Integral Science

Looking again at the Horizon TV programme on Fermat’s Last Theorem proved a very rewarding experience. Unlike the first time I was able to appreciate much more of the fine detail (e.g. with respect to elliptical curves and modular functions). Also it got me thinking again on a number of levels regarding my own mathematical journey. Like Wiles as a child of about 10, I too had heard of Fermat’s Last Theorem. The problem seemed so beguilingly simple that in my naïveté I thought I would be able to solve it. However after many hours of futile endeavour I abandoned this quest in failure. Nevertheless as Mathematics remained my favourite pursuit I hoped to major in the subject at College. However after a troublesome first year when I became greatly disillusioned with the mathematical treatment of the infinite, I dropped out of the class. Many years later I became interested again in Fermat’s Last Theorem from a very different context. I had been paying a great deal of attention to my ne...

It is indeed a pleasure to watch so many beautifully produced programmes highlighting the wonders of the universe. Recently I was viewing "Stephen Hawking's Universe" on Channel 4 and found it fascinating (especially the last episode on the origins of creation). The Big Bang about 13.7 billion years ago has now become so commonly accepted as if it is an established scientific reality not to be questioned. However I always like to take a wider perspective than mere conventional acceptance of present views. Just look at how our worldview has changed so much from even 100 years ago! Is it not reasonable to assume that perhaps even greater changes will take place in the next 100 years making much of what is presently gospel truth seem naive and even foolish! So even if the Big Bang remains the accepted orthodoxy, I am sure that the manner in which it is understood will have changed considerably. Indeed it seems to me somewhat ridiculous to attempt to describe in detail what h...

I have often stated that the current definition of "a string" is somewhat meaningless from any coherent physical perspective. A string is viewed essentially as like a thin elastic band - extremely short in length - that is 1-dimensional (with no other spatial characteristic). However this very definition requires the background existence of space. However it is then also admitted that the dimensions of space and time must in some way arise from strings (as the basic constituent ingredients of everything in the universe). So we clearly have an obvious problem. Indeed there is a strong parallel here with a similar issue in mathematics relating to the prime numbers. The prime numbers are conventionally viewed as the basic (independent) building blocks of the natural number system; however equally the general distribution of the primes intimately depends on the natural numbers. Now I have explained that this issues of the primes in Mathematics ultimately relates to the fact that ...

When doing some research for my articles on the Riemann Hypothesis, I made the interesting discovery that both Darwin's Origin of Species and Riemann's famous article on prime numbers were both published in 1859 (just over 150 years ago). Indeed the historic connection can be shown to be even closer with the publication date of Darwin's book in November of that year while the full text of Riemann's article also appeared in November (in the monthly reports of the Berlin Academy) though Riemann actually had delivered his address on the contents of that article to the Academy in August, 1859. However recently I have come to see an even greater significance to this interesting coincidence of publication dates (of what constituted truly ground breaking initiatives in two different fields). In earlier blogs I addressed the issue that any attempted reconciliation of science and religion would require two key developments. 1) the recognition of an alternative qualitative aspect...

Much is made of the Uncertainty Principle in Quantum Mechanics, whereby it is accepted that both the position and momentum of a sub-atomic particle cannot be precisely determined. So there is a trade-off involved with respect to both aspects with ever greater accuracy with respect to one aspect (e.g. position) inevitably being at the expense of the other (momentum). And this is an inherent problem with respect to the behaviour of such a particle (and not due to practical difficulties with measuring devices). However there is a much wider context to this principle which is not properly recognised (due to the lack of any appropriate qualitative context to Conventional Science). As I have stated before the very basis of Conventional Science is the use of linear rational logic (reflecting in turn the Middle Band of the psychological spectrum). However just as electromagnetic energy has many bands (of varying wavelength and frequency) likewise it is true with the modes of possible rational ...

I have found the odd numbered dimensions more difficult to understand than the even. As we have seen the even are more properly geared for holistic integral interpretation of reality and as my main concern over the years has been to articulate such an approach in appropriate mathematical terms, it is not surprising therefore that the even dimensions held more resonance for me. The clue to understanding however of the higher odd dimension (focusing initially on the positive) is the realisation that human development necessarily entails both differentiation and integration which need to be maintained in healthy balance. So even for one committed to the process of growing in pure contemplative awareness, a certain level of activity at each stage must be maintained. Thus for example when one returns to active involvement (following the development of the more passive even numbered dimension), the next odd numbered dimension will then unfold. Thus arriving at the first of ...

