SOME OF THE MOTIVATION for this volume is the reevaluation of a prominent historiographical orientation of twentieth-century research on the scientific revolution, in light of the proliferation of novel methodological orientations and studies in the last generation of scholars. The historiographical orientation at issue is what is called the mathematization of nature; its exemplary proponents are Alexandre Koyré, Eduard Jan Dijksterhuis, and Edwin Arthur Burtt.1 This should be a welcome reevaluation, especially since the position has held fairly strongly for almost a century. Burtt published the first edition of his Metaphysical Foundations of Modern Science in 1924.2 Dijksterhuis reiterated in large part the historical-philosophical accounts implicit in Burtt’s work in 1950.3 And the views of Burtt and Dijksterhuis found their historiographical champion in Koyré’s Husserlian- and Bachelardian-inspired position.4 In 1950 Dijksterhuis already knew and cited a number of Koyré’s theses from his publications available in the 1940s, such as Koyré’s thesis on the mathematization of physical space and his devaluation of experimental approaches. As Dijksterhuis states with respect to the first thesis, “The substitution of the world-picture of classical physics for that of Aristotle involved a radical change in the conception of space in which the phenomena of nature occur. Without explicitly saying so, scientists had always thought of the latter as physical space to distinguish it from the geometrical space to which the reasonings of mathematics related … In the sixteenth and seventeenth centuries, however, this distinction was becoming blurred … Koyré characterized this by the term ‘mathematization of physical space’ ” (Dijksterhuis 1969, 377).5
The reference Dijksterhuis gives is to Koyré’s 1939 Études galiléennes. There Koyré does state that one of the major changes between classical and modern science is “the geometrisation of space,” that is to say, “the substitution for the concrete space of pre-Galileo physics of the abstract space of Euclidean geometry. It was this substitution that made the invention of the law of inertia possible” (1978, 3).6
The second Koyré thesis referred to by Dijksterhuis is exemplified in his judgment of Francis Bacon’s lack of importance for early modern science. Dijksterhuis cites again from Koyré’s Études galiléennes, in large part approving of Koyré’s view that Bacon “did not make a single positive contribution to science, and in some cases he entirely failed to recognize the merits of others who had done so. This is why Koyré calls it a mauvaise plaisanterie to regard him as one of the founders of modern science” (Dijksterhuis 1969, 396–97).7 And, indeed, Koyré does say that the role of Bacon “in the history of the scientific revolution was completely negligible,” and reinforces this judgment by asserting in a footnote, as Dijksterhuis said, that “ ‘Bacon, the founder of modern science’ is a joke, and a bad one at that.” Koyré continues, stating: “In fact, Bacon understood nothing about science. His manner of thinking was closer to alchemy and magic … to a thinker of the Renaissance than to that of a Galileo or even a Scholastic” (Koyré 1978, 39).
The two theses go hand in hand, as can readily be shown by articles on Galileo published by Koyré in 1943.8 For Koyré, experience is an obstacle in the establishment of modern science. As Koyré says in one article: “We are so well acquainted with, or rather so well accustomed to, the concepts which form the basis of modern science, that it is nearly impossible for us to appreciate rightly either the obstacles that had to be overcome for their establishment, or the difficulties that they imply and encompass” (1968, 3; my emphasis).9 Koyré then states in another article: “One must not forget that observation and experience, in the sense of brute, common-sense experience, did not play a major role—or if it did, it was a negative one, the role of obstacle—in the foundation of modern science” (1968, 18; my emphasis). The connection between Koyré’s two theses—the devaluation of experience and the mathematization of nature—lies in his distinction between experience and experiment:
It is not experience but experiment which played—but only later—a great positive role. Experimentation is the methodical interrogation of nature, an interrogation which presupposes and implies a language in which to formulate the questions, and a dictionary which enables us to read and interpret the answers. For Galileo it was in curves and circles and triangles, in mathematical or even more precisely, in geometrical language—not in the language of common sense or in that of pure symbols—that we must speak to Nature and receive her answers. Yet obviously the choice of language, the decision to employ it, could not be determined by the experience which its use was to make possible. (Koyré 1968, 18–19)
Koyré’s devaluation of experience has the further consequence that Galileo’s observations and experiments are likewise devalued; as Koyré states: “It is obvious that the Galileo experiments are completely worthless: the very perfection of their results is a rigorous proof of their incorrection” (1968, 94).10
So what is one to do; that is, how is one to reevaluate this prominent historiographical orientation? Well, one could point out that Koyré is simply historically wrong about his devaluation of experience, wrong about Francis Bacon, and wrong about Galileo’s experiments.11 Worse yet, his perspective fails to appreciate the rise of scientific societies, the social nature of the epistemology of observation, and so forth. But such an objection is easily countered by simply casting off what Koyré thinks of as an integral part of his view (the necessity of the mathematization of nature as preliminary to any genuine experimental culture) and accepting just a portion of his historiographical orientation, for example, mathematization of nature by itself, as one of the crucial elements of early modern science as contrasted with past science, along with experimental culture by itself. We can see this rejoinder given numerous times, as in Floris Cohen’s review of Joella Yoder’s book Christian Huygens and the Mathematization of Nature. Cohen states: “It is well known that, on the topic of Galileo’s experiments, Koyré has been proved simply wrong—to the extent that he declined to take literally in Galileo’s own statements what we now know should indeed be taken literally. Granting so much does not, however, render all Koyré had to say on the mathematization of nature worthless” (Cohen 1991, 83). Still, the historical studies of the “last generation of scholars” about the culture of experiment require the mathematization of nature to become a less global historiographical thesis, but perhaps to remain an important one.
