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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.