Monday, December 14, 2015

Wholes and Parts (1)

I keep coming back to the most fundamental issue possible with respect to the true nature of number, which unfortunately due to the reduced nature of present mathematical interpretation is completely ignored.

It might help initially to explore this key issue in a concrete manner with a simple practical illustration.

Imagine that we have a cake that is cut into 2 (equal) slices.

Now as each slice comprises one distinct unit we could represent the cake as,

1 + 1 = 2.

In other words the two slices (comprising the cake) entail the addition of the individual (separate) units.

However this represents but a reduced interpretation of the relationship as between whole and parts whereby the (whole) cake is viewed in a merely fragmented manner as the quantitative addition of the individual unit parts.

So therefore from this reduced - merely quantitative - perspective the (whole) cake is represented as 2 (part) units.

However the cake has also its own unique whole identity, which would be represented as 1 (i.e. one whole cake).

So we have the paradox that the cake can be represented as 2 parts or alternatively as 1 whole.

So in the very dynamics of recognition, in order to relate parts and wholes we must implicitly switch as between both part and whole recognition (with respect to objects) or alternatively as between quantitative and qualitative recognition, which are dynamically related to each other in a complementary manner..

Thus again with respect to this example the quantitative recognition of the cake represents its 2 - relatively independent - part slices.

The corresponding qualitative recognition (in this context) then relates to the recognition of the cake as a whole unit (i.e. as interdependent with itself).

Of course the cake could now in turn attain a (part) quantitative status - say - as one of a collection of  cakes!

In our example, we initially treated the slices of the cake as quantitative parts (in relation to the whole cake).

However, each slice in turn has a qualitative identity whereby it is recognised as a whole in its own right. So if for example each slice contained individual components, these would thereby now constitute distinctive parts in relation to the whole slice!

In more general terms, phenomenal reality is necessarily composed of holons (i.e.whole/parts) whereby, in any context, what is whole (from one valid perspective) is equally part from an equally valid related perspective.

And in reverse terms, phenomenal reality is composed of onhols (part/wholes) whereby what is part (from one valid perspective) is equally whole from an equally valid related perspective.

So, in the example above, we illustrated how the whole cake (in relation to its 2 slices) could equally be part (as an individual item in a collection of cakes).

Equally, we saw how the part slices (in relation to the whole cake) could equally serve as unique wholes (in relation to constituent parts of each slice).

I cannot stress enough how important this distinction as between the part and whole status of an item (which is relatively quantitative as to qualitative and qualitative as to quantitative respectively) truly is, for when grasped, it leads to the need for a fundamental new interpretation of the very nature of the number system.

Basically in conventional mathematical terms, a merely reduced quantitative interpretation of number is given, which is of an absolute static nature.

So, for example, though we do indeed refer to a natural number such as 2 as a (whole) integer, in effect it is defined in a merely reduced part manner as quantitative.

Thus 1 + 1 = 2. In other words the whole number (i.e. 2) is treated simply as the quantitative sum of its constituent parts. So again, a fundamental reduction of qualitative in terms of quantitative meaning is thereby directly involved.

However, when we properly allow for the truly distinctive nature of both part and whole meanings in relation to number (which again are - relatively - quantitative as to qualitative and qualitative as to quantitative respectively) we must necessarily move to a new dynamic interactive treatment of the number system.

I will suggest the appropriate manner for achieving this in the next entry.

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