Where Science and Art Coincide (2)

The reduced nature of conventional mathematical interpretation is highlighted by the manner in which ordinal are assumed to be directly implied by their corresponding cardinal notions.

Therefore, 1st and 2nd (in ordinal terms) are assumed to follow directly from the corresponding notions of 1 and 2 (in cardinal terms).

And as the cardinal notion of number is defined in a merely quantitative manner, this reduced interpretation likewise creates the illusion that ordinal notions can likewise be dealt with in the same fashion.

This in fact represents clearly the 1-dimensional nature of Conventional Mathematics where interpretation is based on just one isolated polar reference frame i.e. quantitative.

This means, as I have repeatedly stated, that from this perspective qualitative is always reduced to quantitative meaning.


However, properly understood, cardinal and ordinal relate to two distinct notions of number that are quantitative and qualitative with respect to each other.

The essence of cardinal meaning is that a number is taken as representing a collective whole identity.

So 2 in this cardinal sense represents a collective whole identity (with a quantitative meaning).

Thus the individual units of 2 would be represented  is homogenous terms as 1 + 1. In other words these units, in being exactly similar, lack any qualitative distinction!.

Cardinal numbers are thereby treated in absolute terms as independent (i.e. where their quantitative nature is independent of qualitative meaning).


However the ordinal notion of number is of a uniquely distinct nature.

Once again we have seen that the quantitative notion of number has an independent collective identity (with individual units lacking any unique distinction).

It is quite the reverse with respect to the true qualitative notion of number! Here the number lacks any overall collective identity (as quantitative). However the individual units now are uniquely distinct.

So from this qualitative perspective, the number 2 is uniquely defined by its 1st and 2nd members (in ordinal terms).

The key distinction therefore as between the quantitative and qualitative is that the quantitative is based on the notion of the independence of a collective number group, whereas the qualitative is based on the corresponding notion of the interdependence of the individual members of that group!


Thus again the quantitative notion of 2 is represented through the Type 1 aspect of the number system as 2¹.

This illustrates the pure notion of addition where 1 + 1 = 2 (i.e. 2¹).

Thus the pure notion of addition in this context is directly associated with the quantitative aspect of number.


The corresponding qualitative notion of 2 is represented though the Type 2 aspect of the number system as 1².

This illustrates the pure notion of multiplication where 1 * 1 = 2 (i.e. 1²).

Likewise the pure notion of multiplication is directly associated with the qualitative aspect of number.

Then when both base and dimensional numbers ≠ 1, both quantitative and qualitative transformations of number are involved.

 

Now it is important to understand that once we recognise the two aspects of the number system, that we necessarily move to a dynamic interactive interpretation (where both aspects enjoy a merely relative identity).

For example, if we reflect on it for a moment, the conventional mathematical notion of absolute independent number entities is strictly nonsense. For if numbers were independent in this sense then there would be no possibility of relating then with other numbers!

So we cannot in truth operate satisfactorily with the conventional - merely quantitative - cardinal notion of number.

Likewise we cannot operate with merely the qualitative ordinal notion.

For example, in the pure ordinal interpretation of the number 5, we cannot fix the 1st position with any specific number . Therefore any of the five members can be the first. This likewise means that each member can be 1st, 2nd, 3rd, 4th and 5th respectively. Thus to make unique ordinal distinctions we need to unambiguously fix, in any context, the 1st member (which is indistinguishable from the cardinal notion of 1).

Thus the (quantitative) cardinal notion of number automatically implies the ordinal (in being able to relate numbers to other numbers).

The (qualitative) ordinal notion of number automatically implies the cardinal (in being able to independently fix the 1st number in any relevant context).


Thus in dynamic experiential terms, both cardinal and ordinal notions continually interact and are necessary for each other.

The great significance of this is that our customary mathematical interpretation of the number system is strictly speaking completely untenable (as it is built on absolute cardinal notions).

Once we recognise the dynamic interaction of both quantitative (cardinal) and qualitative (ordinal) notions, the question arises as to the mutual consistency of both types of interpretation!

Alternatively the question arises as to how we can consistently convert qualitative (ordinal) notions in quantitative (cardinal) terms; equally from the other perspective, the questions arises as to how we can consistently convert quantitative (cardinal) notions in qualitative (ordinal) terms.


From another perspective the cardinal notion of number is directly associated with conscious (analytic) type interpretation;
The ordinal notion - by contrast - is directly concerned with unconscious (holistic) type interpretation.

The problem therefore arises as to how the mutual identity of both types of interpretation can be obtained.

Once again this can be approached from two complementary directions:

(i) starting with conscious (analytic), we seek to establish its mutual identity with unconscious (holistic) type interpretation;

(ii) starting with unconscious (holistic), we seek in inverse fashion to establish its mutual identity with conscious (analytic) type interpretation.


Now the significance of the Zeta 1 zeros is that they provide the answer to (i).

The significance of the Zeta 2 zeros is that they provide the answer to (ii).

Of course ultimately both of these aspects are themselves understood as identical.

So ultimate understanding of the number system simultaneously marries both cardinal and ordinal interpretation of number at an analytical with the corresponding understanding of the Zeta 1 and Zeta 2 zeros at a holistic level.

Here at the same time both the unity of both the analytic and holistic aspects of the number system become inseparable from the corresponding unity of both conscious and unconscious aspects of personality.


So the key importance of both the Zeta 1 and Zeta 2 zeros is that they represent the perfect shadow counterpart of our customary cardinal and ordinal appreciation of the number system.

Put in an equivalent manner, they represent the perfect holistic (unconscious) counterpart to our customary analytic (conscious) interpretation of the number system.