Drawing on prior research on indirect proof, this paper reports on a series of exploratory studies that examine the extent to which findings on students’ ways of reasoning about contradiction and contraposition characterize students’ views of indirect existence proofs. Specifically, Study 1 documents students’ comparative selections and selection rationales when asked to choose the “most convincing” proof, given a constructive and nonconstructive existence proof. Study 2 further examines findings from Study 1 by documenting novices’ levels of conviction and interpretations of a nonconstructive existence proof. Findings show that when presented with a nonconstructive proof, students tended to not only find the proof convincing but also interpreted the proof constructively. Moreover, the data indicate students who exhibit an awareness of the nonconstructive structure were divided in terms of their views of which form – constructive or non-constructive – was the most convincing. The discussion considers students’ reactions to the disjunctive structure of nonconstructive existence proofs and use of pragmatic and theoretical modes of thought.
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These definitions of constructive and nonconstructive existence proofs are based on those used in tertiary mathematics texts (see Fig. 1), rather than the philosophies of constructivism or intuitionism.
Advances in programming have led to debates regarding the production of algorithms and, therefore, the constructive / nonconstructive classifications of some proofs (cf. Gray 1994).
Specifically, Hilbert proved that there exists a finite basis for the ring of invariants G = SLn(C) acting on the ring of polynomials R = S(V), when V is a finite dimensional representation of G
It should, however, be noted that Hilbert produced a constructive proof in 1893.
It is also important to note that this classification is mathematical rather than philosophical for those who adhere to the philosophy of constructivism or intuitionism would likely reject the mathematical classification.
A dilemma proof may involve a false premise, as is the case in Argument B. There are other dilemma forms that do not.
Weber and Mejia-Ramos’s definition of relative conviction differs from that offered here and is focused on “the subjective level of probability that one attributes to that claim being true” (p. 16). Their definition is not incompatible with that given above since a higher degree of conviction can be thought of as assigning a higher “subjective level of probability.”
The absence of rationales may be due to the lack of access to mathematical symbols and notations with an online survey environment.
The assumption students would lack familiarity was based on the curricular information available at the university.
Italics are used to indicate a student’s, as opposed to the researcher’s, emphasis in verbalized remarks.
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Brown, S.A. Who’s There? A Study of Students’ Reasoning about a Proof of Existence. Int. J. Res. Undergrad. Math. Ed. 3, 466–495 (2017). https://doi.org/10.1007/s40753-017-0053-6
- Nonconstructive arguments
- Existence proofs
- Proof by contradiction