Clarity, Silence, and the Subjects	Alex Csiszar

Of Descartes's Mathematics	Mar. 24, 2003

I	Writing Mathematical Clarity

These disciplines [arithmetic and geometry] are, therefore, the easiest and most perspicuous of all, and they possess the kind of object that we require, since it would hardly seem human to err in them – except through inadvertence.

—Descartes, Rules for the Direction of the Natural Intelligence

Advertissement

Until now I have tried to render myself intelligible to everyone, but for this treatise, I'm afraid that only those who are familiar with the books on geometry will be capable of reading me. For since they contain many well-demonstrated truths, I thought that it would be superfluous to repeat them here.

—Prefatory note to 1st ed. of Descartes's la Géométrie

How is it that la Géométrie, a text presented by Descartes as exemplary for his method of philosophizing about clear and distinct ideas, was considered to be so unclear by many of his readers (including himself)? It won't do to say that the difficulty arises because Descartes failed to write his mathematical text clearly, or that he was simply a bad writer, for he appears to have designed his Géométrie to be difficult to understand. Perhaps for Descartes clarity is a property of ideas, or of methodology, whereas intelligibility is a property of writing. But in that case what, if any, connection might there be between a method for doing philosophy in a perspicuous way, and a method for writing philosophy intelligibly?

The questions raised by Descartes's seemingly paradoxical statements about the relationship of his mathematical works to the rest of his philosophical program turn out to be significant for the development of certain genres of mathematical writing. While the content of Descartes's la Géométrie was central to the development of certain fields of mathematics, it is more difficult to say in what way the structure or form of the work influenced later mathematics. Descartes was consciously offering an alternative to the classical tradition of presenting theorems with proofs (what he called the synthetic method), by using what he called the analytic method. Of course, Descartes's alternative stylistic precepts eventually lost out to a style of presentation based on theorems, proofs, and axioms, thus making his presentational style appear to be more like an odd aberration than a mainroad in the history of mathematical writing. But still some things stuck: Particularly recognizable is a certain form of argument geared toward producing mathematical readers who are also mathematical writers.

II	Mathematical Subjects

One way to formulate this project is that we are looking to delineate what kind of subject Descartes imagined for the audience of his Géométrie, or that the reading of his Géométrie was meant to call into being. For whom was Descartes's geometry intended to be clear? In an essay titled "Descartes's Geometry as Spiritual Exercise" Matthew Jones has begun just such a project. Jones argues that Descartes intended la Géométrie, and geometry itself, to be a form of spiritual exercise. By this he meant a very general, non-institutional or non-disciplinary form of training in his method of philosophizing which led to the formation of a kind of "elite subject": "Descartes put forth a set of practices for producing an early seventeenth-century elite subject, removed from the corruption of institutions, narratives, proof, and deceptions, and therefore not needing external discipline." 1 The disciplining that the practice of geometry would impart to a reader was meant to stand in for the institutional or disciplinary structure of knowledge itself. But there is something logically fishy about this: Why should one discipline (mathematics) play such a fundamental role in the un-disciplining of knowledge? And this notion of a non-disciplinary elite subject seems strangely empty; surely we could fill it out in more detail. Jones (along with other commentators on la Géométrie2) laments the fact that la Géométrie has almost always been published separate from the Discourse, since the works were published together and thus meant to be read together, the latter as an illustration of the former, and the former as motivation for the latter. But whatever Descartes's intentions might have been, it still remains that this is just not how they ended up being used: the treatise on method attracted a different set of readers from the treatise on geometry. It is worthwhile to investigate whether there might be factors internal to the texts that dictated this bifurcation of readership. For example, despite the joint publication, Descartes does not seem to have expected many of his readers to read la Géométrie. In the "Advertissement" cited above, Descartes explicitly singles out la Géométrie, which has elsewhere served as the paragon of clarity, of the simplest subject matter, as just that treatise that he will not bother making intelligible to everyone. (This is also perhaps why geometry, the science that Descartes insisted one should learn first, came last in the volume of four essays.) Seeing as it was probably mathematicians who were familiar with the "books on geometry", Descartes seems to have imagined a particularly restricted readership for his mathematical text.3 Consider the passage from la Géométrie that provided Jones with his title and central thesis:

And one can always thus reduce all unknown quantities to a single one, whenever the problem can be constructed by circles and straight lines, or by conic sections, or even by some other line that is only one or two degrees more complex. But I will not stop to explain this in more detail, since I would be keeping you from the pleasure of learning it for yourself, as well as the usefulness of cultivating your esprit in exercising yourself in it, which is, in my opinion, the principle thing to be taken from this science. Moreover, I can find nothing here that is so difficult that those who are somewhat versed in common geometry and algebra, and who will pay close attention to all that is in this treatise, would not be able to discover.4

