Wednesday, October 8, 2014

Addendum on Erdős–Kac Theorem

We have seen that the randomness of prime factors (with respect to a given large number) can be encapsulated through the Erdős–Kac theorem, whereby their distribution approaches the perfect bell curve associated with the normal distribution!

Using the accessible manner of description given in Wikipedia, the Erdős–Kac theorem states that if ω(n) is the number of distinct prime factors of n, then, loosely speaking, the probability distribution of
 \frac{\omega(n) - \log\log n}{\sqrt{\log\log n}}
is the standard normal distribution.

So what we are expressing here with respect to the normal distribution is the number of standard deviations that the total of distinct prime factors (for each number) in each case is from the mean.
For example, when n has 10,000 digits each number on average would be composed of about 10 (distinct) prime factors. As the standard deviation (i.e. the square root of log log n) would now be slightly in excess of 3, this entails that about 2/3 of the prime factors would lie within 1 standard deviation on either side (i.e. between 7 and 13).     

It therefore struck me that given that we have a complementary notion of prime randomness with respect to the individual primes in the context of the collective (natural) number system, that it should be possible to also demonstrate this randomness through approximation to a normal distribution.  

If we now let ω(n) refer to the successive gap as between the individual prime numbers, the average (mean) gap is now given as log n. Therefore on the convenient assumption that the standard deviation is again taken as the square root of the mean - though I suspect that this may not be so -  this would entail that

ω(n) – log n
      √log n

is likewise the standard normal distribution in this case. The important thing is that some appropriate expression for the standard deviation should exist, thus enabling the normal distribution to apply.

What we are expressing here by contrast is the number of standard deviations that the gap between successive primes lies from the average (mean) gap. And the normal distribution here is in keeping with the random nature of the individual primes (used to estimate such gaps).

Now by its very nature (which is inherently dynamic and relative), we always remain in the realm of approximation.  

However in principle through sufficiently increasing the size of n, it would be possible to approximate ever more closely to the perfect bell curve associated with the normal (Gaussian) distribution.

It should also be possible in a similar manner to the Erdős–Kac theorem - though it is not my intention to attempt to do so here - to prove this result.  

However the key point to realise is the dynamic interactive context of both notions of prime randomness, so that the normal distributions (associated with such randomness) must be interpreted in a relative - rather than absolute - manner.

Thus from one perspective the (Type 1) individual primes are distributed as randomly as is possible consistent with as much order as as is possible with respect to the (Type 2) prime factors.

Equally from the complementary perspective the (Type 2) prime factors are distributed as randomly as possible consistent with as much order as is possible with respect to the (Type 1) primes.

Thus both the randomness and order with respect to both prime notions (in individual and collective terms) are mutually determined in a holistic synchronistic manner.   

However this very interpretation involving the complementarity of opposite frames of reference, requires a dynamic holistic means of interpretation,which formally remains totally absent from Conventional Mathematics.  

Addendum (3/4/2016)

As I quickly realised after initially wriring this original entry, I was mistaken in applying the complementary (Type 1) application of the Erdos-Kac Theorem to the gaps between the primes. Rather it should have been applied to the frequency of these primes (as arising in similar sized samples taken from within a given region (preferably high up the number system).

I then returned to the correct application in two following entries!

Tuesday, October 7, 2014

Holistic Synchronistic Nature of Number System (5)

In yesterday’s entry, I attempted to illustrate the dynamic interactive context through which the relationship as between analytic and holistic understanding arises in experience.

I then concluded that these two aspects are “real” and “imaginary” with respect to each other.

What is so important to emphasise here is that every mathematical symbol can be given both a real (analytic) and imaginary (holistic) interpretation.

So, as commonly appreciated, complex numbers with both real an imaginary parts are interpreted in an analytic (i.e. quantitative) context.
However complex numbers equally can be given a holistic (i.e. qualitative) meaning and this is the sense which I am now emphasising.

