Conventional physical accounts of the origin of the universe centre on the Big Bang where in an incredibly short period all the fundamental constituents of matter were formed.

However I have argued that the notion of ultimate particles has no strict meaning (in a manifest form) and that matter particles themselves represent more fundamental number interactions with respect to their quantitative and qualitative aspects.

So ultimately all notions of quantity and quality are of a strict mathematical nature. And it is in the dynamic interaction of these two aspects that physical reality - as we know it - arises.

Thus if the Big Bang is to have any true meaning it most fundamentally relates to the manner in which number - again with respect to its specific (quantitative) and holistic (qualitative) aspects - itself arises.

Now of particular interest here would be the primes (and their dynamic relationship with the natural numbers).

As I have demonstrated the manner to which both unfold corresponds to a dynamic interactive process based on an initial binary logical system.

So one again in this system the linear (1) is based on unambiguous positing (+).

The circular (0) is then based on both positing and negating.

So 1 - 1 = 0 (in this system).

The initial prime number is 2 (which holistically represents duality!).

Subsequent (composite) natural numbers are uniquely determined though multiplication of prexisting primes. Subsequent primes are uniquely determined from already (composite) natural numbers through subtraction of 1.

So rather that statically existing, prime and natural numbers are formed through a dynamic interactive process designed to preserve the unique identity of each prime and also the collective interdependence of all primes (with the natural numbers).

As it is a phenomenal process taking place in space and time the primes can never be fully reconciled with natural numbers in this relative manner. However the approximation to the initial state (where quantitative and qualitative aspects are identical) would occur ever closer to the moment of the Number Bang. Before quantitative or qualitative aspects of creation had yet emerged by definition the perfect reconciliation of the number system existed in an ineffable manner.

From one valid perspective, evolution could be looked on as the attempt to fully "know" the nature of this initial ineffable number state.

But once again such knowledge is ultimately also of an ineffable nature. Thus in pure spiritual union where all physical phenomena have been made transparent, one can approach that state in knowledge of the true nature of the primes.

For such knowledge of the mystery inherent in the primes is inseparable from the true nature of existence!

## Saturday, December 31, 2011

## Friday, December 30, 2011

### Changing Our Ideas on the Primes

I have continually asserted the fact that the way we fundamentally look at the primes is very misleading.

Arising from the linear (1-dimensional) nature of Conventional Mathematics, the primes are viewed as the basic (independent) building blocs of the natural number system.

However even momentary reflection on the matter would immediately lead to the realisation that we cannot even begin to think of the primes in the absence of the natural numbers.

For example as soon as we try to rank primes we automatically are required to use the natural numbers in an ordinal sense. So 2 is the 1st, 3 the 2nd, 5 the 3rd, 7 the 4th prime and so on.

But the ordinal natural numbers in turn implies the cardinal use of these numbers. So we cannot give 1st, 2nd, 3rd for example a meaning in the absence of the cardinal numbers 1, 2, 3!

Thus the prime and the natural numbers are mutually interdependent.

This requires adopting the radical view that just as the natural numbers (from one perspective) are determined by the primes, that the primes in turn are determined by the natural numbers.

Now the clue to this new realisation comes from recognition that there are in fact two key logical systems. The linear logical system based on unambiguous either/or distinctions (on which Conventional Mathematics depends) can be represented as + 1.

However the alternative circular logical system based on relative both/and can be represented as 1 - 1 = 0.

Clearly before phenomena can unfold, both the linear and circular systems are identical i.e. as undifferentiated form (which equally is emptiness).

Now the conventional notion that natural numbers are derived from the primes is based on the linear logical approach.

However the reverse view that the primes are derived from the natural numbers springs from appreciation of the circular approach.

Now initially the circular is represented as 1 - 1 = 0.

However with the emergence of natural numbers it is represented as n - 1 (where n is a natural number).

The point about this is that all new primes can be represented uniquely by a natural number (that has been derived from earlier primes - 1).

For example 29 (which is a new prime) can be uniquely derived as 30 - 1.

30 in this case is the composite result of multiplying the earlier primes 2, 3 and 5.

Thus 29 can be uniquely represented as (2 * 3 * 5) - 1.

So the number system starts with 1 and 0 with 0 itself represented as 1 - 1.

Now the origin of the next number 2 (leading to the birth of duality) arises from portraying the circular system in linear terms. Thus rather that preserving the dynamic complementarity of + and - signs, both are represented as +.

Thus instead of 1 - 1 = 0, we now have 1 + 1 = 2.

All subsequent natural and prime numbers are ultimately derived from 2 (and 1).

So 3 the next prime number = (2 * 2) - 1 = 3.

Then the next prime number 5 arises from (2 * 3) - 1 = 5.

So remarkably we have two systems here!

The first (linear) system is based on multiplication of primes (to derive the natural numbers).

The second (circular) system is based on subtraction of 1 (from the composite natural number result of multiplying primes) to uniquely define new prime numbers.

So in actual reality (and experience) both systems are involved. The natural numbers and primes are built up in conjunction with each other in a dynamic interactive manner.

In a very valid sense neither the primes nor natural numbers preexist this dynamic interactive process that takes place in space and time.

Thus the prime numbers and the natural can never be fully reconciled with each other in a formal phenomenal procedure. Such reconciliation takes place in an ineffable manner (as the secret code pre-existing phenomenal creation) and equally the ultimate spiritual realisation of the true mystery of the primes (and natural numbers).

Incidentally the true circular nature of the number 0 is revealed by the fact that is invariant with respect to either positive or negative signs. So + 0 = - 0.

What this really means is that 0 properly embraces both positive and negative signs in a circular manner! And the very symbol we use for 0 signifies this circular nature (just as the symbol we use to represent 1 likewise symbolises its linear nature)!

Arising from the linear (1-dimensional) nature of Conventional Mathematics, the primes are viewed as the basic (independent) building blocs of the natural number system.

However even momentary reflection on the matter would immediately lead to the realisation that we cannot even begin to think of the primes in the absence of the natural numbers.

For example as soon as we try to rank primes we automatically are required to use the natural numbers in an ordinal sense. So 2 is the 1st, 3 the 2nd, 5 the 3rd, 7 the 4th prime and so on.

But the ordinal natural numbers in turn implies the cardinal use of these numbers. So we cannot give 1st, 2nd, 3rd for example a meaning in the absence of the cardinal numbers 1, 2, 3!

Thus the prime and the natural numbers are mutually interdependent.

This requires adopting the radical view that just as the natural numbers (from one perspective) are determined by the primes, that the primes in turn are determined by the natural numbers.

Now the clue to this new realisation comes from recognition that there are in fact two key logical systems. The linear logical system based on unambiguous either/or distinctions (on which Conventional Mathematics depends) can be represented as + 1.

However the alternative circular logical system based on relative both/and can be represented as 1 - 1 = 0.

Clearly before phenomena can unfold, both the linear and circular systems are identical i.e. as undifferentiated form (which equally is emptiness).

Now the conventional notion that natural numbers are derived from the primes is based on the linear logical approach.

However the reverse view that the primes are derived from the natural numbers springs from appreciation of the circular approach.

Now initially the circular is represented as 1 - 1 = 0.

However with the emergence of natural numbers it is represented as n - 1 (where n is a natural number).

The point about this is that all new primes can be represented uniquely by a natural number (that has been derived from earlier primes - 1).

For example 29 (which is a new prime) can be uniquely derived as 30 - 1.

30 in this case is the composite result of multiplying the earlier primes 2, 3 and 5.

Thus 29 can be uniquely represented as (2 * 3 * 5) - 1.

So the number system starts with 1 and 0 with 0 itself represented as 1 - 1.

Now the origin of the next number 2 (leading to the birth of duality) arises from portraying the circular system in linear terms. Thus rather that preserving the dynamic complementarity of + and - signs, both are represented as +.

Thus instead of 1 - 1 = 0, we now have 1 + 1 = 2.

All subsequent natural and prime numbers are ultimately derived from 2 (and 1).

So 3 the next prime number = (2 * 2) - 1 = 3.

Then the next prime number 5 arises from (2 * 3) - 1 = 5.

So remarkably we have two systems here!

The first (linear) system is based on multiplication of primes (to derive the natural numbers).

The second (circular) system is based on subtraction of 1 (from the composite natural number result of multiplying primes) to uniquely define new prime numbers.

So in actual reality (and experience) both systems are involved. The natural numbers and primes are built up in conjunction with each other in a dynamic interactive manner.

In a very valid sense neither the primes nor natural numbers preexist this dynamic interactive process that takes place in space and time.

Thus the prime numbers and the natural can never be fully reconciled with each other in a formal phenomenal procedure. Such reconciliation takes place in an ineffable manner (as the secret code pre-existing phenomenal creation) and equally the ultimate spiritual realisation of the true mystery of the primes (and natural numbers).

Incidentally the true circular nature of the number 0 is revealed by the fact that is invariant with respect to either positive or negative signs. So + 0 = - 0.

What this really means is that 0 properly embraces both positive and negative signs in a circular manner! And the very symbol we use for 0 signifies this circular nature (just as the symbol we use to represent 1 likewise symbolises its linear nature)!

## Tuesday, December 27, 2011

### Vibrating Primes

The primes seemingly give rise to a dynamic vibrating system that is of a physical nature.

Once again there are quantitative (actual) and qualitative (holistic) aspects to the primes, which ultimately are identical in an ineffable formless state (that precedes physical existence).

However as soon as the aspects separate the primes take on a physical existence and become manifest in phenomena with both quantitative and qualitative aspects (that are to a degree separated in space and time).

Physicists now speak of strings as being the ultimate particles. So in this worldview the strings vibrate giving rise to the physical particles of nature.

Now the weakness of this view is that strings are given a merely actual existence. This leads to the mistaken - and highly reductionist - view that the fundamental constituents of matter i.e. strings) are of a uniform nature devoid of any distinctive qualitative features!

Some time ago I had already formed the view that strings must equally possess unique holistic dimensional properties (conforming to a distinctive circular logical system) and that through the interaction of both actual and holistic aspects that physical nature as we know it arises.

However I have now come to realise that ultimately what are termed "strings" relate directly to the prime numbers (with respect to both their quantitative and qualitative characteristics).

Thus what we call physical nature arises directly from the dynamic interaction of the actual and holistic nature of prime numbers (that are utterly unique yet collectively interdependent with each other).

Strictly this vibration of prime numbers is already inherent in fundamental natural particles. In one way this vibration can be seen as an attempt to get back to that ineffable state (where form does not yet exist). From an equally valid perspective it can be seen as the attempt to set the evolution of nature in process where the true understanding of its underlying prime nature can be ultimately attained once again in an ineffable formless manner.

So as we come ever closer to the original state of matter, we equally come ever closer to approaching the secret code contained in the prime numbers which determines the whole subsequent course of evolution.

There are of course no ultimate particles in nature (that are detectable). Rather beyond what is detectable are extremely dynamic physical interactions of an increasingly transient and elusive nature. And these interactions approximate ever more closely to the number code (that perfectly reconciles quantitative and qualitative aspects of the primes) which ultimately is of an ineffable nature.

Once again there are quantitative (actual) and qualitative (holistic) aspects to the primes, which ultimately are identical in an ineffable formless state (that precedes physical existence).

However as soon as the aspects separate the primes take on a physical existence and become manifest in phenomena with both quantitative and qualitative aspects (that are to a degree separated in space and time).

Physicists now speak of strings as being the ultimate particles. So in this worldview the strings vibrate giving rise to the physical particles of nature.

Now the weakness of this view is that strings are given a merely actual existence. This leads to the mistaken - and highly reductionist - view that the fundamental constituents of matter i.e. strings) are of a uniform nature devoid of any distinctive qualitative features!

Some time ago I had already formed the view that strings must equally possess unique holistic dimensional properties (conforming to a distinctive circular logical system) and that through the interaction of both actual and holistic aspects that physical nature as we know it arises.

However I have now come to realise that ultimately what are termed "strings" relate directly to the prime numbers (with respect to both their quantitative and qualitative characteristics).

Thus what we call physical nature arises directly from the dynamic interaction of the actual and holistic nature of prime numbers (that are utterly unique yet collectively interdependent with each other).

Strictly this vibration of prime numbers is already inherent in fundamental natural particles. In one way this vibration can be seen as an attempt to get back to that ineffable state (where form does not yet exist). From an equally valid perspective it can be seen as the attempt to set the evolution of nature in process where the true understanding of its underlying prime nature can be ultimately attained once again in an ineffable formless manner.

So as we come ever closer to the original state of matter, we equally come ever closer to approaching the secret code contained in the prime numbers which determines the whole subsequent course of evolution.

There are of course no ultimate particles in nature (that are detectable). Rather beyond what is detectable are extremely dynamic physical interactions of an increasingly transient and elusive nature. And these interactions approximate ever more closely to the number code (that perfectly reconciles quantitative and qualitative aspects of the primes) which ultimately is of an ineffable nature.

## Wednesday, December 21, 2011

### Sum of Reciprocals of Primes

As is well known the sum of the terms in the harmonic series

1 + 1/2 + 1/3 + 1/4 +.... ~ ln n + γ (where γ = the Euler-Mascheroni constant =.5772..)

It is fascinating therefore that the sum of reciprocals of primes

1/2 + 1/3 + 1/5 + 1/7 + ..... ~ ln ln n + B (where B = Merten's constant = .261497..)

This would of course suggest that the sum of this series diverges for large n!

However just as the harmonic series can be used to calculate the spread as between cardinal prime numbers, likewise this latter series can be used to calculate the spread as between ordinal prime numbers.

In other words all prime numbers can be linked with the ordinal set of natural numbers.

So 2 is the 1st, 3 the 2nd, 5 the 3rd, 7 the 4th prime respectively.

So if we now order these primes in an ordinal prime fashion, then both 3 and 5 are prime (i.e. as the 2nd and 3rd primes).

We could then reorder these surviving primes in an natural number ordinal fashion before selecting once again those surviving numbers that are prime in an ordinal fashion.

Let us take all the primes up to 31 to illustrate,

(1) 2, (2) 3, (3) 5, (4) 7, (5) 11, (6) 13, (7) 17, (8) 19, (9) 23, (10) 29, (11) 31.

These these starting cardinal primes are listed in natural number ordinal fashion (in brackets). We will refer to these as Order 1 Primes. So all prime numbers are Order 1 primes.

Then if we extract the primes (whose ordinal numbers are also prime), we are left with 3, 5 11, 17 and 31.

If we now again rank these ordinally in natural number fashion we have

(1) 3, (2) 5, (3) 11, (4) 17 and (5) 31.

We can refer to this smaller group as Order 2 Primes.

Once again we can then extract only those primes that have a prime number ordinal ranking i.e. 5, 11 and 31.

Then shifting to ordinal natural number ranking we have a new - even smaller - set of surviving primes,

(1) 5, (2) 11 and (3) 31.

We can refer to these then as Order 3 Primes.

Then once more extracting those remaining with an ordinal prime ranking we are left with 11 and 31

So giving natural number ordinal rankings we have,

(1) 11 and (2) 31

These are Order 4 Primes.

Finally extracting the one remaining prime with an ordinal prime ranking we are left with 31.

Finally ranking this as (1) 31, this qualifies as an Order 5 Prime.

It is no accident that 31 corresponds to the Mersenne prime 2^n - 1 (where n = 5).

Indeed if we were to continue up to 127 for example, 127 would then qualify as the one remaining Order 7 Prime. And 127 is the Mersenne prime (where n = 7).

I have suggested at various times that perhaps we could guarantee the generation of Mersenne primes by starting with 2 and then proceeding through switching in an orderly fashion as between quantitative (base) and qualitative (dimensional) use of prime numbers.

So 2^2 - 1 = 3 (as base number).

Then substituting 3 as dimensional number we have,

2^3 - 1 = 7 (which is prime).

Once again substituting 7 as dimension we have,

2^7 - 1 = 127 (which is prime).

Then substituting 127 as dimension we have,

2^127 - 1 = 170141183460469231731687303715884105727 (which is prime).

It is tempting to argue that by using this number as exponent of 2 that we can generate a new Mersenne prime that is incomparably larger than any yet discovered!

Now going back to the harmonic series and sum of the reciprocals of primes.

Once again the sum of the harmonic series for large n approximates to ln n.

The sum of the reciprocals of primes for large n approximates to ln ln n.

Now the prime numbers used as denominators in this series are Order 1 Primes.

It is possible therefore to extend this result for Order 2, Order 3, Order 4 primes etc.

For example ultimately the sum of reciprocals of Order 2 Primes should approximate (for sufficiently large n) to Ln Ln Ln n, Order 3 to Ln Ln Ln Ln n, Order 4 to Ln Ln Ln Ln Ln n etc.

This would suggest that no matter how high the Order of Primes involved that the sum of the series of its reciprocal terms would diverge (for sufficiently large n).

Also by this reckoning the harmonic series could be interpreted as the sum of Reciprocals of Order 0 Primes.

So the natural numbers are prime numbers of Order 0!

What simply this means in effect is that the ordinal ranking of the complete set of primes (i.e. Order 1 Primes) is given by the natural numbers!

As for the spread as between primes, once again for Order 1 Primes the answer approximates to log n.

Clearly the spread will grow for higher Order Primes.

So we can postulate that the number of Order 1 Primes up to 1,000,000 = n/ln n = n1 = 72,283 (approx).

Therefore the number of Order 2 Primes up to 1,000,000 approximates n1/ln n1 = 6469 (approx).

Thus the average gap as between Order 2 Primes approximates n/{n1/ln n1} = 1,000,000/6469.

Therefore in the region of 1,000,000 we would expect the average gap as between Order 2 Primes to approximate 154.6.

Now because n is still of a relatively small magnitude, the actual number of Order 2 Primes would differ significantly from this estimate. However the approximation of estimated to actual would continue to improve (in relative terms) as n increases.

The upshot of what we are doing here is that the prime and natural numbers are in fact completely interdependent with each other.

From one perspective the (individual) natural numbers are derived from the primes; however equally from the complementary perspective, the (general) distribution of the primes is derived from the natural numbers.

1 + 1/2 + 1/3 + 1/4 +.... ~ ln n + γ (where γ = the Euler-Mascheroni constant =.5772..)

It is fascinating therefore that the sum of reciprocals of primes

1/2 + 1/3 + 1/5 + 1/7 + ..... ~ ln ln n + B (where B = Merten's constant = .261497..)

This would of course suggest that the sum of this series diverges for large n!

However just as the harmonic series can be used to calculate the spread as between cardinal prime numbers, likewise this latter series can be used to calculate the spread as between ordinal prime numbers.

In other words all prime numbers can be linked with the ordinal set of natural numbers.

So 2 is the 1st, 3 the 2nd, 5 the 3rd, 7 the 4th prime respectively.

So if we now order these primes in an ordinal prime fashion, then both 3 and 5 are prime (i.e. as the 2nd and 3rd primes).

We could then reorder these surviving primes in an natural number ordinal fashion before selecting once again those surviving numbers that are prime in an ordinal fashion.

Let us take all the primes up to 31 to illustrate,

(1) 2, (2) 3, (3) 5, (4) 7, (5) 11, (6) 13, (7) 17, (8) 19, (9) 23, (10) 29, (11) 31.

These these starting cardinal primes are listed in natural number ordinal fashion (in brackets). We will refer to these as Order 1 Primes. So all prime numbers are Order 1 primes.

Then if we extract the primes (whose ordinal numbers are also prime), we are left with 3, 5 11, 17 and 31.

If we now again rank these ordinally in natural number fashion we have

(1) 3, (2) 5, (3) 11, (4) 17 and (5) 31.

