The Remarkable Secret of the Primes

In yesterday's blog entry, I drew attention to the fact that an important shadow system exists which applies to ordinal - rather than cardinal - interpretation of number (corresponding to the Type 2 aspect of Mathematics).

This number system provides therefore the basis for an equivalent exploration of the relationship of the primes to the natural numbers (that lends itself directly to holistic qualitative interpretation of their nature).


And as this system can be quantitatively represented in complementary fashion in a simple circular fashion (as expressive of the roots of 1), I am frankly amazed that so little apparent research has been carried out from this direction.

Indeed using little more than the most elementary of mathematical techniques, I was able some time ago to derive a highly interesting result that contains within it - I believe - what amounts to an alternative expression of the Riemann Hypothesis (which I will refer to again in a future blog).


Having reflected much on this I have come to accept that perhaps this is not really so surprising!


I have always believed that the nature of the Riemann Hypothesis primarily lends itself to qualitative rather than quantitative interpretation.

Therefore in a mathematical approach that is specifically geared to qualitative type interpretation of the relationship between the primes and the natural numbers, it should in principle be possible to quickly arrive at the central problem to which the Hypothesis relates!


We dealt a little with this alternative system yesterday.

Using a combination of Euler's Identity and de Moivre's Theorem, it is a simple matter to calculate the various roots of 1. Indeed this establishes a direct link - without apparently being recognised - with the (shadow) number system that is appropriate for Type 2 Mathematics.

So e^(iπ) = cos(π) + i sin(π) = - 1; e^(2iπ) = cos(2π) + i sin(2π) = 1, where 2π refers to radians = 360 degrees.

Therefore e^(2kiπ) = cos(2kπ)+ i sin(2kπ)

Thus when k = 1, 2, 3, 4,.... we generate the Type 2 number system, 1^1, 1^2, 1^3, 1^4,..... (which has a direct qualitative interpretation)


We can also use this same formulation to generate the corresponding roots of 1, where they have a complementary quantitative interpretation. So the true significance of Euler's Identity - which is has been voted the most beautiful theorem in Mathematics - is that, like the Riemann Hypothesis, it represents in a very special manner the intersection of mathematical meaning that combines both quantitative and qualitative aspects. And the really sad fact remains that its true beauty cannot therefore be reflected through Conventional Mathematics!


So quite simply, to obtain the nth root of 1, we simply calculate the value of

cos{(2kπ/n)} + i sin{(2kπ/n}.

So for example to obtain the 4th root of 1 where k = 1/4, i.e 1^(1/4) we thereby get

cos{(2π/4)} + i sin{(2π/4}

= cos 90 + i sin 90 = i

Now in this connection of the relationship of prime to natural numbers, we are chiefly interested in the prime roots of 1.

And as we have seen in previous blog entries, a special relationship exists as between a prime number root of an ordinal nature and all the natural numbers up to and including that prime number!


So for example 17 is a prime number. Therefore a special relationship (of a circular nature) exists as between this root of 1 (raised to the power of 1) and all other roots raised to 2, 3, 4, 5, ... 17.


What is truly significant about these roots, is that they generate unique values that can never be replicated from operations entailing other roots.

Indeed this is the true reason for the relational capacity of the primes, whereby any natural number can be expressed as the unique combination of prime factors.

So the answer resides in the amazing unique circular pattern of the prime numbers (as ordinally interpreted through their corresponding roots) which can never be replicated through other roots.


And once again we have the inversion of the normal (i.e. Type 1) manner of looking at the relationship between the primes and the natural numbers.


For example, from the conventional perspective, a key task is to find ways of precisely measuring the general distribution of the primes among the natural numbers. And once again the Riemann Hypothesis is intimately linked to this problem!


However when we approach the issue from an ordinal Type 2 perspective, we are faced with the opposite task of precisely measuring the general distribution of the natural numbers among the primes!


So again to make this more explicit! We have seen that 17 is a prime number and that it can be given an ordinal (circular) interpretation in terms of the 17th root of 1.

However this root is closely associated with its other 16 roots. And properly understood, there is an intimate complementary link as between quantitative and qualitative type interpretation.


So the 17 roots of 1 result from obtaining the 17th root of 1^1, 1^2, 1^3, 1^4,....1^17. So when seen in this light, the roots (in quantitative terms) automatically imply their corresponding dimensions (in qualitative terms).


And if we come back to the generalised expression for Euler Identity, this is precisely how it should be interpreted (in Type 3 mathematical terms).

Here k and 1/k bear a complementary relationship to each other, in qualitative and quantitative terms respectively.

So when k = 1/2, in quantitative terms, k = 2 in corresponding qualitative terms, with respect to the formula.

So calculating 1^(1/2) with respect to the former, we obtain

cos π + i sin π = cos 180 + i sin 180 = - 1.

Then calculating 1^2 with respect to the latter, we get

Cos 4π + i sin 4π

Now from a conventional Type 1 perspective this results in 1.

As there is no recognition here of the qualitative nature of dimension, 1^2 must necessarily be reduced to 1^1.

However when we look more closely at the expression which generates the result, we can see that 4π - literally - represents two full circular revolutions.


This therefore corresponds exactly with the Type 2 meaning of dimensional number as having a circular interpretation. The linear aspect then comes in with respect to the geometrical representation, where radial lines from the centre of the unit circle (in the complex plane) connect to the points on the circumference representing the various roots of the number.

So 1-dimensional implies just one line from the centre (which in qualitative terms means just one pole of reference). Thus circular in this approach is thereby reduced to linear, so that no genuine qualitative aspect can remain.

Then 2-dimensional implies two lines drawn at 180 degrees to each other to the circumference on directly opposite sides (with one positive and the other negative).
Therefore this enables complementarity (with respect to two opposite poles of reference) thus enabling a holistic circular capacity with respect to understanding.
3-dimensional then implies three lines, 4-dimensional 4 lines, and so on.

So for each additional line, the circular circumference once more can be mapped out in corresponding fashion. Thus we get 1 circular, 2 circular, 3 circular, 4 circular rotations of the circumference respectively, matching the corresponding lines in each case.


And then the circular dimensions (which equally incorporate ever more refined multi-directional linear understanding) are used to appropriately interpret the corresponding roots (in inverse fashion). These dimensions therefore possess exactly the same number structure as their corresponding roots but interpreted in a qualitative rather than quantitative manner.

Thus 4 as (qualitative) dimension interprets 1/4 as (quantitative) root; 3 as qualitative dimension interprets 1/3 as quantitative root; 2 as qualitative dimension interprets 1/2 as as quantitative root and in the trivial case 1 as (qualitative) dimension interprets 1 as (quantitative) root. This again means that qualitative is effectively reduced to quantitative in 1-dimensional terms.


