What I am attempting to portray here is a greatly enlarged vision of Mathematics where both standard (quantitative) and the - as yet - unrecognised holistic (qualitative) aspects are involved.
Indeed I outline three broad areas with respect to this enlarged vision.
Type 1 relates to the existing established mathematical approach which is specifically geared to deal in a direct manner with quantitative type relationships,
Type 2 then relates to the unrecognised holistic mathematical approach which is geared to deal with qualitative type relationships to in an indirect rational manner.
Whereas Type 1 Mathematics is based on the linear (1-dimensional) use of reason in an unambiguous either/or fashion, by contrast Type 2 Mathematics is based on circular (higher dimensional) use of reason in a paradoxical both/and manner. So the direct basis of holistic type understanding is intuitive type insight. Now in standard linear rational terms such intuition is merely reduced to reason with Mathematics viewed formally as a merely rational (1-dimensional) discipline.
However in higher dimensional understanding, intuition cannot be reduced in this manner. However quite amazingly all mathematical symbols now acquire a new holistic meaning as a means of indirectly conveying the qualitative relationships involved.
Type 3 is then the most comprehensive form of Mathematics where both Type 1 (quantitative) and the Type 2 (qualitative) aspects dynamically interact.
Now there are a couple of interesting observations that can be made on this New Mathematics.
In a certain sense one can validly claim that all reality is indeed - ultimately - of a mathematical nature i.e. when one includes both the quantitative and qualitative interpretation of its symbols (such as number).
Now this might appear to be reductionist in the sense that affective artistic type appreciation is allowed no role.
However this is not the case! Whereas Type 1 Mathematics is directly of a scientific nature (and indirectly artistic), Type 2 Mathematics is directly of an artistic nature (and indirectly scientific).
In other words the kind of intuitive experience that directly informs holistic mathematical appreciation itself directly arises from artistic type sensibility.
So one cannot successfully hope to encode holistic type appreciation of reality (in an indirect circular rational manner) without already attaining direct affective experience of such reality.
Therefore, science and art are intimately related at a dynamic level of understanding.
Indeed one could also include the religious quest in this mix.
When one understands properly the true nature of Mathematics it becomes clear that is all based on a massive act of faith in the validity of its procedures.
Now this is missed in conventional mathematical terms (as it ignores the true nature of infinite holistic type relationships) this simply entails - as in all mathematical proof - that the infinite is simply reduced to finite interpretation!
However recently, I have been at pains to emphasise that the true implication of the Riemann Hypothesis (which cannot be conventionally proved or disproved) is that for Mathematics to ultimately proceed, a massive act of faith must be placed in the ultimate identity of both the quantitative and qualitative interpretation of its symbols.
So when we look at it in this manner - properly understood - Mathematics in its true enlarged state, entails the balanced mix of both cognitive (scientific), affective (artistic) and volitional (spiritual/religious) meaning.
One other crucial area related to the role of Mathematics and Physics.
As we have seen the ultimate nature of reality is mathematical (in this expanded sense). This is event in the manner in which the mystical traditions speak of such reality as union (1) or emptiness (0).
Physical reality - where phenomena emerge - relates to the breaking of the original perfect mathematical symmetry (where no distinction exists as between the quantitative and qualitative aspects of its symbols).
However as soon as this perfect symmetry is broken (where quantitative considerations become to a degree separated), the physical reality of relative phenomena in space and time comes into existence.
So one way of looking at the goal of physical science is to recognise ultimately its pure mathematical nature (in an ineffable manner where quantitative and qualitative aspects are identical).