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Séminaire DescartesNouvelles recherches sur le cartésianisme et la philosophie moderne

Samedi 23 mai, Amphithéâtre PSL, II. Autour des ouvrages de Frédéric De Buzon (Strasbourg),La Science cartésienne et son objet. Mathesis et phénomène, Paris, Honoré Champion, 2013, et de John A. Schuster (Sydney), Descartes Agonistes: Physico-mathematics, Method and Mechanism 1618-1633, Dordrecht, Springer, 2013.

Roger ARIEW

I enjoyed very much reading and thinking about John Schuster’s and Frédéric de Buzon’s latest monographs; I learned a lot from these works. It may not be surprising, but the two of them actually share many important theses with which I agree. While Schuster is interested in Descartes’ development in the mixed sciences and physico-mathematics in his early years, and de Buzon focuses more on the relations between mathematics and physics throughout Descartes’ career (and beyond), they agree that Descartes’ views on these topics changed over time and are to be contrasted with both scholastic and non-scholastic opinions on the same topics. Descartes no more inherits scholastic views about scientia media and mathesis pura than non-scholastic ones about mathesis universalis. Moreover, both Schuster and de Buzon disagree with the thesis of the mathematization of nature that supposedly links the views of, let us say, Galileo, Descartes, and Newton. Schuster, in fact, proposes instead the physicalization of mathematics for Descartes. I find this thesis intriguing, but ultimately I do not accept it fully, for reasons that will become clearer later on.

The key concept for Schuster is obviously physico-mathematics and for de Buzon, it is mathesis, whether universalis or pura. I will arrange my comments in that order, from physico-mathematics and mixed mathematics, to pure and universal mathematics. Schuster traces Descartes’ development starting in 1618 with him as a practitioner in physico-mathematics, which Schuster takes to be some kind of a “piecemeal corpuscular-mechanism,” or a causal corpuscular account, encompassing both physics and mathematics, in contrast with traditional mixed mathematics. The endpoint of his account is

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1633, with a Descartes who tries to ground both natural science and mathematics in “an overarching dualist metaphysics.” In between are two different Descartes projects, one concerning universal method and the other universal mathematics, the latter, according to Schuster, trying to provide a basis for physico-mathematics and the former attempting to subsume everything, including universal mathematics (pp. 91-93 and elsewhere). There is much to agree with in these claims. However, I am puzzled about the category “physico-mathematics”; as far as I can tell, Descartes himself never used the term. There are only four occurrences of “physico-mathematics” in all of Adam and Tannery. Two of these are by others (van Schooten and Carcavy) referring to Mersenne’s work Cogitata physico-mathematica (AT V, 319 and 413), and one is the title given to a few fragments from Beeckman’s Journal by Adam and Tannery (AT X, 67-77), on the basis of an entry from that journal entitled “Physico-mathematici paucissimi” (AT X, 52). There Beeckman refers to Descartes as someone who “frequented many Jesuits and other students and learned men” and who said that he had never met anyone, beside Beeckman, who investigates nature by joining together physics and mathematics—which delights Beeckman. So the passage, from Beeckman, is about Beeckman himself and only indirectly about Descartes: Beeckman saying that there are only a very few physico-mathematicians (in the plural), and presumably including Descartes in the category. The passage in itself, of course, is much too brief to tell us what Beeckman might be referring to as physico-mathematics, that is, what the nature of that investigation would be.

Now, there are two occurences where Descartes refers to a “Mathematico-Physics,” both occasions being contained in the notorious letter to Beeckman of October 17, 1630. In the first, Descartes tells Beeckman that he has not learned any more from Beeckman’s daydreams of a Mathematico-physics, than from a book of fairy tales. In the other, Descartes 1

complains that Beeckman praises him simply in order to give a boost to his own ego: “you write, after having praised me, that you usually prefer your Mathematico-Physics to my conjectures and allow our mutual friends to understand this. What, I ask, does this mean, other than that you lifted me up so that by comparison a greater share of the glory would come to you? You place a bit higher the stool you want to upend, so that it may stand out more as the throne of your vanity.” Descartes clearly wants to distance himself 2

from Beeckman’s style of research and has been doing so for some time.

“me nihil unquam ex tua illa, quam somnias, Mathematico-Physica magis quam ex 1

Batrachomyomachia didicisse, » AT I, 159. “Sed aperte declarant tuæ literæ qualem habueris laudandi mei causam: scribis 2

enim te solere, postquam me laudasti, Mathematico-Physicam tuam meis conjecturis præferre, idque amicis nostris significare. Quid, quæso, hoc sibi vult, nisi a te idcirco me extolli, ut majorem ex comparatione ista gloriam quæras?” AT I, 164

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When talking about his own work, he uses traditional categories, even for investigations that Schuster most likely would call physico-mathematical. For example, in a letter of May 1637, Descartes indicates that his Dioptrics is a treatise with a mixed subject of philosophy and mathematics; he calls his Geometry an essay in pure mathematics and his Meteors an essay in pure philosophy (AT, l, p. 370). Would Schuster agree with Descartes referring to 3

his Dioptrics, with its corpuscular-mechanistic optical theories, mixed mathematics? —And his Meteors, with its similar analysis of the rainbow, pure philosophy? Perhaps this does not matter; my sense is that Schuster imposes “physico-mathematics” as an analytic category on Descartes. It is an actor’s category in an attenuated fashion, that is, in the sense that others use it at the time, not in the sense that Descartes would use it about himself. But 4

given Schuster’s definition of “physico-mathematics,” it is also not clear that the others are using the concept in the way he defines it. For example, neither Beeckman nor Mersenne, who describe their own work in that fashion, may qualify as “physico-mathematicians” in Schuster’s demarcation of the category. The case of the Compendium musicae might bring this out.

Schuster argues that the Compendium musicae is not a species of physico-mathematics, but just old-fashioned mixed mathematics: “Descartes keeps the Compendium within the realm of mixed mathematics, rather than opening up this potentially physico-mathematical domain.” Schuster knows 5

very well that music theory was part and parcel of Descartes’ and Beeckman’s pursuits in 1618-1619, along with hydrostatics, free fall, navigation, etc. Descartes wrote the Compendium at the end of 1618, while at Breda. He gave that monograph to Beeckman as a New Year’s present to honor his friend’s interest in music. A few months later he wrote to Beeckman, saying: “it was you alone who roused me from my indolence and forced me to recall the learning that had almost escaped from my memory; when my mind strayed so far from serious pursuits, it was you who brought it back to the right path.

“Je propose à cet effet une méthode générale, laquelle véritablement je n'enseigne 3

pas, mais je tâche d'en donner des preuves par les trois traités suivants, que je joins au discours où j'en parle, ayant pour le premier un sujet mêlé de philosophie et de mathématique, pour le second, un tout pur de philosophie; et pour le 3ème, un tout pur de mathématique. » ‘Physico-mathematics’ in single-quotes, as it were, in a Lakatosian fashion.4

p. 103. Also: “Descartes’ Compendium of Music, also composed at this time, in close 5

connection with the tutelage of Beeckman, will not figure in our considerations. The simple reason is that the work is not a serious exercise in physico-mathematics, but, with some tiny exceptions, remained firmly within the traditional conception of mixed mathematics,” p. 20. “The Compendium is not treated in this chapter on Descartes as a physico-mathematician for the simple reason that this early work of Descartes shows hardly any traits of physico-mathematics, staying almost entirely within the realm of traditional mixed mathematics. … At no point do physico-mathematical protocols of the sort we will unpack here make an appearance,” p. 103.

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Thus, if something comes out of my head that may perhaps not to be wholly disdained, you have the right to claim it all as your own,” (AT X 162-63). The letters from those years between Descartes and Beeckman are full of music theory, Beeckman posing questions to Descartes on the subject, as can be seen in the following extract from Descartes: “I received your letter, which I was expecting, and from the first glance I had the pleasure of seeing some notes on music. What clearer display could there be that you have not forgotten me? … As for your question, you resolve it yourself and it could not be done better. There is one thing, in my opinion, you have not sufficiently considered before writing, namely, that in a single voice all the jumps are made by exact consonances.” In fact, the entry in Beeckman’s journal entitled Physico-6

mathematici paucissimi is paired with an entry about Descartes, major and minor chords, and consonances. Again, it is not clear why Beeckman might 7

be referring to himself and Descartes as physico-mathematicians; one possibility is his appreciation of Descartes as someone, as de Buzon says, “who brings an experimental way of distinguishing the major third from the fourth, and as someone who constructs consonances from the continuous division of string segment, in agreement with Beeckman’s own theory of vibration.” The claim that Beeckman considered physico-mathematics to 8

encompass music theory and to include Descartes’ Compendium as a product of the same kind of investigation is at least plausible. The case for Mersenne thinking the same about physico-mathematics is clear: his Cogitata physico-mathematica contains treatises on measures and weights, hydraulics, the art of

AT X, 151-53. The letter continues: “In fact, let note A be distant from note B by an 6

interval of a fifth. It will be necessarily distant from C by the space of a fourth and not perfect, but it will fall short by a schism, as is proved by the attached numbers; and if you use this, it will be easy to find the exact quantity of any kind of tone. And you ought not to say that there should rather be an imperfect fifth between A and D, so that AC is a true and accurate fourth; for a dissonance is best observed in tones that must be heard at the same time than in those heard successively. And in the latter, in my opinion, at least in vocal music and with mathematical elegance, one does not immediately come from the term of one consonance to another, but the passage is made gently by every interval between the two; this would prevent one from distinguishing an error as small as a schism. I remember having noted this in what I have written before about dissonances. If you look carefully there, as in the rest of my music, you will find mathematical demonstrations for all the points I made on the intervals of consonances, scales, and dissonances; but it is undigested, confused, and too briefly explained.” AT X, 52.Chordae majores, intactas minores et consonantes, tactae movent. Observavit 7

RENATUS Picto cordas testudinis inferiores, id est bassiores, pulsas, movere evidenter ipsis consonantes acutiores; acutioribus verà pulsis, graviores non ita evidenter moveri. Quod infertur ex meis upothesibus: crassiores enim globi, quos graves soni edunt, majoribusque intervallis jacti, aptiores sunt tangere fortiterque quiequam impellere (fol. 100v). De Buzon, Roux and Garber ed., p. 144; also p. 160. 8

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navigation and hydrostatics, musical theory and practice, mechanics, and ballistics.