A more promising reevaluation would consist in reconsidering the mathematization of nature from the perspective of “novel methodological orientations,” and in particular, of contextual history. Here one could point out that when we talk about the “mathematization of nature” we mean different things with regard to different thinkers. When one focuses on what mathematizing nature would be for Galileo, for Descartes, Huygens, or Newton, we find that these are radically different activities; what Galileo tried to do in this regard is clearly different from what Descartes tried to do, or what Huygens, Newton, and others tried to do. The mathematization of nature as an account of the scientific revolution or early modern science begins to look like our twentieth-century invention, perhaps a construction we are forcing on the past.12
Such a thesis, however right it seems to me, would be beyond the scope of a single chapter. Instead I propose to examine the mathematization of nature for Descartes and the Cartesians. I will try to show that, from the perspective of a more contextual history, the thesis that mathematization of nature refers to radically different elements in different thinkers can be recovered in a single thinker; it applies to Descartes by himself. Basically, there is no one thing one can call the mathematization of nature in Descartes, perhaps no mathematization of nature at all, if the concept is considered narrowly. I will then corroborate this historical position by demonstrating that various Cartesians in the seventeenth century understood Descartes differently on these issues. The Cartesians have little to say about the mathematization of nature when viewed as a grand narrative for the scientific revolution, though their remarks on the relations between mathematics and physics advance various aspects of Descartes’ understanding about those relations and contrast with the way we conceive of them as part of that grand narrative. But first, I need to discuss the views of Burtt and Dijksterhuis on Descartes and the mathematization of nature.
BURTT AND DIJKSTERHUIS ON DESCARTES’ MATHEMATIZATION OF NATURE
In his Metaphysical Foundations of Modern Science, Burtt’s chapter on Descartes proceeds somewhat chronologically. Burtt refers to Descartes’ early interest in mathematics, including what he calls the “remarkable experience” of November 10, 1619, which confirmed for Descartes the “trend of his previous thinking and gave the inspiration and the guiding principle for his whole life-work,” namely, according to Burtt, the conviction “that mathematics was the sole key needed to unlock the secrets of nature” (Burtt  1954, 105). Burtt follows this introduction with a section called “Mathematics as the Key to Knowledge.” There, Burtt details Descartes’ Rules, which he calls “a series of specific rules for the application of his all-consuming idea,” starting with Rule 1, “that all the sciences form an organic unity” and interpolating something from the Search for Truth that “all the sciences must be studied together and by a method that applies to all” (106–7). He asserts that “the method must be that of mathematics, for all that we know in a science is the order and measurement revealed in its phenomenon” and, citing Rule 4, that “mathematics is just that universal science that deals with order and measurement generally.” Burtt also asserts that “Descartes is at pains carefully to illustrate his thesis that exact knowledge in any science is always mathematical knowledge” (107). He refers to the Rule 3 concepts of intuition and deduction (deduction now considered as mathematical deduction) as two steps of this mathematical method and introduces the simple natures of Rule 14 “as discoveries of intuitions” (108). However this is where Burtt thinks Descartes goes astray: “As he proceeds from this point he is on the verge of the most far-reaching discoveries, but his failure to keep his thought from wandering, and his inability to work out the exceedingly pregnant suggestions that occur to him make them barren for both his later accomplishments and those of science in general … [A]t the crucial points his thoughts wander, and as a consequence Cartesian physics had to be supplanted by that of the Galileo-Newton tradition” (109). The rest of Burtt’s (clearly whiggish) account consists of a generally negative report of Descartes’ views in the Principles—Descartes’ “soaring speculations”—as failing to live up to his initial fundamental mathematical intuition, producing something of mere historical, not scientific, significance.
It is ironic that Burtt is so firmly convinced of his interpretation of Descartes, based on a defective reading of a juvenile unfinished manuscript, that he cannot make much sense of the Principles, Descartes’ mature published treatise, on which his reputation rested during the seventeenth century. The situation, however, is not much better with Dijksterhuis, though the latter is somewhat more sober than the former. Again, we assert that in seventeenth-century science “the structure of the external world was essentially mathematical in character and a natural harmony existed between the universe and the mathematical thought of the human mind.” Dijksterhuis adds that “the standpoint taken by Descartes cannot be better described than by saying that by carrying this conception to its extreme he virtually identified mathematics and natural science” (Dijksterhuis 1969, 404). He then refers to Descartes’ tree of philosophy from the preface to the French edition of the Principles. He recognizes that physics there is depicted as rooted in metaphysics, and that “Mathematics is not referred to,” but he adds that the foundation on which metaphysics is based is also not referred to, and ends with a rhetorical question: “Cannot the explanation of this be that it is mathematical thought, considered not with regard to its contents but to its form, which constitutes this foundation?” (404). I suppose rhetorical questions should not be answered, though the answer is clearly “no.”13 Still, to his credit, Dijksterhuis knows that one cannot find the thesis of the mathematization of nature in Descartes’ first published work, the Discourse on Method. He rightly states that “the formulation of the four famous rules which are recommended as guiding principles for scientific thought is immediately preceded by the statement that the author failed to find the method he needed in the Analysis of the Ancients and the Algebra of the Moderns” (404). He opines that Rule 1 of the Discourse, about evidence, “was apparently inspired by the style of mathematical thought,” and adds that “the other three rules have been kept so vague and general that in the first place they admit of different interpretations and secondly they contain little that is of specifically mathematical character” (405). Dijksterhuis cites with approbation Leibniz’s calling them vacuous and mocking them, describing the method as like advising a chemist to “take what you have to take, do with it what you have to do, and you will get what you desire” (404–5).
