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Mersenne and Mixed Mathematics Mersenne and Mixed Mathematics Antoni Malet Daniele Cozzoli Pompeu Fabra University One of the most fascinating intellectual ªgures of the seventeenth century, Marin Mersenne (1588–1648) is well known for his relationships with many outstanding contemporary scholars as well as for his friendship with Descartes. Moreover, his own contributions to natural philosophy have an interest of their own. Mersenne worked on the main scientiªc questions debated in his time, such as the law of free fall, the principles of Galileo’s mechanics, the law of refraction, the propagation of light, the vacuum problem, the hydrostatic paradox, and the Copernican hypothesis. In his Traité de l’Harmonie Universelle (1627), Mersenne listed and de- scribed the mathematical disciplines: Geometry looks at continuous quantity, pure and deprived from matter and from everything which falls upon the senses; arithmetic contemplates discrete quantities, i.e. numbers; music concerns har- monic numbers, i.e. those numbers which are useful to the sound; cosmography contemplates the continuous quantity of the whole world; optics looks at it jointly with light rays; chronology talks about successive continuous quantity, i.e. past time; and mechanics concerns that quantity which is useful to machines, to the making of instruments and to anything that belongs to our works. Some also adds judiciary astrology. However, proofs of this discipline are The papers collected here were presented at the Workshop, “Mersenne and Mixed Mathe- matics,” we organized at the Universitat Pompeu Fabra (Barcelona), 26 May 2006. We are grateful to the Spanish Ministry of Education and the Catalan Department of Universities for their ªnancial support through projects Hum2005-05107/FISO and 2005SGR-00929. We thank the editors of PoS for their kind welcome to our proposal to publish the papers, with our special gratitude to Roger Ariew for his initial support. Perspectives on Science 2010, vol. 18, no. 1 ©2010 by The Massachusetts Institute of Technology 1 Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc.2010.18.1.1 by guest on 26 September 2021 2 Mersenne and Mixed Mathematics borrowed either from astronomy (that I have comprised under cos- mology) or from other sciences.1 Mersenne dealt with almost every one of the mixed and pure mathe- matical sciences listed above. He ªrst collected a number of treatises on geometry and mixed mathematics in the two editions of his Synopsis (Syn- opsis Mathematica 1626 and Universae geometriae synopsis 1644). He com- posed several treatises on music (Traité de l’harmonie universelle 1627; Ques- tions harmoniques 1634; Les preludes de l’harmonie universelle 1634; Harmonie universelle 1636; Harmonicorum libri XII 1648) and optics (De Natura lucis 1623; Opticae 1644; L’Optique et la catoptrique 1651). Later on he published further collections of essays concerning mechanics, pneumatics, hydro- statics, navigation, and the techniques for establishing weights and mea- sures (Cogitata physico-mathematica 1644; Novarum observationum physico- mathematicarum tomus III 1647). Moreover, Mersenne contributed to spread Galileo’s writings in France (Les nouvelles pensées de Galilée 1639; Les méch- aniques de Galilée 1634) as well as the ideas of thinkers such as Hobbes and Roberval. Mersenne discussed natural philosophical issues not only in his mathematical works but also in a number of philosophical essays, in his huge correspondence, and also in writings whose content is not directly related to science. These are notably the cases of the Quaestiones caeleber- rimae in Genesim, a huge commentary to St. Jerome’s Vulgate that contains contributions to optics and acoustics, of La vérité des sciences, the Harmonic- orum libri XII, and l’Harmonie Universelle, which contains contributions to arithmetic (Knobloch 1974). Many years ago, Robert Lenoble’s groundbreaking Mersenne ou la naissance du mécanisme portrayed the Minim friar as an early and inºuential advocate of the mechanization of the world picture (Lenoble 1943). Lenoble did much to establish Mersenne’s reputation as a Christian thinker who, out of his concern for the attacks the Christian faith was re- ceiving from many quarters, was willing to renounce some Aristotelian tenets in order to make room for philosophical innovations in mechanics, astronomy, music, and so on. In a move that was quite standard in his 1. “[. .] la Géométrie considere la quantité continuë, pure et dénuée de la matière, et de tout qui tombe sous les sens: l’Arithmétique contemple la quantité discrette, à sçavoir les nombres. La Musique considere les nombres harmoniques, c’est à dire qui servent aux sons. La Cosmographie contemple la quantité continuë de tout le monde. L’Optique la considere jointe aux rayons de la lumière. La Chronologie parle de la quantité continuë qui est successive à sçavoir du temps passé: et la Méchanique parle de la quantité, qui sert aux Machines, aux instruments, et à tout ce qui appartient à nos ouvrages. Quelques-uns ajoustent l’Astrologie Judiciaire, mais cette partie n’a nulles demonstrations que celles qu’elle emprunte de l’Astronomie (que j’ai compris sous la Cosmographie) et qu’elle prend des autres sciences” (Mersenne 2003, pp. 38–9). Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc.2010.18.1.1 by guest on 26 September 2021 Perspectives on Science 3 days, Lenoble could easily construct Mersenne’s defense of the value of mathematics and advocacy of the mixed mathematical sciences as straight- forward support for a mechanistic understanding of nature. Richard Popkin substantially reªned Lenoble’s picture by placing Mersenne’s philosophical innovations within the emergence of modern skepticism and by characterizing them in terms of ‘mitigated skepticism’. According to Popkin, while Mersenne partially accepted the skeptics’ critique of the possibility of Aristotelian scientia, and consequently of the knowledge of the real nature of things, he advocated a kind of knowledge consisting “of information about appearances, and hypotheses and predictions about the connections of events and the future course of events” (Popkin 1979, p. 131). Along with the contributions of Lenoble and Popkin, the careful exegetical work of the curators of Mersenne’s correspondence brought to light new facets of Mersenne’s scientiªc work. This includes Mersenne’s view on the question of the motion of the Earth, or his evaluation of the ideas of a number of authors, such as Bruno, Campanella, the atomists and Galileo, whom he condemned in the early 1620s, but that later on evalu- ated more positively. Armand Beaulieu, one of the last editors of Mer- senne’s huge correspondence, has assembled a substantial amount of infor- mation about Mersenne’s biography and his works (Beaulieu 1995). In spite of Mersenne’s many works and his centrality in the seven- teenth-century republic of letters, his writings have not generally received the close attention they deserve. The exceptions are perhaps mathematics proper (as opposed to “mixed” mathematics and to his philosophy of mathematics, on which see below) and music. Mathematics proper greatly interested Mersenne but his contributions to it are not central within his philosophical corpus (Coumet 1972; Knobloch 1974; Engelberg, Gertner 1981; Warusfel 1994). On the other hand, music was perhaps the science that most deeply and continuously interested Mersenne. Crombie has stressed that Mersenne combined a systematic program of measuring acoustical and optical quantities affecting perceptions with the investiga- tion of the nature of light and sound. According to Crombie, the mono- chord was the instrument on which he based his investigations (Crombie 1994, vol. II, p. 822). There have been in recent years a number of sub- stantial contributions to Mersenne’s theory of music, a facet of his thought with which we will not be concerned here (Bailhache 1993 and 1994; Crombie 1994; De Buzon 1994; Knobloch 1994; Fabbri 2003). Scholars have stressed the importance of music in Mersenne’s contribution to set up a post-Aristotelian world picture. Frédéric de Buzon has argued that Mersenne considered the principles of music rooted in those of arithmetic, geometry and physics, whereas Scholastics considered music a science sub- ordinated to arithmetic and Kepler grounded it on geometry. For Mer- Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc.2010.18.1.1 by guest on 26 September 2021 4 Mersenne and Mixed Mathematics senne, music showed the universal harmony of the order of creation; un- derstanding this order needed both an experimental investigation and theological explanations (De Buzon 1994, pp. 123 and 127). In the Colloquium organized in 1988 to commemorate Mersenne’s fourth centenary, a number of substantial papers were read that illuminate points of Mersenne’s biography, the order of Minims to which he be- longed, and his intellectual connections to different philosophers (Con- stant, Fillon 1994). Interestingly, only Pierre Costabel’s essay on Mersenne and cosmology dealt with Mersenne’s scientiªc ideas as opposed to his role as a mathematical and philosophical intelligencer (Costabel 1994). Among recent contributions devoted to Mersenne Peter Dear’s mono- graph stands out (Dear 1988). Stressing Mersenne’s deliberate allegiance to some weak form of Aristotelianism, Dear has shown that Mersenne’s “metamathematics” (to use his word) is largely indebted to orthodox sources of Renaissance scholasticism, and in particular to discussions among Jesuits on the nature of mathematics, its objects, and its “scien- tiªc” status. Dear’s Mersenne stays distant to any sort of causal physics while granting new, central, unprecedented status to the mixed mathe- matical sciences. Recent historiography on Mersenne has been concerned with his reac- tion to Copernicanism and to Galileo’s work. Antonio Nardi has placed Mersenne’s contribution to hydrodynamics within the context of Galilean science (Torricelli, Castelli, and Galileo himself). According to Nardi, even if Mersenne’s work on hydrodynamics was not as cogent as Torri- celli’s, he provided sound and clear explanations of the main hydro- dynamical phenomena (Nardi 1994).
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