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*To*: fom@math.psu.edu*Subject*: FOM: speculations on the future of f.o.m.*From*: Stephen G Simpson <simpson@math.psu.edu>*Date*: Tue, 21 Dec 1999 18:58:41 -0500 (EST)*In-Reply-To*: <199912070051.QAA15441@herbrand.ucsd.edu>*Organization*: Department of Mathematics, Pennsylvania State University*References*: <199912070051.QAA15441@herbrand.ucsd.edu>*Reply-To*: simpson@math.psu.edu*Sender*: owner-fom@math.psu.edu

This FOM posting is a follow-up to Sam Buss's FOM posting of Tue Dec 07 00:51:16 1999. Buss formulates his main point as follows: > My basic assertion was that the real foundations of mathematics is > first-order logic, or the use of rigorous reasoning and > mathematical rigor that can be formalized in first logic. This is > in contrast to the usual point of view that set theory is the > foundations of mathematics. And I think I agree with the underlying sentiment, although I might not formulate it exactly as Buss does. On the eve of the year 2000, let me try to restate the Buss/Simpson point in terms of speculations/predictions about future history. On the one hand, I speculate that the currently dominant, ZFC-style, set-theoretic-foundational orthodoxy may well fall into disrepute fairly soon. This speculation is based on observation of recent directions in f.o.m. research. On the one hand, recent set-theoretical research points to a need for large cardinal axioms going far beyond ZFC, in a somewhat unpcontrollable way. On the other hand, recent research on subsystems of second-order arithmetic indicates that the bulk of current mathematical practice is formalizable in systems much weaker than ZFC. My feeling is that both of these research directions tend to put the ZFC orthodoxy into question, and this could result in widespread rejection of the ZFC orthodoxy within the next 100 years. On the other hand, I predict that our current 20th century idea of mathematical rigor, explicated in terms of formalization in first-order logic, a.k.a. the predicate calculus, will surely remain vital and of the essence in pure mathematics, for at least the next 500 to 1000 years. To me these ideas seem so compelling that I can see no viable alternative. In sum, my feeling is that mathematical rigor and the predicate calculus will endure much, much longer than ZFC set theory as an f.o.m. orthodoxy. FOMers, what are your thoughts on the future of f.o.m. over the next 100 or 1000 years? -- Steve Name: Stephen G. Simpson Position: Professor of Mathematics Institution: Penn State University Research interest: foundations of mathematics More information: http://www.math.psu.edu/simpson/

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