Integral Science

We looked briefly at the qualitative nature of a transcendental number yesterday. Once again it requires the explicit recognition of both linear (discrete) and circular (continuous) notions, with the transcendental aspect relating directly to the necessary (irreducible) relationship as between both. Therefore to stress an important point, if we wish to avoid gross reductionism, we cannot deal with the nature of a transcendental number such as π or e in a merely rational manner! And of course Conventional Mathematics is defined by such reductionism! Thus the value of π properly relates therefore to a mysterious conjunction as between (finite) discrete and (infinite) continuous notions which - literally - transcends the linear interpretation of reason. So the transcendental notion of time (and space) arises from this explicit recognition of the dynamic relationship as between analytic (rational) and holistic (intuitive) type aspects. In the most accurate sense, it reflects the...

Yesterday we looked briefly at the qualitative nature of time (and space) from an (algebraic) irrational perspective. Now an (algebraic) irrational number arises as the solution to a polynomial equation with rational coefficients. The famed square root of 2 - which is the best known example of an irrational number - arises from the simple polynomial expression x^2 = 2! What this implies with respect to the nature of time (and space) is that a hybrid dynamic mix of the two logical systems (linear and circular) is involved, whereby relative notions are continually reduced in somewhat absolute terms and - in reverse - absolute notions quickly transformed in a relative manner. Once again, we see this clearly in nature at the sub-atomic level where energy is continually reduced in terms of mass and mass once more transformed into energy. So in holistic mathematical terms, such interactions properly take place in an environment characterised by irrational notions of time (and space...

As stated so often when properly understood as the very nature of experience, Mathematics has both quantitative and qualitative aspects in dynamic interaction with each other. So from this perspective one does not understand symbols in static terms as absolute forms, but rather in dynamic interactive terms as symbols of transformation! I will now attempt to illustrate one extremely important example of this new understanding (with intimate parallels to the nature of psychological development). As befits the dynamic approach, in a number expression such as a^b, if we designate the base number a in quantitative terms - the dimensional number b is - relatively of a qualitative nature. And it is this interaction as between quantitative and qualitative aspects that can then be used to explain how the nature of number itself evolves to "higher" forms. So for example if we start with the simplest of prime numbers 2 and then raise this to 2 (i.e. 2^2), the result is a natu...

I have commented before on - what I refer to as - the Pythagorean Dilemma. In other words the significance of the discovery that the square root of 2 is an (algebraic) irrational number, was as much of a qualitative as a quantitative nature. As I have stated, the Pythagoreans recognised an important qualitative significance to number. Prior to their discovery of the irrational nature of 2, they had assumed that all number quantities were of a rational nature. Happily this complemented well the scientific paradigm they used to interpret this reality which qualitatively was also of a rational nature. So the true significance of the irrational nature of 2, is that the Pythagoreans lacked the qualitative holistic means to explain how it could arise, thus shattering the harmonious balance they sougth to preserve with respect to mathematical activity. The rational paradigm which still dominates present scientific and mathematical thinking is basically suited to interpretation of me...

To follow the next section requires even subtler understanding of psychological and complementary physical dynamics. My basic starting point with respect to the dynamic understanding of number, is that in any context the base quantity and dimensional number are quantitative as to qualitative (and qualitative as to quantitative) with respect to each other. Thus in the simple expression 1^2, the base number here (1) is understood in quantitative, whereas the corresponding dimensional number (2) is understood - relatively - in a qualitative manner. As we have seen Conventional Mathematics is interpreted in terms of the (default) dimensional number of 1 (as qualitative) whereby qualitative is necessarily reduced to quantitative meaning. Therefore if we take the expression 2^3 to illustrate, the result will be expressed, from this perspective, in reduced quantitative terms as 8 (i.e. 8^1). Now to explore the qualitative nature of mathematical symbols in isolation, we then revers...

As we know from a quantitative perspective rational numbers exist that are not integers i.e. fractions. This applies therefore that from a qualitative perspective, we equally can give meaning to rational numbers as fractions. And as the very nature of time (and space) when appropriately understood is intimately related to the qualitative dimensional notion of number, this likewise applies that we can give meaning to the fractional nature of time (and space) from both complementary physical and psychological perspectives. It perhaps will be easiest in this respect to start with the number 2 (as ordinal dimension). As we have seen this ordinal dimension (from a qualitative perspective) is intimately connected with its corresponding root (in quantitative terms). Thus the 2nd root of 1 can be written as 1^(1/2) = - 1 and in quantitative terms this result matches the corresponding 2nd dimension i.e. 1^2 = - 1 (which here relates to a qualitative interpretation). Thus as we have se...

We will now consider directly the nature of time (and space) associated with the negative integers (as qualitative dimensions). The even integer dimensions (- 2, - 4, - 6, - 8,...) are easier to explain, for here in all cases - from the psychological perspective - time has no phenomenal meaning with experience relating directly a present moment continually renewed. This in turn would be consistent with a pure contemplative state. Of course, because in actual experience, all the varying dimensions co-exist (at least with the potential to exist) we cannot completely isolate the experience of any one dimension. However having said this, at any moment one or more can be especially prominent. So therefore for example if the experience of the negative 2nd dimension is predominant then indeed one will have little consciousness of time (or space) but rather the spiritual awareness of the (absolute) present moment. Once again such experience is of a purely intuitive nature resulting from...

