LINGUIST List 2.573

Fri 27 Sep 1991

Disc: Is Language Finite?

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  1. Jacques Guy, A census of all possible English sentences
  2. Tom Lai, lang not merely fcn ofREAD/NEW air press vs time

Message 1: A census of all possible English sentences

Date: Wed, 25 Sep 91 12:39:43 EST
From: Jacques Guy <j.guytrl.oz.au>
Subject: A census of all possible English sentences
I have already received a couple of remonstrances triggered by my wee
note that the number of possible utterances is finite for any
language. Since a whole book ("The Vastness of Natural Language",
which I have mentioned earlier on) has been devoted to proving the
strange notion that that number is infinite, I suppose it had to
happen. It seems to have escape the authors that the speakers of any
language are endowed with but a finite life span; so that, even if each
were to devote the whole of his [footnote 1] life to constructing a
single sentence (speaking in his sleep, and with his mouth full, too!),
that sentence would be composed of a large but finite number of
elements. The number of elements of any language being finite, the
number of possible sentences is then finite.
Now, just for fun, let me count the number of possible English
sentences (I'll be generous, too, in my estimate). I had mentioned
last time, off the top of my head, that that number was very much
smaller than a googolplex but perhaps greater than a googol. A googol
is 1 followed by 100 noughts, i.e. 10^100. A googolplex is 1 followed
one googol noughts, i.e. 10^(10^100). Let's see.
As you all know, English has one hundred phonemes which combine
freely, without any co-occurrence restrictions, to form words
[footnote 2]. English speakers live 100 years, which they spend
constructing a single sentence, uttered continuously from birth to
death, speaking in their sleep, and with their mouths full, at the
average rate of 10 phonemes per second. How many possible sentences
are there in English?
Workout:
There are 3600 seconds in an hour, 24 hours in a day, 366 days in a year
(being generous again), 100 years in the life of an English speaker. The
length of an English sentence is therefore:
 10*3600*24*366*100 = 31,622,400,000 phonemes
There are 100 phonemes in English. The number of possible English sentences
is then 100 to the power 31,622,400,000 [footnote 3], which is 1 followed
by 63,244,800,000 noughts. Sure, bigger than a googol (which is 1 followed
by a paltry 100 noughts), but what if we compare it to a googolplex?
Call the number of possible English sentences N. Let us be generous
again and say that N is 1 followed by 100,000,000,000 noughts:
 N = 10^(10^11)
Now one googolplex = 10^(10^100)
I won't go into this much. Let's just say that if N were an electron,
the whole visible universe with its billions of stars wouldn't be enough to
make a googolplex, and by a long, long shot too.
Hmm.... suddenly, N seems rather er... puny?
And now for those who will say "but, recursion!":
PROGRAM AnInfiniteSentenceInTurboPascal;
PROCEDURE ChatterOn;
BEGIN write('and talk, ');
 ChatterOn;
END;
PROCEDURE SpeakUp;
BEGIN write('I talk... ');
END;
BEGIN SpeakUp; ChatterOn;
END.
 FOOTNOTES
[1] Yes, "his". Otherwise some would say I write that women spend all
 their lives yakking and how typical of a male chauvinist
 linguistic pig (phalloglottocrate in French, mot de mon invention
 a` ne pas prononcer la bouche pleine).
[2] I said I would be generous, didn't I?
[3] What largesse! I have been incredibly generous again in my
 estimate. Anyone spot it?
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Message 2: lang not merely fcn ofREAD/NEW air press vs time

Date: Thu, 26 Sep 91 10:44 +8
From: Tom Lai <ALTOMLAICPHKVX.BITNET>
Subject: lang not merely fcn ofREAD/NEW air press vs time
I share Allan Wechsler's (and others') view that Jacques Guy's
claim about the (im)possibility of having an infinit language
consisting of finite lengths of strings generated from a finite set of
symbols. But I must also say that Allan Wechsler's claim that
>utterances are, at worst, real-valued functions of air-pressure
>versus time
is wrong. The issue is the cardinality of languages (in the linguist's
sense or in the computer scientist's sense). Strings of symbols have
"structure".
Tom Lai.
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