Riemann Hypothesis: New Perspective (7)

We have seen that when the natural number system is viewed in the conventional analytic manner, that the Riemann zeros thereby represent the (unrecognised) holistic counterpart of this system.

And this can only be properly appreciated in a dynamic interactive context, where analytic and holistic appreciation - representing the independent and interdependent aspects of the number system respectively - are seen as directly complementary with each other.

Looked at from the more psychological perspective, when the natural number system is interpreted in the customary (linear) rational conscious manner, then the Riemann zeros represent the (unrecognised)  unconscious appreciation of this system. Now whereas the direct nature of this appreciation is purely intuitive in nature, indirectly it is then objectively conveyed in a (circular) paradoxical rational fashion. And this again is the precise meaning of "imaginary" from a holistic mathematical perspective!

Therefore the very appreciation of the Riemann zeros requires that the (hidden) unconscious aspect of mathematical understanding be brought fully into the conscious light.
Only then can both the conscious and unconscious aspects of mathematical understanding - which in truth are ultimately truly interdependent - be fully integrated with each other!

And of course, here we face again the most fundamental problem possible with accepted mathematical interpretation in that as a formal discipline of thought it remains totally blind to its deep unconscious aspect.

In other words, everywhere in interpretation the unconscious (intuitive) aspect of understanding is reduced in a merely formal  (linear) rational conscious manner.

Thus in basic terms, in every context the holistic notion of "interdependence" is reduced in an analytic (i.e. independent) manner.

Therefore, as I have repeatedly stated in these blog entries, nothing less than a total revolution is now required in mathematical thinking representing by far the most radical transformation yet in our intellectual history. Not alone will this have the most fundamental repercussions possible for Mathematics (changing and enlarging its scope in an utterly unparalleled manner) but equally it will have the most profound implications for all the sciences and for society generally.

So at present we have just one recognised form of Mathematics i.e. Type 1 Mathematics which is interpreted in an absolute analytical manner.

However in a future golden age of Mathematics we will have at least three interacting forms.

1) Type 1 Mathematics. This will largely equate with present analytic interpretation, though now understood in a relative rather than absolute manner.

2) Type 2 Mathematics. This will be identified with the qualitative interpretation of mathematical symbols, in what I refer to as "Holistic Mathematics".

3) Type 3 Mathematics. This will represent the most comprehensive type of mathematical understanding as the integrated appreciation of both analytic (Type 1) and holistic (Type 2) aspects.

Ultimately both Type 1 and Type aspects can only obtain their true - relatively - separate status through the perspective of integrated Type 3 understanding. In the past I have referred to this as "Radial Mathematics".

In fact these blog entries on the nature of the number system are designed to represent but the most preliminary excursion into Radial Mathematics.


So far however, we have looked at the holistic nature of the Riemann zeros (from the starting appreciation of the standard analytic interpretation of the number system).

However in dynamic interactive terms, these interchange with each other, so that we equally have a holistic interpretation of the natural numbers and an analytic interpretation of the Riemann zeros respectively.

Now one might ask what the holistic appreciation of the natural number system entails!

Whereas with standard analytic interpretation we address the natural number system in collective  terms (i.e. as a  collective whole), here in complementary fashion we view each member of this system in individual terms (where each number now represents a unique whole with respect to its unit members).

Once again let us use the number "3" to illustrate!

From the analytic perspective this number is viewed as composed of independent unit members

i.e. 3 = 1 + 1 + 1 (where each unit is homogeneous and independent).

However from the corresponding holistic perspective this number is now composed of interdependent unit members i.e. 3 = 1 + 1 + 1 (where each unit has a unique meaning through interdependence with the other units).

So what happens in actual experience is that one keeps switching - though not formally recognised in standard interpretation - from the quantitative notion of "3" (representing independent units) to the qualitative notion of "3" (representing "threeness" as the interdependence of units).

This also directly equates with the continual switch from cardinal notions of "3" (where component units are homogeneous) to ordinal notions of "3" (where component units are unique as 1st, 2nd and 3rd respectively).


However just as the analytic interpretation of the (cardinal) natural number system is complemented by the holistic interpretation of the Riemann zeros, equally the holistic interpretation of each natural number (i.e. ultimately with respect to the unique ordinal positions of each prime number) is then complemented by the analytic interpretation of the Riemann zeros.

And this very much conforms with the standard appreciation of the zeros, where they can now be used collectively in a quantitative manner to exactly predict the locations of the primes (having corrected with respect to their initial general frequency).


One of the great problems with the current attempt to appreciate the nature of primes is that they are invariably studied with respect to their cardinal nature. So in effect the ordinal aspect of the primes is simply reduced in cardinal terms.

However properly understood both the cardinal and ordinal aspects of primes - as quantitative to qualitative (and qualitative to quantitative respectively) - are ultimately fully complementary with each other (as in experience) in a dynamic interactive manner.

