- Descartes: methodology
- Descartes: methodology Stephen Gaukroger INTRODUCTION The seventeenth century is often referred to as the century of the Scientific Revolution, a time of fundamental scientific change in which traditional theories were either replaced by new ones or radically transformed. Descartes made contributions to virtually every scientific area of his day. He was one of the founders of algebra, he discovered fundamental laws in geometrical optics, his natural philosophy was the natural philosophy in the seventeenth century before the appearance of Newton’s Principia (Newton himself was a Cartesian before he developed his own natural philosophy) and his work in biology and physiology resulted, amongst other things, in the discovery of reflex action.1 Descartes’s earliest interests were scientific, and he seems to have thought his scientific work of greater importance than his metaphysical writings throughout his career. In a conversation with Burman, recorded in 1648, he remarked: A point to note is that you should not devote so much effort to the Meditations and to metaphysical questions, or give them elaborate treatment in commentaries and the like. Still less should one do what some try to do, and dig more deeply into these questions than the author did: he has dealt with them all quite deeply enough. It is sufficient to have grasped them once in a general way, and then to remember the conclusion. Otherwise they draw the mind too far away from physical and observable things, and make it unfit to study them. Yet it is just these physical studies that it is most desirable for men to pursue, since they would yield abundant benefits for life.2 Despite this, Descartes has often been considered a metaphysician in natural philosophy, deriving physical truths from metaphysical first principles. Indeed, there is still a widespread view that the ‘method’ Descartes espoused is the a priori one of deduction from first principles, where these first principles are truths of reason. This view has two principal sources: an image of Descartes as the de facto founder of a philosophical school—‘rationalism’—in which deduction from truths of reason is, almost by definition, constitutive of epistemology, and a reading of a number of passages in Descartes in which he discusses his project in highly schematic terms as accounts of his method of discovery. The reading of Descartes as founder of a school is largely a nineteenth-century doctrine first set out in detail in Kuno Fischer’s Geschichte der neueren Philosophie in the 1870s. There is a hidden agenda in Fischer which underlies this: he is a Kantian and is keen to show Kant’s philosophy as solving the major problems of modern thought. He sets the background for this by resolving modern preKantian philosophy into two schools, rationalism and empiricism, the first basing everything on truths of reason, the second basing everything on experiential truths. This demarcation displaces the older Platonist/ Aristotelian dichotomy (which Kant himself effectively worked with), marking out the seventeenth century as the beginning of a new era in philosophy, one dominated by epistemological (as opposed to moral or theological) concerns.3 This reading of Descartes is not wholly fanciful, and it has been widely accepted in the twentieth century by philosophers who do not share Fischer’s Kantianism. On the face of it, it has considerable textual support. Article 64 of Part II of Descartes’s Principles of Philosophy is entitled: That I do not accept or desire in physics any principles other than those accepted in geometry or abstract mathematics; because all the phenomena of nature are explained thereby, and demonstrations concerning them which are certain can be given. In elucidation, he writes: For I frankly admit that I know of no material substance other than that which is divisible, has shape, and can move in every possible way, and this the geometers call quantity and take as the object of their demonstrations. Moreover, our concern is exclusively with the division, shape and motions of this substance, and nothing concerning these can be accepted as true unless it be deduced from indubitably true common notions with such certainty that it can be regarded as a mathematical demonstration. And because all natural phenomena can be explained in this way, as one can judge from what follows, I believe that no other physical principles should be accepted or even desired.4 This seems to be as clear a statement as one could wish for of a method which starts from first principles and builds up knowledge deductively. Observation, experiment, hypotheses, induction, the development and use of scientific instruments, all seem to be irrelevant to science as described here. Empirical or factual truths seem to have been transcended, and all science seems to be in the realm of truths of reason. One gains a similar impression from a number of other passages in Descartes, and in the Sixth Meditation, for example, we are presented with a picture of the corporeal world in which—because we are only allowed to ask about the existence of those things of which we have a clear and distinct idea, and because these are effectively restricted to mathematical concepts—it is little more than materialized geometry. The deduction of the features of such a world from first principles is not too hard to envisage. Nevertheless, there are serious problems with the idea that Descartes is advocating an a priori, deductivist method of discovery, and I want to draw attention to four such problems briefly. First, there is the sheer implausibility of the idea that deduction from first principles could generate substantive and specific truths about the physical world. The first principles that Descartes starts from are the cogito and the existence of a good God. These figure in the Meditations and in the Principles of Philosophy as explicit first principles. Now by the end of the Principles of Philosophy he has offered accounts of such phenomena as the distances of the planets from the Sun, the material constitution of the Sun, the motion of comets, the colours of the rainbow, sunspots, solidity and fluidity, why the Moon moves faster than the Earth, the nature of transparency, the rarefaction and condensation of matter, why air and water flow from east to west, the nature of the Earth’s interior, the nature of quicksilver, the nature of bitumen and sulphur, why the water in certain wells is brackish, the nature of glass, magnetism, and static electricity, to name but a few. Could it seriously be advocated that the cogito and the existence of a good God would be sufficient to provide an account of these phenomena? Philosophers, like everyone else, are occasionally subject to delusions, and great claims have been made for various philosophically conceived scientific methods. But it is worth remembering in this connection that Descartes is of a generation where method is not a reflection on the successful work of other scientists but a very practical affair designed to guide one’s own scientific practice. It is also worth remembering that Descartes achieved some lasting results in his scientific work. In the light of this, there is surely something wrong in ascribing an unworkable methodology to him. Second, Descartes’s own contemporaries did not view his work as being apriorist and deductivist, but rather as being committed to a hypothetical mode of reasoning, and, in the wake of Newton’s famous rejection of the use of hypotheses in science, Descartes was criticized for offering mere hypotheses where Newtonian physics offered certainty.5 A picture of Descartes prevailed in his own era exactly contrary to that which has prevailed in ours, and, it might be noted, on the basis of the same texts, that is, above all the Discourse on Method, the Meditations and The Principles of Philosophy. The possibility must therefore be raised that we have misread these texts. Third, if one looks at Descartes’s very sizeable correspondence, the vast bulk (about 90 per cent) of which is on scientific matters, one is left in no doubt as to the amount of empirical and experimental work in which he engaged. For example, in 1626 Descartes began seeking the shape of the ‘anaclastic’, that is, that shape of a refracting surface which would collect parallel rays into one focus. He knew that the standard lens of the time, the biconvex lens, could not do this, and that the refracting telescope constructed with such a lens was subject to serious problems as a result. He was convinced on geometrical grounds that the requisite shape must be a hyperbola but he spent several years pondering the practical problems of grinding aspherical lenses. In two detailed letters to Jean Ferrier (a manufacturer of scientific instruments whom Descartes was trying to attract to Holland to work for him grinding lenses) of October and November 1629, he describes an extremely ingenious grinding machine, with details as to the materials different parts must be constructed of, exact sizes of components, instructions for fixing the machine to rafters and joists to minimize vibration, how to cut the contours of blades, differences between rough-forming and finishing-off, and so on. These letters leave one in no doubt that their author is an extremely practical man, able to devise very large-scale machines with many components, and with an extensive knowledge of materials, grinding and cutting techniques, not to mention a good practical grasp of problems of friction and vibration. The rest of his large correspondence, whether it be on navigation, acoustics, hydrostatics, the theory of machines, the construction of telescopes, anatomy, chemistry or whatever, confirm Descartes’s ability to devise and construct scientific instruments and experiments. Is it really possible that Descartes’s methodological prescriptions should be so far removed from his actual scientific practice? Fourth, Descartes has an extremely low view of deduction. He rejects Aristotelian syllogistic, the only logical formalization