Since the application of Postulate I.2 in the Elements is not uniform, one could wonder in what way should it be applied in Euclid’s plane geometry. Besides legitimizing the question itself from the perspective of a philosophy of mathematical practice, we sketch a general perspective of conceptual analysis of mathematical texts, which involves an extended notion of mathematical theory as system of authorizations, and an audience-dependent notion of proof.

Dado que la aplicación del Postulado I.2 no es uniforme en Elementos, ¿de qué manera debería ser aplicado en la geometría plana de Euclides? Además de legitimar la pregunta misma desde la perspectiva de una filosofía de la práctica matemática, nos proponemos esbozar una perspectiva general de análisis conceptual de textos matemáticos que involucra una noción ampliada de la teoría matemática como sistema de autorizaciones o potestades y una noción de prueba que depende del auditorio.

This paper aims at clarifying the nature of Frege's system of logic, as presented in the first volume of the Grundgesetze . We undertake a rational reconstruction of this system, by distinguishing its propositional and predicate fragments. This allows us to emphasise the differences and similarities between this system and a modern system of classical second-order logic.

O De methodis de Newton (escrito em 1671) é o resultado da revisão de um tratado que ele deixara inacabado por volta de outubro e novembro de 1666 (The october 1666 tract on fluxions, assim intitulado por Whiteside). Nesse período, Newton já havia obtido os principais resultados que seriam expostos cinco anos mais tarde no De methodis, que todos consideram a melhor apresentação da sua teoria das fluxões. Todavia, o próprio termo "fluxão" não ocorre no The october 1666 tract on fluxions, onde o que está em questão são os movimentos ou as velocidades. Do ponto de vista estrito do formalismo matemático, a mudança das velocidades (pontuais) para as fluxões não é muito relevante: os métodos matemáticos do De methodis são essencialmente os mesmos do The October 1666 tract on fluxions. Mas a mudança terminológica é, creio eu, o indício de uma maneira distinta de compreender esses métodos e os objetos aos quais eles se aplicam. Newton diz isso de maneira bastante explícita em uma passagem crucial logo no início do De methodis, ao advertir que o termo "tempo" ali empregado não se refere ao tempo formaliter spectatum, mas a "outra quantidade" por meio "de cuja fluxão o tempo é expresso e mensurado". Neste artigo, comento essa passagem crucial e procuro esclarecer as diferenças essenciais entre a noção de velocidade de 1666 e a de fluxão introduzida em 1671. Isso também me permitirá discutir o papel de Newton na origem da análise do século xviii.

Newton's De methodis (written in 1671) results from a revision of an uncompleted treatise that he had written in October-November 1666 (The October 1666 tract on fluxions, as Whiteside called it). In 1666, Newton already had the main results that he would expound 5 years later in the text that is unanimously considered the best presentation of his theory of fluxions. However, the term "fluxion" itself did not appear however in The October 1666 tract on fluxions, where the question addressed had to do with motions or velocities. From the strict point of view of mathematical formalism, the shift from (punctual) velocities to fluxions is not especially relevant: the mathematical methods of the De methodis are essentially the same as those of The October 1666 tract on fluxions. Still, this terminological change is, I think, the symptom of a different way to understand these methods and the objects they apply to. This is quite explicitly said by Newton in a crucial passage at the beginning of the De methodis, where he claims that the term "time" in his treatise does not refer to the time formaliter spectatum, but to "another quantity" for the "fluxion of which the time is expressed and measured". In this paper, I discuss this passage and try to clarify the essential differences between the 1666 notion of velocity and the new notion of fluxion introduced in 1671. This also enables me to discuss the role of Newton in the origin of 18th century analysis.