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

The project was concerned to build models of the process of building
axiomatisations of mathematical theories, building on notions of mathematical
analogy and transfer. This leads to a marriage between the formalisms
of institution theory and information flow, and the cognitive claims of
Lakoff and Nunez on the formation of mathematical ideas. Lakatos's ideas
on the evolution of mathematical theories was also a strong influence.

Reasoning by analogy between different areas has been a tool in the invention
of scientific theories for a long time, eg "the atom is like the solar system" --
in that electrons/planets are attracted to and move round a central attractor.
The project looked at the role of mathematical analogy -- both analogies
between everyday tasks like measuring and mathematical notions of number;
and analogies between mathematical theories that are already developed.
This opens up new possibilities in creative interactions between computers,
mathematicians, and learners who can explore the possible changes
and variations in inventing and merging new forms of mathematical thinking.

It is possible to build computer models of the understanding of mathematical
domains that reflect both mathematical accuracy, and different particular
ways in which the mathematics reflects individual experience.
For example, there are different notions of number that arise from
different experiences, such as using a measuring stick or looking
at motion long a path; and these ideas come together in notions of
number that developed over a long period.
These models allow computer support for combining such different ideas,
and helping people explore the different ways new mathematical notions
can be formulated, based on existing notions and on new examples, which
suggest new ways of thinking.