Friday, July 17, 2015

Zeta Zeros Made Simple (9)

We have seen that the ordinal nature of number relates directly to its holistic aspect. This is in contrast to the cardinal notion - which by contrast - initially appears as analytic in nature.

What this further means is that whereas the cardinal notion is based directly on the notion of number as independent (from other numbers), the ordinal notion is based directly on the complementary notion of number as interdependent (and thereby related with other numbers).

So for example we view the cardinal number "3" as independent from other numbers in a quantitative manner.

However the ordinal notion of 3rd is necessarily interdependent with a group of other numbers.

So 3rd in the context of the simplest group of 3 related numbers, implies a different relationship than 3rd in the context of  - say - 20 numbers!

Strictly speaking, analytic interpretation is always based on the clear separation of opposite poles.

Thus once again when one refer to a cardinal number in an analytic sense, this implies that the quantitative aspect can be clearly separated from the qualitative! This implies in effect that with such interpretation the qualitative aspect (insofar as it is recognised) is assumed to directly correspond with the quantitative and thereby is reduced to the quantitative.

A mathematician if sufficiently pressed, may even concede that the mental constructs that are internally necessary to interpret the number reality in an objective external manner are strictly of a qualitative nature.

However the subsequent problem of the relation of such psychological constructs with this objective reality is quickly explained away by assuming an absolute correspondence as between both aspects, so that qualitative aspect can thereby be completely ignored.

Holistic interpretation by contrast is based on the dynamic interaction as between opposite poles.

Thus with ordinal recognition, both quantitative and qualitative aspects are necessarily involved.

So, for example, to make the ordinal recognition recognition of 3rd, one must first identify a number group collectively in cardinal terms - say - 3. Then the notion of 3rd implies a qualitative relationship as between the 3 members of this group as 1st, 2nd and 3rd respectively.

Now in the dynamics of experience, both cardinal and ordinal recognition are mutually involved.
Thus cardinal recognition implicitly entails corresponding ordinal recognition; equally ordinal recognition implicitly entails corresponding cardinal recognition. Therefore, properly speaking, both the cardinal and ordinal aspects enjoy a merely relative identity as complementary partners.

However in conventional mathematical terms, the cardinal aspect is misleadingly given an absolute identity with the ordinal thereby reduced to the cardinal.

So for example the relationship of the primes to the natural numbers is viewed in conventional terms solely with respect to their quantitative cardinal identity.

There is indeed however one limiting case where ordinal meaning does indeed reduce directly to cardinal interpretation.

As we have seen, the cardinal definition of 3 = 1 + 1 + 1 (representing homogeneous independent units).

Now, if we continually identify each ordinal number solely with the last member of each number group to which it belongs, then ordinal reduces directly to cardinal meaning.

So for example the last member of a group of 1 unit is clearly that same unit.

So 1st in this context = 1.

If we continue on to identify 2nd with the last member of a group of 2, its meaning is again unambiguous as the last unit of the group.

Then when we identify 3rd with the last member of a group of 3, the meaning is again unambiguous as the last unit.

So 3rd in this context = 1.

Thus with reference to 3, 1st + 2nd + 3rd = 1 + 1 + 1 = 3.

Thus here, ordinal meaning - where each ordinal position is rigidly fixed with the last member of its number group - equates directly with cardinal meaning.

So 1st means 1st in the context of 1; 2nd means 2nd in the context of 2; 3rd means 3rd in the context of 3 and so on.

However, there are innumerable other ways in which each ordinal position could be defined. For example we could define 2nd in the context of 3 members or 2nd in the context of 100 members and so on. Indeed we can identify 2nd in this alternative way with every cardinal group > 2.

Thus an unlimited number of options thereby exist for each ordinal position with a merely relative identity.

If one obtains 2nd place in a competition confined to 3 entrants, this might not seem very impressive. However if it is 2nd in relation to say 10,000 entrants this now - relatively - appears a much greater achievement.

Therefore each ordinal notion (1st, 2nd, 3rd, etc.) can be viewed in two ways!

1) in an absolute fixed rigid manner - amenable to analytic understanding - where it is defined as the last member of its appropriate cardinal number group (i.e. 1st last of 1, 2nd last of 2, 3rd last of 3 and so on).

2) in a relative flexible manner - amenable to holistic understanding - where it can be defined in a relatively distinct manner with reference to an unlimited number of cardinal groups (greater than the minimum size required for the absolute definition).

Properly understood 1) is really just a special limiting case of 2).

Therefore the really big task arises as to how to give expression to the potentially unlimited number of relative options associated with each ordinal position.

We will look at this in the next blog entry, showing how it leads naturally to the complementary - though still largely unrecognised - notion of the Zeta 2 zeros.

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