The "creation" of mathematics spoken of by Ficino, Cardano and others signified a mental production entirely free from material presuppositions, and for that reason not less true but true in a higher sense. It is almost the same sense as that found in Descartes and his followers. Locke asserts the reality of mathematical truths, though he admits that there are in nature no figures corresponding to the archetypes existing in the mind of the geometrician;[44] and Leibniz, commenting on this passage, says that "the ideas of justice and temperance are no more our own invention than those of the circle and the square."[45] Tommaso Cornelio, whom we have quoted on the contrast between physical science and mathematics, also believed that mathematics rested on "certain notions and understandings which nature has put into the minds of men as foundations of science."[46]
Another kind of "creation," and one which seems to have more connexion with Vico's "fingere" is discussed in a passage of Aristotle's Metaphysics which has had a good deal of influence. "We find also," Aristotle says, "geometrical figures by actualising them (ἐνεργεἰα), because they are found by being divided: if they were divided, they would be obvious, but in reality they exist potentially. Why has the triangle two right angles? Because the angles round one point are equal to two right angles. If then we construct the angle along one side, it would become plain to any one looking at it. Why is the angle in the semicircle equal to a right angle? Because if there are three equal lines, two in the base and one drawn perpendicular to it, it is plain to any one who sees it and knows that. Whence it is evident that we discover things that exist potentially by reducing them to actuality. This is because the actuality is understanding, and the potentiality proceeds from the actuality; so we know by making (Greek: καὶ διὰ τοῡτο ποιοῡντες γιγνώσκουσιν)."[47] But these observations belong to the explanations given by Aristotle in this passage of the conceptions of potentiality and actuality; they are not at all opposed to his theory of mathematics as studying the intelligible matter which subsists in sensible matter, and they only explain the difference between potential and actual truth. In the same way we sometimes find in later philosophers the assertion that mathematical truths are demonstrated and problems resolved "by making them." Thus Sarpi writes in the passage mentioned above: "in mathematics, he who constructs knows because he makes, and he who analyses learns because he seeks how the thing is made. The mode of composition then belongs to the inventive faculty and that of analysis to the discursive: the former is that of problems, the latter of theorems; the latter are demonstrated by analysis, the former by composition."[48]
It has also been recently asserted that the Vician philosophy of mathematics reappears bodily in Galileo and his school;[49] an astounding fact when baldly stated, since even though Vico opposes and prefers the great Pisan to Descartes for the moderate use he makes of mathematics in physical science, it is certain that for Galileo as for Leonardo da Vinci mathematics had an objective validity, and the book of nature is written in mathematical characters and geometrical figures. In any case, the passage of Galileo which has been quoted in this reference, on the intensive identity of human with divine knowledge, has nothing to do with the present question, and another passage which asserts that the explanations of terms are free, and it is in the power of every workman to circumscribe and define in his own way the things he is dealing with, without ever being led by this into error or falsehood, and that for instance one may call the bow the stern and the stern the bow, says nothing but a platitude hardly worth saying except by way of adorning a page of controversial rhetoric.[50] In controversy one is often obliged to insist upon platitudes, and the controversy upon which I am now engaged itself presents too many examples.
A passage from the Lezioni accademiche of Galileo's pupil Evangelista Torricelli in which he speaks of the difference between physical and mathematical definitions seems at first sight more convincing. But the critic who has called attention to this passage[51] says too much when he asserts that "it is beyond doubt that Vico had read it," since it is unquestionable that Vico had not read it. The Lezioni accademiche were published first posthumously in 1715[52] and Vico's theory of mathematics is expounded in the De ratione in 1708 and the De antiquissima, 1710. This, it is true, is of secondary importance, for Vico may have known Torricelli's doctrine through indirect channels, through other books or even orally through some Neapolitan friend or pupil of Torricelli; in any case, if the latter's theory though unknown to Vico was really identical with his own, the similarity of ideas between the two would be of the greatest interest. Unfortunately the critic has been too hasty, as it seems to me, even in his study and interpretation of the pages of Torricelli.
