Here’s Terrance Tao on the possible ways in which something might be considered “good” in mathematics (his words):
- Good mathematical problem-solving (e.g. a major breakthrough on an important mathematical problem);
- Good mathematical technique (e.g. a masterful use of existing methods, or the development of new tools);
- Good mathematical theory (e.g. a conceptual framework or choice of notation which systematically unifies and generalizes an existing body of results);
- Good mathematical insight (e.g. a major conceptual simplification, or the realisation of a unifying principle, heuristic, analogy, or theme);
- Good mathematical discovery (e.g. the revelation of an unexpected and intriguing new mathematical phenomenon, connection, or counterexample);
- Good mathematical application (e.g. to important problems in physics, engineering, computer science, statistics, etc., or from one field of mathematics to another);
- Good mathematical exposition (e.g. a detailed and informative survey on a timely mathematical topic, or a clear and well-motivated argument);
- 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);
- Good mathematical vision (e.g. a long-range and fruitful program or set of conjectures);
- Good mathematical taste (e.g. a research goal which is inherently interesting and impacts important topics, themes, or questions);
- Good mathematical public-relations (e.g. an effective showcasing of a mathematical achievement to non-mathematicians, or from one field of mathematics to another);
- Good meta-mathematics (e.g. advances in the foundations, philosophy, history, scholarship, or practice of mathematics);
- Rigorous mathematics (with all details correctly and carefully given in full);
- Beautiful mathematics (e.g. the amazing identities of Ramanujan; results which are easy (and pretty) to state but not to prove);
- Elegant mathematics (e.g. Paul Erdos’ concept of “proofs from the Book”; achieving a difficult result with a minimum of effort);
- Creative mathematics (e.g. a radically new and original technique, viewpoint, or species of result);
- Useful mathematics (e.g. a lemma or method which will be used repeatedly in future work on the subject);
- 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);
- 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);
- Intuitivemathematics (e.g. an argument which is natural and easily visualisable);
- Definitive mathematics (e.g. a classification of all objects of a certain type; the final word on a mathematical topic);
- etc., etc.
With minor tweaking this applies to ALL endeavors.
ALL humans are multifaceted and multi-talented. You can’t be good at everything. But you will be surprised to find that you can be good at many things that, at first, appear unrelated. By pushing your strengths, doing whatever you can to become a master at them, you’ll discover how they link up, and how you can make use of them independently and in conjunction.
I think it’s worth quoting his conclusion in full:
As we can see from the above case study, the very best examples of good mathematics do not merely fulfil one or more of the criteria of mathematical quality listed at the beginning of this article, but are more importantly part of a greater mathematical story, which then unfurls to generate many further pieces of good mathematics of many different types. Indeed, one can view the history of entire fields of mathematics as being primarily generated by a handful of these great stories, their evolution through time, and their interaction with each other. I would thus conclude that good mathematics is not merely measured by one or more of the “local” qualities listed previously (though these are certainly important, and worth pursuing and debating), but also depends on the more “global” question of how it fits in with other pieces of good mathematics, either by building upon earlier achievements or encouraging the development of future breakthroughs. Of course, without the benefit of hindsight it is difficult to predict with certainty what types of mathematics will have such a property. There does however seem to be some undefinable sense that a certain piece of mathematics is “on to something”, that it is a piece of a larger puzzle waiting to be explored further. And it seems to me that it is the pursuit of such intangible promises of future potential are least as important an aspect of mathematical progress than the more concrete and obvious aspects of mathematical quality listed previously. Thus I believe that good mathematics is more than simply the process of solving problems, building theories, and making arguments shorter, stronger, clearer, more elegant, or more rigorous, though these are of course all admirable goals; while achieving all of these tasks (and debating which ones should have higher priority within any given field), we should also be aware of any possible larger context that one’s results could be placed in, as this may well lead to the greatest long-term benefit for the result, for the field, and for mathematics as a whole.
Now go do something “good”,