However, Dijksterhuis is quickly over his disappointment with Descartes’ Discourse: “in order to become really acquainted with the method of Descartes one should not read in the first place the charming Discours, which is a causerie, rather than a treatise, but the … Rules for the Direction of the Mind, which was already composed in 1629” (405).14 And, naturally, we now get the fact that the Rules contains an exposition of Mathesis Universalis, which, Dijksterhuis asserts, “Descartes always regarded as one of his greatest methodological discoveries.” At this juncture Dijksterhuis claims that Descartes wanted to see Mathesis Universalis applied in all the natural sciences, by which he means that Descartes prescribes the application of algebraic methods to all those branches of science that admit of quantitative treatment. He adds that Descartes also admits the possibility of “arranging propositions in deductive chains,” so he concludes that “the aim of the Cartesian method is indeed to cause all scientific thinking to take place in the manner of mathematics, namely by deduction from axioms and by algebraic calculation” (405). Dijksterhuis in this way rejoins the thesis of the mathematization of nature and Burtt’s account. He shares Burtt’s disappointment with Descartes: “Descartes did not get very far in carrying out the concrete program of universal mathematics in science,” though he asserts that “his metaphysical as well as his scientific thinking always followed a mathematical pattern” (406). The rest is a litany of Descartes’ failures, that is, more analyses of Descartes from the perspective of the present: “if Descartes could have foreseen the future of mathematics …”; “Descartes never produced …”; “The modern reader, who is accustomed to find more and more trouble expended on this part of the process of forming scientific concepts, may have some difficulty in looking upon the Cartesian way of studying science as a serious contribution to the methodology of scientific thought.” For Dijksterhuis, as for Burtt, Cartesian physics is of mere historical importance; for Dijksterhuis, it was “an illusion” that enabled Descartes “to put before his contemporaries the transparent ideal of a rational system for the interpretation of nature that was to rely on none but mathematical and mechanical conceptions” (409).
Of course, Descartes did not put before his contemporaries any such ideal as described by Dijksterhuis and Burtt. He put before his contemporaries the arguments of the Discourse, Meditations, and Principles, but not those of the Rules. Simply put, the Rules was not generally available in the seventeenth century, though a few Cartesians had access to various small portions of it, as was obvious in the fourth edition of the Port-Royal Logic (1674, 42) the work itself was first published in Latin in Descartes’ Opuscula Posthuma only in 1701, with a Dutch-language version published in 1684. The main Cartesians published their works before the publication of the Rules, from circa 1654 to circa 1694, without any knowledge of its views. One might be able to argue that an analysis of the Rules in the fashion of Burtt and Dijksterhuis could reveal Descartes’ deepest intuitions, but such an analysis cannot provide any understanding of Descartes’ significance or influence for Cartesians or for anti-Cartesians or for seventeenth-century science in general. This is the important point to make.
A subsidiary point is that the interpretations of Dijksterhuis and Burtt about the Rules are deeply flawed. Take Burtt’s assertion that “all that we know in a science is the order and measurement revealed in its phenomenon” or “Descartes is at pains carefully to illustrate his thesis that exact knowledge in any science is always mathematical knowledge.” When Descartes gives an example of his method in the Rules, he talks about the problem of determining the anaclastic line, in which parallel rays are refracted in such a fashion that they all meet at a point. He does explain that those who limit themselves to mathematics alone cannot investigate the problem, “since it does not belong to mathematics, but to physics” (Descartes 1964–76; AT, 10:394). A person who “looks for the truth in any subject” will not fall into the same difficulty. That person can perceive clearly by intuition both mathematical and physical matters, about the proportion of the angles of incidence and angles of refraction depending on the variation of these angles in virtue of the difference of the media and about the manner in which rays penetrate into a transparent body. The latter presupposes that the nature of illumination is known and what a natural power is in general. As Descartes says: “this is the last and most absolute term in this whole sequence” (AT, 10:395). It is the intuition from which the problem will be solved, from which evident knowledge of the anaclastic line is derived, according to Descartes’ method (see Garber 2001, 85–110). I fail to see how the intuition about the nature of illumination or of a natural power would not be considered knowledge or has to be thought as mathematical knowledge, as Burtt would want it. Nor do I see how any of this can license Dijksterhuis’s claim that “the aim of the Cartesian method is indeed to cause all scientific thinking to take place in the manner of mathematics, namely by deduction from axioms and by algebraic calculation.”15 Burtt and Dijksterhuis are so sure of their general thesis about the mathematization of nature that they construct their own Descartes from an unfinished manuscript that Descartes himself never refers to; they then mostly neglect what he says in his mature published works. Worse yet, they are so sure of the mathematization of nature as the endpoint for physics that they criticize Descartes for failing to see what they think they perceive in present science. In this process, they cannot provide any understanding of Descartes’ views nor of what the Cartesians saw in Descartes. They cannot provide an account of early modern science in relation to what preceded it or in relation to what succeeded it.