Now we come to something that is truly astonishing in its implications.

We saw when discussing the 17 roots of unity, that the 17th root serves as the basis for the calculation of all other roots (relating to the 17 numbers form 1 to 17.

Now in quantitative terms, it is already well known that all these roots are related to each other through remarkable circular type complex number relationships. Thus for the sum of the roots, the answer is always zero and then when we multiply roots we always obtain 1.

Now, interpreting this in corresponding qualitative terms, it implies that these prime roots have an amazing relational capacity in both linear and circular terms. In other words the very basis of the prime roots (when given their complementary qualitative interpretation) is that they possess an unique relational capacity with respect to the reconciliation of individual independence with collective interdependence.


Looked at from a slightly different angle, each prime number - as in this case 17 - embodies itself uniquely in each of the natural numbers from 1 to 17 (in both quantitative and qualitative terms).

Likewise looked at in reverse the natural numbers in turn (from 1 to 17) are uniquely embodied in the primes.

So with every new prime number, the same process takes place once again. Thus with the next prime 19 for example, it again becomes uniquely embodied in every natural number from 1 to 19 in both quantitative and qualitative terms, with the natural numbers in turn becoming collectively embodied in the primes.

This therefore ensures that a unique relationship necessarily exists as between any prime number on the one hand and all the natural numbers up to and including it; and in reverse as between these natural numbers and the primes. And this process is unlimited. So when we reach far out into the number system to discover a new prime - though the natural numbers may already have been transfused through the effects of so many preceding primes - because these influences are unique - they can never be replicated.

So what this entails is that every natural number potentially possesses unlimited unique resonances through relationship with the primes and in reverse the prime numbers too possess an unlimited collective resonance through relationship with the natural numbers. And as always this process relates to both the quantitative and qualitative aspects of these numbers!

And this is ultimately the reason why the primes from a cardinal perspective, while maintaining their individual quantitative identity, yet can collectively generate in a qualitative manner the natural numbers in a amazing holistic manner (without their individual identity being compromised); and equally important, from the reverse ordinal perspective, why the natural numbers while maintaining their individual quantitative identity can collectively generate the primes in a qualitative manner without compromise.

So the secret of the primes is ultimated rooted in the perfect complementarity that characterises the relationship as between both quantitative and qualitative aspects of number.


Indeed when one can fully see simultaneously from both quantitative and qualitative aspects, the prime numbers and natural numbers become in fact identical with each other (with in each case the quantitative aspect perfectly mirroring the qualitative aspect and the qualitative aspect the quantitative aspect respectively). And to arrive at this point is to experience true ineffable mystery where the secret of the primes can at last be properly realised.

Looking Inside the Primes!

I have already introduced two number systems as the basis for Type 1 (quantitative) and Type 2 (qualitative) appreciation respectively.

Initially both Type 1 and Type 2 interpretation can take place in comparative isolation from each other. However ultimately from the Type 3 perspective, their truly relative nature becomes readily apparent so that the Type 1 also - necessarily - has a qualitative and Type 2 a quantitative aspect respectively.

And the reason why this seems reminiscent of the wave/particle nature of particles at the quantum level, is because the deeper root of such complementarity lies in the very foundations of Mathematics.

It has to be stated from the outset that I am already defining Type 1 Mathematics in a - relatively - rather than absolute independent manner (which is the way it appears from the the more comprehensive Type 3 approach). Because no such perspective exists in Conventional Mathematics, it is misleadingly defined in an absolute type manner (which does not lend itself therefore to subsequent integration with the other approaches).


So bearing this important point in mind, the Type 1 is defined in terms of the natural numbers that can vary as quantities and a default number number dimension of 1 (that does not vary). And from this relatively independent perspective, the dimensional number is qualitative, with respect to the base number quantity. So once again, from a qualitative perspective Type 1 Mathematics, is properly defined by its linear (1-dimensional) nature.


Thus in Type 1 terms the natural number system is represented as follows

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

Now Conventional Mathematics attempts to come to grips with the primes through sole reference to this number system (with the dimensional number system ignored).

So in this context we have the individual prime numbers 2, 3, 5, 7,... (which are looked on as the basic building blocks, or atoms, of the natural number system).

Then, we also have the general relationship of the primes to the natural numbers where each (natural) number is expressed as a unique combination of prime factors.

So we have the individual uniqueness of the primes (as separate from the natural number system) and also - what we might call their relational uniqueness (in that all natural numbers entail unique combinations as a product of prime factors).

And both of these aspects are understood - merely - with respect to their quantitative aspect.


However in truth the true picture requires much greater subtlety.

Now, prime numbers, as quantitative, have a cardinal meaning. For example, we could look at the prime number 7 for as representing a set of objects. Thus, the very notion of 7 necessarily implies the prior notion of 1, 2, 3, 4, 5 and 6 respectively.

Now, from a cardinal sense the relationship between these natural numbers (within the primes as it were) have no relational significance. So each natural number can be built up by the addition of 1). So 2 = 1 + 1, 3 = 2 + 1, 4 = 3 + 1 and so on.


The important reason for the lack of relational significance, arises directly from the fact that addition can be carried out from - merely - the 1-dimensional perspective.

So 2^1 = 1^1 + 1^1; 3^1 = 2^1 + 1^1; 4^1 = 3^1 + 1^1 and so on.

Now this clearly suggests that true relational capacity in fact entails the dimensional notion of number (when it varies from 1). And as Conventional Mathematics is qualitatively defined by the 1-dimensional approach, it is thereby ill suited to get to grips with the relational aspect of the primes.


This perhaps becomes clearer when we now look at the relational aspect of the primes (as understood in Type 1 terms).

So the natural numbers are understood in terms of the unique expression of a combination of prime factors.

Therefore for example, 6 = 2 * 3 is uniquely expressed as the product of these two prime numbers (as factors).


Now just as the individual aspect of the primes is understood merely in quantitative terms, likewise this is now also the case with respect to their collective relational aspect.

However when we include the default dimensional number with the primes we get a significant clue, that something vital is missing from the conventional perspective.

Earlier we expressed the two unique factors of 6 as 2 and 3.

However more fully expressed, the product of these two numbers = (2^1) * (3^1). However if we think of this in geometrical terms, we would generate a 2-dimensional rectangle with sides 2 and 3 respectively. So in fact, in saying that 6 is the natural number that is uniquely expressed by the product of its prime factors, 2 and 3, we are in fact giving but a reduced quantitative interpretation (that misses out entirely on the key qualitative significance of their relational aspect).