of deductive inference he would have been familiar with, as being incapable of producing any new truths, on the grounds that the conclusion can never go beyond the premises: We must note that the dialecticians are unable to devise by their rules any syllogism which has a true conclusion, unless they already have the whole syllogism, i.e. unless they have already ascertained in advance the very truth which is deduced in that syllogism.6 Much more surprisingly, for syllogistic was generally reviled in the seventeenth century, he also rejects the mode of deductive inference used by the classical geometers, synthetic proof. In Rule 4 of the Rules for the Direction of Our Native Intelligence, he complains that Pappus and Diophantus, ‘with a kind of low cunning’, kept their method of discovery secret, presenting us with ‘sterile truths’ which they ‘demonstrated deductively’.7 This is especially problematic for the reading of Descartes which holds that his method comprises deduction from first principles, since this is exactly what he is rejecting in the case of Pappus and Diophantus, who proceed like Euclid, working out deductively from indubitable geometrical first principles. Indeed, such a procedure would be the obvious model for his own method if this were as the quotations above from the Principles suggest it is. These considerations are certainly not the only ones, but they are enough to make us question the received view. Before we can provide an alternative reading, however, it will be helpful if we can get a better idea of what exactly Descartes is rejecting in traditional accounts of method, and what kind of thing he is seeking to achieve in his scientific writings. THE REJECTION OF ARISTOTELIAN METHOD Aristotelian syllogistic was widely criticized from the mid-sixteenth century onwards, and by the middle of the seventeenth century had been completely discredited as a method of discovery. This was due more to a misunderstanding of the nature and role of the syllogism, however, than to any compelling criticism of syllogistic. Aristotle had presented scientific demonstrations syllogistically, and he had argued that some forms of demonstration provide explanations or causes whereas others do not. This may occur even where the syllogisms are formally identical. Consider, for example, the following two syllogisms: The planets do not twinkle That which does not twinkle is near The planets are near The planets are near That which is near does not twinkle The planets do not twinkle In Aristotle’s discussion of these syllogisms in his Posterior Analytics (A17), he argues that the first is only a demonstration ‘of fact’, whereas the second is a demonstration of ‘why’ or a scientific explanation. In the latter we are provided with a reason or cause or explanation of the conclusion: the reason why the planets do not twinkle is that they are near. In the former, we have a valid but not a demonstrative argument, since the planets’ not twinkling is hardly a cause or explanation of their being near. So the first syllogism is in some way uninformative compared with the second: the latter produces understanding, the former does not. Now Aristotle had great difficulty in providing a convincing account of what exactly it is that distinguishes the first from the second syllogism, but what he was trying to achieve is clear enough. He was seeking some way of identifying those forms of deductive inference that resulted in epistemic advance, that advanced one’s understanding. Realizing that no purely logical criterion would suffice, he attempted to show that epistemic advance depended on some non-logical but nevertheless internal or structural feature which some deductive inferences possess. This question of the epistemic value of deductive inferences is one we shall return to, as it underlies the whole problem of method. For Aristotle, the epistemic and the consequential directions in demonstrative syllogisms run in opposite directions. That is, it is knowing the premises from which the conclusion is to be deduced that is the important thing as far as providing a deeper scientific understanding is concerned, not discovering what conclusions follow from given premises. The seventeenth-century misunderstanding of the syllogism results largely from a failure to appreciate this. It was assumed that, for Aristotle, the demonstrative syllogism was a method of discovery, a means of deducing novel conclusions from accepted premises. In fact, it was simply a means of presentation of results in a systematic way, one suitable for conveying these to students.8 The conclusions of the syllogisms were known in advance, and what the syllogism provided was a means of relating those conclusions to premises which would explain them. Two features of syllogistic are worth noting in this respect. First, the syllogism is what might be termed a ‘discursive’ device. Consider the case of the demonstrative syllogism. This is effectively a pedagogic device, involving a teacher and a pupil. If it is to be successful, the pupil must accept the conclusion: the conclusion having been accepted, the syllogism shows how it can be generated from premises which are more fundamental, thereby connecting what the pupil accepts as knowledge to basic principles which can act as an explanation for it. This reflects a basic feature of the syllogism, whether demonstrative or not. Generally speaking, syllogistic works by inducing conviction on the basis of shared assumptions or shared knowledge. For this, one needs someone who is convinced and someone who does the convincing. Moreover, the conviction occurs on the basis of shared assumptions, and these assumptions may in fact be false. The kind of process that Descartes and his contemporaries see as occurring in an argument is different from this. Descartes, in particular, requires that arguments be ‘internal’ things: their purpose is to lead one to the truth, not to convince anyone. Correlatively, one can make no appeal to what is generally accepted: one’s premises must simply be true, whether generally accepted or not.9 The discursive conception of argument that Aristotelian syllogistic relies upon requires common ground between oneself and one’s opponents, and in seventeenth-century natural philosophy this would not have been forthcoming. In other words, the case against conceiving of inference in a discursive way links up strongly with the case against appealing in one’s enquiries to what is generally accepted rather than to what is the case. It is from this that the immense polemical strength of Descartes’s attack on syllogistic derives. Second, conceived as a tool of discovery, the demonstrative syllogism does indeed look trivial, but this was never its purpose for Aristotle. Discovery was something to be guided by the ‘topics’, which were procedures for classifying or characterizing problems so that they could be solved using set techniques. More specifically, they were designed to provide the distinctions needed if one was to be able to formulate problems properly, as well as supplying devices enabling one to determine what has to be shown if the conclusion one desires is to be reached. Now the topics were not confined to scientific enquiry, but had an application in ethics, political argument, rhetoric and so on, and indeed they were meant to apply to any area of enquiry. The problem was that, during the Middle Ages, the topics came to be associated very closely and exclusively with rhetoric, and their relevance to scientific discovery became at first obscured and then completely lost. The upshot of this was that, for all intents and purposes, the results of Aristotelian science lost all contact with the procedures of discovery which produced them. While these results remained unchallenged, the problem was not particularly apparent. But when they came to be challenged in a serious and systematic way, as they were from the sixteenth century onwards, they began to take on the appearance of mere dogmas, backed up by circular reasoning. It is this strong connection between Aristotle’s supposed method of discovery and the unsatisfactoriness not only of his scientific results but also of his overall natural philosophy that provoked the intense concern with method in the seventeenth century.10 DESCARTES’S NATURAL PHILOSOPHY Descartes’s account of method is intimately tied to two features of his natural philosophy: his commitment to mechanism, and his commitment to the idea of a mathematical physics. In these respects, his project differs markedly from that of Aristotle, to the extent that what he expects out of a physical explanation is rather different from what Aristotle expects. This shapes his approach to methodological issues to a significant extent, and we must pay some attention to both questions. Mechanism is not an easy doctrine to characterize, but it does have some core theses. Amongst these are the postulates that nature is to be conceived on a mechanical model, that ‘occult qualities’ cannot be accepted as having any explanatory value, that contact action is the only means by which change can be effected, and that matter and motion are the ultimate ingredients in nature.11 Mechanism arose in the first instance not so much as a reaction to scholasticism but as a reaction to a philosophy which was itself largely a reaction to scholasticism, namely Renaissance naturalism.12 Renaissance naturalism undermined the sharp and careful lines that medieval philosophy and theology had drawn between the natural and the supernatural, and it offered a conception of the cosmos as a living organism, as a holistic system whose parts were interconnected by various forces and powers. Such a conception presented a picture of nature as an essentially active realm, containing many ‘occult’ powers which, while they were not manifest, could nevertheless be tapped and exploited if only one could discover them. It was characteristic of such powers that they acted at a distance —magnetic attraction was a favourite example—rather than through contact, and indeed on