These remarks are the keystone for Jones's interpretation of la Géométrie as a means of spiritual exercise, in which the ideal reader transcends disciplinary boundaries, and perhaps explodes the concept of disciplinarity itself. But consider the context of these remarks: the passage occurs in the first few pages of Book 1, capping a very quick discussion (c. 2 pp.) which outlines the method that Descartes recommends for the solution of geometric problems, with the claim that any problem whose solution can be constructed by certain acceptable curves, can be reduced to having to solve an algebraic equation in one unknown. The claim is somewhat problematic since at such an early point in the book, where we know so little about Descartes's notions of constructibility, degrees of complexity, or admissible equations, its scope is difficulty to pin down. He may simply be making the basic claim that it is possible to reduce a given set of equations in the same number of unknown quantities to one equation in one unknown. But it is more likely that the claim is really that any appropriate geometrical problem can be reduced to solving (by means of a geometric construction) a certain algebraic equation in one unknown. This is all the more likely given that the prerequisites that Descartes sets out for understanding the claim include not just knowledge of ordinary geometry and algebra, but an attentive reading of his entire essay! Thus this directive to the reader is not at all the kind that we might have expected, involving perhaps routine operations or lemmas that the writer expects the reader to work out on their own, if they are concerned. Nor does it involve some well-defined geometric problem. Descartes expects his reader to be able to verify this claim after having studied the entire work. It is not just a minor point, but a key claim for the universal applicability of his method in geometry. Sizable chunks of the essay are concerned with just such algebraic reductions for the purposes of eventually constructing solutions to problems, but nowhere in the book is there an explicit proof that this always works. Descartes's analytic method does not proceed from theorem to proof, but instead provides illustrations of relatively general problems which provide the elements, at least ideally, to infer a proof if one is desired. It is the latter task that Descartes leaves to his reader here.

Many things can be said about this strange rhetorical move, but I will single out one in particular. The exercise that the reader would get from such a demonstration is not, as might have been expected, the kind that could take place independently of Descartes's text; rather it must be based on a kind of translation of Descartes's claims into the realm of certain and completely general claims about geometrical problems. It is an exercise that will not simply involve exercising one's esprit in some universalized sense, but will train one to read mathematics, and perhaps philosophy, in a particular way.

The remainder of this essay will explore the issues raised by these initial observations in greater detail. III will briefly offer one important solution to the clarity problem by drawing some conclusions from various scholars' interpretations of the relationship between the philosophical works on methodology and la Géométrie. The next section will problematize this initial conclusion by considering the roles that reading and writing play in Descartes's conceptions of philosophy and mathematics. Finally V will look closely at la Géométrie itself, focusing on Descartes's expository strategies, and especially those moments when Descartes chooses to leave things unsaid.

III	Descartes's Method and la Géométrie

As many scholars have pointed out, there are important ways in which Descartes's work in mathematics and his development of a general philosophical method informed one another. The three documents most obviously relevant to these kinds of parallels (in order of importance) are the Rules for the Direction of the Natural Intelligence, the "Second Set of Replies" accompanying the Meditations, and the Discourse on Method.5 All of these invoke geometry and arithmetic as models worthy of emulation. I will briefly review some of the key ways in which these two Cartesian genres (mathematical and philosophical/methodological) connect to one another. We should keep in mind however that the emphasis historians have placed on tracing the parallels between these two genres has obscured other aspects of the form of Descartes's that don't fit as well with his methodological precepts.

It is in the Rules that Descartes gives his most systematic (although unfinished) treatment of his method, and the central role that geometry is to play in it. Descartes's method for solving philosophical problems involves a reductive step and a constructive step.6 The reductive step starts with a complicated and obscure problem or set of propositions and reduces it in steps to simpler ones, until some simplest idea is reached which can be clearly intuited by the mind. The second, constructive step, builds up from the base level of simple ideas toward a certain answer to the original proposition. The latter thus consists of a deductive procedure guided by the original path taken by the reductive step.

The mathematical analogue to this process is quite specific, and may be summarized as follows:

Given any acceptable geometrical problem:7

1) Assume that the solution is determinate and exists.

2) Reductive step A: Translate the geometrical relations into abstract algebraic relations.

3) Reductive step B: Reduce the algebraic relations (equations) to their simplest standard form. (Descartes discusses certain standard forms of polynomial equations in one unknown.)

4) Constructive step: Geometrically construct the solution(s) of the resulting equation.

The moral for Descartes in using his methodology is that there is no value to proceeding in a random manner, and that the problem one solves should always be as general as possible, for this will make the essence of the problem more apparent. There are two ways in which this universality arises in Cartesian mathematics. First, a geometrical problem based on specific dimensions and angles should be transformed into a general problem involving known and unknown quantities rather than specific numbers. Second, the specificity of geometrical spatial relations should be reduced to abstract algebraic quantities, which could, in principle, stand in for any form of measurement. Once the more generalized problem has been solved, one can go back and fill in the numbers for the specific case.

La Géométrie discusses in detail the ways in which this method can be applied to all acceptable geometrical problems, and classifies and delimits this set of problems based on the kinds of algebraic equations the use of the method calls for in their solution. Thus insofar as the text is concerned with testing and demonstrating the method described above, the latter can be seen as the 'method' of la Géométrie. However, and this is a crucial point, this method for solving geometrical problems cannot possibly be the method that Descartes uses for his own geometrical treatise. Descartes's text is not, in any straightforward sense, a set of geometric problems and solutions employing his method, but is a set of claims and illustrations directed toward justifying the validity and completeness of the method itself. That is to say, la Géométrie is about Descartes's method for solving geometrical problems. Thus, Descartes's method is the content of la Géométrie, but not its form, not its method. Philosophical method has become mathematical content.

