Do Numbers Evolve? (7)

In all that has been said so far, it is plain that we have in fact two distinct notions of number, which dynamically interact in experience.

For example the number 3 can be given an independent (cardinal) existence or alternatively an interdependent (ordinal) type definition.

Thus, from the first perspective, 3 is viewed in quantitative terms, which can be expressed in terms of its component units as 1 + 1 + 1 .So the independent (absolute) units are all homogeneous in nature (thereby lacking any qualitative distinction).

However 3 can also be given an interdependent ordinal type definition, which is expressed in terms of its component members as 1^st + 2^nd + 3^rd. Thus the units here are of an interdependent (relative) nature (thereby lacking quantitative distinction).

I have dealt with this latter ordinal (Type 2) notion of number in my previous entries. This established the vitally important fact that associated with it is a simple function (Zeta 2) with a corresponding set of zeros. These in effect show how to convert Type 2 qualitative type meaning in a (reduced) Type 1 quantitative manner. So the various prime roots of 1 (excluding 1) can thereby be indirectly used to uniquely express its various ordinal members.

So again the 3 roots of 1 can be used to express the unique ordinal nature of 1^st 2^nd and 3^rd members (in the context of 3). However since all ordinal type relationships necessarily entail the fixing of position with respect to one member in an independent fashion, one solution i.e. 1 is thereby trivial in this respect!  

Once again therefore the importance of the (unrecognised) Zeta 2 zeros is that they enable the unique conversion of the  qualitative (Type 2) ordinal nature of number in an indirect quantitative (Type 1) manner.

It is very important to appreciate this fact, as the Zeta 1 (Riemann) zeros can be then shown to play a direct complementary role with respect to number.

The general Zeta 2 equation can be expressed as follows:

ζ₂(s) =  1 + s¹  + s²  + s³  +….. + s^{t – 1}   (with t prime). However ultimately this can be extended to all natural numbers.

Now this can equally be written as

ζ₂(s) =  1 + s^{– 1}  + s^{– 2}  + s^{– 3}  +….. + s^{– (t – 1)}  

Therefore the zeros for this function are given as

ζ₂(s) =  1 + s^{– 1}  + s^{– 2}  + s^{– 3}  +….. + s^{– (t – 1)}   = 0.

The corresponding Zeta 1 equation, i.e. the Riemann zeta function can be expressed by the infinite equation.

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

And the zeros for this function are given by

ζ₁(s) =   1^{– s}  + 2^{–  s} + 3^{– s}  + 4^{– s}   +…..  = 0.

Notice the complementarity as between both expressions!

Whereas the first expression represents a finite, the second represents an infinite series of terms.

Then whereas the natural numbers 1, 2, 3, …. represent dimensional powers in the Zeta 2, they represent base quantities with respect to the Zeta 1 and in reverse fashion whereas the natural numbers represent base quantities with respect to the Zeta 1, they represent dimensional powers with respect to the Zeta 2.

Now in the case of the Riemann (Zeta 1) function we start with the notion of numbers, expressed as the unique product of primes (representing base quantities) as independent entities.

However this begs the obvious question of how to express the corresponding interdependence of these numbers in their consistent interaction with respect to the overall number system!

In other words, in dynamic interactive terms, quantitative independence (with respect to individual numbers) implies the opposite notion of qualitative interdependence (with respect to their overall relationship with each other).

Thus the Riemann (Zeta 1) zeros represent an infinite set of paired numbers that uniquely expresses in Type 2 fashion the qualitative interdependent nature of the number system.

So just as the Zeta 2 zeros, as we have seen convert (Type 2) qualitative ordinal type natural number notions in a (Type 1) quantitative manner, in reverse fashion, the Zeta 1 zeros convert (Type 1) quantitative cardinal type prime notions in a (Type 2) qualitative manner.

Indeed we could rightly say - though Conventional Mathematics has no way of adequately interpreting this notion - that the Zeta 1 zeros express the collective ordinal nature of the primes (with respect to the natural number system).

Indeed in even simpler terms, the Zeta 1 zeros express the holistic basis of the cardinal number system.

Do Numbers Evolve? (6)

As we have seen there are two possible extremes in terms of the appreciation of number.

At one extreme we attempt to separate polarities (such as external and internal, quantitative and qualitative) in an absolute independent manner. This leads to the apparent existence of numbers as absolute fixed entities (of phenomenal form).

This in fact represents the abstract analytic approach to number that characterises conventional mathematical interpretation.

At the other extreme we attempt to view such opposite polarities ultimately as totally interdependent with each other  leading to the appreciation of number as pure energy states (ultimately of an ineffable nature).

For simplicity I refer to the first as the analytic aspect of interpretation (identified with linear reason) and the second to the corresponding holistic aspect (identified with pure intuition, that indirectly has a circular paradoxical interpretation in rational terms).

Actual experience of number is implicitly of a relative nature that necessarily falls between the two extremes. So (absolute) analytic interpretation represents just one limiting perspective that can be approached (but never fully achieved).

Likewise (purely relative) holistic interpretation represents the other limiting perspective that can be approached (but again never fully achieved).

So both aspects are in fact controlled by a fundamental uncertainty principle.

So the attempt to achieve analytic understanding (in an absolute manner) therefore tends to blot out recognition of the equally important holistic aspect; equally the attempt to achieve holistic understanding in a purely relative manner, likewise tends to block out corresponding recognition of the analytic aspect.