a biological model of the cosmos such a mode of action is a characteristic one, since parts of a biological system may affect one another whether in physical contact or not. The other side of the coin was a conception of God as part of nature, as infused in nature, and not as something separate from his creation. This encouraged highly unorthodox doctrines such as pantheism, the modelling of divine powers on natural ones, and so on, and, worst of all, it opened up the very delicate question of whether apparently supernatural phenomena, such as miracles, or phenomena which offered communion with God, such as the sacraments and prayer, could receive purely naturalistic explanations.13 It is important that the conception of nature as essentially active and the attempts to subsume the supernatural within the natural be recognized as part of the same problem. Mechanists such as Mersenne14 saw this clearly, and opposed naturalism so vehemently because they saw its threat to established religion. Mersenne himself also saw that a return to the Aristotelian conception of nature that had served medieval theologians and natural philosophers so well was not going to be successful, for many of the later Renaissance naturalists had based their views on a naturalistic reading of Aristotle which, as an interpretation of Aristotle, was at least as cogent as that offered by Christian apologists like Aquinas. The situation was exacerbated by a correlative naturalistic thesis about the nature of human beings, whereby the soul is not a separate substance but simply the organizing principle of the body. Whether this form of naturalism was advocated in its Averroistic version, where there is only one intellect in the universe because mind or soul, lacking any principle of individuation in its own right, cannot be divided up with the number of bodies, or whether it was advocated in its Alexandrian version, where the soul is conceived in purely functional terms, personal immortality is denied, and its source in both versions is Aristotle himself.15 Such threats to the immortality of the soul were noted by the Fifth Lateran Council, which in 1513 instructed Christian philosophers and theologians to find arguments to defend the orthodox view of personal immortality. Something was needed which was unambiguous in its rejection of nature as an active realm, and which thereby secured a guaranteed role for the supernatural. Moreover, whatever was chosen should also be able to offer a conception of the human body and its functions which left room for something essentially different from the body which distinguished it from those animals who did not share in immortality. Mechanism appeared to Mersenne to answer both these problems, and in a series of books in the 1620s he opposes naturalism in detail and outlines the mechanist response.16 Descartes’s response is effectively the same, but much more radical in its execution. Like Mersenne, he is concerned to defend mechanism, and the idea of a completely inert nature provides the basis for dualism. Dualism in turn is what he considers to be the successful way of meeting the decree of the Lateran Council, as he explains to us in the Dedicatory Letter at the beginning of the Meditations. What he offers is a picture of the corporeal world as completely inert, a separation of mind and body as radical as it could be, and an image of a God who is so supernatural, so transcendent, that there is almost nothing we can say about him. In accord with the mechanist programme, the supernatural and the natural are such polar opposites in Descartes that there is no question of the latter having any degree of activity once activity has been ascribed to the former. His contemporaries were especially concerned to restrict all causal efficacy to the transfer of motion from one body to another in impact but Descartes goes further. While he recognizes the need for such a description at the phenomenal level, at the metaphysical level he does not allow any causation at all in the natural realm.17 Motion is conceived as a mode of a body just as shape is, and strictly speaking it is not something that can be transferred at all, any more than shape can be. The power that causes bodies to be in some determinate state of rest or motion is a power that derives exclusively from God, and not from impact with other bodies. Moreover, this power is simply the power by which God conserves the same amount of motion that he put in the corporeal world at the first instant.18 On Descartes’s account of the persistence of the corporeal world, God is required to recreate it at every instant, because it is so lacking in any power that it does not even have the power to conserve itself in existence. As he puts it in the replies to the first set of objections to the Meditations, we can find in our own bodies, and by implication in other corporeal things as well, no power or force by which they could produce or conserve themselves. Why, one may ask, is such a force or power required? The answer is that causes and their effects must be simultaneous: ‘the concept of cause is, strictly speaking, applicable only for as long as it is producing its effect, and so is not prior to it.’19 My existence at the present instant cannot be due to my existence at the last instant any more than it can itself bring about my existence at the next instant. In sum, there can be no causal connections between instants, so the reason for everything must be sought within the instant:20 and since no such powers are evident in bodies, they must be located in God. Such an inert corporeal world certainly contains none of the powers that naturalists saw as being immanent in nature, it is not a world in which God could be immanent, and it is, for Descartes and virtually all of his contemporaries (Hobbes and Gassendi being possible exceptions), a world quite distinct from what reflection on ourselves tells us is constitutive of our natures, which are essentially spiritual. And it also has another important feature: a world without forces, activities, potentialities and even causation is one which is easily quantifiable. This brings us to the second ingredient in Descartes’s natural philosophy: his commitment to quantitative explanations. This is often seen as if it were a necessary concomitant of mechanism, but in fact mechanism is neither sufficient nor necessary for a mathematical physics. Most mechanists in the early to midseventeenth century offered mechanist explanations which were almost exclusively qualitative: Hobbes and Gassendi are good cases in point. Moreover, Kepler’s thoroughly Neoplatonic conception of the universe as being ultimately a mathematical harmony underlying surface appearances could not have been further from mechanism, yet it enabled him to develop a mathematical account in areas such as astronomy and optics which was well in advance of anything else at his time. But it cannot be denied that the combination of mechanism with a commitment to a mathematical physics was an extremely potent one, and Descartes was the first to offer such a combination to any significant extent. This question must be seen in a broad context. The theoretical justification for the use of mathematical theorems and techniques in the treatment of problems in physical theory is not obvious. To many natural philosophers it was far from clear that such an approach was necessary, justified, or even possible. Aristotle had provided a highly elaborate conception of physical explanation which absolutely precluded the use of mathematics in physical enquiry and it was this conception that dominated physical enquiry until the seventeenth century. Briefly, Aristotle defines physics and mathematics in terms of their subject genera: physics is concerned with those things that change and have an independent existence, mathematics with those things that do not change and have dependent existence (i.e. they are mere abstractions). The aim of scientific enquiry is to determine what kind of thing the subject matter of the science is by establishing its general properties. To explain something is to demonstrate it syllogistically starting from first principles which are expressions of essences, and what one is seeking in a physical explanation is a statement of the essential characteristics of a physical phenomenon—those characteristics which it must possess if it is to be the kind of thing it is. Such a statement can only be derived from principles that are appropriate to the subject genus of the science; in the case of physics, this means principles appropriate to explaining what is changing and has an independent existence. Mathematical principles are not of this kind. They are appropriate to a completely different kind of subject matter, and because of this mathematics is inappropriate to syllogistic demonstrations of physical phenomena, and it is alien to physical explanation. This approach benefited from a well-developed metaphysical account of the different natures of physical and mathematical entities, and it resulted in a physical theory that was not only in close agreement with observation and common sense, but which formed part of a large-scale theory of change which covered organic and inorganic phenomena alike. By the beginning of the seventeenth century, the Aristotelian approach was being challenged on a number of fronts, and Archimedean statics, in particular, was seen by many as the model for a physical theory, with its rigorously geometrical demonstrations of novel and fundamental physical theorems. But there was no straightforward way of extending this approach in statics (where it was often possible to translate the problem into mathematical terms in an intuitive and unproblematic way), to kinematics (where one had to deal with motion, i.e. continuous change of place) and in dynamics (where one had somehow to quantify the forces responsible for changes in motion). Moreover, statics involved a number of simplifying assumptions, such as the Earth’s surface being a true geometrical plane and its being a