Part IV, Chapter 6: Of the Rules which relate to Axioms
EVERY one agrees that there are propositions so clear and so evident in themselves, that they do not need any demonstration; and that all those which are not demonstrated ought to be such, in order to become the principles of a true demonstration. For if they be at all uncertain, it is clear that they cannot be the foundation of a conclusion altogether certain.
But many do not sufficiently comprehend in what this clearness and evidence of a proposition consists. For, in the first place, we must not imagine that a proposition is clear and certain when no one contradicts it; and that we ought to consider it doubtful, or, at least, must be obliged to prove it, when any one denies it. If this were so, there would be nothing certain or clear, since philosophers have been found who have professed to doubt, generally, of everything, and some even who have maintained that there is no proposition at all more probable than its contrary. We ought not, therefore, to judge of certainty and clearness by the disputes of men, for there is nothing that may not be contested, in word, at least; but we must hold as clear that which appears so to all those who will take the trouble to consider things with attention, and who are sincere in the utterance of what their inward conviction is. Hence, what Aristotle says is of most important meaning, that demonstration properly relates to the interior discourse, and not to the exterior; since there is nothing so well demonstrated that it may not be denied by an obstinate man, who undertakes to dispute in words the things even of which he is inwardly persuaded. This is a very ill disposition, and altogether unworthy of a well constituted mind, though it is true that this humour often obtains in the schools of philosophy, through the custom which is introduced among them of disputing about every thing, and making it a point of honour never to yield, he being accounted the man of most mind who is most prompt at discovering evasions for avoiding it; whereas the character of an honourable man is to lay down his arms before the truth as soon as he perceives it, and to love it even in the mouth of his adversary.
Secondly, even those philosophers who hold that all our ideas come from sense, maintain also, that all the certainty and evidence of propositions comes either immediately or mediately from sense. " For,” say they, " even that axiom which is considered as clear and evident as we can possibly desire—the whole is greater than its part—is firmly established in our minds only because that from our infancy we have observed in detail that a man is greater than his head, and a whole house than a chamber, and a whole forest than a tree, and the whole heaven than a star.”
This fancy is as false as that which we have refuted in the First Part, that all our ideas come from sense. For if we were assured of this truth—the whole is greater than its part—only through the different instances in which we had observed it from our infancy, we should have only a probable assurance of it, since induction is only a certain means of knowing a thing when we are assured that the induction is complete;—there being nothing more common than to discover the falsity of what we had believed to be true, on inductions which appeared to us so general, that we could not imagine any exception could be found.
Thus, not long since, it was believed as indubitable that the water contained in a curved vessel, of which one end was much larger than the other, remained always level—being no higher in the small end than in the large—because it had been proved by a multitude of observations. It has been, however, lately found that this is false when one of the ends is extremely narrow, since then the water rises higher in it than necessary.
This shows that inductions alone could never give certain assurance of any truth, before we were assured that they were universal, which is impossible. And, consequently, we could only have a probable assurance of the truth of this axiom, that the whole is greater than its part, if we were only assured of it in consequence of having seen that a man is greater than his head, a forest than a tree, the heaven than a star,—since we should be always open to doubt whether there might not be some other whole, which we had not observed, which was not greater than its part.
It is not, therefore, on these observations which we have made from our infancy that the certainty of this axiom depends. There is, on the contrary, nothing more capable of keeping us in error than the holding fast to these prejudices of our childhood. But this certainty depends solely on this, that the clear and distinct ideas which we have of a whole and of a part manifestly involve that the whole is greater than the part, and that the part is smaller than the whole. And all that could be effected by the different observations which we have made, of a man being greater than his head, a house than a room, has been to furnish us with occasions of paying attention to the ideas of whole and part. But it is positively false that they were the cause of the absolute and immoveable certainty that we have of the truth of this axiom. This, I think, I have demonstrated.
What we have said of this axiom may be said of all others, and thus we believe that the certainty and evidence of human knowledge in natural things depends on this principle, —
All that is contained in the clear and distinct idea of a thing may be affirmed with truth of that thing.
Thus, since the being animal is contained in the idea of man, I may affirm of man that he is animal; since, having all its diameters, equal is contained in the idea of a circle, I may affirm of every circle that all its diameters are equal; since, having all its angles, equal to two right angles is contained in the idea of a triangle, I may affirm this of every triangle.
