From owner-qed Thu Oct 27 09:57:10 1994 Received: from localhost (listserv@localhost) by antares.mcs.anl.gov (8.6.4/8.6.4) id JAA24565 for qed-out; Thu, 27 Oct 1994 09:56:51 -0500 Received: from catseye.idbsu.edu (catseye.idbsu.edu [132.178.200.125]) by antares.mcs.anl.gov (8.6.4/8.6.4) with SMTP id JAA24560 for ; Thu, 27 Oct 1994 09:56:45 -0500 Message-Id: <199410271456.JAA24560@antares.mcs.anl.gov> Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA08637; Thu, 27 Oct 1994 08:57:28 -0600 Date: Thu, 27 Oct 1994 08:57:28 -0600 From: Randall Holmes To: qed@mcs.anl.gov Subject: Semantics Sender: owner-qed@mcs.anl.gov Precedence: bulk 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