Where Science and Art Coincide (3)

Again the general finite expression for the Zeta 2 equation is given as

1 + s¹  + s²  + s³  +….. + s^{t – 1}  (where t is prime)

The simplest of the Zeta 2 zeros (which acts as the important template for all other Zeta 2 zeros) arises in the case where t = 2,

i.e.  1 + s¹  = 0,  so that  s¹  =  – 1. 

Now t in this context refers to the qualitative notion of number (as representing a dimension).

And this is defined by its two ordinal members i.e. 1st and 2nd respectively.

However, as we have seen the 1st dimension is always 1 (and thereby indistinguishable from its cardinal definition). Also as this is necessarily always one of the t roots of 1, in this sense it is not unique.

Therefore when 2 represents a dimensional number, its one unique (i.e. non-trivial) root relates to its 2nd dimension which is given as – 1.


Thus there is an important complementary connection as between the notion of an ordinal dimension and its corresponding root.

So in this case the 2nd dimension (which has a qualitative meaning) is closely associated with the 2nd root of 1 (in quantitative terms).

Thus in quantitative terms – 1 is recognised in Conventional Mathematics as the 2nd root of 1 which geometrically can be represented as a point on the unit circle (in the complex plane).

However what is not at all realised is that – 1 has also a distinct qualitative meaning (which provides the true interpretation of the ordinal notion of 2nd in this context).


It cannot be stressed strongly enough that this qualitative meaning simply cannot be understood in conventional mathematical terms.

Once again Conventional Mathematics is defined in a linear (1-dimensional) rational manner. This implies in the case of fundamental polarities (such as quantitative and qualitative) that just one isolated pole is taken as the exclusive frame of reference.

So in Conventional Mathematics number is solely interpreted with respect to its quantitative aspect. Therefore we cannot incorporate qualitative meaning - except in a grossly reduced fashion - in terms of such 1-dimensional interpretation.

However once we move to 2-dimensional interpretation, it is indeed possible to define both quantitative and qualitative aspects of meaning to number (without reducing one in terms of the other). 

So in the case of 2 (as dimension) the 1st provides us with the quantitative aspect of interpretation. However the 2nd dimension now relates to the true qualitative meaning (implied by the ordinal notion of 2nd).


As we have seen this is given as – 1. Though this indeed does have an indirect analytic quantitative meaning (now in a circular rather than linear fashion) in direct terms this relates holistically to the negation of the form (that is implied through conscious rational understanding).

In other words the true qualitative (dimensional) meaning of – 1 relates to the (literal) negation of 1-dimensional  interpretation (based as it is on just one isolated pole of reference).

Implicitly,  the very ability to successfully negate in this manner, implies the experiential nondual recognition that at an unconscious level both positive and negative poles necessarily co-exist (as identical). Thus the very recognition of the qualitative requires thereby the (temporary) negation of the quantitative aspect (which hitherto had been given an absolute dominance).

Thus – 1 represents the unconscious negation of the quantitative pole (1-dimensional interpretation) which had been hitherto absolutely posited in a merely conscious rational manner.

So once we move to 2-dimensional interpretation, we must explicitly include both conscious (analytic) and unconscious (holistic) type interpretation, representing the interaction of both quantitative and qualitative type meaning.

Now once again  let us be very clear about this! There is no way this process can be short-circuited so as to be understood in a conventional mathematical manner (as this would once again just represent the reduction of qualitative to quantitative meaning)!

And this is the all important message that I am trying to get across here. The very relationship of numbers with each other implies a qualitative aspect (which cannot be properly interpreted in a mere quantitative manner).

So the present interpretation of our treasured number system (and indeed all mathematical relationships) is strictly speaking quite untenable.

Once again we have developed but a highly reduced - and thereby distorted - understanding, which admittedly as a limited extreme case has proved incredibly useful.

However, look at it this way! If this pale reduced version has proven so useful, imagine how much greater the role of Mathematics will be when we begin to understand it properly (without such distortion)!