We cannot dispute this principle without destroying the whole evidence of human knowledge and establishing an absurd Pyrrhonism. For we can judge of things only by the ideas which we have of them, since the only means we have of conceiving them is what we have in our mind, and they are there only through our ideas. Now, if the judgments which we form by considering these ideas do not regard things in themselves, but simply our thoughts—that is to say, if, when I see clearly that the having three angles equal to two right angles is contained in the idea of a triangle, I have no right to conclude, in truth, that every triangle has three angles equal to two right angles, but simply that I think so—it is plain that we could have no knowledge of things, but simply of our thoughts, and, consequently, we should know nothing of the things we are persuaded that we know most certainly; but we should only know that we think them to be so and so, which would manifestly destroy all the sciences.
And it need not be thought that there are any men who seriously acquiesce in these consequences, that we do not know in relation to anything, whether it be true or false in itself. For there are some things so simple and so evident—as, I think, therefore I am; the whole is greater than its part—that it is impossible seriously to doubt whether they are in themselves such as we conceive them. The reason is, that we cannot doubt of them without thinking of them, and that we cannot think of them without believing them true, and, consequently, we cannot doubt them.
Nevertheless, this principle alone is not sufficient to judge of what ought to be believed as an axiom; for there are attributes which are really contained in the idea of things, which, nevertheless, may, and ought, to be demonstrated—as, the equality of all angles of a triangle to two right angles, or of all those of a hexagon to eight right angles. But we must carefully observe whether we need only consider the idea of a thing with a slight attention, in order to see clearly that such an attribute is contained in it, or whether, besides, it is necessary to join to it some other idea, in order to perceive that connection. When it is necessary to consider only the idea, the proposition may be taken as an axiom, especially if that consideration requires only a moderate attention, of which all common minds are capable. But if some other idea be necessary besides the idea of the thing, it is a proposition which needs to be demonstrated. Thus we may give the two following rules for axioms:—1st RULE.
When, in order to see clearly that an attribute belongs to a subject (as that it belongs to a whole to be greater than its part), we need only consider the two ideas of subject and attribute with moderate attention, so that we cannot give this attention without perceiving that the idea of that attribute is truly contained in the idea of the subject. We ought, then, to take this proposition as an axiom which needs no demonstration, because it has, of itself, all the evidence which demonstration could have given to it, since demonstration could do nothing more than show that this attribute belongs to the subject, by employing a third idea to show this connection, which we see already without the aid of any third idea.
But we must not confound a simple exposition (though this should even take the form of an argument) with a true demonstration; for there are axioms which need to be explained, in order that they may be better understood, although they do not need to be demonstrated, the exposition being nothing more than saying in other words, and more at length, what is contained in the axiom, whereas, demonstration requires some new mean which the axiom did not clearly contain.
2d RULE.
When the simple consideration of the idea of the subject and the attribute is not sufficient to enable us to see clearly that the attribute belongs to the subject, the proposition which affirms that it does ought not to be taken as an axiom; but it ought to be demonstrated by employing some other ideas to show that connection, as we employ the idea of parallel lines in order to show that the three angles of a triangle are equal to two right angles.
These two rules are more important than we may think, for it is one of the most common defects among men, that of not consulting themselves in relation to what they affirm or deny,—of referring to what they have heard said, or what they have previously thought, without carefully observing what they would think of them themselves if they were to consider with more attention what passes in their own mind—of confining themselves rather to the sound of the words than to their true ideas—of affirming, as clear and evident, that which it is impossible for them to conceive, and denying, as false, what it would be impossible for them not to believe true, if they would take the trouble to consider it seriously.
For example, those who say that in a piece of wood, besides its parts and their situation, their figure, their motion, or rest, and the pores which enter into their parts—there is still a substantial form distinguished from all this, think they say nothing but what is certain, while, however, they utter a thing which neither themselves nor any one else comprehends, or ever will comprehend.
While, if, on the contrary, we would explain to them the effects of nature, by the insensible parts of which bodies are composed, and by their different situation, size, figure, motion, or rest, and by the pores which traverse these parts, and which allow or arrest the passage of other matters, they believe that we speak to them only of chimeras, although we tell them nothing but what may be conceived very easily; and, by a strange perversion of mind, the facility even with which these things are comprehended induces them to believe that they are not the true causes of natural effects, but that these are more hidden and mysterious; so that they are more disposed to believe those who explain them by principles which they cannot conceive than those who employ only principles which they can understand.
And it is, again, humorous enough, that when we speak to them of insensible parts, they think themselves entitled to reject them, because they can neither see nor touch them; while, however, they rest satisfied with substantial forms, ponderosity, attractive virtue, &c., which they not only never saw or touched, but which they cannot even conceive.