And in this entry, I wish to probe deeply the precise nature of the latter two complementary expressions.
The starting point here is remarkably simple, though quickly becomes much more intricate.
Let us start with the simplest of all expressions viz. x = 1.
Then by the standard laws of conventional algebra, x – 1 = 0.
However if we now square both sides of each expression, something strange happens.
For in the first case, x2 = 1, so that x2 – 1 = 0; however in the second case, (x – 1)2 = 0.
And the latter equation, when expanded is x2 – 2x + 1 = 0.
So though starting with two similar expressions i.e. x = 1 and x – 1 = 0, we quickly find through squaring both, that two new distinct expressions emerge.
And, remarkably what has really happened is that the two final expressions i.e. x2 – 1 = 0 and x2 – 2x + 1 = 0, relate to two distinctive notions of number that are circular and linear with respect to each other. Using a fruitful analogy from quantum physics, they represent thereby the wave and particle aspects, respectively, of number.
However the deeper implications here is that the understanding of number itself can only be properly understand in a truly interactive manner entailing both analytic (quantitative) and holistic (qualitative) appreciation that keep switching in the dynamics of experience.
And Conventional Mathematics is completely unsuited to this new form of understanding as it reduces (in every context) the holistic (qualitative) aspect of understanding in a merely analytic (quantitative) manner.
Therefore to understand properly what happens when we square our original expressions i.e. x = 1 and x – 1 = 0 we must allow for two distinct aspect to the number system, I refer to as Type 1 and Type 2 respectively.
Thus expressed in Type 1 (analytic) terms the natural numbers are defined as,
11, 21, 31, 41, …
In other words, they are defined in linear (1-dimensional) cardinal terms as fixed independent quantitative entities i.e. as points on the real number line.
So the natural numbers are defined with respect to a base number that varies against a fixed (default) dimensional number = 1.
However in Type 2 (holistic) terms the natural numbers are defined in circular (multi-dimensional) ordinal terms as relatively interdependent qualitative relationships entailing the unique sub-units of each number,
11, 12, 13, 14, …
So for example from the Type 1 perspective 2, i.e. 21 = 1 + 1, where the units are considered in quantitative terms as independent and homogeneous, thereby lacking any qualitative distinction.
However from the Type 2 perspective 2, i.e. 12 = 1st + 2nd where the units are considered in qualitative terms as interdependent (i.e. interchangeable) and unique, thereby lacking any quantitative distinction. So what is 1st in one context can be 2nd in another related context (and vice versa).
Now the clue to what truly happens when we square the expression x = 1 (i.e. x1 = 11) is that we now switch directly from the Type 1 to the Type 2 system.
So again in conventional terms, when we square both sides x2 = 1 (i.e. as a number still interpreted in Type 1 quantitative terms). So as this conventional mathematical interpretation ignores the qualitative aspect, 12 is thereby reduced in a quantitative (1-dimensional) manner as 11.
However, properly understood x2 = 12 (i.e. as a number now interpreted in a Type 2 qualitative manner).
Then when we square x – 1 = 0, we now interpret in complementary fashion this relation in a Type 1 quantitative manner.
So (x – 1)2 = 0, i.e. x2 – 2x + 1, has two linear roots i.e. + 1 and + 1 respectively (as the same two points on the real number line).
However x2 = 12, has two circular roots i.e. + 1 and – 1 respectively (as two points on the unit circle).
In fact what we have here are two distinct mathematical notions of dimension that are analytic and holistic with respect to each other.
The 1st linear notion of dimension is the one that is conventionally recognised in mathematics.
So 12, can be geometrically represented in 2-dimensional terms as a square (with side 1 unit).
So if one side represents the length, the other side represents the width.
So the two roots of the equation (x – 1)2 = 0, i.e. + 1 and + 1 represent thereby both the length and width respectively of the square of 1 unit.
And by extension the three roots of the equation of (x – 1)3 = 0, i.e. + 1, + 1 and + 1 represent the length, width and height respectively of a cube of 1 unit.
And though we cannot envisage this in pictorial terms the n roots of the equation (x – 1)n = 0 represent the n sides respectively of a hypercube of I unit.