Conventional Mathematics is however characterised by such an extreme attention on the analytic aspect, that the holistic aspect (which in truth is equally important) is not even formally recognised.

So it must be said – and continually repeated that current Mathematics – despite its admitted great achievements in the quantitative arena is hugely unbalanced and thereby hugely distorted in nature.

Now, properly understood, the zeta zeros (Zeta 1 and Zeta 2) represent the holistic extreme with respect to mathematical interpretation (where it approaches a purely relative state).

Again it might be instructive to illustrate this with respect to the first of the Zeta 2 zeros, indirectly represented by the two roots of 2.

So these two roots, + 1 and – 1, now relate directly to the opposite polarities (such as external and internal) that condition all phenomenal experience.

Now when experience becomes highly refined in an increasingly dynamically interactive manner, one better realises that each pole only has meaning in terms of the other.

So as soon as one posits understanding with respect to one pole e.g. as a number in objective terms, one quickly realises that this has no meaning in the absence of the corresponding perception of number that is opposite and thereby negative. So now one posits the internal perception of number, before again quickly realising (directly through intuition) that is has no meaning independent of its external object.

Thus a ceaseless dynamic interplay takes place in experience as between two opposite poles that momentarily are identified as separate in quantitative terms (in a fixed rational manner). However these poles are then equally experienced as complementary and ultimately identical (in a directly intuitive manner). So through the interplay, the opposite poles continually keep switching as between their positive and negative identities.

Now indirectly this holistic understanding can be represented as + 1 – 1 = 0. And it must be clearly recognised that each pole (external and internal) has both positive and negative states that continually alternate between each other.

So here we combine the momentary quantitative existence of each pole as independent with the combined qualitative existence of both poles as interdependent.

And such quantitative interdependence = 0 (which in holistic terms entails a purely qualitative meaning i.e. without quantitative identity)

  

And if we take any prime number and then express its prime roots, all of these (except 1) will be unique in nature and cannot recur with any other prime.

So in  holistic terms, each prime number is thereby uniquely expressed through its ordinal members indirectly expressed in a quantitative manner by all roots (except 1) .

And the momentary separate identity of each root (as quantitative and independent) is perfectly balanced in each case by the collective identity of all roots (as qualitative and interdependent).

Now the ultimate limit of such understanding approaches a timeless (and spaceless) state where we can no longer distinguish the (separate) quantitative identity of each member from the (collective) qualitative identity of all members. And this represents ineffable reality (of pure emptiness).

So properly understood the evolution of the number system spans the holistic extreme of pure ineffable reality (of emptiness) and the corresponding analytic extreme of absolutely fixed phenomenal reality (of form).

Thus properly understood in experiential terms, both analytic and holistic aspects interact as matter and energy in the ceaseless transformation of number.

Now we have seen in Type 1 terms that all natural numbers are viewed in quantitative terms as the unique product of natural numbers.

So for example 6 is uniquely presented as 2 * 3.

However there is a complementary Type 2 approach to the primes where the natural numbers in ordinal terms are the building blocks of each prime.

Besides prime numbers (as dimensions) we also have natural numbers as dimensions. However the roots of these natural numbers can be directly derived from constituent primes.

So in type 1 terms,

2¹ * 3¹ = 6¹

Equally 1² * ³ = 1⁶

Then when we find the six roots of 1, holistic order is fully preserved in that these roots while preserving a relative quantitative independence can again be collective combined to give a total of zero (representing their qualitative interdependence)

In this way the primes can be seen to be unique in both Type 1 and Type 2 terms (though the order of relationship with the natural numbers is inverted in each case).

In fact both relationships - ultimately expressing the two way interdependence of primes and natural numbers - mutually imply each other.

Do Numbers Evolve? (5)

I will attempt here to provide additional clarification on the holistic - as opposed to the standard analytic - interpretation of number.

Once again analytic interpretation is by its very nature linear (i.e. 1-dimensional) thus enabling numbers to be interpreted with respect to their reduced quantitative values.
This entails interpretation  with single polar reference frames (e.g. as unambiguously objective) in an independent absolute manner.

Thus with 1-dimensional interpretation (i.e. single poles of reference) dynamic interdependence (resulting from the interaction of more than one pole) cannot properly be interpreted and is thereby reduced in an independent manner.

Therefore at a minimum we require at least two interacting polar frames of reference to establish genuine interdependence. And in short holistic appreciation relates to the explicit recognition of the nature of such interdependence.

Now all holistic interdependence necessarily starts with the initial recognition of independence (which is conscious posited as the 1st dimension).

However 2-dimensional appreciation combines this 1st dimension (entailing analytic type appreciation) with a second dimension that entails the negation (of what has been posited).


Now this in fact is deeply relevant to the multiplication of two numbers.

In standard analytic terms when we multiply - say - 3 * 5, the answer is given in a reduced 1-dimensional fashion as 15 (i.e. 15¹). However a simple geometrical representation of this relationship will suggest that through multiplication the nature of the units has changed from linear (1-dimensional) to square (2-dimensional) format.

Thus there is something fundamentally missing from the conventional mathematical treatment of multiplication.

So when we probe more deeply into the nature of this simple operation (i.e. 3 * 5) we find that it cannot be properly explained in the absence of both the quantitative notion of independence and the qualitative notion of interdependence respectively.

So imagine 5 units laid out in 3 separate rows (in a rectangular fashion)! Now this implies that we recognise each unit in a (separate) independent fashion. However to then multiply by 3 we must also recognise that the units in each row share a common identity (thus enabling each row to placed in correspondence  with each other).

