The key problem in reconciling addition with multiplication is that they represent mathematical processes that are quantitative and qualitative with respect to each other. And as Conventional (type 1) Mathematics is based on a merely reduced quantitative approach this creates enormous difficulties in properly appreciating the nature of the problem.
As we have seen properly we have two number systems that are quantitative and qualitative with respect to each other.
1) In the conventional (Type 1) system, the natural numbers 1, 2, 3, 4, 5,.... for example respect quantities are defined with respect to a (default) dimensional value of 1.
So written in full, this system is represented as:
1^1, 2^1, 3^1, 4^1, 5^1,........
2) In the unrecognised (Type 2) system, the same natural numbers 1, 2, 3, 4, 5,.... represent qualitative dimension that are defined with respect to a (default) base quantity of 1.
So written in full, this alternative system is represented as:
1^1, 1^2, 1^3, 1^4, 1^5,........
Now with respect to the first system when we add two numbers, say, 2 + 3,
this is fully represented as
(2^1) + (3^1) = 5^1
However when we add the same two numbers, 2 + 3, with respect to the second system, this is fully represented as
(1^2) * (1^3) = 1^5.
So whereas addition of these two numbers is involved with respect to the first (quantitative) system, multiplication of the same two numbers (now representing dimensions) is entailed with respect to the second (qualitative) system.
This clearly entails that whereas pure addition (i.e. with respect to numbers that are all defined with respect to 1 as default dimension) is of a direct quantitative nature. Pure multiplication (i.e. with respect to numbers that are all defined with respect to 1 as default base) is by contrast of a direct qualitative nature.
Therefore we cannot ultimately hope to reconcile addition and multiplication without equal recognition of both Type 1 (quantitative) and Type 2 (qualitative) numerical systems.
In practice therefore where non-unitary values are given to both (quantitative) base and (qualitative) dimensional numbers, Type 3 Mathematics (representing the coherent interaction of both Type 1 and Type 2 systems) must be used for comprehensive understanding.
A further problem relates to the reconciliation of multiplication and exponentiation.
If multiplication is now treated in a (Type 1) quantitative manner, then
2 * 3 for example is represented as (2^1) * (3^1) = 6^1.
However with respect to the second (Type 2) dimensional system 2 * 3 is represented as
So multiplication with respect to the two numbers in the Type 1 (quantitative) system represents exponentiation with respect to the same two numbers in the second.
And just as (2^1) * (3^1) = (3^1) * (2^1) (with respect to the first)
(1^2)^3 = (1^3)^2 (with respect to the second).