Mathematical Communication is a developing collection of resources for engaging students in writing and speaking about mathematics, whether for the purpose of learning mathematics or of learning to communicate as mathematicians.

Proof Elegance

Context: This lesson plan is from a weekly communication recitation that accompanies M.I.T.’s Real Analysis (18.100C). This week students learn about the Taylor series and the Stieltjes integral (Rudin pp. 120-127). This recitation is often combined with Recitation 9 on proof structure.

Authors: This recitation was developed by Craig Desjardins, Joel B. Lewis, Todd Kemp, Mohammed Abouzaid, Peter Speh, Kyle Ormsby, and Susan Ruff

Communication objectives: Students should begin to develop an awareness of proof elegance as well as an appreciation for revision (even correct proofs can be improved).

Recitation: This recitation varies each term depending on instructor inclinations. Some topics covered include the following:

Discussion of examples of elegant proofs

Recitation instructors: Mohammed Abouzaid and Peter Speh

“We first gave them the proof of the Cauchy-Schwarz inequality from Proofs from the Book, and asked them to compare it to the one we provided: Cauchy-Schwarz.  We then discussed which proof one would discover first, and how it’s a good idea, after having proved something, to think about rewriting it.

“To illustrate what happens when one focuses only on technicality (and not on conceptual ideas), we showed them Rudin’s proof of Theorem 7.18.

“Finally, we asked them to comment on two different ways of proving a formula for the sum of all integers smaller than a given one: Two proofs

“We then discussed preferences, and contexts in which each would be relevant.”

Hardy’s writing on mathematical beauty

Recitation instructor: Kyle Ormsby

“The infamous “elegance” recitation.  This was a difficult one for me to conduct, largely because my own preferences and ideas of mathematical elegance and the importance thereof are by no means set.  I chose to structure the recitation largely around Hardy’s writing on mathematical beauty in A Mathematician’s Apology.  The pertinent Chapters are 10-18, with the take-home sentences appearing in Chapter 10 (“[T]here is no permanent place in the world for ugly mathematics.”) and Chapter 18 (“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.”).

“Hardy’s take is quite alluring and certainly has cache within the mathematics community.  But it’s also snobbish and its modernist perspective on value in mathematics has been surpassed by the postmodern state of contemporary mathematics.  (I could write a lot more about what I mean by that, but this is probably not the appropriate venue.)  I wonder if the Hardy/Proofs from the Book view of elegant math should be updated or modulated by a more current viewpoint.  The answer may well be “no” given that the audience is freshman and sophomore undergraduates…

“One part of the discussion involved asking students for proofs/theorems they thought were elegant.  I was surprised at how many responses I received, and I think it’s a good thing to keep in the recitation.”

Instructor observations

This recitation would benefit from some examples of elegant proofs from real analysis.