Yesterday, we saw how both quantitative and qualitative aspects necessarily attach in a dynamic interactive manner to all natural numbers.
Therefore, through the quantitative aspect, we recognise the distinct independence of each number (from all other numbers). By contrast through the qualitative aspect we recognise the corresponding interdependence of each of these (with all other numbers).
With addition, we are explicitly concerned with the former quantitative notion of number as representing independent units; with multiplication, by contrast, we are now explicitly concerned with the latter qualitative notion of number interdependence with respect to its common identity with various classes (the number of which can be extended without limit).
In addition, no (qualitative) dimensional change in the nature of units takes place; however with multiplication a dimensional - as well as quantitative - change necessarily takes place in units (though this fact is conveniently edited out of the conventional mathematical interpretation of multiplication).
Using the close analogy with quantum physics, number exhibits both particle and wave like characteristics. So which aspect reveals itself depends on the particular context of investigation.
Thus if we associate the operation of addition with its particle like attributes (as independent numbers) the operation of multiplication by contrast will then reveal its corresponding wave like attributes (as interdependent with other numbers).
Clearly therefore, from a dynamic interactive perspective, the nature of number is necessarily relative (with respect to both its independent and interdependent aspects).
So, if we now look at the natural number system as a whole, we have two related perspectives through which it can be viewed.
1) we can view it in the customary analytic manner, where both quantitative and qualitative poles are clearly separated in an absolute type manner (with the qualitative aspect thereby reduced to the quantitative).
Therefore from this perspective, the natural numbers 1, 2, 3, 4,.... are viewed as independent number entities in a quantitative manner.
2) we can view it - in the largely unrecognised - qualitative manner, where the natural numbers 1, 2, 3, 4 in interdependent terms (in the recognition of what is common to different number classes).
So again if we have two rows of items with 3 items in each row, the number "3" now represents what is common to both rows. Therefore, in this sense we are making reference directly to its qualitative - rather than quantitative - meaning.
Of course initially in identifying each row we must recognise "3" in the customary independent manner (as quantitative). However, when we then recognise its commonality with respect to both rows, we switch to the qualitative aspect.
This in fact explains the deeper reason why multiplication necessarily entails both quantitative and qualitative aspects of number transformation.
Thus 2 * 3 = 6 in quantitative terms). However 2 * 3 equally entails a qualitative transformation in a 2-dimensional fashion (though this then is edited out of conventional mathematical interpretation).
Therefore the deeper reason as to why multiplication also entails a qualitative transformation in the nature of units is due to the necessary switch that is involved with respect to the recognition of number (from its independent to its interdependent nature).
There is another helpful way of viewing this qualitative aspect of number.
Once again, to illustrate, we can use "3" to refer to the quantitative aspect of number.
However when we use "3" in the qualitative sense, it can be referred to as "threeness" (i.e. the quality of 3).
Therefore, in operations, a continual switching takes place as between number as representing a quantity, and its corresponding qualitative nature as "numberness".
In psychological terms, quantitative and qualitative aspects respectively are related to the manner in which both conscious and unconscious interact.
In direct terms we can identify the quantitative aspect with (conscious) rational recognition; however the qualitative aspect relates directly to (unconscious) intuitive recognition.
Therefore the customary reduction of qualitative to quantitative interpretation in Conventional Mathematics corresponds directly with the corresponding reduction of intuitive to rational type meaning.
This is certainly true in formal terms, where Mathematics is presented as a body of rational type truths. However indirectly of course, the intuitive aspect must be always involved, as mathematical operations in quantitative terms are strictly speaking meaningless in the absence of the qualitative notion of a common relationship.
The difficulty in dynamic interactive terms is that the qualitative can always be shown to have a corresponding quantitative aspect (just as the quantitative can be shown to have a qualitative aspect).
Again this is directly analogous with quantum physics where particles exhibit wave like characteristics and waves particle like characteristics, respectively.
Thus when one merely concentrates on the quantitative like aspect (associated with qualitative type understanding) then it becomes easy to simply reduce the qualitative in a quantitative manner.
In psychological terms, one switches to the conceptual notion of number through dynamic negation of number perceptions. Now when the initial number perceptions are identified in a quantitative manner, the initial conceptual recognition will be of a qualitative intuitive nature (where it is seen as potentially applying in an infinite manner to all numbers).
However in practice the understanding of the number concept is then quickly reduced in a quantitative rational manner (where it is now understood as actually applying in a finite manner to all numbers).
In reverse fashion, the recognition of distinct number perceptions, implies the dynamic negation of number concepts. Again the initial realisation is implicitly of an intuitive nature. However this is quickly reduced in a rational manner.
Therefore, though in the dynamics of understanding both finite (rational) and infinite (intuitive) notions ceaselessly interact in both quantitative and qualitative terms, standard mathematical interpretation reduces all this in a finite rational fashion relating merely to the quantitative aspect.
In this approach the infinite is then misleadingly viewed as merely a linear extension of the finite notion.
However, when one properly appreciates the unique distinctiveness of both the quantitative and qualitative aspects of number, then the key issue relates as to consistent use with respect to the combination of both aspects.
And this is where the zeta zeros play a crucial role!