I defended my Master’s Thesis on April 30, 2015. The title of it is “Making Sense of Mathematics: The Certitudine Mathematicarum Debate and its Relationship to Plato and Aristotle.” The study focused on the way in which the ontology of mathematics was understood among four individuals in sixteenth-century Italy: Alessandro Piccolomini, Francesco Barozzi, Christoph Clavius, and Benito Pereira. The latter two of these were Jesuits. What became interesting to me was the way in which the Jesuits participated in larger Italian debates regarding the nature of mathematics.
My committee members were Dr. Rienk Vermij (chair), Dr. Kerry Magruder, and Dr. Steven Livesey. What was interesting about all of this is that Dr. Livesey was actually in France on sabbatical and so he participated in the thesis defense via Skype.
Although I performed well enough to warrant a successful defense by my committee, I still think I could improve the argument and its articulation. One day I hope to turn this into an article, but for now I will merely post my conclusion:
“None of the authors analyzed denied the certainty of mathematical demonstration. Moreover, they all believed that the clarity of the methods of mathematical demonstration were its greatest strength. However, what was disagreed upon was the exact nature of mathematics. The significance of this was essential for understanding how mathematics could be applied to the actual world. What exactly was the relationship of arithmetic and geometry with sensible matter? Did mathematical demonstrations describe essential aspects of objects? Or were they merely accidental? In a context in which Aristotelian demonstration was largely regarded as the most authoritative form of reasoning, such questions were of deep significance for understanding the nature of mathematics.
When Alessandro Piccolomini first stated that mathematics was not a demonstratio potissima, his claim was deeply rooted in the fact that he was a logician. His prevailing assumption throughout the entirety of the Commentarium was that the potissima presented the essential form of knowledge. However, since mathematics itself, according to him, were essentially abstractions of the imagination, then, mathematics could not fit the framework of the potissima. A significant part of his argument was the way in which he utilized Aristotle, Plato, and Proclus to demonstrate his point. Aristotle provided for him the nature of the potissima and the rules under which it operated. And Proclus, largely understood as a follower of Plato, provided the nature of mathematics. Beneath the surface of his criticism was the fact that these sources could not be reconciled with each other.
Francesco Barozzi’s response demonstrated this. As indicated, a significant part of Barozzi’s response to Piccolomini was that the methods of mathematics demonstrate their certainty. Equally significant was the fact that Barozzi emphasized that the debate over the certitude of mathematics hinged on the way in which Plato, Aristotle, and Proclus were understood. As he indicates in the Opusculum, his translation of Proclus’s Commentary was intended to help demonstrate the relationship between Plato and Aristotle to mathematics. For Barozzi, Proclus’s Commentary served as a unifier between the two. Moreover, quite clearly in his Opusculum, it was Barozzi’s belief that the only difference between a Platonic and an Aristotelian understanding of the sciences was one of perspective.
Such a divide between the abstraction of mathematics and the clarity of its methods carried on into the Jesuit Order. Benito Pereira borrowed the same basic framework that Piccolomini had presented. However, more so in Pereira than in Piccolomini, it is evident that Pereira considered mathematics to be essentially Platonic in its nature. This was seen in the way that he included the Platonic concept of dialectics in the context of his refutation of mathematics, but also in the way in which he guarded against the theory of reminiscence. For the Jesuit, the way in which he understood mathematics was connected to wider philosophical issues. Christoph Clavius, on the other hand, was most interested in understanding mathematics due to its widespread utility. Quite clearly in his texts as well, it is evident that he understood mathematics as essentially developing from Platonic ideas. And quite provocatively, the clarity of mathematics was best understood in contrast to the obscurity of physics.
What has emerged through this study is the way in which the conflict that ensued within this debate occurred as a result of a divide between an Aristotelian theory of demonstration and a Platonic theory of mathematics. At the time in which the debate was occurring it appears that the prevailing theory of mathematics was one rooted in Plato, and that part of what was involved was how to merge an Aristotelian theory of demonstration and of reality with a Platonic foundation of mathematics. Such an understanding certainly fits with Blancanus’s own understanding. After summarizing the debate, he provided an analysis of Euclid’s Book I in relation to Aristotle and indicated that others desired the same for Plato.
At the beginning of this essay it was pointed out that the prevailing analysis for the debate over the certitude of mathematics was that it shaped the development of Jesuit mathematics. If the thesis of this essay is correct, then Jesuit mathematics ought to be understood within a context of mathematical development in which a significant issue was the reconciliation of Plato and Aristotle. Mathematics was predominately understood as being founded on the texts of Plato, largely Republc VII. As a consequence an important aspect of this debate was how to reconcile the nature of mathematics as established in these texts with an Aristotelian understanding of demonstration. Such reconciliation naturally invoked the classification of the sciences, bringing the systems rooted in Aristotle in comparison with Plato. It was within this context that the nature of mathematics, and its ontology, at least among the University trained, was being analyzed.”
In case you’d like to read the whole thing, you may do so here.