The Meaning of Proofs by Gabriele Lolli

Author:Gabriele Lolli
Language: eng
Format: epub
Tags: math, proofs, narration, Mathematics; proof; literature; storytelling; poetry; ancient rethoric
Publisher: MIT Press

With this definition, together with that of limit, the first modern one, Cauchy began bringing rigor to nineteenth-century analysis. According to the definition, infinitesimals are variables that tend to zero. In his proofs, Cauchy continued to use the language of infinitesimals and (more rarely) of infinities. In the Avertissements of his 1823 and 1829 texts, he added the clarification: “my principal aim has been to reconcile the rigor which I had made a law for myself in my Cours d’analyse, with the simplicity that results from the direct considerations of infinitesimals” (Cauchy 1899, 9; English translation from Cauchy 2013, 124). However, “simplicity” is a vague, relative, and entirely personal idea; it seems more probable that contextualizing the narrative in the language of infinitesimals would allow Cauchy to express the meaning of theorems in a way that seemed simpler to him, which is to say in the way he himself had thought of it. The use of infinitesimal language is held accountable for what became known as “Cauchy errors,” although it is possible that the interpreters do not grasp Cauchy’s particular way of reasoning. Among the often-cited errors is the theorem that a convergent sequence of continuous functions in an interval tends to a continuous function, and the one according to which a continuous function in a finite interval admits the integral. Lolli (2017, ch. 9.8) summarizes the arguments of Detlef Laugwitz (1932–2000) who invites a reinterpretation of Cauchy in terms of his concepts. It would then appear that these did not coincide perfectly with ours; for example, Cauchy did not define continuity in a point but only around a point.

The examples of dynamic terminology could be multiplied. When John Napier (1550–1617) invented logarithms in 1614, he didn’t invent them as they are now taught in schools (those are the refinements of Henry Briggs (1561–1630)). Napier imagined two straight lines, the first divided into intervals of equal length, the second a segment of length r divided by a sequence of points whose distances from the far right formed a decreasing geometric sequence:

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