Authors/Thomas Aquinas/physics/L3/lect6

Lecture 6 Early opinions on the infinite

Latin	English
	LECTURE 6 Early opinions on the infinite
lib. 3 l. 6 n. 1 Postquam philosophus determinavit de motu, hic incipit determinare de infinito. Et primo ostendit quod ad scientiam naturalem pertinet determinare de infinito; secundo incipit determinare, ibi: esse autem infinitum et cetera. Circa primum duo facit: primo ostendit quod ad scientiam naturalem pertinet determinare de infinito; secundo ponit opiniones antiquorum philosophorum de infinito, ibi: et omnes tanquam principium et cetera.	326. After settling motion, the Philosopher now begins to settle the infinite. First he shows that natural science should settle the infinite; Secondly, he begins to determine the infinite, at 336. As to the first he does two things: First he shows that it pertains to natural science to settle the infinite. Secondly, he gives the opinions of the earlier philosophers concerning the infinite, at 329.
lib. 3 l. 6 n. 2 Primum ostendit et ratione et signo. Ratio talis est. Scientia naturalis consistit circa magnitudines et tempus et motum; sed necesse est finitum aut infinitum in his inveniri: omnis enim magnitudo vel motus vel tempus sub altero horum continetur, id est sub finito vel infinito; ergo ad naturalem philosophum pertinet considerare de infinito, an sit et quid sit. Sed quia posset aliquis dicere quod consideratio de infinito pertinet ad philosophum primum ratione suae communitatis, ad hoc excludendum interponit quod non omne ens oportet esse finitum vel infinitum: nam punctus et passio, idest passibilis qualitas, sub nullo horum continetur: ea autem quae pertinent ad considerationem philosophi primi, consequuntur ens inquantum ens est, et non aliquod determinatum genus entis.	327. He proves the first point with an argument and a sign. The argument is as follows: Natural science studies magnitudes and time and motion. But in such things the finite and infinite are necessarily found for every magnitude or motion or time is contained under one or the other, i.e., either the finite or the infinite. Therefore, it pertains to natural science to consider the infinite, namely, as to whether it exists and as to what it is. But because it could be objected that consideration of the infinite pertains to first philosophy, on account of its general character, he counters this by saying that not every being has to be either finite or infinite; for a point and a passion, i.e., passible [sensible] quality, are not contained under either, whereas the objects of consideration in first philosophy are things that follow upon being inasmuch as it is being and not upon some definite genus of being.
lib. 3 l. 6 n. 3 Deinde cum dicit: signum enim quod huius scientiae etc., ostendit idem per signum acceptum a consideratione philosophorum naturalium. Omnes enim qui rationabiliter tractaverunt huiusmodi philosophiam, scilicet naturalem, fecerunt mentionem de infinito. Ex quo colligitur probabile argumentum ab auctoritate sapientum, quod ad philosophiam naturalem pertineat determinare de infinito.	328. Then [225 202 b36] he establishes the same point through a sign taken from the practice of the natural philosophers. For all who have treated this, namely, natural philosophy, according to reason, have mentioned the infinite. This fact is a probable argument, based on the authority of wise men, that it belongs to natural philosophy to settle the infinite.
lib. 3 l. 6 n. 4 Deinde cum dicit: et omnes tanquam principium etc., ponit opiniones antiquorum de infinito. Et primo ostendit in quo diversificabantur; secundo ostendit in quo omnes conveniebant, ibi: rationabiliter autem et cetera. Circa primum duo facit: primo ponit opiniones philosophorum non naturalium de infinito, scilicet Pythagoricorum et Platonicorum; secundo opiniones naturalium, ibi: qui autem de natura omnes et cetera. Circa primum duo facit: primo ostendit in quo conveniebant Pythagorici et Platonici; secundo in quo differebant, ibi: praeter hoc quod Pythagorici et cetera.	329. Then [226 203 a3] he gives the opinion of the earlier philosophers about the infinite. First he shows in what they differ; Secondly, he shows in what they all agreed, at 335. As to the first he does two things: First he gives the opinions on the infinite of those philosophers who were non-natural [i.e., disregarded sense], i.e., the Pythagoreans, and the Platonists; Secondly, he gives the opinions of the natural philosophers, at 333. As to the first he does two things: First he shows the points of agreement between the Pythagoreans and the Platonists; Secondly, their points of disagreement, at 331.