I freely admit that I do not understand Beeckman, Descartes, and Mersenne’s interests in music theory; perhaps their investigations are not properly causal and corpuscularian, but they seem to consider them as a part of physico-mathematics. I am reminded of the first English translation of Descartes’ Compendium in 1653; the Epistle to the reader reveals another reason at least for the seventeenth-century English interest in musical theory, one equally opaque to me, namely, its connection with magic: “[The author] must be so far as a Magician, as to excite Wonder, with reducing into Practice the Thaumaturgical, or admirable Secrets of Musick: I meane the Sympathies and Antipathies betwixt Consounds and Dissounds; the Medico-magical Virtues of Harmonious notes (instanced in the Cure of Sauls Melancholy fitts, and of the prodigious Venome of the Tarantula, etc.) …; the Model of Autophonous, or speaking Statues; and finally, the Cryptological Musick, whereby the secret Conceptions of the mind may be, by the Language of inarticulate Sounds, communicated to a Friend, at a good distance.”

I am likewise puzzled about Schuster’s definition of mixed mathematics. Schuster properly refers to Aristotle for the origin of the distinction: “The term “mixed mathematics” had been framed by Aristotle to refer to a group of disciplines intermediate between natural philosophy, which dealt with those things that change and exist independently of us, and mathematics, which deals with those things that do not change but have no existence independently of us, since numbers and geometrical figures have (contra Plato) an existence only in our minds,” (p. 6). He then brings the term into the seventeenth century by providing the late scholastic implications of Aristotle’s doctrine: “Impressed by the clarity and rigor of mathematics, including practical and mixed mathematics, [Descartes] was also aware (1) that in Aristotelianism no proper explanations, those dealing with matter and cause, could be formulated in mathematical terms; and, (2) that the mixed mathematical disciplines were given a subordinate and non-explanatory role in the accepted Neo-Scholastic map of the domain of natural philosophizing,” (p. 34). All of this is surely right for Thomist scholastics, but, I would argue, there are many variants to the view of the relation between mathematics and natural philosophy, and some of these might possibly blur any distinction between the mixed science and physico-mathematics. A more Augustinian-Platonic (though still scholastic) variant about the classification of the sciences and the status of mathematics can be found in the works of Robert Grosseteste and Roger Bacon. While agreeing with the basic intuition that the higher sciences provide the reason for the lower sciences (the subalternated sciences), Grosseteste disagrees about the status of the composite sciences. He argues that composite sciences have an additional nature about which the higher sciences say nothing; ultimately, he asserts that only mathematics can

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provide the reason for a subalternated science, and even for natural philosophy. Roger Bacon follows him in this. In the Opus Majus and Opus Tertium, Bacon details a view of human knowledge as a hierarchy of knowledge in which mathematics is antecedent to natural philosophy and to metaphysics: "Without mathematics no science can be had." This looks like 9

the forerunner of the mathematization of nature. But perhaps this view was not available as a standard option in the seventeenth-century.

The Jesuit doctrine of the utility of mathematics does look more like what Schuster sketches. Ludovico Carbone, for example, details eleven doubts about the mathematical sciences. Some of these doubts concern the type of abstraction characteristic of mathematics: mathematicians consider bare quantity without any connection to substance; the intelligible matter they arrive at when they set aside sensible matter is merely fictive and cannot be defined in terms of true genus and difference; they abstract from being and the good; they abstract from motion and the natural forces that produce it; and they abstract from all kinds of causes and so cannot use causal reasoning in any of their demonstrations (Carbone 1599, pp. 240-43). One can find analogous doctrines in Jesuits such as Benito Pereira.

It is against this background that the Collegio Romano mathematician Christopher Clavius proposed his reform of mathematics, arguing its importance to natural philosophy. In an essay for the Jesuits on the teaching of mathematical disciplines he wrote: “Physics cannot be understood correctly without [the mathematical disciplines], especially what pertains to that part concerning the number and motion of the celestial orbs, of the multitude of intelligences, of the effect of the stars, which depend on the various conjunctions, oppositions, and other distances between them, of the division of continuous quantities to infinity, of the tides, of the winds, of cornets, the rainbow, halos, and other meteorological matters, of the proportion of motions, qualities, actions, passions, reactions, etc., concerning which the calculatores wrote much” (Clavius 1901, p. 472). Similarly, Clavius disputed the common opinions that the mathematical sciences were too abstract and fictive. In the same work, he wrote: “It will contribute much to this if the teachers of philosophy abstained from those questions which do not help in the understanding of natural things and very much detract from the authority of mathematical disciplines in the eyes of the students, such as those in which they teach that mathematical sciences are not sciences, do not have demonstrations, abstract from being and the good, etc.” (Clavius 1901, p. 471). Obviously Clavius has in mind the kind of views represented by his contemporaries such as Carbone. While Clavius’ positive doctrine about the mixed sciences is not fully revealed, one only has to read his most famous

Opus tertium, in Bacon 1859, p 35; and also Bacon 1928, l, p. 109. For an account of 9

the fortunes of the concept of “composite sciences,” see Livesay 1990.

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work, Sphaera, to see that his heavenly spheres are three-dimensional or solid, and that he thinks his eccentrics and epicycles as real. In the last edition of the textbook, Clavius even calls for a reworking of the arrangement of the celestial orbs in order to save the Galilean novel phenomena. Thus, it is not clear that Clavius and other mixed scientists at the time considered their discipline as “abstracted from all kinds of causes.”

The demarcation between the mixed sciences and physico-mathematics in general does not seem so clear cut to me. One might also adopt de Buzon’s opinion that when Beeckman calls people physico-mathematicians, it involves a “way of harmonizing mathematics and physics, and, in a more particular way, on the ways in which they agree with the small number of philosophical theses that he considers his own and to which he returns again and again,” (Roux and Garber ed., p. 144; p. 159); one may then emphasize that these theses are heterogeneous and include music theory. As de Buzon also says, when Beeckman contrasts his physico-mathematical method and those of Bacon and Stevin, he writes that they do not have the right the right proportion or mix of methamatics and physics, since Stevin is said to be too mathematical and rarely concerned with physics and Bacon not mathematical enough (when dealing with the octave in music theory).

Descartes’ vocabulary does include mathesis pura and mathesis universalis. The references to mathesis pura are mostly clustered around the end of Meditations V and VI (AT VII, 71, 74, and 80) together with the discussions about these doctrines in Objections V by Gassendi and Descartes’ Replies (Rochot ed., 474-75, 518-519 and 522-523). Descartes’ phrase concerns the object of pure mathematics (purae Matheseos objectum); Gassendi complains that these objects, such as the point, line, surface, and indivisibles cannot have real existence, and Descartes replies that, if this were true, there has never existed any triangle nor anything that we conceive as belonging to the essence of the triangle or other geometrical figure, if these essences were never derived from something existent. Interestingly, these passages are translated in the Méditations by the phrase “l’object des demonstrations de Geometrie,” providing a rough equivalency. Still, in the letter to Clerselier 10

that takes the place of Objections V, the term used is “pures mathematiques,” both in the quoted objection and in Descartes’ reply to it (AT IXa, 212-213). Moreover, the Latin phrase has a few earlier occurrences, in letters to and from Ciermans (AT II, 56, 71), and even a single occasion in Rule 6 (AT X, 385). This gives de Buzon plenty of materials for his analyses of the object of mathesis pura, comparing the Cartesian position with the tradition in mathematics, in the Meditations and elsewhere, but also contrasting mathesis pura with mathesis universalis (pp. 19-22 and elsewhere); as he says: “Descartes

Also “l’objet aux demonstrations des Geometres,” and “l’objet de la Geometrie 10

speculative,” (AT IXa, 56, 57, 63),

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puts himself into two relatively distinct traditions, with each of these two concepts, but clearly advances beyond their constraints. Before Descartes, mathesis universalis and mathesis pura et abstracta do not refer to the same object,” (pp. 20-21).

In contrast, mathesis universalis does not play a great role in Descartes’ vocabulary; as far as I can tell, it occurs only in the notorious second part of Rule 4, about method. De Buzon, like Schuster, distinguishes it from the project, that is, the method, of the Rules. As de Buzon represents Schuster, mathesis universalis is a youthful dream, from circa 1619, which is replaced by universal method, circa 1628. De Buzon’s analysis is that the two halves of 11

Rule 4 are two different project, mathesis universalis being concerned with order and measure, while the method of the Rules is concerned with order (p. 71, citing AT X, 451). He concludes that the role played by mathesis universalis for Descartes is that of a philosophical foundation for mathematics (p. 72) and supports it by discussing van Schooten’s work on universal mathematics based on Descartes’ Geometry (Principia matheseos universalis seu introductio ad cartesianae geometricam methodum). This allows de Buzon to put aside mathesis universalis and to consider again the role of mathesis pura in Cartesian philosophy, namely, assertions Descartes makes that “his physics is nothing other than geometry,” and that he has “reduced physics to the laws of 12

mathematics,” having also claimed that his metaphysical demonstrations 13

are more certain than geometrical demonstrations (AT I, 145). De Buzon accomplishes this by considering the object of pure mathematics in the Meditations and then by giving an analysis of mathematics in the Principles.

