Authors/Thomas Aquinas/physics/L3/lect11

Lecture 11 Definition of the infinite

Latin	English
	LECTURE 11 Definition of the infinite
lib. 3 l. 11 n. 1 Postquam philosophus ostendit quomodo est infinitum, hic ostendit quid sit infinitum. Et circa hoc tria facit: primo ostendit quid sit infinitum; secundo ex hoc assignat rationem eorum quae de infinito dicuntur, ibi: secundum rationem autem accidit etc.; tertio solvit rationes quae supra positae sunt, ibi: reliquum autem est et cetera. Circa primum duo facit: primo ostendit quid sit infinitum, excludens quorundam falsam definitionem; secundo excludit quandam falsam opinionem consequentem ex illa falsa definitione, ibi: quoniam hinc accipiunt dignitatem et cetera. Circa primum tria facit: primo proponit quod intendit; secundo manifestat propositum, ibi: signum autem etc.; tertio infert quandam conclusionem ex dictis, ibi: unde melius est opinandum et cetera.	382. After showing how the infinite exists, the Philosopher now explains what it is. About this he does three things: First he shows what the infinite is; Secondly, from this he assigns the reason for the things said of the infinite, at 390 (l.12). Thirdly, he solves the difficulties mentioned earlier, at 400 (l.13). In regard to the first he does two things: First he shows what the infinite is and rejects the false definition of some; Secondly, he rejects a certain false opinion that follows from the above false definition, at 387. About the first he does three things: First, he proposes what he intends; Secondly, he explains the proposition, at 384; Thirdly, he draws a conclusion, at 386.
lib. 3 l. 11 n. 2 Dicit ergo primo quod contrario modo definiendum est infinitum quam sicut quidam definierunt. Dixerunt enim quidam quod infinitum est extra quod nihil est: sed e contra dicendum est quod infinitum est cuius est semper aliquid extra.	383. He says therefore [258 206 b33] that the infinite must be defined in a manner contrary to the way some have defined it. For some have said that infinite is ‘that outside of which there is nothing,” whereas, to the contrary, it should be defined as “that beyond which there is always something.”
lib. 3 l. 11 n. 3 Deinde cum dicit: signum autem est etc., manifestat propositum. Et primo ostendit quod sua assignatio sit bona; secundo quod assignatio antiquorum sit incompetens, ibi: cuius autem nihil est extra et cetera. Ostendit ergo primo quod infinitum sit cuius semper est aliquid extra, per quoddam signum. Dicunt enim quidam quod annuli sunt infiniti, quia per hoc quod habent quandam circulationem, semper est ibi supponere partem ad partem acceptam. Sed hoc dicitur secundum similitudinem, et non proprie: quia ad hoc quod aliquid sit infinitum, requiritur hoc, scilicet quod extra quamlibet partem acceptam sit quaedam alia; ita tamen quod nunquam resumatur illa quae prius fuit accepta. Sed in circulo non est sic, quia pars quae accipitur post aliam partem, est alia solum ab ea quae immediate accepta est, non tamen ab omnibus partibus prius acceptis; quia una pars potest multoties sumi, ut patet in motu circulari. Si igitur annuli dicuntur infiniti propter hanc similitudinem, sequitur quod illud quod est vere infinitum, sit cuius semper sit accipere aliquid extra, si aliquis velit accipere eius quantitatem. Non enim potest comprehendi quantitas infiniti; sed si quis velit eam accipere, accipiet partem post partem in infinitum, ut supra dictum est.	384. Then [259 207 a2] he explains his proposition. First he shows that his description is good; Secondly, that the description of the earlier philosophers is incompetent, at 385. He shows therefore first by an example that the infinite is “that beyond which there is always something.” For some people say that a ring is infinite since, because it has a circular direction, one can always take part after part. But this is to speak analogously and not properly, because to be infinite requires this, namely, that beyond whatever part is taken there be some other part, in such a way, nevertheless, that one never take again a part taken previously. But in a circle this does not happen, because the part which is counted after another happens to be different from that immediately before it, but not from all the parts previously counted, because one part can be counted any number of times, as is evident in a circular motion. Therefore, if rings are called infinite according to this analogy, it follows that that which is truly infinite is something which always has something beyond, if one were to measure its quantity. For it is impossible to measure the quantity of the infinite; but if someone should desire to reckon it, he would take part after part to infinity, as said above.