So once again from a holistic mathematical perspective, to posit (+) simply entails to make conscious.

Corresponding negation (–) then implies to make unconscious.

And just as matter and anti-matter particles will fuse in the generation of physical energy, in like fashion when the positive (conscious) direction of understanding is negated (in an unconscious manner) this leads to the generation of psycho spiritual energy (i.e. intuition).

So just as we can say that 2-dimensional understanding implies the dynamic interaction of both conscious and unconscious aspects (both explicitly recognised), equally we can say that it entails the dynamic interaction of both reason and intuition.

And it is vital to appreciate that in this interaction, that reason and intuition play uniquely distinctive roles.

Basically the rational aspect provides us with (analytic) quantitative recognition (in a finite manner).
The intuitive aspect provides us with (holistic) qualitative recognition (in an infinite manner).

So inevitably just as the qualitative aspect is grossly reduced to the quantitative in conventional mathematical terms, equally the infinite aspect is likewise grossly reduced in a finite manner i.e. where it is misleadingly portrayed as a linear extension of finite notions.

So once again the qualitative dimensional notion of 2nd (in the context of 2) is represented as – 1. Though this can be given indirect quantitative expression (in a circular manner) its direct meaning is qualitative (in a holistic manner).

Furthermore there is always a vertical complementary link as between the qualitative notion of a dimension and its indirect quantitative expression as its corresponding root.

For example the 4th (in the context of 4 as qualitative dimension) has an indirect quantitative expression through the 4th root of 1 as i (again lying on the circle of unit radius). 

However, its direct qualitative meaning is of a  holistic nature.

In fact the great importance of i, in a holistic sense, is that in enables circular holistic meaning (of a paradoxical nature) to be indirectly expressed in a linear rational manner! 


Finally to conclude this entry we can see that the only root of 1 which involves no change (implying a distinct qualitative meaning) is of course the 1st root of 1 which is also 1.

This represents just another way of demonstrating that it is only with 1 as dimension, that qualitative is reduced to quantitative  meaning. For all other numbers as dimensions, a distinctive root value will result (implying likewise a distinctive qualitative meaning).

The great significance of this is that the only value for which the Riemann Zeta function is undefined is where s (the dimensional value) = 1.

From a 2-dimensional mathematical perspective, as the Riemann Zeta Function now entails the mapping as between both quantitative (cardinal) and qualitative (ordinal) type values, this simply becomes meaningless in 1-dimensional terms.

So once again strictly speaking the Riemann Zeta Function (and its associated Riemann Hypothesis) cannot be properly interpreted in conventional (1-dimensional) mathematical terms.

Where Science and Art Coincide (2)

The reduced nature of conventional mathematical interpretation is highlighted by the manner in which ordinal are assumed to be directly implied by their corresponding cardinal notions.

Therefore, 1st and 2nd (in ordinal terms) are assumed to follow directly from the corresponding notions of 1 and 2 (in cardinal terms).

And as the cardinal notion of number is defined in a merely quantitative manner, this reduced interpretation likewise creates the illusion that ordinal notions can likewise be dealt with in the same fashion.

This in fact represents clearly the 1-dimensional nature of Conventional Mathematics where interpretation is based on just one isolated polar reference frame i.e. quantitative.

This means, as I have repeatedly stated, that from this perspective qualitative is always reduced to quantitative meaning.


However, properly understood, cardinal and ordinal relate to two distinct notions of number that are quantitative and qualitative with respect to each other.

The essence of cardinal meaning is that a number is taken as representing a collective whole identity.

So 2 in this cardinal sense represents a collective whole identity (with a quantitative meaning).

Thus the individual units of 2 would be represented  is homogenous terms as 1 + 1. In other words these units, in being exactly similar, lack any qualitative distinction!.

Cardinal numbers are thereby treated in absolute terms as independent (i.e. where their quantitative nature is independent of qualitative meaning).


However the ordinal notion of number is of a uniquely distinct nature.

