From: Chris.Holt@newcastle.ac.uk (Chris Holt)
Newsgroups: alt.individualism
Subject: Re: Prove something (was A = A. Isn't this obvious?)
Date: 5 Oct 92 13:52:41 GMT
Organization: University of Newcastle upon Tyne, UK, NE1 7RU

pjohnson@watserv1.uwaterloo.ca (Peter Johnson) writes:
>In article <BvM2Mu.76o@newcastle.ac.uk> Chris.Holt@newcastle.ac.uk (Chris Holt) writes:

>>"Infinity is the fixed point of the equation x=succ(x)."
>>"John is tall."
>>"Unicorns have exactly one horn."
>>
>>Each of these statements is meaningful, in one way or another; and
>>they refer to infinite, fuzzy, and non-existent things.  Actually,
>>I can understand what you mean if you define "existence" to have
>>only real-world referents; but (not wanting to go back to the
>>question of concepts existing, which we did a couple of years
>>ago) it's a very risky stance to say that there are no fuzzy facts.
>>Why should the universe be precise?

>	(1)	I don't know what the equation x=succ(x) represents, so I
>can't comment on it.

Consider the natural number line generated by the anadic operator
0 (or 1) and a monadic operator succ, the successor function.  Under
Peano's axioms we are guaranteed for all x that the successor of
x is different any of its predecessors.  However, we can take the
limit of the line and call it infinity, which has the property that
the successor of infinity is itself infinity.  Or, if you prefer,
in the complex plane we can talk about taking contour integrals
through infinity (usually via the projection mapping involving
the unit sphere above the origin).  I'm basically just saying that
it is as meaningful to talk about infinity as any other mathematical
concept.  If it was intended to mean that mathematics has no direct
referents in the physical universe, that's pretty trivial.  But
since our domain of discourse includes mathematics, I'm not sure
what the point of it is.

>	(2)	"John is tall." is not fuzzy in context.

On the contrary; we're dealing with fuzzy logic here, not
probability.  We don't set a sharp boundary, such that everyone
over 6'2" (for example) is tall, and everyone under that height
is not.  Rather, we say that someone 6'1" is sort of tall,
someone 6'2" is quite tall, and someone 6'3" is very tall.
The idea is that set membership here doesn't return either
true or false.  See Zadeh a decade ago or so for more.

>	(3)	Unicorns exist as objects in paintings, and you are quite
>correct that it is their nature to have exactly one horn.

But the statement is not meaningful *by virtue of the fact* that
they appear in paintings; they also appear in stories, and one
could imagine that they would only appear in someone's imagination
(when they were invented).  If you wish to say that anything
anyone ever thinks of exists in the domain of concepts, and a
subset of this is mapped to the domain of physical reality
(by people's mental models); and conversely, everything in
the real world exists in physical reality, and a subset of
this is mapped to the domain of mental concepts (again by
mental models), then even people like Quine might start agreeing
with you.  [not exactly, but close enough]

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 Chris.Holt@newcastle.ac.uk      Computing Lab, U of Newcastle upon Tyne, UK
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  Still shone the candlelight on the empty screen, after the power went.