No Longer Fit for Purpose!

The most fundamental distinction as between quantitative and qualitative notions in Mathematics pertains to the relationship as between finite and infinite!

Once again, to properly understand this relationship, finite and infinite must be defined in dynamic interactive terms.

So when the finite aspect such as a number is interpreted in a quantitative manner, then the infinite aspect is thereby of a – relatively – qualitative nature. In our understanding of number both inevitably interact, with the finite aspect directly conveyed through rational and the infinite aspect through intuition respectively. The finite is of an actual localised where the infinite – by contrast – is of a potential holistic nature.

Thus to form knowledge of the number “2” for example, we psychologically form an individual  perception of this number  (as finite) while equally relating it to the universal concept of number which – strictly – is of a potential nature i.e. as potentially applying to all specific numbers.

So both finite and infinite notions are necessarily involved in all number experience from a dynamic interactive perspective.
 

However in a reverse complementary fashion, the finite likewise has an infinite aspect and the infinite a finite aspect respectively.

This in fact is what distinguishes the ordinal from the cardinal notion of number!

So we start by giving a finite number such as “2” a cardinal meaning as a distinct individual entity with an unambiguous localised existence on the number scale.

However when we switch to the corresponding ordinal notion of 2, as 2^nd, this number has no strict meaning in the absence of a more general holistic context (involving its relationship with other numbers).

So when we have just two members of a group, the notion of 2^nd can be given an unambiguous identity in this restricted context.  However its meaning is merely relative as 2^nd in the context of a group of 3 is distinct from 2^nd in the context of 2!. And as we can define 2 in relation to any sized group, the very meaning of 2^nd is therefore potentially unlimited (as we can never exhaust the number system in finite terms).

Thus we started by attempting to interpret the number “2” as absolutely independent in a cardinal manner.

However when we switch to the ordinal notion of “2” we come to realise that its very meaning implies interdependence with other numbers in a merely relative fashion (which is potentially of an infinite nature).

Because cardinal and ordinal notions are necessarily related, this implies that properly speaking the cardinal notion of “2” is independent in a – merely – relative sense.

So in this dynamic context, the cardinal notion of 2 enjoys an independent and the ordinal notion of 2 an interdependent identity in relative terms!

Therefore we see here how each individual number possesses aspects that are quantitative (independent) and qualitative (interdependent) with respect to each other.

However this necessarily applies likewise to the number system as a whole.

Once again we are accustomed to think of the number system in an absolute independent fashion.

So from this conventional perspective, the prime numbers are the building blocks of this system, with every natural number (except 1) representing a unique combination of prime factors.

However the very notion of combining primes necessarily implies a relationship of interdependence. So for example the number 30 can be uniquely expressed as the product of 2 * 3 * 5.

Therefore, proper understanding of this unique combination entails a qualitative - as well as quantitative - aspect of interpretation!


Thus, once again when correctly understood, the number system as a whole possesses aspects that are relatively independent and interdependent with each other.

So from one perspective, the prime numbers are relatively independent in this sense (i.e. distinct from other composite natural numbers).

However they equally are relatively interdependent in the nature of their overall behaviour with respect to the number system.

So just as each individual number has both quantitative (cardinal) and qualitative (ordinal) aspects that are relatively independent and interdependent with respect to each other, the number system as a whole likewise has both quantitative and qualitative aspects that are relatively independent and interdependent with respect to each other.

Thus the prime numbers, as relatively separate entities, enjoy in cardinal terms an independent quantitative existence (as the building blocks of the number system), However, equally the prime numbers, in their unique relationship with each other (as composite natural numbers) enjoy in ordinal terms an interdependent qualitative existence (where each individual is inseparable from the holistic set of factors comprising these numbers).

Therefore, once again in dynamic terms, number independence implies a localised  quantitative existence (without reference to an overall holistic context);  number interdependence - by contrast - implies a holistic qualitative existence (where the interpretation of each individual number is inseparable from the overall holistic context to which its is related). 


And this relationship as between quantitative and qualitative (and independence and interdependence) is mediated in a complementary bi-directional manner as between the prime and natural numbers.   


Again from the Type 1 perspective, the prime numbers reveal themselves as the independent building blocks of the natural number system in cardinal (quantitative) terms.

However from the Type 2 perspective each prime number in turn is revealed as internally composed of an interdependent group of natural numbers in ordinal (qualitative) terms. Thus the prime number group of 3, for example, is necessarily composed of a 1st, 2nd and 3rd member!

So when both perspectives are combined, the direction of causation with respect to each of these partial perspectives is revealed as pardoxical. This clearly implies that ultimately both the quantitative and qualitative aspects are identical with each other in an ineffable manner (where likewise both prime and natural numbers are identical).

