Alt L2 Function

Just as the L1 function can be given two expressions i.e. sum over the positive integers and product over the primes respectively, equally the L2 function can be given two expressions.

In previous blog entries I have referred to this latter expression (which has its roots in the L1 function) as the Alt L2 function.

Now with respect to the Riemann zeta function, a special case occurs where s = 1, yielding the harmonic series,

1/1 + 1/2  + 1/3 + 1/4 + …

This can then be expressed in terms of combinations as

1/1C₀ + 1/2C₁ + 1/3C₂ + 1/4C₃  + …

Now, with respect to the denominator, we can define a new series where the nth term represents the sum of the n terms of the previous series.

So we thereby obtain

1/1 + 1/(1 + 2) + 1/(1 + 2 + 3) + 1/(1 + 2 + 3 + 4) + …

= 1 + 1/3 + 1/6 + 1/10 + …

Now this can equally be expressed in terms of combinations,

i.e. 1/2C₀ + 1/3C₁ + 1/4C₂ + 1/5C₃  + …

So the get this new series, we only need to increase the starting number of each combination (in relation to the previous harmonic series) by 1.

Now if we look at this later combination expression (which starts with 1/2C₀), we find that its value is equal to the corresponding L2 function i.e. 1 + x¹ + x²  + x³  +…, where x = 1/2.

So 1 + 1/2 + 1/2²  + 1/2³ + …   = 2, and  

1 + 1/3 + 1/6 + 1/10 + …  = 2.

And this is universally the case for any integer k (k > 1, where x = 1/k) .

So when x = 1/k, then the L2 function i.e.

1/(1 – 1/k) = 1 + 1/k + 1/k²  + 1/k³ + …  has a corresponding Alt L2 expression as,

1/kC₀ + 1/{(k + 1)C₁}  + 1/{(k + 2)C₂} + 1/{(k + 3)C₃} + …

Therefore when for example k = 4,

1/(1 – 1/4) = 1 + 1/4 + 1/4²  + 1/4³ + …  = 4/3.

So this represents the appropriate L2 function in this case where x = 1/2² = 1/4.

And this can be given an Alt L2 expression as,

1/4C₀ + 1/5C₁ + 1/6C₂ + 1/7C₃ + …

= 1 + 1/5 + 1/15 + 1/35 + …  = 4/3

In fact a fascinating product type expression can also be given for this Alt L2 function.

So when as in this case the starting value of the combination is 4, we let the numerator and denominator of the 1st term = 4 * 3 * 2 * 1.

Thus the 1^st term = (4 * 3 * 2 * 1)/(4 * 3 * 2 * 1) = 1.

Then to get the next term we simply add 1 to each value in the denominator while holding the numerator fixed.

So the 2^nd term = (4 * 3 * 2 * 1)/(5 * 4 * 3 * 2) = 1/5.

And for each subsequent term, we keep repeating by adding 1 to each previous value in the denominator (while holding the numerator fixed).

So the 3^rd term = (4 * 3 * 2 * 1)/(6 * 5 * 4 * 3)  = 1/15,

and the 4^th term = (4 * 3 * 2 * 1)/(7 * 6 * 5 * 4)  = 1/35 and so on.

And the existence here of 4 product terms (in relation to numerator and denominator) coincides with the fact that the value of k (with respect to the L2 function) = 4.

Complementary L1 and L2 Functions Continued

In the last entry, I indicated how each individual term, in the product over primes expression of the L function, can be represented as a complementary L function (where the base and dimensional aspects of number are inverted).

I thereby refer to the former collective function as the L1 function and the latter individual function (representing a single term) as the L2 function.

So the (collective) L1 function is comprised of an infinite series of (individual) L2 functions. And once again these operate in a dynamic complementary manner with respect to each other.

So again in general terms (s > 1), with respect to the Riemann zeta function, the L1 function is customarily represented (in its sum over the positive integers form) as

1^{– s} + 2^{– s} + 3^{– s} + 4^{– s}  + …,

with the corresponding L2 function represented as

1 + (1/p^s)¹  + (1/p^s)²  + (1/p^s)³  + (1/p^s)⁴  + …,

where p is defined over all the primes.  

However it is not only each term of the product over primes that can be represented as an L2 function, but likewise - with minor adjustment - each term of the sum over natural numbers expression.

The 1st term (in the sum over positive integers expression) as 1 stands alone, as it were, without a corresponding L2 expression.

However every other individual term can then be expressed in a general manner as an L2 function as follows.

[1/{1 – 1/(k^s + 1)}] – 1 where k = 1, 2, 3, 4, …

= [{1 + {1/(k^s + 1)}¹ + {1/(k^s + 1)}² + {1/(k^s + 1)}³ + {1/(k^s + 1)}⁴ + …] – 1 = 1/k^s.

For example for ζ(2), where p = 2, the 2^nd term is,

[{1 + {1/(2² + 1)}¹ + {1/(2² + 1)}² + {1/(2² + 1)}³ + {1/(2² + 1)}⁴ + …] – 1,

= [1 + 1/5 + 1/5²  + 1/5³ + 1/5⁴  + …] – 1,

= 5/4 – 1  = 1/2²  = 1/4.

So just as the 1^st term 1, (i.e. + 1) is omitted in terms of a L2 expression, every other term includes the addition of + 1 , in its denominator expression.   

In this context it is fascinating to observe the case where both L1 and L2 functions, with respect to the Riemann zeta function in the analytically continued complex plane, diverge.

For the L1, with s = 1 (representing the dimensional aspect of number as exponent) we have the harmonic series

1/1¹ + 1/2¹ + 1/3¹ + 1/4¹ + … = ∞.

Then with the L2 when s = 1 (representing the corresponding base aspect of number) we have,

1/(1 – 1) = 1 + 1¹ + 1² + 1³  + …  = ∞.

Thus from a dynamic interactive perspective, there is both an L1 and L2 explanation for the one pole in the Riemann zeta function i.e. s = 1 (where the functions are not defined).

So for the L1 function this pole occurs where s (representing the dimensional aspect of number) = 1.

