GEOFFREY GORHAM, BENJAMIN HILL, AND EDWARD SLOWIK

`CONCEPTUAL BACKGROUND`

No other episode in the history of Western science has been as consequential as the rise of the mathematical approach to the natural world, both in terms of its impact on the development of science during the scientific revolution but also in regard to the debates that it has generated among scholars who have striven to understand the history and nature of science. In his recent summary of this “mathematization thesis,” Michael Mahoney recounts the stunningly quick ascendancy of the mathematization of nature, a mere two-hundred-year span that witnessed the overthrow of the Aristotle-inspired Scholastic approach to the relationship between mathematics and natural philosophy that had held sway up through the first half of the Renaissance: “For although astronomy had always been deemed a mathematical science, few in the early sixteenth century would have envisioned a reduction of physics—that is, of nature as motion and change—to mathematics” (1998, 702). Yet, by the end of the seventeenth century this radical change in approach had become dominant. In this introduction, we first summarize and explore some of the main conceptual issues crucial to the mathematization of nature during the scientific revolution. The mathematization thesis signifies above all the transformation of scientific concepts and methods, especially those concerning the nature of matter, space, and time, through the introduction of mathematical (or geometrical) techniques and ideas (Yoder 1989). We next analyze the prominence of mathematization as a historiographical framework within scholarship of the scientific revolution, especially in the twentieth century. Finally, we explain how the contributions to this volume explore, challenge, and reshape these conceptual and historiographical perspectives.

The ideal of mathematization has ancient roots (Bochner 1966). Indeed, as we will see in the next section, modern historiography has emphasized the revival of Platonism in the seventeenth century’s drive to mathematize. The remnants of Plato’s own Pythagoreanism are evident in the Republic, where he advocates an a priori astronomy insofar as the visible motions in the sky “fall short of the true ones—motions that are really fast or slow as measured in true numbers, that trace out true geometrical figures, that are all in relation to one another” (529d1-5; 1997, 1145–46). So Socrates urges: “let’s study astronomy by means of problems, as we do geometry, and leave the things in the sky alone” (530b6-c1; 1146). And in the Timaeus Plato develops an elaborate geometrical cosmology and matter theory, guided by the conviction that the creator, in order to produce the best and most intelligible world, would produce a “symphony of proportion” (32c2; 1237). After Plato, Archimedes’s program of mathematization in the sciences of hydrostatics and mechanics provided a model for Galileo and others (Clagett 1964).

Controversy about the value and limits of mathematization also goes back to the beginnings of philosophy. In Aristotle’s view, Pythagoras and Plato excessively conflated the abstract realm of mathematics with the concrete realm of nature: “the minute accuracy of mathematics is not to be demanded in all cases, but only in the case of things which have no matter. Hence its method is not that of natural science” (995a15–18; 1984, 2:1572). So Aristotle concludes that the student of nature should not simply assume that matter and motion will conform to mathematical principles. Nevertheless, in his own Physics, he acknowledges the importance of “the more physical of the branches of mathematics, such as optics, harmonics, and astronomy” (194a8; 1984, 1:311). And in the methodological treatise Posterior Analytics he indicates that such sciences are subject to geometrical (e.g., mechanics and optics) or arithmetical (e.g., harmonics) demonstration (76a1, 21–25; 1984, 1:123) even though their subject matter is empirical: “it is for the empirical scientist to know the fact, and for the mathematicians to know the reason why” (78b32-3; 1984, 1:128). Aristotle assumed that the theorems of such sciences must be “subordinate” to the theorems of their corresponding mathematical sciences, since he prohibited demonstrations that crossed subject-genera (75b3-20; 1984, 1:122). This way of conceiving the “mixed sciences,” as they came to be known, gained additional influence through the pseudo-Aristotelian treatise on mechanics, whose problems involving wheels, pulleys, and levers were routinely treated geometrically by philosophers through the sixteenth century, including Galileo (Bertoloni Meli 2006). Indeed, arithmetic, geometry, astronomy, and music—already identified as peculiarly mathematical by Plato (Republic Bk 7; 525a–31d; 1997, 1141–47)—were formally and pedagogically grouped together in the classical “quadrivium.” Consequently, the idea that mathematics could be used to directly represent physical phenomena remained an open and contested question through the ancient and medieval periods. In the seventeenth century, the main foci of the ongoing debate can be grouped under three broad conceptual categories: instrumentalism versus realism, types of mathematization, and social context.

`Instrumentalism versus Realism`

Two important sources of skepticism about mathematization can be traced to the Aristotelian strictures mentioned previously, one metaphysical and one methodological. First, it was claimed that matter did not conform to the exactness of mathematics, and second, that the deductive structure of mathematical demonstration was inadequate to capture the causal relationships among natural bodies. Hence, outside of the classical “mixed sciences” of optics, mechanics, and astronomy, the utility of mathematics for understanding nature was severely limited. Based on these concerns, an instrumentalist tradition arose that provided a negative answer to the question, do mathematical objects and their relationships correspond to natural objects and their relationships? Instrumentalism regards the mathematical component of physical theories, for example, the epicycle-deferent system of Ptolemaic astronomy, as a mere calculating device for predicting phenomena (Machamer 1976). And this outlook remained influential through the beginning of the early modern period. It is expressed in Osiander’s preface to Copernicus’s De Revolutionibus (1543), which stipulates that since the astronomer “cannot in any way attain to the true causes, he will adopt whatever suppositions enable the motions to be computed correctly from the principles of geometry … these hypotheses need not be true nor even probable” (1978, xvi).

