Thursday, February 16, 2012

Reason and Intuition (1-Dimensional Interpretation)

I have repeatedly stated now that when appropriately understood, Mathematics represents a dynamic interactive activity entailing both quantitative and qualitative aspects of interpretation.

The quantitative aspect is directly related to "real" reason whereas the qualitative aspect is directly related to holistic intuition (which indirectly can be interpreted in an "imaginary" rational manner).

So we are already demonstrating here the nature of the Type 2 approach in giving the notions of "real" and "imaginary" a holistic qualitative rather than strict quantitative interpretation!

What is truly remarkable is the manner in which all possible configurations with respect to the dynamic interaction of both reason and intuition (i.e. real and imaginary aspects of understanding) can be given a precise holistic mathematical interpretation (in Type 2 terms)!

So in this approach the qualitative notion of a number as representing a dimension is key (which in relation to a base default number quantity of 1, is - relatively - of a qualitative nature).

Thus in the simplest possible case where we have 1^1, the base number is quantitative in nature with the dimensional power or exponent - relatively - of a qualitative nature.

Now we have already seen that there is a direct link as between the corresponding root of the number (in quantitative terms) and its qualitative structure.

So in the case of 1 (when used as a qualitative dimension) there is a close link with the first root of unity (in quantitative terms).

And of course as the number 1 remains unchanged by this process, we can conclude that there is in fact no distinction in 1-dimensional terms as between both qualitative and quantitative interpretation.

Thus, it is the very nature of Conventional (Type 1) Mathematics - which is defined in qualitative terms by its 1-dimensional format - to reduce in any context qualitative to quantitative type meaning.

In other words there is no role at a formal level for holistic qualitative type appreciation in Type 1 terms (though informally it must necessarily be recognised).

Now the very means by which one switches to appropriate intuitive appreciation is through negation of the corresponding positive dimensional number.

So this implies that the appropriate type of intuition in Type 1 Mathematics (where intuition is used to informally confirm the rational connections that are solely recognised in formal terms), corresponds to - 1 as dimension.

This is a very important point which perhaps warrants further elaboration.

For example if I outline the "rational" proof of a proposition such as the Pythagorean Theorem to a student, strictly this has no meaning in the absence of the holistic intuitive insight - whereby he/she is enabled to - literally - see what is implied by the sequential rational connections made. Without this confirming intuition, the penny - as it were - will never drop, with the face of the student, despite my best efforts to rationally explain, maintaining a blank expression.

Indeed a very common problem in attempting to convey Mathematics to a non-specialist audience is that intuitive insight may be lacking to the extent that great difficulties may be experienced in "seeing" the implications of even the most moderate use of abstract reasoning!

We can actually use the negative form of 1 (as dimension) in a very instructive manner to show precisely how intuition operates at the (linear) rational level of understanding.

If in conventional (Type 1) quantitative terms we take the number 2 for example and raise to the power of - 1 i.e. 2^(- 1) we invert the expression to obtain 1/(2^1) i.e 1/2.

We can likewise interpret this same expression from a qualitative (Type 2) perspective. The significance of - 1 (as dimension in this context) is that it - literally - represents the negation of rational linear understanding, thereby enabling holistic intuitive type recognition to take place.

This implies that very means by which we are able to switch from appreciation of the number "2" as a whole integer to the corresponding appreciation of 1/2 as a part fraction, an intuitive holistic component is necessary in understanding to enable the switch to take place.

Now of course once the switch has been made, the corresponding interpretation of the part fraction quickly is reduced in a quantitative rational manner.

It likewise works in reverse. So (1/2)^ (- 1) in Type 1 terms = 2^1 i.e. 2.

So once more from a qualitative (Type 2) perspective, in the reverse switch back from knowledge of the fractional part (1/2) to the corresponding knowledge of the integer whole (2), a holistic intuitive component is required, with the result then interpreted in a (reduced) quantitative manner.

If we generalise this finding, we may quickly appreciate that relating parts to wholes (and whole to parts) in any relevant context, cannot take place without enabling intuition of a holistic kind.

So, like the oil in a car engine, intuition is necessary to lubricate the most fundamental of psychological processes. The relationship between whole and part (and part and whole) is ubiquitous, for by definition we cannot have a specific (part) perception in the absence of its general (whole) concept and vice versa. Thus the dynamic interaction as between perceptions and (related) concepts and concepts and (related) perceptions intimately depends on the holistic intuitive aspect of understanding (pertaining directly to the unconscious).

However in formal terms, the intuitive component is screened out entirely from Type 1 mathematical interpretation.

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