Now when one begins to realise this, one can then appreciate that the relationship between the primes and natural numbers (and natural numbers and primes) is inherently of a paradoxical nature.

In other words, from a dynamic interactive perspective, they mutually depend on each other.

So, for example from the conventional cardinal perspective "3" as a prime is viewed as an independent quantitative "building block" of the natural number system.

However from the complementary ordinal perspective "3" as a prime is already uniquely defined and thereby depends on its 1st, 2nd and 3rd members.

So from the cardinal perspective, each (composite) natural number system appears to depend on individual primes (as building blocks).  However from the ordinal perspective, each individual prime already depends on a unique collection of natural number members.  

Thus there are two complementary forces at work with respect to the relationship between the primes and the natural numbers.

In Type 1 terms the number 6 (as quantitative aspect) is expressed as (2 * 3)¹

However in Type 2 terms the number 6 (as - relatively - qualitative aspect) is expressed as  1^{(2 * 3)}

So we see that both base and dimensional numbers switch in complementary fashion as between the two definitions.

This simply implies that whereas the natural numbers appear to be quantitatively generated from the primes in Type 1 terms that it is the reverse from a Type 2 perspective (i.e. the primes are qualitatively generated - i.e. assume a unique quality of interdependence - through their relationship with the natural numbers).

And in both cases - with respect to both the Type 1 and Type interpretations of the natural number system, we have a (shadow) set of Riemann zeros, which enables the consistent interface of both Type 1 and Type meanings.

So the zeros enable the conversion (1) from quantitative to qualitative meaning and (2) from qualitative to quantitative meaning.

Putting it more simply, underlying our conventional (conscious) understanding of both the cardinal and ordinal nature of number is a hidden (unconscious) system (Riemann zeros) that enables the successful transition as between both aspects.

So the great task in terms of a coherent understanding of the number system is to bring this unconscious system fully into the conscious light.

Riemann Hypothesis: New Perspective (6)

I now will now address the issue of why the Riemann zeros - apparently - all lie on the imaginary line (drawn through .5 on the real axis) with each pair of zeros having the complementary form of a + it and a - it respectively.

As we have seen the conventional number line is based on the quantitative notion of number, where component units are viewed in an independent manner.

So once more, illustrating with respect to the natural numbers, when we refer to "3" for example, implicit in interpretation is the view that 3 = 1 + 1 + 1 (where the homogeneous units are viewed as independent of each other).

However a huge unaddressed problem exists with respect to conventional interpretation in that number can equally be used in an alternative fashion (where component units are now strictly interdependent with each other).

Thus we could equally say for example that a number has 3 factors. In this context, as dealt with yesterday, the (composite) number 6 contains 3 proper factors i.e. 1, 2, and 3.

However the very nature of these factors is that they are all defined with reference to their common relationship to 6!

Therefore, we are using 3 (with respect to the three factors) in a different sense!

So when we maintain here that 3 = 1 + 1 + 1 (i.e. as the sum of its component units) these units must now properly be understood as interdependent with each other.

Thus we have illustrated the remarkably simple, yet entirely overlooked fact, that properly understood, in a dynamic interactive context - which is the only appropriate coherent way to view this matter -  that number keeps switching from its quantitative aspect (where units are relatively independent) to its corresponding qualitative aspect (where units are interdependent) and vice versa.

Alternatively - to borrow from the language of quantum mechanics - number keeps switching as between its particle aspect (as independent) and wave aspect (as interdependent) and alternately also  in reverse manner as between wave and particle aspects.


So if we are to represent the natural numbers as lying on a number line, strictly speaking two distinct lines are required 1) for the quantitative notion of number (where individual units are relatively independent of each other) and 2) for the corresponding qualitative notion of number (where the units share a common interdependence with each other).

This then raises the enormous issue of consistency with respect to both uses.

How, from one perspective, can we be confident that the qualitative (wave) use of number is consistent with the corresponding quantitative (particle) use?

Then, how from the reverse perspective, how can we be confident that the quantitative (particle) use of number is consistent with the corresponding qualitative (wave) use?

And we must keep remembering that in the actual dynamics of number interaction, the quantitative has a likewise (hidden) qualitative and the qualitative a likewise (hidden) quantitative aspect respectively!

Now, when we start from the conventional perspective of the natural number system i.e. as viewed in a quantitative manner, strictly speaking the notion of number, as used with respect to the factors of composites, relates to a different number line (where the unit factors are viewed as interdependent with each other).

So remarkably, we now have the appropriate context for viewing the Riemann zeros i.e. as the (hidden) holistic counterpart of the accepted analytically interpreted natural number system.

And the condition for consistency of the two aspects of the number system (analytic and holistic respectively) is that the Riemann zeros equally lie on a number line.

