Authors/Thomas Aquinas/physics/L3/lect10

Lecture 10 The infinite as existing in potency

Latin	English
	LECTURE 10 The infinite as existing in potency
lib. 3 l. 10 n. 1 Postquam philosophus disputative processit de infinito, hic incipit determinare veritatem. Et primo ostendit an sit infinitum; secundo quid sit, ibi: accidit autem contrarium et cetera. Prima dividitur in duas: in prima ostendit quomodo infinitum sit; in secunda comparat diversa infinita ad invicem, ibi: aliter autem et in tempore et cetera. Circa primum tria facit: primo ostendit quod infinitum quodammodo est, et quodammodo non est; secundo determinat quod est in potentia, et non est sicut actu ens, ibi: dicitur igitur etc.; tertio manifestat quomodo sit in potentia, ibi: non oportet autem potentia ens et cetera.	370. After discussing the infinite dialectically, the Philosopher now begins to determine the truth. First he determines whether there is an infinite; Secondly, what it is, at 382. The first is divided into two parts: In the first, he shows how the infinite exists; In the second, he compares various infinites one to the other, at 374. About the first he does three things: First, he shows that the infinite in a way exists and in a way it does not; Secondly, he shows that it is in potency and is not as a being in act, at 372; Thirdly, he manifests how it is in potency, at 373.
lib. 3 l. 10 n. 2 Dicit ergo primo quod ex praemissis manifestum est, quod non sit aliquod corpus infinitum in actu. Item ex iis quae ante dicta sunt, manifestum est quod si infinitum simpliciter non sit, quod multa impossibilia accidunt. Quorum unum est quod tempus habebit principium et finem: quod reputatur inconveniens secundum ponentes aeternitatem mundi. Et iterum sequetur quod magnitudo non semper sit divisibilis in magnitudines, sed quandoque deveniatur per divisionem magnitudinum ad quaedam quae non sunt magnitudines: sed omnis magnitudo est divisibilis. Item sequetur quod numerus non augeatur in infinitum. Quia igitur secundum determinata neutrum videtur contingere, neque scilicet quod infinitum sit actu, neque quod simpliciter non sit; necesse est dicere quod quodammodo est, quodammodo non est.	371. Accordingly, he says first [246 205 b31] that from the foregoing (ll.8.9) it is manifest that there is no infinite body in act. It is also clear from what has been said (l.7) that if the infinite absolutely does not exist, many impossibilities arise. One is that time will have a beginning and an end, considered impossible by those holding for the eternity of the world. Another is that it would follow that a magnitude would not be always divisible into further magnitudes, but eventually one would arrive through division of magnitudes at certain things which are not magnitudes. But every magnitude is divisible. Likewise, it would follow that number could not increase to infinity. Since therefore, according to what has been said (ll.7-9) neither seems to occur, i.e.,) either an infinite in act or no infinite at all, it must be said that the infinite somehow is and somehow is not.
lib. 3 l. 10 n. 3 Deinde cum dicit: dicitur igitur esse aliud etc., ostendit quod infinitum est sicut potentia ens. Et dicit quod aliquid dicitur esse in actu, et aliquid dicitur esse in potentia. Infinitum autem dicitur esse per appositionem, sicut in numeris, vel per ablationem, sicut in magnitudinibus. Ostensum est enim quod magnitudo non est actu infinita; et sic in magnitudinibus per appositionem infinitum non invenitur, sed per divisionem in eis invenitur infinitum. Non enim est difficile destruere opinionem ponentium indivisibiles esse lineas. Vel, secundum aliam litteram: non est difficile partiri atomos lineas, idest ostendere lineas, quas quidam ponunt indivisibiles, esse partibiles. Dicitur autem infinitum in appositione vel divisione, secundum quod potest apponi vel dividi. Relinquitur igitur quod infinitum sit tanquam in potentia ens.	372. Then [244 205 b18] he shows that the infinite is as a being in potency. And he says that something is said to be in act and something is said to be in potency. Now the infinite is said to come about either by addition, as in numbers, or by subtraction, as in magnitudes. Now it has been shown that magnitude is not infinite in act; hence in magnitudes an infinite through addition is not found, but there is found in them an infinite through division. For it is easy to destroy the opinion that posits lines as indivisibles, or according to another letter, it is easy “to divide indivisible lines,” i.e., to show that lines held indivisible by some, are divisible. Now the infinite, whether in addition or division, is spoken of to the extent of the ability [or potency] to add or divide. It therefore follows that the infinite is as a being in potency.