Now this recognition of interdependence (in a mutual shared identity of each unit) literally entails the (temporary) negation with respect to the (conscious) recognition of a posited independent identity.

Thus the qualitative recognition of a shared identity (through negation of each separate unit) in fact implies the 2nd dimension of understanding in this case.

Thus a comprehensive appreciation of  the multiplication of 3 * 5 entails recognition of the quantitative independence of each individual unit with the qualitative interdependence of all units (i.e. as sharing a common quality).

So comprehensive appreciation is here 2-dimensional, entailing a 1st dimension (relating to independent recognition) and a 2nd dimension (relating to qualitative interdependence in a mutual common recognition).


Now if we were to now properly explain - say - 3 * 5 * 4, this would entail 3-dimensional interpretation. So once again the 1st dimension would relate to the standard analytic appreciation of 60 independent units. However we would now have two layers of interdependence to appreciate. So for example if we arranged 5 units each in 3 rows on a bottom layer, this would entail - as before - the 1st level of interdependence. Then we would could lay each of these rectangles four units high creating a second compounded level of interdependence.

Now remarkably the various roots of 1 (when appropriately interpreted) provide the appropriate means to properly resolve the true nature of multiplication. So the multiplication of 2 numbers requires 2-dimensional interpretation (with a 1st and 2nd dimension applying); the multiplication of 3 numbers would then entail 3-dimensional interpretation (with a 1st 2nd and 3rd dimension applying).

In general the multiplication of n numbers would require n-dimensional interpretation (with a 1st, 2nd, 3rd,....nth dimension applying).


Now - what I refer to as - the Zeta 2 zeros relate to all the dimensions (other than the 1st) which provide the general means for holistically interpreting all such relationships.

So 1 =  xⁿ ;

Thus 1 – xⁿ  = 0.

Therefore (1 – x)(1 + x¹ + x² + x³ + ....+ xⁿ – 1) = 0

Now 1 – x = 0 represents the trivial solution (i.e. x = 1), which relates to the 1st dimension and the initial recognition of the independence of all units.

However 1 + x¹  + x² + x³ + ....+ xⁿ – 1 = 0, provides the equation for establishing the true holistic nature of all higher dimensions.


The simplest possible case (which serves as a holistic template for all others) occurs when n = 2.

So here 1 + x¹ (i.e. 1 + x) = 0; therefore x = – 1.

This is the first of the Zeta 2 zeros and has vitally important role to play.

Basically it serves to express (in an indirect manner) the nature of holistic interdependence in the 2-dimensional case.

Now if we look for the extreme example (of the most highly refined intuitive understanding possible) then – 1 (i.e. the 1st trivial zero) can holistically be understood as representing a pure psycho spiritual energy state (with a complementary interpretation as a pure physical energy state).

So just as anti-matter (when in contact with matter) particles will fuse in a pure physical energy state, likewise this is true of number (which represents the encoded nature of reality in both physical and psychological terms).

Equally all other Zeta 2 zeros can be understood (in their fullest experiential attainment) as representing in holistic terms pure energy states. What this implies is that one can then directly intuit in experience the purely relative nature of an ever increasing number of different frames. This requires therefore great transparency with respect to understanding, where phenomenal rigidity is greatly eroded.


However the Zeta 2 zeros equally play a remarkably important role with respect to our everyday understanding of number (that is not at all well realised).

If we return again to the simplest case of 2, we see that this is identified with a 1st and 2nd dimension that can be holistically represented as + 1 and – 1 respectively.

Now in qualitative terms + 1 simply relates to analytic type understanding (where polar frames are understood as separate). However – 1 represents the unconscious negation of such understanding leading to the directly intuitive realisation (at an unconscious level) of their mutual identity.

Now implicitly such understanding is required to understand the ordinal relationship of 2 members (of a group of 2).

Thus the ordinal identification of 1st and 2nd (with respect to this group of members) implicitly entails corresponding realisation of the first two zeros (trivial and non-trivial).

Likewise the identification of the 1st, 2nd and 3rd (in the context of 3 members) implicitly entails corresponding realisation of the zeros corresponding to n = 3 (with again one trivial corresponding to the root of 1 and the other two non-trivial corresponding to the other two roots).

And in general the ordinal identification of 1st, 2nd, 3rd,....nth (in the context of n members) implicitly entails corresponding realisation of the zeros corresponding to n = n (with again one trivial and the the other n – 1 corresponding to non-trivial solutions).


Thus to put it briefly, the Zeta 2 zeros intimately underlie our everyday analytic appreciation of the ordinal nature of number (as its unrecognised holistic basis). And this unrecognised holistic basis equally implies its unrecognised unconscious basis!

So without implicit interaction of this deepest holistic (unconscious) layer of understanding, the conventional ordinal appreciation of number would simply not be possible.

One important consequence of this is that it demonstrates the merely relative nature of ordinal understanding.
For example we might initially think that the notion of 2nd has an unambiguous identity.

However 2nd (in the context of 2) is distinct from 2nd (in the context of 3) which is distinct from 2nd in the context of 4 and so on!

Thus the notion of 2nd - as indeed all other ordinal number notions - can potentially be given an unlimited number of possible definitions.

Do Numbers Evolve? (4)

I have referred repeatedly to the dynamic interaction as between quantitative and qualitative aspects with respect to number.

Ultimately this interaction relates to the interplay of both the finite (actual) and infinite (potential) notions, which in psychological terms relate to both conscious and unconscious aspects of understanding respectively.