We can refer to this smaller group as Order 2 Primes.

Once again we can then extract only those primes that have a prime number ordinal ranking i.e. 5, 11 and 31.

Then shifting to ordinal natural number ranking we have a new - even smaller - set of surviving primes,

(1) 5, (2) 11 and (3) 31.

We can refer to these then as Order 3 Primes.

Then once more extracting those remaining with an ordinal prime ranking we are left with 11 and 31

So giving natural number ordinal rankings we have,

(1) 11 and (2) 31

These are Order 4 Primes.

Finally extracting the one remaining prime with an ordinal prime ranking we are left with 31.

Finally ranking this as (1) 31, this qualifies as an Order 5 Prime.

It is no accident that 31 corresponds to the Mersenne prime 2^n - 1 (where n = 5).

Indeed if we were to continue up to 127 for example, 127 would then qualify as the one remaining Order 7 Prime. And 127 is the Mersenne prime (where n = 7).

I have suggested at various times that perhaps we could guarantee the generation of Mersenne primes by starting with 2 and then proceeding through switching in an orderly fashion as between quantitative (base) and qualitative (dimensional) use of prime numbers.

So 2^2 - 1 = 3 (as base number).

Then substituting 3 as dimensional number we have,

2^3 - 1 = 7 (which is prime).

Once again substituting 7 as dimension we have,

2^7 - 1 = 127 (which is prime).

Then substituting 127 as dimension we have,

2^127 - 1 = 170141183460469231731687303715884105727 (which is prime).

It is tempting to argue that by using this number as exponent of 2 that we can generate a new Mersenne prime that is incomparably larger than any yet discovered!

Now going back to the harmonic series and sum of the reciprocals of primes.

Once again the sum of the harmonic series for large n approximates to ln n.

The sum of the reciprocals of primes for large n approximates to ln ln n.

Now the prime numbers used as denominators in this series are Order 1 Primes.

It is possible therefore to extend this result for Order 2, Order 3, Order 4 primes etc.

For example ultimately the sum of reciprocals of Order 2 Primes should approximate (for sufficiently large n) to Ln Ln Ln n, Order 3 to Ln Ln Ln Ln n, Order 4 to Ln Ln Ln Ln Ln n etc.

This would suggest that no matter how high the Order of Primes involved that the sum of the series of its reciprocal terms would diverge (for sufficiently large n).

Also by this reckoning the harmonic series could be interpreted as the sum of Reciprocals of Order 0 Primes.

So the natural numbers are prime numbers of Order 0!

What simply this means in effect is that the ordinal ranking of the complete set of primes (i.e. Order 1 Primes) is given by the natural numbers!

As for the spread as between primes, once again for Order 1 Primes the answer approximates to log n.

Clearly the spread will grow for higher Order Primes.

So we can postulate that the number of Order 1 Primes up to 1,000,000 = n/ln n = n1 = 72,283 (approx).

Therefore the number of Order 2 Primes up to 1,000,000 approximates n1/ln n1 = 6469 (approx).

Thus the average gap as between Order 2 Primes approximates n/{n1/ln n1} = 1,000,000/6469.

Therefore in the region of 1,000,000 we would expect the average gap as between Order 2 Primes to approximate 154.6.

Now because n is still of a relatively small magnitude, the actual number of Order 2 Primes would differ significantly from this estimate. However the approximation of estimated to actual would continue to improve (in relative terms) as n increases.

The upshot of what we are doing here is that the prime and natural numbers are in fact completely interdependent with each other.

From one perspective the (individual) natural numbers are derived from the primes; however equally from the complementary perspective, the (general) distribution of the primes is derived from the natural numbers.

### Spiral Waves

I have referred before to Matthew Watkin's book "The Mystery of the Prime Numbers".

One of the features that I especially like about this book is that he successfully converts the standard natural log notion (which has such an important bearing on the general distribution of the primes) into the much more expressive form of an equiangular spiral.

Now in geometric terms equiangular spirals are very suggestive as they combine both linear and circular notions in a systematic ordered manner.

Indeed at the two extremes of such spirals we get - what he refers to as degenerative spirals - of both the straight line and the circle.

Matthew also extends this notion of spirals into the treatment of the famous deviations in the Riemann Prime Counting Function which are intimately related in turn to the non-trivial zeros of the Zeta Function!

I look forward very much to an extended treatment of these "spiral zeros" in the second volume.

It is also worth noting that considerable attention has been given to the Prime Spiral (Ulam's Spiral) where when natural numbers are entered on a grid in spiral fashion that the prime numbers then tend to fall along diagonal lines through the spiral! However my intention here is to focus on the qualitative significance of the relationship of prime numbers to spiral wave forms!

Just as linear and circular notions have a well defined quantitative meaning in conventional (Type 1) mathematical terms, equally they have a well defined qualitative meaning in holistic (Type 2) mathematical terms. However this latter type of interpretation is totally ignored by the mathematics profession.

Thus I would strongly contend that it is the absence of this vital qualitative dimension that is preventing recognition of the true nature of prime numbers.

In other words prime numbers combine both specific (independent) aspects in their individual nature with holistic (interdependent) aspects in their overall distribution.

Actual experience of mathematical reality equally combines specific and holistic elements through the interaction of (conscious) reason and (unconscious) intuition. However once again in conventional terms the qualitative (intuitive) aspect is reduced in a quantitative (rational) manner.

Proper understanding of the Riemann Hypothesis thereby requires both linear (either/or) logic based on the clear separation of polar opposites such as external/ internal and circular (both/and) logic based on the corresponding complementarity of such opposites.

Indeed ultimately the Riemann Hypothesis relates to the vital condition necessary for the consistent relationship of both types of logic!

One of the features that I especially like about this book is that he successfully converts the standard natural log notion (which has such an important bearing on the general distribution of the primes) into the much more expressive form of an equiangular spiral.

Now in geometric terms equiangular spirals are very suggestive as they combine both linear and circular notions in a systematic ordered manner.

Indeed at the two extremes of such spirals we get - what he refers to as degenerative spirals - of both the straight line and the circle.

Matthew also extends this notion of spirals into the treatment of the famous deviations in the Riemann Prime Counting Function which are intimately related in turn to the non-trivial zeros of the Zeta Function!

I look forward very much to an extended treatment of these "spiral zeros" in the second volume.

It is also worth noting that considerable attention has been given to the Prime Spiral (Ulam's Spiral) where when natural numbers are entered on a grid in spiral fashion that the prime numbers then tend to fall along diagonal lines through the spiral! However my intention here is to focus on the qualitative significance of the relationship of prime numbers to spiral wave forms!

Just as linear and circular notions have a well defined quantitative meaning in conventional (Type 1) mathematical terms, equally they have a well defined qualitative meaning in holistic (Type 2) mathematical terms. However this latter type of interpretation is totally ignored by the mathematics profession.

Thus I would strongly contend that it is the absence of this vital qualitative dimension that is preventing recognition of the true nature of prime numbers.

In other words prime numbers combine both specific (independent) aspects in their individual nature with holistic (interdependent) aspects in their overall distribution.

Actual experience of mathematical reality equally combines specific and holistic elements through the interaction of (conscious) reason and (unconscious) intuition. However once again in conventional terms the qualitative (intuitive) aspect is reduced in a quantitative (rational) manner.

Proper understanding of the Riemann Hypothesis thereby requires both linear (either/or) logic based on the clear separation of polar opposites such as external/ internal and circular (both/and) logic based on the corresponding complementarity of such opposites.

Indeed ultimately the Riemann Hypothesis relates to the vital condition necessary for the consistent relationship of both types of logic!

## Sunday, December 18, 2011

### Riemann's Zero

As is well known the Riemann Hypothesis amounts to the statement all the non-trivial zeros of the Riemann Zeta Function lie on the real line whose value = 1/2.

Now in conventional terms, mathematicians have been trying to understand this problem from a merely quantitative perspective.

My persistent point however is that Mathematics equally contains an important (largely unrecognised) qualitative aspect.

Furthermore as the Riemann Hypothesis properly relates to the ultimate reconciliation of both the quantitative and qualitative aspects of mathematical understanding, its significance cannot be appreciated in a merely (reduced) quantitative manner.

Indeed ultimately the Riemann Hypothesis points to the relationship as between dual and nondual meaning.

Imagine a circle that is drawn with unit radius. Now in conventional terms the length of the line diameter of this circle = 2 units. So the midpoint of this line at the centre of the circle from linear perspective divides the line into two equal parts. This point lies exactly halfway (1/2) on the total line.

However if we look at this midpoint from a circular perspective we would represent it as 0. In other words if the radius to the right is + 1, then the radius in the opposite direction is - 1.

Now this behaviour has a direct qualitative significance. Linear understanding is inherently dualistic (which represents the holistic meaning of 2). So the midpoint of the circle at 1 unit represents thereby half of the total line.

However to understand the measurement of the line from the centre of the circle in complementary positive and negative directions we require - literally - nondual insight (so that dual notions are rendered paradoxical). So the midpoint from this perspective is 0.

Properly understanding the Riemann Hypothesis requires both dual and nondual understanding. Indeed it points to the ultimate state where both dual and nondual notions (or alternatively quantitative and qualitative notions) of number are identical. And this state is utterly mysterious and thereby cannot be grasped in a phenomenal manner.

Thus the mysterious order governing the nature of the prime numbers is already inherent in all number behaviour representing a non-phenomenal reality (where qualitative cannot be distinguished from quantitative notions).

The attempt to prove the Riemann Hypothesis in a merely (reduced) quantitative manner is thereby utterly futile.

The significance of the Riemann Hypothesis is ultimately of a truly breath taking order in that - properly understood - phenomenal reality as we know it would not be even possible if it did not hold.

In other words underlying all of visible reality is reality is a secret code ensuring a perfect harmony of the prime numbers with respect to their specific (quantitative) and holistic (qualitative) interaction.

It is this harmony that enables all subsequent phenomenal events (representing varying interactions of this original prime number number code) to unfold.

This once again suggests that ultimately the underlying nature of reality - insofar as it can be phenomenally investigated - is purely mathematical!

In other words at the deepest level, phenomena represent the interaction of a fundamental mathematical code that governs the subsequent behaviour of all natural events. However though we can only come to knowledge of this code through phenomena, its nature - and indeed in reverse fashion what we know as nature - is ultimately ineffable.

So at some stage, Physics will have to abandon the quest for the ultimate particles before arriving at a purely mathematical appreciation of reality that underlies all manifestations of such particles.

Indeed properly understood it has already arrived at this point. As I would see it, string theory represents an elaborate fiction that the ultimate physical particles are strings. Strictly speaking however these have no manifest physical reality but really operate as the vehicle of ever purer mathematical notions.

However my key point all along is that the very scope of Mathematics needs to be radically extended to explicitly include both its qualitative as well as quantitative aspect. In this light all particles ultimately emerge as the dynamic interaction of a prime number mathematical code that is designed to preserve the perfect harmony of both its quantitative and qualitative aspects.

Though this point is largely lost because of the reductionist nature of current scientific understanding, the truly great wonder of reality is how both the finite and infinite - though utterly distinct - yet successfully coexist with each other at all levels of understanding.

Thus before we can even for example engage in conventional mathematical activity, we must already presume this meaningful correspondence of finite and infinite. Indeed as I have frequently pointed out it underlines the very notion of mathematical proof!

Looked at another way, the Riemann Hypothesis is the necessary condition for such correspondence to exist. So it represents in fact a massive act of faith in the subsequent meaning of the whole mathematical enterprise.

As the Riemann Hypothesis thereby already underlies conventional mathematical proof (as the starting condition for its meaningful interpretation), The Riemann Hypothesis cannot therefore be proven (in conventional terms).

However far from this representing a defeat, true realisation of this fact has the capacity to open up appreciation of a greatly enlarged scope for Mathematics where every number, symbol, relationship has a unique qualitative - as well as quantitative - significance. And with this will come an enormous enrichment of the true nature of both Mathematics and Science.

Now in conventional terms, mathematicians have been trying to understand this problem from a merely quantitative perspective.

My persistent point however is that Mathematics equally contains an important (largely unrecognised) qualitative aspect.

Furthermore as the Riemann Hypothesis properly relates to the ultimate reconciliation of both the quantitative and qualitative aspects of mathematical understanding, its significance cannot be appreciated in a merely (reduced) quantitative manner.

Indeed ultimately the Riemann Hypothesis points to the relationship as between dual and nondual meaning.

Imagine a circle that is drawn with unit radius. Now in conventional terms the length of the line diameter of this circle = 2 units. So the midpoint of this line at the centre of the circle from linear perspective divides the line into two equal parts. This point lies exactly halfway (1/2) on the total line.

However if we look at this midpoint from a circular perspective we would represent it as 0. In other words if the radius to the right is + 1, then the radius in the opposite direction is - 1.

Now this behaviour has a direct qualitative significance. Linear understanding is inherently dualistic (which represents the holistic meaning of 2). So the midpoint of the circle at 1 unit represents thereby half of the total line.

However to understand the measurement of the line from the centre of the circle in complementary positive and negative directions we require - literally - nondual insight (so that dual notions are rendered paradoxical). So the midpoint from this perspective is 0.

Properly understanding the Riemann Hypothesis requires both dual and nondual understanding. Indeed it points to the ultimate state where both dual and nondual notions (or alternatively quantitative and qualitative notions) of number are identical. And this state is utterly mysterious and thereby cannot be grasped in a phenomenal manner.

Thus the mysterious order governing the nature of the prime numbers is already inherent in all number behaviour representing a non-phenomenal reality (where qualitative cannot be distinguished from quantitative notions).

The attempt to prove the Riemann Hypothesis in a merely (reduced) quantitative manner is thereby utterly futile.

The significance of the Riemann Hypothesis is ultimately of a truly breath taking order in that - properly understood - phenomenal reality as we know it would not be even possible if it did not hold.

In other words underlying all of visible reality is reality is a secret code ensuring a perfect harmony of the prime numbers with respect to their specific (quantitative) and holistic (qualitative) interaction.

It is this harmony that enables all subsequent phenomenal events (representing varying interactions of this original prime number number code) to unfold.

This once again suggests that ultimately the underlying nature of reality - insofar as it can be phenomenally investigated - is purely mathematical!

In other words at the deepest level, phenomena represent the interaction of a fundamental mathematical code that governs the subsequent behaviour of all natural events. However though we can only come to knowledge of this code through phenomena, its nature - and indeed in reverse fashion what we know as nature - is ultimately ineffable.

So at some stage, Physics will have to abandon the quest for the ultimate particles before arriving at a purely mathematical appreciation of reality that underlies all manifestations of such particles.

Indeed properly understood it has already arrived at this point. As I would see it, string theory represents an elaborate fiction that the ultimate physical particles are strings. Strictly speaking however these have no manifest physical reality but really operate as the vehicle of ever purer mathematical notions.

However my key point all along is that the very scope of Mathematics needs to be radically extended to explicitly include both its qualitative as well as quantitative aspect. In this light all particles ultimately emerge as the dynamic interaction of a prime number mathematical code that is designed to preserve the perfect harmony of both its quantitative and qualitative aspects.

Though this point is largely lost because of the reductionist nature of current scientific understanding, the truly great wonder of reality is how both the finite and infinite - though utterly distinct - yet successfully coexist with each other at all levels of understanding.

Thus before we can even for example engage in conventional mathematical activity, we must already presume this meaningful correspondence of finite and infinite. Indeed as I have frequently pointed out it underlines the very notion of mathematical proof!

Looked at another way, the Riemann Hypothesis is the necessary condition for such correspondence to exist. So it represents in fact a massive act of faith in the subsequent meaning of the whole mathematical enterprise.

As the Riemann Hypothesis thereby already underlies conventional mathematical proof (as the starting condition for its meaningful interpretation), The Riemann Hypothesis cannot therefore be proven (in conventional terms).

However far from this representing a defeat, true realisation of this fact has the capacity to open up appreciation of a greatly enlarged scope for Mathematics where every number, symbol, relationship has a unique qualitative - as well as quantitative - significance. And with this will come an enormous enrichment of the true nature of both Mathematics and Science.

## Thursday, December 15, 2011

### The Harmonic Series Again!

As I have repeatedly stated Conventional (Type 1) Mathematics is formally based on a linear i.e. 1-dimensional rational approach (in qualitative terms).

However the considerable problem that exists is that actual understanding of all mathematical processes entails an interaction of both rational and intuitive type understanding.

Thus the conventional approach simply reduces the intuitive aspect in rational terms. Alternatively it inevitably reduces - in any context - qualitative to quantitative type interpretation.

In qualitative terms, rational understanding always implies the positing of phenomena in a conscious manner.

Intuitive understanding - by contrast - implies the corresponding dynamic negation of such understanding in an unconscious holistic manner.

Thus a full account of the nature of linear understanding (that entails both rational and intuitive type understanding) requires recognition of the 1st dimension in both a positive and negative manner.

I have pointed already to the important fact that the very notion of a fraction implicitly requires transition from the positive (rational) to negative (intuitive) recognition with respect to the 1st dimension.

Thus in quantitative terms 4 is more fully represented as 4^1.

In qualitative terms this represents rational (conscious) understanding with respect to 1st dimension (positive).

However the related fraction 1/4 is initially represented as 4^(- 1)and in qualitative terms this relates to intuitive (unconscious) understanding through negation of the 1st dimension (negative).

So the very process of switching from whole to part recognition in experience implicitly requires switching from rational to intuitive type recognition.

However because Conventional Mathematics is formally defined in qualitative terms with respect to the positive 1st dimension, the significance of this qualitative change in understanding is overlooked with the result interpreted in a merely (reduced) quantitative manner i.e. 1/4 as (1/4)^1. So though a qualitative holistic shift of consciousness is necessary to enable the transformation from whole to part recognition, both whole and part are then interpreted in a merely quantitative manner (i.e. with respect to the positive 1st dimension).

Many indications of what I am saying here are provided by simple mathematical results.

Once again from where conventional rational understanding is of a linear nature, holistic intuitive recognition is qualitatively of a circular nature.

Prime numbers from a rational perspective are the most linear of all numbers and - literally 1-dimensional in nature (with no composite factors).

So 7^1 is truly linear in this regard with no factors.

4 by contrast (as composite) is inherently 2-dimensional in nature i.e. 2^2.

However when we raise a prime number to - 1, a remarkable transformation takes place whereby it now exhibits highly circular characteristics.

So 7^(- 1) = 1/7 = .142857..

Here the 6 digits 142857 have some notable circular properties. For example they recur indefinitely in the same manner. Also when we multiply by 2, 3, 4, 5 and 6 the same digits appear (that maintain the same cyclical order).

So 1/7 is perhaps the best known of the cyclic primes (that exhibit such circular properties).

Then the sum of the natural numbers 1 + 2 + 3 + 4 + 5 +..... represents the archetypal linear series of numbers i.e. as numbers naturally marked off on a straight line.

So all these numbers are implicitly 1-dimensional i.e. raised to the power of 1.

However when we raise these numbers same numbers to - 1

i.e. 1^ (- 1), 2^(- 1), 3^(- 1), 4^(- 1), 5^(- 1) +.... we generate the harmonic series 1 + 1/2 + 1/3 + 1/4 + 1/5 + .....

Now remember my basic starting about regarding the nature of the primes is that they combine extreme characteristics with respect to linear and circular aspects!

So the individual prime numbers are the most independent and linear of all numbers (with no constituent factors).

However the general behaviour of the primes (in the frequency of their overall distribution) involves the other extreme of a holistic circular tendency.

