A proof must, first and foremost, be correct. But even among correct proofs, some proofs are more satisfying, or “elegant,” than others. Even experienced mathematicians aren’t sure quite how to define elegance: elegance is, to some extent, in the eye of the beholder. Although teaching students to write elegant proofs may not be feasible, we can begin to raise student awareness of elegance.
To raise student awareness of elegance, the following strategies have been used in M.I.T.’s communication-intensive offering of Real Analysis.
- Discussing examples, which can be found, for example, in Proofs from the Book.
- Discussing quotes from Hardy’s A Mathematician’s Apology.
- “[T]here is no permanent place in the world for ugly mathematics.” (Chapter 10)
- “In [beautiful] theorems (and in the theorems, of course, I include the proofs) there is a high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions.” (Chapter 18)
- In class discussion, students are asked to volunteer examples of proofs that they find elegant.
- Two mathematicians have debated in front of the class about what characterizes elegance.
- Pair discussion has been tried with sophomore math majors, but these students are not yet sufficiently aware of elegance to have a profitable unstructured small-group discussion about it.
These strategies are elaborated upon in a lesson plan about proof elegance, from M.I.T.’s communication-intensive offering of Real Analysis.
Through discussion, some mathematicians have concluded (contrary to student expectation), that tricks are often not elegant because they often obscure the essence of the proof: an elegant proof should not only be concise, but should also provide some intuition about *why* the statement is true.
Resources
- See the resources for proofs without words on the page Using Visuals
- Proofs from the Book by Martin Aigner and Günter M. Ziegler. AMS review
- Charming Proofs: a Journey into Elegant Mathematics, by Claudi Alsina and Roger B. Nelsen
- The poem “Poet as Mathematician” by Lillian Morrison describes elegant proof. Reprinted in Joanne Growney’s blog Intersections–poetry with mathematics