While this is a distinction that most commentators on la Géométrie don't appear to have made, it is implicit in many of their arguments. What begins as a discussion of philosophical methodology tends to turn into a discussion of curve classification, the logical status of algebra, or, as in Tim Lenoir's apt phrase, the "geometrization of thought."

This discussion suggests a way to explain Descartes's numerous and paradoxical comments about clarity. Following are some instances in which Descartes invokes clarity and distinctness:

. . . we require two things for mental intuition, namely, first, that the proposition be clearly and distinctly understood, and, then, that it as a whole be understood simultaneously, and not successively.9

[The two principal faculties of the natural intelligence are] perspicacity in the distinct intuition of individual things, and sagacity in the artful deduction of the ones from the others.10

But the chief ground of my satisfaction with this method, was the assurance I had of thereby exercising my reason in all matters, if not with absolute perfection, at least with the greatest attainable by me: besides, I was conscious that by its use my mind was becoming gradually habituated to clearer and more distinct conceptions of its objects; and I hoped also, from not having restricted this method to any particular matter, to apply it to the difficulties of the other sciences, with not less success than to those of algebra.11

Clarity and distinctness arise from the intuition of simple objects or propositions. Deduction, which consists in the continuous motion of thought (as in a linked chain) can only share in this clarity and distinctness if the chain can be apprehended simultaneously, rather than successively. In neither the case of intuition, nor of deduction, does this necessarily have anything to do with any act of communication, but is a property of a mental representation or process. As H. J. K. Bos observes, "if we look in La Géométrie for the 'long chains of reasoning' mentioned in the Discourse, we find them not in logical deduction from axiom to theorem but in the chains of motions that combine to trace the curves that are acceptable in geometrical constructions".12	It is this concrete and visual interpretation of clarity that Matt Jones focuses on in his discussion of Cartesian exercise. As Jones puts it:

Descartes apparently saw the connected chain of interdependent motions [of mathematical instruments] as more or less equivalent to a chain of clear and distinct mathematical reasons each yielding something as clear and distinct as the previous . . . The material compass and its products offered, as he suggested, the best exemplar of Descartes's deeply problematic notion of clarity and distinctness and his idiosyncratic view of deduction.13

For Jones, the spiritual exercise that Descartes imagines for his disciples are connected to both literal and imagined constructions of geometrical curves. The clear and distinct ideas in geometry aren't generated by Descartes's geometrical treatise, but by one's actually working out a geometrical problem in the manner that Descartes advocates. Given this reasoning, it makes sense to expect that Descartes would give his readers exercises providing training in his method so that their minds could become "habituated to clearer and more distinct conceptions of its objects."

It is impossible to deny that when Descartes envisioned clarity and distinctness with respect to mathematics he often had in mind these acts of geometric construction. However, there are serious holes in this tidy explanation. First, it turns out that Descartes does not always distinguish between these acts of intuitive clarity and the clarity that might arise from the acts of reading and writing themselves. Second, the kinds of gaps or exercises that Descartes explicitly leaves for his reader are almost never ones that would require an application of his method, and it is often not clear how they would conceivably lead to the intuition of clear and distinct ideas in the sense of the above passages. They are usually not even directly related to the actual construction of geometric solutions.

Moreover, this interpretation leads to a serious contradiction at the heart of Descartes's aims, which no one seems to have noticed. If Descartes wants to provide his disciples with a method that allows them to solve any geometrical problems, then in what sense is what he is offering not a calculative, passive procedure?14 Likewise, if, as Descartes often claimed, the ancients really did conceal their method, then they would presumably have forced their readers to rediscover the method for themselves; the reading experience would therefore not be passive in the least. (However, Descartes's comments strongly suggest that the way in which the ancients intended their texts to be read, or at least the way in which they were currently used in the schools, did indeed discourage active readers, and encouraged passivity.) But if Descartes really has provided a method that can be used to solve any proper geometrical problem (in fact, it wouldn't be an exaggeration to say that Descartes redefines the bounds of geometry in terms of the kinds of problems his method can be used to solve!), and presents this method in such a way that anyone can learn it, then he is very near to having produced a calculus of geometry, which is clearly not his avowed aim.15 In order to solve these problems, we need to look closer at how Descartes himself understood the role of his mathematical writing, as well as the role of writing in general.

IV	Mathematics and Writing

Analysis is a key term Descartes uses when characterizing his mathematical method. Usually (geometrical) analysis is paired with algebra, and even if it's hard to pin down exactly what it refers to, it generally connotes some abstract method of reasoning that the classical geometers may have possessed.

We encounter this [spontaneous fruit] in the easiest of sciences, to wit, in arithmetic and in geometry. For we are well aware that the ancient geometers employed a kind of analysis which they extended to the resolution of all problems, although they may have begrudged leaving it to posterity. And now a certain sort of arithmetic, which one calls 'algebra', is flourishing, aiming to achieve that with respect to number which the ancients did with respect to figures.16

In the early texts on method, Descartes's invocations of "analysis" almost always allude to his belief that the ancient geometers possessed an art of analysis but never wrote it down. Thus, as in the above passage, there is a sharp distinction between the method itself and the presentation of it. However, in the most well-known statement of Descartes's suspicion of the ancients' modes of concealment, there is a hint that their method of analysis is not only a means of doing mathematics, but really is the content of true mathematics. In that case, concealing the art of analysis is as good as concealing the discoveries themselves:

None the less, I have come to believe that [the true] mathematics was later suppressed by [the ancient] writers themselves with a kind of pernicious cunning. For, as many artificers are known to have done with respect to their own inventions, they may perhaps have feared that their mathematics, because it was very easy and simple, would be depreciated if it were divulged to the commonality. In lieu of it, they have preferred to show us some sterile truths cleverly demonstrated from inferences, as it were, the results of their art, in order that we might admire them, rather than to teach us this art itself, which might have completely dispelled the admiration.17

Descartes is here arguing for a completely different focus for what mathematical knowledge should consist in. Instead of consisting of solutions to problems, or clever proofs of theorems, true mathematics should be understood as consisting of descriptions of the methods used to find solutions, or to find proofs of theorems. In these terms, the ancients, by concealing their method, actually failed to achieve anything of mathematical significance.

In the much later "Second Set of Replies" that accompanied the Meditations, Descartes shifts his concept of analysis even more toward issues related to presentation. In responding to a critique of his unorthodox philosophical style in the Meditations, he provides a description of the difference between the analytic and the synthetic method. At first site, this distinction appears to map onto the reduction / construction model of Descartes's method that I described above.18 However this mapping does not fit as well as could be hoped, since the distinction here is explicitly one between modes of writing, or rhetorical strategies, rather than method, plain and simple:

Analysis shows the true way by means of which the thing in question was discovered methodically and as it were a priori, so that if the reader is willing to follow it and give sufficient attention to all points, he will make the thing his own and understand it just as perfectly as if he had discovered it for himself. But this method contains nothing to compel belief in an argumentative or inattentive reader; for if he fails to attend even to the smallest point, he will not see the necessity of the conclusion. Moreover there are many truths which – although it is vital to be aware of them – this method often scarcely mentions, since they are transparently clear to anyone who gives them his attention.

Synthesis, by contrast, employs a directly opposite method where the search is, as it were, a posteriori (though the proof itself is often more a priori than it is in the analytic method). It demonstrates the conclusion clearly and employs a long series of definitions, postulates, axioms, theorems and problems, so that if anyone denies one of the conclusions it can be shown at once that it is contained in what has gone before, and hence the reader, however argumentative or stubborn he may be, is compelled to give assent. However, this method is not as satisfying as the method of analysis, nor does it engage the minds of those who are eager to learn, since it does not show how the thing in question was discovered.

It was synthesis alone that the ancient geometers usually employed in their writings. But in my view this was not because they were utterly ignorant of analysis, but because they had such a high regard for it that they kept it to themselves like a sacred mystery.

Now it is analysis which is the best and truest method of instruction, and it was this method alone which I employed in my Meditations.19

There is much that could be said about this passage, so I will be selective. Although this description is being marshaled in defense of his style of writing in the Meditations, throughout the discussion Descartes makes it clear that he is thinking specifically of models of writing drawn from mathematics. Descartes characterizes these models of writing in part by which elements are to be included, and which are to be excluded. Analysis shows how (in terms of method) a certain truth or object was discovered. But, because they should be clear to an attentive reader, analysis omits mentioning many truths that the reader will need to be aware of. (This kind of clarity is figured as that which is specifically not written.) Synthesis on the other hand demonstrates conclusions clearly by giving all the intermediate steps but conceals the way in which the conclusion was discovered. (This version of clarity is the kind that is necessarily written down.)

Descartes also distinguishes the two methods according to the kinds of readers they are made for: analysis is for those who are eager to learn, and synthesis is for people who are argumentative and perhaps inattentive, and not really concerned with learning. Descartes is claiming to use a rhetoric of pedagogy and subject formation rather than of persuasion. Or better: he uses a different brand of persuasion. Anyone willing to "to join me in meditating and giving the subject attentive consideration" 20 will become convinced on their own through a certain sense of satisfaction and pleasure that the method of analysis imparts. Descartes is willing to forego convincing everyone of the correctness of his claims, in order to convince a certain group of their brilliance.

But even here where Descartes is talking of methods of writing, he refers (toward the end of this passage) to analysis as something that the ancients may have possessed but "kept to themselves." Thus analysis appears to be something that is simultaneously a mode of writing and a method for doing mathematics. There is something about mathematics (at least for Descartes) that appears to promote this blending of what might otherwise seem to be two distinct issues.

This strange merger of meanings is not completely restricted to Descartes's later works. At one point in the Rules Descartes goes so far as to describe geometry and arithmetic as modes of presentation of his philosophical method that he has employed simply to make it more congenial to the reader:

[W]hoever has closely attended to my meaning will still easily perceive that I am here thinking [not at all] of the vulgar mathematics, but rather that I am expounding quite another discipline, of which these examples [from arithmetic and geometry] are the shell rather than the parts . . . Yet I have spoken of a 'shell', not because I want to conceal this doctrine and to cover it up in order to protect it from the public, but rather because I want, in this manner, to clothe and to adorn it, so that it could be more accommodating to human natural intelligence.21

Arithmetic and geometry are figured here as a shell which is meant to "clothe and adorn" his abstract philosophical doctrine. It's a fascinating figure, since metaphors of clothing and adornment are usually applied to certain uses of language, particularly to rhetoric or poetry. Furthermore, we know by this point that geometry and arithmetic play a central role in Descartes's method, and that he himself claims to have studied them in order to perfect his method in the first place. It appears that mathematics is intimately connected not only with Descartes's method, but with his conception of writing.

