What is the triangle in the purely ideal order? A creation of the understanding, which disposes the lines in a triangular form, and, preserving this form, modifies it in a thousand ways. Thus far there is only one postulate and different combinations of it: but the properties of the triangle flow by absolute necessity from the conditions of the postulate: the understanding, however, does not make these properties, it discovers them. The example of the triangle is applicable to all geometry. The understanding takes a postulate; this is its free work, but it must not come in conflict with the principle of contradiction. From this postulate flow absolutely necessary consequences, independent of intellectual action, and involving an absolute truth known by the understanding itself. Consequently it is false to say of them that it makes them. Suppose a man so to place a body, that, left to itself, it will fall to the ground: is it the man who gives it the force to fall? Certainly not, but nature. The man only supplies the condition necessary for the force of gravity to produce its effect: when once the condition is performed, the fall is inevitable. Here, then, is a simile which shows clearly and exactly what happens in the purely ideal order. The understanding performs the conditions; from them flow other truths, not made, but known, by the understanding. This truth is absolute, is as the force of gravity in the order of ideas. Hence we see what is admissible, and what inadmissible in Vico's system. The power of combination, a generally recognized fact, is admissible; the exaggeration of this fact extended to all truths, when it only comprises postulates in their various combinations, is inadmissible.
The rules of algebra are conventional inasmuch as they relate to the expression, for this might evidently have been different. Supposing, however, the expression, the development of the rules, is not conventional, but necessary. In the expression an/an the number of times the quantity has entered as factor might clearly have been expressed in infinite ways; but supposing the present to have been adopted, the rule is not conventional, but absolutely necessary; since whatever the expression, it is always certain that the division of a quantity by itself, with distinct exponents, gives for result the diminution of the number of times it has entered as factor: this is denoted by the remainder of the exponents; and consequently if the number of times be equal in the dividend and the divisor, the result will be = 0. Thus we see that even in algebra, what the understanding has to do, is to perform the conditions, and express them as seems to it best: but here its free work ends, for necessary truths result from these conditions; and these it does not make, but only knows.
311. Vico's merit in this point consists in having expressed a very clear idea of the cause of the greater certainty of the purely ideal sciences. In these the understanding itself performs the conditions upon which it has to build its edifice; it chooses the ground, forms the plan, and raises the construction conformably to it. In the real order this ground is already designated, just as are the plan of the edifice and the materials for its construction. In both cases it is subject to the general laws of reason, but with this difference, that in the purely ideal order, it has to regard these laws and nothing else; but in the real order, it cannot abstract the objects considered in themselves, and is condemned to submit to all the inconveniences they are of a nature to cause. We will explain these ideas by an example. If we would determine the relation of the sides of a triangle under certain conditions, we have only to suppose the conditions and attend to them. The ideal triangle is in our understanding a perfectly exact, and also a fixed, thing. If we suppose it to be an isosceles triangle with the relation of the sides to the base as seven to five, this ratio is absolute, immutable, so long as the supposition remains unchanged. In all our operations upon these data, we are liable to mistakes of calculation, but no error can arise from inexactness of data. The understanding knows, indeed, for what it knows is its own work. If the triangle be not purely ideal, but realized upon paper, or on the ground, the understanding vacillates because those conditions, which, in the purely ideal order, it fixes with all exactness, cannot be transferred in like manner to the real order; and even were they transferred, the understanding would have no means of appreciating them. Therefore, Vico says, with great truth, that our cognitions lose in certainty in the same proportion as they are removed from the ideal order and swallowed up in the reality of things.
312. Dugald Stewart probably had in view this doctrine of Vico when he explained the cause of the greater certainty of mathematical sciences. It does not, he says, depend upon axioms, but upon definitions; that is, he adopts, with a slight modification, the system of the Neapolitan philosopher, that the mathematical are the most certain, because they are an intellectual construction founded upon certain conditions placed by the understanding and expressed by the definition.
This difference between the purely ideal and the real order did not escape the scholastics. They were accustomed to say that there was no science of contingent and particular, but only of necessary and universal things. In the place of contingent substitute reality, since all finite reality is contingent; and instead of universal put ideal, since the purely ideal is all universal; and you will have the same doctrine enunciated in distinct words. It is not easy to show exactly how far modern philosophers have availed themselves of the scholastic doctrine, in so far as the distinction between pure and empirical cognitions is concerned; but it is certain that some very clear passages upon these questions are to be found in the works of the scholastics. It would not be strange if some moderns, particularly Germans, whose laboriousness is proverbial, especially in matters of erudition, had read them.(27)
CHAPTER XXXII.
THE CRITERION OF COMMON SENSE.
313. Common sense is an exceedingly vague expression. It should, like all expressions which contain many and different ideas, be considered under two aspects: that of its etymological, and that of its real value. These two values are not always the same; they are sometimes greatly discrepant; but even in their discrepancy, they usually preserve intimate relations. We must not, in order duly to appreciate the meaning of such expressions, confine ourselves to their philosophical, and contemn their vulgar meaning. In the latter there is often a profound philosophy; for, in such cases, the vulgar sense is a kind of precious sediment left by the flow of reason upon the word during many ages. It thus happens that in measure, as the vulgar sense is understood and analyzed, the philosophical question is determined, and the most intricate questions solved with the greatest facility.
314. It is remarkable, that besides the corporal senses there should be another, called common sense. Sense: This word excludes reflection, all reasoning, all combination; nothing of this kind enters into the meaning of the word to sense. When we sense, the mind is rather passive than active; it does nothing of itself; it does not give, it receives; it suffers, but does not perform, an action. This analysis leads us to a very important result, and this is, the separation from common sense of all that upon which the mind exercises its activity; and the determination of one character of this criterion, which is, with respect to common sense; the understanding has nothing to do but submit itself to a law perceived, to an instinctive and unavoidable necessity.
315. Common: This word excludes all individuality, and shows the object of common sense to be general to all men.
The simple facts of consciousness are facts of sense, but not of common sense; the mind feels them when it abstracts objectiveness and generality; what it experiences within itself is an experience exclusively its own, and one which has no connection with others.
The word common shows the objects of this criterion to be common to all men, and consequently referable to the objective order, since the purely subjective, as such, is limited to the individual, and in no wise affects what is general. So exact is this observation, that in ordinary language no internal phenomenon, however extravagant, is ever said to be opposed to common sense, provided it be expressed simply with abstraction from its relation to the object. If a man says:
