We have seen in the last blog that actual experience entails the dynamic interaction of two notions of number which are quantitative and qualitative with respect to each other.
In other words from one perspective, number has a finite aspect (as actual quantities) which directly corresponds with the rational aspect of interpretation.
However from an equally valid perspective, number has an infinite aspect (as a potential quality for existence). And this corresponds directly with an intuitive holistic aspect of appreciation.
So whereas - from this qualitative perspective - the number concept is understood as potentially applying to all (unspecified) numbers in an infinite manner, from the quantitative perspective, numbers which can be identified in actual terms, are necessarily of a finite nature.
Now of course because Conventional (Type 1) Mathematics in formal terms is based on a mere rational (linear) approach, it thereby in any context must attempt to reduce the potential infinite nature of number in an actual finite manner (treating it - in effect - as a linear extension of finite notions).
So the first task in moving to a more comprehensive mathematical approach - that can properly integrate the quantitative and qualitative aspects of number - is to find a way of giving both aspects (in relative isolation) a distinctive interpretation.
So if we confine ourselves to the natural numbers 1, 2, 3, 4,... these have - in isolation - unique interpretations that are quantitative and qualitative respectively.
Now the clue to what is required here is the recognition that in quantitative terms, these numbers are all - necessarily - defined with respect to a default dimensional value of 1.
So we can write this quantitative system more fully as
1^1, 2^1, 3^1, 4^1,......
As we have seen from the quantitative perspective, when a number is raised to a dimensional power (other than 1), the result is given in a reduced quantitative manner (i.e. defined in terms of the dimension of 1).
So again for example in this approach 3^2 = 9 (i.e. 9^1). In this way the problem of dealing with the qualitative change in dimension involved (through squaring) is thereby avoided.
Exactly the reverse applies to the qualitative system.
Here we are not concerned with the quantitative change in value through raising to a higher dimension, but directly in the nature of the dimensional change.
So therefore we express the natural numbers more fully in this system by raising the default quantity 1, to the natural number dimensions.
So the qualitative system here (entailing the natural numbers) is fully expressed as
1^1, 1^2, 1^3, 1^4,......
Now from the quantitative perspective, this latter system seems somewhat trivial and uninteresting as the default value in each case = 1.
However the very point is that we are not directly concerned here with quantitative - but rather qualitative - interpretation.
And we will show later how this in fact is achieved!
Now we can for example add numbers in both systems.
Thus for example 2 + 3 = 5 in each case.
However when we look more closely something very interesting is happening.
For in the first system 2 + 3 is more fully represented as 2^1 + 3^1 = 5^1.
However in the second system 2 + 3 is more fully represented as 1^2 * 1^3 = 1^5.
So an operation that represents addition with respect to the first system, implies multiplication with respect to the second!
Now look at the following comments of two of the key practitioners in the area of the Riemann Hypothesis.
For example Brian Conrey :
"The Riemann Hypothesis is the most basic connection between addition and multiplication that there is, so I think of it in the simplest terms as something really basic that we don't understand about the link between addition and multiplication."
And Alain Connes has stated in somewhat similar fashion:
"The Riemann Hypothesis is probably the most basic problem in mathematics, in the sense that it is the intertwining of addition and multiplication. It's a gaping hole in our understanding..."
Now I suggest that this gaping hole simply comes down to the fact that comprehensive mathematical interpretation requires distinctive aspects of understanding that are quantitative and qualitative with respect to each other.
And it is in this relationship between the two that the corresponding relationship as between addition and multiplication ultimately resides.
However this will never be discovered while Mathematics remains firmly wedded to a merely (reduced) quantitative method of interpretation!