So mathematical objects such as numbers possess an actual existence from a finite (conscious) perspective directly mediated in rational terms; however equally they possess a potential existence from an infinite (unconscious) perspective that is directly mediated in an intuitive manner. And both of these ceaselessly interact dynamically in experience leading to continual transformation with respect to such objects.

So properly, i.e. in a dynamic interactive manner, number thereby necessarily evolves. And this relates not just to the nature of (internal) psychological understanding, but also to the external objects (both of which - by definition - are now necessarily relative to each other).

However as an alternative to the sole use of  quantitative and qualitative terms, I would suggest the corresponding pairing of analytic and holistic (which perhaps appears a little more scientific).

However it is important to point out that I am using analytic in the broader sense in which the terms is commonly used in science, which equates directly with a reduced quantitative interpretation of relationships!
Now analytic has also a well-defined narrower meaning within Mathematics in relation to the treatment of infinite series and limits. However suffice it to say that within Mathematics, more restricted use of the terms "analytic" (and "analysis") are also analytic in the broader sense of the term (in that they are defined solely within a reduced quantitative context).

Therefore to return to my basic position, properly understood all number has both analytic and holistic aspects (in dynamic relationship with each other).

From one important perspective, this is true internally for each number. So, as we have seen the number "2" for example entails both the analytic aspect of "2" as a specific number quantity in cardinal terms, and the holistic aspect of "2" (i.e. twoness) as collectively applying to all possible instances of "2").
So properly understood these two notions are actual and potential with respect to each other.

And because in the dynamics of experience (like approaching a crossroads from opposite directions) polar reference frames continually switch) there is also an important sense, where "2" now refers to a specific number quality (i.e. in the ordinal notion of 2nd) while "2" now attains a collective meaning in the cardinal notion of dimension that now actually applies to all numbers.

Thus in the dynamics of the experience of each number, there is a ceaseless two-way interplay of both analytic and holistic type understanding, through which we are enabled to switch seamlessly as between cardinal and ordinal type appreciation (with respect to both objects and dimensions).


Then from the other important perspective, similar dynamics apply to the number system as a whole.
This then enables us to consistently combine both the cardinal and ordinal identities of all numbers (not is relative isolation) but in full relationship with other numbers.

Now the precondition for such consistency is that a seamless means exists for switching as between both the Type 1 and Type 2 aspects of the number system.

Thus from one perspective we need to be able to seamlessly convert the Type 2 aspect in a Type 1 manner.
Then equally from the alternative perspective we need to be able to seamlessly convert the Type 1 aspect in a Type 2 manner.


Though its significance seems to me to be completely missed by the mathematical community, I will start with the first of these conversions (which in fact is relatively easy to appreciate).

Now we will illustrate here again for convenience with respect to the number "2".

So the standard analytic definition of "2" (as a specific number quantity) is given through the Type 1 aspect as 2¹. So once again the Type 1 aspect is always defined with respect to the default dimensional value of 1.

The corresponding holistic definition of "2" (as the collective number quality of twoness) is given through the Type 2 aspect as 1². "2" now refers directly to a number dimension (rather than a base quantity).

Thus to convert this Type 2 aspect in Type 2 terms, we need in effect to obtain the square root.

So in general terms xⁿ = 1 with in this case x² = 1. So x = + 1 and – 1.

We have now moved to a circular definition of number (with both + 1 and – 1 lying on the unit circle in the complex plane).

However these two results are given but an analytic quantitative interpretation in conventional mathematical terms.
However the corresponding holistic meaning is highly revealing, requiring in effect a uniquely distinctive manner of mathematical interpretation.

+ in this context entails the psychological notion of positing (i.e. making conscious).
– however entails the corresponding notion of negation (i.e. of what is unconscious) thereby representing unconscious understanding.

When understanding is especially refined, as with the fusion of matter and anti-matter particles in physics, unconscious negation (of what is consciously posited) will approach full attainment resulting in a pure intuitive understanding (representing a psycho spiritual energy state).

So strictly speaking the holistic appreciation of each number represents a pure energy state (with complementary physical and psychological meanings).

Thus in effect we have two extremes with respect to the understanding of number (and remember in dynamic terms number as object has no strict meaning independent of such understanding)!

Thus we can appreciate number in the standard analytic fashion as an absolutely existing quantity form (that never changes). Here it is viewed as nothing in qualitative terms

However from the opposite extreme we can appreciate number in the unrecognised holistic fashion as approaching a pure energy state (where it is nothing in quantitative terms).

However properly understood, number experience entails an interaction somewhere between both extremes, where both quantitative aspects (as form) and qualitative aspects (as energy) ceaselessly interact leading to a continual transformation thereby in the nature of each number.

So once again we have the analytic quantitative extreme (recognised through the Type 1 aspect)

Here 2 = 1 + 1 (Strictly 2¹ =  1¹ + 1¹⁾.

So here the two units are defined in a homogeneous quantitative manner (i.e. without any distinctive quality)

Then in the Type 2 system 2 = 1st + 2nd (so both units are now defined as without any quantitative distinction!)

Then when we convert the Type 2 to the Type 1, we can indirectly represent this important reality consistently in a quantitative manner.

So 1st and 2nd are now represented as + 1 and .– 1 respectively.

And + 1 .– 1 = 0!

So the task of converting consistently from Type 2 to Type 1 implies that we can represent the ordinal members of each group uniquely by a set of circular numbers (lying as roots on the unit circle) that always add up to zero.


