Descartes: methodologyStephen GaukrogerINTRODUCTIONThe seventeenth century is often referred to as the century of the ScientificRevolution, a time of fundamental scientific change in which traditional theorieswere either replaced by new ones or radically transformed. Descartes madecontributions to virtually every scientific area of his day. He was one of thefounders of algebra, he discovered fundamental laws in geometrical optics, hisnatural philosophy was
the natural philosophy in the seventeenth century beforethe appearance of Newton’s
Principia (Newton himself was a Cartesian beforehe developed his own natural philosophy) and his work in biology andphysiology resulted, amongst other things, in the discovery of reflex action.
1Descartes’s earliest interests were scientific, and he seems to have thought hisscientific work of greater importance than his metaphysical writings throughouthis 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 elaboratetreatment in commentaries and the like. Still less should one do what sometry to do, and dig more deeply into these questions than the author did: hehas dealt with them all quite deeply enough. It is sufficient to have graspedthem once in a general way, and then to remember the conclusion.Otherwise they draw the mind too far away from physical and observablethings, and make it unfit to study them. Yet it is just these physical studiesthat it is most desirable for men to pursue, since they would yield abundantbenefits for life.
2Despite this, Descartes has often been considered a metaphysician in naturalphilosophy, deriving physical truths from metaphysical first principles. Indeed,there is still a widespread view that the ‘method’ Descartes espoused is the
apriori one of deduction from first principles, where these first principles aretruths of reason.This view has two principal sources: an image of Descartes asthe
de facto founder of a philosophical school—‘rationalism’—in whichdeduction from truths of reason is, almost by definition, constitutive ofepistemology, and a reading of a number of passages in Descartes in which hediscusses his project in highly schematic terms as accounts of his method ofdiscovery. The reading of Descartes as founder of a school is largely anineteenth-century doctrine first set out in detail in Kuno Fischer’s
Geschichteder neueren Philosophie in the 1870s. There is a hidden agenda in Fischer whichunderlies this: he is a Kantian and is keen to show Kant’s philosophy as solvingthe major problems of modern thought. He sets the background for this byresolving modern preKantian philosophy into two schools, rationalism andempiricism, the first basing everything on truths of reason, the second basingeverything on experiential truths. This demarcation displaces the older Platonist/Aristotelian dichotomy (which Kant himself effectively worked with), markingout the seventeenth century as the beginning of a new era in philosophy, onedominated by epistemological (as opposed to moral or theological) concerns.
3This reading of Descartes is not wholly fanciful, and it has been widelyaccepted in the twentieth century by philosophers who do not share Fischer’sKantianism. On the face of it, it has considerable textual support. Article 64 ofPart II of Descartes’s
Principles of Philosophy is entitled:That I do not accept or desire in physics any principles other than thoseaccepted in geometry or abstract mathematics; because all the phenomenaof nature are explained thereby, and demonstrations concerning themwhich are certain can be given.In elucidation, he writes:For I frankly admit that I know of no material substance other than thatwhich is divisible, has shape, and can move in every possible way, and thisthe geometers call quantity and take as the object of their demonstrations.Moreover, our concern is exclusively with the division, shape and motionsof this substance, and nothing concerning these can be accepted as trueunless it be deduced from indubitably true common notions with suchcertainty that it can be regarded as a mathematical demonstration. Andbecause all natural phenomena can be explained in this way, as one canjudge from what follows, I believe that no other physical principles shouldbe accepted or even desired.
4This seems to be as clear a statement as one could wish for of a method whichstarts from first principles and builds up knowledge deductively. Observation,experiment, hypotheses, induction, the development and use of scientificinstruments, all seem to be irrelevant to science as described here. Empirical orfactual truths seem to have been transcended, and all science seems to be in therealm of truths of reason. One gains a similar impression from a number of otherpassages in Descartes, and in the
Sixth Meditation, for example, we are presentedwith a picture of the corporeal world in which—because we are only allowed toask 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 littlemore than materialized geometry. The deduction of the features of such a worldfrom first principles is not too hard to envisage.Nevertheless, there are serious problems with the idea that Descartes isadvocating an
a priori, deductivist method of discovery, and I want to drawattention to four such problems briefly. First, there is the sheer implausibility ofthe idea that deduction from first principles could generate substantive andspecific truths about the physical world. The first principles that Descartes startsfrom 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. Nowby the end of the
Principles of Philosophy he has offered accounts of suchphenomena as the distances of the planets from the Sun, the material constitutionof the Sun, the motion of comets, the colours of the rainbow, sunspots, solidityand fluidity, why the Moon moves faster than the Earth, the nature oftransparency, the rarefaction and condensation of matter, why air and water flowfrom east to west, the nature of the Earth’s interior, the nature of quicksilver, thenature of bitumen and sulphur, why the water in certain wells is brackish, thenature of glass, magnetism, and static electricity, to name but a few. Could itseriously be advocated that the
cogito and the existence of a good God would besufficient to provide an account of these phenomena? Philosophers, likeeveryone else, are occasionally subject to delusions, and great claims have beenmade for various philosophically conceived scientific methods. But it is worthremembering in this connection that Descartes is of a generation where methodis not a reflection on the successful work of other scientists but a very practicalaffair designed to guide one’s own scientific practice. It is also worthremembering that Descartes achieved some lasting results in his scientific work.In the light of this, there is surely something wrong in ascribing an unworkablemethodology to him.Second, Descartes’s own contemporaries did not view his work as beingapriorist and deductivist, but rather as being committed to a hypothetical modeof reasoning, and, in the wake of Newton’s famous rejection of the use ofhypotheses in science, Descartes was criticized for offering mere hypotheseswhere Newtonian physics offered certainty.
5 A picture of Descartes prevailed inhis own era exactly contrary to that which has prevailed in ours, and, it might benoted, 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 thereforebe 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 tothe amount of empirical and experimental work in which he engaged. Forexample, in 1626 Descartes began seeking the shape of the ‘anaclastic’, that is,that shape of a refracting surface which would collect parallel rays into onefocus. He knew that the standard lens of the time, the biconvex lens, could not dothis, and that the refracting telescope constructed with such a lens was subject toserious problems as a result. He was convinced on geometrical grounds that therequisite shape must be a hyperbola but he spent several years pondering thepractical problems of grinding aspherical lenses. In two detailed letters to JeanFerrier (a manufacturer of scientific instruments whom Descartes was trying toattract to Holland to work for him grinding lenses) of October and November1629, he describes an extremely ingenious grinding machine, with details as tothe 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 andfinishing-off, and so on. These letters leave one in no doubt that their author is anextremely practical man, able to devise very large-scale machines with manycomponents, and with an extensive knowledge of materials, grinding and cuttingtechniques, not to mention a good practical grasp of problems of friction andvibration. 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 andconstruct scientific instruments and experiments. Is it really possible thatDescartes’s methodological prescriptions should be so far removed from hisactual scientific practice?Fourth, Descartes has an extremely low view of deduction. He rejectsAristotelian syllogistic, the only logical formalization of deductive inference hewould 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 anysyllogism which has a true conclusion, unless they already have the wholesyllogism, i.e. unless they have already ascertained in advance the verytruth which is deduced in that syllogism.
