Appendix Iii

THE SOURCES OF VICO'S THEORY OF KNOWLEDGE

My statement, that the criterion of knowledge contained in Vico's formula of the conversion of the true with the created is an original and modern principle, has been contradicted by certain Catholic editors; who state that this doctrine, however true, is not original to Vico, and is indeed far from modern, being a purely Scholastic doctrine. If I thought otherwise, this was only due to my insufficient knowledge of Scholasticism.

I might indeed ask at the outset how such complete ignorance of scholasticism were possible: an ignorance not of its manifold varieties and the tangled forest of its distinctions--that would be comprehensible: but of no less a matter than the fundamental criterion of its theory of knowledge, the starting-point of modern thought and as such, it would seem, inevitably familiar to every student of the elements of philosophy. But since it is always useful to suspect oneself of ignorance, or even to believe oneself more ignorant than one really is, I will make so far as concerns myself a voluntary display of humility. I find it less easy, I confess, to extend the accusation of ignorance to all who, like myself, have failed to run Vico's criterion to earth in the scholastic lumber-room: Jacobi for instance, who on reading it as expressed in the De antiquissima, sees in it the first manifestation of Kantianism and absolute idealism: or the Catholic theologian Baader, who finds its later development in Schelling's philosophy of identity: or the learned and subtle Spanish Thomist, Jaime Balmes, who treats it as a unique idea and attacks it from the scholastic point of view: or the equally learned Catholic Bertini, who accepts and develops Jacobi's observation: or the eminent historian of philosophy Wilhelm Windelband, who, while unacquainted with Vico's doctrines, on coming across indications of a similar thought in Sanchez's Quod nihil scitur was greatly struck by it and endorsed its value by the assertion that it was to bear fruit at a later date and in the hands of a greater philosopher, Immanuel Kant: or again the specialist in the history of scholasticism, Karl Werner, the author of a careful monograph on Vico, who nowhere notices the alleged scholastic character of Vico's theory of knowledge. Scholasticism must indeed be a difficult and mysterious doctrine, if it is inaccessible to all these students, qualified and bound though they are to understand it.

But we cannot pause on the threshold to speculate: we must plunge straight into the argument. In what part of scholasticism can we find Vico's criterion converting knowledge with creation?

The Thomistic saying, "truth and reality are convertible," ens et verum convertuntur, has been quoted: but quotations of this kind are perhaps more calculated to confuse by words than to convince by facts. The same value attaches to the statement that Vico himself confessed the scholastic origin of his principle, since the very first chapter of the De antiquissima begins with the words "in Latin, the truth and the fact reciprocate, or, as the scholastic mob says, convert," "Latinis verum et factum reciprocantur, seu, ut scholarum vulgus loquitur, convertuntur." Here it is perfectly clear to any one on a moment's thought that Vico, Latinist as he was, meant simply to substitute the Ciceronian "reciprocari" for the barbarous "converti."

St. Thomas explained the meaning of his formula quite clearly, especially in the Summa Theologica, Part I. question xvi. art. 3. Here he asks whether the truth and the reality are convertible, utrum verum et ens convertantur; to which he replies as follows: "that as the good is of the nature of the desirable, so the truth has the nature of knowledge. But in so far as a thing has existence in itself, thus far it is knowable. And for this reason it is said in De anima, Bk. III. text. 37 (431 b 21) that 'the soul is in a sense all things' according to sense and intellect. And hence as the good is convertible with the existent, so is the true. But yet as the good adds to existence the nature of the desirable, so also the truth adds a reference to the intellect." (Quod sicut bonum habet rationem appetibilis, ita verum habet ordinem cognitionis. Unumquodque autem in quantum habet de esse, in tantum est cognoscibile. Et propter hoc dicitur in 3 de Anima, text. 37, quod 'anima est quodammodo omnia' secundum sensum et intellectum. Et ideo sicut bonum convertitur cum ente, ita et verum. Sed tamen sicut bonum addit rationem appetibilis supra ens, ita et verum comparationem ad intellectum.) Nothing then can be known except what exists, and nothing can exist but what is good: existence, truth and goodness are all convertible. Thus, too, things are called good in so far as they correspond to the idea in their Creator's mind. "Each single thing partakes of the truth of its own nature in so far as it imitates the knowledge of God, like an artefact in so far as it agrees with the art": "the knowledge of God is the cause of things": "the knowledge of God is the measure of things." (Unumquodque in tantum habet de veritate suae natura, in quantum imitatur Dei scientiam sicut artificiatum in quantum concordat arti I. xiv. 12. Scientia Dei est causa rerum I. xiv. 12. Scientia Dei est mensura rerum I. xiv. 12.) But truth and goodness, the objects of intellect and will respectively, if on the one hand they are "convertible in reality," convertentur secundum rem, on the other they are "distinguishable in thought," diversificantur secundum rationem (I. lix. 2). What have these thoughts in common with Vico's idea that the condition of knowing a truth is to create it? In fact, what is here stated is that the condition of making a thing is to know it, or as St. Thomas says in the same place (I. xiv. 8) in St. Augustine's words (De Trinitate xv. 13) "Universas creaturas et spirituelles et corporales non quia sunt ideo novit Deus, sed ideo sunt quia novit." (God does not know all His creatures corporeal and spiritual because they exist: but they exist because He knows them.)