But, generally speaking, what is Descartes's attitude to writing? In the Discourse, Descartes spends a significant amount of time worrying over what he should publish, and when, and what he should write, and why. He generally makes quite a big display of his nervousness about writing. This hesitancy is marked linguistically by a prevalence of phrases such as "je pense pouvoir dire" (Discourse), "je ne craindrai pas d'introduire" (Géométrie), "ut non verear dicere" (Rules), and "j'ose dire" (often). Such hesitations (which often refer paradoxically to acts of non-hesitation) foreground the fact that Descartes is making very deliberate choices about what to write down and what to conceal. At other times Descartes shows a certain contempt for writing: In la Géométrie he often abstains from demonstrating some claim because it is too facile, and he even cuts a demonstration short, complaining "je m'ennuie desia d'en tant escrire." 22

Nevertheless, Descartes does make a number of more specific comments on the roles that writing can serve in his philosophical program. The following is a list of the advantages of writing things down that Descartes cites at various places in his writings:

(1) He thereby communicates his claims, and the reasons for it, to others. But this is valuable not so much for the sake of verification or falsification (he tells us in the Discourse that he doesn't expect to find anyone whose opinion on his writing is really worthwhile) but so that he might set some others along the right path. (See Discourse, pp. 47-8 & "Replies," p. 112.)

(2) Writing as an aid to memory 1: He is disburdening his mind of ideas so that it can move more freely on to new things. (Rules, pp. 195-7)

(3) Writing as an aid to memory 2: Once Descartes is old, he can return to these writings to remember how he came upon various conclusions. (Rules, pp. 97)

(4) Writing can, occasionally, help to work out one's ideas:

This course [writing thoughts down] recommended itself to me, as well because I thus afforded myself more ample inducement to examine them thoroughly, for doubtless that is always more narrowly scrutinized which we believe will be read by many, than that which is written merely for our private use (and frequently what has seemed to me true when I first conceived it, has appeared false when I have set about committing it to writing) . . . 23

In almost all of these cases, writing is figured as an aid or supplement to a process whose primary seat is in one's thoughts or elsewhere. While Descartes is certainly not suspicious of writing in the classic sense of its usurping one's thoughts, speech, or memory, he does seems to think that writing philosophy is not what doing philosophy primarily consists in. Nevertheless, as we have already seen in the discussion of the key term analysis, this distinction is put under a lot of stress in Descartes's mathematics.

The terms algebra and arithmetic (which refer to nearly the same thing for Descartes) also appear to be subject to this doubling of meaning. When Descartes introduces his idea to use arithmetic to solve geometric problems at the beginning of Book 1 of la Géométrie, he claims to do so for the sake of rendering himself more intelligible: "Et je ne craindray pas d'introduire ces termes d'Arithmetique en la Geometrie, affin de me rendre plus intelligibile." 24 For whom is Descartes making himself more intelligible? The reader, presumably. But algebraic symbolism is obviously not just a device that Descartes has used to make things clearer for his readers, subsequent to the actual work of discovery, for much of what is innovative comes about because of his translation of geometrical operations into algebra. As the Rules show, Descartes is completely aware of the significance of the use of algebra for his properly philosophical method. But even in that work, since algebra is fundamentally a system of abstract symbols for Descartes, it is also a way to write:

. . . because memory is often shaky, and so that we might not be forced to spend some part of our attention on refreshing it while we are engaged with other thoughts, art has most conveniently invented the practice of writing. Relying on this as an aid, we shall here commit nothing at all to memory; rather, relinquishing the fantasy freely and completely to the ideas present to it, we shall put down on paper whatever things have to be retained. And we shall do this by means of very concise symbols, in order that, after we have distinctly inspected individual items, in accordance with the ninth rule, we may be able, in accordance with the eleventh rule, to run through them all with the swiftest movement of thought and to intuit simultaneously as many as possible.25

Since the use of algebra, understood as an abstract mode of calculation, is constitutive of Descartes's method, which he claims, significantly, to be revealing, the invocation of intelligibility may even function as a definition. Geometric intelligibility may be defined by the scope of this new method involving algebra. In that case, the use of arithmetic would not only render him intelligible to other readers, but also to himself. The above passage implies that Descartes really did see algebraic writing as also constitutive of those clear continuous motions of thought (which Bos, Jones, and others attribute solely to the geometric constructions). But it is the act of writing for oneself, rather than reading what has been written by another, that is important for Descartes.

At any rate, the connection or between Descartes's philosophical method, his methods of writing, and the role of mathematics, is more fraught with difficulties than most scholars have noted. At this point it will be useful to look more closely at the text itself: What kinds of reading strategies do the expository strategies of la Géométrie entail? What kinds of exercises does Descartes actually leave for the reader? Just what does he reveal, and what does he conceal?