And this is where the prime numbers can be seen to have an equally valid Type 2 (as well as Type 1) identity.

From the Type 1 perspective, the unique importance of the primes comes from viewing them as the "building blocks" of the natural number system.

So all natural numbers (other than 1) can be uniquely expressed as the product of prime factors.  

However, the primes have an equally important role in Type 2 terms, where however their directional link to the natural numbers is completely reversed.

So from the Type 2 perspective, each prime can be uniquely expressed in an ordinal natural number fashion by its various roots (again except 1).

So for example if we take 5 as a prime, it can be uniquely expressed in terms of its 5 roots (excluding 1 which is common to all roots).

Now these 5 roots provide an indirect (Type 1) means of uniquely expressing in quantitative terms   the various natural number members of 5 (i.e. 1st , 2nd , 3rd, 4th and 5th respectively) in an ordinal manner.

However there is an obvious paradox with respect to the Type 1 and Type 2 approaches.

In the first case, each natural number (except 1) is uniquely defined by its prime members in cardinal terms.

In the 2nd case, each prime is uniquely defined by its natural number members (except 1) in ordinal terms (indirectly expressed in a quantitative manner through its prime roots).

This leads directly to the holistic qualitative recognition of the two-way interdependence of primes and natural numbers in both cardinal and ordinal terms.

In other words a holistic synchronicity entailing the two-way interaction of primes and natural numbers (which is directly qualitative in nature) underlies the deepest workings of the number system.

However though obvious when viewed from the appropriate perspective, the realisation of  this simple fact will permanently elude a mathematical profession that reduces interpretation of number in a merely quantitative fashion.

Do Numbers Evolve? (3)

Just to recap briefly from yesterday's entry!

Every number has two distinctive meanings. So 2, for example represents a specific quantity in cardinal terms; however equally it represents a collective dimensional quality as "twoness" (that potentially applies to all specific quantities).

And both of these meanings in experiential terms are dynamically inseparable from each other.
So every number therefore represents a dynamic interaction with respect to both its quantitative and qualitative aspects (which are complementary).

Then in the dynamics of experience, reference frames continually switch. So 2 now attains a specific quality as 2nd (i.e. the ordinal nature of 2) while the dimensional notion of 2, in complementary fashion, assumes a cardinal identity (which is the conventional meaning associated with a number representing a power or exponent).

So rather than just one unambiguous natural number system that  can be unambiguously defined in rigid absolute terms as;

1, 2, 3, 4,.....,

we now have two complementary aspects of the number system which dynamically interact with each other.

Thus to identify number with its specific quantitative aspect, we assume a default fixed dimensional value of 1.

Therefore, from this perspective, the numerical value of an expression entailing higher powers (i.e. dimensions) is thereby reduced in a 1-dimensional manner.


This quantitative aspect is then represented as:

1¹, 2¹, 3¹, 4¹,.....,


Then in reverse manner to identify number with its collective qualitative aspect, we now in complementary fashion, maintain the base number fixed at 1, while allowing the dimensional value to vary through the natural numbers.

So from this alternative qualitative perspective, the number system is defined as:

1¹, 1², 1³, 1⁴,.....,

I refer to these two aspects as Type 1 and Type 2 respectively.

Both can only be properly understood (thereby mirroring authentic experience) as in dynamic complementary relationship with each other i.e. as quantitative to qualitative (and qualitative as to quantitative respectively).


The quantitative (cardinal) aspect is defined strictly without qualitative meaning.

Thus from this perspective 2 = 1 + 1 (i.e. 2¹ = 1¹ + 1¹).

Thus the two units here are fully homogeneous in quantitative terms (thereby lacking qualitative distinction).

It is the reverse from the opposite ordinal perspective.

Here the two units are represented in qualitative terms as 1st and 2nd (thereby lacking any quantitative distinction).

When one clearly realises that in truth all number operations properly entail both quantitative and qualitative aspects in dynamic relationship with each other, then the key issue arises as to consistency as between both sets of meanings.

This entails that a satisfactory way of converting from quantitative to qualitative (and qualitative to quantitative respectively) necessarily must exist if we are to maintain true confidence in all subsequent operations.

And this is what the zeta zeros essentially relate to, though this is not all yet realised due to the strongly reduced (i.e. merely quantitative) nature of accepted mathematical interpretation.


However just as we have two aspects to the number system, likewise properly we should have two sets of zeta zeros.

I refer to these two two sets as Zeta 1 and Zeta 2 respectively

Now the first set (i.e. Zeta 1) can be identified directly with the Riemann zeros.

However there is an alternative - and simpler - set, whose true function is not properly realised. (which I refer to as Zeta 2).

When we refer back once again to yesterday's blog, I identified two key quantitative/qualitative type relationships with respect to the number system.


So again, illustrating with reference to the number "2", I stated that two notions, which are quantitative and qualitative with respect to each other, are necessarily involved.

Thus we have the specific quantitative notion of 2 in relation to a general collective qualitative notion (as "twoness").

Then when we switch the frame of reference (as in the manner of approaching a crossroads from the opposite direction) we  now obtain a specific qualitative notion of 2 (as the ordinal notion of 2nd) with respect to a collective quantitative notion of 2 (as representing power or dimension).

Note once again how these dynamics are completely short-circuited from the conventional perspective with the quantitative notion of number (both as base number and dimension remaining). So ordinal notions in conventional mathematics are treated in a merely reduced fashion as "rankings" based on cardinal understanding  of  a quantitative nature!


