I explored the qualitative nature of number as dimension in previous blogs especially with respect to 2 and 4.
However the more general point I was making is that associated with every number as dimension is a unique holistic qualitative manner of overall mathematical interpretation.
Now the basic nature of all these dimensions is that both circular and linear understanding are combined in a manner (that is unique for each dimension). Well strictly, I should say unique for prime number dimensions. So we saw in relation to the interpretation of 4 that two of these dimensions (i.e. relating to the real polarities) had already been encountered in dealing with 2 as dimension.
When one considers then that Conventional Mathematics is - in formal terms - confined to interpretation with respect to the qualitative nature of 1 as dimension, one begins to appreciate how limited the scope of such mathematical understanding truly is.
In fact - what we conventionally term - Mathematics represents in fact the extreme special case, where quantitative is divorced from qualitative meaning. For all other dimensions, both quantitative and qualitative are combined with each prime dimensional number representing a unique configuration with respect to the relationship between the two aspects.
However as qualitative and quantitative are necessarily complementary, this means that that we can equally define an unlimited set of alternative number systems where again each prime dimensional number defines a unique system.
So again Conventional Mathematics represents the extreme case where number is defined merely with respect to its quantitative aspect.
However we have seen how number can equally be given an ordinal (qualitative) meaning.
So the initial starting point in coming to appreciation of these alternative number systems (which comprise the LHS of the Riemann Zeta Function for values of s < 0) is that number is now used in a manner where various configurations with respect to both cardinal and ordinal interpretation are used.
So the common sense notion of number (consistent with Type 1 Mathematics) actually implies a pure cardinal system of interpretation. So when we are led to quickly see that the sum of a series such as 1 + 2 + 3 + 4 +..... diverges, it is because we are interpreting number is a pure cardinal (quantitative) manner.
However we could equally give a coherent meaning to the same series as
1 + 2 + 3 + 4 + ..... = 0 by interpreting terms in a pure ordinal sense.
For example is these numbers are now used to represent for example the successive roots of 1, the sum of thee roots (in cardinal terms) = 0 with 1, 2, 3, simply used to qualitatively rank their respective order.
We could also again with respect to roots define the number expression
1 * 2 * 3 * 4 * ... * n = 1 representing the fact that the product of the n roots of 1 = 1.
Indeed if we now view the sum of the natural numbers from a pure ordinal (rather than cardinal perspective) we can validly say that,
1 + 2 + 3 + 4 +.... = 1.
In other words, we are here defining number in a pure Type 2 sense where the expression means that though the numbers are being added, the qualitative dimension from which they are interpreted (i.e. 1) remains the same.
So more fully we are viewing,
1^1 + 2^1 + 3^1 + 4^1 +.... with respect to the unchanging dimensional number (rather than the base quantities)
So in giving these few examples, we have already given the sum of the natural numbers 1 + 2 + 3 + 4 +....., 3 distinctive interpretations.
However in the latter two cases we are dealing with pure ordinal type meaning which universally applies (in these respective contexts). So no variation from a quantitative perspective thereby takes place.
However when we look at the result of the Riemann Zeta Function where s = - 1, we once again get yet another sum of the natural numbers,
i.e. 1 + 2 + 3 + 4 +..... = - 1/12
So in this case a non-trivial quantitative result emerges.
What this suggests therefore is that we are no longer dealing with interpretation of number, in either a pure cardinal or ordinal sense, but rather in a manner where elements of both types of appreciation are combined.
And once again associated with each negative number as dimension is a unique quantitative finite expression that emerges from the Function.
Now interpreted from a strict cardinal perspective, all these results would diverge to infinity.
However the important thing to remember is that this cardinal interpretation is associated solely with 1 as a dimensional number.
So the very reason why we are getting non-intuitive numerical results from the Function on the LHS, is that we are now interpreting number in terms of the exact configurations (of cardinal and ordinal respectively) that are associated with each dimensional number.
This means in effect that each result for the Function (for s < 0) reflects its own unique manner of interpretation.
Thus for example the result of the Function for s = - 3 i.e. 1/120 in this case represents the unique configuration of cardinal and ordinal aspects that are associated with - 3 as a dimension.
Now when we used 3 as a dimensional number on the RHS for s > 1, we were able, in all cases to derive intuitively meaningful finite results by treating number in a merely cardinal manner!
So from a qualitative perspective all these results can be expressed - literally - in a reduced linear manner (as merely quantitative).
However this is clearly not the case on the LHS (when negative values for s are used).
So what we require here is a switch from the Zeta 1 Function to the Zeta 2, which is the true home for the mix of both quantitative and qualitative type appreciation from which the numerical results are derived.
So we will look carefully in future blog entries at how this process unfolds.