Once again we have seen that the quantitative notion of number has an independent collective identity (with individual units lacking any unique distinction).

It is quite the reverse with respect to the true qualitative notion of number! Here the number lacks any overall collective identity (as quantitative). However the individual units now are uniquely distinct.

So from this qualitative perspective, the number 2 is uniquely defined by its 1st and 2nd members (in ordinal terms).

The key distinction therefore as between the quantitative and qualitative is that the quantitative is based on the notion of the independence of a collective number group, whereas the qualitative is based on the corresponding notion of the interdependence of the individual members of that group!


Thus again the quantitative notion of 2 is represented through the Type 1 aspect of the number system as 2¹.

This illustrates the pure notion of addition where 1 + 1 = 2 (i.e. 2¹).

Thus the pure notion of addition in this context is directly associated with the quantitative aspect of number.


The corresponding qualitative notion of 2 is represented though the Type 2 aspect of the number system as 1².

This illustrates the pure notion of multiplication where 1 * 1 = 2 (i.e. 1²).

Likewise the pure notion of multiplication is directly associated with the qualitative aspect of number.

Then when both base and dimensional numbers ≠ 1, both quantitative and qualitative transformations of number are involved.

 

Now it is important to understand that once we recognise the two aspects of the number system, that we necessarily move to a dynamic interactive interpretation (where both aspects enjoy a merely relative identity).

For example, if we reflect on it for a moment, the conventional mathematical notion of absolute independent number entities is strictly nonsense. For if numbers were independent in this sense then there would be no possibility of relating then with other numbers!

So we cannot in truth operate satisfactorily with the conventional - merely quantitative - cardinal notion of number.

Likewise we cannot operate with merely the qualitative ordinal notion.

For example, in the pure ordinal interpretation of the number 5, we cannot fix the 1st position with any specific number . Therefore any of the five members can be the first. This likewise means that each member can be 1st, 2nd, 3rd, 4th and 5th respectively. Thus to make unique ordinal distinctions we need to unambiguously fix, in any context, the 1st member (which is indistinguishable from the cardinal notion of 1).

Thus the (quantitative) cardinal notion of number automatically implies the ordinal (in being able to relate numbers to other numbers).

The (qualitative) ordinal notion of number automatically implies the cardinal (in being able to independently fix the 1st number in any relevant context).


Thus in dynamic experiential terms, both cardinal and ordinal notions continually interact and are necessary for each other.

The great significance of this is that our customary mathematical interpretation of the number system is strictly speaking completely untenable (as it is built on absolute cardinal notions).

Once we recognise the dynamic interaction of both quantitative (cardinal) and qualitative (ordinal) notions, the question arises as to the mutual consistency of both types of interpretation!

Alternatively the question arises as to how we can consistently convert qualitative (ordinal) notions in quantitative (cardinal) terms; equally from the other perspective, the questions arises as to how we can consistently convert quantitative (cardinal) notions in qualitative (ordinal) terms.


From another perspective the cardinal notion of number is directly associated with conscious (analytic) type interpretation;
The ordinal notion - by contrast - is directly concerned with unconscious (holistic) type interpretation.

The problem therefore arises as to how the mutual identity of both types of interpretation can be obtained.

Once again this can be approached from two complementary directions:

(i) starting with conscious (analytic), we seek to establish its mutual identity with unconscious (holistic) type interpretation;

(ii) starting with unconscious (holistic), we seek in inverse fashion to establish its mutual identity with conscious (analytic) type interpretation.


Now the significance of the Zeta 1 zeros is that they provide the answer to (i).

The significance of the Zeta 2 zeros is that they provide the answer to (ii).

Of course ultimately both of these aspects are themselves understood as identical.

So ultimate understanding of the number system simultaneously marries both cardinal and ordinal interpretation of number at an analytical with the corresponding understanding of the Zeta 1 and Zeta 2 zeros at a holistic level.

Here at the same time both the unity of both the analytic and holistic aspects of the number system become inseparable from the corresponding unity of both conscious and unconscious aspects of personality.


