ESCARTES was born at the end of the sixteenth century, a time of enormous
changes in the western intellectual world, largely brought about by the
Reformation. Luther had denied the Church's authority to settle disputes on
matters of faith: it was, he had insisted, the Scriptures alone which carry
authority; pronouncements of the church, even those with long tradition behind
them, were mere opinion, not truth. And so the question was explicitly raised
and debated, how does one determine what the truth is? Protestants
claimed that one had no recourse but to one's own conscience; Catholics tried
to show the unreliability and subjectivity of one person's conscience,
insisting on the need for stability and intersubjectivity which only tradition
could provide. Into this dispute was injected the recently re-discovered works
of the Greek skeptics, which, translated into Latin in the middle of the
sixteenth century, found a wide and receptive audience. Both Catholics
and Protestants found the skeptical arguments a wonderful engine for
attacking their opponents; but the collateral effects of these battles were to
encourage a suspicion of authority generally, including that of the sciences.
And while the skeptical writings were first deployed against superstition,
against the entrenched medieval Aristotelian physics, and against alchemy, once
again they proved a two-edged sword, being eventually directed against
every sort of scientific inquiry, and leading to a general
anti-intellectualism.[1]The widely read Montaigne, enthusiastically endorsing the skeptical writings, argued that neither one's senses nor one's reason can be trusted: the senses are inherently unreliable, while rational conclusions must inevitably depend upon questionable premises. As Richard Popkin paraphrases Montaigne, since "the experts, including the great scientists of [the] time ... offer divergent and incompatible theories, the best we can do is to follow the sage advice of the ancient [Greek] skeptics—suspend judgment on all matters that go beyond the appearance of things ."[2] The Greek skeptics had seemed to argue persuasively that all views are mere opinion, and that there is no valid criterion for selecting any as true. From the prospect of the Greek skeptics, we are like the prisoners in Plato's cave: the best to be hoped for is—in Plato's words—"to be able to remember the order of sequence among the passing shadows and . . . divine their future appearances."[3] Accordingly, a leading experimental scientist of the time, Pierre Gassendi, recommended—in Popkin's paraphrase—that "Religion should be accepted on [simple] faith; while science should be the constructive consequence of recognizing that, in a basic metaphysical sense, nothing can be known. After one has seen this, then one can examine experience, organize it, and use it to predict the future course of events." But a scientific theory is "not a theory about the nature of reality, but only an interpretive model for the world of appearances . . ."[4]
In the seventeenth century, Descartes undertakes to lay a solid foundation for the sciences against such skeptical assault. (But be warned: it is not the medieval "Aristotelian" science, nor yet the emerging experimental sciences, for which Descartes seeks a solid foundation, but, rather, the pure sciences.) Descartes undertakes, in addition, to provide a solid foundation for the basic tenets of religion. He was repulsed, I think, by the view then gaining wide acceptance, that religion must rest upon blind faith, utterly unsupported and unsupportable by rational considerations: such was the low estate to which many contemporary Christians had been brought by the popular skeptical arguments. Descartes stands as defender of absolute truth and knowledge against the ravages of skeptics in much the same posture, I think, as that taken by Plato centuries earlier. The similarity in their views is striking, especially so when we realize that Descartes might well [5] have known of Plato only the few fragmentary things reported by Aristotle and so by the scholastic tradition.
So let us inventory the major parallels:
1. Both Descartes and Plato distinguish knowledge from true belief, characterizing knowledge as excluding even the possibility of being mistaken.One of the important differences, for our purposes, between the Republic and the Meditations is that where Plato is content to sketch his vision of rationalism by means of similes and metaphors, Descartes actually constructs an extended and detailed argument for his view. And that not only helps us appreciate the attractiveness of the theory, it puts us into a better position to consider it critically.
2. Both claim that knowledge requires special objects, different from those of mere belief: the objects of knowledge consist in abstract entities which determine the nature of the universe.
3. Both claim that a special faculty of reason is the means to acquiring knowledge.
4. Both denigrate the world of the senses as a world of mere appearance which, though somehow reflecting imperfectly the real world, is the object merely of belief.