In this context the imaginary aspect relates to the indirect rational attempt to express holistic notions in an analytic type manner. This implies a circular type logic (that appears paradoxical from a linear perspective).

One important expression of such circular logic relates to the complementarity (in any appropriate context) of opposites.

Once again the crossroads illustration can be very helpful.

There is a valid sense - referring to left and right turns - in which both can be given an unambiguous identity (through reference to a single pole of direction).

Thus If I approach the crossroads heading N, I can unambiguously identify for example a left turn.

Then when I approach the crossroads from the opposite direction heading S, I can again unambiguously identify a left turn.

Thus in terms of single independent poles of reference i.e. 1-dimensional (linear) interpretation, both of these turns are designated as left. So this represents conventional analytic type appreciation.

However, clearly in terms of each other where both N and S poles are considered as interdependent, the two turns at the crossroads represent complementary opposites of each other. So if one is designated as left, the other is necessarily right in this context; and if alternatively the first is designated as right, then the other is now necessarily left.

This is 2-dimensional (circular) type interpretation based on the complementarity of opposite poles and is the minimum required for true holistic interpretation.

Now because all mathematical activity - like the crossroad example - necessarily entails opposite polarities (i.e. external and internal and whole and part) both analytic and holistic appreciation are always necessarily involved with respect to comprehensive understanding.

However remarkably we have increasingly attempted to reduce this activity in a solely 1-dimensional (analytic) type manner.

So the position with Conventional Mathematics is exactly akin to one who attempts to maintain that the two opposite turns at the crossroads are both left.
In other words Conventional Mathematics attempts to view all interdependent relationships in a merely quantitative (analytic) manner, where in fact correctly they should be viewed as quantitative (analytic) to qualitative (holistic) respectively.

Now going back to the primes, at the beginning of this present series of entries, I illustrated that the notion of “prime randomness” can in fact be given two complementary interpretations, which thereby implies that the very relationship as between the primes and natural numbers entails both quantitative and qualitative aspects.

However the very recognition of such complementarity requires holistic appreciation. Not surprisingly therefore as such appreciation is formally excluded from conventional interpretation, mathematicians continue vainly to attempt (like one who can only identify left turns at a crossroads) to interpret the relationship between the primes and natural numbers in a merely quantitative - and thereby reduced -  manner.

So the first notion of prime randomness relates to the distribution of the individual primes (as independent) numbers within the collective natural number system.

However the second notion of prime randomness - which has been ably demonstrated through the Erdős–Kac theorem - relates to a complementary notion of prime randomness.

So here we are considering a collection of primes (i.e. distinct prime factors) with respect to each individual natural number.

Once one recognises the complementarity of both of these definitions of prime randomness, then this clearly suggests that we can no longer view the relationship between the primes and natural numbers in a merely quantitative manner. Again this would be akin to persisting in identifying both turns at the crossroads as left!
Rather we now clearly see that the relationship between both is quantitative as to qualitative (and qualitative as to quantitative) respectively.

In other words within each frame of reference (taken separately) we can indeed attempt to interpret the relationship between the primes and natural numbers in a quantitative analytic manner.

However crucially, ultimately both of these interpretations are mutually interdependent. So here we now properly realise the truly holistic synchronistic nature of the number system.

Put another way, the randomness of the (individual) primes with respect to the (collective) natural number system, cannot ultimately be viewed as independent of the randomness of the (collective) prime factors with respect to each (individual) natural number.

In other words both these aspects of random prime behaviour mutually depend on each other so that ultimately the very nature of the number system is seen to be determined in an ineffable synchronistic manner (that is utterly mysterious).   

This equally implies that the very notions of “randomness” and “order” with respect to the number system are themselves fully complementary notions (with a merely relative meaning).

So randomness with respect to individual behaviour implies corresponding order with respect to collective beahaviour and vice versa so that both aspects mutually depend upon each other in a dynamic interactive manner.

