New Interpretations

As I have stated, there are in fact two related aspects to a comprehensive mathematical understanding.

The first is the recognised conventional quantitative approach that is fundamentally based on the linear use of reason (using unambiguous either/or distinctions).

However the second - unrecognised - qualitative approach is based on the circular use of reason (using paradoxical both/and designations).


One way of expressing the difference between both types is that polar opposites (such as positive and negative) are treated as separate in linear terms whereas they are given a complementary meaning (as interdependent) in corresponding circular interpretation.


Fascinatingly however the distinction as between these two types can play a direct role in the interpretation of - what initially appear - as merely quantitative relationships.


My initial recognition of this fact does not stem directly from the Riemann Hypothesis but rather from investigation of the Fibonacci sequence.

The Fibonacci sequence is obtained through starting with 0 and 1 and then successively adding the last term to the previous to obtain a new number in the sequence.
In this way we get 0, 1, 1, 2, 3, 5, 8, 13,.....

Now the ratio of terms in the sequence (again dividing the lst by the previous) approximates to phi = 1.618033... (the golden mean) and steadily improves as the terms in the sequence become larger.

This value in turn for phi is obtained exactly as the expression of the simple quadratic equation

x^2 - x - 1 = 0.


However strictly there are two values for x from this equation. The first is 1.618... and the second is - .618. So the question arises as to the interpretation of this second value.

After consideration of this issue it became clear to me that - correctly understood - both values (which are positive and negative with respect to each other) represent a complementary pairing.
So the correct way of interpreting this is as follows. When we designate the direction of the ratio (i.e. as last term in sequence to previous) as positive, then the corresponding direction (i.e. as previous term to last) is negative.

Therefore whereas 1.618... approximates the ratio of last to previous, - .618 correspondingly approximates the (reverse) ratio of previous to last term in the sequence.

So in this two dimensional interpretation (which befits the 2-dimensional polynomial expression) positive and negative take on a new meaning in the context of a complementary pairing.


I then realised that the use of polynomial equations could be extended to a whole range of sequences (in a similar manner to the Fibonacci).

The natural number sequence {0}, 1, 2, 3, 4,.. then provides one especially interesting example of such an important sequence.

The corresponding polynomial equation is here given as x^2 - 2x + 1 = 0.

The general form of this equation is x^2 + bx + c = 0 and we get corresponding number sequence to which it relates by successively summing - b * (the last term) - c * (the previous term term)

The natural number polynomial then factorises as (x - 1)(x - 1) = 0. So x = 1 is the ratio in both cases.

What this means in effect is that we would treat the natural number sequence in a standard linear manner (as both terms are of the same sign).

Indeed this clearly identifies the natural number sequence as the very symbol (or archetype) if you wish of linear type understanding.

So in this case it does not matter in which direction we take the last two terms of the sequence with the ratio approximating in each case (as the magnitude of the terms increases) to 1.


Having looked at the archetype of a number sequence corresponding to linear type inetrpretation, it then seemed instructive to likwise examine the corresponding archetype of a number sequence corresponding to circular type interpretation.

And the equation for this sequence is given by (x + 1)(x - 1) = 0 containing complementary positive and negative signs.

So the polynomial expression for this sequence is x^2 - 1 = 0.

The corresponding number sequence to which it relates is

0, 1, 0, 1, 0, 1,....

It is the interpretation of the ratio of this sequence that is especially fascinating as it illuminates several of the qualitative features that I have been attempting to illustrate.

1) Whereas interpretation of the natural number sequence entails pure linear understanding, this sequence correspondingly represents pure (2-dimensional) circular understanding.

Note that if we attempt to calculate the ratio with respect to successive terms (based on 1-dimensional linear interpretation) we either get 0/1 = 0 or 1/0 = ∞ (even though the equation suggests that the ratio = 1 or - 1).

However we can eliminate this problem through (correct) 2-dimensional interpretation,

In this case the ratio is either 1/1 or 0/0 (through expressing ratios with respect to number terms separated by a gap of 2).

Now the very essence of 2-dimensional interpretation is that it combines both rational and intuitive elements in equal measure.

So the first ratio 1/1 = 1, corresponds to rational interpretation (of 2-dimensional reality).
The second ratio 0/0 = 1, corresponds to direct intuitive appreciation of the same relationship giving a meaning to a ratio that cannot be properly defined in 1-dimensional terms. So the two elements (rational and intuitive) that are experientially united in 2-dimensional understanding become separated in terms of this linear sequence of numbers.

