The quantitative aspect is where each number is treated as a part i.e. (independent of other numbers).
The qualitative aspect, by contrast, is where each number is given a whole dimensional meaning (as interdependent with all other numbers).
So once more, "3" for example, can be treated in an independent quantitative manner ; however equally it can be used to represent what is common to different classes (which can be increased without limit).
In addition, each number involved is used in the first quantitative sense as for example with 3 + 2; however strictly in multiplication the second qualitative meaning is likewise necessarily involved.
Therefore when we multiply 3 * 2, the use of "3" here signifies a number that is common with respect to the 2 classes involved (as for example 2 rows with 3 items in each row).
And of course we can go on to associate any other natural number in the same manner with "3".
It might help further to refer in this context to the first quantitative use of "3" as the Type 1 interpretation, and then - in relative manner - to the second qualitative use as the Type 2 interpretation.
However this then creates the big problem of how to ensure consistency as between the two distinctive meanings of number (in this case illustrated by "3").
And this problem remains completely unrecognised from the conventional mathematical interpretation (where the qualitative interpretation of number is reduced to the quantitative).
Put another way - though again unrecognised in conventional terms - every number, by its very nature, necessarily speaks in two distinctive languages (that are not directly compatible in terms of each other).
Now n terms of the number system as a whole, this issue can be expressed in the following manner.
From one perspective, each of the natural numbers is independent of every other number (in an absolute quantitative manner).
However all the natural numbers as a collective set, are likewise interdependent with each other (in a relative qualitative manner).
Therefore the fundamental key issue with respect to the number system is to show how both of these meanings (quantitative and qualitative) can be reconciled with each other.
This is where the Riemann zeta function is of such importance.
In its simplest form it is written as
ζ (s) = 1 –
s + 2 – s + 3 – s + 4 – s +…..
Therefore we can see that each natural number is included here in sequential order to be raised to the negative of a dimensional power (written as s).
Then in finding the solutions of the equation for s = 0, i.e.
ζ (s) = 1 – s + 2 – s + 3 – s + 4 – s +….. = 0,
we can derive a crucially important new set of numbers (as solutions for s) which are known as the non-trivial zeta zeros (i.e. the Riemann zeros) which I refer to as the Zeta 1 zeros.
Now the true significance of these zeros is that they provide a set of numbers where both the quantitative and qualitative aspects of the cardinal system of natural numbers are reconciled.
What this means in effect is that at each of these zeros, both the quantitative and qualitative interpretations of number are now identical. However, it is important to appreciate that this can only be understood in a dynamic relative manner where such identity can only be approximated in a phenomenal manner.
The zeros occur as pairs in the form of .5 + it. and .5 – it respectively.
However t represents a transcendental number (which can only be approximated in a rational manner). Likewise the set of zeros is unlimited; however we can always know but a limited finite number of such zeros.
What this entails is that we must imply an entirely distinctive paradoxical type of appreciation when interpreting the zeros.
In conventional mathematical terms, a linear either/or logic is used in appreciation of the quantitative nature of number. Thus a number is either prime or composite; numbers are either random or distributed in an orderly manner; multiplication can be distinguished from addition; finally we can identify either the quantitative or qualitative aspect of number in a separate manner.
However, remarkably the zeta zeros, properly speaking, operate according to a circular both/and logic that is paradoxical from a conventional perspective.
Thus the very nature of the zeros is that with these numbers the distinction between what is prime and composite is eroded; likewise the distinction as between what is random and ordered does not operate; crucially at each zero, addition is reconciled with multiplication and finally and most importantly the distinction as between quantitative and qualitative aspects is likewise now removed.
Though all this seems counter intuitive, it should be easier to appreciate with reference to everyday appreciation of the nature of a crossroads. Thus if one approaches a crossroads heading North, an unambiguous either/or distinction can be made as between left and right turns; likewise if one approaches the crossroads from the opposite direction heading South, again an unambiguous distinction can be made as between left and right turns.
However, when one simultaneously views the turns from both North and South directions, what is left and what is right (separately) is rendered deeply paradoxical. For what is left from one direction is right from the other and vice versa. So depending on context each turn can be left or right!
Fundamentally, the zeros represent the same logic. Though I have not yet dealt with the ordinal nature of number yet in this series of entries, number can indeed be approached from both cardinal and ordinal directions. Though the identification of quantitative and qualitative aspects is ambiguous when taken from just one direction (e.g. cardinal) when both ordinal and cardinal directions are combined, the polar opposites we customarily associate with number interpretation (esp. quantitative and qualitative) are rendered paradoxical.
However, it is simply impossible to grasp this from the conventional mathematical perspective.
In the case of our crossroads, we clearly cannot identify left and right turns, if we only recognise one of these directions! The realisation that the turns can be both left and right (when approached from two opposite directions) would be meaningless in this context.
Likewise, the recognition that a number can simultaneously be both quantitative and qualitative (in an approximate relative manner) is likewise impossible without the clear recognition of separate quantitative (Type 1) and qualitative (Type 2) interpretations in the first place.
So the simultaneous recognition of number as both quantitative and qualitative relates to the most refined form of mathematical understanding (which I refer to as Type 3).
So the zeta zeros from this new perspective, can therefore be expressed succinctly as representing the Type 3 aspect of the natural number system (in cardinal terms).
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