Although studies on students’ difficulties in producing mathematical proofs have been carried out in different countries, few research workers have focussed their attention on the identification of mathematical proof schemes in university students. This information is potentially useful for secondary school teachers and university lecturers. In this article, we study mathematical proof schemes of students starting their studies at the University of Cordoba (Spain) and we relate these schemes to the meanings of mathematical proof in different institutional contexts: daily life, experimental sciences, professional mathematics, logic and foundations of mathematics. The main conclusion of our research is the difficulty of the deductive mathematical proof for these students. Moreover, we suggest that the different institutional meanings of proof might help to explain this difficulty.

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Institutional and personal meanings of mathematical proof

A central question in the philosophy of mathematics is: How do mathematicians know that mathematical propositions are true? The standard answer is that a mathematician knows that a proposition is true because she knows a proof of that proposition. Of course, this answer leaves a couple of important things unexplained. First, how does the mathematician know that the axioms that her proof appeals to (e.g., the axiom of choice) are true? Second, how does the mathematician know that the rules of inference that her proof appeals to (e.g., the law of the excluded middle) are truth preserving? A lot of work in the philosophy of mathematics has gone into trying to give answers to these two questions (see, e.g., Maddy 1988). In this paper, I argue that even if we had completely satisfactory answers to these two questions, we would still not have a completely satisfactory explanation of how mathematicians actually know on the basis of proof that mathematical propositions are true. The reason is that mathematicians often intentionally leave gaps in their proofs.

Intentional Gaps In Mathematical Proofs

En este articulo se considera la demostracion matematica como un objeto complejo que admite distintas interpretaciones y dimensiones. Dentro de la institucion matematica, la interpretacion dominante es la consideracion de la demostracion como demostracion deductiva, formal. Pero esa interpretacion, ademas de limitada desde un punto de vista epistemologico, comporta importantes dificultades para los estudiantes de distintos niveles educativos, incluido el universitario, que manifiestan una gama variada de esquemas personales de demostracion. Atendiendo a dichas razones epistemologicas y didacticas, se propone revisar la interpretacion formalista de la demostracion matematica, flexibilizar su significado, dando cabida en el a otras formas de argumentacion —tambien utilizadas por los matematicos al hacer sus demostraciones—, mas cercanas a las practicas reales de los estudiantes.

https://dialnet.unirioja.es/descarga/articulo/617799.pdf

La demostración en matemática: Una aproximación epistemológica y didáctica

Mathematicians of every age have been seen to criticize the proofs of their predecessors or of their contemporaries as ‘not being rigorous’; and often those which they proposed as replacements for the defective proofs were in their turn considered inadequate by the succeeding generation. This apparent repeated putting into question of mathematicians' arguments at length led some people to think these arguments contained a hidden flaw which meant that a proof could never be found unassailable by any criticism.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2C2CA4C3C0A3CBB82A6F052A1EFDB699/S0025557200158486a.pdf/div-class-title-the-concept-of-rigorous-proof-a-href-fn01-ref-type-fn-span-class-sup-span-a-div.pdf

The concept of ‘rigorous proof’

To categorize the innumerable works which are published every year in mathematics is a superhuman task, on account of the close-knit network, almost impossible to unravel, of links which bind together the various theories, and which are the mainsprings of their progress. The list which follows, then, must not be seen as anything else than a very approximate indication of the various directions in which mathematical research might be taken, with respect to problems which often do not seem to point in these directions.

Jean Dieudonné

Papers

The concept of ‘rigorous proof’

Mathematics of our day