Introduction: The idiosyncratic nature of Renaissance mathematics

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would later become Yet historiographical tendency project onto has led neglect aspects now considered less modern, if outdated.Proclus case point.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