Et ita sunt duae abstractiones intellectus: una quae respondet unioni formae et materiae vel accidentis et subiecti, et haec est abstractio formae a materia sensibili; alia quae respondet unioni totius et partis, et huic respondet abstractio universalis a particulari, quae est abstractio totius in quo consideratur absolute natura aliqua secundum suam rationem essentialem, ab omnibus partibus quae non sunt partes speciei sed sunt partes accidentales. Non autem inveniuntur abstractiones eis oppositae, quibus pars abstrahatur a toto vel materia a forma; quia pars vel non potest abstrahi a toto per intellectum si sit de partibus materiae in quarum diffinitione ponitur totum, vel potest etiam sine toto esse si sit de partibus speciei, sicut linea sine triangulo, vel littera sine syllaba, vel elementum sine mixto. So there are two abstractions of the intellect. One corresponds to the union of form and matter or accident and subject. This is the abstraction of form from sensible matter. The other corresponds to the union of whole and part; and to this corresponds the abstraction of the universal from the particular. This is the abstraction of a whole, in which we consider a nature absolutely, according to its essential character, in independence of all parts that do not belong to the species but are accidental parts. But we do not find abstractions opposed to these, by which a part is abstracted from a whole by the intellect if it is one of the parts of matter in whose definition the whole is included, or it can even exist without the whole if it is one of the parts of the species, for instance, a line without a triangle, a letter without a syllable, or an element without a mixed body. In his autem quae secundum esse possunt esse divisa magis habet locum separatio quam abstractio. Similiter autem cum dicimus formam abstrahi a materia non intelligitur de forma substantiali, quia forma substantialis et materia sibi correspondens dependent ad invicem, ut unum sine alio non possit intelligi, eo quod proprius actus in propria materia fit; sed intelligitur de forma accidentali, quae est quantitas et figura, a qua quidem materia sensibilis per intellectum abstrahi non potest, cum qualitates sensibiles non possint intelligi non praeintellecta quantitate, sicut patet in superficie et colore; nec etiam potest intelligi esse subiectum motus quod non intelligitur quantum. Substantia autem, quae est materia intelligibilis quantitatis, potest esse sine quantitate; unde considerare substantiam sine quantitate magis pertinet ad genus separationis quam abstractionis. But in the case of things that can exist separately, separation rather than abstraction obtains. Similarly, when we say from is abstracted from matter, we do not mean substantial form, because substantial form and the matter correlative to it are interdependent, so that one is not intelligible without the other, because the appropriate act is in its appropriate matter. Rather, we mean the accidental forms of quantity and figure, from which indeed sensible matter cannot be abstracted by the intellect, because sensible qualities cannot be understood unless quantity is presupposed, as is clear in the case of surface and color. And neither can we understand something to be the subject of motion unless we understand it to possess quantity. Substance, however, which is the intelligible matter of quantity, can exist without quantity. Consequently, the consideration of substance without quantity belongs to the order of separation rather than to that of abstraction. Sic ergo in operatione intellectus triplex distinctio invenitur: We conclude that there are three kinds of distinction in the operation of the intellect. una secundum operationem intellectus componentis et dividentis, quae separatio dicitur proprie, et haec competit scientiae divinae sive metaphysicae; There is one through the operation of the intellect joining and dividing which is properly called separation and this belongs to divine science or metaphysics. alia secundum operationem qua formantur quiditates rerum, quae est abstractio formae a materia sensibili, et haec competit mathematicae; There is another through the operation by which the quiddities of things are conceived which is the abstraction of form from sensible matter, and this belongs to mathematics. tertia, secundum eandem operationem, universalis a particulari, et haec competit etiam physicae et est communis omnibus scientiis, quia in omni scientia praetermittitur quod per accidens est et accipitur quod per se est. And there is a third through the same operation which is the abstraction of a universal from a particular, and this belongs to physics and to all the sciences in general, because science disregards accidental features and treats of necessary matters. Et quia quidam non intellexerunt differentiam duarum ultimarum a prima, inciderunt in errorem ut ponerent mathematica et universalia a sensibilibus separata, ut Pythagorici et Platonici. And because certain men (for example, the Pythagoreans and the Platonists) did not understand the difference between the last two kinds of distinction and the first, they fell into error, asserting that the objects of mathematics and universals exist separate from sensible things. Ad primum ergo dicendum, quod mathematicus abstrahens non considerat rem aliter quam sit: non enim intelligit lineam esse sine materia sensibili, sed considerat lineam et eius passiones sine consideratione materiae sensibilis; et sic non est dissonantia inter intellectum et rem, quia etiam secundum rem id quod est de natura lineae non dependet ab eo quod facit materiam esse sensibilem, sed magis e converso. Et sic patet quod abstrahentium non est mendacium, ut dicitur in II Physicorum. Reply Obj. 1: When the mathematician abstracts he does not consider something otherwise than it is. Thus, he does not think that a line exists without sensible matter, but he treats of a line and its properties without considering sensible matter. So there is no disagreement between his intellect and reality, because even in reality what belongs to the nature of a line does not depend upon that which makes matter sensible, but vice versa. Consequently, it is evident that there is no error in the one who abstracts, as is said in the Physics. Ad secundum dicendum, quod materiale dicitur non solum id cuius pars est materia, sed etiam illud quod in materia esse habet. Secundum quem modum linea sensibilis materiale quoddam dici potest, unde per hoc non prohibetur quin linea sine materia intelligi possit: non enim materia sensibilis comparatur ad lineam sicut pars, sed magis sicut subiectum in quo esse habet; et similiter est de superficie et corpore: non enim mathematicus considerat corpus quod est in genere substantiae prout eius pars est materia et forma, sed secundum quod est in genere quantitatis tribus dimensionibus perfectum, et sic comparatur ad corpus quod est in genere substantiae, cuius pars est materia physica, sicut accidens ad subiectum. Reply Obj. 2: By “material” is meant not only that which has matter as a part, but also that which exists in matter; and in this way sensible line can be called something material. So this does not prevent a line from being understood without matter. For sensible matter is not related to a line as a part, but rather as the subject in which it exists, and this is also the case with a surface or body. Obviously, the mathematician does not treat of the kind of body that is in the category of substance, whose parts are matter and form, but rather the body in the category of quantity, constituted by three dimensions. Body, in this sense of the term, is related to body the category of substance (of which physical matter is a part) as accident to its subject. Ad tertium dicendum, quod materia non est principium diversitatis secundum numerum nisi secundum quod in multas partes divisa in singulis partibus formam recipiens, eiusdem rationis plura individua eiusdem speciei constituit. Materia autem dividi non potest nisi praesupposita quantitate, qua remota omnis substantia indivisibilis remanet. Et sic prima ratio diversificandi ea quae sunt unius speciei est penes quantitatem. Quod quidem quantitati competit in quantum in sui ratione situm quasi differentiam constitutivam habet, qui nihil est aliud quam ordo partium; unde etiam abstracta quantitate a materia sensibili per intellectum, adhuc contingit imaginari diversa secundum numerum unius speciei, sicut plures triangulos aequilateros, et plures lineas rectas aequales. Reply Obj. 3: Matter is the principle of numerical diversity or inasmuch as, being divided into many parts, and receiving in each part a form of the same nature, it constitutes many individuals of the same species. Now matter can be divided only if we presuppose quantity in it; if that is taken away, even substance remains indivisible. So the primary reason for the diversification of things of one species lies in quantity. And this is due to quantity because position, which is the arrangement of parts in place, is contained in its notion as a kind of formal difference. So even when the intellect has abstracted quantity from sensible matter, it is still possible to imagine numerically different things in the same species, for example, several equilateral triangles and several equal straight lines. Ad quartum dicendum, quod mathematica non abstrahuntur a qualibet materia, sed solum a materia sensibili. Partes autem quantitatis, a quibus demonstratio sumpta quodammodo a causa materiali videtur sumi, non sunt