lib. 3 l. 10 n. 4 Deinde cum dicit: non oportet autem potentia ens etc., ostendit quomodo infinitum sit in potentia. Dupliciter enim invenitur aliquid in potentia. Uno modo sic quod totum potest reduci in actum, sicut possibile est hoc aes esse statuam, quod aliquando erit statua; non autem sic dicitur esse infinitum in potentia, quod postea totum sit in actu. Alio modo aliquid dicitur in potentia esse, quod postea fit actu ens, non quidem totum simul, sed successive. Multipliciter enim dicitur aliquid esse: vel quia totum est simul, ut homo et domus; vel quia semper una pars eius fit post aliam, per quem modum dicitur esse dies et ludus agonalis. Et hoc modo dicitur infinitum esse simul et in potentia et in actu: omnia enim huiusmodi simul sunt in potentia quantum ad unam partem, et in actu quantum ad aliam. Olympia enim, idest festa agonalia quae celebrabantur in monte Olympo, dicuntur esse et durare secundum agones posse fieri et fieri in actu: quia quamdiu durabant ista festa, aliqua pars illorum ludorum erat in fieri, et aliqua erat ut in futurum fienda.	373. Then [256 206 b20] he shows how the infinite exists in potency. For something is found to be in potency in two ways. In one way, in the sense that the whole can be reduced to act, as it is possible for this bronze to be a statue, because at some time it will be a statue. But the infinite in potency is not so meant as that which later will be entirely in act. In another way, something is said to be in potency in such a way that later it will be in act, not, indeed, all at once, but part after part. For there are many ways in which a thing is said to be: 1) because the whole exists at the same time, as in the case of a man or a house; or 2) because one part of it always comes to be after another part, in the way that a day is said to exist and a competition exists. It is in this latter way that the infinite is said to be at once in potency and in act. For all successive things are at once in potency as to one part and in act as to another part. For the Olympic games, i.e., the festive contests held on Mt. Olympus, are said to be and to continue as long as the contests are scheduled and as long as the schedule is being carried out. For as long as those games lasted, one part of the schedule was taking place at the time and another was to take place later.
lib. 3 l. 10 n. 5 Deinde cum dicit: aliter autem et in tempore etc., comparat diversa infinita ad invicem. Et primo comparat infinitum temporis et generationis, infinito quod est in magnitudinibus; secundo comparat infinitum secundum appositionem et infinitum secundum divisionem in magnitudinibus, ibi: quod autem secundum appositionem et cetera. Circa primum tria facit. Primo proponit quod intendit: et dicit quod aliter manifestatur infinitum in generatione hominum et in tempore, et aliter in divisione magnitudinum.	374. Then [251 206 a25] he compares various infinites one to another: First he compares the infinite of time and of generation to the infinite which is in magnitudes; Secondly, he compares the infinite according to addition to the infinite according to division in the case of magnitudes, at 377. In regard to the first he does three things: First, he proposes his intention and says that the infinite in the generation of man, and in time, must be explained in a manner different from that of the infinite in the division of magnitudes.