Now the very manner in which both linear and circular aspect are involved in experience goes back to the way in which whole and part interact. So to switch from the whole say 4 (4^1) to part 1/4 the dimensional number switches to - 1. So 1/4 = 4^(- 1).

Thus the decisive switch from whole to part requires that conscious (linear) understanding that is defined with respect to the positive 1st dimension be dynamically negated in an unconscious (circular) intuitive manner.

So similar dynamics are involved with respect to the harmonic series (by comparison with the natural number series).

And remarkably the harmonic series gives the simplest answer to the general nature of prime number distribution. In other words the measure of the average spread as between successive prime numbers (which becomes progressively larger as the natural numbers increase) is given by the sum of the harmonic series!

Indeed the change in the average gap between these primes as n i.e n^(+ 1) increases by 1 is given as 1/n i.e. n^(- 1).

So here in the general behaviour of the prime numbers we have an intimate relationship as between the positing and negating respectively (with respect to the 1st dimension) of the number n.

So moving from the specific independent linear notion of an individual prime number to the holistic interdependent circular notion of prime number distribution directly involves the positing (in conscious rational manner) and the corresponding negation (in an unconscious intuitive manner) of linear (1-dimensional) understanding.

So the proper understanding in conventional (Type 1) mathematical terms of the quantitative nature of prime number behaviour is inseparable from corresponding holistic (Type 2) qualitative mathematical interpretation (of such behaviour).

And such understanding requires that we can combine both linear and circular type notions equally in both a quantitative and qualitative manner.

This circular nature of the harmonic series can be demonstrated in yet another striking manner.

Now the harmonic series represents one important example of the Zeta function

i.e. 1/(1^s) + 1/(2^s) + 1/(3^s) + 1/(4^s) + .... where s = 1.

As is well known the for all (positive) even integer values of s the resulting sum of the series can be given in terms of a expression involving pi. And as pi serves as the direct relation of circular and linear quantitative notions in the relationship of the (circular) circumference to its (line) diameter, likewise pi serves as the archetypal relation of corresponding circular and linear notions (understood in a qualitative manner).

For example when s = 2, the sum of the series = {(pi)^2}/6.

The harmonic series can be expressed in terms of the combination of the sum of values corresponding with even integer values for the Zeta Function.

So 1 + 1/2 + 1/3 + 1/4 + ........

= 2{ζ(2)/2 + ζ(4)/4 + ζ(6)/6 + ζ(8)/8 + ......}

And this infinite series can therefore be expressed as the continual sum of terms that consist of powers of pi (that are multiplied by a rational fraction)!

It is again notable that the harmonic series is intimately associated with musical sound. So just as the natural numbers can again be seen as the supreme expression of linear quantitative understanding, the harmonic series can be seen in a sense as a supreme expression of qualitative type appreciation.

And ultimately this is related to the fact that in switching from the positive 1st dimension to its negative one likewise switches from (conscious) rational to (unconscious) intuition, which is the very means by which we switch from quantitative to qualitative type appreciation.

However the considerable problem that exists is that actual understanding of all mathematical processes entails an interaction of both rational and intuitive type understanding.

Thus the conventional approach simply reduces the intuitive aspect in rational terms. Alternatively it inevitably reduces - in any context - qualitative to quantitative type interpretation.

In qualitative terms, rational understanding always implies the positing of phenomena in a conscious manner.

Intuitive understanding - by contrast - implies the corresponding dynamic negation of such understanding in an unconscious holistic manner.

Thus a full account of the nature of linear understanding (that entails both rational and intuitive type understanding) requires recognition of the 1st dimension in both a positive and negative manner.

I have pointed already to the important fact that the very notion of a fraction implicitly requires transition from the positive (rational) to negative (intuitive) recognition with respect to the 1st dimension.

Thus in quantitative terms 4 is more fully represented as 4^1.

In qualitative terms this represents rational (conscious) understanding with respect to 1st dimension (positive).

However the related fraction 1/4 is initially represented as 4^(- 1)and in qualitative terms this relates to intuitive (unconscious) understanding through negation of the 1st dimension (negative).

So the very process of switching from whole to part recognition in experience implicitly requires switching from rational to intuitive type recognition.

However because Conventional Mathematics is formally defined in qualitative terms with respect to the positive 1st dimension, the significance of this qualitative change in understanding is overlooked with the result interpreted in a merely (reduced) quantitative manner i.e. 1/4 as (1/4)^1. So though a qualitative holistic shift of consciousness is necessary to enable the transformation from whole to part recognition, both whole and part are then interpreted in a merely quantitative manner (i.e. with respect to the positive 1st dimension).

Many indications of what I am saying here are provided by simple mathematical results.

Once again from where conventional rational understanding is of a linear nature, holistic intuitive recognition is qualitatively of a circular nature.

Prime numbers from a rational perspective are the most linear of all numbers and - literally 1-dimensional in nature (with no composite factors).

So 7^1 is truly linear in this regard with no factors.

4 by contrast (as composite) is inherently 2-dimensional in nature i.e. 2^2.

However when we raise a prime number to - 1, a remarkable transformation takes place whereby it now exhibits highly circular characteristics.

So 7^(- 1) = 1/7 = .142857..

Here the 6 digits 142857 have some notable circular properties. For example they recur indefinitely in the same manner. Also when we multiply by 2, 3, 4, 5 and 6 the same digits appear (that maintain the same cyclical order).

So 1/7 is perhaps the best known of the cyclic primes (that exhibit such circular properties).

Then the sum of the natural numbers 1 + 2 + 3 + 4 + 5 +..... represents the archetypal linear series of numbers i.e. as numbers naturally marked off on a straight line.

So all these numbers are implicitly 1-dimensional i.e. raised to the power of 1.

However when we raise these numbers same numbers to - 1

i.e. 1^ (- 1), 2^(- 1), 3^(- 1), 4^(- 1), 5^(- 1) +.... we generate the harmonic series 1 + 1/2 + 1/3 + 1/4 + 1/5 + .....

Now remember my basic starting about regarding the nature of the primes is that they combine extreme characteristics with respect to linear and circular aspects!

So the individual prime numbers are the most independent and linear of all numbers (with no constituent factors).

However the general behaviour of the primes (in the frequency of their overall distribution) involves the other extreme of a holistic circular tendency.

Now the very manner in which both linear and circular aspect are involved in experience goes back to the way in which whole and part interact. So to switch from the whole say 4 (4^1) to part 1/4 the dimensional number switches to - 1. So 1/4 = 4^(- 1).

Thus the decisive switch from whole to part requires that conscious (linear) understanding that is defined with respect to the positive 1st dimension be dynamically negated in an unconscious (circular) intuitive manner.

So similar dynamics are involved with respect to the harmonic series (by comparison with the natural number series).

And remarkably the harmonic series gives the simplest answer to the general nature of prime number distribution. In other words the measure of the average spread as between successive prime numbers (which becomes progressively larger as the natural numbers increase) is given by the sum of the harmonic series!

Indeed the change in the average gap between these primes as n i.e n^(+ 1) increases by 1 is given as 1/n i.e. n^(- 1).

So here in the general behaviour of the prime numbers we have an intimate relationship as between the positing and negating respectively (with respect to the 1st dimension) of the number n.

So moving from the specific independent linear notion of an individual prime number to the holistic interdependent circular notion of prime number distribution directly involves the positing (in conscious rational manner) and the corresponding negation (in an unconscious intuitive manner) of linear (1-dimensional) understanding.

So the proper understanding in conventional (Type 1) mathematical terms of the quantitative nature of prime number behaviour is inseparable from corresponding holistic (Type 2) qualitative mathematical interpretation (of such behaviour).

And such understanding requires that we can combine both linear and circular type notions equally in both a quantitative and qualitative manner.

This circular nature of the harmonic series can be demonstrated in yet another striking manner.

Now the harmonic series represents one important example of the Zeta function

i.e. 1/(1^s) + 1/(2^s) + 1/(3^s) + 1/(4^s) + .... where s = 1.

As is well known the for all (positive) even integer values of s the resulting sum of the series can be given in terms of a expression involving pi. And as pi serves as the direct relation of circular and linear quantitative notions in the relationship of the (circular) circumference to its (line) diameter, likewise pi serves as the archetypal relation of corresponding circular and linear notions (understood in a qualitative manner).

For example when s = 2, the sum of the series = {(pi)^2}/6.

The harmonic series can be expressed in terms of the combination of the sum of values corresponding with even integer values for the Zeta Function.

So 1 + 1/2 + 1/3 + 1/4 + ........

= 2{ζ(2)/2 + ζ(4)/4 + ζ(6)/6 + ζ(8)/8 + ......}

And this infinite series can therefore be expressed as the continual sum of terms that consist of powers of pi (that are multiplied by a rational fraction)!

It is again notable that the harmonic series is intimately associated with musical sound. So just as the natural numbers can again be seen as the supreme expression of linear quantitative understanding, the harmonic series can be seen in a sense as a supreme expression of qualitative type appreciation.

And ultimately this is related to the fact that in switching from the positive 1st dimension to its negative one likewise switches from (conscious) rational to (unconscious) intuition, which is the very means by which we switch from quantitative to qualitative type appreciation.

## Wednesday, December 14, 2011

### A Strikingly Simple Prime Number Relationship

As Euler showed the harmonic series 1 + 1/2 + 1/3 + 1/4 +..... + n is approximated as log n + λ (for a finite value of n).

And as λ (the Euler-Masceroni constant) is a constant = .5772 approx this means that for large n, log n is approximated by the harmonic series.

This means therefore that perhaps the simplest expression for the frequency of prime distribution is given as the sum of the harmonic series for n terms divided by n (which becomes increasingly accurate for larger n).

Looked at another way the sum of the harmonic series (for large n) approximates the average spread or gap as between prime numbers in the region of n.

Therefore as the sum of the first million terms for example of the harmonic series = 14.384 (approx). Therefore the average gap as between primes in the region of 1,000,000 is roughly 14. Though this approximation is not yet very accurate, the approximation would greatly improve (in relative terms) as the value of n increases.

Now it is well known that the average spread as between primes continually increases as the value of n increases.

What I find particularly striking in this regard is that the increase in the average spread (or gap) as between primes as we increase n by 1 is given by 1/n.

So for example as we increase n 1,000,000 to 1,000,001 the increase in the average gap as between primes is 1/1,000,000.

(More accurately as we increase n from 999,999.5 to 1,000,000.5 the average gap between primes increases by 1/1,000,000).

This result can easily be demonstrated through differentiation of log n + λ (with respect to n) which results in 1/n.

Now if we multiply the simple expression for the general frequency of primes i.e. n/log n by 1/n we obtain 1/log n (which represents the probability that n is prime).

Thus, we can say that the product of the general frequency of prime distribution and the change in the average gap as between primes (for large n) approximates well the probability that n is prime.

And as λ (the Euler-Masceroni constant) is a constant = .5772 approx this means that for large n, log n is approximated by the harmonic series.

This means therefore that perhaps the simplest expression for the frequency of prime distribution is given as the sum of the harmonic series for n terms divided by n (which becomes increasingly accurate for larger n).

Looked at another way the sum of the harmonic series (for large n) approximates the average spread or gap as between prime numbers in the region of n.

Therefore as the sum of the first million terms for example of the harmonic series = 14.384 (approx). Therefore the average gap as between primes in the region of 1,000,000 is roughly 14. Though this approximation is not yet very accurate, the approximation would greatly improve (in relative terms) as the value of n increases.

Now it is well known that the average spread as between primes continually increases as the value of n increases.

What I find particularly striking in this regard is that the increase in the average spread (or gap) as between primes as we increase n by 1 is given by 1/n.

So for example as we increase n 1,000,000 to 1,000,001 the increase in the average gap as between primes is 1/1,000,000.

(More accurately as we increase n from 999,999.5 to 1,000,000.5 the average gap between primes increases by 1/1,000,000).

This result can easily be demonstrated through differentiation of log n + λ (with respect to n) which results in 1/n.

Now if we multiply the simple expression for the general frequency of primes i.e. n/log n by 1/n we obtain 1/log n (which represents the probability that n is prime).

Thus, we can say that the product of the general frequency of prime distribution and the change in the average gap as between primes (for large n) approximates well the probability that n is prime.

## Friday, November 18, 2011

### Interesting Square Connection

If we just focus on the absolute value of the denominator of the Riemann's Zeta Function for s = - 1, - 3, - 5, - 7 and - 9, which are 12, 120, 252, 240 and 132 respectively we can find an interesting square connection.

So each of these numbers can be expressed as the product of two numbers which differ in ascending order by consecutive powers of 2.

So 12 = 4 * 3 with the difference (1) = 2^0.

120 = 12 * 10 with the difference (2) = 2^1.

252 = 18 * 14 with the difference (4) = 2^2.

240 = 20 * 12 with the difference (8) = 2^3.

132 = 22 * 6 with the difference (16) = 2^4.

After this the pattern begins to break down!

The absolute value corresponding to the denominator for s = - 11 = 32760. This can indeed be expressed as the product of two numbers that differ by a square of 2 but not (but not 2^5).

So 32760 = 182 * 180 with the difference (2) = 2^1.

However with the next number corresponding to s = - 13, no such relationship exists as between the product of two numbers i.e. involving the difference of a square of 2.

So each of these numbers can be expressed as the product of two numbers which differ in ascending order by consecutive powers of 2.

So 12 = 4 * 3 with the difference (1) = 2^0.

120 = 12 * 10 with the difference (2) = 2^1.

252 = 18 * 14 with the difference (4) = 2^2.

240 = 20 * 12 with the difference (8) = 2^3.

132 = 22 * 6 with the difference (16) = 2^4.

After this the pattern begins to break down!

The absolute value corresponding to the denominator for s = - 11 = 32760. This can indeed be expressed as the product of two numbers that differ by a square of 2 but not (but not 2^5).

So 32760 = 182 * 180 with the difference (2) = 2^1.

However with the next number corresponding to s = - 13, no such relationship exists as between the product of two numbers i.e. involving the difference of a square of 2.

## Sunday, November 6, 2011

### Finite and Infinite

I have stated many times that conventional mathematical appreciation is based on a merely reduced notion of the infinite (where effectively it is treated as an extension of the finite).

This for example defines the nature of conventional proof where what is true for the general case (potentially applying to the infinite) is thereby assumed to apply to all actual (finite) cases.

And I have referred to this basic reductionism as interpretation that is linear (i.e. 1-dimensional) in qualitative terms.

This linear approach is also very much to the fore in the treatment of series where in most cases once again a seemingly unambiguous relationship as between finite and infinite emerges.

For example from a finite perspective we can see that a geometrical series such as 1 + 1/2 + 1/4 + ... converges towards some finite limit (getting ever closer to 2 without actually reaching this limiting value).

Therefore when we say that in the limit the value of the series = 2 (where the no. of terms is infinite) this again seems to comply with linear type interpretation (i.e. where the infinite is treated as an extension of finite notions).

However there are other cases where what appears true in finite terms does not readily comply with infinite notions.

For example

1/(1 - x) = 1 + x + x^2 + x^3 + x^4 + ..... (where the number of terms in linear terms is assumed infinite).

This would imply therefore that when x = - 2 that

1/3 = 1 - 2 + 4 - 8 + .....

However when we view the R.H.S in finite terms, we can see that the terms get progressively larger with the value of the series diverging. Now it is true that the terms (and consequent sum of terms) alternate between positive and negative values. However from a linear perspective we cannot say that its sum will converge to a definite finite value.

However the - apparent - equivalent L.H.S expression suggests that this is precisely what happens.

So once again from a linear perspective we obtain a result that is intuitively not in keeping with its rational mode of interpretation.

This strongly suggests that a different form of rational interpretation is required to explain the nature of the result.

However because it is qualitatively defined in terms of 1-dimensional interpretation, Conventional (Type 1) Mathematics is not appropriate for this task.

Now when we return to our example we can see what is the problem

If we consider 1 + x + x^2 + x^3 + x^4 + .... + x^(n - 1), where the series is defined in terms of a finite number (n) of terms,

then 1 - {x ^(n - 1)}/(1 - x) = 1 + x + x^2 + x^3 + x^4 + .... + x^(n - 1)

So the conclusion that

1/(1 - x) = 1 + x + x^2 + x^3 + x^4 + ..... (where n is infinite), is based on the the assumption that n - 1 = n terms (when n is infinite).

Thus the logic that applies for the infinite case i.e. n - 1 = n is directly confused with standard finite logic i.e. n - 1 ≠ n.

This then leads to the non-intuitive result (in linear rational terms) that for example

1 - 2 + 4 - 8 + .... = 1/3

Though mathematicians are of course aware of this anomaly, they attempt to explain it in terms of two results that comply with differing domains of definition.

However this avoids the deeper qualitative question of what such non-intuitive results actually entail!

Facing up to this issue requires accepting the radical conclusion that just as numbers such as 1, 2, 3, 4, ... etc. have a well-defined meaning in quantitative terms, equally they have an - as yet - unrecognised meaning in qualitative terms whereby they refer to unique modes of rational interpretation of symbols.

Once again Conventional (Type 1) Mathematics attempts to confine interpretation to 1-dimensional logic in qualitative terms. However potentially an unlimited set of other qualitative logical interpretations can be given.

And the series that I have used to illustrate this point itself points to the need for a different means of rational interpretation (so that results can then intuitively concur with the correct rational mode adopted).

This for example defines the nature of conventional proof where what is true for the general case (potentially applying to the infinite) is thereby assumed to apply to all actual (finite) cases.

And I have referred to this basic reductionism as interpretation that is linear (i.e. 1-dimensional) in qualitative terms.

This linear approach is also very much to the fore in the treatment of series where in most cases once again a seemingly unambiguous relationship as between finite and infinite emerges.

For example from a finite perspective we can see that a geometrical series such as 1 + 1/2 + 1/4 + ... converges towards some finite limit (getting ever closer to 2 without actually reaching this limiting value).

Therefore when we say that in the limit the value of the series = 2 (where the no. of terms is infinite) this again seems to comply with linear type interpretation (i.e. where the infinite is treated as an extension of finite notions).

However there are other cases where what appears true in finite terms does not readily comply with infinite notions.

For example

1/(1 - x) = 1 + x + x^2 + x^3 + x^4 + ..... (where the number of terms in linear terms is assumed infinite).

This would imply therefore that when x = - 2 that

1/3 = 1 - 2 + 4 - 8 + .....

However when we view the R.H.S in finite terms, we can see that the terms get progressively larger with the value of the series diverging. Now it is true that the terms (and consequent sum of terms) alternate between positive and negative values. However from a linear perspective we cannot say that its sum will converge to a definite finite value.

However the - apparent - equivalent L.H.S expression suggests that this is precisely what happens.

So once again from a linear perspective we obtain a result that is intuitively not in keeping with its rational mode of interpretation.

This strongly suggests that a different form of rational interpretation is required to explain the nature of the result.

However because it is qualitatively defined in terms of 1-dimensional interpretation, Conventional (Type 1) Mathematics is not appropriate for this task.

Now when we return to our example we can see what is the problem

If we consider 1 + x + x^2 + x^3 + x^4 + .... + x^(n - 1), where the series is defined in terms of a finite number (n) of terms,

then 1 - {x ^(n - 1)}/(1 - x) = 1 + x + x^2 + x^3 + x^4 + .... + x^(n - 1)

So the conclusion that

1/(1 - x) = 1 + x + x^2 + x^3 + x^4 + ..... (where n is infinite), is based on the the assumption that n - 1 = n terms (when n is infinite).

Thus the logic that applies for the infinite case i.e. n - 1 = n is directly confused with standard finite logic i.e. n - 1 ≠ n.