materia sensibilis sed pertinent ad materiam intelligibilem, quae etiam in mathematicis invenitur, ut patet in VII Metaphysicae. Reply Obj. 4: Mathematics does not abstract from every kind of matter but only from sensible matter. Now the parts of quantity that seem to be in a way the basis for a demonstration by means of a material cause are not sensible matter; rather, they pertain to intelligible matter, which indeed is found in mathematics, as is clear in the Metaphysics. Ad quintum dicendum, quod motus secundum naturam suam non pertinet ad genus quantitatis sed participat aliquid de natura quantitatis aliunde secundum quod divisio motus sumitur vel ex divisione spatii vel ex divisione mobilis; et ideo considerare motus non pertinet ad mathematicum, sed tamen principia mathematica ad motum applicari possunt. Et ideo secundum hoc quod principia quantitatis ad motum applicantur, naturalis considerat de divisione et continuitate motus, ut patet in VI Physicorum; et in scientiis mediis inter mathematicam et naturalem tractatur de mensuris motuum, sicut in scientia de sphaera mota et in astrologia. Reply Obj. 5: By its very nature motion is not in the category of quantity, but it partakes somewhat of the nature of quantity from another source, namely, according as the division of motion derives from either the division of space or the division of the thing subject to motion. So it does not belong to the mathematician to treat of motion, although mathematical principles can be applied to motion. Therefore, inasmuch as the principles of quantity are applied to motion, the natural scientist treats of the division and continuity of motion, as is clear in the Physics. And the measurements of motions are studied in the intermediate sciences between mathematics and natural science: for instance, in the science of the moved sphere and in astronomy. Ad sextum dicendum, quod in compositis simplicia salvantur et proprietates eorum, licet per alium modum, sicut propriae qualitates elementorum et motus ipsorum proprii inveniuntur in mixto; quod autem est compositorum proprium non invenitur in simplicibus. Et inde est quod quanto aliqua scientia est abstractior et simpliciora considerans, tanto eius principia sunt magis applicabilia aliis scientiis: unde principia mathematicae (sunt) applicabilia naturalibus rebus, non autem e converso, propter quod physica est ex suppositione mathematicae, sed non e converso, ut patet in III Caeli et mundi. Reply Obj. 6: Simple bodies and their properties remain in composite bodies although in a different way, as the proper qualities of the elements and their proper movements are found in a mixed body. What is proper to composite bodies, however, is not found in simple bodies. And so it is that the more abstract and simple the objects of a science are, the more applicable its principles are to the other sciences. Thus the principles of mathematics are applicable to natural things, but not visa versa, because physics presupposes mathematics; but the converse is not true, as is clear in the De caelo et mundo. Et inde est quod de rebus naturalibus et mathematicis tres ordines scientiarum inveniuntur: So there are three levels of sciences concerning natural and mathematical entities. quaedam enim sunt pure naturales, quae considerant proprietates rerum naturalium in quantum huiusmodi, sicut physica et agricultura et huiusmodi; Some are purely natural and treat of the properties of natural things as such, like physics, agriculture, and the like. quaedam vero sunt pure mathematicae, quae determinant de quantitatibus absolute, sicut geometria de magnitudine et arithmetica de numero; Others are purely mathematical and treat of quantities absolutely, as geometry considers magnitude and arithmetic numbers. quaedam vero sunt mediae, quae principia mathematica ad res naturales applicant, ut musica, astrologia, (et) huiusmodi. Quae tamen magis sunt affines mathematicis, quia in earum consideratione id quod est physicum est quasi materiale, quod autem est mathematicum est quasi formale; sicut musica considerat sonos non in quantum sunt soni, sed in quantum sunt secundum numeros proportionabiles, et similiter est in aliis; et propter hoc demonstrant conclusiones suas circa res naturales, sed per media mathematica. Et ideo nihil prohibet si in quantum cum naturali communicant, materiam sensibilem respiciunt: in quantum enim cum mathematica communicant, abstractae sunt. Still others are intermediate, and these apply mathematical principles to natural things; for instance, music, astronomy, and the like. These sciences, however, have a closer affinity to mathematics, because in their thinking that which is physical is, as it were, material, whereas that which is mathematical is, as it were, formal. For example, music