Yet, during the sixteenth and seventeenth centuries the mathematical constructions employed in the new Copernican theory of astronomy began to be accepted by many as providing knowledge of the actual relationships among celestial bodies. Thus, Kepler and Galileo urged that the aim of astronomy was physical truth, not merely to “save the phenomena” via mathematical models (Jardine 1979). And the same realist attitude was extended to the new mathematical work in optics and mechanics. Besides the increase in successful mathematically based approaches, such as Simon Stevin’s work on statics and Galileo’s account of free fall, the main catalyst for the increasing popularity of a realist conception of the link between mathematics and the physical world was almost certainly the rise of the mechanical conception of natural philosophy. By proposing that natural phenomena could be explained by means of machine models, the mathematical relationships that characterize the operation and part-whole relationships of the models offered an obvious and intuitive basis for positing those same mathematical relationships in the natural phenomena themselves. The growing appreciation of the success of mathematical techniques in explaining natural phenomena, combined with the rise of the mechanical philosophy and its realist conception of a hidden world of interacting material particles that have geometrical shapes and volumes, thus encouraged a realist conception of the relation between mathematics and physical reality. As John Henry put it, the “Scientific Revolution saw the replacement of a predominantly instrumentalist attitude to scientific analysis with a more realist outlook” (2008, 8). Galileo’s famous declaration that the book of nature “is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures” (1957, 238–39)—thus turned Osiander’s preface on its head: it was precisely because nature itself was geometrical that mathematical physics had to be true.

`Types of Mathematization`

Galileo’s “book of nature” comment also reveals the type of mathematics that informed much of his work on natural philosophy: geometry, the same approach used by ancient and medieval natural philosophers in the mixed mathematics tradition. At the start of the seventeenth century, geometry could thus lay claim as the most important branch of mathematics for investigating the physical world, especially given the historical precedent of the parallel structure between the synthetic or axiomatic conception of geometry developed in Euclid’s Elements and the deductive methodology of Aristotelian-based Scholastic science. That is to say, axiomatic geometry derives theorems and other elaborate geometric results from a starting point consisting of basic definitions and concepts, and is thus a process that strongly resembles the logical structure of Aristotelian/Scholastic science whose explanatory methodology includes basic metaphysical postulates—“first principles”—as premises, and then goes on to produce specific scientific explanations of various phenomena from that basis (Posterior Analytics; 71b9-78a28; 1984, 1:115–25). Descartes, for example, declared in a 1638 letter to Mersenne that “all my physics are nothing but geometry” (AT 2 268), while Spinoza extended the more geometrico into metaphysics and ethics. Newton’s Principia would constitute one of the last significant examples of this geometrical treatment of physics. A host of mathematical tools would be developed in the seventeenth century that would ultimately transform the acceptable standards of mathematization. While the trigonometric relations embodied in the Snell-Descartes law of refraction and Huygens’s work on harmonic oscillators can be seen as the beginning of this change, the infinitesimal analysis that lay at the heart of the new calculus’s treatment of transcendental curves would mark the most important challenge to the hegemony of the older geometric approach (Mahoney 1998).

The rapid development and increasing usefulness of analytic techniques in mechanics provided a powerful justification for their introduction, no matter how unintuitive or problematic these techniques may have seemed in comparison with the methods derived from classical geometry (Gaukroger 2010). Specifically, the debate about whether geometry should be supplemented by novel algebraic formalisms and techniques reverberated throughout the seventeenth century (Jesseph 1999), culminating in the opposition between Newton’s fluxional version of the differential calculus and the analytical formulation of Leibniz. In this sense, the geometrical character of mathematics that had helped to usher in the scientific revolution, which itself was inspired by the newer mechanical philosophy and ancient geometry, could be seen as limiting the development of mathematization. Defenders of what we might call “geometric fundamentalism,” including Hobbes and Barrow, pointed to the superior intelligibility of geometric proof and to the manifest applicability of geometry to space, time, and matter. Defenders of the new algebraic methods, the so-called mathematical pluralists, pointed to their flexibility and power and to their utility in representing continuous magnitudes, irregular curves, the infinite and infinitesimals, instantaneous velocity, and so on (Mancosu 1996). This is just one example of the way detailed historical scrutiny has recently complicated and enriched the grand narrative of mathematization.

Wholesale mathematization included the aim of extending the mathematical, specifically the geometrical, model of demonstration or method throughout the sciences. In mechanics and other areas of natural philosophy, the aspirational link with mathematics is evident in the many laws of nature, collision rules, and other quantified relationships among material phenomena that were posited by natural philosophers, such as Descartes’ groundbreaking law for the conservation of “quantity of motion” (the product of size and speed; AT 8A 61-2). Given a mathematical formulation, these natural laws could thus be seen as acquiring the same level of necessity and certainty accorded to mathematics as a whole. But the attempts to extend the mathematization of various hypotheses in natural philosophy to the method of natural philosophy itself was never fully realized, despite the obvious success of the former venture by the end of the century. Galileo entitled his last major treatise the Discourses and Mathematical Demonstrations Concerning the Two New Sciences (1638). But Feyerabend and others have noted that this work and the Dialogues rely on a mix of strict demonstration, probabilistic arguments, and rhetoric (Feyerabend 2010; Jardine 1979). Hobbes modeled strictly philosophical method on geometry, but acknowledged the necessity of a hypothetical approach in physics (EW 1 387-8). Descartes hoped his physical principles would be accorded the “absolute certainty” of mathematics but seemed to concede they may possess only “moral certainty” (AT 8A 327). Moreover, although Descartes’ physics aims for mathematical certainty, its content is remarkably free of mathematics, even granting the precedent set by his laws of nature. This split between the mathematization of nature vs. method is perhaps most evident in Spinoza: he hews to the more geometrico in the service of a metaphysical program that is quite unfriendly to mathematization (Schliesser 2014).

Furthermore, while certainty and demonstration were widely heralded, there was also considerable variation among the standards of proof and evidence. Even if mathematical demonstrations delivered certainty, many commentators, especially Aristotelians, denied that they provided substantive (i.e., causal) knowledge of natural processes (Mancosu 1996). The mathematical model of demonstration was not practiced by Bacon and his followers for a slightly different reason, however: they urged the investigation of nature through detailed, immediate “experiments” and the systematic collection of facts or “natural histories” (however, see chapter 2 in this volume).

