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 = n1, this could be written as n1/log n1.
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!