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Overview
Amid the unrest, dislocation, and uncertainty of seventeenthcentury Europe, readers seeking consolation and assurance turned to philosophical and scientific books that offered ways of conquering fears and training the mind—guidance for living a good life.
The Good Life in the Scientific Revolution presents a triptych showing how three key early modern scientists, René Descartes, Blaise Pascal, and Gottfried Leibniz, envisioned their new work as useful for cultivating virtue and for pursuing a good life. Their scientific and philosophical innovations stemmed in part from their understanding of mathematics and science as cognitive and spiritual exercises that could create a truer mental and spiritual nobility. In portraying the rich contexts surrounding Descartes’ geometry, Pascal’s arithmetical triangle, and Leibniz’s calculus, Matthew L. Jones argues that this drive for moral therapeutics guided important developments of early modern philosophy and the Scientific Revolution.
Editorial Reviews
HGerman
Gracefully, and piece by piece, Jones presents the reader with the building blocks of Descartes, Pascal and Leibniz's natural philosophies, identifying their reference points, models and influences, the forms of argumentation they employed . . . and how they circulated (and attempted to popularize) their contributions to knowing, or what some might today call science. Linking these frameworks reveals distinct patters, which historians of all stripes will find quite instructive.— Kelly J. Whitmer
MAA Online
Jones carefully reconstructs the philosophical context of the seventeenth and eighteenth centuries in which these great mathematicians worked. . . . Meticulous and careful scholarship.— Bonnie Shulman
American Historical Review
Jones has done a masterful job of relating the mathematical thought of his three philosophers to their ethical and epistemological concerns.— Lisa T. Sarasohn
Journal of the History of Philosophy
This book sets a fascinating stage for these philosophers, one that not only fits them into their own narrow historical context, but also places them within a broader philosophical context of human attempts to understand just how we think we fit into this strange and sometimes terrifyuing world.— Charlie Huenemann
Isis
The scholarship is formidable, extending way beyond the three principal figures studied to a host of lesserknown thinkers. . . . This is a book that historians of ideas will admire for its meticulous detail and that should also interest less historically oriented philosophers by unfolding a now largely lost world in which science and ethics were integrally interconnected.— John Cottingham
Mario Biagioli
"With an exceptional blend of historical, philosophical, literary, and technical skills, Matt Jones reframes the whole debate about the place of civility in science and philosophy. Politeness has been typically seen as a tool for the social legitimation of the sciences, but Jones demonstrates that Descartes, Pascal, and Leibniz viewed the mathematical sciences as a means to the only civilizing process that really mattered. Civility and mathematics converged in the pursuit of 'the good life'— the refining of the limited cognitive, ethical, and spiritual capacities humans still had in the postlapsarian context. By refusing to reduce civility to either the order of society or nature, Jones’s book shows us the simultaneous production of selves and knowledge.”
Peter Galison
“We have long known two things and yet, only with Matthew Jones’s strikingly original book, have they been put together. On the one hand, the long philosophical tradition, back from the time of the ancients, has stressed the profound connection between the study of philosophy and the conduct of a good life. On the other, Descartes, Pascal, and Leibniz each embedded their science and their philosophy in a common enterprise. What Jones has shown is that, for these pivotal thinkers, the practices of the science was a form of spiritual exercise, a cultivation not only of a right way of thinking but of a right way of being. This is fresh, important work that resituates key works of early modern science—indeed of early modern thought, full stop.”
Lorraine Daston
“This lucid and learned study shows how the quest for virtue intersected with the quest for the truths of mathematics and natural philosophy in the writings of three seminal seventeenthcentury thinkers. The Good Life in the Scientific Revolution can be expected to have reverberations well beyond the history of science and philosophy, for early modern cultural and intellectual historians will surely be challenged by Jones’s new way of making seventeenthcentury moral, religious, political, and scientific thought cohere.”
Stephen Gaukroger
“The book is a tour de force, offering a fundamental reassessment of what drives earlymodern philosophical thought. A wonderfully imaginative and original investigation of the way in which Descartes, Pascal, and Leibniz used mathematical questions to probe human capacity. This is a truly pathbreaking achievement.”