Now of course this is implied by the Riemann Hypothesis, with the requirement that the imaginary line on which all the zeros are postulated to lie is drawn through .5 (on the real axis).

However there is no way that this can be proven using conventional mathematical axioms, as due to the reduced interpretation implied, it is already assumed that the qualitative use of symbols is absolutely identical with the quantitative!


One might query why the zeros should lie on an imaginary line!

And this is where the holistic interpretation of the imaginary notion is of invaluable assistance.

In holistic terms, to posit is to make conscious and to negate is to make unconscious.

In dynamic terms, unconscious negation already implies that one has already posited an object in a conscious manner.

So just as in physics when an anti-matter particle interacts with its matter equivalent, energy is produced, likewise in psycho spiritual terms, when unconscious negation (of what has been already posited) takes place, intuitive energy is generated.

Now we can accurately refer to this - indirectly expressed through the two roots of 1 - as 2-dimensional experience with both positive conscious and negative unconscious directions.

The imaginary notion in analytic terms then entails obtaining the square root of the negative unit.

In corresponding holistic terms this implies the attempt to express the inherently intuitive understanding (where the identity of complementary opposites is directly apprehended) indirectly in an objective rational manner.

So the key point - as I have been demonstrating in my last few entries - is that the holistic appreciation of the zeros implies a high degree of refined intuitive appreciation (where one simultaneously can literally see from two complementary opposite frameworks). In a much more accessible manner, the appreciation of the paradoxical nature of L and R turns at a crossroads implies similar intuitive insight.

So the requirement that all the zeros lie on an imaginary line really points to the fact that these points represent a circular type of paradoxical understanding (when expressed in an indirect rational manner).

Once again we cannot prove that the zeros lie on an imaginary line! What we can say however is that for consistency to be maintained as between the quantitative and qualitative interpretation of mathematical symbols, we must assume that they all lie on an imaginary line (which is the expression of such consistency).

So each Riemann zero represents a point on the imaginary number line, where the quantitative and qualitative aspects of number interpretation are identical. Expressed more accurately in a dynamic interactive manner, each zero holistically expresses a point where both the quantitative and qualitative aspects of number interpretation approach perfect identity!   

So the truth of the Riemann Hypothesis, in this important sense, thereby requires an initial massive act of faith in the subsequent consistency of the whole mathematical enterprise.    

Riemann Hypothesis: New Perspective (5)

Once again, I keep returning to the distinction as between number independence and number interdependence respectively.

It is important to bear in mind that the quantitative notion of number is related directly to the independence of units.

Therefore when one uses for example the number 3 in its accepted quantitative sense, this assumes the identity 3 = 1 + 1 + 1 (where each of the units is considered as  independent relationship of each other).

However one when uses the notion of number in a dimensional sense to represent factors, the very nature of number thereby subtly changes.

So for example if we now say that a number contains 3 factors, this implies that each of the "unit" factors now bears a relationship of (multiplicative) interdependence with the number (of which they are factors).

Thus confining ourselves to proper factors,  1, 2 and 3 are factors of 6.

However this clearly assumes a relationship of interdependence as between 6 and each of its member factors.

Now again we could indeed validly say that in this context that 6 has 3 factors.

However if we wish to express 3 in terms of its unit members 3 = 1 + 1 + 1, this  now implies that the individual members share a relationship of interdependence with each other (through each being a factor of 6).

So in fact - though this observation is missed entirely in conventional mathematical interpretation - we have switched, using the language of quantum mechanics, from the particle aspect of number (implying independence) to its corresponding wave aspect (implying interdependence).

Expressing this more fundamentally, we have switched from the quantitative aspect of number (again implying independence) to its corresponding qualitative aspect (implying interdependence).


Once again, this is vital to understanding the true nature of the Riemann zeros.

As I have stated before a very close relationship connects the Riemann zeros to the factors (i.e. divisors) of the natural numbers.

So once again if we count the (accumulated) factors of the natural numbers to n and then measure the frequency of the Riemann zeros to t (where n = t/2π), the ratio of factors to Riemann zeros → 1, when n and t are sufficiently large.

In fact, the very function of the Riemann zeros is to smooth out the unevenness in occurrence of factors with respect to the primes and composite natural numbers.

We start initially with the primes as "building blocks" of the composite natural numbers (in quantitative terms). 

However, as we have seen, through their relationship as factors of these composite natural numbers, the primes and other natural numbers (representing unique combination of primes) acquire a new relationship of qualitative interdependence with the (composite) natural numbers.  

So in terms of proper factors, the quantitative independence of the primes (as building blocks of the natural numbers) is expressed as 1 (i.e. each prime contains one proper factor = 1).

By contrast the qualitative interdependence of the (composite) natural numbers, as the relationship of primes (or other natural numbers already expressing a unique combination of primes) is expressed through the number containing 2 or more composite factors.

