Authors/Thomas Aquinas/physics/L3/lect12

Lecture 12 Explanations in the light of the definition of the infinite

Latin	English
	LECTURE 12 Explanations in the light of the definition of the infinite
lib. 3 l. 12 n. 1 Posita definitione infiniti, hic ex definitione assignata assignat rationem eorum quae de infinito dicuntur. Et primo eius quod dicitur de appositione et divisione infiniti; secundo eius quod infinitum in diversis secundum ordinem invenitur, ibi: infinitum autem non idem est etc.; tertio eius quod mathematici utuntur infinito, ibi: non removet autem ratio etc.; quarto eius quod infinitum ponitur principium, ibi: quoniam autem causae et cetera. Circa primum duo facit: primo assignat rationem eius quod dicitur de infinito, circa divisionem vel appositionem in magnitudinibus; secundo eius quod dicitur in numeris, per comparationem ad magnitudines, ibi: rationabiliter autem est et cetera.	390. After giving a definition of the infinite, the Philosopher now assigns reasons for the things that are said about the infinite. First, the reason for what is said about addition and division of the infinite; Secondly, the reason for saying that the infinite is found in different things according to a certain order, at 397; Thirdly, the reason for saying that mathematicians use the infinite, at 398; Fourthly, the reason why the infinite is called a principle, at 399. About the first he does two things: First he presents the reason for what is said about the infinite in relation to division and addition in magnitudes; Secondly, the reason for what is said of it in numbers by comparison to magnitudes, at 392.
lib. 3 l. 12 n. 2 Dictum est autem supra quod appositio in infinitum sic invenitur in magnitudinibus, quod tamen non exceditur per hoc quaecumque determinata magnitudo. Sed divisio in infinitum sic invenitur in magnitudinibus, quod dividendo transitur quaecumque quantitas in minus, ut supra expositum est. Hoc autem secundum rationem dicit accidere: quia cum infinitum habeat rationem materiae, continetur intus sicut materia: illud autem quod continet, est species et forma. Manifestum est autem ex iis quae dicta sunt in secundo, quod totum habet rationem formae, partes autem rationem materiae. Cum ergo in magnitudinibus a toto itur ad partes per divisionem, rationabile est quod ibi nullus terminus inveniatur, qui non transcendatur per infinitam divisionem. Sed in additione itur a partibus ad totum, quod habet rationem formae continentis et terminantis: unde rationabile est quod sit aliqua determinata quantitas, quam infinita appositio non transcendat.	391. It was said above (no. 379) that addition to infinity in magnitudes takes place in such a way that the resulting magnitude does not become greater than any given magnitude. But division to infinity in magnitudes results in reaching a quantity that is smaller than any pre-assigned quantity, as was expounded above (no. 379). However, he states [264 207 a32] that this occurs reasonably, for since the infinite is like matter it is contained within just as matter is, while that which contains is the species and form. Now it is clear from what was said in Book II (l.5) that the whole is like form and the parts are like matter. Since, therefore, the division of a magnitude proceeds from the whole to the parts, it is reasonable that no limit be found there which is not transcended through infinite division. But the process of addition goes from the parts to the whole, which is like a form that contains and terminates; hence it is reasonable that there be some definite quantity which infinite addition does not exceed.
lib. 3 l. 12 n. 3 Deinde cum dicit: rationabiliter autem est etc., assignat rationem de infinito in numeris, per comparationem ad magnitudines. Dicitur enim quod in numero invenitur aliquis terminus in minus, quem non est dividendo transcendere: sed non invenitur aliquis terminus in plus; quia quolibet numero est invenire alium maiorem per additionem. In magnitudinibus autem est e converso, ut dictum est. Et huius rationem assignat; et primo quidem quare in numeris aliquis terminus invenitur, qui in minus non transcenditur dividendo. Huius autem ratio est, quia omne unum, inquantum unum, est indivisibile, sicut homo indivisibilis est unus homo et non multi. Quemlibet autem numerum oportet resolvere in unum: quod patet ex ipsa ratione numeri. Numerus enim hoc significat, quod sint aliqua plura uno: quaelibet autem plura excedentia unum plus vel minus, sunt determinatae species numerorum. Unde cum unum sit de ratione numeri, et de ratione unius sit indivisibilitas, sequitur quod divisio numeri stet in termino indivisibili. Quod autem dixerat, quod de ratione numeri sit quod sint plura uno, manifestat per species; quia duo et tria et quilibet alius numerus denominatur ab uno. Unde dicitur in V Metaphys. quod substantia senarii est in hoc quod sit sexies unum, non autem in hoc quod sit bis tria vel ter duo: quia sequeretur quod unius rei essent plures definitiones et plures substantiae; quia ex diversis partibus diversimode consurgit unus numerus.	392. Then [267 207 b15] he explains infinity in numbers by comparison to magnitudes. For it was said that in number there is a smallest terminus below which division does not go; but there is no maximum limit which it cannot exceed, because it is possible through addition to exceed any given number. The opposite however takes place in magnitudes, as was said (no. 391). The reason why is because every unity, inasmuch as it is a unity, is indivisible, as indivisible man is one man and not many men. Now every number can be resolved into unity, as is evident from the nature of number. For number signifies that there are more things than one, and any plurality exceeding one to a greater or lesser degree constitutes a definite species of number. Hence, since unity pertains to the notion of number and indivisibility pertains to the notion of unity, It follows that the division of number should halt at an indivisible terminus. This statement that it is of the nature of number to be more than unity he explains by appealing to the species of number, because 2 and 3 and every other number is denominated by unity. Wherefore it is said in Metaphysics that the substance of 6 consists in its being six times one and not two times three, or three times two. Otherwise, it would follow that of the same thing there would be more than one definition and more than one nature, since, starting from different parts, a same number would come about in different ways.
