Introducción: El
Idiosyncratic Nature of
Renaissance Mathematics
Paolo Rossini
Erasmus School of Philosophy,
Rotterdam, Los países bajos
Ever since its foundation in 1540, the Society of Jesus had had one
mission—to restore order where Luther, Calvin and the other instigators
of the Reformation had brought chaos. To stop the hemorrhage of
believers, the Jesuits needed to form a united front. No signs of internal
disagreement could to be shown to the outside world, lest the congrega-
tion lose its credibility. But in 1570s two prominent Jesuits, Cristophorus
Clavius and Benito Perera, had engaged in a bitter controversy. The issue
at stake had apparently nothing to do with the values on which Ignazio of
Loyola had built the Society of Jesus. And yet the dispute between Clavius
and Perera was matter of concern for the entire Jesuit community. Ellos
were arguing over the certitude of mathematics.
There are many ways of telling the stories of Renaissance mathematics.
Starting with the Quaestio de certitudine mathematicarum—the dispute that
involved Clavius and Perera—is just an example. One may, as Carl Boyer
does in his A History of Mathematics (Merzbach and Boyer 2011), begin by
outlining the conditions that allowed mathematics to reach new heights in
the sixteenth and seventeenth centuries. Chief among these conditions
were the rediscovery of Greek geometry—in particular the works of Euclid
and Apollonius—and the Latin translations of Arabic algebraic and arith-
metic treatises. O, following the example of Klein (1968), one may trace
the transformations undergone by ancient concepts such as that of arithmos
(number in Greek) during Renaissance times. Pero, I believe, no event epit-
omizes the spirit of Renaissance mathematics better than the Quaestio.
This special issue was conceived, assembled, and finalized during the Covid-19 pandemic. I
would like to thank all authors and reviewers for their availability in these uncertain times.
A special thanks goes to Alex Levine for his support throughout the editorial process.
Perspectives on Science 2022, volumen. 30, No. 3
© 2022 by The Massachusetts Institute of Technology
https://doi.org/10.1162/posc_e_00419
353
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354
Introducción
To begin with, the participation of the Jesuits in the Quaestio tells us
eso, in the Renaissance, mathematics was not just the business of math-
ematicians. To be sure, Clavius was a leading mathematician of his time,
but Perera was a theologian and philosopher. Even those scholars who com-
mitted themselves to mathematical research for its own sake—and not as
part of an educational program, like the Jesuits did—had broader interests.
Gerolamo Cardano was an astrologer, natural philosopher, and instrument
maker. Niccolò Tartaglia was a fine humanist and translator. Simon Stevin
was an engineer and a man of letters—Dutch language owes much of its
scientific terminology to Stevin’s neologisms.
But the most important lesson that can be learned from the Quaestio is that
the transition from Renaissance to modern mathematics did not happen over-
night. We see mathematics as the foundation of natural sciences, whereas in
the second half of the sixteenth century the question was whether mathematics
was a science at all. Those who sided with Perera believed that mathematics
did not deserve this title because of the nature of its demonstrations. Mathe-
matical demonstrations, their critics argued, were not syllogisms in the
Aristotelian sense, therefore they could not be said to engender scientific
conocimiento. Renaissance mathematics was a long way from being modern.
Por supuesto, Renaissance mathematics contained the seeds of what would
later become modern mathematics. Yet the historiographical tendency to
project one onto the other has led us to neglect aspects of Renaissance
mathematics now considered to be less modern, if not outdated.
Proclus is a case in point. His Commentary on the First Book of Euclid’s Elements
had a great impact on Renaissance mathematics. In the context of the Quaestio,
advocates of the certitude of mathematics, such as the Padoa professor Francesco
Barozzi, appealed to Proclus’s Commentary as evidence of the robustness of
mathematical demonstrations (De Pace 1993). Perhaps more relevant is the
role of Proclus in the narrative of mathesis universalis—a universal mathematics
capable of producing certain knowledge. In early modern times, many philos-
ophers and mathematicians—from Descartes to Leibniz—embarked on the
quest for mathesis universalis. Como consecuencia, scholars since Heidegger
([1934] 1987) have recognized mathesis universalis—and the related project
of mathematizing nature—as one of the staples of modern philosophy and
ciencia (see also Crapulli 1969). More recent studies have revealed that mathesis
universalis had in fact a longer history, reaching back to Proclus and Aristotle
(Rabouin 2009). Este, Por un lado, has elevated Proclus to the status of
forerunner of modernity. Por otro lado, it seems to have obliterated the
fact that Proclus was writing at a time when mathematics was part of a larger
system of knowledge that included religious elements.
The truth is, Robert Goulding writes in his article in this collection,
that not all of Proclus’ Euclid commentary is about mathematics. Images
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Perspectives on Science
355
of the divine populate various passages of the work, as Proclus uses math-
ematical symbols to invoke the gods and transcend discursive thinking.