A revival of ‘atomistic’ conceptions of nature in the early modern period also encouraged the mathematization trend. Conceiving of nature as discrete, rather than continuous as the Scholastics had typically done, atomism rendered mathematical methods that do not rely on geometric continuity more palatable to natural philosophers. Likewise, the mechanical conception of nature could be seen as helping to sanction the methods of analysis because, just as a machine can be viewed as the sum of its parts so the problem posed by an algebraic equation can be similarly resolved by examination and manipulation of its constituent components. Consequently, at the beginning of the eighteenth century, the art of analysis, exemplified by Pierre Varignon’s refashioning of Newton’s mechanics into the symbolic algebra of Leibniz’s calculus, pointed to the future of nature’s mathematization. Still, the mathematization of nature proceeded on many fronts, prompted by, and in turn stimulating, developments in fields besides mechanics and optics. Astronomy, for example, benefited from the introduction of logarithms, while Fermat and Pascal’s investigations of gambling laid the foundations for probability theory, which would eventually have far-reaching applications in the sciences. Finally, the emphasis on experimentation and observation that gradually developed in the seventeenth century would usher in a growing reliance on quantification and measurement. Unlike the ‘qualitative’ approach to the sciences practiced by the Scholastics, the scientific revolution marked the transition to the quantitative outlook that underlies modern science (Roux 2010; Gingras 2001).

`Social Context`

The transformation of the content of mathematics in the seventeenth century mirrored the changing landscape of the social practices and institutions associated with mathematics. The growing power of Europe—in commerce, navigation, and technology among many other areas—was greatly facilitated by mathematical developments and applications. As the demand for mathematically proficient engineers and craftsmen rose, so their prestige and power in society increased. And with their increasing social and political influence, the authority and value of mathematics in society grew in proportion. The change was most evident in the universities, where mathematicians held an inferior status in comparison to natural philosophers at the beginning of the scientific revolution. Yet, by the end of the seventeenth century, many mathematicians and engineers had elevated their positions within the academy. As mathematical practitioners gained status, their knowledge claims garnered intellectual authority. At the same time, mathematics—previously denigrated by the intellectual elite as the purview of mere calculators, engineers, and merchants—won enhanced status through its increasing association with traditional physics and natural philosophy (Feingold 1984). As Biagioli has shown, Galileo himself was an early and rare example of a mathematician who was able to cross this disciplinary boundary and gain the title (and status) of “philosopher” (1989, 49; see also 1994).1

The efforts of the Medici court in promoting Galileo to the more esteemed rank of philosopher typifies this transition in the standing of mathematicians, a change in no small part explained by the progressively expanding importance placed on mathematical expertise for a host of social projects of wealthy patrons, courts, and institutions. The catalogue of these new or improving craft traditions is extensive; besides the more established mechanical arts of statics, hydrostatics, and kinematics, the practice of civil engineering (e.g., surveying, canal construction, and architecture), navigation, and military construction (e.g., artillery, fortifications) were greatly advanced by the development of mathematical techniques (Dear 1995). For instance, Galileo’s use of geometric methods in his study of parabolic trajectories laid the groundwork for modern ballistics (Hall 1952). The arts were also deeply affected by the advance of mathematics, a trend begun in the Renaissance and exemplified in the use of perspective in painting. Leonardo da Vinci and Albrecht Dürer, whose work extended into the sixteenth century, were not only great artists but also skilled mathematicians and engineers. Many of the artists of the early modern period were inspired by the conviction that the essence of nature is mathematical; hence, the artistic content of their work, as well as the techniques used to produce those works, paralleled the rise of mathematics in the other craft traditions in society (Peterson 2011).

`HISTORIOGRAPHY OF MATHEMATIZATION`

The historiographical thesis that the scientific revolution, and by implication modern science as a whole, is guided by the project of mathematization has a long and controversial history of its own. In the preface to Metaphysical Foundations of Natural Science (1786), Kant asserts that “in any special doctrine of nature there is only as much proper science as there is mathematics therein” (2004, 6). It is the mathematical structure of science, as epitomized by Newtonian mechanics for Kant, that renders its fundamental laws necessary and a priori. The criterion of mathematization also explains why chemistry and psychology are not genuine sciences for Kant, the former because its principles are merely empirical and the latter because mathematics cannot be applied to the laws of inner sense.2 So, already in Kant’s influential reconstruction of modern science, mathematization serves the epistemic authority of certain sciences while marginalizing others. William Whewell, in Philosophy of the Inductive Sciences (1840), evinces a more ambivalent attitude. He acknowledges “how important an office in promoting the progress of the physical sciences belongs to mathematics,” especially those sciences concerned with space, time, and motion (astronomy, optics, and mechanics). But Whewell also emphasizes “other ideas quite as necessary to the progress of exact and real knowledge as an acquaintance with arithmetic and geometry,” especially cause, force, and substance (in sciences such as dynamics and chemistry) (1967, 1:156). More critically, in The Crisis of European Sciences and Transcendental Phenomenology (1936), Husserl claims that with Galileo we “observe the way in which geometry, taken over with the sort of naiveté that keeps every normal geometrical project in motion, determines Galileo’s thinking and guides it to the idea of physics” (1970, 29). From a very different perspective, Joseph Needham opens the third volume of his monumental Science and Civilization in China by observing, “since mathematics and the mathematization of hypotheses has been the backbone of modern science, it seems proper that this subject should precede all others in our attempt to evaluate China’s contributions” (1959, 1). Over several hundred years, and across a diverse range of scholarly perspectives, the assumption has been widely shared that, for better or worse, modern science and mathematics are inextricably linked.

Responsibility for the continuing prominence of mathematization historiography belongs to a trio of early twentieth-century historians, each born in 1892 but hailing from different countries. All three emphasized the increasingly mathematical treatment of problems that had long challenged Aristotelian natural philosophers of the middle ages: the planetary orbits, free fall, collision, and optical phenomena. Each emphasized the debt of modern mathematizers like Kepler and Galileo to ancient precursors, especially Plato, Aristarchus, and Archimedes. But their respective attitudes to mathematization are flavored by distinctive philosophical and normative presuppositions. In The Metaphysical Foundations of Modern Physical Sciences (1924), the American philosopher E. A. Burtt was particularly concerned with the implications of mathematization for “man’s place in the world.” The primary-secondary distinction of Galileo, as well as the rigid mind–body dualism of Descartes, are both portrayed as consequences of wholesale mathematization in science. The former denigrates human subjectivity: “in the course of translating this distinction of primary and secondary into terms suited to the new mathematical interpretation of nature, we have the first stage in the reading of man quite out of the real” (1924, 89). The latter relegates the mind to an obscure, isolated position within the brain: “the universe of the mind, including all experienced qualities that are not mathematically reducible comes to be pictured as locked up … away from the independent, extended realm in a petty series of locations inside of human bodies” (Burtt 1924, 123). For Burtt, then, mathematization is instrumental in the disenchantment of the world that was feared by some early, yet cautious, supporters of the new philosophy like the Cambridge Platonists. So we might say that Burtt was preoccupied with the existential implications of mathematization.

