The basic problem with the conventional approach to Mathematics is quite simple to state.
All experience - including mathematical - is of a dynamic interactive nature entailing the relative independence of distinct phenomena with their overall interdependence in holistic terms.
However as Conventional Mathematics is formally defined in terms of isolated reference frames, e.g. where objective and subjective are clearly separated, it treats mathematical objects abstractly in an independent manner.
So for example numbers (such as prime and natural) are treated in this absolute fashion as possessing an objective independent identity.
However if such numbers did indeed possess such an absolute nature, then strictly it would be impossible to recognise numbers in relation to other numbers as interdependent!
This directly implies therefore implies that such interdependent relationships can only be treated in a reduced manner. This masks therefore problems which on closer examination can be shown to be of the most fundamental nature.
Now when appropriately understood - again in dynamic interactive terms - the relative independent nature of number can be (initially) identified with its cardinal aspect in quantitative terms; the corresponding interdependent aspect, whereby numbers can be related with other numbers, can then be identified with its ordinal aspect in a qualitative manner.
Therefore from this perspective, the quantitative and qualitative aspects of number relate to cardinal and ordinal interpretation respectively.
And because the conventional approach to number - based on absolute notions of independence - is formally defined in a merely quantitative manner, this directly implies that it is not possible to deal with the corresponding ordinal aspect except in a reduced manner that distorts its very meaning.
And we will illustrate the extremely important relevance of this finding shortly!
If we start with the natural number system from the conventional (absolute) mathematical perspective these will be defined directly in a cardinal (quantitative) manner as,
1, 2, 3, 4,.....
However properly understood - in relative terms - all numbers contain two aspects which are quantitative and qualitative with respect to each other.
From this new perspective the cardinal number system is defined in terms of a default (qualitative) dimension of 1,
i.e. 1^1, 2^1, 3^1, 4^1,....
In this context I refer to the first number (that varies) as the base and the fixed invariant number as the dimension and these two numbers are quantitative and qualitative with respect to each other.
The significance of this can be easily illustrated. From the conventional perspective for example when a number is squared we concern ourselves solely in the quantitative transformation thereby involved.
So 2^2 = 4 (i.e. 4^1).
Now if we think of this in geometrical terms, a qualitative change is likewise involved whereby we move from linear (1-dimensional) to square (2-dimensional) units. So a square of 4 square units (with each side 2 units) is qualitatively distinct from a straight line that is 4 units! However from a reduced quantitative perspective this important qualitative distinction is ignored (with literally the square result expressed in 1-dimensional terms).
So when we say that 2*2 = 4, what this implies is that the reduced quantitative value of this number expression = 4. In other words we have ignored the corresponding qualitative change in the number that has thereby occurred.
Now this procedure is valid insofar as the cardinal (quantitative) aspect of number is involved (and strictly only then in a relative sense); however ultimately it leads to total confusion when we explore the corresponding ordinal (qualitative) aspect!
And this leads directly to the most fundamental issue possible with respect to number, for momentary reflection on the matter will immediately make it obvious that we cannot use cardinal notions without ordinal, or ordinal without cardinal. Put simply, we cannot attain a coherent interpretation of number without both quantitative and qualitative aspects equally incorporated in a dynamic relative manner.
So 1, 2. 3. 4 etc. in cardinal terms imply the corresponding notions of 1st, 2nd, 3rd, 4th etc. (from an ordinal perspective).
And as cardinal and ordinal notions are quantitative and qualitative with respect to each other, we therefore cannot properly formulate an interpretation of number (that is consistent) in a merely absolute quantitative manner!
Indeed not alone can we not formulate an ordinal system of number (that is coherent) in this manner, strictly we cannot even formulate a cardinal system that is consistent!
Not surprisingly, these problems lie at the very root of the problem with respect to proper recognition of the relationship between the primes and the natural numbers (and the natural numbers and the primes).
The conventional mathematical approach to looking at this relationship is unbalanced and ultimately untenable.
Approaching the issue from the (absolute) cardinal perspective, it does indeed appear that the relationship is one-way with the primes serving as the building blocks of the natural numbers (excluding 1).
So form this perspective every natural number (again other than 1) can be expressed as the unique product of prime number factors.
So for example 30 = 2*3*5 represents a unique combination of prime number factors and therefore cannot be expressed through any other combination!
However there is a key problem with this approach which is largely overlooked.
When we use a prime factor such as 5 in a cardinal sense it is taken as a single whole. However in any meaningful sense this collective whole set represents the sum of individual number objects.
So 5 = 1 + 1 + 1 + 1 + 1.
However this attempt to define 5 in cardinal terms (i.e. as the sum of individual members that are also cardinal) leads to a crucial problem.
Now clearly in common language we would readily accept that a collection of 5 necessarily includes a 1st, 2nd, 3rd, 4th and 5th member!
However when one reflects on the matter, this represents an ordinal type distinction that cannot be meaningfully derived from 5 = 1 + 1 + 1 + 1 + 1.
In other words the basis of this cardinal definition is to attempt to render number (as without qualitative distinction).
However the very notion of ordinal ranking directly implies that we can somehow distinguish each member (which thereby implies such qualitative distinction).
So, if we insist that all units are homogenous in a merely quantitative manner, then we have no means of attempting any ordinal ranking.
In other words the capacity to make ordinal distinctions comes from the holistic relationship with respect to individual members that are in some sense perceived as possessing a unique quality.
Thus the (absolute) cardinal approach where number is abstractly understood in absolute quantitative terms (as independent) cannot therefore explain the overall holistic relationship as between numbers. Therefore, it has no means within its own definitions of explaining the corresponding ordinal notion of number (without gross reductionism being involved).
And as we cannot even begin to properly deal with cardinal without equally implying ordinal notions, then it cannot provide a coherent interpretation of the cardinal aspect (again without reductionism)!
As I say these represent the most fundamental issues possible with respect to the number system.
Again putting it bluntly I have come to see clearly over the years that the present accepted approach to Mathematics is quite simply not fit for purpose.
So in the next blog entry we will come back to exploring more closely the ordinal nature of number and how it can be given a coherent qualitative interpretation.
No comments:
Post a Comment