Abstract
We examine the influence of word choices on mathematical practice, i.e. in developing definitions, theorems, and proofs. As a case study, we consider Euclid’s and Euler’s word choices in their influential developments of geometry and, in particular, their use of the term ‘polyhedron’. Then, jumping to the twentieth century, we look at word choices surrounding the use of the term ‘polyhedron’ in the work of Coxeter and of Grünbaum. We also consider a recent and explicit conflict of approach between Grünbaum and Shephard on the one hand and that of Hilton and Pedersen on the other, elucidating that the conflict was engendered by disagreement over the proper conceptualization, and so also the appropriate word choices, in the study of polyhedra.
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Notes
A search of “word choice” and “mathematical practice” returns only six articles, all of which are in education studies. A search for “word choice” and “mathematics” returns only 29 works, mostly in education studies and some in linguistics.
We thank an anonymous reviewer for inviting us to state and clarify our methodological assumptions.
We do not discuss here spoken mathematics or visual mathematics. But we conjecture that our remarks about word choices in written mathematics also apply to those in spoken mathematics and to diagram choices in visual mathematics.
We do not explain what makes a subject-matter mathematical on the assumption that any plausible account of this would have as a consequence that solid geometry, including the study of polyhedra, is a mathematical subject-matter.
Confer Mancosu (2008, Introduction, esp. §2, and Chs 6 and 7).
Different notions may also result in the same theorems while developing a subject by distinct paths.
We thank Michael Blaustein of St. John’s College for pointing out to Abrams the importance of motion in Euclid’s work.
When we quote Euclid, we use the standard Thomas L. Heath translation. Emphasis and brackets come from the original text. We do rely on translations alone in our discussions of Greek and Latin texts.
As logicians might put it, Euclid’s meta-theoretic language is the language of motion.
Note that a figure is any thing “contained by any boundary or boundaries,” another use of spatial containment to communicate Euclid’s geometry Euclid (2010, Book 1, Definition 14).
Heath assigns no number to the proposition that only five regular solids can be constructed, preventing us from using the briefer ‘Proposition 19’.
In fact, Euclid proves that only five regular solids exist by showing that only five combinations of triangles or pentagons result in a solid angle: “For a solid angle cannot be constructed with two triangles, or indeed planes. With three triangles the angle of the pyramid is constructed...but a solid angle cannot be formed by six equilateral triangles placed together at one point...” Euclid (2010, p. 480)
In contrast, Francese and Richeson note that there were old and familiar terms for face (hedra) and solid angle (angulus solidus) Francese and Richeson (2007, p. 288).
For Euclid, it is rather that the object of study is the ambient space marked off by the holding pen.
Leonhard Euler, Elementa Doctrinae Solidorum” (“Elements of the Doctrine of Solids”), Novi Commentarii Academiae Scientiarum Petropolitanae 4: pp. 72–93, 1758.
This is different from generalizing a definition within one conception, as here: “the definition of polyhedron can be generalized by allowing non-adjacent faces to intersect...” Coxeter (1973, p. 96) We remark that it is important to explicitly recognize when conceptual shifts occur so as to facilitate understanding. If, say, a student-reader might miss such conceptual shifts, explicitly marking them in one’s treatment seems warranted.
Part of our point is that one can approach the same entities from other vantage points; labeling polytopes as ‘geometric’ may bias one perspective over other possibilities, as Grünbaum’s text shows.
Instead of, say, regions of space or polygonal arrangements. Coxeter does not adopt this convention.
The departure from Euclid here is striking; containment is no longer conveyed by a spatial metaphor.
A non-exhaustive list of examples include the definition of convex hull and Carathéodory’s Theorem on Grünbaum (1967, pp. 14–15), poonem (Hebrew for ‘face’) Grünbaum (1967, p. 20), unboundedness Grünbaum (1967, p. 23), the notion of a polytope’s dual Grünbaum (1967, p. 46), combinatorial scheme Grünbaum (1967, p. 90), and the Euler characteristic of a proper face Grünbaum (1967, p. 138).
Lakatos et al. (1976, p. Legendre, 14, Footnote 1), Lakatos et al. (1976, p. Jonquières, 14, Footnote 2), Lakatos et al. (1976, p. Möbius, 15, Footnote 2), Lakatos et al. (1976, p. Baltzer, 16, Footnote 1), Lakatos et al. (1976, p. Lhuilier, 13), Lakatos et al. (1976, p. Hessel, 15), and Lakatos et al. (1976, p. Kepler, 16, Footnote 2).
Lakatos calls this the “domain of proof,” but here we use the term in a slightly broader sense that corresponds to an area of mathematical inquiry rather than a single proof from such an area Lakatos et al. (1976, p. 64).
One might prefer to say that the conceptual disagreement leads to divergent word choice. Here we stake no claim as to the priority of explanation—though there is no doubt about the close tie between word choice and conceptualization—since our purpose is to examine the vital phenomenon of word choice in mathematical practice, particularly as encoding a conceptualization of mathematical subject-matter.
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Acknowledgements
Elkind was partially supported by a George Washington University Luther Rice Fellowship while a student at The George Washington University. We thank our anonymous reviewers for numerous helpful suggestions and criticisms. This work was undertaken with the support of a Luther Rice Fellowship from George Washington University, and we are grateful for that support.
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Abrams, L., Elkind, L.D.C. Word choice in mathematical practice: a case study in polyhedra. Synthese 198, 3413–3441 (2021). https://doi.org/10.1007/s11229-019-02287-6
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DOI: https://doi.org/10.1007/s11229-019-02287-6