Certitude, the truth of consciousness, is not science; but it is not on that account false. Vico was careful not to call the theories of Descartes false: his intention was only to lower them from complete truth to fragmentary truth, from science to consciousness. Cogito ergo sum is very far from false. That we find it expressed by Plautus's Sosia is an argument not for rejecting it, but for accepting it; only, as a truth of simple consciousness. Thought is not the cause of my existence, and as such is not the ground of scientific knowledge of that existence. If it were, since man, as the Cartesians admitted, consists of body and mind, thought would be the cause of the body: a doctrine which would plunge us into all the mazes of the controversy on the mutual effects of mind and matter. The cogito, then, is a mere sign or indication of my existence, and nothing more. The clear and distinct idea cannot serve as a criterion even of the mind itself, to say nothing of other things; since the mind, though it knows itself, does not create itself, and accordingly is ignorant of the genus or mode by which it has this knowledge. But the clear and distinct idea is all that is granted to human thought, and, as the only wealth it possesses, is beyond price. For Vico, too, metaphysic holds the highest place among the human sciences, and all others depend upon it; but while for Descartes it can proceed by a method of absolute demonstration parallel to that of geometry, for Vico it must be satisfied with probabilities. It is a science not by causes, but of causes. And with probabilities it has been content in its greatest periods, in ancient Greece and in Italy at the Renaissance. Whenever, intoxicated by the arrogance that declares that "a wise man has no opinions" (sapientem nihil opinati), it has sought to abandon the probable, it has set its feet upon the path of confusion and decadence. The existence of God is certain, but not scientifically demonstrable; and any attempt at such a demonstration must be considered a proof not so much of piety as of impiety, since to demonstrate God we must create him: man must become the creator of God. Similarly we must accept as true all that God has revealed; but we must not ask how it comes to be true. That we can never understand. Human science bases itself upon revealed truth and the consciousness of God, and finds there its test of truth; but the foundation itself is not science, but consciousness.
Just as Vico depreciated metaphysics, theology, and physics, the sciences upon which Descartes had bestowed honour and attention, so he reinstated those branches of knowledge which Descartes had in turn despised; namely, history, observation of nature, empirical knowledge of man and society, eloquence and poetry. Or rather, he could vindicate them without reinstating them. Once he had shown that the lofty truths of a geometrically deduced philosophy were themselves brought down to mere probability, to statements having the validity of simple consciousness, the other forms of knowledge were ipso facto conclusively vindicated. All now found themselves upon an equality in the position, whether high or low, which we have described. The idea of a perfect human science, holding itself aloof from another science unworthy of the title, as founded not on reason but on authority, was shown to be illusory. The authority of observations and beliefs, whether one's own or others', public opinion, tradition, the consciousness of mankind, were restored to the position which they had always held: a position which they held even for Descartes himself, who, as often happens, despised the resources in which he was richest and of which he made the greatest use. A conspicuously learned man, he depreciated learning and scholarship, as one who has received nourishment from it might give himself the luxury of speaking with contempt of the common food which by now forms the very blood in his veins.
The Cartesian polemic against authority had proved in some respects beneficial. It put an end to the servile attitude, all too common, of continual appeals to authority. But this error was not more prevalent than that of private judgment, which presumed to reorganise knowledge from top to bottom on the strength of the individual consciousness: a tendency which ultimately, as in the case of Malebranche, leads to prophesying the immolation of all the ancient philosophers and poets, and a return to the nakedness of Adam. It is a fallacy, or at least an excess, which should be avoided by adopting a sound middle course. This course consists in following private judgment with due regard to authority; in a true catholic union of faith with a criticism limited by and helpful to faith; bearing in mind the necessary character of mere probability which is proper to human knowledge or science, and avoiding the tendency of the Reformation which elevates each man's inner consciousness into a divine guide in matters of belief.
