Good Mathematics

Arxiv is where you normally expect preprints of technical papers in mathematics, physics and computer science. However, it was a surprise to run into a general essay on ‘good mathematics’ by Terrance Tao. Tao has considered more than 21 dimensions of what good mathematics can mean. The following is the list from the essay:

1. Good mathematical problem-solving (e.g. a major breakthrough on an important mathematical problem);
2. Good mathematical technique (e.g. a masterful use of existing methods, or the development of new tools)
3. Good mathematical theory (e.g. a conceptual framework or choice of notation which systematically unifies and generalises an existing body of results);
4. Good mathematical insight (e.g. a major conceptual simplification, or the realisation of a unifying principle, heuristic, analogy, or theme);
5. Good mathematical discovery (e.g. the revelation of an unexpected and intriguing new mathematical phenomenon, connection, or counterexample);
6. Good mathematical application (e.g. to important problems in physics, engineering, computer science, statistics, etc., or from one field of mathematics to another);
7. Good mathematical exposition (e.g. a detailed and informative survey on a timely mathematical topic, or a clear and well-motivated argument);
8. Good mathematical pedagogy (e.g. a lecture or writing style which enables others to learn and do mathematics more effectively, or contributions to mathematical education);
9. Good mathematical vision (e.g. a long-range and fruitful program or set of conjectures);
10. Good mathematical taste (e.g. a research goal which is inherently interesting and impacts important topics, themes, or questions);
11. Good mathematical public relations (e.g. an effective showcasing of a mathematical achievement to non-mathematicians, or from one field of mathematics to another);
12. Good meta-mathematics (e.g. advances in the foundations, philosophy, history, scholarship, or practice of mathematics);
13. Rigorous mathematics (with all details correctly and carefully given in full);
14. Beautiful mathematics (e.g. the amazing identities of Ramanujan; results which are easy (and pretty) to state but not to prove);
15. Elegant mathematics (e.g. Paul Erdos’ concept of “proofs from the Book”; achieving a difficult result with a minimum of effort);
16. Creative mathematics (e.g. a radically new and original technique, viewpoint, or species of result);
17. Useful mathematics (e.g. a lemma or method which will be used repeatedly in future work on the subject);
18. Strong mathematics (e.g. a sharp result that matches the known counterexamples, or a result which deduces an unexpectedly strong conclusion from a seemingly weak hypothesis);
19. Deep mathematics (e.g. a result which is manifestly non-trivial, for instance by capturing a subtle phenomenon beyond the reach of more elementary tools);
20. Intuitive mathematics (e.g. an argument which is natural and easily visualisable);
21. Definitive mathematics (e.g. a classification of all objects of a certain type; the final word on a mathematical topic)