[FOM] stopping at ACA_0

[Stephen Pollard]

On Mon, 20 Feb 2006 Nik Weaver wrote:

>Weyl didn't suggest any philosophical reason for stopping at arithmetic
>definability.  If I remember right he was very clear about the possibility
>of going further but felt that for his purposes there was no need to do
>this.  In other words his criterion for stopping at ACA_0 was esthetic.
>Weyl scholars, please correct me if I'm wrong about that.

A relevant text is DAS KONTINUUM p. 23 (p. 32 of the English): "Eine
Analysis 'mit Stufenbildung' ist künstlich und unbrauchbar." To say that a
move to higher types would be "artificial" does sound like an esthetic
judgment. To say that it would be "useless" sounds more like a conjecture
about mathematical pay-offs.

Weyl continues: "Sie verliert ihr eigentliches Erkennntnisobjekt, die Zahl,
aus dem Auge." The point seems to be that analysis is, above all, in the
business of generating knowledge about "number"; a move to higher types
would, he thinks, deflect attention away from this central purpose. In a
note, Weyl recoils at the prospect of mathematicians devoting themselves to
a "profusion of properties and relations." He seems to think that something
essential to the mathematical (or at least analytic) enterprise is at stake
here.

Stephen Pollard
Professor of Philosophy

Division of Social Science
Truman State University
Kirksville, MO 63501

(660) 785-4653