Peter Dear
“The natural world was once at the center of a philosopher’s life, central to answering the question ‘how should one live?’ In this absorbing book, Matthew Jones gives new life to the apparently disparate work of Descartes, Pascal, and Leibniz by showing that these muchstudied figures will only make sense to us when we see them as practical philosophers, for whom the sciences were crucial keys to selfunderstanding in a challenging universe.”
HGerman  Kelly J. Whitmer
"Gracefully, and piece by piece, Jones presents the reader with the building blocks of Descartes, Pascal and Leibniz's natural philosophies, identifying their reference points, models and influences, the forms of argumentation they employed . . . and how they circulated (and attempted to popularize) their contributions to knowing, or what some might today call science. Linking these frameworks reveals distinct patters, which historians of all stripes will find quite instructive."
MAA Online  Bonnie Shulman
"Jones carefully reconstructs the philosophical context of the seventeenth and eighteenth centuries in which these great mathematicians worked. . . . Meticulous and careful scholarship."
American Historical Review  Lisa T. Sarasohn
"Jones has done a masterful job of relating the mathematical thought of his three philosophers to their ethical and epistemological concerns."
Journal of the History of Philosophy  Charlie Huenemann
"This book sets a fascinating stage for these philosophers, one that not only fits them into their own narrow historical context, but also places them within a broader philosophical context of human attempts to understand just how we think we fit into this strange and sometimes terrifyuing world."
Isis  John Cottingham
"The scholarship is formidable, extending way beyond the three principal figures studied to a host of lesserknown thinkers. . . . This is a book that historians of ideas will admire for its meticulous detail and that should also interest less historically oriented philosophers by unfolding a now largely lost world in which science and ethics were integrally interconnected."
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Meet the Author
Matthew L. Jones is associate professor of history at Columbia University.
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DESCARTES, PASCAL, LEIBNIZ, AND THE CULTIVATION OF VIRTUE
By Matthew L. Jones The University of Chicago Press
Copyright © 2006 The University of Chicago
All right reserved.
ISBN: 9780226409559
Chapter One
Geometry as Spiritual Exercise
In some early notes, René Descartes commented on a surprising difference between the health of the body and the health of the soul: "Vices I call maladies of the soul, which are not so easily diagnosed as maladies of the body. While we often have experienced good health of the body, we have never experienced good health of the mind" (AT X:215). The language of spiritual malady harkened back to an ancient tradition, exemplified in Cicero's Tusculan Disputations: "Diseases of the soul are both more dangerous and more numerous than those of the body." Philosophy offered succor. "Assuredly there is an art of healing the soulI mean philosophy, whose aid must be sought not outside ourselves, as in bodily diseases. We must use our own utmost endeavor, with all our resources and strength, to have the power to be ourselves our own physicians." Philosophical exercise could overcome the sickness, the delusions, the sadness inculcated by institutions, tradition, and everyday affairs. Attaining a healthy soul demanded work on oneself, exerciseaskesis.
Descartes' famous quest to devise a superiorphilosophy should be understood within such a therapeutic model of philosophy. Descartes offered his geometry in part as just such a spiritual exercise, able to help one to attain a tranquillity grounded in an intellectual mastery in making all the choices of life. Along with other exercises, the practice of mathematics could help one to steel oneself against outside direction, confusion, and distraction. Influential works of Descartes' time echoed the diagnostic and curative model of philosophy seen in Cicero. In his widely read On Wisdom of 16014, Pierre Charron "calls man to himself, to examine, sound out, and study himself, so that he might know himself and feel his faults and miserable condition and thus render himself capable of salutary and necessary remediesthe advice and teachings of wisdom." At the Jesuit school of La Flèche, Descartes went through a curriculum predicated in part on a model of the ennobling and curative virtues of philosophy.
Descartes denied the therapeutic efficacy of the philosophies and models for spiritual health available in his day. However much he desired a therapy for the soul, the young Descartes could not see how to choose any particular program without means for gauging the health of the soul and the mind. Around 1619, Descartes lacked any certain means for choosing among the many treatments available in the early seventeenth century or for judging his own attempts.
