A remarkable problem of language, mathematics and imagination

According to recent publications within mathematics education research, the meaning of “proof” is still a subject of debate among researchers (e.g. Balacheff, 2008; Cabassut et al., 2012; Mariotti, Durand-Guerrier, & Stylianides, 2018; Reid, 2015; Reid & Knipping, 2010; Stylianides, Bieda, & Morselli, 2016)

A model (a reference epistemological model) proposed by Shinno et al. (2018) consists of three layers —“real-world logic”, “local theory”, and “axiomatic theory”—which are characterized in terms of the epistemological nature of theory, wherein the statement is formulated, and the proof is carried out.

JAPANESE: Some Japanese works distinguish the word ronsho from shomei by referring to their relationship with the system of mathematics. […], such that sh mei is related to deriving consequences from premises for establishing the truth of a proposition, while ronsh is related to the (axiomatic) system in which logical relations between propositions take place. (Shinno et al., 2018, p. 26)

HEBREW: Two Hebrew words are frequently used in the official Israeli mathematics curricula: hohaha (translates as proof) and hanmaka (translates as both argumentation and reasoning). Similarly to Japan, Israeli curricula treat hohaha as a particular case of hanmaka where the former notion presumes mathematical rigor and the latter one is somewhat vague and alludes to providing an argument without strictly prescribing what type of argument may be used. (Koichu, 2018, p. 26)

ENGLISH: Mariotti (2006) states, “it is not possible to grasp the sense of a mathematical proof without linking it to the other two elements: statement and theory” (p. 184).

ENGLISH:  A well-known difficulty is related to the meaning of ‘a’ that can either refer to an individual, a generic element, or an implicit universal quantifier. (Durand-Guerrier et al., 2016, p. 89)

AMERICAN:In a popular undergraduate textbook, Hammack (2013) directly addressed this point: Now we come to the very important point. In mathematics, whenever P(x) and Q(x) are open sentences concerning elements x in some set S (depending on context), an expression of form P(x)⇒Q(x) is understood to be the statement ∀x∈S(P(x)⇒Q(x)). In other words, if a conditional statement is not explicitly quantified then there is an implied universal quantifier in front of it. This is done because statements of the form ∀x∈S, P(x)⇒Q(x) are so common in mathematics that we would get tired of putting the ∀x∈S in front of them. (Hammack, 2013, p. 46). We interpret this excerpt to mean that a focus on universal statements is an invariant part of mathematical practice. (Czocher et al., 2018, p. 25)

RUSSIAN:  there is no article marker distinguishing a from the, the definite from the indefinite. Yet this language contains a sophisticated mathematics register fully capable of distinguishing the meaning ‘there exists’ from ‘there exists a unique’. (Pimm, 1987, p. 81)

Yusuke Shinno, Takeshi Miyakawa, Tatsuya Mizoguchi, Hiroaki Hamanaka, Susumu Kunimune. Some linguistic issues on the teaching of mathematical proof. Eleventh Congress of the European Society for Research in Mathematics Education, Utrecht University, Feb 2019, Utrecht, Netherlands. ffhal02398502




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