We have looked at the positive even numbered dimensions. These are the most suited for pure integral interpretation of reality and are always based on the matching complementarity of opposites. Within the even numbered dimensions 2, 4 and 8 would command a special importance (as suited for integral interpretation of reality). However so far we have only considered the positive numbered even dimensions! So our next task is to give a meaning to the corresponding negative numbers. I have mentioned the philosopher Hegel as proving initially inspirational in terms of formulating the holistic mathematical meaning of 2-dimensional understanding. However I gradually became disenchanted with Hegel's approach. It seemed to me that that he was led into a fundamental error in elevating the mere formal rational interpretation of his logic above the very spiritual intuition that it really illustrated. In other words though one can formulate in rational terms the basic principle that all ...

As one may perhaps appreciate, the qualitative use of numbers (as dimensions) plays a key role in holistic mathematical understanding. So I have spent at this stage more than 40 years in the slow - and often painful - task of unravelling the hidden meaning contained in all numbers (as dimensions). In the attempt here to explain my present position, I will relate how this understanding actually unfolded in development. The first key insight was in recognition of the linear nature of the rational paradigm which underpins interpretation of both mathematics and science. So in qualitative terms, such understanding is defined in holistic mathematical terms by the number 1 (i.e. 1-dimensional interpretation). As we have seen this entails in Conventional Mathematics that all number quantities are ultimately defined in terms of a (default) power of 1. For example 2 ^ 2 = 4 (i.e. 4 ^ 1). In physics it implies for example that object phenomena are unambiguously posited in just one (external) dire...

I have already contributed a number of posts to this personal blog on the subject of String Theory. Though most of the unease within the conventional physics community relates to the difficulties in empirically testing String Theory (for some considerable time to come) I have been concerned with a deeper problem! For in terms of offering a coherent philosophical view of the nature of physical reality the present position is utterly impoverished. Now this failure to properly recognise a key weakness, relates to the traditional bias of science i.e. that is geared merely to quantitative analysis - rather than qualitative synthesis - of reality. Though it is certainly possible that ingenious ways of an indirect nature may be found to remedy the empirical testing deficit, little hope however exists of offering any coherent explanation as to what the theory is supposed to represent. Now scientists may immediately retort that they are not in the business of qualitative (i.e. philosophical) sp...

In the last post I addressed the fact that corresponding to each number in Holistic Mathematics is a unique scientific interpretation of reality. So the range of possible interpretations is infinite with Conventional Science based on just one of these numbers (i.e. 1). This I believe is a truly remarkable finding - which if even remotely grasped - should end all notions of science having reached its zenith! Indeed we can make further remarkable statements based on holistic mathematical interpretation. For not alone does every number possess a unique qualitative dimensional significance (as indicated) but equally every symbol and relationship - with an already established significance in Standard Mathematics - can likewise be given a unique holistic mathematical explanation with intimate relevance for psychological interpretation of reality. And those theories and hypotheses that already have been shown to have a special importance in standard quantitative terms would possess an...

Just as the standard analytic approach to science is ably served by its corresponding mathematical tool i.e. Conventional Mathematics, likewise the (qualitative) integral approach to science is likewise potentially served by its own respective mathematical tool i.e. Holistic Mathematics. A great barrier however that I have continually faced is the recognition that Holistic Mathematics - though using the same symbols - is radically different from what most people understand as Mathematics. So what certainly is not intended here is the standard use of Mathematics to deal with holistic type problems (which represents the old reductionist approach)! Rather it requires a new interpretation of mathematical symbols that inherently depends on corresponding appropriate level of intuitive insight for correct usage. To put this in context, one must appreciate that just as there are many bands on the electromagnetic spectrum (with natural light comprising just one), likewise it is true with respec...

I have been reading again "The End of Science" by John Horgan which I find interesting on several levels. Firstly, whether one agrees or not with its conclusion it puts forward a most provocative hypothesis i.e. that the great era of theoretical scientific discovery is at an end with diminishing returns with respect to further development now to be expected. Secondly it attempts to cover a wide range of different scientific fields conveying in the process some flavour of the rich developments that have already taken place. Finally - and most notably - Horgan managed to do an impressive amount of research in interviewing a significant number of the biggest names associated with these fields (at least at the time of writing in the mid 90's). What I like most about his approach is a certain innate scepticism which prevents him from ever appearing unduly awestruck with their strongly held beliefs. My own approach would be somewhat different to Horgan's and more ...