Basic Problem

As a child I was fascinated with the processes of addition (and subtraction), multiplication (and division) and powers (and roots) in a way that even then hinted at present interests.

Looking back I sensed clearly even then that the latter two processes involved both a qualitative and quantitative transformation of number. However Mathematics in effect ignored the qualitative aspect of this transformation altogether.

This problem is so easy to state that it still amazes me how it is still - seemingly - completely overlooked by the mathematical profession.

Once again if we take the simple case of squaring a number, both quantitative and qualitative aspects are involved.

Thus in the simplest case when we square the number 1, in quantitative terms the result remains unchanged = 1.

However clearly a qualitative transformation has taken place in the nature of units involved. So for example if are to represent the original number 1 i.e. 1^1 we would represent it geometrically by a line segment of 1 unit. However we would then represent 1^2 geometrically as a square (with each side = 1 unit).

So through the process of squaring the number, a qualitative transformation has thereby taken place. In other words we move from a 1-dimensional to a 2-dimensional interpretation!

However, from a qualitative perspective Conventional (i.e. Type 1) Mathematics is defined in linear (1-dimensional) terms whereby the qualitative attributes of numbers (and indeed all symbols) are reduced to mere quantitative interpretation.

So therefore when we obtain the power of a number or - indeed - multiply numbers together, the qualitative transformation in the nature of the numbers is simply ignored with the result expressed in reduced quantitative terms.

So for example 3^3 = 27 (i.e. 27^1).


Thus in a very precise manner I have always referred to the logical nature of Type 1 Mathematics - which misleadingly is viewed as synonymous with all Mathematics - as linear i.e. 1-dimensional from a qualitative perspective. And as repeatedly stated this effectively in any context reduces qualitative to quantitative type interpretation.


Now it can be simply demonstrated that when seen from a more comprehensive perspective that Type 1 Mathematics is characterised by a fundamental logical confusion.

Let us return to our simple example to demonstrate.

Thus 1^2 = 1 (i.e. 1^1) from a Type 1 perspective.

Therefore when we now raise each side to the power of 1/2 we obtain

1^1 = 1^(1/2).

However whereas in Type 1 terms, the RHS yields an unambiguous answer + 1, the LHS has two diametrically opposite possible answers (in Type 1 terms).

So from this perspective the square root of 1 can paradoxically be either + 1 or - 1.


This simple illustration shows that there is a fundamental unrecognised problem with mathematical reasoning (which occurs whenever a qualitative change in the nature of numerical units takes place).

As this problem therefore relates to all multiplication (and division) and all powers (and roots) it is therefore of the utmost significance.


In order to maintain consistency in quantitative terms, Type 1 Mathematics effectively ignores all but the first root (which is thereby referred to as the principle root). So for example if you use a calculator to find the square root of 1 it will give merely this principle root (= + 1). In this way the logical inconsistency that I have demonstrated is thereby conveniently avoided (not solved).

In the next contribution, I will show how the proper solving of this problem requires giving number (as dimension) a new qualitative interpretation.