As we saw yesterday, the Zeta 2 Function provides the means of internally defining the interdependent nature of integers in a consistent manner.
So the individual ordinal members of each integer are thereby designated in a circular fashion through obtaining the corresponding roots of 1. And this serves as the appropriate way of representing such interdependence.
Ultimately as each integer becomes very large, the relationship as between individual quantitative and overall collective components become so dynamic that they approach total identity with each other in a seamless manner.
The primes have a special significance in this context as their individual members are uniquely defined.
So if p is a prime number, the p roots of 1, establish the relationship between its natural number members from 1 to p respectively and the overall relationship between its p members (in a collective qualitative sense).
The relationship here as between each prime and its natural number members is as quantitative to qualitative (and qualitative to quantitative) respectively.
All this is a valuable preparation for understanding the essential role of the non-trivial zeros with respect to the Zeta 1 Function.
Here, we adopt a complementary focus with respect to the relationship between the primes and natural numbers in an external manner where now - in reverse fashion to Zeta 2 - the prime numbers appear as the building blocks of the natural number system (in cardinal terms)
In the Zeta 2 approach each prime number is identified internally with its natural number individual components (in an ordinal manner). Here with the Zeta 1 approach, by contrast, the (entire) natural number system is identified externally with its individual prime number components (in cardinal terms).
So when we understand the relationship between Zeta 1 and Zeta 2 in an appropriate dynamically interactive manner, we can see clearly how they exactly mirror each other as complementary partners.
We saw in earlier blog entries that when the Type 1 approach attempts to “break up” the internal nature of an integer that a crucial problem arises.
For example we attempt to the define the number “3” internally (in terms of its individual number components) as 1 + 1 + 1. However this merely quantitative approach strictly leaves us with no means of providing an ordinal ranking to the numbers. In other words the very ability to clearly distinguish - in any context - a 1st, 2nd and 3rd member, requires giving the numbers some qualitative distinction!
So a new circular number system is required to uniquely define the members of an integer group in a qualitative manner.
A similar problem arises when we try to "break up" the external nature of natural number integers.
For example if we now attempt to define the number 6 externally (in terms of its prime number components) in Type 1 terms, we will be told that it is uniquely defined in terms of its two prime factors i.e. 2 and 3 in a quantitative manner.
So 6 = 2 * 3 in quantitative terms!
However this - apparently simple – multiplication process, once again conceals a crucial difficulty.
The individual prime numbers here i.e. 2 and 3 respectively are originally defined in a linear (1-dimensional) manner. Thus, quite literally we can geometrically represent both numbers as points on the real number line.
However when we now multiply both numbers, an important qualitative - as well as quantitative - transformation takes place.
So in quantitative terms, the answer is indeed 6! However strictly we have now switched from a 1-dimensional to a 2-dimensional number expression!
This can easily be represented by representing the product of the two numbers in terms of a rectangle with side measurements of 2 and 3 (linear) units respectively.
Therefore, whenever we multiply prime numbers (to obtain composite natural numbers) both a quantitative and qualitative transformation is entailed.
However due to the reduced quantitative nature of the Type 1 approach, such qualitative change in the variables is simply ignored.
So in Type 1 terms, 2 * 3 = 6 is interpreted in a reduced quantitative fashion (where implicitly 6 = 6^1). In other words - though strictly a qualitative transformation in the dimensional context has taken place (i.e. 2-dimensional) - the result is misleadingly still interpreted in linear (1-dimensional) terms.
This is why I repeatedly characterise the very nature of conventional Mathematics as 1-dimensional (in qualitative terms).
Of course I recognise that Conventional Mathematics recognises dimensional transformations of number; however it does so merely from a (reduced) quantitative perspective!
Now there is indeed at some level, a recognition of a problem here among professional practitioners.
This is usually expressed in terms of the uneasy relationship that exists as between addition and multiplication.
Thus from the additive perspective we could represent a number such as 6 as
1 + 1+ 1 + 1 + 1 + 1.
However equally we could represent it as 2 * 3.
Unfortunately, there does not appear from this quantitative perspective any obvious way of reconciling the two viewpoints.
However the fundamental issue is so obvious that I am amazed at why it is not clearly seen.
Indeed this very issue caused my earliest disillusionment with the conventional approach when I was 10 years old.
It is that experience that has informed my subsequent development, so that I have never felt unduly bound by conventional mathematical wisdom (which in many ways remains blind to fundamental problems of interpretation).
Putting it simply, wherever two numbers are multiplied (or a number raised to a power or exponent) a qualitative - as well as quantitative - transformation is involved.
So, that in a nutshell is the fundamental issue with respect to reconciling addition with multiplication. It can only be done so in an approach that explicitly recognises both the quantitative and qualitative aspects of mathematical appreciation (which ultimately are fully complementary with each other in a dynamic manner).