To see this more clearly (in qualitative terms) we replace the base number quantities by 1 (thereby switching to the Type 2 approach).

So 1^1 * 1^1 = 1^2.

In other words, the true significance of such a relational capacity, with respect to the primes, is directly due to the manner in which the qualitative dimension, associated with the primes, changes through multiplication.

And again in the conventional mathematical approach, this key subtle distinction - which is vital for comprehension of the relational aspect - can be given no meaning (as it is qualitatively defined within a merely 1-dimensional perspective!)



However to get a fuller picture, we now need to interpret such prime number relationships from within the alternative Type 2 perspective.

So once again the number system (in Type 2 terms) is defined as;


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


This represents the complete inverse of Type 1, where the base quantity is now fixed as 1, and where the natural numbers - relatively - representing qualitative dimensions, can now vary over the natural numbers.


The key to realising what is happening here is that dimensional numbers are now defined with respect to a (qualitative) ordinal - rather than (quantitative) cardinal meaning.

So 7 as a prime number in this context is written more fully as 1^7 and refers to the qualitative notion of 7 as a dimension (i.e. the 7th dimension).


To unlock the significance of this alternative formulation we must switch to a circular rather than linear representation.

The key to this in turn is to recognise that the notion of a dimension in qualitative terms and its corresponding root in a quantitative manner, are inversely related (with the same identical structure).

So 1^7 therefore has its counterpart in 1^(1/7) as the seventh root of 1.

And once again - but now in an ordinal manner - the notion of 7 (as a prime root of 1) likewise implies 6, 5, 4, 3, 2, and 1 as roots of 1 also.

However unlike the previous case where the relationship between the natural numbers in this grouping, allowed for no qualitative distinction, it is quite the opposite here, with all the natural numbers up to 7 i.e. as the 1st, 2nd, 3rd, 4th, 5th, 6th and 7th roots of 1,respectively, possessing a unique qualitative identity. And the reason for this is that the corresponding ordinal numbers as dimensions i.e. 1st, 2nd, 3rd, 4th, 5th, 6th and 7th, all likewise possess a unique qualitative identity (serving as a means of holistic interpretation).


So this illustrates in truly remarkable fashion, how each number, can in truth be given both a quantitative and qualitative interpretation. And the key to this realisation is the fact that the cardinal and ordinal nature of numbers, properly relates to their quantitative and qualitative characteristics respectively.


And just as the the individual primes have a unique identity in Type 1, it is the reverse in Type 2 terms. So the strong circular type relationships that bind the various roots of a prime number and corresponding dimensional interpretations) together, imply that these ordinal roots lack such identity.


We also then can attempt replicate in Type 2 terms the relational aspect of the primes.

So 2 * 3 = 6 from this perspective is represented as 1^(2 * 3) = 1^6

However again a very subtle point arises that is illuminating.

In Type 1 terms when we multiply two numbers, it does not matter in what order we do so.

So 2 * 3 = 3 * 2.


However this is not true with respect to Type 2.

For 1^(2 * 3) = (1^2)^3 and 1^(3 * 2) = (1^3)^2

Now when we invert this in Type 1 terms, the former is:

(2^3)^1 = 8^1 and the latter (3^2)^1 = 9^1


So when we try to represent the relational aspect of the primes (which as a qualitative aspect properly is explained in Type 2 terms), quantitative uniqueness, with respect to generation of the natural numbers, breaks down. So what represents multiplication in the Type 2 case, represents exponentiation with respect to Type 1 (just as addition with respect to Type 2, represents multiplication with respect to Type 1).


This is just another way of saying - complementary to our earlier observation - that this relational capacity of the primes is directly associated with the qualitative nature of number (an is appropriate to Type 2 interpretation).


So let us return once more to our crossroads analogy, though this time in an enlarged manner.

In previous entries, I used this to indicate the crucial difference as between linear frames of reference (using isolated poles) and circular frames (where complementary opposite poles are taken). I did so initially, where the isolated frames related solely to the "real" aspect of quantitative type interpretation.


So I highlighted the fact that the primes can be given a quantitative type interpretation with respect to both their individual and collective attributes (in Type 1 terms) using isolated frames of reference. However when it comes to establishing their two-way interdependence, we need necessarily include the complementary qualitative aspect.

However I did not explicitly elaborate on the direct nature of this qualitative aspect at the time. So I was illustrating the paradox of direction at a crossroads when one can only identify - as it were - a left turn.


However one could equally start from the complementary qualitative direction, attempting to understand, both their individual and collective natures, solely from this perspective. So, this is analgous to being able to identify solely a right turn, which would create the same paradox of direction (when two opposite directions need to be simultaneously recognised at the crossroads).

Using Jungian terminology the qualitative aspect of Mathematics (that I identify with the Type 2 approach) in a very true sense represents the unrecognised shadow of present mathematical activity. Because of extreme specialisation with respect to merely quantitative type interpretation, the Type 2 aspect - though equally necessary for a balanced mathematical understanding - is entirely overlooked.

And because of this, the true nature of the the amazing relational capacity of the primes - which is of a qualitative nature - is continually missed.


Now the shadow of course suggests the unconscious and indeed the holistic capacity to properly appreciate the relational capacity of the primes is directly asssociated with the unconscious in intuitive type terms. Then when the holistic insights emanating from this source are appropriately channeled in an (indirect) rational manner, we are then dealing with the "imaginary" aspect of interpretation (in a precise qualitative manner).


So proper understanding of the nature of the primes requires - as a starting basis - both "real" and "imaginary" aspects of interpretation corresponding to Type 1 and Type 2 Mathematics respectively.


Initially, when both aspects are dealt with in relative isolation, the "real" aspect is directly associated with the (cardinal number) quantitative aspect; by contrast the "imaginary" aspect is associated directly with the (ordinal number) qualitative aspect.

So from this perspective we have seen, that whereas the individual nature of the primes is directly assocaited with the quantitative aspect (as the building blocks of the prime number system), the corresponding general relational nature of the primes is directly associated with the qualitative aspect (where all natural numbers are expressed in terms of a unique combination of prime factors).


Then in the dynamic interactive approach that constitutes the Type 3 mathematical approach, both aspects (Type 1 and Type 2) are united through an interlocking series of complementary type relationships.

And from this perspective, the very distinction of quantitative and qualitative begins to break down, as the quantitative acquires a qualitative aspect and the qualitative in turn a quantitative aspect.


So to illustrate this again in terms of our crossroads analogy! We start off in a relatively unambiguous manner, identifying a left turn (within isolated frames) approaching it in turn from two different directions (north and south) i.e. Type 1. We then in an unambiguous manner, identify right turns (within isolated frames) coming from different directions i.e. Type 2.