For the L2 function, this pole occurs in complementary fashion where s (representing the corresponding base aspect of number) = 1.

Complementary L1 and L2 Functions

In the last blog entry, I have been concentrating attention on the complementary dynamic relative nature of both the infinite sum over the natural numbers and corresponding infinite sum over the primes expressions (which characterise all L-functions).

Therefore what is interpreted in an analytic (quantitative) manner with respect to the right-hand expression (i.e. product over primes) should be interpreted in a corresponding holistic (qualitative) manner with respect to the left-hand expression (i.e. sum over natural numbers).

Equally in reverse, what is interpreted in a holistic (qualitative) manner with respect to the right-hand expression should be interpreted in a corresponding analytic (quantitative) manner with respect to the left-hand expression.

Thus all individual terms (and collection of terms) with respect to both expressions can thereby be given twin analytic (quantitative) and holistic (qualitative) interpretations that interact with each other in a dynamic complementary manner.

This dynamic complementarity can also be illustrated in another closely related manner.

When we look at the Riemann zeta function which expresses a collective infinite sum over the natural numbers i.e.

1^{– s} + 2^{– s} + 3^{– s} + 4^{– s}  + …,

the base aspect of number varies over the positive integers, whereas the corresponding dimensional aspect i.e.  – s remains fixed.

When we now look at each individual term of the corresponding product over primes expression, it is given as 1/(1 – 1/p^s).

However this can be written as an infinite sum,

= 1 + (1/p^s)¹  + (1/p^s)²  + (1/p^s)³  + (1/p^s)⁴  + …

So now here in an inverse complementary manner the dimensional aspect of number varies over the positive integers, whereas the base aspect remains fixed (as 1/p^s).

Thus again when s = 2,

the collective sum over the natural numbers

= 1/1² + 1/2² + 1/3² + 1/4² + …  = 1 + 1/4 + 1/9 + 1/16 + …   = π²/6.

Then for the first individual terms of the product over primes expression (with p = 2),

1/(1 – 1/2²) = 1 + (1/2²)¹  + (1/2²)²  + (1/2²)³  + (1/2²)⁴  + … 

1 + 1/4 + 1/16 + 1/64 + …   = 4/3.

In an earlier blog entry, Dynamic Perspective on Addition and Multiplication, I explained how the dynamic complementary interaction as between addition and multiplication is necessary to explain how the meaning of number itself switches as between base and dimensional aspects (representing in turn the interaction of both quantitative and qualitative aspects).

And now here we see this beautifully exemplified with respect to both the collective (infinite) sum over natural numbers expression and each individual term of the (infinite) product over the primes.

It is because of this dynamic complementarity that I refer to the Riemann zeta function (collective sum over the natural numbers) as the Zeta 1 function and the alternative infinite function, for each individual term in the product over primes expression, as the Zeta 2 function respectively.

Thus the Zeta 1 and the Zeta 2 functions act in a dynamic complementary manner and this in turn is true for all L functions.

So if we refer to the (collective) infinite sum over natural numbers expression as the L1 function, then each individual term in the corresponding infinite product over primes expression operates in a complementary fashion (with respect to base and dimensional aspects) and can be referred to as the L2 function.

Relating Sum over Natural Numbers with Products over Primes

A necessary feature of all L-functions is that they can be expressed as infinite expressions, where a sum over the natural numbers is equal to a corresponding product over the primes for real exponents of s (s > 1).

For example with respect to the Riemann zeta function (which is the best known) an infinite sum over all the natural numbers is equal to an infinite product over all the primes.

     _∞

So ∑1/n^s    =      ∏1/(1 – 1/p^s)

    ^{n = 1                        p}

So when for example s = 2,

1/1² + 1/2² + 1/3² + 1/4² + …       = 4/3 * 8/7 * 25/24 * 49/48 * …   = π²/6

However the key significance of this relationship is missed from the conventional mathematical perspective, where the equations are interpreted in a merely analytic (quantitative) manner.

In fact, properly understood both expressions are dynamically interrelated in a relative manner with twin analytic (quantitative) and holistic (qualitative) aspects, which operate in a complementary fashion.

If for a moment, we concentrate on the first term of the product over primes expression i.e. 4/3, we can view this individual term in an analytic (quantitative) manner, which is directly related to the corresponding first prime i.e. 2.

However this prime relates to the corresponding sum over natural numbers expression in a complementary holistic (qualitative) manner.

So rather than being independent, 2 as a prime is related to all even composite natural numbers as a shared factor. So we move from the individual quantitative notion of 2 (as an independent number) to the collective qualitative notion of that same prime as a shared factor (and thereby interdependent with other factors) with respect to all even natural numbers.

So again when we view each individual prime (in the product over primes expression) in an analytic (quantitative) manner, then in relative terms we must view each corresponding prime (with respect to the sum over natural numbers expression) collectively in a holistic (qualitative) fashion. Here it is seen in each case as a shared factor of a potentially unlimited set of composite natural numbers.

However when we now view the product of all primes with respect to the right hand expression, the frame of reference shifts so that the collective multiplication of these primes in complementary manner now attains a holistic (qualitative) meaning.

Thus the same dynamics are at work here that led us to earlier see that the multiplication of 1 by 1 i.e. 1¹ * 1¹  = 1²_, entails, relative to addition, the holistic qualitative notion of 2. Thus the collective multiplication of the prime related terms changes interpretation from quantitative and thereby independent (in each individual case) to qualitative and thereby relatively interdependent (in the collective situation).

And again in complementary fashion, when we view the addition, in the left hand expression of all the independent natural number terms, this entails the analytic (quantitative) notion of number.

And as always in dynamic interactive terms, the terms of reference can switch so that we can equally view a prime factor such as 2 in independent terms, whereby it attains a direct quantitative significance. However, in this case, the corresponding term in the product over primes expression i.e. 4/3, now acquires a holistic meaning in a relative complementary manner (where the unit members of 2 are understood as interdependent). 

And finally the collective product of all prime terms in the right hand expression can be likewise given a quantitative meaning, with again in relative complementary fashion, the collective sum of the natural number terms (in the left hand expression) now acquiring by contrast a qualitative interpretation.

So the key point is that all number terms both individually and collectively in both expressions can be given dual analytic (quantitative) and holistic (qualitative) interpretations.

And the relationship between the two expressions is of a dynamic interactive nature in a complementary relative manner, where analytic is balanced by holistic and holistic balanced by analytic interpretation respectively.

However because in isolation all number terms can indeed be given an independent quantitative meaning, this leads in conventional mathematical interpretation to the reduction of holistic meaning (with respect to the various number terms) in an absolute quantitative manner.  

As we have seen in our example above ζ(2) = π²/6.

And on careful reflection this very value clearly reflects the dynamic relative nature of number with respect to the Riemann zeta function, where both analytic (linear) and holistic (circular) aspects of understanding interact in two-way fashion with each other.

Again in conventional mathematical terms, the crucially important infinite notion itself becomes reduced in a merely finite linear manner.

So for example with respect to the sum over natural numbers expression, when we add a finite number of terms the sum is a rational number.

Thus again with respect to ζ(2), 1/1² = 1;1/1² + 1/2²  = 5/4; 1/1² + 1/2²  + 1/3²  = 49/36 and so on.

Likewise when from the standard perspective we multiply a finite number of rational terms, we obtain another rational number.

Thus no matter how many finite rational terms are added (or multiplied) in an analytic (quantitative) manner, the sum (or product) is a rational number.

However the fact that the infinite sum is transcendental (and not rational) clearly suggest that a qualitatively distinct notion is required in moving from finite to infinite with respect to the zeta function.

This strongly suggests in turn that very relationship as between finite and infinite is of a relative - rather than absolute - nature.

Thus the reduced notion that somehow the infinite notion can be approached through the successive adding (or multiplying) of finite terms in a linear quantitative manner is without foundation.

So when one starts by defining finite notions in an analytic (linear) manner, then the infinite notion - by contrast - relates to corresponding holistic (circular) interpretation.

Now once again holistic appreciation is directly of an ineffable formless nature; however indirectly in can then be expressed in paradoxical (i.e. circular) rational manner.

When one reflect on the nature of π - which is the most important of the transcendental numbers - from the standard quantitative perspective, it reflects the relationship of its (full) circular circumference to its line diameter.   

Then when one reflects equally on the nature of π from the (unrecognised) holistic qualitative perspective it reflects the dynamic relationship of both pure circular (qualitative) and linear (quantitative) understanding.

Now in my interpretation of  ζ(2), I have attempted to show the nature of such transcendental interpretation from a holistic (qualitative) perspective.

So the holistic nature of transcendental understanding - directly mirrored in this case by the nature of π - is to perfectly balance linear (quantitative) with circular (qualitative) understanding.

Then in this manner the quantitative result of each expression is matched by an appropriate qualitative interpretation of the relationships involved.

It is well known that for all even integer values of ζ(s) where s > 1, that the quantitative results for both the sum over natural numbers and product over primes expressions = kπ^s, where k is a rational number.   

This reflects the fact that the even roots of 1 can always be arranged in a complementary manner, where for every complex root a corresponding negative root will exist.

The fact that such complementarity does not strictly hold for odd numbered roots provides the holistic explanation as to why π is not involved in quantitative results for odd integer values of
ζ(s).     

Relationship of Primes to Natural Numbers: internal and external

We saw yesterday how every prime can be given both an independent quantitative identity (as a “building block”) of the natural number system and a shared qualitative identity (as a unique constituent factor of composite natural numbers).

And once again this key distinction as between the analytic and holistic nature of primes is unrecognised in conventional mathematical terms with the latter holistic meaning (in every relevant context) reduced in a mere analytic fashion.

And this problem by its very nature cannot be remedied within the present accepted mathematical paradigm, which is not geared to deal properly with holistic meaning that is inherently of an unconscious nature (though indirectly capable of expression in a paradoxical rational manner). 

So again from the analytic perspective, a prime such as 2 is given an absolute unambiguous meaning in quantitative terms.

However when 2 is used as a factor of an even composite number, it then enjoys a relative shared meaning, that is holistic and qualitative in nature.

In other words, through the fact that 2 is thereby shared with other prime factors, it acquires a unique qualitative resonance (through this shared relationship).

So from this relative context, 2 resonates in a distinctive qualitative manner whenever it is used with any other - or combination of other - prime factors.

Thus in the simplest case, 2 for example attains a unique qualitative shared meaning when combined again with 2 (to derive the composite natural number 4).

However it then attains a distinctly unique shared meaning when combined with 3 (to derive the composite natural number 6). And of course this can be continued on indefinitely with 2 - and by extension - every prime number.

Therefore when one properly accepts this new dynamic interactive manner of understanding the primes, the very way one looks on their relationship with the natural numbers is fundamentally changed.

From the reduced analytic perspective, one starts with the primes as pre-given quantitative entities in an absolute manner.

Then one attempts to explain the derivation of the natural number system in one-way fashion as resulting from the unique relationship of prime factors (that are still misleadingly viewed in an absolute quantitative manner).

Thus the natural numbers are themselves then viewed as absolute entities in a merely quantitative manner.

However from the dynamic interactive perspective - where both analytic and qualitative aspects of number are explicitly recognised - it is all somewhat different.

Thus internally each prime is viewed in quantitative terms as composed of independent homogeneous units (that thereby lack qualitative distinction); however equally each prime is viewed in a qualitative manner as composed of uniquely distinct natural number ordinal members that are fully interdependent - and thereby interchangeable - with each other.

So in this sense each unit lacks quantitative distinction.

However from a dynamic perspective, these aspects (quantitative and qualitative) are viewed as complementary in a relative manner. 

From the quantitative perspective, a prime is seen as a “building block” of the natural numbers.

However from the corresponding qualitative perspective, a prime is seen as composed of a unique set of ordinal natural number members.

Thus because of this inherent complementarity, both the quantitative and qualitative aspects of the primes can only find their appropriate interpretation within a dynamic relative framework.