The significance of the non-trivial zeros - both for the Zeta 1 and Zeta 2 Functions - arise clearly from this dynamic interactive context.

 The Zeta 2 zeros provide - in an indirect circular number fashion - a ready means to give quantitative meaning to the ordinal notion of number. So for example 2nd  (in the context of  a group of 2 numbers) is obtained through the 2nd of the two roots of 1 (i.e. 1^1/2).

It thereby can be given quantitative expression as a point, – 1, on the circle of unit radius. Then 2nd (in the context of 3) is given by the 2nd of the three roots of 3 i.e. 1^2/3 and so on. Thus we can give in this manner a potentially unlimited number of quantitative interpretations to the ordinal notion of 2nd.

However these roots strictly have no meaning in the absence of the overall circular group of members to which they are related. Holistic interdependence then arises through summing the n prime roots of 1. So in the context of 2 roots, this implies + 1 – 1 = 0.

And in similar fashion we can give quantitative expression to all other ordinal numbers such as 3rd, 4th etc. together with corresponding holistic expressions of the qualitative interdependence of the sum of roots (where the number of individual members of a prime group = n and the sum of its n roots = 0). 

Thus in dynamic interactive terms, both the quantitative analytic independence of each individual member ultimately approaches full coincidence with the qualitative holistic interdependence of the overall (circular) group 0f members.
And this is what dynamically is meant by the notion of interdependence (which must necessarily be of an approximate relative nature in phenomenal number terms).


Then the Zeta 1 zeros provide - in an imaginary linear number fashion - a ready means to give a corresponding qualitative ordinal meaning to the cardinal number system (as a whole). Now more correctly in a dynamic interactive context, the notion of number interdependence here entails no remaining distinction as between both quantitative and qualitative interpretation!

So quantitative independence implies complete separation from its qualitative aspect (which however can only be approximated in a dynamic relative context). Then quantitative/qualitative interdependence implies that both meanings simultaneously co-exist (though again necessarily in an approximate relative manner from a phenomenal number perspective).


As we have seen multiplication gives rise to a qualitative aspect of number transformation (though this is formally ignored in conventional mathematical terms).

Thus the quantitative independence of the prime numbers (with respect to the natural number system) entails that these can be expressed without need for prior multiplication of other numbers).

So the Zeta 1 non-trivial zeros (in this dynamic interactive context) express the other extreme where the independence of each zero in isolation is (relatively) inseparable from the combined distribution of all these zeros on the imaginary number line.

And in a recent blog entry "Stunning Accuracy", I illustrated this remarkable feature of interdependence with respect to the (Zeta 1) non-trivial zeros, in the manner in which their absolute frequency can be predicted to an incredible degree of accuracy by a very simple general formula!

We saw how in the circular context of the Zeta 2 zeros, the quantitative independence of each individual nth root must be seen in conjunction with the overall sum of n roots (which displays their combined qualitative interdependence).


Then in the corresponding linear imaginary context of the Zeta 1 zeros, the quantitative independence of each zero (as a distinct point on the imaginary number line through 1/2), must be seen in conjunction with the combined qualitative interdependence of all zeros on the line.

Once again, this interdependence is demonstrated through the consistent predictive accuracy in absolute terms with respect to their overall distribution and then the consequent manner in which they can be used to smooth out deviations with respect to the corresponding general prediction of frequency of the primes.


So ultimately both sets of zeros are interdependent in a complementary manner (as likewise are the primes and natural numbers).

In fact the Zeta 1 and Zeta 2 zeros simply express the quantitative/qualitative interdependence of the primes and the natural numbers from two opposite directions.

In the context of the Zeta 1, this interdependence, in the unlimited set of non-trivial zeros on the imaginary number line through 1/2, is expressed with respect to the overall number system (where every natural number represents a unique composition of primes in cardinal terms).

In the context of Zeta 2, this interdependence, in a second unlimited set of non-trivial zeros on the circle of unit radius in the complex plane, is expressed with respect to each prime number (representing a unique composition of natural numbers in ordinal terms). 


Perhaps the biggest lesson we must learn is that proper interpretation of both sets of zeros cannot take place in the absence of a truly dynamic interactive approach to numerical relationships (which entails giving equal emphasis to both its quantitative and qualitative aspects).

In this context the conventional mathematical approach with sole emphasis on its quantitative aspect (in an absolute manner) is clearly no longer fit for purpose!

Indeed it never was properly fit for purpose. However, considerable success with respect to the specialised - and necessarily limited quantitative aspect of mathematical development - has blinded us to this fact now for several milennia.