In Principles, Part II (art. 64), Descartes says: “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.” What the text of the article explains is that Descartes recognizes “no 14

Indeed Schuster still holds that mathesis Universalis is the older project. Knowing 11

that the Cambridge manuscript is missing the second part of Rule 4, he conjectures that “The abridged Cambridge ms. looks to be a version concocted as a holding action, after the discovery of the law of refraction in 1626 and before the final abandonment of the project in 1628. The ms. seems to have been sculpted to avoid overt revelation of the obvious difficulties arising from rules 17–22, … so that it might seem to a reader that the project was still on course” (p. 389). Serjeantson is working on the hypothesis that the Cambridge ms is the older version and that Rule 4b is a later addition, which would support de Buzon’s position.

To Mersenne, 27 July 1638, AT II, p. 268.12

To Mersenne, 11 March 1640, AT III, p. 39.13

AT VII, 78, IX, 101. The French version is almost the same: “I do not accept any 14

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.”

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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: “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 motion 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 of the Principles, representing the Meditations, Descartes asserts that extended substance can be clearly and distinctly understood as constituting the nature of body and that extension as a mode of substance can be no less clearly and 15

distinctly understood as substance itself. Descartes then lists the properties 16

or attributes of extension as their shapes, the situation of their parts, and their motions. It happens that these properties are what “the geometers call 17

quantity.” But that’s 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 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, as is mathematics. Thus, unlike 18

Grosseteste and Bacon, Descartes does not accept mathematics as the foundation for knowledge underlying both physics and metaphysics.

Principles I, art. 63.15



Let me put the same point somewhat differently. Descartes is no atomist, but 18

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.

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This is how Descartes was understood by the Cartesians. Régis has an exemplary exposition of this, delineating carefully among metaphysics, mathematics, and physics:19

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

As does Le Grand. On the standard question of the certainty of natural philosophy 19

(what he also calls physiology), Le Grand proceeds very much in the spirit of a scholastic, substituting Cartesian terminology and doctrines. The usual objection to natural philosophy being a science is that it treats material things as changeable this 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.” (Entire Body of Philosophy I, p. 92). Le Grand 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,” (I, p. 92).

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must have diverse efficient causes that correspond to them and can only be the particular bodies that have resulted from the division of matter. 20

In the end, we don’t really have an instance of the mathematization of nature, but of the metaphysicalization of mathematics. We need to put Descartes’ metaphysical mathematics alongside Descartes’ metaphysical physics.

Régis, Système général, p. 64: “La Metaphysique ne sert pas seulement à l’ame pour 20

se connoitre elle-même, elle luy est encore necessaire pour commoitre les choses qui sont hors d’elle, toutes les Sciences naturelles dependent de la Metaphysique; la Mathematique, la Physique et la Morale sont fondés sur ses principes: En effect, si les Geometres sont assurez que les trois angles d’un triangle sont égaux à deux droits, ils ont receu cette certitude de la Metaphysique, qui leur a enseigné que tout ce qu’ils conçoivent clairement est vray, et qu’il est tel, parce que toutes leur idées doivent avoir une cause exemplaire qui contient formellement toutes les les proprietez que ces idées representent. Si les Physiciens sont assurez que la substance étendue existe et qu’elle est divisée en plusieur corps, ils sçavent cela par la Metaphysique, qui leur apprend, non seulement que l’idée qu’ils ont de l’étendue, doit avoir une cause exemplaire, qui ne peut estre que l’étendue même; mais encore que les differentes sensations qu’ils ont, doivent avoir des causes efficienttes diverses qui leur repondent, et qui ne peuvent estre que les corps particuliers qui ont resulté de la division de la Matiere.”

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Les Archives du



Daniel GARBER

I am very pleased to participate in this discussion, and have an opportunity to reflect on two very important contributions to our understanding of Descartes and his intellectual project. As we shall see, the two projects are somewhat different, and offer interesting contrasts. But, at the same time, they have enough in common to create an interesting conversation. Both are historical, and both are philosophical (sorry, John, but it’s true…), though Schuster leans more strongly toward the historical and de Buzon leans more strongly toward the philosophical. But both involve the interaction between Descartes’ ideas in mathematics and his other projects, including his method and his natural philosophy.

While Schuster’s book goes as far as a consideration of the Principia philosophiae (1644), his real focus is on Descartes’ thought up to and including Le monde. And while de Buzon’s book begins with a chapter on the mathesis universalis of the early years, his focus is really on the later years. For that reason, I will begin with some remarks on Schuster’s Descartes-Agonistes.

There are a number of themes in Schuster’s magisterial book. For one, Schuster is interested in tracing the beginnings and development of Descartes’ program for physico-mathematics. As he puts it in the preface to the book, Schuster wants to trace the development of Descartes “from being a physico-mathematician who was seeking piecemeal corpuscular-mechanical grounding to his work, to becoming a corpuscular-mechanical systematiser with recognizable physico-mathematical conceptual stitches holding together large parts of the system.” [vi] In his earliest period, during his apprenticeship with Isaac Beeckman, Descartes’ (and Beeckman's) project was

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the derivation of mathematical characterizations of particular phenomena through an understanding of an underlying mechanical-corpuscular model. Examples of this include Descartes’ treatment of the hydrostatic paradox, the free-fall of bodies, and the refraction of light. The mathematical characterization of these phenomena was important for Descartes, but more important still was an understanding of the underlying corpuscular-mechanical causal mechanism. This is contrasted with Descartes’ later physico-mathematical project, where what was important was not just individual explanations of individual phenomena, but a systematic natural philosophy to bind all of the individual explanations together. On Schuster’s view, correct in my opinion, with the project of Le monde Descartes moves from the domain of a problem solver to that of a natural philosopher. [11-12] Even so, the physico-mathematical project remains with him for the rest of his career.

But there is another theme on Schuster’s book that I would like to discuss. This is his treatment of the mathesis universalis in the context of the Regulae. For Schuster, the method of the Regulae and the mathesis universalis are deeply intertwined, and both are connected with the physico-mathematical project. Schuster writes:

…much of our story of Descartes agonistes is precisely the story of the intended and unintended entanglements of these two trajectories—in physico-mathematical natural philosophy, and in analytical mathematics, promoted to fantasy programs in universal mathematics and method… [103]

The only text where Descartes discusses the mathesis universalis is in the second half of Rule 4 of the Regulae, what Schuster calls Rule 4B; the first part of Rule 4 is 4A. On Schuster’s reading, as I understand it, the discussion of the mathesis universalis in 4B represents the genesis of the Regulae project. According to Schuster, Rule 4B dates from between March and November 1619. Rule 4A was written after November 1619, and was modeled on 4B. [230, 237] In Rule 4A Descartes discusses the method itself: “We need a method if we are to investigate the truth of things.” Schuster writes:

…the method discussed in rule 4A and the surrounding rules arose from an attempted analogical extension of that only partly elaborated universal mathematics, and that this enthusiastic and hopeful extension arguably did take place from November 1619, hard on the heels of the inscription of the notion of universal mathematics in rule 4B. [238; cf. 248, 257]

In this way, Descartes’ method of the Regulae is represented as a direct extension of the mathesis universalis.

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In the early stages of the composition of the Regulae, Schuster sees Descartes going from the mathesis universalis to the method. When he comes back to the project later in the 1620s, between 1626 and 1628 under the influence of Mersenne it goes in the opposite direction, from method to mathesis universalis. [Quotation: “He decided to return…”] In particular, on Schuster’s reading, Descartes’ project in the later 1620s involved a legitimation of the mathesis universalis. How did this work?

By Rule 14, all well-defined problems are construed as dealing with extension and their relations. Now, in Rule 12 Descartes presents a physiological sketch of sense perception and imagination, what Schuster calls the o-p-p nexus [‘ optics-psychology-physiology nexus ’]. On this view, the imagination is simply a mechanist/corpuscularian configuration of bodies with certain shapes and sizes. He thus argues that “it is on the basis of the o-p-p nexus that the truth of the operations of universal mathematics will be grounded, as well as the ontological reference of its objects of inquiry.” [318] Now,

…even in the imaginative rendering of quantity, one is dealing with a real body. The imagination and the patterns in it are corporeal entities, bearing the same ontological certification as the mechanical deliverances of sensation registered in the common sense. [321]

In this way the mathesis universalis takes the physical structures in the imagination as its objects:

The corporeal world is indeed the ultimate object of universal mathematics; but, it is known only under the category of the two-dimensional shapes and patterns registered in sensation and delivered up to the validating gaze of the intellect. … As a corporeal locus, the imagination is an ontologically suitable ‘screen’ upon which extension-symbols can be manipulated; and the operations performed on the symbols have the certification of being clearly intuited in the ‘real extension of bodies’—they are true and true of the world. [Schuster, 325]

In this last stage of the Regulae, physico-mathematics is integrated into the mathesis universalis, by way of Descartes’ conception of a dimension. In Rule 14 Descartes writes: “By ‘dimension’ we mean some mode or aspect in respect of which some subject is considered measureable.” [AT X 447] Length, breadth and depth are all dimensions, but so is weight, speed, or anything else that can be subjected to measure. And, Schuster argues, the notion of dimension allows us to integrate the physico-mathematical project into mathesis universalis:

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As regards physico-mathematics, it will now be that part of universal mathematics which deals with problems about the relations holding between given and sought dimensions of physical properties measured in or between bodies. [Schuster, 323]

And so we have a neat package: mathesis universalis, method, and now physico-mathematics all bound up into a single program.