2) The actual equation gives rise to two answers + 1 and - 1. So as befits 2-dimensional interpretation, the direction in which terms are taken is important. So if with each second term we take the direction of last to previous term as positive then the corresponding direction of previous to last is thereby negative. And this is precisely what we would expect with 2-dimensional interpretation!

In fact what I have illustrated above is a simple illustration of the general equation x^n = 1 (where n = 2).

When n = 3 for example we would derive the sequence

0, 0, 1, 0, 0, 1,... so that we can only define the initial ratio (where x = 1) as the ratio of terms (separated by a gap of 3).


It would be tempting to push this even further in the attempted qualitative appreciation of imaginary numbers.

As we know the slutions to the equation x^2 + 1 = 0 are x = + i and x = - i, (where i is the square root of - 1).

Now the sequence of numbers to which this relates is

0, 1, 0 - 1, 0, 1, 0 - 1,....

Now again as this represents a 2-dimensional equation (with positive and negative polarities) a 2-dimensional qualitative interpretation is required.

In this case again taking ratios of each second term we get a ratio of 1/- 1 or alternatively - 1/1, and also 0/0.

So the clear implication is that the very essence of the qualitative interpretation of the imaginary number is that the notion of positive and negative directions is now rendered entirely ambiguous.
Also the ratio (which in each case is - 1) points to unconscious pure 2-dimensional rather than conscious reality.

So the very notion of an imaginary number qualitatively relates to the attempt to express a purely (2-dimensional) circular notion - that literally entails the dynamic negation of conscious understanding - indirectly in linear (1-dimensional) terms.

And the corresponding quantitative definition directly parallels this interpretation entailing the square root of a negative unit.


The importance of this demonstration is that similar problems of interpretation arise in defining values for the Riemann Zeta Function.

In standard linear terms the value of the sequence

1/(1^s) + 1/(2^s) + 1/(3^s) +... only converges for values of s > 1.

However Riemann extended the domain of definition for all complex values of s (except 1).

This then leads to the interesting problem where the sum of sequences (which clearly diverge in standard linear terms with no attainable sum) are now given a specific finite result.

For example when s = 0, we generate the following series 1 + 1 + 1 + 1 +.....

Now clearly in conventional terms the sum of this this series is infinite (i.e. does not converge to a finite limit).

However by a remarkable piece of alchemy we can in fact give a finite value to the series

This entails defining a corresponding eta function which generates the alternating series

1 - 1 + 1 - 1 + 1 ....

Remarkably this again relates to the simple equation we discussed above (in case of Fibonacci type sequences) where now x^n = 0.

So if x^n = 0, then - x^n = 0

Therefore 1 - x^n = 1

Thus (1 - x)(1 + x + x^2 + x^3 +......x^[n - 1]) = 1

So (1 + x + x^2 + x^3 +......x^[n - 1]) = 1((1 - x)

Letting x = - 1 as n → ∞

Then 1 - 1 + 1 - 1 + .... = 1/2

The question then arises as to the interpretation of this alternating series.

If we view these terms in 2-dimensional (circular) complementary fashion (where successive positive and negative terms form a pair) then the sum = 0.

However if we consider it in linear (1-dimensional) terms (where we break such a pairing) the sum will always relate to the remaining odd term = 1.

It would seem therefore that since two possible answers can arise that we should take the mean of both = 1/2.

So in fact the value of this eta function (s = 0) = 1/2.

However in qualitative terms the important point to observe is that this actually expresses the average of both linear and circular type calculations.

So the key thing in this context to understand is that the famed Riemann Hypothesis involves in qualitative terms precisely the same explanation.

In other words the Riemann Hypothesis can only be properly explained in terms of a condition enabling the balanced consistency of both linear (quantitative) and (circular) qualitative understanding.

Incidentally by using the eta function (for s = 0) it is then possible using a simple transformation to calculate a value for the corresponding zeta function = - 1/2.

However this involves deliberately combining finite with infinite notions.
Therefore the resulting value cannot be given a meaning in merely linear terms (suited to finite considerations).

However when we properly allow for the qualitative distinctions as between finite and infinite notions (using linear and circular interpretation respectively) then we can give this (and other negative integer values of the zeta function) an intelligible meaning.