There are in fact two complementary relationships as between the primes and the natural numbers.
From the standard well-known perspective, this relates to the quantitative (base) aspect of such numbers.
However from the little regarded alternative perspective it relates to qualitative (dimensional) aspect of such numbers i.e. as represented by factors.
So for example, from the former external perspective we might seek to calculate the frequency of primes up to a given natural number.
From the latter internal perspective, we might then seek to calculate the ratio of natural to prime factors within this given number.
In both contexts, log n is of special importance.
In the former external case, it measures the average gap as between prime numbers. In the latter internal case, it measures the average frequency of natural number factors.
Thus again in the former case, n/log n measure the average frequency of primes to n. In the latter case n/log n measures the average gap as between the natural number factors of n.
So one can see clearly, even in these simple illustrations, how two complementary measurements with respect to the primes and natural numbers, are at play which - relatively - are quantitative and qualitative with respect to each other.
So there is a (base) quantitative aspect which relates to the fundamental notion of such numbers as points on the (1-dimensional) line. However there is also a qualitative dimensional aspect aspect, which relates to the equally important aspect establishing the spatial - and indeed temporal - relationship as between such numbers.
And this is vitally important to emphasise because numbers (like physical matter) strictly can have no meaning in the absence of space and time dimensional characteristics.
It is therefore only the gross fallacy of the continual reduction of the qualitative (relational) aspects of number in quantitative terms that has led to the utterly mistaken assumption that numbers can have an absolute abstract existence (independent of space and time).
So without a qualitative (relational) aspect, numbers can enjoy no meaningful existence as quantities. And without a quantitative (independent) aspect, no meaningful relationship can be established as between numbers.
However clearly both of these aspects can only be appropriately understood in a dynamic interactive manner, where both (quantitative) independence and (qualitative) interdependence
are understood in a truly relative fashion.
This is all very pertinent to interpretation of the true meaning of Riemann's Hypothesis.
The assumption here is that all the Zeta 1 (non-trivial) zeros lie on an imaginary line drawn through 1/2.
Now bear in mind what I have been repeatedly saying in these blog entries regarding the truly complementary nature of both quantitative and qualitative aspects of the number system!
And remember that I have also strongly maintained that the Riemann zeros in fact indirectly represent the hidden qualitative aspect of this system (in cardinal terms)!
However because of the reduced nature of Conventional Mathematics, where qualitative considerations are reduced in an absolute quantitative manner, the misleading rational assumption is made that all real number quantities already lie on the (1-dimensional) number line.
However when we view the number system appropriately in a truly dynamic interactive manner, then we are no longer entitled to make this assumption regarding the horizontal linear nature of the quantitative aspect independent of the qualitative aspect (represented by the non-trivial zeros).
Equally we are not entitled to make the assumption regarding the vertical linear nature of the qualitative aspect (represented by the non-trivial zeros) independent of the quantitative aspect (represented by the real line).
In particular there is no way of proving the truth regarding the vertical nature (on an imaginary line) of the non-trivial zeros i.e. of proving Riemann's Hypothesis in conventional mathematical terms, as this very assumption is already implicit in the acceptance of the horizontal nature of the real numbers (on the real line).
In other words, in making the assumption that all real numbers can be consistently expressed as lying on a (1-dimensional) line, we already assume total consistency as between the independent aspect of number (in quantitative terms) and the interdependent aspect of number (in qualitative terms), whereby numbers can be consistently related with each other.
So once again the truth regarding the qualitative aspect of number, which the non-trivial zeros indirectly express, is already unwittingly assumed in the very assumptions regrading the real number line.
Thus the truly fundamental issue which the Riemann Hypothesis - when appropriately interpreted - raises, is the ultimate consistency with respect to both the quantitative (analytic) and qualitative (holistic) use of mathematical symbols.
In more psychological terms, it relates to the ultimate consistency of both the conscious (rational) and unconscious (intuitive) interpretation of these same symbols.
And once again this issue cannot remotely be solved in an absolute conventional mathematical manner, as it already blindly assumes that the qualitative aspect is consistent with the quantitative.
So properly understood, ultimate belief in such underlying consistency represents a giant act of faith in the entire subsequent mathematical enterprise.
What we can therefore say in a dynamic relative manner, is that if our quantitative assumptions regarding the number line are to be valid, then equally the assumption that all the non-trivial zeros lie on an imaginary line (through 1/2) must also be valid.
Equally, if the assumption that all the non-trivial zeros must lie on an imaginary line (through 1/2) is true, then the assumption that all real numbers lie on the number line must also be true.
