Abstract
This article examines the role and function of so-called quasi-empirical methods in mathematics, with reference to some historical examples and some examples from my own personal mathematical experience, in order to provide a conceptual frame of reference for educational practice. The following functions are identified, illustrated, and discussed: conjecturing, verification, global refutation, heuristic refutation, and understanding. After some fundamental limitations of quasi-empirical methods have been pointed out, it is argued that, in genuine mathematical practice, quasi-empirical methods and more logically rigorous methods complement each other. The challenge for curriculum designers is, therefore, to develop meaningful activities that not only illustrate the above functions of quasi-empirical methods but also accurately reflect an authentic view of the complex, interrelated nature of quasi-empiricism and deductive reasoning.
Résumé
Cet article analyse le rôle et la fonction des méthodes dites « quasi-empiriques » en mathématiques, par le biais de certains exemples historiques et d’autres provenant de ma propre expérience, dans le but de fournir un cadre de référence conceptuel l’enseignement. Les fonctions identifiées, illustrées et analysées sont les suivantes:
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• la conjecture (recherche par induction, généralisation, analogie, etc.)
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• la vérification (tentative d’obtenir des certitudes sur la vérité ou la validité d’une affirmation ou d’une hypothèse)
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• la réfutation globale (démonstration du fait qu’une affirmation est fausse grâce à la génération d’un contre-exemple)
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• la réfutation heuristique (reformulation, affinement ou perfectionnement d’une affirmation essentiellement vraie par le biais de contre-exemples ponctuels)
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• la compréhension (compréhension d’un théorème, d’un concept, d’une définition ou d’une démonstration, ou encore contribution à la découverte d’une preuve ou à la formulation précise d’une définition).
Nous nous intéressons en particulier à l’utilisation de plus en plus courante de l’informatique pour explorer les différents sujets, car l’ordinateur fournit des images visuelles et d’autres stimuli qui alimentent les intuitions susceptibles de contribuer à une meilleure compréhension d’un secteur de recherche donné en mathématiques. Nous soulignons la distinction importante qui existe entre la réfutation globale et la réfutation heuristique. En effet, si la première vise à démontrer la fausseté des résultats mathématiques, la seconde contribue à raffiner et à reformuler aussi bien les résultats que leur démonstration. Après une brève analyse des avantages que présentent les méthodes quasi-empiriques dans chacune des catégories citées plus haut, nous donnons des exemples qui soulignent les limites de ces méthodes pour ce qui est des certitudes (certains résultats, par exemple, résistent à de nombreuses épreuves avant de céder). De plus, il est rare que les méthodes quasi-empiriques servent à approfondir le niveau de compréhension (par exemple à comprendre pourquoi les résultats sont vrais), et il est également rare qu’elles contribuent à une systématisation des mathématiques chez les étudiants (par exemple qu’elles servent à établir des liens, etc.)
Dans la pratique des mathématiques, nous estimons donc que, loin de s’opposer les unes aux autres, les méthodes quasi-empiriques et les méthodes plus rigoureuses sur le plan de la logique se complètent. Le défi à relever dans la mise au point des curriculums consiste à créer des activités significatives capables non seulement d’illustrer les fonctions citées plus haut, mais également de fournir une vision authentique de la complexité des liens qui caractérisent les raisonnements quasi-empiriques et les raisonnements déductifs.
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de Villiers, M. The Role and Function of Quasi-empirical Methods in Mathematics. Can J Sci Math Techn 4, 397–418 (2004). https://doi.org/10.1080/14926150409556621
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DOI: https://doi.org/10.1080/14926150409556621