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Article 5259 of comp.ai.philosophy:
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>From: ramsay@unixg.ubc.ca (Keith Ramsay)
Newsgroups: comp.ai.philosophy,sci.philosophy.tech
Subject: Re: Peano and the commerce of ideas and representatio
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Date: 25 Apr 92 03:57:04 GMT
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In article <556@trwacs.fp.trw.com> erwin@trwacs.fp.trw.com 
(Harry Erwin) writes:
>[...] Much of what has passed on this board is
>passe' for the category theorists, which is why I invite one to join this
>discussion. 

I am not a category theorist, but I think I know enough about category
theory to raise serious doubts whether it has the relevance you're
proposing for it.

>The existence of the universal object "3" is a consequence of
>the Axiom of Choice. 

"Universal object" has a technical meaning in category theory. I'm not
convinced it is connected in a useful way with the concept of "object"
(or "universal") used in a philosophical context.

I wasn't able to find your reference in the library, but as I recall
the typical use of the axiom of choice in category theory would
specialize, in this case, to using it to show that given a 3-element
set {a,b,c}, there is a function assigning to each 3-element set in a
category of sets, a specified one-to-one correspondence with {a,b,c}.
For each such 3-element set there are, of course, 6 such 1-1
correspondences. The only point is to have a chosen one for all the
sets ("at the same time"). This would be a prelude to making a
universal construction using {a,b,c} as "universal object".

I don't see what bearing this, or anything resembling this, could have
on the philosophical issues. Further explanation is welcome.

>Cohen's demonstration that the Axiom of Choice is in
>the same category as Euclid's 5th Postulate, i.e., independent of the
>remaining axioms, undermined my belief in Platonic ideals, particularly
>after Robinson demonstrated that there were non-standard models of
>arithmetic where the Axiom of Choice did not hold. 

I think the implications of independence results need to be examined
more carefully.

What "status" does the parallel axiom have? We have various
interpretations which we can apply to the terms "line" and "point"; by
some the parallel axiom becomes a true statement, and by some it
becomes a false one.

The hidden suggestion behind many arguments which say that a
proposition has "the same status as the parallel axiom" is something
like that the statement in question (the axiom of choice, for example)
*cannot* be given a fixed interpretation at all (or at least that not
*all* of the statements in an undecidable theory can simultaneously be
given a fixed interpretation.) This is a very different story, and
requires much more argument than simply to say that a different
interpretation can be imposed on them. *Anything* can be
reinterpreted!

By the way, I also find it somewhat weird that a claim such as "there
are non-standard models of (some axiom system for) arithmetic" can be
invoked *against* the existence of abstract structures (of which
models are some!).

>Thus I've regarded much
>of what has passed on this board as being insufficiently grounded in the
>Foundations of Mathematics.

I suspect it is, though perhaps for different reasons than you.

I think this belongs more in sci.philosophy.tech, which is why I've
prolonged the cross-post. Follow up appropriately.
-- 
Keith Ramsay            Even if a proof from given axioms can, in
ramsay@unixg.ubc.ca     principle, be found mechanically, is it not
                        often found in ways not all that different
                        from finding a new axiom? -Georg Kreisel


