Spectrum of Development

The two aspects of the number system that I have identified i.e. Type 1 and Type 2 respectively, are intimately related to the distinction as between addition and multiplication. To see this clearly, I concentrate on - what I refer to as - pure addition and pure multiplication. Pure addition simply relates to the successive adding of 1 to the previous unit (starting with 1). Pure multiplication by contrast relates to the successive multiplication by 1 of the previous unit (again starting with 1). In this way, we are enabled to generate both the Type 1 and Type 2 aspects of the number system. Thus through addition of 1 we obtain, 1 1 , 2 1 , 3 1 , 4 1 , ...... So in this Type 1 aspect, the base are the natural numbers, which are all defined with respect to the (default) dimensional number of 1. This defines the conventional mathematical approach to numbers where the natural numbers - and indeed all reals - are represented as lying on the (1-dimensional) number lin

As we have seen, with the linear (analytic) interpretation of number, the ordinal notion is effectively reduced in cardinal terms. Therefore we must move to a circular (holistic) interpretation, so that the ordinal aspect - while necessarily existing in relationship to the cardinal - yet preserves its own distinct identity. When one reflects on it, the ordinal notion can only have meaning with respect to a group of numbers that are defined in a cardinal manner. So for example one can validly seek to interpret the ordinal notions of 1st and 2nd (in the context of 2), the ordinal notions of 1st, 2nd and 3rd (in the context of 3), the ordinal notions of 1st, 2nd, 3rd and 4th (in the context of 4), the ordinal notions of 1st, 2nd, 3rd, 4th and 5th (in the context of 5) and so on. So again to illustrate, if we have a class of 20 pupils (in cardinal terms), then we can provide ordinal rankings in an examination for 1st, 2nd, 3rd, .....20th positions. What is fascinating here is tha

To recap again briefly, during the 3 levels that characterise Band 3 (in my map of development), I had come to realise the extraordinary potential importance of the holistic dimensional notion of number (according to its Type 2 aspect). Through this holistic mathematical interpretation of number, I could see how the dynamic interactive nature of experience is implicitly structured in mathematical terms, with each holistic dimension relating to a unique configuration with respect to the fundamental polarities of experience (internal/external, whole/part and form/emptiness).  Indeed as the 3rd polarity set here represents but a special requirement with respect to the first two sets, this entails that phenomenal reality (in physical and psychological terms) is fundamentally conditioned by the dynamic interaction of just two sets of polarities (i.e. internal/external and whole/part). In psycho spiritual terms the "higher" dimensions (> 1) represent increasingly refined int

Before proceeding further, I will once again attempt to make clear the distinction that I make as between the two notions of number - in effect particle and wave aspects - that continually interact in experience as analytic to holistic (and holistic to analytic) respectively. I define firstly the Type 1 aspect (where the number in question is referred to as the base). So every base number here is defined with respect to a default dimensional number of 1. Thus 2, for example in Type 1 terms is defined as 2 1 . I then define the Type 2 aspect (where the number in question is referred to as the dimensional number). Every dimensional number here is now defined, in inverse fashion with respect to a default dimensional base of 1. Thus 2 now in Type 2 terms is defined as 1 2 . Now it is vital to appreciate these two aspects of the number system operate in a dynamic relative fashion as complementary, where they keep switching (both in base and dimensional terms) as between analyt

I have spoken many times before how Band 3 development, in my experience, was characterised by a keen realisation of the holistic mathematical nature of number (as relating directly to dimensions). Now Band 2 - on which conventional scientific notions are firmly based is strictly 1-dimensional in this qualitative sense. This entails the absolute separation - in formal terms - of the key polarities of experience i.e. external/internal, whole/part and form/emptiness respectively, so that in any context involving these polarities, interpretation takes place with respect to just one fixed pole. So conventional mathematical and scientific interpretation is based on the attempt to abstract objective (external) reality from mental (internal) interpretation. Likewise it is based on a mere quantitative relationship as between whole and parts (whereby the whole is viewed in reduced terms as the sum of its parts). Finally it is based on the consideration of phenomenal notions of material form

I would like to convey here the true significance of the Riemann Hypothesis. However it is vital to appreciate from the outset that this significance cannot be properly understood within the accepted confines of Conventional Mathematics, which solely recognises, in formal terms, the quantitative (analytic) interpretation of its symbols. Indeed from one important perspective, the Riemann Hypothesis - when adequately interpreted - points directly to a critical limitation in the overall accepted framework of Mathematics.   The number system is fundamental to the reality we call Mathematics. And what in turn is central to the number system is the relationship between the primes and the natural numbers. Now a prime has no factors (other than itself and 1). So 7 for example is an early example of a prime. All other natural numbers (i.e. composites) can then be uniquely expressed as the product of primes. So for example 30 ( = 2 * 3 * 5) is therefore composed of 3 prime factors