DESCARTES ON THE RELATIONS BETWEEN MATHEMATICS AND PHYSICS
It is not as if Descartes does not issue enough statements about what he considers the relation among physics, metaphysics, and mathematics in his published writings, the Discourse (1637), Meditations (1641–42), and Principles (1644–47), as well as in his Correspondence (published posthumously in three volumes, 1657–67). There are numerous pronouncements that Descartes is looking for certainty at least equal to that of mathematics; in the Discourse, he intimates that the real use that can be made of mathematics is to extend its method into other realms (AT, 6:7), and to prepare the mind to follow the real philosophical method, which mathematics presupposes (19–22). This last statement can lead one to consider that Descartes does not accept mathematics as the foundation for all knowledge. In fact, early on he claims that his metaphysical demonstrations are more certain than geometrical demonstrations (AT, 1:145). But Descartes does say that all his “physics is nothing other than geometry” (AT, 2:68), and he speaks of “having reduced physics to the laws of mathematics” (AT, 3:39). That is how Descartes’ anonymous correspondent in the letters used as preface to the Passions of the Soul seems to have understood Descartes: “the [Scholastic] Philosophers accept mathematics as part of their physics, because almost all of them are ignorant of it; but on the contrary, the true physics is a part of mathematics” (AT, 11:314–15).16
There is also the last article of Principles (II, art. 64, AT, 8:78): “The only principles I accept or desire in physics are those of geometry or abstract mathematics, because these explain all natural phenomena and enable us to provide certain demonstrations of them” (AT, 7:78; 11:101).17 It is an important article indeed, which Descartes published in a prominent work. What the text of the article explains is that Descartes recognizes “no matter in corporeal things apart from what can be divided, shaped, and moved in all sorts of ways, that is, the one the geometers call quantity”—that “he considers in such matter only its divisions, shapes, and motions”—because he does not want to admit anything as true “other than what has been deduced from [these] indubitable common notions so evidently that it can stand for a mathematical demonstration.” Descartes ends his article by asserting that “since all natural phenomena can be explained in this way, I do not think that any other principles are either admissible or desirable in physics [than the ones that are here explained].” It is important to note that the properties of matter that Descartes accepts, the divisions, shapes, and motions of corporeal things, are not accepted because they are geometrical or mathematical, but because they are the modes of extension that can be distinctly known. In Part I of the Principles, that is, the “metaphysical” portion, representing the Meditations, Descartes asserts that extended substance can be clearly and distinctly understood as constituting the nature of body (Principles I, art. 63, AT, 8:30-1) and that extension as a mode of substance can be no less clearly and distinctly understood as substance itself (Principles I, art. 64, AT, 8:31). Descartes then lists the properties or attributes of extension as their shapes, the situation of their parts, and their motions (Principles I, art. 65, AT, 8:32). It happens that these properties are what “the geometers call quantity.” But that is because mathematicians rely on some of the same clear and distinct perceptions as natural philosophers do. Descartes roots his physics in a metaphysics that produces, at first,18 a physics that looks the same as mathematics, not because it is rooted in mathematics, but because it is rooted in a metaphysics of clear and distinct ideas.19 But I do not think many scholars would have been tempted to call this the mathematization of nature or have considered it as an integral part of the scientific revolution.
CARTESIANS AND THE RELATIONS BETWEEN PHYSICS AND MATHEMATICS
It may be one thing to write about Descartes’ deepest intuitions as we understand them and another to explicate his influence on his followers; that is, how he was understood by others. When the issue is the scientific revolution, one’s account should resonate with the latter. It thus becomes relevant to understand how Descartes was understood by his followers. The Cartesians are a diverse group. Let me limit myself to a few representative thinkers: Du Roure, Rohault, Le Grand, and Régis. Le Grand and Régis are famous for their attempts to publish multivolume Cartesian textbooks that would mirror what was taught in the schools, containing treatises on Cartesian logic, metaphysics, physics, and ethics. Le Grand initially published a popular version of Descartes’ philosophy in the form of a scholastic textbook (1671), expanding it in the 1670s and 1680s. The work, Institution of Philosophy, as it was called then, was then translated into English together with other texts by Le Grand and printed in two large volumes as part of An Entire Body of Philosophy according to the Principles of the famous Renate Des Cartes (1694). On the continent, Régis issued his three-volume Système Général selon les Principes de Descartes at about the same time (1691). The difficulties Régis encountered in obtaining permission to publish considerably delayed its publication. The various portions of this work embody Régis’s adaptations of diverse works, both Cartesian and non-Cartesian: Antoine Arnauld’s Port-Royal logic (mostly excerpted); Robert Desgabet’s peculiar metaphysics;20 Rohault’s physics; and an amalgam of Gassendist, Hobbesian, and especially Pufendorfian ethics.21 Ultimately, Régis’s unsystematic (and very often un-Cartesian) Système set the standard for Cartesian textbooks. Early attempts at setting out a complete Cartesian system before those of Le Grand and Régis included Du Roure’s multivolume La Philosophie divisée en toutes ses parties (1654) and its (less Cartesian) successor Abrégé de la vraye philosophie (1665). Du Roure was one of the first followers of Descartes, belonging to the group that formed around Descartes’ literary executor, Claude Clerselier.