6Much more surprisingly, for syllogistic was generally reviled in the seventeenthcentury, he also rejects the mode of deductive inference used by the classicalgeometers, synthetic proof. In Rule 4 of the
Rules for the Direction of Our NativeIntelligence, he complains that Pappus and Diophantus, ‘with a kind of lowcunning’, kept their method of discovery secret, presenting us with ‘steriletruths’ which they ‘demonstrated deductively’.
7 This is especially problematicfor the reading of Descartes which holds that his method comprises deductionfrom first principles, since this is exactly what he is rejecting in the case ofPappus and Diophantus, who proceed like Euclid, working out deductively fromindubitable geometrical first principles. Indeed, such a procedure would be theobvious 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 tomake us question the received view. Before we can provide an alternativereading, however, it will be helpful if we can get a better idea of what exactlyDescartes is rejecting in traditional accounts of method, and what kind of thinghe is seeking to achieve in his scientific writings.
THE REJECTION OF ARISTOTELIAN METHODAristotelian syllogistic was widely criticized from the mid-sixteenth centuryonwards, and by the middle of the seventeenth century had been completelydiscredited as a method of discovery. This was due more to a misunderstandingof the nature and role of the syllogism, however, than to any compelling criticismof syllogistic.Aristotle had presented scientific demonstrations syllogistically, and he hadargued that some forms of demonstration provide explanations or causes whereasothers 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 nearThe planets are nearThe planets are near
That which is near does not twinkleThe planets do not twinkleIn Aristotle’s discussion of these syllogisms in his
Posterior Analytics (
A17), heargues that the first is only a demonstration ‘of fact’, whereas the second is ademonstration of ‘why’ or a scientific explanation. In the latter we are providedwith a reason or cause or explanation of the conclusion: the reason why theplanets do not twinkle is that they are near. In the former, we have a valid but nota demonstrative argument, since the planets’ not twinkling is hardly a cause orexplanation of their being near. So the first syllogism is in some wayuninformative compared with the second: the latter produces understanding, theformer does not. Now Aristotle had great difficulty in providing a convincingaccount 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 ofidentifying those forms of deductive inference that resulted in epistemicadvance, that advanced one’s understanding. Realizing that no purely logicalcriterion would suffice, he attempted to show that epistemic advance dependedon some non-logical but nevertheless internal or structural feature which somedeductive inferences possess. This question of the epistemic value of deductiveinferences is one we shall return to, as it underlies the whole problem of method.For Aristotle, the epistemic and the consequential directions in demonstrativesyllogisms run in opposite directions. That is, it is knowing the premises fromwhich the conclusion is to be deduced that is the important thing as far asproviding a deeper scientific understanding is concerned, not discovering whatconclusions follow from given premises. The seventeenth-centurymisunderstanding of the syllogism results largely from a failure to appreciatethis. It was assumed that, for Aristotle, the demonstrative syllogism was amethod of discovery, a means of deducing novel conclusions from acceptedpremises. In fact, it was simply a means of presentation of results in a systematicway, one suitable for conveying these to students.
8 The conclusions of thesyllogisms were known in advance, and what the syllogism provided was ameans of relating those conclusions to premises which would explain them. Twofeatures of syllogistic are worth noting in this respect.First, the syllogism is what might be termed a ‘discursive’ device. Considerthe 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 theconclusion: the conclusion having been accepted, the syllogism shows how it canbe generated from premises which are more fundamental, thereby connecting whatthe pupil accepts as knowledge to basic principles which can act as anexplanation for it. This reflects a basic feature of the syllogism, whetherdemonstrative or not. Generally speaking, syllogistic works by inducingconviction on the basis of shared assumptions or shared knowledge. For this, oneneeds someone who is convinced and someone who does the convincing.Moreover, the conviction occurs on the basis of shared assumptions, and theseassumptions may in fact be false. The kind of process that Descartes and hiscontemporaries see as occurring in an argument is different from this. Descartes,in particular, requires that arguments be ‘internal’ things: their purpose is to leadone to the truth, not to convince anyone. Correlatively, one can make no appealto what is generally accepted: one’s premises must simply be true, whethergenerally accepted or not.
9 The discursive conception of argument thatAristotelian syllogistic relies upon requires common ground between oneself andone’s opponents, and in seventeenth-century natural philosophy this would nothave been forthcoming. In other words, the case against conceiving of inferencein a discursive way links up strongly with the case against appealing in one’senquiries to what is generally accepted rather than to what is the case. It is fromthis that the immense polemical strength of Descartes’s attack on syllogisticderives.Second, conceived as a tool of discovery, the demonstrative syllogism doesindeed look trivial, but this was never its purpose for Aristotle. Discovery wassomething to be guided by the ‘topics’, which were procedures for classifying orcharacterizing problems so that they could be solved using set techniques. Morespecifically, they were designed to provide the distinctions needed if one was tobe able to formulate problems properly, as well as supplying devices enabling oneto 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 inethics, political argument, rhetoric and so on, and indeed they were meant toapply to any area of enquiry. The problem was that, during the Middle Ages, thetopics came to be associated very closely and exclusively with rhetoric, and theirrelevance to scientific discovery became at first obscured and then completelylost. The upshot of this was that, for all intents and purposes, the results ofAristotelian science lost all contact with the procedures of discovery whichproduced them. While these results remained unchallenged, the problem was notparticularly apparent. But when they came to be challenged in a serious andsystematic way, as they were from the sixteenth century onwards, they began totake on the appearance of mere dogmas, backed up by circular reasoning. It isthis strong connection between Aristotle’s supposed method of discovery and theunsatisfactoriness not only of his scientific results but also of his overall naturalphilosophy that provoked the intense concern with method in the seventeenthcentury.
10DESCARTES’S NATURAL PHILOSOPHYDescartes’s account of method is intimately tied to two features of his naturalphilosophy: his commitment to mechanism, and his commitment to the idea of amathematical physics. In these respects, his project differs markedly from that ofAristotle, to the extent that what he expects out of a physical explanation israther different from what Aristotle expects. This shapes his approach tomethodological issues to a significant extent, and we must pay some attention toboth questions.Mechanism is not an easy doctrine to characterize, but it does have some coretheses. Amongst these are the postulates that nature is to be conceived on amechanical model, that ‘occult qualities’ cannot be accepted as having anyexplanatory value, that contact action is the only means by which change can beeffected, and that matter and motion are the ultimate ingredients in nature.
11Mechanism arose in the first instance not so much as a reaction to scholasticismbut as a reaction to a philosophy which was itself largely a reaction toscholasticism, namely Renaissance naturalism.
