Where Science and Art Coincide (9)

I have been writing over the past number of blogs regarding the nature of the Zeta 2 zeros which basically refer to the non-trivial roots (i.e. the roots other than 1) corresponding to prime numbered dimensions.

Once again these zeros initially arise in the context of finite solutions for

ζ₂(s) =  1 + s¹  + s²  + s³  +….. + s^{t – 1} (with t prime) = 0.

So for example when t = 3, the relevant equation is

1 + s¹  + s² = 0, which has two unique solutions

i.e. s = – 1/2 + .866i and s – 1/2 – .866i


Then when extended in the (conventional) infinite manner,

 ζ₂(s) =  1 + s¹  + s²  + s³  +…..  = 0, provided we group terms in regular cycles of t.

If however we increase terms as is normally the case one at a time then the value for ζ₂(s) = 1/2 for all (non-trivial) prime numbered root values of t.

And as we have seen this value of 1/2 expresses a probable value (in the same manner as the likelihood of Heads on the toss of an unbiased coin).

Now with respect to the Zeta 1 Function, according to the Riemann Hypothesis the non-trivial zeros are all postulated to lie on the imaginary line drawn through 1/2. However strictly this value expresses a probability (with an inevitable circularity in interpretation applying).

Ultimately such probabilities reflect an act of faith that actual finite behaviour is ultimately  consistent with potential infinite behaviour.

So once again potentially for the general infinite case, the probability of Heads (on tossing an unbiased coin) = 1/2.

Now the relative frequency of the actual number of  Heads in any finite number of trials is likely to deviate from this potential result.

Thus in effect the probability notion reflects the belief that ultimate consistency as between both sets of behaviour i.e. potential (qualitative) and actual (quantitative) will be maintained in a relative approximate manner.

However we clearly cannot prove this in the standard absolute fashion (based on merely reduced quantitative notions).


Now I wish again to emphasise the huge potential significance of these higher dimensions.

Firstly they open up an entirely new holistic appreciation of mathematical symbols (based directly on qualitative rather than quantitative recognition).

Secondly they provide the crucial means by which we are enabled to make ordinal distinctions as between numbers.

Again we must be clear from the outset of two distinct notions of number.

The Type 1 aspect relates to the cardinal interpretation of number in a collective quantitative manner (where individual members lack qualitative distinction).

Thus for example, 3 from a cardinal perspective reflects a collective number quantity (as a whole unit).

Though we can represent 3 in quantitative terms as 1 + 1 + 1, these individual units (as homogeneous) thereby lack any qualitative distinction.


The Type 2 aspect by contrast relates to the ordinal interpretation of number as unique qualitative units (where the collective sum of these units now lacks any quantitative distinction).

Now the Type 2 aspect is indirectly expressed in a quantitative manner through its corresponding roots of 1 (all of which - except 1 are uniquely defined for prime roots).

So the unique qualitative units of 3 are its 1st, 2nd and 3rd members (which arise through a mutual interdependence with each other).

Expressed in an indirect quantitative manner, these are 1, – 1/2 + .866i and – 1/2 – .866i

Then when the mutual interdependence of these is expressed through their sum, the quantitative value = 0.

Thus the Type 1 aspect is based on the quantitative notion of number as independent; the Type 2 aspect - by contrast - is based on the qualitative notion of number as interdependent.


However in dynamic interactive terms independence necessarily implies interdependence and interdependence, independence respectively.

Thus we cannot meaningfully place cardinal numbers quantitatively in relation to each other without an implied ordinal aspect; likewise we cannot meaningfully rank numbers ordinally without an implied cardinal aspect. So for example we must always fix the first member of a group in cardinal terms before other members can be meaningfully related to each other.


So the enormous significance of the Zeta 2 (non-trivial roots) is that they provide the true qualitative basis for the ordinal interpretation of number.

Thus implicit in our customary ordinal interpretation of number at the conscious level of quantitative interpretation is an unrecognised unconscious aspect that enables true recognition of number interdependence in holistic terms.

Thus the Zeta 2 zeros represent the (unrecognised) unconscious basis of the ordinal number system.


Properly understood the experience of number represents the dynamic interaction therefore of conscious (analytic) and unconscious (holistic) aspects.

So the natural numbers represent the conscious (analytic) aspect; the Zeta 2 zeros represent the unconscious (holistic) aspect with respect to its ordinal interpretation.

However because of our merely reduced conventional interpretation of the number system in quantitative terms, the vital role of the Zeta 2 zeros still remains completely unknown.