This is in 1628. But it doesn’t last. There is a tension between the mathesis universalis and closely connected method on the one hand, and the physico-mathematical program. The physico-mathematics is discursive, “the explanations being always verbal, qualitative and discursive.” [334] This is in tension with the idea of mathematicization. Schuster writes:

Descartes was soon to shift gears to escape these binds. He would jettison the universal mathematics he had just worked out in partial detail, along with any serious claim that what he was now doing derived from the method. Instead, he would begin to work on the composition of Le Monde, a system of inevitably qualitative and discursive corpuscular-mechanical natural philosophy and cosmology. [338]

At this moment, Descartes abandoned the method as a real project: it became simply as show project, a bit of “public packaging,” his “veritable intellectual trademark on the public stage,” but not his real intellectual program. [See a bunch of quotations on the handout.]

And with this change came a different kind of validation for the project—both the physico-mathematical project and the new, cleaner and more abstract project: a metaphysical justification of the sort that we find in the Meditationes.

This, then, is Schuster’s reading. On his reading it is the mathesis universalis that is fundamental: the mathesis universalis gives rise to the project of the method, which then also becomes integrated with the project of the physico-mathematics. The whole project is validate by its link to actual bodies with actual sizes and shapes as they exist in the imagination. As this project fails, in 1628 or so, the mathesis universalis and the method are abandoned, replaced by a more abstract mathematics and by a new more systematic version of the physico-mathematical project, now grounded not in Descartes’ physiological and materialistic account of the imagination but in metaphysics, in the veracity of God.

The picture is rich and complicated, and in its way plausible. But I’m still skeptical. Briefly, here are my worries.

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For Schuster, the mathesis universalis and the method are deeply intertwined: according to Schuster, the method is a kind of generalization of the mathesis universalis in late 1619, and later, after 1626, the method gives rise to the mathesis universalis. But can that be right? Here are the definitions that Descartes gives of method and mathesis universalis:

By a method I mean reliable rules which are easy to apply, and such that if one follows them exactly one will ever take what is false to be true or fruitlessly expend one’s mental effort, but will gradually and constantly increase one’s knowledge until one arrives at a true understanding of everything within one’s capacity. [AT X 371-2]

When I considered the matter more closely, I came to see that the exclusive concern of mathematics is with questions of order or measure and that it is irrelevant whether the measure in question involves numbers, shapes, stars, sounds, or any other object whatever. This made me realize that there must be a general science which explains all the points that can be raised concerning order and measure irrespective of the subject-matter, and that this science should be termed mathesis universalis –a venerable term with a well-established meaning - for it covers everything that entitles these other sciences to be called branches of mathematics. [AT X 377-78]

I find it very difficult to imagine that method and mathesis universalis could be related as closely as Schuster wants them to be: mathesis universalis is a science, a scientia, a set of propositions that is endorsed as true, while method is a set of rules for arriving at scientia, something on its face altogether different. I simply do not see how one can pass so easily from mathesis universalis to method and back again. How can you derive a set of rules for investigating nature from a scientia, or a scientia from a set of rules?

But there is another reason that I am skeptical about Schuster’s thesis. Schuster’s reading of the Regulae is based on the assumption that the mathesis universalis in Rule 4B is the oldest part of the text, and that the text of the Regulae is centered around expanding the mathesis universalis into the method of the Regulae. But recently Richard Serjeantson of Trinity College Cambridge discovered a new manuscript of the Regulae. Let us call the new manuscript the “Cambridge manuscript” and the older one that we have had up until now the “Posthumous manuscript.” Now, the Cambridge manuscript is about 40% shorter than the posthumous manuscript. There are many small and subtle differences between the Cambridge manuscript and the posthumous manuscript. But most strikingly, the Cambridge manuscript entirely lacks the discussion of the mathesis universalis. Now, the general consensus of those who have studied the Cambridge manuscript is that it is an earlier version of the text, and that the posthumous manuscript is an expansion of the Cambridge manuscript, a later version. But the Cambridge manuscript

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already contains the definition of method, and much of the development of the method found in the posthumous manuscript. If the Cambridge manuscript is, indeed earlier than the posthumous manuscript, the method must have been developed independently of the mathesis universalis, and the mathesis universalis must have been a later addition to the project.

Now, Schuster knew of the Cambridge manuscript when he published the book. His argument, though, is that the Cambridge manuscript represents a later abridgment of the fuller text found in the posthumous manuscript. [See handout for the argument.] But I don’t buy the reading. As I said, there are other differences between the Cambridge manuscript and the posthumous manuscript that suggest that the Cambridge manuscript must be earlier. The most striking one is this. Now, in one passage of Rule 14, the posthumous manuscript reads as follows:

…the power through which we properly cognize a thing, is purely spiritual, and no less distinct from the whole body than is blood from bone, or the hand from the eye…

In this passage, Descartes is clearly alluding to his views on mind/body distinction, and the mind as a spiritual thing, distinct from the body. But in the Cambridge manuscript, it is interestingly different:

…the power through which we cognize a thing is something in us no less distinct from the phantasy than is the eye or the hand…

Here there is no apparent reference to Descartes’ dualism. This suggests that the Cambridge manuscript may have been written before Descartes came to be a dualist, and the posthumous manuscript written afterward. There are many other features that suggest that the posthumous manuscript is later, particularly the much fuller development of Rules 6, 8, and 12. If so, it would be a serious problem for Schuster’s reading. If, as this dating suggests, the Cambridge manuscript is later than the posthumous manuscript, it is highly unlikely that the mathesis universalis is as intimately connected with the method as Schuster’s story suggests.

Connected with this is some skepticism about the justificatory project of the Regulae. Schuster argues that the discussion of what he calls the o-p-p network, the physiology of sensation and imagination, primarily in Rule 12 is meant to give a validation of the mathesis universalis. Again, the Cambridge manuscript calls this into question. Though it lacks the mathesis universalis, the Cambridge manuscript contains at least the rudiments of the account of perception and imagination found in the posthumous manuscript. There is a kind of justificatory project in the Regulae, to be sure: but I would argue that it is not the mathesis universalis that is being justified, but the claim that intuition

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and deduction, the two modes of acquiring genuine knowledge Descartes introduces in Rule 3 are, indeed, the only paths to genuine knowledge.

There is a great deal to admire in Schuster’s book. I think his account of Descartes’ early explorations into physico-mathematics are brilliant and illuminating, as are his accounts of the later Le monde and Principia philosophiae. This, for me, is the heart of the book. But unlike Schuster, I find the role of the mathesis universalis in Descartes’ early thought marginal.

And in that respect, I think I agree with Frédéric de Buzon. In his interesting and original new book, La Science cartésienne et son objet : Mathesis et phénomène, de Buzon deals with some central questions in Descartes’ natural philosophy. The book puts forward a number of striking new interpretations, as well as offering an ingenious attempt to solve some knotty problems that have been plaguing Descartes’ readers for many years.

The title indicates the main focus of the book: the roles of mathesis and phenomenon in Descartes’ science of nature. De Buzon’s argument is set out in a sequence of core chapters. He begins by distinguishing between two notions related to mathesis: the mathesis universalis and mathesis pura et abstracta, or mathesis for short. The mathesis universalis of the Regulae is the subject matter of chapt. 1. There seems little innovative in Descartes’ conception of the mathesis universalis in relation to what went before, de Buzon suggests. But even though this is the longest chapter in the book, it is mathesis pura et abstracta and not mathesis universalis that dominates the book. Unlike Schuster, de Buzon’s judgment seems to be that the mathesis universalis is not really a central issue in Descartes’ thought.

De Buzon argues that unlike Descartes’ conception of mathesis universalis, his conception of mathesis pura does constitute a radical departure from earlier conception of the notion. Unlike earlier conceptions of mathesis, which include both arithmetic and geometry, arithmetic and number seem to drop out altogether, according to de Buzon. Furthermore, number is replaced by motion, which together with three-dimensional extension now make up the basic constituents of reality. (117, 124) In this respect the mathesis of the Meditations is also a radical departure from the mathesis universalis of the Regulae. (123)

With this new conception of mathesis in place, in chapt. 3 de Buzon now confronts one of the central puzzles of Descartes’ natural philosophy, the claim in Principia philosophiae II.64 that “the only principles which I accept, or require in physics are those of geometry and pure mathematics; these principles explain all natural phenomena, and enable us to provide quite certain demonstrations regarding them.” This claim also brings in for the first time the second major notion at play in the book, that of phenomena. Phenomena involve sense, and sense involves matter in motion. Indeed, the contents of sensation, for Descartes, can only be matter in motion as well, de

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Buzon argues. Since motion is now at the center of mathesis for Descartes, we can know that all phenomena can be explained mathematically, on this new conception of mathematics. (143-46)

(It is interesting here that Schuster doesn’t address this passage directly. But, as I indicated earlier, he seems to dismiss all such claims identifying physics with mathematics in Descartes’ later writing as mythic speech, not reflective of Descartes’ true position or practice. With this we can see the opposition between Schuster and de Buzon quite clearly: de Buzon sets aside the early mathesis universalis project which Schuster sees as central to Descartes, while Schuster sets aside the later mathematical account of nature which de Buzon sees as central. )

De Buzon extends the argument by discussing Descartes’ laws of nature. (Chapt. 4) His claim is that the major innovation in Descartes’ laws is the fact that the laws of nature are the laws of motion, laws that are not just general (and vague) framework principles, but principles that can be used in the explanation of specific phenomena. He claims that since motion now pertains to mathesis pura, the laws of nature are a part of mathesis pura. (190ff).

The focus on mathesis as a central notion in Descartes’ natural philosophy is interesting and original, but I am somewhat skeptical that it fully succeeds. Key to de Buzon’s argument is the claim that on Descartes’ novel conception of mathesis, arithmetic and number drop out and are replaced by motion. (117, 122-26, 132). I’m not convinced about the elimination of arithmetic, which, in any case, plays no real role in de Buzon’s argument, but the inclusion of motion in mathesis allows him to say, for example, that all his physics is mathematics and that all the phenomena are explicable in terms of mathesis. Furthermore, it is what allows de Buzon to claim that even the laws of nature themselves reduce to mathesis.

De Buzon wants to understand Descartes’ project in natural philosophy in terms of mathesis and motion in a relatively straightforward way. But the story has to be more complicated than the one he tells. De Buzon doesn’t take into account an important distinction that Descartes introduces. In a letter to Mersenne, 27 July 1638, Descartes writes:

I am obliged to M. Desargues’ for taking the trouble to make it clear that he is sorry I no longer wish to continue my studies in geometry. But I have decided to give up only abstract geometry, that is to say, the investigation of problems which function merely as mental exercises. My aim is to have more time to devote to another sort of geometry where the problems have to do with the explanation of natural phenomena. If he cares to think about what I wrote about salt, snow, rainbows, etc., he will see that my entire physics is nothing but geometry. [AT II 268]

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Here Descartes claims that he is giving up “la géométrie abstraite,” though he plans to pursue “une autre sorte de géométrie,” a geometry that deals with “l’explication des phénomènes de la nature.” Nor are the demonstrations appropriate to this other sort of geometry the same as those found in abstract geometry. As he wrote to Mersenne on 27 May 1638, just a few months earlier:

You ask if I regard what I have written about refraction as a demonstration. I think it is, in so far as one can be given in this field without a previous demonstration of the principles of physics by metaphysics - which is something I hope to do some day but which has not yet been done—and so far as it has ever been possible to demonstrate the solution to any problem of mechanics, or optics, or astronomy, or anything else which is not pure geometry or arithmetic. But to require me to give geometrical demonstrations on a topic that depends on physics is to ask me to do the impossible. [AT II 141-42]

Though the concepts he uses in physical explanations are broadly geometrical, Descartes claims here that manner of demonstration is not.

Closely connected with this ambiguity in Descartes’ conception of geometry is an ambiguity in his conception of motion. In a certain sense it is not surprising to find motion among the basic concepts of geometry. There is certainly a tradition of defining lines in terms of the ideal motion of a point, surfaces in terms of the ideal motion of a line, and volumes in terms of the ideal motion of a surface. This is also the kind of motion that seems in use in Descartes’ Géométrie, when he is talking about the generation of proper geometrical curves though his compass, or the generation of mechanical curves such as the quadratrix or the spiral through certain simultaneous motions of points and lines. The motion in question in abstract geometry is not the real physical motion treated in physics, but the ideal motion of geometrical objects, a kind of motion for which considerations of velocity and force are irrelevant .1

When Descartes is concerned about physics, it seems to me that it must be physical motion that is at issue. Consider, for example, the laws of motion in part II of the Principia philosophiae. While motion is in question, it

On the distinction between geometrical and physical motion and its place in the 1

history of geometry, see, e.g., R. Rashed, “Le mouvement en géométrie classique,” Al-Mukhatabat 7 (2013), p. 58-68; P. Mancosu, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (Oxford, Oxford University Press, 1996), chapt. 4.1; H. Bos, Redefining Geometrical Exactness: Descartes' Transformation of the Early Modern Concept of Construction (New York: Springer, 2001), passim. While this distinction was generally made, Hobbes seems to have been an exception. For Hobbes, the object of geometry is extended body, and its motion is physical motion. On this, see Douglas Jesseph, “Hobbesian Mechanics,” in Oxford Studies in Early Modern Philosophy III (Oxford: Oxford University Press, 2006), 119-152.

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cannot be the abstract geometrical motion normally treated in geometry, the motion that geometers use to generate lines from points, or surfaces from line, or the motion that Descartes used in his Géométrie to generate curves using his compass. The derivation of the laws of motion for Descartes involves the real motion of real bodies, which are sustained by the continual activity of an immutable God . The ideal motion of abstract geometry does not satisfy these 2

laws of motion.To talk about geometry or mathematics in Descartes is thus

ambiguous between talking about abstract, pure mathematics, with an ideal conception of the motion of geometrical objects, and the kind of mathematics appropriate for physics, with its robust physical conception of motion that pertains to concrete bodies, sustained over time by an immutable God. If de Buzon’s thesis is that Descartes’ physics is mathematical in the second sense, then it is uncontroversially true, but not very interesting. If, on the other hand, the claim is that it is mathematical in the first sense, then the thesis is much more surprising, but it is also much more difficult to understand how it could be true. Descartes’ notorious claim in Principia philosophiae II 64 seems to indicate that it is abstract mathematics that he has in mind: “the only principles which I accept, or require in physics are those of geometry and pure mathematics.” If the claim is that the principles of physics derive from abstract geometry and mathematics, then it is not obvious how it is could be true, or even consistent with the view Descartes seems to have taken in his letters in 1638.

If we are to understand the way in which Descartes’ project in natural philosophy can be understood to be mathematical, we must take into account the complexity of Descartes’ thought about mathematics and the different kinds of motion at issue in that domain. Elegant as Frédéric de Buzon’s proposed solution may be, Descartes’ texts do not articulate as clearly or distinctly as he is trying to make them speak.

I would like to end by thanking both John Schuster and Frédéric de Buzon for giving us two excellent books, which will continue to stimulate discussion for a long time.

Principia II 36, 39, 42.2

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Les Archives du



Sophie ROUX

À supposer que l’histoire de la philosophie et l’histoire des sciences existent encore dans quelques siècles, si d’aventure une historienne des interprétations de la pensée de Descartes en vient à mettre la main sur les ouvrages de Frédéric de Buzon et de John Schuster, elle sera d’emblée frappée par leur diversité :

- Diversité de volume tout d’abord. L’ouvrage de Frédéric de Buzon, que j’appellerai désormais comme Mathesis, est un petit ouvrage écrit en assez gros caractères ; l’ouvrage de John Schuster, désigné par la suite comme Agonistes, est au contraire un très gros ouvrage écrit en tout petits caractères.

- Diversité dans la composition ensuite. Alors que Mathesis résulte de la réunion lâche, autour de la question directrice de la mathesis, de neuf articles publiés entre 1995 et 2013 puis retravaillés pour l’occasion, les différents chapitres d’Agonistes, même si eux aussi ont été en partie l’objet de publications séparées dans les trente dernières années, raconte véritablement une histoire marquée par des approfondissements, mais aussi par des impasses et des revirements, l’histoire de la naissance d’un philosophe naturel systématique.

- En troisième lieu, diversité quant aux œuvres de Descartes qui sont analysées. À l’exception du premier chapitre consacré à la mathesis, Mathesis porte sur ses œuvres publiées, du Discours de la méthode aux Passions de l’âme ; Agonistes en revanche, à l’exception du dernier chapitre consacré à la comparaison de la cosmologie des Principes et de la cosmologie du Monde, se concentre sur des œuvres de jeunesse que Descartes n’a pas publiées, les Règles, le Monde, et de nombreux fragments.

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- En quatrième et dernier lieu, diversité quant aux thèses soutenues. Bien sûr, toute diversité se dit sur fond d’identité. Le terrain commun à Mathesis et à Agonistes est celui de la physique — ou de la philosophie naturelle si, comme c’est le cas d’Agonistes, on préfère utiliser ce que les sociologues ont coutume d’appeler les « catégories des acteurs ». Indépendamment de cette différence de dénomination, Mathesis et Agonistes ne voient pas la philosophie naturelle de la même manière, en grande partie en raison de la manière dont ils caractérisent leur situation disciplinaire. C’est par cette différence que je commencerai ma présentation. Je continuerai en présentant les thèses principales de Mathesis. Finalement, j’examinerai plus spécifiquement les premiers chapitres d’Agonistes.

1. Histoire des sciences et histoire de la philosophieTandis qu’Agonistes met à distance l’histoire de la philosophie,

Mathesis distingue sa démarche de celle qu’adopte l’historien des sciences.a) D’entrée de jeu, Agonistes nous avertit solennellement :« I do not, in general, find amongst the cohorts of professional

historians of early modern philosophy treatment of the same sorts of properly historical questions […] that one finds amongst at least some historians of science […]. I cannot think of more than a small number of occasions when an historiographical insight or problem of some import to this project has been stimulated by such a practitioner [c’est-à-dire un historien de la philosophie]. Nor do I expect that the kinds of categories explored and deployed in this work will be of particular interest or relevance within the empire of history of philosophy » (Schuster 2013, p. viii).

Autrement dit, ce que racontent les historiens de la philosophie n’a pas eu d’influence sur le projet d’Agonistes, parce que ce dernier pose des « questions proprement historiques » ; et, pour la même raison, Agonistes ne devrait en général pas intéresser les historiens de la philosophie. Selon Schuster, non seulement l’histoire des sciences diffère de l’histoire de la philosophie, mais elle lui est supérieure.

