zeta 1 Function) can be given a quantitative interpretation in accordance with Conventional (Type 1) Mathematics (that is defined qualitatively in default 1-dimensional terms). And once again, this always implies that qualitative type considerations are reduced in a quantitative manner.
However when one allows for a uniquely distinctive (Type 2) aspect to Mathematics, this implicitly implies a (hidden) alternative Zeta 2 Function that indirectly can provide a distinctive qualitative interpretation (for Type 1 quantitative results).
So just as π in quantitative terms implies the pure relationship of the circular circumference to its line diameter, in corresponding qualitative (Type 2) terms, π equally implies the pure relationship of circular and linear type meaning.
What this in turn implies - as I have illustrated so many times before with respect to the crossroads analogy - is the ability to give opposite poles in understanding both an independent meaning (as separate) and then a truly interdependent meaning (as complementary and - indeed - ultimately - identical).
And with respect to the simplest example (where s = 2) this implies in Zeta 2 terms, qualitative appreciation of 2 as a dimensional number.
This then quantitatively corresponds with the 2 roots unity which can can be geometrically portrayed as the circle of unit radius, drawn in the complex plane (with a single line diameter that is positive to the right of centre and negative to the left).
Now it has to be understood that when s is positive, understanding is conducted in a rational linear manner. This therefore leads to paradox in terms of the qualitative interpretation of all dimensions (other than 1).
Thus, in attempting to convey the very nature of interdependence in rational terms, one must to a degree keep both poles separate. So therefore the best one can do from a rational perspective is to directly imply that both poles (positive and negative) are indeed ultimately identical (when properly understood as complementary).
Though it is simplest to understand the nature of such circular complementarity (in paradoxical rational terms) for the case where s = 2, the same basic principle applies for all even integer values of s.
What this implies from a quantitative perspective is that roots of 1, for all such even values can be arranged in a complementary manner. So for example if s = 6, this means that 3 roots can be chosen that can then be identically matched in complementary fashion (i.e. by multiplying by - 1) with the 3 remaining roots.
Therefore when we understood the Zeta Function properly in qualitative terms (i.e. through recognition of the alternative Zeta 2 aspect that qualitatively interprets Zeta 1 results) we can then recognise - in a necessarily indirect rational manner - that for all even integer values of s, a perfect complementarity can ultimately be achieved with respect to both independence and interdependence.
In other words, polarised interpretations that arise through initially interpreting in a relatively independent manner (with respect to one isolated frame of reference) are ultimately understood to be fully interdependent with each other when these frames are related.
So again if we take the simplest example (for s = 2) of the two turns at a crossroads, one can initially understand using - relatively - isolated reference frames with respect to one fixed direction of movement, that both left and right turns have an unambiguous meaning.
However when we then relate frames of reference simultaneously (as interdependent) one now appreciate that left and right turns have a purely relative meaning, depending on context.
Thus, what is a left turn travelling up the road will be right when travelling down in the opposite direction; likewise what is right when travelling up the road will be left when approached in the opposite (down) direction.
Now the deeper understanding of such a relationship implies that whereas - in direct terms - independent type distinctions are provided through reason, interdependent appreciation - in direct terms - comes from holistic type intuition.
So though one can indeed indirectly come to a very refined rational appreciation of the nature of interdependence in a circular paradoxical manner, the full realisation of such interdependence can only come directly through (holistic) intuition.
Therefore, to put it another way, though the very nature of Conventional (Type 1) Mathematics - defined as its is by the linear (1-dimensional) approach - is to attempt to understand number with respect to its mere cardinal (quantitative) nature, in truth both cardinal and ordinal aspects always interact (reflecting both the quantitative and qualitative aspects of number respectively).
And though the qualitative (ordinal) aspect can be indirectly conveyed in a refined (circular) rational manner, its direct understanding requires holistic intuitive appreciation!
The shocking indictment therefore that must be made against Conventional (Type 1 Mathematics) is that it cannot possibly deal with both the cardinal and ordinal aspects of number in a consistent manner (as it formally defined solely with respect to the cardinal i.e. quantitative aspect)!
However in the more comprehensive Type 3 approach (which I am attempting to demonstrate here) even when explicitly recognising the quantitative numerical validity of mathematical expressions from the Type 1 perspective (as with the Riemann Function for values of s > 1), one implicitly still recognises the - indirect - need for a corresponding Type 2 interpretation (of a distinctive qualitative nature).
This provides the appropriate perspective to realise that when one then switches to the LHS of the Function, that it is the qualitative aspect that now is given a direct expression (with corresponding numerical values arising, having a merely indirect quantitative interpretation)!
Stating it in an equivalent alternative fashion, though on the RHS of the Function (for s > 1) the interaction of both cardinal and ordinal aspects of number does lend itself explicitly to quantitative type interpretation (and implicitly qualitative), with respect to the LHS of the Function (for s < 0) the position is reversed.
Here the interaction of both cardinal and ordinal aspects of number now lends itself explicitly to qualitative type interpretation of an intuitive kind (with the rational numbers arsising implicitly possessing merely an indirect quantitative meaning).
Appreciation of this very point is truly vital! For, when properly grasped, this entails that Riemann's famous Functional Equation - when properly interpreted - always establishes a complementary type connection as between quantitative and qualitative (and - in reverse - qualitative and quantitative) type interpretation with respect to RHS and LHS values of the Function!
And this in turn means that we cannot possibly interpret the Function in a comprehensive manner, without recognition of its two vital aspects i.e. the Zeta 1 Function and the Zeta 2 Function respectively!
So again on the RHS (for s > 1) the Zeta 1 Function is explicitly used for interpretation in quantitative terms (with the Zeta 2 Function implicitly used from a qualitative perspective).
Then on the LHS (for s < 0), it is the Zeta 2 that is now explicitly used to interpret numerical values from a qualitative perspective (with the Zeta 1 implicitly used in a quantitative manner).
Now it should be patently obvious that from the conventional perspective, quantitative values of the Riemann Zeta Function for s < 0 make no sense!
For example when s = - 2,
ζ(- 2) = 1 + 4 + 9 + 16 + ....., which clearly diverges from a Type 1 (linear) mathematical perspective. Yet according to the Riemann Functional Equation,
ζ(- 2) = 0.