lib. 3 l. 6 n. 5 Dicit ergo primo quod omnes philosophi posuerunt infinitum esse sicut quoddam principium entium; sed hoc fuit proprium Pythagoricis et Platonicis, quod ponerent infinitum non esse accidens alicui alteri naturae, sed esse quoddam per se existens. Et hoc competebat eorum opinioni, quia ponebant numeros et quantitates esse substantias rerum; infinitum autem in quantitate est; unde et infinitum per se existens ponebant.	330. He says therefore that while all the philosophers posited the infinite as a certain principle of things, only the Pythagoreans and Platonists asserted that the infinite is not something accidental to some nature but something existing of itself. This is not surprising, because it is in keeping with their claim that numbers and quantities are the substances of things. Now the infinite is found in quantity; hence they posited that the infinite exists of itself.
lib. 3 l. 6 n. 6 Deinde cum dicit: praeter hoc quod Pythagorici etc., ostendit differentiam inter Platonem et Pythagoricos: et primo quantum ad positionem infiniti; secundo quantum ad radicem ipsius, ibi: et hi quidem infinitum esse et cetera. Quantum autem ad positionem infiniti, in duobus differebat Plato a Pythagoricis. Pythagorici enim non ponebant infinitum nisi in sensibilibus: cum enim infinitum competat quantitati, prima autem quantitas est numerus, Pythagorici non ponebant numerum separatum a sensibilibus, sed dicebant numerum esse substantiam rerum sensibilium; et per consequens neque infinitum erat nisi in sensibilibus. Item Pythagoras considerabat quod sensibilia quae sunt infra caelum, sunt circumclausa caelo, unde in eis non potest esse infinitum: et propter hoc ponebat quod infinitum esset in sensibilibus extra caelum. Sed Plato e contrario ponebat quod nihil est extra caelum: neque enim dicebat esse extra caelum aliquod corpus sensibile, quia caelum dicebat esse continens omnia sensibilia; neque etiam ideas et species rerum, quas ponebat esse separatas, dicebat esse extra caelum, quia intus et extra significant locum; ideae vero secundum ipsum non sunt in aliquo loco, quia locus corporalium est. Item dicebat Plato quod infinitum non solum est in rebus sensibilibus, sed etiam in illis, idest in ideis separatis; quia etiam in ipsis numeris separatis est aliquid formale, ut unum, et aliquid materiale, ut duo, ex quibus omnes numeri componuntur.	331. Then [227 203 a6] he shows the difference between Plato and the Pythagorean, first, as to the laying down of the infinite; secondly, as to the basis thereof (no. 332). Regarding the laying down of the infinite, Plato differed in two respects from the Pythagoreans. For the Pythagoreans did not lay down an infinite except in sensible things. Since the infinite belongs to quantity, and the first quantity is number, the Pythagoreans, not laying down number to be separated from sensible things, but stating number to be, rather, the substance of sensible things, consequently did not lay down any infinite except in sensible things. Likewise Pythagoras considered that the sensible beings which are within the confines of the heavens are circumscribed by the heavens—whence the infinite cannot be in them—hence he laid down that the infinite was in the sensible things outside the heavens, But Plato by contrast laid down that nothing is outside the heavens. For neither did he say that there was outside the heavens any sensible body, since he maintained that the heavens contained all sensible things; nor did he say that the ideas and species of things, which he laid down as being separated, were outside the heavens, since “inside of” and “outside of” signify place, while the ideas, according to him, are not in any place, place being of corporeal things. Plato likewise said that the infinite is not only in sensible things, but also in “them”, i.e., the separated ideas, there being, even in the separated numbers something formal, such as unity, and something material, such as duality, out of which all numbers are composed.
