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Article 4908 of comp.ai.philosophy:
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>From: holmes@opal.idbsu.edu (Randall Holmes)
Newsgroups: comp.ai.philosophy
Subject: Re: ai
Keywords: ai,realism,Goedel,platonism
Message-ID: <1992Apr3.200218.26197@guinness.idbsu.edu>
Date: 3 Apr 92 20:02:18 GMT
References: <atten.702298596@groucho.phil.ruu.nl> <ktp7d0INNgna@exodus.Eng.Sun.COM>
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In article <ktp7d0INNgna@exodus.Eng.Sun.COM> silber@orfeo.Eng.Sun.COM (Eric Silber) writes:
>In article <atten.702298596@groucho.phil.ruu.nl> atten@phil.ruu.nl (Mark van Atten) writes:
>>There is an ongoing discussion right now on mathematical realism in the ai
>>group. Some people question the relevance of this to ai. In this article,
>>I want to discuss
>>1 The objective existence of mathematical objects
>>2 The significance of this for ai
>>
... stuff deleted...
>>Further, even if names are arbitrary, yet once they have been imposed their
> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>>consequences are necessary and certain truths arise which, though they depend
> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>>on the symbols imposed, are nevertheless real. For example, the rule of nine
> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> This only establishes 'local-truth' insofar as the naming and inference
> apparatus are bound to the anatomy of an individual mind or class of minds.
> ( The genetically encoded base-layout of the human brain circumscribes how
> any instance of a human brain/mind can 'name' and 'infer' )
..stuff deleted...
>>I.2 Goedel on objectivity
>>'The same possibilities of thought are open to everyone, so the world of
>  ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>>possible forms is objective and absolute. Possibility, then, is not dependent
> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>>on an observer; it is therefore real because it is not subject to our will.'
> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> These propositions suffer from the same error as Leibniz above, viz.
> they assume that because a certain class of minds is subject to the same
> constraints, ALL minds, and even the Universal Mind (if any) must admit
> the existence of the same 'fundamental forms'. 
> Well, I may be grossly wrong , if so, please show me how, but I think that
> all members of Goedel's class named 'everyone' are humans, and he cannot use
> the inherited common architecture of the class of human brains
> as a hook upon which to hang the existence of "absolute possible forms".
>....

This is an argument which is frequently used.  The problem with it is
that there is absolutely nothing to indicate that mathematics or logic
has anything to do with "the anatomy of a class of minds".  There is
no neurological basis for 17 being prime!  The "inherited common
architecture of the human brain" is irrelevant to the issue.  There
are excellent reasons to believe that the sensory capabilities of
other intelligent beings (if any) would be profoundly different from
ours, and so would their emotional make-up or "temperament" (emotions
they might lack entirely).  There is no reason at all to believe that
their _logic_ or _mathematics_ would be different from ours (except
insofar as they might have differing aptitudes and interests, and so
study different areas or have more or less success in establishing
results).  An example: we can grasp the cardinalities of sets
directly--up to about 7.  After that we have to count.  A being who
could grasp cardinalities directly up to about 100 would be better at
certain kinds of mathematical manipulations than we are--but it would
agree with us that 17 is prime.  Since Silber brought the Universal
Mind into it, let's consider one who grasps aleph-null objects all at
once (probably not an inhabitant of our physical universe!)  Such a
being would be able to answer questions directly that we cannot answer
_at all_ (it can decide all questions in Peano arithmetic, including
consistency for all of our theories, by inspection--but it would have
its own puzzles!).  Nonetheless, it would agree that 17 is prime.
Either of these beings might have no interest in prime numbers; but if
the question were posed, either would agree.
	The underlying problem is the notion that truths of logic and
mathematics are "constraints on our minds".  I don't think that these
subjects actually have anything to do with psychology at all.  (From
my standpoint, Godel is making the same mistake in referring to
"possibilities of _thought_[my italics]").  I maintain that truths of
logic and mathematics govern the possible arrangements of objects,
properties and relations in all domains.  The reason psychology comes
into it is that to do mathematics _we_ have to think about it.  17 is
prime independently of our thinking about it, and its primality is not
a fact of psychology.  Frequently we have to fight our psychology in
order to do mathematical or (technical) logical thinking at all (a lot
of us _lose_).  The best analogy I can think of is to claim that
because we study the stars by looking at them, the stars belong in the
domain of the study of vision!  Certainly the stars are objects of
vision, but we do not learn about them by studying our eyes, nor will
the universe of stars be different for a being with different eyes
(this being might have more or less information about the stars, but
not _contradictory_ information about the stars).
	What are the "fundamental forms" at issue?  To see that 17 is
prime, all that is necessary (speaking informally) is that there be
objects, that any pair of objects be either equal or unequal, and that
there be enough of them!  I don't agree that there is or can be a
class of minds which would disagree with the conclusion that 17 is
prime; there certainly are classes of minds which don't care, but that
is a separate issue.  Moreover, I think that the issues in _all_ of
mathematics and logic are finally combinatorial issues on the same
kind of level (speaking _very informally_!--I am not saying that all of
mathematics is combinatorics in a technical sense, and I note that
some of it is _infinite_ combinatorics in this loose sense), and that
any "constraints" implied thereby do indeed apply to everything and
everyone (in the broadest sense) without exception.  I put
"constraints" in quotes, because I don't think that any real
constraints are involved.  It is not a deprivation of liberty to be
forbidden to implement logical impossibilities.


-- 
				--Sincerely,
				M. Randall Holmes
				Math. Dept., Boise State Univ.
				holmes@opal.idbsu.edu


