I have already stated that the conventional interpretation of the infinite leads to a reduced notion whereby it is viewed as an extension of the finite.
For example we still routinely refer to series as infinite e.g. the natural number series. Here the mistaken impression is given that with the terms getting progressively larger that "in the limit" they become infinite.
However strictly this is just nonsense. As terms get larger they always remain finite and at no stage pass over to a so-called infinite state.
It would be valid to say that we cannot set a limit to the size of possible terms in the series (while still accepting that they always remain finite). But this is quite different from saying that they become infinite!
A similar problem results in the treatment of "infinitesmals" which again is a meaningless notion.
As a quantity becomes progressively smaller, it still remains finite and at no stage becomes infinitesmal.
Though it may be quite appropriate to approximate such quantities as 0 when obtaining the numerical value of a calculation, an important qualitative distinction remains in that such calculations are always of a merely relative nature.
Put yet another way, the distinction as between infinite and finite concepts relates to the corresponding distinction as between potential and actual notions.
For example the number concept is of an infinite nature potentially relating to any number (not actually specified). However a specific number perception e.g. 2 is then actualised in a finite manner. Thus the dynamic interaction that necessarily exists in experience as between the number concept and related perceptions inevitably combines both finite and infinite notions.
Thus to avoid confusing the infinite with the finite we need to recognise that - again in dynamic terms - the finite determination of any number always implies that other finite numbers thereby remain indeterminate.
Thus what is loosely - and inaccurately - referred to as an infinite series relates to a situation where the progressive determination of certain terms implies that other terms remain actually indeterminate!
Therefore when correctly understood in dynamic experiential terms, the uncertainty principle applies to all mathematical interpretation.
Once again this relates to the fact that (discrete) rational and (continuous) intuitive aspects are always entailed with respect to understanding. Thus we can only make greater clarity with respect to one aspect through accepting corresponding fuzziness with respect to the other.
And as I have been demonstrating the apparent absolute nature of Conventional Mathematics stems from the - mistaken - view that mathematical interpretation is merely of a rational nature!
Once again when we properly accept the true interactive nature of mathematical possibility of "proof" in an absolute sense disappears.
The "proof" of a general hypothesis strictly relates to its merely potential nature as applying to all (non-specified) cases which is of an infinite continuous nature. However the application of the "proof" to any specific case entails actualisation in a finite discrete manner. So to maintain that the general "proof" logically applies to the specific case entails a basic confusion (whereby the infinite is reduced to finite notions).
This is not to suggest that mathematical "proof" is thereby of no value. Rather it is to suggest that it is subject to the uncertainty principle and thereby of a merely probable nature. Indeed momentary reflection on the issue will reveal that the very acceptance of mathematical "proof" entails a certain form of social consensus that can later transpire to have been mistaken. For example Andrew Wiles first proof of "Fermat's Last Theorem" was found to be in error after it had already passed through a rigorous process of verification. So the present status of the revised proof is strictly of a probable nature. In other words as time goes by with no further errors being found we can accept its probable truth with an ever greater degree of confidence!
So in dynamic experiential terms (which represents the true nature of mathematical understanding) all "proof" is subject to the uncertainty principle.
However even within the reduced assumptions of Conventional Mathematics there are certain problems that in principle can be shown to have no proof (or disproof).
And chief among these problems is the Riemann Hypothesis.
Wednesday, October 27, 2010
Tuesday, October 26, 2010
The Harmonic Series
The Pythgoreans have a crucial input into true appreciation of the Riemann Hypothesis in at least two important ways.
We have already discussed the first of these relating to the square root of 2 in "The Pythagorean Dilemma".
The Pythagoreans realised to their dismay that this number was irrational. It did not suffice for them to simply prove in quantitative terms that the number was indeed irrational. More importantly they were seeking a qualitative appreciation as to how such a number could arise.
The essence of an irrational number is that it contains both finite and infinite aspects. In the subsequent mathematical understanding however of irrational numbers a solely reduced quantitative interpretation is provided. Thus the real need, which the Pythagoreans clearly realised, to provide both quantitative an qualitative interpretation has thereby been avoided.
The square root of 2 also points to a circular appreciation of number where - paradoxically - both positive and negative interpretations can both be correct.
So correct to 4 decimal places the square root of 2 can be expressed as + 1.4142 or - 1.4142.
Now the relevance of all this to the Riemann Hypothesis is that we obtain the square root by raising 2 to the dimensional power of 1/2.
And of course the Riemann Hypothesis states that all the non-trivial zeros of the zeta function will relate to values of s i.e dimensional powers with real part = 1/2.
However the Pythagoreans became equally famous for the discovery of the significance of what has come to be known (in their memory) as the harmonic series.
This is the simple sequence comprising the reciprocals of the natural nos.
1 + 1/2 + 1/3 + 1/4 +.....
Now the Pythagoreans were able to connect this series with musical harmonics. They seemed to realise that if a vessel full of water was struck and then successively also struck when half full, a third full, a quarter full etc. that the musical notes generated would appear harmonious to the ear.
Indeed this provided for them striking confirmation of the overriding importance of the natural numbers in explaining nature's secrets. So the very terms "music of the spheres" derives from this discovery.
However the harmonic series provides the starting base for - what is known as - Riemann's Zeta Function where each of the natural number reciprocals can be raised not alone to the dimensional power of 1 but to any complex number power.
However the harmonic series in itself can be shown to have a striking relevance for understanding the behaviour of the primes.
The prime number theorem is often stated in its simplest form as where the ratio of n/log n to the true frequency of primes approaches 1 as n becomes progressively larger.
This implies that log n provides a very good measurement (especially for large values of n) of the average gap as between successive prime numbers.
However there is a close connection as between the harmonic series and log n.
In fact the harmonic series (where the denominator ranges over the natural numbers from 1 to n) = log n + k (where k is known as Euler's constant = .5772 approx).
Thus for very large values of n the sum of the harmonic series itself provides a very accurate estimate of the average gap between successive prime numbers.
So this provides just one striking example as to the close connection as between the primes and natural numbers!
It is also remarkable in another sense. Once again the sum of the harmonic series (where n is finite) = log n + k.
Thus when we differentiate with respect to n we get 1/n.
This therefore implies that the average gap as between primes increases by 1/n (as n increases by 1)
This would imply for example that as move from n = 1,000,000 to 1,000,001 that the average gap between primes itself increases by 1/1,000,000 (or perhaps more accurately 1/1,000,000.5).
This is truly remarkable in that it links an important aspect of prime number behaviour in an extremely simple manner to the reciprocals of the natural numbers!
It is also fascinating that the famous formula n/log n relating to the frequency of the primes can be expressed as the ratio of the two zeta functions (where n is taken over a finite range) for s = o and s = 1 respectively.
When s = o, the zeta function
= 1/1^0 + 1/2^0 + 1/3^0 + 1/4^0 + .....
= 1 + 1 + 1 + 1 +...... = n
When s = 1, the zeta function
= 1/1^1 + 1/2^1 + 1/3^1 + 1/4^1 + ..... (which approximates to log n when n is suitably large)
Therefore we can approximate n/log n as
zeta (0)/zeta (1) where the range of values for n is finite (and values of series calculated in the conventional linear manner).
So perhaps (though more inaccurate for lower values of n) this could provide the simplest formulation of the prime number theorem,
i.e. {zeta (0)/zeta (1)}/{actual occurrence of primes from 1 to n} approximates 1(when n is sufficiently large in magnitude).
We have already discussed the first of these relating to the square root of 2 in "The Pythagorean Dilemma".
The Pythagoreans realised to their dismay that this number was irrational. It did not suffice for them to simply prove in quantitative terms that the number was indeed irrational. More importantly they were seeking a qualitative appreciation as to how such a number could arise.
The essence of an irrational number is that it contains both finite and infinite aspects. In the subsequent mathematical understanding however of irrational numbers a solely reduced quantitative interpretation is provided. Thus the real need, which the Pythagoreans clearly realised, to provide both quantitative an qualitative interpretation has thereby been avoided.
The square root of 2 also points to a circular appreciation of number where - paradoxically - both positive and negative interpretations can both be correct.
So correct to 4 decimal places the square root of 2 can be expressed as + 1.4142 or - 1.4142.
Now the relevance of all this to the Riemann Hypothesis is that we obtain the square root by raising 2 to the dimensional power of 1/2.
And of course the Riemann Hypothesis states that all the non-trivial zeros of the zeta function will relate to values of s i.e dimensional powers with real part = 1/2.
However the Pythagoreans became equally famous for the discovery of the significance of what has come to be known (in their memory) as the harmonic series.
This is the simple sequence comprising the reciprocals of the natural nos.
1 + 1/2 + 1/3 + 1/4 +.....
Now the Pythagoreans were able to connect this series with musical harmonics. They seemed to realise that if a vessel full of water was struck and then successively also struck when half full, a third full, a quarter full etc. that the musical notes generated would appear harmonious to the ear.
Indeed this provided for them striking confirmation of the overriding importance of the natural numbers in explaining nature's secrets. So the very terms "music of the spheres" derives from this discovery.
However the harmonic series provides the starting base for - what is known as - Riemann's Zeta Function where each of the natural number reciprocals can be raised not alone to the dimensional power of 1 but to any complex number power.
However the harmonic series in itself can be shown to have a striking relevance for understanding the behaviour of the primes.
The prime number theorem is often stated in its simplest form as where the ratio of n/log n to the true frequency of primes approaches 1 as n becomes progressively larger.
This implies that log n provides a very good measurement (especially for large values of n) of the average gap as between successive prime numbers.
However there is a close connection as between the harmonic series and log n.
In fact the harmonic series (where the denominator ranges over the natural numbers from 1 to n) = log n + k (where k is known as Euler's constant = .5772 approx).
Thus for very large values of n the sum of the harmonic series itself provides a very accurate estimate of the average gap between successive prime numbers.
So this provides just one striking example as to the close connection as between the primes and natural numbers!
It is also remarkable in another sense. Once again the sum of the harmonic series (where n is finite) = log n + k.
Thus when we differentiate with respect to n we get 1/n.
This therefore implies that the average gap as between primes increases by 1/n (as n increases by 1)
This would imply for example that as move from n = 1,000,000 to 1,000,001 that the average gap between primes itself increases by 1/1,000,000 (or perhaps more accurately 1/1,000,000.5).
This is truly remarkable in that it links an important aspect of prime number behaviour in an extremely simple manner to the reciprocals of the natural numbers!
It is also fascinating that the famous formula n/log n relating to the frequency of the primes can be expressed as the ratio of the two zeta functions (where n is taken over a finite range) for s = o and s = 1 respectively.
When s = o, the zeta function
= 1/1^0 + 1/2^0 + 1/3^0 + 1/4^0 + .....
= 1 + 1 + 1 + 1 +...... = n
When s = 1, the zeta function
= 1/1^1 + 1/2^1 + 1/3^1 + 1/4^1 + ..... (which approximates to log n when n is suitably large)
Therefore we can approximate n/log n as
zeta (0)/zeta (1) where the range of values for n is finite (and values of series calculated in the conventional linear manner).
So perhaps (though more inaccurate for lower values of n) this could provide the simplest formulation of the prime number theorem,
i.e. {zeta (0)/zeta (1)}/{actual occurrence of primes from 1 to n} approximates 1(when n is sufficiently large in magnitude).
Sunday, October 24, 2010
Finite and Infinite
Ultimately the true nature of the Riemann Hypothesis pertains to the key relationship as between finite and infinite (not just in Mathematics but with respect to all living experience).
Indeed the various ways in which I have already expressed this relationship represent the finite/infinite connection.
For example we can formulate this as the essential relationship of quantitative and qualitative, linear and circular, discrete and continuous, classical and quantum mechanical, order and chaos etc.
However conventional mathematical understanding is properly geared solely for (rational) finite interpretation. Though infinite notions are of course recognised they are treated in a strictly reduced manner (i.e. that is amenable to rational type analysis).
In other words the true meaning of the infinite is thereby lost in Mathematics.
This remains the - largely unrecognised - elephant in the room as its key overriding central problem.
To understand what is involved here we have to once again recognise that actual mathematical understanding is never strictly rational (though explicitly in formal terms it is indeed represented as rational!)
Rather, such understanding always involves a dynamic interaction of both rational (conscious) and intuitive (unconscious) elements.
Put quite simply, whereas the rational element of this interaction pertains directly to appreciation of the finite, by contrast the intuitive aspect pertains directly to the infinite.
Though implicitly mathematicians may indeed recognise the importance of intuition - especially in the generation of creative insights - explicitly however Conventional Mathematics is interpreted in a merely reduced rational manner (where in effect the holistic notion of the infinite is reduced in a finite manner).
Thus if we are to properly incorporate the infinite with finite notions then we must include a new qualitative type of mathematical understanding (which I call Holistic Mathematics).
However it is important to recognise that Holistic Mathematics operates in terms of a distinctive logical system (which is circular in nature).
It is only through the appropriate circular use of logic that intuition - pertaining directly to appreciation of the infinite in experience - can be indirectly translated in rational terms.
Once again I will briefly express the key difference here as between the two logical systems.
With linear logic - which defines conventional mathematical experience - the key polarities of experience are treated as separate and independent.
For example internal and external - which necessarily underpin all phenomenal experience - represents one important example of such key polarities.
So in Mathematics the external (objective) aspect is treated as independent of the internal (subjective) aspect. This thereby creates the impression of a strict objective validity to mathematical truths (which ultimately is unwarranted).
Likewise in Mathematics, whole notions are treated as independent of parts so that we can study either aspect in isolation from each other. In fact perhaps the most characteristic feature of the linear approach is the manner in which wholes are reduced to parts. In this way both aspects can be treated in a merely quantitative manner (where the whole is literally seen as the sum of the parts).
Thus the key qualitative distinction as between wholes and parts is thereby lost through such interpretation!
With circular logic the key polarities are treated as interdependent (rather than separate). Ultimately of course this means that with full interdependence any notion of polarities as phenomenally separate disappears. We are then left in experience with pure formless appreciation (which is the very nature of intuitive awareness).
However we can characterise the nature of such awareness in an indirect rational manner through the complementarity of opposites. This leads to the recognition - like left and right turns on a road - that opposites have a merely arbitrary definition in any context (depending on the polar frame of reference). So for example what is considered as (quantitatively) whole in one context could be considered as part in another (and vice versa). Now the mysterious dynamic enabling this switching of reference poles is of a qualitative (intuitive) nature and not thereby confused with rational interpretation in isolated contexts.
Put another way - deliberately using holistic mathematical language - there are two key tasks in all understanding (which thereby includes Mathematics). First we must successfully differentiate phenomenal symbols (as independent); secondly we must successfully integrate those same symbols (as interdependent).
In actual mathematical experience, the first of these tasks (of differentiation) is properly achieved through (linear) reason using either/or logic; however the second equally important task (of integration) is properly achieved though intuition that is indirectly represented in a paradoxical manner through circular reason (using both/and logic).
Once again there is a huge unrecognised problem with conventional mathematical interpretation in that is based solely on (linear) reason. Therefore it can only deal with the qualitative task of integral interpretation in a reduced manner.
The relevance of all this for the Riemann Hypothesis is that - properly understood - it actually points to the fundamental condition for the successful harmonisation of both (analytic) quantitative and (holistic) qualitative type understanding.
Imagine in geometrical terms a straight line diameter = 1 unit circumscribed by its circular circumference. Now the midpoint which is common to both circle and line occurs at the midpoint of the line (i.e. at 1/2).
In a nutshell this is what the Riemann Hypothesis is all about i.e. the central condition that is necessary to reconcile both the quantitative (linear) and qualitative (circular) aspects of mathematical understanding.
Indeed the various ways in which I have already expressed this relationship represent the finite/infinite connection.
For example we can formulate this as the essential relationship of quantitative and qualitative, linear and circular, discrete and continuous, classical and quantum mechanical, order and chaos etc.
However conventional mathematical understanding is properly geared solely for (rational) finite interpretation. Though infinite notions are of course recognised they are treated in a strictly reduced manner (i.e. that is amenable to rational type analysis).
In other words the true meaning of the infinite is thereby lost in Mathematics.
This remains the - largely unrecognised - elephant in the room as its key overriding central problem.
To understand what is involved here we have to once again recognise that actual mathematical understanding is never strictly rational (though explicitly in formal terms it is indeed represented as rational!)
Rather, such understanding always involves a dynamic interaction of both rational (conscious) and intuitive (unconscious) elements.
Put quite simply, whereas the rational element of this interaction pertains directly to appreciation of the finite, by contrast the intuitive aspect pertains directly to the infinite.
Though implicitly mathematicians may indeed recognise the importance of intuition - especially in the generation of creative insights - explicitly however Conventional Mathematics is interpreted in a merely reduced rational manner (where in effect the holistic notion of the infinite is reduced in a finite manner).
Thus if we are to properly incorporate the infinite with finite notions then we must include a new qualitative type of mathematical understanding (which I call Holistic Mathematics).
However it is important to recognise that Holistic Mathematics operates in terms of a distinctive logical system (which is circular in nature).
It is only through the appropriate circular use of logic that intuition - pertaining directly to appreciation of the infinite in experience - can be indirectly translated in rational terms.
Once again I will briefly express the key difference here as between the two logical systems.
With linear logic - which defines conventional mathematical experience - the key polarities of experience are treated as separate and independent.
For example internal and external - which necessarily underpin all phenomenal experience - represents one important example of such key polarities.
So in Mathematics the external (objective) aspect is treated as independent of the internal (subjective) aspect. This thereby creates the impression of a strict objective validity to mathematical truths (which ultimately is unwarranted).
Likewise in Mathematics, whole notions are treated as independent of parts so that we can study either aspect in isolation from each other. In fact perhaps the most characteristic feature of the linear approach is the manner in which wholes are reduced to parts. In this way both aspects can be treated in a merely quantitative manner (where the whole is literally seen as the sum of the parts).
Thus the key qualitative distinction as between wholes and parts is thereby lost through such interpretation!
With circular logic the key polarities are treated as interdependent (rather than separate). Ultimately of course this means that with full interdependence any notion of polarities as phenomenally separate disappears. We are then left in experience with pure formless appreciation (which is the very nature of intuitive awareness).
However we can characterise the nature of such awareness in an indirect rational manner through the complementarity of opposites. This leads to the recognition - like left and right turns on a road - that opposites have a merely arbitrary definition in any context (depending on the polar frame of reference). So for example what is considered as (quantitatively) whole in one context could be considered as part in another (and vice versa). Now the mysterious dynamic enabling this switching of reference poles is of a qualitative (intuitive) nature and not thereby confused with rational interpretation in isolated contexts.
Put another way - deliberately using holistic mathematical language - there are two key tasks in all understanding (which thereby includes Mathematics). First we must successfully differentiate phenomenal symbols (as independent); secondly we must successfully integrate those same symbols (as interdependent).
In actual mathematical experience, the first of these tasks (of differentiation) is properly achieved through (linear) reason using either/or logic; however the second equally important task (of integration) is properly achieved though intuition that is indirectly represented in a paradoxical manner through circular reason (using both/and logic).
Once again there is a huge unrecognised problem with conventional mathematical interpretation in that is based solely on (linear) reason. Therefore it can only deal with the qualitative task of integral interpretation in a reduced manner.
The relevance of all this for the Riemann Hypothesis is that - properly understood - it actually points to the fundamental condition for the successful harmonisation of both (analytic) quantitative and (holistic) qualitative type understanding.
Imagine in geometrical terms a straight line diameter = 1 unit circumscribed by its circular circumference. Now the midpoint which is common to both circle and line occurs at the midpoint of the line (i.e. at 1/2).
In a nutshell this is what the Riemann Hypothesis is all about i.e. the central condition that is necessary to reconcile both the quantitative (linear) and qualitative (circular) aspects of mathematical understanding.
Thursday, October 14, 2010
Riemann Hypothesis and Physical Connections
When I first read about connections as between the Riemann non-trivial zeros and certain physical energy states in quantum physics, I was not at all surprised.
For I had long believed that important physical applications of the Riemann zeros would necessarily exist.
Indeed I would go considerably further. Not alone do the zeros have implications for physics but equally they have a deep relevance for psychological understanding of various spiritual energy states. Furthermore in the holistic mathematical understanding of what is involved the two forms of understanding (physical and psychological) are fully complementary.
Though I have always recognised - in principle - the potential physical relevance of the zeros (and of course the associated Riemann Hypothesis) my main focus has been on the psychological relevance of the Riemann zeros.
And here there is an interesting twist! For in attempting to provide a coherent interpretation of what the Hypothesis actually entails (which then leads to a simple resolution of the Hypothesis), I have found the trivial zeros to be of greater significance.
Indeed my initial interest in the Riemann Hypothesis was sparked by the desire to provide a coherent explanation for the trivial zeros.
For example if the take the first of these for the zeta function (where s = -2) this would result in the series
1 + 4 + 9 + 16 +.....
Now clearly from a conventional linear quantitative perspective, this series diverges so that it has no finite sum.
However according to Riemann's Zeta Function, the sum of this infinite series = 0.
So it was through attempting to explain the nature of this unexpected value that I realised that it referred directly - not to a quantitative but rather - to a qualitative interpretation of number.
Basically in qualitative terms, 2-dimensional understanding refers to the complementarity of (real) polar opposites in experience such as internal and external. Though in linear (1-dimensional) terms these are clearly separated in experience enabling for example the unambiguous objective interpretation of mathematical calculations, in circular (2-dimensional) terms these are seen as complementary and interdependent (with ultimately no division possible between them).
Strictly 2-dimensional understanding (as positive) relates to the rational paradoxical interpretation of this circular type relationship.
However 2-dimensional understanding (as negative) relates to the purely intuitive appreciation of the same relationship (where secondary rational distinctions are negated).
Put another way when s = - 2, correct qualitative understanding relates to a pure contemplative i.e. spiritual energy state. (It would certainly have resonated with the Pythagoreans before the unfortunate subsequent split in mathematical understanding!) Though it is correctly represented as zero, it relates directly in this instance to a qualitative - rather than quantitative - meaning.
Therefore we give a coherent numerical explanation to the first of these trivial zeros through interpreting the relationship directly in qualitative terms. Equally all the other trivial zeros can be explained as "higher" intuitive contemplative states of understanding (that are nothing in phenomenal terms).
Riemann also provided a fascinating transformation formula enabling one to calculate from conventional values for s > 1, corresponding non-conventional values for s < 0. So the clear implication here is that the transformation formula - when correctly understood - provides a clear (indissoluble) relationship as between numerical results with a direct quantitative value (for s > 1) on the right hand side of his equation and corresponding results with a direct qualitative holistic value (for s < 0) on the left hand side.
We can also continue this procedure in the critical strip (0 < s < 1) where now values for the zeta series represent a hybrid mix of both quantitative and qualitative aspects. The exact matching of both left and right hand side values requires that s = 1/2 where - by definition - quantitative and qualitative aspects exactly match with each other.
And this in short is what the Riemann Hypothesis is all about i.e. the crucial condition enabling the consistency of both quantitative and qualitative aspects of mathematical interpretation.
Unfortunately in a Mathematics that only formally recognises the quantitative aspect of understanding the true nature of the Riemann Hypothesis will always prove elusive.
Incidentally attempts to "prove" the Riemann Hypothesis with respect to establishing an exact correspondence with certain physical energy states seems to me somewhat confused.
Though it would indeed be exciting and important to conclusively establish such a link this would only demonstrate an especially strong correlation as between corresponding mathematical and physical systems.
However perfect correlation - even if demonstrated to exist - does not constitute proof!
For I had long believed that important physical applications of the Riemann zeros would necessarily exist.
Indeed I would go considerably further. Not alone do the zeros have implications for physics but equally they have a deep relevance for psychological understanding of various spiritual energy states. Furthermore in the holistic mathematical understanding of what is involved the two forms of understanding (physical and psychological) are fully complementary.
Though I have always recognised - in principle - the potential physical relevance of the zeros (and of course the associated Riemann Hypothesis) my main focus has been on the psychological relevance of the Riemann zeros.
And here there is an interesting twist! For in attempting to provide a coherent interpretation of what the Hypothesis actually entails (which then leads to a simple resolution of the Hypothesis), I have found the trivial zeros to be of greater significance.
Indeed my initial interest in the Riemann Hypothesis was sparked by the desire to provide a coherent explanation for the trivial zeros.
For example if the take the first of these for the zeta function (where s = -2) this would result in the series
1 + 4 + 9 + 16 +.....
Now clearly from a conventional linear quantitative perspective, this series diverges so that it has no finite sum.
However according to Riemann's Zeta Function, the sum of this infinite series = 0.
So it was through attempting to explain the nature of this unexpected value that I realised that it referred directly - not to a quantitative but rather - to a qualitative interpretation of number.
Basically in qualitative terms, 2-dimensional understanding refers to the complementarity of (real) polar opposites in experience such as internal and external. Though in linear (1-dimensional) terms these are clearly separated in experience enabling for example the unambiguous objective interpretation of mathematical calculations, in circular (2-dimensional) terms these are seen as complementary and interdependent (with ultimately no division possible between them).
Strictly 2-dimensional understanding (as positive) relates to the rational paradoxical interpretation of this circular type relationship.
However 2-dimensional understanding (as negative) relates to the purely intuitive appreciation of the same relationship (where secondary rational distinctions are negated).
Put another way when s = - 2, correct qualitative understanding relates to a pure contemplative i.e. spiritual energy state. (It would certainly have resonated with the Pythagoreans before the unfortunate subsequent split in mathematical understanding!) Though it is correctly represented as zero, it relates directly in this instance to a qualitative - rather than quantitative - meaning.
Therefore we give a coherent numerical explanation to the first of these trivial zeros through interpreting the relationship directly in qualitative terms. Equally all the other trivial zeros can be explained as "higher" intuitive contemplative states of understanding (that are nothing in phenomenal terms).
Riemann also provided a fascinating transformation formula enabling one to calculate from conventional values for s > 1, corresponding non-conventional values for s < 0. So the clear implication here is that the transformation formula - when correctly understood - provides a clear (indissoluble) relationship as between numerical results with a direct quantitative value (for s > 1) on the right hand side of his equation and corresponding results with a direct qualitative holistic value (for s < 0) on the left hand side.
We can also continue this procedure in the critical strip (0 < s < 1) where now values for the zeta series represent a hybrid mix of both quantitative and qualitative aspects. The exact matching of both left and right hand side values requires that s = 1/2 where - by definition - quantitative and qualitative aspects exactly match with each other.
And this in short is what the Riemann Hypothesis is all about i.e. the crucial condition enabling the consistency of both quantitative and qualitative aspects of mathematical interpretation.
Unfortunately in a Mathematics that only formally recognises the quantitative aspect of understanding the true nature of the Riemann Hypothesis will always prove elusive.
Incidentally attempts to "prove" the Riemann Hypothesis with respect to establishing an exact correspondence with certain physical energy states seems to me somewhat confused.
Though it would indeed be exciting and important to conclusively establish such a link this would only demonstrate an especially strong correlation as between corresponding mathematical and physical systems.
However perfect correlation - even if demonstrated to exist - does not constitute proof!
Wednesday, October 13, 2010
The Invisible Gap
In the most fundamental sense, the Riemann Hypothesis relates to the invisible gap, as it were, that divides quantitative (linear) and qualitative (circular) type interpretation of reality. In psychological terms this relates to the inevitable interaction that necessarily takes place with respect to all understanding (including of course mathematical) between rational (analytic) and intuitive (holistic) type processes of understanding.
Unfortunately the problem for Conventional Mathematics is that it in formal terms it recognises solely the role of rational interpretation. Therefore it can only deal with this interaction in a reduced manner i.e. by attempting to explain - what properly relates to - the rational and intuitive, in merely rational terms.
Alternatively, it can only attempt to deal with the relationship of both quantitative and qualitative type mathematical understanding in a reduced quantitative manner.
And this is in a nutshell is the very reason why a satisfactory "proof" of the Riemann Hypothesis has proven so elusive.
Quite simply - when the nature of the problem is properly appreciated - the Riemann Hypothesis can have no solution in conventional mathematical terms.
This same problem can be expressed in other ways that directly impinge on the understanding of the true nature of the Hypothesis.
For example the Riemann Hypothesis lies in that invisible gap where the discrete and continuous interpretation of number is united. And this is central to appreciation of the true nature of prime numbers in the attempt to successfully unite their discete individual identities with the continuous nature of their overall frequency among the natural numbers.
There is a key problem here which again is not properly recognised. The study of individual primes and their overall general frequency have both quantitative and qualitative aspects that are of a (conscious) analytic and (unconscious) holistic nature with respect to each other. Just we can choose in isolation to investigate an atomic particle with respect to either its particle or wave aspect in quantitative terms, likewise we can attempt to study in isolation both individual primes and their general distribution with respect to their quantitative characteristics.
However this approach will inevitably break down in the simultaneous integration of both aspects (which are now complementary). So here we must incorporate both quantitative and qualitative type appreciation. Conventional Mathematics by its very nature is not geared for this task. Not alone can it not resolve the Hypothesis, it is not even capable of providing a coherent explanation of its true nature.
Another way of expressing the same problem is that all classical systems have counterpart systems that are quantum mechanical in nature. Once again we can indeed attempt to study both systems in a separate manner with respect to their - mere - quantitative aspects. However if we wish to properly relate both types once again we have to broaden appreciation to include both quantitative and qualitative modes of interpretation.
Finally, as is now evident from particle physics, the Riemann zeros can be given a coherent explanation in terms of the energy states of certain chaotic quantum mechanical systems. This again clearly points in my mind to an inevitable interaction as between two distinct modes of behaviour with properties that are analytic and holistic with respect to each other.
However through all this the elephant in the room is ignored.
There is an entirely distinctive holistic interpretation that can be given to every mathematical symbol (in what I term Holistic Mathematics) that constitutes the missing qualitative aspect of mathematical understanding.
It is in the very relationship of the two aspects of Mathematics (quantitative and qualitative) that the simple resolution of the Riemann Hypothesis is found not as a proof but rather as a fundamental axiom that underlies all the lesser axioms on which conventional mathematical interpretation is based.
When asked once what was the most important problem in Mathematics, the great mathematician Hilbert - as claimed in Constance Reid's book "Hilbert" - replied.
"The problem of the zeros of the zeta function. Not only in mathematics. But absolutely most important"
And in fact Hilbert was right!
There is a famous sutra in Buddhism which ultimately is equivalent to the Riemann Hypothesis
"Form is not other than Void; Void is not other than Form".
One could equally say with respect to Mathematics.
"The quantitative aspect is not other than the qualitative; the qualitative is not other than the quantitative."
So it is in this mysterious intersection of quantitative and qualitative aspects that the Riemann Hypothesis resides.
So the end the resolution of the Hypothesis is attained not through reason but in that ineffable spiritual experience where quantitative and qualitative distinctions are no longer necessary (nor even can be made).
Unfortunately the problem for Conventional Mathematics is that it in formal terms it recognises solely the role of rational interpretation. Therefore it can only deal with this interaction in a reduced manner i.e. by attempting to explain - what properly relates to - the rational and intuitive, in merely rational terms.
Alternatively, it can only attempt to deal with the relationship of both quantitative and qualitative type mathematical understanding in a reduced quantitative manner.
And this is in a nutshell is the very reason why a satisfactory "proof" of the Riemann Hypothesis has proven so elusive.
Quite simply - when the nature of the problem is properly appreciated - the Riemann Hypothesis can have no solution in conventional mathematical terms.
This same problem can be expressed in other ways that directly impinge on the understanding of the true nature of the Hypothesis.
For example the Riemann Hypothesis lies in that invisible gap where the discrete and continuous interpretation of number is united. And this is central to appreciation of the true nature of prime numbers in the attempt to successfully unite their discete individual identities with the continuous nature of their overall frequency among the natural numbers.
There is a key problem here which again is not properly recognised. The study of individual primes and their overall general frequency have both quantitative and qualitative aspects that are of a (conscious) analytic and (unconscious) holistic nature with respect to each other. Just we can choose in isolation to investigate an atomic particle with respect to either its particle or wave aspect in quantitative terms, likewise we can attempt to study in isolation both individual primes and their general distribution with respect to their quantitative characteristics.
However this approach will inevitably break down in the simultaneous integration of both aspects (which are now complementary). So here we must incorporate both quantitative and qualitative type appreciation. Conventional Mathematics by its very nature is not geared for this task. Not alone can it not resolve the Hypothesis, it is not even capable of providing a coherent explanation of its true nature.
Another way of expressing the same problem is that all classical systems have counterpart systems that are quantum mechanical in nature. Once again we can indeed attempt to study both systems in a separate manner with respect to their - mere - quantitative aspects. However if we wish to properly relate both types once again we have to broaden appreciation to include both quantitative and qualitative modes of interpretation.
Finally, as is now evident from particle physics, the Riemann zeros can be given a coherent explanation in terms of the energy states of certain chaotic quantum mechanical systems. This again clearly points in my mind to an inevitable interaction as between two distinct modes of behaviour with properties that are analytic and holistic with respect to each other.
However through all this the elephant in the room is ignored.
There is an entirely distinctive holistic interpretation that can be given to every mathematical symbol (in what I term Holistic Mathematics) that constitutes the missing qualitative aspect of mathematical understanding.
It is in the very relationship of the two aspects of Mathematics (quantitative and qualitative) that the simple resolution of the Riemann Hypothesis is found not as a proof but rather as a fundamental axiom that underlies all the lesser axioms on which conventional mathematical interpretation is based.
When asked once what was the most important problem in Mathematics, the great mathematician Hilbert - as claimed in Constance Reid's book "Hilbert" - replied.
"The problem of the zeros of the zeta function. Not only in mathematics. But absolutely most important"
And in fact Hilbert was right!
There is a famous sutra in Buddhism which ultimately is equivalent to the Riemann Hypothesis
"Form is not other than Void; Void is not other than Form".
One could equally say with respect to Mathematics.
"The quantitative aspect is not other than the qualitative; the qualitative is not other than the quantitative."
So it is in this mysterious intersection of quantitative and qualitative aspects that the Riemann Hypothesis resides.
So the end the resolution of the Hypothesis is attained not through reason but in that ineffable spiritual experience where quantitative and qualitative distinctions are no longer necessary (nor even can be made).
Thursday, October 7, 2010
New Interpretations
As I have stated, there are in fact two related aspects to a comprehensive mathematical understanding.
The first is the recognised conventional quantitative approach that is fundamentally based on the linear use of reason (using unambiguous either/or distinctions).
However the second - unrecognised - qualitative approach is based on the circular use of reason (using paradoxical both/and designations).
One way of expressing the difference between both types is that polar opposites (such as positive and negative) are treated as separate in linear terms whereas they are given a complementary meaning (as interdependent) in corresponding circular interpretation.
Fascinatingly however the distinction as between these two types can play a direct role in the interpretation of - what initially appear - as merely quantitative relationships.
My initial recognition of this fact does not stem directly from the Riemann Hypothesis but rather from investigation of the Fibonacci sequence.
The Fibonacci sequence is obtained through starting with 0 and 1 and then successively adding the last term to the previous to obtain a new number in the sequence.
In this way we get 0, 1, 1, 2, 3, 5, 8, 13,.....
Now the ratio of terms in the sequence (again dividing the lst by the previous) approximates to phi = 1.618033... (the golden mean) and steadily improves as the terms in the sequence become larger.
This value in turn for phi is obtained exactly as the expression of the simple quadratic equation
x^2 - x - 1 = 0.
However strictly there are two values for x from this equation. The first is 1.618... and the second is - .618. So the question arises as to the interpretation of this second value.
After consideration of this issue it became clear to me that - correctly understood - both values (which are positive and negative with respect to each other) represent a complementary pairing.
So the correct way of interpreting this is as follows. When we designate the direction of the ratio (i.e. as last term in sequence to previous) as positive, then the corresponding direction (i.e. as previous term to last) is negative.
Therefore whereas 1.618... approximates the ratio of last to previous, - .618 correspondingly approximates the (reverse) ratio of previous to last term in the sequence.
So in this two dimensional interpretation (which befits the 2-dimensional polynomial expression) positive and negative take on a new meaning in the context of a complementary pairing.
I then realised that the use of polynomial equations could be extended to a whole range of sequences (in a similar manner to the Fibonacci).
The natural number sequence {0}, 1, 2, 3, 4,.. then provides one especially interesting example of such an important sequence.
The corresponding polynomial equation is here given as x^2 - 2x + 1 = 0.
The general form of this equation is x^2 + bx + c = 0 and we get corresponding number sequence to which it relates by successively summing - b * (the last term) - c * (the previous term term)
The natural number polynomial then factorises as (x - 1)(x - 1) = 0. So x = 1 is the ratio in both cases.
What this means in effect is that we would treat the natural number sequence in a standard linear manner (as both terms are of the same sign).
Indeed this clearly identifies the natural number sequence as the very symbol (or archetype) if you wish of linear type understanding.
So in this case it does not matter in which direction we take the last two terms of the sequence with the ratio approximating in each case (as the magnitude of the terms increases) to 1.
Having looked at the archetype of a number sequence corresponding to linear type inetrpretation, it then seemed instructive to likwise examine the corresponding archetype of a number sequence corresponding to circular type interpretation.
And the equation for this sequence is given by (x + 1)(x - 1) = 0 containing complementary positive and negative signs.
So the polynomial expression for this sequence is x^2 - 1 = 0.
The corresponding number sequence to which it relates is
0, 1, 0, 1, 0, 1,....
It is the interpretation of the ratio of this sequence that is especially fascinating as it illuminates several of the qualitative features that I have been attempting to illustrate.
1) Whereas interpretation of the natural number sequence entails pure linear understanding, this sequence correspondingly represents pure (2-dimensional) circular understanding.
Note that if we attempt to calculate the ratio with respect to successive terms (based on 1-dimensional linear interpretation) we either get 0/1 = 0 or 1/0 = ∞ (even though the equation suggests that the ratio = 1 or - 1).
However we can eliminate this problem through (correct) 2-dimensional interpretation,
In this case the ratio is either 1/1 or 0/0 (through expressing ratios with respect to number terms separated by a gap of 2).
Now the very essence of 2-dimensional interpretation is that it combines both rational and intuitive elements in equal measure.
So the first ratio 1/1 = 1, corresponds to rational interpretation (of 2-dimensional reality).
The second ratio 0/0 = 1, corresponds to direct intuitive appreciation of the same relationship giving a meaning to a ratio that cannot be properly defined in 1-dimensional terms. So the two elements (rational and intuitive) that are experientially united in 2-dimensional understanding become separated in terms of this linear sequence of numbers.
2) The actual equation gives rise to two answers + 1 and - 1. So as befits 2-dimensional interpretation, the direction in which terms are taken is important. So if with each second term we take the direction of last to previous term as positive then the corresponding direction of previous to last is thereby negative. And this is precisely what we would expect with 2-dimensional interpretation!
In fact what I have illustrated above is a simple illustration of the general equation x^n = 1 (where n = 2).
When n = 3 for example we would derive the sequence
0, 0, 1, 0, 0, 1,... so that we can only define the initial ratio (where x = 1) as the ratio of terms (separated by a gap of 3).
It would be tempting to push this even further in the attempted qualitative appreciation of imaginary numbers.
As we know the slutions to the equation x^2 + 1 = 0 are x = + i and x = - i, (where i is the square root of - 1).
Now the sequence of numbers to which this relates is
0, 1, 0 - 1, 0, 1, 0 - 1,....
Now again as this represents a 2-dimensional equation (with positive and negative polarities) a 2-dimensional qualitative interpretation is required.
In this case again taking ratios of each second term we get a ratio of 1/- 1 or alternatively - 1/1, and also 0/0.
So the clear implication is that the very essence of the qualitative interpretation of the imaginary number is that the notion of positive and negative directions is now rendered entirely ambiguous.
Also the ratio (which in each case is - 1) points to unconscious pure 2-dimensional rather than conscious reality.
So the very notion of an imaginary number qualitatively relates to the attempt to express a purely (2-dimensional) circular notion - that literally entails the dynamic negation of conscious understanding - indirectly in linear (1-dimensional) terms.
And the corresponding quantitative definition directly parallels this interpretation entailing the square root of a negative unit.
The importance of this demonstration is that similar problems of interpretation arise in defining values for the Riemann Zeta Function.
In standard linear terms the value of the sequence
1/(1^s) + 1/(2^s) + 1/(3^s) +... only converges for values of s > 1.
However Riemann extended the domain of definition for all complex values of s (except 1).
This then leads to the interesting problem where the sum of sequences (which clearly diverge in standard linear terms with no attainable sum) are now given a specific finite result.
For example when s = 0, we generate the following series 1 + 1 + 1 + 1 +.....
Now clearly in conventional terms the sum of this this series is infinite (i.e. does not converge to a finite limit).
However by a remarkable piece of alchemy we can in fact give a finite value to the series
This entails defining a corresponding eta function which generates the alternating series
1 - 1 + 1 - 1 + 1 ....
Remarkably this again relates to the simple equation we discussed above (in case of Fibonacci type sequences) where now x^n = 0.
So if x^n = 0, then - x^n = 0
Therefore 1 - x^n = 1
Thus (1 - x)(1 + x + x^2 + x^3 +......x^[n - 1]) = 1
So (1 + x + x^2 + x^3 +......x^[n - 1]) = 1((1 - x)
Letting x = - 1 as n → ∞
Then 1 - 1 + 1 - 1 + .... = 1/2
The question then arises as to the interpretation of this alternating series.
If we view these terms in 2-dimensional (circular) complementary fashion (where successive positive and negative terms form a pair) then the sum = 0.
However if we consider it in linear (1-dimensional) terms (where we break such a pairing) the sum will always relate to the remaining odd term = 1.
It would seem therefore that since two possible answers can arise that we should take the mean of both = 1/2.
So in fact the value of this eta function (s = 0) = 1/2.
However in qualitative terms the important point to observe is that this actually expresses the average of both linear and circular type calculations.
So the key thing in this context to understand is that the famed Riemann Hypothesis involves in qualitative terms precisely the same explanation.
In other words the Riemann Hypothesis can only be properly explained in terms of a condition enabling the balanced consistency of both linear (quantitative) and (circular) qualitative understanding.
Incidentally by using the eta function (for s = 0) it is then possible using a simple transformation to calculate a value for the corresponding zeta function = - 1/2.
However this involves deliberately combining finite with infinite notions.
Therefore the resulting value cannot be given a meaning in merely linear terms (suited to finite considerations).
However when we properly allow for the qualitative distinctions as between finite and infinite notions (using linear and circular interpretation respectively) then we can give this (and other negative integer values of the zeta function) an intelligible meaning.
The first is the recognised conventional quantitative approach that is fundamentally based on the linear use of reason (using unambiguous either/or distinctions).
However the second - unrecognised - qualitative approach is based on the circular use of reason (using paradoxical both/and designations).
One way of expressing the difference between both types is that polar opposites (such as positive and negative) are treated as separate in linear terms whereas they are given a complementary meaning (as interdependent) in corresponding circular interpretation.
Fascinatingly however the distinction as between these two types can play a direct role in the interpretation of - what initially appear - as merely quantitative relationships.
My initial recognition of this fact does not stem directly from the Riemann Hypothesis but rather from investigation of the Fibonacci sequence.
The Fibonacci sequence is obtained through starting with 0 and 1 and then successively adding the last term to the previous to obtain a new number in the sequence.
In this way we get 0, 1, 1, 2, 3, 5, 8, 13,.....
Now the ratio of terms in the sequence (again dividing the lst by the previous) approximates to phi = 1.618033... (the golden mean) and steadily improves as the terms in the sequence become larger.
This value in turn for phi is obtained exactly as the expression of the simple quadratic equation
x^2 - x - 1 = 0.
However strictly there are two values for x from this equation. The first is 1.618... and the second is - .618. So the question arises as to the interpretation of this second value.
After consideration of this issue it became clear to me that - correctly understood - both values (which are positive and negative with respect to each other) represent a complementary pairing.
So the correct way of interpreting this is as follows. When we designate the direction of the ratio (i.e. as last term in sequence to previous) as positive, then the corresponding direction (i.e. as previous term to last) is negative.
Therefore whereas 1.618... approximates the ratio of last to previous, - .618 correspondingly approximates the (reverse) ratio of previous to last term in the sequence.
So in this two dimensional interpretation (which befits the 2-dimensional polynomial expression) positive and negative take on a new meaning in the context of a complementary pairing.
I then realised that the use of polynomial equations could be extended to a whole range of sequences (in a similar manner to the Fibonacci).
The natural number sequence {0}, 1, 2, 3, 4,.. then provides one especially interesting example of such an important sequence.
The corresponding polynomial equation is here given as x^2 - 2x + 1 = 0.
The general form of this equation is x^2 + bx + c = 0 and we get corresponding number sequence to which it relates by successively summing - b * (the last term) - c * (the previous term term)
The natural number polynomial then factorises as (x - 1)(x - 1) = 0. So x = 1 is the ratio in both cases.
What this means in effect is that we would treat the natural number sequence in a standard linear manner (as both terms are of the same sign).
Indeed this clearly identifies the natural number sequence as the very symbol (or archetype) if you wish of linear type understanding.
So in this case it does not matter in which direction we take the last two terms of the sequence with the ratio approximating in each case (as the magnitude of the terms increases) to 1.
Having looked at the archetype of a number sequence corresponding to linear type inetrpretation, it then seemed instructive to likwise examine the corresponding archetype of a number sequence corresponding to circular type interpretation.
And the equation for this sequence is given by (x + 1)(x - 1) = 0 containing complementary positive and negative signs.
So the polynomial expression for this sequence is x^2 - 1 = 0.
The corresponding number sequence to which it relates is
0, 1, 0, 1, 0, 1,....
It is the interpretation of the ratio of this sequence that is especially fascinating as it illuminates several of the qualitative features that I have been attempting to illustrate.
1) Whereas interpretation of the natural number sequence entails pure linear understanding, this sequence correspondingly represents pure (2-dimensional) circular understanding.
Note that if we attempt to calculate the ratio with respect to successive terms (based on 1-dimensional linear interpretation) we either get 0/1 = 0 or 1/0 = ∞ (even though the equation suggests that the ratio = 1 or - 1).
However we can eliminate this problem through (correct) 2-dimensional interpretation,
In this case the ratio is either 1/1 or 0/0 (through expressing ratios with respect to number terms separated by a gap of 2).
Now the very essence of 2-dimensional interpretation is that it combines both rational and intuitive elements in equal measure.
So the first ratio 1/1 = 1, corresponds to rational interpretation (of 2-dimensional reality).
The second ratio 0/0 = 1, corresponds to direct intuitive appreciation of the same relationship giving a meaning to a ratio that cannot be properly defined in 1-dimensional terms. So the two elements (rational and intuitive) that are experientially united in 2-dimensional understanding become separated in terms of this linear sequence of numbers.
2) The actual equation gives rise to two answers + 1 and - 1. So as befits 2-dimensional interpretation, the direction in which terms are taken is important. So if with each second term we take the direction of last to previous term as positive then the corresponding direction of previous to last is thereby negative. And this is precisely what we would expect with 2-dimensional interpretation!
In fact what I have illustrated above is a simple illustration of the general equation x^n = 1 (where n = 2).
When n = 3 for example we would derive the sequence
0, 0, 1, 0, 0, 1,... so that we can only define the initial ratio (where x = 1) as the ratio of terms (separated by a gap of 3).
It would be tempting to push this even further in the attempted qualitative appreciation of imaginary numbers.
As we know the slutions to the equation x^2 + 1 = 0 are x = + i and x = - i, (where i is the square root of - 1).
Now the sequence of numbers to which this relates is
0, 1, 0 - 1, 0, 1, 0 - 1,....
Now again as this represents a 2-dimensional equation (with positive and negative polarities) a 2-dimensional qualitative interpretation is required.
In this case again taking ratios of each second term we get a ratio of 1/- 1 or alternatively - 1/1, and also 0/0.
So the clear implication is that the very essence of the qualitative interpretation of the imaginary number is that the notion of positive and negative directions is now rendered entirely ambiguous.
Also the ratio (which in each case is - 1) points to unconscious pure 2-dimensional rather than conscious reality.
So the very notion of an imaginary number qualitatively relates to the attempt to express a purely (2-dimensional) circular notion - that literally entails the dynamic negation of conscious understanding - indirectly in linear (1-dimensional) terms.
And the corresponding quantitative definition directly parallels this interpretation entailing the square root of a negative unit.
The importance of this demonstration is that similar problems of interpretation arise in defining values for the Riemann Zeta Function.
In standard linear terms the value of the sequence
1/(1^s) + 1/(2^s) + 1/(3^s) +... only converges for values of s > 1.
However Riemann extended the domain of definition for all complex values of s (except 1).
This then leads to the interesting problem where the sum of sequences (which clearly diverge in standard linear terms with no attainable sum) are now given a specific finite result.
For example when s = 0, we generate the following series 1 + 1 + 1 + 1 +.....
Now clearly in conventional terms the sum of this this series is infinite (i.e. does not converge to a finite limit).
However by a remarkable piece of alchemy we can in fact give a finite value to the series
This entails defining a corresponding eta function which generates the alternating series
1 - 1 + 1 - 1 + 1 ....
Remarkably this again relates to the simple equation we discussed above (in case of Fibonacci type sequences) where now x^n = 0.
So if x^n = 0, then - x^n = 0
Therefore 1 - x^n = 1
Thus (1 - x)(1 + x + x^2 + x^3 +......x^[n - 1]) = 1
So (1 + x + x^2 + x^3 +......x^[n - 1]) = 1((1 - x)
Letting x = - 1 as n → ∞
Then 1 - 1 + 1 - 1 + .... = 1/2
The question then arises as to the interpretation of this alternating series.
If we view these terms in 2-dimensional (circular) complementary fashion (where successive positive and negative terms form a pair) then the sum = 0.
However if we consider it in linear (1-dimensional) terms (where we break such a pairing) the sum will always relate to the remaining odd term = 1.
It would seem therefore that since two possible answers can arise that we should take the mean of both = 1/2.
So in fact the value of this eta function (s = 0) = 1/2.
However in qualitative terms the important point to observe is that this actually expresses the average of both linear and circular type calculations.
So the key thing in this context to understand is that the famed Riemann Hypothesis involves in qualitative terms precisely the same explanation.
In other words the Riemann Hypothesis can only be properly explained in terms of a condition enabling the balanced consistency of both linear (quantitative) and (circular) qualitative understanding.
Incidentally by using the eta function (for s = 0) it is then possible using a simple transformation to calculate a value for the corresponding zeta function = - 1/2.
However this involves deliberately combining finite with infinite notions.
Therefore the resulting value cannot be given a meaning in merely linear terms (suited to finite considerations).
However when we properly allow for the qualitative distinctions as between finite and infinite notions (using linear and circular interpretation respectively) then we can give this (and other negative integer values of the zeta function) an intelligible meaning.
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