I am working with Simon Colton and John Charnley to develop a rigorous, computationally detailed
and plausible account of how creation by software could occur. We use examples and theories of human creativity, particularly in the visual arts and in mathematics, to inspire our development of formalisms to describe and extend the notion of creativity in software.
I am on the Steering Committee and Senior Programme Committee for the Third International Conference on Computational Creativity.
Researching Mathematical Research
I am working with Ursula Martin to investigate current mathematical practice; in particular, ways in which mathematicians are using web 2.0 technology. We are interested in whether Lakatos's theory of how mathematicians communicate and use counterexamples to refine concepts, conjectures and proofs was an accurate description of evolution in mathematics, and whether Polya's problem solving heuristics are used in research mathematics.
I am currently organising the Symposium on Mathematical Practice and Cognition II with Brendan Larvor.
The Wheelbarrow Project (April 2008 - May 2011)
I worked on this project with Alan
Clark, Simon Colton, Andrew Ireland,
Maria Teresa Llano Rodriguez and Ramin Ramezani. We developed computational and theoretical models of conceptual blending, analogical reasoning, and metaphors in mathematics, and investigated how theories of informal reasoning can be applied to mathematical reasoning.
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
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