Indeed these features can be even more dramatically illustrated!

Though the individual primes are distributed as randomly (as is relatively possible) within the collective natural number system, a corresponding order applies to the general nature of this prime distribution.

Most simply this can be expressed as n/log n which estimates the frequency of primes among the natural numbers (which becomes relatively ever more accurate with sufficiently large n).

However there is an equally important complementary expression of this relationship which is not properly recognised.

Just as we can have primes and natural numbers (as Type 1 base quantities) equally we can have primes and natural numbers (as Type 2 dimensional qualities).
In other words when primes and natural number numbers are used to represent factors they relate strictly to the dimensional aspect of number (with a relatively qualitative as to quantitative relationship with respect to corresponding base numbers).

Some time ago I began to investigate the relationship as between the (distinct) prime and natural factors of a number.

Now again to illustrate with the number 24, the distinct prime factors are 2 and 3, whereas the natural factors are 2, 3, 4, 6, 12, and 24. So in effect the natural number factors represent all the natural numbers (except 1) that will divide into a number. However, with respect to primes we do not admit any natural number factors!.

Now I quickly discovered that the average distribution of prime factors per number is given as log n. (Indeed log n – 1 is more accurate but for large n, this distinction can be ignored!)

Also the Hardy-Ramanujan Theorem postulates that the average number of distinct factors for a number n would approach log log n (which approximation would improve for large n).

This entails that if we were to attempt to obtain the ratio of natural number to (distinct) prime factors, it would be given as log n/log log n (for large n).

Thus if we let log n = n1, this could be written as n1/log n1.

Thus we have now two prime number theorems with a similar format. The first expresses the approximate number of primes up to any natural number; the other expresses the ratio of natural number factors to (distinct) prime factors for any number.

Now once again, properly these must be understand in a dynamic complementary manner.
So the distribution of primes (as numbers and factors) in each case with respect to the natural numbers are intimately dependent on each other. So once again the ultimate two-way relationship between the primes and natural numbers is if a holistic synchronistic nature.

Another fascinating observation can be made!
In the first case i.e. frequency of primes, an additive relationship connects the primes with the natural numbers.
Therefore the natural numbers comprise the sum of the primes and the remaining (composite) natural numbers.
However in the second case a multiplicative relationship connects the prime factors with natural number factors. So the latter formula expresses the times we must multiply the number of (distinct) primes to obtain the corresponding number of natural (number) factors.

So this beautifully expresses the point that the key relationship as between addition and multiplication is itself as quantitative as to qualitative (and qualitative as to quantitative) respectively!

Monday, October 6, 2014

Holistic Synchronous Nature of Number System (4)

The second of the two key polarity sets relates to whole and part (collective and individual) which is vitally important with respect to all mathematical understanding.

Again however conventional interpretation leads to basic reductionism whereby whole and part are both understood in a merely quantitative manner with the whole thereby indistinguishable from the sum of its parts.  

However when we allow for the complementary interaction as between the poles, both quantitative and qualitative aspects of interpretation are involved. Then once again like the two turns at a crossroads, when one turn is designated left, the other - relatively - is necessarily right (and vice versa), likewise, in any dynamic context, if the whole is viewed in a quantitative manner, then the parts - relatively - are qualitative in nature; likewise when the whole is viewed in a qualitative manner, the parts are - relatively - quantitative in nature.

Thus whole and part interactions entail both quantitative and qualitative aspects that keep alternating between each other in a dynamic interactive manner.

Again it may be instructive to probe the psychological dynamics through which both quantitative and qualitative aspects arise in experience.

We have already dealt with the external and internal polarity set.

Therefore for example when we form knowledge of a number as object (in external terms) this is necessarily balanced to a degree by a corresponding mental perception that is - relatively - of an internal nature.

Though in formal terms these aspects are considered as separate and reduced in terms of each other in an absolute conscious manner, implicitly some degree of (unconscious) awareness necessarily must also exist of the complementarity of both poles (thereby enabling the mental interaction with number objects to take place).