Thus the Fundamental Theorem of Arithmetic i.e. that very natural number integer other than 1 is either prime or can be uniquely expressed as the product of prime number factors, is misleadingly portrayed through the (conventional) Type 1 approach in a merely reduced quantitative manner. However properly understood there is also - literally - a qualitative dimension to this Fundamental Theorem (which is continually misrepresented in conventional terms).
So we once again come back to the key issue of interdependence! Because Conventional Mathematics in formal terms is defined strictly within independent polar reference frames (i.e. where objective is abstracted from the subjective pole and quantitative abstracted from the qualitative pole) it has no means within its interpretations of dealing with the key issue of (holistic) interdependence (except in a grossly reduced fashion).
However once we understand clearly that every composite natural number (as representing the unique product two or more primes) involves both quantitative and qualitative transformation then we are led directly into this key notion of holistic interdependence (as the means of reconciling the dynamic relationship of both aspects).
So this is the context in which the non-trivial zeros properly arise.
In the Zeta 2, we saw how the non-trivial zeros (as the unique values for the prime numbered roots of 1), provide the means of solving the internal problem (of the relationship of the primes to the natural numbers).
So once again for example from the (standard) reduced Type 1 perspective
5 = 1 + 1 + 1 + 1 + 1.
Because these units are all homogenous in quantitative terms, we are left with no way of making a qualitative distinction.
However through the 5 roots of 1 we can give natural number member an individual unique distinction (on a circular number scale) in - relative - quantitative terms, while also maintaining a holistic interdependence as between the collective group of prime members.
So the very meaning of interdependence in this context is that it provides the means of reconciling quantitative with qualitative Type interpretation.
And - quite magically - the non-trivial zeros of Zeta 2 (as the solutions to the equation) when appropriately interpreted, provide the ready solution for this internal relationship between the natural numbers and primes (in all possible cases).
So in a direct complementary manner we can now perhaps appreciate that the (famed) non-trivial zeros of the Zeta 1 equation, likewise provide the solution for the external relationship as between the primes and natural numbers (in all possible cases).
In other words - at one fell swoop as it were - these non-trivial zeros provide the ready means of reconciling each individual prime on the one hand with their overall collective relationship with the natural numbers.
In short - though again it is not clearly recognised as such - the non-trivial zeros represent the solution to this problem of interdependence with respect to the entire natural number system.
So just as the prime numbers represent the independent extreme, the composite natural numbers (as the product of prime numbers) entail an interdependent identity with quantitative and qualitative aspects.
The formula for the non-trivial zeros clearly demonstrates this as it represents the circular equivalent of the standard linear formulas with respect to both the gaps as between primes and the general frequency of primes.
So as the composite numbers (in inverse fashion to the primes) become ever more prevalent as we ascend the natural number scale, one would expect their frequency to increase in inverse fashion to the primes (which indeed is what happens).
So in short the non-trivial zeros represent the interdependent relationship of primes to natural numbers with the whole set of non-trivial zeros completely establishing this interdependence.
So at one extreme we have the independent nature of the primes and natural numbers (where quantitative and qualitative notions are clearly separated). Then at the other extreme we have the interdependent nature in the entire set of zeros where the primes and natural numbers are understood as identical.
And in a comprehensive understanding neither of these aspects can be understood in the absence of the other. They are in fact ultimately fully complementary.
We perhaps can put it even more simply.
Both the primes and natural can be defined in both a quantitative and qualitative manner (i.e. in Type 1 and Type 2 terms).
However, in dynamic terms, the relationship between both primes and natural numbers and (natural numbers and primes) is as quantitative to qualitative (and qualitative as to quantitative) respectively.
The non-trivial zeros in each case (from complementary opposite standpoints) establish the means of reconciling both interpretations i.e. through showing that quantitative and qualitative aspects are ultimately identical (in a - necessarily - relative approximate manner).
As I have repeatedly stated, Conventional Mathematics attempts to view quantitative and qualitative poles as completely separate (purely independent).
However quantitative and qualitative are dynamically related in phenomenal terms and ultimately identical (purely interdependent).
And properly understood, we cannot begin to understand the nature of our everyday – seemingly independent - number system in the absence of the interdependent dynamics operating at its very core, as the fundamental requirement for its - relative - consistency.
The non-trivial zeros (from both perspectives) where both its quantitative and qualitative aspects can be understood as ultimately interdependent, represent the opposite polar extreme to current mathematical interpretation of the nature of number.
The non-trivial zeros therefore simply cannot be properly reflected through the conventional mathematical paradigm.
As it formally has no recognition for the qualitative aspect, it therefore can have no proper notion of interdependence (which implies both aspects).
In short, the non-trivial zeros point to the need for a completely new understanding of mathematical - and indeed all scientific - reality (where quantitative and qualitative aspects of understanding can be dynamically related at every level).