Once again when I maintain that every mathematical notion has a qualitative as well as (recognised) quantitative interpretation this relates to a dynamic holistic form of understanding that is directly based on intuitive type recognition. However the mathematical basis of such understanding comes from the indirect rational means through which such understanding is then appropriately conveyed. And as we have seen such understanding is of a circular logical nature.

Now of course all this might appear very strange to anyone accustomed to look at Mathematics from a merely Type 1 perspective. And it is has to be said that specialisation in this respect has been so strong that it has all but blotted out recognition of the alternative Type 2 perspective.

However the simple fact remains that it is not possible - for example - to properly understand the nature of prime numbers in the absence of Type 2 understanding.

Clearly imaginary numbers are of great importance in conventional Type 1 terms.

However once again - because of the lack of any adequate Type 2 understanding - a merely reduced form of interpretation operates (leaving us completely blind to their true philosophical significance).

In fact imaginary numbers entail the incorporation of the alternative circular logical approach that is expressed in a reduced linear rational manner. In this way imaginary numbers seemingly can be successfully incorporated within the standard Type 1 approach (that is characterised by the linear logical approach).

In conventional (Type 1) terms, the imaginary number i is defined as the square root of - 1. Thus when we square i (to get a 2-dimensional expression of 1^2 which is - 1)) me move back into the world of real numbers.

So what is imaginary here i.e. i actually results from attempting to express - 1 (which is of a real nature and pertaining to a 2-dimensional interpretation) in a reduced 1-dimensional manner.

Thus in this way we can see directly how the imaginary amounts to a linear (1-dimensional) way of expressing a circular (2-dimensional) notion.

In this way the 2nd dimension - and by extension all higher dimensional expressions - can be successfully incorporated through the use of complex numbers in the standard linear Type 1 framework that characterises the conventional approach to Mathematics.

However the imaginary notion - as with all mathematical concepts - can equally be given a true holistic interpretation in a qualitative (Type 2) manner.

As we seen we have already defined the 2nd dimension in qualitative holistic terms as - 1 (literally relating in this context to the dynamic negation of the unitary notion of form). In other words the very means through which we move from (linear) conscious understanding (that is characterised by the separation of polar opposites in experience) to (circular) unconscious appreciation (where opposite polarities are understood as complementary) is through the negation of the positive pole (identified as conscious).

So properly understood in a true qualitative sense the imaginary concept in Mathematics points directly to the vital role of unconscious (as well as conscious) appreciation in Mathematics.

Informally this is to a degree recognised. For example one could attempt for example to rationally explain the Pythagorean theorem to a student. However without supporting intuition, that student would lack - literally - to see what is implied by the rational connections.

Likewise in any mathematical work of a creative nature, intuition is a vital ingredient in developing the key insights necessary to sustain the whole endeavour.

However when in comes to formal interpretation in conventional terms, the role of intuition in mathematical understanding is completely ignored. Thus the standard - merely - rational interpretations that are offered, in a very important sense misrepresent the true nature of mathematical experience.

So we can now perhaps appreciate how the imaginary concept - as used in conventional terms - actually represents an indirect way of incorporating holistic mathematical notions withing the standard analytic framework.

However when appreciated from a full qualitative perspective, the imaginary notion points directly to the need for inclusion of an entirely distinctive form of understanding (which I refer to as Type 2).

Let me clarify here this very important point.

Conventional (Type 1) Mathematics does indeed include both real and imaginary notions. However it can only deal with both in a merely reduced quantitative manner.

Thus in qualitative terms, Type 1 Mathematics confines itself to the merely real (conscious rational) aspect of mathematical understanding.

Thus Type 2 Mathematics - as I define it - is directly designed to deal with the imaginary aspect of mathematical understanding in qualitative terms.

So again in this qualitative context we can define Type 1 as related to the real aspect and Type 2 to the imaginary aspect of mathematical understanding respectively.

Clearly in the most comprehensive context both the real and imaginary aspects of mathematical understanding will be combined in both a quantitative and qualitative manner.

And I refer to this most comprehensive form as Type 3 Mathematics!

However before leaving, I wish to point to a key difference as between the quantitative and qualitative notion of the imaginary!

Because in linear terms opposite polarities are clearly separated this implies that positive (+) is of course clearly separated from (-).

However in circular terms (for the 2-dimensional case) opposite polarities exhibit perfect complementarity. So in this context the positive (+) is interdependent and indeed ultimately identical with the negative (-) pole.

Thus whereas the imaginary in quantitative terms is defined as the square root of - 1, in the corresponding qualitative interpretation - 1 has a dynamic meaning that implies negation of what is positive. So the negative in this context necessarily requires the positive pole to dynamically operate!

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