Authors/Thomas Aquinas/physics/L3/lect7

Lecture 7 Arguments for and against the infinite

Latin	English
	LECTURE 7 (203 b15) Arguments for and against the infinite
lib. 3 l. 7 n. 1 Positis opinionibus antiquorum de infinito, hic incipit inquirere veritatem. Et primo obiicit ad utramque partem; secundo solvit, ibi: quod quidem igitur actu corpus et cetera. Circa primum duo facit: primo ponit rationes ad ostendendum quod infinitum sit; secundo ad ostendendum quod non sit, ibi: habet autem dubitationem et cetera.	336. Having listed the opinions of the earlier philosophers on the infinite, Aristotle now begins to inquire into the truth of the matter. First he objects to both sides of the question; Secondly, he solves the objections, in Lecture 10. About the first he does two things: First he gives reasons to show that the infinite exists; Secondly, to show that it does not exist, at 342.
lib. 3 l. 7 n. 2 Circa primum ponit quinque rationes. Quarum prima sumitur ex tempore, quod secundum communem opinionem antiquorum infinitum erat: solus enim Plato generavit tempus, ut in octavo huius dicetur. Dicit ergo primo quod ad ostendendum infinitum esse, ex quinque rationibus accipi potest: et primo quidem ex tempore, quod est infinitum secundum illos qui dicebant tempus semper fuisse et semper futurum esse.	337. In regard to the first he gives five reasons to show that the infinite exists. The first of these is based on time, which according to the common opinion of the earlier philosophers was infinite [i.e., always was and always will be]. For Plato alone supposed that time was generated, as will be shown in Book VIII (l.2). The first of these is taken from time, which, according to the common opinion of the ancients, was infinite. Indeed, Plato alone generated time, as will be said in Book VIII (l.2). He says therefore first that the infinite is shown to exist by five arguments. Abe first of these is taken from time, which is infinite according to those who held that time always was and always will be.
lib. 3 l. 7 n. 3 Secunda ratio sumitur ex divisione magnitudinum in infinitum. Infinito enim in magnitudinibus utuntur etiam mathematici in suis demonstrationibus: quod non esset si infinitum totaliter tolleretur a rebus: oportet igitur ponere infinitum.	338. The second reason is taken from the infinite divisibility of magnitude. For even mathematicians use the infinite in their demonstrations. This, however, would not happen, if there were no infinite at all; hence the infinite exists.
lib. 3 l. 7 n. 4 Tertia ratio sumitur ex perpetuitate generationis et corruptionis, secundum plurium opinionem. Si enim totaliter tolleretur infinitum, non posset dici quod generatio et corruptio in infinitum durarent; unde oporteret dicere quod quandoque totaliter generatio cessaret, quod est contra multorum opinionem. Oportet igitur ponere infinitum.	339. The third reason is based on the perpetual processes of generation and. corruption according to the opinion of many; for if the infinite were denied, generation and corruption could not endure indefinitely; hence, it would have to be admitted that generation would sometime cease, which is against the opinion of many. Therefore, it is necessary to posit the infinite.
lib. 3 l. 7 n. 5 Quarta ratio sumitur ex apparenti ratione finiti. Videtur enim pluribus quod de ratione finiti sit, quod semper includatur ab aliquo alio: quia videmus apud nos omne finitum extendi usque ad aliquid. Demonstrato igitur aliquo corpore, si illud sit infinitum, habetur propositum; si autem sit finitum, oportebit quod terminetur ad aliquid aliud, et iterum illud, si sit finitum, ad aliquid aliud. Aut ergo erit procedere in infinitum, aut devenietur ad aliquod corpus infinitum; et utroque modo ponitur infinitum. Unde necesse est quod nullus sit terminus corporum, si semper oportet quod omne finitum includatur ab aliquo altero.	340. The fourth reason is based on the apparent nature of the infinite, to many seems to consist in this that it is something always included by something else, because we observe that every finite reaches into something else. Let a body be pointed out; if it be infinite, then the infinite exists; if it be finite, it must be terminated at something else, and this latter, if it in turn be finite, at something else. We must either proceed thus to infinity or come to a body that is infinite. In either case, the infinite exists. Hence there can be no end to bodies, if every finite body is always included by some other.
