Responsio. Dicendum, quod propter difficultatem huius quaestionis coactus est Plato ad ponendum ideas: cum enim, ut dicit Philosophus in I Metaphysicae, crederet omnia sensibilia semper esse in fluxu secundum opinionem Cratyli et Heracliti, et ita existimaret de eis non posse esse scientiam, posuit quasdam substantias a sensibilibus separatas, de quibus essent scientiae et darentur diffinitiones. Sed hic defectus accidit ex eo quod non distinxit quod est per se ab eo quod est secundum accidens: nam secundum accidens falluntur plerumque etiam sapientes, ut dicitur in I Elenchorum. I answer that it was the difficulty of this problem that drove Plato to posit ideas. Believing that all sensible things were always in flux, as Cratylus and Heraclitus taught, he thought there can be no science concerning them, as the Philosopher says in the Metaphysics. So he claimed that there were substances separated from the sense world, which might serve as the objects of science and of definitions. He made this mistake because he failed to distinguish what is essential from what is accidental. For it happens that by accident even the wise often fall into error, as is said in the Sophistic Refutations. Ut autem probatur in VII Metaphysicae, cum in substantia sensibili inveniatur et ipsum integrum, id est compositum, et ratio id est forma eius, per se quidem generatur et corrumpitur compositum, non autem ratio sive forma, sed solum per accidens: non enim fit domum esse, ut ibidem dicitur, sed hanc domum. Unumquodque autem potest considerari sine omnibus his quae ei non per se comparantur, et ideo formae et rationes rerum quamvis in motu exsistentium, prout in se considerantur absque motu sunt; et sic de eis sunt scientiae et diffinitiones, ut ibidem Philosophus dicit. Non autem scientiae sensibilium substantiarum fundantur super cognitione aliquarum substantiarum a sensibilibus separatarum, ut ibidem probatur. Now, as is shown in the Metaphysics, we find in a sensible substance both the whole or the composite itself, and also its nature (ratio) or form; and it is the composite that is essentially generated and corrupted and not the nature or form, except accidentally. As the Metaphysics says, it is not house that is made, but this house. Now anything can be thought of without all the items that are not essentially related to it. Consequently, forms and natures, though belonging to things existing in motion, are without motion when they are considered in themselves; and so they can be the objects of sciences and of definitions, as the Philosopher says. As he proves, the sciences of sensible reality are not based upon the knowledge of certain substances separated from the sense world. Huiusmodi autem rationes quas considerant scientiae quae sunt de rebus, considerantur absque motu; (et) sic oportet quod considerentur absque illis secundum quae competit motus rebus mobilibus. Cum autem omnis motus tempore mensuretur, et primus motus sit motus localis, quo remoto nullus alius motus inest, oportet quod secundum hoc aliquid sit mobile, quod est hic et nunc; hoc autem consequitur rem ipsam mobilem secundum quod est individuata per materiam exsistentem sub dimensionibus signatis; Natures of this kind, which are the objects of the sciences of real beings, are thought of without motion; and so they must be thought of without those conditions by reason of which motion belongs to mobile things. Now, because every motion is measured by time, and the primary motion is local motion (for without it there is no other motion), a thing must be subject to motion inasmuch as it exists here and now; and it exists under these conditions insofar as it is individuated by matter having determinate dimensions. unde oportet quod huiusmodi rationes, secundum quas de rebus mobilibus possunt esse scientiae, considerentur absque materia signata et absque omnibus his quae consequuntur materiam signatam, non autem absque materia non signata, quia ex eius notione dependet notio formae, quae determinat sibi materiam; et ideo ratio hominis, quam significat diffinitio et secundum quam procedit scientia, consideratur sine his carnibus et sine his ossibus, non autem sine carnibus et ossibus absolute. Et quia singularia includunt in sui ratione materiam signatam, universalia vero materiam communem, ut dicitur in VII Metaphysicae, ideo praedicta abstractio non dicitur formae a materia absolute, sed universalis a particulari. Possunt ergo huiusmodi rationes sic abstractae considerari dupliciter: Consequently, natures of this kind, which make possible sciences of things subject to motion, must be thought of without determinate matter and everything following upon such matter; but not without indeterminate matter, because on its notion depends the notion of form that determines matter to itself. Thus the nature of man, which his definition signifies and which is the object of science, is considered without this flesh and these bones, but not absolutely without flesh and bones. And because individuals include determinate matter in their nature, whereas universals include common matter, as is said in the Metaphysics, the above-mentioned abstraction is not said to be the abstraction of form from matter absolutely, but the abstraction of the universal from the particular. Natures of this sort, thus abstracted, can be considered in two ways. uno modo secundum se, et sic considerantur sine motu et materia signata, et hoc non invenitur in eis nisi secundum esse quod habent in intellectu; First, in themselves; and then they are thought of without motion and determinate matter. This happens to them only by reason of the being they have in the intellect. alio modo secundum quod comparantur ad res quarum sunt rationes; quae quidem res sunt in materia et motu, et sic (sunt) principia cognoscendi illa, quia omnis res cognoscitur per suam formam. Second, they can be viewed in relation to the things of which they are the natures; and these things exist with matter and motion. Thus they are principles by which we know these things, for everything is known through its form. Et ita per huiusmodi rationes immobiles et sine materia particulari consideratas habetur cognitio in scientia naturali de rebus mobilibus et materialibus extra animam exsistentibus. Consequently, in natural science we know mutable and material things existing outside the soul through natures of this kind; that is to say, natures that are immobile and considered without particular matter. Ad primum ergo dicendum, quod materia non est individuationis principium nisi secundum quod est sub dimensionibus signatis exsistens; et sic etiam scientia naturalis a materia abstrahit. Reply Obj. 1: Matter is the principle of individuation only insofar as it exists with determinate dimensions, and in this sense natural science indeed abstracts from matter. Ad secundum dicendum, quod forma intelligibilis est quiditas rei: obiectum enim intellectus est quid, ut dicitur in III De anima. Quiditas autem compositi universalis, ut hominis aut animalis, includit in se materiam universalem, non autem particularem, ut dicitur in VII Metaphysicae; unde intellectus communiter abstrahit a materia signata et condicionibus eius, non autem a materia communi in scientia naturali; quamvis etiam in scientia naturali non consideretur materia nisi in ordine ad formam, unde etiam forma per prius est de consideratione naturalis quam materia. Reply Obj. 2: The intelligible form is a thing’s quiddity, for, as the De anima says, the object of the intellect is the quiddity of a thing. Now, as is said in the Metaphysics, the quiddity of a universal composite, like man or animal, includes within itself common but not particular matter. So the intellect regularly abstracts from determinate matter and its conditions; but in natural science it does not abstract from common matter, although matter itself is considered in natural science only in relation to form. For this reason the natural scientist is more concerned with form than with matter. Ad tertium dicendum, quod de primo motore non agitur in scientia naturali tamquam de subiecto vel de parte subiecti, sed tamquam de termino ad quem scientia naturalis perducit. Terminus autem non est de natura rei cuius est terminus, sed habet aliquam habitudinem ad rem illam, sicut terminus lineae non est linea sed habet ad eam aliquam habitudinem. Ita etiam et primus motor est alterius naturae a rebus naturalibus, habet tamen ad eas aliquam habitudinem in quantum influit eis motum, et sic cadit in consideratione naturalis, scilicet non secundum ipsum sed in quantum est motor. Reply Obj. 3: Natural science does not treat of the first mover as its subject or as part of its subject, but as the end to which natural science leads. Now the end does not belong to the nature of the thing of which it is the end, but it has a relation to it; as the end of a line is not the line but is related to it. So also the first mover is of a different nature from natural things, but it is related to them because it moves them. So it falls under the consideration of natural science, not in itself, but insofar as it is a mover. Ad quartum dicendum, quod scientia est de aliquo dupliciter: uno modo primo et principaliter, et sic scientia est de rationibus universalibus supra quas fundatur; alio modo est de aliquibus secundario et quasi per reflexionem quandam, et sic de illis rebus quarum sunt illae rationes, in quantum illas rationes applicat ad res etiam particulares, quarum sunt adminiculo inferiorum virium: ratione enim universali utitur sciens et ut re scita et ut medio sciendi, per universalem enim hominis rationem possum iudicare de hoc vel de illo. Rationes autem universales rerum omnes sunt immobiles, et ideo quantum ad hoc omnis scientia de necessariis est; sed rerum quarum sunt illae rationes quaedam sunt necessariae et immobiles, quaedam contingentes et mobiles, et quantum ad hoc de rebus contingentibus et mobilibus dicuntur esse scientiae. Reply Obj. 4: Science treats of something in two ways: in one way, primarily and principally; and in this sense science is concerned universal natures, which are its very foundation. In another way it treats of something secondarily, as by a sort of reflection; and in this sense it is concerned with the things whose natures they are, inasmuch as, using the lower powers, it relates those natures to the particular things possessing them. For a knower uses a universal nature both as a thing known and as a means of knowing. Thus, through the universal nature of man we can judge of this or that particular man. Now, all universal natures of things are immutable; and so, in this respect, all science is concerned with what is necessary. But some of the things possessing these natures are necessary and immutable, whereas others are contingent and subject to movement, and in this respect sciences are said to be concerned with the contingent and mutable. Ad quintum dicendum, quod quamvis universale non moveatur, est tamen ratio rei mobilis. Reply Obj. 5: Although a universal is not mutable, it is nevertheless the nature of a mutable thing. Ad sextum dicendum, quod anima et aliae formae naturales quamvis non moveantur per se, moventur tamen per accidens; et insuper sunt perfectiones rerum mobilium, et secundum hoc cadunt in consideratione naturalis. Terra vero quamvis secundum totum non moveatur, quod accidit ei in quantum est in suo loco naturali, in quo aliquid quiescit per eandem naturam per quam movetur ad locum, tamen partes eius moventur ad locum cum sunt extra locum proprium; et sic terra et ratione quietis totius, et ratione motus partium, cadit in considerationem naturalis. Reply Obj. 6: Although the soul and other natural forms are not themselves subject to motion, they are moved accidentally, and they are, moreover, the perfections of mutable things; and for this reason they come within the domain of natural science. But even though the earth as a whole is not moved (for it happens to be in its natural place, where a thing is at rest in virtue of the same nature through which it is moved to a place), nevertheless, when its parts are outside their proper place, they are moved to a place. Thus the earth falls within the domain of natural science both by reason of the immobility of the whole earth and by reason of the movement of its parts. Ad septimum dicendum, quod mutabilitas illa quae competit omni creaturae, non est secundum aliquem motum naturalem sed secundum dependentiam ad Deum; a quo si sibi deserentur, deficerent ab eo quod est. Dependentia autem ista pertinet ad considerationem metaphysici potius quam naturalis. Creaturae etiam spirituales non sunt mutabiles nisi secundum electionem; et talis mutatio non pertinet ad naturalem sed magis ad divinum. Reply Obj. 7: The mutability characteristic of all creatures is not with respect to any natural motion, but with respect to their dependence on God, separation from whom entails destruction of their very being. And that dependence falls under the consideration of metaphysics rather than under that of natural philosophy. Spiritual creatures, moreover, are mutable only with regard to choice; and this sort of motion is not the concern of the natural philosopher but rather of the metaphysician. Articulus III Article 3 Does mathematics treat, without motion and matter, of what exists in matter? Ad tertium sic proceditur: videtur quod mathematica consideratio non sit sine materia de his quae habent esse in materia. Cum enim veritas consistat in adaequatione rei ad intellectum, oportet esse falsitatem quandocumque res consideratur aliter quam sit. Si ergo res quae sunt in materia sine materia considerat mathematica, eius consideratio erit falsa, et sic non erit scientia, cum omnis scientia sit verorum. Obj. 1: We proceed as follows to the third article: it seems that mathematical thinking does not treat, without motion and matter, of what exists in matter, for since truth consists in the conformity of thing to intellect, there must be falsehood whenever we think of something otherwise than it is. If then in mathematics we consider what is in matter in abstraction from matter, we will consider it falsely; and so mathematics will not be a science, for every science is concerned with what is true. Praeterea. Secundum Philosophum in I Posteriorum cuiuslibet scientiae est considerare subiectum et partes subiecti. Sed omnium materialium secundum esse materia pars est. Ergo non potest esse quod aliqua scientia consideret de his quae sunt in materia absque hoc quod materiam consideret. Obj. 2: Again, as the Philosopher states, every science has the task of considering a subject and the parts of the subject. Now in actual existence matter is a part of all material things. So it is impossible for a science to treat of what is in matter without treating of matter. Praeterea. Omnes lineae rectae sunt eiusdem speciei. Sed mathematicus considerat lineas rectas numerando eas, alias non consideraret triangulum et quadratum; ergo considerat lineas secundum quod differunt numero et conveniunt specie. Sed principium differendi his quae secundum speciem conveniunt est materia, ut ex supra dictis patet. Ergo materia consideratur a mathematico. Obj. 3: Again, all straight lines are specifically the same. But the mathematician treats of straight lines by numbering them; otherwise he would not treat of the triangle and the square. It follows that he considers lines as specifically the same and numerically different. But it is clear from the above that matter is the principle differentiating things specifically the same. So the mathematician treats of matter. Praeterea. Nulla scientia quae penitus abstrahit a materia demonstrat per causam materialem. Sed in mathematica fiunt aliquae demonstrationes quae non possunt reduci nisi ad causam materialem, sicut cum demonstratur aliquid de toto ex partibus: partes enim sunt materia totius, ut dicitur in II Physicorum; unde et in II Posteriorum reducitur ad causam materialem demonstratio qua demonstratur quod angulus qui est in semicirculo est rectus ex hoc quod utraque pars eius est semirectus. Ergo mathematica non omnino abstrahit a materia. Obj. 4: Again, no science completely abstracting from matter demonstrates through a material cause. But in mathematics some demonstrations are made which can only be reduced to a material cause, as when we demonstrate something about a whole by its parts. For, as the Physics says, parts are the matter of the whole. Thus in the Posterior Analytics the demonstration that the angle in a semi-circle is a right angle from the fact that each of its two parts is half of a right angle, is reduced to a material cause. Therefore, mathematics does not entirely abstract from matter. Praeterea. Motus non potest esse sine materia. Sed mathematicus debet considerare motum; quia cum motus mensuretur secundum spatium, eiusdem rationis et scientiae videtur esse considerare quantitatem spatii, quod pertinet ad mathematicum, et quantitatem motus. Ergo mathematicus non omnino dimittit considerationem materiae. Obj. 5: Again, motion cannot exist without matter. But the mathematician ought to consider motion, because, since motion is measured relative to space, to consider the quantity of space, which pertains to the mathematician, and the quantity of motion, has the same nature and belongs to the same science. Therefore, the mathematician does not entirely leave matter out of consideration. Praeterea. Astrologia quaedam pars mathematicae est, et similiter scientia de sphaera mota, et scientia de ponderibus, et musica, in quibus omnibus fit consideratio de motu et rebus mobilibus. Ergo mathematica non abstrahit totaliter a materia et motu. Obj. 6: Again, astronomy is a part of mathematics, and so too is the science of the moved sphere, the science of weights, and music, all of which treat of motion and mobile things So mathematics does not entirely abstract from matter and motion. Praeterea. Naturalis consideratio tota est circa materiam et motum. Sed quaedam conclusiones demonstrantur communiter a mathematico et naturali, ut utrum terra sit rotunda, et utrum sit in medio caeli. Ergo non potest esse quod mathematica omnino abstrahat a materia.—Si dicatur quod abstrahit tantum a materia sensibili, contra: materia sensibilis videtur esse materia particularis, quia sensus particularium est, a qua omnes scientiae abstrahunt; ergo mathematica consideratio non debet dici magis abstracta quam aliqua aliarum scientiarum. Obj. 7: Again, natural science is entirely concerned with matter and motion. But some conclusions are demonstrated alike by the mathematician and the natural scientist, for instance, whether the earth is round and whether it is in the middle of the universe. Therefore, mathematics cannot entirely abstract from matter. If it be said that mathematics abstracts only from sensible matter, the contrary seems true. Sensible matter seems to be particular matter, because what the senses perceive are particular things, and all the sciences abstract from this kind of matter. So mathematical thinking should not be called more abstract than that of the other sciences. Praeterea. Philosophus in II Physicorum dicit tria esse negotia: primum est de mobili et corruptibili, secundum de mobili et incorruptibili, tertium de immobili et incorruptibili. Primum autem est naturale, tertium divinum, secundum mathematicum, ut Ptolemaeus exponit in principio Almagesti. Ergo mathematica est de mobilibus. Obj. 8: Again, the Philosopher says that there are three branches of study: the first concerns what is mutable and corruptible, the second what is mutable and incorruptible, and the third what is immutable and incorruptible. As Ptolemy explains, the first is natural science the third divine science, and the second mathematics. Therefore mathematics concerns what is mutable. Sed contra est quod Philosophus dicit in VI Metaphysicae. On the contrary is the Philosopher’s statement in the Metaphysics. Praeterea. Quaedam res sunt quae quamvis sint in materia, tamen non recipiunt in sui diffinitione materiam, ut curvum, et in hoc differt a simo. Sed philosophia debet de omnibus entibus considerare. Ergo oportet de huiusmodi esse aliquam partem philosophiae, et haec est mathematica, cum ad nullam aliam pertineat. Moreover, some things, although existing in matter, do not contain matter in their definition; for instance, curve, which differs in this respect from snub. Now philosophy should treat of all beings. Hence some part of philosophy must consider beings of this sort; and this is mathematics, for this does not belong to any other part. Praeterea. Ea quae sunt priora secundum intellectum possunt sine posterioribus considerari. Sed mathematica sunt priora naturalibus, quae sunt in materia et motu: habent enim se ex additione ad mathematica, ut dicitur in III Caeli et mundi. Ergo mathematica consideratio potest esse sine materia et motu. Moreover, what is prior from the point of view of the intellect can be considered without what is posterior. Now mathematicals are prior to natural things existing in matter and motion, for the latter are so related to mathematicals that they add something to them, as is said in the De caelo et mundo. Therefore, mathematical investigation can be without matter and motion. Responsio. Dicendum, quod ad evidentiam huius quaestionis oportet (videre) qua(liter) intellectus secundum suam operationem abstrahere possit. Sciendum est igitur quod secundum Philosophum in III De anima duplex est operatio intellectus: una quae dicitur intelligentia indivisibilium, qua cognoscit de unoquoque quid est, alia vero qua componit et dividit, scilicet enuntiationem affirmativam vel negativam formando. I answer that in order to throw light on this question we must understand how the intellect in its operation is able to abstract. We must realize that, as the Philosopher says, the intellect has two operations, one called the understanding of indivisibles, by which it knows what a thing is, and another by which it joins and divides, that is to say, by forming affirmative and negative statements. Et hae quidem duae operationes duobus quae sunt in rebus respondent. Now these two operations correspond to two principles in things. Prima quidem operatio respicit ipsam naturam rei, secundum quam res intellecta aliquem gradum in entibus obtinet, sive sit res completa, ut totum aliquod, sive res incompleta, ut pars vel accidens. The first operation concerns the nature itself of a thing, in virtue of which the object known holds a certain rank among beings, whether it be a complete thing, like some whole, or an incomplete thing, like a part or an accident. Secunda vero operatio respicit ipsum esse rei; quod quidem resultat ex congregatione principiorum rei in compositis, vel ipsam simplicem naturam rei concomitatur, ut in substantiis simplicibus. The second operation has to do with a thing’s being (esse), which results from the union of the principles of a thing in composite substances, or, as in the case of simple substances, accompanies the thing’s simple nature. Et quia veritas intellectus est ex hoc quod conformatur (rei), patet quod secundum hanc secundam operationem intellectus non potest vere abstrahere quod secundum rem coniunctum est; quia in abstrahendo significaretur esse separatio secundum ipsum esse rei, sicut si abstraho hominem ab albedine dicendo homo non est albus, significo esse separationem in re, unde si secundum rem homo et albedo non sint separata, erit intellectus falsus. Hac ergo operatione intellectus vere abstrahere non potest nisi ea quae sunt secundum rem separata, ut cum dicitur homo non est asinus. Now, since the truth of the intellect results from its conformity with reality, it is clear that in this second operation the intellect cannot truthfully abstract what is united in reality, because the abstraction would signify a separation with regard to the very being of the thing. For example, if I abstract man from whiteness by saying, “Man is not white,” I signify that there is a separation in reality. So if in reality man and whiteness are not separate, the intellect will be false. Through this operation, then, the intellect can truthfully abstract only those things that are separate in reality, as when we say, “Man is not an ass.” Sed secundum primam operationem potest abstrahere ea quae secundum rem separata non sunt, non tamen omnia, sed aliqua. Cum enim unaquaeque res sit intelligibilis