However the true nature of the 2nd circular notion remarkably, is not properly understood in conventional mathematical terms as it is in fact directly associated with an entirely distinctive holistic form of dimension.
Of course, circular notions e.g. with respect to the various roots of 1 are indeed recognised, but invariably in a merely analytic manner (where they are considered as separate from each other).
However the true holistic notion of dimension requires that the various roots of 1 - as indirect quantitative representations of qualitative notions - be interpreted in an interdependent manner (where they are understood as interchangeable with each other).
I will illustrate this again briefly with an oft-quoted example regarding the interpretation of a crossroads.
Now when one approaches a crossroads along a straight road - say heading N - then a left turn for example has an absolute unambiguous meaning.
This is because the frame of reference i.e. the direction of movement, is one-dimensional. So there is only one direction considered here in terms of approaching the crossroads i.e. N.
So we can unambiguously identify the left turn in this context as + 1 with the other right turn (which by definition is not a left turn) thereby as – 1.
Thus + 1 (a left turn) and – 1, as the two conventional roots of 1, carry here a strictly analytic meaning.
Now, if alternatively we were to approach the crossroads from the other direction (heading S) then again left and right turns can be given an unambiguous meaning represented as + 1 and – 1 respectively (as the interpretation is again 1-dimensional with only one direction of approach to the crossroads considered).
However if now consider the approach to the crossroads simultaneously from both N and S directions, then circular paradox is clearly involved for what is left from one direction is right from the other; and what is right from one direction is left from the other.
So in numerical turns what is + 1 from one direction (e.g. a left turn), continually switches to – 1 from the other (i.e. right) turn and vice versa.
Thus what we have here is a holistic 2-dimensional interpretation of left and right (i.e. + 1 and – 1) which are fully relative and thereby interchangeable with each other.
Now whereas 1-dimensional interpretation from one fixed reference frame is absolute and analytic, 2-dimensional interpretation (from two polar reference frames simultaneously) is by contrast relative and holistic in nature.
We could validly equate then 1-dimensional with (linear) rational and 2-dimensional with (circular) intuitive interpretation respectively.
However indirectly we can give intuitive appreciation an indirect rational interpretation in a paradoxical logical fashion.
So whereas with linear logic opposite polarities such as + and – are clearly separated, with circular logic, + and – are understood as fully interdependent (and thereby interchangeable) with each other.
Now the importance of this is that all experience is conditioned by polarities such as external and internal and whole and part that continually interact in dynamic fashion with each other.
This intimately applies also to mathematical understanding.
Therefore through conventional mathematical interpretation is based on the assumption of the abstract independent existence of “objects” (such as number), strictly these have no meaning apart from subjective mental constructs that are used in their interpretation. And both objective and subjective aspects are thereby external and internal with respect to each other.
However once again conventional mathematical interpretation is based on the misguided belief that “objects” such as numbers can be properly understood in an external (1-dimensional) manner possessing an absolute quantitative identity.
And in general, n-dimensional interpretation from the holistic perspective entails highly refined interdependent relationships entailing n interchanging reference frames.
However, though not strictly valid, the corresponding analytic approach attempt to give independently viewed objects a succession of higher dimensions in space.
Though in experiential terms, it is not possible to go beyond 3 space dimensions in this manner, the extension of the linear notion of dimensions can then be abstractly extended to n dimensions.
So the key point again regarding the two complementary expressions of the Zeta 2 function is that they relate to the analytic and holistic notion of dimension respectively.
Thus the infinite sum of reciprocals of the unique numbers associated with (x – 1) n corresponds to the analytic (linear) notion of dimension (envisaged as n independent linear directions in space).
The complementary approach relates to xn – 1 or 1 – xn = 0. Then to get rid of the one linear dimension, we divide by 1 – x to obtain 1 + x1 + x2 + x3 + … + xn – 1 = 0.
This then represent the finite expression of the Zeta 2 function.
And then for the geometric series expressions used to define each term of the Zeta 1 (Riemann) function, we use the infinite version of this function, i.e.
1 + x1 + x2 + x3 + …