L M Carolino's conclusions on João Delgado's contribution

Luís Miguel Carolino had written a number of fascination papers on João Delgado and the status of mathematics in 17th centuary Portugal.
These include: João Delgado SJ e a "Quaestio de Certitudine Mathematicarum" em inícios do século XVII, Revista Brasileira de História da Matemática 6 (11) (2006), 17-49; and Cristoforo Borri and the epistemological status of mathematics in seventeenth-century Portugal, Historia Mathematica 34 (2) (2007), 187-205.
We give below a version of Carolino's conclusions in the first of these two papers.

João Delgado and the "Quaestio de Certitudine Mathematicarum"

The position of João Delgado, defending the scientific character of mathematics in Aristotelian terms, bears witness to the change that was observed at the beginning of the seventeenth century in the level of the argumentation used by Jesuit mathematicians in the context of the famous controversy over "Quaestio de Certitudine Mathematicarum". Both Clavius and his disciples Biancani and Delgado agreed that mathematics should be considered a science. However, while Clavius based the scientificity of mathematics on the epistemological specificity of this discipline, which made it possible to affirm its superiority in relation to "natural science", both Delgado and Biancani sought to affirm the place of mathematics in the context of Aristotelian classification. For them, mathematics was not only a science, but, more than that, an Aristotelian science. That is, it was a necessary knowledge of a causal nature, which was expressed through the syllogism.

João Delgado defended the Aristotelian nature of mathematical knowledge, proposing a broader understanding of what were the causes on which science was based. In this context, he argued that the type of causes considered by science also included causes "without physical movement and without existence". In other words, in Delgado's way of thinking, the discussion was no longer restricted to the physical plane, as happened with the case of philosophers who rejected the scientificity of mathematics. Science was characterized, then, as a kind of certain knowledge, evident and based on syllogism. Since mathematics obeyed these requirements, it was unequivocally an Aristotelian science.

In the years that followed, Delgado's position on the certainty and scientificity of mathematics was not the dominant one among Jesuit mathematicians. Biancani, whose thesis became more accepted by the Jesuit mathematicians, sought instead to establish mathematics as an Aristotelian science through the foundation of the ontological status of mathematical ideas. Nevertheless, there are some arguments proposed by Delgado, such as the role of definition as the "average term" in mathematical demonstrations, or the defence that in mathematics we resort to formal and material causes, that came to be developed in later works such as those of Biancani. It is not intended, therefore, to argue that Delgado was a "precursor" to the Italian mathematician. We intended to illustrate how complex and gradual the approach of Jesuit mathematicians was to the celebrated "Quaestio de Certitudine Mathematicarum".

JOC/EFR February 2017

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