12 Renaissance naturalismundermined the sharp and careful lines that medieval philosophy and theologyhad drawn between the natural and the supernatural, and it offered a conceptionof the cosmos as a living organism, as a holistic system whose parts wereinterconnected by various forces and powers. Such a conception presented apicture of nature as an essentially active realm, containing many ‘occult’ powerswhich, while they were not manifest, could nevertheless be tapped and exploitedif only one could discover them. It was characteristic of such powers that theyacted at a distance —magnetic attraction was a favourite example—rather thanthrough contact, and indeed on a biological model of the cosmos such a mode ofaction is a characteristic one, since parts of a biological system may affect oneanother whether in physical contact or not. The other side of the coin was aconception of God as part of nature, as infused in nature, and not as somethingseparate from his creation. This encouraged highly unorthodox doctrines such aspantheism, the modelling of divine powers on natural ones, and so on, and, worstof all, it opened up the very delicate question of whether apparently supernaturalphenomena, such as miracles, or phenomena which offered communion withGod, such as the sacraments and prayer, could receive purely naturalisticexplanations.
13 It is important that the conception of nature as essentially activeand the attempts to subsume the supernatural within the natural be recognized aspart of the same problem. Mechanists such as Mersenne14 saw this clearly, andopposed naturalism so vehemently because they saw its threat to establishedreligion. Mersenne himself also saw that a return to the Aristotelian conception ofnature that had served medieval theologians and natural philosophers so well wasnot going to be successful, for many of the later Renaissance naturalists hadbased their views on a naturalistic reading of Aristotle which, as an interpretationof Aristotle, was at least as cogent as that offered by Christian apologists likeAquinas. The situation was exacerbated by a correlative naturalistic thesis aboutthe nature of human beings, whereby the soul is not a separate substance butsimply the organizing principle of the body. Whether this form of naturalism wasadvocated in its Averroistic version, where there is only one intellect in theuniverse because mind or soul, lacking any principle of individuation in its ownright, cannot be divided up with the number of bodies, or whether it wasadvocated in its Alexandrian version, where the soul is conceived in purelyfunctional terms, personal immortality is denied, and its source in both versionsis Aristotle himself.
15 Such threats to the immortality of the soul were noted bythe Fifth Lateran Council, which in 1513 instructed Christian philosophers andtheologians to find arguments to defend the orthodox view of personalimmortality.Something was needed which was unambiguous in its rejection of nature as anactive realm, and which thereby secured a guaranteed role for the supernatural.Moreover, whatever was chosen should also be able to offer a conception of thehuman body and its functions which left room for something essentially differentfrom the body which distinguished it from those animals who did not share inimmortality. Mechanism appeared to Mersenne to answer both these problems,and in a series of books in the 1620s he opposes naturalism in detail and outlinesthe mechanist response.
16Descartes’s response is effectively the same, but much more radical in itsexecution. Like Mersenne, he is concerned to defend mechanism, and the idea ofa completely inert nature provides the basis for dualism. Dualism in turn is whathe considers to be the successful way of meeting the decree of the LateranCouncil, 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 completelyinert, a separation of mind and body as radical as it could be, and an image of aGod who is so supernatural, so transcendent, that there is almost nothing we cansay about him.In accord with the mechanist programme, the supernatural and the natural aresuch polar opposites in Descartes that there is no question of the latter havingany degree of activity once activity has been ascribed to the former. Hiscontemporaries were especially concerned to restrict all causal efficacy to thetransfer of motion from one body to another in impact but Descartes goesfurther. While he recognizes the need for such a description at the phenomenallevel, at the metaphysical level he does not allow any causation at all in thenatural realm.
17 Motion is conceived as a mode of a body just as shape is, andstrictly speaking it is not something that can be transferred at all, any more thanshape can be. The power that causes bodies to be in some determinate state ofrest or motion is a power that derives exclusively from God, and not from impactwith other bodies. Moreover, this power is simply the power by which Godconserves the same amount of motion that he put in the corporeal world at thefirst 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 anypower that it does not even have the power to conserve itself in existence. As heputs it in the replies to the first set of objections to the
Meditations, we can findin our own bodies, and by implication in other corporeal things as well, no poweror 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 effectsmust be simultaneous: ‘the concept of cause is, strictly speaking, applicable onlyfor as long as it is producing its effect, and so is not prior to it.’
19 My existence atthe present instant cannot be due to my existence at the last instant any more thanit can itself bring about my existence at the next instant. In sum, there can be nocausal connections between instants, so the reason for everything must be soughtwithin the instant:
20 and since no such powers are evident in bodies, they must belocated in God. Such an inert corporeal world certainly contains none of thepowers that naturalists saw as being immanent in nature, it is not a world inwhich God could be immanent, and it is, for Descartes and virtually all of hiscontemporaries (Hobbes and Gassendi being possible exceptions), a world quitedistinct from what reflection on ourselves tells us is constitutive of our natures,which are essentially spiritual. And it also has another important feature: a worldwithout forces, activities, potentialities and even causation is one which is easilyquantifiable.This brings us to the second ingredient in Descartes’s natural philosophy: hiscommitment to quantitative explanations. This is often seen as if it were anecessary concomitant of mechanism, but in fact mechanism is neither sufficientnor necessary for a mathematical physics. Most mechanists in the early to midseventeenthcentury offered mechanist explanations which were almostexclusively qualitative: Hobbes and Gassendi are good cases in point. Moreover,Kepler’s thoroughly Neoplatonic conception of the universe as being ultimately amathematical harmony underlying surface appearances could not have beenfurther from mechanism, yet it enabled him to develop a mathematical account inareas such as astronomy and optics which was well in advance of anything elseat his time. But it cannot be denied that the combination of mechanism with acommitment to a mathematical physics was an extremely potent one, andDescartes was the first to offer such a combination to any significant extent.This question must be seen in a broad context. The theoretical justification forthe use of mathematical theorems and techniques in the treatment of problems inphysical theory is not obvious. To many natural philosophers it was far fromclear that such an approach was necessary, justified, or even possible. Aristotlehad provided a highly elaborate conception of physical explanation whichabsolutely precluded the use of mathematics in physical enquiry and it was thisconception that dominated physical enquiry until the seventeenth century.Briefly, Aristotle defines physics and mathematics in terms of their subjectgenera: physics is concerned with those things that change and have anindependent existence, mathematics with those things that do not change andhave dependent existence (i.e. they are mere abstractions). The aim of scientificenquiry is to determine what kind of thing the subject matter of the science is byestablishing its general properties. To explain something is to demonstrate itsyllogistically starting from first principles which are expressions of essences,and what one is seeking in a physical explanation is a statement of the essentialcharacteristics of a physical phenomenon—those characteristics which it mustpossess if it is to be the kind of thing it is. Such a statement can only be derivedfrom principles that are appropriate to the subject genus of the science; in thecase of physics, this means principles appropriate to explaining what is changingand has an independent existence. Mathematical principles are not of this kind.They are appropriate to a completely different kind of subject matter, andbecause of this mathematics is inappropriate to syllogistic demonstrations ofphysical phenomena, and it is alien to physical explanation. This approachbenefited from a well-developed metaphysical account of the different natures ofphysical and mathematical entities, and it resulted in a physical theory that wasnot only in close agreement with observation and common sense, but whichformed part of a large-scale theory of change which covered organic andinorganic phenomena alike.By the beginning of the seventeenth century, the Aristotelian approach wasbeing challenged on a number of fronts, and Archimedean statics, in particular,was seen by many as the model for a physical theory, with its rigorouslygeometrical demonstrations of novel and fundamental physical theorems. Butthere was no straightforward way of extending this approach in statics (where itwas often possible to translate the problem into mathematical terms in anintuitive and unproblematic way), to kinematics (where one had to deal withmotion, i.e. continuous change of place) and in dynamics (where one hadsomehow to quantify the forces responsible for changes in motion). Moreover,statics involved a number of simplifying assumptions, such as the Earth’s surfacebeing a true geometrical plane and its being a parallel force field. Thesesimplifying assumptions generate all kinds of problems once one leaves thedomain of statics, and the kinds of conceptual problems faced by naturalphilosophers wishing to provide a mathematical physics in the seventeenthcentury were immense.
