It is significant how the musical analogy is so often invoked in relation to the primes.
I have referred before to the fact that the Chapter in Keith Devlin's "The Millennium Problems" on the Riemann Hypothesis is entitled "The Music of the Primes".
Then the very same title is used for Marcus de Sautoy's popular book on the primes.
And Michael Berry who has done extensive work in recent years with respect to the physical implications of the non-trivial zeros says
“... we can give a one-line nontechnical statement of the Riemann hypothesis: The primes have music in them."
Indeed this musical analogy extends right back to the time of Pythagoras who discovered intimate connections as between terms in the - since titled - harmonic series and musical sounds.
And the harmonic series in turn serve as the basic foundation stone as it were on which the subsequent Euler and Riemann Zeta Functions have been built.
However what I find quite baffling, is why the obvious implications of such links as between music and the primes have been repeatedly missed in a formal mathematical context.
Probably the most universal way in which the qualitative dimension of experience is conveyed is through music. So this should immediately suggest therefore a marked qualitative dimension to the mathematical study of the primes.
There is indeed also a marked quantitative aspect with respect to the structure of music; however which however is not to be confused with its qualitative appreciation. It should perhaps further suggest that with respect to the primes that both quantitative and qualitative aspects are closely linked.
Also it is interesting to note that musical often goes with mathematical ability.
Euler for example, who made such ground breaking discoveries with respect to the primes wrote extensively about music, manufactured instruments and was also a practitioner of some talent.
However the simple fact remains that mathematicians have never really made the important connection that the "music of the primes" implies an important qualitative dimension to the very nature of Mathematics. Rather, with ever more specialised technical developments rapidly taking place in the discipline, it has become increasingly defined within a hermetically-sealed chamber (that allows no intrusion of the all-important qualitative dimension).
For some reason from an early age I was naturally able to make this qualitative connection as an obvious consequence of the multiplication process.
I have related before that in doing multiplication problems as a young schoolboy, as for example in calculating the area of a field, I quickly appreciated that a qualitative as well as quantitative dimension was involved.
So with a rectangular field we start with two sides measured in linear (1-dimensional) units. However after multiplication the area is then correctly expressed in square i.e. (2-dimensional) units.
However in the more abstract use of number in pure Mathematics, this qualitative dimensional aspect is ignored altogether with mathematical results interpreted in a merely reduced quantitative manner.
That such a fundamental issue can be glossed over with such apparent ease, despite the so-called "rigour” of pure Mathematics, therefore led me to be highly sceptical regarding the nature of mathematical interpretation. So I have since operated with an expectation – arising from this early experience - that I would find equally important fundamental issues in Mathematics that are completely overlooked.
Thus in the context of mathematics generally - and most especially the primes – there are in truth are two equally important aspects to Mathematics i.e. quantitative and qualitative with the qualitative aspect entirely censored out - in formal terms - of what conventionally passes as Mathematics.
So having accepted that there is indeed an (unrecognised) qualitative dimension to Mathematics, the next task was to see how this could be successfully incorporated in a manner going beyond mere philosophical conviction regarding its true nature.
And it eventually dawned on me that the qualitative aspect is intimately tied up with the dimensional notion of number (when given an appropriate circular type interpretation).
In short this qualitative notion of a dimensional number fits exactly from with the inverse notion of the corresponding roots of that number (in quantitative terms).
Thus, right away we can pinpoint exactly the nature of Conventional Mathematics. In qualitative terms this is defined by its linear (1-dimensional) nature and the corresponding 1 root of unity is of course equally 1. So – quite literally – from a 1-dimensional perspective, the qualitative is not distinguished from quantitative meaning with the result that the qualitative is necessarily reduced – in any given context – to mere quantitative interpretation!
And it should be already patently obvious – though it still clearly eludes the Mathematics profession - that we are never going to understand “the music of the primes” from this perspective. We may indeed obtain a reduced quantitative interpretation with respect to many features of the primes, but it will always remain like one though having an unrivalled technical mastery – say – of the structure of Beethoven’s music yet has never listened to his work (and worse still refuses to listen when invited to do so!).
So when we next move on to the number 2 (as the qualitative aspect of dimension) its structure is inversely related to the two roots of unity.
Now in quantitative terms this relates to the two separate results + 1 and – 1. (So linear always implies separation of opposites).
However circular interpretation by contrast entails interdependence. So
2-dimensional interpretation is thereby defined as the complementary of polar opposites (i.e. with positive and negative aspects).
This again directly implies that the very appreciation of interdependence requires from a qualitative perspective circular rather linear logical understanding. And once more – in formal terms – this is totally excluded in Conventional Mathematics.
So when mathematicians attempt to deal with interdependence they do so within uni-polar (separate) frames of reference, So like one who understands the two turns at a crossroads as both left, Conventional Mathematics deals with the interdependence of important opposite poles such as the general and particular (as for example the relationship of a theoretical result to individual cases) in merely quantitative terms.