Finally we attempt to understand both left and right as interdependent with each other, in a two-way complementary manner.

This leads to a continual shifting of isolated reference frames, so that what was left becomes right, and what was right left, in quick succession i.e. Type 3.

So when we give the two axes in the complex plane, which do indeed intersect at a
central point like a crossroads, a Type 3 interpretation, both real and imaginary axes (representing Type 1 and Type 2 mathematical understanding) keep shifting so as to approximate ever closer to the mutual ultimate identity of both the quantitative and qualitative aspects of the primes.

So once again just as real and imaginary can be given a quantitative interpretation with respect to the Riemann Zeta Function Function, equally they can - and of course should - be equally given a corresponding qualitative interpretation.


So once again, I have arrivedly at the same conclusion - reflected from yet another angle - that the key issue, points to the condition for the ultimate identity of both quantitaive and qualitative aspects of the primes (which lies at the very centre). And this of course is what the Riemann Hypothesis is all about. At the centre of the crossroads we are neither turning left or right (from a linear perspective) or equally from the other circular perspective, turning both left and right (and right and left) simultaneously.

So here both line and circle (and linear and circular) are finally reconciled.

From Triviality To Ineffable Mystery

In many ways this represents a return to yesterday's blog entry.

However my intention here is to highlight the enormous gap that I would see as between a truly comprehensive mathematical approach (that coherently combines both quantitative and qualitative aspects of understanding) and the highly reduced conventional model that spuriously attempts to maintain its monopoly with respect to mathematical truth.


If we take the - apparently - simple expression 1^4, this seems somewhat trivial from a conventional (Type 1) mathematical perspective.

Here the numbers are viewed in a merely cardinal (quantitative) sense.

This again means that each number unit is treated in a totally homogeneous fashion (devoid of any qualitative distinction). So from the perspective 4 can be broken down as 1 + 1 + 1 + 1 (where each cardinal unit is considered as identical).

Likewise also with respect to the multiplication 1^4 = 1 * 1 * 1 * 1 (with each unit again treated as identical in quantitative terms).

And then again because it ignores any qualitative distinction attaching to each unit the 4 (as dimension with respect to 1 as base number) itself can have no significance from a merely quantitative perspective.


So form a Type 1 perspective 1^4 is thereby quickly reduced as indistinguishable in quantitative terms from 1^1.

So, as I said from such a perspective, 1^4 seems a somewhat trivial expression.


Contrast this now with Type 3 interpretation.

The starting point here is that each number can be given both a quantitative and qualitative meaning, Yet such meanings (which - are - relatively distinct) yet can then be also combined in a dynamically interactive manner.

Thus 4 as dimensional number now also can be given an ordinal (qualitative) interpretation which is inversely related with its corresponding root.

Therefore 4 as ordinal dimension i.e. 1^4 (where it serves as a holistic means of interpretation with respect to the 4th dimension) has its quantitative complementary counterpart as 1^(1/4).

Now this quantitative root = i is a number that simultaneously has both a linear and circular significance (in quantitative terms). In other words from a geometrical perspective, it lies as the extreme point on the imaginary line drawn from the centre of the circle of unit radius (in the complex plane) where it meets the circular circumference. So here at this point the very notion of line and circle coincide.

Then in corresponding qualitative terms, it relates to the qualitative interpretation of the imaginary number i, which equally combines an intersection of both linear and circular type understanding.

So from the ordinal (qualitative) perspective each of the "units" comprising 4 are distinct.

So in this context of 4, the 1st, 2nd, 3rd, and 4th units are of a distinctive qualitative nature, which are "seen" through conversion from the linear to the corresponding circular aspect of the number system.

This implies that we cannot properly understand the qualitative meaning associated with the 4th dimension without - in this context - providing meaning to the 1st, 2nd, and 3rd dimensions also.

Once again, this has its quantitative counterpart equivalent in the fact that we have - again in this context - 4 roots of units i, - 1, - i and + 1 respectively.

Thterefore, the deeper significance of multiple roots of a number is that they inevitably refer to underlying ordinal qualitative type distinctions (which cannot be reduced in a quantitative manner).

And because Type 1 Mathematics makes no provision for qualitative distinctions, it can give multiple solutions to polynomial equations no proper interpretation!

So in understanding 4 properly in qualitative terms we also include the understanding 1, 2, and 3 in a distinctive manner.

However Type 3 enables the combination of both cardinal and ordinal type interpretation.

So from this context 1^4 also requires 1^3, 1^2 and 1^1 which inversely are related to 1^1, 1^(3/4), 1^(2/4) and ^(1/4) as the counterpart quantitative roots.

And here is where the cardinal notion of 4 as dimension comes in, for when we raise each of the root expressions to the power of 4 we obtain 1^4, 1^3, 1^2 and 1^1 thereby directly switching - relatively - from a quantitative to a qualitative context.

So we have now combined the cardinal notion of dimension with its corresponding ordinal equivalent.

Put another way, we have combined linear with circular understanding (which is implied by the very geometrical notion of roots) which then must be given both linear and circular interpretation in qualitative terms.


Finally the true meaning of 1^4 strictly implies a direct coincidence of linear and circular type notions (from both the quantitative and qualitative perspectives).

So the true meaning of the expression is one of ineffable mystery (as here both quantitative and qualitative aspects are identical).

However in the very attempt to unravel such mystery we must necessarily attempt to separate to a degree both aspects (quantitative and qualitative).

So in the actual realm of phenomenal type understanding, a degree of uncertainty necessarily attaches to any mathematical expression (including the "simple" one here i.e. 1^4).

So one can perhaps see how the rich mystery inherent in very simplest of mathematical expressions evaporates completely from a Type 1 perspective (where it is treated as utterly trivial).

And such extreme reductionism of truth is what we still misleadingly refer to as Mathematics.

Type 3 can thus indeed provide the appropriate perspective to appreciate why Type 1 Mathematics can reach such a conclusion.

So it provides the context to show that while Type 1 certainly has the appearance of being "absolute", that this in fact represents but an extreme distortion of the much more authentic nature of what properly constitues mathematical truth.

Issues with Respect to Cardinal and Ordinal Numbers

It might helpful to clarify here in greater detail the precise distinction as between the quantitative and qualitative interpretation of numbers respectively.

Our starting point - though ultimately it is not quite so simple - is to consider with respect to any expression (comprising both base and dimensional numbers) the base number as quantitative and the dimensional number as qualitative respectively.

