In yesterday's blog entry, I referred to the experiential basis of an inherently dynamic appreciation of the number system, whereby concepts of a (potential) infinite and perceptions of an (actual) finite nature continually interact in a relative manner.
Of course concepts can likewise be given a (reduced) actual and perceptions a (transformed) potential meaning. So the important point to grasp is that relative to each other, number concepts and perceptions are infinite as to finite (and finite as to infinite) in a dynamic complementary manner.
In psycho spiritual terms this dynamic appreciation is then associated with the two-way interaction of both intuition and reason in experience.
Unfortunately however, a much reduced interpretation of such experience is given in Conventional Mathematics, whereby the (potential) infinite is formally reduced in an (actual) finite manner with corresponding (holistic) intuition likewise reduced in (analytic) rational terms.
Thus instead of an inherently dynamic relative appreciation of number, a static absolute interpretation arises, which ultimately distorts the very nature of number (and by extension all mathematical relationships).
Another important aspect of this dynamic appreciation of number is that it is intimately related to our experience of space and time.
Though I had not yet progressed sufficiently to properly articulate the appropriate nature of time (and space) that would correspond to this dynamic appreciation of number, at least I was clearly aware that it would likewise be of a - relative - rather than absolute nature and would in effect require a new understanding of the true holistic nature of such dimensions.
After dropping Mathematics at the end of the 1st year, I continued my degree taking Politics and Statistics in its place. So in this sense I still kept one foot as it were in the mathematical camp.
However in an unexpected manner, the choice of Politics was to ultimately open the door to making the new holistic mathematical connections that I was seeking.
A major part of the course related to the political thinking of the great philosophical figures such as Plato, Aristotle, Hobbes, Locke, Rousseau and Hegel. As I had now become keenly interested in philosophy, I started to explore deeply their overall positions.
It seemed apparent to me that the various schools of philosophy were based too much on the dualistic emphasis (e.g. empiricism and idealism) of just one limited aspect of overall experience.
So I was searching for a much more holistic vision which could naturally synthesise all these various schools as constituent parts.
In this regard I found that the last mentioned Hegel in many ways to my taste with his sweeping evolutionary view of history. This interest owed much in fact to the brilliance of our lecturer - a Jesuit priest - who clearly was a big fan and well versed in his philosophy. In fact he devoted an entire year to the study of Hegel and these lectures quickly became the favourite hours in my week.
Now I may add that I was far from an uncritical admirer. I found him - unfortunately like many academic philosophers - a very poor communicator. Also - possibly in part due to his comfortable state position - he espoused an unacceptable form of nationalism which may well have contributed to the terrible rise of Nazism in the 20th century.
However, as I became attuned to his notion of the dialectic, I began slowly to see connections with my emerging holistic mathematical notions.
In particular I mentioned in the last blog entry how at an earlier stage, I had pondered deeply the two roots of 1, trying to make sense of what seemed to me a paradoxical situation.
Well now I believed I could give a coherent explanation to this problem.
I would put it like this! As the conventional treatment of number is 1-dimensional, this means that opposite poles of experience, which inherently interact in experience, must be reduced in terms of each other in a static absolute manner.
So in this context, our experience of number necessarily entails both external and internal aspects. The external aspect treats the number as an object (out there somewhere in abstract space).
The internal aspect relates to the mental construct i.e. perception of number which necessarily must co-exist with the external objective experience.
So in experiential terms, with respect to number, we have the interaction of both an (external) object and the corresponding (internal) perception.
Now conscious awareness of object and perception implies that they be posited (+) in experience. (So I am using + now in a holistic rather than analytic manner).
However to switch from object to perception (and perception to object) we must dynamically negate (in unconscious manner) what has been formerly posited.
So with respect to these two polarities, dynamic experience entails the continual positing (+ 1) and corresponding negation (– 1) of single individual reference frames.
I realised that this was the very essence of 2-dimensional - as opposed to 1-dimensional - interpretation. In other words, when we attempt to express the dynamic nature of experience, with respect to 2 polar reference frames, in a (1-dimensional) analytic manner, it appears as paradoxical, where what is + 1 (as a posited pole) can equally from the opposite perspective be represented as – 1 (a negated pole).
