The germanic development of the pre-modern notion of number. From c. 1750 to Bolzano’s "Rein analytischer Beweis"

Resumen

The historiography and philosophy of mathematics usually characterize the educational reforms undertaken in France and the Germanic territories during the last decade of the 18th century and the first decade of the 19th century as follows: while in France such reforms were carried out after the French Revolution and in particular with the creation of the École polytechnique (in 1794) and the reform of the École normale supérieure (in 1795) in Paris, the Germanic reforms began around 1810 with the foundation of the Universität zu Berlin. This characterization, although it might be useful to summarize the aforementioned period when the interest lies in what happened before –e.g. the development of mathematical analysis in 18th century– or later –e.g. the development of real analysis and set theory in 19th century–, has contributed to a lack of awareness, if not a misunderstanding, of what happened in mathematics not only in between such developments but also during the 18th and 19th centuries. According to the thesis here defended, both the general proposal and the particular results contained in Bolzano’s early works (1804- 1817) continue to be misread: by 1817 Bolzano’s mathematical notions and practices, though they hinted some ground-breaking concerns and features, were heavily deviant from later Weierstrassian arithmetizing notions and practices, insofar as Bolzano’s work still had essential traits of a semantic-ontological conception of analysis. In fact, Bolzano’s account as a transitional mathematician, that is, as one whose early works were aimed in some way towards the land on which the 19th century real analysis was developed, yet deeply rooted in inherited views and practices which were common among Germanic mathematicians around 1800, is a thesis closely linked to the one stated at the outset on the biased narrative of Germanic educational reforms of the early 19th century.