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*To*: fom@math.psu.edu*Subject*: FOM: Scope of FOM?*From*: John Pais <paisj@medicine.wustl.edu>*Date*: Sun, 06 Jun 1999 11:17:50 -0700*CC*: paisj@medicine.wustl.edu*References*: <v03110725b3784da58fce@[24.93.99.130]>*Sender*: owner-fom@math.psu.edu

FOM is a list addressing Foundations of Mathematics (f.o.m.). However, it is not clear to me (a fairly new list member) from recent posting exactly how broad the f.o.m. scope is. It seems that two important main threads involve: 1. What follows from what--intuitively or "naturally" (as indicated by R. Tragesser). 2. What follows from what--formally (e.g. as indicated by H. Friedman and S. Simpson). So, assuming 1 is permitted, then we also have: 3. Given a candidate of type 1, i.e. an intuitive representation of a piece of mathematics, and a candidate of type 2, i.e. a formal representation of the same piece of mathematics, then set evaluative criteria through which both types of representations can be compared and contrasted in terms of how 'effectively' and 'faithfully' each captures the mathematics itself (presumeably representation independent). 4. Create and evaluate hybrid representations 'optimized' in various ways and for different purposes on their type 1 and type 2 features. Based on my limited FOM experience it seems that 4. has been addressed only very weakly by restricting attention (FOM list scope?) to representations of type 2, with some discussion of inevitably emerging type 1 features. In my opinion, serious attention to 4. is very important for *communicating* mathematics within the community of mathematicians: researchers, teachers, and learners. My intent is not to try to diminish the importance of precisely and formally codifying mathematics, but to specifically focus on Mac Lane's qualification (in Mathematics: Form and Function, p. 377) that doing mathematics involves "not the *practice* of absolute rigor, but [the maintenance of] a *standard* of absolute rigor." p. 378 "Moreover, there are good reasons why Mathematicians do not usually present their proofs in fully formal style. It is because proofs are not only a means to certainty, but also a means to *understanding* [my italics]. Behind each substantial formal proof their lies an idea, or perhaps several ideas. The idea, initially perhaps tenuous [e.g. intuitive], explains why the result holds. The idea becomes Mathematics only when it *can be* [my italics] formally expressed, but that expression must be so couched as to reveal the idea; it will not do to bury the idea under the formalism." p. 379 "...Proofs serve both to convince and to explain--and they should be so presented." So, let me respectfully ask Harvey and Steve, is the 'foundational' activity I describe above in 3 and 4 within the scope of the FOM list? Thanks, John Pais

**Follow-Ups**:**Stephen G Simpson**- FOM: formalization; Pais/Gonshor confusion

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