The Transition from Formula-Centered to Concept-Centered Analysis Bolzano's Purely Analytic Proof. as a Case Study

Philosophia Scientiae 18 (1):113-129 (2014)
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Abstract

Deux transitions ont eu lieu aux xviiie et xixe siècles dans le déve­loppement de l'analyse mathématique : de l'approche géométrique à l'approche axée sur des formules d'une part ; de l'approche axée sur les formules à l'ap­proche conceptuelle d'autre part. En nous appuyant sur la Preuve purement analytique de Bolzano, nous montrons qu'il adopte une approche que l'on peut qualifier de conceptuelle. Nous parvenons à la conclusion selon laquelle Bolzano n'adopte pas la même attitude selon qu'il se rapporte à l'approche géométrique d'une part, à l'approche axée sur des formules d'autre part ; dans le premier cas, il est question de rejet, dans le second cas de non-participation. Bolzano appuie sa méthodologie conceptuelle sur des opinions philosophiques partagées en partie par certains mathématiciens partisans d'une approche de l'analyse axée sur les formules.In the 18th and 19th centuries two transitions took place in the development of mathematical analysis: a shift from the geometric approach to the formula-centered approach, followed by a shift from the formula-centered approach to the concept-centered approach. We identify, on the basis of Bolzano's Purely Analytic Proof [Bolzano 1817], the ways in which Bolzano's approach can be said to be concept-centered. Moreover, we conclude that Bolzano's attitude towards the geometric approach on the one hand and the formula-centered approach on the other were of a different nature; the for­mer being one of rejection, the latter of non-participation. Bolzano supports his concept-centered methodology by philosophical views, which were partially shared by mathematicians with a formula-centered approach to analysis.

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Author Profiles

Iris Loeb
VU University Amsterdam
Stefan Roski
Universität Hamburg

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