V	Writing and Re-Writing la Géométrie

Since there is not enough space to review the numerous instances of tasks left to the reader in la Géométrie, I will summarize my reading of the text with the following list which characterizes the kinds of things Descartes leaves unsaid:

• Descartes does not leave his reader specific or well-defined problems that can be solved by a straightforward application of his method.26

• He sometimes leaves the enumeration of certain special cases unfinished.27

• It is rare for intermediate steps in a demonstration to be left to the reader. (Descartes does regularly leave out intermediate steps but he doesn't usually call attention to having done so.) Descartes is more likely to end an argument by remarking that the remaining demonstrations (which would establish the generality of his original claim) are easy if one follows the same general plan of attack. (Of course this is often not quite true!)

• The kinds of tasks Descartes explicitly leaves to the reader often relate to the justification of the method itself, rather than to its efficacious employment. As we have already seen, some of the most famous instances of Descartes's leaving a task to the reader are of this kind.

Rather than going into each of these points in detail, I will try to do justice to the intricacy of Descartes's text by focusing in on one section of it in detail.

In Book 3 (pp. 369-413) Descartes uses some particularly strange strategies for the presentation of his material. He begins the Book with a short commentary (2.5 pp) on the importance of simplicity in the choice of construction curves for problems, but then takes a long detour into a theory of equations (pp 371-89, nearly half the length of Book 3) finishing with its application to a certain geometric problem. Descartes introduces this section with the following remarks: "Or affin que je puisse icy donner quelques reigles, pour eviter l'une et l'autre de ces deux fautes, il faut que je die quelque chose en general de la nature des Equations." 28 The mistakes he is referring to are the use of overly complex, and of overly simple, curves to solve a given geometric problem. The long section on equations that follows is not addressed to this question, and in the way that Descartes presents things, it doesn't even seem to have much to do with geometry at all. However, Descartes inserts rare reminders that these algebraic equations are only interesting insofar as they can be applied to geometric problems:

Or quand pour trouver la construction de quelque problesme, on vient a une Equation, en laquelle la quantité inconnue a trois dimensions; premierement . . . 29

Et pourcequ'on ne trouve aucune racine, ny vraye, ny fausse, en ces deux dernieres Equations, on connoist de la que les quatre de l'Equation dont elles procedent sont imaginaires; & que le Problesme, pour lequel on l'a trouvée, est plan de sa nature; mais qu'il ne sçavroit en aucune façon estre construit, a cause que les quantités données ne peuvent se joindre.30

Despite these occasional and rather surprising reminders, this long section of Book 3 is taken up with manipulating polynomial equations, reducing them to certain standard forms, and finding their roots. However, the form that these demonstrations take imposes serious challenges on our ability to determine the scope of Descartes's intentions. The major demonstrations or procedures that Descartes takes up are, in order:31

1) "Descartes's rule of signs" for determining the number of positive roots of a polynomial equation (not proven).

2) A procedure for increasing or diminishing all the roots of a polynomial by a given amount.

3) A procedure for removing the second term of any polynomial equation (by a substitution of variables).

4) A procedure for converting all the roots of an equation into positive roots (which Descartes calls "true" roots).

5) A procedure for multiplying the roots of a polynomial by a given amount.

6) How to solve a polynomial equations of degree 3, when it has a rational root.

7) A procedure for solving certain polynomial equations of degree 4, as follows:

• Remove the second term of the equation as described earlier.

• Convert the resulting equation into a 3rd degree equation (by a substitution of variables).

• Attempt to find a root of the 3rd degree equation.

• If successful, use the root of the 3rd degree equation to rewrite the 4th degree equation (with no 2nd term) as two quadratic equations.

• Solve the resulting quadratic equations.

• Resubstitute the variable from the original 4th degree equation to obtain its root(s).

What is striking about this inventory is that Descartes demonstrates some facts about equations in full generality, others only in part, and completely leaves out numerous other useful tools for equation solving, but he makes little or no comment (until much later) as to why he chooses to include certain facts and not others. This already creates some uncertainty for the reader, and there are other specific ways in which the reading experience is rendered somewhat disorienting. Rather than making his demonstrations using abstract examples or general theorems, Descartes proceeds in large measure via examples. The order of the demonstrations proceeds in a relatively regular fashion, cycling through the following three tasks:

(1) A general claim is made.

(2) The claim is illustrated with a particular example equation (involving numbers).

(3) A second example of greater (although not by any means full) generality is worked out.

It is this third step which is the strangest, for these more general examples that Descartes chooses to use are quite complicated and seemingly randomly so. Later it turns out that most of these examples are not as random as they originally appeared. But they are introduced in such a strange way that it is worth going into some detail to determine why Descartes chose this style of presentation.

In the following remarks I will be referring to certain equations from the text; for the sake of clarity and ease of reference I will simply label them from the outset:

A:

B:

C:

These are some of the example equations Descartes uses to illustrate the various general claims made in his theory of equations. A is introduced (p. 377) to illustrate the derivation of an equivalent equation missing the third term. (There is no indication given why this awkward example is chosen.) B is derived from A using the substitution . A little later on, C is introduced (p. 382) as an example of a 3rd degree polynomial that has a rational root. Not only is this a strangely complicated example for Descartes to choose apparently out of thin air, but for no apparent reason he uses a 6th degree polynomial in his illustration of a rule for 3rd degree polynomials. (It can be treated as a 3rd degree polynomial in terms of y2.)

Further on, Descartes reintroduces B (p. 384) and shows that C may be derived from it using his newly introduced method for reducing equations of 4th degree with no second term to equations of degree 3. It is not likely that the reader will notice that this new example is the same equation that came up earlier in connection with A on p. 377, since Descartes again gives no indication that we have seen it earlier (and when we did see it, all the focus was on equation A).

