Zeta Zeros and the Changing Nature of Number (3)

Yesterday we looked at the Type 1 notion of number with respect to our example of 5 chairs.

Again in this context 5 has a (reduced) quantitative meaning as 5 = 1 + 1 + 1 + 1 + 1.

However in the dynamics of understanding, 5 keeps switching from its "part" notion of 5 individual items to its "whole" notion of  1 collective group of items (and vice versa). And these are strictly quantitative as to qualitative (and qualitative as to quantitative) with respect to each other.

In this way we are able to recognise the chairs both as whole units in their own right and yet parts with respect to the single group!

Once again in conventional interpretation, this dynamic two -way interactive relationship as between whole and parts (in quantitative and qualitative terms) is reduced in an absolute quantitative manner.


So in Type 1 terms, when we say,

 5 = 1 + 1 + 1 + 1 + 1,

each of the individual units is homogeneous in nature and thereby lacking any qualitative distinction!


However there is an alternative Type 2 complementary manner of defining this relationship as,

5 = 1st + 2nd + 3rd + 4th + 5th.

In this case, whereas each of the individual units now possesses a unique qualitative distinction in ordinal terms, the collective sum of the units lacks any quantitative distinction!

Thus 5 - as indeed all numbers and mathematical symbols -  has a Type 1 analytic meaning (without qualitative distinction) and a Type 2 holistic meaning (without quantitative distinction).


Indirectly this Type 2 meaning can be converted in a Type 1 quantitative manner.

So in Type 2 terms the 5 fractions 1/5, 2/5, 3/5, 4/5 and 5/5 are expressed as,

1^1/5, 1^2/5, 1^3/5, 1^4/5 and 1^5/5 representing the corresponding meaning of 1st, 2nd, 3rd, 4th and 5th respectively (in the context of a group of 5).

Now the reason we divide by 5 is because we are attempting to express a 5-dimensional notion (i.e. the related notion of 5) in a 1-dimensional (linear) manner through which the conventional independent notion of number is interpreted!  

Now apart from the last 1^5/5 = 1, all the others have a merely relative meaning.

For example, if I identify again a group of 5 chairs and identify 4 of these chairs as the 1st, 2nd, 3rd and 4th respectively, then - by definition - the one remaining chair is unambiguously the 5th member in this case.

Therefore whenever we identify a member of a group as the nth (of a group of n) ordinal meaning is reduced in a cardinal manner.

So now 5 =  1 + 1 + 1 + 1 + 1 = 1st + 2nd + 3rd + 4th + 5th!

However if we leave the initial choice open, all ordinal positions - depending on context - can be associated with each of the 5 members.

As I explained in an earlier blog entry (the 1st of this series), when we isolate this last case as the one trivial solution, the other 4 non-trivial roots will be expressed by the equation;

1  +  s¹ +  s²  +  s³  +  s⁴ = 0

These 4 solutions, .309 + .951 i,  – .809 + .588 i, – .809 + .588 i and .309  – 951 i (correct to 3 decimal places), thereby express (indirectly in quantitative manner) the 1st, 2nd, 3rd and 4th relative ordinal positions (in the context of 5)

What is amazing here is that number is now serving a - holistic - rather than analytic role or alternatively a relative rather than absolute meaning.

Depending on the choices made with respect to position, any of the 4 results can be chosen for each of any 4 members of the group (with the 5th = 1), with the others interchanging in circular manner as required so that the overall sum of the 5 = 0.


The relative nature of what is involved can be most easily understood in the case of a number group of just 2 members.

Now what is 1st or 2nd in this group is purely arbitrary before the initial choice is made! In this sense it is similar to the quantum world  whereby a particle can exist as a superposition of states before its actual existence is determined  through making an arbitrary decision as to location.

So in potential terms, if one particular item is chosen as the 1st (i.e. whereby it is posited as the 1st) this thereby negates the 2nd item (as 1st).

However equally if the other item is now posited as 1st, then the remaining item is thereby negated with respect to this position.


We could say therefore that the two positions are represented by the two roots of 1 i.e. + 1 and – 1 .

And the sum of these roots = 0, which expresses the merely relative notions of these positions!


Now I am already using + 1 and – 1 in a holistic manner that relates directly to the (intuitive) unconscious aspect of  relative interdependence . This is in striking contrast to the corresponding analytic manner that relates directly to the (rational) conscious aspect of  absolute independence.

Though of course we can never in experience totally separate both poles as the notion of interdependence can only meaningfully start from what is already seen as independent (and vice versa).


Though the unconscious dynamics are harder to appreciate, with respect to our example of a number group of 5, we would now determine the relative ordinal positions of the 5 members through obtaining the 5  roots of 1.

In experiential terms, this enables one to give a relatively independent meaning to each ordinal position, while recognising that in collective interdependent terms they cancel each other out!

In this sense therefore ordinal meaning lies at the other extreme from cardinal.

Whereas cardinal meaning is understood in an analytic quantitative manner with numbers independent of each other, ordinal meaning is by contrast understood in a holistic qualitative manner with all numbers strictly interdependent with each other!