considers sounds, not inasmuch as they are sounds, but inasmuch as they are proportionable according to numbers; and the same holds in other sciences. Thus they demonstrate their conclusions concerning natural things, but by means of mathematics. Therefore nothing prevents their being concerned with sensible matter insofar as they have something in common with natural science, but insofar as they have something in common with mathematics they are abstract. Ad septimum dicendum, quod quia scientiae mediae de quibus dictum est, communicant cum naturali secundum id quod in earum consideratione est materiale, differunt autem secundum id quod in earum consideratione est formale, ideo nihil prohibet has scientias cum naturali habere interdum easdem conclusiones, non tamen per eadem demonstrant, nisi secundum quod scientiae sunt immixtae et una interdum utitur eo quod est alterius; sicut rotunditatem terrae naturalis probat ex motu gravium, astrologus autem per considerationem lunarium eclipsium. Reply Obj. 7: Because the intermediate sciences mentioned above have something in common with natural science as regards what is material in their procedure, but differ from it as regards what is formal in it, nothing prevents these sciences from occasionally having the same conclusions as natural science. Nevertheless, they do not use the same means of demonstration, unless the sciences are mixed and one occasionally uses what belongs to another, as the natural scientist proves that the earth is round from the movement of heavy bodies, while the astronomer proves it by considering eclipses of the moon. Ad octavum dicendum, quod sicut dicit Commentator ibidem, Philosophus non intendit ibi distinguere scientias speculativas, quia de quolibet mobili, sive sit corruptibile sive incorruptibile, determinat naturalis, mathematicus autem in quantum huiusmodi non considerat aliquod mobile. Intendit autem distinguere res de quibus scientiae speculativae determinant, de quibus seorsum et secundum ordinem agendum est, quamvis illa tria genera rerum tribus scientiis appropriari possint. Entia enim incorruptibilia et immobilia praecise ad metaphysicum pertinent, entia vero mobilia et incorruptibilia propter sui uniformitatem et regularitatem possunt determinari quantum ad suos motus per principia mathematica; quod de mobilibus corruptibilibus dici non potest. Et ideo secundum genus entium attribuitur mathematicae ratione astrologiae, tertium vero remanet proprium soli naturali. Et sic loquitur Ptolemaeus. Reply Obj. 8: As the Commentator says, the Philosopher there did not intend to distinguish between the speculative sciences, because the natural scientist treats of everything subject to motion, whether it be corruptible or incorruptible, while the mathematician as such does not treat of anything subject to motion. But he intended to distinguish between the things studied by the speculative sciences, which must be treated separately and in order, although these three sorts of things can be apportioned to the three sciences. For incorruptible and immobile beings pertain precisely to the metaphysician. However, mobile and incorruptible beings, owing to their uniformity and regularity, can be determined in their movements by mathematical principles; this cannot be said of beings that are mobile and corruptible. Therefore, as Ptolemy says, the second kind of beings is ascribed to mathematics through astronomy, while the third kind remains the proper domain of natural science alone. Articulus IV Article 4 Does divine science treat of what exists without matter and motion? Ad quartum sic proceditur: videtur quod scientia divina non sit de rebus a motu et materia separatis. Scientia enim divina maxime videtur esse de Deo. Sed ad Dei cognitionem pervenire non possumus nisi per effectus visibiles, qui sunt in materia et motu constituti: Rom. I: invisibilia enim ipsius etc. Ergo scientia divina non abstrahit a materia et motu. Obj. 1: We proceed as follows to the fourth article: it seems that divine science does not treat of things separate from motion and matter, for divine science seems to he especially concerned with God. Now we can come to know God only by way of his visible effects, which are created in matter and motion, as it is said in the Epistle to the Romans, the invisible things of him, from the creation of the world, are clearly seen, being understood by the things that are made (Rom 1:20). Therefore, divine science does not abstract from matter and motion. Praeterea. Illud cui aliquo modo motus convenit non est omnino a motu et materia separatum. Sed motus aliquo modo Deo convenit; unde dicitur Sap. VII de spiritu sapientiae quod est mobilis et mobilior omnibus mobilibus, et Augustinus dicit, VIII Super Genesim, quod Deus movet (se) sine tempore et loco, et Plato posuit primum movens movere se ipsum. Ergo scientia divina, quae de Deo determinat, non est omnino a motu separata. Obj. 2: Again, that to which motion in some way belongs is not entirely separate from motion and matter. But motion in some way belongs to God. Thus it is said in Wisdom 7:24 that the spirit of wisdom is mobile and more mobile than all mobile things. And Augustine says that God moves himself without time and place. Plato also asserted that the first mover moves itself. Therefore divine science, which treats of God, is not entirely separate from motion. Praeterea. Scientia divina non solum habet considerare de Deo, sed etiam de angelis. Sed angeli moventur et secundum electionem, quia de bonis facti sunt mali, et secundum locum, ut patet in illis qui mittuntur. Ergo illa de quibus scientia divina considerat non sunt omnino a motu separata. Obj. 3: Again, divine science must treat not only of God but also of angels. But angels change both with regard to choice, because they became bad after having been good, and also with regard to place, as is evident in the case of those who are sent as messengers. So the objects of divine science are not entirely separated from motion. Praeterea. Ut videtur Commentator dicere in principio Physicorum, omne quod est, vel est materia pura, vel forma pura, vel compositum ex materia et forma. Sed angelus non est forma pura, quia sic esset actus purus, quod solius Dei est; nec iterum est materia pura; ergo est compositus ex materia et forma. Et sic scientia divina non abstrahit a materia. Obj. 4: Again, as the Commentator seems to say in the beginning of the Physics, every being is either pure matter, or pure form, or a composite of matter and form. But an angel is not a pure form, because then he would be pure act, which is true of God alone. Neither is he pure matter. So he is a composite of matter and form. Therefore divine science does not abstract from matter. Praeterea. Scientia divina, quae ponitur tertia pars speculativae philosophiae, est idem quod metaphysica, cuius subiectum est ens, et principaliter ens quod est substantia, ut patet in IV Metaphysicae. Sed ens et substantia non abstrahit a materia, alias nullum ens inveniretur quod haberet materiam. Ergo scientia divina non est a materia abstrahens. Obj. 5: Again, divine science, the third part of speculative philosophy, is the same as metaphysics, whose subject is being, and especially substantial being. This is clear in the Metaphysics. But being and substance do not abstract from matter; otherwise there would be no material being. So divine science does not abstract from matter. Praeterea. Secundum Philosophum in I Posteriorum ad scientiam pertinet considerare non solum subiectum, sed partes et passiones subiecti; sed ens est subiectum scientiae divinae, ut dictum est; ergo ad ipsam pertinet considerare de omnibus entibus. Sed materia et motus sunt quaedam entia. Ergo pertinent ad considerationem metaphysicae, et sic scientia divina ab eis non abstrahit. Obj. 6: Again, according to the Philosopher, it is the business of a science to consider not only a subject but also the divisions and attributes of that subject. Now, as we have said, being is the subject of divine science. Therefore it is the business of this science to treat of all beings. But matter and motion are beings. Therefore they come under the consideration of metaphysics, and so divine science does not abstract from them. Praeterea. Sicut dicit Commentator in I Physicorum, scientia divina demonstrat per tres causas, scilicet efficientem, formalem et finalem. Sed causa efficiens non potest considerari sine consideratione motus, similiter nec finis, ut dicitur in III Metaphysicae; unde in mathematicis propter hoc quod sunt immobilia, nulla demonstratio per huiusmodi causas datur. Ergo scientia divina non abstrahit a motu. Obj. 7: Again, divine science demonstrates by means of three causes: efficient, formal, and final, as the Commentator says. But we cannot consider an efficient cause without taking motion into account; and the same thing is true of a final cause, as the Metaphysics says. Thus, because the objects of mathematics are immobile, there are no demonstrations through these causes in that science. Consequently, divine science does not abstract from motion. Praeterea. In theologia determinatur de creatione caeli et terrae et actibus hominum et multa huiusmodi quae in se materiam et motum continent. Ergo non videtur theologia a materia et motu abstrahere. Obj. 8: Again, in theology we treat of the creation of the heavens and the earth, of acts of men, and many similar things that involve matter and motion. So theology does not seem to abstract from matter and motion. Sed contra est quod Philosophus