lib. 3 l. 6 n. 7 Deinde cum dicit: et hi quidem infinitum esse parem etc., ostendit differentiam eorum quantum ad radicem infiniti. Et dicit quod Pythagorici attribuebant infinitum uni radici, scilicet numero pari. Et hoc manifestabant dupliciter. Primo per rationem: quia id quod concluditur ab alio et per aliud terminatur, quantum est de se, habet rationem infiniti; quod autem concludit et terminat, habet rationem termini. Par autem numerus comprehenditur et concluditur sub impari. Si enim proponitur aliquis numerus par, undique divisibilis apparet; cum vero addita unitate ad imparem numerum reducitur, iam quandam indivisionem consequitur, ac si par sub impari constringatur: unde videtur quod par sit per se infinitum, et causet in aliis infinitatem. Ostendit etiam idem per signum. Ad cuius evidentiam sciendum est quod in geometricis, gnomon dicitur quadratum super diametrum consistens cum duobus supplementis: huiusmodi igitur gnomon circumpositus quadrato, constituit quadratum. Ex huius ergo similitudine in numeris gnomones dici possunt numeri qui aliquibus numeris adduntur. Est autem hic observandum, quod si aliquis accipiat numeros impares secundum ordinem progressionis naturalis, et unitati, quae est quadratum virtute (inquantum semel unum est unum), addat primum numerum imparem, scilicet ternarium, constituetur quaternarius, qui est numerus quadratus; nam bis duo sunt quatuor. Si vero huic secundo quadrato addatur secundus impar scilicet quinarius, consurgit novenarius, qui est quadratum ternarii; nam ter tria sunt novem. Si autem huic tertio quadrato addatur tertius impar, scilicet septenarius, consurgit sedecim, qui est quadratum quaternarii: et sic semper per ordinatam additionem numerorum imparium resultat eadem forma in numeris, scilicet quadratum. Per additionem autem parium, semper resultat diversa figura. Nam si primus par, scilicet duo, addantur unitati, consurgit ternarius, qui est figurae trilaterae; si autem huic addatur secundus par, scilicet quaternarius, consurgit septenarius, qui est figurae heptagonae: et sic semper variatur figura numerorum ex additione parium. Et hoc videtur esse signum quod uniformitas pertinet ad numerum imparem, difformitas autem et varietas et infinitum pertinent ad numerum parem. Et hoc est quod dicit: signum huius, scilicet quod infinitum sequatur numerum parem, est hoc quod contingit in numeris: circumpositis enim gnomonibus, idest numeris additis, circa unum, idest circa unitatem, et extra, idest circa alios numeros, aliquando quidem fit alia species, idest alia forma numeralis, scilicet per additionem numeri paris; aliquando autem fit una species, scilicet per additionem numeri imparis. Et sic patet quare Pythagoras numero pari attribuerit infinitatem. Plato autem attribuebat duabus radicibus, scilicet magno et parvo: haec enim duo secundum ipsum sunt ex parte materiae, cui competit infinitum.	332. Then [228 203 a10] he shows the difference between them as to the basis of the infinite. And he says that the Pythagoreans attributed the infinite to a basis which was “even number.” And they demonstrated this in two ways. The first was an argument. That which is enclosed by another, and is terminated by another has the nature of the infinite; whereas that which encloses and terminated has the nature of a term. Now even number is comprehended and included under odd number. For if some even number is proposed, it is seen as in every way divisible. But when by the addition of unity it is reduced to an odd number, it now takes on a certain indivisibility, as though even was compressed under odd. Hence it seems as though “even” is infinite in itself, and causes infinity in others. Secondly, the same is shown by an example. To follow it one must know that in geometry a “gnomon” is the name for a square on the diameter with two supplements [i.e., three squares put together to form the shape of an “L”]. If a square is added to this gnomon, a square is constituted. From this likeness those numbers may be called “gnomons” which are added to certain numbers. Here one should notice that if one takes the odd numbers according to the order of natural progression, and to unity, which is a square as to power (since one times one is one), one adds the first odd number, namely, three, there will be constituted four, which is a squared number since twice two is four. If now to this second square there is added the second odd number, namely, five, one obtains nine, which is the square of three, since three times three is nine. Then if to this third square there is added the third odd number, namely, seven, one obtains sixteen, which is the square of four. And thus, following the ordered addition of odd numbers, there always arises the same form in those numbers, namely, a square. By the addition of even numbers, however, there is always produced a different shape. For if the first even number, namely, two, be added to unity, there arises three, which has a triangular figure; if then to this there be added the second even number, namely, four, one has seven, which is in the shape of a heptagon. And thus, in this wise the figure of the resulting numbers constantly varies with the addition of even numbers. And this appears to be a sign that uniformity belongs to odd number, while difformity and variation and the infinite belong to even number. Hence he says, namely, that a sign of this, i.e., that infinity follows even number, is what occurs in numbers. For by the addition of gnomons, i.e., numbers, to one, i.e., to unity, and outside, i.e., to other numbers, sometimes there occurs another species, i.e., another natural form, namely, when one adds an even number; sometimes there occurs a single species, namely, when one adds an odd number. From this it is evident why Pythagoras attributed infinity to even number. But Plato attributed it to two roots, namely, to the “large” and the “small.” Tor these two, according to him, belong to matter, to which in turn the infinite belongs.
