Hi, siam-ga:
The prefixes "in-" and "ex-" have come down to us in both English and
French from Latin prepositions.
Some discussion of "ex-" was already given in an earlier Google Answers thread:
[Q: What does the "ex-" stand for? (Google Answers)]
http://answers.google.com/answers/threadview?id=437977
where Google Answers Researcher gentryunderwood-ga noted that in Latin
"ex" means "out(side) of":
[ex- (one entry, Online Etymological Dictionary)]
http://www.etymonline.com/index.php?term=ex-
Especially in Latin the preposition has a secondary connotation of
"from". For example, creation "ex nihilo", literally "out of
nothing", intends to convey creation "from nothing".
The English meaning "in(side)" or "(with)in" of prefix "in-"
(sometimes "im-") is familiar as the preposition still means largely
the same thing in English as it did in Latin, although in French the
modified spelling "en-" (or "em-") is more common (and we have
"embraced" many such words in English also).
[in- (two entries, Online Etymological Dictionary)]
http://www.etymonline.com/index.php?term=in-
A confusing thing is that "in-" can also mean "not", in both Latin and
English, derived from the Greek prefix "a(n)-", as well "un-".
However to be brief, "include" and "exclude" are derived by prefixing
from the Latin verb "claudere", meaning to close or shut, to meaning
"keep in" or "keep out":
[include]
http://www.etymonline.com/index.php?term=include
[exclude]
http://www.etymonline.com/index.php?term=exclude
* * * * * * * * * * * * * * * * * * *
The distinction between inclusion and exclusion presents one with the
seminal boundary betwee what is accepted and what is rejected. Such
boundaries, both in absolute and relative contexts, form the
environment for logic and reason. So it is reasonable to double check
for hidden connections as we encounter them in programming and
mathematics.
* * * * * * * * * * * * * * * * * * *
The words minimum and maximum (plurals minima and maximum) also have
evident Latin origins. From the Latin root "min-" meaning small, we
get "minus" for a deficit, the comparative "minor" for smaller, and
the superlative "minimus" (neut. "minimum") meaning least:
[minus]
http://www.etymonline.com/index.php?term=minus
[minimum]
http://www.etymonline.com/index.php?term=minimum
Similarly maximum is the Latin neutral superlative to a root "magnus"
meaning great or large:
[magnum]
http://www.etymonline.com/index.php?term=magnum
[maximum]
http://www.etymonline.com/index.php?term=maximum
* * * * * * * * * * * * * * * * * * *
In an earlier Comment I pointed out that while inclusion and exclusion
are dichotomous ("you're either in or out"), the relationship between
minimum and maximum is not that of dichotomy, as in particular
something could in theory be both at the same time. I referred
instead to these as dual concepts, an approach admittedly influence by
mathematical training, since one obtains one from the other by
reversing the "sense" (direction) of the implied notion of inequality
(comparison).
At the level of mathematical logic one can often make out, as here,
that the "dual" is a non-trivial form of double negative.
Suppose for simplicity that our context is a linearly ordered finite set.
The "included" maximum is that value in the set which is greater than
any other element. It is at the same time the only element of the set
which is _not_ less than some other element of the set.
Likewise the "included" minimum can be defined as the value in the set
which is less than any other element, or alternatively as the only
element of the set which is _not_ bigger than some other element of
the set.
If we switch to a discussion or definition of "excluded" maximum or
minimum, we find that there isn't much more that can be said to
generally connect these with their "included" counterparts (except to
say in a finite context as here that they are necessarily unequal,
assuming a value cannot both be in and out of the set).
In mathematical analysis there are contexts, inherently infinite such
as the real numbers/Dedekind cuts on the rationals, in which it is
important to prove equality between the least upper bound of a
strictly increasing sequence {x_i} and the minimum (greatest lower
bound) of the complement of the union of half open intervals (-oo,
x_i). To fully explore these possibilities would lead us on a
tangential discussion from which we'd not soon return.
* * * * * * * * * * * * * * * * * * *
We can push the derivations of words further back into the
Indo-European or proto-Indo-European family of languages. The
Latin/Greek roots "mag-"/"mega-" trace back to the Sanskrit "mahat"
for great:
[m:ht:a (mahataa) = great (Sanskrit)]
http://www.alkhemy.com/sanskrit/dict/dictall.html
or even these apparently related roots from the Dravidic family of languages:
[235. magh = might, power (Dravidian)]
http://www.datanumeric.com/dravidian/page060.html
[247. me III, megh = great (Dravidian)]
http://www.datanumeric.com/dravidian/page061.html
The Latin/Greek roots "min-"/"mei-" may well trace back to the
Dravidian "mei" for little, lessen:
[250. mei I = little, lessen (Dravidian)]
http://www.datanumeric.com/dravidian/page062.html
It is easy to find echoes of our many modern prefixes for "not" in
these ancient languages, niH in Sanskrit and ne in Dravidian:
[en:H (niH) = without (Sanskrit)]
http://www.alkhemy.com/sanskrit/dict/dictall.html
[285. ne = negation (Dravidian)]
http://www.datanumeric.com/dravidian/page065.html
However I found no evidence that the dichotomy of in-/ex- prefixes or
prepositions would "unify" in some way with the distinction between
small and great at any stage of recorded language, short of
degenerating to a common connection with not/negation/without.
regards, mathtalk-ga |