This then leads to an intuitive fusion of opposites whereby the (independent) specific actual nature of the number (both as object and mental perception is thereby cancelled out). So what happens - though we explicitly awareness of the nature of this process may well be absent, we move to a potential - rather than actual - knowledge of interdependence that carries an infinite  meaning.

So in effect with respect to our example of the number “2”, we switch from the actual specific experience of this number (both as separate object and perception respectively) to a new holistic appreciation of the interdependent notion of “twoness” that potentially applies in any context to “2”. 
Without such implicit appreciation of this  notion of “twoness” (that is potentially infinite in scope) we would be unable to recognise specific examples of “2” in an actual context.

Thus in the understanding of any number - and indeed every mathematical phenomenon - the dynamic relationship between actual (conscious) and potential (unconscious) notions necessarily takes place.

The conscious appreciation relates directly to rational (analytic) type interpretation; the unconscious appreciation, by contrast relates directly to intuitive (holistic) type interpretation.

Though mathematicians may informally recognise the importance of intuitive understanding (especially for creative work) explicitly Conventional Mathematicians is presented as a set of absolute rational type connections which gravely distorts its true nature.

So we have moved from the actual notions of external and internal polarities (relating to the number “2” as object and mental perception respectively) to the potential appreciation of the notion of “twoness” as holistically applying to all possible specific cases of “2”.

In the dynamics of understanding this causes a decisive shift from the notion of number as a specific object (and corresponding perception) to the conceptual notion of the holistic nature of number as applying in all possible cases.

However this intuitive holistic understanding then becomes quickly reduced in an actual manner. So one quickly moves from the potential (intuitive) holistic notion of number (and corresponding mental concept) as applying in an infinite manner, to the rational notion of the general notion of number as specifically applying in all actual cases.

Indeed the conventional notion of mathematical proof is based on this merely reduced interpretation. Thus when a theorem e.g. is generally proved (e.g. the Pythagorean Theorem) strictly this has an infinite - merely potential - application.
Then when we identify a specific application in an actual finite context it is assumed that this directly corresponds with the general proof.

Now I am not arguing that conventional mathematical proof has no value! However I am saying is that it is based on the direct reduced assumption whereby what is potentially true is assumed to be equally true in an actual manner.

Thus the mighty assumption of a direct coherence as between quantitative (finite) and qualitative (infinite) notions is thereby made.

However this assumption of coherence which underlies all conventional mathematical relationships, cannot be itself addressed within this framework (which simply reduces one to the other).

This is why it is imperative to move to a dynamic interactive interpretation of mathematical relationships that can embrace both quantitative (analytic) and qualitative (holistic) aspects as equal partners.

So in all mathematical understanding we have the dynamic process through which specific objects (and mental perceptions) are related to whole objects (and mental concepts).

And in the very dynamics of such understanding the unconscious (based on complementary recognition of both poles) is vitally necessary in enabling the switching as between specific and whole objects (on the one hand) and perceptions and concepts on the other.

Now the very nature of conventional mathematical interpretation leads to an extreme degree of rigidity in the nature of such interaction whereby actual and potential objects (and corresponding perceptions and concepts) are assumed to confirm each other in an absolute manner.

However we understand more appropriately carefully distinguishing rational (analytic) from intuitive (holistic) type appreciation, mathematical interpretation is revealed to be strictly of a - merely - arbitrary relative understanding which paradoxically leads to a greatly enhanced appreciation of its true unlimited nature.

Now once the true holistic nature of mathematical understanding (as qualitatively distinct from its analytic counterpart) is properly realised a key problem that relates to how this can be effectively translated in a phenomenal manner.

In everday experience when we experience phenomena both conscious (analytic) and unconscious (holistic) aspects are involved.