Unlike Du Roure, Le Grand, and Régis, who tried to publish complete “systems” of Cartesian philosophy, Rohault limited himself to natural philosophy. He was the foremost proponent of Cartesian physics in the decades immediately following the death of Descartes. In the mid-1650s he began to hold weekly lectures at his house in Paris; these “mercredis de Rohault” brought him to the attention of prominent Cartesians. He became Régis’s teacher and won him over to the cause of Cartesianism. Rohault was best known for his 1671 Traité de physique, which went through numerous editions and remained a standard textbook in Cartesian natural philosophy well into the eighteenth century, long after Rohault’s death in 1672. The Traité de physique was initially translated into Latin in 1682 and then again, with annotations by Samuel Clarke, in 1697. Clarke’s Latin edition was translated into English in 1723 by his younger brother John and published as Rohault’s System of Natural Philosophy. As the work went through multiple editions, Samuel Clarke increasingly “illustrated” it with “notes taken mostly out of Sr. Isaac Newton’s Philosophy.”
First, I need to touch on the question of what the Cartesians take as Descartes’ method in general. The answer is as expected: What they consider as method varies somewhat, but does not involve Mathesis Universalis or anything from the Rules. For example, Du Roure’s section of the multivolume Philosophy on Cartesian logic consists of a summary of Discourse Part II, with a commentary on Descartes’ rules of method, in succeeding chapters. Du Roure’s view of the usefulness of those precepts is influenced by Descartes’ preface to the French edition of the Principles. He recommends Descartes’ logic in order for us to conduct our reason well: “But because it depends considerably on usage, it is extremely advantageous to practice the rules on simple and easy questions, such as those of mathematics. And when we will have acquired some habit in discovering the truth, we must apply ourselves with care to Philosophy” (Du Roure 1654, 183–84).22 This is the view of logic and mathematics as tools for sharpening the mind, much like solving crossword puzzles, not as the foundation of physics.
The Port-Royal Logic, which dominated in the second half of the seventeenth century, thus also Régis’s Logique and Le Grand’s Logick, all end with a section called “Method.” By method, however, these writers mean analysis and synthesis—which does not have to be anything particularly Cartesian23—though we do find Descartes’ rules of method enumerated in the chapters on analysis. The Port-Royal Logic lists Descartes’ four rules, saying that they are “general to all sorts of methods and not particular to the method of analysis alone” (Arnauld 1674, 375) but then moves on to give five rules of composition, focusing on these and enlarging them by chapter 10 to eight principal ones (428–31). Régis follows suit. He adds a chapter on “the advantages we draw from observing the four precepts of analysis” (Régis 1691, 152–54) and abbreviates the lengthy Port-Royal discussion of synthesis into a single small chapter and just three brief rules: leave no term ambiguous; use clear and evident principles; and demonstrate all propositions (56). Part IV of Le Grand’s Logick, “Concerning Method, or the Orderly Disposition of Thoughts,” deals with the analytic and synthetic methods, that is, resolution and composition. As part of the analytic method, Le Grand asserts that this method is the art that guides reason in the search for truth; because we cannot proceed to something unknown except by means of something known and questions are propositions that include something known and something unknown, whenever the nature or cause of anything is proposed, we must
in the first place accurately examine all the Conditions of the question propounded, without minding things as are Extraneous, and do not belong to the Question. Secondly, We are to separate those things which are certain and manifest from those that include any thing of Confusion or Doubt … Thirdly, Every Difficulty we meet with is to be divided into Parts … Fourthly, We are orderly to dispose of our Perceptions, and the Judgments we frame thence; so that beginning from the most easie, we may proceed by degrees to those that are more difficult … Fifthly, That the Thing in question, be furnished with some Note or other that may determine it, and make us judge it to be the same, whenever we meet with it. (Le Grand 1694, 1:46)
This seems to be Le Grand’s version of Descartes’ four rules of method, restricted to what is useful to analysis. Le Grand ends his Logick with chapters on composition, giving various rules of definition, axiom, and demonstration similar to the ones given by the Port-Royal Logic. As could be expected, the seventeenth-century Cartesians construct accounts of Descartes’ method based on their understanding of Descartes’ assertions in the Discourse and Principles.
The Cartesians also integrate Descartes’ various comments about the relation between mathematics and physics into their accounts. Rohault discusses some of these relations in the preface to his Treatise on Physics, beginning with a rebuke of scholastics for not teaching mathematics in their schools: “The Fourth Defect that I observed in the Method of the [School] Philosophers, is the neglecting Mathematicks to that Degree, that the very first Elements therof are not so much as taught in their Schools. And yet, which I very much wonder at, in the Division which they make of a Body of Philosophy, they never fail to make Mathematicks one Part of it” (Rohault 1729, n.p.; roughly 13–16). He then formulates the argument we have already seen in Du Roure about the use of mathematics in general:
Now this Part of Philosophy is perhaps the most useful of all others, at least it is capable of being apply’d more Ways than all others: For besides that Mathematicks teach us a very great number of truths which may be of great Use to those who know how to apply them: They have this further very considerable advantage, that by exercising the Mind in a Multitude of Demonstrations, they form it by Degrees and accustom it to discern Truth from Falsehood infinitely better, than all the Precepts of Logick without Use can do. And thus those who study Mathematicks find themselves perpetually convinced by such Arguments as it is impossible to resist, and learn insensibly to know Truth and to yield to Reason.
