Monday, November 24, 2014

Do Numbers Evolve? (2)

In the last blog entry I argued that the conventional belief in the absolute existence of number is untenable from an experiential perspective.

So all numbers possess both external (objective) and internal (mental) aspects which dynamically interact.

Thus the conventional view of number represents but a special limiting case where both poles are fully abstracted from each other. Now this cannot of course completely occur in experiential terms (which would render understanding of number impossible); however it can be approached in a relative manner.

Thus the conventional absolute view of number (as rigid unchanging entities) is then appropriately understood as just one special - though admittedly important - limiting case with respect to interpretation.

As I have frequently stated this is directly associated with linear (1-dimensional) understanding based on interpretation within single isolated polar reference frames.

So for example the conventional treatment of number in merely quantitative terms - rather than a relationship entailing both quantitative and qualitative aspects - represents such linear interpretation.

However when we recognise the truly relative nature of mathematical understanding, as the interaction of opposite poles such as external and internal, this opens up entirely new vistas where the number can be given a potentially unlimited series of dimensional interpretations.

So we move here from the extremely restricted default position of Conventional Mathematics i.e. as 1-dimensional in absolute terms, to an unlimited number of partial relative interpretations, where each number represents a unique dynamic configuration

Therefore from this relative perspective if the interpretation can change as between differing numbers (representing dimensions) then the objective reality then likewise necessarily changes with respect to all these numbers. So from this enhanced dynamic perspective the dimensional notion of number represents perpetual evolution with respect to its very nature.

Now once again, due to the restricted quantitative bias of Conventional Mathematics, this dynamic notion of number evolution is entirely edited out of the picture.

So to give a simple example, when one raises a number 2 to a non-unitary power (i.e. dimension) such as 2, the result is given in a merely reduced quantitative fashion (i.e. as 1-dimensional)!

Thus 22 = 4 (i.e. 41).

Now one can easily appreciate, that when seen in geometrical terms, that 22 represents square rather than linear units. However this qualitative change in the nature of units involved is simply ignored in conventional mathematical terms (with a merely reduced quantitative interpretation remaining).

In fact this reduced view is graphically illustrated in the following quote from Alain Connes (from Karl Sabbagh's "Dr. Riemann's Zeros" P. 205).

“It really is a fantastic step to understand that the square of a number - which is just a geometrical square - and the cube, which is just a geometrical cube - can be added together, even though you would say, "But one has dimension the length squared and the other the length cubed" and you would never add things which have different dimensions. So algebra is an amazing achievement, and once you have formulated things in algebraic terms then they take on a life of their own.”

This brings me directly to consideration of the second fundamental set of polarities that govern all mathematical experience, i.e. whole (collective) and part (individual) which in a very direct way determine this key relationship as between quantitative and qualitative.

So, one recognises a number, as for example "2", both individual and collective aspects are necessarily involved (which are quantitative and qualitative with respect to each other)

Thus in external terms, the individual number object "2" that has an actual existence, has no meaning in the absence of the collective number notion of "2" (that potentially applies to all specific instances of "2").

Put another way the recognition of "2", in any specific case, requires the corresponding notion of "twoness" (that collectively apples to all such possible cases).

Then from the corresponding internal perspective, the individual number perception of "2" - again with an actual existence - has no meaning in the absence of the corresponding concept of "2" (i.e. twoness) with a general potential applicability to all possible cases of "2".

So when the individual recognition of "2" is quantitative (in actual terms), the corresponding collective recognition of "2" (or twoness) is - relatively - of a qualitative potential nature.

However, as always in the dynamics of experience, reference frames can switch, with the individual recognition qualitative and the collective recognition now of a quantitative nature. In effect this qualitative recognition corresponds with the ordinal notion of "2" (as 2nd).

Likewise from this perspective, the collective recognition of "2" (as twoness) is now - relatively quantitative (applying to all actual instances of "2").

Therefore in the dynamics of experience, one keeps switching as between both the quantitative and qualitative notions of "2" in individual terms and equally the quantitative and qualitative notions of "2" (as twoness) with respect to both cardinal and ordinal usage. And this happens both externally with respect to objective recognition and internally (with respect to perception and corresponding concept).

And a similar dynamic interaction is involved with respect to the recognition of any specific number.

We then move on to consideration of the general recognition of number.

Once again this will combine both internal (mental) and external (objective) aspects.

And again the general recognition of an individual number integer in a cardinal quantitative manner has no strict meaning in the absence of the collective qualitative notion of number (as "numberness") that potentially applies in all specific cases.

And then when the reference frames switch we attain the individual recognition of that number in a corresponding ordinal manner (which is qualitative in manner), Then - in relative terms - the collective notion of number attains a quantitative interpretation (as applying to all actual numbers).

Now with respect to conventional mathematical interpretation, all these mutually interacting dynamics are short-circuited in a grossly reduced fashion.

Thus as we have already seen,the external/internal interaction is disregarded with numbers viewed absolutely in objective terms (with a corresponding absolute mental interpretation).

Likewise the individual number "2" is interpreted strictly with respect to its quantitative nature, while the general notion of "2" (insofar as it is recognised) is treated merely with respect to actual occurrences (that are likewise interpreted in a merely quantitative manner).

Then it is somewhat similar with respect to the general recognition of a number with both individual and collective aspects treated in a merely reduced quantitative manner.

However if we are to properly understand the key role that the zeta zeros (both Zeta 1 and Zeta 2) play with respect to the number system, we have to inherently appreciate it in a dynamic interactive manner (where both quantitative and qualitative aspects are equally recognised).

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