5. Both claim that there is one supreme entity which underlies the existence of everything else and makes them intelligible.
6. Both view mathematics as special among the sciences.
7. And finally, both see the soul as different in kind from the body, and as embodying (so to speak) the principle of the truly real.
Now with that bit of framework, I want to turn to the text. But I want to begin by making a few remarks on how to read this very nicely written work. The first thing to note is that Descartes often advances views for our consideration which he then rejects. I should like the words, "nevertheless," "however," and "but" to jump out at you as you read the text.[6] Look at page 76 for an example of what I am talking about. On the first paragraph, Descartes offers as a reason for rejecting all his opinions, that they have been acquired by the senses and that "it is prudent never to trust wholly those things which have once deceived us." Yet notice that the second paragraph begins with "But" and goes on to characterize such indiscriminate distrust of the senses as lunatic. In other words, in this second paragraph, Descartes is completely taking back what seemed to be his assertion of the previous paragraph. And the first word of the second paragraph, the crucial "but" gives his warning that he might either modify or even retract his earlier claim. Looking further down the page, the last paragraph begins with "Nevertheless," to signal that here at last Descartes will find reasonable grounds on which to doubt opinions which depend upon the senses. For a second example, turn the page to page 78: in the top paragraph, Descartes has just been explaining why our belief in the truths of mathematics and geometry can escape the ground for doubting the senses, and so perhaps can be claimed as knowledge. In the middle of the page, in the paragraph beginning (surprise!) with "Nevertheless," he finds that he cannot claim to know such things as mathematics just yet. Throughout the Meditations Descartes repeatedly advances tentative claims which he then takes back or modifies almost immediately, each time warning us of what he is doing with one of those magic words. So watch out for the "nevertheless" or you will accuse him of saying what he explicitly denies.
The second note I'd like to make on the text is this: Throughout the Meditations, Descartes is quite careful to remind you what has been proved and to prepare you for where the argument is going from there. Perhaps the most impressive illustration of this is found in the sixth meditation. Look at page 128. Having just drawn a distinction between two sorts of idea, which we may characterize as "abstract" ones and "sensory" ones, Descartes tells us that he will consider whether from sensory ideas he can "derive some certain proof of the existence of corporeal things." In the next to last paragraph, he gives us an inventory of the next series of pages: "First," he says he will recall what he formerly believed; "Afterward" he will examine the reasons which, in meditation one, forced him to consider these opinions doubtful; and "finally" he will consider what he ought now to believe. This inventory is followed by four paragraphs written in the past tense, in which Descartes tells us what he formerly believed. Then, in the next several paragraphs, also in the past tense, he discusses the reasons for doubting these former opinions, explicitly referring to two general reasons for doubting which "I have recently added." Then, at the bottom of page 131, Descartes shifts out of the past tense in order to talk about what we ought now to believe: and it is, of course, here where one should look for the promised proof. What I want you to see, then, is that not only does Descartes use the topic sentence, he uses tenses and other grammatical devices to help the reader find his way. When you read and re-read the Meditations, look for these things and let them help you: you will find that they make the argument much easier to follow.