dicit in VI Metaphysicae quod prima philosophia est circa separabilia, scilicet a materia, et immobilia. Prima autem philosophia est scientia divina, ut ibidem dicitur. Ergo scientia divina est abstracta a materia et motu. On the contrary, the Philosopher says in the Metaphysics that first philosophy deals with things that can exist separately, that is, from matter, and with immobile things. Now first philosophy is divine science, as he says in the same place. Therefore divine science abstracts from matter and motion. Praeterea. Nobilissima scientia est de nobilissimis entibus. Sed scientia divina est nobilissima. Cum ergo entia immaterialia et immobilia sint nobilissima, de eis erit scientia divina. Moreover, the most excellent science deals with the most excellent beings. But the most excellent science is divine science. Therefore, since immaterial and immobile beings are the most excellent, divine science will treat of them. Praeterea. Philosophus dicit in principio Metaphysicae quod scientia divina est de primis principiis et causis. Huiusmodi autem sunt immaterialia et immobilia. Ergo de talibus est scientia divina. Moreover, the Philosopher says in the beginning of the Metaphysics that divine science concerns first principles and causes. Now these are immaterial and immobile. Therefore things of this sort are the objects of divine science. Responsio. Dicendum, quod ad evidentiam huius quaestionis scire oportet quae scientia divina scientia dici debeat. Sciendum siquidem est quod quaecumque scientia considerat aliquod genus subiectum, oportet quod consideret principia illius generis, cum scientia non perficiatur nisi per cognitionem principiorum, ut patet per Philosophum in principio Physicorum. Sed principiorum duo sunt genera. I answer that in order to throw light on this question we must understand what science should be called divine science. We must realize indeed that if a science considers a subject-genus, it must investigate the principles of that genus, since science is perfected only through knowledge of principles, as the Philosopher explains in the beginning of the Physics. Now there are two kinds of principles. Quaedam enim sunt quae et sunt in se ipsis quaedam naturae completae, et sunt nihilominus principia aliorum, sicut corpora caelestia sunt quaedam principia inferiorum corporum, et corpora simplicia corporum mixtorum; et ideo ista non solum considerantur in scientiis ut principia sunt, sed etiam ut sunt in se ipsis res quaedam. Et propter hoc de eis non solum tractatur in scientia quae considerat ipsa principiata, sed etiam habent per se scientiam separatam, sicut de corporibus caelestibus est quaedam pars scientiae naturalis praeter illam in qua determinatur de corporibus inferioribus, et de elementis praeter illam in qua tractatur de corporibus mixtis. Some are complete natures in themselves and nevertheless they are the principles of other things, as the heavenly bodies are principles of lower bodies and simple bodies are principles of mixed bodies. In the sciences, therefore, we study them not only insofar as they are principles, but also insofar as they are certain things in themselves. And for this reason they are considered not only in the science of the beings of which they are the principles, but also in a separate science. Thus there is a branch of natural science treating of heavenly bodies distinct from that treating of lower bodies, and there is one treating of the elements distinct from that treating of mixed bodies. Quaedam autem sunt principia quae non sunt naturae completae in se ipsis, sed solum sunt principia naturarum, sicut unitas numeri, et punctus lineae, et forma et materia corporis physici; unde huiusmodi principia non tractantur nisi in scientia in qua de principiatis agitur. There are some principles, however, that are not complete natures in themselves, but only principles of natures, as unity is the principle of number, point the principle of line, and form and matter principles of natural bodies. Principles of this sort, then, are investigated only in the science dealing with the things of which they are principles. Sicut autem uniuscuiusque determinati generis sunt quaedam communia principia quae se extendunt ad omnia principia illius generis, ita etiam et omnia entia secundum quod in ente communicant, habent quaedam principia quae sunt principia omnium entium. Quae quidem principia possunt dici communia dupliciter secundum Avicennam in sua Sufficientia: Now just as there are certain common principles of any particular genus extending to all the principles of that genus, so too all beings, inasmuch as they share in being, have certain principles that are the principles of all beings. And as Avicenna says, these principles can be called common in two ways: