Just to briefly recap!
When we start out (internally) with each individual prime, the corresponding ordinal notion of that prime remains holistically undefined by the Zeta 2 zeros.
Then, when we start out (externally) with the collection of all the primes, the corresponding cardinal notion of each individual prime remains holistically undefined by the Zeta 1 (Riemann) zeros.
Therefore in truth the two sets of zeros (along with both the cardinal and ordinal nature of the primes) are intimately related in bi-directional fashion. And it is this combined appreciation of the dynamic interaction of both sets of zeros, that I refer to as the Zeta 3 zeros.
In other words, just as internally the two-way relationship as between prime and natural numbers, indirectly expressed though the Zeta 2 zeros, arises in a synchronistic manner (that is ultimately ineffable), likewise externally, the two-way relationship as between prime and natural numbers, now indirectly expressed through the Zeta 1 zeros, likewise arises in a synchronistic manner (that is ultimately ineffable).
So therefore the internal synchronicity (expressed through the Zeta 2 zeros), dynamically implies the external synchronicity (implied by the Zeta 1 zeros). Likewise the external synchronicity (expressed through the Zeta 1 zeros) dynamically implies the internal synchronicity (expressed through the Zeta 2 zeros).
Thus in respect to (internal) Zeta 2 zeros, we start with the initial requirement that each number is prime so that only the prime roots of 1 (except 1) are extracted.
Therefore in order to extend the Zeta 2 zeros to encompass all the composite natural numbers (as well as primes) we require the external relationship implied by the Zeta 1 zeros (where the primes are seamlessly integrated in collective manner with the natural numbers).
And in turn the "circular" nature of the Zeta 1 zeros intimately depends on the initial default definition (with respect to the standard unit radius) in Zeta 2 terms.
Therefore the great fundamental question of how both quantitative and qualitative aspects of understanding can be consistently combined in the understanding of number - and indeed every other possible mathematical relationship - ultimately dissolves in total mystery.
So underlying the entire mathematical edifice, inescapably lies an initial act of faith in the subsequent consistency of all its relationships.
This prior consistency - as between its quantitative and qualitative aspects - cannot be proven from within its axioms. Rather the subsequent use of such axioms depends on this initial prior assumption of consistency.
And this is as close as we can come in the mathematical realm to a pure TOE. However it is one that renders all - merely - analytic attempts to grasp the ultimate nature of reality as futile.
Though it has been greatly forgotten in recent centuries, the true goal of Mathematics - as the ancients better realised - is to combine both the analytic and holistic aspects of understanding in the marriage of reason and contemplative wisdom (as refined intuition). Here, science ultimately ends in comprehension of the pure mystery of all existence.
So mathematical symbols, especially with respect to number - rightly interpreted - ultimately represent the last phenomenal veil that bridges the phenomenal world of form from pure ineffable emptiness.
However - and this point requires to be repeatedly stressed in our present mathematical culture - all such symbols have both an analytic (quantitative) and holistic (qualitative) meaning so that strictly, in dynamic interactive terms, these symbols entail the relationship of both aspects.
I have commented before e.g. "Buddhist Mathematics" on the famous heart sutra;
"Form is not other than Emptiness
Emptiness is not other than Form"
Now translating this in holistic mathematical language;
Oneness (1) is not other than Nothingness (0)
Nothingness (0) is not other than Oneness (1)
In other words, the unity of all form i.e. in the holistic appreciation of what is spiritually inherent in all creation, is ultimately inseparable from the corresponding appreciation of (spiritual) emptiness (as not being identifiable with phenomenal forms).
Now an important feature of the present IT revolution is that it owes so much to the rational analytic interpretation of 1 and 0 as separate binary digits.
However the corresponding holistic realisation is that 1 (as oneness) and 0 (as nothingness) are ultimately identical (in a spiritually intuitive manner). Therefore, just as all information can be potentially encoded using the analytic interpretation of the binary digits, likewise all (dynamic) transformation processes can likewise be potentially encoded using the holistic interpretation of the same two digits (relating to linear and circular type logic respectively).
