Once again we return to the important distinction as between analytic and holistic type understanding that, potentially, equally apply to (all) mathematical symbols and relationships.
Analytic interpretation essentially is based on the clear separation of polar opposites in experience.
So again with respect to the simple example of road directions, North (N) and South (S) are polar opposites.
When we then approach a crossroads heading in just one direction - say - N, then we can unambiguously identify left and right turns with respect to this clearly separated polar reference frame (i.e. N).
Equally when we approach the crossroads from the opposite direction heading S, again we can unambiguously identify left and right turns with respect to this clearly separated polar reference frame (i.e. S).
So in terms of either reference frame (considered separately in an independent fashion) left and right turns have an unambiguous meaning where a turn is either left or right in an absolute manner!
This represents in fact a good example of analytic type interpretation (as I define it) where meaning takes place within independent polar reference frames that can be clearly separated.
Now all mathematical understanding is necessarily conditioned by polar reference frames. For example all such understanding necessarily entails an interactive relationship as between external (objective) and internal (subjective) poles.
So analytic interpretation in this context is based on the assumption that the external (objective) aspect can be clearly separated from the corresponding internal (subjective) aspect in an absolute type manner. In this manner a direct correspondence is thereby necessarily assumed as between objective truth and subjective (mental) interpretation.
Likewise all mathematical understanding necessarily entails an interactive relationship as between individual (part) and collective (whole) notions that are quantitative and qualitative with respect to each other.
So once again analytic interpretation is based on the assumption that quantitative and qualitative aspects can be again be clearly separated in absolute type manner from each other.
Thus formally Conventional Mathematics is viewed as the objective interpretation of quantitative type relationships (in an absolute unambiguous type manner).
In other words Conventional Mathematics is formally associated with merely analytic (Type 1) interpretation of all relationships. I equally refer to this as linear type interpretation (which literally commences from the view that all the real numbers lie on a number line!)
However when we return to our crossroads example we can perhaps appreciate that a distinctive circular type of interpretation - that is paradoxical in terms of linear - equally applies.
Therefore when we consider the two polar directions of N and S simultaneously as interdependent with each other, then left and right turns at a crossroads are rendered as circular and paradoxical (in terms of conventional linear interpretation).
So what is designated as a left turn (heading N) is equally a right turn (heading S); likewise what is designated a right turn (heading N) is equally a left turn (heading S).
Thus when we view both polar reference frames simultaneously (as interdependent) each turn is understood as both left and right (depending on context).
So linear logic (based on independent polar reference frames) is associated with unambiguous either/or distinctions; circular logic (based simultaneously on interdependent reference frames) is based on paradoxical both/and distinctions.
Now it is this latter type of understanding, where the fundamental polar reference frames that necessarily govern all mathematical experience are viewed simultaneously as interdependent, that I term holistic.
So from one important perspective, in holistic terms we cannot hope to clearly separate the external pole of objective mathematical recognition from the corresponding internal pole of mental interpretation!
Likewise from an equally important perspective, we cannot hope to clearly separate the quantitative pole from corresponding qualitative pole of mathematical understanding.
In other words there are base and dimensional aspects to all number recognition which are quantitative and qualitative with respect to each other.
Thus when I define 3 for example in a quantitative base fashion, it already assumes the default dimensional aspect of 1.
So 3 (in Type 1) terms is expressed as 31.
However when I define 3 - relatively - in a dimensional qualitative fashion, it already assumes the default quantitative base of 1.
So 3 (in Type 2) terms is expressed as 13.
Thus all numbers - such as 3 in this example - continually switch as between their Type 1 and Type 2 aspects (that are quantitative and qualitative with respect to each other).
Like with the turns at the crossroads, when we attempt to understand either aspect (Type 1 or Type 2) in isolation, unambiguous analytic type interpretation is possible.
However, when we attempt to understand these two aspects as interdependent, then just as with the turns at the crossroads, interpretation now becomes deeply circular and paradoxical in a holistic type manner.
Now just as comprehensive interpretation of the turns at a crossroads ultimately combines both analytic (linear) and holistic (circular) type understanding equally this is so in mathematical terms.
In fact, properly understand, the Type 1 and Type 2 aspects correspond to the distinction as between cardinal and ordinal interpretation of number (that are quantitative and qualitative with respect to each other).
Ultimately with respect to the crossroads example, we come to the realisation, by combining both analytic and holistic type interpretation, that left and right turns have merely a relative meaning (depending on an arbitrary context).
Likewise all mathematical interpretation of relationships - in what I term Type 3 understanding, combines analytic and holistic type interpretation - has a merely relative meaning (depending on context).
In conclusion, in this entry, I wish to demonstrate the paradoxical nature of the Riemann (Zeta 1) zeros from the holistic mathematical perspective.
Once again, using analytic type distinctions, in Conventional Mathematics, a clear absolute distinction is made as between the primes and the composite natural numbers which are connected with each other through multiplication.