parallel force field. These simplifying assumptions generate all kinds of problems once one leaves the domain of statics, and the kinds of conceptual problems faced by natural philosophers wishing to provide a mathematical physics in the seventeenth century were immense.21 In the Rules for the Direction of Our Native Intelligence, Descartes outlined a number of methodological and epistemological proposals for a mathematical physics. The Rules, the writing of which was abandoned in 1628, is now thought to be a composite text, some parts deriving from 1619–20 (Rules 1–3, part of Rule 4, Rules 5–7, part of Rule 8, possibly Rules 9–11) and some dating from 1626–8 (part of Rule 4, part of Rule 8, and Rules 12–21).22 The earlier parts describe a rather grandiose reductionist programme in which mathematics is simply ‘applied’ to the natural world: 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 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.23 This project for a ‘universal mathematics’ is not mentioned again in Descartes’s writings and, although the question is a disputed one,24 there is a strong case to be made that he abandoned this kind of attempt to provide a basis for a mathematical physics. The later Rules set out an account of how our comprehension of the corporeal world is essentially mathematical in nature, but it is one which centres on a theory about how perceptual cognition occurs. Throughout the Rules, Descartes insists that knowledge must begin with ‘simple natures’, that is, with those things which are not further analysable and can be grasped by a direct ‘intuition’ (intuitus). These simple natures can only be grasped by the intellect—pure mind, for all intents and purposes—although in the case of perceptual cognition the corporeal faculties of sense perception, memory and imagination are also called upon. The imagin ation is located in the pineal gland (chosen because it was believed to be at the geometrical focus of the brain and its only non-duplicated organ), it is the point to which all perceptual information is transmitted, and it acts as a kind of meeting place between mind and body, although Descartes is understandably coy about this last point. In Rule 14, Descartes argues that the proper objects of the intellect are completely abstract entities, which are free of images or ‘bodily representations’, and this is why, when the intellect turns into itself it beholds those things which are purely intellectual such as thought and doubt, as well as those ‘simple natures’ which are common to both mind and body, such as existence, unity and duration. However, the intellect requires the imagination if there is to be any knowledge of the external world, for the imagination is its point of contact with the external world. The imagination functions, in fact, like a meeting place between the corporeal world and the mind. The corporeal world is represented in the intellect in terms of spatially extended magnitudes. Since, Descartes argues, the corporeal world is nothing but spatially extended body, with the experience of secondary qualities resulting from the mind’s interaction with matter moving in various distinctive ways, the representation of the world geometrically in the imagination is an entirely natural and appropriate mode of representation. But the contents of the mind must also be represented in the imagination and, in so far as the mind is engaged in a quantitative understanding, the imagination is needed in order that the mathematical entities on which the intellect works can be rendered determinate. For example, the intellect understands ‘fiveness’ as something distinct from five objects (or line segments, or points or whatever), and hence the imagination is required if this ‘fiveness’ is to correspond to something in the world. In fact, ‘fiveness’ is represented as a line comprising five equal segments which is then mapped onto the geometrical representations of the corporeal world. In this way, our understanding of the corporeal world, an understanding that necessarily involves sense perception, is thoroughly mathematical. It should be noted that the intellect or mind, working by itself, could never even represent the corporeal world to itself, and a fortiori could never understand it. DESCARTES’S METHOD OF DISCOVERY In the Discourse on Method, Descartes describes the procedure by which he has proceeded in the Dioptrics and the Meteors in the following terms: The order which I have followed in this regard is as follows. First, I have attempted generally to discover the principles or first causes of everything which is or could be in the world, without in this connection considering anything but God alone, who has created the world, and without drawing them from any source except certain seeds of truth which are naturally in our minds. Next I considered what were the first and most common effects that could be deduced from these causes, and it seems to me that in this way I found the heavens, the stars, an earth, and even on the earth, water, air, fire, the minerals and a few other such things which are the most common and simple of all that exist, and consequently the easiest to understand. Then, when I wished to descend to those that were more particular, there were so many objects of various kinds that I did not believe it possible for the human mind to distinguish the forms or species of body which are on the earth from the infinity of others which might have been, had it been God’s will to put them there, or consequently to make them of use to us, if it were not that one arrives at the causes through the effects and avails oneself of many specific experiments. In subsequently passing over in my mind all the objects which have been presented to my senses, I dare to say that I have not noticed anything that I could not easily explain in terms of the principles that I have discovered. But I must also admit that the power of nature is so great and so extensive, and these principles so simple and general, that I hardly observed any effect that I did not immediately realize could be deduced from the principles in many different ways. The greatest difficulty is usually to discover in which of these ways the effect depends on them. In this situation, so far as I know the only thing that can be done is to try and find experiments which are such that their result varies depending upon which of them provides the correct explanation.25 But what exactly is Descartes describing here? We cannot simply assume it is a method of discovery. In a letter to Antoine Vatier of 22 February 1638, Descartes writes: I must say first that my purpose was not to teach the whole of my Method in the Discourse in which I propound it, but only to say enough to show that the new views in the Dioptrics and the Meteors were not random notions, and were perhaps worth the trouble of examining. I could not demonstrate the use of this Method in the three treatises which I gave, because it prescribes an order of research which is quite different from the one I thought proper for exposition. I have however given a brief sample of it in my account of the rainbow, and if you take the trouble to reread it, I hope it will satisfy you more than it did the first time; the matter is, after all, quite difficult in itself. I attached these three treatises [the Geometry, the Dioptrics and the Meteors] to the discourse which precedes them because I am convinced that if people examine them carefully and compare them with what has previously been written on the same topics, they will have grounds for judging that the Method I adopt is no ordinary one and is perhaps better than some others.26 What is more, in the Meteors itself, Descartes tells us that his account of the rainbow is the most appropriate example ‘to show how, by means of the method which I use, one can attain knowledge which was not available to those whose writings we possess’.27 This account is, then, clearly worth looking at. The Meteors does not start from first principles but from problems to be solved, and Descartes then uses the solution of the problem to exemplify his method. The central problem in the Meteors, to which Book 8 is devoted, is that of explaining the angle at which the bows of the rainbow appear in the sky. He begins by noting that rainbows are not only formed in the sky, but also in fountains and showers in the presence of sunlight. This leads him to formulate the hypothesis that the phenomenon is caused by light reacting on drops of water. To test this hypothesis, he constructs a glass model of the raindrop, comprising a large glass sphere filled with water and, standing with his back to the Sun, he holds up the sphere in the Sun’s light, moving it up and down so that colours are produced. Then, if we let the light from the Sun come from the part of the sky marked AFZ, and my eye be at point E, then when I put this globe at the place BCD, the part of it at D seems to me wholly red and incomparably more brilliant than the rest. And whether I move towards it or step back from it, or move it to the right or to the left, or even turn it in a circle around my head, then provided the line DE always marks an angle of around 42° with the line EM, which one must imagine to extend from the centre of the eye to the centre of the sun, D always appears equally red. But as soon as I made this angle DEM the slightest bit smaller it did not disappear completely in the one stroke but first divided as into two less brilliant parts in which could be seen yellow, blue, and other colours. Then, looking towards the place marked K on the globe, I perceived that, making the angle KEM around 52°, K also seemed to be coloured red, but not so brilliant…28 Descartes then describes how he covered the globe at all points except B and D. The ray still emerged, showing that the primary and secondary bows are caused by two refractions and one or two internal reflections of the incident ray. He next describes how the same effect can be produced with a prism, and this indicates that neither a curved