@MISC{Tao07whatis, author = {Terence Tao}, title = {What is good mathematics?}, year = {2007} }

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Abstract

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);