We are perhaps already at the beginning of a truly momentous shift in mathematical - and by extension - all scientific understanding where the prevailing paradigm will change in a radical and unprecedented fashion.
We saw in the past two blogs how our very understanding of prime numbers needs to be altered in the light of Riemann's ground breaking discoveries.
So - again using the analogy from Quantum Mechanics - prime numbers can no longer be understood as static fixed entities (with specific integer values). Rather they need to be considered in relative terms as dynamic interactive entities (possessing complementary wave and particle aspects).
Putting it another way they possess both specific and holistic aspects (or - in the terms that I customarily use - quantitative and qualitative).
As I say, when appropriately interpreted this is already inherent in Riemann's discoveries with respect to his Zeta Function where both knowledge of specific prime numbers and their corresponding holistic aspect through the non-trivial zeros (relating to the unlimited solutions of the complex Zeta function) can be seen to be mutually encoded in each other.
Now why the obvious connection here with Quantum Mechanics is not clearly seen owes a great deal I believe to the limited - merely quantitative - approach that is employed in Conventional (Type 1) Mathematics.
To see the problem involved we will digress once again to my favourite example of road directions.
If I am travelling up a straight (vertical) road and come to a crossroads I can unambiguously identify - for example - a left turn.
Now if I later travel down the same road from the opposite direction, I can again unambiguously identify a left turn.
However we will now have designated both turns unambiguously as left.
And this results from using isolated independent frames of reference.
So in the former case the frame of reference was unambiguously "up" while in the latter it was unambiguously "down".
Now clearly of course when we use simultaneous frames of reference the two turns must necessarily be left and right (in relation to each other).
The relevance of this for the conventional mathematical approach is striking (as it solely recognises the quantitative approach).
Therefore when for example we approach the primes from one direction with respect to their specific attributes their quantitative nature seems apparent. Likewise when we then approach them from the opposite direction with respect to their general attributes, again their quantitative nature seems readily apparent.
So in effect the quantitative approach is used with respect to both specific and holistic appreciation of the primes. And admittedly when using isolated frames of reference in studying their aspects independently, both indeed exhibit quantitative features.
However when we bring both aspects together (simultaneously, as it were) to understand how both are interrelated, then the specific and holistic nature of the primes are quantitative and qualitative with respect to each other. Thus we cannot hope to properly understand - in this context - the relationship between quantitative and qualitative aspects while using a merely quantitative type approach!
This is why I have consistently maintained that the Riemann Hypothesis cannot be satisfactorily understood - not alone resolved - in the absence of the complementary qualitative aspect of mathematical understanding.
And this problem is not just isolated to the Riemann Hypothesis but is in fact endemic in the most common of mathematical processes.
Even as a child of 10, I had marked difficulties with the mathematical treatment of squares and the corresponding treatment of square roots.
I could see then that whenever a number is squared that a qualitative - as well as quantitative - transformation takes place. So when for example we square 3 (3^2) in quantitative terms a transformation takes place (to 9) and in qualitative terms from 1-dimensional to 2-dimensional (i.e. square) units. So the quantitative aspect is tied up with the base unit (3) and the qualitative aspect with the dimensional aspect of number (in this case 2).
So we have here the same old problem. When for example when we raise a number to a certain power e.g. a^b, both the base number a and the dimensional power b do indeed have a quantitative aspect (when considered in isolation from each other). However when we combine both aspects, then a and b are actually quantitative and qualitative with respect to each other.
So as I would see it, the very way that powers and roots are handled in Conventional (Type 1) Mathematics is grossly reductionist.
Now, remedying this problem requires that an alternative (Type 2) Mathematics be put in place to deal with the qualitative aspect (initially in isolation). Then when both are combined we need to move to Type 3 Mathematics for a comprehensive understanding of what is involved through interaction of both types. However, there is still precious little recognition of the truly fundamental nature of this problem in Mathematics.
Thus my own attempts for 40 years have been largely concerned with formulating the bones of this Type 2 Mathematics (which I call Holistic Mathematics). It is only in recent times that I have attempted to combine Type 1 and Type 2 in a very preliminary manner with reference to the most beautiful formula in Mathematics (Euler's Identity) and then the greatest unsolved problem in Mathematics (the Riemann Hypothesis). So what I have been attempting - again in a necessarily preliminary manner - serves as an introduction to Type 3 Mathematics.
Thus, through approaching the problem from the background of immersion in an entirely distinctive qualitative type approach, it quickly became apparent to me that the Riemann Hypothesis is not in fact a problem that can be solved in conventional (Type 1) terms. It properly relates to the basic condition necessary for reconciling both the quantitative (specific) and holistic (qualitative) aspects of the primes thus making it in effect a central axiom for Type 3 Mathematics.
When we look at Riemann's Zeta Function, we can see that this relationship as between base natural numbers and varying dimensional powers (s) is central (with the added complication that the dimensional powers are formulated in a complex manner).
It should also be obvious that the Riemann Zeta Function throws up a whole series
of values for s (with real part < 1) that have no strict meaning from a standard linear rational perspective. For example 1 + 2 + 3 + ..... diverges to infinity (in standard terms) yet acquires a value of - 1/12 (in the context of Riemann's Zeta function). The question then arises as to how we can reconcile both quantitative interpretations (which I have never seen satisfactorily answered).
And now further we see - in the light of the Function - that the primes exhibit two complementary aspects (of a specific and holistic nature) respectively.
However though both of these aspects can be investigated in isolation from a merely quantitative perspective, relative to each other they are quantitative and qualitative (and qualitative and quantitative) respectively.
The deeper meaning of all this is that the prime numbers as quantities have a mirror side as a qualitative means of interpreting reality. So we cannot solve the physical mystery of the primes without equally solving in mirror terms the psychological mystery of their appropriate interpretation (a task that greatly transcends the methods of Conventional Mathematics).
In psychological terms we could say, from a Jungian perspective, that they possess both conscious aspects (as distinct entities) and unconscious aspects (as archetypes of a universal reality) with both ultimately identical. In complementary physical terms they possess specific attributes and holistic communication abilities (reflecting their quantitative and qualitative nature).
These were originally identical as mere potential for existence and yet have now considerably evolved to yield the wonderful universe we inhabit.
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