En premier lieu, l’histoire de la philosophie telle qu’elle a été pratiquée dans le monde anglophone depuis une cinquantaine d’année aurait en effet à tort transformé Descartes en « an eerily contextless and anonymous author of atomic, isolated, putatively timelessly relevant texts, the Discourse on Method and the Meditations, the Father of Modern Philosophy (epistemology) with whom their philosophy instructors were, apparently, in constant, but critical discussion » (Schuster 2013, p. 3). La limite de cette réponse est cependant que, comme le note d’ailleurs rapidement Schuster, cette situation anglo-saxonne a évolué, en particulier sous l’influence de collègues comme Dan Garber, Roger Ariew, Desmond Clarke ou Peter Machamer.

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L’explication de la supériorité de l’histoire des sciences par rapport à l’histoire de la philosophie a donc une autre source. Elle tient à ce que, selon Schuster, les historiens de la philosophie n’auraient pas placé pas le centre de gravité de l’œuvre de Descartes là où il se trouverait en réalité, à savoir, dans la philosophie naturelle. Cette réponse présente la même limite que la précédente, mais, cette fois, Agonistes ne le signale pas. On remarquera de surcroît qu’il hésite entre deux positions, l’une plus faible et l’autre plus forte.

Selon la plus faible de ces positions, il est possible d’écrire un livre sur Descartes le philosophe naturel sans se préoccuper de sa métaphysique ou plus généralement des parties de sa philosophie qui ne seraient pas liées plus ou moins directement aux sciences. Schuster écrit ainsi : « whatever other scholars and studies may define as the mature Descartes, in this work the mature Descartes is the mature natural philosopher » (Schuster 2013, p. 2), ou encore, « Not every documentable concern of Descartes in the years 1618–1633 will be examined, only those bearing on his agendas, products and arguable self-images or identities in relation to natural philosophy, physico- and mixed mathematics and method » (Schuster 2013, p. 7). Autrement dit, selon cette position faible, il s’agit d’affirmer qu’une perspective centrée sur la philosophie naturelle est légitime, sans exclure pourtant que d’autres perspectives soient possibles.

Selon la plus forte de ces positions en revanche, il est impossible de ne pas écrire un livre sur Descartes le philosophe naturel, parce que Descartes a d’abord et avant tout été un philosophe naturel. Ce n’est alors plus une question de perspective, mais la vérité même de ce que Descartes a été, wie e[r] eigentlich gewesen ist. Schuster écrit ainsi : « Descartes was from beginning to end a natural philosopher » (Schuster 2013, p. 11) ou encore, « Descartes’ education rendered him an adherent of natural philosophical. […] We shall see that it was only in two brief, and unsuccessfully consummated moments, that he ever envisioned leaving behind and marginalizing the culture or, as we shall term it, the field of natural philosophizing culture » (Schuster 2013, p. 36).

Il n’est pas possible de savoir laquelle de ces deux positions Schuster a choisi. L’essentiel est que, en plaçant la philosophie naturelle au cœur de l’entreprise cartésienne, il estime s’opposer de front à l’histoire sans histoire qu’auraient racontée les historiens de la philosophie.

b) On ne trouve pas dans Mathesis de jugement aussi catégorique sur le partage des disciplines. Alors qu’Agonistes consacre ses deux premiers chapitres, c’est-à-dire près de cent pages, à des questions de méthode -- tout en soutenant que la méthode n’est pas à proprement parler un objet épistémologique --, Mathesis garde une réserve de bon aloi sur ces questions. Il est cependant amusant de constater que les quelques mentions qui y sont faites de ce qu’on appellera faute de mieux « la figure de l’historien des

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sciences » dessinent un positionnement disciplinaire exactement opposé à celui d’Agonistes.

La première mention de cette figure se trouve dans le chapitre 1. Deux traditions interprétatives concernant la mathesis universalis de la Règle IV-B y sont opposées : d’une part les « historiens des sciences inspirés peu ou prou par le positivisme » comme Louis Liard, Gaston Milhaud ou Eduard Jan Dijksterhuis, qui estiment que « la mathesis n’est autre chose que l’algèbre “rêvée” de 1619 et se confond ainsi avec la mathématique universelle » ; d’autre part les historiens de la philosophie, c’est-à-dire Ernst Cassirer qui, à la suite de Paul Natorp, « souligne les aspects les plus essentiels de la mathesis universalis » et « note avec exactitude que la méthode des Regulae fonde l’arithmétique et la géométrie » (de Buzon 2013, p. 60, note 64). Les historiens des sciences proposent donc une interprétation fautive là où les historiens de la philosophie ont choisi la bonne interprétation. La suite du même chapitre confirme d’ailleurs cette conclusion : j’y reviendrai, il s’agit de défendre la lecture que Règles avait proposée Jean-Luc Marion.

Selon le deuxième ensemble d’occurrences de cette figure, qui interviennent à la fin du chapitre 4 consacré aux lois de la nature, l’historien des sciences est non seulement celui qui interprète mal la mathesis universalis, mais celui qui, ayant de fausses attentes, projette indûment sur les textes ce qui ne s’y trouve pas : « Descartes n’emploie pas le terme loi là où l’historien des sciences pourrait l’attendre » (de Buzon 2013, p. 192), et surtout : « Les historiens des sciences ont parfois tendance à assimiler la notion de loi de la nature telle qu’elle est présente chez Descartes à des formules qui portent seulement la marque d’une universalité, voire d’une probabilité, permettant d’expliquer des faits déjà connus et d’en prédire de nouveaux » (de Buzon 2013, p. 195). Il convient en effet de rappeler que Descartes appelle « lois » des énoncés caractérisés non seulement par leur universalité, mais par le fait d’avoir été formulés en tant que commandements d’un supérieur -- dans le cas des lois de la nature, en tant que commandements de Dieu. On se demande ici à quels historiens des sciences de Buzon peut bien penser ; de manière étonnante, c’est au manuel de Rudolph Carnap, Les fondements philosophiques de la physique, qu’il se réfère alors.

Finalement, le chapitre 6 consacré à la Dioptrique s’ouvre lui aussi par une confrontation des lectures de l’historien des sciences et de l’historien de la philosophie. « Aux yeux de l’historien des sciences […], il s’agit d’un texte technique sur le fonctionnement des lentilles et leur fabrication, ainsi que sur les lois physiques qui les permettent. Mais, vue par l’historien de la philosophie, la Dioptrique est le texte où Descartes reprend la question fondamentale de la perception visuelle en la généralisant aux autres sens et en lui donnant une explication absolument nouvelle. La Dioptrique est ainsi un moment décisif dans l’évolution de la pensée cartésienne » (de Buzon 2013, p. 221). Là encore, on a une opposition frontale entre l’historien des

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sciences et l’historien de la philosophie ; et, on l’aura compris, la perspective adoptée dans ce chapitre est celle du second de ces personnages.

Contrairement à Agonistes, Mathesis se positionne donc comme un ouvrage écrit par un historien de la philosophie pour les historiens de la philosophie. Il ne s’agit pas ici de prétendre qu’on aurait ainsi identifié les deux points de vue à partir desquels ces deux ouvrages se déploieraient tout entiers : comme tous les ouvrages riches, ils multiplient les points de vue. Mais, comme on va maintenant le montrer, le partage disciplinaire entre histoire des sciences et histoire de la philosophie est important aussi bien pour Mathesis que pour Agonistes.

2. MathesisMathesis a pour cœur deux anciennes questions du commentaire

cartésien, deux questions déjà si anciennes qu’il est bien possible qu’elles ne soient jamais complètement résolues. La première question est de déterminer quel est le rapport entre la physique de Descartes et sa métaphysique ; la seconde est de comprendre ce que Descartes, qui n’a pas mis en place une physique mathématique du même ordre que celle de Galilée, a bien pu vouloir dire en écrivant dans sa lettre à Mersenne du 27 juillet 1638 : « toute ma physique n’est autre chose que géométrie » (AT II 268, cité in de Buzon 2013, p. 13 ; voir également AT I 199, passim).

Comme l’abréviation que j’ai choisie pour désigner cet ouvrage l’indique, la notion de mathesis est centrale pour répondre à ces deux questions. Encore faut-il préciser que, selon de Buzon, la mathesis relève du registre de la fondation métaphysique, cette fondation ayant des effets réels sur ce qui est fondé, en ce sens qu’elle détermine l’objet même de la science cartésienne. Les premiers paragraphes de l’Avant-Propos l’affirment très fermement : « On représente parfois la mathématique cartésienne et la philosophie qui l’accompagne comme constituées définitivement avant l’interrogation métaphysique apparaissant vraisemblablement en 1629-1630, tandis que la fondation métaphysique postérieure des science [sic] semble devoir garantir de l’extérieur la validité de la mathématique dans la connaissance des corps […]. [L]a métaphysique aurait comme seul but de garantir les vérités mathématiques […] et d’assurer la stabilité du monde. […] La question pourrait se résumer en se demandant simplement si la métaphysique de Descartes […] se borne à produire une double garantie […] ou bien si elle a une visée plus forte dans la détermination des objets de connaissance. Comme on peut s’en douter, nous choisissons la deuxième hypothèse, qui implique que la métaphysique cartésienne ait pour fonction de définir, voire de redéfinir, la mathématique et la physique dans leurs premiers concepts » (de Buzon 2013, p. 11-12 ; id., p. 97- 99, résume la première hypothèse en disant qu’elle revient à faire tenir à la métaphysique le rôle

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d’une police d’assurances et l’attribue à Martial Gueroult et à Desmond Clarke).