Alexandre Koyré is undoubtedly the historian most responsible for the twentieth-century embrace of the mathematization thesis. Koyré (1978) traced seventeenth-century mathematization to the influence of Plato (and Renaissance Platonism), especially in the case of Galileo who “argues for the superiority of Platonist mathematicism over abstract empiricism” (37). Galileo’s Platonism is particularly evident for Koyré in his attitude toward the use of idealization and approximation in science. Not only was Galileo a skilled deviser of “thought experiments,” even the experiments he actually carried out often required “mathematical license” or idealization about the hardness of surfaces, the sphericality of balls, and the parallel orientation of gravitational lines of force, and so on. In Dialogues Concerning the Two Chief World Systems, the Aristotelian Simplicio quips that “mathematical subtleties do very well in the abstract, but they do not work out when applied to sensible and physical matters” (Galileo 2001, 236). According to Koyré, Galileo’s response was in effect that “the real and the material are homogeneous and that a geometrical figure can exist in a material form” (1978, 204).3 Like Burtt, Koyré sees mathematization as key to a fundamental shift in man’s conception of the universe. But unlike Burtt he is less concerned with man’s place in the world than the world’s place in the universe. In From the Closed World to the Infinite Universe (Koyré 1957), modern science, particularly through the work of Copernicus, Kepler, and Galileo, replaces the qualitative, closed, and finite cosmos of Christianized Aristotelianism with a purely quantitative, open, and infinite universe. Integral to this process is what Koyré calls the “geometrization of space”: “the substitution of homogeneous and abstract—however now considered as real—dimension space of the Euclidean geometry for the concrete and differentiated place-continuum of pre-Galilean physics and astronomy” (1968, 6–7). Space and time are no longer framed in relation to privileged places (like the center of the earth) and events (like creation ex nihilo) but rather homogenized and extended infinitely in all directions. With the geometrization of cosmic space-time, matter and motion were also mathematized.4 Matter, now assumed identical in the terrestrial and celestial spheres, becomes either pure res extensa, as in Descartes, or corpuscular, as in Boyle and Newton. Finally, motion and rest are conceived as equally real and are represented geometrically as curves and trajectories or symbolically as algebraic formulas. So we might say that Koyré is preoccupied with the cosmological side of mathematization.

Finally E. J. Dijksterhuis’s 1961 The Mechanization of the World Picture, originally published in Dutch in 1950, offered in certain respects a corrective to the already influential writings of Koyré. For example, whereas Koyré held that revolution or “relative discontinuity” was typical of scientific change—a thesis later strengthened by Thomas Kuhn—Dijksterhuis held that “the” scientific revolution of the seventeenth century was unique in the severity of the conceptual rupture it involved. But he agreed that the revolutionary innovation of seventeenth-century science consisted precisely in “mathematization” (a near slogan in Dijksterhuis’s writings): “the treatment of natural phenomena in words had to be abandoned in favor of mathematical formulation of the relation observed between them. In the present century, functional thinking with its essential mathematical mode of expression has not only been maintained, but has even come to dominate science” (1961, 501).

Dijksterhuis closely linked mathematization with the other major innovation commonly associated with the scientific revolution: mechanization. The pristine “mechanical philosophy” of Descartes conceived the functioning of natural processes by analogy with simple machines: levers, pulleys, and wheels. But such analogies could not provide an irreducible role for “force” and “attraction,” which increasingly figured in analyses of impact, free fall, and projectile motion. “Even the most skilled mechanic,” Dijksterhuis observed, “is unable to construct apparatuses in which material objects move in consequence of their mutual gravitation” (1961, 497). On Dijksterhuis’s analysis, such concepts began to be associated with purely “functional” (i.e., mathematical) relationships rather than mechanical interactions in the traditional analogies.5 Once mathematized, mechanics freed itself from intuitive metaphysical constraints, like no action at a distance, no velocity at an instant, and so on, that had confounded the early mechanical philosophy: “the science called mechanics had emancipated itself in the seventeenth century from its origin in the study of machines, and had developed an independent branch of mathematical physics dealing with the motion of material objects” (1961, 498). And so the heretofore “mixed” or “subordinate” science of mechanics, when finally given a rigorous mathematical formulation by Newton and his followers, became identified with the ancient Aristotelian science of physics itself. So we might say that Dijksterhuis is preoccupied with the mechanical side of mathematization.

The mathematization thesis was criticized for its emphasis on physics and astronomy at the expense of biology and medicine, and for its neglect of important natural philosophers like Bacon and Boyle. This led to a revised “two traditions” model of early modern science. According to Thomas Kuhn’s influential version of the model, the “classical” tradition includes the familiar “mixed sciences” and, beginning in the fourteenth century, the science of local motion. This tradition—mathematical, rationalistic, and abstract—was practiced by Kepler, Galileo, and Descartes. The newer “Baconian” tradition, comprising sciences like chemistry, magnetism, and early electrical theory, was experimental, empiricist, and concrete. Given this methodological split, the Baconian approach had little impact on the rapid advance of sciences like astronomy and optics in the seventeenth century: “For a person schooled to find geometry in nature, a few relatively accessible and mostly qualitative observations were sufficient to confirm and elaborate theory” (Kuhn 1977, 38). Conversely, mathematization came to the Baconian sciences of chemistry and electricity much later than the classical sciences. Seen in this light, the “two traditions” historiography did not challenge, but rather confirmed, the dominant narrative of mathematization, privileging traditional physical sciences while reinforcing the conception of the chemical and life sciences as immature “fact-gathering.” This is reflected in the historiography of the eminent scholar Richard Westfall. Emphasizing metaphysics, rather than methodology, his version of the two traditions thesis contrasts the “Pythagorean-Platonic” tradition, “which looked on nature in geometric terms, convinced that the cosmos was constructed according to principles of mathematical order,” and the “mechanical philosophy,” “which conceived of nature as a huge machine and sought to explain the hidden mechanism behind the phenomena” (Westfall 1978, 1). While acknowledging the “combined influence” of the two approaches, Westfall generally portrays the mechanical philosophy as “an obstacle to the full mathematization of nature” (42) that was eventually achieved by Newton (cf. Guicciardini 2009). In a relatively recent article, Westfall asserts that “the geometrization of nature is perhaps our most distinctive legacy from the scientific revolution,” noting that it came first in physics but later spread to chemistry and molecular biology to such an extent that “to be a scientist today it is necessary to understand and do mathematics” (1990, 59).

