It struck me today that it may be misleading to continue referring to the Riemann Zeta Function and what it means, as I am offering an enlarged and - ultimately - very different interpretation than that proposed by Riemann.
Both Euler and Riemann of course operated within the accepted standard quantitative framework of Mathematics (which I refer to as Type 1 Mathematics).
However my basic standpoint is that Mathematics - when appropriately understood - equally possesses an (unrecognised) qualitative aspect. So every number, symbol, sign relationship, hypothesis etc. that can be given a quantitative type interpretation in Type 1 terms, can equally be given a coherent qualitative interpretation from a holistic mathematical perspective (which I term Type 2 Mathematics).
Then, finally my position is that truly comprehensive mathematical understanding requires the harmonious integration of both Type 1 and Type 2 aspects (which I refer to as Type 3 Mathematics).
So my new treatment of the Riemann Zeta Function - which perhaps more accurately could be referred to as the Zeta 3 Function - is intended as representing a preliminary introduction to Type 3 Mathematics.
So Euler first defined the Zeta Function in quantitative (Type 1) with respect to real values of the function (where s > 1).
Riemann then extended the Zeta Function – again in quantitative (Type 1) terms - with respect to all complex values of the function (except where s = 1).
However I am now proposing a radical further extension – incorporating both quantitative (Type 1) and qualitative (Type 2) aspects, with respect to all complex values of the Function (except where s = 1). And as subtle complementary relationships on a variety of different levels connect both quantitative and qualitative interpretations of s, then in effect this thereby represents an exercise in Type 3 Mathematics.
So to put some perspective on where we have travelled with the Zeta Function, we have started on the real axis with the intention of explaining the nature of both quantitative and qualitative type interpretation for values of the Function (where s > 1).
Though from a quantitative perspective we can readily accept the results already obtained by Euler for the Function, even here a more careful subtle interpretation is required, where in actual terms all values obtained correctly represent relative finite approximations (to an unknowable “true” value). This is precisely because the “true” value now is seen to represent the dynamic interaction of both quantitative and qualitative aspects of interpretation).
However our main focus here is – necessarily – on the unrecognised qualitative aspect of interpretation.
So, all quantitative values of s (on the RHS for s > 1) are initially defined in a linear manner i.e. in Type 1 terms, which accords in qualitative terms with a default 1-dimensional interpretation.
However the appropriate qualitative interpretation of s must be taken in accordance with the dimensional value of s that applies in each case to the Function.
So in yesterday’s blog entry I spent some time clarifying the nature for s = 2, of what 2-dimensional interpretation in qualitative terms precisely implies.
One startling conclusion that immediately arises, in the context of the newly defined Zeta Function, is that Type 1 Mathematics is uniquely unsuited to appropriate clarification of what is involved.
Because Type 1 Mathematics – in formal terms – is solely defined with respect to its quantitative aspect, it therefore has no means of correctly representing an interpretation that – by definition – is based on the incorporation of both quantitative and qualitative aspects.
So, just as The Riemann Zeta Function remains undefined for s = 1, in quantitative terms, the new Zeta Function remains undefined for s = 1, in a corresponding qualitative manner.
And what this simply means is that Type 1 Mathematics again is uniquely unsuited to interpretation of this newly defined Function.
This further implies that it is futile to seek a proof of the Riemann Hypothesis using the standard Type 1 approach.
In the new Zeta Function, the Riemann Hypothesis represents the key condition for obtaining the mutual identity of both quantitative and qualitative aspects of interpretation. Obviously an approach that does not recognise the qualitative aspect is thereby not fit for purpose!
Indeed just as we demonstrated yesterday that the key limitation of Type 1 Mathematics ultimately relates to an unsatisfactory interpretation of finite and infinite notions, this same problem is endemic in all conventional (Type 1) mathematical proof.
Correctly understood finite and infinite represent distinctive notions which cannot be reduced in terms of each other. The infinite represents a holistic notion relating to a (universal) potential meaning whereas the finite represents a specific notion relating to a (partial) actual meaning. However though they cannot be properly reduced in terms of each other, they do necessarily continually interact in dynamic fashion.
So when appropriately viewed the infinite is seen as always inherent within actual phenomena (which of course includes numbers). Now, with respect to the new Zeta Function, the only case where this not occur is where s = 1 (in both quantitative and qualitative terms). So here, there is a disassociation of both finite and infinite, where from one perspective the infinite notion is treated as a linear extension of the finite, or from the opposite perspective the finite notion treated as a linear extension of the infinite.
And it is this latter case that intimately applies to the nature of mathematical proof.
If for example I say that the Pythagorean Theorem has been proved, this correctly implies that this has been established in general terms (as potentially applying to “all” cases). So, essentially this relates to a qualitative type conclusion that is infinite in scope.
However application of a proof must necessarily apply in a limited number of actual cases (entailing quantitative meaning of a finite nature).
So right at the heart of mathematical proof (in Type 1 terms) we have a basic confusion of quantitative with qualitative meaning. And this is inevitably the case as the very basis of such Mathematics operates on the reduction – in any context – of qualitative to quantitative type interpretation.
If we look at our crossroads analogy again we can perhaps see the problem more clearly.
Within isolated reference frames we can give a general proof a valid quantitative interpretation; then again in a particular situation we can give its application in particular cases a valid quantitative type interpretation. However in mutual relation to each other, both the general and particular are as quantitative and qualitative (and qualitative and quantitative). So the situation in Type 1 Mathematics is like the person who cannot see that the turns at a (horizontal) crossroads are necessarily left and right and right and left with respect to each other. And the problem again relates to isolated frames of reference (which defines the 1-dimensional approach).
Thus we can conclude from a Type 3 perspective that all mathematical proof is of a relative nature (and subject to uncertainty). And this uncertainty, which is akin to the particle and wave nature of sub-atomic particles, arises out of the complementary quantitative and qualitative type nature of mathematical understanding.
Indeed the deeper root of the Uncertainty Principle within Quantum Mechanics arises from this more fundamental uncertainty at the heart of Mathematics!
Mathematical proof when correctly viewed represents but a special form of social consensus among the mathematical community who agree to share the same basic assumptions. However the very fact that I am strongly questioning here the nature of such assumptions bears testament to the fact that it is of a merely relative (rather than absolute) nature.
However if one accepts the limited assumptions of Type 1 Mathematics (which is indeed valid for dimensional 1 interpretation) then most propositions such as the Pythagorean Theorem do indeed have a valid proof that appear absolute from that perspective.
However one key implication of the new Zeta Function is that Type 1 Mathematics is uniquely unsuited to clarification of the very nature - not alone proof - of the Riemann Hypothesis. Of course it is still highly valuable as a means of clarification of the quantitative features with respect to the Zeta Function. However as the central issue properly relates to a prior relationship as between quantitative and qualitative type meaning it is clearly not appropriate for this task.
So all mathematical proof within a Type 3 mathematical perspective is of a merely relative nature (and subject to uncertainty).
Within the narrow confines of Type 1 Mathematics, most propositions are indeed capable of proof. However the Riemann Hypothesis which requires a Type 3 perspective for its adequate interpretation, clearly does not belong to this category!