Vico makes no kind of mention of the formula ens et verum convertuntur, though he knows and quotes--a fact which has escaped my critics--the analogous phrase "the true and the good are convertible," verum et bonum convertuntur: a formula which he diverts to his own purposes, or rather unites it with his own. "In the first place," he writes, "I establish a truth which is convertible with the created, and in this sense I understand the good of the schools, convertible with existence: and hence I infer that the one and only truth is in God, since in Him is contained all Creation." This union is reached quite openly by identifying verum with factum, then factum with ens, and finally the verum-factum-ens with the bonum: by substituting the doctrine of Vico for that of the schools. By such a method of interpretation one could reduce all doctrines to a single one, a perennis philosophia. I do not say that it would be a method entirely devoid of truth; but it is certainly not a historical method.

That Vico's criterion is not only different from but inconsistent with Thomism was shown, as I have already said, by Balmes; who pronounced it "specious but devoid of solid foundation." He uses St. Thomas's statements to controvert Vico's theological doctrine that God understands because He creates, opposing to it the Scholastic view that He creates because He understands. He denies, that the Word was conceived by the mere knowledge of what is contained in the divine omnipotence, for it is conceived not simply by creatures but also and chiefly by the cognition of the divine essence ("for the Father by understanding himself and the Son and the Holy Ghost and all other things embraced by His knowledge conceives the Word, so that thus the whole Trinity is implied in the Word, and also every creature": Pater enim intellegendo se et Filium et Spiritum sanctum et omnia alia quae ejus scientia continentur concipit Verbum, ut sic tota Trinitas Verbo dicatur, et etiam omnis creatura); he objects that, granting this criterion, God could never know himself, because He is not His own cause. He denies that intelligence is only possible through causality, inasmuch as it is also possible through identity. He accuses Vico's criterion of involving scepticism: in a word he maintains that the facts of knowledge are known by reason, even if they are not the products of reason. I am not concerned to ask whether Balmes is right, or whether Vico's criterion can be reconciled with Christian theology. I am concerned merely with establishing, not only by quotation from St. Thomas but also by the help of the judgment of an authoritative interpreter of his system that this doctrine is not Thomistic.

Even granting that the criterion in question is irreconcilable with Thomism but not with an improved Christian theology, it is certainly irreconcilable with both in the form it adopts in what I have called "Vico's second theory of knowledge," in the Scienza Nuova, which Balmes either did not know or omitted to mention, and is passed over by my critics with a light-heartedness that is not particularly enviable. One of them asserts that "the alleged distinction" (the distinction that is drawn by myself) "between Vico's first and second theories of knowledge does not in point of fact exist, and produces no effects of any kind." What? Has it no effects, when those historical studies and sciences of mind, which in the De antiquissima occupied the lowest position among mere probabilities became in the Scienza Nuova the truest of all--true even in a higher degree than mathematics itself as dealing with the human world which "is man's creation?" when their form is found "in the modifications of the actual human mind itself?" when they have "a reality as much greater, as the reality of the laws of human affairs is greater than that of points, lines, areas and figures?" Is there no distinction, when we pass from the scepticism of the De antiquissima to the rationalism of the statement that these "proofs are of a divine nature," and must produce "a divine pleasure, since in God to know and to create are one and the same"?

