Articulus 3 Article 3 Utrum possit esse aliquid infinitum actu secundum magnitudinem Whether an actually infinite magnitude can exist? Ad tertium sic proceditur. Videtur quod possit esse aliquid infinitum actu secundum magnitudinem. In scientiis enim mathematicis non invenitur falsum, quia abstrahentium non est mendacium, ut dicitur in II Physic. Sed scientiae mathematicae utuntur infinito secundum magnitudinem, dicit enim geometra in suis demonstrationibus, sit linea talis infinita. Ergo non est impossibile aliquid esse infinitum secundum magnitudinem. Objection 1: It seems that there can be something actually infinite in magnitude. For in mathematics there is no error, since there is no lie in things abstract, as the Philosopher says (Phys. ii). But mathematics uses the infinite in magnitude; thus, the geometrician in his demonstrations says, Let this line be infinite. Therefore it is not impossible for a thing to be infinite in magnitude. Praeterea, id quod non est contra rationem alicuius, non est impossibile convenire sibi. Sed esse infinitum non est contra rationem magnitudinis, sed magis finitum et infinitum videntur esse passiones quantitatis. Ergo non est impossibile aliquam magnitudinem esse infinitam. Obj. 2: Further, what is not against the nature of anything, can agree with it. Now to be infinite is not against the nature of magnitude; but rather both the finite and the infinite seem to be properties of quantity. Therefore it is not impossible for some magnitude to be infinite. Praeterea, magnitudo divisibilis est in infinitum, sic enim definitur continuum, quod est in infinitum divisibile, ut patet in III Physic. Sed contraria nata sunt fieri circa idem. Cum ergo divisioni opponatur additio, et diminutioni augmentum, videtur quod magnitudo possit crescere in infinitum. Ergo possibile est esse magnitudinem infinitam. Obj. 3: Further, magnitude is infinitely divisible, for the continuous is defined as that which is infinitely divisible, as is clear from Phys. iii. But contraries are concerned about one and the same thing. Since therefore addition is opposed to division, and increase opposed to diminution, it appears that magnitude can be increased to infinity. Therefore it is possible for magnitude to be infinite. Praeterea, motus et tempus habent quantitatem et continuitatem a magnitudine super quam transit motus, ut dicitur in IV Physic. Sed non est contra rationem temporis et motus quod sint infinita, cum unumquodque indivisibile signatum in tempore et motu circulari, sit principium et finis. Ergo nec contra rationem magnitudinis erit quod sit infinita. Obj. 4: Further, movement and time have quantity and continuity derived from the magnitude over which movement passes, as is said in Phys. iv. But it is not against the nature of time and movement to be infinite, since every determinate indivisible in time and circular movement is both a beginning and an end. Therefore neither is it against the nature of magnitude to be infinite. Sed contra, omne corpus superficiem habet. Sed omne corpus superficiem habens est finitum, quia superficies est terminus corporis finiti. Ergo omne corpus est finitum. Et similiter potest dici de superficie et linea. Nihil est ergo infinitum secundum magnitudinem. On the contrary, Every body has a surface. But every body which has a surface is finite; because surface is the term of a finite body. Therefore all bodies are finite. The same applies both to surface and to a line. Therefore nothing is infinite in magnitude. Respondeo dicendum quod aliud est esse infinitum secundum suam essentiam, et secundum magnitudinem. Dato enim quod esset aliquod corpus infinitum secundum magnitudinem, utpote ignis vel aer, non tamen esset infinitum secundum essentiam, quia essentia sua esset terminata ad aliquam speciem per formam, et ad aliquod individuum per materiam. Et ideo, habito ex praemissis quod nulla creatura est infinita secundum essentiam, adhuc restat inquirere utrum aliquid creatum sit infinitum secundum magnitudinem. I answer that, It is one thing to be infinite in essence, and another to be infinite in magnitude. For granted that a body exists infinite in magnitude, as fire or air, yet this could not be infinite in essence, because its essence would be terminated in a species by its form, and confined to individuality by matter. And so assuming from these premises that no creature is infinite in essence, it still remains to inquire whether any creature can be infinite in magnitude. Sciendum est igitur quod corpus, quod est magnitudo completa, dupliciter sumitur, scilicet mathematice, secundum quod consideratur in eo sola quantitas; et naturaliter, secundum quod consideratur in eo materia et forma. We must therefore observe that a body, which is a complete magnitude, can be considered in two ways; mathematically, in respect to its quantity only; and naturally, as regards its matter and form. Et de corpore