With that advice out of the way, I want to turn to Descartes' arguments. It would be boring to talk about things which you can perfectly well find without my help, so I want to concentrate on some of the things which you might overlook or have particular difficulty with. Of course, this might prove boring also; but my defense will be that something which is boring can nevertheless be useful. I want to begin with the first page of the first meditation, page 75; it is, unfortunately, the most quickly read page in the whole text. In the very first sentence, in speaking of his former opinions as "badly assured premises," Descartes is suggesting that a body of beliefs is not merely a collection of independent assertions. but a system of premises, from which new beliefs are inferred almost automatically—something like the system of axioms and theorems of geometry. Although that comparison might not be surprising—coming as it does from an expert in geometry—it really does seem to be helpful. It's precisely because beliefs, like rabbits, tend to breed, that in courts of law, witnesses are asked to recount only what they saw; judges realize that various inferences depending not only on the relevant observations but also upon other beliefs, may otherwise be imparted as authoritative. What might seem a bit naive is the supposition of the law that one could somehow extract and then report just his bare observations without also bringing to bear other beliefs. Descartes would deny the possibility of doing such a thing; since one's beliefs are something like a system of geometry, if you eventually find a mistake, you cannot then repair the damage merely by retracting the errant belief. For in the meantime, it might have spawned numerous other beliefs—hidden progeny, since they don't carry marks identifying their parents. If we retract the parent, its children nevertheless remain in the system. Thus, if you find a mistake in a system of beliefs, the only way to avoid the possibility of error is to dismantle the entire system and to begin again. Toward the bottom of the page, Descartes says that not only should he avoid beliefs which he actually doubts, he must even "abstain from the belief in things which are not entirely certain and indubitable;" in other words, he is saying that knowledge must exclude even the possibility of mistake. And so, the only way one can have knowledge of a system of assertions is to know each of its premises beyond any possibility of doubt. Precisely that is required for "firm and constant knowledge in the sciences."
Perhaps we should pause to notice just how radical, how utterly Platonic, this conception of knowledge is. Descartes is saying that for you to know something, it is not enough that you have no doubts about it; it is not even enough that your substantial evidence gives you no reasons to doubt it. What is required for knowledge is that it not be possible to doubt it. Our courts of law require that a matter be proved beyond a reasonable doubt: in Descartes' court, it must be proved beyond even an unreasonable doubt. If it is possible, conceivable, even imaginable, that this talk is being delivered by a Martian clone, grown from a pod in Boardman's garage, then you cannot know that a human being is now speaking to you. (But perhaps that's an unfortunate example.) Anyway, it is this radical notion that people are talking about when they speak of Plato's, and Descartes', concept of absolute knowledge.
In his second meditation, when Descartes pushes the method of doubt to its fullest extent, several truths survive; since these cannot be doubted, Descartes must know them. The first of these is, of course, that "I exist." But it is the second truth to which I want to turn your attention. When Descartes asks, "what am I?" it might have struck you as odd that he tests potential answers by asking whether he can doubt them. The test appears strange because one's ability to doubt something doesn't normally show that it is false. If I can doubt that I'm the baldest teacher at Lawrence, does it follow that I'm not? It is only when we get to the language used on page 84, when Descartes rephrases his answer, and so his question, in terms of "what is inseparable from my nature," that we realize that the question was special. When he proceeds to say, "I am now admitting nothing except what is necessarily true," we finally see that the apparently ordinary question was really a question about what is essential to my being what I am. But what is the test for a characteristic's being essential to some kind of thing? Well, if we can consistently imagine the thing without that characteristic, then it isn't essential. If you could coherently imagine a unicorn without a horn on its forehead, then having a horn would not be essential to being a unicorn; for if it were, you couldn't have imagined it. Similarly, if you can consistently doubt whether you have a body, then having a body is not essential to your being what you are.
Thus, when on page 84 Descartes claims to know "I am . . . only a thinking being," what he says he knows is an abstract truth about his nature. And so this piece of knowledge is similar to almost all of the other things which he will subsequently find that it knows: it is a necessary truth regarding a certain kind of thing's having a particular nature or essence. In the third meditation, Descartes will begin with an abstract truth about the nature of God—if there be one—to the conclusion that God must exist. In the fifth meditation, he will give numerous examples concerning the nature of geometrical objects. Such necessary truths, and the conclusions which follow from them, are virtually the only possible objects of knowledge for Descartes. Notice the similarity to Plato.