La seconde précision importante est que, selon de Buzon, il y a non pas une, mais deux notions de mathesis qui jalonnent l’itinéraire de Descartes : d’une part la mathesis universalis de la Règle IV-B ; d’autre part, la pura atque abstracta mathesis de la Cinquième Méditation que prolonge la mathesis abstracta des Principes II 64. Dans l’un et l’autre cas, Descartes reprend des éléments de traditions qu’il modifie.

Dans les Règles, plus exactement dans la Règle IV-B, la mathesis universalis qui interviendrait serait, selon l’expression de Jean-Luc Marion que Mathesis reprend à son compte, une « “méta-mathématique non–mathématique”, par la position d’une “mathématicité non-mathématique des mathématiques” » (p. 68), ou encore, plus simplement, « une fondation philosophique de la mathématique » (p. 72, voir également p. 94). La mathesis universalis ainsi comprise, en donnant un fondement à toutes les disciplines mathématiques, principalement aux mathématiques pures (arithmétique et géométrie, ayant pour objet respectifs les nombres et les grandeurs continues) mais aussi de manière dérivée, aux mathématiques mixtes (musique, optique, mécanique et astronomie).

Contrairement à la mathesis universalis, la pura atque abstracta mathesis qui intervient au terme de la Cinquième Méditation aurait pour objet une grandeur continue la figure et surtout le mouvement, le nombre disparaissant de son orbite (de Buzon 2013, p. 13, p. 117, p. 123–124). Ce changement d’objet serait confirmé aussi bien dans la lettre à Ciermans du 23 mars 1638 présentant le mouvement comme le principal objet de la mathematica pura que par la mathesis abstracta des Principes, II 64. D’où viendrait la raison pour laquelle « la physique de Descartes ne se sert pas de la mathématique comme d’un simple instrument intellectuel » (de Buzon 2013, p. 176), la physique toute entière géométrique de Descartes n’étant en rien « une physique mathématique au sens moderne du terme » (de Buzon 2013, p. 126).

Pour étayer cette dernière affirmation, Mathesis se réfère à un passage de Cassirer. Traduisons ce dernier intégralement: « [La signification [du concept de grandeur] n’est pas épuisée en ce qu’il donne une unité aux approches purement mathématiques. Ce concept découle également de la “mathématique universelle”, qui, à côté de la géométrie et de l’arithmétique, comprend tout le domaine de la connaissance du réel – avant tout la mécanique, l’astronomie et la physique.] Le concept systématique des mathématiques est tout aussi bien le concept systématique des sciences de la nature. C’est guidée par cette pensée que la physique de Descartes se constitue. [Son progrès décisif ne vient pas de ce qu’elle utilise les mathématiques comme un moyen pour ordonner et pour identifier les processus naturels qui sont supposés réels.] Les mathématiques sont pour Descartes plus qu’un instrument exact logiquement pour dominer en pensée

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la réalité présente. Elles constituent la supposition que nous devons poser au fondement pour définir la réalité en général. La “nature” n’est pas un être qui préexiste indépendamment de la connaissance ; c’est un concept qui doit être d’abord être déterminé à partir des conditions de la connaissance. C’est dans la poursuite radicale de cette pensée que réside l’originalité proprement philosophique de la physique de Descartes. Selon cette dernière, ce qui est réel, c’est ce qui obéit à la condition de pouvoir être exactement connu » (Cassirer 1902, p. 16–17, cité, à l’exception de ce qui est entre crochets, in de Buzon 2013, p. 176).

Outre que Cassirer ne parle pas ici de la pura atque abstracta mathesis des Méditations, mais de la mathesis universalis des Règles, son éclairage semble de prime abord trop kantien. Mais selon de Buzon, ce qu’écrit Cassirer doit être pris tout à fait au sérieux car « on voit mal comment ne pas donner un rôle transcendantal à la mathesis » (de Buzon 2013, p. 177). C’est que, de même que Cassirer estime que les mathématiques constituent la nature cartésienne, de même de Buzon juge que la pura atque abstracta mathesis des Méditations détermine l’objet de la science cartésienne. L’application du terme « transcendantal » à la notion de mathesis indique ainsi l’importance de cette notion comme le caractère décisif de la transformation fondamentale qu’elle connaît dans les Méditations, dont le passage de « universalis » à « pura atque abstracta » serait le signe. On ne s’étonnera dès lors pas que les réponses données par de Buzon aux deux questions initiales s’enracinent dans cette transformation de la mathesis. En premier lieu, avec l’élaboration de la pura atque abstracta mathesis, le rapport de la métaphysique à la physique devient un rapport de fondation actif, un rapport grâce auquel la métaphysique détermine l’objet de la physique. En second lieu, c’est en tant que l’objet de la physique est une quantité continue douée de figure et de mouvement que la physique de Descartes est géométrique, et seulement en tant que cela.

La reconstruction proposée par Mathesis est une reconstruction magistrale. Comme toute reconstruction toutefois, elle peut laisser quelques doutes. On en distinguera trois, qu’on présentera du moins important au plus important.

Le premier doute vient de ce que de Buzon n’est pas le seul à avoir proposé une distinction entre deux formes de mathesis. À ma connaissance, le premier à l’avoir fait est Léon Brunschvicg, dans un article où il commente l’édition qu’avait donnée Étienne Gilson du Discours de la méthode. Gilson avait suivi Milhauld qui, faisant l’hypothèse d’une continuité entre les Règles et le Discours de la méthode, avait affirmé que la mathesis des premières correspond à la méthode du second (Milhaud 1921). Brunschvicg oppose à Gilson que la différence est grande entre la mathesis universalis et la mathesis pura : alors que la première, étant du côté de la géométrie de l’espace, serit liée la physique, la seconde serait avant tout une algèbre. Bien plus, ce serait

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une réflexion sur des problèmes mathématiques, en particulier le problème de Pappus, qui déterminerait le passage de l’une à l’autre, de sorte que l’évolution des concepts philosophiques proviendrait chez Descartes d’une modification des concepts mathématiques (Brunschvicg 1927, p. 280, p. 285–286). Je ne crois pas que la lecture de Brunschvicg soit préférable à celle de Mathesis : néanmoins, il me semble qu’elle devrait être discutée, ne serait-ce que parce qu’elle donne l’exemple d’une autre manière de penser la distinction entre mathesis universalis et mathesis pura, et de situer l’origine de l’évolution de Descartes sur ce point.

Le second doute vient peut-être plus d’une différence de tempérament philosophique que de ce qu’on pourrait effectivement prouver et objectiver, mais, de cette différence également, il faut tenir compte. Il est certain selon moi qu’il existe des changements entre les Règles et les textes que Descartes a ultérieurement consacrés à la physique. Mais ces changements dépendent-ils seulement ou même principalement du travail du texte de la Cinquième Méditation ? Et le passage de l’expression « mathesis universalis » à l’expression « pura atque abstracta mathesis » est-il à ce point significatif ? J’aurais pour ma part tendance à penser que la physique de Descartes a toujours été un peu plus indépendante de sa métaphysique qu’il ne l’aurait souhaité ; et que, s’il a toujours écrit avec une grande maîtrise, il est arrivé qu’il n’ajuste pas parfaitement les uns aux autres les ensembles lexicaux complexes qu’il manipulait.

Le troisième et dernier doute tient à la considération de l’ensemble de l’ouvrage. Une fois affirmé que la mise en place de la pura atque abstracta mathesis constitue le moment proprement cartésien, Mathesis l’applique dans différents domaines ou la décline à propos de différents objets. Mais il faut bien dire que les modalités d’application ou de déclinaison sont alors très variables. Si la pura atque abstracta mathesis est déterminante dans le chapitre sur les lois de la nature, puisque c’est elle qui fait que Descartes, qui donne la prééminence au mouvement par rapport à la forme, est plus physicien que Beeckman (de Buzon 2013, p. 175–176, p. 188–189), elle n’intervient pas du tout dans les chapitres intitulés « Image et imagination » et « Figures et pensées », d’autant que le second porte sur la Dioptrique, antérieure à l’élaboration de cette nouvelle mathesis. Quant aux problèmes de l’atomisme, de l’individualité des corps et des passions, qui sont traités dans les derniers chapitres, ils ne sont pas seulement étrangers à la mathesis pura, mais à la mathesis en général. On pourrait se dire que cette relative distension du lien entre la mathesis et ses objets vient du mode de composition de l’ouvrage, qui rassemble plusieurs articles. Mais peut-être aussi la distension est-elle plus profonde : même une fois déterminé en général l’objet de la physique, la plus grande partie du travail reste à faire, ce qui diminue le rôle que peut avoir la détermination de son objet par la métaphysique. De sorte que, si tous ces chapitres présentent un intérêt propre aussi bien que des interprétations

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exactes et stimulantes, il n’était peut-être pas nécessaire de les enrégimenter sous la bannière de la mathesis.

3. AgonistesIl est plus difficile de dégager les questions centrales du livre touffu

qu’est Agonistes que de les dégager dans le cas de Mathesis. On peut néanmoins s’appuyer sur le titre, un titre intriguant pour ceux qui connaissent les thèses de mathématiques que les jésuites du collège de Clermont faisaient défendre dans les années 1630-1650 sous le nom générique d’Agones mathematici ou d’Exercitatio mathematica ad agones panegyricos. Il ne semble toutefois pas qu’Agonistes contienne quelque allusion que ce soit au contexte jésuite. L’idée est plutôt, comme l’explique longuement les deux premiers chapitres, que, dans ce qu’il faut bien appeler en termes bourdieusiens le champ particulièrement compétitif de la philosophie naturelle, et à un moment où ce champ était l’objet d’une remise en cause particulièrement aigüe (on trouve aux pages 77-80 d’Agonistes des expressions comme celles de « critical period », « unique conjuncture », « desperate competition », « climatic struggle »), Descartes a été, sinon ce cavalier français qui selon Charles Péguy était parti d’un si bon pas, du moins un compétiteur audacieux, comme l’affirme un passage qui, à ma connaissance, est le seul à suggérer une explication du titre : « No wonder René Descartes, as a radical and bold player in the natural philosophical contest of the age, in our view deserves the epithet, Descartes agonistes » (Schuster 2013, p. 82).