It is true that upon this point my attention has been recalled to a well-known passage of Galileo (Dialogo dei massimi sistemi), an especial favourite of our own Spaventa, where we find the thought that the human intellect differs from the divine extensivè, but not intensivè, and that if the divine intellect knows infinitely more about mathematical propositions because it knows them all, yet "of these few facts known by the human intellect, its knowledge is equal to that of the divine in objective certainty, since it attains comprehension of the necessity than which no greater certainty, it seems, can exist." But in any case Galileo was not a Schoolman, and moreover this pronouncement of his seemed so dangerous to the Christian theory of ideas, that he himself was obliged to alter it by admitting that while "so far as the truth of which mathematical proof gives no knowledge is concerned, this is identical with that which the divine wisdom knows," yet "the manner, in which God knows the infinitely numerous propositions of which we know a few, is immensely superior to our own, which proceeds discursively from one conclusion to another, while His is simple intuition." It is important too not to forget that this very statement figures among the heads of the accusation in Galileo's trial.

If the formula of the conversion of the true with the created is not found in Thomism, it may perhaps be found, at least in its original, sceptical or mystical, intention, in other tendencies of scholasticism or mediaeval philosophy generally. With Thomism Vico seems to have had neither acquaintance nor sympathy: but from his autobiography it is plain that he studied nominalism and the summaries of Petrus Hispanus and Paulus Venetus, though with little profit, and later also, much more profitably, the Scotist philosophy; which he considered the most Platonic of the Scholastic systems. Traces of this appear in several views expressed in the De antiquissima, especially in those dealing with universals and ideas. In this direction, the direction that is to say of the Scotist system and the closely allied system of Occamism, I have attempted various researches, without attaining any remarkable results: further, I have applied for assistance to various specialists in Scholasticism, but in vain; they would do nothing but express their own superficial impressions or lose themselves in idle disputation. In general it seems possible to say that Duns Scotus's theory of knowledge presents points of affinity to that of Vico: for example, in the polemic against the Thomistic doctrine of the adaequatio intellectus et rei, which he refutes by applying it to the divine knowledge, since God knows objects as willed by Him, and they exist because He wills their existence without His being necessitated by them. For Occam again the thought of objects has no reality and objectivity (or subjectivity according to the usage of Scholastic terminology, which is the reverse of modern) in God, and is nothing else than the objects themselves, known by God according to the possibility of creating them, in virtue of which they are thinkable to the divine mind. But the question for Vico is not merely the priority of creation to knowledge or knowledge to creation, but the convertibility or identity of knowledge and creation.

In certain recently published philosophical observations by Paolo Sarpi, a nominalist of Occam's school, the following statements are to be found. They are the more notable because standing as they do without any results in Sarpi's thought and being undeveloped in subsequent philosophy, they seem to be not his own invention but a mere repetition of scholastic dicta. "We have certain knowledge both of the existence and of the cause of those things which we understand fully how to create: of those which we know by experience, we know the existence, but not the cause. We can however guess at it, and look simply for a possible cause: but out of many found to be possible we cannot be certain which is the true one. This fact may be seen in descriptions of astronomical theories, and would also be true in the case of a man who saw a clock for the first time. Of the various guesses, that of a man who knew how to make similar objects would be nearest the truth, e.g. one who understood the construction of machinery when he saw a different kind of machine: but none the less he will never on that account know for certain. There are then three kinds of knowledge: first, knowledge how to make the object: secondly, experience of it: and thirdly, guessing at possibilities." This thought, then, namely that objects are known by their creator, and that God knows objects because He creates them, seems to have been current in the schools: and this explains the fact of its reappearing in an incidental manner and as an obvious truth in Francesco Sanchez's Quod nihil scitur (1581) where it is declared impossible "perfecte cognoscere quis quae non creavit; nec Deus creare potuisset nec creata regere quae non perfecte precognovisset" (that one should know perfectly things which he has not created: nor could God have been able to create nor after creating them to control things which He had not perfectly foreknown).

