Wednesday, June 29, 2016

Clarifying the Nature of Ordinal Numbers (1)

To recap again briefly, during the 3 levels that characterise Band 3 (in my map of development), I had come to realise the extraordinary potential importance of the holistic dimensional notion of number (according to its Type 2 aspect).

Through this holistic mathematical interpretation of number, I could see how the dynamic interactive nature of experience is implicitly structured in mathematical terms, with each holistic dimension relating to a unique configuration with respect to the fundamental polarities of experience (internal/external, whole/part and form/emptiness).  Indeed as the 3rd polarity set here represents but a special requirement with respect to the first two sets, this entails that phenomenal reality (in physical and psychological terms) is fundamentally conditioned by the dynamic interaction of just two sets of polarities (i.e. internal/external and whole/part).

In psycho spiritual terms the "higher" dimensions (> 1) represent increasingly refined intuitive states, such as unfold through the process of authentic contemplative development. Then associated with these states are corresponding refined cognitive (and affective) structures of a circular paradoxical nature, which are precisely configured in a mathematical fashion as the holistic interpretation of number.

These holistic states and structures are however all initially grounded in the holistic "1" as the 1st dimension, which represents analytic understanding of a linear rational kind. So therefore, before nondual understanding of a true holistic nature can unfold, we must first initially understand phenomena in a dualistic (analytic) fashion.
Unfortunately in our present culture - especially in mathematical and scientific terms - an enormous degree of reductionism takes place, where the holistic nature of understanding is in effect completely disregarded!

I felt then that I had already made considerable progress with respect to this new holistic interpretation of number. I could readily appreciate how it was especially applicable to the "higher" levels of psychological development and how it would also form the basis for true integral scientific understanding.

However I had not yet reached the stage where I could properly relate this appreciation of mathematical symbols with the conventional accepted analytic interpretation. In other words, I had not yet properly attained to a radial interpretation of mathematical symbols (which always remained my true goal).

However as the descent through the various levels of Band 5 unfolded, this situation radically changed (especially with the onset of Level 2).

I had long puzzled over the precise nature of cardinal and ordinal numbers.

For example 2 (as a cardinal number) = 1 + 1.

So, in this definition of number, each unit is given a homogeneous impersonal identity (in a merely quantitative manner). Put another way, each unit here lacks any distinctive qualitative identity!

However the ordinal definition of number reverses the situation.

So, for example we could now refer to 2 = 1st + 2nd. However 1st and 2nd only have meaning in the context of interdependence. So ordinal notions necessarily express a relationship between numbers, which inherently is of a qualitative nature.

And it then became clear to me on reflection that the cardinal definition implicitly implied the ordinal and the ordinal in turn implicitly implied the cardinal.

Therefore when we use 2 in the customary quantitative sense (i.e. as a cardinal number) implicitly we realise that it is composed of a 1st and 2nd unit (in ordinal terms).

Likewise when we use 2 in the combined ordinal sense as comprising a 1st and 2nd unit, implicitly we must equally interpret these units as independent (in cardinal terms).

However this poses a huge problem, which is completely ignored in conventional mathematical interpretation.

Once again, the cardinal notion of number implies the assumption of "independent" units; the ordinal notion however implies the complementary notion of "interdependent" units. So properly considered, independence (in the numbers to be related) and interdependence (as the corresponding relationship between numbers) are complementary terms.  Therefore for cardinal and ordinal to be successfully combined, both the independence and interdependence of numbers must be properly interpreted in a dynamic relative - rather than static absolute - manner.

However in Conventional Mathematics,  the ordinal notion becomes effectively reduced to cardinal, with a static absolute interpretation of number (of a merely quantitative nature) resulting.

I then spent a considerable amount of time in properly clarifying how this reduction of ordinal to cardinal meaning occurs.

Let us take the simple example - which I have used before - of ranking two cars in size (with larger car ranked 1st). So say the choice is between  a BMW and a Panda. Then the BMW is ranked 1st and the Panda 2nd.

However say we now use a different ranking criterion e.g. age, with the newest car ranked 1st, and we are informed that the Panda was purchased in 2015 and the BMW in 2010. So the Panda is now ranked 1st and the BMW 2nd.

So the important point here is that implicit in the very notion of ordinal rankings is that the rankings can change (depending on relative context).

So what is 1st in one context, can be 2nd in another context and vice versa (as we saw with the ranking of the cars).

Therefore the very nature of ordinal positions from a holistic perspective - representing the potential or possibility of what can happen - is that in any grouping of numbers, ordinal positions can be fully interchanged with each other. We have already demonstrated this in the case of 2 members of a group where 1st and 2nd can be freely interchanged with each other. With 3 members of a group 1st, 2nd and 3rd can be interchanged with each other. Then more generally, with n members 1st, 2nd, 3rd,....nth position can be freely interchanged with each other.

Now in (analytic) cardinal terms, each number has a fixed identity (which is non-interchangeable). So 3 cannot for example be changed to 5 in some other context.

So we can perhaps see how the (analytic) cardinal and (holistic) ordinal notion of number are complementary with each other in an opposite manner. Whereas the cardinal can be represented in linear, the ordinal is best represented by contrast in a circular manner.

However in practise the ordinal is reduced to the cardinal, because in any actual context, no ambiguity will arise, due to the consideration of just one reference frame for ranking purposes. In other words, when we apply linear (1-dimensional) interpretation to ordinal numbers, they are effectively reduced in cardinal terms.

So in this reduced sense, each ordinal position is measured in static fashion as the last of the cardinal group in question.

So 1st equates with the last member of 1 (= 1). 2nd then equates with the last member of 2 (= 1).
3rd equates with the last member of 3 (with the other two positions already filled (= 1) and so on.

Therefore the ordinal identity, for example, of 3 = 1st + 2nd + 3rd = the cardinal identity of 1 + 1 + 1.

However implicit in the very ability to rank numbers (according to some set criterion) is the  appreciation that these rankings are unique for the criterion adopted. Thus we have no difficulty in accepting the "paradox" of the car rankings in the example above, because we implicitly recognise that 1st and 2nd can interchange (depending on context).

In other words, underlying our customary analytic interpretation of ordinal numbers, where they are given an absolute fixed identity in linear terms, is an unrecognised holistic appreciation, where all positions are interchangeable with each other.

And this is vital in the very ability to understand 1st, 2nd, 3rd .... in a qualitatively unique manner.
However, remarkably, the fundamental appreciation that these ordinal numbers represent qualitative - rather than strict quantitative - distinctions has been all but lost in Conventional Mathematics. Thus the true qualitative nature of ordinal numbers is carefully concealed through the use of the more neutral terminology of "ordinal rankings". 

Indeed we had another example of this ordinal dilemma before in the example of the crossroads where what is unambiguously left or right (when the crossroads is approached from just one direction) is paradoxically both left and right (when we admit an approach simultaneously from both directions).

Therefore I clearly realised now that the fundamental interpretation of ordinal numbers in Conventional Mathematics is strictly untenable in any meaningful sense. In other words a limited special case of a more general phenomenon is used in a way that reduces interpretation in a highly distorted manner.

Therefore though in truth number is of a dynamic relative nature, it is represented in a merely static absolute fashion; likewise though both quantitative and qualitative aspects define the interaction of all numbers (in cardinal and ordinal terms), conventional interpretation is subsequently reduced in a merely quantitative manner.

So the next stage of my investigation was to seek a means of "converting" the inherent qualitative nature of ordinal numbers in an indirect quantitative fashion.

Alternatively, this could be expressed as the attempt to express the Type 2 aspect of the number system in a conventional Type 1 manner.  

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