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Date: Thu, 27 Oct 1994 08:57:28 -0600
From: Randall Holmes
To: qed@mcs.anl.gov
Subject: Semantics
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The point Dahn makes about the undecidability of questions about
Diophantine equations over the integers relative to any given axiom
set can be taken in more than one way. I think it should be
interpreted naively: we simply don't know everything about the actual
integers, and there is no reason we should. This does not mean that
we aren't perfectly well aware of what the actual integers are (the
way to know what the actual integers are is not to list them all!). I
know what Patagonia is, but I don't know all about it; the same
applies to N.
The question of what is true in the actual integers certainly does
lie within mathematics, even when we can't answer it. The concept of
the "standard model" of arithmetic (as opposed to all the other models)
is a perfectly legitimate mathematical concept. (of course, the wool
can be pulled further over our eyes by considering the "standard model"
of arithmetic as defined in a nonstandard model of set theory, which
might not be standard :-( )
I don't buy into a dichotomy between syntax and semantics. On one
level, we are manipulating syntactical objects. But our objective is
to manipulate syntactical objects in such a way as to be faithful to
what we take to be the reference of these objects (even if we suppose
the referents to be fictitious). I like to think of the syntactical
objects as an implementation of the class of mathematical objects
referred to, with the allowed operations on the syntactical objects
serving as an ADT interface for the type of abstract object which is
being manipulated. I don't think that much sense can be made of
syntactical manipulations considered purely as such.
The opinions expressed | --Sincerely,
above are not the "official" | M. Randall Holmes
opinions of any person | Math. Dept., Boise State Univ.
or institution. | holmes@math.idbsu.edu