lib. 3 l. 7 n. 6 Quinta ratio sumitur ab apprehensione intellectus vel imaginationis. Unde dicit quod illud quod maxime facit communem dubitationem inducentem homines ad ponendum infinitum, est ex hoc, quod intellectus nunquam deficit, quin super quodlibet finitum datum possit aliquid addere. Existimabant autem antiqui philosophi quod res responderent apprehensioni intellectus et sensus: unde dicebant quod omne quod videtur, est verum, ut dicitur in IV Metaphys.: et propter hoc credebant quod etiam in rebus esset infinitum. Inde est enim quod videtur numerus esse infinitus: quia intellectus cuilibet numero dato unitatem addendo, facit aliam speciem. Et eadem ratione videntur magnitudines mathematicae, quae in imaginatione consistunt, esse infinitae: quia qualibet magnitudine data, possumus imaginari maiorem. Et eadem ratione videtur esse extra caelum quoddam spatium infinitum: quia possumus imaginari extra caelum in infinitum quasdam dimensiones. Si autem est infinitum spatium extra caelum, necesse videtur quod sit corpus infinitum, et quod sint mundi infiniti. Et hoc duplici ratione. Prima ratio est, quia si consideretur totum spatium infinitum, totum secundum se consideratum est uniforme: non est ergo assignare rationem quare magis in una parte illud spatium sit vacuum a corpore quam in alia. Si ergo in aliqua parte illius spatii invenitur magnitudo corporalis huius mundi, oportet quod in qualibet parte illius spatii inveniatur aliqua magnitudo corporalis sicut quae est huius mundi: et sic oportet corpus esse infinitum sicut et spatium: vel etiam oportet mundos esse infinitos, ut Democritus posuit. Alia ratio est ad idem ostendendum; quia si est infinitum spatium, aut est vacuum aut est plenum. Si est plenum, habetur propositum, quod sit corpus infinitum: si autem est vacuum, cum vacuum nihil aliud sit quam locus non repletus corpore, possibilis tamen repleri, necesse est quod si est spatium infinitum, sit etiam locus infinitus, qui possit repleri corpore. Et ita oportebit esse corpus infinitum, quia in perpetuis non differt contingere et esse. Unde si contingit locum infinitum repleri corpore, oportet dicere quod sit repletus corpore infinito. Necesse ergo videtur dicere quod sit corpus infinitum.	341. The fifth reason is taken from the apprehension of the intellect or of the imagination. Hence, he says that that which chiefly constitutes the common difficulty which induces men to posit the infinite is that the intellect never is exhausted but can always add something to any given finite amount. Now the earlier philosophers supposed that things corresponded to the intellect’s or senses’ apprehension of them; hence because they said that whatever appeared to be is true, as stated in Metaphysics IV (l.11), they believed that even in reality there exists an infinite. Hence number seems to be infinite, because the intellect can always create a new number, simply by adding unity to a given number. For the same reason mathematical magnitudes, which exist in the imagination, seem to be infinite, because, given any definite magnitude, we can imagine a greater. And for the same reason there seems to be an infinite space beyond the heavens, because we can imagine certain dimensions existing beyond the heavens to infinity. Now if there be infinite space beyond the heavens, it seems that there is an infinite body and even infinite worlds. This for two reasons. The first is that if the totality of space be considered infinite, that totality will be uniform; hence, there is no reason why that space should be devoid of body in one part rather than in another, Therefore, if there is found in one part of that space the bodily magnitude of this world, then there should be found in each part of that space some bodily magnitude comparable to that of this world. Thus body must be infinite in the same way as space or there must even exist infinite worlds, as Democritus supposed. Another reason proving the same point is that if there be infinite space, it is either empty or full. If it is full, we have our point that there is infinite body; but if it is empty, then since the empty is a place not filled with a body but capable of being so filled, it follows that if space is infinite, there is infinite place capable of being filled with body. Thus there must be infinite body, because in perpetual matters, there is no difference between what can be and what is. Hence, if infinite place can be filled with body, it must be admitted that it is filled with infinite body. Therefore, it seems necessary to say that there is infinite body.
lib. 3 l. 7 n. 7 Deinde cum dicit: habet autem dubitationem etc., obiicit in contrarium. Et circa hoc tria facit. Primo ostendit quaestionem esse dubitabilem, ne rationes praemissae omnino verum concludere videantur; secundo ostendit quot modis dicitur infinitum, ibi: primum ergo determinandum etc.; tertio ponit rationes ad ostendendum infinitum non esse, ibi: separabile quidem igitur esse et cetera.	342. Then [232 203 b30] he takes the opposite position. And in regard to this he does three things: First, he shows that the matter is debatable, lest anyone suppose that the afore -mentioned reasons are unassailable; Secondly, he gives the various meanings of the word “infinite,” at 344; Thirdly, he gives reasons showing that the infinite does not exist, at 345.