This then leads to the non-intuitive result (in linear rational terms) that for example

1 - 2 + 4 - 8 + .... = 1/3

Though mathematicians are of course aware of this anomaly, they attempt to explain it in terms of two results that comply with differing domains of definition.

However this avoids the deeper qualitative question of what such non-intuitive results actually entail!

Facing up to this issue requires accepting the radical conclusion that just as numbers such as 1, 2, 3, 4, ... etc. have a well-defined meaning in quantitative terms, equally they have an - as yet - unrecognised meaning in qualitative terms whereby they refer to unique modes of rational interpretation of symbols.

Once again Conventional (Type 1) Mathematics attempts to confine interpretation to 1-dimensional logic in qualitative terms. However potentially an unlimited set of other qualitative logical interpretations can be given.

And the series that I have used to illustrate this point itself points to the need for a different means of rational interpretation (so that results can then intuitively concur with the correct rational mode adopted).

## Wednesday, November 2, 2011

### The Strange Case of η( - 1)

The Eta Series for s = - 1 is,

η( - 1) = 1/1^(- 1) - 1/2^(- 1) + 1/3^(- 1) - 1/4^(- 1) + ....

Therefore

η( - 1) = 1 - 2 + 3 - 4 + 5 - 6 + ....

Withe reference to the Riemann Zeta Function the value for this Eta series = 1/4.

The question then arises as what meaning can we give this result!

It is perhaps better in illustrating to start with η(0) = 1 - 1 + 1 - 1 + 1 - 1 + ...

The value of this alternating series = 1/2

It is easy enough in this case to see how this value might arise!

If we take an even number of terms, the sum of the series = 0.

However if we take an odd number of terms the sum = 1.

Therefore it seems reasonable - where the number of terms is unspecified - to average the two values as 1/2. However in illustrating this we need to consider a finite number of terms.

In more general terms the answer here is n/2 (where the nth term when it is odd = 1).

However what is interesting is that when we now consider the series (in Type 1 mathematical terms) as infinite

i.e. 1/(1 + x) = 1 + x + x^2 + x^3 + x^4 + ...,

by setting x = - 1 we obtain 1/2 unambiguously as the correct answer.

i.e. 1/2 = 1 - 1 + 1 - 1 + 1 - ..

So the value here from the infinite perspective as 1/2 can be simply considered as n/2 (where the quantitative value of n is set at a default value of 1).

What I am getting at here is very significant indeed in terms of properly interpreting the nature of the Riemann Hypothesis!

Conventional (Type 1) Mathematics is inherently defined with respect to a default dimensional (qualitative) value of 1. In other words the nature of Type 1 Mathematics is qualitatively linear (1-dimensional) where all quantitative values are ultimately reduced in 1-dimensional terms.

So for example 2^2 = 4^1 (in Type 1 terms).

However Holistic (Type 2) Mathematics - in inverse fashion - is inherently defined with respect to a default base quantitative value of 1. So in concentrating on the nature of qualitative transformation (as with the switch from finite to infinite series) Type 2 interpretation is not directly concerned with the quantitative nature of number but rather as its representation of a holistic transformation (where dimensional powers other than + 1 are entailed)!

Now this will perhaps become clearer when we look at both the finite and infinite interpretation of terms corresponding to η(- 1).

η(- 1) = 1 - 2 + 3 - 4 + .....

We consider this series initially as finite and attempt to sum its value in linear (Type 1) terms.

For example we will initially derive η(0) with 10 terms.

i.e. y = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9

Then when we differentiate y with respect to x we obtain 9 remaining terms on the RHS

i.e. 1 - 2x + 3(x^2) - 4(x^3) + 5(x^4) - 6(x^5) + 7(x^6) - 8(x^7) + 9(x^8)

Setting x = - 1 we obtain

1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9

Then in summing this finite series by grouping terms in pairs

the sum of the first 8 terms is - 1 - 1 - 1 - 1

So, if the number of terms is even the sum of the series = - (n - 2)/2

Thus as we originally started with n = 10 the sum of the first 8 terms = - (8/2) = - 4.

However if we sum the first n - 1 terms (which is now odd) we obtain - (n - 2)/2 + (n - 1).

Thus the sum of the first 9 terms = - 4 + 9 = 5.

As we did before for η(0), since there is a 50:50 chance of obtaining the positive value for the sum (associated with an odd number of terms). So the average = {(n - 1) - (n - 2)/2}/2 = (2n - 2 - n + 2)/4 = n/4

So again with originally n = 10, the sum of the first 9 terms = 5. As there is a 50:50 chance of getting this value, the expected value is therefore 2.5. And this is the value corresponding to n/4 (where n = 10).

What is fascinating is that when we then consider the series in infinite terms through differentiating both sides of original expression we get

1/(1 + x)^2 = 1 + 2^x + 3(x^2) + 4(x^3) + .....

By setting x = - 1,

1/4 = 1 - 2 + 3 - 4 + 5 - .....

So once again the sum for the infinite series is the same as derived for the finite, with the important difference that n is here given a default value of 1.

This strongly suggests that in transforming from finite to infinite expressions the very representation of number itself switches from a specific (quantitative) to a holistic (qualitative) meaning. In other words the expected value of the nth term (when n is odd) is n/4. So when referring to the general ratio (rather than its specific quantitative value) we get 1/4!

η( - 1) = 1/1^(- 1) - 1/2^(- 1) + 1/3^(- 1) - 1/4^(- 1) + ....

Therefore

η( - 1) = 1 - 2 + 3 - 4 + 5 - 6 + ....

Withe reference to the Riemann Zeta Function the value for this Eta series = 1/4.

The question then arises as what meaning can we give this result!

It is perhaps better in illustrating to start with η(0) = 1 - 1 + 1 - 1 + 1 - 1 + ...

The value of this alternating series = 1/2

It is easy enough in this case to see how this value might arise!

If we take an even number of terms, the sum of the series = 0.

However if we take an odd number of terms the sum = 1.

Therefore it seems reasonable - where the number of terms is unspecified - to average the two values as 1/2. However in illustrating this we need to consider a finite number of terms.

In more general terms the answer here is n/2 (where the nth term when it is odd = 1).

However what is interesting is that when we now consider the series (in Type 1 mathematical terms) as infinite

i.e. 1/(1 + x) = 1 + x + x^2 + x^3 + x^4 + ...,

by setting x = - 1 we obtain 1/2 unambiguously as the correct answer.

i.e. 1/2 = 1 - 1 + 1 - 1 + 1 - ..

So the value here from the infinite perspective as 1/2 can be simply considered as n/2 (where the quantitative value of n is set at a default value of 1).

What I am getting at here is very significant indeed in terms of properly interpreting the nature of the Riemann Hypothesis!

Conventional (Type 1) Mathematics is inherently defined with respect to a default dimensional (qualitative) value of 1. In other words the nature of Type 1 Mathematics is qualitatively linear (1-dimensional) where all quantitative values are ultimately reduced in 1-dimensional terms.

So for example 2^2 = 4^1 (in Type 1 terms).

However Holistic (Type 2) Mathematics - in inverse fashion - is inherently defined with respect to a default base quantitative value of 1. So in concentrating on the nature of qualitative transformation (as with the switch from finite to infinite series) Type 2 interpretation is not directly concerned with the quantitative nature of number but rather as its representation of a holistic transformation (where dimensional powers other than + 1 are entailed)!

Now this will perhaps become clearer when we look at both the finite and infinite interpretation of terms corresponding to η(- 1).

η(- 1) = 1 - 2 + 3 - 4 + .....

We consider this series initially as finite and attempt to sum its value in linear (Type 1) terms.

For example we will initially derive η(0) with 10 terms.

i.e. y = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9

Then when we differentiate y with respect to x we obtain 9 remaining terms on the RHS

i.e. 1 - 2x + 3(x^2) - 4(x^3) + 5(x^4) - 6(x^5) + 7(x^6) - 8(x^7) + 9(x^8)

Setting x = - 1 we obtain

1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9

Then in summing this finite series by grouping terms in pairs

the sum of the first 8 terms is - 1 - 1 - 1 - 1

So, if the number of terms is even the sum of the series = - (n - 2)/2

Thus as we originally started with n = 10 the sum of the first 8 terms = - (8/2) = - 4.

However if we sum the first n - 1 terms (which is now odd) we obtain - (n - 2)/2 + (n - 1).

Thus the sum of the first 9 terms = - 4 + 9 = 5.

As we did before for η(0), since there is a 50:50 chance of obtaining the positive value for the sum (associated with an odd number of terms). So the average = {(n - 1) - (n - 2)/2}/2 = (2n - 2 - n + 2)/4 = n/4

So again with originally n = 10, the sum of the first 9 terms = 5. As there is a 50:50 chance of getting this value, the expected value is therefore 2.5. And this is the value corresponding to n/4 (where n = 10).

What is fascinating is that when we then consider the series in infinite terms through differentiating both sides of original expression we get

1/(1 + x)^2 = 1 + 2^x + 3(x^2) + 4(x^3) + .....

By setting x = - 1,

1/4 = 1 - 2 + 3 - 4 + 5 - .....

So once again the sum for the infinite series is the same as derived for the finite, with the important difference that n is here given a default value of 1.

This strongly suggests that in transforming from finite to infinite expressions the very representation of number itself switches from a specific (quantitative) to a holistic (qualitative) meaning. In other words the expected value of the nth term (when n is odd) is n/4. So when referring to the general ratio (rather than its specific quantitative value) we get 1/4!

## Monday, October 31, 2011

### Calculating ζ (- 1) and ζ (- 3)

Without attempting to use analytic continuation on the complex plane, we will now calculate the first two values for negative odd integer values of s i.e. - 1 and - 3, for the Riemann Zeta Function.

1/(1 - s) = 1 + s + s^2 + s^3 + s^4 + s^5 + s^6 + .......

Differentiating both sides,

1/{(1 - s)^2} = 1 + 2s + 3(s^2) + 4(s^3) + 5(s^4) + 6(s^5) + .....

Setting s = - 1,

1/4 = 1 - 2 + 3 - 4 + 5 - 6 + ......

Therefore 1/4 = η(- 1)

ζ(- 1) = η(- 1)/{1 - 2^(s - 1)} = (1/4)/(- 3) = - 1/12

If we differentiate both sides again

2/{1 - s)^3 = 2 + 6s + 12(s^2) + 20(s^3) + 30(s^4) +....

Then differentiating once more,

6/(1 - s)^4 = 6 + 24s + 60(s^2) + 120(s^3) +.......

Setting s = - 1,

6/16 = 6 - 24 + 60 - 120 + .....

η(- 3) = 1 - 2^3 + 3^3 - 4^3 + 5^4 - ......

= 1 - 8 + 27 - 64 + 125 - ....

So η(- 3) + 6/16

= (1 - 8 + 27 - 64 + 125 - ....) + (6 - 24 + 60 - 120 + .....)

= 1 - 2 + 3 - 4 + 5 - ......

Thus η(- 3) + 6/16 = η(- 1)

Therefore η(- 3) = η(- 1) - 6/16

= 1/4 - 6/16 = 4/16 - 6/16 = - 2/16 = - 1/8

ζ(- 3) = η(- 3)/{1 - 2^(s - 3)} = (- 1/8)/(- 15) = 1/120

This would suggest that in principle we should be able to work out all the odd negative integer values for s through a similar process of combining already attained values for s with the varied series that arise from continued differentiation of both sides of the equation.

The question then arises as to why similar attempts with respect to even integer values of s do not hold.

As we have seen,

2/{1 - s)^3 = 2 + 6s + 12(s^2) + 20(s^3) + 30(s^4) - 42(s^5) + ....

Thus setting s = - 1,

2/8 = 2 - 6 + 12 - 20 + 30 - 42 + ....

Therefore by combining terms in successive pairs 2/8 = - 4 - 8 - 12 - ......

= - 4(1 + 2 + 3 + ....)

As the term inside the bracket is ζ(- 1), this would imply that

2/8 = - 4(- 1/12) = 1/3 which is meaningless.

However what is interesting in this case is that instead of obtaining an eta expression on the RHS (as with odd integer values for s) we obtain a zeta value.

So clearly whereas we can derive zeta values from eta, we cannot here derive a zeta value from another zeta value!

The reason is that this value for ζ(- 1) arises when we set s = + 1 on the RHS which means that the value for 2/{1 - s)^3 is thereby infinite!

Once again

when s = - 1,

2/8 = 2 - 6 + 12 - 20 + 30 - 42 + ....

η(- 1) = 1 - 2 + 3 - 4 + 5 - 6 + ....

Therefore 2/8 - η(- 1) = 1 - 4 + 9 - 16 + 25 - 36 + ...

So 2/8 - η(- 1) = η(- 2)

Therefore η(- 2) = 2/8 - 1/4 = 0.

And as ζ(- 2) = η(- 2)/{1 - 2^(s - 1)} this implies that ζ(- 2)= 0.

Thus we derive the correct answer in this case.

And a similar result in principle would emerge whenever we use an negative even integer value (i.e. - 4, - 6, -8, etc.) for s.

1/(1 - s) = 1 + s + s^2 + s^3 + s^4 + s^5 + s^6 + .......

Differentiating both sides,

1/{(1 - s)^2} = 1 + 2s + 3(s^2) + 4(s^3) + 5(s^4) + 6(s^5) + .....

Setting s = - 1,

1/4 = 1 - 2 + 3 - 4 + 5 - 6 + ......

Therefore 1/4 = η(- 1)

ζ(- 1) = η(- 1)/{1 - 2^(s - 1)} = (1/4)/(- 3) = - 1/12

If we differentiate both sides again

2/{1 - s)^3 = 2 + 6s + 12(s^2) + 20(s^3) + 30(s^4) +....

Then differentiating once more,

6/(1 - s)^4 = 6 + 24s + 60(s^2) + 120(s^3) +.......

Setting s = - 1,

6/16 = 6 - 24 + 60 - 120 + .....

η(- 3) = 1 - 2^3 + 3^3 - 4^3 + 5^4 - ......

= 1 - 8 + 27 - 64 + 125 - ....

So η(- 3) + 6/16

= (1 - 8 + 27 - 64 + 125 - ....) + (6 - 24 + 60 - 120 + .....)

= 1 - 2 + 3 - 4 + 5 - ......

Thus η(- 3) + 6/16 = η(- 1)

Therefore η(- 3) = η(- 1) - 6/16

= 1/4 - 6/16 = 4/16 - 6/16 = - 2/16 = - 1/8

ζ(- 3) = η(- 3)/{1 - 2^(s - 3)} = (- 1/8)/(- 15) = 1/120

This would suggest that in principle we should be able to work out all the odd negative integer values for s through a similar process of combining already attained values for s with the varied series that arise from continued differentiation of both sides of the equation.

The question then arises as to why similar attempts with respect to even integer values of s do not hold.

As we have seen,

2/{1 - s)^3 = 2 + 6s + 12(s^2) + 20(s^3) + 30(s^4) - 42(s^5) + ....

Thus setting s = - 1,

2/8 = 2 - 6 + 12 - 20 + 30 - 42 + ....

Therefore by combining terms in successive pairs 2/8 = - 4 - 8 - 12 - ......

= - 4(1 + 2 + 3 + ....)

As the term inside the bracket is ζ(- 1), this would imply that

2/8 = - 4(- 1/12) = 1/3 which is meaningless.

However what is interesting in this case is that instead of obtaining an eta expression on the RHS (as with odd integer values for s) we obtain a zeta value.

So clearly whereas we can derive zeta values from eta, we cannot here derive a zeta value from another zeta value!

The reason is that this value for ζ(- 1) arises when we set s = + 1 on the RHS which means that the value for 2/{1 - s)^3 is thereby infinite!

Once again

when s = - 1,

2/8 = 2 - 6 + 12 - 20 + 30 - 42 + ....

η(- 1) = 1 - 2 + 3 - 4 + 5 - 6 + ....

Therefore 2/8 - η(- 1) = 1 - 4 + 9 - 16 + 25 - 36 + ...

So 2/8 - η(- 1) = η(- 2)

Therefore η(- 2) = 2/8 - 1/4 = 0.

And as ζ(- 2) = η(- 2)/{1 - 2^(s - 1)} this implies that ζ(- 2)= 0.

Thus we derive the correct answer in this case.

And a similar result in principle would emerge whenever we use an negative even integer value (i.e. - 4, - 6, -8, etc.) for s.

## Wednesday, September 28, 2011

### New Perspective on Mathematical Proof

I have argued before that correctly speaking - in dynamic experiential terms - that all mathematical proof is subject to the uncertainty principle.

Thus it represents but an especially important form of social consensus that can never be absolute. In fact in certain terms such consensus can prove especially flawed.

Thus for a period in 1993, it was believed that that Fermat's Last theorem had been proved (only for a fatal flaw in reasoning later being discovered). Now the prevailing consensus since 1995 is that this remaining problem has been satisfactorily resolved. So as time goes by - with no further flaws being discovered - we can accept with an ever greater degree of confidence that Fermat's Last Theorem has been proven. However this conviction always remains of a merely probabilistic nature (that is necessarily subject to a degree of uncertainty).

At a deeper level I have challenged the conventional notion of mathematical proof in that it represents just one limited form of interpretation (where the qualitative aspect of understanding is necessarily reduced in quantitative terms).

More formally, in holistic mathematical terms conventional proof corresponds to a linear (1-dimensional) mode of qualitative interpretation (where again meaning is reduced in quantitative terms). However correctly understood we can have potentially an infinite set of mathematical interpretations (corresponding to all other dimensional numbers) where quantitative and qualitative aspects of interpretation - though necessarily related - preserve a distinctive aspect. And in the inevitable relative interaction between both aspects a degree of uncertainty necessarily exists.

So therefore a comprehensive proof requires both qualitative and quantitative aspects (that are necessarily relative). Conventional proof only appears therefore of an absolute nature because the qualitative aspect is entirely neglected.

However recently I have come to realise that there is yet another way in which the uncertainty principle necessarily applies to all mathematical proof.

As we have seen the Riemann Hypothesis represents the important starting axiom whereby the quantitative and qualitative aspects of mathematical understanding can be directly reconciled.

So the Riemann Hypothesis is already necessarily inherent in conventional mathematical aspects. Because of the reduced nature of understanding (where the qualitative aspect of truth is reduced to the quantitative), it is already assumed that if a theorem is proved in general (holistic) terms that it thereby necessarily applies to each individual case (within its class). So for example one we accept that the Pythagorean Theorem is true for the general case (establishing

However this implies confusing the potentially infinite nature of the holistic general proposition with the actual finite nature of individual examples. In other words the qualitative aspect of understanding is reduced thereby to the quantitative.

Thus when we properly preserve the unique qualitative distinction of (potential) infinite and (actual) finite notions, we cannot in the absence of a higher authority as it were, automatically infer the truth of the specific from the corresponding truth of the general case.

Now acceptance of the Riemann Hypothesis is necessary to correctly make such an inference. However, as we have seen, because the Riemann Hypothesis supersedes conventional mathematical axioms it cannot be proved (or disproved) from a conventional mathematical perspective.

Therefore the acceptance that a general (qualitative) proof applies in specific (quantitative) terms is correctly based on the validity of the Riemann Hypothesis (which itself cannot be proved or disproved).

That means therefore that mathematical proof is ultimately based on an act of faith (and is thereby subject to uncertainty).

In the highly unlikely case that a non-trivial zero of the Riemann Zeta Function is ever found off the line (with real part = .5) this will pose an interesting dilemma.

We may be tempted to initially maintain that the Riemann Hypothesis has in fact been proven to be untrue.

However this would raise a much deeper problem. For if the Riemann Hypothesis is untrue, then we are no longer entitled to maintain the connection between an actual specific case and the general truth (for the potentially infinite case).