We made the distinction yesterday as between implicit qualitative recognition of the 1st dimension as negative (where it remains completely ignored in formal mathematical interpretation), and full explicit recognition which inevitably leads to a redefinition of the nature of Mathematics (whereby both quantitative and qualitative aspects are recognised). So once again, a mathematician may well recognise the important role of intuition with respect to important new discoveries. And this inherently requires to a degree - sometimes marked - the temporary negation of customary rational understanding. This then allows deeper holistic insight to incubate in the unconscious which is essential in enabling an important new breakthrough. But unfortunately such a mathematician will then formally interpret this new finding in a merely reduced rational manner (with the 1st dimension as positive solely recognised). As I live in Dublin I can identify with the inscription on Brougham Bridge in ho...

As we have seen, Conventional Science is based on rational understanding of a linear logical kind which directly conforms with the qualitative interpretation of 1 (as a number dimension). And of course it is the very nature of such interpretation that qualitative notions are thereby reduced (for any relevant context) in a merely quantitative manner! Also, as we have seen, directly associated with this approach is the interpretation of time also as 1-dimensional (where it moves in a single positive direction). However, once we recognise that associated with all numbers is a corresponding qualitative - as well as recognised - quantitative interpretation, then potentially we can have an unlimited number of mathematical interpretations (all of which assume a certain limited validity within their appropriate relative context). This likewise entails that time (and space) itself - when appropriately understood - possesses a potentially unlimited number of possible directions (associated...

So far we have investigated how a unique experience of time (and space) is associated with all the positive integers (representing dimensional numbers). And as such an experience directly complements the nature of physical reality, this implies likewise that a corresponding structure of time (and space) likewise exists in the physical world with respect to all positive integers. And once again the very reason why this is not readily apparent is that the conventional scientific paradigm is based solely on the linear use of reason (with 1 as the default dimensional number). This then explains the customary interpretation of time as 1-dimensional (where it is believed to move in a single forward direction). Likewise we have seen that a fundamental distinction divides, as it were, interpretation of time (and space) with respect to the even and corresponding interpretation with respect to the odd dimensional numbers. Basically - when understood appropriately in a dynamic relative ma...

In dealing with the nature of 2, 4 and 8 dimensions of time (and space) respectively, we saw how that in each case they are characterised by the complementarity of opposite poles. So 2-dimensional reality is characterised by the complementarity of the "real" poles (i.e. external and internal). 4-dimensional is then characterised by the additional complementarity of the "imaginary" poles (i.e. whole and part). 8-dimensional is finally chracterised by the additional complementarity of special complex poles where both real and imaginary parts are of equal magnitude. This can then be understood as relating to the ultimate interaction as between form and emptiness (i.e. where the dynamic interaction of phnomena are so refined that they do not even appear to arise). So in this sense the dynamic nature of form becomes inseparable from emptiness. Now it must be understood that further distinct structures with respect to the nature of time (and space) are associated...

We have already looked at the 4-dimensional nature of time, relating ultimately to the dynamic complementary interaction with respect to (i) internal and external and (ii) whole and part aspects. This notion of time strictly applies to all phenomenal processes (both physical and psychological). In my own writings I have concentrated on the 1, 2 and 4-dimensional interpretation of relationships culminating with the even more refined 8-dimensional structure. Now before going on to look at the crucial distinction as between the even and odd integer dimensions with respect to the interpretation of time, we will now look briefly at the nature of this 8-dimensional interpretation. Once again the 8 dimensions of time (and space) as qualitatively interpreted are inversely related to the 8 roots of 1 (in quantitative terms). We have already encountered the first 4 of these roots 1, - 1, i and - i when looking at 4-dimensional interpretation. However four additional roots arise i.e. 1/k(i +...

Perhaps before moving on further it might be instructive to elaborate a little more on what the imaginary nature of time entails. As we have seen in psychological terms the notion of imaginary time arises due to the unconscious aspect of human experience. This then accords in complementary fashion with the holistic aspect of physical phenomena. Now this holistic aspect remains very difficult for conventional scientists to accept as it implies that - what can only be understood as - holistic intelligence strictly applies with respect to all physical processes. Last night while watching a TV programme on slime mold, a superb example was given of this holistic intelligence. So even though the slime mold represents a simple organic system comprising a single cell it can create complicated networks that are as efficient (if not more efficient) than those designed by professional engineers. So I would describe this as an innate holistic capacity that has emerged through evolution. How...

We have already looked at both the linear (1-dimensional) and circular (2-dimensional) perspectives on time. Once again from the 1-dimensional perspective, time is conceived in somewhat absolute fashion as having one positive direction. Then from the corresponding 2-dimensional perspective, time is conceived in relative terms as having two (complementary) directions that are positive and negative with respect to each other. So the the nature of time conforms (in physical and psychological terms) directly with the holistic mathematical notion of dimensions, whereby each integer (representing the number of qualitative dimensions) bears an inverse complementary relationship with its corresponding number of roots with respect to unity, in quantitative terms. Thus the 2-dimensional qualitative structure (of both time and space) is inversely related to the 2 roots of 1 i.e. + 1 and - 1 (in quantitative terms). However this mathematical notion of dimensions can then be generalised for a...