Part IV, Chapter 1: Of Knowledge (with the Preface to the Part)
IT remains that we explain the last part of logic—that relating to method—which is, without doubt, one of the most useful and most important. We have thought it right to unite with it what belongs to demonstration, because this does not commonly consist of a single argument, but of a series of several reasonings, by which we incontrovertibly prove some truth; and, moreover, because, in order to demonstrate well, it is indeed of little avail to know the rules of syllogism, which we rarely transgress, while it is of the first importance to arrange our thoughts clearly, and to avail ourselves of those which are clear and evident, to penetrate into what may appear more obscure.
And since demonstration has knowledge for its end, it is necessary first to say something of it.
IF, when we consider any maxim, we recognise the truth of it in itself, and by an evidence which we perceive without the aid of any other reason, this kind of knowledge is called intelligence; and it is thus that we know first principles.
But if it is not convincing of itself, some other motive is necessary to render it so, and this motive is either authority or reason. If it is authority which leads the mind to embrace what is proposed to it, this is what is called faith. If it is reason, then either this reason does not produce complete conviction, but leaves still some doubt, and this acquiescence of the mind, accompanied with doubt, is what is called opinion.
Or this reason produces complete conviction; and then, either it is clear only in appearance, and requires attention, and the persuasion which it produces is an error, if it be really false; or, at least, a rash judgment, if, being true in itself, we, nevertheless, had not sufficient reason for believing it to be true.
But if this reason is not only apparent, but weighty and true, which we recognise by a longer and more minute attention, by a stronger persuasion, and by a quality of clearness, which is more vivid and penetrating, than the conviction which this reason produces,— is called knowledge, in relation to which many questions arise.
The first is—Whether there be such a thing? that is to say, whether we have cognitions founded on clear and certain reasons, or, in general, whether we have clear and certain cognitions, for this question relates as much to intelligence as to knowledge.
There are some philosophers who have made denying their profession, and who have even established on that foundation the whole of their philosophy; and amongst these philosophers some are satisfied with denying certainty, admitting, at the same time, probability, and these are the new Academics; the others, who are the Pyrrhonists, have denied even this probability, and have maintained that all things are equally obscure and uncertain.
But the truth is, that all these opinions, which have made so much noise in the world, have never existed any where, save in discourses, disputes, or writings, and no one has ever been seriously convinced of them. They were only the sport and amusement of unoccupied and ingenious persons; but never the feelings of which they were inwardly and deeply conscious, and by which they endeavoured to conduct their life. Hence the best means of convincing these philosophers would be to refer them to their conscience and good faith, and to require from them, whether, after all these discourses, in which they had laboured to prove that it is impossible to distinguish sleep from waking, or madness from sound mindedness, they were not persuaded, despite their argument, that they did not sleep, and were of a sound mind. And if they had had any sincerity, they would have denied all their vain subtilties, by avowing freely that they had never been able to believe these things when they had tried to do so.
And if any one were found who could entertain a doubt as to whether he were awake or sane, or able even to believe that the existence of all external things was uncertain,—being in doubt as to the existence of a sun, a moon, or of matter,—no one could, however, be found to doubt, as St Augustine says, that he is, that he thinks, that he lives. For whether he were asleep or awake, whether he were of a diseased or sound mind, whether he were deceived or not deceived, he is at all events certain, inasmuch as he thinks, that he exists, and that he lives; since it is impossible to separate being and life from thought, and to believe that what thinks neither exists nor lives. And from this clear, certain, and indubitable knowledge, he may form a rule for accepting as true all thoughts which he may find as clear as this one appears to be.
It is equally impossible to doubt our perceptions when we separate them from their objects. Thus, whether there be such things as the sun and the earth or not, I am certain that I imagine I see them. I am certain that I doubt when I doubt,—that I believe I see, when I believe I see,—that I believe I hear, when I believe I hear, and so of the rest. So that, restricting ourselves to the mind alone, and considering its modifications, we find a vast number of clear cognitions, whose truth it is impossible to doubt.
This consideration may enable us to decide another question which has arisen in relation to this subject,—to wit, whether the things which we know only through the mind are more or less certain than those which we know through the senses? For it is clear from what we have said above, that we are more assured of those perceptions and ideas which we discover only by a mental reflection, than we are of any of the objects of sense. We may say further, that while the senses do not always deceive us in the report which they give, our assurance, nevertheless, that they do not deceive, arises, not from the senses themselves, but from a reflection of the mind, through which we discern when we ought to believe, and when we ought not to believe, the senses.
And hence it must be confessed that St Augustine had good ground to maintain, after Plato, that the determination of truth, and the rule for its discernment, belong not to the senses, but to the mind:—Non est judicium veritatis in sensibus; and also, that the certainty which may be derived from the senses is of no great extent,—there being many things which we imagine ourselves to know through sense, of which we cannot affirm that we have a complete assurance.