So in this context therefore, when we take the number expression 3^2 for example, the (base) number 3 refers to a quantitative and the (dimensional) number 2 - relatively - to a qualitative meaning.


Now this interpretation of quantitative and qualitative bears a very close relationship in turn with the corresponding interpretation of cardinal and ordinal.

So when we consider 3 as a number quantity, we have the cardinal use of number in mind. 3 here conveys the notion of multiple homogeneous units. So when we say 1 + 1 + 1 = 3, the very point is that no qualitative distinction is made as between the units, thereby enabling them to be treated in a merely quantitative manner.


However when we consider by contrast 3 as a (dimensional) number quality, we have the ordinal use of number in mind. So for example, I spent some time in the context of the Riemann Zeta Function, clarifying the qualitative nature of the 2nd dimension, and then later the 4th dimension. Therefore the nos. 2 and 4 in this ordinal sense i.e. as representing the 2nd and 4th dimensions respectively, clearly should not be confused with their cardinal use.


And then it becomes quickly apparent that ordinal always to a degree implies cardinal (and cardinal, ordinal) suggesting strongly therefore, that numbers in dynamic interactive terms, necessarily keep switching as between cardinal and ordinal meaning.


In truth a high degree of subtlety is required to properly clarify, in any relevant context, the precise relationship as between quantitative (cardinal) and qualitative (ordinal) interpretation. And this is what defines the Type 3 mathematical approach.


However, again because of the explicit lack of any qualitative dimension, Conventional (Type 1) Mathematics must necessarily confuse ordinal with cardinal meaning.


I was already aware of an important example of this key issue from childhood. Even then it struck me as distinctly odd that the two roots of 1 could be expressed as + 1 and - 1.

I tended to always look at Mathematics in a holistic manner. So for example I reckoned that from a qualitative perspective, mathematicians would not accept that the truth value of a proposition could be true and false at the same time. For this would be tantamount for example to accepting that the Riemann Hypothesis could be proved true one day and then equally proved false the next.
So Conventional Mathematics is strongly based on the premise as regards proof that the positive pole necessarily excludes its negative (and vice versa)!


However yet in the parallel world of quantitative truth, in the simple case of the square root of 1, the positive includes the negative (and the negative the positive).

So again even from a young age, I could see that here was a fundamental issue of great importance that pointed to a glaring inconsistency. For given that Mathematics is defined in a merely quantitative manner, there should thereby be an automatic correspondence regarding the logical approach with respect to the behaviour of specific mathematical quantities on the one hand and generalised mathematical proof on the other!


Now the key to unlocking this problem is the recognition that when we refer, for example, to the two roots of 1, that both the cardinal and ordinal use of number are necessarily involved.


So the 2 roots (in cardinal terms) relate to both the 1st and 2nd roots (from an ordinal perspective).

Therefore the key to dealing with the matter consistently is to recognise that the qualitative aspect of the number system is also required and that the 1st and 2nd roots (understood in quantitative manner) have complementary qualitative interpretations as the 1st and 2nd dimensions of 1.


So the 1st dimension, is written as 1^1 and the 2nd dimension as 1^2.

Then the 1st answer + 1 (for the two roots) properly represents the square root of 1^2 (i.e. 1^1) whereas the second root - 1, properly represents the square root of 1^1 i.e. 1^(1/2)!


So we can now see clearly that an important qualitative distinction actually applies to the two roots, with the positive root defined with respect to the 2nd dimension and the negative with respect to the 1st.

We could of course express these roots as simple equations.

+ 1 is the solution to x2 (i.e. x squared) = 1^2.

- 1 is the solution to x2 = 1^1.

So the first root results directly in the customary linear notion of 1 (i.e. the 1st root of 1). Not surprisingly this is taken as the principle root in Conventional Mathematics.

However from a qualitative perspective the true square root of 1 relates to its second root i.e. - 1.

And though these roots do indeed have a quantitative meaning, properly they cannot be divorced from corresponding qualitative interpretation. And to obtain the appropriate qualitative dimension that can provide such interpretation, we find the reciprocal of the dimensional number used to obtain the root.


Now for the 1st root this is trivial as the reciprocal of 1 is 1. So the positive root in quantitative terms, corresponds directly with linear (1-dimensional) interpretation.

However the 2nd root, represented by 1/2 (as dimensional power of 1 in quantitative terms) is thereby correctly interpreted in a qualitative manner as 1^2. This implies that the 2nd dimension - in an ordinal sense - is required to properly interpret any expression with the dimensional power of 1/2 (in quantitative terms). And all of this is deeply central to proper interpretation of the Riemann Zeta Function (where s as a dimensional number, can assume any value in the complex plane)!


There are however subtle difficulties regarding the precise definition of an ordinal number.

So when I defined the ordinal dimension 2 and then later the ordinal dimension 4, these were tied up with the cardinal numbers 2 and 4 respectively.

In other words, we cannot define the ordinal number 2 in the absence of the cardinal number 2. So we require 2 in cardinal terms to give a 2nd unit meaning (from an ordinal perspective).

Likewise we cannot define the ordinal number 4 (which we also dealt with in earlier blog entries), in the absence of the cardinal number 4. So the 1st time we encounter 2 in an ordinal sense is with respect to 2 (in cardinal terms) and the first time again we encounter 4 in an ordinal sense is with respect to the cardinal number 4!


And this is precisely the context in which we thereby give meaning to 2 and 4 (as ordinal numbers).

However, we also could give a potentially unlimited meaning to the 4th dimension with respect to cardinal numbers 5, 6, 7....and do on. However these would not be written as integer values.


To illustrate let us briefly deal with the 4 roots of 1

i relates to the 4th root of 1^1 i.e. 1^(1/4) i.e. the true 4th root.
- 1 relates to the 4th root of 1^2 i.e. 1^(2/4) = 1^(1/2)
- i relates to the 4th root of 1^3 i.e. 1^(3/4)
+ 1 relates to the 4th root of 1^4 i.e 1^(4/4) = 1^1.


Now the corresponding dimensional numbers with which these are qualitatively associated, are 4/1 = 4, 4/2 = 2, 4/3, and 4/4 = 1.

So interestingly when we switch from quantitative to qualitative interpretation, the order sequence is reversed, so for example the 1st root is associated with the last dimension!

However the 3rd dimension in the context of 4 (as cardinal number) is not an integer! And this is because the integer value only applies when both cardinal number and ordinal ranking coincide!

Then, as we have been repeatedly saying with Type 3 interpretation, it can keep alternating as between both the quantitative and qualitative meanings associated with each number!