Now in basic terms, as we shall see, the Zeta 2 zeros refer directly to conversion as between quantitative and qualitative interpretation with respect to individual numbers (such as "2").

However we also saw that a more general problem exists with respect to the collective relationship of numbers to the number system (in quantitative and qualitative terms).

Therefore the quantitative general notion of "a number" has strictly no meaning in the absence of the corresponding qualitative notion of "numberness" (that potentially applies in all cases).

Now the famed Riemann zeros (which I refer to as the Zeta 1 zeros) properly relate to this more general problem with respect to the number system as a whole, of ensuring consistency with respect to both the quantitative and qualitative interpretation of number. In direct terms they provide a means of converting as between quantitative and qualitative type usage.

Put another way - both with respect to any specific number and numbers generally - the notion of quantitative independence has no meaning in the absence of the corresponding notion of qualitative interdependence  (that thereby enables numbers to be related to each other).

Therefore we can only properly understand the number system in a dynamic relative manner (entailing the complementary notions of independence and interdependence respectively).

Once again even momentary reflection on the matter should immediately suggest to one that there is something fundamentally wrong with conventional mathematical interpretation.

We insist on interpreting numbers in an absolute independent manner (i.e. with respect to their mere quantitative characteristics). However this begs the obvious question of how numbers can then be related with each other (which assumes some quality of interdependence).

However because such reduced interpretation has now become so ingrained due to an unquestioned consensus, the mathematical community remains blind as to this must fundamental of all issues!

Do Numbers Evolve? (2)

In the last blog entry I argued that the conventional belief in the absolute existence of number is untenable from an experiential perspective.

So all numbers possess both external (objective) and internal (mental) aspects which dynamically interact.

Thus the conventional view of number represents but a special limiting case where both poles are fully abstracted from each other. Now this cannot of course completely occur in experiential terms (which would render understanding of number impossible); however it can be approached in a relative manner.

Thus the conventional absolute view of number (as rigid unchanging entities) is then appropriately understood as just one special - though admittedly important - limiting case with respect to interpretation.

As I have frequently stated this is directly associated with linear (1-dimensional) understanding based on interpretation within single isolated polar reference frames.

So for example the conventional treatment of number in merely quantitative terms - rather than a relationship entailing both quantitative and qualitative aspects - represents such linear interpretation.

However when we recognise the truly relative nature of mathematical understanding, as the interaction of opposite poles such as external and internal, this opens up entirely new vistas where the number can be given a potentially unlimited series of dimensional interpretations.


So we move here from the extremely restricted default position of Conventional Mathematics i.e. as 1-dimensional in absolute terms, to an unlimited number of partial relative interpretations, where each number represents a unique dynamic configuration

Therefore from this relative perspective if the interpretation can change as between differing numbers (representing dimensions) then the objective reality then likewise necessarily changes with respect to all these numbers. So from this enhanced dynamic perspective the dimensional notion of number represents perpetual evolution with respect to its very nature.

Now once again, due to the restricted quantitative bias of Conventional Mathematics, this dynamic notion of number evolution is entirely edited out of the picture.

So to give a simple example, when one raises a number 2 to a non-unitary power (i.e. dimension) such as 2, the result is given in a merely reduced quantitative fashion (i.e. as 1-dimensional)!

Thus 2² = 4 (i.e. 4¹).

Now one can easily appreciate, that when seen in geometrical terms, that 2² represents square rather than linear units. However this qualitative change in the nature of units involved is simply ignored in conventional mathematical terms (with a merely reduced quantitative interpretation remaining).


In fact this reduced view is graphically illustrated in the following quote from Alain Connes (from Karl Sabbagh's "Dr. Riemann's Zeros" P. 205).


“It really is a fantastic step to understand that the square of a number - which is just a geometrical square - and the cube, which is just a geometrical cube - can be added together, even though you would say, "But one has dimension the length squared and the other the length cubed" and you would never add things which have different dimensions. So algebra is an amazing achievement, and once you have formulated things in algebraic terms then they take on a life of their own.”


This brings me directly to consideration of the second fundamental set of polarities that govern all mathematical experience, i.e. whole (collective) and part (individual) which in a very direct way determine this key relationship as between quantitative and qualitative.

So, one recognises a number, as for example "2", both individual and collective aspects are necessarily involved (which are quantitative and qualitative with respect to each other)

Thus in external terms, the individual number object "2" that has an actual existence, has no meaning in the absence of the collective number notion of "2" (that potentially applies to all specific instances of "2").

Put another way the recognition of "2", in any specific case, requires the corresponding notion of "twoness" (that collectively apples to all such possible cases).

Then from the corresponding internal perspective, the individual number perception of "2" - again with an actual existence - has no meaning in the absence of the corresponding concept of "2" (i.e. twoness) with a general potential applicability to all possible cases of "2".


So when the individual recognition of "2" is quantitative (in actual terms), the corresponding collective recognition of "2" (or twoness) is - relatively - of a qualitative potential nature.

However, as always in the dynamics of experience, reference frames can switch, with the individual recognition qualitative and the collective recognition now of a quantitative nature. In effect this qualitative recognition corresponds with the ordinal notion of "2" (as 2nd).

Likewise from this perspective, the collective recognition of "2" (as twoness) is now - relatively quantitative (applying to all actual instances of "2").