In other words we would no longer be entitled to infer that the demonstrated finding of our errant zero undermines the truth of the general proposition.

In fact it would be even much worse in that we would no longer be able to trust any proposition that has been proved in conventional terms.

So once again the mathematical edifice is ultimately dependent on a supreme act of faith (in the correspondence of infinite with finite notions).

For this reason anyone who believes in Mathematics should hope - and indeed pray - that no errant zero ever crops up for this would rightly undermine faith in the whole enterprise!

Thus it represents but an especially important form of social consensus that can never be absolute. In fact in certain terms such consensus can prove especially flawed.

Thus for a period in 1993, it was believed that that Fermat's Last theorem had been proved (only for a fatal flaw in reasoning later being discovered). Now the prevailing consensus since 1995 is that this remaining problem has been satisfactorily resolved. So as time goes by - with no further flaws being discovered - we can accept with an ever greater degree of confidence that Fermat's Last Theorem has been proven. However this conviction always remains of a merely probabilistic nature (that is necessarily subject to a degree of uncertainty).

At a deeper level I have challenged the conventional notion of mathematical proof in that it represents just one limited form of interpretation (where the qualitative aspect of understanding is necessarily reduced in quantitative terms).

More formally, in holistic mathematical terms conventional proof corresponds to a linear (1-dimensional) mode of qualitative interpretation (where again meaning is reduced in quantitative terms). However correctly understood we can have potentially an infinite set of mathematical interpretations (corresponding to all other dimensional numbers) where quantitative and qualitative aspects of interpretation - though necessarily related - preserve a distinctive aspect. And in the inevitable relative interaction between both aspects a degree of uncertainty necessarily exists.

So therefore a comprehensive proof requires both qualitative and quantitative aspects (that are necessarily relative). Conventional proof only appears therefore of an absolute nature because the qualitative aspect is entirely neglected.

However recently I have come to realise that there is yet another way in which the uncertainty principle necessarily applies to all mathematical proof.

As we have seen the Riemann Hypothesis represents the important starting axiom whereby the quantitative and qualitative aspects of mathematical understanding can be directly reconciled.

So the Riemann Hypothesis is already necessarily inherent in conventional mathematical aspects. Because of the reduced nature of understanding (where the qualitative aspect of truth is reduced to the quantitative), it is already assumed that if a theorem is proved in general (holistic) terms that it thereby necessarily applies to each individual case (within its class). So for example one we accept that the Pythagorean Theorem is true for the general case (establishing

*potentially*that in any right angled triangle the square on the hypotenuse is equal to the sum of squares on the other two sides) that this necessarily applies in any*actual*case.However this implies confusing the potentially infinite nature of the holistic general proposition with the actual finite nature of individual examples. In other words the qualitative aspect of understanding is reduced thereby to the quantitative.

Thus when we properly preserve the unique qualitative distinction of (potential) infinite and (actual) finite notions, we cannot in the absence of a higher authority as it were, automatically infer the truth of the specific from the corresponding truth of the general case.

Now acceptance of the Riemann Hypothesis is necessary to correctly make such an inference. However, as we have seen, because the Riemann Hypothesis supersedes conventional mathematical axioms it cannot be proved (or disproved) from a conventional mathematical perspective.

Therefore the acceptance that a general (qualitative) proof applies in specific (quantitative) terms is correctly based on the validity of the Riemann Hypothesis (which itself cannot be proved or disproved).

That means therefore that mathematical proof is ultimately based on an act of faith (and is thereby subject to uncertainty).

In the highly unlikely case that a non-trivial zero of the Riemann Zeta Function is ever found off the line (with real part = .5) this will pose an interesting dilemma.

We may be tempted to initially maintain that the Riemann Hypothesis has in fact been proven to be untrue.

However this would raise a much deeper problem. For if the Riemann Hypothesis is untrue, then we are no longer entitled to maintain the connection between an actual specific case and the general truth (for the potentially infinite case).

In other words we would no longer be entitled to infer that the demonstrated finding of our errant zero undermines the truth of the general proposition.

In fact it would be even much worse in that we would no longer be able to trust any proposition that has been proved in conventional terms.

So once again the mathematical edifice is ultimately dependent on a supreme act of faith (in the correspondence of infinite with finite notions).

For this reason anyone who believes in Mathematics should hope - and indeed pray - that no errant zero ever crops up for this would rightly undermine faith in the whole enterprise!

## Wednesday, September 21, 2011

### The Mystery of the Primes

I have been reading Matthew Watkins' book "The Mystery of the Primes" (the first volume of a proposed trilogy) recently with much interest.

Though Matthew is clearly a qualified mathematician and therefore well able to deal with issues relating to the primes in the accepted specialised language that fellow practitioners employ, he opts here for a user friendly approach that would be accessible to most lay people (with little or no grounding in Mathematics).

What I like about this approach is that he clearly appreciates how apparently simple mathematical concepts lead to profound problems of a philosophical nature. So philosophical, psychological, religious and even economic observations are introduced early on in a much more wide ranging approach than is conventional, to unveiling the mystery of the primes.

I look forward to following Matthew's trail to see where it will lead in the further two volumes (yet to be published). I am sure that it will be interesting!

Having introduced - what he refers to as - "spiral waves" he then intends to probe more deeply into their relationship to the famed Riemann zeros in the second volume. These, magically have the capacity to eliminate the deviations associated with the generalised prediction of prime number frequency (relating to a continuous function) so that ultimately we can precisely obtain the actual frequency of such primes (which is of a discrete nature).

Such deviations seemingly correspond precisely to the vibrations of some unknown system (with a tantalising relationship to known quantum chaotic systems of a physical nature).

So a fundamental question relates to what in fact is vibrating!

My own considerations on this issue have led to the conclusion that it is the prime numbers themselves that are vibrating.

We are still far too accustomed to think of prime numbers in discrete independent terms as the building blocks of the number system. However, even momentary reflection on the matter reveals that they can have no strict meaning in the absence of the natural numbers.

To resolve this issue one must therefore look at both the prime and the natural numbers in a dynamic interactive manner. Implicit in this observation is the clear recognition that numbers have both quantitative and qualitative aspects. So rather than having a static identity (of an abstract nature) all numbers - from this new perspective - represent dynamic interaction patterns (with both quantitative and qualitative characteristics).

To be precise, the prime number code - which governs subsequent phenomenal interactions - is of a potential kind and thereby already inherent in the physical phenomena that emerge in reality. However the very unfolding of such phenomena in space and time implies some degree of separation of the quantitative from the qualitative aspect of the primes. Once again therefore the ultimate state where quantitative and qualitative (or specific and holistic) aspects are perfectly reconciled (in direct spiritual union of reality) is of an absolute ineffable nature.

And it is to this ultimate state that the Riemann Hypothesis directly points! In other words the Riemann Hypothesis relates to the fundamental condition for maintaining consistency with respect to both the quantitative and qualitative behaviour of the primes. Without such consistency (which unfortunately is the case if the Riemann Hypothesis does not hold) we would have no basis for either trusting the validity of our mathematical axioms or our holistic intuitions as to nature of mathematical truth.

As the condition, to which the Riemann Hypothesis relates, precedes such axioms and intuitions (and is already inherent in their very use) there is no way in which the Riemann Hypothesis can be proved (or disproved) in conventional mathematical terms.

It is therefore truly of the most fundamental nature possible, with its acceptance representing an act of faith in the subsequent consistency of the whole mathematical enterprise!

So in a qualified sense the prime numbers themselves vibrate! More correctly it is the interaction of both the prime and natural numbers that leads to the vibrations corresponding to the Riemann non-trivial zeros. Once again though the starting code is set in an absolute ineffable manner, these vibrations can only become manifest in a phenomenal physical manner (where some degree of separation of both quantitative and qualitative aspects has already taken place).

So the fascinating conclusion that we can draw from this is that manifest physical reality itself represents, at a more fundamental level, vibrations corresponding to number patterns (which are thereby inherent in all physical forms).

Now this might appear ludicrous when we think of numbers with respect to their mere quantitative characteristics. However when we equally recognise the qualitative dimensional characteristics (associated with number) it seems - at least to me - a somewhat obvious conclusion.

And this is the precise nub of the matter. Properly understood the dimensional aspect (i.e. power or exponent) of a number is of a qualitative nature (with respect to its number quantity). However this simple truth is - seemingly - entirely missed in conventional Type 1 appreciation. So when the number a is raised to a corresponding dimensional number b (as power or exponent) i.e. a^b, then - properly understood a and b are quantitative and qualitative with respect to each other.

A simple clue to this truth is given by the fact that when 1 is raised to a dimension that is a rational fraction such as 1/3, a circular number results (i.e. lying on the circle of unit radius in the complex plane). So though 1 and 1/3 are conventionally interpreted as linear quantities (i.e. as points on the real number line) clearly a profound transformation results (from linear to circular form) in relating the base quantity 1 with its dimensional exponent 1/3. So the simple truth is that these numbers - which can indeed be given a quantitative definition in isolation - are actually quantitative and qualitative in relation to each other!

And Type 2 i.e. Holistic Mathematics (which is directly of a qualitative nature) is based on circular rather than linear logical notions!

One important implication of this finding - which I was even dimly aware of as a child - is that one cannot hope to give a coherent explanation of an apparently simple relationship such as the square root of 1 in the absence of Type 2 (qualitative) mathematical understanding!

So remarkably, because of the unquestioned quantitative bias of Conventional Mathematics (which is thereby quite unbalanced) the most obvious problems of mathematical interpretation are persistently overlooked!

Therefore, for example when a number is raised to a whole number dimension (other than 1) in conventional terms a merely reduced quantitative interpretation is given of the result.

Thus 2^3 in Type 1 terms = 8 (i.e. 8^1).

Now by looking at this in a geometrical manner we can easily appreciate that a dimensional transformation of a qualitative nature is equally involved. In other words 2^3 represents cube (3-dimensional) rather than linear (1-dimensional) units.

So, as I have repeatedly expressed, Conventional (Type 1) Mathematics necessarily reduces the qualitative notion of dimension in a merely linear (1-dimensional) fashion.

However, when we begin to appreciate the true qualitative nature of dimension, we perhaps can recognise that physical dimensions - when correctly understood - are intimately related to holistic mathematical notions (of a qualitative Type 2 nature).

I have expressed for some time - see my Integral Science blog - dissatisfaction with the philosophical notion of a string (which has no strict physical meaning).

However once we accept that physical phenomena represent a certain rigid reduction with respect to original vibration patterns of a purely mathematical nature, then we can perhaps appreciate the ultimate nature of strings as the prime number constituents of natural reality with the relationship collectively between strings equally representing the prime number dimensional aspects of this same natural reality.

Reality at its most fundamental level is written in a double binary code (1 and 0) where both quantitative and qualitative aspects interact. This provides a means therefore of potentially encoding all (quantitative) information as is well recognised in the present digital age; however what is not equally recognised is that the same digits when holistically interpreted in a qualitative manner can potentially encode all transformation processes!

Then at the next most fundamental level, reality is written in the prime number code (again with respect to its quantitative and qualitative aspects). And from this code ultimately all phenomenal reality, as we know it, is derived!

Though Matthew is clearly a qualified mathematician and therefore well able to deal with issues relating to the primes in the accepted specialised language that fellow practitioners employ, he opts here for a user friendly approach that would be accessible to most lay people (with little or no grounding in Mathematics).

What I like about this approach is that he clearly appreciates how apparently simple mathematical concepts lead to profound problems of a philosophical nature. So philosophical, psychological, religious and even economic observations are introduced early on in a much more wide ranging approach than is conventional, to unveiling the mystery of the primes.

I look forward to following Matthew's trail to see where it will lead in the further two volumes (yet to be published). I am sure that it will be interesting!

Having introduced - what he refers to as - "spiral waves" he then intends to probe more deeply into their relationship to the famed Riemann zeros in the second volume. These, magically have the capacity to eliminate the deviations associated with the generalised prediction of prime number frequency (relating to a continuous function) so that ultimately we can precisely obtain the actual frequency of such primes (which is of a discrete nature).

Such deviations seemingly correspond precisely to the vibrations of some unknown system (with a tantalising relationship to known quantum chaotic systems of a physical nature).

So a fundamental question relates to what in fact is vibrating!

My own considerations on this issue have led to the conclusion that it is the prime numbers themselves that are vibrating.

We are still far too accustomed to think of prime numbers in discrete independent terms as the building blocks of the number system. However, even momentary reflection on the matter reveals that they can have no strict meaning in the absence of the natural numbers.

To resolve this issue one must therefore look at both the prime and the natural numbers in a dynamic interactive manner. Implicit in this observation is the clear recognition that numbers have both quantitative and qualitative aspects. So rather than having a static identity (of an abstract nature) all numbers - from this new perspective - represent dynamic interaction patterns (with both quantitative and qualitative characteristics).

To be precise, the prime number code - which governs subsequent phenomenal interactions - is of a potential kind and thereby already inherent in the physical phenomena that emerge in reality. However the very unfolding of such phenomena in space and time implies some degree of separation of the quantitative from the qualitative aspect of the primes. Once again therefore the ultimate state where quantitative and qualitative (or specific and holistic) aspects are perfectly reconciled (in direct spiritual union of reality) is of an absolute ineffable nature.

And it is to this ultimate state that the Riemann Hypothesis directly points! In other words the Riemann Hypothesis relates to the fundamental condition for maintaining consistency with respect to both the quantitative and qualitative behaviour of the primes. Without such consistency (which unfortunately is the case if the Riemann Hypothesis does not hold) we would have no basis for either trusting the validity of our mathematical axioms or our holistic intuitions as to nature of mathematical truth.

As the condition, to which the Riemann Hypothesis relates, precedes such axioms and intuitions (and is already inherent in their very use) there is no way in which the Riemann Hypothesis can be proved (or disproved) in conventional mathematical terms.

It is therefore truly of the most fundamental nature possible, with its acceptance representing an act of faith in the subsequent consistency of the whole mathematical enterprise!

So in a qualified sense the prime numbers themselves vibrate! More correctly it is the interaction of both the prime and natural numbers that leads to the vibrations corresponding to the Riemann non-trivial zeros. Once again though the starting code is set in an absolute ineffable manner, these vibrations can only become manifest in a phenomenal physical manner (where some degree of separation of both quantitative and qualitative aspects has already taken place).

So the fascinating conclusion that we can draw from this is that manifest physical reality itself represents, at a more fundamental level, vibrations corresponding to number patterns (which are thereby inherent in all physical forms).

Now this might appear ludicrous when we think of numbers with respect to their mere quantitative characteristics. However when we equally recognise the qualitative dimensional characteristics (associated with number) it seems - at least to me - a somewhat obvious conclusion.

And this is the precise nub of the matter. Properly understood the dimensional aspect (i.e. power or exponent) of a number is of a qualitative nature (with respect to its number quantity). However this simple truth is - seemingly - entirely missed in conventional Type 1 appreciation. So when the number a is raised to a corresponding dimensional number b (as power or exponent) i.e. a^b, then - properly understood a and b are quantitative and qualitative with respect to each other.

A simple clue to this truth is given by the fact that when 1 is raised to a dimension that is a rational fraction such as 1/3, a circular number results (i.e. lying on the circle of unit radius in the complex plane). So though 1 and 1/3 are conventionally interpreted as linear quantities (i.e. as points on the real number line) clearly a profound transformation results (from linear to circular form) in relating the base quantity 1 with its dimensional exponent 1/3. So the simple truth is that these numbers - which can indeed be given a quantitative definition in isolation - are actually quantitative and qualitative in relation to each other!

And Type 2 i.e. Holistic Mathematics (which is directly of a qualitative nature) is based on circular rather than linear logical notions!

One important implication of this finding - which I was even dimly aware of as a child - is that one cannot hope to give a coherent explanation of an apparently simple relationship such as the square root of 1 in the absence of Type 2 (qualitative) mathematical understanding!

So remarkably, because of the unquestioned quantitative bias of Conventional Mathematics (which is thereby quite unbalanced) the most obvious problems of mathematical interpretation are persistently overlooked!

Therefore, for example when a number is raised to a whole number dimension (other than 1) in conventional terms a merely reduced quantitative interpretation is given of the result.

Thus 2^3 in Type 1 terms = 8 (i.e. 8^1).

Now by looking at this in a geometrical manner we can easily appreciate that a dimensional transformation of a qualitative nature is equally involved. In other words 2^3 represents cube (3-dimensional) rather than linear (1-dimensional) units.

So, as I have repeatedly expressed, Conventional (Type 1) Mathematics necessarily reduces the qualitative notion of dimension in a merely linear (1-dimensional) fashion.

However, when we begin to appreciate the true qualitative nature of dimension, we perhaps can recognise that physical dimensions - when correctly understood - are intimately related to holistic mathematical notions (of a qualitative Type 2 nature).

I have expressed for some time - see my Integral Science blog - dissatisfaction with the philosophical notion of a string (which has no strict physical meaning).

However once we accept that physical phenomena represent a certain rigid reduction with respect to original vibration patterns of a purely mathematical nature, then we can perhaps appreciate the ultimate nature of strings as the prime number constituents of natural reality with the relationship collectively between strings equally representing the prime number dimensional aspects of this same natural reality.

Reality at its most fundamental level is written in a double binary code (1 and 0) where both quantitative and qualitative aspects interact. This provides a means therefore of potentially encoding all (quantitative) information as is well recognised in the present digital age; however what is not equally recognised is that the same digits when holistically interpreted in a qualitative manner can potentially encode all transformation processes!

Then at the next most fundamental level, reality is written in the prime number code (again with respect to its quantitative and qualitative aspects). And from this code ultimately all phenomenal reality, as we know it, is derived!

## Wednesday, August 24, 2011

### Critical Region (3)

It has to be clearly recognised that the infinite is a qualitatively distinct concept from that of the finite. Ultimately it points to the fact that with respect to reality we have both actual and potential aspects. The actual is always made manifest in finite terms whereas the infinite more correctly pertains to the potential (from which the actual emerges). Again in direct scientific terms, the finite is appropriated in a (conscious) rational manner whereas the infinite is appropriated in an (unconscious) intuitive manner.

This in turn poses great difficulties for the conventional (Type 1) mathematical approach which in formal terms is confined to solely (linear) rational modes of interpretation.

Therefore the notion of the infinite that applies in Type 1 Mathematics is but a reduced quantitative notion that in many respects is entirely inadequate.

Somehow the mistaken view is perpetuated - as for example with series - that if we keep increasing the number of terms that we eventually accumulate an "infinite" no. (of such terms). However this is strictly speaking nonsense! Once again the infinite is a qualitatively distinct notion from the finite and cannot be "reached" therefore in finite manner. So if we keep increasing the terms of - for example - the natural number series without limit as might be said, we always obtain a finite number of terms. At no stage does the actual number of terms become infinite!

There are direct indications - even in Type 1 Mathematics - that the infinite is indeed qualitatively different from the finite. For example if we add two numbers in finite terms (say 1 + 2) a quantitative transformation is involved. Likewise if we multiply two non unitary numbers (say 2 * 3) again a quantitative transformation takes place.

However if we add two infinite numbers (or indeed a finite to a infinite number) no quantitative transformation is involved. Again if we multiply two infinite numbers or (an infinite by a finite) again no transformation is involved. Now, Cantor did indeed explore the notion of different types of infinities (within the confines of Type 1 Mathematics) but essentially his work - though presented in a reduced quantitative manner - draws attention to the fact that there is inevitably a qualitative aspect also involved. So ultimately the conclusion for example that the set of transcendental numbers is "bigger" than than that of the rationals is a reduced quantitative way of expressing the fact that the transcendentals are qualitatively distinct from the rationals (combining both discrete and continuous notions) whereas the rational are based on merely discrete notions!

All of this is of vital importance with respect to values for the Riemann Zeta Function which once again is defined (in Type 1 terms) as the infinite series

1/(1^s) + 1/(2^s) + 1/(3^s) + 1/(4^s) + ...... where s is any complex number a + bi.

Now it must be stressed again that this use of the infinite - which defines Type 1 Mathematics - represents but a reduced quantitative notion (i.e. where the potential notion of the infinite is reduced in actual finite terms).

In this context this approach does seem to work in certain cases. As is well known Euler had earlier shown that the Function indeed converges for all real values of s > 1. and this result can be exended to all complex numbers where again the real part a > 1.

Riemann however managed to extend the domain of definition for the Function to all values of s (except 1).

However this creates the immediate problem that the results for a wide range of these values make no sense from the conventional linear perspective.

My main concern here is not to show precisely how Riemann managed to achieve this "magical" transformation. What I am more concerned to deal with are the key philosophical implications that are involved which ultimately requires that an entirely distinctive type of Mathematics (Type 2) is used.

Ultimately this dramatically changes the very nature of the Riemann Hypothesis from a hypothesis in Type 1 Mathematics to a new fundamental hypothesis that serves as the cornerstone of the reconciliation of Type 1 and Type 2 Mathematics (which is the basis for the more comprehensive Type 3 Mathematics).

In other words all mathematical symbols possess both quantitative and qualitative aspects corresponding to two distinct types of interpretation. So the fundamental requirement is that consistency can be maintained with respect to both requirements. And the Riemann Hypothesis provides the very condition necessary for such consistency.

This of course implies that prime numbers inherently combine such quantitative and qualitative aspects in a very special manner!

Now we are looking here at the critical region of the Riemann Zeta Hypothesis for all values of s between 0 and 1.

Now the Function is undefined for s = 1 where the harmonic series results which in conventional terms sums to infinity.

The question then arises as to why the domain of definition cannot be stretched so as to include s = 1 (when in fact it can include every other value).

Now the qualitative mathematical reason (Type 2) is very illuminating in this regard.

Type 1 Mathematics is qualitatively based on the linear rational approach which is - literally - 1-dimensional in nature. Thus when for example we square the number 2 a qualitative as well as quantitative transformation takes place. So strictly the answer is now 4 square (i.e. 2-dimensional) units. However in Type 1 terms we simply ignore this qualitative dimensional change and express the result in reduced - merely - quantitative terms. So in Type 1 calculations all dimensional charges in the units involved are reduced to 1. Thus 2 * 2 = 4 (i.e. 4 ^ 1).

Now once we accept the true qualitative nature of a dimension, then numerical calculations (involving transformational changes) can be given an alternative result based on qualitative rather than quantitative considerations.

In the context of the Riemann Zeta Function therefore we give the Function - when it diverges from a Type 1 perspective - a finite Type 2 interpretation (where the result converges).

Remember Type 1 and Type 2 are finite and infinite with respect to each other! Therefore a finite result in quantitative terms is infinite from a qualitative perspective. Likewise what appears infinite in quantitative terms will now appear finite from a qualitative perspective. Put another way, if the Zeta Function diverges in standard Type 1 terms it necessarily converges from the alternative Type 2 perspective.

However the one obvious exception here is where s = 1. Because this series is already defined in terms of the default qualitative dimension used in Type 1 Mathematics, it therefore has no alternative meaning from a Type 2 perspective. However, for all other dimensional values of s, an alternative does necessarily exist.

This in turn poses great difficulties for the conventional (Type 1) mathematical approach which in formal terms is confined to solely (linear) rational modes of interpretation.

Therefore the notion of the infinite that applies in Type 1 Mathematics is but a reduced quantitative notion that in many respects is entirely inadequate.

Somehow the mistaken view is perpetuated - as for example with series - that if we keep increasing the number of terms that we eventually accumulate an "infinite" no. (of such terms). However this is strictly speaking nonsense! Once again the infinite is a qualitatively distinct notion from the finite and cannot be "reached" therefore in finite manner. So if we keep increasing the terms of - for example - the natural number series without limit as might be said, we always obtain a finite number of terms. At no stage does the actual number of terms become infinite!

There are direct indications - even in Type 1 Mathematics - that the infinite is indeed qualitatively different from the finite. For example if we add two numbers in finite terms (say 1 + 2) a quantitative transformation is involved. Likewise if we multiply two non unitary numbers (say 2 * 3) again a quantitative transformation takes place.

However if we add two infinite numbers (or indeed a finite to a infinite number) no quantitative transformation is involved. Again if we multiply two infinite numbers or (an infinite by a finite) again no transformation is involved. Now, Cantor did indeed explore the notion of different types of infinities (within the confines of Type 1 Mathematics) but essentially his work - though presented in a reduced quantitative manner - draws attention to the fact that there is inevitably a qualitative aspect also involved. So ultimately the conclusion for example that the set of transcendental numbers is "bigger" than than that of the rationals is a reduced quantitative way of expressing the fact that the transcendentals are qualitatively distinct from the rationals (combining both discrete and continuous notions) whereas the rational are based on merely discrete notions!

All of this is of vital importance with respect to values for the Riemann Zeta Function which once again is defined (in Type 1 terms) as the infinite series

1/(1^s) + 1/(2^s) + 1/(3^s) + 1/(4^s) + ...... where s is any complex number a + bi.

Now it must be stressed again that this use of the infinite - which defines Type 1 Mathematics - represents but a reduced quantitative notion (i.e. where the potential notion of the infinite is reduced in actual finite terms).

In this context this approach does seem to work in certain cases. As is well known Euler had earlier shown that the Function indeed converges for all real values of s > 1. and this result can be exended to all complex numbers where again the real part a > 1.

Riemann however managed to extend the domain of definition for the Function to all values of s (except 1).

However this creates the immediate problem that the results for a wide range of these values make no sense from the conventional linear perspective.

My main concern here is not to show precisely how Riemann managed to achieve this "magical" transformation. What I am more concerned to deal with are the key philosophical implications that are involved which ultimately requires that an entirely distinctive type of Mathematics (Type 2) is used.

Ultimately this dramatically changes the very nature of the Riemann Hypothesis from a hypothesis in Type 1 Mathematics to a new fundamental hypothesis that serves as the cornerstone of the reconciliation of Type 1 and Type 2 Mathematics (which is the basis for the more comprehensive Type 3 Mathematics).

In other words all mathematical symbols possess both quantitative and qualitative aspects corresponding to two distinct types of interpretation. So the fundamental requirement is that consistency can be maintained with respect to both requirements. And the Riemann Hypothesis provides the very condition necessary for such consistency.

This of course implies that prime numbers inherently combine such quantitative and qualitative aspects in a very special manner!

Now we are looking here at the critical region of the Riemann Zeta Hypothesis for all values of s between 0 and 1.

Now the Function is undefined for s = 1 where the harmonic series results which in conventional terms sums to infinity.

The question then arises as to why the domain of definition cannot be stretched so as to include s = 1 (when in fact it can include every other value).

Now the qualitative mathematical reason (Type 2) is very illuminating in this regard.

Type 1 Mathematics is qualitatively based on the linear rational approach which is - literally - 1-dimensional in nature. Thus when for example we square the number 2 a qualitative as well as quantitative transformation takes place. So strictly the answer is now 4 square (i.e. 2-dimensional) units. However in Type 1 terms we simply ignore this qualitative dimensional change and express the result in reduced - merely - quantitative terms. So in Type 1 calculations all dimensional charges in the units involved are reduced to 1. Thus 2 * 2 = 4 (i.e. 4 ^ 1).

Now once we accept the true qualitative nature of a dimension, then numerical calculations (involving transformational changes) can be given an alternative result based on qualitative rather than quantitative considerations.

In the context of the Riemann Zeta Function therefore we give the Function - when it diverges from a Type 1 perspective - a finite Type 2 interpretation (where the result converges).

Remember Type 1 and Type 2 are finite and infinite with respect to each other! Therefore a finite result in quantitative terms is infinite from a qualitative perspective. Likewise what appears infinite in quantitative terms will now appear finite from a qualitative perspective. Put another way, if the Zeta Function diverges in standard Type 1 terms it necessarily converges from the alternative Type 2 perspective.

However the one obvious exception here is where s = 1. Because this series is already defined in terms of the default qualitative dimension used in Type 1 Mathematics, it therefore has no alternative meaning from a Type 2 perspective. However, for all other dimensional values of s, an alternative does necessarily exist.

## Thursday, August 18, 2011

### The Critical Region (2)

We can perhaps illustrate more the nature of qualitative - as opposed to quantitative - type interpretation of numerical values with respect to the Riemann Zeta function for the case where s = 1.

This results again in the well known harmonic series,

1 + 1/2 + 1/3 + 1/4 + ..... which in standard interpretation diverges to infinity.

Now by subtracting the even terms (*2) from the original series we come up with the corresponding Eta series,

1 - 1/2 + 1/3 - 1/4 + ..... which does indeed converge in conventional terms giving the well known result i.e. Ln 2.

However on closer expression what we have subtracted to derive the Eta result seemingly results in the original harmonic series!

So when we multiply the even terms i.e. 1/2 + 1/4 + 1/6 + 1/8 +... etc. by 2 we thereby obtain 1 + 1/2 + 1/3 + 1/4 +... which is the original harmonic series.

Therefore the Eta series which sums to Ln 2 - on this basis - has been derived by subtracting the harmonic series from itself.

So on this logic Ln 2 = 0.

Well what does this actually mean?

Once again, we have obtained this seemingly nonsensical result through mixing 1-dimensional and 2-dimensional notions of terms. When all the terms are of the same sign, linear 1-dimensional notions apply. However when alternating positive and negative terms are involved 2-dimensional notions (in the balancing of successive positive and negative terms) apply.

So on closer inspection the quantitative result of ln 2, corresponding to the Eta series, requires that we follow one set way of ordering terms. For example if we reordered the terms consistently adding two successive even terms before subtracting

i.e. (1/2 + 1/4) - 1/3 + (1/6 + 1/8) - 1/5 +..... we would get a different result!

So when we initially subtracted the even terms (* 2) from the original harmonic series, we were using a linear (1-dimensional) logic. However in interpreting the resulting Eta Series we are using a strict 2-dimensional logic where successive terms are taken as complementary pairings.

Now the very essence of non-dimensional understanding is that it entails the direct confusion of linear and circular notions. In psychological terms we would say that such understanding remains both undifferentiated and non integrated representing mere potential for the subsequent unfolding in experience of both aspects i.e. (linear) differentiation and (circular) integration. So with successful development the first main requirement is the specialised development of linear (1-dimensional) understanding which dominates conventional scientific and mathematical understanding. Then where authentic spiritual contemplative development unfolds higher dimesnions of understanding can thereby take place.

Geometrically we could envisage the 0th dimension as the point at the centre of a circle (which equally is the centre of its line diameter).

So when we say that ln 2 = 0, we are referring to its qualitative dimensional nature (rather than its quantitative value). In other words 0 in dimensional terms relates to the direct embedding of linear (single term) interpretation that is unambiguously of one sign, with complementary (two term) interpretation where values are taken as a pair where they alternate with successive positive and negative values.

This results again in the well known harmonic series,

1 + 1/2 + 1/3 + 1/4 + ..... which in standard interpretation diverges to infinity.

Now by subtracting the even terms (*2) from the original series we come up with the corresponding Eta series,

1 - 1/2 + 1/3 - 1/4 + ..... which does indeed converge in conventional terms giving the well known result i.e. Ln 2.

However on closer expression what we have subtracted to derive the Eta result seemingly results in the original harmonic series!

So when we multiply the even terms i.e. 1/2 + 1/4 + 1/6 + 1/8 +... etc. by 2 we thereby obtain 1 + 1/2 + 1/3 + 1/4 +... which is the original harmonic series.

Therefore the Eta series which sums to Ln 2 - on this basis - has been derived by subtracting the harmonic series from itself.

So on this logic Ln 2 = 0.

Well what does this actually mean?

Once again, we have obtained this seemingly nonsensical result through mixing 1-dimensional and 2-dimensional notions of terms. When all the terms are of the same sign, linear 1-dimensional notions apply. However when alternating positive and negative terms are involved 2-dimensional notions (in the balancing of successive positive and negative terms) apply.

So on closer inspection the quantitative result of ln 2, corresponding to the Eta series, requires that we follow one set way of ordering terms. For example if we reordered the terms consistently adding two successive even terms before subtracting

i.e. (1/2 + 1/4) - 1/3 + (1/6 + 1/8) - 1/5 +..... we would get a different result!

So when we initially subtracted the even terms (* 2) from the original harmonic series, we were using a linear (1-dimensional) logic. However in interpreting the resulting Eta Series we are using a strict 2-dimensional logic where successive terms are taken as complementary pairings.

Now the very essence of non-dimensional understanding is that it entails the direct confusion of linear and circular notions. In psychological terms we would say that such understanding remains both undifferentiated and non integrated representing mere potential for the subsequent unfolding in experience of both aspects i.e. (linear) differentiation and (circular) integration. So with successful development the first main requirement is the specialised development of linear (1-dimensional) understanding which dominates conventional scientific and mathematical understanding. Then where authentic spiritual contemplative development unfolds higher dimesnions of understanding can thereby take place.

Geometrically we could envisage the 0th dimension as the point at the centre of a circle (which equally is the centre of its line diameter).

So when we say that ln 2 = 0, we are referring to its qualitative dimensional nature (rather than its quantitative value). In other words 0 in dimensional terms relates to the direct embedding of linear (single term) interpretation that is unambiguously of one sign, with complementary (two term) interpretation where values are taken as a pair where they alternate with successive positive and negative values.

## Wednesday, August 17, 2011

### The Critical Region (1)

As we have seen the Euler Zeta Function

1/(1^s) + 1/(2^s) + 1/(3^s) + 1/(4^s) +..... is defined for all values of s > 1.

However in standard linear terms we cannot give numerical meaning to the function for other values of s.

For example when s = 0, we generate the series 1 + 1 + 1 + 1 +.... which - again in conventional terms - diverges to infinity.

However Riemann showed that in his treatment of the function where s can take on any complex value that it is possible to extend the domain of definition of the Function for all values of x (except 1).

The critical region involves values of s from 0 to 1. It has long been known that all the non-trivial zeros must lie in this region (with the Riemann Hypothesis suggesting that they all lie on the line (for real part of s = .5)

Now this is where qualitative - as opposed to mere quantitative - interpretation of numerical becomes extremely important.

Once again it is not possible to give a finite meaning to the sum of a series such as 1 + 1 + 1 + 1 +.... which clearly gets larger and larger and in conventional terminology diverges to infinity.

However in the Riemann Zeta Function the sum of this series (and a vast range of other divergent series) are indeed given a definite finite value. So this raises the very obvious question as to what such a result can mean. And the fascinating answer is that it points in all cases to an additional holistic qualitative interpretation in accordance with Type 2 Mathematics.

The ultimate implication is that we cannot properly understand the very meaning of the Riemann Hypothesis in the absence of Type 2 mathematical understanding. And once we do establish the true meaning of the Hypothesis it becomes readily apparent that it can neither be proved nor disproved in standard mathematical terms (i.e. in accordance with Type 1 interpretation).

It may be helpful at this stage to raise an area in relation to Fibonacci type number sequences that initially provided for me many of the insights regarding qualitative interpretation of numerical values that are so useful with respect to the Riemann Zeta function.

Yesterday, in my related blog on "The Spectrum of Mathematics" I briefly mentioned these. So I will myself here:

For example the Fibonacci Sequence can be obtained with reference to the simple quadratic equation x^2 - x - 1 = 0.

What we do here is to start with 0 and 1 and then combine the second term (* 1) with with the first term (*1) to get 1. Now these two values are obtained as the negative of the coefficients of the last 2 terms in the quadratic expression. So the last 2 terms in the sequence are now 1 and 1. So again combining the second of these (*1) with the first (*1) we now obtain the next term in the sequence i.e. 2 So the final 2 terms are now 1 and 2 and we continue on in the same manner to obtain further terms.

Now a fascinating aspect of such sequences is that we can then approximate the positive value for x in the original equation (i.e. phi) through the ratio of the last 2 terms in the sequence (taking the larger over the smaller).

The equation x^2 - 1 = 0 gives the correspondent to the pure 2-dimensional case where the values for x = + 1 and - 1.

This corresponds to the general quadratic equation x^2 + bx + c = 0 where b = 0 and c = - 1.

So in starting with 0 and 1 we keep adding zero times the second term to 1 times the first to get 0 as the next term. And it continues in this fashion so that we get 0, 1, 0, 1, 0, 1,....

Now what is interesting here is that we cannot approximate the (positive) value of x directly through getting the ratio of successive terms which will give us either 0/1 or alternatively 1/0.

However we can obtain the value directly through concentrating on the ratios of terms (occuring as each second term in sequence). In this we get either 1/1 or 0/0. The first would give us the conventional rational quantitative interpretation using linear (1-dimensional) logic. However the second actually corresponds to the qualitative holistic interpretation according to circular (2-dimensional) logic. This can be expressed as the complementarity of opposite poles so that 0, which numerically is given here could equally be represented as 1 - 1 where both aspects must be taken as a pairing.

So to sum up:

Thee initial equation x^2 - 1 = 0 i.e. x^2 = 1, is of a pure 2-dimensional nature. Therefore we can only obtain meaningful quantitative solutions by taking the ratio - not of successive terms as in linear terms - but rather the ratio of every second term (as befits 2-dimensional interpretation).

Two results now arise. The first, 1/1 gives us the (positive) quantitative result of the equation (i.e. the square root of 1).

The second 0/0 gives the qualitative basis of this result based on circular 2-dimensional understanding (that entails the complementarity of opposite poles).

When we attempt to obtain the root in linear terms through the ratio of opposite terms we get either 1/0 or 0/1. What both of these indicate is a relationship between two different interpretations, 1/0 (as between 1-dimensional and 2-dimensional and in reverse fashion 0/1 (as between 2-dimensional and 1-dimensional).

Both of these result from the attempt to split up what is inherently of a 2-dimensional nature (in qualitative terms) in a manner amenable to 1-dimensional linear understanding (which is not appropriate in this context).

So we cannot interpret the behaviour of such a sequence without reference to its qualitative dimensional characteristics. Because of the merely reduced quantitative interpretation of symbols employed in Type 1 Mathematics, these qualitative aspects are never properly investigated.

So when s = 0 the Riemann Zeta Function results in the sum of terms,

1 + 1 + 1 + 1 +...... which diverges in linear (1-dimensional) quantitative terms.

However it is possible to provide a finite value for this series. In my piece on "Holistic Values" on "The Spectrum of Mathematics" blog I explain how this is done:

The Zeta Function is defined as

1 + 1/(2^s) + 1/(3^s) + 1/(4^s) +......

If we consider just the even values terms and subtract double of each of these terms from the original series we obtain the well known Eta Function which is defined in terms of alternating terms

1 - 1/(2^s) + 1/(3^s) - 1/(4^s) +......

Now through multiplying each of the even valued terms by 2^s we can derive the original terms in the Zeta Function.

This therefore enables us to establish a simple relationship as between the two Functions so that the Zeta Function = Eta Function divided by {1 -1/[2^(s - 1)]}

When s = 0 the Eta function results in the alternating sequence of terms

1 - 1 + 1 - 1 + 1 - ...

Now the sum of this sequence does not properly converge in conventional terms.

When we add up an even number of terms the value = 0; however when we add an odd number the value = 1. Thus by taking the average of these two results we can come up with a single answer = 1/2.

And then from this Eta value the corresponding Zeta value can be easily calculated = -1/2.

There is in fact another way of doing this: If we attempt to obtain the value of 1/(1 - x) we generate the infinite series:

1 + x + x^2 + x^3 + ....

Clearly this latter series only converges for values of x between - 1 and + 1.

But the former expression can be defined for all values of x (except where x = 1).

Now when x = - 1, 1/(1 -x) = 1/2;

If we attempt to express the equivalent series in terms of x = - 1, we obtain

1 - 1 + 1 - 1 +.... which gives us the result that we have already calculated through another means.

However we have already used this Eta value to calculate the corresponding value for the Zeta series where s = 0

i.e. 1 + 1 + 1 + 1 +..... = - 1/2

Now this series can equally be generated by letting x = 1 in our series

1 + x + x^2 + x^3 + ....

So 1 + x + x^2 + x^3 + .... = - 1/2;

However 1/(1 - x) = 1 + x + x^2 + x^3 + .... = - 1/2 (when x = 1).

However 1/(1 - x) = 1/0 (when x = 1.

What this establishes therefore is that the famed result for the Riemann Zeta Function where s = 0 involves the relationship as between linear and circular type interpretation.

In other words the result, - 1/2 is actually the attempt to express the qualitative nature of circular (2-dimensional) understanding in a linear (1-dimensional) manner. Once again 2-dimensional interpretation involves two poles as an inherent pairing that are positive and negative with respect to each other. So if we take the negative pole and attempt to express it as a fraction of the pairing we get - 1/2.

1/(1^s) + 1/(2^s) + 1/(3^s) + 1/(4^s) +..... is defined for all values of s > 1.

However in standard linear terms we cannot give numerical meaning to the function for other values of s.

For example when s = 0, we generate the series 1 + 1 + 1 + 1 +.... which - again in conventional terms - diverges to infinity.

However Riemann showed that in his treatment of the function where s can take on any complex value that it is possible to extend the domain of definition of the Function for all values of x (except 1).

The critical region involves values of s from 0 to 1. It has long been known that all the non-trivial zeros must lie in this region (with the Riemann Hypothesis suggesting that they all lie on the line (for real part of s = .5)

Now this is where qualitative - as opposed to mere quantitative - interpretation of numerical becomes extremely important.

Once again it is not possible to give a finite meaning to the sum of a series such as 1 + 1 + 1 + 1 +.... which clearly gets larger and larger and in conventional terminology diverges to infinity.

However in the Riemann Zeta Function the sum of this series (and a vast range of other divergent series) are indeed given a definite finite value. So this raises the very obvious question as to what such a result can mean. And the fascinating answer is that it points in all cases to an additional holistic qualitative interpretation in accordance with Type 2 Mathematics.

The ultimate implication is that we cannot properly understand the very meaning of the Riemann Hypothesis in the absence of Type 2 mathematical understanding. And once we do establish the true meaning of the Hypothesis it becomes readily apparent that it can neither be proved nor disproved in standard mathematical terms (i.e. in accordance with Type 1 interpretation).

It may be helpful at this stage to raise an area in relation to Fibonacci type number sequences that initially provided for me many of the insights regarding qualitative interpretation of numerical values that are so useful with respect to the Riemann Zeta function.

Yesterday, in my related blog on "The Spectrum of Mathematics" I briefly mentioned these. So I will myself here:

For example the Fibonacci Sequence can be obtained with reference to the simple quadratic equation x^2 - x - 1 = 0.

What we do here is to start with 0 and 1 and then combine the second term (* 1) with with the first term (*1) to get 1. Now these two values are obtained as the negative of the coefficients of the last 2 terms in the quadratic expression. So the last 2 terms in the sequence are now 1 and 1. So again combining the second of these (*1) with the first (*1) we now obtain the next term in the sequence i.e. 2 So the final 2 terms are now 1 and 2 and we continue on in the same manner to obtain further terms.

Now a fascinating aspect of such sequences is that we can then approximate the positive value for x in the original equation (i.e. phi) through the ratio of the last 2 terms in the sequence (taking the larger over the smaller).

The equation x^2 - 1 = 0 gives the correspondent to the pure 2-dimensional case where the values for x = + 1 and - 1.

This corresponds to the general quadratic equation x^2 + bx + c = 0 where b = 0 and c = - 1.

So in starting with 0 and 1 we keep adding zero times the second term to 1 times the first to get 0 as the next term. And it continues in this fashion so that we get 0, 1, 0, 1, 0, 1,....

Now what is interesting here is that we cannot approximate the (positive) value of x directly through getting the ratio of successive terms which will give us either 0/1 or alternatively 1/0.

However we can obtain the value directly through concentrating on the ratios of terms (occuring as each second term in sequence). In this we get either 1/1 or 0/0. The first would give us the conventional rational quantitative interpretation using linear (1-dimensional) logic. However the second actually corresponds to the qualitative holistic interpretation according to circular (2-dimensional) logic. This can be expressed as the complementarity of opposite poles so that 0, which numerically is given here could equally be represented as 1 - 1 where both aspects must be taken as a pairing.

So to sum up:

Thee initial equation x^2 - 1 = 0 i.e. x^2 = 1, is of a pure 2-dimensional nature. Therefore we can only obtain meaningful quantitative solutions by taking the ratio - not of successive terms as in linear terms - but rather the ratio of every second term (as befits 2-dimensional interpretation).

Two results now arise. The first, 1/1 gives us the (positive) quantitative result of the equation (i.e. the square root of 1).

The second 0/0 gives the qualitative basis of this result based on circular 2-dimensional understanding (that entails the complementarity of opposite poles).

When we attempt to obtain the root in linear terms through the ratio of opposite terms we get either 1/0 or 0/1. What both of these indicate is a relationship between two different interpretations, 1/0 (as between 1-dimensional and 2-dimensional and in reverse fashion 0/1 (as between 2-dimensional and 1-dimensional).

Both of these result from the attempt to split up what is inherently of a 2-dimensional nature (in qualitative terms) in a manner amenable to 1-dimensional linear understanding (which is not appropriate in this context).

So we cannot interpret the behaviour of such a sequence without reference to its qualitative dimensional characteristics. Because of the merely reduced quantitative interpretation of symbols employed in Type 1 Mathematics, these qualitative aspects are never properly investigated.

So when s = 0 the Riemann Zeta Function results in the sum of terms,

1 + 1 + 1 + 1 +...... which diverges in linear (1-dimensional) quantitative terms.

However it is possible to provide a finite value for this series. In my piece on "Holistic Values" on "The Spectrum of Mathematics" blog I explain how this is done:

The Zeta Function is defined as

1 + 1/(2^s) + 1/(3^s) + 1/(4^s) +......

If we consider just the even values terms and subtract double of each of these terms from the original series we obtain the well known Eta Function which is defined in terms of alternating terms

1 - 1/(2^s) + 1/(3^s) - 1/(4^s) +......

Now through multiplying each of the even valued terms by 2^s we can derive the original terms in the Zeta Function.

This therefore enables us to establish a simple relationship as between the two Functions so that the Zeta Function = Eta Function divided by {1 -1/[2^(s - 1)]}

When s = 0 the Eta function results in the alternating sequence of terms

1 - 1 + 1 - 1 + 1 - ...

Now the sum of this sequence does not properly converge in conventional terms.

When we add up an even number of terms the value = 0; however when we add an odd number the value = 1. Thus by taking the average of these two results we can come up with a single answer = 1/2.

And then from this Eta value the corresponding Zeta value can be easily calculated = -1/2.

There is in fact another way of doing this: If we attempt to obtain the value of 1/(1 - x) we generate the infinite series:

1 + x + x^2 + x^3 + ....

Clearly this latter series only converges for values of x between - 1 and + 1.

But the former expression can be defined for all values of x (except where x = 1).

Now when x = - 1, 1/(1 -x) = 1/2;

If we attempt to express the equivalent series in terms of x = - 1, we obtain

1 - 1 + 1 - 1 +.... which gives us the result that we have already calculated through another means.

However we have already used this Eta value to calculate the corresponding value for the Zeta series where s = 0

i.e. 1 + 1 + 1 + 1 +..... = - 1/2

Now this series can equally be generated by letting x = 1 in our series

1 + x + x^2 + x^3 + ....

So 1 + x + x^2 + x^3 + .... = - 1/2;

However 1/(1 - x) = 1 + x + x^2 + x^3 + .... = - 1/2 (when x = 1).

However 1/(1 - x) = 1/0 (when x = 1.

What this establishes therefore is that the famed result for the Riemann Zeta Function where s = 0 involves the relationship as between linear and circular type interpretation.

In other words the result, - 1/2 is actually the attempt to express the qualitative nature of circular (2-dimensional) understanding in a linear (1-dimensional) manner. Once again 2-dimensional interpretation involves two poles as an inherent pairing that are positive and negative with respect to each other. So if we take the negative pole and attempt to express it as a fraction of the pairing we get - 1/2.

## Thursday, July 28, 2011

### A Deeper Code

Marcus du Sautoy is back on BBC with a three part series where he argues that mathematics is the secret code behind nature's secrets. Though Maths is apparently abstract, yet the behaviour of nature is wonderfully written in mathematical language.

Though du Sautoy confines himself to the conventional quantitative aspect of Mathematics, of even greater wonder to me at present is the realisation that nature (and indeed all life) is likewise written mathematically in a qualitative code which we do not yet even properly realise.

Indeed recently I have come to the view that the true nature of reality is indeed mathematical in both quantitative and qualitative terms.

One implication of this is that nature's ultimate physical secrets cannot be discovered merely through phenomenal investigation of reality, for these very phenomena already embody dynamic mathematical configurations with respect to both its quantitative and qualitative aspects.

One of the reasons why the qualitative aspect of Mathematics is missed is because it does not initially conform to rational investigation of the standard logical kind. Rather it conforms to an appreciation of interdependence (rather than independence) which is then indirectly conveyed through paradoxical interpretation of a circular kind.

So every mathematical symbol can be given a valid interpretation according to two logical systems that are linear and circular with respect to each other. Whereas the former corresponds with quantitative appreciation, the latter relates to qualitative appreciation (indirectly conveyed through mathematical symbols).

Put another way, whereas we now realise that numbers in quantitative terms can be both real and imaginary, the corresponding corollary in qualitative terms is that mathematical logical interpretation can likewise be both real and imaginary. So once again real in this context corresponds with linear type rational appreciation, whereas imaginary relates to circular or paradoxical type rational awareness.

One remarkable implication arising from this perspective is that qualitative type appreciation is directly of an affective kind, that indirectly can then be given a valid mathematical interpretation.

This would imply that ultimately a comprehensive mathematical appreciation implies substantial balance being maintained as between artistic (affective) and scientific (cognitive) type awareness, though the language of mathematics will be be rightly couched in cognitive terms.

In one valid sense, qualitative mathematical appreciation relates to the subtle appreciation of the the true dimensional nature of reality that - apart from some developments in string theory - is not currently recognised.

This would again imply that affective experience provides the direct means of appreciating such dimensions, which embody all phenomena with their unique qualitative characteristics.

Finally it struck me forcibly today that all this gives a new meaning to my interpretation of the Riemann Hypothesis as a statement regarding the ultimate reconciliation of both the quantitative and qualitative aspects of mathematical experience.

So before phenomena can even come into being, a crucial condition regarding the nature of prime numbers must be fulfilled guaranteeing their consistency according to two sets of logic that must necessarily diverge somewhat with respect to the phenomenal world. Therefore the subsequent manifestation of phenomena in nature is already based on deep mathematical principles that precede and ultimately also transcend their very existence in actual form. Therefore the Riemann Hypothesis can have no proof in conventional terms, for the very truth to which it relates already precedes any partial logical investigation either in standard (linear) or unrecognised (circular) terms.

Thus when we probe nature to its very limits, we must eventually leave the world of the merely physical to embrace what is truly mathematical. Indeed the very rigidity that defines phenomenal objects already implies a degree of reduction in the - ultimately ineffable - mathematical principles governing their nature. And this applies most readily to the qualitative aspect (the mathematical nature of which is not yet even recognised).

So the true nature of Mathematics - with respect to its quantitative and (unrecognised) qualitative aspects - lies at the very bridge that serves to connect the phenomenal world of physical form with the ineffable world of spiritual emptiness.

Though du Sautoy confines himself to the conventional quantitative aspect of Mathematics, of even greater wonder to me at present is the realisation that nature (and indeed all life) is likewise written mathematically in a qualitative code which we do not yet even properly realise.

Indeed recently I have come to the view that the true nature of reality is indeed mathematical in both quantitative and qualitative terms.

One implication of this is that nature's ultimate physical secrets cannot be discovered merely through phenomenal investigation of reality, for these very phenomena already embody dynamic mathematical configurations with respect to both its quantitative and qualitative aspects.

One of the reasons why the qualitative aspect of Mathematics is missed is because it does not initially conform to rational investigation of the standard logical kind. Rather it conforms to an appreciation of interdependence (rather than independence) which is then indirectly conveyed through paradoxical interpretation of a circular kind.

So every mathematical symbol can be given a valid interpretation according to two logical systems that are linear and circular with respect to each other. Whereas the former corresponds with quantitative appreciation, the latter relates to qualitative appreciation (indirectly conveyed through mathematical symbols).

Put another way, whereas we now realise that numbers in quantitative terms can be both real and imaginary, the corresponding corollary in qualitative terms is that mathematical logical interpretation can likewise be both real and imaginary. So once again real in this context corresponds with linear type rational appreciation, whereas imaginary relates to circular or paradoxical type rational awareness.

One remarkable implication arising from this perspective is that qualitative type appreciation is directly of an affective kind, that indirectly can then be given a valid mathematical interpretation.

This would imply that ultimately a comprehensive mathematical appreciation implies substantial balance being maintained as between artistic (affective) and scientific (cognitive) type awareness, though the language of mathematics will be be rightly couched in cognitive terms.

In one valid sense, qualitative mathematical appreciation relates to the subtle appreciation of the the true dimensional nature of reality that - apart from some developments in string theory - is not currently recognised.

This would again imply that affective experience provides the direct means of appreciating such dimensions, which embody all phenomena with their unique qualitative characteristics.

Finally it struck me forcibly today that all this gives a new meaning to my interpretation of the Riemann Hypothesis as a statement regarding the ultimate reconciliation of both the quantitative and qualitative aspects of mathematical experience.

So before phenomena can even come into being, a crucial condition regarding the nature of prime numbers must be fulfilled guaranteeing their consistency according to two sets of logic that must necessarily diverge somewhat with respect to the phenomenal world. Therefore the subsequent manifestation of phenomena in nature is already based on deep mathematical principles that precede and ultimately also transcend their very existence in actual form. Therefore the Riemann Hypothesis can have no proof in conventional terms, for the very truth to which it relates already precedes any partial logical investigation either in standard (linear) or unrecognised (circular) terms.

Thus when we probe nature to its very limits, we must eventually leave the world of the merely physical to embrace what is truly mathematical. Indeed the very rigidity that defines phenomenal objects already implies a degree of reduction in the - ultimately ineffable - mathematical principles governing their nature. And this applies most readily to the qualitative aspect (the mathematical nature of which is not yet even recognised).

So the true nature of Mathematics - with respect to its quantitative and (unrecognised) qualitative aspects - lies at the very bridge that serves to connect the phenomenal world of physical form with the ineffable world of spiritual emptiness.

## Thursday, July 21, 2011

### Odd Numbered Integers (9)

Unexpectedly this morning, while trying out an insight that struck me yesterday, I seemed to have detected a very interesting pattern that governs the denominators of values of the Riemann Zeta Function for negative odd integer values.

This pattern relates to divisibility of the denominator by the first two perfect numbers 6 and 28 and can be stated succinctly as follows.

(i) The denominator of such values is always divisible by 6.

(ii) in every 3rd case the denominator is divisible by both 6 and 28 (and only in such a case).

(iii) The denominator does not appear to be divisible by any other perfect numbers.

For example the 1st zeta result where s = - 1 is - 1/12 and the denominator is clearly divisible by 6.

The 2nd zeta result where s = - 3 is 1/120 and again the denominator is divisible by 6.

The 3rd zeta result where s = - 5 is - 1/252 and the denominator here is divisible by both 6 and 28.

And this trend continues. So the denominators (240 and 132) for both s = - 7 and s = - 9 are divisible by 6, whereas the denominator for s = - 11 (32760) which is the 3rd in the sequence, is divisible by both 6 and 28. Indeed in this case it is divisble by 6 * 28. However that is not generally the case!

Using zeta results compiled in Mc Gill University this trend can be verified for the first relevant 100 zeta values (i.e. up to s = -199).

It should be also stated that this numerical behavioural characteristic does not extend to denominators of the zeta function for positive even integer values of s!

However with respect to any qualitative interpretation of the meaning of such results it is perhaps too early to speculate.

Indeed even more dramatic numerical patterns exist with respect to the denominator of these zeta values (for negative odd integers).

As we have seen each successive value is divisible by 6 (i.e. 2 * 3).

Then every 2nd successive denominator value is divisible by 5; every 3rd succesive denominator nvalue is divisible by 7; every 5th successive denominator value is divisible by by 11; every 6th successive denominator value is divisible by 13; every 8th successive denominator value is divisible by 17; every 9th successive denominator value by 19; every 11th successive denominator value is divisible by 23; every 14th successive denominator value is divisible by 29 and so on.

In other words where the absolute value of s is prime, every {|s - 1|/2)th denominator value in the zeta sequence for negative odd integral values is thereby divisible by |s|.

Put another way if therefore every {|s - 1|/2)th denominator value is divisible by |s|, then |s| is prime.

For example if |- 31| is prime then every 15th value in the zeta sequence should be divisible by 31

Now the absolute value of the denominator is the first of these cases (15th value in sequence) is 85932 which is divisible 31.

Now for every further 15th value in sequence the absolute value of denominator will be divisible by 31.

For example the absolute value for the denominator of 30th value is 3407203800 which once again is divisible by 31.

Therefore we could conclude from this that 31 is a prime number!

Not alone does this pattern appear to hold unbiversally but equally for all absolute prime values of s > 3, the only time when the demominator is divisible by |s| is for the {|s - 1|/2)th denominator value in the sequence.

Therefore we could safely conclude for this value of |s| where = 31, that 31 is indeed a prime number.

And of course this would hold for all other values of |s| where the same principle applies!

Put finally yet another way if s = 2, 4 ,6, 8,....

then for zeta (1 - s), where the value of (s + 1) is prime, the absolute value of denominator will be divisible by all factors of s where with the addition of 1 are prime (and only these factors).

So one again for example when s = 36, zeta (1 - s) is zeta (- 35).

In this case s + 1 = 37 which is prime. Now all the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.

Now with addition of 1 in each case we get 2, 3, 4, 5, 7, 10, 13, 19 and 37.

However since 4 and 10 are not prime we can exclude these numbers.

Theerfore the denominator of zeta (-35) i.e. 69090840 is divisible by 2, 3, 5, 7, 13, 19 and 37 (and only these prime numbers).

And what is remarkable is that when the denominator is divided by the product of all these prime factors the result is s.

Therfore 69090840/(2 * 3 * 5 * 7 * 13 * 19 * 37) = 36.

Though this final result does not universally hold it does so in some cases i.e. when the denominator is divided by product of all prime factors the result is s.

This pattern relates to divisibility of the denominator by the first two perfect numbers 6 and 28 and can be stated succinctly as follows.

