Once again, if we take the number 3 to illustrate in cardinal terms this is treated unambiguously in quantitative terms as representing a collective whole integer. Thus if we wish to break it into constituent units i.e. 1 + 1 + 1 this must be rendered in homogeneous terms (without qualitative distinction).
However if we look on 3 in ordinal terms, it takes on a very distinctive qualitative type meaning based on its relationship with other numbers in a group.
The simplest case would involve a group of 3 members. So the ordinal notion of 3rd therefore implicitly entails setting up a relationship with the two other members in the group, which thereby can be designated in this context as 1st and 2nd.
To do this we must initially fix the position of the 1st member. Now the fascinating thing about 1st is that ordinal and cardinal meanings necessarily coincide! In other words with the 1st member we have by definition no outside context yet (with other members). So in this sense the ordinal notion of 1st must coincide with the cardinal notion of 1.
Once again this is precisely why the 1-dimensional paradigm employed in Conventional Mathematics ordinal and cardinal meanings coincide! So in effect ordinal notions are effectively reduced in cardinal terms!
So the fixing of position of the 1st member in a group implies the cardinal notion of 1.
Now this can be done in any of 3 different ways (as there is no distinction as between the three cardinal units of 3).
This means that in true circular terms we can have three distinctive arrangements of 1st, 2nd and 3rd.
In other words this ordinal approach demonstrates the true interdependence of the group (in a merely relative manner).
Therefore, each of the 3 members of the group can be 1st, 2nd and 3rd (depending on context).
So we have moved quickly here from one extreme to another.
In the cardinal definition, 3 has an unambiguous (linear) quantitative meaning that is independent of other numbers.
In the ordinal definition 3 (as 3rd) has a merely (circular) qualitative meaning that is intimately dependent on its relative relationship with other numbers (as interdependent).
So depending on context, any of the 3 individual members of the number group can be designated as the 3rd!
In 1-dimensional interpretation, as the ordinal relationship is confined to switching as between one member of a group, effectively it becomes indistinguishable from its cardinal identity.
Thus in 1-dimensional interpretation the cardinal numbers are literally represented as points (drawn at an equal distance from each other) with their corresponding ordinal identities assumed to follow from their cardinal positions.
This equally coincides with the fact that when the dimension of a number is 1, the corresponding (one) root of the number is thereby identical. When the dimension (i.e. exponent) > 1, then the structure of the corresponding roots becomes more complex. So for example when we have 3 roots, these serve, in an indirect quantitative manner, to provide the ordinal relationships between the 3 members of a prime group!
Therefore, once a number is defined with respect to any other dimensional number other than 1, we then have to draw a clear distinction as between cardinal and ordinal type interpretations with respect to number.
Once again this is of supreme importance with respect to interpretation of the Riemann Zeta Function which of course entails the natural numbers defined with respect to varying dimensional powers (i.e. s).
Now the one dimensional value for which the Riemann Zeta function is undefined is where s = 1.
From a holistic perspective, this is precisely because no distinction can be drawn here as between cardinal (quantitative) and ordinal (qualitative) type interpretations.
So this immediately suggest that - when appropriately defined - the Riemann Zeta Function, via the Functional Equation, relates to the intimate connections as between quantitative and qualitative type meaning. The significance of .5 in the context of the Riemann Hypothesis thereby relates to the condition necessary for the mutual identity of both cardinal and ordinal interpretations.
Therefore, the importance of .5 is that it represents the dimensional value relating to the Zeta 1 Function (with all the non-trivial zeta zeros presumed to lie on the imaginary line drawn through this point).
However we can demonstrate an equal remarkable significance to .5 in the context of the Zeta 2 Function (where it now represents a corresponding quantitative value).
As mentioned in the two previous blog entries we can initially define the Zeta 2 Function initially as,
ζ2(s) = 1 + s1 + s2 + s3 +….. + st – 1 (with t prime)
And for the zeta zero solutions we set,
ζ2(s) = 1 + s1 + s2 + s3 +….. + st – 1 = 0
The question then arises as to what happens is we attempt to extend the Zeta 2 series in the(conventional) infinite manner!
1 + s1 + s2 + s3 +…..
As we have seen in the simplest finite case case (the non-trivial second root of 1)
1 + s1 = 0 with s = – 1.
Thus if we insert this value in the infinite series we get
1 – 1 + 1 – 1 + 1 –……
Now clearly there are only two options here for the value of the infinite series!
If we have an even number of terms then the sum = 0!
If we have an odd number of terms the sum = 1.
As the chance of an even or odd number of terms is similar, then we can say that the probable value of the series = .5.
There is a simple identity formula we can use to obtain this value
1/(1 – s) = 1 + s1 + s2 + s3 +…..
So when we insert the value of s = – 1 on the LHS, the value on the RHS = ½ (i.e. .5)
What is remarkable here, is that the result of the formula for the first of these zeta zeros, strictly represents - not an actual - but rather a probable value!
What is even more remarkable is that the probable value of the infinite series for all Zeta 2 zeros is likewise .5.
One way of expressing this is that the sum of all non-trivial roots of 1 = – 1 and when we insert this value in the formula we get .5.
However I will demonstrate it more fully for the two non-trivial roots of the 3 roots of 1 (correct to 3 decimal places),
i.e. the solutions for 1 + s1 + s2 = 0 i.e. s = – .5 +.866i and – .5 +.866i.
Now there three possibilities with respect to the number of terms in the series here here
3n, 3n + 1 and 3n + 2. (where n = 1, 2, 3, 4,....)
If we have 3n terms the value of series in both cases = 0 (for each root value used). .
However if we have 3n + 1 terms there are two possibilities (for the two root values) with the answer in each case = 1
If we have 3n + 2 terms there are again two possibilities.
For the 1st root value, the sum of series = 1 – .5 +.866i = .5 +.866i
For the 2nd root value the sum of series = 1 – .5 –.866i = .5 - .866i
Thus the sum of these two possibilities = 1
Therefore for 6 different options (i.e. 2 root values applies to 3 different terms of series) the overall expected value = 2 + 1 = 3.
Therefore the average expected value = .5.
So remarkably for all Zeta 2 zeros (and indeed all non-trivial root values for the natural numbers), the expected value (or probable value) of the infinite Zeta 2 series = .5.
And this is what determines the real part of s for non-trivial zeros (in the complementary Zeta 1 series) where,
ζ1(s) = 0.So what constrains the non-trivial zeros of the Zeta 1 Function to lie on the imaginary line through .5 is the fact that the (probable) value of the Zeta 2 infinite Function for all its non-trivial zeros = .5.
And of course there is a circular relationship involved here, as the very reason why this value of .5 holds with respect to the Zeta 2 Function is because of its application in turn to the Zeta 1 Function.
So correctly understood, in dynamic interactive terms the value of .5 is simultaneously co-determined with respect to both Functions!
However what is important to point out is that this value of .5 cannot be interpreted in the standard absolute analytic manner.
In truth it represents a merely probable value. This in turn reflects an inherently dynamic interpretation of the series where both quantitative and qualitative aspects necessarily interact.
It is very reminiscent of the nature of interpretation within Quantum Mechanics.
Indeed at a deeper level, the very behaviour manifested by particles in quantum physical terms is rooted in the (true) inherent dynamic nature of the number system.