lib. 3 l. 6 n. 8 Deinde cum dicit: qui autem de natura etc., ponit opiniones naturalium philosophorum de infinito. Sciendum est ergo quod omnes naturales philosophi, qui scilicet naturaliter principia rerum tradiderunt, dixerunt quod infinitum non est per se subsistens, sicut supra dictum est; sed ponunt infinitum esse accidens alicuius naturae ei suppositae. Qui ergo posuerunt unum principium tantum materiale, quodcumque sit, de numero eorum quae dicuntur elementa, sive aer sive aqua sive aliquid medium, dixerunt illud esse infinitum. Qui vero fecerunt plura elementa sed finita secundum numerum, nullus eorum posuit quod elementa essent infinita secundum quantitatem: ipsa enim distinctio elementorum contrariari videbatur infinitati utriusque eorum. Sed illi qui fecerunt infinita secundum numerum, dicunt ex omnibus illis infinitis fieri quoddam unum infinitum per contactum.	333. Then [229 203 a16] he gives the opinions of the natural philosophers about the infinite. He says that all the natural philosophers, those, namely, who gave natural [i.e., sensible principles for things, taught that the infinite does not subsist by itself, as said above (no. 330), but is an accident of some nature. Hence those who posited just one material principle (some member of the list of things called elements, i.e., air or water or something intermediate) said it was infinite. But of those who posited a finite number of principles, none supposed them to be infinite in quantity: for the very distinction of the elements seemed to conflict with the notion that they could be infinite. But those who posited. an infinitude of principles said that from all those infinites was formed one infinite through contact.
lib. 3 l. 6 n. 9 Et hi fuerunt Anaxagoras et Democritus: qui in duobus differebant. Primo quidem in quidditate principiorum infinitorum: nam Anaxagoras posuit illa infinita principia esse infinitas similes partes, ut carnis et ossis et huiusmodi; Democritus autem posuit huiusmodi infinita principia esse indivisibilia corpora, differentia secundum figuras; quae quidem corpora dicebat esse semina totius naturae. Alia differentia est quantum ad habitudinem horum principiorum ad invicem. Anaxagoras enim dixit quod quaelibet harum partium infinitarum esset commixta cuilibet, sicut quod in qualibet parte carnis esset os et e converso, et similiter de aliis. Et hoc ideo, quia vidit quod quodlibet fit ex quolibet; et cum crederet quod omne quod fit ex aliquo, est in eo, syllogizavit quod quodlibet sit in quolibet. Et ex hoc videtur ipse affirmare quod aliquando omnes res erant simul confusae ad invicem, et nihil erat distinctum ab alio. Sicut enim haec caro et hoc os commiscentur ad invicem, quod demonstratur per generationem eorum ad invicem, sic etiam est de quolibet alio. Omnia igitur aliquando fuerunt simul. Est enim accipere principium disgregationis non solum in aliquo uno, sed in omnibus simul: quod sic probabat. Quod enim fit ex alio, erat prius ei commixtum, et per hoc fit, quod segregatur ab eo; sed omnia fiunt, licet non simul; oportet igitur ponere unum principium generationis omnium, non solum uniuscuiusque. Et hoc unum principium vocavit intellectum, cui soli competit distinguere et congregare, propter hoc quod est immixtus. Quod autem fit per intellectum, videtur habere quoddam principium; quia intellectus a determinato principio incipiens operatur. Si ergo segregatio fit ab intellectu, oportet dicere quod segregatio habeat quoddam principium; unde concludebat quod aliquando omnia fuerint simul, et quod motus quo segregantur res ab invicem, aliquando incoeperit, cum prius non fuerit. Sic igitur Anaxagoras posuit unum principium fieri ex altero. Sed Democritus dicit quod unum principium non fit ex altero: sed tamen natura corporis, quae est communis omnibus indivisibilibus corporibus, differens secundum partes et figuras, est principium omnium secundum magnitudinem, inquantum ex indivisibilibus ponebat componi omnes magnitudines divisibiles. Et sic concludit quod ad philosophum naturalem pertinet considerare de infinito.	334. Those who taught this were Anaxagoras and Democritus, who differed in two respects. They differed first as to the nature of the infinite principles: for Anaxagoras taught that the infinite principles were infinite similar parts of flesh and of bone and so on; but Democritus taught that-they were indivisible bodies differing in figure. He said these bodies were the seeds of all of nature. Another difference was as in the relation of these principles one to the other. For Anaxagoras said that each of these parts was a mixture of all the others, so that in each part of flesh there was bone and vice versa and the same for the other parts. He came to this opinion because he saw that anything came from anything; and, hence, since he believed that whatever comes to be from something is in it, he concluded that everything is in everything. And from this he seems to assert that at some time all things were commingled and nothing was distinct from anything else. Just as this flesh and this bone are commingled (which is proved by the generation of one from the other) so is everything else commingled. Therefore at one time all things were together. For it is necessary to posit a principle of separation not only in one single thing but in all things simultaneously. He proved this thus: Whatever comes to be from something other was previously commingled with it and is produced by being separated from it; but all things are produced, though not all at the same time; therefore, there must be some one principle generating not only each thing but all things. This one principle he called “intellect,” which alone has the capacity to separate and bring together because it is itself uncommingled. Now whatever comes to be through intellect seems to have a principle; because intellect acts by starting from a definite principle. Therefore, if separation is brought about by intellect, separation must have a principle; hence, he concluded, at some time all things were together and the motion by which things were separated one from the other began in time, and did not previously exist. Thus Anaxagoras laid down one principle as producing another. But Democritus said that one principle is not derived from another, but that the nature of body which is common to all indivisible bodies, though different in parts and figure, is the principle of all things according to magnitude, for he posited that all divisible magnitudes are composed of indivisibles. And thus does Aristotle conclude that to consider the infinite pertains to the natural philosopher.
