Analytic and Holistic Interpretation of Mathematical Dimensions

Just as there are two complementary ways of expressing the Zeta 1 (Riemann) function i.e. as a sum over (all) the natural numbers and sum over (all) the primes, equally there are two complementary ways of expressing the Zeta 2 function.

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 = 1^st + 2^nd where the units are considered in qualitative terms as interdependent (i.e. interchangeable) and unique, thereby lacking any quantitative distinction. So what is 1^st in one context can be 2^nd 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 2^nd 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 + … + x^{n – 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  + …

Missing Piece of the Jigsaw

In recent posts on my related blog-site “Spectrum of Mathematics” I have drawn attention to an important “missing piece of the jig-saw” with respect to a full explanation of the nature of the Riemann zeta function.

From a conventional perspective the Riemann Zeta function is identified solely with - what I refer to as - the Zeta 1 function. 

And as is well known this function can be expressed in two ways, both as a sum over natural numbers and a product over primes.

So in general terms,

ζ₁(s) = ∑ 1/n^s   = ∏ 1/(1 – p^–s)   
            ^{n = 1            p}   

So for example, when s = 2,

ζ₁(2)  = 1/1² + 1/2² + 1/3² + …    = 1/(1 – 2^{– 2}) * 1/(1 – 3^{– 2}) * 1/(1 – 5^{– 2}) * …

= 1 + 1/4 + 1/9 + …  = 4/3 * 9/8 * 25/24 * …   = π²/6 .

However my strong contention throughout is that the Zeta function can only be properly understood in a dynamic relative manner, entailing the dynamic interaction of two related aspects, which I refer to as Zeta 1 and Zeta 2 functions.

Properly understood, this also requires two distinctive types of mathematical understanding that are analytic (quantitative) and holistic (qualitative) with respect to each other.

And the analytic aspect (in this newly defined context) relates to the notion of number as relatively independent (of other numbers) whereas the holistic aspect relates to the complementary appreciation of number as relatively interdependent (with other numbers).

So in the dynamics of understanding, the very nature of mathematical symbols keeps switching as between analytic and holistic appreciation, i.e. their particle and wave aspects - which are complementary opposite in nature.   

Therefore in conventional mathematical interpretation, a crucial distortion is at work, whereby the holistic (qualitative) aspect - in every formal context - is reduced in a merely analytic (quantitative) manner. 

When one fully grasps the significance of this observation, then it becomes apparent that the conventional understanding of number, despite all the admitted great advances that have been made, fundamentally cannot be fit for purpose.  

Whereas I refer to the Zeta 1 as ζ₁(s) - strictly ζ₁(s₁) - I refer to the Zeta 2 function as ζ₂(s), or again more accurately as ζ₂(s₂).

In general terms ζ₂(s₂) in its infinite expression  = 1 + s¹₂ + s₂² + s₂³ + …   = 1/(1 –  s₂ ).

So in fact it represents an infinite geometric series with common ratio = s₂ .

However the significance here is that each of the individual terms in both the sum over natural numbers and product over primes expressions of the Zeta 1 function, can be expressed in terms of the corresponding Zeta 2 aspect.  

In this way the Zeta 1 function, seen from one important perspective can be viewed as representing a collective sum (over all the natural numbers) or alternatively a collective product (over all the primes) of individual Zeta 2 functions.

So again to briefly illustrate, let us take the 3^rd term of the Zeta 1 function above (for s₁ = 2)!

Now, in the sum over natural numbers expression, this is given as 1/9, which can be stated in terms of the Zeta 2 function as,

{1 + 1/10 + (1/10)² + (1/10)³ + …} – 1 = {1/(1 – 1/10)} – 1  = 10/9 – 1 = 1/9.

The link here with the 3^rd natural number 3 can be shown through rearranging the denominator of each term of the expression in the following manner,

{1 + 1/(3² + 1) + 1/(3² + 1)² + 1/(3² + 1)³ + …} – 1.

And this approach can be fully generalised.

Therefore the 4^th term of the Zeta 1 (sum over natural numbers expression) = 1/16.

And {1 + 1/17 + (1/17)² + (1/17)³ + …} – 1 = {1/(1 – 1/17)} – 1  = 17/16 – 1 = 1/16.              

Again {1 + 1/17 + (1/17)² + (1/17)³ + …} – 1 

=  {1 + 1/(4² + 1) + 1/(4² + 1)² + 1/(4² + 1)³ + …} – 1.

In this way the 4^th natural number (i.e. 4) is directly associated with the Zeta 2 expression for this 4^th term (of the Zeta 1 function).

And the Zeta 2 function can equally be used in place of each individual term of the product over primes expression of the Zeta 1.

So again when s = 2, the 3^rd term of the product over primes expression (for the Zeta 1) = 25/24.

Then in terms of the Zeta 2,

1 + 1/25 + 1/25² + 1/25³ + …  = 1/(1 – 1/25) = 25/24.

And we can then show directly the link here in the Zeta 2 (with respect to the 3^rd prime = 5) by rewriting the expression in the following manner i.e.

1 + 1/5² + 1/5⁴ + 1/5⁶   + …  = 1/(1 – 1/5²) = 25/24.

Thus once again, the Zeta 1 function - both with respect to its sum over natural numbers and product over primes expressions - can be completely written as the collective sum and product respectively of individual Zeta 2 functions.