So the primes have no factors (other than 1 and the prime number itself) while the composites necessarily contain two or more prime factors.
This of course also also entails that the composites necessarily contain two or more natural number factors.
Then I make the following definitions
With all primes both 1 and the prime itself are excluded as factors; however with composites, though 1 is again excluded, the natural number itself is included as a factor.
So from this perspective, 5 as prime, has no factors, while 4 as composite has 2 natural number factors (i.e. 2 and 4).
Therefore in general terms, while all primes have no factors, then all composites have 2 or more factors!
In this way a clear analytic distinction separates the primes and the composites.
If we now raise the intriguing issue of numbers that contain just 1 factor (as it were) this then confounds the analytic logic that clearly separates the primes and composites.
However, from a holistic perspective, this has an unexpected meaning, whereby the Riemann (Zeta 1) zeros are now seen as the entities that paradoxically bridge the divide that separates the primes from the composites.
So again in analytic terms, a number is either prime or composite; however in holistic terms a number is both prime and composite (when Type 1 and Type 2 aspects of number are related).
Now of course we cannot hope to understand this in a conventional mathematical manner as it is based solely on Type 1 analytic type interpretation. However we can hope to understand this in Type 3 terms (which requires the ability to recognise both the Type 1 and Type 2 aspects of number definition and then to simultaneously relate them as interdependent).
Essentially what this entails is the ability to recognise that the relationship of primes to natural numbers (and natural numbers to primes) is directly opposite with respect to both reference frames (Type 1 and Type 2).
Therefore in holistic terms (where both are now seen as interdependent) the primes and composites are seen as perfect mirrors of each other and thereby ultimately identical in an ineffable manner.
This is then beautifully reflected in the fact that the Riemann (Zeta 1) zeros, in holistic terms can be seen as those numbers with just one factor (thereby bridging the analytic divide as between the primes and composites).
Let us see more clearly how this operates.
Once again in analytic terms we start with the primes 2 and 3 (with no factors) before encountering 4 (with two factors). Then we encounter the prime 5 (with no factors) and then 6 as composite (with 3 natural number factors). Next we encounter 7 as prime (with no factor) and then 8 as composite (with 3 factors).
Thus there is considerable discontinuity here as between the primes (with no factors) and the composites (with multiple factors).
The Riemann (Zeta 1 zeros) can then be fruitfully seen as the attempt to reconcile these factor disparities as between primes and composites.
Now for proper comparison we divide each Riemann (Zeta 1) zero by 2π.
Thus the positions of the first 8 zeros (to 2 decimal places) are,
2.25, 3.35, 3.98, 4.84, 5.24, 5.98, 6.51, 6.89
Now compare this with the corresponding accumulation (up to the number 8) of the first eight natural number factors (of the composites).
So when a composite has factors we repeat that number (in accordance with its total number of factors). So 4 has 2 factors, 6 has 3 factors and 8 also has 3 factors.
So we record this as
4, 4, 6, 6, 6, 8, 8, 8,
Though we have only considered the first 8 values here, one can see how they represent a sort of smoothing of the discontinuous nature of the primes (with no factors) and the composites (with multiple factors).
So whereas the primes (with 0 factors) and the composites (with multiple factors) alternate in a somewhat uneven manner, the Riemann Zeros represent the perfect harmonisation, with each zero occurring just once.
When we match the Riemann (Zeta 1) zeros up to t and the corresponding accumulation of natural number factors up to n (where n = t/ 2π), two remarkable features are in evidence.
1) the frequency of Riemann Zeros matches very closely the corresponding frequency of natural number factors. For example up to t = 628, we have 361 zeros, whereas up to n = 100 (where n = 628/2π) we have the accumulation of 357 natural number factors.
2) the accumulated sum of the product of each composite number by its total number of factors (up to n) matches very closely the corresponding sum of the linearly adjusted Riemann zeros up to t (where again n = t/ 2π). For example, I have calculated that that this accumulated sum up to n = 100 is 201367, whereas the corresponding sum of linearly adjusted Riemann zeros (up to 628) is 20133 (an accuracy of nearly 99%).
See "Estimating Sum of Riemann Zeros (2)".
Thus this offers valuable empirical evidence for the fact that the Riemann zeros represent in fact a smoothing out of the discontinuous manner in which factors occur as between the primes and the composites.
So to repeat, the crucial feature of the holistic interpretation of the Riemann (Zeta 1) zeros is that they represent paradox in terms of analytic mathematical notions.
Thus again in analytic terms, we clearly separate the primes and composites; with the Riemann zeros, these features are holistically reconciled. Alternatively in analytic terms we clearly separate randomness and order with respect to the number system. Again through the zeros, these features are holistically reconciled. Likewise in analytic terms we separate the operations of addition and multiplication. Again through the zeros, these two operations are dynamically reconciled.