The first glimmers of a new way forward appear elsewhere in his notebook where Descartes discussed a number of new mathematical discoveries involving instruments and machines. Before long he applied these mathematical discoveries to the vexing question of the health of the soul. New forms of exercise, including geometric practice, could provide a way to measure and produce spiritual health. Descartes underscored the concreteness of these exercises a few years later when he advised the study of "the simplest and least exalted arts, and especially those in which order prevailssuch as those of artisans who weave and make carpets, or the feminine arts of embroidery, in which threads are interwoven in an infinitely varied pattern." The same held for arithmetic and games with numbers: "It is astonishing how the practice of all these things exercises the mind," so long as we do not borrow their discovery from others but invent them ourselves (AT X:404). For Descartes, such study habituated one to experiencing clear and distinct order: "We must therefore practice these easier tasks first, and above all methodically, so that by following accessible and familiar paths we may grow accustomed, just as if we were playing a game, to penetrating always to the deeper truth of things" (AT X:405; CSM I:36). Boldly Descartes announced, "Human discernment [sagacitas] consists almost entirely in the proper observance of such order" (AT X:404; CSM I:35). Upon such discernment of order rested the ability to make the will capable of accepting and following the guidance of the intellect. Much as Cicero had maintained, individual endeavor was necessary to foster the dependence on oneself central to spiritual health.
No activity could develop this discernment better than mathematics: "[T]hese rules are so useful in the pursuit of deeper wisdom that I have no hesitation in saying that this part of our method was designed not just for the sake of mathematical problems; our intention was, rather, that the mathematical problems should be studied almost exclusively for the sake of the excellent practice which they give us in the method" (AT X:442; CSM I:59). With his new geometry, Descartes did not offer algorithms that mechanically yielded certain knowledge about mathematical objects and a world governed by mathematical laws. His geometry offered exemplary practice in seeing and thinking clearly, in experiencing with a healthy soul. Such practice was essential both to improve mathematics and to heal the self. Geometrical exercise could help cultivate the discernment needed for choosing among philosophical doctrines and practices, for finding a purpose for one's life, and for prescribing the means necessary to achieve that purpose: in sum, for pursuing the good life.
This chapter studies one set of exercises Descartes deemed propaedeutic to a better life and better knowledge: his famous, if too little known, geometry. Commentators have often stressed how Descartes offered a mathematical model for knowledge, but they have not often inquired into the rather odd mathematics he set forth as this model. His geometry, neither Euclidean nor algebraic, has its own standards, its own rigor, and its own limitations. These characteristics ought to modify our view of Descartes' much ballyhooed creation of the "modern subject" of knowledge. While the technical details of his geometry might seem interesting and comprehensible only to historians of mathematics, the essential features of his mathematics can be made readily comprehensible to anyone with highschool algebra. These fundamental characteristics reveal much about his concerns about what prevents and produces the right sort of subjectivity. More than simply providing a philosophical account of the modern, Cartesian subject in his Meditations, Descartes offered techniques for becoming a new kind of thinking, knowing, and acting subject. To pursue his philosophy was nothing less than to cultivate and to organize one's self. He offered his revolutionary but peculiar mathematics as a fundamental practice in this philosophy pursued as a way of life.
What does all this potentially anecdotal detail have to do with the substance of Descartes' mathematics? Descartes certainly may have held mathematics to be good for some sort of pedagogical exercise, but how could that possibly make the content, the essence, of that mathematics any different? After all, since Euclid had exemplified mathematical rigor in antiquity, what could be more obvious than the idea that mathematics retains an unchanging core despite all its varied uses, its manifold representations, and the motivations for doing it?
