Wednesday, March 21, 2012

Number Inconsistency (3)

I have been at pains to highlight in recent blog entries the truly fundamental problem that exists, with respect to how relationships between objects are dealt with in Conventional Mathematics.

So once again, when an object is viewed in absolute terms as having a fixed independent identity (in quantitative terms), then strictly this rules out the possibility of establishing relationships (which necessarily implies interdependence) with other objects (in a qualitative manner).

Thus in order to proceed, Conventional Mathematics must necessarily reduce - in any context - relational meaning (which is qualitative) to mere quantitative interpretation.

And in a very precise manner, I have defined such Mathematics as 1-dimensional in qualitative terms (i.e. where the qualitative aspect is reduced to the quantitative).

Now I have always recognised this linear rational (1-dimensional) approach to Mathematics as representing an extremely important special case!

However, appropriate appreciation of this special case requires placing it in a much wider context, where the full range of dimensional interpretations (as qualitative) can be employed.

Therefore corresponding to each number (as dimension) is a unique manner of interpreting mathematical symbols. And in every other case (except 1), such a number entails a dynamic configuration entailing quantitative and qualitative meaning.

What this entails in turn is recognition that in all other cases, ordinal is clearly distinguished from cardinal meaning.

Then in Type 2 terms, one comes to appreciate - which we will be demonstrating later - that an unlimited set of interpretations exist for any number that is defined in ordinal terms (which correspond precisely with the qualitative meaning of number as dimension). Thus, we gradually begin to appreciate that from this perspective, the ordinal nature of number as truly relational (i.e. qualitative) is precisely the opposite to that of cardinal meaning (as independently fixed in a quantitative manner).

However an even greater surprise is then revealed in Type 3 terms where both cardinal and ordinal interpretation can interact in a dynamic relative manner. For here, the cardinal nature of number - which was formerly viewed as fixed - now can take on a limitless number of possible alternative interpretations; whereas in reverse the ordinal can now be given a - relatively - fixed identity!

And if you want to know briefly where the full variety of all these possible interpretations are contained then we need to look no further than the Riemann Zeta Function (which of course we will be returning to again!).

So we are investigating here the Type 2 approach to ordinal number (which will entail a circular - rather - than linear approach to meaning).

The first clue to the ambiguity inherent in number as ordinal comes from the attempt to apply relative rankings.

Now when we confine ourselves to 1 object, we can see that both cardinal and ordinal meaning directly coincide. In other words in a set or collection of 1 object, 1 (as cardinal representing the number of objects in the set) directly coincides with 1 in ordinal terms (as 1st member of this set). In fact paradox is avoided here - as by definition what - is 1st in this case coincides with what is last. And this is the essence of the linear (1-dimensional) approach!

Now when we move to 2 objects, ambiguity in ordinal terms begins to arise.

For here what is defined as 1st is now the 1st (of 2 objects), whereas in the former case it represented the 1st (of 1 object).

From an ordinal perspective 2 now represents the 2nd of 2 objects (which we could define as the default ordinal definition of 2).

However when we examine 1 and 2 in an ordinal sense with respect to 3 objects, the relative meaning once again changes (with now 1 representing the 1st of 3 objects, and 2 representing the 2nd of 3 objects respectively). And here the new number 3 acquires its default ordinal definition (i.e. as the 3rd of 3 objects).

Thus, the general picture should be clear.

Once we increase our finite set of objects by 1, all previous ordinal relationships of numbers are thereby changed. (Strictly speaking this does not apply to 1, which serves as the reference point from which all other relations are based!)

So in this manner we are indeed led to quickly see that the ordinal (relational) nature of number is truly relative - rather than absolute.

However to properly interpret all of this in a true mathematical manner we need to switch to a Type 2 (circular) perspective.

So crucially here to give the numbers 1, 2, 3, ...their true ordinal meaning, we define them in dimensional terms (with respect to 1 as default base quantity).

So once again the in Type 1 natural number system i.e.

1^1, 2^1, 3^1, 4^1,.....,

the natural numbers - as defined in (quantitative) terms - are all expressed with respect to a default (qualitative) dimension of 1.

This system is therefore suited to the interpretation of cardinal - rather than ordinal - number meaning!

In the Type 2 natural number system i.e.

1^1, 1^2, 1^3, 1^4,.....,

the natural numbers - as defined in (qualitative) terms - are all expressed with respect to a default base quantity of 1.

This alternative system is thereby suited to the interpretation of ordinal - rather than cardinal - number meaning.

Now it can again be quickly see that the only case where cardinal and ordinal meaning coincide with respect to both systems is the first term (where the dimensional number = 1). and once again this defines the conventional mathematical approach!.

However to explore the true nature of ordinal meaning we must shift to the circular representation of the 2nd number system.

Once again - as I have expressed many times before - a complementary relationship exists as between a number D (representing a qualitative dimension) and its corresponding reciprocal 1/D (representing - relatively - in quantitative terms its corresponding root).

Now once again when we start with 1 object, we cannot meaningfully distinguish qualitative from quantitative as D = 1/D (= 1).

So the first meaningful exploration of ordinal type meaning occurs when we move to a set of 2 objects.

So here D = 2 in ordinal terms (as representing the 2nd dimension).

1/D = 1/2 in ordinal terms (as representing the 2nd root of 1).

So in terms of the Type 2 system 2 (as its default ordinal definition) = 1^2 (in qualitative terms). This is then directly complementary with 1^(1/2) in a quantitative manner = - 1.

