Spectrum of Development

The finding that every number - indeed every mathematical - symbol can be given a coherent holistic as well as analytic interpretation is of the very first magnitude. In fact it reveals a fundamental unrecognised problem that lies at the very heart of Mathematics i.e. that its symbols cannot in fact be coherently understood in a static absolute fashion. Rather the number system - and indeed all mathematical relationships - should be properly interpreted in an inherently dynamic manner reflecting the interaction of complementary aspects. Now for some considerable time, I had been aware that we can define the natural number system in this complementary fashion (using Type 1 and Type 2 formulations that are the inverse of each other). So again the (analytic) Type 1 aspect is defined as, 1 1 , 2 1 , 3 1 , 4 1 ,……… Here the base ranges over the natural number sequence (with respect to the default dimensional value of 1). What this means in fact is that number is viewed

In yesterday's blog entry, I dealt with the 2nd level (Band 4), which can indeed extend over many years. Basically it once again relates to the continuous detachment from an over-identification with the "higher" spiritual aspect of personality, so as to eventually allow equal emphasis to be given to the corresponding "lower" physical aspect. For in truth both are fully complementary - and ultimately identical - with each other. So in an equivalent manner, one is here recognising the need to fully harmonise transcendence (as spirit beyond all form) with immanence (as spirit prior to and thereby inherent in all such form). Now, one might indeed query to what extent distinct levels of development are even appropriate at this stage of development. It is important to bear in mind that I always use the notion of discrete stages in a relative dynamic manner, where they remain necessarily continuous with all other stages of development. At this point, the continu

I identified in yesterday's blog entry the (fundamental) Euler Identity as the appropriate holistic mathematical expression of Level 1 (Band 4). However though this indeed  corresponds to the pure holistic expression of oneness (in transcendent terms) because this in turn relates to the "higher" spiritual aspect of personality that still to a degree dominates the "lower" physical aspect, one is not yet enabled  to equally experience the pure holistic meaning of oneness (in complementary immanent terms). So a subtle volitional attachment to the "higher" spiritual aspect remains, which results in a compensatory involuntary projection  of - still - repressed primitive elements with respect to the "lower" physical aspect. Thus the next level (Level 2) relates to the gradual negation of excess attachment to the "higher" spiritual, which in turn enables one to engage more fully with the unrecognised projections with respect to the &

I will elaborate here on the holistic mathematical significance of the symbols 2iπ, which play a central role with respect to the (fundamental) Euler Identity. Now 2π represents the circumference of the (real) unit circle (i.e. with radius 1). Therefore, 2iπ can be expressed as the circumference of the corresponding "imaginary" circle (with radius i). However what does one holistically mean by "imaginary" in this context? Well, the holistic meaning of "imaginary" represents the indirect attempt to express the dynamic complementary notion of the union of opposites (i.e. + 1 and   – 1) in the standard rational manner. What this in effect means is that when the pure intuitive notion (represented by the complementarity of opposites) is fully realised  in experience, any  remaining  linear notion of  + 1 and   – 1 as representing separate notions of distance are dissolved. This can be easily seen through drawing the unit circle with its line diameter ho

I have stated that the (fundamental) Euler Identity can be effectively used to clarify the holistic mathematical nature of the 3 Levels of Band 4. However, this of course entails that we can appropriately interpret its symbols in a true holistic manner (that is directly dependent on the interaction of nondual intuition with a refined paradoxical appreciation in rational terms). Now the starting formulation of the (fundamental) Euler Identity can be directly associated with Band 4 (Level 1). So, e 2iπ   = 1. However, as explained in the last blog entry to distinguish both the Type 1 and Type 2 aspects of number, each is now presented more fully with respect to both a base and dimensional number. Therefore e 2iπ   = 1 1 . However, uniquely in the natural number system 1 1 appears as the first number in both Type 1 and Type 2 aspects. Therefore we need a more refined way here to distinguish the (true) Type 2  from the corresponding (conventional) Type 1  understanding.

For some time I remained deeply puzzled by a very interesting facet of the (fundamental) Euler Identity. So once again, e 2iπ = 1. However, it is well known in conventional mathematical terms that when we raise any number to 0 that the resulting answer = 1. Therefore e 0 = 1. This then seemed to suggest that as  e 2iπ = e 0   that therefore 2iπ = 0, which would seem absurd! Indeed again in conventional terms,  e – 2iπ = 1/e 2iπ = 1/1 = 1. So this would now seem to suggest that 2iπ  = – 2iπ, which again would be absurd from this conventional perspective. In fact, e 2kiπ = 1, where k = 1, 2, 3, .......n Therefore, seemingly,  e 2iπ  =  e 4iπ  =  e 6iπ  = .......= 1, implying that 2iπ = 4iπ = 6iπ etc. It was only that I began to clearly realise something with truly enormous mathematical implications in that all numbers have two aspects, which depending on context, keep switching in the dynamics of experience. Now I refer to these two aspects as analytic (quantitativ

As we have seen, the Euler Identity can be used in a fascinating manner to provide a holistic mathematical interpretation of the nature of contemplative spiritual union. The deeper implication of this is that - properly understood - Mathematics entails a dynamic interaction between analytic and holistic aspects that lie at opposite extremes of each other. Thus from the conventional analytic (Type 1) extreme, mathematical symbols are understood in terms of absolutely rigid symbols of form, interpreted in a linear (1-dimensional) manner. However at the opposite holistic (Type 2) extreme, these "same" symbols are now understood in a completely fluid interactive sense, ultimately representing pure energy states in a circular (0-dimensional) fashion. Thus the paradox inherent in the analytic, can then only be resolved through the corresponding holistic interpretation. Therefore in order to achieve this, one must move from initial linear rational understanding (of the mat