In large part, this is Rohault’s take on Descartes’ justification for mathematics outside the tree of philosophy: exercising the mind—crossword puzzles and all that. But Rohault goes a bit further, justifying the use of mathematics in natural philosophy—indeed, in all arts—with two additional arguments:
First, that as there is a natural Logick in all Men, so is there also natural Mathematicks, which according as their Genius’s are disposed, make them more or less capable of Invention. Secondly, That if their Genius alone, conducted only by natural Light, will carry them so far, we cannot but hope Greater Things from the same Genius if the study of Mathematicks be added to its natural Light, than if that study be neglected. And indeed all the propositions in Mathematicks are only so many truths, which those, who apply themselves to them, come to the Knowledge of by good Sense.
This offers a positive role for mathematics that does not refer expressly to Cartesian metaphysics. It demonstrates Rohault’s recognition (shared by Descartes) that mathematics and physics rely on the same intellectual faculties (Du Roure [1654, 188] expresses a similar sentiment). But it is not an argument to the effect that the method of physics is the same as the method of mathematics or that mathematical truth or mathematical properties are the basis for physical truth or physical properties.
Rohault’s generally positive view is not reflected in the work of his follower. Régis (1691) demarcates between mathematics and physics, specifically asserting that he has avoided all mathematical questions in his philosophy:
Those who read this book will more easily experience its flaws if they do not stop at equivocal words, ambiguous definitions, or any idea that is foreign to Philosophy, given that we have even purposely avoided Mathematical questions, both because they are little understood by the majority of those who want to apply themselves to Philosophy, and because we all too often confuse them with purely Physical questions, though they are of an entirely different nature. For one is not satisfied in Mathematics by knowing that some things have greater magnitude than some other things; we claim also to know with evidence the exact ratios holding between them, or precisely by how much they are greater, which does not at all concern Physics. (Régis 1691, preface)
Régis continues his demarcation between physics and mathematics, accepting the usefulness but denying the importance of mathematics to physics, stressing the experiential basis of physics, in contrast with how geometry is usually practiced: “one can be a good Physicist without being a great Geometer, but one cannot be a great Geometer without being a good Physicist, at least if we have Geometry consist (as we must) in demonstrations based upon facts, or on constant truths; for if we have it consist (as is usually done) in demonstrations based on arbitrary assumptions, nothing prevents a bad physicist from being a good geometer” (n.p.).
Unlike Rohault and Régis, who emphasize the empirical aspects of natural philosophy (see Ariew 2013; Dobre 2013), Le Grand is interested in the standard question of the certainty of natural philosophy (what he also calls physiology). He proceeds very much in the spirit of a scholastic, substituting Cartesian terminology and doctrines. He has considered the nature of God and inquired into his attributes: “Physiology comes next to be considered by us, which contemplates Natural Things, and deduceth their Causes from the first Original … Now that Physiology is a Species of Science, and is conversant with things that are True and Necessary, appears from the Demonstrations that are made of Natural Things; the Certainty whereof depends on the Stability of Things that are defined, and supposeth their determinate Essence” (Le Grand 1694, 1:91).
Le Grand then attempts to answer the objection: since bodies are only perceived by the senses and the senses may represent false things to the understanding, how can the certainty required for science be had in natural things? His answer is that “It is False that Material Things are known by the Senses … to speak properly, nothing is conveigh’d from things without us, by the Organs of Sense, to our Minds, save only some Bodily motions, by which the Ideas of Objects are offer’d to them … Wherefore, Bodily things are not known by the Senses, but by the Understanding alone: So that to be sensible of a Material Substance, is nothing else, but to have an Idea of it, which is not the work of the outward Senses, but of Cogitation” (Le Grand 1694, 1:92).
The further objection is that natural philosophy treats material things as changeable, which seems inconsistent with the notion of science as certain and perpetual knowledge. Le Grand’s answer is that “Nevertheless we must say, that Natural Philosophy is indeed a Science, because the Nature of a Science is not consider’d with respect to the things it treats of, but according to its Axioms of an undoubted Eternal Truth. For tho’ the things which Physiology handles, be changeable; yet the Judgments we make of them are stable and firm; and consequently the Truth we have of them is Eternal and unchangeable” (ibid.). Le Grand (1694) gives as examples of these indubitable and constant truths propositions such as “all that is bodily is changeable” and “every mixed body is dissoluble.” In this way, he rejoins here Descartes’ view from in the end of Principles, Part II:
Forasmuch as every Science hath a Subject, about which it is conversant, and to which, whatsoever is handled in the same may be attributed either as Principles, Parts or Affections, we say that the Material Subjects of Physiology, are natural things, and that Magnitudes, Figures, Situation, Motion, and Rest are the Formal Subject of it;… Wherefore, if a Natural Philosopher considers nothing in matter besides these Divisions, Figures and Motions, and admit nothing for Truth concerning them, which is not evidently deducible from common Notions, whose Truth is unquestionable, it is altogether manifest, that no other Principles are to be looked for in Natural Philosophy, than in Geometry or abstract Mathematicks; and consequently that we may have as well Demonstrations of Natural Things, as of Mathematical. (1:92)
Let us repeat the last thought: as long as we limit ourselves to what is deducible from common notions, we may have demonstrations of natural things as well as those of mathematical things. Régis (1691) has an exemplary exposition of the same Cartesian view, delineating carefully among metaphysics, mathematics, and physics:
Metaphysics not only serves the soul to make itself known to itself, it is also necessary for it in order to know things outside it, all natural sciences depending on metaphysics: mathematics, Physics, and Morals are founded on its principles. In fact, if Geometers are certain that the three angles of a triangle are equal to two right angles, they received this certainty from Metaphysics, which has taught them that everything they conceive clearly is true and that it is so because all their ideas must have an exemplary cause that contains formally all the properties these ideas represent. If Physicists are certain that extended substance exists and that it is divided into several bodies, they know this through Metaphysics, which teaches them not only that the idea they have of extension must have an exemplary cause, which can only be extension itself, but also that the different sensations they have must have diverse efficient causes that correspond to them and can only be the particular bodies that have resulted from the division of matter. (64)24
The Cartesians found Descartes’ philosophy enormously important for the seventeenth century. The verso of Du Roure’s title page from his Philosophy tells the story very well; while he appreciates Gassendi and Hobbes and quotes them at times, his admiration for Descartes knows no bounds: “One can oppose Hobbes, Gassendi, and Descartes against all those whom are glorified by Rome and Greece … Those who would take the trouble to read this philosophy will find numerous opinions of these three wise philosophers, but principally those of Descartes. This is why I want to show the extent he is esteemed by the following testimony.” The six subsequent paragraphs are superlative praise for Descartes, including: “Descartes is the premier philosopher of all time.” When we tell the story of the seventeenth century, we need to capture what these thinkers found so appealing about Descartes (and what the anti-Cartesians found so dangerous). And when we do so, we find also many different views about the relations between mathematics and natural philosophy: that natural philosophy can develop a method similar to that of mathematics; that propositions in natural philosophy can be as certain as those of mathematics; that mathematics can be of use in sharpening one’s mind for the practice of philosophy; that mathematics has a mode of exposition that is particularly persuasive; that philosophy can be based on the same clear and distinct ideas as those on which mathematics are based. But we do not find the view that the method of philosophy is reducible to the method of mathematics or that philosophy is founded in mathematics. The generally positive views of mathematics in Descartes and the Cartesians do not legitimate a historical or historiographical thesis of the mathematization of nature in the fashion of Burtt, Dijksterhuis, and Koyré.
Descartes, R. 1964–76. Oeuvres de Descartes
1. I take the language of the motivation for this volume from the original prospectus of the workshop on which the volume is based.
2. The work had a second edition in 1932. Burtt indicates that the second edition contains no changes in his narrative before Newton: “No historical researches during the last six years with which I have become acquainted seem to require any essential changes in the survey here embodied, so far as it reaches” ( 1954, preface).
3. Mechanisering van het wereldbeeld, 1950; I will be citing the English translation by Dikshoorn (Dijksterhuis 1969).
4. Koyré studied with Husserl at Gottingen. As Sophie Roux (2010) states: “Husserl claimed that Galileo was the first to mathematize nature, i.e., according to Husserl, to surreptitiously substitute mathematical idealities for the concrete things of the intuitively given surrounding world” (1n). For Koyré’s Bachelardian inspiration, see, for example, Iliffe (1995).
5. See also Koyré’s (1962, 11–12) reiteration of this view. The 1957 English version of that work and Koyré’s A Documentary History of the Problem of Fall from Kepler to Newton (1955) find their way into the bibliography of Dijksterhuis’s 1961 English translation, but, obviously, not in the text itself.
6. For Koyré, the second major change was the “dissolution of the cosmos,” with all that entails. An aside: To the extent that I think that the law of inertia was first formulated by Descartes in his 1632 Le Monde, I do not think correct Koyré’s view that the mathematization of nature made the law of inertia possible. But I will not pursue this train of thought here.
7. Dijksterhuis also cites with approbation another opinion referring to Bacon as one of the “great creative spirits of the seventeenth century.” He then asserts that both “parties are right up to a point. Perhaps more the first than the second: if Bacon with all his writings were to be removed from history, not a single scientific result would be lost” (1969, 397).
8. “Galileo and the scientific revolution of the seventeenth century” and “Galileo and Plato” reprinted as chapters 1 and 2 of Koyré (1968), translated into French and reprinted in Koyré (1966). Dijksterhuis also knows Koyré’s “Galileo and Plato.”
9. Iliffe refers to “obstacle” and “mutation” as Bachelardian concepts.
10. From “An experiment in measurement” of 1953. In this way Koyré positions himself against both Pierre Duhem’s internal continuous and Marxist external social accounts.
11. The literature showing Koyré wrong about Galileo’s experiments is large and now fairly old; see the works of Stillman Drake et al. A more historical appreciation of Francis Bacon’s scientific method and views on experiments is perhaps as extensive, but more recent; see the essays of Dana Jalobeanu et al. and chapter 2 in this book.
12. See also the excellent other suggestions about contextualizing the mathematization of nature in Roux (2010).
13. I will not go into the details of this answer. It should suffice to refer to Descartes on the creation of the eternal truths and the fact that, for Descartes, metaphysical truths are more certain than mathematical truths.
14. Dijksterhuis actually knows that the Rules was not published until 1701 in the Opera Posthuma.
15. A word about Mathesis Universalis: It has been pointed out (Weber 1964) that Rule 4 has two dissonant parts, the second of which contains Descartes’ views on Mathesis Universalis. While some able commentators (Marion 1975, for example) have argued that one can provide a reading of Rule 4 that takes both parts into account, others have argued that Mathesis Universalis is either a later or an earlier version of Rule 4 or even that it does not belong at all in the manuscript. I think that these issues can be settled in favor of Mathesis Universalis being a later interpolation and I am confirmed in this by the fact that the recently discovered Cambridge manuscript of the Rules is missing part 2 of Rule 4, containing Mathesis Universalis. See the edition of the Rules by Serjeantson and Edwards (forthcoming). Thus, for Descartes, Mathesis Universalis is not the “guiding principle for his whole life-work,” and mathematics was not “the sole key needed to unlock the secrets of nature.”
16. The last few assertions differ from the first few in that they reveal something like a metaphysical thesis about the relations between mathematics and physics, as opposed to an epistemological or methodological one. There is, of course, also the notion of geometric order (more geometrico) in Descartes’ appendix to Second Replies. But as Descartes makes clear, this is not a method, but a mode of exposition not applying solely to narratives proceeding by axioms, postulates, and theorems. For a development of this view in a Cartesian, see Lodewijk Meyer’s preface to Spinoza’s Descartes’ Principles of Philosophy. The issue is complex, exemplars of it span such diverse thinkers as Jean-Baptiste Morin’s Quod Deus sit and Nicolaus Steno’s Elementorum myologiae specimen seu Musculi description geometrica.
17. The French version is almost the same: “I do not accept any principles in physics that are not also accepted in mathematics, so that I may prove by demonstration everything I would deduce from them; these principles are sufficient, inasmuch as all natural phenomena can be explained by means of them” (AT, 9:101).
18. By Book 3 of the Principles, Descartes will be invoking hypotheses or supposition that he knows cannot be reduced to the principles of Part I or their deductions in Part II; as he asserts: “I dare say that you would find at least some logical connection and coherence in it, such that everything contained in the last two parts [that is, Principles 3 and 4] would have to be rejected and taken only as a pure hypothesis or even as a fable, or else it all has to be accepted. And even if it were taken only as a hypothesis, as I have proposed, nevertheless it seems to me that, until another is found more capable of explaining all the phenomena of nature, it should not be rejected” (AT, 4:216–17). See Ariew (2010, 31–46).
19. Let me put the same point somewhat differently. Descartes is no atomist, but supposing he was, he would refer all natural phenomena to his two fundamental principles, atoms and the void. The properties of bodies then would be “what the geometers call quantity,” namely, size, shape, and motion.
20. For an account of the peculiarities of the Cartesian metaphysics of Desgabets and Régis, see Schmaltz (2002).
21. A contemporary description of the work, from a letter by Simon Foucher to Leibniz, confirms this impression of eclecticism: “You know that I think Regis has given the public a great system of philosophy in 3 quarto volumes with several figures. This work contains many very important treatises, such as the one on percussion by Mariotte, chemistry by l’Emeri, medicine by Vieuxsang and by d’Uvernai. He even speaks of my treatise on Hygrometers, although he does not name it. There is in it a good portion of the physics of Rohault and he refutes there Malbranche, Perraut, Varignon—the first concerning ideas, the second concerning weight, and the third, who has recently been received by the Académie Royale des Sciences, also concerning weight. The Metheores of l’Ami also in part adorn this work, and the remainder is from Descartes. Regis conducted himself rather skillfully in his system, especially in his ethics” (Leibniz 1890, 1:398–400).
22. In the prelude to his tree of Philosophy, Descartes asserts: “a man who as yet has merely the common and imperfect knowledge … should above all try to form for himself a code of morals sufficient to regulate the actions of his life.… After that he should likewise study … the logic that teaches us how best to direct our reason in order to discover those truths of which we are ignorant. And since this is very dependent on custom, it is good for him to practice the rules for a long time on easy and simple questions such as those of mathematics. Then, when he has acquired a certain skill in discovering the truth in these questions, he should begin seriously to apply himself to the true philosophy” (AT, 10b:13–14). This is, of course, related to the statement in the Discourse cited earlier that “the real use that can be made of mathematics is … to prepare the mind to follow the real philosophical method.”
23. There are numerous methods called analysis and synthesis in early modern philosophy, most of which have nothing to do with the various things Descartes called analysis and synthesis: resolution and composition within the method of the Rules, the two modes of demonstration of the Second Replies, or the analysis (and synthesis) of the ancients. The notion originates at least from Aristotle’s Posterior Analytics and is found in seventeenth-century scholastic textbooks in the portion of their Logic texts dealing with scientific method. Du Roure’s analysis and synthesis follow the same lines as scholastic authors such as Scipion Dupleix: “Method is the order of the sciences and of their discourse: where one makes several things out of one, which is called the analytic method, or from several one, which is called the synthetic or compositional method” (Du Roure 1665, section 2). For more on Scholastic and Cartesian Logic, see Ariew (2014).
24. Régis continues: “Metaphysics not only serves as foundation for all natural Sciences, it is yet simpler and easier to acquire than all of them; the mind’s access to this science is common to all kinds of native intelligences, because there is nothing in life or in the society of men which does not dispose or lead itself to it. Every occasion all needs contribute incessantly to the material of Metaphysics which concerns the knowledge of the soul and we experience in ourselves all the proofs of the things that are the object of this knowledge. In contrast, with the other sciences we are required to go out from ourselves in order to consider the objects we examine. For example, we go out from ourselves in Geometry in order to contemplate shapes, we go out from ourselves in Physics to consider motions, and we go out from ourselves in Morals in order to observe the conduct of other men.”
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