quidem naturali, quod non possit esse infinitum in actu, manifestum est. Nam omne corpus naturale aliquam formam substantialem habet determinatam, cum igitur ad formam substantialem consequantur accidentia, necesse est quod ad determinatam formam consequantur determinata accidentia; inter quae est quantitas. Unde omne corpus naturale habet determinatam quantitatem et in maius et in minus. Unde impossibile est aliquod corpus naturale infinitum esse. Now it is manifest that a natural body cannot be actually infinite. For every natural body has some determined substantial form. Since therefore the accidents follow upon the substantial form, it is necessary that determinate accidents should follow upon a determinate form; and among these accidents is quantity. So every natural body has a greater or smaller determinate quantity. Hence it is impossible for a natural body to be infinite. Hoc etiam ex motu patet. Quia omne corpus naturale habet aliquem motum naturalem. Corpus autem infinitum non posset habere aliquem motum naturalem, nec rectum, quia nihil movetur naturaliter motu recto, nisi cum est extra suum locum, quod corpori infinito accidere non posset; occuparet enim omnia loca, et sic indifferenter quilibet locus esset locus eius. Et similiter etiam neque secundum motum circularem. Quia in motu circulari oportet quod una pars corporis transferatur ad locum in quo fuit alia pars; quod in corpore circulari, si ponatur infinitum, esse non posset, quia duae lineae protractae a centro, quanto longius protrahuntur a centro, tanto longius distant ab invicem; si ergo corpus esset infinitum, in infinitum lineae distarent ab invicem, et sic una nunquam posset pervenire ad locum alterius. The same appears from movement; because every natural body has some natural movement; whereas an infinite body could not have any natural movement; neither direct, because nothing moves naturally by a direct movement unless it is out of its place; and this could not happen to an infinite body, for it would occupy every place, and thus every place would be indifferently its own place. Neither could it move circularly; forasmuch as circular motion requires that one part of the body is necessarily transferred to a place occupied by another part, and this could not happen as regards an infinite circular body: for if two lines be drawn from the center, the farther they extend from the center, the farther they are from each other; therefore, if a body were infinite, the lines would be infinitely distant from each other; and thus one could never occupy the place belonging to any other. De corpore etiam mathematico eadem ratio est. Quia si imaginemur corpus mathematicum existens actu, oportet quod imaginemur ipsum sub aliqua forma, quia nihil est actu nisi per suam formam. Unde, cum forma quanti, inquantum huiusmodi, sit figura, oportebit quod habeat aliquam figuram. Et sic erit finitum, est enim figura, quae termino vel terminis comprehenditur. The same applies to a mathematical body. For if we imagine a mathematical body actually existing, we must imagine it under some form, because nothing is actual except by its form; hence, since the form of quantity as such is figure, such a body must have some figure, and so would be finite; for figure is confined by a term or boundary. Ad primum ergo dicendum quod geometer non indiget sumere aliquam lineam esse infinitam actu, sed indiget accipere aliquam lineam finitam actu, a qua possit subtrahi quantum necesse est, et hanc nominat lineam infinitam. Reply Obj. 1: A geometrician does not need to assume a line actually infinite, but takes some actually finite line, from which he subtracts whatever he finds necessary; which line he calls infinite. Ad secundum dicendum quod, licet infinitum non sit contra rationem magnitudinis in communi, est tamen contra rationem cuiuslibet speciei eius, scilicet contra rationem magnitudinis bicubitae vel tricubitae, sive circularis vel triangularis, et similium. Non autem est possibile in genere esse quod in nulla specie est. Unde non est possibile esse aliquam magnitudinem infinitam, cum nulla species magnitudinis sit infinita. Reply Obj. 2: Although the infinite is not against the nature of magnitude in general, still it is against the nature of any species of it; thus, for instance, it is against the nature of a bicubical or tricubical magnitude, whether circular or triangular, and so on. Now what is not possible in any species cannot exist in the genus; hence there cannot be any infinite magnitude, since no species of magnitude is infinite. Ad tertium dicendum quod infinitum quod convenit quantitati, ut dictum est, se tenet ex parte materiae. Per divisionem autem totius acceditur ad materiam, nam partes se habent in ratione materiae, per additionem autem acceditur ad totum, quod se habet in ratione formae. Et ideo non invenitur infinitum in additione magnitudinis, sed in divisione tantum. Reply