The Importance of Mathematical Conceptualisation

Mathematicians typically invoke a wide range of reasons as to why their research is valuable. These reveal considerable differences between their personal images of mathematics. One of the most interesting of these concerns the relative importance accorded to conceptual reformulation and development compared with that accorded to the achievement of concrete results. Here I explore the conceptualists' claim that the scales are tilted too much in favour of the latter. I do so by taking as a case study the debate surrounding the question as to whether groupoids are significantly more powerful than groups at capturing the symmetry of a mathematical situation. The introduction of groupoids provides a suitable case as they score highly according to criteria relating to theory-building rather than problem-solving.

Several of the arguments for the adoption of the groupoid concept are outlined, including claims as to its capacity for reformulating existing theory, its ability to measure symmetry more systematically, and its ‘naturalness’. This last notion is given an extensive treatment.