A cognitive model of axiom formulation and reformulation with application to AI and software engineering

Mathematical and scientific theories rest on foundations which areassumed in order to create a paradigm within which to work. Thesefoundations sometimes shift. We want to investigate where foundationscome from, how they change, and how AI researchers can use these ideasto create more flexible systems. For instance, Euclid formulatedgeometric axioms which were thought to describe the physicalworld. These were the foundations on which concepts, theorems andproofs in Euclidean geometry rested. Euclidean geometry was latermodified by rejecting the parallel postulate, and non-Euclideangeometries were formed, along with new sets of concepts andtheorems. Another example of axiomatic change is in Hilbert'sformalisation of geometry: initially his axioms contained hiddenassumptions which were soon discovered and made explicit. Paradoxesfound in Frege's axiomatisation of number theory led to Zermelo andFraenkel modifying some of his axioms in order to prevent problem setsfrom being constructed. On a less celebrated, but equally remarkable,level children are able to formulate mathematical rules about theirenvironment such as transitivity or the commutativity of arithmetic,and to modify these rules if necessary. It is astonishing that humansare able to form mathematical concepts, to abstract mathematicalrules, to explore the space that these rules define, and to modify therules in the face of counterexamples or other problems. Recent workin cognitive science by Lakoff and Nunez and in the philosophy ofmathematics by Lakatos suggests ways in which this may be done. Weintend to construct and evaluate a computational theory and model ofthis process and to explore the application of our model to AI andsoftware engineering. This is an ambitious project, with thepotential to bring together and deeply influence diverse fieldsincluding cognitive science, automated mathematical reasoning,situated embodied agents, and AI problem solving domains which wouldbenefit from a more flexible approach. Developing a set of automatedtechniques which are able to take a problem and change it into adifferent, more interesting problem could have great impact on thesedomains. In particular, we aim to explore the application of ourtheory and model to constraint satisfaction problems and softwarespecifications requirements. A general theory of how constraints,specifications or goals can be formulated and reformulated could leadto a communal set of powerful new AI techniques