From the standard well-known perspective, this relates to the quantitative (base) aspect of such numbers.
However from the little regarded alternative perspective it relates to qualitative (dimensional) aspect of such numbers i.e. as represented by factors.
So for example, from the former external perspective we might seek to calculate the frequency of primes up to a given natural number.
From the latter internal perspective, we might then seek to calculate the ratio of natural to prime factors within this given number.
In both contexts, log n is of special importance.
In the former external case, it measures the average gap as between prime numbers. In the latter internal case, it measures the average frequency of natural number factors.
Thus again in the former case, n/log n measure the average frequency of primes to n. In the latter case n/log n measures the average gap as between the natural number factors of n.
So one can see clearly, even in these simple illustrations, how two complementary measurements with respect to the primes and natural numbers, are at play which - relatively - are quantitative and qualitative with respect to each other.
So there is a (base) quantitative aspect which relates to the fundamental notion of such numbers as points on the (1-dimensional) line. However there is also a qualitative dimensional aspect aspect, which relates to the equally important aspect establishing the spatial - and indeed temporal - relationship as between such numbers.
And this is vitally important to emphasise because numbers (like physical matter) strictly can have no meaning in the absence of space and time dimensional characteristics.
It is therefore only the gross fallacy of the continual reduction of the qualitative (relational) aspects of number in quantitative terms that has led to the utterly mistaken assumption that numbers can have an absolute abstract existence (independent of space and time).
So without a qualitative (relational) aspect, numbers can enjoy no meaningful existence as quantities. And without a quantitative (independent) aspect, no meaningful relationship can be established as between numbers.
However clearly both of these aspects can only be appropriately understood in a dynamic interactive manner, where both (quantitative) independence and (qualitative) interdependence
are understood in a truly relative fashion.
This is all very pertinent to interpretation of the true meaning of Riemann's Hypothesis.
The assumption here is that all the Zeta 1 (non-trivial) zeros lie on an imaginary line drawn through 1/2.
Now bear in mind what I have been repeatedly saying in these blog entries regarding the truly complementary nature of both quantitative and qualitative aspects of the number system!
And remember that I have also strongly maintained that the Riemann zeros in fact indirectly represent the hidden qualitative aspect of this system (in cardinal terms)!
However because of the reduced nature of Conventional Mathematics, where qualitative considerations are reduced in an absolute quantitative manner, the misleading rational assumption is made that all real number quantities already lie on the (1-dimensional) number line.
However when we view the number system appropriately in a truly dynamic interactive manner, then we are no longer entitled to make this assumption regarding the horizontal linear nature of the quantitative aspect independent of the qualitative aspect (represented by the non-trivial zeros).
Equally we are not entitled to make the assumption regarding the vertical linear nature of the qualitative aspect (represented by the non-trivial zeros) independent of the quantitative aspect (represented by the real line).
In particular there is no way of proving the truth regarding the vertical nature (on an imaginary line) of the non-trivial zeros i.e. of proving Riemann's Hypothesis in conventional mathematical terms, as this very assumption is already implicit in the acceptance of the horizontal nature of the real numbers (on the real line).
In other words, in making the assumption that all real numbers can be consistently expressed as lying on a (1-dimensional) line, we already assume total consistency as between the independent aspect of number (in quantitative terms) and the interdependent aspect of number (in qualitative terms), whereby numbers can be consistently related with each other.
So once again the truth regarding the qualitative aspect of number, which the non-trivial zeros indirectly express, is already unwittingly assumed in the very assumptions regrading the real number line.
Thus the truly fundamental issue which the Riemann Hypothesis - when appropriately interpreted - raises, is the ultimate consistency with respect to both the quantitative (analytic) and qualitative (holistic) use of mathematical symbols.
In more psychological terms, it relates to the ultimate consistency of both the conscious (rational) and unconscious (intuitive) interpretation of these same symbols.
And once again this issue cannot remotely be solved in an absolute conventional mathematical manner, as it already blindly assumes that the qualitative aspect is consistent with the quantitative.
So properly understood, ultimate belief in such underlying consistency represents a giant act of faith in the entire subsequent mathematical enterprise.
What we can therefore say in a dynamic relative manner, is that if our quantitative assumptions regarding the number line are to be valid, then equally the assumption that all the non-trivial zeros lie on an imaginary line (through 1/2) must also be valid.
Equally, if the assumption that all the non-trivial zeros must lie on an imaginary line (through 1/2) is true, then the assumption that all real numbers lie on the number line must also be true.
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