Two pages later, B is introduced as an example again, this time of the method of finding a root of a 4th degree polynomial with no second term. This time, although Descartes doesn't refer explicitly to the previous uses of B, he clearly does expect the reader to remember them. In order to understand the calculation that he undertakes here, the reader must use the results of literally all of the example cases I have enumerated above. And although equation A does not get explicitly mentioned, in turns out that x, the independent variable from equation A, is what Descartes (unbeknownst to the reader) has been looking for all along:

Now we already have , and therefore x, the quantity in the search for which we have performed all of these operations, is

. (387)

It should be emphasized that this remark is the first explicit cue to the reader that all of these operations have been directed toward some specific end. Of course, as the examples were piling up, and as certain equations seemed to keep arising, the notion that all of this was directed toward some particular purpose slowly became manifest. But it did so only through vague implicatures. We might say that the principle of relevance32 had been put under severe stress throughout this series of demonstrations and the reader would have had to make various desperate attempts to rationalize Descartes's peculiar illustrative choices. But still, not everything has been explained: why would we want to know the solution of equation A anyway?

Descartes doesn't answer (nor does he raise) this question immediately, but moves straight into a geometrical problem:

To emphasize the value of this rule, I shall apply

it to a problem. Given the square AD and the line	[diagram here: see hard copy – AC]

BN, to prolong the side AC to E, so that EF, laid

off from E on EB, shall be equal to NB.33

He goes on to explain that Pappus had posed and solved this problem, but that it would be very unlikely for anyone to come up with Pappus's solution on their own. On the other hand, with the tools Descartes has provided, a solution can be worked out that does not require the seemingly intuitive leap that Pappus's solution seems to require. Using his algebraic method, Descartes derives the equation , whose solution for x will give us the length of DF. As it happens, the latter equation is exactly equation A, which Descartes has recently solved, and which he introduced nearly twenty pages earlier, at that time giving no indication of its significance.

Descartes completes this discussion with a characteristic invitation to the reader to finish the job:

I could also add other rules for equations up to the supersolid, or the sixth degree, or higher, but I would prefer to include them in one rule, and say in general, that when we have tried to reduce the equations to the same form as one having as many dimensions as would result from the multiplication of two of lesser degree, and having tried all the means by which this multiplication is possible, and none having been successful, then one can be assured that the equations cannot be reduced to simpler ones. Thus if the unknown quantity is of the 3rd or 4th degree, the problem one is trying to solve is solid, and if it is of 5th or 6th degree, it is one degree higher, and so on.

Moreover, I have here omitted the demonstrations of what I have said because they seem to me so easy that, as long as you take the pains to examine methodically whether I have been mistaken, they will present themselves to you, and it would be more useful to learn them in this fashion, than by reading them.34

As usual, the demonstrations of the claims that Descartes has just made go far beyond the solution of a few simple problems, and may or may not even be true (depending on how we interpret Descartes; see Bos's Redefining Geometrical Exactness35). The significance of this lengthy analysis may be summarized in the following points:

1) Although Descartes claims to have applied his rules to this problem, in retrospect it turns out that the solution of the problem is what has motivated much of Descartes's "general" discussion of equations, and has been the deciding factor for what to include, and what not to include. Thus the order in which Descartes made his discoveries, and what caused him to make the ones he did, is quite distinct from the way in which his text describes it.

2) Descartes's writing strategy is anything but a straightforward representation of "a continuous motion of thought."

3) Similarly, Descartes's reader is required to employ a highly discontinuous and nonlinear reading strategy, moving backwards and forwards in order to reconstruct the logic of the argument, particularly to understand why Descartes supplies the techniques and methods that he does, and not others.

None of this is to suggest that Descartes blundered in the writing of his mathematical work. There are numerous indications that Descartes imagined something just like this repetitive back and forth movement of the reader over the text.36 We have already seen that a number of the so-called tasks left to the reader are of the variety that require a very attentive and necessarily discontinuous form of reading.37 Moreover, by foregoing a continuous argument in the text, Descartes forces the reader to exercise their mathematical faculty in constructing one for themselves. And because of the rather global character of this non-linearity (the fact that many of Descartes's so-called exercises require an attentive reading of the whole book is a prime example), a successful reader of Descartes's Géométrie is almost certainly also a writer of geometry. Recall Descartes's conception of the continuous movement of thought:

For this reason, I shall run through these relations several times, in a certain continuous movement of thought, simultaneously intuiting individual ones and passing on to the others, until I have learned to pass from the first to the last so swiftly that, leaving room for almost no functions of the memory, I seem to intuit the whole at once. For, in that the memory is relieved in this fashion, the sluggishness of the natural intelligence is also emended, and, in a ceratin manner, the capacity of this intelligence is extended.38

The text does not reproduce this continuous motion, but precisely by withholding it, it forces the reader to produce it for themselves. What Descartes says of the writing style of the Meditations, that it demands strict attentiveness, applies a fortiori to his Géométrie:

I am therefore right to require particularly careful attention from my readers; and the style of writing that I selected was one which I thought would be most capable of generating such attention. I am convinced that my readers will derive more benefit form this than they will themselves realize; for when the synthetic method of writing is used, people generally think that they have learned more than is in fact the case.39

