Using analogical representations for mathematical concept formation

Alison Pease

Simon Colton

Ramin Ramezani

Alan Smaill

Markus Guhe

pp. 301-314

in: Lorenzo Magnani, Walter Carnielli, Claudio Pizzi (eds), Model-based reasoning in science and technology, Berlin, Springer, 2010

Abstract

We argue that visual, analogical representations of mathematical concepts can be used by automated theory formation systems to develop further concepts and conjectures in mathematics. We consider the role of visual reasoning in human development of mathematics, and consider some aspects of the relationship between mathematics and the visual, including artists using mathematics as inspiration for their art (which may then feed back into mathematical development), the idea of using visual beauty to evaluate mathematics, mathematics which is visually pleasing, and ways of using the visual to develop mathematical concepts. We motivate an analogical representation of number types with examples of "visual" concepts and conjectures, and present an automated case study in which we enable an automated theory formation program to read this type of visual, analogical representation.