But need we continue to look for it in the guise of a casual remark or an isolated proposition, devoid of philosophical connexion, in the works of philosophers or the lecture-rooms of the schools? Did it not simply form a part of the common thought which daily declares that the man who has made a thing knows it better than he who has not made it? Probably a little attention would reveal it in many and dissimilar treatises; and for my own part, while reading the Chronicon of Otto of Freising the other day, I came across it in the introduction to the third book, where the chronicler, writing as is well known under the influence of St. Augustine's Civitas Dei, is arrested by the objection that God's designs in history are inscrutable, and delivers himself of the following reflections: "What then shall we do? If we cannot understand, shall we hold our peace? Then who will reply to those who flatter, repel those who attack, and by the reason and might of his words confute those who would destroy the faith that is in us? So we cannot understand the secret counsels of God, and yet we are often compelled to give a reasonable account of these things. What? Shall we reason about things which we do not understand? We can give reasons, but human reasons, when yet we cannot understand the divine reasons. And thus it happens that when we speak of theological matters, lacking the right words for them, we being men use our own words; and in speaking of so great a God in human language, we use our words the more boldly quo ipsum figmentum nostrum cognoscere non dubitamus, because we never doubt that we know the thing we have ourselves formed: quis enim melius cognoscit quam qui creavit? for who knows a thing better than he who has created it?" The logic of the Abbot of Freising at this point may be thought a trifle sophistical: but the fact remains that he refers to a common opinion that he knows things who has made them.

But probably Vico was stimulated to the establishment of his criterion less by certain tendencies of Scotism or by current opinions than by the philosophers of the Renaissance, which he considered the golden age of metaphysical study, when shone, as he says, "Marsilio Ficino, Pico della Mirandola, Augustino Nifo and Augustino Steuco, Jacopo Mazzoni, Alessandro Piccolomini, Matteo Acquaviva and Francesco Patrizio." In Ficino, whose name he couples with those of Plato and Plotinus, and especially in his Theologia Platonica, Vico could read a magnificent description of the productive character of the divine wisdom and its parallelism with that of the geometrician. Nature, says Ficino, which is divine art, differs from human art in that it produces its creations from within, by living reasons: and "it does not touch the surface of matter by means of a hand or any other external instrument, as the soul of a geometer touches the dust when he describes figures upon the earth, but perinde ut geometrica mens materiam intrinsecus phantasticam fabricat, it operates like the mind of a geometer creating an imaginary matter from within itself. For as the geometer's mind, while it considers within itself the nature of figures, forms internally by pictures the image of figures, and by means of this image forms an imaginary spirit without any toil or design, so in the divine art of nature a wisdom of some kind by means of intellectual processes endows with natural seeds the life-giving and motive force itself which is its companion." Vico must have recalled this passage in Ficino when in his inaugural lecture of 1699 he compared God, "the artist of nature," to the human mind which "we may without impiety call the God of art," just as he must have remembered it in the De antiquissima where he compares God to the geometrician. Vico might however have found thoughts of this kind in various Renaissance philosophers, not only in Ficino: among others, in Girolamo Cardano, who contrasts divine and human knowledge, though with a different conclusion; and restricts the one to finite objects ("for understanding is brought about by a kind of proportion, proportione quaderni fit, and there is no proportion between the infinite and the finite"), denying that man can know God, for as Vico said later in almost the same words, "if I knew God, I should be God," si scirem Deus essem. Thus he postulated "other sciences, and other modes of understanding, entirely different from this of ours; more true, more solid, more firm, as a body is than its shadow: and again other principles which we can by no reason apprehend." And not only did he postulate them, but among the human sciences he observed one which as opposed to the natural sciences reached not merely the surfaces of things but almost the things themselves, namely mathematics. "The human soul, situated in the body, cannot attain to the substances of things, but wanders about upon their surfaces by the help of the senses, examining measurements, actions, resemblances and doctrines. But the knowledge of the mind, which creates the fact, is in a sense itself the fact, just as even among human sciences the knowledge of a triangle, that it has three angles equal to two right angles, is practically identical with the truth itself (scientia vero mentis, quae res facit, est quasi ipsa res, veluti etiam in humanis scientia trigoni, quod habeat tres angulos duobus rectis aequales, eadem ferme est ipsi veritati), whence it is clear that there is in us a natural science of a different kind from true science." Here, in the definition of divine knowledge and of the procedure of human knowledge in the case of mathematics, as opposed to that of physical science, is implicit the principle that true knowledge consists in the identity of thought with its object.