lib. 3 l. 10 n. 6 Secundo ibi: omnino quidem enim sic est etc., ostendit quid sit commune omnibus infinitis. Et dicit quod hoc omnino et universaliter in omnibus infinitis invenitur, quod infinitum est in semper aliud et aliud accipiendo secundum quandam successionem, ita tamen quod quidquid accipitur in actu de infinito, totum sit finitum. Unde non oportet accipere quod infinitum sit aliquid totum simul existens, sicut hoc aliquid demonstratum, sicut accipimus hominem vel domum; sed sicut sunt successiva, ut dies et ludus agonalis, quorum esse non est hoc modo quod aliquid eorum sit sicut quaedam substantia perfecta tota actu existens. In generatione autem et corruptione, etsi in infinitum procedatur, semper illud quod accipitur in actu, est finitum. In toto enim decursu generationis, etiam si procedatur in infinitum, et omnes homines qui simul actu accipiuntur, sunt finiti secundum numerum, et huiusmodi finitum oportet accipere alterum et alterum, secundum quod quidam homines succedunt quibusdam.	375. Secondly [252 206 a26] he shows what is common to all infinites, saying in all of them it is universally found that the infinite consists in always taking one thing followed by another according to some certain succession, in such away that the whole of whatever is taken, be finite. Hence one must not suppose the infinite to be some whole existing all at once, as a substance that can be pointed out, e.g., a man or a house. Rather the infinite must be taken as in the case of successive things, such as a day or a tournament, whose existence is not that of a perfect substance actually existing as a complete whole all at once. Now, in generation and corruption, even though the process continue to infinity, whatever is taken in act is finite. For in the whole course of generation, even should it proceed to infinity, both all the men existing at a given time are finite in number, and this finite amount must be taken as other and other, accordingly as men succeed one another in time.
lib. 3 l. 10 n. 7 Tertio ibi: sed in magnitudinibus etc., ostendit differentiam. Et dicit quod illud finitum quod accipimus in magnitudinibus, vel apponendo vel dividendo, permanet et non corrumpitur: sed illa finita quae accipiuntur in infinito decursu temporis et generationis humanae corrumpuntur; ita quod per istum modum non contingat tempus et generationem deficere.	376. Thirdly, [253 206 a33] he shows how they differ, saying that the finite actually present in magnitudes as a result of adding or of dividing is permanent and is not corrupted, but the finites considered in the infinite course of time and of human generation are corrupted, although in such a way that time and generation themselves do not fail.
lib. 3 l. 10 n. 8 Deinde cum dicit: quod autem secundum appositionem etc., comparat duo infinita quae sunt in magnitudinibus, scilicet secundum appositionem et secundum divisionem. Et circa hoc tria facit: primo ponit convenientiam inter utrumque infinitum; secundo ostendit differentiam, ibi: non tamen excellit etc.; tertio infert quandam conclusionem ex dictis, ibi: quare excellere et cetera. Dicit ergo primo quod quodammodo infinitum secundum appositionem est idem cum infinito secundum divisionem; quia infinitum secundum appositionem fit e converso cum infinito secundum divisionem. Secundum enim quod aliquid dividitur in infinitum, secundum hoc in infinitum videtur posse apponi ad aliquam determinatam quantitatem.	377. Then [254 206 b3] he compares the two types of infinite which are found in magnitudes; namely, the infinite according to addition and the infinite according to division. About this he does three things: First he shows their points of agreement; Secondly, he shows wherein they differ, at 379; Thirdly, he draws a conclusion from what has been said, at 380. In regard to the first, [254] he says that in some sense the infinite resulting from addition is the same as the one resulting from division, because the former comes to be as a converse of the latter. For it is accordingly as something is divided to infinity, that additions to infinity seem to be able to be made to some determinate quantity.
