Over the past few years I have been engaged in reassessing computational logic on the light of the needs of working mathematicians and users of mathematics: I argue that a new approach is needed if we are to be of value to this community.
As an example I present recent work on computational logic tools to support the symbolic analysis of differential equations in computer algebra systems like MAPLE or Mathematica. Computer algebra systems implement algorithms for differential rings and fields, rather than notions involving limits, and hence are unreliable in matters involving analysis rather than algebra, particularly where parameters are involved. We consider the problems inherent in using a computer algebra system to investigate differential equations, and describe how we have solved some of them by means of calls from MAPLE to a continuity checker and a library of facts about elementary functions which we have implemented in PVS.
A model of computation is abstract if, when applied to any algebra, the resulting programs for computable functions and sets on that algebra are invariant under isomorphisms, and hence do not depend on a representation for the algebra. Otherwise it is concrete. Intuitively, concrete models depend on the implementation of the algebra.
The difference is particularly striking in the case of topological partial algebras, and notably in algebras over the reals. We investigate the relationship between abstract and concrete models of partial metric algebras. In the course of this investigation, interesting aspects of continuity, extensionality and non-determinism are uncovered.
This is joint work with J.V. Tucker (Swansea, Wales).
Gödel has argued that we can cultivate the intuition or 'perception' of abstract concepts in mathematics and logic. His ideas about the intuition of concepts are related to many other themes in his work, and especially to his reflections on the incompleteness theorems. I will describe briefly how Gödel's claims about the intuition of abstract concepts are related to some other themes in his philosophy of mathematics. I will then focus on a central question that has been raised in the literature on Gödel: what kind of account could be given of the intuition of abstract concepts? I sketch an answer to this question that is based on the work of a philosopher to whom Gödel also turned in this connection: Edmund Husserl.
This talk is drawn from work-in-progress on Gödel's philosophy of mathematics, some of which has been published recently in the Bulletin of Symbolic Logic 4, 2 (1998), 181-203.
The talk should be of interest to philosophers, logicians, mathematicians and computer scientists.