Therefore in the dynamics of experience, one keeps switching as between both the quantitative and qualitative notions of "2" in individual terms and equally the quantitative and qualitative notions of "2" (as twoness) with respect to both cardinal and ordinal usage. And this happens both externally with respect to objective recognition and internally (with respect to perception and corresponding concept).

And a similar dynamic interaction is involved with respect to the recognition of any specific number.

We then move on to consideration of the general recognition of number.

Once again this will combine both internal (mental) and external (objective) aspects.

And again the general recognition of an individual number integer in a cardinal quantitative manner has no strict meaning in the absence of the collective qualitative notion of number (as "numberness") that potentially applies in all specific cases.

And then when the reference frames switch we attain the individual recognition of that number in a corresponding ordinal manner (which is qualitative in manner), Then - in relative terms - the collective notion of number attains a quantitative interpretation (as applying to all actual numbers).


Now with respect to conventional mathematical interpretation, all these mutually interacting dynamics are short-circuited in a grossly reduced fashion.

Thus as we have already seen,the external/internal interaction is disregarded with numbers viewed absolutely in objective terms (with a corresponding absolute mental interpretation).

Likewise the individual number "2" is interpreted strictly with respect to its quantitative nature, while the general notion of "2" (insofar as it is recognised) is treated merely with respect to actual occurrences (that are likewise interpreted in a merely quantitative manner).

Then it is somewhat similar with respect to the general recognition of a number with both individual and collective aspects treated in a merely reduced quantitative manner.

However if we are to properly understand the key role that the zeta zeros (both Zeta 1 and Zeta 2) play with respect to the number system, we have to inherently appreciate it in a dynamic interactive manner (where both quantitative and qualitative aspects are equally recognised).

Do Numbers Evolve? (1)

On first impression, this might seem to most people as a somewhat ridiculous question.

Indeed the conventional view - which gives great comfort to so many practitioners in our fast changing world - is that number represents the only thing we can rely on to remain absolutely the same. So from this perspective, the prime numbers for example were the same yesterday, today and will forever remain so here and indeed anywhere else in the Universe (where intelligent beings exist to discover them)!

However on closer examination, strictly this conventional view can be convincingly shown to represent but an illusion (which admittedly however in a reduced quantitative sense has proved of enormous benefit).

In physical terms to accurately classify any object we must be able to identify it with a universal class to which it belongs.
To give  a trivial example, to speak unambiguously with respect to a fruit such as a strawberry, we need to be able to define accurately a universal class to which all strawberries belong.

However we will eventually discover that at the margins difficult problems of identification will exist with a certain degree of arbitrariness as to whether a particular example correctly falls into the relevant class. So the boundaries of our definition are necessarily vague and approximate.

Now we might initially think that this problem does not exist in the mathematical world of "abstract" objects that thereby free us from such physical restraints.

However, paradoxically on closer reflection a much greater degree of mystery attaches to the universal class constituting "number" than any physical class (such as strawberries).

So to unambiguously recognise a particular number we should be able to define the universal class of "number". However this is a far more difficult task that one might imagine.

So for example the development of Mathematics has seen a steady increase in the somewhat exotic objects that are universally recognised as numbers.

Initially, number was identified solely with the natural (counting) numbers which are solely positive.. Then gradually, after much resistance their negative counterparts also cam to be included.
A further advance then led to the inclusion of rational fractions (such as 1/2) in the number system. Then the Pythagoreans in investigating the square root of 2 were, to their horror, to discover a new type of irrational number. Such irrational numbers have now been further refined to include both algebraic (such as √2) and transcendental numbers (such as π).

Another major development with respect to the solutions of polynomial equations led to the recognition of imaginary numbers (based on i as the square root of  – 1).

And in more recent times, further developments led to the inclusion of transfinite numbers and a whole new strange class of numbers (based on the primes) referred to as p-adic numbers.


So over the millennia we have seen a remarkable evolution in the objects that are now recognised as legitimately belonging to the number system. Thus it seems to me reasonable to assume that further extension is likely to take place in the future with as yet unknown number objects becoming included.

Thus there is clearly very fuzzy boundaries existing as to what might be considered as number. To put it more bluntly we are unable to properly define what is number and yet attempt to claim an absolute unambiguous identity for every specific number object encountered.

Now though there is no proper (epistemological) justification for such certainty.
So what really characterises - as I hope to presently  demonstrate at length - the apparent absolute nature of mathematical objects such as numbers, is a largely unquestioned mass consensus, which at bottom is geared to the preservation of a considerable illusion regarding their true nature and indeed the true nature of Mathematics generally.

Let me illustrate this now with respect to one of the simplest, most important and best known numbers i.e. "2".

Now again according to the general mathematical consensus, 2 has an absolute rigid identity that can be successfully abstracted from our changing everyday physical world.

This view, expressed in its extreme fashion for example by G. H. Hardy, looks on numbers as eternally existing in some kind of mathematical Heaven (ungoverned by the laws of space and time).

However on closer reflection this view can be shown to be quite untenable.

The starting point here for more authentic understanding is the recognition that Mathematics is intimately bound up with experience. So therefore we start by examining how the recognition of number experientially unfolds.

Now all experience - including of course mathematical - is governed by twin sets of fundamental polarities that dynamically interact.

The first of these relates to external (objective) and internal (mental subjective) polarities.

Therefore the experience of the number "2" entails both an external pole (i.e. as object) and a corresponding internal pole (as mental perception).

So the experience of the number "2" entails a dynamic interaction of both object and perception (which cannot be meaningfully abstracted in absolute manner from each other).