lib. 3 l. 11 n. 4 Deinde cum dicit: cuius autem nihil est etc., probat quod definitio antiquorum sit incompetens, tali ratione. Id cuius nihil est extra est definitio perfecti et totius. Quod sic probat. Definitur enim unumquodque totum esse cui nihil deest: sicut dicimus hominem totum aut arcam totam, quibus nihil deest eorum quae debent habere. Et sicut hoc dicimus in aliquo singulari toto, ut est hoc particulare vel illud, ita etiam haec ratio competit in eo quod est vere et proprie totum, scilicet in universo, extra quod simpliciter nihil est. Cum autem aliquid desit per absentiam alicuius intrinseci, tunc non est totum. Sic igitur manifestum est quod haec est definitio totius: totum est cuius nihil est extra. Sed totum et perfectum vel sunt penitus idem, vel sunt propinqua secundum naturam. Et hoc ideo dicit, quia totum non invenitur in simplicibus, quae non habent partes: in quibus tamen utimur nomine perfecti. Per hoc igitur manifestum est quod perfectum est cuius nihil est extra ipsum. Sed nullum carens fine est perfectum; quia finis est perfectio uniuscuiusque. Finis autem est terminus eius cuius est finis: nullum igitur infinitum et interminatum est perfectum. Non ergo competit infinito definitio perfecti, cuius scilicet nihil est extra.	385. Then [260 207 a8] he proves that the definition of the earlier philosophers is since “that outside of which there is nothing” is a definition of the perfect and a whole thing. Here is his proof. Every whole is defined as “that to which nothing to lacking”—as we speak of a whole man, or of a whole box, if they lack nothing which they ought to have. And just as we speak thus in regard to some individual whole, as in the case of this or that particular, so too this notion holds in regard to what is truly and perfectly whole, namely, that outside of which there is absolutely nothing. But when something is lacking through the absence of something intrinsic, then such a thing is not a whole. So it is evident that this is the definition of a whole: “a whole is that nothing of which is outside of it.” But a whole thing and a perfect thing are either entirely the same or of a proximate nature. He says this, because “whole” is not found in simple things which have no parts; in which things, nevertheless, we use the word “perfect.” This shows that the perfect is “that which has nothing of itself outside of it.” But nothing that lacks an end is perfect, because the end is the perfection of each thing. For the end is the term of that of which it is the end. Nothing infinite, therefore, and unterminated, is perfect. Hence the definition of the perfect as that, namely, which has nothing of itself outside itself, does not apply to the infinite.
lib. 3 l. 11 n. 5 Deinde cum dicit: unde melius est opinandum etc., inducit quandam conclusionem ex dictis. Quia enim infinito non competit definitio totius, manifestum est quod melius dixit Parmenides quam Melissus. Melissus enim dixit totum universum esse infinitum; Parmenides vero dixit quod totum finitur per aeque pugnans a medio, in quo designavit corpus universi esse sphaericum. In sphaerica enim figura lineae a medio usque ad terminum, scilicet circumferentiam, ducuntur secundum aequalitatem, quasi aeque pugnantes sibi invicem. Et recte dicitur quod totum universum sit finitum, quia totum et infinitum non se invicem consequuntur quasi sibi continuata, sicut lino continuatur linum in filando. Erat enim proverbium, ut ea quae se consequuntur, dicerentur sibi continuari sicut linum lino.	386. Then [261 207 a15] he draws a conclusion from the foregoing. Since the definition of a “whole” does not apply to the infinite, it is clear that the position of Parmenides is better than that of Melissus. For Melissus said that the whole universe was infinite. Parmenides said the whole is terminated by what is “striving equally from the middle,” by which he designated the body of the universe as spherical. For in a spherical figure, lines from the center to the term, i.e., the circumference, are drawn according to equality, as though “striving equally” with each other. And it is rightly stated that the whole universe is finite, for to be a whole and to be infinite are not reciprocally connected, i.e., not continuous as thread follows thread in spinning. For there was a proverb that things which follow one upon the other should be said to be continuous as thread following thread.