We return now to a personal account of the journey through the 3 main levels of Band 5. Once again these bear a complementary relationship (on the spiritual descent) to the same three levels (on the spiritual ascent). However whereas as the direction on the ascent is primarily of a transcendent nature, where nondual spiritual development is - to a considerable extent - at the expense of customary dualistic understanding, here a reverse immanent emphasis applies, where one now gradually attempts to incorporate the dual in a balanced manner with the nondual. And once again, these 3 levels (of Band 5) are characterised by a special emphasis on each of the corresponding key polarity sets. And just as with Band 3 the journey initially commences at Level 1 with intense exposure to the (horizontal) interaction of external and internal polarities, equally it is similar at Band 5. However whereas at Band 3, this largely represents the gradual breakdown of rational analytic type apprec

We have looked now briefly at the modes of development both with respect to the primary "colours" (cognitive, affective and volitional) and the many - and somewhat arbitrary - list of secondary modes (i.e. "multiple intelligences"). And we have seen that as far as successful integration of the personality, the primary modes are vital. In fact the complete mature development of the personality requires that cognitive, affective and volitional be both differentiated successfully in a sequential manner with respect to each of the major bands (and accompanying levels) on the spectrum, while also being simultaneously integrated with each other. Now, while imbalances are indeed possible with respect to development of the three primary modes (up to and including Band 5), the final two radial bands would however require a high degree of integration with respect to all three modes. As we have seen, in terms of the secondary manifestations, a person may show special

An important omission from Howard Gardner's list of "multiple intelligences" is any mention of humour. I have always found it significant that humour is generally omitted from the story of development, which is indeed very odd as it constitutes (in all its varied forms) a very important intelligence. Some years ago I gave special attention to this neglected issue submitting an article to the "Integral World" site entitled " Humour and Related Experience ". I was subsequently gratified when it then later received unexpected attention from some very interesting correspondents. Without attempting to properly summarise what I have already written in that article, I will make the following comments. Just as I have been emphasising with respect to musical and kinisthetic intelligence the dynamic nature in which they can provide temporary "peak" experience (esp. with respect to emotional experience), this is even more true with respect to

We return now to bodily-kinisthetic intelligence. With regards to this intelligence it is especially difficult to see how it might develop though all the major bands (and accompanying levels) of development. There is a sense in which the overall perspective with which one views kinisthetic intelligence   can itself undergo change. However, as we all know special proficiency in this regard is not itself a necessary prerequisite for full development. However directly and indirectly it is indeed very significant in our culture. As we have seen sporting success in any field requires a special form of kinisthetic intelligence (that is in a sense unique to that sport). This is probably the first intelligence to properly unfold in development, where it is tied to early infant sensori-motor ability. This then can later blossom into a special talent for particular sports. Now it is certainly true that the existence of the mere raw ability for a particular sport does not guarantee

Let us now return to "musical" which as we have of seen is one of the seven multiple intelligences referred to by Gardner. I am somewhat puzzled by the nature of this inclusion. Clearly music is very important in many people's lives (both in terms of performance and appreciation). But it refers to just one of many possible artistic talents. For example one could have a special gift e.g. as a painter, an actor, for fashion or interior design, for writing (e.g. plays, novels or poetry). And this by no means exhausts the possible list of artistic fields. So clearly though music does indeed represent one highly important artistic domain, there are of course many others, where individuals can demonstrate a special ability. Though one could perhaps, at a stretch provide a full spectrum model, especially in terms of different levels of appreciation of - say - a Wagner symphony, this perhaps remains a bit exaggerated. Certainly the appreciation of music entails the affect

I mentioned in yesterday's entry that the "multiple intelligences", as outlined by Gardner do not necessarily develop through the entire spectrum. Most do, but not necessarily all! One especially interesting case here relates to logico-mathematical ability. As conventionally understood, this attains full development through specialised rational structures that are associated with Band 2 (in my model of the spectrum). And this equates directly with linear (i.e.1-dimensional) interpretation. What this simply entails is that the fundamental polar opposites, which dynamically condition all phenomenal reality, are separated in an absolute manner. Chiefly this entails the absolute separation of external and internal and whole and part polarities  with quantitative and qualitative poles, which in dynamic experiential terms are necessarily complementary with one another. Thus with linear rational understanding, interpretation - in every context - takes place with respect