Various strategies can be used to teach precision, rigor, and their importance.
Examples of seemingly correct arguments that are wrong
- In The Man who Knew Infinity: A Life of the Genius Ramanujan, Kanigel gives as an example Ramanujan’s claim of having “found a function which exactly represents the number of prime numbers less than x.” Using this example, Kanigel clearly explains for a lay audience why rigorous proof is essential in mathematics, even (especially) for those with a strong intuitive understanding of mathematics. These pages could be a suitable reading assignment for students. (By Robert Kanigel 1991, pp. 215-224; Readers are expected to know what an infinite series is, but other necessary math terms and concepts are introduced.)
- In the excerpt mentioned above, Kanigel presents a familiar proof that 2 = 1. Here is an interactive version of that fallacious proof.
- The following example of a seemingly intuitively obvious claim was suggested by M.I.T.’s Katrin Wehrheim. This claim presents a continuous version of proof by induction:Suppose that, if a statement is true for some real number x, then it’s true for all real numbers within a neighborhood of x. Then if the statement is true for 0, it must be true for all real numbers.The problem with this claim can be illustrated by counterexample: consider the open interval (-1, 1). If a statement is true on this interval only, then the conditions of the claim are satisfied, but the statement is not true for all reals.Note that the claim can be made true by adding the assumption that, if the statement is true for a subset of the reals, then it’s also true for the supremum and infimum of this subset. A rigorous proof can be given with basic real analysis (using the fact that the reals are complete) or topology (rephrasing the assumptions as the subset being open and closed, and using the fact that the reals are connected).
The importance of rigor may be argued from a historical context
- In his 1928 paper “Mathematical Rigor, Past and Present,” James Piermont summarizes the historical improvements in rigor from Newton and Leibnitz, through Euler, Cauchy, and Weierstrass, and the subsequent questions raised by “the Mengenlehre of Cantor, or the theory of aggregates (sets), [which] has brought to light a number of paradoxes or antimonies which have profoundly disturbed the mathematical community…” This article will be accessible only to those who have completed a course in Real Analysis.
- More accessible to a general audience is Logicomix: An epic search for truth, a graphic novel by Apostolos Doxiadis and Christos H. Papadimitriou, art by Alecos Papadatos, color by Annie di Donna, Bloomsbury, 2009, New York Times book review, by Jim Holt, Sept 27, 2009 In the remarkably engaging form of a graphic novel, Logicomix presents the quest for the foundations of mathematics. The story follows Russell, who is disappointed as a student to realize that Euclid’s work depends on unproven postulates and that terms like “infinitesimal” are not rigorously defined, and who devotes his life to trying to create a solid foundation for mathematics. This “quest” meets formidable obstacles, including the paradox “Does the set of all sets contain itself?” and Godel’s incompleteness theorem, which states that not everything can be proved. The story engagingly focuses on the characters involved, including the historical context of the two world wars. The thesis seems to be that, although logic is a powerful tool and has lead to the development of computers, for true wisdom, we need not only logic but also those things of which logic cannot speak. The graphic novel can give students an appreciation for the importance of rigor, but does so in a way that also makes clear its limitations. Particularly useful excerpts for a discussion of rigor and precision appear on pages 120 (illustrating precision of language) and 150-152 (after the discovery of non-Euclidean geometries, a call from Hilbert for mathematicians to use only rigorous proof and to strengthen the foundations of mathematics).
- Quinn, F., “A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today,” Notices of the AMS Vol. 59, No. 1, Jan 2012, pp. 31-37. Quinn uses a historical perspective to explain why rigorous “core mathematics” is under-appreciated and gives contemporary arguments to counter historical objections. These objections include the following:
- Proof by contradiction (“the law of the excluded middle”) is unreliable.
- Axiomatic definitions and detailed logical arguments lack reality and meaning, are artificially rigid, and lead to content-free manipulation.
- Consistency of a logical system cannot be proved from within the system (Gödel)
Quinn’s responses to these objections reveal the importance and power of the methods of “core mathematics,” such as precise definitions and logically complete proofs.
The importance of rigor within the context of mathematics
- Terry Tao’s blog post “There’s more to mathematics than rigour and proofs” places the importance of rigour within the greater context of stages in the development of mathematical thinking during undergraduate and graduate education. The “rigour” stage is important and necessary, but it’s also important to move on to the next stage.
Engage students in identifying and writing precise text
- In “Learning the Language of Mathematics,” Robert Jamison uses examples and counterexamples of good definitions to illustrate how he teaches students to use mathematical language precisely. From Language and Learning Across the Disciplines, Vol 4, No 1, May 2000.
- In MIT’s communication-intensive offering of Real Analysis, students begin the term by generating multiple definitions of the same notion and examining the consequences of the various definitions. The class discussion enables the recitation instructor to comment on precision as needed, and the instructor also provides feedback on the resulting writing assignment.
- Dean Peterson’s “Writing Precise Explanations of Graphic/Tabular Display of Economic Data is an assignment that asks students for short, precise explanations of economic data displayed in graphs and tables
See also the resources listed on the MathDL Mathematical Communication page on wording and punctuation. Please contribute additional materials for helping students to communicate precisely and rigorously.