Again from my standpoint it should be patently obvious that a qualitative aspect is also necessarily involved; but because of accepted (unquestioned) orthodoxy going back now more than two millennia this is again missed.
Now in a previous blog we dealt with the nature of pi. It is worth returning here to illustrate a few more significant points.
From a conventional (linear) perspective, pi is treated in a merely quantitative manner. So from this perspective pi represents the ratio of the circular circumference to its line diameter (in quantitative terms).
However as always we can give these symbols a corresponding qualitative (holistic) interpretation. So from this context pi now expresses the relationship as between circular and linear type understanding.
The two roots of unity can be geometrically expressed through drawing a circle with its line diameter with the direction to the right marked positive and that to the left negative.
This diagram likewise serves as a good illustration of 2-dimensional understanding (in qualitative terms).
So initially using linear understanding, we deal with the two opposite poles as separate. Understanding at this level does indeed seem unambiguous (with both interpreted in absolute terms as positive). However once we relate both poles as interdependent, paradox immediately arises (with respect to linear interpretation). Thus this requires a movement to a distinctive type of holistic understanding that can embrace paradox (i.e. that is circular in nature).
Therefore the very interpretation of pi itself has both a coherent qualitative as well as quantitative interpretation.
So when we understand pi correctly from a qualitative perspective, we realise that it entails both linear and circular aspects of interpretation.
And just as in quantitative terms pi is recognised as perhaps the best known example of a transcendental number, we can now likewise perhaps appreciate the qualitative notion of transcendental, which represents the dynamic relationship as between both linear and circular aspects of understanding (necessarily expressed in a reduced rational manner).
Now pi represents the pure relationship here; however all transcendental numbers entail the same basic relationship! By contrast an algebraic irrational number such as the square root of 2 represents merely the circular aspect (interpreted in a linear reduced manner).
It is not crucial at this stage to appreciate the implications of everything stated here. Rather the key requirement is to get some notion that there is indeed a valid alternative aspect to Mathematics (where every symbol that already has a defined quantitative meaning can likewise be given a distinctive qualitative interpretation).
So having defined the nature of pi in both (Type 1) quantitative and (Type 2) qualitative terms (separately as it were), we then bring both aspects together in Type 3 understanding.
And the Riemann Zeta Function requires such combined (Type 3) understanding for its proper comprehension.
Again when we define pi in 1-dimensional terms it has merely a quantitative interpretation.
Pi is referred to as a constant though its numerical value cannot be written down precisely. Though it is recognised that it is not a rational number (but rather a transcendental) this too is defined in a merely reduced quantitative manner (i.e. that cannot be the solution of a polynomial equation with rational coefficients). So though clearly rational and transcendental type numbers possess a profound qualitative dimension – even the very word transcendental would suggest as much – by the very nature of the linear (1-dimensional) approach, this aspect cannot be approached.
However when we interpret pi in a 2-dimensional qualitative context, both quantitative and qualitative aspects dynamically co-exist (in the same number as it were).
Not alone does this imply a new qualitative interpretation entailing both linear and circular elements, but - because quantitative and qualitative are now related - it also implies an important subtle change in the interpretation of the quantitative nature of pi.
Again, the two linear frames of reference come from the more refined recognition that as a particular number, pi is necessarily in relationship with its general number concept. And the circular aspect comes when we recognise both aspects as interdependent.
However, as I say, this then subtly changes the quantitative interpretation of pi.
So from this quantitative perspective, we understand pi in dynamic relative terms, as having a merely approximate value! (So strictly pi cannot now be a constant in quantitative terms).
What this means is that the actual value of pi can never be fully known, with an inherent uncertainty attaching to its sequence of digits. So to keep it simple, if we write down the digits in binary form, with even trillions of these digits calculated already, we only have a 50/50 chance of correctly predicting the next digit!
Now this might seem reminiscent of Quantum Mechanics and indeed it is! In fact the deeper roots of the Uncertainty Principle in Quantum Mechanics, spring from a more fundamental uncertainty in Mathematics (which ultimately relates to the complementarity of both its quantitative and qualitative aspects).
And when we begin to understand Mathematics from all other dimensions (except 1) the same uncertainty applies to everything.
It is indeed like moving from a Newtonian to a Quantum Mechanical Universe except that it is much more fundamental than that!.
For example when interpreted in a 2-dimensional sense (which is the most accessible of the alternative dimensions) all mathematical proof is subject to the Uncertainty Principle.
So in what sense is pi now a constant!
Well! remember at the 2-dimensional level of appreciation, pi represents a dynamic interaction of both quantitative and qualitative aspects. So it is in this combined entity as it were (that the constant value resides).
However there is always something ineffable about this situation, as we can never know what it is exactly, once we separate quantitative and qualitative aspects. And in phenomenal terms quantitative and qualitative aspects must necessarily be separated to a degree.
So its quantitative value – as befits dynamic interaction – can only be known in a relative approximate fashion.
Indeed the deeper root of the ineffable nature of pi lies – ultimately – in the ineffable nature of the Riemann Hypothesis.