Therefore, a very uneven pattern is in evidence with respect to the occurrence of factors of the natural numbers.

Thus looking at numbers up to 10, starting with 2, as prime this represents 1 factor; 3 also as prime represents 1 factor; 4 then as the first composite number represent 2 factors; 5 as prime then represents again 1 factor; 6 as composite now represents 3 factors,; 7 as prime represents 1; 8 as composite represents 3; 9 as composite represents 2 and 10 as composite represents 3.

The Riemann zeros represent the attempt to harmonise both the quantitative and qualitative aspects of the number system i.e. with respect to the unitary nature of factors associated with the primes and the multiple nature of factors associated with the composite natural numbers.   

Now just like the recognition that left and right turns can interchange with each other in paradoxical fashion at a crossroads (depending on the reference frame N or S from which the crossroads is approached), likewise the recognition of the true paradoxical nature of the Riemann zeros (in holistic terms) is derived from the ability to simultaneously view number with respect to both its quantitative and qualitative aspects (i.e. in Type 1 and Type 2 terms). 

Thus aspects clearly separated with respect to number in an analytic manner (as distinct) are yet seen from a holistic perspective to approach mutual identity (in a dynamic interactive manner).

Therefore in proper analytic terms, the aspect of number as representing the base is clearly separated from the corresponding aspect representing the dimensional aspect of number.  Because of the gross reductionism of conventional mathematical interpretation, this however is not attempted (with both aspects confused with each other). However I have demonstrated proper separation is achieved through defining both the Type 1 and Type 2 aspects of the number system! 

However in holistic terms, both of these aspects are then seen as identical with each other - or more accurately as approaching perfect identity with each other - in a dynamic interactive manner.

Thus from this perspective, each Riemann zero represents a point (on the imaginary number line) where the Type 1 and Type 2 aspects of the number system approach identity.

Again. in analytic terms, the notion of primes and natural numbers are clearly separated.

So in conventional (Type 1) terms the (composite) natural numbers clearly depend for their quantitative identity on the primes (as fundamental "building blocks"). However in unrecognised (Type 2) terms, the primes (and unique combinations of primes) obtain their qualitative identity through their relationship with the (composite) natural numbers as factors.

So from this perspective, the Riemann zeros holistically represent the points on the imaginary line where both their quantitative and qualitative aspects approach dynamic identity.

Also in analytic terms, addition and multiplication are clearly separated through the Type 1 and Type 2 aspects of the number system (where multiplication with respect to the Type 1 represents addition with respect to the Type 2 aspect and multiplication with respect to the Type 2, addition with respect to the Type 1 aspect  respectively.  

From this perspective, the Riemann zeros holistically represent the points on the imaginary number line, where both addition and multiplication approach a common identity.

Riemann Hypothesis: New Perspective (4)

To properly understand the nature of the Riemann zeros, one requires the ability the be able to look at the relationship between the primes and natural numbers simultaneously from two reference frames (which are paradoxical in terms of each other).

Now a simple example of where this ability is required related to the nature of a crossroads, where one can readily appreciate that left and right turns have a purely relative meaning (depending on context).

Thus when travelling N on a straight path, one can unambiguously define left and right turns at the crossroads; equally when travelling S in the opposite direction towards the crossroads, one can again unambiguously define left and right turns.

Thus within each single polar reference frame (i.e. either N and S considered separately) one can give an unambiguous meaning to L and R turns. However when we now consider these turns simultaneously from both N and S directions, paradox is involved. For what is L (from the N direction) is R (from the S); equally what is R (from the N direction) is L (from the S).

Therefore, in dynamic experiential terms, where both N and S directions (representing polar reference frames) are simultaneously combined, L and R turns then have a merely relative meaning (depending on arbitrary context).

Now with respect to this simple crossroads, most people will have little difficulty in intuitively appreciating the paradox that what is L (from one perspective) is R (from the opposite perspective); and what is R (from one perspective) is L (from the opposite).

However an altogether much greater difficulty is likely to arise, when applying the same kind of understanding to appreciation of the Riemann zeros.

Now here instead of N and S directions, we have quantitative and qualitative type interpretations of mathematical number symbols. Once again the quantitative aspect relates to their separate independence (from each other); the qualitative aspect then relates to their mutual interdependence (with each other). Then in place of L and R turns we have numbers representing base and dimensional values respectively.

So again in general terms with respect to aⁿ, a is the base and n the dimensional number respectively.

And both the Type 1 and Type 2 aspects of the natural number system are defined with respect to base and dimensional numbers..

With the Type 1 aspect, the base number varies over all the natural numbers, while the dimensional number remains fixed as 1.

With the Type 2 aspect, in inverse complementary terms, the dimensional number varies over the natural numbers, while the base number remains fixed as 1.

