I have already outlined in earlier blogs details of the the two number systems corresponding with the Type 1 (quantitative) and Type 2 (qualitative) aspects of Mathematics.
So once again, Type 1 is is defined with respect to a fixed number 1 (as dimensional quality), with base number quantities varying. So the natural number system here is:
1^1, 2^1, 3^1, 4^1,.....
By contrast Type 2 is defined with respect to a fixed number 1 (as base quantity), with dimensional numbers varying. So the natural number system here is:
1^1, 1^2, 1^3, 1^4,.....
Addition (can take place in both systems (as separate).
So in the Type 1 system 2 + 3 = 5;
So written out in full, this expression is
2^1 + 3^1 = 5^1
Here a quantitative transformation with respect to the variables takes place (without any corresponding qualitative transformation).
Likewise in the Type 2 system 2 + 3 = 5.
So from a Type 2 perspective 2 + 3 = 5, written out in full is,
1^2 * 1^3 = 1 ^(2 + 3) = 1^5
So here by contrast a qualitative transformation with respect to the variables takes place without any corresponding quantitative transformation.
So we here can resolve quickly the mystery of the nature of addition and multiplication.
What implies addition with respect to the Type 1 (quantitative) system, implies multiplication with respect to the 2nd (qualitative) system.
This explains why both Alain Connes and Brian Conrey (writing in the context of the Riemann Hypothesis) have admitted that there is something fundamental regarding the relationship of addition to multiplication that is not yet grasped (i.e. in Type 1 Mathematics).
And the essence of the problem is that multiplication strictly relates to qualitative - rather than quantitative - transformation. And in a system that is defined solely by its quantitative aspect, it is therefore not possible to deal adequately with the nature of multiplication.
So, multiplication is necessarily given a merely reduced quantitative interpretation in Type 1 terms (where it is not properly distinguished from addition).
So for example 2 * 3 can be translated (in Type 1 terms) as 2 + 2 + 2 (i.e. 2^1 + 2^1 = 2^1 where we add 2^1 to itself 3 times) which clearly demonstrates the lack of dimensional change).
Thus again, the very nature of addition is tied up with quantitative transformation, whereas by contrast the nature of multiplication is tied up with qualitative (i.e. dimensional) transformation. And I have already explained in recent blog entries the nature of interpretation for dimensional values 2 and 4 of the Riemann Zeta Function as the (circular) qualitative nature of dimensional transformation.
In a Type 3 mathematical approach, both the base and dimensional numbers can vary with both numbers always interpreted in a complementary fashion.
So, if starting from the base number (as representing a quantity) then - relatively - from this perspective, the now varying dimensional number is understood in corresponding qualitative terms as representing a quantitative value;
Now in isolation, this will be true when we vary the number representing qualitative dimension (which was fixed at 1 for the Type 1 system), or alternatively vary the base quantity (which was fixed at 1 for the Type 2 system).
For example in the Type 1 system when we start for example with 3^1, the base number is a quantity and the dimensional number - realtively - of a qualitative nature.
So if we allow the dimensional number to change to 2, 3 is a base quantity raised to 2 as a qualitative dimension. So we now have 3^2.
Likewise in the Type 2 system, when we start, in reverse fashion, for example with 1^2, the base number represents a quantity and the dimensional number has a qualitative interpretation.
So if we now allow the base quantity to vary to 3, while the dimensional number remains, we again obtain 3^2.
Now in isolation, the same expression is obtained for both systems!
However just as in relation to each other at a crossroads. two left turns (in isolation) necessarily are left and right (and right and left) in relation to each other, likewise when we combine both Type 1 and Type 2 expressions for the "same" numbers, as in this example 3^2, what was quantitive (from one perspective) becomes qualitative from the other; and what was qualitative from the other perspective, becomes quantitative from the first.
So in Type 3 terms both the numbers 3 and 2 in the expression 3^2, have both quantitative and qualitative aspects, which continually alternate in understanding.
Therefore - in relative terms - if 3 is quantitative, then the exponent 2 is quantitative; also in reverse, in relative terms, if 3 is qualitative, then 2 is quantitative.
So to conclude:
Type 1 Mathematics is based on the pure quantitative extreme (which is defined in terms of a default dimensional number of 1). Therefore when higher dimensional numbers are involved (as powers), the result is ultimately expressed in terms of the invariant dimensional value of 1 (thus enabling reduced quantitative interpretation).
Type 2 Mathematics, is based on the complementary pure qualitative extreme (which is defined in terms of a default base number of 1). Therefore, when higher number dimensions are involved, the result is expressed in terms of an invariant base quantity of of 1 (thereby enabling reduced qualitative interpretation).
Using these extreme cases we can isolate the nature of addition and multiplication respectively. So what represents addition with respect to Type 1, represents multiplication with respect to Type 2 (clearly indicating the quantitative and qualitative nature of addition and multiplication respectively).
Now Type 3 Mathematics (which is by far the most comprehensive) allows both base quantity and dimension to vary (thereby necessitating exponentiation with respect to values other than 1).
What we have concluded is quite remarkable, in that a true account of non-trivial exponentiation (i.e. for values other than 1 with respect to both base quantity and dimensional number) necessarily requires Type 3 Mathematics.
And from this perspective the two numbers (base and dimension) acquire both quantitative and qualitative (and qualitative and quantitative) interpretations in two-way relationship with each other.
Now the Riemann Zeta Function clearly requires both base numbers (i.e. the natural numbers) and dimensional numbers for any numbers in the complex plane, where both can vary..
This clearly implies that the Riemann Zeta Function requires a Type 3 mathematical approach (for proper interpretation).