The other root of 2 in this case (relating to 1 as ordinal number) is expressed with respect to 2/2 which is taken as {1^2}1/2 = + 1.

So we now have 2 objects which are + 1 and - 1 (in relation to each other).

To understand what this means in a qualitative relational sense, it may be helpful to return to our example of the two turns at a crossroads.

Now the first task in terms of meaningful interpretation is to understand directions from the linear (1-dimensional) perspective.

So fixing our polar frame of reference with - say - moving up the road, I can give a left turn (in isolation) an unambiguous (i.e. positive) meaning (i.e. + 1, where 1 simply means in this context 1 direction).

Likewise I can give a right turn (in isolation) an unambiguous direction (again as + 1).

However when I now consider both turns in relation to each other, two polar directions need to be simultaneously involved. So we move here from 1-directional to 2-directional understanding. And as the literal geometrical interpretation of dimension implies direction, then this means in turn that we have moved from 1-dimensional to 2-dimensional interpretation.

Thus in relation to each other, both turns at the crossroads are now complementary opposites. So in mathematical terms if the left direction = + 1, then - relatively - the right = - 1; likewise if right = + 1, then left = - 1.

So in precise terms the 2nd dimension here relates to the recognition of the complementary interdependence as between 2 objects (which relates directly to 2 as ordinal number). And this literally requires the ability to dynamically negate the (unambiguous) recognition of each single object (as independent). Therefore the 2nd dimension equates directly with - 1 (in holistic quantitative terms).

However before we can establish such interdependence we must be able to - literally - posit each object as independent (from a 1-dimensional perspective). So this relates directly to the corresponding 1st dimension (i.e. + 1 as holistically interpreted in quantitative terms).

Thus the full interpretation of the relationship between two objects, necessarily entails the 2 roots of unity, with the 1st root (relating in qualitative terms to the 1st dimension) establishing initial object independence and the 2nd root (relating in corresponding qualitative terms to the 2nd root) establishing the interdependence as between both objects.

The remarkable generalisation that follows from this is that with respect to any finite set of objects n, the full ordinal interpretation with respect to all possible relationships as between these n objects contained within that set, is provided through calculation of the corresponding n roots of 1, when each of these roots is given its appropriate qualitative interpretation.

And as we have seen the correct ordinal interpretation of number is directly associated with such dimensional interpretations.

Now one can perhaps realise how - literally - complex - such interpretations become
when we move beyond 2 objects, as all now involve imaginary - as well - as real numbers.

We will return again to the precisely meaning of imaginary in an ordinal context! However I want to demonstrate initially how the ordinal nature of a number continually changes as we define it with respect to a larger set of numbers (which likewise implies higher dimensional interpretation).

In the context of 2 objects (implying 2 corresponding dimensions) 2, as ordinal dimensional number (2nd dimension) corresponds with 1/2 as quantitative power.

Then, in the context of 3 dimensions, 3 as ordinal dimensional number, corresponds with 1/3 as quantitative power; 2 however as ordinal dimensional number corresponds with 2/3 as quantitative power i.e. {(1^2)^1/3). And as each dimensional number as qualitative (D) ìs the inverse of (1/D) as quantitative root, this implies that the correct dimensional number (for 2 in the context of 3) is 3/2.

In the case of 4 objects (requiring 4 corresponding dimensions of interpretation) 2 as ordinal number corresponds with 2/4 as quantitative power (i.e. 1/2). And this in turn corresponds with 2 as dimension.

Now it will be seen that this is not unique as it corresponds exactly with the ordinal interpretation of 2 (in the context of 2).

In other words the 2nd of the 4 roots of 1 (i - 1, - i and 1) i.e. 1^(1/4), {(1^2)^1/4}, {(1^3)^1/4} and {(1/4)^1/4} is identical with the default ordinal definition of 2.

The reason for this overlap is due to the fact that 4 (as ordinal dimensional number) is not prime!

So the significance of a prime number p in the context of ordinal dimensional interpretation, is that it is always associated with a unique natural number set of roots (from 1 up to and including p)!

One final important point which will be developed in a future blog!

The purely relative nature of ordinal numbers (other than 1) with respect to finite sets, is reversed completely in the context of a (potentially) infinite set.

So in this context of infinity, the ranking of any number appears as absolute. So 2 in the context of what is (potentially) infinite has a - relatively - fixed meaning!

Thus, as we will see we have now come full circle which ends with a surprising revelation.

We started by considering how cardinal numbers appear as absolute and independent in the context of a finite set of numbers. So for example if we obtain the sum of numbers with respect to any finite set, the manner in which we rank the numbers before adding makes no difference to the final result.

We then demonstrated how ordinal numbers appear as merely relative (and interdependent) in the context of a finite set.

Next we saw how in the context of a (potentially) infinite set, ordinal numbers now appear as - relatively - independent.

The final conclusion therefore - following from such complementary type connections - is that in the context of infinite sets, cardinal numbers now are of a merely relative nature.

In other words in the context of an infinite set the actual sum of a series can have a - potentially - unlimited number of possible answers!

Put another way, with respect to sum of an infinite series of cardinal numbers, the precise ordering of terms can be crucial to the outcome!

Though this feature of infinite series has indeed been long observed in Conventional Mathematics, we can now see that its deeper explanation arises from the two-way integration of both the quantitative and qualitative (or cardinal and ordinal) aspects of number.

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