lib. 3 l. 12 n. 4 Deinde cum dicit: in plus autem semper est intelligere etc., assignat causam quare in numeris additio excedit omnem determinatam multitudinem. Et dicit quod possumus semper intelligere quolibet numero dato alium maiorem, per hoc quod magnitudo dividitur in infinitum. Manifestum est enim quod divisio causat multitudinem: unde quanto plus dividitur magnitudo, tanto maior multitudo consurgit; et ideo ad infinitam divisionem magnitudinum sequitur infinita additio numerorum. Et ideo sicut infinita divisio magnitudinis non est in actu sed in potentia, et excedit omne determinatum in minus, ut dictum est; ita additio numerorum infinita non est in actu sed in potentia, et excedit omnem determinatam multitudinem. Sed hic numerus, qui sic in infinitum multiplicatur, non est numerus separatus a decisione magnitudinum.	393. Then [266 207 b10] he gives the reason why in numbers addition exceeds any predetermined multitude. And he says that we can always think of a number greater than any given number, for the reason that magnitude is divided to infinity. For it is plain that division causes multitude; hence the more magnitude is divided the greater is the multitude that results, and upon the infinite division of magnitudes there follows the infinite addition of numbers. Therefore just as infinite division of magnitude is not in act but in potency, and exceeds every determinate quantity in smallness, as was said (nos. 391,392), so the infinite addition of numbers is not in act but in potency, and exceeds every determinate multitude. But this number which is thus multiplied to infinity is not a number independent of the division of magnitudes.
lib. 3 l. 12 n. 5 Circa quod sciendum est quod divisio, ut dictum est, multitudinem causat. Est autem duplex divisio: una formalis, quae est per opposita; et alia secundum quantitatem. Prima autem divisio causat multitudinem, quae est de transcendentibus, secundum quod ens dividitur per unum et multa: sed divisio continuae quantitatis causat numerum, qui est species quantitatis, inquantum habet rationem mensurae. Et hic numerus multiplicabilis est in infinitum, sicut et magnitudo divisibilis est in infinitum: sed multitudo quae sequitur divisionem formalem rerum, non multiplicatur in infinitum; sunt enim determinatae species rerum, sicut et determinata quantitas universi. Et ideo dicit quod hic numerus, qui multiplicatur in infinitum, non separatur a divisione continui. Neque tamen hic numerus sic est infinitus, sicut aliquid permanens, sed sicut semper in fieri existens, inquantum successive additur supra quemlibet numerum datum; sicuti etiam est de tempore et de numero temporis. Numerus enim temporis crescit successive per additionem diei ad diem, non quod omnes dies sint simul.	394. On this point it must be remembered that division, as was stated (no. 393), causes multitude. But division is of two kinds: one is formal, which is through opposites; the other is according to quantity. Now the first division causes that multitude which is a transcendental, accordingly as being is divided into “one” and “many”; but the division of continuous quantity causes number, which is a species of quantity, insofar as it has the notion of measure. And this number can grow to infinity, just as magnitude is divisible to infinity. But the multitude which arises from formal division cannot grow to infinity. For the species of things are determined, just as there is a determined quantity of the universe. That is why he says that the number which grows to infinity is not separated from the division of the continuum. Nor is this number infinite in the sense of something permanent. Rather it is as something always in a state of becoming, inasmuch as, to any given number, additions may be successively made, as is evident in the case of time and the number of time. For the number of time increases successively by the addition of day to day but not all days existed at once.