21In the
Rules for the Direction of Our Native Intelligence, Descartes outlined anumber of methodological and epistemological proposals for a mathematicalphysics. The
Rules, the writing of which was abandoned in 1628, is now thoughtto be a composite text, some parts deriving from 1619–20 (Rules 1–3, part ofRule 4, Rules 5–7, part of Rule 8, possibly Rules 9–11) and some dating from1626–8 (part of Rule 4, part of Rule 8, and Rules 12–21).
22 The earlier partsdescribe a rather grandiose reductionist programme in which mathematics issimply ‘applied’ to the natural world:When I considered the matter more closely, I came to see that the exclusiveconcern of mathematics is with questions of order or measure and that it isirrelevant whether the measure in question involves numbers, shapes, stars,sounds, or any object whatever. This made me realize that there must be ageneral science which explains all the points that can be raised concerningorder and measure irrespective of the subject-matter, and that this scienceshould be termed
mathesis universalis.
23This project for a ‘universal mathematics’ is not mentioned again in Descartes’swritings and, although the question is a disputed one,
24 there is a strong case tobe made that he abandoned this kind of attempt to provide a basis for amathematical physics. The later Rules set out an account of how ourcomprehension of the corporeal world is essentially mathematical in nature, butit is one which centres on a theory about how perceptual cognition occurs.Throughout the
Rules, Descartes insists that knowledge must begin with ‘simplenatures’, that is, with those things which are not further analysable and can begrasped by a direct ‘intuition’
(intuitus). These simple natures can only begrasped by the intellect—pure mind, for all intents and purposes—although inthe case of perceptual cognition the corporeal faculties of sense perception,memory and imagination are also called upon. The imagin ation is located in thepineal gland (chosen because it was believed to be at the geometrical focus of thebrain and its only non-duplicated organ), it is the point to which all perceptualinformation is transmitted, and it acts as a kind of meeting place between mindand body, although Descartes is understandably coy about this last point. In Rule14, Descartes argues that the proper objects of the intellect are completelyabstract entities, which are free of images or ‘bodily representations’, and this iswhy, when the intellect turns into itself it beholds those things which are purelyintellectual such as thought and doubt, as well as those ‘simple natures’ whichare 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 ofthe external world, for the imagination is its point of contact with the externalworld. The imagination functions, in fact, like a meeting place between thecorporeal world and the mind. The corporeal world is represented in the intellectin terms of spatially extended magnitudes. Since, Descartes argues, the corporealworld is nothing but spatially extended body, with the experience of secondaryqualities resulting from the mind’s interaction with matter moving in variousdistinctive ways, the representation of the world geometrically in the imaginationis an entirely natural and appropriate mode of representation. But the contents ofthe mind must also be represented in the imagination and, in so far as the mind isengaged in a quantitative understanding, the imagination is needed in order thatthe mathematical entities on which the intellect works can be rendereddeterminate. For example, the intellect understands ‘fiveness’ as somethingdistinct from five objects (or line segments, or points or whatever), and hence theimagination 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 whichis then mapped onto the geometrical representations of the corporeal world.In this way, our understanding of the corporeal world, an understanding thatnecessarily involves sense perception, is thoroughly mathematical. It should benoted that the intellect or mind, working by itself, could never even represent thecorporeal world to itself, and
a fortiori could never understand it.
DESCARTES’S METHOD OF DISCOVERYIn the
Discourse on Method, Descartes describes the procedure by which he hasproceeded in the
Dioptrics and the
Meteors in the following terms:The order which I have followed in this regard is as follows. First, I haveattempted generally to discover the principles or first causes of everythingwhich is or could be in the world, without in this connection consideringanything but God alone, who has created the world, and without drawingthem from any source except certain seeds of truth which are naturally inour minds. Next I considered what were the first and most common effectsthat could be deduced from these causes, and it seems to me that in thisway 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 mostcommon and simple of all that exist, and consequently the easiest tounderstand. Then, when I wished to descend to those that were moreparticular, there were so many objects of various kinds that I did notbelieve it possible for the human mind to distinguish the forms or species ofbody which are on the earth from the infinity of others which might havebeen, had it been God’s will to put them there, or consequently to makethem of use to us, if it were not that one arrives at the causes through theeffects and avails oneself of many specific experiments. In subsequentlypassing over in my mind all the objects which have been presented to mysenses, I dare to say that I have not noticed anything that I could not easilyexplain in terms of the principles that I have discovered. But I must alsoadmit that the power of nature is so great and so extensive, and theseprinciples so simple and general, that I hardly observed any effect that I didnot immediately realize could be deduced from the principles in manydifferent ways. The greatest difficulty is usually to discover in which ofthese ways the effect depends on them. In this situation, so far as I know theonly thing that can be done is to try and find experiments which are suchthat their result varies depending upon which of them provides the correctexplanation.
25But what exactly is Descartes describing here? We cannot simply assume it is amethod 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 Methodin the Discourse in which I propound it, but only to say enough to showthat the new views in the
Dioptrics and the
Meteors were not randomnotions, and were perhaps worth the trouble of examining. I could notdemonstrate the use of this Method in the three treatises which I gave,because it prescribes an order of research which is quite different fromthe one I thought proper for exposition. I have however given a briefsample of it in my account of the rainbow, and if you take the trouble to rereadit, I hope it will satisfy you more than it did the first time; the matteris, after all, quite difficult in itself. I attached these three treatises [the
Geometry, the
Dioptrics and the
Meteors] to the discourse which precedesthem because I am convinced that if people examine them carefully andcompare them with what has previously been written on the same topics,they will have grounds for judging that the Method I adopt is no ordinaryone and is perhaps better than some others.