All of this is potentially extremely rich territory for further investigation. And again because Type 1 Mathematics has no qualitative dimension, not alone can these issues not be properly interpreted, they cannot even be properly approached from that limited perspective!.


The very clue to the interaction of quantitative and qualitative with respect to the roots of unity, is provided through their geometrical representation. Now, quantitative always implies linear and qualitative circular, respectively!


And this is perfectly mirrored in geometrical terms. Here the coordinates of each root of 1 (in the complex plane) simultaneously lie on the extremity of a radial line (drawn from the centre of the circle) which also lie on the circular circumference. So at this point of intersection, both line and circular circumference coincide!


In like manner, corresponding qualitative interpretation of such roots (where the root bears an inverse relationship with its dimensional number), always entails the intersection of linear (quantitative) with circular (qualitative) type interpretation where both ultimately coincide. And this ultimately can only be dealt with satisfactorily in a comprehensive Type 3 mathematical approach.

An appropriate Type 1 approach deals with the quantitative aspect in relative isolation; an appropriate Type 2 approach then deals with the qualitative aspect in relative isolation; finally the comprehensive Type 3 approach deals with both aspects, as relatively independent and interdependent respectively, in a dynamic interactive manner.

In this context, not alone is the conventional mathematical approach confined to the Type 1 approach, but unfortunately in a manner that greatly hinders access to the other two approaches. This is due to the absolute nature of its assumptions (which do not accord with authentic experience of mathematical reality).

So even with respect to specialisation in Type 1 Mathematics, the appropriate explanatory manner properly should recognise the important role of the other approaches. And this patently is not the case at present!

Addition, Multiplication and Exponentiation

I have already outlined in earlier blogs details of the the two number systems corresponding with the Type 1 (quantitative) and Type 2 (qualitative) aspects of Mathematics.

So once again, Type 1 is is defined with respect to a fixed number 1 (as dimensional quality), with base number quantities varying. So the natural number system here is:

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


By contrast Type 2 is defined with respect to a fixed number 1 (as base quantity), with dimensional numbers varying. So the natural number system here is:

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


Addition (can take place in both systems (as separate).


So in the Type 1 system 2 + 3 = 5;

So written out in full, this expression is

2^1 + 3^1 = 5^1

Here a quantitative transformation with respect to the variables takes place (without any corresponding qualitative transformation).



Likewise in the Type 2 system 2 + 3 = 5.

So from a Type 2 perspective 2 + 3 = 5, written out in full is,

1^2 * 1^3 = 1 ^(2 + 3) = 1^5

So here by contrast a qualitative transformation with respect to the variables takes place without any corresponding quantitative transformation.


So we here can resolve quickly the mystery of the nature of addition and multiplication.

What implies addition with respect to the Type 1 (quantitative) system, implies multiplication with respect to the 2nd (qualitative) system.

This explains why both Alain Connes and Brian Conrey (writing in the context of the Riemann Hypothesis) have admitted that there is something fundamental regarding the relationship of addition to multiplication that is not yet grasped (i.e. in Type 1 Mathematics).

And the essence of the problem is that multiplication strictly relates to qualitative - rather than quantitative - transformation. And in a system that is defined solely by its quantitative aspect, it is therefore not possible to deal adequately with the nature of multiplication.

So, multiplication is necessarily given a merely reduced quantitative interpretation in Type 1 terms (where it is not properly distinguished from addition).


So for example 2 * 3 can be translated (in Type 1 terms) as 2 + 2 + 2 (i.e. 2^1 + 2^1 = 2^1 where we add 2^1 to itself 3 times) which clearly demonstrates the lack of dimensional change).

Thus again, the very nature of addition is tied up with quantitative transformation, whereas by contrast the nature of multiplication is tied up with qualitative (i.e. dimensional) transformation. And I have already explained in recent blog entries the nature of interpretation for dimensional values 2 and 4 of the Riemann Zeta Function as the (circular) qualitative nature of dimensional transformation.


In a Type 3 mathematical approach, both the base and dimensional numbers can vary with both numbers always interpreted in a complementary fashion.

So, if starting from the base number (as representing a quantity) then - relatively - from this perspective, the now varying dimensional number is understood in corresponding qualitative terms as representing a quantitative value;

Now in isolation, this will be true when we vary the number representing qualitative dimension (which was fixed at 1 for the Type 1 system), or alternatively vary the base quantity (which was fixed at 1 for the Type 2 system).


For example in the Type 1 system when we start for example with 3^1, the base number is a quantity and the dimensional number - realtively - of a qualitative nature.

So if we allow the dimensional number to change to 2, 3 is a base quantity raised to 2 as a qualitative dimension. So we now have 3^2.


Likewise in the Type 2 system, when we start, in reverse fashion, for example with 1^2, the base number represents a quantity and the dimensional number has a qualitative interpretation.

So if we now allow the base quantity to vary to 3, while the dimensional number remains, we again obtain 3^2.


Now in isolation, the same expression is obtained for both systems!

However just as in relation to each other at a crossroads. two left turns (in isolation) necessarily are left and right (and right and left) in relation to each other, likewise when we combine both Type 1 and Type 2 expressions for the "same" numbers, as in this example 3^2, what was quantitive (from one perspective) becomes qualitative from the other; and what was qualitative from the other perspective, becomes quantitative from the first.

So in Type 3 terms both the numbers 3 and 2 in the expression 3^2, have both quantitative and qualitative aspects, which continually alternate in understanding.

Therefore - in relative terms - if 3 is quantitative, then the exponent 2 is quantitative; also in reverse, in relative terms, if 3 is qualitative, then 2 is quantitative.


So to conclude:

Type 1 Mathematics is based on the pure quantitative extreme (which is defined in terms of a default dimensional number of 1). Therefore when higher dimensional numbers are involved (as powers), the result is ultimately expressed in terms of the invariant dimensional value of 1 (thus enabling reduced quantitative interpretation).

Type 2 Mathematics, is based on the complementary pure qualitative extreme (which is defined in terms of a default base number of 1). Therefore, when higher number dimensions are involved, the result is expressed in terms of an invariant base quantity of of 1 (thereby enabling reduced qualitative interpretation).


Using these extreme cases we can isolate the nature of addition and multiplication respectively. So what represents addition with respect to Type 1, represents multiplication with respect to Type 2 (clearly indicating the quantitative and qualitative nature of addition and multiplication respectively).

Now Type 3 Mathematics (which is by far the most comprehensive) allows both base quantity and dimension to vary (thereby necessitating exponentiation with respect to values other than 1).


What we have concluded is quite remarkable, in that a true account of non-trivial exponentiation (i.e. for values other than 1 with respect to both base quantity and dimensional number) necessarily requires Type 3 Mathematics.

