In my previous entries, I have stressed that every number can be given both an analytic and holistic interpretation respectively and that in the dynamics of experience, both aspects are inevitably intertwined, with one made explicit in conscious manner, with the other remaining - relatively - implicit in unconscious fashion. .
And each number can likewise be given a base identity or a dimensional identity respectively.
So once more illustrating with respect to the dimensional aspect, the number 3 has an analytic interpretation with respect to 3 (referring to 3 dimensions in a quantitative manner). However 3 equally has a qualitative interpretation as the "threeness" or the quality of 3, which thereby enables the common identification of all members relating to a class of 3 dimensions (such as the length width and height measurements of different rooms).
However one might wish to probe further as to the precise difference as between the quantitative and qualitative interpretations.
So again, if I for example refer to the 3 dimensions with respect to the room of a house (length, width and height) this represents the accepted quantitative view.
However in conventional terms the distinct identity of a number (such as 3) used with respect to objects is not properly distinguished from what is used for dimensions.
But there are crucial differences. 3 as used for objects has a finite specific meaning i.e. as 3 unit objects). i.e. 3 = 1 + 1 + 1
However 3 as used for dimensions has by contrast a collective general meaning. Here each unit (i.e. separate dimension) applies potentially to every possible natural number in an infinite manner). So one more, length, width and height measurements could apply to 1, 2, 3, 4,.......rooms.
There is also another key difference:
When we use 3 in the restricted finite sense (where each unit applies to just one actual object) the units are treated as independent and homogeneous.
So when 3 = 1 + 1 + 1, the relationship between units is not considered.
However as far as dimensional "units" are concerned, this is not really the case. Here the units are not in fact independent but are related to each other in an ordered fashion as length, width and height respectively.
Thus treating the units as independent gives them a reduced meaning. Now it is true that from a quantitative perspective, that if we have 3 dimensions for a room, as length width and height respectively, the total volume will be the same (irrespective of the order in which they are taken).
So in fact when we multiply numbers a dimensional aspect is always involved. However in reduced quantitative terms this is ignored so that 2 * 3 * 5 for example = 30 (with no reference to the dimensional change involved).
In other words when we multiply 2 * 3 * 5 in this way, it is as if we accept that these measurements thereby belong to to the same dimension. So for example if we recognise the length as the only dimension, then 2 * 3 * 5 thereby represents the 3 numbers multiplied with respect to the same dimension. So the answer is thereby given in 1-dimensional terms.
So the very key to recognising higher dimensions (> 1) is that such dimensions by their very nature are not absolutely independent of each other, but must exist with respect to each other in an orderly manner.
So what we are faced with all the time is a constant dialectic as between notions of independence and interdependence respectively.
With independence, we view the units as quantitative in a cardinal manner.
So again 3 = 1 + 1 + 1.
However with interdependence, we view the units as qualitative in an ordinal fashion.
So here 3 = 1st + 2nd + 3rd.
And in experiential terms with respect to understanding, these two notions are necessarily of a relative nature in a dynamic complementary manner.
Thus we can only explicitly recognise cardinal units as independent (in an explicit quantitative manner), if we already implicitly recognise a corresponding ordinal relationship between units (in a qualitative manner).
Likewise we can only explicitly recognise ordinal units as interdependent (in a qualitative manner) if we already implicitly recognise a corresponding cardinal relationship between units (in a quantitative manner).
So the key issue then relates to how we can successfully convert as between qualitative and quantitative notions respectively.