26What is more, in the
Meteors itself, Descartes tells us that his account of therainbow is the most appropriate example ‘to show how, by means of the methodwhich I use, one can attain knowledge which was not available to those whosewritings we possess’.
27 This account is, then, clearly worth looking at.The
Meteors does not start from first principles but from problems to besolved, and Descartes then uses the solution of the problem to exemplify hismethod. The central problem in the
Meteors, to which Book 8 is devoted, is thatof explaining the angle at which the bows of the rainbow appear in the sky. Hebegins by noting that rainbows are not only formed in the sky, but also infountains and showers in the presence of sunlight. This leads him to formulatethe hypothesis that the phenomenon is caused by light reacting on drops ofwater. 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 tothe Sun, he holds up the sphere in the Sun’s light, moving it up and down so thatcolours are produced. Then, if we let the light from the Sun comefrom the part of the sky marked AFZ, and my eye be at point E, then whenI put this globe at the place BCD, the part of it at D seems to me wholly redand incomparably more brilliant than the rest. And whether I movetowards it or step back from it, or move it to the right or to the left, or eventurn it in a circle around my head, then provided the line DE always marksan angle of around 42° with the line EM, which one must imagine toextend from the centre of the eye to the centre of the sun, D always appearsequally red. But as soon as I made this angle DEM the slightest bit smallerit did not disappear completely in the one stroke but first divided as into twoless 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, butnot so brilliant…
28Descartes 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 causedby two refractions and one or two internal reflections of the incident ray. He nextdescribes how the same effect can be produced with a prism, and this indicatesthat neither a curved surface nor reflection are necessary for colour dispersion.Moreover, the prism experiment shows that the effect does not depend on theangle of incidence and that one refraction is sufficient for its production. Finally,Descartes calculates from the refractive index of rainwater what an observer wouldsee when light strikes a drop of water at varying angles of incidence, and findsthat the optimum difference for visibility between incident and refracted rays isfor the former to be viewed at an angle of 41°–42° and the latter at an angle of51°–52°,
29 which is exactly what the hypothesis predicts.This procedure is similar to that followed in the
Dioptrics, and in someinspects 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 inDescartes’s work in mathematics. Indeed, the later parts of the
Rules turntowards specifically mathematical considerations, and Rules 16–21 have suchclose parallels with the
Geometry that one can only conclude that they containthe early parts of that work in an embryonic form. Rule 16 advises us to use ‘thebriefest possible symbols’ in dealing with problems, and one of the first thingsthe
Geometry does is to provide us with the algebraic signs necessary for dealingwith 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; andagain one of the first directives in the
Geometry is that we label all linesnecessary for the geometrical construction, whether these be known or unknown.Finally, Rules 18–21 are formulated in almost identical terms in the
Geometry.
30There is something ironic in this, for one would normally associate amathematical 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
OnConic Sections, for example—one finds lists of definitions and postulates anddeductive proofs of theorems relying solely on these. If one now turns toDescartes’s
Geometry, one finds something completely different. After a fewpages of introduction, mainly on the geometrical representation of thearithmetical operations of multiplication, division and finding roots, we arethrown into one of the great unsolved problems bequeathed by Antiquity—Pappus’s locus problem for four or more lines, which Descartes then proceeds toprovide us with a method of solving.Descartes’s solution to the Pappus problem is an ‘analytic’ one. In ancientmathematics, a sharp distinction was made between analysis and synthesis.Pappus, one of the greatest of the Alexandrian mathematicians, had distinguishedbetween two kinds of analysis: ‘theoretical’ analysis, in which one attempts todiscover the truth of a theorem, and ‘problematical analysis’, in which oneattempts 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, theproposed procedure fails to yield what one is seeking, or one can show theproblem to be insoluble, then synthesis is not needed, and analysis is complete inits own right. In the case of positive results, however, synthesis is needed, albeitfor different reasons. Synthesis is a difficult notion to specify, and it appears tohave been used with slightly different meanings by different writers, but it isbasically that part of the mathematical process in which one proves deductively,perhaps from first principles, what one has discovered or shown the truth ofby analysis. In the case of theoretical analysis, one needs synthesis, because inthe analysis what we have done is to show that a true theorem follows from atheorem whose truth we wish to establish, and what we must now do is to showthat the converse is also the case, that the theorem whose truth we wish toestablish follows from the theorem we know to be true. The latter demonstration,whose most obvious form is demonstration from first principles, is synthesis. Asynthetic proof is, in fact, the ‘natural order’ for Greek and Alexandrianmathematicians, the analysis being only a ‘solution backwards’. So what we areinvariably presented with are the ‘naturally ordered’ synthetic demonstrations:there is no need to present the analysis as well. Descartes objects to suchprocedures. He accuses the Alexandrian mathematicians Pappus and Diophantusof presenting only the synthesis from ulterior motives:I have come to think that these writers themselves, with a kind ofpernicious cunning, later suppressed this mathematics as, notoriously,many inventors are known to have done where their own discoveries wereconcerned. They may have feared that their method, just because it was soeasy and simple, would be depreciated if it were divulged; so to gain ouradmiration, they may have shown us, as the fruits of their method, somebarren truths proved by clever arguments, instead of teaching us themethod itself, which might have dispelled our admiration.
31In other words, analysis is a method of discovery, whereas synthesis is merely amethod 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 verystraightforward once one has the analytic demonstration, and indeed in manycases the latter is simply a reversal of the former. Moreover, all equations havevalid converses by definition, so if one is dealing with equations, as Descartes isfor example, then there is no special problem about converses holding. But thesynthetic demonstration, unlike the analytic one, is a deductively valid proof, andthis, for the ancients and for the vast majority of mathematicians since then, isthe only real form of proof. Descartes does not accept this, because he does notaccept that deduction can have any value in its own right. We shall return to thisissue below.The case of problematical analysis with a positive outcome is morecomplicated, for here there was traditionally considered to be an extra reasonwhy synthesis was needed, namely the production of a ‘determinate’ solution. Inrejecting synthesis in this context, Descartes is on far stronger ground. Indeed,one of the most crucial stages in the development of algebra consists precisely ingoing beyond the call for determinate solutions. In the case of geometry, analysisprovides one with a general procedure, but it does not in itself produce aparticular 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 anindeterminate solution, and we need a final synthetic stage corresponding to thegeometrical solution; this is the numerical exploitation of the indeterminatesolution, where we compute determinate numbers. Now in traditional terms the
Geometry is an exercise in problematical analysis, but Descartes completelyrejects the traditional requirement that, following such an analysis, synthesis isneeded to construct or compute a determinate figure or number. For themathematicians of Antiquity this was the point of the exercise, and it was only ifsuch a determinate figure or number could be constructed or computed that onecould be said to have solved the problem. Towards the end of the Alexandrianera, most notably in Diophantus’s
Arithmetica, we do begin to find the search forproblems and solutions concerned with general magnitudes, but these are neverconsidered an end in themselves, and they are regarded as auxiliary techniquesallowing the computation of a determinate number, which is the ultimate point ofthe 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 procedureand the traditional one:It must be pointed out that while arithmeticians have usually designatedeach magnitude by several units, i.e. by a number, we on the contraryabstract from numbers themselves just as we did above [Rule 14] fromgeometrical figures, or from anything else. Our reason for doing this ispartly to avoid the tedium of a long and unnecessary calculation, butmainly to see that those parts of the problem which are the essential onesalways remain distinct and are not obscured by useless numbers. If forexample we are trying to find the hypotenuse of a right-angled trianglewhose 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, andfind that the hypotenuse is ′ (
a2
+ b2), leaving the two parts of theexpression,
a2 and
b2, distinct, whereas in the number they are runtogether…. We who seek to develop a clear and distinct knowledge ofthese 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 itdepends on what has been given, but in fact it is in knowledge of this kindalone that science consists.
