Wednesday, January 13, 2010

More on Dimensions in String Theory

As stated so frequently in these blogs, the standard linear (1-dimensional) approach employed in Conventional Science is unable to maintain the key qualitative distinction (in any appropriate context) as between whole and part. It can only operate therefore by essentially reducing whole to part notions (in real conscious terms) or alternatively part to whole notions (in a likewise real conscious manner).

Fundamental therefore to maintaining the proper qualitative distinction as between both aspects is the recognition that unconscious (circular) and conscious (linear) notions must be formally incorporated with each other in interpretation.

As we have seen, whereas in a direct sense the conscious aspect of understanding relates to (real) rational interpretation (corresponding to either/or logic)), the unconscious relates directly to corresponding intuitive recognition (relating to complementary both/and logic).

However this latter intuitive aspect can be given an indirect rational interpretation in - what in precise holistic mathematical fashion is - an imaginary fashion.

So Conventional Science deals essentially with "reality" that is amenable to real rational interpretation. Though it must be readily admitted that complex mathematical notions are also widely used, these are restricted to merely quantitative type analysis. Thus the qualitative method of interpretation remains decidedly real.

However to deal adequately with "reality" requires that we incorporate likewise in qualitative terms an approach that is both real and imaginary.
Here essentially in terms of rational understanding we give expression to two distinct types of logical understanding. The real aspect requires standard either/or logic (based on the clear separation of polar opposites in experience); the imaginary aspect uses an alternative both/and logic (based on the dynamic complementarity of these same polar opposites in experience).

Thus, whereas the analytic aspect of scientific understanding (suited to differentiated interpretation) is directly associated with the real aspect, the corresponding holistic aspect (suited to integral interpretaton) is associated with the imaginary aspect.

So in qualitative terms, science needs to incorporate a complex rational approach (combining both real and imaginary aspects of understanding).

Now this is all deeply relevant to the very understanding of dimension in science.
Current accepted understanding - which is heavily based on solely real interpretation - is but a reduced notion that does not represent its true meaning.

Quite remarkably the true notion of dimension (with respect to both physical and psychological reality) corresponds exactly with the holistic mathematical notion (when interpreted in its qualitative fashion).

The very relationship as between object phenomena and dimensions relates in turn to the fundamental relationship of whole and part.

Now to properly preserve the true qualitative distinction as between whole and part (thereby avoiding reductionism) we must understand both aspects as real and imaginary with respect to each other.

In the very dynamics of experience, our actual appreciation of object phenomena and related dimensions (of space and time) requires the interaction of conscious and unconscious. Whereas the appreciation of objects relates directly to conscious recognition, dimensional appreciation (in providing a whole background context) is by contrast provided directly by the unconscious. Unfortunately, subsequently in rational interpretation, dimensions are simply reduced in terms of the parts (so that the whole - in any context - tends to be viewed as the sum of its quantitative parts).

To appreciate the true relationship of objects and dimensions, we need to consider the simple mathematical relationship whereby 1 is raised to a rational fractional power (say 1/4).

So if start with the base (object) number quantity as 1 and the dimension (i.e. power or exponent) as 1/4.

Now both of these numbers are linear (i.e. lie on the straight line).

However when we allow for interaction as between object quantity and dimensional quantity (through raising the former to the latter) a remarkable transformation takes place.

So when we raise 1 to 1/4 we obtain the result i (the square root of -1).

Now this new number is not linear (i.e. does not lie on the straight line). However it is circular (i.e. lies on the circle of unit radius drawn in the complex plane).

Thus whenever we raise a rational number to a rational fractional dimension (provided that the expression cannot be reduced further) a transformation takes place from linear to circular (in quantitative terms).

(In reverse manner when we now raise 1 to the circular dimension i, once again a transformation takes place this time from circular to linear.
So the result .207879... is linear lying on the straight line number scale).

Now what is completely missed in conventional mathematical understanding is that when we now raise the same number to a whole number dimension that a transformation likewise takes place from linear to circular (but now in qualitative terms).

So in holistic mathematical terms when we raise 1 to 4 i.e. express the number 1 with respect to 4-dimensional interpretation, the result is imaginary (corresponding to circular understanding).

Likewise when we interpret any relationship in an integer dimension (other than 1) a transformation takes place in the nature of understanding from linear to circular.

Now the very reason why this important finding is missed in conventional interpretation it solely operates in a linear manner (and thereby unable to preserve the uniqueness of qualitative distinctions).

So when we raise 1 to 4, though the qualitative nature of the relationship has now changed, in quantitative terms the result remains the same. In other words 1 (raised to the power of 4) is indistinguishable from 1 (raised to the power of 1) in standard linear interpretation. This is why - in a very precise manner - standard mathematical - and by extension standard scientific - interpretation is literally 1-dimensional!

So the true nature of object phenomena to dimensions (both with respect to physical reality and corresponding psychological interpretation) is as linear to circular which in rational translation is as real to imaginary.

Thus when "reality" is properly understood in dynamic interactive terms it is necessarily of a complex nature (with interacting real and imaginary components).

Now you may wonder where this all gets us with respect to string theory!

In the last blog, I made the important observation - that properly speaking - the very notion of a string can be given two distinct interpretations (which ultimately are complementary).

Once again in standard physical terms the 1-dimensional string is looked on as the fundamental building block of matter. The dynamic vibrations of the string then lead to the physical particles that comprise reality.

However there is equally another (unrecognised) qualitative psychological notion of a string. Here the 1-dimensional string (i.e. linear type understanding) is looked on as the basic building block for understanding of reality. However when such understanding dynamically vibrates it can then generate all the "higher" dimensions of understanding that comprise our potential interpretation of reality.

In this context in terms of our second notion of a string, conventional scientific appreciation (which is decidedly 1-dimensional in nature) simply represents the lowest possible energy state of the string (where very little dynamic vibration or interaction) is allowed.

One of the big "eureka" moments for me with respect to my complementary approach to stings came when reading Brian Greene's excellent book "The Elegant Universe".

On P. 144, in Figure 6.2 he gives some examples of vibrating strings, where just as the different vibrational patterns of a violin string give rise to different musical notes, the different patterns of a fundamental string give rise to different mass and force charges.

Now the three diagrams presented in this figure resemble pretty exactly the geometrical interpretation of the 2, 4 and 8 roots of unity respectively (which in holistic mathematical terms corresponds with the 2, 4 and 8 dimensions of scientific interpretation). And these are the very dimensions of interpretation that I have long advocated as essential in the integral scientific approach!

In "The Fabric of the Cosmos" Greene then repeats this diagram on p.357, now including an extra diagram (geometrically resembing the 16 roots of unity). And recently I have started to qualitatively incorporate 16-dimensional understanding in my interpretative model!

So the point once again is that in a more comprehensive understanding, the notion of a string needs to be defined in two complementary ways (corresponding to two distinct logical systems).

Once again the existing definition corresponds to attempted analytic understanding of physical reality (i.e. the real aspect of interpretation).

However the alternative definition, that I am providing, corresponds - by contrast - to attempted holistic understanding of both physical and psychological reality as complementary (i.e. the imaginary aspect of interpretation).

Adn then this is the crucial point!

Just as the relationship as between object phenomena to dimensions is correctly complex (as real to imaginary) then this clearly implies that we cannot hope to properly incorporate dimensions in string theory without likewise incorporating this alternative holistic understanding.

So to sum up, in qualitative holistic mathematical terms:

Conventional (Analytic) Science comprises the real part of (scientific) interpretation;

Integral (Holistic) Science comprises the imaginary part;

Radial (Comprehensive) Science combines both parts in a complex rational approach.

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