(I have reworded the headings on the three blogs – that referred to the non-trivial zeros - to reflect more accurately the scope of enquiry contained therein).
We mentioned how in deriving finite values for the Riemann Zeta Function for s > 1, that a basic reductionist process is involved whereby the infinite notion is in effect treated as a linear extension of the finite.
We also commented on the fact that the value of ζ(2) = (pi^2)/6.
Now in a reduced quantitative manner, this certainly appears to be the case. However when we look more closely, we begin to realise that there is necessarily also a qualitative aspect to this result with the value of pi in truth representing a dynamic interaction as between both quantitative and qualitative aspects (with an inherent uncertainty attached to its value).
ζ(2) represents the sum of terms of 1 + 1/4 + 1/9 + 1/16 + …
If we take the actual sum of any finite number of terms in this series, we clearly get a rational answer and by taking a sufficiently large number of terms we can indeed approximate very closely the “true” value for (pi^2)/6.
However pi does not represent a rational - but rather an irrational (transcendental) - number.
So clearly a qualitative change in the nature of the number is involved in moving from a finite to an “infinite” number of terms.
In truth the infinite aspect here does not relate to quantitative measurement but rather to a potential quality that is inherent in the number quantity.
So the very nature of pi - and by extension (pi^2)/6 - is that it necessarily combines both quantitative and qualitative aspects in its very nature. Indeed this is strictly true of all numbers (when appropriately interpreted).
We can never precisely ascertain the quantitative value of pi (in actual terms). We can indeed approximate this value to any required degree of accuracy by treating it in a rational manner. However it is not in fact a rational number and this is because of a qualitative aspect (which is inherent in its transcendental status).
So though Conventional Mathematics attempts to treat pi solely as a number quantity, in truth it represents a relationship between finite (quantitative) and potential (infinite) aspects with an inevitable uncertainty thereby attached to its actual nature.
This is indirectly revealed in quantitative terms by the fact that its decimal sequence continues indefinitely (with no fixed pattern).
We can indeed see more precisely into the nature of pi by looking initially at its definition from a quantitative perspective.
So in this context pi represents the relationship between the circular circumference and its line diameter.
Now if we place a point at the centre to spearate both axes, we would create two poles with measurement to the right positive and to the left negative respectively.
However the very essence of the linear approach is to treat both poles as positive. So both directions of the radius – from the centre of the line diameter - are thereby treated in uni-polar fashion as positive, with pi then representing the ratio of the length of the circumference to its line diameter.
However both line and circle have equally a qualitative meaning where they relate to linear and circular understanding respectively.
It would be useful here to once again employ our crossroads analogy with the centre of the line representing the intersection of the two roads.
Now Conventional Mathematics – in the same manner as the treatment of the diameter in quantitative terms - gives the same designation in both directions.
So on this occasion if I travel up a road through the centre of a horizontal crossroads, I can unambiguously identify a right turn; then when later coming down the same road, I can again unambiguously identify a right turn. So using uni-polar reference frames, both turns at the crossroads are designated as right!
And again this is precisely how Conventional Mathematics necessarily operates (recognising in formal terms solely a quantitative aspect). So for example in relation to the primes when studying their individual nature, it adopts a quantitative interpretation; then when it switches direction with respect to general frequency of the primes, again it adopts a quantitative interpretation.
However when we view both turns as interdependent, they are necessarily right and left and left and right with respect to each other respectively.
Likewise when we view the two directions of mathematical enquiry as interdependent, they must necessarily be quantitative and qualitative (and qualitative and quantitative) with respect to each other.
The devastating implications of this – when properly appreciated – is that by definition, the very notion of interdependence cannot be properly interpreted within the accepted mathematical framework!
Therefore it is strictly futile even attempting to understand, for example, the two-way relationship of the prime numbers to the non-trivial zeros within the conventional linear framework of Mathematics.
And of course this implies that the very attempt to approach the Riemann Hypothesis from the standpoint of Conventional (Type 1) Mathematics is equally futile.
So the circular notion from a qualitative (Type 2) mathematical perspective relates to interdependent polar reference frames, which in their simplest form requires two poles!