Few scholars of earlymodern science and philosophy would now doubt the need to study rigorously the contingent constitutions of systems of thought or sets of epistemic practices and the embedding of those constitutions within their cultural and social roles. In nearly every study of such systems, however, a cordonedoff core of logic and rigorously argued philosophy remains relatively unexamined. So, one ought rightly to contend, however variable the cultural uses of philosophy and mathematics, there remains an invariable, autonomous essence in each to be studied in itself, an essence that can be abstracted from those uses. In the earlymodern period, mathematics, logic, and natural philosophy were widely supposed to be rigorous, coherent, and certain, but there was considerable debate on what all those terms meant and implied. After the criticisms of humanists, skeptics, and scholastics of the sixteenth century, the very definition and centrality of proof was contested. For his part, Descartes rejected standard mathematical proof. He was hardly alone. In his time, no consensus existed on the objects of mathematics, its proof techniques, its proper institutional settings, its place in the hierarchy of disciplines, or its relationship to "mathematical practitioners." Recognizing these historical qualities of mathematics hardly implies the sophomoric relativism of "anything goes." It means that the very things that make a technical history of an object technical, systematic, or rigorous are themselves historical. The contingent history of logic and proof must be made evident and then explained. Each competitor in the variety of competing programs demands nonanachronistic technical inquiry. None is reducible to modern logic or proof, at least not without ignoring the historical understandings of rigor and proof and the historical motivations for pursuing mathematics and seeking certainty.
With his account of mathematics, Descartes called for changes in the objects of mathematics: so he limited proof processes, expanded the kinds of allowable curves, and added algebraic representation. Equally he called for altering the practitioners, the subjects of mathematics. Recent historical attention to the selffashioning of mathematicians has paid insufficient attention to the variety of mathematical practices (and their metamathematical embedding) that were to help effect such fashioning. To assess correctly the contingent nature of mathematics demands examination of the changing social roles and statuses of its practitioners. Likewise, to evaluate its practitioners and their social roles requires careful inquiry into technical content and practices, into the range of objects, tools, and allowed proof techniques of their particular mathematics.
What then is a spiritual exercise? As noted in the introduction, I take spiritual exercises to be practices held to be useful for cultivating the self. Specifying a spiritual exercise means something like determining (1) a set of practices, (2) a conception of the self, where the self need not be reduced exclusively to a mind or an intellect, and (3) the people the exercises are for, that is, the social field of the application of the exercises, either explicit or implicit. For Descartes, his geometry offered a set of cognitive exercises that were extraordinarily useful, if not quite necessary, for achieving a selfregulated life grounded in apprehending natural and moral truths.
This chapter studies Descartes' Geometry of 1637, on its own terms, using his repetitive statements of its purpose, its contents, its foundation. In his writings from the 1620s until his death, he asserted regularly that mathematics is an exercise, perhaps the best that we have to practice thinking well. Taking this claim seriously helps to clarify problematic aspects of his geometry and philosophy of mathematics. Looking at both his mathematics and his understanding of its purpose, we will escape a long tradition of equating the subject Descartes aimed to create through his exercise with the socalled Cartesian and modern subjects.
A number of insightful scholars have rightly stressed the need to examine Descartes' works as forms of exemplary practice and not only as philosophical doctrine to be rationally reconstructed. They have largely avoided his mathematics. His curious mathematics offers a key for understanding how Descartes' intended to have his philosophy practiced. I focus on the laborious nature of mathematics: while not the corporeal toil of a contemporary artisan or farmer, it is exercise, hard exercisea point obvious enough to mathematicians but too often absent from histories and philosophies of the subject. Work using geometry as exemplar could produce the focused intelligence that Descartes thought was the heart of cultivation and was necessary for discerning and pursuing the good life.
ENVISIONING THE ANCIENTS
In Discourse on Method (1637), Descartes explained that the mathematics he experienced in the Jesuit school of La Flèche led him to conclude that mathematics was good only for clever tricks and mean trades. Why then had Plato's Academy refused to admit anyone ignorant of mathematics, a science that seemed so "puerile and hollow"? The ancients, Descartes decided, must have had a "mathematics altogether different from the mathematics of our time" (AT X:376; cf. CSM I:18). Descartes caught glimpses of this true mathematics in the ancient mathematicians. Why only glimpses? In a remarkable piece of historical reconstruction, Descartes asserted that the ancients feared "that their method, being so easy and simple, would become cheapened if it were divulged, and so, to make us wonder, they put in its place sterile truths deductively demonstrated with some ingenuity, as the effects of their art, rather than teaching us this art itself, which might have dispelled our admiration" (AT X:37677; cf. CSM I:19). Misled by the ancients' "ruse" of formal proof, Descartes' amazed contemporaries memorized the sterile truths of the ancients and failed to seek the relationships behind those truths and the methods used to discover them.
