I have identified a central problem with Conventional Mathematics in that within its own terms of reference it has no way of satisfactorily distinguishing the cardinal and ordinal aspects of number (which relate to the quantitative and qualitative aspects of number respectively).
Once again if we attempt to give numbers an absolute quantitative identity (as independent entities) which is the rationale of Conventional Mathematics, then this leaves - by definition - no role for a corresponding qualitative identity (whereby numbers can be appropriately viewed within a relational context of interdependence with each other).
So, Conventional Mathematics can only proceed through gross reductionism (whereby the qualitative relational aspect of number is effectively reduced to the quantitative)! And this is inevitably associated - as I have been continually pointing out - by a litany of confused thinking (especially in relation to treatment of the infinite notion).
Thus, the starting point for a more comprehensive mathematical vision is that mathematical entities (including of course numbers) have both independent (quantitative) and interdependent (qualitative) aspects that are relative.
This of course implies that the cardinal and ordinal aspects of number (when properly interpreted) are necessarily of a dynamic relative nature.
One important consequence of this new relative definition (comprising complementary aspects) is that it quickly leads to corresponding recognition that numbers have both particle and wave aspects (that again are of a relative nature).
Now it might be helpful initially to identify the cardinal aspect with the (independent) particle nature of number and the ordinal aspect with its (relational) wave aspect. However ultimately - in what I refer to as Type 3 mathematical understanding - the particle is clearly seen (from a complementary perspective) to be wave like, and equally the wave aspect (again from a complementary perspective) as particle like!
However it must be stressed once again that the relationship between both of these aspects cannot be properly interpreted from a merely quantitative perspective, as these distinctions relate to the twin quantitative and qualitative aspects, (the very interaction of which properly constitutes the true nature of number)!
So, as I have repeatedly stated on this blog since the very first entry, the clear message is that its long neglected qualitative aspect now urgently needs to be incorporated within Mathematics.
Indeed it is exactly the same message for Physics as the deeper understanding of wave/particle duality in Quantum Mechanics points equally to the fact that physical reality itself cannot be properly interpreted with respect to merely its quantitative aspect!
To conclude this entry, in light of our recent discussions we can provide a new way of stating the Riemann Hypothesis as the fundamental condition necessary for maintaining consistency as between both the cardinal and ordinal aspects of number.
However once again as this relates to the ultimate reconciliation of quantitative with qualitative meaning, it is thereby futile attempting to seek a proof in - mere - quantitative terms! Indeed as I have repeatedly stated, the essential nature of the Riemann Hypothesis cannot be properly grasped in conventional mathematical terms!