In the passage in question, a lecture Della leggerezza, read to the Accademia della Crusca, Torricelli controverts, as based on mere appearances and not confirmed by facts and reasoning, Aristotle's definition in the De coelo: "heavy is that which has a natural property of going towards the centre." He remarks upon this: "The definitions of Physics differ from those of Mathematics in that the former are obliged to adapt and adjust themselves to the object defined, while the latter mathematical definitions are free and can be formed at the will of the geometrician who is defining. The reason is perfectly plain: the things defined in Physics do not come into being with the definition, they exist already by themselves and are found in nature previously. But the things defined by geometry, that is by the science of abstraction, have no existence in the universe of the world other than that which definition gives to them in the universe of intelligence. Thus whatever objects of Mathematics are defined, the same objects will come into existence simultaneously with the definition."[53]
The arbitrary character of mathematics seems here to be clearly stated. But let us reserve our judgment and read on. "If I were to say, the circle is a plane figure with four equal sides and four right angles, this is not at all a false definition; but for the rest of my book I should have to mean, whenever I spoke of a circle, a certain figure which others have called a square. But if a man should say in Physics, 'the horse is a rational animal,' should we not be justified in calling him the horse? We must first look very carefully to see whether the horse is a rational animal or not and then define it as it is, in order that the physical definition may conform to the object and not be counted defective." Here we see that what appeared to be a profound thought has turned out to be a platitude; it is indifferent whether we call the bow the stern or the stern the bow, said Galileo, or, says Torricelli in his turn, whether we call a square a circle or a circle a square; while it does not seem to him an indifferent matter whether we call a horse a rational animal. But even this does not prevent him from admitting later some degree of arbitrariness in physical terminology, when he says, "since then it is not demonstrated that the intrinsic principle of downward motion exists upon the earth, I will accept this definition, if the tests will allow me, as the simple imposition of a name, and, replacing the verb 'to be' by the verb 'to be called,' I will adapt the definition to my own requirements thus: That is called heavy which descends to the centre. Whenever any one says, the earth is heavy, I will agree, but always with the interpretation that the word 'heavy' only signifies descending in a lighter medium."[54]
It seems to me then that the difference which he begins by laying down between mathematics and physical science is considerably obscured in the sequel. And indeed how could Torricelli have seriously thought that the foundation of mathematics was a "fiction," when among his lectures one heard the title "in Praise of Mathematics"? In this lecture he says, quite in the Galilean style: "That to read the great Book of the Universe, the book on whose pages may be found the true philosophy written by God, mathematics are indispensable, will be seen by any one who with noble thoughts aspires to the science of the integral parts and greatest members of this huge body we call the World. The one alphabet, the only characters with which we can read the great manuscript of the divine philosophy in the book of the Universe are those poor figures you see in the text-books of geometry."[55] The most we can see in these statements is a vague and hazy presentment of the profound difference between physical truths and the so-called truths of mathematics.
In conclusion, until for the third of my three points we can discover much more obvious "sources" than those suggested up till now, I shall see no cause to modify my verdict upon the originality of Vico's conception of mathematics. This originality is further proved by the important consequences drawn by Vico from his theory of mathematics for his philosophical method; for every one knows that a thought taken over bodily from another remains inert and sterile, while an original idea is always active and fruitful.
Note.—I have selected, of the various criticisms directed against my book on Vico, that concerning his "originality," because this gave me opportunities for researches and explanations of some value. But my book has been subjected to two general criticisms which do not lend themselves to the same treatment.
It has been said that in my exposition of Vico's philosophy I have followed my personal philosophical convictions: and sermons and epistles have been showered upon me preaching the duty of casting off prejudices, etc., and narrating the history of philosophy in an objective manner, etc. But I should like my critics to believe that my "convictions" cannot have, to my mind, the character of prejudices, but precisely that of liberation from prejudice, which is what they demand: that detachment and purity of understanding which is necessary for the comprehension of historical facts, and is not, as some fancy, a primeval innocence, but the fruit of laborious cultivation. To grasp Vico historically in his strict reality I have been compelled to undergo a catharsis of prejudices, consisting in my case of the philosophy to which my own efforts had led me. My ideas may be untrue, but that is another question; and that means that if their falsity is proved I am bound to clear and purify my mind by means of less false ideas; but these in their turn must always be ideas and become convictions. In point of abstract method, no objection at all can be made to any one who looks at Vico through the spectacles of scholasticism if he thinks they make his sight more distinct and penetrating; the most we can do is to try and persuade him that there are better spectacles on the market. But we certainly have the right to smile if this same scholastic goes on to warn us that "in studying a philosopher, in investigating and reconstructing his thought, it is absolutely necessary to bring to the task a mind free from preconceptions and hostile to prejudices"; while all the time he is trying to pass off his scholastic opinions and religious beliefs under the banner of objectivity, sincerity and freedom from prejudice. "Philosophers"—I have seen this assertion too—"are unfitted for writing the history of philosophy, because they have ideas of their own." And who is fitted for it? People who are not philosophers? Does not Vico teach us precisely this, that where he who makes the facts (as the philosopher makes philosophy) himself narrates them, there history reaches its highest certainty?