Thus in general terms, 

∑ 1/n^s   = ∏ 1/(1 – p^–s)  = ζ₁(s)  =   ∑{ζ₂(1/n) – 1}^s =   ∏{ζ₂(1/p)^s}
^{n = 1            p                                     n = 2                                p}

However, though in its own way remarkable, this formulation of the Zeta 1 function (as the collective sum and product respectively of individual Zeta 2 functions) is not yet complete.

Though we have been able to express the Zeta 1 function in two related manners (again as both the sum over natural numbers and product over primes respectively), so far internally, we have expressed the Zeta 2 function in just one way as the sum of repeated multiplied terms.

However to complete the picture, we need to show an alternative formulation for the Zeta 2 function where the value of the function, in complementary fashion, results from the sum of repeated added terms.

And this is where the recent entries on the Spectrum of Mathematics web-site have borne fruit, as they have finally made this latter piece of the jig-saw readily apparent.

In those entries, I consider - rather in the manner of the Fibonacci sequence - the unique infinite number sequences associated with the general polynomial equation,

(x – 1)ⁿ = 0.

Now in the case of the Fibonacci, the unique number sequence,

0, 1, 1, 2, 3, 5, 8, 12, 21, … is associated with the equation x² – x – 1 = 0.

Corresponding unique infinite number sequences are likewise associated with (x – 1)ⁿ = 0.

For example when n = 2, we obtain (x – 1)² = 0, i.e. x² – 2 x + 1 = 0.

And the unique number sequence associated with this equation is the set of natural numbers, i.e. 0, 1, 2, 3, 4, 5, ….

Then when n = 3, we obtain (x – 1)³ = 0, i.e. x³ – 3x²  + 3x – 1 = 0.

And the unique number sequence associated with this equation is the set of triangular numbers,

0, 0, 1, 3, 6, 10, 15, ….

There are strong links here with the binomial theorem, and indeed the diagonal rows (and columns) of Pascal’s triangle can be used as an alternative way of determining the unique number sequence associated with (x – 1)ⁿ for each natural number value of n.

So for n = 4, we obtain the so-called tetrahedral numbers,

0, 0, 0, 1, 4, 10, 20, 35, ….

Now the significance of all these number sequences in the present context, is with respect to the their (infinite) sum of their reciprocals.

So for example, in the first case when we sum the reciprocals of the natural numbers, we obtain

1 + 1/2 + 1/3 + 1/4 + …

And of course, this represents the well-known harmonic series, which is the value of the Zeta 1 function i.e. ζ₁(s₁), where s₁  = 1.

Though the sum of this series diverges to infinity, in all other cases for (x – 1)ⁿ where n > 2, the sum of reciprocals of the unique number sequences involved, converge to a finite rational number.

Furthermore a simple general pattern relates to these sums, with the value depending solely on n and given by the simple expression (n – 1)/(n – 2).

Therefore the sum of reciprocals of the triangular numbers associated with (x – 1)³, i.e.

1 + 1/3 + 1/6 + 1/10 + 1/15 + …  =  (3 – 1)/(3 – 2) = 2/1.

Now to show that this sum of reciprocals involves all the natural numbers, we can rewrite it as follows

1 + 1/(1 + 2) + 1/(1 + 2 + 3) + 1/(1 + 2 + 3 + 4) + 1/( 1 + 2 + 3 + 4 + 5) + …

Therefore the n^th term in this manner contains the sum of the 1st n natural numbers!

Then the denominators in reciprocals of number sequences for (x – 1)ⁿ, where n > 3, contain compound combinations of all the natural numbers (to n) for the nth term.

The importance (in this context) of these sums of reciprocals is that they can then be used as the alternative Zeta 2 expressions, where each individual term of the Zeta 1 - both in its sum over natural numbers and product over primes expressions - now represents the sum of additive terms with respect to the Zeta 2 infinite series.     

So again with respect to the Zeta 1 function, where s = 2, the 3^rd term of the sum over natural numbers expression = 1/9.

This can now be expressed through the alternative formulation of the Zeta 2 (representing the sum of compound natural number terms).

So we use here the unique digit sequence associated with (x – 1)¹¹ = 0,

i.e. 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 11, 66, 286, 1001, 3003, 8008, …

There is a relatively quick way of working out the terms in all these sequences based on the universal ratio of the (t + 1)^th to the t^th term = (t + n – 1)/t.

Therefore as 8008 (the last number given) is the 7^th  term, the next term = 8003 * (7 + 11 – 1)/7 = 19448.

These sequences can all be found, listed to a large number of terms at

The On-line Encylopeadia of Integer Sequences.

So the sum of reciprocals of the sequence associated with (x – 1)¹¹ = 0, is

1 + 1/11 + 1/66 + 1/286 + 1/1001 + 1/3003 + 1/8008 + 1/19448 + …

= (11 – 1)/11– 2) = 10/9.

So the 3^rd term in the Zeta 1 sum over natural numbers expression (where s = 2) =

(1 + 1/11 + 1/66 + 1/286 + …) – 1

And the denominators 11, 66, 286 represent in turn, ordered compound combinations of the first 2, first 3 and first 4 natural numbers respectively.

Then the corresponding 3^rd term in the Zeta 1 product over primes expression (where s = 2) = 25/24

This turn is associated with the alternative Zeta 2 functions based relating to the sum of reciprocals of the unique number sequence associated with (x – 1)²⁶ = 0,

i.e. 1 + 1/26 + 1/351 + 1/3276 + …

= (26 – 1)/26– 2) = 25/24.

And alternative Zeta 2 functions are available for the individual terms in the corresponding Zeta 1 functions (both in the sum over natural numbers and product over primes expressions) for all integer values of s ≥ 2.