So If I am looking to buy a house, clearly the house has an actual existence that can be consciously identified. However the house will equally embody the more holistic (unconscious) desire for meaning. So in a sense I will be searching for my “dream” house than embodies these deeper holistic desires.

Strictly this is true of all phenomena and indeed of all mathematical phenomena.

Insofar as the object (and corresponding perception) has an analytic identification this corresponds to “real” conscious meaning.

However insofar as the object has a holistic identification this corresponds to an “imaginary” unconscious meaning.

So just as in quantitative terms we now recognise that complex numbers have  both “real” and “imaginary” aspects, likewise in qualitative terms we  realise that the enhanced complex interpretation of mathematical relationships (combining both analytic and holistic aspects) equally contains both “real” and “imaginary” aspects.

So the “imaginary” aspect of mathematical interpretation strictly represents an indirect rational way of communicating holistic meaning.

Thus in the very dynamics of understanding in the very way that mathematical perceptions and concepts interact, we keep switching as between “real” actual and “imaginary” potential meaning.

However in formal terms this process in interpreted in a merely reduced rational manner as solely “real” from a qualitative perspective.

So Conventional Mathematics is formally confined to merely Type 1 appreciation (of a quantitative analytic kind) i.e. the real aspect of mathematical appreciation.

Holistic Mathematics is formally identified with merely Type 2 appreciation (of a qualitative holistic kind) i.e. the imaginary aspect of mathematical appreciation.

Radial Mathematics is then explicitly identified with both Type 1 and Type 2 (in mutual conjunction with each other) i.e. the complex aspect of mathematical appreciation

Thursday, October 2, 2014

Holistic Synchronous Nature of Number System (3)

In this entry, I will probe further the deeper psychological basis for the holistic synchronistic aspect of number.

All experience - including of course mathematical - is conditioned by fundamental sets of twin dynamically interacting polarities.

The simplest of these relates to the objective/subjective (alternatively external/internal or outside inside) set of polarities.

Therefore in psychological terms, whenever we recognise a number - say the number 2 - both external and internal aspects are necessarily involved.

Thus corresponding to the external number object, is a corresponding subjective mental perception of a - relatively - internal nature.

Strictly speaking therefore, recognition of a number is a dynamic experience entailing the interaction of both an external aspect (as object) and corresponding internal aspect (as mental perception). And properly understood, like the left and right designations of a turn at a crossroads, external and internal likewise have a purely relative meaning!   

However normal experience - especially with mathematical interpretation - is considerably reduced in a conscious rational manner .

What this entails is the explicit separation of the two polarities involved.
Thus rather than the dynamic recognition of an interactive process, phenomenal experience - in this case of the number 2 - is rigidly reduced in a static manner with respect to just one pole.

So in this reduced manner, there are now two absolute ways in expressing the number experience.

1) We can identify the number absolutely with its (external) objective aspect in the (erroneous) belief that the number objectively exists (independent of our subjective mental relationship with it). Here, the implicit assumption is then made that the (internal) mental perception directly corresponds with the number object (i.e. as an exact copy as it were of the object).

2)  We can identify the number absolutely with its (internal) subjective aspect, again in the (erroneous) belief that the number has a mental existence (independent of any objective reality). When this is the case, again an implicit assumption is made that the (external) number object directly corresponds with its mental perception (i.e. as an exact projection of this perception).

In effect in does not matter from this absolute perspective whether one adopts the realistic (objective) or idealist (subjective) perspective as to the nature of number existence for the interaction is thereby reduced in an absolute in terms of just one pole. So we can reduce the internal aspect in terms of the external; alternatively we can reduce the external in terms of the internal resulting effectively in the same absolute interpretation of number existing.

Such absolute interpretation of number (as independently existing) likewise explicitly coincides with the equal reduction of unconscious type intuitive appreciation in a conscious rational manner.

Equally this could be expressed as the reduction of the holistic aspect of appreciation in a merely analytic manner.