Readers of Proclus, then and now, have treated these passages as if they
were relics of a bygone era—embellishments rather than integral parts
del texto. But what would happen if we took Proclus’ godly interludes
seriously? We would find out, Goulding argues, that the objective of the
Commentary lies beyond the borders of mathematics, in the pious lands of
“inner theurgy”—the complex of rituals and practices developed by the
Neoplatonists to ascend towards the One. By exploring the theurgic
dimension of Proclus’s Commentary, Goulding adds a puzzling layer to
our understanding of this text. His essay is a lesson in the richness of math-
ematical history.
To tell the untold stories of Renaissance mathematics, we also need new
herramientas. En años recientes, Matteo Valleriani and his team at the Max Planck Insti-
tute for the History of Science have demonstrated that the study of astro-
nomical treatises—such as the Sphere of Johannes de Sacrobosco—can be a
test bed for the application of statistical methods to the humanities. Aquí,
together with Beate Federau and Olya Nicolaeva, Valleriani shows how qual-
itative and quantitative approaches are both essential in unearthing a hidden
fragment of Georg Joachim Rheticus’s intellectual biography. Despite being
a disciple of Copernicus and one of the editors of his De revolutionibus orbium
coelestium (On the Revolutions of Heavenly Spheres, 1543), Rheticus, the authors
afirmar, advocated for geocentricism—the opposite view of his master. He did
so by authoring or editing texts that were reused in several printed editions
of Sacrobosco’s Sphere over almost a century (1538–1629). This was in line
with what Westman (1975) has called the “Wittenberg interpretation” of
the Copernican theory—an interpretation proposed by the members of
the Melanchthon circle to fit the Copernican models into a geostatic view
of the universe. Rheticus, sin embargo, published his texts anonymously, cual
is why his geocentric sympathies have gone unnoticed until now. In their
artículo, Valleriani, Federau, and Nicolaeva force Rheticus out of the shadows
and reveal how his silent influence allowed the earth to stay at the center of
the universe a little longer.
Both the cases of Proclus and Rheticus prove that misconceptions about
authors and texts—resulting from the habit of analyzing them through the
lens of modernity—have impoverished our knowledge of Renaissance math-
ematics. Another source of misconceptions has been the anachronistic dis-
tinction between pure and applied science. For a long time, historians have
been under the impression that Renaissance mathematics was a pure science,
the mathematician a scholar whose only job was to solve abstract problems
(Roux 2010). Few scholars have done more to debunk this myth than Mario
Biagioli. He has shown us that even the most skilled mathematician needed
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356
Introducción
a patron in Renaissance times (Biagioli 1993); and that theoretical ambi-
tions must be accompanied by a penchant for practical matters, como
designing mathematical instruments and, como consecuencia, claiming author-
ship for them (Biagioli 2006). In this special issue, Biagioli outlines a pecu-
liar way of defending intellectual property. When accused of not being the
inventor of his compass, Galileo did not reply to these charges by patenting
his instrument. En cambio, he let his students speak for him. As part of the same
defensive strategy, he also printed an instruction manual titled Operazioni del
compasso geometrico et militare (Operations of the Geometric and Military Compass,
1606). Tomados juntos, the students’ testimonies and the instructions
demonstrated that only Galileo could teach how to build and operate his
compass—a fact that established him as its inventor.
By the time Galileo had secured the rights to his compass, the Quaestio
was also fading off—although echoes of it could be heard into the seven-
teenth century (see Malet 1997; Mancosu 1996). At the Collegio Romano—
the headquarters of the Society of Jesus in Rome—Clavius had gained the
upper hand over Perera. Como resultado, Euclid’s Elements had become a compul-
sory reading for the students of hundreds of Jesuit colleges over the world.
A few of those students would turn out to be brilliant mathematicians:
Grégoire de Saint-Vincent, Honoré Fabri, André Tacquet. This was the
intellectual climate in which Claude François Milliet Dechales—another
disciple of Ignatius of Loyola—published his monumental history of
mathematics in 1674. In the article that concludes this volume, Antoni
Malet gives a thorough account of what he sees as “a late contribution
to the genre, probably the last substantial one before histories inspired
by ‘enlightened’ perspectives appeared.” Jean-Etienne Montucla’s 1758
Histoire des mathématiques has long been recognized as the starting point
of mathematical historiography. Malet’s study adds to the evidence that
histories of mathematics were written well before Montucla—in fact as
early as the sixteenth century (see also Goulding 2010). By means of a close
reading of Milliet Dechales’ treatise, Malet reveals how the French Jesuit
saw the history of mathematics as an irresistible progress. Discovery after
discovery, mathematics was bound to replace speculative natural philoso-
phy as the queen of sciences. Clavius would have been happy.
Referencias
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Absolutism. chicago: University of Chicago Press. https://doi.org/10
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Biagioli, Mario. 2006. Galileo’s Instruments of Credit: Telescopes, Images, Secrecy.
chicago: University of Chicago Press. https://doi.org/10.7208/chicago
/9780226045634.001.0001
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