lib. 3 l. 6 n. 10 Deinde cum dicit: rationabiliter autem et principium etc., ponit quatuor, in quibus antiqui philosophi concordabant circa infinitum. Quorum primum est, quod omnes posuerunt infinitum esse principium; et hoc rationabiliter, idest per probabilem rationem. Non enim possibile est, si infinitum est, quod sit frustra, idest quod non habeat aliquem determinatum gradum in entibus. Nec potest habere aliam virtutem nisi principii: quia omnia quae sunt in mundo, vel sunt principia vel ex principiis; infinito autem non competit habere principium, quia quod habet principium, habet finem. Unde relinquitur quod infinitum sit principium. Sed attendendum est quod in hac ratione utuntur aequivoce principio et fine: nam quod est ex principio, habet principium originis; infinito autem repugnat principium et finis quantitatis vel magnitudinis. Secundum autem quod attribuebant infinito est, quod sit ingenitum et incorruptibile. Et hoc sequitur ex eo quod est principium. Omne enim quod fit, necesse est quod accipiat finem, sicut et habet principium; et etiam cuiuslibet corruptionis est aliquis finis: finis autem repugnat infinito; unde esse generabile et corruptibile repugnat infinito. Et sic patet quod non est aliquod principium infiniti, sed magis infinitum est principium aliorum. Et in hoc etiam aequivoce sumebant principium et finem, sicut et supra. Tertium autem quod attribuebant infinito erat, quod contineret et gubernaret omnia: hoc enim videtur esse primi principii. Et hoc dixerunt quicumque non posuerunt praeter materiam, quam dicebant infinitam, alias causas, scilicet agentes, ut intellectum posuit Anaxagoras et concordiam Empedocles. Continere enim et gubernare magis pertinet ad principium agens, quam ad materiam. Quartum autem quod infinito attribuebant est, quod esset quoddam divinum: omne enim quod est immortale aut incorruptibile, divinum appellabant: et hoc posuit Anaximander et plures antiquorum philosophorum naturalium.	335. Then [230 203 b4] he outlines four points of agreement among the early philosophers in regard to the infinite. The first of which is that all posited the infinite as a principle, and this “reasonably” i.e., for the following reason: If the infinite exists, it is impossible for It to be in vain, i.e., that it lack some definite standing among the beings of reality. But it can have no power other than that of a principle. For all things in the world are either principles, or derived from principles. But it is not fitting for the infinite to have a principle, because what has a principle has an end. Hence it follows that the infinite is a principle. Note, however, that in this reasoning, “principle” and “end” are both used equivocally; for that which is derived from a principle has a principle of origin whereas it is to have a principle and end of its quantity or size which is incompatible with infinite. The second point of agreement is that they denied coming into existence and ceasing to exist to the infinite. This follows from the fact that it is a principle. For whatever is produced must have an ending just as it has a principle; and likewise any process of corruption has an end. But “end” and “infinite” are incompatible; hence the infinite can neither be generated nor corrupted. Hence it is clear that the infinite has no principle, but that the infinite is the principle of everything else. This argument, too, uses “principle” and “end” equivocally, as above. The third point of agreement is that they attributed to the infinite the prerogative of containing and governing all things, for this seems to belong to a first principle. And this was the opinion of those who did not grant in addition to matter, which they said was infinite, other causes, namely, agent causes, as Anaxagoras posited an intellect and Empedocles concord. For to contain and to govern pertain more to an active principle than to matter. The fourth point of agreement was to attribute divinity to the infinite; for whatever is immortal or incorruptible they called divine. This was the doctrine of Anaximander and a number of the ancient natural philosophers.