To another group of the Cartesian sciences, however, Vico seems to grant a privileged position, one, that is, not of consciousness but of science strictly so called, in the sphere not of certitude but of truth; namely, the mathematical sciences. These, according to him, form the only region in which man's knowledge is identical in character with God's, perfect and demonstrative. This is not due, as Descartes supposed, to their self-evident character. Self-evidence, when employed in physical science and in matters of action, does not yield truth of the same conclusiveness as in mathematics. Nor is mathematics in itself self-evident. What clear and distinct idea can lead, for instance, to the conception of a line as composed of points having no parts? But the indivisible point which cannot be conceived in the world of reality, can be nevertheless denned. By defining certain names, man creates the elements of mathematics; by the postulates, he carries them on to infinity; by the axioms, he establishes certain eternal truths; and, disposing these elements with the help of these infinities and this eternity, he creates the truth which he teaches. The validity of mathematics then arises not from the Cartesian principle, but precisely from Vico's other proposition, the conversion of knowledge with creation. "We demonstrate mathematics, because we create their truth" (mathematica demonstramus, quia verum facimus). Man assumes unity and multiplicity, points and figures, and creates numbers and quantities which he knows perfectly because they are his own work. Mathematics is a constructive science; not only in its problems, but even in its theorems, which are commonly supposed to be mere objects of contemplation. For this reason it is a science which demonstrates per causas, in opposition to that other common view which excludes from mathematics the concept of causation. It is in fact the only one among all the human sciences which truly demonstrates by causes. Hence its extraordinary accuracy. The whole secret of the geometrical method lies first in defining the terms, that is, creating the concepts which are to be the subject matter of our reasoning; secondly, in establishing certain common principles by mutual consent of the disputants; and lastly, if required, in making certain postulates of such a nature that they can be granted, to enable us to proceed with our deductions, which without such an agreement could make no progress; then, upon these principles, to advance by degrees from the demonstration of the simplest truths to the most complex, and not to affirm the complex propositions before examining singly their component parts.
It might be said that, as to the validity of mathematics, Vico is in agreement with Descartes; he differs from him only in his reason for this validity. And, admitting that Vico's reason must be thought the more profound, this would only enhance and strengthen the mathematical ideal which Descartes had set before science. If mathematics is the one perfect form of knowledge attained by the human mind, obviously we must found the others upon it, and either remodel or condemn them according to its pattern. Vico, in short, was hasty in declaring Descartes wrong: he had found a better argument whose existence the latter had not suspected. But, however strongly this may appear at first sight (and so it has appeared to some commentators), on a closer examination it is seen that the high perfection attributed by Vico to mathematics is more apparent than real; that the vaunted conclusiveness of its method is by his own confession gained at the expense of truth: in a word, that the stress of his theory falls less on the truth of mathematics than on its arbitrary nature.
The fact is, that man, while occupying himself with the investigation of the nature of things, and ultimately realising his total inability to attain it, not having in himself the elements of which they are composed, which are indeed all external to his nature, is led by degrees to the intention of profiting by this very fault of his mind. By means of abstraction—not, be it remembered, abstraction from material things, for Vico is opposed to the empirical origin of mathematics, but abstraction brought to bear on metaphysical entities—he creates two fictions, duo sibi confingit: the point in geometrical figures, and unity in multiplication. Each is a fiction, utrumque fictum, because the point when drawn is no longer a point, and the unit when multiplied is no longer one. Then, from these fictions, by his own arbitrary fiat, proprio iure, he assumes an infinite process, so that lines may be produced or the unit multiplied ad infinitum. Thus he constructs for his own purposes a world of forms and numbers, all of which he embraces within himself; and by lengthening, shortening, and combining the lines and adding and subtracting the numbers, he performs infinite operations and learns infinite truths. Since he cannot define things, he defines names; since he cannot reach the elements of reality he satisfies himself with imaginary elements, the ideas arising from which admit of no dispute. Like God, ad Dei instar, from no material substrate and, as it were, out of nothing he creates the point, the line, and the surface; the point, assumed as that which has no parts; the line, as the locus of a point, or as length without breadth or depth; and the surface, as the meeting of two different lines in one point, that is, length and breadth without depth. Thus mathematics overcomes the failing of human knowledge, that its objects are always external to itself, and that the mind which endeavours to know them has not created them. Mathematics creates what it knows; it contains in itself its own elements, and thus forms a perfect copy of the divine knowledge (scientiae divinae similes evadunt).
The reader of these and other similar descriptions and praises by Vico of the processes of mathematics seems to observe in them something like a tinge of irony; which, if not actually intentional, certainly results from the facts of the case. The brilliant truth of mathematics arises, it appears, from despair of attaining truth; its tremendous power from the knowledge of impotence. The similarity of the mathematician to God is not altogether unlike that of the imitator of an object to its creator. What God is in the universe of reality, man is in the universe of quantity and number,—a universe indeed, but one peopled by abstractions and fictions. The divinity which has been conferred upon man is only, so to speak, a Twelfth-night Godhead.
