Spectrum of Development

There are in fact two complementary relationships as between the primes and the natural numbers. From the standard well-known perspective, this relates to the quantitative (base) aspect of such numbers. However from the little regarded alternative perspective it relates to qualitative (dimensional) aspect of such numbers i.e. as represented by factors. So for example, from the former external perspective we might seek to calculate the frequency of primes up to a given natural number. From the latter internal perspective, we might then seek to calculate the ratio of natural to prime  factors within this given number. In both contexts, log n is of special importance. In the former external case, it measures the average gap as between prime numbers. In the latter internal case, it measures the average frequency of natural number factors. Thus again in the former case, n/log n measure the average frequency of primes to n. In the latter case n/log n measures the average ga...

I ended the last blog entry by attempting to succinctly explain the true significance of the famed Riemann (i.e. Zeta 1) zeros. In fact what I said there requires just a little more clarification. Remember the fruits of this understanding arise from a dynamic interactive manner of understanding number relationships (which always involves complementary opposite poles)! So therefore when we start with the customary analytic view of the natural number system i.e. as independent cardinal numbers in quantitative terms, the Zeta 1 (Riemann) zeros then operate as the qualitative (holistic) counterpart of this system i.e. where the interdependence of these numbers, through their unique prime factor combinations, can be indirectly represented in a numerical fashion. However because in dynamic terms reference frames continually switch, we can equally start with the (unrecognised) holistic view of the natural number system (where one is directly aware in an intuitive manner of the interde...

We are accustomed through conventional mathematical training to view the primes in quantitative terms as the independent "building blocks" of the natural number system. However what is not all realised is that very nature of the primes changes when they exist - not individually - but rather as factor components of the unique product combinations that generate the composite numbers. It is in this manner therefore that the qualitative nature of the primes arises i.e. through their interdependence with other prime factors. So for example both 2 and 3 (as separate individual primes) can in a valid - though strictly relative - sense be viewed as independent "building blocks" in quantitative terms. However when 2 and 3 are then combined through multiplication to uniquely generate the composite number 6, i.e. 2 * 3 = 6, both 2 and 3 now acquire a relative interdependent meaning in this context, which is thereby of a qualitative (holistic) nature. And of course th...

It may be useful again at this point to emphasise the key significance of what I have been articulating in these blog entries. We are accustomed to thing of Mathematics - especially in its treatment of number - in an absolute unambiguous manner (where the meaning of symbols remains fixed). However in truth an unlimited number of relative type interpretations can potentially apply, with the standard conventional approach representing just one special limiting case. Put another way the standard interpretation is of a 1-dimensional nature, whereby qualitative type considerations with respect to mathematical symbols are reduced in a merely quantitative manner (within a rigidly fixed framework). However associated with every mathematical symbol is a unique qualitative manner of interpretation. Thus Analytic i.e. Conventional Mathematics relates to the quantitative interpretation of mathematical symbols whereas Holistic Mathematics relates to their qualitative inetrpretation.  A...

I will start this entry by contrasting the respective meanings in Type 1 and Type 2 terms of the fractions 1/3, 2/3 and 3/3 respectively. In Type 1 terms these would be given as (1/3) 1 , (2/3) 1 and (3/3) 1 respectively. So from the standard quantitative (analytic)  perspective (1/3) 1 represents 1 of 3 (equal) parts i.e. one third; (2/3) 1 represents 2 of 3 (equal) parts i.e. two thirds. (3/3) 1 represents 3 of 3 (equal) parts i.e. 1 as a whole unit,   However in Type 2 term, these fractions would be given as 1 1/3 , 1 2/3 and 1 3/3 respectively. Now in standard quantitative terms, these represent the 3 roots of 1 i.e. – .5 + .866i, – .5 – .866i and 1 respectively. However these also have an important qualitative (holistic) interpretation. And in dynamic interactive terms, when the Type 1 interpretation relates to the quantitative (analytic) aspect, then the corresponding Type 2 interpretation relates to the corresponding qualitative (holistic) ...

As we have seen, there are two aspects with respect to the understanding of all numbers - and indeed by extension all mathematical relationships - that are quantitative (analytic) and qualitative (holistic) with respect to each other. Both of these aspects can only be properly understood in a dynamic relative context, where each type of understanding  implies the other in a complementary manner. Now, from a psychological perspective, the analytic aspect is directly related to rational type appreciation (of a conscious kind); by contrast the holistic aspect is directly related to intuitive type appreciation (of an unconscious nature). Therefore, when understood appropriately, the role of intuition with respect to mathematical understanding is utterly distinct and cannot be confused with reason. However, because in effect conventional mathematical understanding entails the reduction of holistic type meaning (in an analytic manner), equally this entails the corresponding ...

There is an important (unappreciated) paradox with respect to the quantitative definition of any number. For example, if we take the cardinal number "3"to illustrate, it can be defined in the conventional mathematical manner as, 3 = 1 + 1 + 1. This represents - what I term - analytic interpretation, whereby the (whole) sum i.e. 3, is treated in an actual quantitative manner as the sum of its independent (part) units. So here again, each of its three (sub) units is defined in an independent homogeneous manner i.e. without qualitative distinction. Therefore, from an ordinal perspective, there is no way to distinguish (with respect to dimensions of space and time) which units are 1st, 2nd and 3rd respectively. So in this ordinal context, each unit can potentially qualify as both 1st, 2nd and 3rd respectively. In other words in - what I term - holistic interpretation, each (part) unit is treated in a qualitative manner as potentially representing the interdependence...

In integral terms, the levels of Band 3 properly constitute both the new emerging "higher" stages of that band and the continually revisited "lower" stages of Band 1 (with which they are - in horizontal, vertical and diagonal terms, dynamically complementary). Therefore from a true integral perspective, we do not have here individual stages (in a discrete separate manner) but rather the growing interpenetration of all the stages of Band 1 and Band 3. However because integration is not yet fully balanced, typically more emphasis is placed initially on the differentiation of the new "higher" stages of Band 3 (without full consideration of the consequent need for integration of these with the corresponding complementary stages of Band 1). So in this differentiated sense, it is still correct to give each new stage of Band 3 a relatively distinct independent identity. In terms of my own journey through these stages (of Band 3), I was indeed well aware ...

I have drawn attention to the great holistic mathematical significance of 2, 4, and 8 (representing dimensional nos.) in terms of the developmental task of integration (with complementary applications in physical and psychological terms). Once again, 2 directly relates to horizontal bi-directional integration (within a given level). 4, relates to horizontal and vertical bi-directional integration (within and between levels), where both aspects of integration are still pursued in a - relatively - separate manner. 8, then relates to diagonal bi-directional integration simultaneously within and between levels. This is required to fully integrate the "higher" levels of Band 3 with the complementary "lower" levels of Band 1 (within and between levels). However in dynamic experiential terms, the task of integration cannot be properly considered in the absence of the corresponding requirement for sufficient differentiation of the major levels of each band. So we ...

In my approach, 2-dimensional understanding - associated with Band 3 (Level 1) - is mainly geared towards the bi-directional  horizontal integration of external and internal polarities (within a given level). So from the external perspective, each stage represents a new understanding with respect to the physical world; then from the internal perspective each stage represents corresponding new understanding with respect to psychological reality. In general with respect to human development, an unbalanced emphasis is placed on stages (solely with respect to their internal psychological characteristics). What is not all properly realised however is that each new stage of psychological development is equally associated with a new stage of scientific - and of course mathematical - understanding. So the present accepted scientific paradigm simply reflects the understanding associated with one limited band (i.e. Band 2) of the overall spectrum. 4-dimensional un...