It is important to realise that Conventional Mathematics is explicitly based  in every formal context on such gross reductionism.

So what properly represents the dynamic interaction of external and internal poles of experience (both of which properly enjoy a distinct relative validity) is explicitly interpreted in a reduced - and thereby distorted - static manner, whereby they are separated as absolutely independent of each other.

Thus an interactive experience that properly entails both conscious (analytic) and unconscious (holistic) modes of understanding is formally interpreted in a merely conscious (analytic) manner.

Putting it simply therefore, the conventional belief in the abstract nature of number, which has come to dominate accepted thinking over the past few millennia, fundamentally distorts its true nature.

So in truth number - and by extension all mathematical constructs - entails a dynamic interactive process with a merely relative validity.

From this perspective opposite poles cannot be directly reduced in terms of each other in an absolute manner. Rather they enjoy an existence of relative independence (where opposite poles are considered as separate) together with relative interdependence (where the poles are considered as complementary).

The independent aspect corresponds directly to rational (conscious) understanding of an analytic nature; the interdependent aspect corresponds directly with intuitive (unconscious) understanding that is by contrast holistic.

Now looking at the psychological process through which we understand phenomena such as number, we can indeed go more deeply into the dynamics involved.
What is vital to appreciate here is that unconscious is necessary to enable switching as between opposite poles.

So for example when we form knowledge of a number e.g. 2 as an (external) object, the conscious rational mind is directly involved. However the switch to the corresponding recognition of the (internal) perception of 2 requires the intervention of the unconscious. Now as the unconscious is inherently based on the complementarity and ultimate unity of opposite poles such as external and internal, it thereby acts to counteract the unbalanced emphasis on just one pole. So the positing of 2 as an (external) object is thereby to a degree negated, enabling the switch to the corresponding (internal) recognition of 2 as a mental perception. Once again unbalanced identification with the positing of this pole (in a rigid phenomenal manner), will gain be counteracted to a degree by the unconscious leading to its corresponding negation.

Thus when the holistic unconscious in sufficiently developed very flexible switching continually takes place as between both external and internal polarities, which are identified in an increasingly relative manner.

However when the unconscious is not recognised understanding becomes increasingly rigid with both external and internal poles confirming each other in an absolute manner.

Thus the distorted interpretation of mathematical symbols such as numbers (which represents accepted mathematical understanding in our culture) inevitably leads to confirmation of its reduced findings in an increasingly absolute manner.

Looking at this from a Jungian perspective, this reveals that an enormous collective shadow hangs over the Mathematics profession in a total failure to recognise its own unconscious identity.

Objective truth does not exist in abstraction (even in Mathematics); rather what we understand as objectively true (in external terms), always reflects a corresponding means of mental interpretation (in an internal manner).

So the apparent absolute nature of mathematical truth simply confirms the distorted nature of conventional interpretation (where external and internal poles are reduced in terms of each other).

However once we properly allow for the dynamic interaction of poles an unlimited number of mathematical interpretations - each with a limited relative validity - are seen to exist.

So Conventional Mathematics is decidedly 1-dimensional in nature (based in any context on just one absolute pole of reference).

However when we allow for the interaction of opposite poles Mathematics becomes multi-dimensional (with a potentially unlimited number of relative interpretations).  

Wednesday, October 1, 2014

Holistic Synchronous Nature of Number System (2)

In yesterday’s entry, I concluded by demonstrating the holistic synchronistic nature of the number system.

In other words a two-way mutually interdependent relationship exists as between the primes and natural numbers, which is of a holistic synchronistic nature where ultimately both number aspects are simultaneously determined in an ineffable manner.

Now the great problem with the conventional mathematical approach is that it is 1-dimensional in nature, thereby unambiguously seeking solely analytic interpretation of relationships in a one-directional manner.

So again, the conventional mathematical approach, by viewing numbers in cardinal terms, attempts to explain natural numbers solely in terms of the primes.

