From owner-qed Wed Nov 2 09:07:46 1994 Received: from localhost (listserv@localhost) by antares.mcs.anl.gov (8.6.4/8.6.4) id JAA17401 for qed-out; Wed, 2 Nov 1994 09:05:47 -0600 Received: from life.ai.mit.edu (life.ai.mit.edu [128.52.32.80]) by antares.mcs.anl.gov (8.6.4/8.6.4) with SMTP id JAA17396 for ; Wed, 2 Nov 1994 09:05:41 -0600 Received: from spock (spock.ai.mit.edu) by life.ai.mit.edu (4.1/AI-4.10) for qed@mcs.anl.gov id AA25234; Wed, 2 Nov 94 10:05:25 EST From: dam@ai.mit.edu (David McAllester) Received: by spock (4.1/AI-4.10) id AA00416; Wed, 2 Nov 94 10:05:21 EST Date: Wed, 2 Nov 94 10:05:21 EST Message-Id: <9411021505.AA00416@spock> To: podnieks@mii.lu.lv Cc: qed@mcs.anl.gov In-Reply-To: Karlis Podnieks's message of Tue, 1 Nov 1994 11:01:50 +0200 (EET) Subject: Semantics Sender: owner-qed@mcs.anl.gov Precedence: bulk I agree that "it is too narrow a view of mathematics to consider it as only concerning the consequences of an initially chosen set of axioms". This is only the first (and the main) component of mathematics. The second one is the art of inventing interesting, powerful, wonderful etc. sets of axioms. The point I have been trying to make (perhaps too often now) is that thinking seems to require thinking Platonically, even for syntacticists who, I claim, think Platonically about syntax. It may be possible to "compile" different axiom systems into one's subconscious so that one can shift one's Platonic thinking between different axiom systems. I am skeptical however. It seems more likely that we think more effectively if we commit our Platonic thinking to a single system. For example, even set theorists who spend their time considering a wide variety of axiom systems are generally fierce Platonisists in their commitment to the existence of V. When they think about other systems then think Platonically (presumably in the same fixed metatheory) about various MODELS of the alternate systems (such as L where the axiom of choice is indeed true). David