Saturday, August 25, 2012

Multidimensional Nature of Time and Space (13)

As we know from a quantitative perspective rational numbers exist that are not integers i.e. fractions.

This applies therefore that from a qualitative perspective, we equally can give meaning to rational numbers as fractions.

And as the very nature of time (and space) when appropriately understood is intimately related to the qualitative dimensional notion of number, this likewise applies that we can give meaning to the fractional nature of time (and space) from both complementary physical and psychological perspectives.

It perhaps will be easiest in this respect to start with the number 2 (as ordinal dimension). As we have seen this ordinal dimension (from a qualitative perspective) is intimately connected with its corresponding root (in quantitative terms).

Thus the 2nd root of 1 can be written as 1^(1/2) = - 1 and in quantitative terms this result matches the corresponding 2nd dimension i.e. 1^2 = - 1 (which here relates to a qualitative interpretation).

Thus as we have seen the 2nd dimension in qualitative terms relates to the - relative - negative direction of the nature of time (and space) in switching as between polar opposites in experience. And we already have seen how this dimension is implicitly involved in all scientific interpretation (though explicitly completely ignored).

However because each whole number (as dimension) is intimately linked with its reciprocal (in quantitative terms) this implies that we can now give a meaning to 1/2 with respect to the nature of time (in quantitative terms).

What this simply means is that because now one explicitly recognises the existence of two dimensions with respect to the qualitative nature of time (that are positive and negative with respect to each other) then if we isolate just one of these directions (in absolute terms) it thereby represents 1/2 of the total number of dimensions.

Once again let us illustrate with the simple example of a straight road. So starting from a given point, I can move up or down the road. Now if I separate the two reference frames (considering movement with either "up" or "down" as independent), movement along the road will take place positively in space and time. So clearly because there are two possible directions, one of these in isolation represents 1/2 (of the total number of possible directions).

However if I now consider the two directions as interdependent (befitting the qualitative approach) movement is of a merely relative nature. So positive movement up the road, thereby - relatively - implies negative movement with respect to the corresponding "down" direction. And it is this relatively negative movement that the 2nd dimension directly implies (in qualitative terms).

So an integer number (in qualitative terms) is necessarily associated with a corresponding fraction (from a quantitative perspective). So in this restricted quantitative sense, we can thereby give a fractional meaning to time (and space).

Let us further illustrate with respect to the especially important case of 4 dimensions. The 4 dimensions of 1 correspond in turn to the 4 roots of 1^1, 1^2, 1^3 and 1^4 respectively.

Therefore the 4 roots of 1 in quantitative terms are 1^(1/4), 1^(2/4) = 1^(1/2), 1^(3/4) and 1^(4/4) = 1^1.

The corresponding dimensions in qualitative terms are provided through the reciprocals of these powers i.e. 4, 2, 4/3 and 1.

Now three of these are integral dimensions relating to the 1st, 2nd and 4th dimensions respectively.

However the 3rd dimension (in this context of 4 dimensions) is already expressed as a fractional number. And in this case the fractional number has a qualitative rather than quantitative meaning!

Underlying this is a very deep issue indeed with enormous consequences for the very nature of Mathematics which seems to me entirely overlooked in conventional understanding.

Putting it simply, an unavoidable ambiguity attaches to the ordinal notion of number.

For example we might consider that 3 is an unambiguous number. However 3 can be given both a cardinal and ordinal meaning.
And when we look at 3 in an ordinal sense its meaning is entirely relative. In other words the 3rd of a group of 4 items is quite distinct from the 3rd of a group of 5.

Equally the 3rd dimension (as the 3rd of 4) is quite distinct from the 3rd (as the 3rd of 5).

In other words, properly understood, the ordinal nature of number is merely relative. And as the ordinal itself is inseparably linked with its corresponding cardinal meaning, this implies that the cardinal notion of number - when properly understood - is likewise of a merely relative nature.

This is just another way of recognising that the number system itself represents - when appropriately understood - a dynamic interaction as between quantitative and qualitative aspects (which are - relatively - cardinal and ordinal with respect to each other).

Furthermore, Riemann's finding that a harmonic system of wavelike numbers (the non-trivial zeros) underlines the number system is simply evidence - again when correctly understood in dynamic terms - of the dual relative nature of number.

Therefore in considering higher numbered dimensions (in ordinal terms) we are inevitably led to the generation of fractional numbers for most of these dimensions. And the integer dimensions represent but a special case of these fractional dimensions.

In other words, if we limit ourselves to n members (in cardinal terms) the nth member (as ordinal) can be given an unambiguous interpretation. Thus the 4th dimension (of 4 dimensions) can be written with the integer 4 (in qualitative terms). However the 4th member of any higher number of dimensions will be represented as a fraction (in qualitative terms).

Thus from this perspective, the dimensions of time (and space) can be given meaning in terms of rational fractions both directly in qualitative and indirectly in quantitative terms.

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