Here Descartes is clearly thinking of his method of writing as analysis, since it is being opposed to the "synthetic method of writing." The method of argument that I have just analyzed in detail is likely a good example of this analytic method since it certainly does seem designed to generate "particularly careful attention." But this method of writing is not quite the same thing as the philosophical method that la Géométrie describes. In the analysis of the little Pappus problem above Descartes opposes his solution to Pappus's because the latter requires some kind of intuitive leap, and his own only requires the use of the right methodological tools. Essentially, while Pappus's solution is more elegant ("it would be much more difficult to arrive at the equation, but it would be simpler to solve"40), Descartes's is more brute force, but it will usually work. The kinds of attentive readers that Descartes imagines for his Géométrie are not ones that would be satisfied solving certain well-delimited problems by means of a method that Descartes has kindly provided.

VI	Conclusions

Matthew Jones states in his essay that he hopes to ". . . escape the long tradition of equating the subject Descartes aimed to create through his exercise with the so-called Cartesian and modern subjects," and thereby to "get at the subjects and objects of Descartes's mathematics and early natural philosophy".41 I couldn't agree more with making this distinction, but I don't think it goes far enough. Descartes imagined more than one kind of subject of natural philosophy and mathematics. There would be those who would exercise themselves in geometrical, and eventually philosophical, problems by using his method, perhaps in just the way that Jones describes. But this is quite distinct from the kind of reading subject Descartes envisioned would engage with his actual geometrical text, a subject who would extend and perfect the mathematical method itself by reading and rewriting it. Of course, since Descartes never explicitly distinguished between these different kinds of subjects, there is reason to suspect that his notion of the ideal mathematical subject was simply overdetermined:

We would be glad, at this point, to encounter a reader favorably disposed toward the study of arithmetic and of geometry, even though I would rather that he had not yet been involved in these disciplines than that he had been taught them in the vulgar manner.42

. . . what makes my treatise so difficult to understand is that I have attempted to put in it nothing that I believed to be known by [Viete], or by anyone else.43

I have only omitted a number of things that would have served to render [la Géométrie] clearer, which I did by design, and I would not have it otherwise.44

It is not clear in just what way Descartes expected his non-mathematically-trained readers to learn his method, since he did not seem to expect many to read la Géométrie, and the Discourse only gave hints of the method. La Géométrie must have figured for those readers as a signifier of a body of knowledge to which they would not have access but which was still necessary if they wanted to engage with Descartes on an even footing in the fields of knowledge. La Géométrie is a book written for a Cartesian mathematical expert, who demonstrates their expertise through specialized acts of reading and writing. And this new subject is emphatically one who has already, in some sense, entered within a certain disciplinary space, constituted by these forms of literary expertise.

Finally, a word on the problem of clarity. Although clarity may sometimes be invoked independently of writing, we have seen that it can't be wholly separated from it. Any claims about clarity necessarily imply a restriction of audience, since we can always ask: Clarity for whom? As we have seen, the clarity that Descartes envisions for his analytic mathematical writing involves strategically placed silences This version of clarity has both an aesthetic45 and a disciplinary component. When Descartes leaves the reader exercises, he does so in part for the sake of their cultivation, and in part for the sake of their pleasure. The last line of his book returns to the latter: "And I hope that posterity will be kind to me, not only for those things that I have explained, but also for those that I have intentionally omitted, in order to leave them the pleasure of discovery."46

This ability to take pleasure in difficulty is constitutive for the Cartesian mathematical subject. While Descartes's mode of writing mathematics is quite different in many ways from what came before him, as well as from what exists today, Descartes's notion that mathematical reading should constitute a kind of pleasurable exercise has survived. The following sampling of contemporary mathematicians' remarks on the nature of their discipline suggests that the issues of clarity, aesthetics, and disciplinarity are deeply intertwined:

Mathematicians are not concerned merely with finding and proving theorems; they also want to arrange and assemble the theorems so that they appear not only correct but evident and compelling. Such a goal, I feel, is aesthetic rather than epistemological.47

I already mentioned the idea of mathematics as an art, a poetry of ideas. . . . in order for one to appreciate mathematics, to enjoy it, one needs a unique feeling for intellectual elegance and beauty of ideas in a very special world of thought. It is not surprising that this can hardly be shared with non-mathematicians: Our poems are written in a highly specialized language, the mathematical language . . . unfortunately, these poems can only be understood in the original. The resemblance to an art is clear.48

For generations mathematicians have used taste to select those results that are worthy of public scrutiny. And while some people might show more taste than others, the selection was part of the process of doing mathematics. This is more than etiquette: it's an essential feature of our subject. Without taste, mathematics ceases to be Art. Like scientists, we should continue to perform experiments, by any means available. But like artists, sculptors, and composers, we must exercise judgement about what should be placed on public display.49

I doubt that the notion that a certain aesthetics of reading and writing is constitutive for the cultivation of a disciplinary self begins with Descartes, but it certainly is a prominent aspect of his approach to mathematics. Thus while he may indeed have envisioned a kind of non-disciplinary "elite subject" for his philosophical writings and mathematical method, this claim needs to be balanced by a realization that Descartes's mathematical work was also a key moment in the formation of a certain expert subject of mathematics.