32In sum, for Descartes, concern with general magnitudes is constitutive of themathematical enterprise.Descartes’s algebra transcends the need to establish converses, because he isdealing with equations, whose converses always hold, and it transcends thetraditional view that one solves a problem only when one has constructed adeterminate figure or computed a determinate number. But Descartes goes furtherthan this, rejecting the need for deductive proof altogether. The reason why hedoes this lies ultimately not so much in his rejection of synthetic demonstrationsin mathematics but in his conception of the nature of inference. Before we look atthis question, however, it is worth looking briefly at what role deduction doesplay in Descartes’s overall account.
METHODS OF DISCOVERY AND PRESENTATIONIn Article 64 of Part II of
The Principles of Philosophy, Descartes writes:I know of no material substance other than that which is divisible, hasshape, and can move in every possible way, and this the geometers callquantity and take as the object of their demonstrations. Moreover, ourconcern is exclusively with the divisions, shape and motions of thissubstance, and nothing concerning these can be accepted as true unless itbe deduced
(deducatur) from indubitably true common notions with suchcertainty that it can be regarded as a mathematical demonstration. Andbecause all natural phenomena can be explained in this way, as one canjudge from what follows, I believe that no other physical principles shouldbe accepted or even desired.Like the passage from the
Discourse on Method that I quoted above, there is asuggestion here that deduction from first principles is Descartes’s method ofdiscovery. Can we reconcile these and many passages similar to them withDescartes’s rejection of deductive forms of inference, such as synthesis inmathematics 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 problemshave to be posed in quantitative terms and there are a number of constraints on whatform an acceptable solution takes: one cannot posit ‘occult qualities’, one mustseek ‘simple natures’, and so on. The solution is then tested experimentallyto determine how well it holds up compared with other possible explanationsmeeting the same constraints which also appear to account for the facts. Finally,the solution is incorporated into a system of natural philosophy, and the principalaim of a work like the
Principles of Philosophy is to set out this naturalphilosophy in detail. The
Principles is a textbook, best compared not with workslike the
Optics and the
Meteors, which purport to show one how the empiricalresults were arrived at, but with the many scholastic textbooks on naturalphilosophy which were around in Descartes’s time, and from which he himselflearnt whilst a student.
33 Such a textbook gives one a systematic overview of thesubject, presenting its ultimate foundations, and showing how the parts of thesubject are connected. Ultimately, the empirically verified results have to befitted into this system, which in Descartes’s case is a rigorously mechanist systempresented with metaphysical foundations. But the empirical results themselvesare not justified by their incorporation within this system: they are justifiedpurely 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 knowledgeto us, and this is its role in the
Principles.Again, there is something of an irony here, for the kind of misunderstandingof Descartes’s methodological concerns which has resulted in the view that hemakes deduction from first principles the source of all knowledge is rathersimilar to the kind of misunderstanding that Descartes himself fosters in the caseof Aristotle on the one hand and the Alexandrian mathematicians on the other. Inthe case of Aristotle, he takes a method of presentation of results which havealready been established to be a method of discovery. In the case of Pappus andDiophantus, he maintains that a method of presentation is passed off as a methodof discovery. Yet both followers and critics of Descartes have said exactly thesame of him; taking his method of presentation as if it were a method ofdiscovery, they have often then complained that there is a discrepancy betweenwhat he claims his method is and the procedure he actually follows in hisscientific work.
34This suggests that there may be something inherently problematic in the ideaof a ‘method of discovery’. If one compares the kind of presentation one finds inthe
Geometry with what one finds in the
Principles of Philosophy, there is, on theface of it, much less evidence of anything one would call ‘method’ in the formerthan in the latter. Certain basic maxims are adhered to, and basic techniquesdeveloped, in the first few pages of the
Geometry, but the former are really toorudimentary to be graced with the name of ‘method’, and the latter arespecifically mathematical. In the very early days (from around 1619 to the early1620s), when Descartes was contemplating his grand scheme of a ‘universalmathematics’, there was some prospect of a really general method of discovery,for universal mathematics was a programme in which, ultimately, everythingwas reduced to mathematics. But once this was (wisely) abandoned, and themathematical rules were made specifically mathematical, the general content ofthe ‘method’ becomes rather empty. Here, for example, are the rules of methodas they are set out in the
Discourse on Method:The first was never to accept anything as true if I did not have evidentknowledge of its truth: that is, carefully to avoid precipitate conclusionsand preconceptions, and to include nothing more in my judgements thanwhat presented itself to my mind so clearly and so distinctly that I had nooccasion to doubt it. The second, to divide each of the difficulties Iexamined into as many parts as possible and as may be required in order toresolve them better. The third, to direct my thoughts in an orderly manner,by beginning with the simplest and most easily known objects in order toascend little by little, step by step, to knowledge of the most complex, andby supposing some order even among objects that have no natural order ofprecedence.
35There is surely little that is radical or even novel here, and the list is more in thenature of common-sense hints rather than something offering deepenlightenment (unless it is specifically interpreted as a some-what crypticstatement of an algebraic approach to mathematics, in which case it is novel, butit then becomes very restricted in scope and can no longer have any claim to be ageneral statement of ‘method’). The same could be said of Aristotle’s ‘topics’: theytoo offer no systematic method of discovery, and certainly nothing that wouldguarantee success in a scientific enterprise, but rather general and open-endedguidance. But ‘methods of discovery’ do not perform even this modest roleunaided.It is interesting in this respect that, in this passage as in others, Descartes findsit so difficult to present his ‘method of discovery’ without at the same timementioning features appropriate to his method of presentation. The reason forthis lies in the deep connections between the two enterprises, connections whichDescartes seems reluctant to investigate. While it is legitimate to present thedeductive structure of the
Principles of Philosophy as a method of presentationas opposed to a method of discovery, it must be appreciated that the structureexhibited in, or perhaps revealed by, the method of presentation is a structurethat will inevitably guide one in one’s research. It will not enable one to solvespecific problems, but it will indicate where the problems lie, so to speak, andwhich are the important ones to solve: which are the fundamental ones and whichthe peripheral ones. Leibniz was to realize this much more clearly than Descartesever 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 problematicareas in a systematic and thorough way.
