Integral Science

This latest contribution in many ways entails a summary of the points that I have made in recent blog entries and in discussion with Anthony Judge in relation to “Transforming the Art of Conversation - conversing as the transformative science of development” at his hugely impressive “Laetus in Praesens” site. I have been long fascinated by the fact that the two binary digits (1 and 0) when used in a quantitative manner can potentially encode all information processes. I am therefore of the opinion that the same two digits when used in an appropriate qualitative manner can likewise potentially encode all transformation processes. So transformation itself (in all its manifestations) is basically encoded in number when appreciated in a qualitative manner. Now as geometrical symbols, 1 can be identified with the straight line and 0 with a circular circumference. So the relationship of 1 and 0 in qualitative terms implies the relationship between (rational) linear and (intuitive) ...

I have always felt a special affinity to Taoism where the basic nature of reality is explained in a manner that readily lends itself to holistic mathematical understanding. So the Tao represents the ineffable undivided unity (which equally is a nothingness in phenomenal terms). Then phenomenal reality arises from the splitting of this unity into polar opposites that are understood as separate from each other. However a deeper understanding of the nature of these opposites leads to the realisation that they are complementary (and ultimately identical in nondual terms) as yin and yang. So it is this latter realisation that enables the process of harmonising phenomenal reality with the original absolute nature of Tao. Last night I was briefly reading the section on Taoism in that wonderful little book on "Mysticism" by F.C. Happold. There, I saw the seeds of an even closer relationship in its thought to my recent notions expressed in these blogs on the all important role o...

We have to be careful here. It is very hard in practice to distinguish numbers from the symbols used for their representation. And the very nature of such representation is that we thereby give a distinct phenomenal identity to number (as represented by its symbol). So when I use the symbol "1" to represent the notion of one, it thereby assumes this phenomenal identity. Furthermore, because understanding of number in our culture is dominated by its quantitative aspect, numbers thereby become misleadingly identified as abstract phenomenal objects (with an absolute identity). However in truth the meaning of number is much more elusive. As I have been at pains to illustrate, every number has both a qualitative as well as recognised quantitative aspect. Basically, the quantitative aspect relates to the notion of number as independent (i.e. where phenomenal poles such as external and internal are separated). The corresponding qualitative aspect relates to the correspond...

We are already well aware of the great significance of number from the conventional quantitative perspective. However what we have not yet recognised yet is the equal significance of number from the greatly neglected qualitative dimensional perspective. And when we combine these two aspects of number in interactive terms, then it is but a short leap to the recognition that - at its most fundamental - phenomenal reality is but the dynamic representation of number configurations (in both quantitative and qualitative terms). Now a widespread view in contemporary physics is that reality is fundamentally composed of tiny 1-dimensional strings, the unique vibrations of which give rise to all the particles from which more conventional material forms are composed. However the notion of "physical strings" in any meaningful philosophical sense is but a fiction arising from the reductionist quantitative viewpoint that matter must ultimately be composed of smaller constituent par...

We now come back to highlighting the significance of the prime numbers. Just as the prime numbers are recognised in quantitative terms as the building blocks of the natural number system, likewise the prime numbers - though conventionally unrecognised - are equally the building blocks in qualitative terms of the natural number system. What this again implies is that all numbers (as dimensions) are built up from prime number constituents. Then, as the number dimensions directly relate to the dimensions of space and time (physically and psychologically) these likewise are built from prime numbers (in qualitative terms). Furthermore, as the qualitative characteristics that are inherent in natural phenomena are but manifestations of such space and time configurations, the prime numbers can then be clearly seen - in literal terms - as the fundamental basis of all qualitative characteristics in nature. Thus, looked at from these two distinct perspectives (in isolation) the prime ...

We now come back to highlighting the significance of the prime numbers. Just as the prime numbers are recognised in quantitative terms as the building blocks of the natural number system, likewise the prime numbers - though conventionally unrecognised - are equally the building blocks in qualitative terms of the natural number system. What this again implies is that all numbers (as dimensions) are built up from prime number constituents. Then, as the number dimensions directly relate to the dimensions of space and time (physically and psychologically) these likewise are built from prime numbers (in qualitative terms). Furthermore, as the qualitative characteristics that are inherent in natural phenomena are but manifestations of such space and time configurations, the prime numbers can then be clearly seen - in literal terms - as the fundamental basis of all qualitative characteristics in nature. Thus, looked at from these two distinct perspectives (in isolation) the prime ...

Yesterday, I briefly attempted to explain the qualitative significance of the Euler Identity which essentially represents a holistic mathematical description of the precise nature of spiritual transformation where emptiness and form (and form and emptiness) are united. Here the contemplative journey - literally - comes full circle with the transcendent goal (beyond all phenomenal form) finally revealed as identical with its immanent source (as already inherent within such form). Last week I was looking at a fascinating programme on the mapping of the Universe. We are of course accustomed to the mapping of planet Earth and to a lesser extent our planetary system. But ambitious attempts have already been made to provide a map of the entire Milky Way galaxy. And even beyond that considerable progress has been made with respect to the mapping of the visible universe (made up of countless billions of galaxies). Even some are already attempting to know what lies beyond the visible univers...

Yesterday we considered how number (as qualitative dimension) can be given an imaginary (as well as real) meaning and that this thereby also applies to time (and space) in both physical and psychological terms. Basically what this entails is that development can take two complementary directions that are transcendent and immanent with respect to each other. Therefore if we associate real numbers (as dimensions) with the transcendent aspect, then the corresponding imaginary numbers are then - relatively - associated with the immanent aspect. Though all these numbers (as dimensions) are implicit in actual human experience, remarkably little progress has yet been made with respect to any coherent explicit appreciation. And as I have stated repeatedly the conventional paradigm of Science and Mathematics as we know is merely of a 1-dimensional nature (in qualitative terms). Now, when appropriately interpreted, the other dimensional numbers do unfold in varying degrees through the pr...

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...