Sunday, August 21, 2011

The True Nature of Prime Numbers

It is essential to recognise from the onset that prime numbers have both quantitative and qualitative aspects that are of equal importance.

Now the quantitative aspect is readily recognised in Type 1 Mathematics, where prime numbers are viewed as the building blocks of the number system.
So from this perspective, we have a one way link whereby the natural numbers are derived from the primes.

However the primes have an equally important qualitative aspect in their overall holistic relationship to the natural numbers. So from this perspective we have the reverse direction whereby the primes with respect to their general distribution intimately depend on the natural numbers.

Now it might be argued that Type 1 Mathematics is intimately concerned likewise with this latter aspect to which for example the Prime Number Theorem and Riemann Hypothesis directly relate!

However as befits Type 1 Mathematics, both the individual and collective aspect of primes is approached from a merely (reduced) quantitative perspective.

Whereas in isolation both the individual prime numbers and their general distribution do indeed have a valid quantitative aspect, in dynamic relation to each other the relationship between them is necessarily quantitative to qualitative (and qualitative to quantitative) respectively.

I have illustrated this important point countless times before. However as it is so vital in grasping the essential nature of the primes, I will do so again.

If one travels up a road a turn will have an unambiguous designation (as left or right). So say for example we come to a crossroads where we have a left turn (in the Western direction) and a right turn (in the Eastern direction).

Now it is possible here to operate through recognising both turns as left in the following manner. If we mark one turn as left travelling up the road, by switching reference frames and now moving down the road, what was right from the up direction will now be left from the down direction.

So by switching reference frames and treating each situation in isolation, a merely left designation now applies to both turns.

It is precisely the same with Type 1 Mathematics. When behaviour with respect to the individual primes is adopted, a quantitative approach applies. Then when attention switches to the general nature of the primes again a - merely - quantitative approach is adopted.

Now of course where both aspects are studied in isolation this is perfectly valid.

However when we attempt to combine both aspects to show the necessary interdependence as between individual and general behaviour, then both quantitative and qualitative aspects must be formally incorporated. And this requires Type 1 and Type 2 Mathematics.

So the behaviour of primes is properly of a dynamic interactive nature, where both the individual primes and their collective behaviour (with respect to the natural numbers) are simultaneously determined.

And this clearly implies the equal importance of both the quantitative and qualitative aspects of the primes.

So the differentiation of the individual primes (in a discrete manner) cannot be abstracted from their corresponding integration with the natural numbers (that is continuous).

So the behaviour of prime numbers in correct dynamic terms is intimately related to natural life processes (which have both complementary physical and psychological aspects).

However the ultimate secrets governing behaviour are already inherent in such processes prior to all such phenomenal investigation.

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