In recent years, the mathematization thesis has been subject to varied criticism and analysis. From the perspective of the sociology of scientific knowledge, writers have explored the social and disciplinary implications of the increasing prestige of mathematics. Yves Gingras, for instance, documents how the mathematization of physics served to isolate emerging fields like “rational mechanics” and magnetism from public discussion (and hence criticism). Mathematization “had the effect of excluding actors from legitimately participating in the discourses on natural philosophy” (Gingras 2001, 385), thereby galvanizing the professional status of the new science. As Steven Shapin has discussed, such exclusion was one of Boyle’s major concerns about excessive mathematization. While acknowledging the “usefulness” of arithmetic in experiment, and the elegance of geometrical proof, Boyle wrote in one of his own works on hydrostatics, “I had rather geometricians should not commend the shortness of my proofs than that those other readers, whom I chiefly designed to gratify, should not thoroughly apprehend the meaning of them” (Shapin 1994, 337). Whether because of, or despite, its exclusivity, geometry became part and parcel of physics in the wake of Newton’s Principia. In the “Preface to the Reader,” Newton explicitly uses mathematical precision to set the boundary between “rational” and merely “practical” mechanics: “rational mechanics will be the sense, expressed in exact proportions and demonstrations, of the motions that result from any forces whatever” (1999, 382).6

Even more recently, several philosophically oriented scholars have subjected the standard mathematization historiography to close scrutiny. In a searching critique of Koyré’s “Platonist” Galileo, Gary Hatfield has argued that Galileo’s mathematical approach (in contrast with Kepler’s and Descartes’) is not guided by any metaphysical presuppositions—Platonist, Aristotelian, or otherwise. Rather the application of geometry to nature by Galileo was vindicated simply by its successes, in a variety of theoretical contexts and experimental practices: “his achievement was to show how a mathematical approach to nature could be justified by its successes in practice, and specifically how it might be sufficiently justified by numerous local instances of application” (Hatfield 1990, 139). Lorraine Daston has taken a parallel line against Burtt, complaining that his fixation on metaphysics—especially the allegedly alienating metaphysics of the mechanical philosophy—and his disregard of social and political context, blinded him to the immense complexity of mathematization. He therefore failed to solve the central epistemological problem why mathematization was so compelling to so many: “Burtt’s answer was that the new science required it, but this claim does not carry conviction: there were too many versions of the new science, with and without mathematics; too many versions of mathematized nature, with and without the mechanical philosophy; and too many versions of why nature should be mathematized to warrant any straightforward connection” (Daston 1991, 525; see also contributions to Garber and Roux 2012). Similarly, and even more recently, Sophie Roux has argued that if mathematization involves the application of mathematics to other fields of knowledge, there was little agreement in the seventeenth century about the meaning and aims of such cross-disciplinary application nor even about the nature of mathematics itself (2010, 324). Roux tentatively concludes, as Hatfield did, that “the grand narrative of mathematization has to be enriched by the dense spectrum of various mathematical practices” (327).

`OVERVIEW OF THIS VOLUME`

As a historiographical thesis, mathematization has many virtues. Its longevity attests to that. It is an elegant thesis, and highlights what seems to be a constitutive element of modern scientific practice, providing mathematically precise and rigorous explanations of natural phenomena. It also appears to neatly demarcate mature from immature sciences, thereby offering a framework for following the transformation of a discipline into a properly scientific one. And the thesis is strongly unifying: it unifies individual thinkers and disparate groups into a single movement; in other words, it prompts us to see how Cartesians and Newtonians, mechanists and iatrochymists, Galileo and Leibniz, all shared a philosophical outlook about the nature and aims of science. It also makes good sense of how the unification of phenomena was achieved during the period, such as the unification of celestial and terrestrial movements or the mechanistic unification of living and nonliving bodies, namely under a common mathematical formulation. Furthermore, the mathematization thesis respects the early moderns’ own claims to be mathematizing nature. Finally, it is a very simple thesis. The idea of applying mathematics to nature in order to generate scientific understanding is as easy to formulate as it is to grasp. So it is thus not difficult to appreciate the perennial allure of mathematization as a historical thesis.

Recent historical work and historiographical trends, however, have put considerable pressure on the mathematization thesis, and in many cases have begun to undercut its power and plausibility as a narrative of the scientific revolution. It has always been recognized that there were outliers to the mathematization story: Gassendi and Charleton, Locke and Boyle, and Sydenham and La Mettrie should figure into the narrative of the scientific revolution, but they displayed little real interest in mathematics or quantitative explanations of natural phenomena. And it has always been recognized that within certain domains of natural philosophy (medicine, biology, and psychology, for example) very little mathematization was successful or even attempted. Although previously it may have been easier to view these as exceptions that prove the rule, they begin to look anomalous when taken in conjunction with more recent trends.

More detailed and careful historical work has begun to emphasize the importance of experience and experiment for the early stages of the scientific revolution as well as the central roles played by other conceptual moves and intellectual trends (Dobre and Nyden 2013). The rise of contextualist history of science and philosophy has also begun to highlight the many additional factors propelling the scientific revolution, such as the importance of scholarly societies, the impact of the discovery of the Americas, and sixteenth-century developments in economics and statecraft. The importance of developments within other, more “practical” disciplines, such as navigation and geography, art, anatomy, and pharmacology, are also being identified and explored. To say the least, the story of the emergence of modern science is much more complicated than the mathematization thesis generally suggests. Add to this the growing trend to deny that this emergence is revolutionary, as opposed to gradual or halting, and there seems to be no place for the mathematization thesis in current history of science.

