Note 3 - Holistic Mathematics and the dynamic interaction between perceptions and concepts.
It might be instructive here to clarify the nature of oneness as applying to perceptions and concepts respectively.
In dynamic experiential terms perceptions and concepts continually interact.
Starting from a dualistic perspective we can identify perceptions directly with the knowledge of individual (or particular) units.
Thus the perception of "a house" relates to an individual notion of one.
However concepts - by contrast - are more universal and collective. Therefore the concept of "house" is completely general in that it applies - potentially - to any particular house.
Thus the general concept of oneness here applies to any individual unit.
So we have the individual notion of "one" and the collective notion of "oneness". Strictly speaking the individual "one" - in the context we have adopted - is actual, whereas the collective "oneness" is potential i.e. applies potentially to any individual unit.
When these are related in dynamic terms - which is the very nature of experience - the one gives rise to the many and the many gives rise to the one.
So the collective "oneness" embodied in conceptual understanding - when identified with actual individual perceptions - becomes many individual "ones".
Thus when we apply the concept of "house" to separate individual perceptions of "a house" we become aware of many houses.
Likewise when separate individual perceptions of "houses" are related to the concept of "house" we become aware of a collective notion of oneness i.e. the quality of "houseness" (embodied in each individual house).
So the relationship between the one and the many and the many and the one reflects the dynamic interaction of perceptions with its (class) concept and (class) concept with its corresponding perceptions.
Alternatively, it represents the dynamic interaction of both quantitative and qualitative (and qualitative and quantitative) notions of oneness.
This interaction of perceptions with concept (and concept with perceptions) implicitly represents the dynamic interaction of form and emptiness (and emptiness and form).
If we identify a perception in quantitative terms e.g. the perception of "a house", the corresponding concept is qualitative and strictly without quantity. So the concept of "house", representing the general quality of "houseness", cannot be directly identified with any actual house. In other words it potentially applies to all houses without actually representing any (particular) house.
So in this case where we identify in quantitative terms, the perception represents form and the concept (strictly) represents emptiness (i.e. of form).
However if we reverse this procedure by identifying in qualitative terms, the concept now represents form and the perception represents (strictly) emptiness (i.e. without quality).
When we look more deeply at this interaction we can see the importance of our fundamental (mathematical) operations of positing and negating.
In dynamic terms once we posit a perception, it must to a degree be negated, before we can switch to experience of its corresponding concept.
So to move in experience to appreciation of the concept of "house" we must dynamically negate individual perceptions of (separate) " houses".
Likewise when we posit a concept, in turn it must to some degree be negated, before we can switch to experience of corresponding perceptions.
So to move from the concept of "house" to the experience of individual perceptions of separate "houses" we must dynamically negate this concept.
Now when - as is customary - dynamic negation remains merely implicit and somewhat weakly developed, a good deal of rigidity can attach to perceptions and concepts.
One extreme is represented by the researcher who has an ability to posit perceptions without a corresponding ability to organise them properly in a conceptual fashion.
The other extreme is represented by the theorist, who can readily posit concepts (within a given field), without corresponding ability to relate them properly to actual perceptions.
However the biggest problem arising from lack of dynamic negation is that creative understanding is greatly impeded. So where positing greatly dominates negating, structures of conceptual organisation and the nature of perceived data tend to reconfirm each other within rigidly set patterns of experience.
However where dynamic negation is suffiently developed in a balanced manner, perceptual and conceptual experience can take place in an unreduced fashion.
Here (quantitative) form (1) continually gives way to (qualitative) emptiness (0) and (qualitative) form (1) continually gives way to (quantitative) emptiness (0).