When one thinks about it for a moment, the two-way relationship of the primes and natural numbers does not include - what I refer to as - the two original numbers i.e. 1 and 0.
So in an important sense, at an even more fundamental level of mathematical reality, 1 and 0 themselves originate in a synchronistic manner and then remain deeply implicit in all other number notions such as the primes and remaining natural numbers.
Thus one obvious weakness with the conventional approach to explaining the natural number system is that in postulating the primes as the independent "building blocks" of that system, 1 (the most important natural number) is automatically excluded, even though using the Peano-based approach, the entire natural number system can be derived from 0 (through the successive adding of 1).
So a clear benefit of the dynamic approach I advocate, is that this necessarily entails the two-way interaction of the Peano-based addition and the prime-based multiplication approaches respectively.
Indeed it is only within this dynamic interactive approach that the true complementary relationships as between addition and multiplication can become evident.
However already underlying the additive and multiplicative approaches to derivation of the natural numbers, is the prior two-way relationship as between the original numbers 1 and 0.
In fact in its most basic form, 1 and 0 are necessarily always involved in the dynamic interaction of the quantitative and qualitative aspects of number.
For what is distinctly appreciated as 1 (in a quantitative manner) entails the corresponding realisation that it thereby represents 0 (in a qualitative fashion).
In this sense, what is distinctly recognised as a unit (1) in a quantitative analytic manner, thereby cannot be distinctly recognised as a corresponding unit (1) in qualitative holistic terms.
So what represents 1 in analytic terms, represents 0 in a holistic manner.
And likewise what is then generally recognised as a general unit (1) i.e. in the overall appreciation of oneness in a qualitative holistic manner, is thereby not distinctly recognised as a part unit (1) in quantitative analytic terms.
So, in reverse manner, what represents 1 in holistic terms, represents 0 in an analytic manner.
Therefore, in experience, one continually switches in two-way fashion as between its quantitative and qualitative aspects through the mutual two-way switching as between 1 and 0.
And this is intimately tied up with the proper appreciation of number.
Now I have repeatedly emphasised that there are two aspects to number, which I refer to as Type 1 and Type 2 respectively.
So the number "2" for example is defined as 21 in the
Type 1 and 12 in the
Type 2 aspects respectively, with 2 being identified as the base in the former and the dimensional number in the latter aspect respectively.
Now, from an additive approach, we can represent 2 as 1 + 1 (with respect to both aspects).
However, what is not recognised in conventional mathematical terms is that the very meaning has now switched from a - relatively - analytic perspective, in the Type 1, to a holistic perspective in Type 2 terms.
In other words, Type 1 relates to independent units, whereas Type 2 relates to interdependent units respectively. Put another way - reminiscent of quantum mechanics - we have thereby switched from a particle to a wave definition of number.
So when the use of 2 (as base number) refers to separate individual units (in an analytic manner) the corresponding use of 2 (as dimensional number) properly refers - in dynamic relative terms - to the holistic relationship of these units with the whole.
For example - say - in a more concrete manner, we divide a cake into 2 (equal) slices. Each of the slices now represent independent units in analytic terms. However the overall relationship of these (part) slices to the (whole) cake, represents interdependent units in a holistic manner.
So the proper use of 1 in this context - referring to the qualitative whole - is 0 (in dynamic relation to a quantifiable slice).
Likewise the use of 1 - now referring to each quantifiable part - is 0 (in dynamic relation to the qualitative whole).
So again just as 1 and 0 - as statically understood in analytic terms - are so important in terms of encoding information, 1 and 0, as dynamically understood in holistic terms, are equally important in terms of encoding - potentially - all transformation processes. This then serves as the fundamental means through which switching in both physical and psychological terms takes place as between quantitative (differentiated) and qualitative (integrated) phenomena respectively.