Now I want to show where Descartes takes a further step toward Plato, arguing that we acquire such knowledge through reason alone. Turn to page 94. Descartes here draws a distinction between kinds of idea. There is first the category of sensory idea, which has two subclasses: (1a) sensory ideas which are "alien"—he means not that they are Martian clones, but that they are imposed on the mind by some corporeal thing; and (1b) sensory ideas which are made or "invented by myself" (as with the things we create in our imaginations). But it will turn out to be the second category of idea—those which are "born with me," that is, "innate ideas"—which are almost exclusively involved in knowledge. It is connections between innate ideas which may be clearly and distinctly conceived. Note the parallel with Plato: Descartes' two basic categories of idea correspond to Plato's distinction between one's having mere sensory appearances, and one's focussing on the abstract and eternal things. For Descartes, innate ideas represent essences which exist outside of, and independently of, human minds.
Recall again the first two pieces of knowledge which Descartes finds in the second meditation. Their real significance to Descartes' overall argument is revealed in the third paragraph of the third meditation, beginning at the bottom of page 91. Here Descartes argues that since he has some pieces of knowledge, he must then already know how to identify infallible knowledge: he need only make his means of indentification explicit. And so Descartes establishes the "general principle that everything which we conceive very clearly and very distinctly is wholly true." Think of this principle as a "touchstone" of knowledge, an infallible indicator that a given assertion can be known. You need to realize that this touchstone is going to be used as a premise in every important argument in the remaining meditations. The first time it gets used, however, it is traveling under an alias: turn to page 95. In the last full paragraph on this page, you will find, in an almost parenthetical explanation, that the phrase, "revealed by the light of nature," means the same thing as "clearly and distinctly conceived," Consider the imagery in this pair of phrases: you can see that the metaphor of light, here in Descartes, is closely parallel to the metaphor of the sun which Plato uses to characterize the Good. Plato's Good illuminates truths which one may see with his mind's eye; in Descartes, one's God-given faculty of the understanding, through an inspection of the mind (here, recall the piece of wax discussion from page 88), apprehends truth illuminated by the light of nature.
Thus far, I've suggested that the senses—that is, sensory ideas—have no major role in acquiring knowledge. But now I want to bring to your attention a passage in which Descartes quite explicitly disparages the senses. If we turn to page 134, Descartes has just proved the existence of a corporeal world. But as you might not have expected, what we can know about this world is limited to what we learn through mathematics and geometry. On the first full paragraph of page 134, Descartes now asks why God permitted us to have beliefs based on the senses, doubtful and uncertain as beliefs from this source inevitably are; since God is not a deceiver, they must presumably convey some truth. The senses seem, he says at the bottom of the page, to be implicated in the connection of the mind to the body. Continuing the discussion for several pages, Descartes finally answers the question on the first full paragraph of page 137: "Thus I see that both here and in many other similar cases I am accustomed to misunderstand and misconstrue the order of nature, because although these sensations were given to me only to indicate to my mind which objects are useful or harmful to the composite body of which it is a part, and are for that purpose sufficiently clear and distinct, I nevertheless use them as though they could obtain direct information about the essence and the nature of external objects, about which they can of course give me no information except very obscurely and confusedly." He then goes on to argue that our liability to mistakes when using the senses does not show God to lack goodness, since the arrangement—rickety as it is—is on the whole effective in the preservation of the body.
What I want you to notice, then, is that the role of the senses is utterly inferior and subordinate to that of the faculty of the understanding; that is, the senses are subordinate to reason. While the senses may be useful to preserve the body, they are not the vehicles by which we can obtain knowledge. Like Plato, then, Descartes depreciates the senses; knowledge comes from the eye of reason, not through the senses. The senses are useful only because, and so long as, the mind is harnessed to the body; and since, as both Descartes and Plato speculate, the mind might well be immortal, the senses could well prove a temporary expedient. (For Descartes' speculation on this matter, see the Discourse, at the end of Part Five on page 44: "when we know how different they are, we understand more fully the reasons which prove that our soul is by nature entirely independent of the body, and consequently does not have to die with it. Therefore, as long as we see no other causes which might destroy it, we are naturally led to conclude that it is immortal.")