So much as modern mathematical interpretation attempts to cover over this important issue through technical abstract explanations with respect to analytic continuation, domains of definition, holomorphic functions etc., it cannot properly explain within its own terms such an ambiguous result.
And the true reason for this is that proper interpretation requires recognition of the qualitative - as well as the quantitative - aspect of mathematical interpretation! So quite simply, the result for ζ(- 2) seems counter-intuitive when interpreted in conventional linear terms (where by definition the qualitative nature of such intuition is reduced in a quantitative rational manner).
Indeed the first real indication as to the truly qualitative (ordinal) nature of the Riemann Zeta Function (where s < 0) is provided through appropriate dimensional interpretation of the negative sign for values of s.
I have explained before how dynamic negation (of rational interpretation) is the very means by which holistic intuitive appreciation arises in experience.
So if we wish to truly appreciate the interdependent nature of qualitative (ordinal) type number relationships, then we must switch directly to this intuitive mode (which only indirectly can be given rational expression).
So the important point to grasp is that the trivial zeros of the Riemann Zeta Function directly relate to the qualitative - rather than the quantitative - appreciation of number.
So what the trivial zeros thereby demonstrate is the nature of perfect ordinal interdependence (where number relationships are involved).
Expressed more precisely, the trivial zeros represent a direct qualitative (ordinal) appreciation of the interaction of both the cardinal and ordinal features of number (where perfect integration of both is achieved).
In other words when one appreciates number relationships in such a manner, one does so - not with respect to the initial independence of the numbers involved - but rather from the resulting interdependence arising in the relationship.
So, let us explain again this carefully with respect to the first of the trivial zeros i.e. s = - 2.
Now we already have dealt with the rational counterpart (for s = 2) where the relationship is given its geometric expression (in complementary quantitative terms) with respect to the 2 roots of 1. So we have a circle and a line diameter with the radius portion to the right of the centre positive (and the radius to the left negative). Now of course in dynamic interactive terms, what is positive or negative here is merely of an arbitrary relative nature!
So once again one can - literally - posit a left or right turn at a crossroads in unambiguous linear terms using relatively independent frames of reference i.e. when the direction of approach to the crossroads is fixed in one-way.
So when one moves up the road and approaches the crossroads from this direction, what is left and right (in this linear context) will have an unambiguous meaning!
Then, when having gone through the crossroads one switches direction and heads back down the road, one can again fix left and right turns in an unambiguous manner (from this alternative independent direction).
However when one now simultaneously relates both turns as interdependent, deep paradox arises (from a rational linear perspective). For what is left from one direction is also right from the alternative direction; likewise what is right from one direction is also left from the alternative:
So rational appreciation of interdependence is fully paradoxical (in linear terms). And this is the best one can manage in terms of indirect RHS appreciation of the qualitative nature of the 2-dimensional result.
However when one now switches to the LHS and considers the same dimensional number qualitatively in a negative fashion, this implies direct holistic intuitive realisation of the true nature of such interdependence.
Alternatively it represents the direct appreciation of the pure ordinal nature of number (which implies complete relational interdependence)!
So the startling implication is that the trivial zeros for the Riemann Zeta Function do not - directly - represent a quantitative but rather a qualitative interpretation of number.
And just as the quantitative (cardinal) interpretation of number relates directly to rational appreciation (of a linear kind), the corresponding qualitative interpretation relates directly to holistic intuitive appreciation (which appears as circular when indirectly expressed in rational terms).
The true qualitative nature of results for the non-trivial zeros is even symbolically indicated trough the very use of 0 as the symbol for zero (which directly indicates its circular nature)!
Now this might indeed seem strange (and out of bounds of what normally is considered Mathematics).
But this is precisely my point! For what is normally considered as Mathematics can be seen on close examination to be of an extremely reduced nature and quite frankly not fit for purpose.
And Conventional (Type 1) Mathematics is certainly not suited for proper interpretation of the Riemann Zeta Function.
In fact I now realise that for many years I have been paying Conventional Mathematics a deference which is not in truth warranted. Thus, while recognising the distinctive qualitative aspect for several decades now, I believed that Conventional Mathematics could still preserve a valid place (within its own terms of reference). However I now clearly realise that this position is not in fact tenable!
As it is ultimately impossible to properly understand the quantitative aspect of Mathematics (in abstraction from the qualitative) without confusion upon confusion being heaped on each other, I can now see that whole mathematical enterprise is in urgent need of rebuilding from the ground up (which is exactly what I am now, in my own way, attempting to do).
Of course this does not mean that the very considerable achievements of Conventional Mathematics have all been in vain. Rather it means that they will have to undergo radical reinterpretation, so that they can then be placed in their proper context (as one aspect of a much more comprehensive vision of Mathematics).
Indeed in the very term used for these zeros of the Zeta Function, the customary bias of Conventional Mathematics is clearly demonstrated.
They are referred to as the trivial zeros (implying that - unlike the non-trivial -they have no important role from a quantitative perspective (with respect to clarification of the distribution of primes).
However if one was to obtain a clearer understanding of the true nature of even the first of the trivial zeros (for s = - 2), then one's appreciation of both prime numbers and the Riemann Zeta Function would be changed forever!
Indeed it was precisely this insight, obtained many years ago, that gave me the initial confidence to pursue the Riemann Hypothesis with a view to unlocking its true nature.
From my earlier explorations in Holistic Mathematics - where I was seeking a satisfactory qualitative mathematical manner of mapping out the nature of contemplative development - I had already reached an understanding that the first of these "trivial zeros" related properly to a contemplative state of understanding (rather than rational interpretation).
In this regard I was deeply influenced by the writings of the Spanish mystic St. John of the Cross (which I have recounted in other blog entries). Now St. John is the most profound spiritual writer on the nature of the dynamic negations leading to contemplative development (of a pure intuitive kind). And he literally refers to the nature of such negation as nada i.e. nothing! In other words when one directly obtains a pure realisation of the nature of interdependence, then nothing - by definition - remains in phenomenal terms!