Obj. 3: The infinite in quantity, as was shown above, belongs to matter. Now by division of the whole we approach to matter, forasmuch as parts have the aspect of matter; but by addition we approach to the whole which has the aspect of a form. Therefore the infinite is not in the addition of magnitude, but only in division. Ad quartum dicendum quod motus et tempus non sunt secundum totum in actu, sed successive, unde habent potentiam permixtam actui. Sed magnitudo est tota in actu. Et ideo infinitum quod convenit quantitati, et se tenet ex parte materiae, repugnat totalitati magnitudinis, non autem totalitati temporis vel motus, esse enim in potentia convenit materiae. Reply Obj. 4: Movement and time are whole, not actually but successively; hence they have potentiality mixed with actuality. But magnitude is an actual whole; therefore the infinite in quantity refers to matter, and does not agree with the totality of magnitude; yet it agrees with the totality of time and movement: for it is proper to matter to be in potentiality. Articulus 4 Article 4 Utrum possibile sit esse multitudinem infinitam secundum actum Whether an infinite multitude can exist? Ad quartum sic proceditur. Videtur quod possibile sit esse multitudinem infinitam secundum actum. Non enim est impossibile id quod est in potentia reduci ad actum. Sed numerus est in infinitum multiplicabilis. Ergo non est impossibile esse multitudinem infinitam in actu. Objection 1: It seems that an actually infinite multitude is possible. For it is not impossible for a potentiality to be made actual. But number can be multiplied to infinity. Therefore it is possible for an infinite multitude actually to exist. Praeterea, cuiuslibet speciei possibile est esse aliquod individuum in actu. Sed species figurae sunt infinitae. Ergo possibile est esse infinitas figuras in actu. Obj. 2: Further, it is possible for any individual of any species to be made actual. But the species of figures are infinite. Therefore an infinite number of actual figures is possible. Praeterea, ea quae non opponuntur ad invicem, non impediunt se invicem. Sed, posita aliqua multitudine rerum, adhuc possunt fieri alia multa quae eis non opponuntur, ergo non est impossibile aliqua iterum simul esse cum eis, et sic in infinitum. Ergo possibile est esse infinita in actu. Obj. 3: Further, things not opposed to each other do not obstruct each other. But supposing a multitude of things to exist, there can still be many others not opposed to them. Therefore it is not impossible for others also to coexist with them, and so on to infinitude; therefore an actual infinite number of things is possible. Sed contra est quod dicitur Sap. XI, omnia in pondere, numero et mensura disposuisti. On the contrary, It is written, Thou hast ordered all things in measure, and number, and weight (Wis 11:21). Respondeo dicendum quod circa hoc fuit duplex opinio. Quidam enim, sicut Avicenna et Algazel, dixerunt quod impossibile est esse multitudinem actu infinitam per se, sed infinitam per accidens multitudinem esse, non est impossibile. Dicitur enim multitudo esse infinita per se, quando requiritur ad aliquid ut multitudo infinita sit. I answer that, A twofold opinion exists on this subject. Some, as Avicenna and Algazel, said that it was impossible for an actually infinite multitude to exist absolutely; but that an accidentally infinite multitude was not impossible. A multitude is said to be infinite absolutely, when an infinite multitude is necessary that something may exist. Et hoc est impossibile esse, quia sic oporteret quod aliquid dependeret ex infinitis; unde eius generatio nunquam compleretur, cum non sit infinita pertransire. Now this is impossible; because it would entail something dependent on an infinity for its existence; and hence its generation could never come to be, because it is impossible to pass through an infinite medium. Per accidens autem dicitur multitudo infinita, quando non requiritur ad aliquid infinitas multitudinis, sed accidit ita esse. Et hoc sic manifestari potest in operatione fabri, ad quam quaedam multitudo requiritur per se, scilicet quod sit ars in anima, et manus movens, et martellus. Et si haec in infinitum multiplicarentur, nunquam opus fabrile compleretur, quia dependeret ex infinitis causis. Sed multitudo martellorum quae accidit ex hoc quod unum frangitur et accipitur aliud, est multitudo per accidens, accidit enim quod multis martellis operetur; et nihil differt utrum uno vel duobus vel pluribus operetur, vel infinitis, si infinito tempore operaretur. Per hunc igitur modum, posuerunt quod possibile est esse actu multitudinem infinitam per accidens. A multitude is said to be accidentally infinite when its existence as such is not necessary, but accidental. This can be shown, for example, in the work of a carpenter requiring a certain absolute multitude; namely, art in the soul, the movement of the hand, and