A friend of Descartes', the writer JeanLouis Guez de Balzac, made a similar point about Cicero. Cicero's codified rhetoric for swaying the mob had been mistaken as his true rhetoric and dialectic and then made into strict rules. The contemporaries of Guez de Balzac and Descartes had confused instantiations of technique with the essence of mathematics and rhetoric. While these ancient techniques certainly could deceive, move, and direct, they could neither dispel wonder nor produce novel mathematicians or genuine orators. In misapprehending the grounds of these techniques, their contemporaries had made themselves into people needing external discipline and rules, a mob to be swayedwhen, presumably, they ought to have been swaying the mob. They lost the true mathematics and rhetoric and, with them, genuine knowledge and the ability to invent and discover on their own. Slaves to rules and outward appearances, they were not masters of their own disciplines, or of themselves.
Descartes and Guez de Balzac connected this slavish dependence on technique to the institutionalization of mathematics and rhetoric. Like many of their contemporaries, they envisioned the ancients as successful, stable, and productive precisely because they were honnêtes hommescultivated gentlemenoutside stultifying institutions. Blaise Pascal captured this common seventeenthcentury view well: "We imagine Plato and Aristotle only in the long robes of pedants. They were honnêtes gens, much like the others, laughing with their friends. And when they amused themselves by writing their Laws and Politics, they did so without trying. [Writing these books] was the least philosophical and serious part of their lives; the most philosophical was to live simply and tranquilly. If they wrote of politics, it was to regulate a hospital of madmen" (S457). Generations had mistaken institutionalized technique useful for regulating others as essential philosophical doctrine and considered ways of living. The outer garments of rhetoric, mathematics, and philosophy had been substituted for the real methods of these disciplines. Like many of their contemporaries, Descartes, Guez de Balzac, and Pascal assimilated the ancients to the seventeenthcentury ideal of honnêteté. Early in the century, the notion referred primarily to a vision of judgment and taste among the nobility of the robe. Honnêteté offered a normative ideal of selfcultivation that combined a genteel nonspecialization, proper manners, and truth making outside formal institutions with elements of taste and judgment more broadly conceived.
Whereas his contemporaries were trapped by final results and codified procedures, Descartes argued that the "same light of mind that allowed [the ancients] to see that one must prefer virtue to pleasure and the honorable to the useful ... also gave them true ideas in philosophy and the method." Descartes echoed Cicero's famous critique of Aristotle in phrase and intent: "Whereas Aristotle is content to regard utilitas or advantage as the aim of deliberative oratory, it seems to me that our aim should be honestas and utilitas." The genuine knowledge and true skill of the ancients allowed them to be productive mathematicians and to recognize the proper basis of the good life, a life of virtue and duty toward others.
Moderns could surpass the ancients. Descartes and Guez de Balzac maintained that the ancient gentlemenphilosophers had failed to achieve their own ideal, whether in mathematics or in rhetoric. That they took the trouble "of writing so many vast books about" geometry showed that "they did not have the true method for finding all" the solutions (AT VI:376). While the ancients had the seeds of true method, Descartes argued, "they did not know it perfectly" (AT X:376; cf. CSM I:18). Guez de Balzac and Descartes likewise claimed that the ancients had the seeds of true rhetoric but lacked true rhetoric itself. Ancient mathematics was far from perfect. Descartes' evidence? "[T]heir extravagant transports of joy and the sacrifices they offered to celebrate discoveries of little weight demonstrate clearly how rude they were" (AT X:376; cf. CSM I:18). The ancients' lack of selfcontrol proved that some mathematical discoveries came as surprisesdiscovered not through methodic comprehension but with miraculous genius or just plain luck. Even if the lucky ancients had the right informal social structures, they lacked the complementary exercises necessary for eliminating imitation and surprise. They lacked, in sum, the true method for autonomous mathematical discovery and selfmastery.
