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From: costello@DEC-Lite.Stanford.EDU (Tom Costello)
Newsgroups: comp.ai.philosophy
Subject: Re: how many distinct thoughts can a person have?
Message-ID: <1991Jun19.233743.27733@neon.Stanford.EDU>
Date: 19 Jun 91 23:37:43 GMT
Article-I.D.: neon.1991Jun19.233743.27733
References: <1991Jun19.033316.18773@athena.mit.edu> <314@trwacs.UUCP> <1991Jun19.173307.10704@athena.mit.edu>
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Reply-To: costello@DEC-Lite.Stanford.EDU (Tom Costello)
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In article <1991Jun19.173307.10704@athena.mit.edu>, mlevin@athena.mit.edu (Mike Levin) writes:
|> In article <314@trwacs.UUCP> erwin@trwacs.UUCP (Harry Erwin) writes:
|> >mlevin@jade.tufts.edu writes:
|> >
|> >
|> >>thoughts is uncountable." Does anyone have any arguments for or
|> >>against the idea that the number of possible distinct human thoughts
|> >>(or mental states) is uncountably infinite? 
|> >
|> >The number of distinct human thoughts isn't even countably infinite
|> >in a quantum-mechanical universe,
||> >[stuff deleted]
|> >-- 
|> >Harry Erwin
|> >Internet: erwin@trwacs.fp.trw.com
|> 
|>   I am sorry - I didn't specify the following in my original message.
|> I am not looking for arguments from physics or neurobiology (but if
|> you have an interesting one, let's hear it).
|
|> mental state because ...". So I am more interested in arguments
|> stemming from the higher-level concepts of "idea" or "thought" rather
|> than the bottom-up approcah of trying to figure out how many states
|> the brain can have, etc. In asking this question I realize that I am
|> disregarding the behaviorist objection, and also (in trying to make
|> this substratum- and theory-independent) appealing to each person's
|> knowlege of what an idea or thought is.
|> 
|> Mike Levin


This sounds like Russell's quote, that ``there is a simple
arithmetical proof that there are less things in heaven and earth than 
are dreamt of in my philosophy''.  This is based on the cardinality
of the power set of all things being greater than the set of all things.

To say the number of possible thoughts is uncountable is stretching
the idea of countability from math to philosophy.  In mathematics the 
term means there exists no one to one correspondence in the system,
of the set to the set of natural numbers.  The term is relative to
the system doing the correspondence, a set can be countable to 
one system and uncountable to another.

  The fact that we have the
concept of uncountable is not enough to prove we have uncountable thoughts.
This follows simply from the fact there are countable models of ZFC.


In fact given that there is presumably a model of all possible thoughts,
of some infinite cardinality,that is there is 
a model of all concepts and all relations on concepts
, there must therefore be a model of cardinality aleph 0, by
the downward Lowen Skolem theorem, this of course assumes that
the language of thought is countable, but this seems reasonable even to
dualists on purely behavourialist grounds.

The great danger in using concepts like uncountable in philosophy,
or ai, is that the person using them usually has an agenda.
That is something along the line of, ``there are an uncountable number of
possible thoughts therefore the physical symbol hypothesis is false.''

Using uncountable for arguments like this is pure rhetoric, and shows
a lackl of understanding of the term.  In fact I cannot think of any
particular use of uncountable in ai that wuld not be entirely equivalent
to its use in pure logic, or formal systems, and thus its use
in describing thoughts assumes rather than rejects the
physical symbol hypothesis.

Tom.