Once again however, I emphasise two sets of zeros i.e. Zeta 1 and Zeta 2, relating to both the cardinal and ordinal aspects of the number system respectively.
The first of these - frequently referred to as the Riemann zeros - relating to the cardinal aspect, is intimately tied up with the distinction as between addition and multiplication.
Because of the reductionist (1-dimensional) rational nature of Conventional Mathematics, no proper means exists for the distinction of the quantitative and qualitative aspects of number, with the qualitative aspect in every context necessarily reduced in a merely quantitative manner.
So for example, the natural numbers 1, 2, 3,..., can be used in two distinctive senses that are quantitative and qualitative with respect to each other.
The first sense is the accepted quantitative meaning.
So, again if I take the number "3" to illustrate, in quantitative terms this is understood unambiguously in an absolute manner as independent.
This then could be represented as 1 + 1 + 1, i.e. as composed in quantitative terms of homogeneous units (that literally lack any qualitative distinction).
When we we are engaged in addition, this first notion of number is directly involved.
Thus 3 + 2 for example = (1 + 1 + 1) + (1 + 1) = 1 + 1+ 1 + 1 + 1 i,e 5!
Now what is often forgotten however with respect to conventional addition is that strictly speaking, no number can have an absolute independent identity. Rather implicitly - even with respect to addition - the recognition must exist that each number necessarily has a qualitative relationship to the common notion of number (as dimension).
In other words all the natural numbers lie on a common line (which is thereby 1-dimensional).
Therefore it is through the common membership of each natural number with this common line that we are thereby enabled to relate numbers with each other (as in addition).
However this common relationship with the number line, whereby we are enabled to meaningfully give order to numbers, is strictly of a qualitative, rather than quantitative nature.
Therefore to preserve the relative distinction as between quantitative and qualitative aspects, each number is now defined with respect to this default 1st dimension.
Therefore we now define the number "3" with respect to its quantitative meaning as 31..
Because the explicit focus is on the quantitative aspect of number, it is highlighted in black, whereas the dimensional aspect, which is relatively qualitative and merely implicit, is shown in a light grey colour.
Therefore, in a more refined manner, we see that though the quantitative and qualitative aspects of number are now both recognised, that addition entails that explicit emphasis be placed on the quantitative aspect.
However the significance of this is that - strictly - even addition must now be understood in a dynamic relative manner. This is very much analogous to the designation of a turn at a crossroads as either left and right (which necessarily depends on the direction from which the crossroads is approached).
In other words, we can still make unambiguous distinctions as to a turn i.e. as either left or right, but now this clearly enjoys but an arbitrary relative meaning (depending on context).
Now what is not all clearly recognised from the conventional mathematical perspective is that when we switch to multiplication, the sense in which number is used, likewise changes.
We have already explained the addition of 3 and 2 i.e. 3 + 2.
We now look at the corresponding multiplication of 3 and 2, i.e. 3 * 2.
To illustrate this operation we can imagine two rows (with 3 items in each row).
If we treated this operation as just an extended form of addition then clearly we could represent 3 * 2 as 3 + 3 = 6.
However, this entails a hidden form of reductionism that completely conceals the true distinction of multiplication from addition.
So we have two rows (with 3 times in each row).
Therefore what is crucial with respect to multiplication is the recognition that these 3 items are common to both rows. So therefore instead of adding all individual items (as independent of each other), through this recognition of mutual commonality (with respect to the items in each row) we can thereby multiply 3 by the number of rows involved (which in this case is 2).
Therefore crucially the operation of multiplication entails the opposite complementary notion of number, as what is common (i.e. interdependent) as between different classes.
So in this case we have two classes (i.e rows) with 3 the number that is common in both cases.
However we are now using the number "3" is an altogether different sense than before - not in an independent quantitative manner - but rather in the complementary qualitative interdependent sense (where its relationship with other similar classes is explicitly recognised).
The clear recognition of this point is of the greatest utmost importance, for when one grasps its significance one then clearly realises that the entire conventional edifice of Mathematics is simply not fit for purpose.
And this is ably demonstrated by the fact that Conventional Mathematics - by its very nature - lacks the means for clearly recognisiing the complementary quantitative and qualitative aspects of number, which alternate when one switches from the operation of addition to multiplication.
Again n this simple example that I have used, we have associated the number "3" with 2 (in the recognition of two common classes of 3 members).
However the number "3" - which I am using here to illustrate - can be associated with any other natural number (on the dimensional line).
So what really has happened here is that the relative emphasis on base number (3) and dimensional number (1) has now switched in complementary fashion.
Therefore in the case of multiplication, we now place explicit emphasis on the dimensional aspect of 1 (as representing the common relationship between numbers) while the emphasis on the quantitative nature of 3 is now merely implicit, though of course we must be able to implicitly recognise 3 in quantitative terms, before we can explicitly realise its common identity with other classes of 3 (in a qualitative manner).
Therefore we represent this latter qualitative meaning of 3 as 31.
Therefore we have now demonstrated how the operations of addition and multiplication entail two different notions of number (that in dynamic interactive terms are complementary in experience).
With addition the meaning of a number such as "3" is explicitly quantitative in nature, while the 1-dimensional line, that is relatively of a qualitative nature - establishing its relationship with other numbers - remains merely implicit.
Once again this notion of 3 is represented as 31.
With multiplication, by contrast the meaning of "3" has now switched in complementary fashion, so that it is explicitly qualitative in nature (relating directly to its dimensional nature), while the quantitative meaning of "3" is now merely implicit.
This notion of 3 is represented as 31 (with the dimensional number of 1 now highlighted in black and the number "3" represented in contrasting light grey fashion).
In the next entry, I will establish why these distinctions are crucial for appreciation of the true role of the Riemann (i.e. Zeta 1) zeros.