lib. 3 l. 11 n. 6 Deinde cum dicit: quoniam hinc accipiunt etc., excludit quandam falsam opinionem ex praedicta definitione falsa exortam. Et primo communiter quantum ad omnes; secundo specialiter quantum ad Platonem, ibi: quoniam si continet et cetera. Dicit ergo primo quod quia aestimaverunt infinitum coniungi toti, hinc acceperunt quasi dignitatem, idest rem per se notam, de infinito, quod omnia contineret et omnia in se haberet; propter hoc quod habet quandam similitudinem cum toto, sicut id quod est in potentia habet similitudinem cum actu. Infinitum enim inquantum est in potentia, est sicut materia respectu perfectionis magnitudinis: et est sicut totum in potentia, non autem in actu. Quod patet ex hoc, quod infinitum dicitur secundum quod possibile est aliquid dividi in minus, et secundum quod ex opposito divisioni potest fieri appositio, ut supra dictum est. Sic igitur infinitum secundum se, idest secundum propriam rationem, est in potentia totum: et est imperfectum, sicut materia non habens perfectionem. Non autem est totum et finitum secundum se, idest secundum propriam rationem qua est infinitum; sed secundum aliud, idest secundum finem et totum, ad quod est in potentia. Divisio enim quae est possibilis in infinitum, secundum quod ad aliquid terminatur, dicitur esse perfecta: et secundum quod vadit in infinitum, est imperfecta. Et manifestum est quod, cum totius sit continere, materiae autem contineri, quod infinitum inquantum huiusmodi non continet, sed continetur: inquantum scilicet id quod de infinito est in actu, semper continetur ab aliquo maiori, secundum quod possibile est aliquid extra accipere.	387. Then [262 207 a18] he rejects a false opinion that arose from the aforesaid definition, and first in a general way, covering all variations; secondly the opinion of Plato, at 389. He says therefore first that because some thought that whole and infinite were mutually connected, they consequently took it as a “dignity” [axium], i.e., something self-evident, that the infinite contains all things and that it has all things in itself. This was due to the fact that the infinite has a likeness to a whole, as what is in potency has a likeness to act. For the infinite, inasmuch as it is in potency, is as matter in respect to the perfection of magnitude, and it is as a whole in potency, not as a whole in act. This is proved by the fact that the infinite is based on the possibility of dividing things into what is smaller and of making, by a contrasting division, continual additions, as was said above (l.10). Consequently, the infinite in itself, according to its proper nature, is a whole in potency only; and it is something imperfect, comparable to matter not having perfection. For it is not whole and infinite [or finished] according to itself, i.e., according to proper notion by which it is infinite, but according to something other, i.e., according to end and whole, to which it is in potency. For division, which is possible ad infinitum, is called “perfect” insofar as it is, whereas the division that goes on ad infinitum is imperfect. And it is clear, since it is the whole that contains but matter that is contained, that the infinite an such does not contain but is contained. This is true, insofar, namely, as whatever is in act of the infinite is always contained by something greater, accordingly as it is possible to take something beyond.
lib. 3 l. 11 n. 7 Ex hoc autem quod est sicut ens in potentia, non solum hoc sequitur, quod infinitum contineatur et non contineat: sed etiam sequuntur duae aliae conclusiones. Quarum una est, quod infinitum inquantum huiusmodi est ignotum, quia est sicut materia non habens speciem, idest formam, ut dictum est; materia autem non cognoscitur nisi per formam. Alia conclusio est, quae ex eodem sequitur, quod infinitum magis habet rationem partis quam totius, quia materia comparatur ad totum ut pars. Et recte infinitum se habet ut pars, inquantum non est de ipso accipere nisi aliquam partem in actu.	388. Now from the fact that the infinite is as a being in potency, not only does it follow that the infinite is contained and does not contain, but two other conclusions also follow. One is that the infinite, as such, is unknown, because it is as matter without species, i.e., form, and matter is not known except through form. The other conclusion, which has the same source, is that the infinite has more the notion of a part than that of a whole, since matter is compared to the whole as a part. And it is not a surprise that the infinite conducts itself as a part, inasmuch as only a part of it is ever actual.
lib. 3 l. 11 n. 8 Deinde cum dicit: quoniam si continet etc., excludit opinionem Platonis, qui ponebat infinitum tam in sensibilibus quam in intelligibilibus. Et dicit quod ex hoc manifestum est etiam quod, si magnum et parvum, quibus Plato attribuit infinitum, sunt in sensibilibus et in intelligibilibus tanquam continentia (propter hoc quod continere attribuitur infinito); sequitur quod infinitum contineat intelligibilia. Sed hoc videtur esse inconveniens et impossibile, quod infinitum, cum sit ignotum et indeterminatum, contineat et determinet intelligibilia. Non enim determinantur nota per ignota, sed magis e converso.	389. Then [263 207 a28] he rejects an opinion of Plato who posited an infinite both in sensible and in intelligible things. And he states that from this it is plain also that if “the large” and “the small.” to which Plato attributed infinity, are in sensible and intelligible things as containing (by virtue of containment being attributed to the infinite), it follows that the infinite contains the intelligible things. But this seems unfitting and impossible, namely, that the infinite, since it is unknown and undetermined, should contain and determine intelligible things. For the known is not determined by the unknown, but rather the converse is true.