As readers of this blog will know, I have been posting now for some time on the Riemann Hypothesis.
I had already become convinced before the first posting that the Riemann Hypothesis pointed to an - as yet - important unaddressed issue with relation to the true nature of number.
Indeed so fundamental is this issue that it had already become very clear to me then that not only is the Riemann Hypothesis incapable of proof within the standard form of accepted Mathematics but that its very nature cannot be properly interpreted from this perspective!
So for me the Riemann Hypothesis has served as an invaluable pathway towards a deeper understanding of the true nature of the number system. And at last perhaps the basic explanation of this nature can now be given.
The crucial starting point is that - properly understood - all mathematical notions can be given distinctive analytic and holistic interpretations. As I have repeatedly stated this is really what the Riemann Hypothesis is about i.e. the ultimate reconciliation of both the quantitative (analytic) and qualitative (holistic) aspects of number!
Therefore this distinction as between quantitative and qualitative aspects intimately applies to the very nature of number. So rather that viewing the number system in a somewhat fixed absolute manner in merely quantitative terms, we must dramatically change our perspective so as to view it in as inherently dynamic, representing the interaction of twin complementary analytic (quantitative) and holistic (qualitative) aspects.
The quantitative (analytic) aspect basically relates to the understanding of number as separate and independent, Because of the dominance of this aspect we have come to view numbers - misleadingly - as absolute fixed entities. However though this perspective is so deeply ingrained in our culture and accepted largely without question, in truth it represents but a reduced and ultimately distorted viewpoint.
In order to be related, numbers must enjoy a certain interdependence with other numbers. So numbers necessarily enjoy both an independent aspect (as separate from) and interdependent (as shared with) other members. So both aspects are of a relative - rather than absolute - nature! And it is in the recognition of this interdependence that the holistic (qualitative) aspect relates. Therefore to equally recognise both the independent and interdependent aspects of number, we must employ two distinctive means of mathematical interpretation.
Thus in the language I customarily use, we have therefore both Type 1 (analytic) and Type 2 (holistic) aspects to all mathematical interpretation.
Initially both aspects can be developed in relative isolation from each other. However, comprehensive mathematical understanding requires the combined dynamic interaction of the two aspects (in what I refer to as the Type 3 approach).
Though Type 1 and Type 2 aspects are - relatively - analytic and holistic with respect to each other, both can be given (in their isolated separate contexts) an analytic presentation which readily concurs with common sense understanding.
Because from a conventional mathematical perspective, no clear distinction is made as between analytic and holistic aspects, the natural number system is uniformly defined for example as 1, 2, 3, 4,......
However properly understood this number system has two distinct aspects (which highlights the distinction as between addition and multiplication).
So from the Type 1 perspective all natural numbers are defined with respect to a (default) dimensional value of 1
i.e. 1^1, 2^1, 3^1, 4^1,......
Thus the natural number 3 from this aspect implies 3^1. Then when we attempt to define it with respect to its individual units we get 3 = 1 + 1 + 1.
From the Type 2 perspective all natural numbers are defined - in an inverse manner - as dimensional powers with respect to a (default) base number quantity of 1.
i.e. 1^1, 1^2, 1^3, 1^4,......
Then the natural 3 from this alternative aspect imples 1^3. So when we attempt to define it with respect to its individual units we get 3 = 1 * 1 * 1.
So right away we see that the fundamental distinction as between addition and multiplication relates respectively to the two differing aspects of the number system (Type 1 and Type 2).
However as Conventional Mathematics does not recognise in formal terms the distinction as between analytic (quantitative) and holistic (qualitative) meaning it has no means of adequately dealing with this key issue.
Indeed I have come to firmly hold what might seem an outrageous position i.e. that Conventional Mathematics is simply not fit for purpose as - by its very definitions - it can provide no satisfactory means of reconciling the key notion of (qualitative) interdependence, with respect to any relationship, with that of (quantitative) independence.
The fact that mathematicians in practice do not recognise this as the no. 1 issue again indicates how deeply ingrained the reduced (i.e. merely quantitative) approach to Mathematics has now become!
Of course it is well recognised - in the context of the Riemann Hypothesis - that a deep problem exists in terms of reconciling the nature of addition with multiplication. However mathematicians attempt to view this in a merely quantitative manner whereas - correctly understood - it fundamentally relates to a prior distinction as between the quantitative and qualitative means of mathematical interpretation.
However what is truly remarkable is the existence for both the Type 1 and Type 2 analytic aspects of the number system of two hidden interpenetrating holistic systems (the full nature of which is just mindboggling in its implications).
These two systems derive from the non-trivial solutions for both the Zeta 1 and Zeta 2 equations.
Now again because Conventional Mathematics does not formally recognise the Type 2 number system, not surprisingly it can provide no real appreciation of - what I refer to as - the Zeta 2 non-trivial solutions.
However recently I have come to realise that our very ability to make consistent ordinal distinctions as between numbers, intimately depends on these Type 2 solutions (which manifest themselves as successive roots of 1). And as we have seen the prime numbered roots of 1 lead to a unique form of circular interdependence, in relational terms, that is the exact opposite of the Type 1 appreciation of primes as comprising the unique (independent) building blocks of the number system.
So the holistic (circular) complex number system as the non-trivial unique solutions (except 1) for prime numbered roots of 1, is vitally necessary in order to make consistent ordinal distinctions as between numbers. Thus the logical sequence of 1st, 2nd, 3rd, 4th etc. would have no meaning in the absence of this unique number system.
In complementary fashion the non-trivial zeros to the Zeta 1 function as an infinite set of special complex numbers is necessary to enable the very identification of natural numbers in finite terms. Without this equally unique holistic system, it would not be possible to use numbers consistently in a cardinal sense.
So putting it simply there is one holistic system of zeta zeros for the Zeta 1 Function that underlies the cardinal number system ensuring its consistency; there is an alternative holistic system of zeta zeros - whose function is yet completely unrecognised - that likewise underlies the ordinal system ensuring its consistency. Using quantum mechanical terminology, the number system therefore comprises two distinct aspects with particle (analytic) and wave (holistic) manifestations in both cases.
However in Type 3 terms - representing the most comprehensive type of mathematical appreciation - these two systems are simultaneously co-determined with respect to both their analytic and holistic aspects. So implicit in the emergence of starting cardinal notions of order are corresponding ordinal notions; and implicit in the starting emergence of ordinal notions are corresponding cardinal notions. So underlying our customary cardinal and ordinal notions of order (in any context) are two hidden holistic number systems (of complex form) that ultimately are mutually co-determined in an ineffable manner.
And herein lies the great secret of life underlying all phenomenal meaning in both quantitative and qualitative terms. However this will never be properly appreciated while Mathematics remains deeply rooted in its distorted (i.e. merely one-sided) quantitative perspective.