LINGUIST List 2.595

Mon 30 Sep 1991

Disc: Is Language Finite?

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Directory

1. , 2.582 Is Linguist Infinite?
2. "Michael Kac", Re: 2.582 Is Linguist Infinite?
3. Pete Humphrey, Is Language Infinite
4. , 2.582 Is Linguist Infinite?

Michael Kac writes:
 Anyone interested in this question might find it worthwhile to look at
 the paper by Rounds et al. in *Mathematics of Language*, ed. A. Manaster-
 Ramer (Benjamins, 1987), particularly the last para. of sec. 4 (p. 354).
As one of the authors of the article, I followed the instructions and
read the last para. of sec. 4. It turns out that we pointed out that
it is indeed reasonable to take languages as finite. However, this
only makes sense if we treat them not as fixed finite sets but as
families of finite sets with no fixed bound on size. As far as I know,
Yuri Gurevich at U. of Michigan and some other people have developed
a formal model of this sort to serve as a model of the behavior
of computers. But I should caution that not much rides on the distinction
between infinite sets and such families of finite sets, although there
are some arguments for the latter view as more realistic (both for
computers and for human beings).
I would also add that
I am surprised by the vehemence with which several correspondence assert
as fact or as scientific truth something that I would regard as a matter
for indirect argument at best and perhaps only of mathematical convenience.
The same applies, of course, the Langendoen/Postal work: I still cannot
believe that Terry and Paul could seriously claim that the issue of
whether NL sentences are merely of unbounded length or of infinite
length (and whether the collection of English sentences was countable
or not) a factual question. But by the same token I cannot understand
the self-righteous dismissal of their proposals by those who somehow
possess the certitude (that I so notably lack) that NL sentences are
only of finite length.
The same applies to the question of whether there are any truly
analog phenomena in nature. I do not feel at all certain that nature
is ultimately digital as some seem to.
Finally, in response to Tom Lai's query, the following is mathematically
correct (this we can be certain of): If we assume that there is no
upper bound on the length of English sentences, then there is an infinite
(but only countable) set of these, since they can then be placed in
one-to-one correspondence with the natural numbers. If we assume that
English sentences can be of actually infinite length, then (depending
on what else we assume), they may form either a countable or an uncountable
set (it is crucial that we assume something more, because you can have
a finite (even a singleton) set of infinite-length sentences).

Message 2: Re: 2.582 Is Linguist Infinite?

Footnote to my reference to Rounds et al.: The question addressed in
the relevant parts of the paper (one of which I explicitly mentioned)
is the question of whether NL's are infinite, not that of whether they're
denumerable. Post facto, I found my own wording confusing.
MK

It seems that any language, in written form, would be countable. The n
symbols of the writing system can be taken as the digits of a base-n number
and every sentence in the language will be a finite string of these digits.
This gives a simple one-to-one correspondence to the natural numbers.
- Pete Humphrey

Kac says:
 Langendoen and Postal's *The Vastness of Natural Languages* ...
 argue that NL's are NONDENUMERABLY infinite -- indeed, that they
 are maximally so (that is, that the number of sentences in a NL is
 greater than any cardinal number).
Assuming finite-length sentences and finite numbers of combining
elements (whether phonemes or something else), set theory won't get
you anywhere beyond countable infinity. How do they get anything
bigger?
By the way, although mathematics talks <<about>> vast cardinals, only
a countable number of mathematical objects can ever be individually
named. Is this the paradox they play on, perhaps?
	-s