Indeed this is the very situation that arises at a crossroads. When using a single polar frame of reference (N or S) a turn at a crossroads has an unambiguous meaning. So if heading N up a road, one can clearly designate a left turn and right turn respectively. However when one switches the frame of reference by heading S, again one can designate left and right turns at the crossroads in an unambiguous manner. So both of these cases of unambiguous identification entail 1-dimensional interpretation (where just one pole of reference is used).
However when one now considers N and S simultaneously as interdependent (i.e. 2-dimensional interpretation), deep paradox arises, for what is left from one perspective is right from the other and what is right from one is left from the other.
So expressed in a reduced 1-dimensional manner, 2-dimensional interpretation implies a result that can be + 1 (a left turn) or – 1 (not a left turn i.e. a right turn) depending on the single pole of reference (N or S) used.
Now it must be remembered that our very experience of number, necessarily entails the continual dynamic switching of reference poles (such as external and internal)
So the deeper connection I was now able to make - which is mind-boggling in terms of its consequences - is that the proper interpretation of 2 (as a dimensional power or number) requires a unique form of interactive understanding based on the complementarity of opposite poles.
Furthermore, as Conventional Mathematics by its very nature is 1-dimensional (based on single reference poles) it - literally - cannot even recognise this reality.
Put another way, Conventional Mathematics can only incorporate analytic type interpretation (based on independent poles of reference).
However there is another aspect of mathematical interpretation that is inherently dynamic i.e. Holistic Mathematics, which is based on poles of reference that are understood as interdependent.
The simplest - and indeed most important - example of such interdependent holistic appreciation entails 2 complementary poles i.e. that are + 1 and – 1 with respect to each other.
However in principle we can extend this indefinitely. So for example in the important case where we simultaneously consider the interdependence of 4 poles, 4-dimensional interpretation is used.
And in more general terms where n poles are considered simultaneously, n-dimensional interpretation is used.
I mentioned before that the problem of multiplication relates directly to the fact that it entails interdependent, whereas addition entails independent notions.
So in additive terms,
1 + 1 = 2, i.e. 11 + 11 = 21. (This again represents the Type 1 definition of number based on a default 1-dimensional interpretation implying single independent frames of reference).
However in multiplicative terms,
11 * 11 = 12 (By contrast, this represents the Type 2 definition of number based in this case on the simultaneous interdependence of two reference frames).
The startling implication of all this is that we cannot therefore properly understand the multiplication of 1 by 1 in conventional mathematical terms. Conventional interpretation gives but a reduced perspective that distorts its very nature! In other words, the key notion of interdependence, which it entails, is necessarily reduced in an analytic (i.e. independent) manner.
So I had clearly shown (from my perspective) that a coherent approach to Mathematics necessarily entails both Type 1 and Type 2 aspects of the number system. Associated with Type 1 is the standard quantitative approach to number based on analytic notions of absolute independence. By contrast, associated with the Type 2 is an alternative qualitative approach based on holistic notions of interdependence. Both are clearly necessary in a coherent comprehensive approach.
Finally in this entry, I will answer my earlier childhood query, which implied that from the 2-dimensional perspective a theorem (such as the Pythagorean) could be true and false at the same time.
Now in 1-dimensional terms. external and internal poles are reduced in terms of each other (and treated as one). So therefore when a theorem proven as absolutely true, this thereby rules out the opposite that it is false. However from a 2-dimensional perspective, both external and internal poles are recognised.
So we could now interpret the proof of a theorem in two ways:
1) as externally true (in an objective sense):
2) as internally true (in terms of mental interpretation).
Now in 2-dimensional terms, both of these interactively arise in a dynamic relative manner. So from this perspective, the truth of any proposition is strictly of a relative approximate nature.
Thus if we now attempt to reduce this 2-dimensional relative truth in an absolute 1-dimensional manner, then two positions are possible.
1. We can absolutely affirm the truth of the theorem in an external objective manner. (+ 1). This implies therefore the opposite (i.e.absolute falsehood) with respect to the truth of the theorem as internal interpretation (– 1).
2. We can absolutely affirm the truth of the theorem in an internal subject manner (as mental interpretation). This implies therefore the opposite (i.e. absolute falsehood) with respect to the truth of the theorem in an external objective manner (– 1).
Again, what is truly remarkable is that a parallel form of interpretation applies to the two roots of 1 (in quantitative terms). These two roots represent therefore the reduced linear (1-dimensional) attempt to represent the "higher" 2-dimensional behaviour of number (that entails the dynamic interaction of opposite poles).