I have already dealt with the important issue of the square root of 1 commenting on how but a reduced explanation is given through the conventional quantitative approach of Type 1 Mathematics. Strictly speaking this leads to logical inconsistency that amazingly is just brushed aside (as it cannot be resolved from a Type 1 perspective).
So using the Type 2 qualitative holistic treatment of mathematical symbols it is possible to demonstrate that corresponding inversely with the quantitative notion of the 2nd root of unity is a unique qualitative interpretation (relating to a differing logical system).
So remarkably corresponding then with each number as dimension is a unique logical system of interpretation. Therefore there are an infinite number of possible interpretations with the default approach of Type 1 Mathematics corresponding with the use of 1 (as dimension). And as we have seen this in turn corresponds with the standard linear rational approach that seeks to make unambiguous either/or distinctions.
However an even more direct example of the confusion lurking in the Type 1 approach arises in the context of the Riemann Zeta Function that gives rise to the most important unsolved problem in Type 1 Mathematics i.e. the Riemann Hypothesis.
Again from a Type 1 perspective when one squares a number a single valued result with no ambiguity results.
So for example the square of 1 is 1, the square of 2 is 4 and so on.
Therefore when one adds the squares of the natural number series i.e. 1 + 4 + 9 + 16 + ... the result should clearly diverge (from a Type 1 perspective) to infinity.
However remarkably according to the Riemann Zeta Function (in what represents the first of the so called trivial zeros) 1 + 4 + 9 + 16 +.... = 0
Now, no satisfactory explanation can be offered within Type 1 mathematical appreciation as to to the legitimacy of such a result. One can of course attempt to explain how it arises as the result of extending through analytic continuation a function to all regions of the complex plane.
However this in itself does not deal with the direct problem of how two results (that are contradictory from a Type 1 perspective) can arise. It is like asking someone to believe that now 1 + 1 = 3 (rather than the accepted result of 2) though perhaps even more ridiculous.
The startling resolution of this problem is however provided through Type 2 appreciation (where one now appreciates through the logical structure of the 2nd rather than the 1st dimension).
Therefore to properly understand the Riemann Zeta Function (applying to both positive and negative values of the dimensional power s) requires a combination of both Type 1 and Type 2 Mathematics.
Indeed the important Riemann Functional Equation can then be correctly seen as showing the relationship as between Type 1 (quantitative) and Type 2 (qualitative) numerical values.
The importance of this finding in turn for the Riemann Hypothesis is that it establishes the key condition for ensuring the consistency of both Type 1 and Type 2 mathematical understanding.
This then serves as the key axiom for the pursuit of the more comprehensive Type 3 Mathematics (where both quantitative and qualitative aspects continually interpenetrate in interpretation).
Of course this finding thereby removes the possibility of proving the Riemann Hypothesis within the accepted Type 1 mathematical interpretation.
In other words the truth to which the Riemann Hypothesis relates already precedes the axioms of Type 1 Mathematics. This provides the initial condition - literally of faith - underlying both Type 1 and Type 2 Mathematics i.e. that the truths derived from both types of understanding can be trusted as meaningful in their distinctive domains.
So clearly this initial axiom underlying belief in the logical consistency of Type 1 Mathematics cannot itself be proven from Type 1 interpretation!
For anyone wishing to see there are other ample hints showing that prime numbers cannot be understood in merely quantitative terms.
Resulting from Riemann's work is the finding that associated with each of the non-trivial zeros of the Zeta Function is a characteristic wave pattern. The accumulation of thse wave patterns can in turn enable a more exact distribution of the no. of primes (within a given natural number magnitude).
These waves patterns are therefore essential in appreciating the true harmony of the primes. Indeed the Zeta Function in itself is ultimately rooted in the harmonic series (which Pythagoras demonstrated has close connections with the harmony we experience in musical sounds).
However though there is a marked quantitative basis to music, clearly it entails also (in the overall relationship of different notes to each other) a true qualitative appreciation.
Likewise though obviously there is a marked quantitative basis to individual prime numbers, there is likewise a distinctive qualitative basis in the overall holistic relationship which the primes bear to each other.
The big limitation in Type 1 Mathematics is that both the individual and collective nature of primes can only be investigated in a merely reduced quantitative manner.
However in dynamic terms the correct relationship as between the individual and collective aspects is as quantitative to qualitative (and qualitative to quantitative) respectively.
And this is the central truth embodied in the Riemann Hypothesis!
There are other interesting issues worthy of investigation e.g. as to why a Type 1 approach in the context of the Riemann Zeta Function can generate results that are applicable to Type 2! So I will return to this in a future contriubution.
This reveals an even deeper truth regarding the nature of Mathematics in that it essentially entails a dynamic living interactive process (pertaining to both the physical and psychological realms).
Thus the truths embodied in prime numbers already reflect the fundmamental manner in which wholes and parts are related to each other in nature.
Prime numbers are therefore equally of both a quantitative (analytic) and qualitative (holistic) nature. So we have prime numbers as base quantities (that can be raised to dimensions or powers that are - relatively - qualitative in nature).
The big limitation of Type 1 Mathematics is that it has no means of dealing with primes as representing dimensional numbers except in a reduced quantitative manner!
Ultimately this likewise has huge implications for physics in that the starting point for the emergence of material phenommena derives from the dynamic relationship of the prime numbers (with respect to both their quantitative and qualitative aspects).
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