307. The understanding knows what it makes; but this is not all that it knows; for it has truths which neither are nor can be its works, since they are the basis of all its works, as, for example, the principle of contradiction. Can the impossibility of a thing being and not being at the same time be said to be the work of our reason? Assuredly not. Reason itself is impossible if this principle be not supposed; the understanding finds it in itself as an absolutely necessary law, as a condition sine qua non of all its acts. Here, then, Vico's criterion fails: "the understanding knows only the truth it makes:" and yet the understanding knows but does not make the truth of the principle of contradiction.
308. Facts of consciousness are known by reason, although they are not its production. These facts are not only present to consciousness, but are also objects of the combinations of reason: here, then, Vico's criterion again fails.
309. Although in those things that are a purely intellectual work, the understanding knows what it makes, it does not make whatever it chooses; for then we should have to say that science is perfectly arbitrary: instead of the geometrical results we now have, we might have others as numerous as the individuals who deal in lines, surfaces, and solids. This shows reason to be subject to certain laws, its constructions to be connected with conditions which it cannot abstract. One of these conditions is the principle of contradiction, which would, were it to fail, annihilate all knowledge. True, by a series of intellectual constructions one may ascertain the size of a sphere; but can two understandings obtain two different values of it? They cannot, for that would be an absurdity: they may choose different ways, or express their demonstrations and conclusions in different terms; but the value is the same: if there be any discrepancy, it is because one or the other has fallen into an error.
310. If we thoroughly examine this matter, we shall perceive that the intellectual construction, of which Vico speaks, is a fact generally admitted. There are in this philosopher's system two new things, the one good, the other bad; the good, is to have indicated one reason of the certainty of mathematics; the bad, is to have exaggerated the value of his criterion.
We have said that his system expressed a fact generally recognized, but exaggerated by him. The understanding undoubtedly creates, in some sense, ideal sciences; but in what sense? Solely by taking postulates, and combining its data in various ways. Here ends its creative power, for in these postulates and combinations it discovers truths not placed there by itself.
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.