And of course this insight equally applies to the pure nature of numerical interdependence within Mathematics (which is - literally - nothing from an independent quantitative perspective)!
So here we are at the other extreme of the pure ordinal appreciation of number (which is nothing from an independent cardinal perspective)!
Once again the treatment of number in Conventional Mathematics is in terms of its - merely - independent (cardinal) identity, which is misleadingly interpreted in absolute terms.
However if numbers have an absolutely fixed quantitative identity, then there is no way without gross reductionism of giving such numbers a corresponding ordinal (i.e. relational) identity!
So to avoid such confused reductionism, in truth numbers have to be redefined with both independent (cardinal) and interdependent (ordinal) aspects which are now understood in relative terms.
Thus at one extreme we have the quantitative (cardinal) appreciation of numbers (in rational terms) where their - relatively - independent nature is highlighted.
However at the other extreme, we have the qualitative (ordinal) appreciation of numbers (in intuitive terms) where their - relatively - interdependent relational nature is highlighted.
Thus in the context of the Riemann Zeta Function, the trivial zeros represent a state of pure relational number interdependence.
So again if we take the simplest case where s = - 2, this would imply in the context of our crossroads illustration, a pure intuitive realisation of the interdependence of left and right directions (when opposite frames of reference are simultaneously considered).
Though the nature of such interdependence does indeed become more complex for the other negative even integer values of s, in principle the basic position remains the same.
Once again for any even integer dimensional value, we can through the Type 2 Zeta Function, provide a complementary matching set of roots (in quantitative terms) so that they always - literally - cancel out in this manner.
Thus the corresponding negative values for these integers relate to the pure intuitive (ordinal) interpretation of such interdependence (that is directly of a qualitative nature).
So far from being trivial, the true nature of the "trivial zeros" provides a fundamental key for proper comprehension of both the Riemann Zeta Function and its associated Riemann Hypothesis.
For the clear realisation that the trivial zeros relate to a pure qualitative (ordinal) rather than cardinal (quantitative) appreciation of number, immediately suggests the complementary nature of quantitative and qualitative type interpretation that exists with respect to RHS and LHS of the Function.
And the Riemann Hypothesis is then simply seen in this context as the central condition required for preserving the harmonious distribution of the primes (i.e. where quantitative and qualitative interpretations are identical).
Wednesday, March 28, 2012
Number Inconsistency (6)
We have seen that an imaginary number can be interpreted in two different ways (that are ultimately complementary).
(a) As the manner of expressing what is properly an ordinal notion in an indirect cardinal manner (for incorporation in a Type 1 interpretation of number).
Considerable use is now made of complex numbers in Conventional Mathematics; however both parts are are treated strictly as quantities within this approach. Thus the true nature of the imaginary part (as representative of an alternative qualitative relational number system that is ordinal) remains completely unrecognised when treated in an absolute manner.
However when the Type 1 approach is understood in a relative fashion with complex nos. again treated in quantitative terms, implicit in such understanding is the recognition that the imaginary aspect relates properly to the alternative qualitative relational aspect that is now - in a Type 1 context - indirectly given a quantitative expression. And it is this latter type of understanding that is properly consistent with the most comprehensive Type 3 mathematical interpretation.
(b) As the manner of expressing a cardinal notion in an indirect ordinal manner (for incorporation in the Type 2 interpretation of number).
Again in this approach – though still almost entirely unrecognised – complex nos. are interpreted strictly with respect to appreciation of their qualitative (ordinal) nature. However implicit in this is a recognition of the quantitative meaning of the real aspect (within the Type 1 system) that is now indirectly given a qualitative expression within Type 2.
What is again clear from this is that the true significance of complex nos. is entirely missed within the absolute Type 1 framework of Conventional Mathematics. The essential point is that number has both cardinal (independent) and ordinal (relational) meanings which are - relatively - distinct. Therefore to incorporate the ordinal aspect within a real quantitative type approach, it must be treated in an imaginary fashion.
Likewise from the complementary perspective to incorporate cardinal aspect within a - relatively - real ordinal interpretation, the quantitative must be treated as imaginary.
This in Type 3 terms what is imaginary from a quantitative perspective is equally real from the corresponding qualitative perspective; and what is real from a qualitative perspective is imaginary from the corresponding quantitative perspective.
So in this context real and imaginary have ultimately a purely relative meaning.
Higher dimensional interpretations (s > 2), combine both real and imaginary aspects. This entails that 3-dimensional and all higher dimensional interpretations entail number configurations with both cardinal (quantitative) and ordinal (qualitative) features that are properly distinguished.
From a quantitative perspective (again for s > 2) the roots of 1 will entail complex values (with real and imaginary parts) that serve as the quantitative counterpart of real and imaginary interpretation in qualitative terms.
This entails that each dimension is associated with a unique configuration with respect to both analytic (quantitative) and holistic (qualitative) type appreciation. This would mean in turn from a psychological perspective, a unique configuration with respect to both rational and intuitive type processes.
And properly understood both the quantitative and qualitative aspects are complementary.
Thus to properly interpret the quantitative nature of the roots of 1, we need the complementary Type 2 higher dimensional interpretation.
Equally in deriving the structure of these dimensions we require the complementary Type 1 appreciation of corresponding roots.
However though complex values occur at s = 3, the most important occurs for s = 4.
Now looking at the 4 roots of 1 we have 2 real and 2 imaginary.
From a Type 2 perspective this implies a perfect integration of both cardinal and ordinal type meaning. And looking at it from a Type 1 perspective, we have two real and two imaginary roots. However these imaginary roots – though expressed in quantitative terms - are understood as representative of ordinal relationships pertaining to the Type 2 system.
Likewise from the Type 2 perspective, these real and imaginary roots are now given a qualitative Type 2 interpretation with respect to 4–dimensional appreciation, with again perfect matching symmetry.
So then from a Type 3 perspective what is real in one system is imaginary in the other and what is imaginary is real; likewise what is positive in one is negative in the other and vice versa.
We live in a world of 4 dimensions. The deeper understanding of this implies that all reality is subject to opposite polarities in real and imaginary terms, with what is imaginary in terms of one system real in terms of the other and vice versa.