Given these historiographical trends of the last thirty years, it is time to undertake a systematic reevaluation and potential reconceptualization of mathematization as a historical thesis. That is the aim of this volume. In calling for and offering some steps toward a reconceptualization, we are suggesting that the time is right to rethink (1) what mathematization does or should consist in; (2) how it squares with recent scholarship; and (3) its overall value as a historical framework for the emergence of science in the seventeenth century.

As one might expect, scholarly opinion varies on such questions, and the chapters presented here discuss a wide variety of issues and reconceptualizations. Possibilities range from the more extreme view that the mathematization thesis is not only historically useless but outright misleading to the very conservative position that it is historically accurate and requires little or no modification. Between these poles, there are positions that seek to rehabilitate the thesis by recharacterizing it in alternative terms,7 limiting it to one of a variety of contributing factors,8 or restricting it to particular individuals or sciences. If we could draw any generalization from this volume, and the current state of scholarship, it would be that the mathematization thesis as usually conceived is overly simplistic. While simplicity can be a virtue of historical explanation, in this case it depends on excessive vagueness in central concepts such as “mathematics,” “nature,” and “application.” As they are made more determinate, the unity imposed by the mathematization thesis risks falling apart, and its usefulness as a grand narrative framework is compromised. As scholarship proceeds, we should expect the picture regarding mathematization to become even more nuanced and complicated.

In chapter 1, Carla Rita Palmerino (“Reading the Book of Nature: The Ontological and Epistemological Underpinnings of Galileo’s Mathematical Realism”) investigates Galileo’s reputation as the “godfather” of mathematization. She investigates how Galileo’s idealizations are part of a coherent and sophisticated realist ontology and epistemology of mathematics. According to Palmerino, for Galileo mathematical entities are mind-independent and rooted in the physical world, not unlike Barrow’s geometrized notion of mathematics. But the mathematical structures of reality are too complex to be properly grasped on their own, thus necessitating the simplifying analyses in scientific models. Palmerino’s essay closes the gap between Galileo’s mathematics and his realist science by establishing how he conceived of mathematics in realist terms. This relieves the mathematization thesis of one of its persistent objections, which is that Galileo’s mathematical science illicitly assumed the applicability of mathematics to nature. While it might be possible to see Palmerino’s contribution as opening the door to a more sophisticated and fatal critique of the mathematization thesis, Palmerino herself does not invite this. A more natural understanding of her contribution seems to be to take it as part of a larger defense of the traditional mathematization thesis.

Dana Jalobeanu (chapter 2, “ ‘The Marriage of Physics with Mathematics’: Francis Bacon on Measurement, Mathematics, and the Construction of a Mathematical Physics”) challenges the common view that Bacon is antagonistic to mathematization in natural philosophy. Jalobeanu argues, to the contrary, that there was an important quantitative aspect to Baconian natural histories. Other scholars have discussed the importance of careful and precise measurements within Baconian natural histories. But Jalobeanu is particularly interested in Bacon’s claim that “a good marriage of Physics with Mathematics begets Practice,” in which she perceives the seeds of a distinctively Baconian mathematization of nature. The linchpin of her position is an analysis of Bacon’s “reductive experiments,” which were aimed at reducing imperceptible powers or qualities to perceptible ones that could be precisely measured. These experiments were then to be expanded and multiplied until complete tables of the quantities, in every experimental configuration, are recorded. Jalobeanu’s vision of a Baconian mathematization of physics suggests a relatively moderate revision of the mathematization thesis, making room for Bacon within the traditional narrative. A more critical reader, however, may view her analysis as further evidence against any univocal notion of mathematization.

Richard T. W. Arthur’s chapter (“On the Mathematization of Free Fall: Galileo, Descartes, and a History of Misconstrual”) illustrates how the mathematization historiography has distorted our understanding of how key figures of the scientific revolution struggled with one of its central problems. Arthur focuses on one of the supposed crowning achievements of mathematization, the law of free fall, and suggests that we have anachronistically mischaracterized and misunderstood the episode and its significance for mathematizing nature. Arthur’s analysis centers on the concept of instantaneous velocity, which has been regarded as instrumental in both Galileo’s and Descartes’ accounts of free fall. But this was an impossible concept for them, Arthur argues, because they understood velocity as an affection of motion, which could never occur in an instant. By recognizing this anachronism, we can now explain why Galileo and Descartes can seem confused when they are not. On the nature of mathematization, Arthur indicates that “the process was nowhere near as smooth as it would appear” and only one of several drivers of philosophical development (which he terms “epistemic vectors”) that simultaneously propel and constrain scientific thought. But Arthur’s analysis may also suggest stronger conclusions. It undercuts the idea that physical space was geometrized in the way Koyré suggested, because motion was not yet conceived as a continuous magnitude. But it also suggests that there really was not a univocal conception of mathematics, motion, or nature extending from Galileo and Descartes through Leibniz and Newton and Clarke.

Roger Ariew (chapter 4, “The Mathematization of Nature in Descartes and the First Cartesians”) provides the volume’s most vigorous and sustained critique of the oversimplification of the mathematization thesis. Ariew concentrates on one special but important question—how the Cartesians viewed the relationship between mathematics and physics, especially when they thought they were articulating Descartes’ own account. He shows that there was no consensus about the meaning and value of mathematization among the Cartesians. They variously held that mathematics was a tool for sharpening the mind and nothing more (Du Roure and Rohault); that mathematics employed the same faculty of the mind as physics but was not the basis for physical truth or method (Rohault); and that mathematics and physics are not even relevant to one another (Le Grand and Regis). But nowhere, Ariew emphasizes, did they embrace Descartes’ notion of a mathesis universalis, which figures so prominently in the mathematization historiography. Indeed, why would they, Ariew asks, since Descartes’ Regulae was unknown to them? Moreover, the relations between Descartes’ own mathematics and physics were not due to the mathesis universalis, nor to the idea that physics was fundamentally mathematics, but to their both being rooted in the metaphysics of clear and distinct ideas. The disparate and often negative views of mathematization among the Cartesians undermines any wholesale or univocal version of the mathematization thesis. Ariew concludes that mathematization might well be a “twentieth-century invention, perhaps a construction we are forcing on the past.”