At this point, I want to turn to the first of Descartes' arguments for God's existence, because it uses some technical jargon which might give you trouble. Turn to page 97, where Descartes is spelling out his important causal principle. I'll make two preliminary remarks. The first is to underscore how well this text is written: although I haven't the time to prove it, the text itself, even if you have no prior knowledge of the traditional distinctions between "actual or formal reality" and "objective reality", will allow you to tease out of these pages what Descartes is saying; it takes time, patience, and an enjoyment of sleuthing, but the text really is complete in itself. The second thing to note is that the technical machinery of this argument comes straight from the Scholastic tradition; that is interesting because Descartes, the father of modern philosophy, is sometimes portrayed as an intellectual revolutionary who swept aside the earlier Scholastic tradition in order to begin afresh. But Descartes' wholesale acceptance of Scholastic concepts and assumptions both here and elsewhere shows—as the translator points out in his introduction—that Descartes is not only the father of modern philosophy, but also the child of the medieval Scholastic tradition.
Now let's look at page 97. Descartes supposes that everything which is capable of existing would have a certain degree of reality because of the kind of thing it is. Every kind of thing can be located on a single scale of reality—the scale of actual reality. The more perfect kinds of thing will have more reality than the less perfect kinds of thing. At the top of the scale will be God: if He exists, then by definition, He will have an infinite amount of reality or perfection; lower on the scale will be angels; lower still will be human minds; even lower will be external, corporeal objects; and lower still will be ideas. Now the first part of the causal principle is that something lower on this scale of actual reality cannot cause something higher on the scale: a cause must have at least as much actual reality as its effect does. That is why something cannot come from nothing. (It's also why a Cartesian would say that evolution without God's intervention would be unthinkable.) This first part of the causal principle, then compares the actual reality of the cause to the actual reality of the effect.
So far we have only spoken about the most basic sort of reality, actual reality. But in addition to their actual reality, some things have also a second kind of reality: this second kind of reality belongs to anything which represents something else. Consider, for example, a painting of an angel: the oil and canvas surface has the degree of actual reality which any material thing would have; but because it represents something else, the painting also has a second kind of reality, the reflected reality of the thing which it represents. Thus, there is a way in which a picture of an angel is more real than a picture of a bird. Because things have this second kind of reality in virtue of the object represented, our translator styles it "objective" reality; but because it is confusing to ignore the present-day connotations of the word, "objective," I think it's better to call it representational reality. Thus, if a thing represents something else, it has an amount of representational reality which depends upon the actual reality of the thing being represented. So now we can state the second part of the causal principle: a cause must have as much actual reality as its effect has representational reality. Thus, since I am a puny creature compared to an angel, not only can I not create an angel—that would violate the first part of the causal principle—I cannot even create, utterly by myself, a picture of an angel. Only something as perfect as an angel would be capable of causing a picture of an angel to exist. So, if a human artist paints a picture of an angel, although the human has sufficient reality to have caused the oil and canvas to exist, he would not have enough reality to account for the high degree of representational reality of the painting; clearly he required, as we may put it, "inspiration." (For similar reasons, I think that the second part of the causal principle would imply that a monkey could not create the works of Shakespeare—no matter how long it labored at the keyboard.) At any rate, notice one last point which will help you understand the argument for God's existence: since an idea—for instance, an idea of an angel—represents something—remember, ideas are like images of objects[7]—the second part of the causal principle must apply to ideas as well as to portraits.
Now I want to explore with you a problem in interpretation. In your reading you have already discovered or guessed that proving the existence of a non-deceiving God is going to remove the only obstacle to having knowledge of mathematics. On pages 124 and 125, Descartes draws the conclusion that he can now have such knowledge. In doing so, however, he uses what might appear to be a circular argument: on the first sentence of page 125, Descartes says "I can infer as a consequence that everything which I conceive clearly and distinctly is true." As Descartes' contemporary, Arnauld, objected,[8] it is circular first to use the touchstone to prove the existence of God and afterwards to use the existence of God to prove the touchstone. It must be conceded that there is some internal evidence that Descartes might have considered the touchstone to be merely provisional when he first introduces it in meditation three. Back in meditation three, on page 92, Descartes worded his conclusion this way: "And therefore IT SEEMS that I can already establish as a general principle that everything which we conceive very clearly and very distinctly is wholly true." Nevertheless, I think that we must look for a more credible way to understand what Descartes was trying to do. For it is difficult to believe that this mathematical genius, the creator of Cartesian coordinates and the founder of analytical geometry, would fall prey to what, on Arnauld's interpretation, is a simple and obvious logical blunder. And if he had originally blundered, it would be puzzling why he subsequently passed up several opportunities to correct himself: this charge of circularity was made on three separate occasions, and on each of them Descartes hotly replied that his critic failed to understand the plain meaning of the text.[9] Whether or not that meaning is as plain as Descartes supposed, it seems reasonable to search for a way to understand these pages and that passage as not asserting the circularity. And this will require us to explore the details of the problem which Descartes had raised in the first meditation.
Let's consider that earlier possibility: how exactly was an evil God supposed to be able to bring it about that a person is mistaken about something as simple as that 2+3=5? In order for me to make a mistake, I've actually got to be using the arithmetical concepts: a parrot which vocalizes, "two plus three is six—awk," is not making a mistake, since its vocalizations express no corresponding mathematical belief. And on the other hand, if I have somehow learned the series of natural numbers as "one, two, three, four, six, five, seven," then the mathematical belief expressed by my saying "two plus three is six" is not mistaken, but correct. (Of course I betray an unfortunate mistake about English usage, but I do not make a mistake of mathematics.) The possibility which Descartes envisions must be different from these.
At the bottom of page 124, in discussing an example of the doubt which might otherwise arise without our knowledge of God's existence, Descartes explicitly locates the problem in one's turning his attention away from the details of a mathematical demonstration. A demonstration is, of course, an occasion on which one conceives of something clearly and distinctly, and then draws inferences from that truth. If I've just been concentrating on a demonstration of some mathematical truth, there can be no room for doubt until after I have turned my attention away from the details of the demonstration. Suppose, for example, I demonstrate to myself that two plus three equals five by thinking of five marks as being grouped into a pair and a triplet. So long as I am concentrating on this, I continue to see it clearly and distinctly, and so—evidently—cannot make a mistake. But if I turn my attention away from this demonstration, so that I now merely remember its having been a convincing demonstration, then I am liable to mistake if God be an evil genius. An evil genius could not make me "see" that two plus three equals six through the details of a demonstration since no such attempted demonstration could support my intuitions. I could not—consistently with Descartes' theory—clearly and distinctly conceive what is false. But an evil genius might make me believe that I had just earlier considered a successful demonstration that two plus three equals six, even though I hadn't. [10] Notice that my memory of what was earlier passing through my mind is not something which, in Descartes' theory, I clearly and distinctly conceive: for one's memories are not represented by innate ideas (and, as I pointed out earlier in my talk, it is only innate ideas which can be clearly and distinctly conceived). Thus, an evil genius might make me mistaken in what I believe at a later time about what I had accomplished at some earlier time. And that would raise havoc with anyone who wished to comprehend rather long proofs having a variety of parts, all of which could not be grasped at one sitting; and it would also raise havoc if one took too long in applying the results of a simple demonstration. For example, I might successfully demonstrate that two plus three equals five, but then by the time I came to apply the results in order to add the number of new hairs I've grown this last decade, I would be at the mercy of the evil genius. So I suggest reading the first sentence of page 125 as a somewhat clumsy attempt to say that since we know that it is not possible for there to be an evil God, therefore there is no general difficulty in trusting at a later time those truths we recall having conceived clearly and distinctly. (Note, by the way, that the impossibility of an evil genius does not guarantee that one will never suffer lapses and mistakes of memory; it only guarantees that there can be no insurmountable, systematic flaws in one's memory. Presumably, we can overcome particular, local, problems of memory by resort to individualized strategies.) The things which Descartes says immediately before and after the troublesome passage support this interpretation. And while Descartes' argument, so interpreted, may still be liable to criticism, at least it won't be prey to the charge of depending upon an elementary logical blunder.