Indeed a clue is given to this in the work of Jung who saw the number 4 as extraordinarily important (from this qualitative perspective).
He also drew attention to the most common forms of mandalas which so often are based on ornate pictorial representations corresponding to the geometrical representation of the four dimensions (four roots of 1) and eight dimensions (eight roots of 1) respectively.
And here we can see the precise mathematical nature for such integration where both the ordinal (relational) and cardinal (independent) nature of number are seen as ultimately identical!
Now one of the important practical implications of this understanding is that it provides an entirely new perspective with which to deal with the Riemann Zeta Function where through using two systems of interpretation, both real and imaginary values become interchangeable in both systems!
For example, the non-trivial zeros (in Type 1 terms) combine a constant real part of 1/2 with varying imaginary values!
This directly implies that we can use these values in Type 2 terms, where now the imaginary aspect is constant at 1/2 and the imaginary parts are now treated as real.
As we have seen we have seen that average mean value of roots of 1 of both cos and sin parts (for any prime number p) approaches 2/pi. However the deviations of actual computed values from this value need to be explained. So just as the non-trivial zeros have a role in Type 1 terms (with reference to their imaginary parts) in precisely predicting the (cardinal) distribution of primes, they likewise have a role in Type 2 terms in precisely predicting this complementary (ordinal) distribution of the primes, with the imaginary aspect now interpreted as real.
So the non-trivial zeros in this context take on an entirely new significance which throws significant light on their true nature!
And the process works both ways as in reverse fashion the Type 2 approach can be used to highlight the wave nature - not of the general distribution of primes - but rather of each individual prime number in Type 1 terms.!
(a) As the manner of expressing what is properly an ordinal notion in an indirect cardinal manner (for incorporation in a Type 1 interpretation of number).
Considerable use is now made of complex numbers in Conventional Mathematics; however both parts are are treated strictly as quantities within this approach. Thus the true nature of the imaginary part (as representative of an alternative qualitative relational number system that is ordinal) remains completely unrecognised when treated in an absolute manner.
However when the Type 1 approach is understood in a relative fashion with complex nos. again treated in quantitative terms, implicit in such understanding is the recognition that the imaginary aspect relates properly to the alternative qualitative relational aspect that is now - in a Type 1 context - indirectly given a quantitative expression. And it is this latter type of understanding that is properly consistent with the most comprehensive Type 3 mathematical interpretation.
(b) As the manner of expressing a cardinal notion in an indirect ordinal manner (for incorporation in the Type 2 interpretation of number).
Again in this approach – though still almost entirely unrecognised – complex nos. are interpreted strictly with respect to appreciation of their qualitative (ordinal) nature. However implicit in this is a recognition of the quantitative meaning of the real aspect (within the Type 1 system) that is now indirectly given a qualitative expression within Type 2.
What is again clear from this is that the true significance of complex nos. is entirely missed within the absolute Type 1 framework of Conventional Mathematics. The essential point is that number has both cardinal (independent) and ordinal (relational) meanings which are - relatively - distinct. Therefore to incorporate the ordinal aspect within a real quantitative type approach, it must be treated in an imaginary fashion.
Likewise from the complementary perspective to incorporate cardinal aspect within a - relatively - real ordinal interpretation, the quantitative must be treated as imaginary.
This in Type 3 terms what is imaginary from a quantitative perspective is equally real from the corresponding qualitative perspective; and what is real from a qualitative perspective is imaginary from the corresponding quantitative perspective.
So in this context real and imaginary have ultimately a purely relative meaning.
Higher dimensional interpretations (s > 2), combine both real and imaginary aspects. This entails that 3-dimensional and all higher dimensional interpretations entail number configurations with both cardinal (quantitative) and ordinal (qualitative) features that are properly distinguished.
From a quantitative perspective (again for s > 2) the roots of 1 will entail complex values (with real and imaginary parts) that serve as the quantitative counterpart of real and imaginary interpretation in qualitative terms.
This entails that each dimension is associated with a unique configuration with respect to both analytic (quantitative) and holistic (qualitative) type appreciation. This would mean in turn from a psychological perspective, a unique configuration with respect to both rational and intuitive type processes.
And properly understood both the quantitative and qualitative aspects are complementary.
Thus to properly interpret the quantitative nature of the roots of 1, we need the complementary Type 2 higher dimensional interpretation.
Equally in deriving the structure of these dimensions we require the complementary Type 1 appreciation of corresponding roots.
However though complex values occur at s = 3, the most important occurs for s = 4.
Now looking at the 4 roots of 1 we have 2 real and 2 imaginary.
From a Type 2 perspective this implies a perfect integration of both cardinal and ordinal type meaning. And looking at it from a Type 1 perspective, we have two real and two imaginary roots. However these imaginary roots – though expressed in quantitative terms - are understood as representative of ordinal relationships pertaining to the Type 2 system.
Likewise from the Type 2 perspective, these real and imaginary roots are now given a qualitative Type 2 interpretation with respect to 4–dimensional appreciation, with again perfect matching symmetry.
So then from a Type 3 perspective what is real in one system is imaginary in the other and what is imaginary is real; likewise what is positive in one is negative in the other and vice versa.
We live in a world of 4 dimensions. The deeper understanding of this implies that all reality is subject to opposite polarities in real and imaginary terms, with what is imaginary in terms of one system real in terms of the other and vice versa.
Indeed a clue is given to this in the work of Jung who saw the number 4 as extraordinarily important (from this qualitative perspective).
He also drew attention to the most common forms of mandalas which so often are based on ornate pictorial representations corresponding to the geometrical representation of the four dimensions (four roots of 1) and eight dimensions (eight roots of 1) respectively.
And here we can see the precise mathematical nature for such integration where both the ordinal (relational) and cardinal (independent) nature of number are seen as ultimately identical!
Now one of the important practical implications of this understanding is that it provides an entirely new perspective with which to deal with the Riemann Zeta Function where through using two systems of interpretation, both real and imaginary values become interchangeable in both systems!
For example, the non-trivial zeros (in Type 1 terms) combine a constant real part of 1/2 with varying imaginary values!