Daniel Garber’s chapter (“Laws of Nature and the Mathematics of Motion”) proposes a more moderate reconceptualization of the mathematization thesis. His analysis of laws suggests that mathematization ought to be reconceived as one of many factors motivating and animating the scientific revolution. It may seem that mathematics and laws naturally coincide as mathematics captures the necessity and universality of laws. But Garber argues that for three key figures in the mathematization story—Galileo, Descartes, and Hobbes—the notions of law and the notions of mathematical representations of nature remained quite separate. In Descartes, the laws of nature are not formulated mathematically and there is little that is overtly mathematical in his natural philosophy as a whole. In Galileo, mathematics is everywhere, but the concept of law is missing from his physics and his overall scientific vision. And although Hobbes modeled philosophy and physics on geometry and offered important inertial principles, he was very careful not to label these statements “laws.” Garber’s analysis challenges the assumption that mathematization promoted scientific understanding via laws of nature, as well as the view that mathematization helped secure certainty and necessity in an otherwise voluntarist theological context. But Garber’s analysis leaves open the possibility that the link between mathematization and laws was forged later, by Newton or Leibniz, for example, and that mathematization and the search for laws are parallel factors driving the early scientific revolution (perhaps among many others).

Yet another moderate strategy for reconceiving mathematization is developed by Douglas Jesseph (chapter 6, “Ratios, Quotients, and the Language of Nature”), which explores the debate between Wallis and Barrow about the conceptual foundations of mathematics. According to Jesseph, the disagreement between the arithmetically minded Wallis and the geometrically minded Barrow gives rise to different conceptions of the mechanistic project. Wallis embraced an arithmetic or numerical treatment by which ratios could be compounded and compared. Barrow, in contrast, adopted a geometrically grounded, relational analysis of ratios, which precluded compounding and numerical comparison. According to Jesseph, these divergent models of ratios influenced their respective conceptions of the relationship between mechanical motions and mathematics. For Wallis, mechanics was part of the mixed mathematical sciences in that it consisted of the application of general mathematics to motion. For Barrow, however, mechanical motion is geometrical because it formed the epistemic and ontological basis for geometry itself. One lesson of Jesseph’s analysis is that historians of mathematization must attend to the complex seventeenth-century debates over the foundations of mathematics itself, particularly geometrical versus arithmetic constructions. On the whole, we can discern two parallel forms of mathematization emerging from the Wallis-Barrow dispute. The arithmetically based form fed into the Leibnizian calculus while the geometrically based form culminated in the fluxions of Newton’s Principia.

Eileen Reeves’s chapter (“Color by Numbers: The Harmonious Palette in Early Modern Painting”) is yet another contribution suggesting that mathematization is not sufficiently unified to support a grand narrative of the scientific revolution. Reeves examines a very interesting case of the failure of mathematization—late sixteenth- and early seventeenth-century neo-Pythagorean efforts to explain colors as varying ratios of blackness and whiteness. Interestingly, their neo-Pythagoreanism itself seemed to function as an epistemological obstacle to the development of any coherent and plausible color theory. It was not until the differentiation of primary and secondary colors in the work of François Aguilon, who was anxious to distance himself from neo-Pythagorean predecessors, that color theory was able to fully develop. But, Reeves emphasizes, it was not through Aguilon’s theoretical presentation that the theory of primary and secondary colors was able to spread, but rather through the work of painters and artists who worked with pigments and hues. Reeves neatly presents two of Diego Valázquez’s paintings as visual commentaries on Aguilon’s color theory. The significance of Reeves’s analysis for the historiographical thesis of mathematization is twofold. Although neo-Pythagoreanism is often seen as part of the story of mathematization, here it is presented as an epistemological obstacle inhibiting the science of color theory. Furthermore, it underscores the diversity among domains of mathematization, since Pythagorean harmonics have little to do with the geometrization of space or the Euclidean model of scientific demonstration. There seems to be, at best, only a family resemblance among practices of seventeenth-century mathematization.9

Lesley B. Cormack (chapter 8, “The Role of Mathematical Practitioners and Mathematical Practice in Developing Mathematics as the Language of Nature”) presents additional serious problems for a unified mathematization doctrine. As noted earlier, a key plank in Koyré’s influential version of the mathematization thesis is the geometrization of space that supposedly began in the fourteenth century. Cormack emphasizes that, owing to the prominence of humanism, natural philosophy during the fifteenth and sixteenth centuries was for the most part not mathematically oriented. This suggests that even if physical space was geometrized in the fourteenth century, this played little role in the (alleged) mathematization of nature in the seventeenth century.10 Rather than being in the hands of the natural philosophers or the universities, Cormack argues that mathematics was in the hands of “mathematical practitioners”: painters, mapmakers, instrument designers, military consultants, navigators, and merchants. While it might seem this analysis demands only a corrective about the timing and transmission of the geometrization of space, a stronger conclusion more damaging to the mathematization thesis also seems possible—there was no geometrization of space prior to the rise of mathematical natural philosophy. Like the Egyptians and Mesopotamians who originally developed geometry and arithmetic for building monuments and conducting state business, the sixteenth-century mathematical practitioners were driven by the practical necessities of trade, war, and finance. With lives and fortunes on the line, they had no need for a philosophical breakthrough like the geometrization of physical space. Whether one draws the weaker or the stronger conclusion from Cormack’s essay, at the very least she shows how the mathematization thesis needs to be augmented by including mathematical practitioners in the story.

The next three chapters examine the influential and complex role of the great polymath Leibniz in the process of mathematization. Kurt Smith (chapter 9, “Leibniz on Order, Harmony, and the Notion of Substance: Mathematizing the Science of Metaphysics and Physics”) supports a reentrenchment of the traditional mathematization thesis, advocating its extension into the realm of metaphysics, particularly within the Leibnizian tradition. Smith shows how Leibniz used the idea of a mathematical procedure to make sense of his central metaphysical notions of order and harmony. The primary procedure in question is Leibniz’s early version of what would become known as Cramer’s Rule. He argues that the fact that there is a mathematical solution to a set of equations describing the various forces at work in a substance suggests that Leibniz conceived of harmony and order as the convergence on a determinate. So Leibniz’s version of mathematization was not limited to his mechanics or natural philosophy, it affected even his ontology itself.

