What Is Good Mathematics?

Some personal thoughts and opinions on what “good quality mathematics” is and whether one should try to define this term rigorously. As a case study, the story of Szemerédi’s theorem is presented. 1. The many aspects of mathematical quality We all agree that mathematicians should strive to produce good mathematics. But how does one define “good mathematics”, and should one even dare to try at all? Let us first consider the former question. Almost immediately one realises that there are many different types of mathematics which could be designated “good”. For instance, “good mathematics” could refer (in no particular order) to (i) Good mathematical problem solving (e.g. a major breakthrough on an important mathematical problem); (ii) Good mathematical technique (e.g. a masterful use of existing methods or the development of new tools); (iii) Good mathematical theory (e.g. a conceptual framework or choice of notation which systematically unifies and generalises an existing body of results); (iv) Good mathematical insight (e.g. a major conceptual simplification or the realisation of a unifying principle, heuristic, analogy, or theme); (v) Good mathematical discovery (e.g. the revelation of an unexpected and intriguing new mathematical phenomenon, connection, or counterexample); Received by the editors February 7, 2007. 2000 Mathematics Subject Classification. Primary 00A30. c ©2007 American Mathematical Society Reverts to public domain 28 years from publication