However, when one properly appreciates the corresponding ordinal approach to number, the relationship as between the primes and natural numbers is inverted with each prime now uniquely defined in a natural number manner.

So the position with the conventional mathematical approach is - as I have stated before – akin to one who can only identify left turns at a crossing.

So when one is travelling north up a straight road and encounters a crossing, the left turn (at this crossing) can be unambiguously identified; likewise when one travelling S from the opposite direction and again encounters the same crossing, once more the left turn can be unambiguously identified.

Therefore, through operating in an analytic manner with respect to independent polar reference frames in both cases,  the two turns at the crossroads can be unambiguously identified as left!

However when we view these crossroad directions in a holistic manner (where N and S directions are simultaneously viewed as interdependent), then they are understood as directly opposite to each other. So of one turn is designated as left, then the other is thereby (relatively) right and alternatively if a turn is designated as right the other turn is then necessarily left in this context.

So analytic understanding is identified with the attempt to understand mathematical relationships in a static 1-dimensional manner (i.e. within an isolated fixed polar frame of reference).

Holistic understanding (at a minimum) entails the corresponding attempt to understand such relationships in a dynamic 2-dimensional manner (i.e. within two interacting frames of reference).

Just as we can attempt to approach the crossroads from two isolated directions (N and S) that are initially considered independent of each other, likewise we can approach the relationship between the primes and the natural numbers from two opposite directions - which I term Type 1 and Type 2 - that again are initially considered independent of each other.

However, when we attempt to view both directions in a holistic manner as interdependent, then we appreciate the purely paradoxical nature of both (analytic) sets of findings. It is at this point that we can then clearly appreciate that the ultimate relationship between both is of a purely synchronistic nature that is ineffable.

Now again with respect to the crossroads example, it is obvious how both analytic and holistic type appreciation are involved. So provided that the crossroads is approached from just one direction, we can unambiguously identify in analytic terms both left and right turns in an absolute manner.

However when we view the crossroads holistically as being approached from two directions simultaneously, what is absolutely left and right is rendered purely paradoxical. So combining both analytic and holistic appreciation, both left and right directions now have a merely arbitrary meaning depending on the relative context. 

Likewise in truth it is similar with all mathematical relationships.
These can be given an - apparent - absolute interpretation within single isolated poles of reference in an analytic manner.

However, what is not properly appreciated within the present conventional approach to Mathematics is that all such absolute relationships possess a mirror image alternative explanation within an equally valid opposite polar frame of reference.

Then when both of these opposite interpretations are then properly realised as dynamically interdependent with each other, true holistic appreciation of an intuitive synchronistic nature emerges. 

The most fundamental mathematical relationship relates to that as between the primes and natural numbers.

When properly interpreted in analytic terms we can approach this relationship in quantitative terms from two opposite directions (Type 1 and Type 2 respectively).

Then when we realise the dynamic interdependence of both sets of relationships, true holistic appreciation emerges of their mutual synchronicity.

Expressed in perhaps an even simpler manner, analytic interpretation relates to the quantitative aspect of mathematical relationships, which can be of either a Type 1 or Type 2 nature considered within single isolated poles of reference.

Holistic appreciation by contrast relates directly to the qualitative aspect of mathematical relationships, and arises from clear appreciation of the mutual dynamic relative interdependence of both Type 1 and Type 2, within complementary reference frames. 

When one views Conventional Mathematics from this perspective, it is clearly seen - despite all its great achievements - as simply not fit for purpose.

Not alone is the vitally important holistic aspect of appreciation completely unrecognised (in formal terms) but even from an analytic perspective, the Type 2 aspect of quantitative appreciation is likewise effectively ignored.

One day it will be clearly recognised that both the analytic (quantitative) and holistic (qualitative) appreciation of all mathematical relationships are of equal importance and strictly have no meaning in the absence of each other.

And nowhere is this realisation more important than in appreciation of the true dynamic nature of the number system.