Justin E. H. Smith (chapter 10, “Leibniz’s Harlequinade: Nature, Infinity, and the Limits of Mathematization”) supports a similar restriction on the mathematization thesis as urged by Jesseph, but from a very different perspective. He focuses on Leibniz’s reception of the iatromechanical tradition, and his biology and physiology more broadly, rather than Leibnizian mechanics. The iatromechanists, Smith emphasizes, conceived their project as fundamentally mathematical. Leibniz embraced this conception and supplemented it with his own theory of actual infinities. The notion of infinite structures within natural bodies allowed Leibniz to extend his solution to the problem of the continuum to the physiology of living things. This reflected Leibniz’s ambitious hope to account for all of nature in a unified, broadly mathematical way. Although this mathematizing project was unsuccessful in the end (perhaps was even doomed from the start), Smith argues that it should be seen as an important, albeit unorthodox, part of the mathematization project of the seventeenth century. Like Jesseph, Smith can be seen as localizing the mathematization thesis, in this case to iatromechanism, while calling attention to mathematization programs far from mechanics and astronomy. The vast differences between traditional, geometrical forms of mathematization and Leibniz’s infinity-based approach suggest moreover that Leibniz’s approach represented a distinct and unique form of mathematization.

Ursula Goldenbaum (chapter 11, “The Geometrical Method as a New Standard of Truth, Based on the Mathematization of Nature”) suggests that within the rationalist tradition mathematization extends into the realm of philosophical methodology. For her, the key is the geometrical method, which she views as an extension of the mathematization project into philosophy as a whole. This method was applied globally—even, as in Spinoza, to human emotions and ethics itself. The primary attraction of this method was that it promised to secure the certainty of philosophical knowledge the same way that mathematization promised to ground the certainty of scientific knowledge. So as for Kurt Smith, mathematization is not so much critiqued but rather extended into the rationalist philosophy of the seventeenth century.

Finally, Christopher Smeenk (chapter 12, “Philosophical Geometers and Geometrical Philosophers”) explores the use Newton made of Barrow’s geometrized conception of mathematics to bridge the gap between the mathematically ideal and the physically real. Smeenk shows how Newton used Barrow’s geometrical conception of mathematics to overcome a prima facie barrier to the adoption of a mathematical methodology in physics: the tension between the universality and certainty of mathematics versus the particularity and uncertainty of real systems. According to Smeenk, Newton rejected the idea that mathematics is especially abstract or ideal but held instead that its proper domain of application is material objects rather than ideal entities. However, he did not promulgate a crude empiricist epistemology of mathematics in nature. Rather, he held that quantitative descriptions are rationally reconstructed on the basis of observations taken from within an appropriate conceptual framework. Like Jesseph, Smeenk’s analysis of Newton’s geometrically based mathematics suggests a way of reconceiving the mathematization thesis where different models of mathematization in Newton, Barrow, and Wallis develop independently or even in competition. But Smeenk’s and Jesseph’s points can also be taken in the more restrictive sense, whereby the mathematization of nature is seen as a feature only of the later, Newtonian and Leibnizian stage of the scientific revolution. In any case, taking their work in either of these directions requires a revision of the mathematization thesis, dethroning it as an overarching narrative of the scientific revolution.

This volume presents a variety of possible reconceptualizations of the mathematization historiography. One possibility is to reject it completely as oversimplified and misleading, both for historical and pedagogical purposes. More moderate revisions are also possible: mathematization might be one factor of many necessary to explain the scientific revolution; or it may be of limited historical utility helpful for explaining (1) the contributions of specific individuals or schools; (2) the unfolding of particular fields within natural philosophy; (3) the progress in certain subperiods of the scientific revolution. Finally, certain conservative positions seek to retain the substance of the mathematization thesis and even extend it beyond its traditional scope of natural philosophy to philosophical concerns more broadly. Regardless of what the scholarly consensus turns out to be, there is no doubt that the mathematization thesis has played an important role in shaping our conceptions of the scientific revolution and it deserves to be carefully reexamined in this new era of historical methodologies and frameworks.

`ABBREVIATIONS`

AT | Descartes, R. 1976. Oeuvres de Descartes (citation by volume and page number) |

EW | Hobbes, T. 1839–45. English Works (citation by volume and page number) |

`NOTES`

1. See also Henry (2008), chapter 2, for an overview of recent work. Westman (1980) has urged a similar point about the emerging authority of astronomy in the sixteenth century, one of the traditional mixed sciences.

2. See further Friedman (1992) and Massimi (2010).

3. The Platonist origin of Galileo’s geometrical approach has been forcefully challenged by some historians. For example, Wallace (1984) has explored the influence of the Jesuit mathematicians of the Collegio Romano, especially Christopher Clavius, on Galileo’s early philosophy. Nevertheless Koyré’s version of the mathematization thesis gained wide acceptance within mainstream historiography of science in the twentieth century. The British historian Rupert Hall, while underscoring the complexity of the scientific revolution, and the dramatic transformations within mathematics itself, strongly reaffirms Koyré’s thesis in numerous works, including the late Revolution in Science 1500–1750: “The most eloquent and full defense of this process [the mathematization of nature] was given by Galileo whose mathematization of the science of the motions of real bodies furnished a model for physical science general during the following century” (1983, 12).

4. On the geometrization of space, time, matter, and motion, see more recently McGuire (1983), Jalobeanu (2007), and Palmerino (2011).

5. Koyré puts the same point about gravitation in terms of Galileo’s famous saying: “a mathematical stricture that lays down the rule of syntax in God’s book of nature” (1968, 13).

6. See further the recent articles by Domski (2013) and Dunlop (2013).

7. Sophie Roux suggests that it can be recast by focusing on the kinds of mathematical practices used during the period (2010, 319–37) or in terms of their polemical stances toward the Schoolmen (2013). Craig Martin (2014) develops this approach.

8. As H. Floris Cohen has done (2010).

9. For an incisive critique of a “family resemblance” approach to reconceptualizing mathematization, see Roux (2013, 57–58).

10. This can be related to suggestions by Noel Swerdlow (1993) and H. Floris Cohen (1994; 2010) that an ancient mathematical form of natural philosophy was rediscovered in the fifteenth and sixteenth centuries rather than being transmitted to them via a fourteenth-century discovery.

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