lib. 3 l. 7 n. 8 Dicit ergo primo quod dubitatio est circa infinitum, utrum sit vel non sit: multa enim impossibilia consequuntur iis qui ponunt infinitum omnino non esse, sicut ex praemissis patet; et etiam iis qui ponunt infinitum esse, multa accidunt impossibilia, ut ex consequentibus rationibus patebit. Est etiam dubitatio qualiter infinitum sit, utrum scilicet sit aliquid per se existens, sicut quaedam substantia; vel sicut aliquod accidens per se conveniens alicui naturae; aut neutro modo sit (scilicet neque per se existens, sicut substantia, neque sicut accidens per se), sed nihilominus, si est accidens, est aliquod infinitum continuum, et aliqua infinita secundum multitudinem. Sed maxime pertinet ad considerationem philosophi naturalis, si est aliqua magnitudo sensibilis infinita: nam magnitudo sensibilis est magnitudo naturalis.	343. He says therefore [232 203 b30] that there is a question about whether the infinite exists or not. For, on the one hand, many impossibilities follow upon holding that it does not; those, for example, listed in 337 ff. On the other hand, there are also difficulties attendant upon holding that the infinite does exist, as will be clear subsequently (no. 345 ff). There is doubt also as to its manner of existence. Does it exist as a substance does, or as an accident belonging essentially to some nature? Of if neither as a substance nor as an essential accident, but as an accident nevertheless, is there some infinite continuum and are there things infinite in number? Now it very much pertains to the philosopher of nature to discuss whether there exists such a thing as an infinite sensible magnitude, for a sensible magnitude is a natural magnitude.
lib. 3 l. 7 n. 9 Deinde cum dicit: primum ergo determinandum est etc., ostendit quot modis dicitur infinitum: et ponit duas divisiones infiniti. Quarum prima est communis infinito et omnibus privative dictis. Nam invisibile dicitur tripliciter, vel quod non est aptum natum videri, ut vox, quae non est de genere visibilium; vel quod male videtur, sicut quod videtur in obscuro aut a remotis; vel quod natum est videri et non videtur, sicut quod est omnino in tenebris. Sic igitur et uno modo dicitur infinitum, quod non est natum transiri (nam infinitum idem est quod intransibile): et hoc est quia est de genere intransibilium, sicut indivisibilia ut punctus et forma; per quem etiam modum dicitur vox invisibilis. Alio modo dicitur infinitum, quod quantum est de se, transiri potest, sed eius transitus non potest perfici a nobis, sicut si dicatur profunditas maris esse infinita: vel si potest perfici, tamen vix et cum difficultate, sicut si dicamus quod iter usque in Indiam est infinitum. Et utrumque istorum pertinet ad hoc quod est esse male transibile. Tertio modo dicitur infinitum, quod est natum transiri quasi de genere transibilium existens, quod tamen non habet transitum ad finem; ut si esset aliqua linea non habens terminum, vel quaecumque alia quantitas: et sic proprie dicitur infinitum. Aliam divisionem propriam infiniti ponit ibi: amplius infinitum etc., dicens quod infinitum dicitur vel per appositionem, sicut in numeris; aut secundum divisionem, sicut in magnitudinibus; aut utroque modo, sicut in tempore.	344. Then [233 204 a2] he shows in how many ways “infinite” is said, and lists two divisions of the infinite. The first division is con on to the infinite and to all things said privatively. For “invisible” is said in three ways: either as denoting 1) what of its very nature 13 not apt to be seen, for example, a sound which is not in the genus of visible things; or 2) what is difficult to see, as what is seen in the dark or from a distance; 3) what is apt to be seen but is not, as something in total darkness. Correspondingly, what of its very nature is not apt to be passed over is called “infinite” (for the infinite is the same as that which cannot be passed over)—and this is because it belongs to the genus of intraversable things, as are indivisibles, such as a point and a form; this is the way that a sound was called invisible. In a second way, infinity is ascribed to what could be passed over but its passage is impossible for us; thus, we say that the depth of the sea is infinite; or if it could be passed through, it would be with difficulty, as if we should say that a trip to India is infinite. Both of these belong to that which is “difficultly traversable.” In a third way, infinity is ascribed to what is passable but there is no passage to its terminus; for example, a line without an end or any other such quantity without limits; this is the proper cense of the word “infinite.” He then gives the other division of infinite, [233 bis], saying that infinity is spoken either by addition, as in numbers, or according to division, as in magnitudes, or both ways, as in time.
