In our present culture, a huge split separates the domain of reason from that of emotion exemplified by a corresponding clear split as between the sciences and the arts.
And nowhere is that split more clearly evident than in the conventional interpretation of Mathematics, which formally is based exclusively on a reduced form of rational interpretation.
I have already identified in earlier blog entries with respect to the basic operations of addition and multiplication (and subtraction and division) that two distinct types are involved in each case (which are not properly distinguished in conventional terms).
So once again for example when we say that 3 (as a cardinal number) = 1 + 1 + 1, implicit in this definition is the assumption that these units are of a homogeneous independent quantitative nature (thereby lacking any qualitative identity).
However when we express 3 in the alternative ordinal fashion, 3 = 1st + 2nd + 3rd, implicit in this definition is the assumption that each of the units is now unique in an individual qualitative manner where, strictly, the sum of the units lacks any quantitative identity!
So properly understood, we have now switched from the quantitative to the qualitative notion of 3 (as "threeness").
Therefore once we accept that both cardinal and ordinal notions of number must be coherently related, we have to abandon completely the conventional assumption of natural numbers as possessing an absolute independent identity.
Rather, number is now understood in a dynamic interactive manner, which entails notions of both relative independence and relative interdependence respectively.
So now number is clearly seen as composed of two complementary aspects that interact with each other in a relative - rather than absolute - manner.
Therefore with respect to the cardinal (quantitative) aspect, we can now say that addition of the component units of 3 (i.e. 1 + 1 + 1) relates to their independent status (in relative terms).
However, the corresponding addition with respect to the three ordinal units (1st + 2nd + 3rd) relates to their interdependent status (again in relative terms).
So we have now properly recognised the two distinct aspects of addition in relative terms (as relating to independent and interdependent units respectively).
Once more - and it is vital to properly grasp this key point - the abstract stance of conventional mathematical interpretation is fatally flawed in that it attempts to view number reality in an absolute rigid manner. This thereby reduces in every context meaning that is truly qualitative in a merely quantitative manner. Alternatively expressed this implies the corresponding reduction of ordinal notions in cardinal terms.
Properly understood however, number reality - which is inherently dynamic and interactive - is always relates in experience to both its complementary physical and psychological expressions.
The deeper implication of what I am stating here is that we can no longer hope, from this dynamic perspective, to view Mathematics in merely cognitive terms as relating to rational truth!
We have already seen that Mathematics contains both positive (conscious) and negative (unconscious) aspects.
Whereas one may indeed initially identify the positive aspect with rational understanding, the corresponding negative aspect relates direct to intuitive type appreciation (that indirectly can be conveyed in a circular rational manner).
So once more, mathematicians may indeed informally accept the importance of intuition (especially for creative work). However, because of the rigid framework from which interpretation is attempted, they have no way of properly distinguishing intuition from reason. In effect intuition is simply reduced to reason, with all proof formally expressed in rational terms.
Likewise however - perhaps more surprisingly - one eventually discovers that Mathematics also contains cognitive (real) and affective (imaginary) aspects.
And in a comprehensive mathematical understanding, both cognitive and affective aspects must be properly harmonised with each other
So again with respect to our simple example, when one recognises in quantitative terms that 3 = 1 + 1 + 1, this directly entails cognitive type understanding (of a rational impersonal nature).
However implicitly when one recognises in qualitative terms that 3 (as "threeness" = 1st + 2nd + 3rd) this entails affective type understanding (where the senses are involved in a personal manner).
Unfortunately, appreciation of this latter type of affective recognition has all but been completely lost in the modern understanding of Mathematics.
Though implicitly it still requires at least some small degree of affective sense recognition to successfully make ordinal type distinctions, at an explicit level this recognition is then reduced in a merely rational impersonal manner.
So one of the truly damaging effects of the great cultural influence of mathematical - and by extension all scientific - understanding is that this is exercising an enormous influence in reducing qualitative to quantitative meaning in so many areas of our lives.
And the root of this very problem lies at the very heart of Mathematics itself in the attempted preservation of but a limited - and ultimately hugely distorted - rigid understanding of the true nature of all its relationships.