And from this perspective the two numbers (base and dimension) acquire both quantitative and qualitative (and qualitative and quantitative) interpretations in two-way relationship with each other.


Now the Riemann Zeta Function clearly requires both base numbers (i.e. the natural numbers) and dimensional numbers for any numbers in the complex plane, where both can vary..

This clearly implies that the Riemann Zeta Function requires a Type 3 mathematical approach (for proper interpretation).

Implications for Mathematical Proof

From a Type 3 mathematical perspective - quite literally - every mathematical symbol, relationship, theorem etc. that can be given a Type 1 interpretation (in quantitative terms) must be interpreted in a complementary interactive manner with respect to both quantitative and qualitative aspects.

Indeed strictly speaking mathematical meaning now resides in the combined dynamic relationship as between aspects (rather than in either aspect as separate).


This therefore has great implications for the mathematical notion of proof.

From a (linear) quantitative perspective, a proposition is accepted as "proved" once the appropriate Type 1 explanation has been given.

So from a Type 1 perspective therefore, the Pythagorean Theorem has indeed been proved. In fact a wide variety of different valid proofs are available in this regard.


However, as we have seen there is also - as I have been demonstrating - with respect to the Riemann Zeta Function - an equally valid qualitative significance associated with every mathematical entity.

So in principle for every Type 1 proposition that has been proved in Type 1 terms, a coherent holistic qualitative interpretation can be given.

In broad terms that might be helpful, the Type 1 aspect essentially deals with "how" a proposition is true (or untrue) as the case may be.

The Type 2 aspect then uncovers the deeper "why" of the truth value of the proposition.

And then finally the Type 3 approach (which combines the previous two aspects as interdependent) marries both the "how" and the "why" in a manner that is truly meaningful in both a rational and holistic intuitive manner.


Now it is important to stress that the very manner of understanding from the rational and intuitive perspectives requires very distinctive capacities.

Therefore one who specialises in an unduly narrow fashion in linear type reason is highly unlikely to possess a developed holistic capacity (at a Type 2 level); likewise one who naturally operates in a holistic manner with respect to understanding again is very unlikely to possess sufficient rigour with respect to the linear rational (Type 1) aspect.


Indeed the true position is much worse than that; because of the extreme specialisation of Mathematics as a discipline, it is highly unlikely that most mathematicians would even want to recognise the (hitherto) unrecognised Type 2 aspect (even when shown to them)!


Thankfully, there will always be some notable exceptions at the margins, as it were, who possess significant mathematical talent while remaining open to a variety of other influences. So I would expect that most "specialists" would simply dismiss a blog such as this as quite irrelevant in terms of what they consider "Mathematics". However there may be yet a few generous souls who can recognise that I am indeed seriously addressing key fundamental mathematical issues, while offering a radical vision of a greatly enhanced mathematical future. And it is that small audience of open minded individuals that I principally wish to address in these blogs!


So to sum up all propositions, that can be proved in Type 1 terms, can be given a corresponding holistic rational (in Type 2 terms) which essentially implies a distinctive type of "proof".

Now, proof in Type 1 terms implies a linear sequence of logic steps (based on an initial set of axioms) so as to eventually arrive at the desired conclusion in this manner.

So the proof in effect is shown to be implied by the initial axioms, as demonstrated through independent logical steps taken to arrive at the conclusion.


However "proof" in Type 2 terms operates in a distinctive circular manner, where the interdependence of all aspects contained is holistically demonstrated through a dynamic set of two-way complementary type relationships. Ultimately arriving at such a "proof" relates to a required level of intuitive capacity (that can enable such connections). Indirectly however the connections can then be rationally demonstrated in a circular logical fashion.


Now it must be clarified right away, that we are not here speaking about the intuitive approach as understood in Conventional Mathematics. Riemann and Ramanujan for example were both highly intuitive mathematicians, but they still operated within the confines of Type 1 (quantitative) Mathematics.

I am speaking about the intuitive capacities required for true qualitative (i.e. Type 2) appreciation which is of an entirely different order altogether. Type 1 Mathematics is strictly 1-dimensional from a qualitative perspective. So I am talking about the intuition that potentially applies to all dimensions other than 1.


So the Type 1 aspect is designed to provide the quantitative aspect of mathematical proof (based on a linear type logical approach).


The Type 2 aspect is designed to provide the corresponding holistic qualitative aspect of mathematical proof (based on a circular type logical approach).


The Type 3 aspect is then designed to integrate both aspects of proof in a dynamic interactive manner.


The possibilities here are limitless! However ultimately I suspect that when propositions are proved in an appropriate manner (in accordance with the Type 3 approach) that two two aspects of proof should match in a synchronous manner. Then one should simultaneously see the relationship of the rational Type 1 connections to the holistic Type intuitive type connections (indirectly translated in a rational circular manner).


This would then open up immensely creative possibilities for quickly moving from a (recognised) Type 1 to an (as yet unrecognised) Type 2 aspect of proof; or alternatively for moving from a (recognised Type 2) to an (as yet unrecognised) Type 1 aspect.


So one of my present goals is to prove some mathematical proposition, which I am not presently aware has been achieved in Type 1 terms, from a corresponding proof of the Type 2 aspect (that I can intuitively "see" first in holistic terms).


I did succeed (at least to my own satisfaction) in one of my earliest mathematical goals, which was to resolve - what I term - the Pythagorean Dilemma.

The Pythagoreans did indeed recognise that Mathematics possessed both quantitative and qualitative aspects. It is only the subsequent specialisation in a merely quantitative direction, that has led to such a significant loss of the qualitative dimension!

So in their discovery that the square root of 2 was irrational, they were able apparently to provide a (Type 1) quantitative proof. However their dilemma was that they were not able to provide the corresponding qualitative explanation that would make this new finding intuitively accessible. So the all important holistic appreciation of the true nature of an irrational number was missing. In other words they lacked the Type 2 aspect of proof!


Therefore, after some time developing holistic mathematical notions, I set the task of providing this missing Type 2 aspect using my own insights and terminology to resolve the issue.

Recently, I briefly explored this issue again in Type 3 terms and to a considerable degree was able to establish the mutual correspondence as between both aspects of proof.


So in this context, the Type 1 aspect relates to proof of why - or rather how - the square root of 2 is irrational (in a specific quantitative manner).


The Type 2 aspect relates to proof of why (philosophically) the square of root of 2 is irrational (in a holistic qualitative manner).


The Type 3 aspect relates to establishing a mutual correspondence of both quantitative and qualitative aspects through matching explanations so that the linear can be readily adopted to circular, and the circular to linear appreciation respectively. In this way the two aspects are seen as reflecting the "same" interactive truth reality (containing both quantitative and qualitative aspects) in a complementary fashion.

Number Uncertainty

There is an additional point following on from yesterday's blog contribution.

I had mentioned - again from the Type 1 conventional mathematical perspective (which defines its quantitative nature) that Euler had proved that for all even values of s > 1, that the numerical result for the Zeta Function can be expressed in the form of k *(pi^s) where k represents a rational number (< 1).

So the remaining task is to demonstrate why this is equally true from a Type 2 qualitative perspective.

As we have seen there is an inverse relationship as between the the qualitative number s (representing dimensions) and its corresponding s roots (in quantitative terms).

Now where we have an even number of roots these can always be paired off in a complementary manner.

Geometrically this could be illustrated by plotting the s roots on the circle of unit radius (in the complex plane). Now, if from each point we draw a a diameter line it will connect with an opposite point on the circle. So for example in the case of the 6 roots of unity we would have three diameter lines with the coordinate points at the end of each line representing the same numbers with opposite signs (with respect to both real and imaginary parts).

So though the numbers can now be complex, they still are arranged in a complementary manner (with positive and negative poles).

Thus, they therefore represent an extension of the basic relationship of the circumference to its line diameter (with for each even value of s, s/2 complementary pairs existing).


Having said this, a special significance attaches to the complementary pairings of s (where s represents a power of 2). And with respect to such powers, 2, 4 and 8 are the most important.

So in the case of 2 we get positive and negative real poles.

With 4 we get positive and negative poles (in both real and imaginary directions). Then with 8 we get positive and negative poles in real, imaginary and complex directions (where the 4 complex cases have the special property of representing null lines).


So likewise in an inverse qualitative fashion (representing dimensions) when s is even, we again can arrange opposite poles in the complex plane in a complementary fashion (so that they all literally lie on diameter lines through the centre of the circle).


Thus the same basic relationship of circular to linear is maintained (which represents the qualitative definition of pi).

Therefore we can have a 2-dimensional interpretation of pi in qualitative terms (representing one pair with 2 opposite poles), a 4-dimensional interpretation of pi (representing two pairs, each with two poles), a 6-dimensional (representing 3 pairs. each with 2 poles) and so on.

But for those dimensions relating to powers of 2, a special significance attaches, with again 2, 4 and 8 the most important, as these fundamentally dictate the dynamic nature of all phenomenal relationships.


In this qualitative context, 2 represents the complementarity of opposite real polarities (such as external and internal); 4 represents the combined complementarity of both external and internal and also whole and part that are real and imaginary with respect to each other; 8 represents the more intricate combined complementarity of external and internal, whole and part, and form and emptiness in ways that are - relatively - real, imaginary and complex (as null lines) with respect to each other.

And once again remember we are now using all these mathematical notions in a Type 2 qualitative - rather than quantitative - sense.


Again the special significance of even dimensional values s (that are powers of 2) is demonstrated through the denominator of the rational value by which the relevant expression in pi is multiplied.

As we have seen in such cases the denominator can be expressed as the product (where the same value can be used more than once) of all primes from 2 to s + 1 (and only these primes). For all other even values the denominator can be expressed as the product of all primes from 3 to s + 1 (and only these primes).

So in qualitative terms a stronger relational aspect with respect to the primes attaches to denominator values associated with s (representing a power of 2).


I have mentioned before that Jungian Psychology eespecially lends itself to mathematical interpretation of a qualitative nature.

Jung drew attention to the role of mandalas (pictorial symbols) serving as key archetypes of integration (i.e. with a strong relational value).

Now, the most frequently used mandalas are based on 4-fold or 8-fold ornate pictorial patterns, which are based on the geometrical structure of 4 and 8 (as dimensions) resprectively.

Both 4 and 8 are rich enough to include both real and imaginary aspects (which in qualitative terms would represent both conscious and unconscious). A 2-fold pattern however would be confined to real aspects (which therefore would not adequately include the unconscious).

So the mandalas therefore can be seen in psychological terms as expressing the harmonious integration of conscious and unconscious aspects of the personality. And the most popular forms are based on the use of 4 and 8 as numbers (understood in a qualitative dimensional context).


When correctly interpreted from a Type 3 mathematical perspective, all numbers are subject to the Uncertainty Principle.

We demonstrated this already in relation to pi. One can perhaps appreciate, because of the nature of pi as a transcendental number, how such undertainty might apply.


However, strictly speaking it is in fact true of all numbers (in Type 3 terms). So even with the individual prime numbers, a necessary uncertainty attaches to their nature in quantitative terms.

This is due - as always - to the fact that at the Type 3 level, every mathematical symbol represents a dynamic interaction of both quantitative and qualitative aspects in its inherent identity.

So again a number e.g. an individual prime is properly defined in terms of such a 2-way interaction. Therefore when we attempt to abstract the quantitative aspect it ncessarily has an approximate - merely relative - status.


Thus to illustrate, the (absolute) identity of 5 (as representative of an individual prime number) properly relates in Type 3 terms, to the dynamic interaction of both its quantitative and qualitative aspects.

So when we try to look at 5 in merely quantitative terms, we meet the old problem that the finite number system can never be fully determined in a actual manner. Once again the potential infinite nature of number properly relates to the qualitative aspect, which cannot be reduced in a finite actual manner!

Now an individual number properly only has meaning in relation to the overall concept of number as infinite (which is always to a degree indeterminate in finite actual terms).

So in this sense as we are understanding 5 in relation to what is ultimately of a finite indeterminate nature, its value is likewise approximate. Or to put it another way, though we define 5 as a number, we cannot properly define what number is (in an actual - merely - quantitative manner).

Once again there are great similarities here with the nature of Quantum Mechanics. Though quantum mechanical efefcts are most easily demonstrated at the level of sub-atomic matter, strictly speaking - as is now accepted in Physics - all objects are subject to the same effects.

However because these effects are scarcely noticable at the macro level of everyday experience, they can be ignored for all practical purposes (where the unambiguous Newtonian viewpoint serves as an excellent approximation).


And it is the very same with number (and indeed all mathematical relationships). The apparent absolute nature of a number such as 5, in strict terms represents an approximation (which however can be effectively ignored in the context of Type 1 Mathematics).


So the relationship of Type 1 to Type 3 Mathematics is somewhat similar to the corresponding relationship as between Newtonian Physics and Quantum Mechanics.

In truth the approximate nature of quantum mechanical relationships is itself rooted in the more fundamental uncertainty that attaches to all mathematical relationships.