surface nor reflection are necessary for colour dispersion. Moreover, the prism experiment shows that the effect does not depend on the angle of incidence and that one refraction is sufficient for its production. Finally, Descartes calculates from the refractive index of rainwater what an observer would see when light strikes a drop of water at varying angles of incidence, and finds that the optimum difference for visibility between incident and refracted rays is for the former to be viewed at an angle of 41°–42° and the latter at an angle of 51°–52°,29 which is exactly what the hypothesis predicts. This procedure is similar to that followed in the Dioptrics, and in some inspects to that followed in the Geometry. It is above all an exercise in problemsolving, and the precedent for such an exercise seems to have been developed in Descartes’s work in mathematics. Indeed, the later parts of the Rules turn towards specifically mathematical considerations, and Rules 16–21 have such close parallels with the Geometry that one can only conclude that they contain the early parts of that work in an embryonic form. Rule 16 advises us to use ‘the briefest possible symbols’ in dealing with problems, and one of the first things the Geometry does is to provide us with the algebraic signs necessary for dealing with geometrical problems. Rule 17 tells us that, in dealing with a new problem, we must ignore the fact that some terms are known and some unknown; and again one of the first directives in the Geometry is that we label all lines necessary for the geometrical construction, whether these be known or unknown. Finally, Rules 18–21 are formulated in almost identical terms in the Geometry.30 There is something ironic in this, for one would normally associate a mathematical model with a method which was axiomatic and deductive. Certainly, if one looks at the great mathematical texts of Antiquity —Euclid’s Elements or Archimedes’s On the Sphere and the Cylinder or Apollonius’s On Conic Sections, for example—one finds lists of definitions and postulates and deductive proofs of theorems relying solely on these. If one now turns to Descartes’s Geometry, one finds something completely different. After a few pages of introduction, mainly on the geometrical representation of the arithmetical operations of multiplication, division and finding roots, we are thrown into one of the great unsolved problems bequeathed by Antiquity— Pappus’s locus problem for four or more lines, which Descartes then proceeds to provide us with a method of solving. Descartes’s solution to the Pappus problem is an ‘analytic’ one. In ancient mathematics, a sharp distinction was made between analysis and synthesis. Pappus, one of the greatest of the Alexandrian mathematicians, had distinguished between two kinds of analysis: ‘theoretical’ analysis, in which one attempts to discover the truth of a theorem, and ‘problematical analysis’, in which one attempts to discover something unknown. If, in the case of theoretical analysis, one finds that the theorem is false or if, in the case of problematical analysis, the proposed procedure fails to yield what one is seeking, or one can show the problem to be insoluble, then synthesis is not needed, and analysis is complete in its own right. In the case of positive results, however, synthesis is needed, albeit for different reasons. Synthesis is a difficult notion to specify, and it appears to have been used with slightly different meanings by different writers, but it is basically that part of the mathematical process in which one proves deductively, perhaps from first principles, what one has discovered or shown the truth of by analysis. In the case of theoretical analysis, one needs synthesis, because in the analysis what we have done is to show that a true theorem follows from a theorem whose truth we wish to establish, and what we must now do is to show that the converse is also the case, that the theorem whose truth we wish to establish follows from the theorem we know to be true. The latter demonstration, whose most obvious form is demonstration from first principles, is synthesis. A synthetic proof is, in fact, the ‘natural order’ for Greek and Alexandrian mathematicians, the analysis being only a ‘solution backwards’. So what we are invariably presented with are the ‘naturally ordered’ synthetic demonstrations: there is no need to present the analysis as well. Descartes objects to such procedures. He accuses the Alexandrian mathematicians Pappus and Diophantus of presenting only the synthesis from ulterior motives: I have come to think that these writers themselves, with a kind of pernicious cunning, later suppressed this mathematics as, notoriously, many inventors are known to have done where their own discoveries were concerned. They may have feared that their method, just because it was so easy and simple, would be depreciated if it were divulged; so to gain our admiration, they may have shown us, as the fruits of their method, some barren truths proved by clever arguments, instead of teaching us the method itself, which might have dispelled our admiration.31 In other words, analysis is a method of discovery, whereas synthesis is merely a method of presentation of one’s results by deriving them from first principles. Now it is true that in many cases the synthetic demonstration will be very straightforward once one has the analytic demonstration, and indeed in many cases the latter is simply a reversal of the former. Moreover, all equations have valid converses by definition, so if one is dealing with equations, as Descartes is for example, then there is no special problem about converses holding. But the synthetic demonstration, unlike the analytic one, is a deductively valid proof, and this, for the ancients and for the vast majority of mathematicians since then, is the only real form of proof. Descartes does not accept this, because he does not accept that deduction can have any value in its own right. We shall return to this issue below. The case of problematical analysis with a positive outcome is more complicated, for here there was traditionally considered to be an extra reason why synthesis was needed, namely the production of a ‘determinate’ solution. In rejecting synthesis in this context, Descartes is on far stronger ground. Indeed, one of the most crucial stages in the development of algebra consists precisely in going beyond the call for determinate solutions. In the case of geometry, analysis provides one with a general procedure, but it does not in itself produce a particular geometrical figure or construction as the solution to a problem and, until this is done, the ancients considered that the problem had not been solved. Parallel constraints applied in arithmetic. Arithmetical analysis yields only an indeterminate solution, and we need a final synthetic stage corresponding to the geometrical solution; this is the numerical exploitation of the indeterminate solution, where we compute determinate numbers. Now in traditional terms the Geometry is an exercise in problematical analysis, but Descartes completely rejects the traditional requirement that, following such an analysis, synthesis is needed to construct or compute a determinate figure or number. For the mathematicians of Antiquity this was the point of the exercise, and it was only if such a determinate figure or number could be constructed or computed that one could be said to have solved the problem. Towards the end of the Alexandrian era, most notably in Diophantus’s Arithmetica, we do begin to find the search for problems and solutions concerned with general magnitudes, but these are never considered an end in themselves, and they are regarded as auxiliary techniques allowing the computation of a determinate number, which is the ultimate point of the exercise. Descartes’s approach is completely and explicitly at odds with this. As early as Rule 16 of the Rules he spells out the contrast between his procedure and the traditional one: It must be pointed out that while arithmeticians have usually designated each magnitude by several units, i.e. by a number, we on the contrary abstract from numbers themselves just as we did above [Rule 14] from geometrical figures, or from anything else. Our reason for doing this is partly to avoid the tedium of a long and unnecessary calculation, but mainly to see that those parts of the problem which are the essential ones always remain distinct and are not obscured by useless numbers. If for example we are trying to find the hypotenuse of a right-angled triangle whose given sides are 9 and 12, the arithmeticians will say that it is ′ (225), i.e. 15. We, on the other hand, will write a and b for 9 and 12, and find that the hypotenuse is ′ (a2 + b2), leaving the two parts of the expression, a2 and b2, distinct, whereas in the number they are run together…. We who seek to develop a clear and distinct knowledge of these things insist on these distinctions. Arithmeticians, on the other hand, are satisfied if the required result turns up, even if they do not see how it depends on what has been given, but in fact it is in knowledge of this kind alone that science consists.32 In sum, for Descartes, concern with general magnitudes is constitutive of the mathematical enterprise. Descartes’s algebra transcends the need to establish converses, because he is dealing with equations, whose converses always hold, and it transcends the traditional view that one solves a problem only when one has constructed a determinate figure or computed a determinate number. But Descartes goes further than this, rejecting the need for deductive proof altogether. The reason why he does this lies ultimately not so much in his rejection of synthetic demonstrations in mathematics but in his conception of the nature of inference. Before we look at this question, however, it is worth looking briefly at what role deduction does play in Descartes’s overall account. METHODS OF DISCOVERY AND PRESENTATION In Article 64 of Part II of The Principles of Philosophy, Descartes