In
yesterday’s entry, I attempted to illustrate the dynamic interactive context
through which the relationship as between analytic and holistic understanding
arises in experience.

I then concluded
that these two aspects are “real” and “imaginary” with respect to each other.

What is so
important to emphasise here is that every mathematical symbol can be given both
a real (analytic) and imaginary (holistic) interpretation.

So, as
commonly appreciated, complex numbers with both real an imaginary parts are interpreted
in an analytic (i.e. quantitative) context.

However
complex numbers equally can be given a holistic (i.e. qualitative) meaning and
this is the sense which I am now emphasising.

In this
context the imaginary aspect relates to the indirect rational attempt to
express holistic notions in an analytic type manner. This implies a circular type
logic (that appears paradoxical from a linear perspective).

One important
expression of such circular logic relates to the complementarity (in any
appropriate context) of opposites.

Once again
the crossroads illustration can be very helpful.

There is a
valid sense - referring to left and right turns - in which both can be given an
unambiguous identity (through reference to a single pole of direction).

Thus If I
approach the crossroads heading N, I can unambiguously identify for example a
left turn.

Then when I
approach the crossroads from the opposite direction heading S, I can again
unambiguously identify a left turn.

Thus in
terms of single independent poles of reference i.e. 1-dimensional (linear) interpretation, both of these turns are designated as left. So this represents conventional
analytic type appreciation.

However,
clearly in terms of each other where both N and S poles are considered as interdependent,
the two turns at the crossroads represent complementary opposites of each
other. So if one is designated as left, the other is necessarily right in this
context; and if alternatively the first is designated as right, then the other
is now necessarily left.

This is
2-dimensional (circular) type interpretation based on the complementarity of
opposite poles and is the minimum required for true holistic interpretation.

Now because
all mathematical activity - like the crossroad example - necessarily entails opposite
polarities (i.e. external and internal and whole and part) both analytic and
holistic appreciation are always necessarily involved with respect to
comprehensive understanding.

However
remarkably we have increasingly attempted to reduce this activity in a solely
1-dimensional (analytic) type manner.

So the
position with Conventional Mathematics is exactly akin to one who attempts to
maintain that the two opposite turns at the crossroads are both left.

In other
words Conventional Mathematics attempts to view all interdependent
relationships in a merely quantitative (analytic) manner, where in fact correctly
they should be viewed as quantitative (analytic) to qualitative (holistic)
respectively.

Now going
back to the primes, at the beginning of this present series of entries, I
illustrated that the notion of “prime randomness” can in fact be given two
complementary interpretations, which thereby implies that the very relationship
as between the primes and natural numbers entails both quantitative and
qualitative aspects.

However the
very recognition of such complementarity requires holistic appreciation. Not
surprisingly therefore as such appreciation is formally excluded from
conventional interpretation, mathematicians continue vainly to attempt (like
one who can only identify left turns at a crossroads) to interpret the relationship
between the primes and natural numbers in a merely quantitative - and thereby
reduced - manner.

So the first
notion of prime randomness relates to the distribution of the individual primes
(as independent) numbers within the collective natural number system.

However the
second notion of prime randomness - which has been ably demonstrated through
the Erdős–Kac
theorem - relates to a complementary notion
of prime randomness.

So here we
are considering a collection of primes (i.e. distinct prime factors) with
respect to each individual natural number.

Once
one recognises the complementarity of both of these definitions of prime
randomness, then this clearly suggests that we can no longer view the
relationship between the primes and natural numbers in a merely quantitative
manner. Again this would be akin to persisting in identifying both turns at the
crossroads as left!

Rather we
now clearly see that the relationship between both is quantitative as to
qualitative (and qualitative as to quantitative) respectively.

In other
words within each frame of reference (taken separately) we can indeed attempt
to interpret the relationship between the primes and natural numbers in a
quantitative analytic manner.

However crucially,
ultimately both of these interpretations are mutually interdependent. So here
we now properly realise the truly holistic synchronistic nature of the number
system.

Put another
way, the randomness of the (individual) primes with respect to the (collective)
natural number system, cannot ultimately be viewed as independent of the
randomness of the (collective) prime factors with respect to each (individual) natural
number.

In other
words both these aspects of random prime behaviour mutually depend on
each other so that ultimately the very nature of the number system is seen to be determined in an ineffable synchronistic manner (that is utterly mysterious).

This equally
implies that the very notions of “randomness” and “order” with respect to the
number system are themselves fully complementary notions (with a merely relative
meaning).

So randomness
with respect to individual behaviour implies corresponding order with respect
to collective beahaviour and vice versa so that both aspects mutually depend
upon each other in a dynamic interactive manner.

Indeed these
features can be even more dramatically illustrated!

Though the
individual primes are distributed as randomly (as is relatively possible)
within the collective natural number system, a corresponding order applies to
the general nature of this prime distribution.

Most simply
this can be expressed as n/log n which estimates the frequency of primes among
the natural numbers (which becomes relatively ever more accurate with
sufficiently large n).

However
there is an equally important complementary expression of this relationship
which is not properly recognised.

Just as we can
have primes and natural numbers (as Type 1 base quantities) equally we can have
primes and natural numbers (as Type 2 dimensional qualities).

In other
words when primes and natural number numbers are used to represent factors they
relate strictly to the dimensional aspect of number (with a relatively qualitative
as to quantitative relationship with respect to corresponding base numbers).

Some time
ago I began to investigate the relationship as between the (distinct) prime and
natural factors of a number.

Now again
to illustrate with the number 24, the distinct prime factors are 2 and 3, whereas
the natural factors are 2, 3, 4, 6, 12, and 24. So in effect the natural number
factors represent all the natural numbers (except 1) that will divide into a
number. However, with respect to primes we do not admit any natural number
factors!.

Now I
quickly discovered that the average distribution of prime factors per number is
given as log n. (Indeed log n – 1 is more accurate but for large n, this
distinction can be ignored!)

Also the Hardy-Ramanujan Theorem postulates that the average number
of distinct factors for a number n would approach log log n (which
approximation would improve for large n).

This entails that if we were to attempt to obtain the ratio
of natural number to (distinct) prime factors, it would be given as log n/log
log n (for large n).

Thus if we let log n = n

_{1}, this could be written as n_{1}/log n_{1}.
Thus we have now two prime number theorems with a similar
format. The first expresses the approximate number of primes up to any natural
number; the other expresses the ratio of natural number factors to (distinct)
prime factors for any number.

Now once again, properly these must be understand in a
dynamic complementary manner.

So the distribution of primes (as numbers and factors) in
each case with respect to the natural numbers are intimately dependent on each
other. So once again the ultimate two-way relationship between the primes and
natural numbers is if a holistic synchronistic nature.

Another fascinating observation can be made!

In the first case i.e. frequency of primes, an additive
relationship connects the primes with the natural numbers.

Therefore the natural numbers comprise the sum of the primes
and the remaining (composite) natural numbers.

However in the second case a multiplicative relationship
connects the prime factors with natural number factors. So the latter formula expresses
the times we must multiply the number of (distinct) primes to obtain the corresponding number of natural (number) factors.

So this beautifully expresses the point that the key relationship as between addition and multiplication is itself as quantitative as
to qualitative (and qualitative as to quantitative) respectively!

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