lib. 3 l. 12 n. 6 Deinde cum dicit: in magnitudinibus autem etc., ostendit quod e contrario est in magnitudinibus. Dividitur enim continuum in infinitum, ut dictum est. Sed in maius non procedit in infinitum etiam secundum potentiam, quia quantum unumquodque est in potentia, tantum potest esse in actu. Si igitur esset in potentia naturae quod cresceret aliqua magnitudo in infinitum, sequeretur quod esset aliqua magnitudo sensibilis infinita; quod est falsum, ut supra dictum est. Relinquitur igitur quod non est in potentia additio magnitudinum in infinitum sic quod excedatur omnis determinata quantitas: quia sequeretur quod esset aliquid maius caelo.	395. Then [267 207 b15] he shows that the opposite occurs in magnitudes. For although a continuum be divided to infinity, as was said (nos. 393,394), the size cannot grow indefinitely even potentially. For as great as a thing is in potency, so great can it be in act. If, therefore, it were in the potency of nature that a magnitude grow to infinity, it would follow that there would actually be some infinite sensible magnitude—which is false, as stated (ll.8.9). The consequence is, therefore, that addition of magnitudes cannot go on to infinity so as to exceed every pre-determined quantity; for otherwise there would be something greater than the heavens.
lib. 3 l. 12 n. 7 Ex quo patet falsum esse, quod quidam dicunt, quod in materia prima est potentia ad omnem quantitatem: non enim est in materia prima potentia nisi ad determinatam quantitatem. Patet etiam ex praemissis ratio quare non oportet numerum tantum esse in actu, quantum est in potentia, sicuti hic dicitur de magnitudine: quia additio numeri sequitur divisionem continui, per quam a toto itur ad id quod est in potentia ad numerum. Unde non oportet devenire ad aliquem actum finientem potentiam. Sed additio magnitudinis ducit in actum, ut dictum est. Commentator autem assignat aliam rationem: quia potentia ad additionem magnitudinis est in una et eadem magnitudine; sed potentia ad additionem numerorum est in diversis numeris, inquantum cuilibet numero potest aliquid addi. Sed haec ratio parum valet, quia sicut per additionem est alia et alia species numeri, ita alia et alia species mensurae, secundum quod bicubitum et tricubitum dicuntur species quantitatis. Et etiam quidquid additur superiori numero, additur inferiori; et secundum hoc in uno et eodem numero, scilicet binario vel ternario, est potentia ad infinitam additionem.	396. From the foregoing it is plain that the claim of some that in prime matter there is a potency to every quantity is false; for in prime matter there is a potency only to determined quantity. It Is plain also from the foregoing why number does not have to be as great in act as it is potentially, as is said here of magnitude: for addition occurs in number as a consequence of the division of the continuum, by which one passes from a whole to what is in potency to number. Hence one need not arrive at some act terminating the potency. But the addition of magnitudes arrives at act, as was said (no. 391). The Commentator [Averroes], however, assigns another reason: namely, that potency to addition in magnitude is in one and the same magnitude but the potency to addition in numbers is in various numbers inasmuch as to any number something can be added. But this reason has little value because just as addition produces varying species of number, so also varying species of measure, as, for example, “two cubits long” and “three cubits long” are called species of quantity. Moreover, whatever is added to a higher number is added to the lower. Accordingly, there is in one and the same number, e.g., two or three, a potency to infinite addition.
lib. 3 l. 12 n. 8 Deinde cum dicit: infinitum autem non idem est etc., ostendit quomodo infinitum inveniatur diversimode in diversis. Et dicit quod infinitum non est secundum eandem rationem in motu et in magnitudine et tempore, ac si esset una natura univoce praedicata de eis: sed dicitur de posteriori eorum secundum prius, sicut de motu propter magnitudinem, in qua est motus, vel localis vel alterationis vel augmenti; de tempore autem propter motum. Et hoc ideo quia infinitum competit quantitati, motus autem est quantus secundum magnitudinem, et tempus propter motum, ut infra patebit. Et ideo dicit quod nunc utimur his, sed posterius manifestabitur de unoquoque eorum quid sit, et quod omnis magnitudo sit divisibilis in magnitudines.	397. Then [268 207 b21] he shows how the infinite is found in diverse ways in diverse things. And he says that the infinite is not found according to the same aspect in motion and magnitude and time, as if it were one nature being predicated univocally in all three cases. Rather it is said of the subsequent member in terms of its antecedent, for example, of motion by reference to the magnitude in which notion takes place (whether it be local motion, alteration or augmentation;) and of time by reference to notion. This happens because the infinite pertains to quantity, and notion is quantified by reference to magnitude, while time is quantified by reference to motion, as will be evident below (Bk. IV, l.17). And therefore he says that we are now mentioning these, but later what each of them is will be explained, as well as that every magnitude is divisible into magnitudes (Bk. VI, l.1).