Then when we extend this thinking externally to the relationship as between all the primes and the natural number system, again there are two aspects which interact with each other in a dynamic relative manner. 

So from the quantitative perspective, we see the primes as the “building blocks” of the entire natural number system (with each composite natural number composed of a unique combination of prime factors).

However from the qualitative perspective it looks very different with the unique spacing as between each prime determined through the combined relationship of the primes with the natural numbers.

And when one reflects on the matter both of these aspects are necessarily interdependent.

Thus we cannot give an exact location to each prime in quantitative terms, without establishing the overall relationship of the primes to the natural numbers (in a quantitative manner).

Likewise we cannot establish an overall relationship of the primes to the natural numbers (in qualitative terms) without knowledge of the individual identity of each prime (in a quantitative manner).

Therefore the clear conclusion from this paradox - which only becomes properly apparent when viewed dynamically - is that we can neither pre-determine the individual quantitative identity of each prime nor the collective qualitative relationship of all the primes with the natural numbers.

Rather both of these features simultaneously arise in a synchronistic fashion (which is ultimately ineffable).

So the highest knowledge of the relationship between the primes and natural numbers is the clear realisation that their ultimate nature is indivisible with both mirroring each other in a perfect manner.       

Independent and Shared Nature of Primes

We saw how every number can be equally given a holistic (qualitative) as well as analytic (quantitative) meaning.

In direct terms, the latter holistic meaning is equivalent in psycho-spiritual terms to an intuitive energy state.

And as psychological and physical aspects of reality are dynamically complementary, this entails that number can equally be given a holistic meaning in terms of physical energy states.

Because of the close correspondence as between the Riemann zeros and certain data representing the excited energy states of atomic particles, this clearly suggests that the zeros relate directly to the holistic - rather than analytic - aspect of number.

However speaking still somewhat in general terms, a fascinating and crucially important point can be made regarding the holistic nature of number.

As we know modern science, with its strong quantitative bias is rooted in analytic interpretation.

However what is not all yet realized in our culture is that likewise all of the various arts are intrinsically rooted in the holistic interpretation of number.

In other words all the qualitative attributes universally manifest in created phenomena are encoded in the notion of number (when given its appropriate holistic interpretation).

So again every number, as it were, has its own unique holistic signature, which forms the fundamental basis for the qualitative aspects of all phenomena.

Therefore sometime in the future for example it will be readily acknowledged that that true aesthetic appreciation (in all its forms) is rooted in enhanced holistic mathematical appreciation.  

And sadly we still live in an age where the holistic aspect of mathematics is not even formally recognised!

It must be emphatically stress however that holistic mathematical appreciation cannot be considered as an optional add-on to present analytic understanding, as it requires a radically distinct mind-set, where increasing specialisation in a contemplative type vision of reality is required.

In the past, such specialisation in contemplative training was related to advanced spiritual practice (associated with the various religious movements East and West).

However rarely was sustained attention given - with the possible exception of the Pythagoreans - to the implication of advanced contemplative states for mathematical interpretation. So perhaps it is only now that the true need is emerging in our culture for a radically new mathematical approach.
  

Perhaps in making this point I could usefully relate my own experience where holistic understanding naturally emerged while attending University following on profound disillusionment with the conventional mathematical approach.

Then as the holistic aspect underwent considerable development, for many years I suffered a sharp decline in ability to follow the established abstract approach to mathematical problems.

And it is now only in recent years - following several decades of holistic training - that I have been gradually able to look at a fundamental problem such as the Riemann Hypothesis from a dynamic perspective (entailing both analytic and holistic aspects) where it now appears in an entirely different light.    

In some future golden age, I suggest that mathematics by its very nature will entail the balanced integration of both its analytic and holistic aspects. However we are still very far away from that day due to a completely blindness at present to the need for true holistic appreciation.

So to return to the primes, we have seen that each prime can be given both an analytic (quantitative) and holistic (qualitative) interpretation.

Again from the former perspective, each prime is unique it that it has no constituent factors (other than itself and 1). Thus it thereby serves as a quantitative “building block” of the natural number system.

However from the latter perspective, each prime is unique as it is composed in ordinal terms of a group of natural number members, which are themselves (apart from the last) unique.

Thus again for example 5 as a prime is unique from the former quantitative perspective in that it has no factors (other than 5 and 1).

Then from the latter qualitative perspective, 5 is unique in that its 1^st, 2^nd, 3^rd, and 4^th members are distinct. Indirectly this is demonstrated through the corresponding four (of 5) roots of 1, i.e. 1^1/5, 1^2/5, 1^3/5, 1^4/5 which cannot repeat for any other prime.

The final root i.e. 1^5/5, by definition = 1, and this then provides the means by which the ordinal notion is reduced in conventional mathematical terms.

So in conventional usage the ordinal positions are not treated as interchangeable, but rather fixed with the last unit member of each number group.  

Now conventionally, as we know the natural numbers (apart from 1) are obtained through a unique combination of prime factors.

So again from this perspective 6 = 2 * 3.

6 therefore is uniquely composed in quantitative terms of its two component prime “building blocks” i.e. 2 and 3.

So from the analytic perspective 6 is now likewise considered as an independent quantity in an absolute manner.

However though 2 and 3 initially can indeed be considered in isolation as independent “building blocks”, the very fact of combining them creates a new unique holistic interdependent identity. And this is a vitally important point.

Once again in isolation 2 and 3 have an independent identity as prime quantities.

However when combined together as factors of a new composite natural number i.e. 6, they now attain a shared status as prime factors of that number.

Thus there is a clear distinction to be made as between the analytic (quantitative) notion of a prime as an independent “building block” (in isolation from other primes) and the corresponding holistic (qualitative) notion of that same prime as a shared factor of a composite natural number. 

We have already seen in a previous blog entry the distinction as between the analytic and holistic aspects of number intimately relates to the complementary nature of the operations of addition and multiplication respectively.

And if one is to understand the true nature of dual sum over natural numbers and product over primes expressions (which are common to all L-functions) then this distinction is vital.

Paradoxical Nature of the Primes

We have already looked in detail at the number “2” distinguishing clearly both its analytic (quantitative) and holistic (qualitative) aspects of interpretation.

So from the former perspective, 2 can be viewed in quantitative terms as composed of independent homogeneous units.

However from the latter perspective 2 can equally be viewed in qualitative terms as uniquely composed of 1^st and 2^nd units which are interdependent (i.e. interchangeable with each other).

And to properly preserve both distinctions, 2 must be viewed in a dynamic interactive manner where quantitative and qualitative aspects are understood as complementary.

And what is true above in relation to the number “2” is equally true of all numbers.

So rather than being understood in abstract terms as absolute quantitative entities, as for example with the natural numbers - all numbers rightly represent dynamic interacting patterns, entailing the complementary action of both analytic (quantitative) and holistic (qualitative) aspects of behaviour.

This of course intimately applies therefore to the fundamental nature of the primes (with 2 the first member).

However when one looks at the conventional mathematical interpretation of the primes the analytic (quantitative) aspect is solely emphasised.

So the primes are thereby viewed in quantitative linear terms as the fundamental “building blocks” of the natural number system.

In other words, every natural number (except 1) represents a unique combination of one or more prime factors and all these natural numbers are then viewed in reduced linear terms (i.e. as points on the real line).

However each prime equally has an important circular identity, representing its (unrecognised) qualitative nature.

We can see this perhaps most simply in relation to the number 2.

As already stated the holistic (qualitative) appreciation of this number relates to recognition of 1^st and 2^nd members that are fully interchangeable with each other.

Indeed, once again this is the very recognition that is implicitly is involved when one recognises that both left and right turns at a crossroads are interchangeable depending on which direction (N or S) the crossroads is approached.

However the question that then occupied my mind for some time was how to give this holistic type recognition a satisfactory mathematical interpretation.

Returning to the crossroads left and right turns are necessarily opposites of each other.

So if approaching the crossroads (heading N) we denote a left turn as + 1, then a right turn (in this context) is – 1.

However if now approaching the crossroads (heading S) we now denote the right turn as + 1, then a left turn (in this alternative context) is – 1.

So the pure holistic recognition resides in the paradoxical appreciation that what is + 1 can equally be – 1 and what is – 1 can be + 1.

And this recognition, which is directly intuitive in origin, represents a psycho-spiritual energy state.

So far from intuitive insight directly corresponding with rational understanding it is quite the opposite in that it can only be indirectly expressed in rational terms in a paradoxical (i.e. circular) manner.

If we take the two roots of 1, we once again get + 1 and – 1. However on this occasion both results are clearly separated in a linear manner.

So for example when one approaches a crossroads from just one direction - say heading N - left and right turns have an unambiguous meaning. So if one designates a left turn as + 1, then a right turn is clearly –  1  (in this actual context).

However if one attempts to designate positive and negative signs to left and right turns without specifying the direction from which the crossroads is approached then clearly paradox is involved (in this potential context).

And this in a nutshell raises the key distinction as between analytic and holistic type recognition.

Analytic recognition relating to actual reality is unambiguous (amenable to linear rational interpretation) , whereas holistic recognition relating to potential reality is clearly paradoxical when conveyed in rational terms. So psychological intuition can only be indirectly conveyed in rational terms in a circular (i.e. paradoxical) manner.

However this raises enormous issues for conventional mathematical interpretation.

In other words professional mathematicians may indeed recognise the importance of intuitive insight (especially for creative new findings).

However there is then the mistaken belief that such intuition can be assumed to directly correspond - and thereby be formally reduced - in a strictly linear rational manner.

The key issue with respect to all mathematical interpretation is thereby completely overlooked i.e. to establish a consistent correspondence as between rational type understanding (of a linear nature) and intuitive recognition (that is paradoxical in rational terms).

And as we shall see this is the key issue relating to appreciation of the nature of the original Riemann Hypothesis and the generalised Riemann Hypothesis (applying to all L-functions).

A parallel way of expressing the dual nature of both the analytic and holistic aspects of number is to say that all numbers can be given interpretations as both structures of form and energy states respectively.

Thus the quantitative extreme, which is emphasised in conventional mathematical interpretation, views a prime (such as 2) as an absolute number form in an analytic manner. This corresponds directly with linear reason. And then the primes as structural forms are viewed as the “building blocks” of the natural number system. So the natural numbers are likewise viewed as structures of form uniquely comprising prime components as factors.

However the corresponding qualitative extreme is to view a prime (again such as 2) as a purely relative energy state in a holistic manner (with complementary psychological and physical attributes). This corresponds directly in psychological terms with intuitive insight (that indirectly can be conveyed in a circular i.e. paradoxical rational manner).   

Therefore in dynamic experiential terms, the understanding of number entails the interaction of notions of form with corresponding energy states resulting in a ceaseless transformation with respect to its inherent nature.   

So again from the analytic perspective a prime is viewed in quantitative terms as an independent “building block” of the natural number system in linear terms. So all composite natural numbers are assumed to lie on the same real line as the primes from which they are derived.

However from the corresponding holistic perspective a prime is viewed in qualitative terms as representing a unique circle of interdependence with respect to its group of natural number members (in ordinal terms).

So for example from this perspective, 5 as a prime is viewed as uniquely composed of 1^st, 2^nd, 3^rd, 4^th and 5^th members.

This uniqueness is then indirectly expressed in a circular number fashion through obtaining the corresponding 5 roots of 1 (which lie as equidistant points on the circle of unit radius in the complex plane).

Now while it is true that mathematicians have long been aware of this circular number system, once again their interpretation of its nature has been solely with respect to its analytic (quantitative) features.

However the true holistic significance in this context of the 5 roots of 1 is that they indirectly express the notions of 1^st, 2^nd, 3^rd, 4^th and 5^th (with respect to a group of 5 members) where positions are interchangeable.

And in this context the uniqueness of each prime is revealed through the fact that all its prime roots (with the exception of the default root 1), cannot be repeated with respect to any other prime.  