writes: I know of no material substance other than that which is divisible, has shape, and can move in every possible way, and this the geometers call quantity and take as the object of their demonstrations. Moreover, our concern is exclusively with the divisions, shape and motions of this substance, and nothing concerning these can be accepted as true unless it be deduced (deducatur) from indubitably true common notions with such certainty that it can be regarded as a mathematical demonstration. And because all natural phenomena can be explained in this way, as one can judge from what follows, I believe that no other physical principles should be accepted or even desired. Like the passage from the Discourse on Method that I quoted above, there is a suggestion here that deduction from first principles is Descartes’s method of discovery. Can we reconcile these and many passages similar to them with Descartes’s rejection of deductive forms of inference, such as synthesis in mathematics and syllogistic in logic? I believe we can. Descartes’s procedure in natural philosophy is to start from problem-solving, and his ‘method’ is designed to facilitate such problem-solving. The problems have to be posed in quantitative terms and there are a number of constraints on what form an acceptable solution takes: one cannot posit ‘occult qualities’, one must seek ‘simple natures’, and so on. The solution is then tested experimentally to determine how well it holds up compared with other possible explanations meeting the same constraints which also appear to account for the facts. Finally, the solution is incorporated into a system of natural philosophy, and the principal aim of a work like the Principles of Philosophy is to set out this natural philosophy in detail. The Principles is a textbook, best compared not with works like the Optics and the Meteors, which purport to show one how the empirical results were arrived at, but with the many scholastic textbooks on natural philosophy which were around in Descartes’s time, and from which he himself learnt whilst a student.33 Such a textbook gives one a systematic overview of the subject, presenting its ultimate foundations, and showing how the parts of the subject are connected. Ultimately, the empirically verified results have to be fitted into this system, which in Descartes’s case is a rigorously mechanist system presented with metaphysical foundations. But the empirical results themselves are not justified by their incorporation within this system: they are justified purely in observational and experimental terms. It is important to realize this, because it is fundamental to Descartes’s whole approach that deduction cannot justify anything. What it can do is display the systematic structure of knowledge to us, and this is its role in the Principles. Again, there is something of an irony here, for the kind of misunderstanding of Descartes’s methodological concerns which has resulted in the view that he makes deduction from first principles the source of all knowledge is rather similar to the kind of misunderstanding that Descartes himself fosters in the case of Aristotle on the one hand and the Alexandrian mathematicians on the other. In the case of Aristotle, he takes a method of presentation of results which have already been established to be a method of discovery. In the case of Pappus and Diophantus, he maintains that a method of presentation is passed off as a method of discovery. Yet both followers and critics of Descartes have said exactly the same of him; taking his method of presentation as if it were a method of discovery, they have often then complained that there is a discrepancy between what he claims his method is and the procedure he actually follows in his scientific work.34 This suggests that there may be something inherently problematic in the idea of a ‘method of discovery’. If one compares the kind of presentation one finds in the Geometry with what one finds in the Principles of Philosophy, there is, on the face of it, much less evidence of anything one would call ‘method’ in the former than in the latter. Certain basic maxims are adhered to, and basic techniques developed, in the first few pages of the Geometry, but the former are really too rudimentary to be graced with the name of ‘method’, and the latter are specifically mathematical. In the very early days (from around 1619 to the early 1620s), when Descartes was contemplating his grand scheme of a ‘universal mathematics’, there was some prospect of a really general method of discovery, for universal mathematics was a programme in which, ultimately, everything was reduced to mathematics. But once this was (wisely) abandoned, and the mathematical rules were made specifically mathematical, the general content of the ‘method’ becomes rather empty. Here, for example, are the rules of method as they are set out in the Discourse on Method: The first was never to accept anything as true if I did not have evident knowledge of its truth: that is, carefully to avoid precipitate conclusions and preconceptions, and to include nothing more in my judgements than what presented itself to my mind so clearly and so distinctly that I had no occasion to doubt it. The second, to divide each of the difficulties I examined into as many parts as possible and as may be required in order to resolve them better. The third, to direct my thoughts in an orderly manner, by beginning with the simplest and most easily known objects in order to ascend little by little, step by step, to knowledge of the most complex, and by supposing some order even among objects that have no natural order of precedence.35 There is surely little that is radical or even novel here, and the list is more in the nature of common-sense hints rather than something offering deep enlightenment (unless it is specifically interpreted as a some-what cryptic statement of an algebraic approach to mathematics, in which case it is novel, but it then becomes very restricted in scope and can no longer have any claim to be a general statement of ‘method’). The same could be said of Aristotle’s ‘topics’: they too offer no systematic method of discovery, and certainly nothing that would guarantee success in a scientific enterprise, but rather general and open-ended guidance. But ‘methods of discovery’ do not perform even this modest role unaided. It is interesting in this respect that, in this passage as in others, Descartes finds it so difficult to present his ‘method of discovery’ without at the same time mentioning features appropriate to his method of presentation. The reason for this lies in the deep connections between the two enterprises, connections which Descartes seems reluctant to investigate. While it is legitimate to present the deductive structure of the Principles of Philosophy as a method of presentation as opposed to a method of discovery, it must be appreciated that the structure exhibited in, or perhaps revealed by, the method of presentation is a structure that will inevitably guide one in one’s research. It will not enable one to solve specific problems, but it will indicate where the problems lie, so to speak, and which are the important ones to solve: which are the fundamental ones and which the peripheral ones. Leibniz was to realize this much more clearly than Descartes ever did, arguing that we use deductive structure to impose order on information, and by using the order discerned we are able to identify gaps and problematic areas in a systematic and thorough way.36 Failure to appreciate this crucial feature of deductive structure will inevitably result in a misleading picture in which the empirical results are established first and then, when this is done, incorporated into a system whose only role is the ordering of these results. But such a procedure would result in problem-solving of a completely unsystematic and aimless kind, and this is certainly not what Descartes is advocating. The method of presentation does, then, have a role in discovery: it complements discovery procedures by guiding their application. The extent to which Descartes explicitly recognizes this role is problematic, but there can be no doubt that his account of method presupposes it. THE FUNDAMENTAL PROBLEM OF METHOD: EPISTEMIC ADVANCE The heart of the philosophical problem of method in Descartes lies not in reconciling his general statements on method with his more specific recommendations on how to proceed in scientific investigation, or in clarifying the relation between his ‘method of discovery’ and his ‘method of presentation’, but in an altogether deeper and more intractable question about how inference can be informative. Inference is necessarily involved in every kind of scientific enterprise, from logic and mathematics to natural philosophy, and the whole point of these enterprises is to produce new knowledge, but the canonical form of inference, for Descartes and all his predecessors, is deductive inference, and it is a highly problematic question whether deductive inference can advance knowledge. The question became highlighted in the sixteenth century when there was intense discussion of the Aristotelian distinction between knowledge how and knowledge why, and the ways in which the latter could be achieved. Turnebus,37 writing in 1565, tells us that the (Aristotelian) question of method was the most discussed philosophical topic of the day. These debates were conducted in the context of the theory of the syllogism, and although, with the demise of syllogistic, the explicitly logical context is missing from seventeenth-century discussions of method, there is always an undercurrent of logical questions. Descartes raises the question of method in the context of considerations about the nature of inference in the following way in Rule 4 of the Rules: But if our method rightly explains how intellectual intuition should be used, so as not to fall into error contrary to truth, and how one must find deductive paths so that we might arrive at knowledge of all things, I cannot see anything