lib. 3 l. 12 n. 9 Deinde cum dicit: non removet autem ratio mathematicos etc., ostendit quomodo mathematici utuntur infinito. Et dicit quod ratio praedicta, qua ponimus non esse magnitudinem infinitam in actu, non removet considerationem mathematicorum, qui utuntur infinito; puta cum geometra dicit, sit talis linea infinita. Non enim indigent ad suam demonstrationem infinito in actu, neque eo utuntur: sed solum indigent quod sit aliqua linea finita tanta quanta est eis necessaria, ut ex ea possint subtrahere quod volunt. Et ad hoc sufficit quod aliqua maxima magnitudo sit; quia alicui maximae magnitudini competit, quod possit dividi secundum quantamcumque proportionem respectu alterius magnitudinis datae. Unde ad demonstrandum non differt utrum sit hoc modo vel illo, scilicet vel infinita vel finita maxima quantitas. Sed quantum ad esse rei multum differt, utrum sit vel non sit.	398. Then [269 207 b27] he explains how mathematicians make use of the infinite, and says that the argument that there is no actually infinite magnitude,(ll.8,9) does not destroy the consideration of the mathematicians, who use the infinite, as, for example, when the geometer says, “Let this line be infinite.” For they do not need for their demonstrations the infinite in act, nor do they use it, but they need only some finite line of sufficient quantity for their needs, so as to be able to subtract from it so much as they wish. For their purpose it is enough that there exist some maximum magnitude which can be divided according to any proportion in respect to another given magnitude. Hence, for purposes of demonstration, it makes no difference whether this maximum magnitude be one way or the other, i.e., finite or infinite; but as to the being of things, it makes a great difference whether it is one or the other.
lib. 3 l. 12 n. 10 Deinde cum dicit: quoniam autem causae divisae sunt etc., ostendit quomodo infinitum sit principium. Et dicit quod cum sint quatuor genera causarum, ut supra dictum est, patet ex praemissis quod infinitum est causa sicut materia: infinitum enim habet esse in potentia, quod est proprium materiae. Sed materia quidem quandoque est sub forma, quandoque autem sub privatione. Infinito autem non competit ratio materiae secundum quod est sub forma, sed secundum quod est sub privatione: quia scilicet infinitum dicitur per remotionem perfectionis et termini. Et propter hoc subiungit quod ipsi infinito esse est privatio, idest ratio infiniti in privatione consistit. Et ne aliquis intelligat quod infinitum est materia sicut materia prima, subiungit quod per se subiectum privationis, quae constituit rationem infiniti, est continuum sensibile. Et hoc apparet, quia infinitum quod est in numeris causatur ex infinita divisione magnitudinis; et similiter infinitum in tempore et motu causatur ex magnitudine: unde relinquitur quod primum subiectum infiniti sit continuum. Et quia magnitudo secundum esse non est separata a sensibilibus, sequitur quod subiectum infiniti sit sensibile. Et in hoc etiam concordant omnes antiqui, qui utuntur infinito sicut principio materiali. Unde inconveniens fuit quod attribuerunt infinito continere, cum materiae non sit continere, sed magis contineri.	399. Then [270 207 b34] he shows how the infinite is a principle. And he says that since there are four genera of causes, as was said above (Bk. II), the infinite is a cause in the manner of matter. For the infinite has being in potency, which is proper to matter. Now matter is sometimes under a form and sometimes under privation. The infinite, however, has the notion of matter, not insofar as matter lies under a form but inasmuch as matter has privation—for the infinite implies the lack of perfection and term. That is why the Philosopher adds that the being of the infinite is privation, i.e., the notion of the infinite consists in privation. And lest anyone suppose that the infinite is matter like prime matter, he adds that the per se subject of the privation which constitutes the nature of the infinite is the sensible continuum. That this is so is clear from the fact that the infinite found in numbers is caused from the infinite division of magnitude; and similarly, the infinite in time and notion are caused by magnitude. Hence, the first subject of the infinite in the continuum. And since really existing magnitude is not separated from sensible things, it follows that the subject of the infinite is sensible. And on this point all the earlier philosophers agree who use the infinite as a material principle. Wherefore they improperly attributed to the infinite the capacity to contain, for matter does not contain but rather is contained.