36 Failure to appreciate this crucialfeature of deductive structure will inevitably result in a misleading picture inwhich 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. Butsuch a procedure would result in problem-solving of a completely unsystematicand aimless kind, and this is certainly not what Descartes is advocating. Themethod of presentation does, then, have a role in discovery: it complementsdiscovery procedures by guiding their application. The extent to which Descartesexplicitly recognizes this role is problematic, but there can be no doubt that hisaccount of method presupposes it.
THE FUNDAMENTAL PROBLEM OF METHOD: EPISTEMIC ADVANCEThe heart of the philosophical problem of method in Descartes lies not inreconciling his general statements on method with his more specificrecommendations on how to proceed in scientific investigation, or in clarifyingthe relation between his ‘method of discovery’ and his ‘method of presentation’,but in an altogether deeper and more intractable question about how inferencecan be informative. Inference is necessarily involved in every kind of scientificenterprise, from logic and mathematics to natural philosophy, and the wholepoint of these enterprises is to produce new knowledge, but the canonical formof inference, for Descartes and all his predecessors, is deductive inference, and itis a highly problematic question whether deductive inference can advanceknowledge.The question became highlighted in the sixteenth century when there wasintense discussion of the Aristotelian distinction between knowledge how andknowledge why, and the ways in which the latter could be achieved. Turnebus,
37writing in 1565, tells us that the (Aristotelian) question of method was the mostdiscussed philosophical topic of the day. These debates were conducted in thecontext of the theory of the syllogism, and although, with the demise ofsyllogistic, the explicitly logical context is missing from seventeenth-centurydiscussions of method, there is always an undercurrent of logical questions.Descartes raises the question of method in the context of considerations about thenature of inference in the following way in Rule 4 of the
Rules:But if our method rightly explains how intellectual intuition should beused, so as not to fall into error contrary to truth, and how one must finddeductive paths so that we might arrive at knowledge of all things, I cannotsee anything else is needed to make it complete; for I have already saidthat the only way science is to be acquired is by intellectual intuition ordeduction.
38Intellectual intuition is simply the grasp of a clear and distinct idea. But what isdeduction? In Rule 7 it is described in a way which makes one suspect that it isnot necessary in its own right:Thus, if, for example, I have found out, by distinct mental operations, whatrelation 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 willsee what the relation is between
A and
E, nor can the truths previouslylearned give me a precise idea of it unless I recall them all. To remedy thisI would run over them many times, by a continuous movement of theimagination, in such a way that it has an intuition of each term at the sametime that it passes on to the others, and this I would do until I have learnedto pass from the first relation to the last so quickly that there was almost norole left for memory and I seemed to have the whole before me at the sametime.
39This suspicion is confirmed in Rule 14:In every train of reasoning it is merely by comparison that we attain to aprecise 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 aresearching for and what we are given, viz.
A and
C, in respect of the fact thateach 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 truthabout things, it will be better for the reader to reject them altogether and toconceive that all knowledge whatsoever, other than that which consists inthe simple and pure intuition of single independent objects, is a matter ofthe comparison of two things or more with each other. In fact practicallythe 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 torely upon the light of nature alone, in beholding the truth whichcomparison gives us.
40The difference between intuition and deduction lies in the fact that whereas thelatter 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 thededuction so quickly that we no longer have to rely on memory, with the resultthat we grasp the whole in a single intuition at a single time. The core ofDescartes’s position is that by compacting inferential steps until we come to adirect comparison between premises and conclusion we put ourselves in aposition 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 wemust 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 newknowledge, that it result in episte-mic advance. The second is a question aboutwhether deductive inference can be further analysed or explained: it is a questionabout the justification of deduction, but not one which refers us to its epistemicworth for, as Dummett has rightly pointed out, our aim ‘is not to persuadeanyone, not even ourselves, to employ deductive arguments: it is to find asatisfactory explanation for the role of such arguments in our use of language.’
41Now these two kinds of question were not always clearly distinguished inDescartes’s time, and it was a prevalent assumption in the seventeenth century thatsyllogistic, in both its logical and its heuristic aspects, could be justified if andonly if it could show its epistemic worth. But the basis for the distinction wascertainly there, and while the questions are related in Descartes, we can find setsof considerations much more relevant to the one than the other.The first question, that of epistemic informativeness, concerns the use offormalized deductive arguments, especially the syllogism, in the discovery ofnew results in natural philosophy. Earlier, we looked very briefly at howAristotle tried to deal with this question, by distinguishing two different forms ofsyllogism, one scientific because it provided us with knowledge why somethingwas the case, the other non-scientific because it only provided us withknowledge that something was the case. The logicians of antiquity were cruciallyconcerned with the epistemic informativeness of various kinds of deductiveargument, and both Aristotle and the Stoics (founders of the two logical systemsof Antiquity) realized that there may be no logical or formal difference betweenan informative and an uninformative argument, so they tried to capture thedifference in non-logical terms, but in a way which still relied on structuralfeatures of arguments, for example the way in which the premises werearranged. All these attempts failed, and the question of whether deductivearguments can be informative, and if so what makes them informative, remainedunresolved.The prevalent seventeenth-century response to this failure was to argue thatdeductive arguments can never be epistemically informative. Many critics oflogic right up to the nineteenth century criticized syllogistic arguments for failingto yield anything new, where what is meant by ‘new’ effectively amounts to‘logically independent of the premises’. But of course a deductive argument isprecisely 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 inmathematics or in natural philosophy, does not genuinely produce those results.Deduction is merely a mode of presentation of results which have already beenreached by analytic, problem-solving means. This is hard to reconcile, however,with, say, our learning of some geometrical theorem by following through theproof from first principles in a textbook. Even if Descartes could show that onecan never come to know new theorems in the sense of inventing them by goingthrough some deductive process, this does not mean that one could not come toknow them, in the sense of learning something one did not previously know, bydeductive means. Indeed, it is hard to understand what the point of the
Principlescould be if Descartes denied the latter. But in that case his argument againstdeduction as a means of discovery is a much more restricted one than he appearsto think. Moreover, I have already indicated that deduction seems to play aguiding role in discovery, in the sense of invention or ‘genuine’ discovery, inDescartes, because his procedures for problem-solving are quite blind as far asthe ultimate point of the exercise is concerned. Finally, the way in which he setsup the argument in the first place is somewhat question-begging. We arepresented with two alternatives: using his ‘method’, or deduction from firstprinciples. But someone who has a commitment to the value of deductiveinference in discovery, as Leibniz was to have, will not necessarily want to tiethis to demonstration purely from first principles: Leibniz’s view was thatdeduction only comes into use as a means of discovery once one has a verysubstantial body of information (discovered by non-deductive means). This is apossibility that Descartes simply does not account for.On the second question, Descartes’s view is expressed admirably in Rule 4 ofthe
Rules for the Direction of Our Native Intelligence, when he says that ‘nothingcan 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 ofintuition, is unanalysable, simple and primitive. Like the
cogito, which is thecanonical example of an intuition, no further question can be raised about it,whether in justification or explanation. This raises distinctive problems for anytreatment of the nature of deduction. It is interesting to note here the wide gulfbetween Aristotle’s classic account of the justification of deductive principlesand Descartes’s approach. In the
Metaphysics, Aristotle points out that proofsmust come to an end somewhere, for otherwise we would be involved in aninfinite regress. Hence there must be something that we can rely on withoutproof, and he takes as his example the law of non-contradiction. The law isjustified by showing that an opponent who denies it must, in denying it, actuallyassume 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 andnot have a particular property depending on who is perceiving the thing, cannotbe sustained. Descartes can offer nothing so compelling. It is somethingambiguously psychological—the ‘light of reason’ or the ‘light of nature’—thatstops the regress on Descartes’s conception. Whereas Aristotle was concerned, inhis justification, to find a form of argument which was irresistible to anopponent, all Descartes can do is postulate some form of psychological clarityexperienced by the knowing subject. Nevertheless, it was Descartes’s conceptionthat held sway, being adopted in two extremely influential works of the laterseventeenth century: Arnauld and Nicole’s
Port-Royal Logic, and Locke’s
EssayConcerning Human Understanding.
