Intuitive acceptance of proof by contradiction

Abstract

The formal acceptance of a mathematical proof is based on its logical correctness but, from a cognitive point of view, this form of acceptance is not always naturally associated with the feeling that the proof has necessarily proved the statement. This is the case, in particular, for proof by contradiction in geometry, which can be linked to a loss of evidence in various ways, owing to its particular logical structure and to the difficulty in managing geometrical figures with contradictory properties. In this paper, we observe that students produce argumentation by starting with the assumption that the claim is false (indirect argumentation), and that they seem to accept this as more evident than the proofs by contradiction. On the basis of the notion of intuitive knowledge developed by Fischbein and through the analysis of task-based interviews, we investigate the intuitive acceptance of proof by contradiction and of indirect argumentation, underlining, in particular, that indirect argumentation can be produced as a compromise between a proof by contradiction and the need for a more evident argument.

A Hypothesis Framework for Students’ Difficulties with Proof By Contradiction

“…For example, Brown (2016) noted that students' difficulties with understanding how PBC works ("Foundational Hypotheses") were partly the result of textbooks not adequately training students on formal logic ("Training Hypotheses"). Antonini (2019) argued that "even if a person formally knows that a statement has been proved, this knowledge is not always associated with the feeling that the statement is necessarily true" (p. 794). That is, logical clarity ("Foundational Hypotheses") is not a sufficient condition for instilling conviction ("Conviction Hypothesis").…”

“…1 suggests that the situation is incredibly complex, and the research literature backs this up. Antonini (2019) provided an important example of this using the case of Fabio, a senior undergraduate (and probably the most-cited student in the entire PBC literature). Despite understanding the structure of PBC ("Foundational Hypotheses") and producing PBCs ("Operational Hypotheses"), he appeared to have strong affective reservations with the technique, noting "the absurdity is ... at least embarrassing.…”

“…Historical trends are believed to be one force behind this hypothesis. Antonini (2019) noted "in many cases throughout history some mathematicians have discussed its [PBC] acceptability and have proposed to exclude proof by contradiction from proving methods" (p. 794). Gasser (1992) outlined some of these historical moments dating back to at least 1662, many of which argued that while PBC might show that something is true, PBC often falls short of explaining why that thing is true.…”

“…In other research, students will seek out new objects to bridge the impossible and possible worlds. In some cases, this is done through a process known as "abduction" (Antonini 2019;Antonini and Mariotti 2009;Mariotti and Antonini 2009), and in other cases, through a "pseudo-object" (Baccaglini-Frank et al 2013, 2018Leung and Lopez-Real (2002). Finally, Koichu (2012) found that "definitions and axioms of geometry can be intellectually necessitated for students by means of the exploration of impossible objects" (p. 2).…”

Generating mathematical knowledge in the classroom through proof, refutation, and abductive reasoning

“…We consider abduction because, according to Peirce (1932Peirce ( , 1935, who coined the term, this type of reasoning involves explanation and discovery, which are relevant to our focal mathematical activity. Mathematics education research has recently explored the role of abductive reasoning in the context of conjecturing and proving (e.g., Antonini, 2019;Arzarello & Sabena, 2011;Baccaglini-Frank, 2019;Meyer, 2010;Pedemonte & Reid, 2011). Our study also aims to 3 We are aware that the mathematical process described by Lakatos does not represent all aspects of mathematical disciplinary practice (Hanna, 1995), and we also do not think that every aspect of mathematical practice (including Lakatos's description) should inform the teaching of mathematics in classrooms (Weber et al, 2020).…”