This directly implies that we can use these values in Type 2 terms, where now the imaginary aspect is constant at 1/2 and the imaginary parts are now treated as real.
As we have seen we have seen that average mean value of roots of 1 of both cos and sin parts (for any prime number p) approaches 2/pi. However the deviations of actual computed values from this value need to be explained. So just as the non-trivial zeros have a role in Type 1 terms (with reference to their imaginary parts) in precisely predicting the (cardinal) distribution of primes, they likewise have a role in Type 2 terms in precisely predicting this complementary (ordinal) distribution of the primes, with the imaginary aspect now interpreted as real.
So the non-trivial zeros in this context take on an entirely new significance which throws significant light on their true nature!
And the process works both ways as in reverse fashion the Type 2 approach can be used to highlight the wave nature - not of the general distribution of primes - but rather of each individual prime number in Type 1 terms.!
Friday, March 23, 2012
Number Inconsistency (5)
We have seen how dealing appropriately with the ordinal (relational) nature of number requires going beyond the 1-dimensional qualitative approach (within which Conventional Mathematics is defined).
Again this is is necessary as thc customary approach has no means - within its own terms of reference - of satisfactorily distinguishing qualitative from quantitative type meaning.
Alternatively this entails that Conventional Mathematics cannot properly deal with the ordinal (qualitative) nature of number. Worse still because - in dynamic experiential terms - both cardinal and ordinal meaning are interrelated, it implies that Conventional Mathematics ultimately cannot even properly deal with its own chosen area of the quantitative nature of number (and by extension all quantitative notions)!
I have defined on numerous occasions the nature of the 3 Types of Mathematics (which are necessary in an overall comprehensive framework).
Once again Type 1 refers to the quantitative aspect (which is the sole specialisation of Conventional Mathematics). However even here there are two distinct approaches.
Unfortunately - as I would see it - Conventional Mathematics is very much rooted in an absolute Type 1 approach that is - qualitatively - of a linear logical (i.e. 1-dimensional) nature. At present it shows no openness whatever to a counterbalancing qualitative approach (that ultimatekly must be incorporated for full mathematical comprehension).
The alternative Type 1 approach - which I strongly advocate - is defined in a strictly relative manner. Though specialisation of the quantitative aspect of mathematical symbols is certainly legitimate in this approach, implicitly it recognises that a complementary qualitative type treatment of the same symbols is equally possible!
So the quantitatave aspect of Mathematics is understood here in a - relatively - independent manner.
And if we are to proceed to a truly comprehensive mathematical understanding (involving all 3 types) then the Type 1 must be defined in a relative - rather than absolute - fashion.
Just as the 1st dimension of interpretation (in this relative context) initially provides the (Type 1) standard for quantitative type cardinal interpretation of number (based on independence), as I demonstrated in a recent blog the 2nd dimension (Type 2) likewise provides the standard basis for qualitative type ordinal interpretation of number (based on interdependence).
Then combining the 1st and 2nd dimensions of interpretation - corresponding to the linear and circular use of logic respectively - one for example can give a complete explanation of the relationship which two objects (i.e. numbers) have with each other, where both cardinal (quantitative) and ordinal (qualitative) distinctions are both preserved. I used once more the - apparently - simple illustration of the two turns at a crossroads to illustrate this point!
So putting it simply! The 1st dimension of interpretation can be clearly associated with cardinal (quantitative) meaning (in a relatively independent sense); the 2nd dimension can be clearly associated with ordinal (qualitative) type meaning (in a relatively interdependent sense).
However rather like the particle and wave nature of sub-atomic particles, once we bring both aspects together, the wave aspect becomes also particle like, and the particle aspect wave-like: likewise with respect to number: once we attempt to combine both the cardinal (quantitative) and ordinal (qualitative) nature of numbers the cardinal nature acquires ordinal like features, whereas the ordinal acquires cardinal like features.
And the fascinating clue to what all this means is given by the higher dimensional numbers with respect to 1 (> 2) with their corresponding roots of 1 (> 2).
In other words once we go higher than 2, all roots of 1 combine both real and imaginary parts.
So the fascinating and important question then arises as to what the imaginary aspect means in this context (of cardinal and ordinal interpretation).
To appreciate this we need to go back to the 2nd root of 1, which is the quantitative counterpart of the number 2 as dimension (in qualitative terms).
This was defined as - 1. Now the corresponding qualitative interpretation was as the negation of independent conscious type understanding of a rational nature. This equally represents the manner through which interdependent unconscious type holistic appreciation of an intuitive kind takes place.
Therefore in all relationships, whereas understanding of the independent aspect of such relationships is strictly provided through (conscious) reason (1st dimension), appreciation of interdependence by contrast is provided through (unconscious) intuition (2nd dimension). However indirectly this latter aspect can be interpreted in a circular rational fashion (that is paradoxical in terms of conventional reason).
So - 1 in this rational context is given a 2-dimensional (ordinal) interpretation as both positive and negative (+ and -) which can also be expressed as the complementarity of opposites (i.e. opposite poles). It must be remembered that negating in a dynamic interactive context already presupposes a positive element (like anti-matter fusing with matter particles).
Thus we have established in qualitative terms, how - 1 thereby represents the fundamental nature of the 2nd dimension (through which interdependence with respect to opposite poles takes place).
However as we have qualitatively defined it here (in Type 2 terms), this represents the 2nd - rather than the 1st - dimension.
So of we are to reduce this notion appropriately so that it can now be defined in Type 1 terms, we thereby obtain the square root.
Thus the imaginary no. i = the square root of - 1, serves as the (reduced) Type 1 way of incorporating the ordinal (qualitative) aspect of Type 2 Mathematics in an accepted Type 1 cardinal (quantitative) context.
Equally from the opposite perspective of Type 2 Mathematics, i serves as the (reduced) Type 2 manner of incorporating the cardinal (quantitative) aspect of Type 1 Mathematics in an accepted Type 2 ordinal (qualitative) context.
The deeper implications of all this is that complex numbers - when properly viewed from a Type 1 or Type 2 perspective - necessarily incorporate both quantitative and qualitative type aspects.
However in each case one of these aspects remains masked (with its true nature hidden).
Thus from the Type 1 perspective, though we attempt to view both the real and imaginary aspects of number (in a merely quantitative manner), the imaginary aspect in fact represents the alternative qualitative aspect of Mathematics (that remains hidden however through being veiled in a quantitative mask).
Likewise from the Type 2 perspective, though we again may attempt to now view both the real and imaginary aspects of number (in a merely qualitative manner), the imaginary aspect now in fact represents the corresponding quantitative aspect of Mathematics (that remains hidden however in a qualitative mask)!
And once one clearly realises this dilemma, from the relatively isolated stances of both Type 1 and Type 2 Mathematics respectively, then one necessarily must start moving to the Type 3 approach (where both quantitative and qualitative aspects can be properly integrated).
All of this of course is deeply relevant to proper understanding of the Riemann Zeta Function. As it is defined with respect to the complex plane, with both (matching) quantitative and qualitative type interpretations, a Type 3 mathematical approach is very much required for its proper comprehension.
Again this is is necessary as thc customary approach has no means - within its own terms of reference - of satisfactorily distinguishing qualitative from quantitative type meaning.
Alternatively this entails that Conventional Mathematics cannot properly deal with the ordinal (qualitative) nature of number. Worse still because - in dynamic experiential terms - both cardinal and ordinal meaning are interrelated, it implies that Conventional Mathematics ultimately cannot even properly deal with its own chosen area of the quantitative nature of number (and by extension all quantitative notions)!
I have defined on numerous occasions the nature of the 3 Types of Mathematics (which are necessary in an overall comprehensive framework).
Once again Type 1 refers to the quantitative aspect (which is the sole specialisation of Conventional Mathematics). However even here there are two distinct approaches.
Unfortunately - as I would see it - Conventional Mathematics is very much rooted in an absolute Type 1 approach that is - qualitatively - of a linear logical (i.e. 1-dimensional) nature. At present it shows no openness whatever to a counterbalancing qualitative approach (that ultimatekly must be incorporated for full mathematical comprehension).
The alternative Type 1 approach - which I strongly advocate - is defined in a strictly relative manner. Though specialisation of the quantitative aspect of mathematical symbols is certainly legitimate in this approach, implicitly it recognises that a complementary qualitative type treatment of the same symbols is equally possible!
So the quantitatave aspect of Mathematics is understood here in a - relatively - independent manner.
And if we are to proceed to a truly comprehensive mathematical understanding (involving all 3 types) then the Type 1 must be defined in a relative - rather than absolute - fashion.
Just as the 1st dimension of interpretation (in this relative context) initially provides the (Type 1) standard for quantitative type cardinal interpretation of number (based on independence), as I demonstrated in a recent blog the 2nd dimension (Type 2) likewise provides the standard basis for qualitative type ordinal interpretation of number (based on interdependence).
Then combining the 1st and 2nd dimensions of interpretation - corresponding to the linear and circular use of logic respectively - one for example can give a complete explanation of the relationship which two objects (i.e. numbers) have with each other, where both cardinal (quantitative) and ordinal (qualitative) distinctions are both preserved. I used once more the - apparently - simple illustration of the two turns at a crossroads to illustrate this point!
So putting it simply! The 1st dimension of interpretation can be clearly associated with cardinal (quantitative) meaning (in a relatively independent sense); the 2nd dimension can be clearly associated with ordinal (qualitative) type meaning (in a relatively interdependent sense).
However rather like the particle and wave nature of sub-atomic particles, once we bring both aspects together, the wave aspect becomes also particle like, and the particle aspect wave-like: likewise with respect to number: once we attempt to combine both the cardinal (quantitative) and ordinal (qualitative) nature of numbers the cardinal nature acquires ordinal like features, whereas the ordinal acquires cardinal like features.
And the fascinating clue to what all this means is given by the higher dimensional numbers with respect to 1 (> 2) with their corresponding roots of 1 (> 2).
In other words once we go higher than 2, all roots of 1 combine both real and imaginary parts.
So the fascinating and important question then arises as to what the imaginary aspect means in this context (of cardinal and ordinal interpretation).
To appreciate this we need to go back to the 2nd root of 1, which is the quantitative counterpart of the number 2 as dimension (in qualitative terms).
This was defined as - 1. Now the corresponding qualitative interpretation was as the negation of independent conscious type understanding of a rational nature. This equally represents the manner through which interdependent unconscious type holistic appreciation of an intuitive kind takes place.
Therefore in all relationships, whereas understanding of the independent aspect of such relationships is strictly provided through (conscious) reason (1st dimension), appreciation of interdependence by contrast is provided through (unconscious) intuition (2nd dimension). However indirectly this latter aspect can be interpreted in a circular rational fashion (that is paradoxical in terms of conventional reason).
So - 1 in this rational context is given a 2-dimensional (ordinal) interpretation as both positive and negative (+ and -) which can also be expressed as the complementarity of opposites (i.e. opposite poles). It must be remembered that negating in a dynamic interactive context already presupposes a positive element (like anti-matter fusing with matter particles).
Thus we have established in qualitative terms, how - 1 thereby represents the fundamental nature of the 2nd dimension (through which interdependence with respect to opposite poles takes place).
However as we have qualitatively defined it here (in Type 2 terms), this represents the 2nd - rather than the 1st - dimension.
So of we are to reduce this notion appropriately so that it can now be defined in Type 1 terms, we thereby obtain the square root.
Thus the imaginary no. i = the square root of - 1, serves as the (reduced) Type 1 way of incorporating the ordinal (qualitative) aspect of Type 2 Mathematics in an accepted Type 1 cardinal (quantitative) context.
Equally from the opposite perspective of Type 2 Mathematics, i serves as the (reduced) Type 2 manner of incorporating the cardinal (quantitative) aspect of Type 1 Mathematics in an accepted Type 2 ordinal (qualitative) context.
The deeper implications of all this is that complex numbers - when properly viewed from a Type 1 or Type 2 perspective - necessarily incorporate both quantitative and qualitative type aspects.
However in each case one of these aspects remains masked (with its true nature hidden).
