It has to be clearly recognised that the infinite is a qualitatively distinct concept from that of the finite. Ultimately it points to the fact that with respect to reality we have both actual and potential aspects. The actual is always made manifest in finite terms whereas the infinite more correctly pertains to the potential (from which the actual emerges). Again in direct scientific terms, the finite is appropriated in a (conscious) rational manner whereas the infinite is appropriated in an (unconscious) intuitive manner.
This in turn poses great difficulties for the conventional (Type 1) mathematical approach which in formal terms is confined to solely (linear) rational modes of interpretation.
Therefore the notion of the infinite that applies in Type 1 Mathematics is but a reduced quantitative notion that in many respects is entirely inadequate.
Somehow the mistaken view is perpetuated - as for example with series - that if we keep increasing the number of terms that we eventually accumulate an "infinite" no. (of such terms). However this is strictly speaking nonsense! Once again the infinite is a qualitatively distinct notion from the finite and cannot be "reached" therefore in finite manner. So if we keep increasing the terms of - for example - the natural number series without limit as might be said, we always obtain a finite number of terms. At no stage does the actual number of terms become infinite!
There are direct indications - even in Type 1 Mathematics - that the infinite is indeed qualitatively different from the finite. For example if we add two numbers in finite terms (say 1 + 2) a quantitative transformation is involved. Likewise if we multiply two non unitary numbers (say 2 * 3) again a quantitative transformation takes place.
However if we add two infinite numbers (or indeed a finite to a infinite number) no quantitative transformation is involved. Again if we multiply two infinite numbers or (an infinite by a finite) again no transformation is involved. Now, Cantor did indeed explore the notion of different types of infinities (within the confines of Type 1 Mathematics) but essentially his work - though presented in a reduced quantitative manner - draws attention to the fact that there is inevitably a qualitative aspect also involved. So ultimately the conclusion for example that the set of transcendental numbers is "bigger" than than that of the rationals is a reduced quantitative way of expressing the fact that the transcendentals are qualitatively distinct from the rationals (combining both discrete and continuous notions) whereas the rational are based on merely discrete notions!
All of this is of vital importance with respect to values for the Riemann Zeta Function which once again is defined (in Type 1 terms) as the infinite series
1/(1^s) + 1/(2^s) + 1/(3^s) + 1/(4^s) + ...... where s is any complex number a + bi.
Now it must be stressed again that this use of the infinite - which defines Type 1 Mathematics - represents but a reduced quantitative notion (i.e. where the potential notion of the infinite is reduced in actual finite terms).
In this context this approach does seem to work in certain cases. As is well known Euler had earlier shown that the Function indeed converges for all real values of s > 1. and this result can be exended to all complex numbers where again the real part a > 1.
Riemann however managed to extend the domain of definition for the Function to all values of s (except 1).
However this creates the immediate problem that the results for a wide range of these values make no sense from the conventional linear perspective.
My main concern here is not to show precisely how Riemann managed to achieve this "magical" transformation. What I am more concerned to deal with are the key philosophical implications that are involved which ultimately requires that an entirely distinctive type of Mathematics (Type 2) is used.
Ultimately this dramatically changes the very nature of the Riemann Hypothesis from a hypothesis in Type 1 Mathematics to a new fundamental hypothesis that serves as the cornerstone of the reconciliation of Type 1 and Type 2 Mathematics (which is the basis for the more comprehensive Type 3 Mathematics).
In other words all mathematical symbols possess both quantitative and qualitative aspects corresponding to two distinct types of interpretation. So the fundamental requirement is that consistency can be maintained with respect to both requirements. And the Riemann Hypothesis provides the very condition necessary for such consistency.
This of course implies that prime numbers inherently combine such quantitative and qualitative aspects in a very special manner!
Now we are looking here at the critical region of the Riemann Zeta Hypothesis for all values of s between 0 and 1.
Now the Function is undefined for s = 1 where the harmonic series results which in conventional terms sums to infinity.
The question then arises as to why the domain of definition cannot be stretched so as to include s = 1 (when in fact it can include every other value).
Now the qualitative mathematical reason (Type 2) is very illuminating in this regard.
Type 1 Mathematics is qualitatively based on the linear rational approach which is - literally - 1-dimensional in nature. Thus when for example we square the number 2 a qualitative as well as quantitative transformation takes place. So strictly the answer is now 4 square (i.e. 2-dimensional) units. However in Type 1 terms we simply ignore this qualitative dimensional change and express the result in reduced - merely - quantitative terms. So in Type 1 calculations all dimensional charges in the units involved are reduced to 1. Thus 2 * 2 = 4 (i.e. 4 ^ 1).
Now once we accept the true qualitative nature of a dimension, then numerical calculations (involving transformational changes) can be given an alternative result based on qualitative rather than quantitative considerations.
In the context of the Riemann Zeta Function therefore we give the Function - when it diverges from a Type 1 perspective - a finite Type 2 interpretation (where the result converges).
Remember Type 1 and Type 2 are finite and infinite with respect to each other! Therefore a finite result in quantitative terms is infinite from a qualitative perspective. Likewise what appears infinite in quantitative terms will now appear finite from a qualitative perspective. Put another way, if the Zeta Function diverges in standard Type 1 terms it necessarily converges from the alternative Type 2 perspective.
However the one obvious exception here is where s = 1. Because this series is already defined in terms of the default qualitative dimension used in Type 1 Mathematics, it therefore has no alternative meaning from a Type 2 perspective. However, for all other dimensional values of s, an alternative does necessarily exist.