else is needed to make it complete; for I have already said that the only way science is to be acquired is by intellectual intuition or deduction.38 Intellectual intuition is simply the grasp of a clear and distinct idea. But what is deduction? In Rule 7 it is described in a way which makes one suspect that it is not necessary in its own right: Thus, if, for example, I have found out, by distinct mental operations, what relation exists between magnitudes A and B, then what between B and C, between C and D, and finally between D and E, that does not entail that I will see what the relation is between A and E, nor can the truths previously learned give me a precise idea of it unless I recall them all. To remedy this I would run over them many times, by a continuous movement of the imagination, in such a way that it has an intuition of each term at the same time that it passes on to the others, and this I would do until I have learned to pass from the first relation to the last so quickly that there was almost no role left for memory and I seemed to have the whole before me at the same time.39 This suspicion is confirmed in Rule 14: In every train of reasoning it is merely by comparison that we attain to a precise knowledge of the truth. Here is an example: all A is B, all B is C, therefore all A is C. Here we compare with one another what we are searching for and what we are given, viz. A and C, in respect of the fact that each is B, and so on. But, as we have pointed out on a number of occasions, because the forms of the syllogism are of no aid in perceiving the truth about things, it will be better for the reader to reject them altogether and to conceive that all knowledge whatsoever, other than that which consists in the simple and pure intuition of single independent objects, is a matter of the comparison of two things or more with each other. In fact practically the whole task set the human reason consists in preparing for this operation; for when it is open and simple, we need no aid from art, but are bound to rely upon the light of nature alone, in beholding the truth which comparison gives us.40 The difference between intuition and deduction lies in the fact that whereas the latter consists in grasping the relations between a number of propositions, intuition consists in grasping a necessary connection between two propositions. But in the limiting case, deduction reduces to intuition: we run through the deduction so quickly that we no longer have to rely on memory, with the result that we grasp the whole in a single intuition at a single time. The core of Descartes’s position is that by compacting inferential steps until we come to a direct comparison between premises and conclusion we put ourselves in a position where we are able to have a clear and distinct idea of the connection, and this provides us with a guarantee of certainty. What is at issue here is the question of the justification of deduction, but we must be careful to separate out two different kinds of demand for justification. The first is a demand that deductive inference show itself to be productive of new knowledge, that it result in episte-mic advance. The second is a question about whether deductive inference can be further analysed or explained: it is a question about the justification of deduction, but not one which refers us to its epistemic worth for, as Dummett has rightly pointed out, our aim ‘is not to persuade anyone, not even ourselves, to employ deductive arguments: it is to find a satisfactory explanation for the role of such arguments in our use of language.’41 Now these two kinds of question were not always clearly distinguished in Descartes’s time, and it was a prevalent assumption in the seventeenth century that syllogistic, in both its logical and its heuristic aspects, could be justified if and only if it could show its epistemic worth. But the basis for the distinction was certainly there, and while the questions are related in Descartes, we can find sets of considerations much more relevant to the one than the other. The first question, that of epistemic informativeness, concerns the use of formalized deductive arguments, especially the syllogism, in the discovery of new results in natural philosophy. Earlier, we looked very briefly at how Aristotle tried to deal with this question, by distinguishing two different forms of syllogism, one scientific because it provided us with knowledge why something was the case, the other non-scientific because it only provided us with knowledge that something was the case. The logicians of antiquity were crucially concerned with the epistemic informativeness of various kinds of deductive argument, and both Aristotle and the Stoics (founders of the two logical systems of Antiquity) realized that there may be no logical or formal difference between an informative and an uninformative argument, so they tried to capture the difference in non-logical terms, but in a way which still relied on structural features of arguments, for example the way in which the premises were arranged. All these attempts failed, and the question of whether deductive arguments can be informative, and if so what makes them informative, remained unresolved. The prevalent seventeenth-century response to this failure was to argue that deductive arguments can never be epistemically informative. Many critics of logic right up to the nineteenth century criticized syllogistic arguments for failing to yield anything new, where what is meant by ‘new’ effectively amounts to ‘logically independent of the premises’. But of course a deductive argument is precisely designed to show the logical dependence of the conclusion on premises, and so the demand is simply misguided. Descartes’s response is rather different. It consists in the idea that the deduction of scientific results, whether in mathematics or in natural philosophy, does not genuinely produce those results. Deduction is merely a mode of presentation of results which have already been reached by analytic, problem-solving means. This is hard to reconcile, however, with, say, our learning of some geometrical theorem by following through the proof from first principles in a textbook. Even if Descartes could show that one can never come to know new theorems in the sense of inventing them by going through some deductive process, this does not mean that one could not come to know them, in the sense of learning something one did not previously know, by deductive means. Indeed, it is hard to understand what the point of the Principles could be if Descartes denied the latter. But in that case his argument against deduction as a means of discovery is a much more restricted one than he appears to think. Moreover, I have already indicated that deduction seems to play a guiding role in discovery, in the sense of invention or ‘genuine’ discovery, in Descartes, because his procedures for problem-solving are quite blind as far as the ultimate point of the exercise is concerned. Finally, the way in which he sets up the argument in the first place is somewhat question-begging. We are presented with two alternatives: using his ‘method’, or deduction from first principles. But someone who has a commitment to the value of deductive inference in discovery, as Leibniz was to have, will not necessarily want to tie this to demonstration purely from first principles: Leibniz’s view was that deduction only comes into use as a means of discovery once one has a very substantial body of information (discovered by non-deductive means). This is a possibility that Descartes simply does not account for. On the second question, Descartes’s view is expressed admirably in Rule 4 of the Rules for the Direction of Our Native Intelligence, when he says that ‘nothing can be added to the pure light of reason which does not in some way obscure it.’ Intuition, and the deductive inference that must ultimately reduce to a form of intuition, is unanalysable, simple and primitive. Like the cogito, which is the canonical example of an intuition, no further question can be raised about it, whether in justification or explanation. This raises distinctive problems for any treatment of the nature of deduction. It is interesting to note here the wide gulf between Aristotle’s classic account of the justification of deductive principles and Descartes’s approach. In the Metaphysics, Aristotle points out that proofs must come to an end somewhere, for otherwise we would be involved in an infinite regress. Hence there must be something that we can rely on without proof, and he takes as his example the law of non-contradiction. The law is justified by showing that an opponent who denies it must, in denying it, actually assume its truth, and by showing that arguments that apparently tell against it, such as relativist arguments purporting to show that a thing may both have and not have a particular property depending on who is perceiving the thing, cannot be sustained. Descartes can offer nothing so compelling. It is something ambiguously psychological—the ‘light of reason’ or the ‘light of nature’—that stops the regress on Descartes’s conception. Whereas Aristotle was concerned, in his justification, to find a form of argument which was irresistible to an opponent, all Descartes can do is postulate some form of psychological clarity experienced by the knowing subject. Nevertheless, it was Descartes’s conception that held sway, being adopted in two extremely influential works of the later seventeenth century: Arnauld and Nicole’s Port-Royal Logic, and Locke’s Essay Concerning Human Understanding.42 The reason for this is not hard to find. The Aristotelian procedure relies upon a discursive conception of inference, whereby one induces an opponent to accept what one is arguing on the basis of accepting certain shared premises: such a mode of argument works both at the ordinary level of convincing someone of some factual matter, and at the metalevel of justifying the deductive principles used to take one from premises to conclusion. But as I indicated