And in dynamic experiential terms - as with the crossroads - both quantitative  and qualitative appreciation of number symbols is involved with respect to base and dimensional values.

Now the zeros of Riemann zeta function entail that

1^{– s} + 2^{– s} + 3^{– s} + 4^{– s} + ..........  = 0

So The Riemann zeros establish a direct relationship as between the natural number system (as base number symbols) with complex numbers of the form a +/– it (as corresponding dimensional number symbols).

And when we start by viewing the base natural numbers in a quantitative manner, then the dimensional numbers (representing the Riemann zeros) must be viewed in a corresponding qualitative manner.

Going back to the crossroads example, analytic interpretation is unambiguous in an absolute linear rational manner. Therefore from this perspective a turn at a crossroads is either L or R.

However, corresponding holistic interpretation of the crossroads requires a circular (paradoxical) type logic, where both turns are simultaneously viewed as both L and R (depending on relative context).

Therefore the clear implication here is that when we initially view the natural number system in a customary analytic, the Riemann zeros must be interpreted in a complementary holistic fashion.

Now again in analytic terms, Type 1 and Type 2 aspects of the number system are clearly separated; however in holistic terms, Type 1 and Type 2 are simultaneously integrated with each other. 

So holistic appreciation of the Riemann zeros requires that one simultaneously views the number system from both its Type 1 and Type 2 aspects. Indeed, crucially the Riemann zeros entail those points in the number system where both Type 1 (quantitative) and Type 2 (qualitative) interpretations directly coincide with each other. 

We could equally say that they represent the points where randomness and order with respect to the number system coincide; we could also say that they represent points where the primes and natural numbers coincide; finally we could say that they represent points where both addition and multiplication coincide. 

Once again these statements have no meaning from the (linear) analytic perspective, which defines conventional mathematical interpretation. However they do they indeed have a very deep significance from the (circular) holistic perspective, which unfortunately is completely overlooked in conventional terms. And the holistic interpretation inherently requires a dynamic interactive mode of appreciating mathematical symbols that combines both quantitative (as independent) and qualitative aspects (as interdependent) respectively. So in this dynamic environment, both independence and interdependence acquire a merely relative meaning (depending on context).
  

So once more, we cannot hope in the present context to appreciate the nature of the Riemann zeros in a solely analytic fashion (employing independent reference frames).

Thus it is strictly speaking impossible to properly appreciate the Riemann zeros from a mere quantitative perspective without explicit recognition of the qualitative aspect of mathematical understanding!

 

Therefore the apparent absolute nature of the natural number system in accepted analytic terms,  where all natural numbers can be represented as unique combinations of primes, itself is intimately dependent on an equally important holistic number system (represented by the Riemann zeros) where the dynamic relative independence and interdependence of both its quantitative and qualitative aspects is given expression.

So from this perspective, the Riemann zeros simply represent the (hidden) holistic basis of the natural number system (as analytically interpreted in conventional mathematical terms).

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

So from an equally valid perspective, the Riemann zeros represent the (hidden) analytic basis of the natural number system (as holistically appreciated with respect to the interdependence of all such numbers).

Then from a more comprehensive perspective, it can be clearly seen that both the natural number system and the Riemann zeros are dynamically interdependent with each other with respect to both analytic and holistic aspects, ultimately approaching a state of pure synchronicity in an ineffable manner.

Riemann Hypothesis: New Perspective (3)

We have seen the important basis for the existence of the Riemann zeros in the previous entry.

Once more, this relates to the unrecognised qualitative nature of number, whereby the natural numbers (representing factors)  acquire a unique form of interdependence, through their relationship with other natural numbers.

And the frequency of these Riemann zeros are intimately related to the corresponding accumulated frequency of the (proper) factors of the natural numbers.

So we can now see that the Riemann zeros are directly connected with the attempt to give an indirect quantitative meaning to mathematical; relationships that are - directly - of a qualitative nature.

However in dynamic interactive terms - which is the only appropriate way for viewing such number relationships - it is vital that balance be maintained as between both relative independence and relative interdependence respectively. Alternatively we could express this as the requirement to maintain dynamic balance as between both analytic (quantitative) and holistic (qualitative) aspects of the number system.

Thus once again we start by viewing the primes as the independent building blocks of the natural number system (in quantitative terms). However we then must come to the equal appreciation of the composite natural numbers as representing the unique interdependence of primes (in a qualitative fashion).

Therefore for example, 2 and 3 (as uniquely distinct primes) represent independent building blocks of the quantitative aspect of the number system. However the number 6 (which is composite in nature) now represents, through multiplication, the interdependence of these two primes.

And this interdependence is of a strictly qualitative nature, representing the fact that through multiplication of these two numbers a transformation in their dimensional nature takes place.


So once again - as separate distinct primes in quantitative terms, both 2 and 3 are defined in a 1-dimensional manner i.e. represented by two distinct points on the number line.

However the product of 2 and 3, i.e. 2 * 3, now entails (as in concrete terms with a rectangular table) a transformation in their dimensional nature, which is strictly - in relative terms - qualitative in nature. 

Put another way, in the context of the composite number 6, both 2 and 3 acquire through multiplicative interdependence, a qualitative resonance (reflecting a new shared dimensional nature).

So the question then arises as to how an (indirect) quantitative meaning can be given for all the interdependent relationships that arise through the multiplication of primes.

And just as - in the context of 6 - the qualitative identity of the primes can be expressed through the fact that 2 and 3 now represent unique factors of 6, this equally applies to the factors of all composite numbers.

Also, because natural number divisors (that are not prime) can be factors of a composite number, these must also be included. However as these natural numbers themselves necessarily reflect a unique combination of primes, the factors of the (composite) natural numbers - to which the qualitative identity of the primes relates ultimately relate - reflect factors that are either prime (or represent unique combinations of primes).

So for example, both 2 and 3 represent unique prime factors of 12 (as well as 6) whereby they acquire a distinctive qualitative resonance. However 6 is also a factor of 12 (now representing a composite natural number).

But 6 itself represents a prior unique combination of primes i.e. 2 * 3. So not alone do 2 and 3, as individual factors of 12, acquire a new qualitative resonance, but also 6 (= 2 * 3) which already reflects a prior unique combination of primes.

And in terms of the balanced dynamic appreciation of the number system, both the relative independence of the primes (as quantitative building blocks) and the relative interdependence of the primes (through their qualitative relationship to the composite natural numbers), must be given equal priority.

In other words, they must be viewed ultimately as fully synchronous with each other in an ineffable manner (that defies linear rational explanation).


And this is vital to appreciate because the Riemann zeros in effect represent the perfect balance as between Type 1 (quantitative) and Type 2 (qualitative) notions of the primes i.e. where from one perspective the independence of the primes (in quantitative terms) is maintained and yet from the other the interdependence of the primes (in corresponding qualitative terms) is likewise preserved.

Riemann Hypothesis: New Perspective (2)

I have shown how the Type 1 and Type 2 natural number systems are related to successive addition and multiplication respectively with respect to 1.

So these represent the positive integers.

However, we can equally show how the Type 1 and Type 2 systems for negative integers are related to subtraction and division with respect to 1.

So in Type 1 terms, 1¹ – 1¹  = 0;  0 – 1¹  = – 1¹;  – 1¹ – 1¹  = – 2¹; – 2¹ – 1¹  = – 3¹ and so on!

Then in Type 2 terms 1¹ / 1¹  = 1⁰;  1⁰ / 1¹  = 1^{– 1};  1^{– 1} / 1¹  = 1^{– 2}; 1^{– 2} / 1¹   = 1^{– 3} and so on!

Therefore all the integers (positive, negative and 0) are defined in Type 1 terms through addition and subtraction respectively; equally all the integers are then defined in Type 2 terms through multiplication and division respectively.  


So returning to the positive integers, once again in conventional mathematical terms, the Type 1 quantitative aspect of multiplication is solely recognised (whereby in effect multiplication is reduced to addition).

In particular in conventional mathematical terms, the primes are considered as the quantitative "building blocks" of the natural number system. So every natural number thereby represents a unique combination of primes.

For example in conventional terms 6 = 2 * 3 (which represents the unique combination of primes for this number).

However, when one carefully reflects on the matter, this notion of multiplication represents but a shorthand reduced form of addition.

So the operator 3 here serves to indicate that the number "2" be added to itself 3 times.

Therefore the multiplication operation 2 * 3 in effect is reduced to the addition operation of 2 + 2 + 2.

The "proof" that this reductionism in fact occurs is that in conventional mathematical terms, the result 6 (from multiplying 2 * 3) is treated as just another number (on the 1-dimensional line).

So therefore in effect, in Type 1 terms, the multiplication operation (2 * 3) = 6¹ is inseparable from the addition operation 2 + 2 + 2  i.e. 2¹ + 2¹ + 2¹  = 6¹.

However if we think of  2 * 3 in geometrical terms - say as a table top with length 3 units and width 2 units (representing metres) - clearly the result will now be expressed in 2-dimensional terms (as square units).

So therefore through multiplication, a transformation has thereby taken place in the qualitative nature of the units involved!


This point is in fact central to the appreciation of what the famed Riemann zeros in fact represent!

When I started to seriously investigate the frequency of occurrence (on the imaginary number line) of these zeros, I began to realise that they bear a remarkably close relationship to the proper factors of the natural numbers.

Now the proper factors include all divisors of a number (other than the number itself)!

So for example with respect to the number 12, its proper factors (i.e. natural numbers which divide evenly into this number) are 1, 2, 3, 4 and 6 (So 12 is here excluded as a proper factor).

Basically if one accumulates the total of all  proper factors numbers up to n, they will match to ever higher degrees of percentage accuracy the corresponding frequency of the Riemann zeros up to t, where n = t/2π.

Now the reason why n = t/2π is that frequency of factors is measured on a linear scale (the number line) whereas the Riemann zeros are measured on a circular scale (based on the unit circle). And of course where the radius of the unit circle = 1, therefore its corresponding circumference = 2π. Therefore to convert the "circular" units to which the frequency of Riemann zeros relate to the "linear" units (representing the accumulated total of factors) we must divide by 2π.


One might now perhaps ask why there should be such a close relationship as between the accumulated factors of the natural numbers and the Riemann zeros!

And the answer gets to the very heart of this key distinction that I was making (in my previous blog entry) as between the quantitative notion of number independence and the qualitative notion of number interdependence respectively.

So we can indeed - from one valid perspective - start off by viewing the primes as the "independent" building blocks of the natural number system.

However as I was at pains to point out in my previous entry, the very process of multiplication causes a switch from the quantitative notion of number "independence" to the qualitative notion of number "interdependence ". 

Therefore though we can initially look - with reference to our illustration - at the two primes "2" and "3" as "independent" numbers in quantitative terms, through the very process of multiplication, a new relationship of number interdependence is created (in a qualitative manner). In other words, in the context of "6", both "2" and "3" (as proper divisors of this number) acquire a new qualitative resonance (which is unique for each number).

So putting it simply, once we uncover a (proper) factor of a number, we thereby identify the qualitative identity of that factor (through its unique relationship with that number).  


So again to illustrate with reference to "12" which we have already mentioned, the (proper) factors 1, 2, 3, 4 and 6 all acquire a distinctive qualitative identity (through their unique relationship with the number 12).

Now these factors include composite numbers (viz. 4 and 6) as well as primes. 

However, as we know the composites - though depending on the primes - can then be given a Type 1 quantitative identity (as numbers on the 1-dimensional number line). However when considered in relation to other numbers (as factors) they too now acquire a qualitative relationship (through their unique interdependence as factors with those numbers). 
However, it is equally true that all such composite numbers already express a  relationship with respect to a unique combination of primes. So these unique combinations of primes themselves find a new qualitative significance through their subsequent relationship to other natural numbers (as factors). 

Thus it is utterly fallacious - as in conventional mathematical terms - to attempt to view the number system in a merely reduced (Type 1) quantitative manner composed of independent building blocks represented by the primes.

Properly understood, the number system represents a dynamic interactive relationship of both quantitative "independence" and qualitative "interdependence" respectively.

Therefore from one valid perspective, the quantitative ordering of the primes (with respect to the natural numbers) has no strict meaning apart from the corresponding qualitative nature of the natural numbers (through the combined relationship of the primes as factors).

Equally from the opposite perspective, the qualitative nature of the natural numbers (in their unique combinations of factors) has no strict meaning apart from the  "independent" nature of the primes   (as quantitative building blocks).


Therefore from this dynamic perspective, the ultimate nature of the number system approaches pure synchronicity in an ineffable manner (where the primes and natural numbers mutually reflect each other in an identical fashion).

Riemann Hypothesis: New Perspective (1)

Recently on my companion blog "Spectrum of Development", I have been highlighting a fundamental unrecognised problem with conventional mathematical interpretation, which in fact is the important basis for the proper distinction of both addition and multiplication respectively.

This problem relates to the fact that in conventional terms no distinction is made as between independent and interdependent units respectively.

Now whereas the independent aspect is identified with the quantitative nature of number in cardinal fashion, the corresponding interdependent aspect is then properly identified with its unrecognised qualitative aspect in an ordinal manner.

So to preserve this distinction as between quantitative (i.e. analytic) and qualitative (i.e. holistic) aspects, we must now define number in a dynamic interactive manner (combining notions of both relative independence and relative interdependence). And I refer to these two complementary aspects of the number system as Type 1 and Type 2 respectively.


The natural numbers are then defined in Type 1 terms with respect to the default dimension of 1.

So from a quantitative perspective, the natural numbers - and by extension all real numbers - are viewed as existing on the (1-dimensional) number line.

Once again in Type 1 terms, if we take any natural number, it will appear to be composed of independent units (of a strictly homogeneous nature).

So for example, in quantitative terms, 3 = 1 + 1 + 1. So these independent units (which are completely interchangeable with each other)  thereby lack any qualitative distinction!


However it is all subtly different from a Type 2 perspective.

Here, with respect to each natural number - in inverse terms - the dimensional number varies with respect to a default base number of 1.

And as we shall presently see, this Type 2 aspect now properly defines the true qualitative nature of number (where each number is viewed as being potentially related to all other numbers).

And it is this distinction as between the Type 1 and Type 2 aspects that properly defines the corresponding distinction as between addition and multiplication respectively.

So again in Type 1 terms, 3 is defined with respect to the addition of its 3 "independent" units. It is important in this context to appreciate that "independence" is understood in a relative - rather than absolute - manner!

Thus, 3¹ = 1¹ + 1¹ + 1¹.

However by contrast in Type 2 terms, 3 is defined with respect to the multiplication of its 3 "independent" units.

Thus 1³ = 1¹ * 1¹ * 1¹.

Now we can easily envisage this Type 2 aspect in geometrical terms, by considering a simple cube (with side 1 unit).

So we can readily appreciate that this represents a 3-dimensional object with its (base) side = 1 unit.

However, when one carefully reflects on the matter, the nature of the units comprising 3 in Type 2 is quite distinct from their corresponding nature in Type 1 terms.

Once again whereas these units are relatively independent (and fully interchangeable with each other) in Type 1 terms, this is not the case from the Type 2 perspective (where they are clearly related to each other in an ordered manner). So the 3 units here represent the length, width and height of the cube respectively (which can only be appreciated through their ordered relationship with each other).

So therefore in the Type 1 case, we consider the units in a quantitative manner as relatively independent of each other; then in Type 2 terms we properly consider the units in a qualitative manner with respect to their relative interdependence (i.e. their shared relationship with each other). 

We could equally refer to the Type 2 aspect as comprising the ordinal relationship between the units.

So again in Type 1 (cardinal) terms, 3 = 1 + 1 + 1 (in a quantitative manner).

Then in Type 2 (ordinal) terms, 3 = 1st + 2nd + 3rd (in a qualitative manner).

Now once again the (separate) three cardinal units, which are homogeneous in Type 1 terms, thereby lack qualitative distinction; however in reverse fashion the (combined) three ordinal units in Type 2 terms, lack quantitative distinction. This can then be indirectly demonstrated in a quantitative manner through adding the 3 roots of 1 (= 0).

Now what is fascinating to observe here is that what represents multiplication (from a Type 1 perspective), equally represents addition (in Type 2 terms).

So again in Type 1 terms,  1¹ * 1¹ * 1¹ = 1³.

However in Type 2 terms, 1³ = 1^{1 + 1 + 1}.

Therefore what we have really shown here is that there are two distinct forms of addition (which relate to independent and interdependent units respectively).

Thus with reference to a simple example, if we have a cake comprising 3 equal slices, one can readily appreciate that in quantitative (cardinal) terms the cake = the sum of its 3 (independent) slices.

So properly, we are viewing the cake here in quantitative terms as the sum of its 3 parts.

However we can equally view the cake in qualitative (ordinal) terms as the sum of the 3 (interdependent) slices  i.e. as comprising 1st, 2nd and 3rd slices. 

So properly we are  now viewing the cake here in true qualitative, i.e. whole terms (as the relationship between its three slices).

Of course, in dynamic experiential terms, both quantitative (part) and qualitative (whole) aspects necessarily interact. So explicit recognition of the quantitative aspect requires corresponding implicit recognition of the qualitative; likewise explicit recognition of the qualitative implies implicit recognition of the quantitative. 

However the crucial reductionism in conventional mathematical terms entails that no distinction is possible as between quantitative (part) notions and qualitative (whole) notions.

In other words in every context, the (qualitative) whole is absolutely reduced in terms of its mere (quantitative) parts.

And just as we have two type of addition (relating to Type 1 and Type 2 units respectively), equally we have two types of multiplication.

So in Type 1 terms 1¹ * 1¹ * 1¹ = 1³. However in Type 2 terms, 1^{1 *  1 *  1} = 1³. However the dimensional number 3 now relates to the sum of (relatively) independent, whereas previously it represented (relatively) interdependent units.

In other words, when we perform multiplication with respect to (relatively) independent units in Type 1 terms, they are thereby transformed to their (relatively) interdependent status (from a Type 2 perspective).

However when we perform multiplication with respect to (relatively) interdependent units in Type 2 terms, they are likewise transformed - in reverse manner - to their (relatively) independent status (from a Type 1 perspective).

Thus we properly have two types of addition with respect to relative independence and relative interdependence respectively (where the status in each context - as independent or interdependent respectively - remains unchanged).

However, equally we properly have two types of multiplication with respect to relative independence and relative interdependence respectively (where the status in each context - as independent or interdependent respectively - is altered). 

The huge question then arises as to the consistency with respect to both types of addition and both types of multiplication!

In conventional mathematical terms, because these distinctions (ultimately reflecting the quantitative and qualitative aspects of number) are completely overlooked, in effect this consistency is blindly assumed to exist.

However, when properly understood, the Riemann Hypothesis can be viewed as a fundamental requirement for preserving such consistency (with respect to both addition and multiplication).