The Language of Mathematics
By: David Bressoud @dbressoud
David Bressoud is DeWitt Wallace Professor Emeritus at Macalester College and former Director of the Conference Board of the Mathematical Sciences. davidbressoud.org
In my July 2022 column, My Mathematical Journey: The Alternating Sign Matrix Theorem, I briefly recounted the influence of Lakatos’s Proofs and Refutations on my own thinking about mathematics. This month I want to delve more deeply into the significance of that book and how mathematicians use language.
Lakatos illustrates examples of mathematical results that had been proven but were then followed by counterexamples. The main example is Euler’s formula for polyhedra, v – e + f = 2. The second example, which appears in an appendix, is Cauchy’s result that every convergent infinite sum of continuous functions is continuous. In both cases, the results are essentially correct. The counterexamples arise from inadequate definitions. What exactly do we mean by a polyhedron? What exactly do we mean by convergence? In the latter case, what Cauchy undoubtedly intended was uniform convergence. But as I stated last July, the distinction between convergence and uniform convergence was not explicitly recognized until several decades after Cauchy stated his theorem.
But there is a broader issue with the language of mathematics. Mathematicians have learned to assign very precise and often non-intuitive definitions to the words they use. I love how Lewis Carroll put it:
“When I use a word,” Humpty Dumpty said, in a rather scornful tone, “it means just what I choose it to mean - neither more nor less.” “The question is,” said Alice, “whether you can make words mean so many different things.” “The question is,” said Humpty Dumpty, “which is to be master - that's all.”— Lewis Carroll, Through the Looking Glass.
An excellent example is when a mathematician speaks of a limit, and even worse when the talk is of “approaching a limit.” A limit is a boundary. When one approaches a limit, the image is of getting progressively closer to that boundary. But the fact is that a value can still be a limit even when we have oscillation across this limit. We can repeatedly leap over this boundary, moving away from it half of the time.
Barbara Edwards and Michael Ward published a wonderful article in the Monthly in 2004, “Surprises from Mathematics Education Research: Student (Mis)use of Mathematical Definitions.” Drawing on the work of Vinner and Tall, they distinguish the concept definition, the actual definition of a mathematical term, and the concept image, the image that the concept draws upon.
I doubt that anyone has ever given you a definition of the word “cat” that has actually helped you to identify a cat. Instead, we have a lifetime of experiencing animals that we have learned are or are not cats. We have a concept image of a cat. That is how we learn most of our language. Students enter calculus with a concept image of a limit. The idea of ignoring that and instead paying attention to the concept definition is foreign.
Figure 1. Proofs and Refutations
Indeed, that is the way that mathematicians work, creating precise concept definitions only when forced to do so. I am certain that Euler stated his formula in total confidence that he understood what was meant by a polyhedron and that Cauchy stated his theorem with the assumption that convergence was necessarily what today we describe as uniform.
Few students come to us with an appreciation for the distinction between concept definition and concept image. It is one of the many habits of mind that must be instilled.
That raises an interesting question. Rather than presenting students with a term such as “limit” that is used in a precise and unfamiliar way, should mathematicians have invented a totally new word? I am reminded of E.J. Dijksterhuis’s critique of Newton’s use of the old vocabulary of force, mass, and gravity with new and quite different meanings. Particularly problematic is the word “inertia,” coined by Kepler to describe resistance to motion. Today we still speak of the need to “overcome inertia.” Not just must a force be applied, but one sufficient to overcome the body’s inherent inertia.
“It was Newton’s task to create order in this chaos of terms and notions. The best method for him to follow would have been that of Hercules cleansing the Augean stables, i.e. radical rejection of the old and subsequent reconstruction from the bottom. In this case it would have meant placing mechanics on a new foundation with the aid of sharply defined terms, preferably not taken from everyday speech, so that they were not yet charged with misleading associations.” (Dijksterhuis, p. 465)
And yet, do we really want to load students with totally unfamiliar words? While many of the connotations implicit in the word “limit” are misleading, others are helpful. Nevertheless, we as teachers need to be aware that such traps do lie in wait for our students.
Figure 2. The Great Stellated Dodecahedron, image from Robert Webb’s Stella software. http://www.software3d.com/Stella.php
Before closing, I want to suggest to those who have not read Proofs and Refutations to do so. It is fun to read, written as a play with the players denoted by Greek letters. The extensive footnotes indicate the historical figures who raised each of the various objections. In the process, the definition of what constitutes a polyhedron is gradually refined until, by page 117 we get the proof given by Poincaré in 1899 in which a polyhedron is defined as a collection of objects of 0, 1, 2, or 3 dimensions together with matrices that indicate which objects of dimension k–1 are adjacent to which objects of dimension k and in which every closed circuit is a bounding circuit. Poincaré translated the definition of a polyhedron into a collection of statements about vector spaces. The proof of Euler’s formula is now just a series of observations about linear transformations of vector spaces (see Figures 3 through 7 following the references).
One of the side benefits of Poincaré’s proof is that it readily generalizes to polyhedra of any dimension and expands the notion of a polyhedron. The Great Stellated Dodecahedron (Figure 2) has edges and faces that intersect. There are twelve faces, each of which is a pentagram (five-pointed star) with five vertices and five edges, yielding 30 edges. Three faces meet at each of the 20 vertices: 12-30+20 = 2. It is not convex, but it does satisfy Poincaré’s definition of a polyhedron.
References
Lewis Carroll. Through the Looking Glass. Collins-World, Cleveland. OH, 1974.
E.J. Dijkserhuis. The Mechanization of the World Picture: Pythagoras to Newton. Princeton University Press, 1986.
Barbara S. Edwards and Michael B. Ward. Surprises from Mathematics Education Research: Student (Mis)use of Mathematical Definitions. The American Mathematical Monthly, Vol. 111, No. 5 (May, 2004), pp. 411-424. https://www.jstor.org/stable/4145268
Imre Lakatos. Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press, 1976. https://www.cambridge.org/core/books/proofs-and-refutations/575FC8A6B4FAB79E649EDF5FBB9C6E10#
Henri Poincaré. Complément à l’Analysis Situs. Rendiconti del Circolo Matematico di Palermo, 13 (1899), pp.285–343.
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Figure 7.