The idea of the opposition of mathematics to physical science, in the certainty of the one and the uncertainty of the other, persisted in the Neapolitan philosophers and scientists of Vico's youth, even if they lost sight of the reason of this opposition. Tommaso Cornelio in his "progymnasma" De ratione philosophandi (1661) after reviewing the errors produced by the illusions of sense in physical science, says, "the contemplations of mathematics are not subjected to errors of this kind, dealing as they do with things whose images are not introduced into the mind by the senses; for the mind can by itself adequately conceive figures and numbers, whose properties and analogies are examined by mathematicians, without aid from sense." This ought to be emphasised, since it seems highly probable that Vico was stimulated to the establishment of his general theory of knowledge by reflection upon mathematics and the contrast between it and physical science. In fact the Latin speeches, our earliest documents for his studies, though they show the influence of Ficino and a certain amount of Cartesianism, are never dominated by this general criterion. It is only in the last of these speeches, that of 1707, that the distinction between mathematics and natural science begins to appear; in the next year it is clearly stated in the De ratione studiorum, where it takes the form of a general criterion. "We demonstrate geometry because we make it: if we could demonstrate physical facts, we should be creating them. For the true forms of things exist only in God the greatest and best, and to these the nature of them conforms" (geometrica demonstramus quia facimus: si physica demonstrare possemus, faceremus. In uno enim Deo Opt. Max. sunt verae rerum formae, quibus earundem est conformata natura). And this theory attained its full development in 1710 in the De antiquissima.

Such are the probable precedents, or as the common but inaccurate metaphor expresses it, "sources" of Vico's theory of knowledge. I do not think that the formation of this theory can have been influenced by the propositions of Geulinx and Malebranche which have been pointed out to me, namely that "no one can make that which he does not know," and that "God alone knows his works, because he foreknows his action." In these propositions the old Thomistic doctrine is substantially summarised. Much more tenable would be a connexion or at least a comparison with Spinoza; we may recall the Spinozistic identification of the ordo et connexio idearum and the ordo et connexio rerum. Another ingenious, but I think, inaccurate view is that "the analytic geometry of Descartes was the introduction of the genetic principle into the study of geometrical objects," so that the principle verum ipsum factum "before being formulated by Vico had been practised by Descartes"; and Vico in the De antiquissima "adopted the scientific method of Descartes, which he stated as the convertibility of the true with the created," raising it "from a certitude to a criterion." We are dealing here not with practice, but simply with the theory of method: for this method, conceived in its universality, just so far as it is practical has always been practised; not by Descartes alone, and not only by analytic geometry.

We should certainly be better informed as to the precedents of Vico's criterion if we knew more of his studies preparatory to the De antiquissima, and if in general we had more literary evidence about his youth. Perhaps even the precedents I have indicated, which I only called probable, are not quite free from an element of chance; they may be connexions only imagined by myself and non-existent for Vico's mind, while others not accidental may perhaps be still unknown, or await discovery by a student more fortunate than myself. But it may not be out of place to remark that the search for "precedents" does nothing to explain the new thought that followed them; much less does it detract from the value of that thought. Such information, though on the one hand it enriches our knowledge of the history of philosophy, on the other hand has absolutely no effect upon the determinate thought under examination. It is valuable in the biography of a philosopher, but valueless for the comprehension of the proper meaning of the new theory, which must be sought essentially in the new problem which it faces and attempts to solve. In the history of philosophy the same principles hold good as in that of literature. Take for example the episode of Argante and Tancred in canto xix. of "Jerusalem Delivered"; Argante, while taking up his position for the fight with his adversary, turns "as if in doubt" to the "afflicted city," towards Jerusalem attacked by the crusaders; and when Tancred brutally mocks him, asking whether he does this out of fear, he replies:--

I was but thinking how this city, The immemorial green of Juda's realm, Is falling, vanquished; whose unhappy fate I have in vain endeavoured to repel.

Here the precedents are easily found; Hector parting from Andromache and foreseeing the unhappy fate of Ilion, Priam and all his people (ἒσσεται ἧμαρ, etc., II. vi. 448-9); or Aeneas as he gazes upon its downfall (ruit alto a culmine Troia: ... si Pergama dextra, etc., Aen. ii. 290-92). And yet the tragic melancholy of Argante is an entirely new creation, and altogether original to Tasso.