For example, we may know through sense that one body is larger than another body, but we cannot know with certainty what is the true and natural size of each body. To understand this, it is only necessary to consider that if we had never seen external objects in any other way than through the medium of magnifying glasses, it is certain that we should have figured to ourselves bodies, and all the measurements of bodies, according to that size only in which they had appeared to us through these glasses. Now our eyes themselves are glasses, and we do not know exactly whether they may not diminish or augment the objects which we behold, or whether these artificial glasses, which we imagine diminish or augment them, may not, on the contrary, represent their true size. And, therefore, we do not know the natural and absolute size of any body.
We do not know, either, whether our perception of the size of objects is the same as that of others; for although two persons may agree together in their measurement, that a given body, for example, is only five feet, yet, nevertheless, that which the one conceives to be a foot may not be the same as that which the other does; for they each conceive what their eyes severally represent to them. Now it may be that the eyes of one do not represent the same thing to him which the eyes of others do to them, because they are glasses differently cut.
This diversity, however, is probably not great, because we do not perceive any difference in the conformation of the eye sufficient to produce any remarkable change; besides which, though our eyes are glasses, they are, however, glasses cut by the hand of God: so that we have good ground for believing that they represent, for the most part, the truth of objects, except when their natural figure is injured or disturbed by some defect.
However this may be, though the judgment of the size of objects be to some extent uncertain, this is not very important, and we are not from it to conclude that there is no certainty in any of the other representations of sense; for, though I may not know exactly, as I have said, what is the natural and absolute size of an elephant, I do know, however, that he is greater than a horse, and less than a whale; which is sufficient for all the purposes of life.
There is, therefore, certainty and uncertainty both in the mind and in the senses; and it would be an equal mistake to maintain that all things should be considered either as certain or uncertain.
Reason, on the other hand, compels us to acknowledge, in relation to this, three degrees.
For there are some things which we may know clearly and certainly. There are others which we cannot know with the clearness of truth, but to the knowledge of which we may hope to arrive. And, finally, there are some which it is impossible to know with certainty, either because we have not the principles which would lead us to them, or because they are too disproportionate to our minds.
The first kind comprehends all that we know through demonstration, or through intelligence.
The second is the matter of the study of philosophers. But they may spend their time uselessly, if they do not know how to distinguish these from the third,—that is to say, if they cannot discern the things at the knowledge of which the mind may arrive, from those which it is incapable of reaching.
The shortest method which can be found in the study of the sciences, is that of never engaging in the search after any of those things which are above us, and which we cannot reasonably hope to be able to comprehend. Of this kind are all the questions which relate to the power of God, and generally all that belongs to the infinite, which it is absurd to attempt to reduce within the limits of our mind; for our mind, being finite, is lost and confounded in the infinite, and remains overwhelmed with the multitude of conflicting thoughts which it furnishes.
This is the shortest and most convenient solution which can be given of a great number of questions, on which we may dispute for ever, because we can never attain to any knowledge of them sufficiently clear to fix and hold our minds. Is it possible for a creature to have been created from eternity? Can God make a body infinite in size?—a movement infinite in swiftness?—a multitude infinite in number? Is an infinite number even, or uneven? Is one infinite greater than another? He who should say at once, I know nothing about these things, will have advanced as far in a moment, as he who should have spent twenty years in reasoning on them; and the only difference there would be between them is, that he who had laboured to solve these questions is in danger of falling into a lower state than that of simple ignorance, which is that of believing himself to know what he does not.
There are also a great number of metaphysical questions, which are too vague, too abstract, and too far removed from clear and well-known principles, to be ever resolved; and the best way is for us to have as little to do with them as we can; and, after having learned, in general, what they are, to resolve boldly to be ignorant of them.
Nescire quaedam, magna pars sapientiae.
In this way, by freeing ourselves from inquiries in which it is impossible to succeed, we shall be able to make more progress in those which are adapted to the capacity of our mind.
But it must be remarked that there are some things which are incomprehensible in their manner, but which are certain in their existence. We are unable to conceive how they can be, while it is certain, nevertheless, that they are.
What is more incomprehensible than eternity, and what, at the same time, is more certain? So that those even, who, through an awful blindness, have destroyed in their mind the knowledge of God, are obliged to attribute it to the most vile and contemptible of all things, which is matter.
How can we comprehend that the smallest grain of matter is infinitely divisible, and that we can never reach a part so small, but that it not only contains many others, but also an infinity; that the smallest grain of wheat contains in itself as many parts, though proportionally smaller, as the whole world,—that all imaginable forms are actually found in it, and that it contains in itself a small world, with all its parts—a sun, a heaven, stars, planets—a world with admirable exactness of proportions,—and that there are none of the parts of that grain which do not still themselves contain a proportional world? What must be the part in so small a world which answers to the size of a grain of wheat? and what a tremendous difference must there be, in order that we may be able to say truly, that what a grain of wheat is in relation to the whole world, that part is in relation to a grain of wheat? Nevertheless that part, whose littleness is already incomprehensible to us, contains still another world proportional; and so on to infinity, without our being able to find any which has not as many relative parts as the whole world, however numerous these may be.
All these things are inconceivable; and they must, nevertheless, necessarily be so, since we can demonstrate the divisibility of matter to infinity, and since geometry has furnished us with proofs of it, as plain as those of any of the truths which it reveals to us.
For this science shows us that there are certain lines which have no common measure, and which are called, for this reason, incommensurable, as the diagonal of a square, and the sides. Now, if this diagonal and the sides were composed of a certain number of indivisible parts, one of these indivisible parts would be the common measure of these two lines, and, consequently, these two lines cannot be composed of a certain number of indivisible parts.
It is demonstrated, again, by this science, that it is impossible for a square number to be double of another square number, while, however, it is very possible that an extended square may be double of another extended square. Now, if these two extended squares were composed of a certain number of ultimate parts, the large square would contain double the parts of the small one, and both being squares, there would be a square number double another square number, which is impossible.
Finally, there is nothing more clear than this principle, that two non-extensions cannot form an extension, and that an extended whole has parts. Now, taking two of these parts, which we assume to be indivisible, I ask, whether these have extension, or whether they have not? If they have, they are therefore divisible, and have many parts; if they have not, they are two negations of extension, and thus it is impossible for them to constitute an extension.
We must renounce human certainty before we can doubt the truth of these demonstrations; but to help us to conceive, as far as is possible, this infinite divisibility of matter, I have added yet another proof, which shows us at the same time a division to infinity, and a motion which slackens to infinity, without ever arriving at rest.
It is certain that, though we may doubt whether extension be divisible to infinity, we cannot, at all events, doubt that it may be augmented to infinity, and that to a plain of a hundred thousand leagues we may join another of a hundred thousand leagues, and so on to infinity. Now this infinite augmentation of extension proves its infinite divisibility; and in order to comprehend this, we have only to imagine a level sea which extends infinitely in length, and a vessel on the shore of that sea, which sets out from port in a straight line. It is certain that to any one looking from the port at the hull of the vessel reflected through a glass, or any other diaphanous body, the ray which terminates at the base of that vessel will pass through a certain point of the glass, and that the horizontal ray will pass through another point of the glass higher than the first. Now, in proportion as the vessel moves away, the point of the ray which terminated at the base of the vessel will always ascend, and will infinitely divide the space which is between the two points; and the farther the vessel goes, the slower it will ascend, without ever ceasing to rise, and without ever arriving at the point of the horizontal ray, because the two lines, intersecting each other in the eye, could never be either parallel or in the same line. Thus this example furnishes at once the proof of a division to infinity of extension, and of a diminution to infinity of motion.
It is through this infinite diminution of extension, which arises from its divisibility, that we are able to prove these problems, which appear impossible from the terms:—To find an infinite space equal to a finite space, or which may be only the half or the third, &c., of a finite space. We may resolve them in different ways; and the following is one, clumsy enough, but very easy:—If we take the half of a square, and the half of that half, and so on to infinity, and then join all these halves together by their longest line, we shall form from them an area of an irregular figure, which will always diminish to infinity at one of the ends, and which will be equal to the whole square; for the half, and the half of that half plus the half of that second half, and so on to infinity; the third, and the third of the third, and so on to infinity, constitute a half. The fourths, taken in the same way, make the third, and the fifths the fourth. By joining the ends of these thirds or these fourths, we shall make from them a figure which will contain the half or the third of the whole area, which will be infinite in length on one side, while diminishing continually in breadth.
The advantage which may be derived from these speculations is not simply the acquisition of these knowledges, which are in themselves barren enough, but in teaching us to know the true limits of our mind, and in making us confess, whether we will or no, that there are some things which exist although we are not able to comprehend them; and hence it is well for a man to weary himself with these subtilties, in order to check his presumption, and to take away from him the boldness which would lead him to oppose his feeble intelligence to the truths which the church proposes to him, under the pretext that he cannot understand them; for, since the strength of the human mind is compelled to bow before the smallest atom of matter, and to confess that it clearly sees that it is infinitely divisible, without being able to comprehend how this can be, it is manifest that we sin against reason in refusing to believe the marvellous effects of the omnipotence of God (which is in itself incomprehensible), because our mind is unable to comprehend them.
But as it is profitable for the mind sometimes to be led to feel its own feebleness, through the consideration of those objects which are above it, and which, being above it, abase and humble it, it is certain, also, that we must endeavour to choose, for our ordinary occupation, subjects and matters which may be more adapted to our capacity, and whose truth we may be able to discover and comprehend. This is done, either by proving effects through their causes, which is called proving a priori, or by demonstrating, on the contrary, causes through their effects, which is called proving a posteriori. It is necessary to extend these terms a little, in order to bring under them all kinds of demonstrations; but it was well to notice them in passing, that we may understand them, and that we may not be surprised when we meet with them in the books or in the discourses of philosophy; and since these reasons are commonly composed of many parts, it is necessary, in order to render them clear and conclusive, to dispose them in a certain order and method. Of this method we shall treat in the greater part of the present book.