Wednesday, October 12, 2011

Three Levels of Appreciation of Transcendental Numbers

I have mentioned on many occasions that three type of mathematics exist i.e Type 1, Type 2 and Type 3 respectively.

I will now interpret the meaning of a transcendental number such as e according to the three types.

Now in conventional Type 1 appreciation, e represents e^1. So when one concentrates merely on the quantitative interpretation of a number, it is always defined with respect to the (default) 1st dimension. So here we have - literally - a linear rational interpretation of number.

Properly speaking as we will see, a number such as e transcends mere rational interpretation. In qualitative terms, rational (linear) is suited to the interpretation of rational discrete numbers that are finite in nature.

However the very essence of e - and indeed every transcendental number - is that they combine both finite and infinite aspects in their very nature. Thus though the quantitative value of e can be approximated to any required degree of accuracy, its true value always remains unknown (leading potentially to an unlimited number of terms in its decimal sequence).

Though there are many ingenious ways of representing e, as with all transcendental numbers it cannot be the solution to a polynomial equation.

In Type 2 appreciation, in a somewhat inverse fashion e represents the dimension to which the (default) no. 1 is raised.

So e in Type 2 terms is represented as 1^e.Therefore as a dimension, it now takes on a holistic qualitative significance (with respect to its default base quantity).

This is a crucial point that is not addressed in Type 1 interpretation. From Type 1 perspective, when e is used as a power it still represents a number quantity. However properly speaking the relationship between the base quantity and its dimensional power is always quantitative as to qualitative (and qualitative as to quantitative).

So e in Type 2 terms takes on an appropriate qualitative holistic meaning. Now a transcendental number in qualitative always expresses a relationship between rational (discrete) and intuitive (continuous) notions.
In this context e has a special significance as the number which uniquely reconciles both aspects within its own nature.
So in psychological (and corresponding physical terms) with e both the processes of (discrete) differentiation and (continuous) integration are reconciled.
It serves therefore as an especially advanced qualitative symbol of such differentiation and integration (where both aspects are indistinguishable).

In Type 3 terms, understanding keeps switching as between both quantitative and qualitative interpretation with respect to e as both base and dimensional number respectively.

So we understand e as a (base) quantity. Then attention switches to understanding e in holistic terms as a dimensional quality; then attention switches again to now understanding e in holistic terms also as a base quality; and finally in this cycle attention switches to understanding e also as a dimensional quality.

So in this dynamic interactive manner, e whether representing a base or dimensional number, possesses both quantitative and qualitative aspects which keep alternating in experience.

This means that in Type 3 terms a proof always entails both quantitative and qualitative aspects and is subject to the Uncertainty Principle.

For example the Type 1 proof that any algebraic number raised to an algebraic power can in truth be given both a quantitative (Type 1) and qualitative (Type 2) interpretation. (In qualitative terms this amounts to a holistic mathematical interpretation of how development proceeds from the psychic/subtle to the causal level).

Then in Type 3 terms the Uncertainty Principle necessarily applies to the dynamic interpretation that combines both. In this manner, in comprehensive Type 3 terms, all mathematical proof is subject to the Uncertainty Principle.

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