The Holistic Nature of the Number System (1)

The obsession with the number line - as the means of representing real numbers – simply reflects the 1-dimensional (i.e. linear) paradigm on which Conventional Mathematics is based.

Now as I have expressed repeatedly in these blog entries the deeper basis of this (1-dimensional) paradigm is rooted in an approach whereby mathematical interpretation is conducted absolutely in terms of just one polar reference frame. This in turn leads (i) to the notion of mathematical “objects” that are unaffected through (subjective) mental interaction and (ii) an exclusive focus on the quantitative aspect of mathematical relationships (with no reference to a qualitative context of meaning).

Of course momentary reflection on the matter will quickly show that our actual experience of Mathematics is inherently of a dynamic relative nature entailing the continual interaction of both internal and external aspects and likewise the continual interaction of part (quantitative) and whole (qualitative) notions.

In psychological terms this equally entails both conscious and unconscious aspects of understanding through the corresponding interaction of rational (conscious) and intuitive (unconscious) processes.

Momentary reflection on the matter should also show that the accepted method adopted by Conventional Mathematics reduces the truly dynamic nature of actual mathematical experience in a greatly reduced absolute static manner.

So the necessary dynamic interaction of opposite poles of experience is formally expressed statically in terms of just one pole!

Likewise the dynamic interaction psychologically of (conscious) reason and (unconscious) intuition is rigidly expressed solely in terms of (conscious) reason.

 

It is therefore hardly surprising in this context that our present understanding of the number system should be fatally flawed.

Certainly from my perspective it has long been obvious that - when appropriately understood i.e. in accordance with actual experience - that the number system is necessarily of a dynamic interactive nature.

And of course this equally implies that it should be interpreted in a relative - rather than absolute - manner.

So we cannot therefore ultimately divorce the (apparent) objective nature of number from the subjective mental means of its interpretation.

Therefore in this relative sense when we change the nature of interpretation the objective nature of number objects likewise changes!

 

Likewise - and perhaps even more tellingly - we cannot ultimately divorce the (apparent) quantitative aspect of number from its corresponding relational context (which is of a qualitative dimensional nature).

As I expressed in yesterday’s entry, once we accept that numbers can indeed be related to each other then this implies that their independent status (in specific cardinal terms) is necessarily of a relative - rather than absolute - nature.

So in this dynamic context the quantitative nature of number relates to its cardinal status as relatively independent.

In other words, without an implicit acceptance that to have meaning, numbers must be placed in an ordered manner with respect to other numbers, it would be impossible to locate cardinals on the number scale! So the quantitative definition of number thereby implicitly implies a distinctive qualitative aspect.

 

Also, in this dynamic context, the qualitative nature of number relates to its ordinal status as relatively interdependent (with respect to other numbers).

So once again the very notion of ordinal requires placing a number in a group context (with respect to other numbers) with its identity thereby defined with respect to the group members involved.

Thus for example to define 2^nd I must place the number 2 in relation to other group members. The simplest case would involve a group of 2 members. Therefore in this context, I can unambiguously define a 2^nd member. However the definition of 2^nd clearly changes when the number of group members increases. So 2^nd in the context of 200 members is clearly distinct from 2^nd in the context of 2.

So the ordinal identity of a number (reflecting its qualitative identity) springs from a relationship of interdependence with a wider group of numbers.

However, once again we must define interdependence in a merely relative sense here, as we start with the cardinal notion before we can establish its ordinal identity.

 

So the notion of 2^nd (as ordinal) already dynamically implies 2 (as cardinal). Likewise the notion of 2 (as cardinal) already dynamically implies 2^nd (as ordinal).

However dynamic interaction necessarily implies a phenomenal context of space and time. Though we may appreciate from what has been said, that ultimately cardinal and ordinal identity must be identical, clearly this cannot be the case in a phenomenal context (which necessarily implies a degree of relative separation). Thus total unification of both cardinal and ordinal aspects of the number system points to an ineffable state.

Now this is all deeply relevant for appreciation of the true nature of the Riemann Hypothesis (which postulates the very condition for this ultimate identity)!

The vital fact to grasp is that cardinal and ordinal aspects require two distinctive interpretations respectively.

Thus great confusion pervades Conventional Mathematics. Because it is 1-dimensional in nature, it must necessarily attempt to interpret both cardinal and ordinal aspects from the same quantitative perspective.

Therefore though the ordinal aspect properly relates to the qualitative aspect of number, it is mistakenly dealt with in a merely quantitative manner.

This is a key problem of the very first magnitude with respect to the number system, which is at present just totally ignored!

As I was stating yesterday the quantitative (cardinal) aspect directly relates to the analytic interpretation of number; however the qualitative (ordinal) aspect properly relates to the (as yet unrecognised) holistic interpretation.    

Whereas the cardinal notion relates to the number line, the ordinal notion - by contrast - relates to the circle.   

So cardinal and ordinal notions are linear and circular with respect to each other.

Solving the ordinal problem (i.e. relating to the qualitative nature of ordinal numbers) comes through the Type 2 number system.

Therefore the qualitative i.e. dimensional notion of 2 is expressed as 1².   

As we have seen the corresponding cardinal notion is expressed in terms of the Type 1 number system as 2¹.

However to give indirect quantitative expression to the qualitative notion of number we convert to a circular format through obtaining the corresponding 2 roots of 1.

So + 1 and – 1 are these 2 roots which again - in this context - provide the indirect quantitative means of expressing the two ordinal members of a group of 2.

However with holistic understanding both quantitative and qualitative appreciation must be balanced with each other.

So the true qualitative appreciation requires intuitively being able to see + 1 and – 1 as interdependent with each other.

Put another way, this is the appreciation of ordinal rankings as purely relative (depending on context).
 

I have explained before how implicitly this is what one does for example in recognising that turns at crossroads can be either left or right (depending on context).

So if approaching to crossroads from one direction we label the 1^st turn + 1 (a left turn) and the 2^nd, – 1 (i.e. not a left turn) then clearly if the crossroads is approached from the opposite direction the 2^nd will now be labelled ,+ 1 (a left turn) and the 1^st, – 1 (i.e. not a left turn).

So each turn can be both left and right (depending on context).

This understanding therefore illustrates the very nature of interdependence which requires the ability to see simultaneously from - at a minimum - two opposite reference frames.

Now clearly we cannot make such paradoxical connections in the context of just one polar reference frame.

However Conventional Mathematics is formally defined in terms of just one such frame (where relationships appear unambiguous and linear).

Therefore by its very nature, Conventional Mathematics is not equipped to deal with the key notion of interdependence (except in a misleading reduced sense).

And as ordinal meaning directly relates to number interdependence, Conventional Mathematics cannot properly deal with the ordinal notion.

It is quite remarkable. Though the use of the circular number system (defined in the complex plane with circle of unit radius) provides the appropriate means for defining the ordinal nature of number, I have yet to see it mentioned anywhere!

We have something simple, right under our noses as it were and we fail to see it! This is directly due to the pronounced linear bias of Conventional Mathematics (based on mere analytic type understanding).

Now of course the unit circle is well recognised in Conventional Mathematics. However appreciation of its true role remains greatly limited due to the inevitable attempt to understand its nature in a merely linear manner. However to properly understand the circle, we require circular type interpretation!

Thus to appreciate the nature of ordinal interpretation we must adopt holistic understanding that requires at a minimum 2-dimensional interpretation (entailing two interacting polar reference frames).

And again such holistic understanding is totally missing from what is formally accepted as Mathematics!