Thus from the Type 1 perspective, though we attempt to view both the real and imaginary aspects of number (in a merely quantitative manner), the imaginary aspect in fact represents the alternative qualitative aspect of Mathematics (that remains hidden however through being veiled in a quantitative mask).
Likewise from the Type 2 perspective, though we again may attempt to now view both the real and imaginary aspects of number (in a merely qualitative manner), the imaginary aspect now in fact represents the corresponding quantitative aspect of Mathematics (that remains hidden however in a qualitative mask)!
And once one clearly realises this dilemma, from the relatively isolated stances of both Type 1 and Type 2 Mathematics respectively, then one necessarily must start moving to the Type 3 approach (where both quantitative and qualitative aspects can be properly integrated).
All of this of course is deeply relevant to proper understanding of the Riemann Zeta Function. As it is defined with respect to the complex plane, with both (matching) quantitative and qualitative type interpretations, a Type 3 mathematical approach is very much required for its proper comprehension.
Thursday, March 22, 2012
Number Inconsistency (4)
We have arrived at the point where even the cardinal notion of number - in the context of a (potentially) infinite series - can have a purely relative meaning.
And what is fascinating about this situation is that it cannot be properly explained in the absence of the complementary qualitative (ordinal) notion of number meaning.
As we have seen, within Conventional Mathematics, the qualitative (ordinal) notion of number is reduced in quantitative terms.
This is likewise associated as we have seen with the treatment of infinite series - in effect - as an extension of linear finite notions.
Now if we have a series of positive terms, such linear extension with respect to the finite, seemingly leads to an unambiguous result (in infinite terms).
So from this perspective the sum of an an infinite series series will appear to either converge or diverge in an unambiguous manner.
So for example for in the case of the well-known geometric series 1 + 1/2 + 1/4 + 1/8 + ....., this seemingly converges to the value of 2. So as the sum consistently approximates ever closer to 2 (over a finite range), then by the logic of linear extension, if we were to take a sufficient (i.e. infinite) number of terms the answer would be 2.
Thus from this perspective (where all the terms are of the same sign) an unambiguous answer (2) results for the sum of the infinite series.
Now in the case of the harmonic series
1 + 1/2 + 1/3 + 1/4 + 1/5 +....,
though initially it may not appear obvious, it is easy enough to demonstrate that this series will diverge.
Therefore as we keep increasing the number of terms, the sum shows no sign of approaching a limiting value. Therefore once again by the reductionist process of linear extension (of what is true for the finite) we conclude unambiguously that the sum of terms of the harmonic series diverges to infinity.
However, the deeper reason why these seemingly unambiguous results arise, is of a qualitative nature.
The very definition of 1-dimensional in qualitative terms (which is complementary with its reciprocal as 1st root of 1) is that both coincide as + 1. So there is an identity of qualitative dimension (and quantitative reciprocal) with respect to the Type 2 number system as 1^1. Thus, when all terms of a series are defined unambiguously (with respect merely to the positive sign) both finite and infinite interpretation - which properly are of a quantitative and qualitative nature respectively - can seemingly be successfully reduced in terms of each other.
So significantly within Conventional Mathematics, when we use the sign for addition (+), it is given a merely quantitative interpretation!
However if we now look at the alternating version of the harmonic series we get,
1 - 1/2 + 1/3 - 1/4 + 1/5 -......
What is significant now is that we are using both positive (+) and negative (-) signs.
Now once again these are given a merely quantitative interpretation in Conventional Mathematics (defined as it is in 1-dimensional terms).
However from an appropriate 2-dimensional perspective, whereas the 1st dimension again provides the same quantitative interpretation, the 2nd dimension now provides the corresponding qualitative (ordinal) interpretation.
So, + in this context means positing (of finite meaning); - however implies the negation of such finite meaning in what is qualitatively infinite. Now in psychological terms, the finite aspect will be directly associated with (conscious)reason (using linear logic); the infinite aspect will however be associated with (unconscious) intuition which then indirectly can be expressed in a rational fashion (as circular logic).
Much as conventional mathematicians may wish to avoid this issue, the actual behaviour of the infinite alternating series now incorporates both 1-dimensional (quantitative) and 2-dimensional (qualitative) aspects, with the resulting sum of terms having a merely relative value, that crucially depends on the qualitative i.e. ordinal ranking of terms.
Now if we order the terms in a systematic way so that we add up sequentially, i.e. 1st, followed by 2nd, 3rd, 4th term and so on, the sum of the series will indeed appear to converge to a definite value i.e. the natural log of 2 (.693417..).
However as it is an infinite series, an unlimited number of other ordered arrangements are possible.
For example we could proceed by taking the first positive term and then subtracting the the first two negative terms, then adding the next positive term before again subtracting the next two negative terms and so on.
So here we would have
1 - 1/2 - 1/4 + 1/3 - 1/6 - 1/8 + 1/5 - 1/10 - 1/12 +....
= (1 - 1/2) - 1/4 + (1/3 - 1/6) - 1/8 + (1/5 - 1/10) - 1/12 +....
= 1/2 - 1/4 + 1/6 - 1/8 + 1/10 - 1/12 +...
(1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 +...)/2 = (log 2)/2
So the sum of the series again converges when uniquely ordered in this manner to another finite value (that is exactly half the first).
And there are an unlimited number of other possible arrangements with in some cases the series converging to a distinct finite value and in other cases diverging to infinity!
Now it is important to observe that when we sum up this alternating harmonic series over a finite range, an unambiguous result emerges (approximating to log 2). Here the precise ordering of terms has no impact on the eventual result (which is the same in all cases).
However this clearly is not the case for the series now continued over a (potentially) infinite range.
What this implies therefore is that the very process of linear type extension (i.e. where the infinite is treated as a quantitative extension of finite notions) is inapplicable in this case.
And if its behaviour cannot be explained through Conventional Mathematics (in 1-dimensional terms), then this clearly implies that a deeper more appropriate level of interpretation is required.
Furthermore, we have showed that its behaviour can be properly explained from a 2-dimensional perspective (combining both quantitative and qualitative interpretations of mathematical symbols).