lib. 3 l. 10 n. 9 Manifestat igitur quomodo sit infinitum divisione in magnitudine. Et dicit quod si aliquis in aliqua magnitudine finita, accepta aliqua parte determinata per divisionem, semper accipiat dividendo alias partes secundum eandem rationem, idest proportionem, sed non secundum eandem quantitatem in eadem proportione, non pertransibit dividendo illud finitum; puta si a linea cubitali accipiat medietatem, et iterum a residuo medietatem; et sic in infinitum procedere potest. Servabitur enim in subtrahendo eadem proportio, sed non eadem quantitas subtracti; minus est enim secundum quantitatem dimidium dimidii quam dimidium totius. Sed si semper sumeret eandem quantitatem, oporteret quod semper magis ac magis augeretur proportio. Puta si a quantitate decem cubitorum subtrahatur unus cubitus, subtractum se habet ad totum in subdecupla proportione: si autem iterum a residuo subtrahatur unus cubitus, subtractum se habebit in maiori proportione; minus enim unus cubitus exceditur a novem quam a decem. Sicut igitur servando eandem proportionem diminuitur quantitas, ita sumendo eandem quantitatem augetur proportio. Si ergo aliquis sic subtrahendo ab aliqua magnitudine finita, semper augeat proportionem sumendo eandem quantitatem, transibit dividendo magnitudinem finitam; puta si a linea centum cubitorum semper subtrahat unum cubitum. Et hoc ideo est, quia omne finitum consumitur quocumque finito semper accepto. Aliter igitur infinitum non est secundum divisionem, nisi in potentia, quod tamen simul est actu cum potentia, sicut dictum est de die et de agone. Et cum infinitum sit semper in potentia, assimilatur materiae, quae est semper in potentia; et non est per se existens in actu totum, sicut finitum est in actu. Et sicut infinitum secundum divisionem est in potentia cum actu simul, similiter dicendum est de infinito secundum appositionem, quod quodammodo est idem cum infinito secundum divisionem, ut dictum est. Inde autem manifestum est quod infinitum per appositionem est in potentia, quia semper contingit aliquid aliud accipere apponendo.	378. He demonstrates, therefore, how the infinite in division exists in magnitude. Thus he states that if someone, in some infinite magnitude, having taken some determinate part by division, should then continue to take other parts by division, always maintaining the same ratio, I.e., proportion, he will not go through that finite magnitude by means of division. For example, from a line of one cubit we may take one half, and from the remainder one-half again. We can proceed in this process to infinity. For the same proportion will be maintained in subtracting, but not the same amount of what is subtracted. The half of the half is less, according to quantity, than the half of the whole. But if we were to take away always the same amount, the proportion taken away would be continually growing. For example, if from a quantity of ten cubits, we take away one cubit, the ratio of the part removed to the original is one-tenth. If we take from the remainder another inch, the ratio between the part removed and that which remains will be in a greater proportion [i.e., one-ninth]. For one cubit is less exceeded by 9 than by 10. Just as, by preserving the same proportion throughout, the quantity subtracted is continually smaller, so, by taking away the same amount each time, the proportion gets continually larger. If, therefore, by so subtracting from some finite magnitude, we continually increase the proportion by taking away the same amount, the original magnitude will be exhausted. For example, if from a line of 10 cubits we always subtract one cubit. This will happen because every finite thing will be exhausted by continually removing the same finite amount. The infinite that depends on division does not exist, therefore, except in potency, but with this potency there exists always something in act, as was said of a day or of a tournament. And since the infinite is always in potency, it is assimilated to matter, which likewise is always in potency; and it never exists in act in its entirety, the way that the finite is in act. And just as the infinite according to division is at once in potency and act, so too is the infinite according to addition, which has been shown to be in some sense the same as the infinite according to division, as was said (no. 377). And the reason why the infinite according to addition is in potency is that it can always grow through addition.
lib. 3 l. 10 n. 10 Deinde cum dicit: non tamen excellit etc., ostendit differentiam inter infinitum secundum appositionem et infinitum secundum divisionem. Et dicit quod infinitum per appositionem non excedit in maius omnem magnitudinem finitam datam; sed infinitum secundum divisionem excedit omnem determinatam parvitatem in minus. Accipiamus enim aliquam determinatam parvitatem, puta unius digiti: si lineam centum cubitorum dividam in infinitum, accipiendo semper dimidium, venietur ad aliquid minus uno digito. Sed apponendo in infinitum, e contrario divisioni, erit dare aliquam quantitatem finitam quae nunquam pertransibitur. Dentur enim duae magnitudines, quarum utraque sit decem cubitorum, et tertia quae sit viginti. Si igitur id quod subtraho in infinitum, accipiendo semper dimidium ab una magnitudine decem cubitorum, addatur alteri quae etiam est decem cubitorum, nunquam pervenietur in infinitum apponendo ad mensuram quantitatis quae est viginti cubitorum: quia quantum remanebit in magnitudine cui subtrahitur, tantum deficiet a data mensura in quantitate cui addetur.	379. Then [255 206 b18] he shows the difference between the infinite according to addition and the infinite according to division. And he says that the former does not exceed any given finite magnitude, whereas the latter diminishes beyond any pre-determined smallness. For If we take any predetermined smallness, for example, the width of a finger, we can, by repeated halving of a line of 10 cubits, arrive at a remainder which is less than the width of a finger. But in adding to infinity, in distinction to division, there will exist some given finite quantity which will never be gone through. Take two magnitudes each of 10 cubits, and a third one of 20 cubits. If what I subtract to infinity from one magnitude of 10 cubits, always taking a half, is added to the other, which is also of 10 cubits, I shall never reach, by adding to infinity, the measure of the quantity of 20 cubits, since as much as remains in the quantity being divided will be lacking from the given measure in the quantity being added to.
