We discover nothing in this refutation of skepticism calculated to destroy it. Even supposing all to admit the principle of causality, which all do not admit, what aid can he draw from this principle, when he makes the work of that very understanding, which must make use of it, the only criterion? If causality be the only criterion of truth, the understanding is isolated, and cannot, in the order of effects, take one step beyond what it has itself produced; and, in the order of causes, it cannot ascend higher than itself; for, were it so to ascend, it would know things not made by itself, would know its own cause. With this supposition the skeptics must triumph; cognition is confined to the internal world, to simple appearances; and when one would go out of these, he stumbles against the only criterion which opposes the cognition of all not made by the understanding itself. We do then see reality, but are separated from it by an impassable abyss. The world in itself may be any thing we choose to suppose it; but with respect to us, it will be nothing. This law applies to every intelligence, so that vitality can only be known by the first cause.
These consequences, inadmissible as they are, if we would not throw ourselves unreservedly into the tide of skepticism, are nevertheless inevitable in Vico's system. An original way truly of combatting skepticism, thus to throw open its widest gates!
CHAPTER XXXI.
CONTINUATION OF THE SAME SUBJECT.
304. If the Neapolitan philosopher's criterion be anywhere admissible, it can only be in ideal truths; for as these are absolutely cut off from existence, we may well suppose them to be known even by an understanding which has not in reality produced them. So far as known by the understanding they involve no reality, and consequently no condition that exacts any productive force not referable to a purely ideal order. In this order the human reason seems really to produce. If we, for example, take geometry, we shall readily perceive that, even in its profoundest parts and in its greatest complications, it is only a kind of intellectual construction, wherein that only is to be found which reason has placed there.
Reason it is which by force of perseverance has succeeded in uniting elements and so disposing them as to attain that wonderful result, of which it may say with truth: this is my work.
If we carefully observe the development of the science of geometry, we shall perceive that the extended series of axioms, theorems, problems, demonstrations and solutions, begins with a few postulates, and that it goes on with the aid of the same, or others discovered by reason according to the demands of necessity or utility.
What is a line? A series of points. The line, then, is an intellectual construction, and involves only the successive fluxions of a point. What is a triangle? An intellectual construction wherein the extremities of three lines are united. What is a circle? Also an intellectual construction; the space enclosed by a circumference formed by the extremity of a line revolved around a point. What are all other curves? Lines described by the movement of a point governed by a certain law of inflexion. What is a surface? Is not its idea generated by the motion of a line, just as that of a solid is generated by the motion of a surface? And what are all the objects of geometry but lines, surfaces, and solids of various kinds, combined in various ways? Universal arithmetic, whether arithmetic properly so called, or algebra, is a creation of the understanding. Number is a collection of units, and it is the understanding that collects them. Two is only one and one, and three only two and one; and thus with all numerical values. The ideas expressing these values consequently contain a creation of our mind, are its work, and include nothing not placed there by it.
We have already observed that algebra is a kind of language. Its rules are partly conventional, and its most complicated formulas may be reduced to a conventional principle. Take one of the simplest: a0 = 1: but why is it? Because a0 = an-n; why? Because there is a conventional usage to mark division by the remainder of the exponents; and consequently an/an, which is evidently equal to one, may be expressed an/an = an-n = 1.
305. These observations seem to prove Vico's system to be really true, so far as pure mathematics, that is, science of the purely ideal order, is concerned. Possibly also the same may be said of it in relation to other science, as for example, metaphysics; but we shall not follow it farther, since it is not easy to find a ground free from conflicting opinions. Moreover, having shown how far Vico's system is admissible in mathematics, we have thereby given a solution to difficulties to which it is subject in its other branches.
306. That in a purely ideal order the understanding constructs is undeniable, and the schools agree in this. There is no doubt that reason supposes, combines, compares, deduces; operations which are inconceivable without some kind of intellectual construction. The understanding in this case knows what it makes, because its work is present to it: when it combines it knows that it combines; when it compares or deduces, it knows that it compares or deduces; when it builds upon certain suppositions, which it has itself established, it knows in what they consist, since it rests upon them.
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,