Ficino, Cardano, Tommaso Cornelio, Scotus and Occam, and any others who have been or shall be added to the list, have or may have anticipated this or that element of Vico's formula: and yet when we turn from their statements to the De antiquissima and the polemics that follow it, and read the definition of science, of true science, as the conversion of the true with the created, it strikes us as an entirely original theory. The fact is that Vico had not to face the same opponents and to solve the same problems that were faced and solved by the schoolmen, nominalists and mystics of the Middle Ages or by the Platonists and naturalists of the Renaissance, nor yet those of Descartes in his Discours sur la méthode; and the saying that "he alone knows things who creates them" acquires a new value, a new meaning (and this is its proper meaning) from its being used to refute the Cartesian cogito and the doctrine of immediate knowledge. Vico takes an old rusty sword and makes of it at least a glittering and trenchant weapon. For the same reason the phrase is no longer a mere accident or incident, but the starting-point of a special study, the foundation of a new philosophy, and Vico could quite well describe it as something not learnt from another but thought out and established by himself. And when he wants to find some original for it, he invents a history which is really a fiction or a myth; namely the history of ancient Italian wisdom which used this criterion as its supreme guide and left a trace of it in the Latin language in the synonymity of the words verum and factum.

The refutation of the Cartesian criterion (which De Sanctis thought "complete," the "last word of criticism") is the negative aspect of Vico's theory of knowledge. Its positive side, absent in the De antiquissima, is developed as we have said in the Scienza Nuova, where the human knowledge of the mind and of history is raised to the level of divine knowledge. And since some critics have not only chosen to ignore the obvious difference between these two phases of Vico's thought but have spoken of a too easy transition from the one to the other, it will be well to observe that this transition was for Vico if not entirely conscious at least very slow and very difficult. He must at one time have shared Descartes' and Malebranche's contempt for history; in the speech of 1701 he even echoed a saying of Descartes against philologists:--"You, Philologist, boast of knowing everything about the furniture and clothing of the Romans and of being more intimate with the quarters, tribes and streets of Rome than with those of your own city. Why this pride? You know no more than did the potter, the cook, the cobbler, the summoner, the auctioneer of Rome." But eleven years later, in the second reply to the Giornale dei letterati, Vico refers to the same phrase with the contrary conclusion, and deplores that "the study of languages is to-day considered profitless, thanks to the authority of Descartes, who says that to know Latin is to know no more than did Cicero's servant-girl." Vico had in the meantime become conscious of the importance of the "probable" knowledge of history and politics. He refers to his former anti-historical Cartesianism in a passage of the De constantia philologiae which has generally escaped notice. Speaking of philology he says: "I, who have all my life delighted in the use of reason more than in memory, seem to myself the more ignorant the more facts I know in philology. Whence René Descartes and Malebranche were not far wrong when they said that it was alien to the philosopher to work much and for long at philology." But he adds that later he perceived that "these two most notable philosophers ought, it they had been zealous for the common glory of Christendom, not for the private glory of philosophers, so to have pressed forward the study of philology as to see whether philology could be attached to the principles of philosophy (ut viderent philosophi an philologiam ad philosophiae principia revocare possent)." The elevation of philology to the rank of philosophy, of the knowledge of the world of man to the level of divine knowledge, is the positive aspect of Vico's theory of knowledge. It is this that is developed in the Scienza Nuova, towards which the De antiquissima, with the indication of the historical sciences as against Cartesianism, only prepared the way.

Thus of the three points in which I placed the originality and value of Vico's first theory of knowledge, two, namely the criterion of knowledge opposed to that of Descartes and the defence of concrete as opposed to abstract sciences, are not only left intact by the inquiries into their sources which I have just described, but are actually reinforced.

There remains the third of my points: the Vician theory of the arbitrary nature of mathematics, the originality of which has also been impugned by arguments which seem to me to have even less foundation than those I have examined above.

Do we find the doctrine that the fundamental objects of mathematics, the unit of arithmetic and the point of geometry, are unreal or fictitious, propounded before Vico's time? Do we find it--this is the chief point--propounded not as a casual remark or an intuition of a truth, but as a consciously reasoned concept from which legitimate consequences are drawn as to the limitations of mathematics and its inability to furnish real knowledge of mind, nature and history?

