De fundamento matheraaticae, et exercitio
The Dangers of False Mathematics
The author warns against the deceptive use of reason and nature in mathematics, which can lead the mind away from truth.
On the senses, the powers of the soul, the progress of reason, and the effectiveness of the liberal arts. If only the errors of mathematicians could be as easily removed from superior minds as the demons of these illusions are quieted in the presence of true faith and a sound conscience. Yet they err all the more dangerously because they seem to ground their error in the solidity of nature and the vigor of reason. It seems reckless to everyone to violate the law of nature, and foolish to disagree without cause from what reason has persuaded us of. They start from truths, therefore, so that by walking in truths for a longer time, they might cast themselves and their followers into the snare and pit of falsehood. They lay a foundation for their doctrine using a 'probable' mathematics—one stated in short, penultimate terms, which nature suggests, reason proves, and the experience of utility confirms—so that from there, through the slippery slope of their own opinions and a kind of phantom of reason, they might fall most disastrously into a 'reprobate' mathematics presented in extended, penultimate terms. For first they examine the things that nature created and investigate them in many ways, inquiring now how they consist of their parts, and now how they consist of matter and form. To do this more easily, they weigh the powers of the senses and measure the effectiveness of the intellect.
The Hierarchy of Perception
The soul ascends from sensory experience to imagination, and finally to the intellect, which acts as the citadel of understanding.
And because the dullness of the senses does not go beyond the nature of physical things, they rise little by little, with the help of others, to more subtle things. For sight, when it is in the body alone and that body is present, examines only colors, quantities, and shapes; sound alone reaches the hearing; taste judges flavors; the power of smell is entirely occupied with odors; and touch discerns what is hard, soft, smooth, rough, heavy, light, hot or cold, wet or dry. It is sometimes an explorer of shape, and conscious of quantity and weight; it also perceives pain and pleasure. It is diffused through almost all parts of the sensing body, and is so connected to the soul that when it departs, the whole life of the body seems to depart as well. But if you are inquiring about the aforementioned properties of absent bodies, the imagination can present them to you by drawing a likeness from those things the senses have already known, and it will be the more faithful the more expressive the likeness is. That is why Virgil’s Tityrus complains that his imagination was misled by an inaccurate likeness: 'Meliboeus, I thought the city they call Rome was like this one of ours, where we shepherds are used to driving our tender lambs.' 'But this one has raised its head as far above other cities as cypresses are accustomed to do above the bending viburnum shrubs.' But if it expresses the likeness of a thing more familiarly, the imagination is faithful and true, such as the one Andromache uses in Virgil: 'The only image of my Astyanax remained for me; he carried his eyes, his hands, and his face just like this, and now he would be growing up with you at the same age.' However, if one must turn to incorporeal things, reason and intellect are needed, since without understanding these cannot be comprehended, and there cannot be a true judgment about them without reason. The intellect, therefore, exerts its power when other faculties fail; established as if in the citadel of the soul, it embraces all lower things, since higher things cannot be comprehended by lower ones.
The Intellect as a Workshop
The intellect functions by composing and dividing concepts, serving as the essential instrument for all philosophy and the liberal arts.
It views things sometimes as they are, sometimes differently; sometimes it considers them simply, sometimes in combination; sometimes it joins what is separate, and sometimes it pulls apart and separates what is joined. The intellect proceeds simply when it contemplates any given thing—for instance, when it conceives of a man or a horse. But when it grasps several things in stages, it is subject to composition; for example, when it mentally grasps the concept of a white man or a running horse. It joins what is separate, as if it were to attach a horse's neck to a human head, bringing in feathers from all sides so that, as the poet says, 'a woman beautiful above ends in a foul, black fish.' Poets pass this on to their listeners in words when they describe a goat-stag, a centaur, or a chimera. It separates what is joined, as if it were to hold a form without matter, even though it cannot exist at all without that form, except for the form of being and the forms of forms that adhere to it—those from which the things that are in matter and make up a body have flowed. Furthermore, when it views things by composing them in a way other than they are, because it is empty and lacking in the truth of things, it veers toward the error of opinion; and if it asserts that something is or is not, it is opinion in the full sense of the word. But although it may join things in a way other than they are, as long as it does so simply, the conception won't be empty, for it produces the most efficient path for the entire investigation of wisdom. This is, in fact, the instrument of all philosophy, which both sharpens the mind with remarkable subtlety and distinguishes individual things from one another by the property of their nature. If you take away the abstracting intellect, the workshop of the liberal arts will perish, since without its work, none of them can be properly grasped or taught.
Universals and the Secrets of Nature
Through abstraction, the mind moves from individual substances to universal concepts, allowing for a deeper understanding of nature's hidden secrets.
Just as it approaches form without matter, it also approaches matter without form; and what it cannot grasp by its own inherent power, it sometimes comprehends through a certain deficiency of its own—much like seeing darkness by not seeing, or hearing silence by not hearing. Look, there isn't a human who isn't white, black, or some shade in between, and a human can't exist without immediately being a specific human, since for someone to exist and to be a single individual are the same thing. Nevertheless, the intellect reaches toward the human in such a way that its gaze doesn't descend to any particular human, but rather views things generally, since it can only exist as an individual. For just as there is a diversity in speaking and signifying, there is also in understanding, which transcends the modes of existence by its own multiplicity; and the human, who can't exist except as an individual, is enclosed within the universal conception of the mind. Reason therefore defines what the intellect conceives—a mortal, rational animal—and no wise person would rightly doubt that this applies only to those subject to it. As it gathers the similarities and differences of things, as it probes more deeply into the agreements of things that differ and the differences of things that agree, as it diligently investigates what each thing has in common with many or with few, and as it contemplates more sharply what must be present in individual things and what cannot be absent, it finds within itself many states of things—some universal, others individual. By defining these at its own discretion and dividing them in many ways, it transmits the mind's gaze to the hidden secrets of nature, so that nothing in the natural world remains fully hidden from its eyes. First, it looks more sharply at the substance that underlies all things, in which the hand of the Creator is proven, as it clothes that substance in various properties and forms—like its own garments—and informs it with the perceptible qualities of the senses, so that it might be more easily grasped by the human mind. Therefore, what the senses perceive and what is subject to forms is singular and primary substance. However, that without which it can neither exist nor be understood is substantial to it, and is generally called secondary substance. But that which is present to a substance and yet can be absent while the substance remains is grouped among accidents. A thing is singular if there's only one of it; it's universal if, though not by nature, it's common to many by conformity. It's perhaps easier to find in the intellect than in the nature of things—where it's clear that genera, species, differences, properties, and accidents (which are called universals) are to be found—since it's a small reward and an infinite labor to seek the subsistence of universals in the actual world, whereas in the mind they're found usefully and quite easily. For if someone considers in his mind the substantial similarity of things that differ only in number, he grasps the species; but if the agreement of things that differ even in species occurs to the mind, the breadth of the genus is unfolded in the mind. Finally, while the intellect perceives the conformity of things that nature has assimilated either substantially or accidentally, it's moved in the comprehension of universals; but when it examines the differences of similar things, the more precisely it does so, the more familiarly it approaches the works of nature, which are singular things, and if it inspects a substance clothed in its properties, it doesn't depart from the condition of nature. But if, with the species removed, it strips it as if of a garment of forms, it exercises its own sharpness and contemplates the nature of things—what, namely, they are in themselves and what they are in others—more freely and faithfully, while it discusses and discerns one by one the substance of things, quantity, relation, quality, position, place, time, having, doing, and suffering.
The Four Pillars of Wisdom
The liberal arts—arithmetic, music, geometry, and astronomy—form the four boundaries of worldly wisdom, guiding the soul toward truth.
Although these things cannot exist individually, they can be investigated individually; and this speculation, in which the nature of magnitude and multitude—the two things that encompass and surround the entire world—is discussed, is most useful as a summary of all philosophy. Is the intellect idle or useless when it abstracts while doing this, when it is through this that the soul climbs by the steps of the liberal arts to the throne of consummate philosophy? Therefore, it divides multitude—which grows to infinity by its own power, just as magnitude decreases to infinity—into two parts, contemplating it now simply and in itself, and now as related to something else, assigning one part to arithmetic and reserving the other by full right to music. It also cuts magnitude into two species, subjecting the immobile one to geometers, and the mobile one, of course, to those who profess the science of the stars and celestial things. Mathesis—that is, the doctrinal—consists entirely of these four species, and it attains the perfection of worldly wisdom as if by the four boundaries of philosophy. The first step, therefore, is to borrow the power of numbers from arithmetic; the second is to draw the grace of proportions from music; the third is to obtain the knowledge of measurements from geometry; the fourth and final step attains the true position of the stars and investigates the power of celestial things. Of those who profess the science of the stars, some fall into the error of opinion and drift toward fables—as Hyginus does—while others are content with the power of imagination alone, reserving what is true for the judgment of the more learned, satisfied as long as they hold to a likeness of the truth. Astrology, in one way or another, welcomes the professors of both into its own field of learning. There are others who, in their study of the stars, do indeed watch for the truth, content only if they can accurately grasp the true position and movement of the stars and the logic of the signs.
Read the original Latin
sensuum, et uiribus animae, et profectu rationis, et efficacia liheralium disciplinarum. Possit utinam tam facile mathematicorum error a praestantioribus animis amoueri quam leuiter in conspectu uerae fidei et sanae conscientiae istarum illusionum demonia conquiescunt. Verumtamen eo periculosius errant, quo in soliditate naturae et uigore rationis suum fundare uidentur errorem. Temerarium siquidem uidetur uniuersis in legem naturae praeuaricari, et stultum gratis dissentire ab eo quod ratio persuasit. A ueris ergo sumunt initium, ut per uera diutius gradientes se et sequipedas suos praecipitent in laqueum et foueam falsitatis. Mathesim ergo probabilem quae penultima breui enuntiatur, quam et natura indueit, ratio probat, et utilitatis experientia approbat, quasi quoddam doctrinae suae iaciunt fundamentum, ut exinde opinionum suarum lubrico quasi quadam imagine rationis in mathesim reprobam, quae profertur extensa penultima, pemiciosissime prolabantur. Primo namque res quas natura creauit discutiunt, easque multipliciter inuestigant, nunc quomodo ex partibus suis, nunc quomodo ex materia et forma constant inquirentes. Quod ut facilius possint, sensuum uires pensant et intellectus efficaciam metiuntur.
Et quia sensuum hebetudo rerum corporearum naturam non transgreditur, paulatim aliorum beneficio ad a subtiliora consurgunt. Visus etenim in solo corpore eoque praesente colores tantum quantitates examinat et figuras: sonus solus contingit auditum: gustus de saporibus iudicat: olfaciendi uis in odoribus tota uersatur: quid durum, quid moUe sit, quid lene, quid asperum, quid ponderosum, quid leue, quid calidum aut frigidum, quid humidum sit aut siccum, tactus discemit; est interdum figurae explorator, et quantitatis et ponderis conscius; dolorem quoque percipit et uoluptatem; isque per omnes fere sensati corporis partes difiunditur, adeoque animae coherens est ut eo discedente uideatur et tota corporis uita discedere. Si uero corporum absentium praefatas proprietates inquiris, eas tibi tracta similitudine ab his quas sensus agnouit, poterit imaginatio praesentare, quae tanto erit fidelior, quanto expressior similitudo. Vnde et Titirus apud Maronem non assecutae simihtudinis uitio lapsam sibi queritur imaginationem: Vrbem quam dicunt Romam, Melibee, putaui stultus ego huic nostrae similem, quo saepe solemus pastores ouium teneros depellere fetus. Verum haec tantum alias inter caput extulit urbes, quantum lenta solent inter uiburna cupressi. Sin autem familiarius expresserit rei similitudinem, fidelis et uera est imaginatio, quali utitur Andromache apud Virgilium: michi sola mei super Astianactis imago; sic oculos, sic ille manus, sic ora ferebat, et nunc equali tecum pubesceret euo. Verum si ad incorporea diuertendum est, ratione opus est et intellectu, cum absque intelligentia haec non ualeant comprehendi, et uerum non possit esse de his sine ratione iudicium. Intellectus itaque aliis deficientibus exerit uires suas, et quasi in arce animae constitutus omnia inferiora complectitur, cum ab inferioribus superiora nequeant comprehendi.
Et nunc quidem res ut sunt,nunc aliter intuetur, nunc simpliciter, nunc composite, nunc disiuncta coniungit, nunc coniuncta distrahit et disiungit. Simpliciter ergo incedit intellectus, dum rem quamlibet contemplatur, puta hominem equumue concipiens. Cum uero res plures gradatim complectitur, compositioni obnoxius est; puta hoc intellectum, hominem album equumue currere, mente complectens. Disiuncta coniungit, ut si humano capiti ceruicem iungat equinam, uarias inducens undique plumas ut, iuxta poetam, turpiter atrum desinat in piscem mulier formosa supeme Hunc uero ad auditores suos uerbo traiciunt poetae, cum hircoceruum, centaurum describunt et chimeram. Coniuncta uero disiungit, ut si formam teneat absque materia, cum tamen sine ea forma omnino esse non possit, nisi forma essendi et ei adherentes formae formarum, ex quibus illae fluxerunt, quae in materia sunt et corpus efficiunt. Porro a cum res aliter quam sint componendo inspicit, eo quod cassus est et a rerum ueritate deficiens, ad opinionis errorem uergit, et si esse asserit uel non esse, pleno nomine opinio est. Sed licet aliter quam sint, dum tamen simpliciter, coniuncta disiungat, non inanis erit conceptio, quae totius inuestigationis sapientiae expeditissimam parit uiam, Hic est enim totius philosophiae instrumentum, quod et mentem mira subtilitate exacuit et res singulas a se inuicem naturae suae proprietate distinguit. Si abstrahentem tuleris intellectum, liberalium artium officina peribit, cum citra ipsius operam nulla earum rite haberi ualeat aut doceri.
Hic itaque sicut formam sine materia, sic et materiam aggreditur sine forma; et quod propriae uirtutis potentia tenere non sufficit, suo quodam defectu interdum comprehendit, ut si uideantur tenebrae non uidendo et non audiendo silentium audiatur. Ecce non est homo qui non sit albus aut niger aut medii coloris particeps, nec potest esse homo, qui non statim sit aliquis homo, cum idem sit alicui esse et unum numero esse. Verumtamen sic hominem intellectus attingit, ut ad neminem hominem aspectus illius descendat, generaliter intuens, quod non nisi singulariter esse potest. Sicut enim dicendi et significandi, sic et intelligendi diuersitas, subsistendi modos sui numerositate transcendit, et homo, qui non potest nisi singulariter esse, uniuersali clauditur mentis conceptione. Diffinit ergo ratio quod concipit intellectus, animal rationale mortale, quod in solos subditos cadere nemini recte sapientium ambiguum est. Dum itaque rerum similitudines et dissimilitudines colligit, dum difierentium conuenientias et conuenientium difierentias altius perscrutatur, dum quid singula cum pluribus, quid cum paucioribus commune habeant, diligentius inuestigat, quaeue rebus singulis adesse necesse sit, quae abesse non possint, perspicacius contemplatur, multos apud se rerum inuenit status, alios quidem uniuersales, alios singulares. Quos pro arbitrio suo diffiniens et multiphariam diuidens ad ipsius archana naturae, mentis transmittit aspectum ut nichil naturalium plene absconditum sit ab oculis eius. Et primo substantiam, quae omnibus subest, acutius intuetur, in qua manus naturae probatur artificis, dum eam uariis proprietatibus et formis quasi suis quibusdam uestibus induit et suis sensuum perceptibilibus informat, quo possit aptius humano ingenio comprehendi.
Quod igitur sensus percipit, formisque subiectum est, singularis et prima substantia est. Id uero sine quo illa nec esse nec intelligi potest, ei substantiale est, et plerumque secunda substantia nominatur. Hoc autem quod adest quidem substantiae et ea manente abesse potest, accidentibus aggregatur. Singulare quidem si unus tantum sit; uniuersale, si, licet non natura, conformitate tamen sit commune multorum. Quod forte facilius in intellectu quam in natura rerum poterit inueniri, in quo genera et species, differentias propria et accidentia, quae uniuersaliter dicuntur, planum est inuenire,cum in actu rerum subsistentiam a uniuersalium quaerere exiguus fructus sit et labor infinitus, in mente uero utiliter et facillime reperiuntur. Si enim rerum solo numero differentium substantialem similitudinem quis mente pertractet, speciem tenet; si uero etiam specie differentium conuenientia menti occurrat, generis latitudo mente diffiinditur. Denique dum rerum, quas natura substantialiter uel accidentaliter assimilauit, conformitatem percipit intellectus, in uniuersalium comprehensione mouetur; cum uero simUium differentias agitat, quanto expressius, tanto ad opera naturae, quae singularia sunt, familiarius accedit, si substantiam suis uestitam proprietatibus inspicit, a naturae conditione non diuertit. Sin autem detracta specie, eam quasi formarum exuit uestimento, acumen quidem suum exercet, et rerum naturam, quid scilicet in se, quid in aliis sint, liberius et fidelius contemplatur, dum rerum substantiam, quantitatem, ad aliquid, qualitatem, situm esse, ubi, quando, habere, facere, pati, singillatim discutit et discemit.
Quae etsi singillatim esse non possint, singillatim tamen inuestigari queunt; et ad totius philosophiae compendium, utilissima est haec speculatio, in qua magnitudinis et multitudinis, quae duo totum orbem complectuntur et ambiunt, natura discutitur. Numquid abstrahens intellectus, dum haec agit, otiosus est aut inutilis, per quem animus honestarum artium gradibus ad thronum consummatae philosophiae conscendit? Multitudinem ergo quae potentia sui in infinitum crescit, sicut e contra magnitudo decrescit in infinitum, bimembri diuisione partitur, dum eam nunc simpliciter et per se, nunc ad aliud relatam contuetur, alteram demandans arithmeticae, alteram musicae pleno iure reseruans. Magnitudinem quoque secat in duas species, alteram immobilem subiciens geometris, alteram scilicet mobilem his qui astrorum et celestium scientiam profitentur. In his uero quattuor speciebus mathesis, id est, doctrinalis tota consistit, et quasi quattuor philosophiae limitibus mundanae sapientiae perfectionem assequitur. Primus itaque gradus est ab arithmetica numerorum uirtutem mutuare: secundus proportionum gratiam a musica trahere: tertius obtinere scientiam a geometria mensurarum: quartus idemque nouissimus ueram positionem siderum assequitur et uim celestium perscrutatur. Eorum uero qui scientiam profitentur astrorum, alii opinionis errore prolabuntur ad fabulas, in quo deprehenditur et Iginus, alii sola imaginationis uirtute contenti sunt, quid uerum sit doctiorum iudicio reseruantes, satis agentes sibi, dummodo ueri similitudinem teneant. Vtrosque alterutra astrologia eruditionis suae recipit professores.
Sunt alii, qui in professione astrorum ueritati quidem inuigilant, eo solo contenti, si positionem ueram motumque siderum rationemque signorum ueraciter assequuntur.
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