(i) The denominator of such values is always divisible by 6.

(ii) in every 3rd case the denominator is divisible by both 6 and 28 (and only in such a case).

(iii) The denominator does not appear to be divisible by any other perfect numbers.

For example the 1st zeta result where s = - 1 is - 1/12 and the denominator is clearly divisible by 6.

The 2nd zeta result where s = - 3 is 1/120 and again the denominator is divisible by 6.

The 3rd zeta result where s = - 5 is - 1/252 and the denominator here is divisible by both 6 and 28.

And this trend continues. So the denominators (240 and 132) for both s = - 7 and s = - 9 are divisible by 6, whereas the denominator for s = - 11 (32760) which is the 3rd in the sequence, is divisible by both 6 and 28. Indeed in this case it is divisble by 6 * 28. However that is not generally the case!

Using zeta results compiled in Mc Gill University this trend can be verified for the first relevant 100 zeta values (i.e. up to s = -199).

It should be also stated that this numerical behavioural characteristic does not extend to denominators of the zeta function for positive even integer values of s!

However with respect to any qualitative interpretation of the meaning of such results it is perhaps too early to speculate.

Indeed even more dramatic numerical patterns exist with respect to the denominator of these zeta values (for negative odd integers).

As we have seen each successive value is divisible by 6 (i.e. 2 * 3).

Then every 2nd successive denominator value is divisible by 5; every 3rd succesive denominator nvalue is divisible by 7; every 5th successive denominator value is divisible by by 11; every 6th successive denominator value is divisible by 13; every 8th successive denominator value is divisible by 17; every 9th successive denominator value by 19; every 11th successive denominator value is divisible by 23; every 14th successive denominator value is divisible by 29 and so on.

In other words where the absolute value of s is prime, every {|s - 1|/2)th denominator value in the zeta sequence for negative odd integral values is thereby divisible by |s|.

Put another way if therefore every {|s - 1|/2)th denominator value is divisible by |s|, then |s| is prime.

For example if |- 31| is prime then every 15th value in the zeta sequence should be divisible by 31

Now the absolute value of the denominator is the first of these cases (15th value in sequence) is 85932 which is divisible 31.

Now for every further 15th value in sequence the absolute value of denominator will be divisible by 31.

For example the absolute value for the denominator of 30th value is 3407203800 which once again is divisible by 31.

Therefore we could conclude from this that 31 is a prime number!

Not alone does this pattern appear to hold unbiversally but equally for all absolute prime values of s > 3, the only time when the demominator is divisible by |s| is for the {|s - 1|/2)th denominator value in the sequence.

Therefore we could safely conclude for this value of |s| where = 31, that 31 is indeed a prime number.

And of course this would hold for all other values of |s| where the same principle applies!

Put finally yet another way if s = 2, 4 ,6, 8,....

then for zeta (1 - s), where the value of (s + 1) is prime, the absolute value of denominator will be divisible by all factors of s where with the addition of 1 are prime (and only these factors).

So one again for example when s = 36, zeta (1 - s) is zeta (- 35).

In this case s + 1 = 37 which is prime. Now all the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.

Now with addition of 1 in each case we get 2, 3, 4, 5, 7, 10, 13, 19 and 37.

However since 4 and 10 are not prime we can exclude these numbers.

Theerfore the denominator of zeta (-35) i.e. 69090840 is divisible by 2, 3, 5, 7, 13, 19 and 37 (and only these prime numbers).

And what is remarkable is that when the denominator is divided by the product of all these prime factors the result is s.

Therfore 69090840/(2 * 3 * 5 * 7 * 13 * 19 * 37) = 36.

Though this final result does not universally hold it does so in some cases i.e. when the denominator is divided by product of all prime factors the result is s.

## Friday, June 3, 2011

### Odd Numbered Integers (8)

It might help to summarise the rationale of what has been involved in making these qualitative connections with the Riemann Zeta Function (for negative odd integer values).

Once again it is vital to appreciate that standard (unambiguous) quantitative type interpretation of numbers is associated with a default dimensional value of 1. So for example when numbers are raised to a power (other than 1), the attempt is made to obtain a reduced numerical result (in terms of 1).

Thus in this context 2^2 = 4 (i.e. 4^1). Therefore, though a qualitative change in the nature of the number takes place (through raising to the power of 2) the result is expressed in a reduced merely quantitative manner.

However the Riemann Zeta Function diverges for negative odd integer values of the dimensional power s.

So when s = - 1, we obtain in this context the series 1 + 2 + 3 +..... which clearly in standard terms sums to infinity.

However through the process of analytic continuation the Riemann Zeta Function can be given an alternative finite interpretation for all negative odd integers of s as a rational number.

Now the fascinating explanation for this alternative behaviour is that the finite numerical result now obtained relates to a circular rather than linear interpretation.

For simplicity, I have referred to this result as qualitative (rather than quantitative). However strictly speaking this is not properly correct.

So in a more complete fashion we have distinguished two distinct types of explanations for numerical results of the Riemann Zeta Function.

For values of s > 1, they can be given the standard linear (absolute) type interpretation. This results from the assumed absolute separation here of qualitative interpretation from the objective numerical result obtained.

However for values of s < 0, they can only be given an alternative circular (relative) type interpretation inherently in terms of the actual value for s utilised.

Here we operate according to the different assumption that both qualitative interpretation with resulting numerical values are always in dynamic interaction with each other. Thus the rational values that result from the Riemann Zeta Function thereby reflect the circular interrelationship of aspects of understanding that are always qualitative and quantitative with respect to each other. Furthermore this relationship tends towards complementarity (whereby both aspects are perfect mirrors of each other).

Now strictly this perfect complementarity properly only applies for negative even integer values of s. Thus the resulting value of 0 reflects that we can no longer separate any phenomenal result (in merely quantitative terms). So we have here pure intuitive realisation (in psychological terms) which perfectly mirrors appreciation of the empty holistic ground (underlying all phenomenal awareness of physical type reality)

However in a provisional limited sense for the negative odd integers - even though interdependence again characterises the relationship between qualitative interpretation and quantitative type results - a certain relative degree of separation can take place.

Thus from the psychological perspective the numerical result of the Function has a qualitative interpretation (as a certain mode of rationality).

Meanwhile from the physical perspective the same numerical result can be given a quantitative interpretation (i.e. representing a certain quantum relationship).

However as the numerical magnitude of dimensions increases a progressively higher level of pure energy characterises all relationships.

Thus from the psychological perspective, rationality becomes so refined that it ultimately cannot be separated from the spiritual intuitive energy with which it interacts.

Likewise from the corresponding physical perspective, quantitative phenomena become so unstable and short-lived that they can no longer be distinguished from the pure physical energy with which they are associated.

And this pure energy itself ultimately becomes inseparable from an empty holistic ground of nature (as the source of all physical reality) that is complemented by a pure empty spiritual experience (as the goal or realisation of all reality).

I will draw attention to another crucial distinction.

From the standard linear perspective, all numerical values are given an abstract identity (as essentially independent of all physical and psychological behaviour).

However from the corresponding holistic circular perspective, all numerical values necessarily express fundamental phenomenal relationships (with both physical and psychological aspects) that ultimately tend to full complementarity.

This suggests therefore that the results of the Riemann Zeta Function - as I have been explaining - have a direct relevance to both physical and psychological reality.

Likewise this is true of the famed non-trivial zeros which now have dual interpretations (both in the standard abstract sense and the new holistic sense as intimately related to both physical and psychological reality).

Whereas there is now some recognition that these zeros may indeed have a direct physical interpretation, there is no recognition as yet of their corresponding psychological relevance (as representing various refined states of interpretation).

And ultimately - as I have repeatedly stated - the very message of the Riemann Hypothesis is that physical and psychological aspects (through both quantitative and qualitative type interpretation) are ultimately inseparable.

Once again it is vital to appreciate that standard (unambiguous) quantitative type interpretation of numbers is associated with a default dimensional value of 1. So for example when numbers are raised to a power (other than 1), the attempt is made to obtain a reduced numerical result (in terms of 1).

Thus in this context 2^2 = 4 (i.e. 4^1). Therefore, though a qualitative change in the nature of the number takes place (through raising to the power of 2) the result is expressed in a reduced merely quantitative manner.

However the Riemann Zeta Function diverges for negative odd integer values of the dimensional power s.

So when s = - 1, we obtain in this context the series 1 + 2 + 3 +..... which clearly in standard terms sums to infinity.

However through the process of analytic continuation the Riemann Zeta Function can be given an alternative finite interpretation for all negative odd integers of s as a rational number.

Now the fascinating explanation for this alternative behaviour is that the finite numerical result now obtained relates to a circular rather than linear interpretation.

For simplicity, I have referred to this result as qualitative (rather than quantitative). However strictly speaking this is not properly correct.

So in a more complete fashion we have distinguished two distinct types of explanations for numerical results of the Riemann Zeta Function.

For values of s > 1, they can be given the standard linear (absolute) type interpretation. This results from the assumed absolute separation here of qualitative interpretation from the objective numerical result obtained.

However for values of s < 0, they can only be given an alternative circular (relative) type interpretation inherently in terms of the actual value for s utilised.

Here we operate according to the different assumption that both qualitative interpretation with resulting numerical values are always in dynamic interaction with each other. Thus the rational values that result from the Riemann Zeta Function thereby reflect the circular interrelationship of aspects of understanding that are always qualitative and quantitative with respect to each other. Furthermore this relationship tends towards complementarity (whereby both aspects are perfect mirrors of each other).

Now strictly this perfect complementarity properly only applies for negative even integer values of s. Thus the resulting value of 0 reflects that we can no longer separate any phenomenal result (in merely quantitative terms). So we have here pure intuitive realisation (in psychological terms) which perfectly mirrors appreciation of the empty holistic ground (underlying all phenomenal awareness of physical type reality)

However in a provisional limited sense for the negative odd integers - even though interdependence again characterises the relationship between qualitative interpretation and quantitative type results - a certain relative degree of separation can take place.

Thus from the psychological perspective the numerical result of the Function has a qualitative interpretation (as a certain mode of rationality).

Meanwhile from the physical perspective the same numerical result can be given a quantitative interpretation (i.e. representing a certain quantum relationship).

However as the numerical magnitude of dimensions increases a progressively higher level of pure energy characterises all relationships.

Thus from the psychological perspective, rationality becomes so refined that it ultimately cannot be separated from the spiritual intuitive energy with which it interacts.

Likewise from the corresponding physical perspective, quantitative phenomena become so unstable and short-lived that they can no longer be distinguished from the pure physical energy with which they are associated.

And this pure energy itself ultimately becomes inseparable from an empty holistic ground of nature (as the source of all physical reality) that is complemented by a pure empty spiritual experience (as the goal or realisation of all reality).

I will draw attention to another crucial distinction.

From the standard linear perspective, all numerical values are given an abstract identity (as essentially independent of all physical and psychological behaviour).

However from the corresponding holistic circular perspective, all numerical values necessarily express fundamental phenomenal relationships (with both physical and psychological aspects) that ultimately tend to full complementarity.

This suggests therefore that the results of the Riemann Zeta Function - as I have been explaining - have a direct relevance to both physical and psychological reality.

Likewise this is true of the famed non-trivial zeros which now have dual interpretations (both in the standard abstract sense and the new holistic sense as intimately related to both physical and psychological reality).

Whereas there is now some recognition that these zeros may indeed have a direct physical interpretation, there is no recognition as yet of their corresponding psychological relevance (as representing various refined states of interpretation).

And ultimately - as I have repeatedly stated - the very message of the Riemann Hypothesis is that physical and psychological aspects (through both quantitative and qualitative type interpretation) are ultimately inseparable.

## Tuesday, May 31, 2011

### Odd Numbered Integers (7)

There are further fascinating aspects in qualitative terms associated with the rational fractions for negative odd integers of the Riemann Zeta Function.

Again - before any association with the Riemann Function - I had become aware of a certain important pattern associated with contemplative type development.

In my writings I distinguish carefully as between contemplative and radial develolopment.

Now the essence of the former type of development is that it represents an unfolding of intuitive type understanding. And I refer to this as Band 3 on the Spectrum of Development.

Just as the (differentiated) rational unfolds at the earlier Bands through a - literal - reduction in intuitive type type development, now in reverse this (integral) intuitive type awareness unfolds through a similar reduction in rational understanding.

So initially in development as the more contemplative type stages unfold, one learns to economise greatly on the use of associated rational faculties. So the task is to learn to give expression to this new intuitive understanding through temporarily allowing the rational faculties become largely dormant.

Now this qualitative notion of a reduction in rational type understanding is replicated by the magnituide of the rational fraction associated with the Riemann Zeta Function (for the negative odd integers). So it starts with - 1/12 for s = - 1 and then quickly drops (in absolute terms) so that for s = - 3 the result is 1/120.

So we could say therefore that by the 3rd dimension a great reduction (and corresponding) refinement has already taken place in the use of reason. This in turn is largely facilitated by a substantial degree of unconscious darkness in experience (so that one does not have the freedom to readily exercise the rational faculties).

For s = - 5, where the result is 1/252, the absolute value reaches its minimum. This is turn is associated with the darkest period of mystical development (in what St. John would refer to as the passive night of spirit). Indeed typically a crisis emerges here due to the considerable absence of associated intuitive light to support the rational structures. This in turn is usually due to an extreme transcendent focus to development at this time. So we have here the start of a rebalancing whereby immanence as well as transcendence is recognised in development. And associated with this growing immanence is a corresponding recovery in rational type development. So for s = - 7, the result is 1/240 (which is now slightly larger in absolute terms).

In my earlier writings I then associated the attainment of an extreme in pure contemplative type development with the 8th dimension.

Now gradually I came to realise that a further Band of development would now unfold whereby the specialisation of this intuitive type development could take place (just as the earlier specialisation of rational understanding took place with the second Band).

This in turn - as is emphasised in the mystical literature - is likewise associated with attaining proper balance as between emphasis on form and emptiness (i.e. immanence and transcendence).

This in fact requires the unfolding of a further 8 dimensions. So that radial development does not properly commence till the 16th dimension has been traversed.

Fascinatingly between s = - 15 and s = -17, the absolute value of the rational fraction for the Riemann Zeta function finally exceeds 1.

So this likewise signals in qualitative terms the full incorporation of form with emptiness.

I had been impressed in earlier years by the fact that the great exponents in my own Christian tradition of radial type development had so often been people of considerable action. They were in a sense superhuman with respect to the degree of their commitment to changing society, as they saw it, especially in religious terms. So this progressive ability to engage with phenomenal form is a key feature of ongoing development in the radial life (which in turn corresponds with the unfolding of dimensions of understanding > 16). Put another way because of a growing balanced immersion in spirit (as central) they maintained the progressing capacity to endure with equanimity the extremes of dealing with phenomenal type activities.

Seen in this light therefore the numerical results for the negative odd integers of the Riemann Zeta Function have a direct qualitative relevance in terms of the nature of rational development itself (at the higher dimensions). And again with the negative dimensions such rational development obtains its greatest refinement through the significant removal of associated intuitive support!

And once again because psychological and physical reality are complementary this means that the same rational values have a direct physical relevance for high energy physics.

What one could thereby postulate is at extremely high energy levels (that would still be way beyond present technological capacity to generate) progressively heavier phenomenal particles would be associated with energy. However the very detection of these particles would require that we can isolate them from their associated high-energy environment. Therefore because of the difficulty of doing this these heavier particles (of which there is no upward limit) would become increasing unstable.

So even here we can see the complementarity of physical and psychological. Whereas in physical terms these "heavy" particles are associated with instability, in corresponding psychological terms the ability to deal with increasing "heaviness", in the increasing demands of phenomenal activity, corresponds with an ever-growing stability in personality terms.

Again - before any association with the Riemann Function - I had become aware of a certain important pattern associated with contemplative type development.

In my writings I distinguish carefully as between contemplative and radial develolopment.

Now the essence of the former type of development is that it represents an unfolding of intuitive type understanding. And I refer to this as Band 3 on the Spectrum of Development.

Just as the (differentiated) rational unfolds at the earlier Bands through a - literal - reduction in intuitive type type development, now in reverse this (integral) intuitive type awareness unfolds through a similar reduction in rational understanding.

So initially in development as the more contemplative type stages unfold, one learns to economise greatly on the use of associated rational faculties. So the task is to learn to give expression to this new intuitive understanding through temporarily allowing the rational faculties become largely dormant.

Now this qualitative notion of a reduction in rational type understanding is replicated by the magnituide of the rational fraction associated with the Riemann Zeta Function (for the negative odd integers). So it starts with - 1/12 for s = - 1 and then quickly drops (in absolute terms) so that for s = - 3 the result is 1/120.

So we could say therefore that by the 3rd dimension a great reduction (and corresponding) refinement has already taken place in the use of reason. This in turn is largely facilitated by a substantial degree of unconscious darkness in experience (so that one does not have the freedom to readily exercise the rational faculties).

For s = - 5, where the result is 1/252, the absolute value reaches its minimum. This is turn is associated with the darkest period of mystical development (in what St. John would refer to as the passive night of spirit). Indeed typically a crisis emerges here due to the considerable absence of associated intuitive light to support the rational structures. This in turn is usually due to an extreme transcendent focus to development at this time. So we have here the start of a rebalancing whereby immanence as well as transcendence is recognised in development. And associated with this growing immanence is a corresponding recovery in rational type development. So for s = - 7, the result is 1/240 (which is now slightly larger in absolute terms).

In my earlier writings I then associated the attainment of an extreme in pure contemplative type development with the 8th dimension.

Now gradually I came to realise that a further Band of development would now unfold whereby the specialisation of this intuitive type development could take place (just as the earlier specialisation of rational understanding took place with the second Band).

This in turn - as is emphasised in the mystical literature - is likewise associated with attaining proper balance as between emphasis on form and emptiness (i.e. immanence and transcendence).

This in fact requires the unfolding of a further 8 dimensions. So that radial development does not properly commence till the 16th dimension has been traversed.

Fascinatingly between s = - 15 and s = -17, the absolute value of the rational fraction for the Riemann Zeta function finally exceeds 1.

So this likewise signals in qualitative terms the full incorporation of form with emptiness.

I had been impressed in earlier years by the fact that the great exponents in my own Christian tradition of radial type development had so often been people of considerable action. They were in a sense superhuman with respect to the degree of their commitment to changing society, as they saw it, especially in religious terms. So this progressive ability to engage with phenomenal form is a key feature of ongoing development in the radial life (which in turn corresponds with the unfolding of dimensions of understanding > 16). Put another way because of a growing balanced immersion in spirit (as central) they maintained the progressing capacity to endure with equanimity the extremes of dealing with phenomenal type activities.

Seen in this light therefore the numerical results for the negative odd integers of the Riemann Zeta Function have a direct qualitative relevance in terms of the nature of rational development itself (at the higher dimensions). And again with the negative dimensions such rational development obtains its greatest refinement through the significant removal of associated intuitive support!

And once again because psychological and physical reality are complementary this means that the same rational values have a direct physical relevance for high energy physics.

What one could thereby postulate is at extremely high energy levels (that would still be way beyond present technological capacity to generate) progressively heavier phenomenal particles would be associated with energy. However the very detection of these particles would require that we can isolate them from their associated high-energy environment. Therefore because of the difficulty of doing this these heavier particles (of which there is no upward limit) would become increasing unstable.

So even here we can see the complementarity of physical and psychological. Whereas in physical terms these "heavy" particles are associated with instability, in corresponding psychological terms the ability to deal with increasing "heaviness", in the increasing demands of phenomenal activity, corresponds with an ever-growing stability in personality terms.

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