All through the Middle Ages the Aristotelian theory of mathematics is continually enunciated. According to this theory mathematics is the most certain of the sciences because the simplest; it abstracts from all sensible matter, but not from intelligible matter (ὖλη νοητή) which exists in sensible objects but not qua sensible (ἐν τοῑς ἀἰσθητοῑς ὑπάρχουσα μὴ ᾖ ἀἰσθητά) According to Cassiodorus it constituted the body of doctrinalis as opposed to naturalis (physical) science and divina. Albertus Magnus followed Aristotle in defining mathematical entities as separable "in imagination," "in thought" but not "in reality" (in phantasmate, secundum rationem, non secundum esse) from the sensible matter to which "they are conjoined by existence" (per esse sunt coniunctae); and St. Thomas said that mathematics "though the objects it considers are not separate, yet considers them in so far as they are separate" (etsi sunt non separata ea quae considerat, tamen considerat ea in quantum sunt separata). The arbitrary character of its foundations was never suspected. Dante, when he wished to indicate "the things which not being subject to our power we can only contemplate and not create," enumerated "the objects of mathematics, physical science and divinity" (mathematica, physica et divina).

Just as mathematics was not always equally valued in antiquity, so, and much more so, after the Renaissance of learning, it was variously exalted or despised. Giordano Bruno satirised the abuse of it, and said that without physical science "to be able to calculate and measure, to understand geometry and perspective, is but a pastime of ingenious fools," and warned his readers against confusing mathematical "signs" and real "causes": "a reflected or direct ray, an acute or obtuse angle, a perpendicular, incident or straight line, a greater or smaller are of a circle, such and such an aspect, are mathematical circumstances and not natural causes. To play with geometry is one thing, to prove by means of nature is another. It is not lines and angles that make the fire more or less hot, but near and far situations, short and long spaces of time." Campanella flatly denied Aristotle's assertion of the superiority of mathematics to physical science, declaring that its alleged purity was really weakness (debilitas), its simplicity was inability to include more things (plura accipere), its universality a contradiction against the nature of true science which is always of particulars (de singularibus), its demonstrative method by signs not by causes (per signa, non per causas); and finally that it is not a science investigated for its own sake and is valueless unless it is applied to physical matters (nisi applicentur physicis rebus). Bacon is of the same opinion, that mathematics taken by itself is useless, and is useful only as an "auxiliary science," a "great appendix" to the physical sciences. These definitions and restrictions, and others like them, might have yielded as a conclusion the entirely instrumental and practical character of mathematical science: but the conclusion was not drawn, so far as I know; and Bacon himself considered mathematics as in itself too exclusively and uselessly theoretical. "For since," he goes on in the passage above quoted, "it is a fact of human nature, no doubt to the great detriment of science, that it rejoices in the open plains of generalities, so to speak, rather than in the forests and closes of the particular, no discovery is more pleasant and gratifying than mathematics wherewith to sate this love of wandering and of meditation."

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; 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." 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."

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 (ἐνεργεία), 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 (καὶ διὰ τοῡτο ποιοῡντες γιγνώσκουσιν)." 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."

It has also been recently asserted that the Vician philosophy of mathematics reappears bodily in Galileo and his school; 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. 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 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 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."

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."

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." 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?

The other criticism concerns the idealistic interpretation which I have given to some of Vico's doctrines. It is contended that Vico was a Catholic, and that fact is supposed to prove that he could not have entertained the ideas which I find in his works. But that Vico professed himself an entirely orthodox Catholic, and that he clung to Catholicism with all the strength and zeal of his mind I have myself said again and again: I have even defended him against the accusations or praises dealt out to him by other critics for deceit or prudence in his attitude to the Church. But is it really so amazing, so unheard-of a thing, to find heterodox ideas in an orthodox writer? Are they not found in the Early Fathers and the Schoolmen, in mediaeval and modern theologians and mystics? To take an example of the many that occur, an example for a double reason above suspicion: Nicholas of Cusa was a Catholic and in fact a Cardinal of Holy Church, and in his lifetime the intimate friend of three popes. And yet the Catholic historian of Scholasticism, De Wulf, wrote of him "Le Cardinal catholique est-il donc panthéiste?... Il s'en défend vivement dans son Apologia doctae ignorantiae, mais on peut dire de lui comme d'Eckehart: 'il fait fléchir la logique au profit de son orthodoxie et retient de force les conséquences de ses prémisses'" (Hist. de la philos. médiévale, p. 389). If this happened to the Cardinal of Cusa or the Franciscan Master Eckehart, could it not happen to the Catholic Vico? M. de Wulf the Catholic historian is allowed to use this admirable method of criticism and to distinguish intention and action, will and logic. Why should it be denied to me? But enough.