I am on the Steering Committee and Senior Programme Committee for the Third International Conference on Computational Creativity.

I am currently organising the Symposium on Mathematical Practice and Cognition II with Brendan Larvor.

Mathematical and scientific theories rest on foundations which are assumed in order to create a paradigm within which to work. These foundations sometimes shift. We investigated where foundations come from, how they change, and how AI researchers can use these ideas to create more flexible systems. For instance, Euclid formulated geometric axioms which were thought to describe the physical world. These were the foundations on which concepts, theorems and proofs in Euclidean geometry rested. Euclidean geometry was later modified by rejecting the parallel postulate, and non-Euclidean geometries were formed, along with new sets of concepts and theorems. Another example of axiomatic change is in Hilbert's formalisation of geometry: initially his axioms contained hidden assumptions which were soon discovered and made explicit. Paradoxes found in Frege's axiomatisation of number theory led to Zermelo and Fraenkel modifying some of his axioms in order to prevent problem sets from being constructed. On a less celebrated, but equally remarkable, level children are able to formulate mathematical rules about their environment such as transitivity or the commutativity of arithmetic, and to modify these rules if necessary. Recent work in cognitive science by Lakoff and Núñez and in the philosophy of mathematics by Lakatos suggests ways in which this may be done. We constructed and evaluated computational theories and models of aspects of this process and explored the application of our model to AI and software engineering.

The case for support is here. This work was supported by EPSRC grant EP/F035594/1.

During the project, we organised the Symposium on Mathematical Practice and Cognition