So what I am demonstrating here is that even in the apparent context of merely quantitative type meaning (i.e. with respect to the summing of terms of an infinite series) that such behaviour cannot be properly explained in the absence of corresponding qualitative type interpretation of mathematical symbols.
So when I say that Conventional Mathematics is not fit for purpose, I mean precisely what I say.
Not alone does it totally fail to deal with the the amazingly rich (but unrecognised) world of the qualitative meaning of mathematical symbols; it cannot explain properly the nature of its own recognised domain of the quantitative.
And the underlying reason for this is that ultimately both the quantitative and qualitative aspects of mathematical understanding - with both equally important - are inseparable.
And what is fascinating about this situation is that it cannot be properly explained in the absence of the complementary qualitative (ordinal) notion of number meaning.
As we have seen, within Conventional Mathematics, the qualitative (ordinal) notion of number is reduced in quantitative terms.
This is likewise associated as we have seen with the treatment of infinite series - in effect - as an extension of linear finite notions.
Now if we have a series of positive terms, such linear extension with respect to the finite, seemingly leads to an unambiguous result (in infinite terms).
So from this perspective the sum of an an infinite series series will appear to either converge or diverge in an unambiguous manner.
So for example for in the case of the well-known geometric series 1 + 1/2 + 1/4 + 1/8 + ....., this seemingly converges to the value of 2. So as the sum consistently approximates ever closer to 2 (over a finite range), then by the logic of linear extension, if we were to take a sufficient (i.e. infinite) number of terms the answer would be 2.
Thus from this perspective (where all the terms are of the same sign) an unambiguous answer (2) results for the sum of the infinite series.
Now in the case of the harmonic series
1 + 1/2 + 1/3 + 1/4 + 1/5 +....,
though initially it may not appear obvious, it is easy enough to demonstrate that this series will diverge.
Therefore as we keep increasing the number of terms, the sum shows no sign of approaching a limiting value. Therefore once again by the reductionist process of linear extension (of what is true for the finite) we conclude unambiguously that the sum of terms of the harmonic series diverges to infinity.
However, the deeper reason why these seemingly unambiguous results arise, is of a qualitative nature.
The very definition of 1-dimensional in qualitative terms (which is complementary with its reciprocal as 1st root of 1) is that both coincide as + 1. So there is an identity of qualitative dimension (and quantitative reciprocal) with respect to the Type 2 number system as 1^1. Thus, when all terms of a series are defined unambiguously (with respect merely to the positive sign) both finite and infinite interpretation - which properly are of a quantitative and qualitative nature respectively - can seemingly be successfully reduced in terms of each other.
So significantly within Conventional Mathematics, when we use the sign for addition (+), it is given a merely quantitative interpretation!
However if we now look at the alternating version of the harmonic series we get,
1 - 1/2 + 1/3 - 1/4 + 1/5 -......
What is significant now is that we are using both positive (+) and negative (-) signs.
Now once again these are given a merely quantitative interpretation in Conventional Mathematics (defined as it is in 1-dimensional terms).
However from an appropriate 2-dimensional perspective, whereas the 1st dimension again provides the same quantitative interpretation, the 2nd dimension now provides the corresponding qualitative (ordinal) interpretation.
So, + in this context means positing (of finite meaning); - however implies the negation of such finite meaning in what is qualitatively infinite. Now in psychological terms, the finite aspect will be directly associated with (conscious)reason (using linear logic); the infinite aspect will however be associated with (unconscious) intuition which then indirectly can be expressed in a rational fashion (as circular logic).
Much as conventional mathematicians may wish to avoid this issue, the actual behaviour of the infinite alternating series now incorporates both 1-dimensional (quantitative) and 2-dimensional (qualitative) aspects, with the resulting sum of terms having a merely relative value, that crucially depends on the qualitative i.e. ordinal ranking of terms.
Now if we order the terms in a systematic way so that we add up sequentially, i.e. 1st, followed by 2nd, 3rd, 4th term and so on, the sum of the series will indeed appear to converge to a definite value i.e. the natural log of 2 (.693417..).
However as it is an infinite series, an unlimited number of other ordered arrangements are possible.
For example we could proceed by taking the first positive term and then subtracting the the first two negative terms, then adding the next positive term before again subtracting the next two negative terms and so on.
So here we would have
1 - 1/2 - 1/4 + 1/3 - 1/6 - 1/8 + 1/5 - 1/10 - 1/12 +....
= (1 - 1/2) - 1/4 + (1/3 - 1/6) - 1/8 + (1/5 - 1/10) - 1/12 +....
= 1/2 - 1/4 + 1/6 - 1/8 + 1/10 - 1/12 +...
(1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 +...)/2 = (log 2)/2
So the sum of the series again converges when uniquely ordered in this manner to another finite value (that is exactly half the first).
And there are an unlimited number of other possible arrangements with in some cases the series converging to a distinct finite value and in other cases diverging to infinity!
Now it is important to observe that when we sum up this alternating harmonic series over a finite range, an unambiguous result emerges (approximating to log 2). Here the precise ordering of terms has no impact on the eventual result (which is the same in all cases).
However this clearly is not the case for the series now continued over a (potentially) infinite range.
What this implies therefore is that the very process of linear type extension (i.e. where the infinite is treated as a quantitative extension of finite notions) is inapplicable in this case.
And if its behaviour cannot be explained through Conventional Mathematics (in 1-dimensional terms), then this clearly implies that a deeper more appropriate level of interpretation is required.
Furthermore, we have showed that its behaviour can be properly explained from a 2-dimensional perspective (combining both quantitative and qualitative interpretations of mathematical symbols).
So what I am demonstrating here is that even in the apparent context of merely quantitative type meaning (i.e. with respect to the summing of terms of an infinite series) that such behaviour cannot be properly explained in the absence of corresponding qualitative type interpretation of mathematical symbols.
So when I say that Conventional Mathematics is not fit for purpose, I mean precisely what I say.
Not alone does it totally fail to deal with the the amazingly rich (but unrecognised) world of the qualitative meaning of mathematical symbols; it cannot explain properly the nature of its own recognised domain of the quantitative.
And the underlying reason for this is that ultimately both the quantitative and qualitative aspects of mathematical understanding - with both equally important - are inseparable.
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