Dear Koichiro,
please allow me to contribute to your points:
" ...
Theoretical questions ... mostly refer to what, when, where
> and how. Although the aspects of what and how, may be properly abstracted
> depending upon the questions to be phrased, the other two aspects of when
> and where remain concrete particular throughout without suffering from
> abstractions. ...
> Unless we have a prior framework of space and time, it would be hard to
> raise questions of when and where. A most weakest framework must be
> homogeneous space and time. ... "
in the following fashion:
The where and when (as opposed to what and whow) have in common that they cause a complete
enumeration on the elements 1..n. There is not one single element of a spatial or temporal series
which would not be there, the first till the last. Their order can change or the position in the
sequence on which something specific will happen, takes place, is to be found, etc. is the relevant
info: but the fact remains that each element of a sequence (temporal or spatial sequence) has a
rigid neighbourhood situation among all elements 1..n.
With “how and what” the situation is different. The subject of “how” and
“what” is a quality property which can get an interpretation only by being contrasted
to other elements of the set. (“red” can be only pointed out if there are non-red
elements also.) Therefore, the enumeration of elements answering to a quality question can not be
complete (as the non-red elements have also to number at least 1). Here, we speak of an incomplete
enumeration.
Complete enumeration is used if one supposes rigid neighbourhood situations over a length of at
least n. This is the case in Euclid geometry and Descartes models, it also underlies the concept of
N.
Incomplete enumeration is used, among others, in the case of only segment-wise rigidity of the
neighbourhood relations. A description of a quality has the form of an addition (n1 are red, n2 are
green, n3 are blue and together they number n is n1+n2+n3=n). There, the enumeration starts afresh
within each summand. We have several enumerations 1..k, where k is the extent of a summand (or the
thotal number of summands). In this case, the longest rigid neighbourhood is implicitly presupposed
only to the extent of max(no of summands, extent of biggest summand).
Understanding spatial/temporal and qualitative descriptions can greatly be helped if one
investigates, how long the implicitly presupposed rigid stretch of a continuum actually stretches.
In biology, it does not stretch that far as, e.g. in mechanics.
Hope that you will find this helpful.
Karl
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Received on Thu Oct 20 12:18:02 2005