42 The reason for this is not hard to find. TheAristotelian procedure relies upon a discursive conception of inference, wherebyone induces an opponent to accept what one is arguing on the basis of acceptingcertain shared premises: such a mode of argument works both at the ordinarylevel of convincing someone of some factual matter, and at the metalevel ofjustifying the deductive principles used to take one from premises to conclusion.But as I indicated earlier, the discursive conception was generally discredited inthe seventeenth century. It requires common ground between oneself and one’sopponents, and Descartes and others saw such common ground as the root of theproblem 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 whatis the case. The ‘natural light of reason’, on the other hand, provided internalresources by which to begin afresh and reject tradition.The acceptance of this view had disastrous consequences for the study ofdeductive logic. Descartes’s algebra contained the key to a new understanding oflogic. Just as Descartes had insisted that, in mathematics, one must abstract fromparticular numbers and focus on the structural features of equations, soanalogously one could argue that one should abstract from particular truths andexplore the relation between them in abstract terms. Such a move would havebeen tantamount to the algebraic construal of logic, something which isconstitutive of modern logic. But Descartes did not even contemplate such amove, not because of the level of abstraction involved, which would not haveworried him if his work in mathematics is any guide, but because he was unableto see any point in deductive inference.
CONCLUSIONDescartes’s approach to philosophical questions of method was extremelyinfluential from the seventeenth to the nineteenth centuries, and it replacedAristotelianism very quickly. It was part of a general anti-deductivist movement,whether this took the form of a defence of hypotheses (in the seventeenthcentury) or of induction (in the eighteenth and nineteenth centuries). Thisinfluence was transmitted indirectly through Locke, however, and with theinterpretation of seventeenth- and eighteenth-century philosophy in terms of twoopposed schools of thought, rationalism and empiricism, this aspect ofDescartes’s thought tended to become forgotten, and his more programmaticstatements about his system were taken out of context and an apriorist anddeductivist methodology ascribed to him. The irony in this is that Descartes notonly vehemently rejected such an approach, but his rejection goes too far. Iteffectively rules out deduction having any epistemic value, and this is somethinghe not only could not establish but which, if true, would have completelyundermined his own
Principles of Philosophy. But this is not a simple oversighton Descartes’s part. It reflects a serious and especially intractable problem, orrather set of problems, about how deductive inference can be informative, whichDescartes was never able to resolve and which had deep ramifications for hisaccount of method.
NOTES1 This is, at least, the usual view. For a dissenting view see G.Canguilhem,
Laformation du concept de réflexe an XVIIe et XVIIIe siècles (Paris, PressesUniversitaires 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 toHume: 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 argumentwith a distinctively discursive structure: in his treatment of scepticism. Scepticalarguments have a distinctive non-logical but nevertheless structural feature. Theyrely upon the interlocutor of the sceptic to provide both the knowledge claims andthe definition of knowledge (i.e. the premises of the argument). The sceptic thenattempts to show a discrepancy between these two. Were the sceptic to provide thedefinition of knowledge, or to provide knowledge claims or denials that certainthings are known, the ingenious dialectical structure of sceptical arguments wouldbe undermined. This is a very traditional feature of sceptical arguments, and it is asign of Descartes’s ability to handle it that he not only uses it to undermineknowledge claims in the
First Meditation, but he also uses it to destroy scepticismin making the sceptic provide the premise of the
cogito, by putting it in the form ‘Idoubt, therefore I exist’. In other words he is able to turn the tables on the scepticby using the same form of argument that makes scepticism so successful. But, ofcourse, once he has arrived at his foundation, he shuns this form of argument, fornow he has premises he can be certain of and so he no longer has any need to useforms 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 thefocus of the sixteenth-century discussion is different, and it derives from Aristotle’sdistinction between scientific and non-scientific demonstration. On the sixteenthcenturydisputes, see Neal W.Gilbert,
Renaissance Concepts of Method (NewYork, Columbia University Press, 1960).11 For more detail on the problems of defining mechanism, see J.E.McGuire, ‘Boyle’sConception of Nature’,
Journal of the History of Ideas 33 (1972) 523–42; and AlanGabbey, ‘The Mechanical Philosophy and its Problems: Mechanical Explanations,Impenetrability, and Perpetual Motion’, in J.C.Pitt (ed.)
Change and Progress inModern Science (Dordrecht, Reidel, 1985), pp. 9–84.12 See K.Hutchison, ‘Supernaturalism and the Mechanical Philosophy’,
History ofScience 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 andtheology he played a major role in co-ordinating and making known current workin natural philosophy from the mid-1620s onwards. He attended the same school asDescartes but their friendship, which was to be a lifelong one, began only in themid-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 MiddleAges and Renaissance. The naturalism of Alexander of Aphrodisias (fl. AD 200),the greatest of the Greek commentators on Aristotle, does not seem to have beentaken seriously until the Paduan philosopher Pietro Pomponazzi (1462–1525) tookit up, and even then it was never explicitly advocated as a doctrine that presents thewhole 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 ofPhilosophy 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 inDescartes’, 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 alienablein [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: Conceptsof 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, seeGilson [5.20]. On the textbook tradition more generally, see Patricia Reif, ‘TheTextbook Tradition in Natural Philosophy, 1600–1650’,
Journal of the History ofIdeas 30 (1969) 17–32.34 It is rare to find anyone who not only takes the deductive approach at face valueand also believes it is viable, but there is at least one notable example of such aninterpretation, 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 translatedin 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 deFrance. He was one of the leading humanist translators of his day, and had anextensive 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, theBritish 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).
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