a hammer; and supposing that such things were infinitely multiplied, the carpentering work would never be finished, forasmuch as it would depend on an infinite number of causes. But the multitude of hammers, inasmuch as one may be broken and another used, is an accidental multitude; for it happens by accident that many hammers are used, and it matters little whether one or two, or many are used, or an infinite number, if the work is carried on for an infinite time. In this way they said that there can be an accidentally infinite multitude. Sed hoc est impossibile. Quia omnem multitudinem oportet esse in aliqua specie multitudinis. Species autem multitudinis sunt secundum species numerorum. Nulla autem species numeri est infinita, quia quilibet numerus est multitudo mensurata per unum. Unde impossibile est esse multitudinem infinitam actu, sive per se, sive per accidens. This, however, is impossible; since every kind of multitude must belong to a species of multitude. Now the species of multitude are to be reckoned by the species of numbers. But no species of number is infinite; for every number is multitude measured by one. Hence it is impossible for there to be an actually infinite multitude, either absolute or accidental. Item, multitudo in rerum natura existens est creata, et omne creatum sub aliqua certa intentione creantis comprehenditur, non enim in vanum agens aliquod operatur. Unde necesse est quod sub certo numero omnia creata comprehendantur. Impossibile est ergo esse multitudinem infinitam in actu, etiam per accidens. Likewise multitude in nature is created; and everything created is comprehended under some clear intention of the Creator; for no agent acts aimlessly. Hence everything created must be comprehended in a certain number. Therefore it is impossible for an actually infinite multitude to exist, even accidentally. Sed esse multitudinem infinitam in potentia, possibile est. Quia augmentum multitudinis consequitur divisionem magnitudinis, quanto enim aliquid plus dividitur, tanto plura secundum numerum resultant. Unde, sicut infinitum invenitur in potentia in divisione continui, quia proceditur ad materiam, ut supra ostensum est; eadem ratione etiam infinitum invenitur in potentia in additione multitudinis. But a potentially infinite multitude is possible; because the increase of multitude follows upon the division of magnitude; since the more a thing is divided, the greater number of things result. Hence, as the infinite is to be found potentially in the division of the continuous, because we thus approach matter, as was shown in the preceding article, by the same rule, the infinite can be also found potentially in the addition of multitude. Ad primum ergo dicendum quod unumquodque quod est in potentia, reducitur in actum secundum modum sui esse, dies enim non reducitur in actum ut sit tota simul, sed successive. Et similiter infinitum multitudinis non reducitur in actum ut sit totum simul, sed successive, quia post quamlibet multitudinem, potest sumi alia multitudo in infinitum. Reply Obj. 1: Every potentiality is made actual according to its mode of being; for instance, a day is reduced to act successively, and not all at once. Likewise the infinite in multitude is reduced to act successively, and not all at once; because every multitude can be succeeded by another multitude to infinity. Ad secundum dicendum quod species figurarum habent infinitatem ex infinitate numeri, sunt enim species figurarum, trilaterum, quadrilaterum, et sic inde. Unde, sicut multitudo infinita numerabilis non reducitur in actum quod sit tota simul, ita nec multitudo figurarum. Reply Obj. 2: Species of figures are infinite by infinitude of number. Now there are various species of figures, such as trilateral, quadrilateral and so on; and as an infinitely numerable multitude is not all at once reduced to act, so neither is the multitude of figures. Ad tertium dicendum quod, licet, quibusdam positis, alia poni non sit eis oppositum; tamen infinita poni opponitur cuilibet speciei multitudinis. Unde non est possibile esse aliquam multitudinem actu infinitam. Reply Obj. 3: Although the supposition of some things does not preclude the supposition of others, still the supposition of an infinite number is opposed to any single species of multitude. Hence it is not possible for an actually infinite multitude to exist. Quaestio 8 Question 8 De existentia Dei in rebus The Existence of God in Things Quia vero infinito convenire videtur quod ubique et in omnibus sit, considerandum est utrum hoc Deo conveniat. Et circa hoc quaeruntur quatuor. Since it evidently belongs to the infinite to be present everywhere, and in all things, we now consider whether this belongs to God; and concerning this there arise four points of inquiry: Primo, utrum Deus sit in omnibus rebus. (1) Whether God is in all things? Secundo, utrum Deus sit ubique. (2) Whether God is everywhere?