secundum quod est in actu, ut dicitur in IX Metaphysicae, oportet quod ipsa natura sive quiditas rei intelligatur vel secundum quod est actus quidam, sicut accidit de ipsis formis et substantiis simplicibus, vel secundum id quod est actus eius, sicut substantiae compositae per suas formas, vel secundum id quod est ei loco actus, sicut materia prima per habitudinem ad formam et vacuum per privationem locati; et hoc est illud ex quo unaquaeque natura suam rationem sortitur. Through the first operation, however, we can abstract things that are not separate in reality; not all, it is true, but some. For, since everything is intelligible insofar as it is in act, as the Metaphysics says, we must understand the nature itself or the quiddity of a thing either inasmuch as it is a certain act (as happens in the case of forms themselves and simple substances); or through that which is its act (as we know composite substances through their forms); or through that which takes the place of act in it (as we know prime matter through its relation to form, and a vacuum through the absence of a body in place). And it is from this that each nature is given its definition. Quando ergo secundum hoc per quod constituitur ratio naturae et per quod ipsa natura intelligitur, natura ipsa habet ordinem et dependentiam ad aliquid aliud, tunc constat quod natura illa sine illo alio intelligi non potest, sive sint coniuncta coniunctione illa qua pars coniungitur toti, sicut pes non potest intelligi sine intellectu animalis, quia illud a quo pes habet rationem pedis dependet ab eo a quo animal est animal; sive sint coniuncta per modum quo forma coniungitur materiae, ut pars comparti vel accidens subiecto, sicut simum non potest intelligi sine naso; sive etiam sint secundum rem separata, sicut pater non potest intelligi sine intellectu filii, quamvis istae relationes inveniantur in diversis rebus. Therefore, when the nature itself is related to, and depends on something else, with regard to that which forms the definition (ratio) of the nature, and through which the nature itself is understood, clearly we cannot know the nature without that other thing. This is true whether they are connected as a part is united to a whole (as we cannot know foot without knowing animal, because that whereby foot has the nature of foot depends on that whereby animal is animal); or whether they are connected as form is united to matter, or as one part to another part, or as accident to subject (as we cannot know snub without nose); or even whether they are separated in reality (as we cannot know father without knowing son, although these relationships are found in different things). Si vero unum ab altero non dependeat secundum id quod constituit rationem naturae, tunc unum potest ab altero abstrahi per intellectum ut sine eo intelligatur non solum si sint separata secundum rem, sicut homo et lapis, sed etiam si secundum rem coniuncta sint, sive ea coniunctione qua pars et totum coniunguntur, sicut littera potest intelligi sine syllaba sed non e converso, et animal sine pede sed non e converso; sive etiam sint coniuncta per modum quo forma coniungitur materiae et accidens subiecto, sicut albedo potest intelligi sine homine et e converso. But if one thing does not depend on another with regard to that which forms the definition of the nature, then the intellect can abstract the one from the other so as to know it without the other. This is true not only if they are separated in reality, like man and stone, but also if they are united in reality, whether they are joined as part and whole (as letter can be understood without syllable, but not vice versa, and animal without foot, but not conversely); or even if they are joined as form is united to matter and accident to subject (as whiteness can be understood without man and vice versa). Sic ergo intellectus distinguit unum ab altero aliter et aliter secundum diversas operationes: quia secundum operationem qua componit et dividit distinguit unum ab alio per hoc quod intelligit unum alii non inesse, in operatione vero qua intelligit quid est unumquodque, distinguit unum ab alio dum intelligit quid est hoc, nihil intelligendo de alio, neque quod sit cum eo, neque quod sit ab eo separatum; unde ista distinctio non proprie habet nomen separationis, sed prima tantum. Accordingly, through its various operations the intellect distinguishes one thing from another in different ways. Through the operation by which it composes and divides, it distinguishes one thing from another by understanding that the one does not exist in the other. Through the operation, however, by which it understands what a thing is, it distinguishes one thing from another by knowing what one is without knowing anything of the other, either that it is united to it or separated from it. So this distinction is not properly called separation, but only the first.