So the key importance of both the Zeta 1 and Zeta 2 zeros is that they represent the perfect shadow counterpart of our customary cardinal and ordinal appreciation of the number system.

Put in an equivalent manner, they represent the perfect holistic (unconscious) counterpart to our customary analytic (conscious) interpretation of the number system.

Where Science and Art Coincide (1)

We return again to the (non-trivial) zeta zeros.

However we must keep reminding ourselves that there are in fact two complementary sets of these zeros both of which are vitally necessary for true appreciation of their very nature.

So I refer to these two sets as Zeta 1 and Zeta 2.

The Zeta 1 correspond to the customary set of zeros recognised (though not their true nature) in conventional mathematical terms.

Once again I refer to this infinite series as ζ₁(s), where

ζ₁(s) = 1^–s  + 2^–s  + 3^–s  + 4^–s  +……..,

 

So the (non-trivial zeros) are solutions for the equation,

1^–s  + 2^–s  + 3^–s  + 4^–s  +…….. = 0,

Assuming the Riemann Hypothesis is true these solutions occur in pairs and are all of the form

s = a + it and s = a – it respectively.


The Zeta 2 corresponds to an alternative set of zeros corresponding to a finite series where both base (quantitative) and dimensional (qualitative) numbers are inverted with respect to the Zeta 1.

So whereas s represents a dimensional value with respect to the Zeta 1, it represents a base value with respect to the Zeta 2; likewise, whereas the natural numbers 1, 2, 3, 4,.... represent successive base numbers with respect to the Zeta 1, they represent successive dimensional numbers with respect to the Zeta 2.

So the Zeta 2 is (initially) defined as the finite series

ζ₂(s) =  1 + s¹  + s²  + s³  +….. + s^{t – 1} (with t prime), .

So the (non-trivial) zeros arise in this case as solutions for ζ₂(s) = 0,

i.e. 1 + s¹  + s²  + s³  +….. + s^{t – 1}  = 0.

Now these solutions in fact conform exactly to the t roots of 1 (with the omission of the non-unique root of 1).

In other words as 1 is always one of the t roots of 1, in this sense it represents a trivial root.

So the other roots - which are uniquely defined for all prime number roots - thereby represent the non-trivial roots.

In certain respects, I had already become aware of the crucial starting significance of this Zeta 2 Function from about the age of 10 and it may help to illuminate its significance by reciting it again.

When we square the number 1, this can be written as 1².

Now clearly no quantitative change takes place in the expression which is still 1.

However, a qualitative change has taken place in the nature of units involved. In other words we have moved from linear (1-dimensional) to square (2-dimensional) units.

Now I use "base" here to refer simply to a number (which is then raised to a certain value).  

So in this context whenever the base number remains fixed as 1, the corresponding dimensional number takes on a purely qualitative meaning.

Therefore in the expression 1², 2 has a purely qualitative meaning.

The upshot of this is that every number can be given both a quantitative and a qualitative definition (according to what I refer to as the Type 1 and Type 2 aspects of the number systems respectively).

The Type 1 aspect is geared to the standard quantitative definition where every base number (as a variable quantity) is defined with respect to a fixed dimensional value of 1.

So 2 from this quantitative perspective is defined as 2¹.

The Type 2 aspect, by contrast is geared to the (unrecognised) qualitative definition, where 1 (as fixed base number) is defined with respect to a variable dimensional quality.

 So 2 from this complementary qualitative perspective is defined as 1².


Now the really important thing to grasp is that Conventional Mathematics is solely defined in terms of the Type 1 aspect of the number system.

So for example, when we have an expression (where the dimensional value ≠ 1) a merely reduced quantitative result will be given.

Therefore to illustrate from the Type 1 perspective, the result of the numerical expression 2² is given in a reduced quantitative manner as 4 i.e. 4¹. 

Now if we looked on this expression geometrically we would represent it as a square (of side 2) which is qualitatively different from 4 linear (1-dimensional) units.

However because of the grossly reductionist nature of Conventional Mathematics, this crucial qualitative distinction is completely overlooked.

In other words, in a very precise qualitative manner, the fundamental paradigm of Conventional Mathematics is 1-dimensional.

I have gone into the deeper nature of what this means in several places.

Basically what it entails is that in any context that qualitative meaning is reduced to quantitative interpretation. This is turn results from using just one isolated polar reference frame.

In other words, in dynamic terms all experience is necessarily conditioned by the interaction of fundamental poles. These relate (i) to the interaction of external (objective) and internal (subjective) aspects and (ii) quantitative (analytic)) and qualitative (holistic) aspects.

Therefore to avoid such relative interaction, from a 1-dimensional perspective, in every context,  interpretation is rigidly frozen in terms of just 1 pole.

So for example numbers are treated as absolute objective entities (thus avoiding dynamic interaction with the internal subjective aspect); Likewise numbers are viewed in a merely quantitative analytic manner thereby avoiding dynamic holistic interaction (of a qualitative kind).


As I say, even as a 10 year old, I could see that there was something fundamentally wrong with mathematical interpretation.

Though of course I had not yet the intellectual capacity to properly articulate the nature of this problem, my conviction was so strong that I never subsequently accepted the highly reduced nature of Mathematics.

So I was soon after determined - even if I never was to receive any support - to reformulate basic mathematical notions such as number, in a properly coherent dynamic interactive manner (as I saw it).

It initially baffled me as to why everyone was not likewise identifying this obvious reductionism as the most significant basic problem intimately affecting all mathematical interpretation!

Unfortunately I eventually came to realise that  established mathematical conventions are so strong, that aspiring mathematicians in effect quickly learn to conspire with accepted practice. Then this mode of thinking becomes so habitual that its crucial limitations become quite impossible to see clearly.


We have already come a long way!

So we see that every number - and by extension every mathematical notion - has distinctive quantitative and qualitative aspects (in terms of  the Type 1 and Type 2 systems of interpretation).

Secondly, as both of these aspects necessarily interact in experience, we need to move to a dynamic  interpretation of number that is relative - rather than absolute - in nature.


Just as we have properly have Type 1 and Type 2 aspects to the number system, this equally implies that we need Zeta 1 and Zeta 2 aspects to properly define the Riemann Zeta Function (and of course its associated Riemann Hypothesis).

And as the Type 2 refers directly to the qualitative aspect of the number system, the Zeta 2 refers directly to the qualitative aspect of interpretation (with respect to the Riemann Zeta Function).

Now the Zeta 2 Function - when appropriately interpreted in a qualitative manner (i.e. where a qualitative interpretation is given to the notion of number as dimension) leads to the startling realisation that potentially an unlimited number of possible mathematical interpretations (all with a limited relative validity) exist.

So once again Conventional Mathematics is defined in a fixed absolute type manner (even when dealing with dynamic interactions).

This reflects that fact that is defined solely in a 1-dimensional fashion (always using isolated independent poles of reference).

This then leads to the inevitable reduction (in any context) of qualitative to quantitative meaning.

Now I am not for a moment denying the great value of this type of Mathematics! However it is vital to realise that it represents just one special limiting case.

In all other mathematical interpretations (corresponding to dimensional numbers ≠ 1), a dynamic relative interpretation is taken of mathematical symbols, where quantitative and qualitative aspects (while preserving a certain relative independence) are seen to interact with each other (in a relatively interdependent manner).


A truly startling deficiency regarding Conventional Mathematics is that by its very means of interpretation, it is totally lacking any genuine notion of interdependence.

Just reflect on this simple observation for a moment! If numbers can be absolutely defined as independent entities, then how can we then establish their interdependence with other numbers?

Alternatively, in an equivalent manner, how can we derive  the ordinal nature of number from its cardinal identity?

Well! the answer to both questions is simply that we can't (without gross reductionism).

If you can appreciate this, then you have already taken the first step towards realising the crucial role of the zeta zeros in the number system.