earlier, the discursive conception was generally discredited in the seventeenth century. It requires common ground between oneself and one’s opponents, and Descartes and others saw such common ground as the root of the problem of lack of scientific progress within the scholastic-Aristotelian tradition. It was seen to rest on an appeal to what is generally accepted rather than to what is the case. The ‘natural light of reason’, on the other hand, provided internal resources by which to begin afresh and reject tradition. The acceptance of this view had disastrous consequences for the study of deductive logic. Descartes’s algebra contained the key to a new understanding of logic. Just as Descartes had insisted that, in mathematics, one must abstract from particular numbers and focus on the structural features of equations, so analogously one could argue that one should abstract from particular truths and explore the relation between them in abstract terms. Such a move would have been tantamount to the algebraic construal of logic, something which is constitutive of modern logic. But Descartes did not even contemplate such a move, not because of the level of abstraction involved, which would not have worried him if his work in mathematics is any guide, but because he was unable to see any point in deductive inference. CONCLUSION Descartes’s approach to philosophical questions of method was extremely influential from the seventeenth to the nineteenth centuries, and it replaced Aristotelianism very quickly. It was part of a general anti-deductivist movement, whether this took the form of a defence of hypotheses (in the seventeenth century) or of induction (in the eighteenth and nineteenth centuries). This influence was transmitted indirectly through Locke, however, and with the interpretation of seventeenth- and eighteenth-century philosophy in terms of two opposed schools of thought, rationalism and empiricism, this aspect of Descartes’s thought tended to become forgotten, and his more programmatic statements about his system were taken out of context and an apriorist and deductivist methodology ascribed to him. The irony in this is that Descartes not only vehemently rejected such an approach, but his rejection goes too far. It effectively rules out deduction having any epistemic value, and this is something he not only could not establish but which, if true, would have completely undermined his own Principles of Philosophy. But this is not a simple oversight on Descartes’s part. It reflects a serious and especially intractable problem, or rather set of problems, about how deductive inference can be informative, which Descartes was never able to resolve and which had deep ramifications for his account of method. NOTES 1 This is, at least, the usual view. For a dissenting view see G.Canguilhem, La formation du concept de réflexe an XVIIe et XVIIIe siècles (Paris, Presses Universitaires de France, 1955), pp. 27–57. 2 (Nottingham [5.10], 30. 3 On the issue of rationalism versus empiricism see Louis E.Loeb, From Descartes to Hume: Continental Metaphysics and the Development of Modern Philosophy (Ithaca, N.Y., Cornell University Press, 1981), ch. 1. 4 [5.1], vol. 8 (1), 78–9. 5 See L.Laudan, Science and Hypothesis (Dordrecht, Reidel, 1981), ch. 4. 6 Rule 10, [5.1], vol. 10, 406. 7 [5.1], vol. 10, 376–7. 8 See Jonathan Barnes, ‘Aristotle’s Theory of Demonstration’, in J.Barnes, M. Schofield, and R.Sorabji (eds) Articles on Aristotle, vol. 1, Science (London, Duckworth, 1975), pp. 65–87. 9 There is one occasion on which Descartes does in fact make use of a form of argument with a distinctively discursive structure: in his treatment of scepticism. Sceptical arguments have a distinctive non-logical but nevertheless structural feature. They rely upon the interlocutor of the sceptic to provide both the knowledge claims and the definition of knowledge (i.e. the premises of the argument). The sceptic then attempts to show a discrepancy between these two. Were the sceptic to provide the definition of knowledge, or to provide knowledge claims or denials that certain things are known, the ingenious dialectical structure of sceptical arguments would be undermined. This is a very traditional feature of sceptical arguments, and it is a sign of Descartes’s ability to handle it that he not only uses it to undermine knowledge claims in the First Meditation, but he also uses it to destroy scepticism in making the sceptic provide the premise of the cogito, by putting it in the form ‘I doubt, therefore I exist’. In other words he is able to turn the tables on the sceptic by using the same form of argument that makes scepticism so successful. But, of course, once he has arrived at his foundation, he shuns this form of argument, for now he has premises he can be certain of and so he no longer has any need to use forms of argument which demand shared (but possibly false) premises. 10 It should be noted that ‘method’ was a central topic in the sixteenth century, but the focus of the sixteenth-century discussion is different, and it derives from Aristotle’s distinction between scientific and non-scientific demonstration. On the sixteenthcentury disputes, see Neal W.Gilbert, Renaissance Concepts of Method (New York, Columbia University Press, 1960). 11 For more detail on the problems of defining mechanism, see J.E.McGuire, ‘Boyle’s Conception of Nature’, Journal of the History of Ideas 33 (1972) 523–42; and Alan Gabbey, ‘The Mechanical Philosophy and its Problems: Mechanical Explanations, Impenetrability, and Perpetual Motion’, in J.C.Pitt (ed.) Change and Progress in Modern Science (Dordrecht, Reidel, 1985), pp. 9–84. 12 See K.Hutchison, ‘Supernaturalism and the Mechanical Philosophy’, History of Science 21 (1983) 297–333. 13 See D.P.Walker, Spiritual and Demonic Magic from Ficino to Campanella (London, the Warburg Institute, 1958). 14 Marin Mersenne (1588–1648) was one of the foremost advocates of mechanism, and as well as writing extensively on a number of topics in natural philosophy and theology he played a major role in co-ordinating and making known current work in natural philosophy from the mid-1620s onwards. He attended the same school as Descartes but their friendship, which was to be a lifelong one, began only in the mid-1620s, during Descartes’s stay in Paris. 15 Averroes (c. 1126–c. 1198), the greatest of the medieval Islamic philosophers, developed what was to become the principal form of naturalism in the later Middle Ages and Renaissance. The naturalism of Alexander of Aphrodisias (fl. AD 200), the greatest of the Greek commentators on Aristotle, does not seem to have been taken seriously until the Paduan philosopher Pietro Pomponazzi (1462–1525) took it up, and even then it was never explicitly advocated as a doctrine that presents the whole truth. On the Paduan debates over the nature of the soul see Harold Skulsky, ‘Paduan Epistemology and the Doctrine of One Mind’, Journal of the History of Philosophy 6 (1968) 341–61. 16 See Robert Lenoble, Mersenne on la naissance de mécanisme (Paris, Vrin, 2nd edn, 1971). 17 On this question see M.Gueroult, ‘The Metaphysics and Physics of Force in Descartes’, and A.Gabbey, ‘Force and Inertia in the Seventeenth Century: Descartes and Newton’, both in [5.27], 196–229 and 230–320 respectively. 18 See, for example, his reply to Henry More’s objection that modes are not alienable in [5.1], vol. 5, 403–4. 19 [5.1], vol. 7, 108; [5.5], vol. 2, 78. 20 See Wahl [5.73]. 21 For details of this issue see Stephen Gaukroger, Explanatory Structures: Concepts of Explanation in Early Physics and Philosophy (Brighton, Harvester, 1978), ch. 6. 22 For details see Jean-Paul Weber, La Constitution du texte des Regulae (Paris, 1964). 23 [5.1], vol. 10, 377–8; [5.5], vol. 1, 19. 24 See John Schuster, ‘Descartes’ Mathesis Universalis 1619–28’, in [5.27], 41–96. 25 [5.1], vol. 6, 63–5. 26 [5.1], vol. 1, 559–60. 27 [5.1], vol. 5, 325. 28 [5.1], vol. 6, 326–7. 29 [5.1], vol. 6, 336. 30 See the discussion in Beck [5.39], 176ff. 31 Rule 4, [5.1], vol. 10, 276–7; [5.1], vol. 5, 19. 32 [5.1], vol. 10, 455–6, 458. 33 On the authors whom Descartes would have studied at his college, La Flèche, see Gilson [5.20]. On the textbook tradition more generally, see Patricia Reif, ‘The Textbook Tradition in Natural Philosophy, 1600–1650’, Journal of the History of Ideas 30 (1969) 17–32. 34 It is rare to find anyone who not only takes the deductive approach at face value and also believes it is viable, but there is at least one notable example of such an interpretation, namely Spinoza’s Principles of the Philosophy of René Descartes (1663). 35 [5–1], vol. 6, 18–19; [5.5], vol. 1, 120. 36 See, for example, Leibniz’s letter to Gabriel Wagner of 1696, in C.I.Gerhardt (ed.) Die philosophischen Schriften von Gottfried Wilhelm Leibniz (Berlin, Weidman, 1875–90), vol. 7, 514–27, especially Comment 3 on p. 523. The letter is translated in L.E.Loemker, Gottfried Wilhelm Leibniz: Philosophical Papers and Letters (Dordrecht, Reidel, 1969), pp. 462–71, with Comment 3 on p. 468. 37 Adrianus Turnebus (1512–65) was Royal Reader in Greek at the Collège de France. He was one of the leading humanist translators of his day, and had an extensive knowledge of Greek philosophy. 38 [5.1], vol. 10, 372; [5.5], vol. 1, 16. 39 [5.1], vol. 10, 387–8; [5.5], vol. 1, 25. 40 [5.1], vol. 10, 439–40; [5–5], vol. 1, 57. 41 M.Dummett, ‘The Justification of Deduction’, in his Truth and Other Enigmas (London, Duckworth, 1978), p. 296. 42 On this influence see, specifically in the case of Locke, J.Passmore, ‘Descartes, the British Empiricists and Formal Logic’, Philosophical Review 62 (1953), 545–53; and more generally, W.S.Howell, Eighteenth-Century British Logic and Rhetoric (Princeton, N.J., Princeton University Press, 1971). BIBLIOGRAPHY Original language editions 5.1 Adam, C. and Tannery, P. (eds) Oeuvres de Descartes, Paris, Vrin, 12 vols, 1974–86. 5.2 Alquié, F. Oeuvres Philosophiques, Paris, Garnier, 3 vols, 1963–73. 5.3 Crapulli, G. (ed.) 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