Moderns needed better social forms and better exercises to renew and exceed the virtues of the ancients. Descartes' friend Guez de Balzac spearheaded a movement to civilize the unruly texts and practices of the Renaissance, to take the fruits of humanism and strip them of the extravagance and pedantry exemplified in theto his minduncontrolled works of Montaigne and classical scholars such as Girolamo Cardano. Only then could humanism be properly deinstitutionalized, its essence revealed, and the true potential of ancient learning nourished. This return to the urbanité of noninstitutionalized higher philosophy needed new forms of writing and printing, which Guez de Balzac worked to produce.
Returning to the true mathematics that the devious ancients had obscured meant creating new exercises capable of producing knowledge and civility alike. In the debate surrounding his Geometry, Descartes became enveloped in controversy with the famous Toulousian mathematician and lawyer Pierre de Fermat. Descartes condemned Fermat's mathematics as wondrous and uncivil: "Mr. Fermat is a Gascon; I am not. It is true that he has found numerous particular beautiful things and that he is a man with a great mind. But, as for me, I have always worked to consider things with extreme generality, to the end of being able to discern rules that also have utility elsewhere." "Gascon" was a welltargeted snub: it suggested provincialism, extravagance, amusement, and a quest for advancementin sum, incivility and disorder. Descartes attacked how Fermat arrived at results: "without industry and by chance, one can easily fall onto the path one must take to encounter" a solution. Fermat was doubly particular: he found pretty trinkets and he did so through geniuschancy genius. For Descartes, chance findings typified mathematics as then practiced: such practices were an inferior means toward a lower form of mathematical knowledge, one that, nonetheless, often produced results. The aleatory nature of such discovery produced wonder, dependence, and extravagance, not clarity, independence, and selfcontrol.
Fermat's mathematics was inferior as mathematics because it failed as an activity appropriate for cultivating the mind more generally. Descartes claimed that his own general geometric method would offer true understanding because an orderly constructive process produced all of its results. Mathematical problem solving was not to proceed by some new ad hoc technique dependent on individual instantiations of skill, expertise, and genius for every problem. For Descartes, real mathematical knowledgeknowledge capable of properly exercising the minddemanded that chance and genius be eliminated. Mathematical practices that made the assent to inferential stepsdeductive proofsthe essence of mathematics had to be abandoned, although Descartes called for the temporary use and then abandonment of such inferential steps. Descartes worked to sever a true mathematics of cultivation, the true method, from specious pretenders, which offered little but calculation, passive procedures, and, too often, deception.
DESCARTES' GEOMETRY OF 1637
The sixteenth and seventeenth centuries witnessed countless mathematical "duels," in which mathematicians tried to best one another with their ingenuity in solving particular problems. Here is a very simple example of a typical problem: Given a triangle ABC and a point D outside the triangle, one must construct a line through D that divides the triangle in two equal parts (fig. 1.1). All solutions must comprise a series of constructions of circles and lines produced by standard compasses and straightedges. Usual solutions to such problems included no information about how to arrive at the solution or how to go about solving a similar problem. Besting someone in a mathematical duel typically meant making others marvel at your ingenuity in solving but did not include offering them heuristic instruction. Descartes criticized his rivals, such as Fermat, for finding "particular things" instead of producing rules useful in geometry and elsewhere.
Descartes aimed to give a set of general tools offering a certain method for problem solving. His Geometry begins audaciously. "All problems in Geometry can easily be reduced to such terms that there is no need from then on to know more than the lengths of certain straight lines to construct them" (AT VI:369). By assigning letters to line lengths, Descartes could translate relationships among geometric objects into algebraic formulas. His geometry gained a powerful toolbox of algebralargely why the book is so famous. In principle, although not in practice, every algebraic manipulation corresponded to a geometric construction. In Descartes' mathematics, geometric problems get geometrical constructions as solutions.
(Continues...)
Table of Contents
List of Illustrations
Acknowledgements
List of Abbreviations
A Note on Conventions
Introduction
Part 1: Descartes
1. Geometry as Spiritual Exercise
2. A Rhetorical History of Truth
Part 2: Pascal
3. Mathematical Liaisons
4. The Anthropology of Disproportion
Part 3: Leibniz
5. Forms of Expression
6. Seeing All at Once
Epilogue
Notes
Bibliography
Index