Put another way, strictly speaking a mathematical object such as "2" has no meaning independent of the corresponding mental perception of "2" with both poles in tandem properly constituting an interactive dialogue of number meaning.

In other words all number understanding has a merely relative validity.

So Conventional mathematics is in fact directly based on a reduced interpretation of such experience. Here the two poles are viewed with respect to their absolute separation (though implicitly in experiential terms this is not possible). Thus the number object (in this case"2") is misleadingly given an abstract absolute objective identity.
Interpretation is then misleadingly viewed as simply mirroring in mental terms (again in an absolute manner) this absolute identity.

In this way in conventional mathematical terms, interaction as between opposite poles (external and internal) is thereby completely edited out of the picture (in explicit terms).

This then is misleadingly associated with the considerable illusion that numbers thereby enjoy an absolute rigid identity (unrelated to time).

However because such reductionism is so entrenched in our mathematical thought processes (conditioned now though several millennia) it is extremely difficult to get mathematicians to address this issue.

Even on the rare occasions when I have seen mathematicians seriously question the  basis of such procedures (in philosophical reflection on their discipline), they still seemed in a sense to operate with split personalities, readily accepting all such reductionism (without question) when operating as mathematicians.
And I accept that there is enormous pressure on professional mathematicians (in maintaining the respect of their peers) to operate precisely in this manner.

This is why I have long considered that paradoxically the blunt message that Mathematics (as presently understood) is not in fact fit for purpose can only be properly preached by someone standing outside the profession altogether (while still remaining deeply interested in Mathematics).

A Simple Example

In a recent blog, I suggested that in principle the Erdős–Kac Theorem should have a complementary application with respect to the distribution of prime numbers where the normal (Gaussian) distribution can be used to explain behaviour. To demonstrate this important aspect, we take repeated samples (of same size) within a relatively restricted region of the number system, where changes in the average gap as between primes is so small as to be discounted.

So starting with the n = 1,000,000,000 I took 100 samples of size t = 1000 up to n = 1,000,100,000.

For convenience in identifying the number of primes in each sample, I took them in strict sequence with the accompanying values listed below. As the primes are themselves distributed in a random fashion this would seem permissible in this instance.
Alternatively - at least in theory - 100 repeated random samples of 1000 (with replacement) could be taken within the same range i.e. 100,000,000 to 100,100,000. However in practice this would be more difficult.

So the first row for example represents the 10 samples in sequence from 100,000,000 to 100,010,000 with the last row, for example representing the final 10 sample values from 100,090,000 to 100,100,000.

54	56	57	55	57	61	57	56	47	51
61	55	57	49	43	54	56	43	58	54
51	56	51	49	43	54	56	43	58	54
60	56	55	57	54	52	59	56	51	56
49	43	59	64	55	63	62	53	49	51
63	54	40	54	51	56	52	54	42	57
53	73	55	50	54	53	61	49	52	56
54	59	44	57	50	56	56	53	52	54
57	56	52	54	63	43	54	52	51	48
49	55	57	52	54	52	56	48	56	66

Now in general terms, the mean number of primes in each sample can be approximated as t/log n 

= 1000/18.42 = 54.29 (approx)

This equates well with actual value averaged over the 100 sample results above = 54.11.

Much more problematic however is the provision of a general formula to approximate the standard deviation (for all values of n).

Though I experienced doubts on several occasions with respect to my initial "hunch", repeated empirical testing seems to suggest it as perhaps the simplest and best estimate,

i.e. √{t/(2log n)}

This would give the  standard deviation as 5.21 and compares well enough with the estimated standard deviation (based on the 100 sample values) = 5.46. This does not of course constitute a proof, and indeed a much greater degree of sampling would be required to truly establish it as the most likely estimate.

However in principle by now using the normal distribution, we could estimate the probabilities associated for example of sample values lying within any prescribed distance from the mean (on both sides).

For example we would expect for the above a little in excess of 2/3 of sample values to lie within 1 standard deviation of the mean value.

This would suggest therefore the probability that 2/3 of sample values (for frequency of primes occurring) would lie in the range of 49 - 59 (approx).



Addendum (5/3/2016).  Having returned to this issue in recent days, I feel I can bring more clarity to a situation that I did not really feel had been properly dealt with, first time around.

Though I was hoping that the standard deviation would correspond to √{t/(log n)}, I was led - largely through the empirical evidence of a small sample - to adjust it somewhat to fit the data.

However on reflection this was not warranted. Even just a few stray "outliers" with respect to this data would have a key large influence on the standard deviation. Therefore it was unrealistic to expect that the empirical example would fit in with theoretical explanations.

The  Erdős–Kac Theorem states that if ω(n) is the number of (distinct) prime factors of n, the probability distribution of



is the normal distribution.

In like manner I am suggesting that if  ω(n) is the number of primes in a sample of size t. (i.e. where samples are taken in the region of n) the probabilility distribution of

ω(n) - (t/log t)

√{(t/logt)}

is the normal distribution. 

The Erdős–Kac Theorem would suggest that where the average number of (distinct) primes is approximately 100 (with standard deviation 10), then we would expect just a little more than 2/3 of all factors to lie within 1 standard deviation of 100 either side of the mean (i.e. between 90 and 110).

In like manner if the average mean value of the number of primes in each sample is 100 (where samples are taken in the region of n) then we would likewise expect again that in a little more than 2/3 of samples, the number of primes would lie within 1 standard deviation (i.e. 10) of  100 (i.e. between 90 and 110).