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In article <971@kodak.UUCP> bayers@kodak.UUCP (mitch bayersdorfer) writes:
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| I don't mean to make a reductio ad adsurdum, but can any alphabet which
| a given human can perceive be truely infinite?  Given that human beings have 
| a finite (but very large) number of neurons in their visual and cerebral
| cortex, and that any distinct alphabetic character would exceed the 
| thresholds of an enumerable permutation of those neurons, mustn't there
| be only a finite number of characters (and concepts?).  I am assuming that 
| neural thresholds for a certain learned concept are relatively constant.  If
| another concept produces the same permutation, but only to a greater 
| degree, then couldn't that be reduced down to two concepts-- one a measure 
| of the concept, and the other a measure of its degree?  
|               - Mitch Bayersdorfer
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You are addressing a problem that linguists have had to wrestle
with for a long time. Back in the 1950's, the linguist Noam
Chomsky did some original research on the properties of formal
grammars that might be used to generate sentences in a natural
language. Among other things, he discovered that any reasonable
grammar (or production system) to generate sentences in natural
language would have recursive properties that would, in effect,
make the set of sentences of the language infinite in size. The
corollary to this, of course, is that there is no "longest
sentence" in a natural language. Realizing that it is somewhat
silly to claim that a human can utter or comprehend a sentence
that might be a few giga-centuries in duration, Chomsky fudged
around and invented a distinction between linguistic
"competence" and linguistic "performance". Actually, it's a
reasonable fudge.

A good analogy for this is simple arithmetic. If you know the
rules of, say, multiplication, there is no reason why you can't
multiply two numbers that require a few giga-light-years worth of
paper to write down (in 10-point type). You are theoretically
competent to do this. Your performance abilities (life span,
attention span, etc.) will, however, undoubtedly keep you from
achieving the end product.

Cantor discovered several kinds of infinities. Perhaps what we
need to discover is a kind of sub-infinity. The actual number of
sentences in any natural language is "infinite" in the same
sense that the number of grains of sand on a beach is infinite,
i.e. "not really".

Are alphabets like sentences? Uh, no, not at all! I have stayed out
of this discussion about "infinite alphabets", since I do not
understand the issue. I enter reluctantly. Here goes...

An alphabet is a set (decidedly finite) of glyphs (e.g., little
marks on paper). A language adopts an alphabet, and tries to map
its sound system (its phonology) onto the alphabet. Sometimes
this works well, sometimes not so well. For example, both
Italian and English use the Latin alphabet. Neither language's
phonology maps directly to the alphabet, but Italian maps better
than English. In Italian there are only a few little problems
(the letters "c", "g", "e", and "o" do not map one-to-one to
Italian phonemes). In English, there are many, many problems
(which we are all aware of).

The sound system of every language contains a finite number of
phonemes, and a finite (but much larger) number of syllables.
Therefore, any alphabet (phomeme-glyph mapping), or any
syllabary (syllable-glyph mapping) should be finite.

Ideographic writing systems, such as Japanese "kanji", are not
alphabets, and they are not syllabaries. If we assume that the
languages that use ideographic writing systems allow free
creation of new glyphs, then these writing systems have an
infinite number of glyphs, in exactly the same sense that there
is an infinite number of sentences in English, or that there is
an infinite number of art works that can be created by mankind.

-- 
Tim Smith
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