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*To*: fom@math.psu.edu, simpson@math.psu.edu*Subject*: Re: FOM: role of formalization in f.o.m.*From*: kanovei@wmwap1.math.uni-wuppertal.de (Kanovei)*Date*: Tue, 1 Jun 99 08:15:41 +0200*Cc*: kanovei@cs.utk.edu*Sender*: owner-fom@math.psu.edu

From: Stephen G Simpson <simpson@math.psu.edu> Date: Mon, 31 May 1999 19:36:19 -0400 (EDT) > As a preliminary scheme that seems appropriate for discussing the > current scene in f.o.m., I would propose to distinguish among: > > 1. rigorous mathematics, referring to normal 20th century standards > of mathematical rigor. > > 2. metamathematics, i.e. rigorous arguments showing showing how > various pieces of rigorous mathematics can be codified in the > predicate calculus. > .............. There is another well defined case yet not fully classified by this scheme. 1*. Rigorous mathematics which a) starts with an EXPLICITLY DEFINED list L of "postulates" b) predends that what follows is based on L and on nothing else, c) but, differently from 2, does not claim any adherence to any deduction system and may not mention any formal deduction at all. Example: Euclid, Hilbert. This is equal to Simpson's 1 IF we add to 1 that L=ZFC, say. This is also equal to Simpson's 2, IF we note that the rigogous mathematical reasoning is exactly the predicate calculus. Strangely, these two IFs appear to be very important for some mathematicians of undisputable greatness. V.Kanovei

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