Abstract
This paper analyzes some of the ambiguities that arise among statements with the copular verb “is” in the mathematical language of textbooks as compared to day-to-day English language. We identify patterns in the construction and meaning of “is” statements using randomly selected examples from corpora representing the two linguistic registers. We categorize these examples according to the part of speech of the object word in the grammatical form “[subject] is [object].” In each such grammatical category, we compare the relative frequencies of the subcategories of logical relations conveyed by that construction. Within some categories we observe that the same grammatical structure alternatively conveys different logical relations and that the intended logical relation can only sometimes be inferred from the grammatical cues in the statement itself. This means that one can only interpret the intended logical relation by already knowing the relation among the semantic categories in question. Such ambiguity clearly poses a communicative challenge for teachers and students. We discuss the pedagogical significance of these patterns in mathematical language and consider the relationship between these patterns and mathematical practices.
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Notes
We somewhat alternate between calling these logical relations and semantic relations. These terms are by no means synonymous. We identify them here because, from a mathematical standpoint, conditional or biconditional relations are parts of logic, but, from a reading standpoint, students may have to use semantic understanding to infer the logical relation.
Indeed, one of our philosopher colleagues argued that such claims are not conditionals, but rather universals (L. Clapp, personal communication December, 2016; c.f. Durand-Guerrier, 1996).
Note that this search differs from our previous analysis because it only considers the word directly to the right of is rather than the object word we coded in our random samples. This search then would not capture instances like “is strictly increasing” since the gerund is not directly after is.
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Appendix. Sample statements with mathematical symbols reintroduced
Appendix. Sample statements with mathematical symbols reintroduced
Example 1: \( \overrightarrow{e_n} \) is the standard basis for \( {\mathfrak{R}}^{n.} \)
Example 7: If a is a type 1 integer and b is a type 2 integer, then a ∙ b is a type 2 integer.
Example 8: It can be shown that the best strategy is to pass over the first k − 1 candidates where k is the smallest integer for which \( \frac{1}{k}+\frac{1}{k+1}+\dots +\frac{1}{n-1}\le 1 \).
Example 9: If T is a complete binary tree of height h, then T has 2h + 1 − 1 vertices.
Example 10: If A ⊆ M, we say that A is a compact subset of M; if regarded as a subspace of M, it is a compact metric space.
Example 15: The graph of f is concave up on I if f′ is increasing.
Example 16: It turns out (see Appendix B) that the direction of v × w is given by the right-hand rule.
Example 18: Suppose S is connected (so also nonempty).
Example 20: Every element of x is in the set Cx by reflexivity.
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Dawkins, P.C., Inglis, M. & Wasserman, N. The use(s) of is in mathematics. Educ Stud Math 100, 117–137 (2019). https://doi.org/10.1007/s10649-018-9868-6
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DOI: https://doi.org/10.1007/s10649-018-9868-6