The Role of Axiomatisation in Mathematical Discovery

Alison Pease

Dirk Schlimm has argued that the role of axiomatisation in mathematical discovery and the evolution of contemporary mathematics has been undervalued in philosophy of science and mathematics. We describe some examples of mathematical invention that illustrate evolution of axiomatisations in historical case studies, and argue that heuristics suggested by Lakatos in his rational reconstructions and Boden in her work on creativity can be used to describe some axiomatic development in mathematics. We further illustrate this with a simple computational representation of some of these heuristics in a system based on Colton's HR program. This can take in a set of axioms in group theory, construct a theory from the axioms and evaluate the theory based on quality and difficulty, and then modify its axiom set by dropping, negating, or adding new axioms.