Sunday, February 5, 2012

Approximate Nature of Prime Numbers

I stated in my last blog that the Uncertainty Principle lies at the heart of the number system (and my extension all mathematical procedures). So my intention here is to clarify in a little more detail what this situation actually entails.

Though initially it might seem very hard to accept, the implications of Riemann's discoveries in relation to the primes is that at a complex level of investigation, our knowledge is necessarily of a merely approximate nature.

Now it might be helpful in this regard to keep relating to the analogous situation in physics. Here at the "real" everyday level of existence, objects appear to possess an objective unambiguous identity. However at the deeper subatomic levels of reality, revealed through Quantum Mechanics, objects have a merely relative approximate existence. From one perspective at this level, we can no longer clearly divorce the (subjective) observer from what is (objectively) observed. So "objects" are now necessarily seen in a dynamic interactive manner (where both polarities - external and internal - are involved).

Likewise "objects" now possess both wave and particle aspects with again both necessarily interacting in a dynamic relative manner.

So in an analogous manner at the customary "real" level of mathematical experience, numbers - such as the primes - appear to possess a clear unambiguous objective identity (in discrete terms).

However at the deeper complex level of analytical investigation this clear picture irretrievably breaks down (with the full implications of this yet to be addressed by the mathematical community).

It is customarily portrayed that Riemann somehow has managed to support the traditional "real" view of the primes (as having a discrete integer identity).

His famous formula for eliminating deviations - so we are told - opens up the way in principle to exactly calculate both the general frequency of the primes (up to any required natural number) while also precisely providing the location of each prime number.

However this is merely valid in an approximate manner and it is in the clear realisation of this point that we can only begin to appreciate the true nature of the primes at this deeper level - literally - as approximate values.

Riemann's method - which is indeed truly ingenious - provides a means from moving from the general distribution of the primes (which initially is represented by a smooth continuous curve) to gradually approximate - principally through adjustments based on the non-trivial zeros - to the original step function nature of the actual primes.

So having catered for the other small adjustments, seemingly all that is required so as to exactly pinpoint the location of the primes is to progressively add in each contribution that is made by the non-trivial zeros and through doing this zone in ever more closely on their precise location.

However the problem with this procedure is that - no matter how many non-trivial zeros are allowed to make their contribution - we will never arrive at the discrete integer values (that we are so familiar with from our "real" everyday understanding).

Now it might be said that if in principle we were to add in the contributions of the "infinite" number of non-trivial zeros that exist, then we would indeed finally obtain the integer values of the primes.

However with respect to a finite procedure we cannot - by definition - ever add in the contributions of all the non-trivial zeros, for no matter how many are included an unlimited amount must necessarily always remain excluded! In other words we can only determine any set of non-trivial primes by leaving an unlimited set of other non-trivial primes indeterminate!

So once again it can be seen here that the very belief that we can calculate integer prime values from Riemann's method represents the old fallacy of confusing infinite with finite notions (i.e. qualitative with quantitative). And as I have said all along ultimately this is what the Riemann Hypothesis is all about i.e. the need to reconcile finite with infinite notions (which properly belong to two distinctive domains of understanding).

So to be strictly accurate, the relationship which Riemann establishes as between primes and non-trivial zeros, is of a dynamic relative nature.

Thus from one perspective, starting with Riemann's general continuous function for prime distribution - after allowing for two other small adjustments - we can use the non-trivial zeros to move ever more closely to the actual location of the individual primes (and the overall frequency of such primes).

However strictly this is always of an approximate relative nature (and subject to uncertainty). So the prime numbers can never obtain a discrete integer identity through this process!

Likewise from the other perspective we can move from the knowledge of these approximate values obtained through the process to more precisely measure the frequency and precise location of the non-trivial zeros.

And of course the more we concentrate on fixing one aspect e.g. the location of individual primes, the more fuzzy our knowledge of non-trivial zeros will become. If on the other hand we try to concentrate on fixing the precise location of non-trivial zeros, the more fuzzy our knowledge of the precise location of the primes.

Part of the problem that we have in appreciating this point is that we are always starting from a pre-conceived knowledge of individual integer primes on the one hand (and a preconceived knowledge of the location of non-trivial zeros on the other).

Thus in testing Riemann's procedures we already know the answers we are seeking to obtain from the procedure.

So for example if we use his methods to predict the location of all primes up to 1,000,000 we already know the location of all these primes. So when the approximations get close to these - already known - prime integers we automatically round off the approximate values to conclude that we have exactly located the primes!

However this is essentially to miss the crucial point that the two levels of primes are of a distinct nature.

The first (i.e. of already known discrete integer values) reflects the absolute type notions of number that characterises the conventional linear approach.

Once again this is directly analogous to our identification of objects in unambiguous terms at the everyday macro level of existence (in accordance with Newtonian type physics).

The second (i.e. of merely approximate values) reflects the relative type notions that - properly understood - characterises the complex analytic view of the primes.

And of course here the analogy with the quantum level of behaviour of sub-atomic particles is especially relevant.

And as I have demonstrated, uncertainty naturally reigns here as we must always compromise to a degree as between our "particle" knowledge of the location of individual prime numbers and our "wave" knowledge of the "momentum" (or wavelength) of a non-trivial zero!

Physicists found the findings of Quantum Mechanics extremely difficult to accept (and still do). However largely on experimental grounds they were eventually led to accept its validity.

I would strongly imagine that mathematicians will find the merely approximate nature of prime numbers (at the corresponding "sub-atomic" level of mathematical investigation) even more difficult to accept. For one thing the - ultimately mistaken - belief in the absolute validity of mathematical procedures has led to a mind set that will create enormous barriers to such new acceptance.

Furthermore the same kind of experimental evidence will not be available as in physics.

So ultimately I believe a slow philosophical process will be involved where it will become gradually apparent that much mathematical interpretation in many crucial respects leaves a great deal to be desired. And in all of this the need to to recognise the unacceptable manner in which the infinite notion is persistently reduced in finite terms will be key.

And really, as I see it, this is the message that lies at the heart of the Riemann Hypothesis.

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