lib. 3 l. 7 n. 10 Deinde cum dicit: separabile quidem igitur etc., ponit rationes ad excludendum infinitum: et primo ad excludendum infinitum separatum, quod Platonici posuerunt; secundo ad excludendum infinitum a rebus sensibilibus, ibi: rationabiliter quidem igitur et cetera. Circa primum ponit tres rationes. Circa quarum primam dicit quod impossibile est infinitum esse separatum a sensibilibus, ita quod ipsum infinitum sit aliquid per se existens, sicut Platonici posuerunt. Quia si ponitur infinitum esse aliquid separatum, aut habet aliquam quantitatem (scilicet continuam quae est magnitudo, aut discretam quae est multitudo), aut non. Si est substantia sine accidente quod est magnitudo vel multitudo, oportet quod infinitum sit indivisibile: quia omne divisibile vel est numerus vel magnitudo. Si autem aliquid est indivisibile, non erit infinitum nisi primo modo, scilicet prout dicitur aliquid infinitum quod non est aptum natum transiri, sicut dicitur vox invisibilis: sed hoc est praeter intentionem praesentis quaestionis, qua quaerimus de infinito, et praeter intentionem eorum qui posuerunt infinitum; non enim intenderunt ponere infinitum sicut indivisibile, sed sicut intransibile, idest quod natum est transiri et non habet transitum. Si vero infinitum non sit solum substantia, sed etiam habeat accidens quod est magnitudo et multitudo cui competit infinitum, et sic infinitum insit substantiae secundum illud accidens; non erit infinitum inquantum huiusmodi principium eorum quae sunt, sicut antiqui posuerunt; sicut etiam non dicimus invisibile esse principium locutionis, quamvis accidat voci, quae est principium locutionis.	345. Then [234 204 a8] he lays down the arguments leading to an exclusion of the infinite: First those excluding a separated infinite, such as laid down by the Platonists; Secondly, those excluding the infinite from sensible things, at no. 349. With respect to the first he lays down three reasons. As to the first of these he says that it is impossible for the infinite to be separated from sensible things, in such a way that the infinite should be something existing of itself, as the Platonists laid down. For if the infinite is laid down as something separated, either it has a certain quantity (namely, continuous, which is size, or discrete, which is number), or not. If it is a substance without either the accident of size or that of number, then the infinite must be indivisible—since whatever is divisible is either number or size. But if something is indivisible, it will not be infinite except in the first way, namely, as something is called “infinite” which is not by nature susceptible to being passed through, in the same way that a sound is said to be “invisible” [as not being by nature susceptible to being seen], but this is not what is intended in the present inquiry concerning the infinite, nor by those who laid down the infinite. For they did not intend to lay down the infinite as something indivisible, but as something that could not be passed through, i.e., as being susceptible to such, but with the passage having no completion. If, however, the infinite should not only be a substance, but also should have an accident which is size or number to which the infinite belongs, in such a way that the infinite would be inherent in the substance in the manner of that accident, then the principle of existing things will not be infinite as such the ancients laid down, just as we do not say that the principle of speech is invisible, although such a thing is an accident of sound, which is the principle of speech.
lib. 3 l. 7 n. 11 Secundam rationem ponit ibi: amplius quomodo contingit etc.: et est talis. Minus est separabile et per se existens passio quam subiectum; sed infinitum est passio magnitudinis et numeri; sed magnitudo et numerus non possunt separari et per se existere, ut in metaphysica probatum est; ergo neque infinitum.	346. The second reason [235 204 a17] is as follows. A passion is less separable and able to exist of itself than a subject. But the infinite is a passion of size and number—which cannot be separated and exist of themselves, as is proved in the Metaphysics [XI, l.10]. Therefore neither can this be so of the infinite.