lib. 3 l. 10 n. 11 Deinde cum dicit: quare excellere omne etc., inducit conclusionem ex dictis. Et primo inducit eam; secundo manifestat per dictum Platonis, ibi: quoniam et Plato et cetera. Dicit ergo primo quod ex quo appositio in infinitum non facit transcendere omnem determinatam quantitatem, non est possibile esse, nec etiam in potentia, quod excellatur omnis determinata quantitas per appositionem. Quia si esset in natura potentia ad appositionem transcendentem omnem quantitatem, sequeretur quod esset actu infinitum; sic quod infinitum esset accidens alicui naturae, sicut naturales philosophi extra corpus huius mundi quod videmus, ponunt quod est quoddam infinitum, cuius substantia est aer vel aliquid aliud huiusmodi. Si ergo non est possibile esse corpus sensibile actu infinitum, ut ostensum est, sequitur quod non sit potentia in natura ad appositionem transcendentem omnem magnitudinem; sed solum ad appositionem infinitam per contrarium divisioni, ut dictum est. Quare autem si esset potentia ad infinitam additionem transcendentem omnem magnitudinem, sequatur esse corpus infinitum in actu, non autem ad additionem infinitam in numeris, transcendentem omnem numerum, sequatur esse numerum infinitum in actu, infra ostendetur.	380. Then [256 206 b20] he draws a conclusion from the foregoing. First he draws it; secondly he explains it by a saying of Plato, at 381. He says therefore first [256] that since addition to infinity never actually transcends every determined quantity, it is not possible, even in potency, to transcend every determined quantity by addition. For if there were in nature potency for addition transcending every quantity, it would follow that something actually infinite exists; such an infinite would be an accident of some nature, in the same way that the natural philosophers posit, outside the world we see, some sort of infinite, whose substance is air or something similar. But if, as was shown (ll.8,9), no infinite sensible body exists in act, it follows that there is in nature no potency to transcend every magnitude by addition, but only a potency to the infinite addition which is in contrast to [and derived from] division, as was said above (no. 379). Why the existence of a potency to infinite addition transcending every magnitude would imply a body infinite in act, whereas in numbers infinite addition transcending every number does not imply an actually infinite umber will be explained below(in Lecture 12).
lib. 3 l. 10 n. 12 Deinde cum dicit: quoniam et Plato propter hoc etc., manifestat quod dixerat per dictum Platonis. Et dicit quod quia infinitum in appositione magnitudinum est per oppositum divisioni, propter hoc Plato duo fecit infinita, scilicet magnum, quod pertinet ad additionem, et parvum quod pertinet ad divisionem; quia scilicet infinitum videtur excellere et per additionem in augmentum, et per divisionem in decrementum, vel tendendo in nihil. Sed cum ipse Plato faciat duo infinita, non tamen utitur eis: quia cum numerum poneret substantiam esse omnium rerum, in numeris non invenitur infinitum per divisionem, quia in eis est minimum unitas; neque etiam per additionem secundum ipsum, quia dicebat quod species numerorum non variantur nisi usque ad decem, et postea reditur ad unitatem, computando undecim et duodecim et cetera.	381. Then [257 206 b27] he confirms what he has said by a dictum of Plato, saying that because the infinite resulting from the addition of magnitudes is the reverse of division, Plato therefore posited two infinites: “the large,” which pertains to addition, and “the small,” which pertains to division—for the finite seems to excel both by addition unto increase, and by division unto decrease or towards nothing. Yet although Plato makes two infinites, he does not use them. For in number, which he posited to be the substance of all things, there is no infinite by division since there is among them something smallest, which is unity; nor is there according to him an infinite according by addition, since he said that the species of number vary only up to ten and then a return is made to unity when we count eleven, twelve, and so on.