In yesterday's blog entry, I discusses - using the number 3 to illustrate - how its four distinctive meanings could be represented in mathematical terms.
This implies a dynamic interactive form of understanding where both a base and dimensional number are used in conjunction with each other. When the number i.e. 3 in this case is used to denote the base aspect, the corresponding dimensional aspect is given as 1; however when 3 is then used to denote the dimensional aspect, the corresponding base aspect is now 1.
Then in each case either the base or dimensional number is explicit in conscious understanding, with the other aspect - which is dynamically complementary - playing a merely implicit role in an unconscious manner.
So understanding keeps switching as between both conscious and unconscious recognition with respect to base and dimensional number respectively.
Here I wish to trace out more precisely the psychological dynamics that are involved with respect to such number recognition.
First we have the rational analytic perception of "3", which equates directly with 1), in yesterday's blog entry. This corresponds to the specific recognition in an explicit conscious quantitative manner of the number object "3" (as having a distinct individual identity).
Again I have denoted this as 31. This entails however the implicit unconscious recognition of 1 representing a dimension. In other words without this implicit recognition of the 1st dimension as potentially applying to all possible natural numbers, we would not be able to identify "3" in an explicit manner.
Next we have the intuitive holistic perception of "3", which equates directly with 2), in yesterday's entry. This corresponds to the explicit general recognition in an unconscious qualitative manner of the number object "3" (as applying to all actual classes of 3).
I have identified this in inverse fashion as 31. This again entails the implicit conscious recognition of 1 as representing the 1st dimension now identified in rational fashion as applying to all finite natural numbers. So once again without this implicit recognition of the 1st dimension, we could not collectively identify 3 with different classes (of 3 objects).
Then we have the rational analytic concept of "3" which equates directly with 3) in yesterday's entry.
This corresponds to the explicit general recognition in a conscious quantitative manner of the number "3" as representing dimension (i.e. as comprising 3 linear dimensions).
I have denoted this as 13. This again entails the explicit conscious recognition of "3" representing 3 (linear) dimensions. However once again without the implicit recognition of 1, where 1 now has a qualitative holistic meaning giving each unit an individual uniqueness this recognition of 3 dimensions would not be possible (as this requires uniquely identifying each dimension as length width and height respectively
Finally we have the intuitive holistic concept of "3"which equates directly with 4) in yesterday's entry. This corresponds to the explicit general recognition in an unconscious qualitative manner of "3" representing dimension of potentially applying in an infinite fashion to each of its 3 directions.
I have denoted this as 13. This again entails the explicit unconscious recognition of 3 representing 3 (circular) dimensions. However without the implicit recognition of 1, the arbitrary relative position of each dimension (as 1st, 2nd and 3rd respectively) would not be possible. In other words the 1st dimension must be fixed in a relatively - independent fashion before the other two dimensions can be related to it.
Therefore to conclude each number continually alternates in dynamic interactive fashion as between its analytic and holistic expression with respect to both base and dimensional aspects; this corresponds directly in complementary fashion with the likewise interaction as between both rational and intuitive expression with respect to both perceptions and concepts respectively.