lib. 3 l. 7 n. 12 Tertiam rationem ponit ibi: manifestum autem est et cetera. Et dicit manifestum esse quod non potest poni, quod infinitum sit in actu, et quod sit sicut substantia quaedam, et sicut principium rerum. Aut enim infinitum erit partibile, aut impartibile. Si quidem erit partibile, necesse est quod quaelibet pars eius sit infinitum, si infinitum est substantia: quia si infinitum est substantia, et non dicitur de aliquo subiecto ut accidens, oportebit quod idem sit infinitum et infinito esse, idest essentia et ratio infiniti. Non enim idem est id quod est album et natura albi: sed id quod est homo, est hoc quod est natura hominis. Unde oportebit quod si infinitum sit substantia, aut sit indivisibile, aut dividatur in partes infinitas, quod est impossibile; quia ex multis infinitis componi aliquid idem est impossibile, quia oporteret infinitum terminari ad aliud infinitum. Apparet etiam non solum ex ratione sed etiam ex similitudine, quod si infinitum sit substantia et dividatur, oportet quod quaelibet pars eius sit infinita. Sicut enim quaelibet pars aeris est aer, ita et quaelibet pars infiniti erit infinita, si infinitum sit substantia et principium. Quia si sit principium, oportet infinitum esse substantiam simplicem, non compositam ex partibus difformibus, sicut homo, cuius non quaelibet pars est homo. Cum ergo impossibile sit alicuius infiniti quamlibet partem esse infinitam, oportet quod infinitum sit impartibile et indivisibile. Sed illud quod est indivisibile, non potest esse infinitum in actu: quia quod est infinitum in actu est quantum, et omne quantum est divisibile. Sequitur ergo quod si est infinitum in actu, non sit sicut substantia, sed sub ratione accidentis quod est quantitas. Et si hoc sit infinitum, non erit principium, sed illud cui accidit infinitum; sive illud sit aliqua substantia sensibilis, ut aer, sicut posuerunt philosophi naturales; sive sit aliqua substantia intelligibilis, ut par, sicut posuerunt Pythagorici. Unde manifestum est quod inconvenienter dixerunt Pythagorici, ponentes infinitum esse substantiam, et simul cum hoc ponentes ipsum esse divisibile: quia sequitur quod quaelibet pars eius sit infinita; quod est impossibile, ut supra dictum est.	347. The third argument [236 204 a20] is as follows. He [Aristotle] states that it is clear that the infinite cannot be laid down as being in act, and as being a certain substance, and as being the principle of things. For the infinite is either divisible, or indivisible. If indeed it is divisible, every one of its parts will have to be infinite, on the supposition that the infinite is a substance. For if it is a substance, and is not predicated of any subject as an accident, then that which is infinite and the nature of the infinite, i.e., the essence and notion of the infinite, will have to be the same. For that which is white and the nature of white are not the same, but that which is man, and the nature of man, are. Whence it will be necessary that the infinite, if it be a substance, be either indivisible, or divided into parts which are infinite—which is impossible, since it is impossible to compose some same thing out of many infinities, as this would involve one infinite’s being terminated by another infinite. It likewise appears not only from argument but also from an analogy that if the infinite is a substance and is divided, it is necessary that each and every part of it be infinite. For just as every part of air is air, so too every part of the infinite will be infinite, if the infinite is a substance and a principle. For if it is a principle, the infinite has to be a simple substance, not composed out of differing parts, as in the case of man whose every part is not man. Since, therefore, it is impossible for every part of some infinite to be infinite, the infinite must then be unable to be reduced to parts, and indivisible. But what is indivisible cannot be infinite in act—since whatever is infinite in act is quantified, and everything quantified is divisible. It follows, therefore, that if there be any infinite in act, it is not after the manner of substance, but has the reason of the accident which is quantity. And if this be infinite, it will not be a principle, but that to which the infinite occurs, whether it be some sensible substance, such as air; or some intelligible substance, such as “even,” as the Pythagoreans laid down. Whence it is plain that the Pythagoreans did not speak sensibly, positing the infinite as a substance, and the same time holding it as divisible—since it follows that every part of it would be infinite, which is impossible, as said above.
lib. 3 l. 7 n. 13 Ultimo autem dicit quod ista quaestio, quae est: an infinitum sit in mathematicis quantitatibus et in rebus intelligibilibus non habentibus magnitudinem, est magis universalis quam sit praesens consideratio. Nos enim intendimus ad praesens de rebus sensibilibus, de quibus tradimus scientiam naturalem: utrum in ipsis sit corpus infinitum in augmentum, ut antiqui naturales posuerunt.	348. Finally, he says that this question “whether there be an infinite in mathematical quantities and in intelligible things not having magnitude” is a more general one than the present question. For our question concerns sensible things about which natural science treats: “Whether among natural things there be a body infinite in size, such as the early philosophers posited.