The different origin assigned by Vico to mathematics results in a correspondingly profound change in the validity of its truth. Mathematics no longer, as with Descartes, stands at the summit of human knowledge, an aristocratic science, destined to reclaim and to rule over the inferior sciences. It occupies a field as strictly limited as it is unique, beyond which if it ever attempts to pass it loses in a moment its magical virtue.
The power of mathematics is met by obstacles both a parte ante and a parte post, in its foundations and in the superstructure which in its turn it is to support. In its foundations, because if it creates its own elements, that is to say, the initial fictions, it does not create the matter of which they are formed, which is given to it no less than to the other human sciences by metaphysics, which while it cannot supply it with its true subject matter, supplies it with definite images of it. From metaphysics, geometry takes the point by drawing it, that is by annihilating it as a point, and arithmetic the unit by multiplying it, that is by destroying it qua unit. But since metaphysical truth, however certain it may seem to consciousness, is indemonstrable, mathematics itself rests in the last resort upon authority and probability. This is enough to expose the fallaciousness of any mathematical treatise which makes use of metaphysics. Vico seems to be involved in a kind of circle between geometry and metaphysics, of which the former, according to him, owes its truth to the latter, and after receiving it gives it back again to metaphysics, thus in turn supporting the human science by the divine. But this conception, the truth of which is more than doubtful, indeed we may frankly call it inconsistent and contradictory, recalls, whatever its value, the metaphysical or rather poetical or symbolic use made of mathematics by Pythagoras and other philosophers of antiquity and the Renaissance, and has no resemblance to a mathematically-treated philosophy like the Cartesian. Geometry in Vico's opinion is the one hypothesis by which metaphysics passes over into physical science. But while making this advance it remains a hypothesis, a probability, something intermediate between faith and criticism, imagination and reason; which indeed is the eternal character of metaphysics and human science in general according to Vico's point of view in this first phase of his theory of knowledge.
Just as mathematics cannot be the basis of metaphysics, the science from which it is itself derived, so it cannot provide a foundation for the other sciences, although they follow it in order of derivation. All objects other than number and size are beyond the reach of the geometrical method. Physical science is indemonstrable: if we could demonstrate the physical world, we should be creating it (si physica demonstrare possemus, faceremus): but we do not create it, and are accordingly unable to demonstrate it. The introduction of the mathematical method into natural science has not helped it. Without the mathematical method, science makes great discoveries; by its means it has made none, whether great or small. The physical science of to-day is in fact like a house, sumptuously furnished by former owners, to which their heirs have added nothing, but have occupied themselves merely in moving and rearranging the furniture. Accordingly, we must reintroduce and maintain the experimental method in physical science, as opposed to this mathematical method; the English tendency as opposed to the French; the cautious use made of mathematics by Galileo and his school, as against the Cartesians' reckless and presumptuous employment of it. The English are right in not allowing physical science to be taught in the mathematical style. Such a style admits of progress only when the terms are defined, the axioms established, and the postulates granted. In physical science we have to define not terms but things: we can make no unchallenged statements; and the complexity of nature forbids our forming any postulates. Thus in the more favourable instances this method results in a mere harmless verbalism. Observations of nature are expounded with the phrases: "By definition IV.," "By postulate II.," "By axiom III.," and concluded with the pompous abbreviation "Q.E.D." But all this carries no demonstrative conviction. The mind retains as much freedom of opinion as it had before listening to such noisy methods. In these circumstances Vico could not refrain from satirical comparisons. The geometrical method, he says, in its proper sphere works unnoticed; when it makes a noise, it shows that it is doing no work; just as a coward's attack consists of much shouting and no blows, while a brave man holds his tongue and strikes home. Again, the man who upholds the geometrical method in subjects where it fails to carry conviction, when he pronounces this to be an axiom, or that to be a demonstrated truth, is like a man who draws amorphous pictures, quite unrecognisable without assistance, and then writes underneath "This is a man," or "This is a satyr," or "This is a lion," or the like. Hence it happens that the very same geometrical method served Proclus to demonstrate the principles of Aristotelian science, and Descartes to demonstrate his own, though totally distinct from, if not diametrically opposed to them. Yet each was a great geometrician, whom no one could accuse of inability to use the method. What ought to be introduced into natural science is not the method of geometry, but its conclusiveness; which is precisely what can never be done. Still less is it possible in other sciences, in proportion as they are more material and concrete; least of all in ethical science. For this reason, where the reality cannot be used, the name is misused instead; till, just as the title "Master," which Tiberius once refused as too haughty, is given to-day to the humblest man, so the name "demonstration," applied as it is to arguments at best probable, sometimes patently fallacious, has impaired the respect due to truth.