So from the analytic (quantitative) perspective, each prime is unique as it contains no factors (other than itself and 1).

However from the corresponding (qualitative) perspective, each prime is unique because each of its natural number ordinal members is unique (with the exception of the default last member).

And again this is highly revealing for the very interpretation of ordinal numbers in conventional mathematical terms is to treat them in fixed terms as the last member of its number group.

So 1^st is the last - and indeed only - member of  a group of 1; 2^nd is then the last member of a group of 2; 3^rd is the last of  a group of 3; 4^th is the last of a group of 4 and so on.

In this way the ordinal notion, which is inherently of a qualitative nature can be successfully reduced in a merely quantitative manner.  

However even brief reflection on the dual nature of a prime i.e. with both analytic and holistic properties, reveals its paradoxical nature.

Thus again from the quantitative perspective, each prime is unambiguously viewed in a one-way manner as a “building block” of the natural numbers.

However from the corresponding qualitative perspective, each prime is already seen to be composed uniquely of a group of natural number members in an ordinal manner.

So from the former cardinal perspective the natural numbers appear to uniquely depend on the primes. However from the corresponding ordinal perspective, each prime is already uniquely composed of a group of natural number members.

The clear implication of this paradoxical situation is that the primes and natural numbers are completely interdependent in two-way fashion with each other - ultimately in an ineffable manner - with respect to both quantitative and qualitative characteristics.

More on Dynamic Nature of Addition and Multiplication

We saw in the last blog entry how the number 2 can be given both analytic (quantitative) and holistic (qualitative) interpretations that are dynamically interdependent in complementary fashion with each other.

And this inherent feature can in principle be extended to all numbers.

So far from representing abstract independent entities with an absolute interpretation in quantitative terms, the very nature of number is inherently dynamic and relative, representing the two-way interaction of both its quantitative and qualitative aspects.

And the fundamental nature of addition and multiplication is revealed through this new understanding, where these two operations likewise operate in a dynamic complementary fashion enabling understanding to ceaselessly switch as between the quantitative and qualitative type aspects of number. 

In a way I find it utterly surprising that this fact is not more widely appreciated for I have been keenly aware from childhood of an obvious problem (that is steadfastly avoided in conventional mathematical interpretation). 

Thus in the most basic sense, there is a valid sense in which we can quantitatively distinguish (as independent members) the individual units of number.

However equally there is a valid sense in which we must likewise qualitatively relate (as shared members) these same units.

And to put it bluntly in conventional mathematical terms this latter qualitative aspect of number is inevitably reduced in a merely quantitative manner.

In other words from this limited perspective, no clear distinction can be made as between the twin notions of number independence (where units have a separate identity) and number interdependence (where units have a shared identity) respectively.

And it is this fundamental problem that lies at the root of the problem of properly relating the operations of addition and multiplication.

In an attempt to clarify this point further let us look at the simple example where we multiply the first two primes i.e. 2 and 3. 

So in conventional terms 2 * 3 = 6. Now expressed more fully we could represent this

as  2¹ * 3¹ = 6¹.with the three numbers represented as points on the same real line.

However, as well as a quantitative transformation through multiplication (i.e. to 6) a qualitative (dimensional) change in the nature of the units also takes place.

So if we imagine a rectangular table of width 2 and length 3 metres respectively, then the area of the table will be expressed in square metres.

So we move from 1-dimensional to 2-dimensional units. However in the standard treatment of multiplication, this qualitative transformation is then reduced in a quantitative manner, with the result i.e. 6, represented in 1-dimensional terms (as a point on the real number line).

Though in dynamic relative terms, base and dimensional numbers are quantitative and qualitative with respect to each other, both are understood however through this geometrical representation in a somewhat linear manner.

So we switch in other words from the analytic use of number representing (actual) finite quantities to the corresponding analytic use of number representing (actual) finite dimensions.

However in dynamic interactive terms, the switch is always of a truly complementary nature (i.e. from analytic to holistic and holistic to analytic aspect, respectively).

Thus there is a hugely important holistic aspect to the interpretation of multiplication (without which it has no strict meaning).

To more easily illustrate this point, imagine that we have 3 objects (say coins) placed in two rectangular rows!

Using addition in standard analytic fashion, we could add up the 3 independent coins in row 1, = 3 and then proceed to add up the 3 independent coins in row 2, = 3.

And as both rows would likewise be considered as independent, the total number of coins represents the sum of the 2 rows i.e. 3 + 3 = 6.

From the conventional mathematical perspective, multiplication can then be used to short-circuit the process of laboriously adding separate rows, through the recognition that a common similarity exists as between each row. So now, the realisation that we have two similar rows leads to the use of the operator 2, which is then multiplied by the number of objects in each row.

So from a multiplicative perspective, the total number of objects is 2 * 3 = 6.

Thus in conventional mathematical terms, multiplication is represented as a form of short-hand addition.

So, 2 * 3 = 3 + 3

Of course, with just 2 similar rows, multiplication does not offer any real benefit over addition. However, say with 100 similar rows, multiplication would then offer a much simpler way of expressing the total no. of objects (than the successive addition of many rows).

However there is a hugely important - though largely unrecognised - flaw to this interpretation of multiplication.

Again, the conventional mathematical approach to cardinal number is based on the assumption that unit members are independent of each other in an absolute fashion.

However, the very process of multiplication entails the recognition - as in the example above - of the common similarity as between the various rows (or alternatively various columns) and likewise the common similarity as between members of each row (and column).

Thus in our example, the use of 2 as an operator depends on recognition that the two rows are similar in number terms, i.e. share a mutual interdependence.

And from an ordinal perspective, it does not matter which row is identified as 1^st or 2^nd, for by definition they are mutually interchangeable.

Therefore, in this example, the recognition of the similarity of the two rows implies the holistic - rather than the analytic - interpretation of 2, where the mutual interdependence of both unit rows is recognised. 

Expressed in an equivalent manner the recognition of the similarity of the two rows, whereby they are recognised as mutually interdependent, entails the qualitative aspect of 2 (as “twoness”). By contrast the recognition of the independent nature of each row entails the quantitative aspect of 2 (as two).

This recognition of similarity as between the two rows equally implies recognition of the similarity i.e. shared interdependence of the 3 items within each row (where 1^st, 2^nd and 3^rd units are potentially interchangeable with each other). Equally it implies recognition of the similarity of the 2 items within each column (where 1^st and 2^nd units are potentially interchangeable with each other).

So all in all, the operation of multiplication implies recognition of both the analytic independence (where units are viewed as separate) and the holistic interdependence of number (where units are viewed as similar).

And this operation can only be properly understood in a dynamic interactive manner, where both the analytic (quantitative) aspects of number as independent and the holistic (qualitative) aspects of number as interdependent, are viewed as complementary.

This likewise applies to addition.

It is all very well treating numbers as independent in an analytic manner, but the consequent result from addition requires a transformation to a new holistic identity (not apparent in the independent units).

So, for example, we can start by viewing 1 + 1 as independent units in individual quantitative terms. However the very act of recognition, enabling the new whole collective identity of 2, requires corresponding realisation of the interdependence of unique units (as interchangeable). So addition and multiplication properly entail both analytic and holistic aspects.

Thus the crucial point here is that addition and multiplication are analytic (quantitative) and holistic (qualitative) with respect to each other with both operations complementary in dynamic interactive terms.

This implies that when - as is customarily the case - addition is identified directly with the quantitative transformation of number (in default 1-dimensional terms), multiplication is then - relatively - identified with its qualitative transformation (where the dimensional aspect changes) and vice versa.

Thus, in the simplest case, from the former perspective of addition, 1¹ + 1¹ = 2¹. So here we have a quantitative change in number in linear (1-dimensional) terms. Thus the base aspect of number changes while the dimensional aspect (implicitly) remains fixed.

However, from the latter perspective of multiplication, we have 1¹ * 1¹ = 1². Here in inverse relative fashion, a qualitative change in number occurs. So in a complementary manner, the dimensional aspect of number now changes while the base aspect remains (implicitly) fixed.

In conventional mathematical terms, it is customary to represent the natural numbers (without reference to base or dimensional characteristics), once again reflecting the reduction of qualitative to quantitative characteristics.

Therefore the natural numbers are listed, representing solely number quantities, as 1, 2, 3, 4, …

However from the enhanced interactive perspective, number possesses two distinct aspects (base and dimension respectively) both of which alternate as between analytic (quantitative) and holistic (qualitative) meanings in the dynamics of understanding.

Alternatively, we could say, using a very close physical analogy, that number keeps switching as between its particle (independent) and wave (interdependent) aspects, with respect to both base and dimensional characteristics.

So again rather than the natural numbers being given in absolute fashion as 1, 2, 3, 4, …, we now have two twin interacting aspects,

1¹, 2¹, 3¹, 4¹, … and 1¹, 1², 1³, 1⁴, …, which I customarily refer to - in relative terms - as the Type 1 and Type 2 aspects of the number system respectively.

And when understood in the appropriate dynamic manner, experience of these two aspects keep dynamically switching in complementary fashion, as between analytic (quantitative) and holistic (qualitative) aspects with respect to both base and dimensional characteristics, depending on which reference frame is employed. 

So, if the base aspect of number is interpreted in a quantitative manner, then the dimensional aspect, relatively, is qualitative; however if the base aspect is interpreted in a qualitative manner, then in this context, relatively, the dimensional number is now quantitative. 

Thus all numbers can be given - depending on relative context - analytic and holistic interpretations with respect to both base and dimensional characteristics.

Both Type 1 and Type 2 aspects, in this interactive appreciation, are intimately related to the key distinction as between addition and multiplication as complementary opposites.

When, for example, we multiply 1¹ * 1¹ to obtain 1², a qualitative change in the nature of number as dimension takes place. So 2 in this context, represents the holistic notion of dimension, relating to the interdependence of both unit members. Though in direct terms the transformation applies to dimensional units, implicitly this likewise entails the unit base numbers.

Then, when we add 1¹ + 1¹ to obtain 2¹, a quantitative change, relatively, in the nature of the base units takes place.  So 2, by contrast, now represents the analytic notion of number (as base), relating to the independence of both unit members. However, though in direct terms the transformation now applies to the base units, implicitly this likewise entails the dimensional notion of 2.

Therefore the remarkable fact remains that whenever we use any number, with respect to base or dimensional aspect, in the standard analytic quantitative manner, this likewise implicitly requires the corresponding holistic notion of that same number, which has now become implicitly embodied, as it were, in the number.

So from a psychological perspective, the conscious rational interpretation of number always carries the distant echo of its unconscious holistic counterpart.

However, in the main, we remain completely deaf to this unconscious echo.

So we cannot explicitly define addition (without implicitly requiring multiplication); and we cannot explicitly define multiplication (without implicitly requiring addition). 

Dynamic Perspective on Addition and Multiplication

Things are not always what they seem!

I mentioned this yesterday with reference to the standard analytic treatment of number 1 + 1 = 2.

Indeed based on our common-sense intuitions of reality this statement seems so obvious that it is frequently used as a metaphor for self-evident truth.

In fact however it reveals the reduced nature of accepted mathematical understanding, which through millennia now of unchallenged usage has become so deeply ingrained in our thinking that we have become completely blind as to its limitations.

The conventional understanding of this simple number relationship is based on mere analytic appreciation (where the “whole” number 2, represents the quantitative sum of its part unit members).

So the unit numbers here are viewed as absolutely independent of each other (in a homogeneous manner). In other words, as there is no way to distinguish as between the two units, they thereby lack qualitative distinction.

However if the individual units were indeed absolutely independent in such a manner, there would be no way of combining them to obtain the new collective identity of 2! 

Therefore the very ability to relate the two unit numbers presupposes a distinct qualitative identity of relational interdependence.

In the conventional treatment of addition, this key fact is completely overlooked.

In other words the qualitative aspect of number (i.e. relational interdependence) is simply reduced in an absolute quantitative manner.

This then enables the convenient fiction to be maintained that numbers can be successfully combined with each other (while preserving their absolute quantitative identity).

Thus from this perspective 1 + 1  = 2 (understood in absolute quantitative terms).

However when one addresses the gross inconsistency entailed here, then the very nature of number is thereby radically changed.

For when we accept that all numbers necessarily possess a qualitative aspect of relational interdependence, then the corresponding quantitative aspect (of distinct identity) can only be properly conceived in a relatively independent fashion.

Thus all numbers necessarily possess both an analytic aspect (of quantitative independence) and a holistic aspect (of qualitative interdependence) respectively, which are dynamically related to each other in complementary fashion.

So let us now return to our simple identity, 1 + 1 = 2.

Again, in terms of appropriate dynamic appreciation, each unit is relatively independent in a quantitative manner, while equally possessing a qualitative identity of shared relational interdependence.

Whereas the quantitative aspect is directly related to the cardinal, the qualitative aspect is directly related to the corresponding ordinal nature of number respectively.

So again from the quantitative perspective, the two units are given a separate individual identity (in actual terms); however from the qualitative perspective, they are given a shared collective identity (in a potential manner).

So with respect to number recognition, each individual unit must be quantitatively distinguished (in an actual manner).

However equally, the two units must be given a qualitative identity of shared relational interdependence (in potential terms). From this perspective ordinal notions of 1^st and 2^nd with respect to the two units are viewed as fully interchangeable with each other.

At a deeper psychological level, whereas the analytic (quantitative) aspect of number relates directly to rational (conscious) recognition, the holistic (qualitative) aspect relates by contrast directly to intuitive (unconscious) appreciation.

So in conventional mathematical practice, the reduction of the holistic (qualitative) aspect of number recognition, in analytic (quantitative) terms, parallels precisely the corresponding psychological reduction of intuitive (unconscious) appreciation in a rational (conscious) manner.

Properly therefore, a comprehensive understanding of number entails both conscious and unconscious aspects of appreciation where both need to be explicitly recognised.

However once more, in conventional mathematical terms, the holistic (unconscious) aspect is formally reduced in an analytic (conscious) manner.

Therefore the true nature of number is inherently dynamic with quantitative notions of relative independence continually interacting with qualitative notions of relative interdependence respectively.

And in this new dynamic context, the two relative notions of number independence and interdependence are in turn intimately linked with the mathematical operations of addition and multiplication respectively.

Thus when we start with the quantitative aspect using the operation of addition,

1 + 1 = 2.

More fully we can express this relationship as,

1¹ + 1¹ = 2¹.

And using the Peano postulates all the natural numbers can ultimately be derived from successively adding 1 to a previous natural number (starting with 1).

So to combine numbers in a quantitative manner (through the operation of addition) we must explicitly define them with respect to the default dimension of 1 (which in this context remains implicit in understanding). In this way all natural numbers are viewed as points on the same real number line.

Thus in this context, 2 (i.e. 2¹) represents the quantitative sum of the two base units. 

However when we now, in relative terms, combine the two units through multiplication, a different qualitative notion of number emerges, which can be expressed as follows,

1¹ * 1¹ = 1².

So in dynamic complementary fashion the focus is explicitly on the dimensional units (with the corresponding base units now remaining implicit).

Thus in this dynamically related context 2 (i.e. 1²) represents the qualitative product of the two dimensional units).

What this means in effect is that these two units are now understood in holistic ordinal terms (as relatively interdependent) whereby 1^st and 2^nd positions are freely interchangeable with each other.

So just as we recognise that what are designated right and left turns at a crossroads can switch with each other depending on the direction from which the crossroads is approached, we likewise recognise that the two dimensional units can likewise switch ordinal positions (depending on context).

However as always in dynamic interactive terms, reference frames can switch.

Thus there is a valid sense in which addition can likewise be identified in qualitative terms with the interdependence of its unit members.

So 1¹ + 1¹ = 2¹. So here we are explicitly recognising in a holistic manner the qualitative interdependence of the two base units (with quantitative recognition now implicit).

And finally there is equally a valid sense in complementary fashion, through which multiplication can be likewise identified with the quantitative independence of its dimensional units (i.e. where each dimensional unit is recognised as separate).

Thus 1¹ * 1¹ = 1².

Therefore what we have demonstrated here is the following: 

1) The number 2 - and by extension all natural numbers - can be given both analytic (quantitative) and holistic (qualitative) interpretations with respect to both base and dimensional usage. Whereas the analytic aspect corresponds directly with rational, the holistic aspect by contrast corresponds directly with intuitive type recognition.   

2) The relationship between both is of a dynamic interactive nature with quantitative and qualitative aspects understood in complementary fashion.

So when the analytic (quantitative) aspect of number is made explicit, the corresponding  holistic (qualitative) aspects remains, in this context implicit.

However, when the holistic (qualitative) aspect of number is now in turn made explicit, the corresponding analytic (quantitative) aspect is now implicit. 

3) What is extremely important to bear in mind is that in this new dynamic framework, addition and multiplication become the very means by which the switching from quantitative to qualitative (and in turn qualitative to quantitative) recognition takes place.

So addition and multiplication operate as dynamic complementary partners in the very recognition of number.

Thus if we return to the number 2 for a moment, there is of course a valid sense in which we can recognise its two units as quantitatively independent of each other; however there is an equally valid sense in which we can recognise that these two units are likewise interdependent (and thereby interchangeable with each other) in an ordinal manner. So in this context 2 necessarily contains unique 1^st and 2^nd units.

And both of these interpretations are necessarily related in two-way fashion with each other.

Thus we cannot explicitly recognise the quantitative independence of the two units (without implicit recognition of their corresponding qualitative interdependence).

And we cannot explicit recognise the qualitative interdependence of the two units as, interchangeably, 1^st and 2^nd respectively (without implicit recognition of their quantitative independence). 

Thus properly understood, the number 2 - and by extension every other number - is necessarily of a dynamic interactive nature, comprising analytic (quantitative) and holistic (qualitative) aspects in a complementary manner.