So whereas the lines drawn from the centre represent linear interpretation (based on uni-polar reference frames (as independent), the circumference represents circular appreciation (based on bi-polar reference frames) as interdependent.
Now clearly both approaches are needed in Mathematics. Before we can relate two aspects together as interdependent, we must recognise them separately (as independent).
However once again by its very nature, Conventional Mathematics is inherently unsuited to the treatment of interdependence.
In fact this can be illustrated in a manner which is utterly devastating with respect to the nature of Conventional Mathematics.
The only dimensional value for which the Riemann Zeta Function is undefined is s = 1. And this is the very dimension that defines the conventional mathematical approach. So for every other value of s the Function is indeed defined.
So Mathematics can in truth be defined from an unlimited number of dimensions (each of which represents a unique manner of interpreting its symbols).
Now all these alternative dimensional interpretations entail a distinctive relationship as between quantitative and qualitative aspects of interpretation.
And this in turn is precisely why the Zeta Function can be defined for these values!
This clearly establishes – when correctly interpreted – that the Riemann Zeta Function is really about establishing a precise complementary relationship as between quantitative and qualitative type aspects, with the Riemann Hypothesis lying right at its centre as the condition necessary for establishing the ultimate identity of both aspects.
So the one interpretative system that is uniquely unsuited to proper comprehension of the Riemann Zeta Function (and its associated Riemann Hypothesis) is the 1-dimensional approach (that defines Conventional Mathematics).
So we can validly say without a hint of hyperbole, that just as in quantitative terms the Riemann Zeta Function remains (uniquely) undefined where s = 1, likewise in qualitative terms it remains (uniquely) undefined where s = 1.
And as we have seen, this once again means that the conventional mathematical approach – defined by its 1-dimensional approach - is by its very nature unsuited to the unlocking of the rich secrets of the Riemann Zeta Function (and of course the Riemann Hypothesis).
At a more fundamental level, this points directly to the unsatisfactory manner in which finite and infinite notions are treated. Though these notions represent two distinct poles (with respect to understanding), they are necessarily reduced in terms of each other from a conventional mathematical perspective.
So from one direction the infinite is treated as a linear extension of the finite as with its treatment of convergent series on the Riemann Zeta Function (where s > 1).
Alternatively the finite is treated as a linear extension of the infinite (which has great relevance for treatment of corresponding Zeta values where s < 0).
So therefore when we attempt to add a finite to an infinite number in conventional (reduced) terms the infinite number remains unchanged!
This in fact represents a fundamental mistreatment of the very notions of finite and infinite!
Properly understood the finite aspect always has an actual (specific) meaning, whereas the infinite has a potential (holistic) meaning.
And in dynamic interactive terms, the potential aspect always remains inherent within actual numbers.
We demonstrated this earlier with respect to pi.
Pi has indeed a quantitative aspect (which in actual terms is subject to uncertainty and therefore not fully determinate); however equally inherent within it is a qualitative holistic aspect, where in Jungian terms it serves as a number archetype.
So the true transcendental nature of pi is meaningless in the absence of both quantitative and qualitative aspects.
Indeed just as pi is so important from a quantitative perspective, equally - when appropriately viewed - it should likewise be seen as an important key number archetype pointing to the fundamental relationship as between linear and circular notions (i.e. independence and interdependence).
Finally it should be apparent that when we define the Zeta Function with respect to s = 2, we do indeed obtain a finite quantitative result that in actual terms is - relatively - determinate. However the expression (pi^2)/6 also necessarily contains a qualitative aspect as the relationship of circular to linear understanding.
And because we are dealing with 2-dimensional understanding, only two poles are involved.
So inherent in the very result for ζ(2) is a significant clue as to the qualitative nature of interpretation that is appropriate in this case. In other words the proper interpretation of the result for ζ(2) i.e. (pi^2)/6, entails both a linear (quantitative) and a circular (qualitative) relationship. And in this case it entails just two (real) poles of understanding (which is the precise meaning of 2-dimensional).
We will return to further exploration of these issues in future blogs.