Descartes s’est donc battu, mais pour quoi exactement ? La réponse donnée à cette question par Agonistes est totalement différente de celle que donne Mathesis, ne serait-ce que pour des questions de date. Alors que, pour Mathesis, le tournant essentiel a lieu quand s’écrivent les Méditations, pour Agonistes, il a lieu en 1629 ; c’est la raison pour laquelle, alors que le premier de ces ouvrages porte principalement sur les œuvres publiées, le second s’arrête avec la publication du Monde. De surcroît, ce tournant n’a pas pour cause une conquête nouvelle, mais, plutôt, l’abandon et l’échec, l’effritement ou l’effondrement – si Mathesis raconte l’histoire d’une découverte et d’une conquête, Agonistes, c’est un peu Les illusions perdues : Descartes en viendrait à faire de la métaphysique et à mettre sa physique en système après avoir échoué partout ailleurs.

Agonistes brosse en effet un contraste saisissant entre deux périodes cartésiennes. Jusqu’en 1629, c’est-à-dire jusqu’à la deuxième rédaction des Règles, le jeune Descartes est un philosophie naturel se proposant de réformer les sciences mixtes de l’intérieur mais aussi un mathématicien qui en est venu à entretenir deux rêves qui seront ensuite abandonnés, le rêve d’une mathesis universalis, le rêve d’une méthode générale. Après 1629, l’impuissance à réaliser ces rêves, ainsi que d’autres événements qui sont mentionnés mais ne semblent pas être considérés comme déterminants (Schuster 2013, p. 384 sqq.),

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conduisent Descartes à embrasser un nouveau projet : il cherche désormais à construire un système de philosophie naturelle, en expliquant absolument tous les phénomènes en termes causaux. Dans ce qui suit, je me consacre au début de la première de ces périodes, c’est-à-dire aux chapitres 3–5, en posant les questions au fur et à mesure qu’elles se présentent.

Dans le chapitre 3, Agonistes commence par souligner la différence entre les sciences mixtes de la tradition et la manière dont Descartes a pratiqué des exercices « physico-mathématiques » quand il était en compagnie de Beeckman en 1618-1619. Indépendamment de ce qu’inclut la catégorie de physico-mathématique, « a vague, but trendy concept » (Schuster 2013, p. 599), Schuster montre que, dans ces exercices, Descartes a souhaité expliquer en termes physiques certains résultats déjà connus et complètement décrits mathématiquement. L’analyse du manuscrit portant sur le paradoxe hydrostatique de 1619 est ici exemplaire. Le paradoxe hydrostatique est une reformulation de la loi selon laquelle la pression exercée par un liquide sur le fond du vase dans lequel il se trouve dépend seulement de la hauteur de ce liquide et de la surface du fond du vase. Reformulée sous forme de paradoxe, cela revient à dire que, quelles que soient les formes biscornues des vases considérés, la pression exercée au fond des vases est la même à la condition que la hauteur du liquide et la surface du fond du vase soient identiques. Stevin avait donné une démonstration par l’absurde de cette loi. Descartes propose quant à lui une démonstration physique de cette loi, consistant à appliquer à des atomes pesants des concepts mécaniques, en particulier des concepts impliquant des tendances :

« For Stevin’s formally rigorous and conclusive geometrical demonstration, Descartes substitutes a very different kind of account. Descartes did not and could not have denied the rigor of Stevin’s account. If he was conceding rigor to Stevin’s analysis, what was Descartes seeking to accomplish? The answer is that he was seeking proper explanation, meaning explanation in terms of natural philosophy or physics. For all its oddness to us, this little exercise seemed to Descartes to bespeak the possibility of some new sort of agenda in the mixed mathematical science of hydrostatics and between it and a corpuscular or atomistic natural philosophy, or more generally between the mixed mathematical sciences, plural, and a new kind of natural philosophical discourse - atomist and mechanist » (Schuster 2013, p. 119-120).

Cette analyse est également appliquée aux fragments bien connus concernant la loi de la chute des corps, dont Agonistes renouvelle l’interprétation en montrant les limites qu’atteint alors Descartes, ce dernier ne pouvant pas trouver dans les figures qu’il dessine les causes qu’il suppose (Schuster 2013, p. 130) et hésitant entre plusieurs lois possibles (Schuster 2013, p. 152). Elle est également appliquée à un fragment moins bien connu

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concernant la lumière, dont l’analyse sera reprise dans la reconstruction magistrale de la découverte de la loi de la réfraction qui fait l’objet du chapitre 4. L’essentiel est que, dès le début, Agonistes dégage plusieurs fils directeurs qu’il suivra ensuite. Un de ces fils est l’idée que les tendances instantanées dans des milieux fluides ne sont pas un élément explicatif parmi d’autres, mais l’élément absolument central de la physique cartésienne (Schuster 2013, p. 119 sqq.). L’objection est cependant qu’à l’évidence, on trouve d’autres choses dans cette physique, par exemple des lois du choc et des corpuscules figurés.

Un autre fil directeur est que cette physique vise, non à donner une description mathématique phénoménale, mais à rendre compte de résultats mathématiques en recourant à des causes microscopiques. Dans ces conditions, un des leitmotiv de l’ouvrage est que, plutôt que dire que Descartes a mathématisé la physique, on devrait dire qu’il a physicalisé les mathématiques : alors que les mathématiques mixtes traditionnelles ne s’engageaient pas sur les questions ontologiques et aïtologiques, Descartes les y aurait introduites, les transformant de l’intérieur (Schuster 2013, p. 12, p. 56 sqq., p. 592, passim). Que les mathématiques soient physicalisées ou que la physique soit mathématisée, il n’y a plus beaucoup de place, pour une physique sans mathématique ; en d’autres termes, la physico-mathématique cesse d’être subordonnée à la philosophie naturelle et acquiert une véritable autonomie (Schuster 2013, p. 102). S’il y a un avantage à parler de physicalisation des mathématiques, c’est à mon sens qu’on comprend mieux le rôle qu’on pu avoir les sciences mixtes de la tradition aristotélicienne dans la constitution de la physique cartésienne, sans pourtant y réduire cette dernière.

Mais le jeune Descartes ne se contente pas de physicaliser les mathématiques. C’est également un mathématicien, et, d’après le chapitre 5 d’Agonistes, un mathématicien conduit par sa pratique à deux excroissances théoriques, d’une part, au milieu de l’année 1619, la mathesis universalis de la Règle 4-B, qui ne concernerait que les disciplines mathématiques, d’autre part, vers la mi-novembre 1619, le projet d’une méthode absolument universelle, correspondant à la Règle 4-A.

Agonistes présente le projet de mathesis universalis comme une extension malheureuse de la mathématique analytique de Descartes. Mais, alors que le détail des textes de physique est examiné de très près, seulement 4 pages sont consacrées aux mathématiques effectives pratiquées par Descartes, en l’occurrence à ses compas, c’est-à-dire dispositifs matériels permettant de résoudre des problèmes de proportions. La question est donc simple : si on veut comprendre un peu mieux la mathesis universalis, ne faut-il pas en revenir aux mathématiques cartésiennes et produire à leur propos une reconstitution analogue à celle que propose Agonistes à propos de la

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physique ? Ou, d’une manière un peu différente : une fois admis que la mathesis universalis a été un horizon qui est toujours resté un peu lointain, faut-il rapporter cet horizon à l’histoire des discours sur les mathématiques (particulièrement les classifications scolaires des disciplines mathématiques et les prologues faisant l’éloge des mathématiques) ou bien plutôt à l’histoire des mathématiques elles-mêmes ?

Concernant la méthode, la thèse d’Agonistes se situe entre la thèse positiviste selon laquelle la méthode guiderait l’enquête scientifique, voire la justifierait, et la thèse constructiviste, selon laquelle la méthode serait de part en part idéologique : cette thèse est que tout discours de la méthode est un discours mythique qui a une fonction « politique » dans l’organisation des sciences (Schuster 2013, p. 293 sqq.). Là encore, je dois reconnaître ma perplexité. Nous savons qu’il y a une politique des sciences, souvent terriblement agressive par les temps qui courent. Nous savons aussi que le Discours de la méthode comporte dans ses dernières parties une réflexion sur la communication de la philosophie et un appel aux pouvoirs du temps, en particulier pour organiser les pratiques expérimentales qu’un particulier ne peut à lui seul prendre en charge. On peut bien appeler cela « politique des sciences » ; mais on peut se demander si le lien avec la méthode que la deuxième partie décrit comme la prise de conscience progressive de la démarche d’un esprit singulier est si direct qu’on puisse effectivement en parler comme d’une politique des sciences.

RéférencesBrunschvicg, Léon, « Mathématique et Métaphysique chez Descartes », Revue

de métaphysique et de morale, 34, 1927, pp. 277-324Cassirer, Ernst, Leibniz’s System in seinen wissenschaftlichen Grundlagen,

Marburg, N.G. Elwert’sche Verlagsbuchhandlung, 1902.Milhaud, Gaston, Descartes savant, Paris, Alcan, 1921.

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