Robert A. (Bob) Morris ram at CS.UMB.EDU
Tue Jan 8 01:07:42 CST 2002

```Ken Kinman writes:
> Date:         Mon, 7 Jan 2002 16:07:35 -0600
> From: Ken Kinman <kinman at hotmail.com>
> To: TAXACOM at usobi.org
> Subject:      Ashlock was treated badly

> [...]
>               ------- Ken
> P.S.  And it strikes me as odd that Ashlock's attempt at precision
>should be
> regarded as illogical or unwarranted, and yet precision suddenly
>becomes
>"directionality"
> of character states.  In most fields of human endeavour, "ordering"
> encompasses both adjacency and directionality.  I think Mr. Spock
> why you don't consider your "polarization" as a part of "ordering".
> ...

Umm, as a computer scientist and former mathematician, I am clueless
"directionality" in this context. However, as normally used in
mathematics, adjacency typically has little meaning except for
discrete sets. For example a continuous numeric quantity would
normally be said to be ordered, have directionality, but no
adjacency. Among the numbers between 0 and 1, 1/3 is smaller than 1/2
but 1/3 is not adjacent to any number at all. Between 1/3 and any
number you choose there always lies another number. Many other
numbers. Many, many, really.(*).

An ordered set in which between any two elements always lies
another---i.e. an ordered set with no adjacency relation---is said by
mathematicians to be "dense".

Bob Morris

(*)inside joke. See any text that discusses the uncountablity of the
real numbers and the countability of the rationals. In fact, pursuing
such a book will show you that there actually is an ordering of the
fractional numbers between 0 and 1 that does have an adjacency
relation on it---but it is not the usual ordering (for example, in
this ordering 1/2 comes before 1/3 and 2/3 before 1/4....) and its
interest is largely unrelated to the familiar arithmetic or
measurement aspects of fractions. But if you make it through the book,
you will be convinced that no matter how cleverly contrived, there can
not be any adjacency relation on the set of all real numbers,
including the irrationals like pi, sqrt(2), etc. Those irrationals,
they're tricky, but ya gotta love 'em. Without them we could not have
calculus as we know it, and then where would biology be? Hah, hah,
just serious.

```