Before I conclude my talk, I want to try to help you appreciate the attractiveness of Descartes' theory. Suppose that we are reflecting on what knowledge is and how it can be acquired. It occurs to us to consider the clearest example of something which we know. Simple truths of mathematics would seem to be the things that we can know with greater certainty and sureness than anything else. A person can't make a mistake about them: for if he appears to have made a mistake, it must turn out either to have been a slip of the tongue or to reveal that since he has evidently failed to comprehend the relevant mathematical concepts, he didn't have a belief about mathematics at all. A young child who triumphantly recites "two plus three equals six" to his grandmother is not expressing a mathematical belief, and so, not a false mathematical belief, although he might have the false non-mathematical belief that this is the sentence which his grandfather had taught him earlier. In contrast, for all the other kinds of thing which we commonly say we know in daily life—for example, that we are presently in Appleton or that there are, alas, neither mountains nor ocean nearby—we can always think of conceivable circumstances in which we'd have to concede that we were wrong. So, if simple mathematical truths will be our clearest examples of what we know, all those other things shouldn't be put into the same class: we might believe them; we might even have a very good evidence for confidently believing them; still, they can't really measure up to the standards set by "two plus three equals five." They aren't things we genuinely know.
And upon further reflection, we can find other important ways in which our knowledge of simple mathematical truths seems to be special:
1. They don't require evidence: the fact that when we put two coins in a pocket and then three more coins in the pocket, we subsequently pull five coins out of the pocket, isn't relevant to proving that two plus three equals five is true; and the fact that when we mix two liters of alcohol with three liters of water, we subsequently find something less than five liters of liquid, isn't relevant to disproving the mathematical truth. Mathematical truths seem to be known independently of what we see and hear and touch and taste and smell.
2. Mathematical assertions are true, and so, evidently, they are true of something. But true of what? Numbers are purely mathematical concepts, and so are operations such as addition. It looks as though we must postulate a special world, a world containing numbers, which mathematical assertions are about.
3. Although different linguistic communities attach different signs and names to the numbers and to mathematical operations, the truths of mathematics are objectively true quite independently of human decision. A United Nations agreement could not make two plus three equal six—although, of course, an agreement could change the signs and words we use to refer to various numbers and operations.
4. We seem to gain knowledge of mathematical truths by an act of intuition: our reason reaches out to the independent world of mathematics and discovers pre-existing truths. And there are always more truths there, awaiting discovery by some clever mathematician.
Descartes' touchstone, his clarity and distinctness of conception, is his attempt to characterize the features of that special kind of intuition which allows one to know a mathematical truth. But even if you find his characterization unhelpful, you will not yet have any reason to deny the existence of a special, perfectly familiar, act of intuition by which we gain knowledge of mathematical truths. Descartes supposes that he can know some non-mathematical truths through the same act of intuition. But even if you reject that claim, you will still be left with the question, how do you account for those special features of mathematical knowledge? One may find Descartes' theory distasteful, but what has required philosophers to take it seriously is that it does provide a coherent explanation of those four special features. The difficulty of finding a coherent alternative, which is as persuasive as Descartes' explanation, has been the hard nut on which empiricists have chewed for centuries. [11]
Finally, it will perhaps be useful to consider Descartes in historical context once again, this time looking at what followed. Descartes apparently saw himself as defender not only of science but of the Catholic Church; yet in this first-person monologue which begins from one person's own ideas, and proceeds under the determination to settle every question without resort to outside authority and tradition, it certainly looks as though Descartes' example would have given greater comfort to the Protestant cause than to the Catholic. Looking beyond the religious conflict which helped motivate his attempt to find an answer to skeptical doubt, Descartes' view of science as essentially a mathematical enterprise proved irresistible in the history of science which followed.[12] And his "way of ideas"—the method of beginning from one's own ideas, and constructing on their basis whatever one can know, proved equally irresistible in the history of philosophy. And even if he was not successful in stopping the onslaught of skepticism, Descartes ensured that skepticism would be taken seriously by those philosophers who followed him.
William S. Boardman
Department of Philosophy
Lawrence University