Authors/Thomas Aquinas/physics/L2/lect3

Lecture 3 HOW PHYSICS AND MATHEMATICS DIFFER IN THEIR CONSIDERATION OF THE SAME THING

Latin	English
	LECTURE 3 (193 b 22-194 a 11) HOW PHYSICS AND MATHEMATICS DIFFER IN THEIR CONSIDERATION OF THE SAME THING
lib. 2 l. 3 n. 1Postquam philosophus ostendit quid sit natura et quot modis dicitur, hic consequenter intendit ostendere de quibus considerat scientia naturalis. Et dividitur in partes duas: in prima ostendit quomodo differat naturalis a mathematico; in secunda ostendit ad quae se extendat consideratio scientiae naturalis, ibi: quoniam autem natura et cetera. Circa primum tria facit: primo movet quaestionem; secundo ponit rationes ad quaestionem, ibi: etenim plana etc.; tertio solvit quaestionem, ibi: de his igitur negotiatur et cetera. Dicit ergo primo quod postquam determinatum est quot modis natura dicitur, considerandum est in quo differat mathematicus a naturali philosopho.	157. After the Philosopher has explained what nature is and how many ways the name is used, he here intends to show what it is that natural science considers. This section is divided into two parts. First he shows how natural science differs from mathematics. Secondly, where he says, ‘Since nature has...’ (194 a 12; L4 #166), he designates that to which the consideration of natural science extends. Concerning the first part he makes three points. First he states the question. Secondly, where he says, ‘Obviously physical bodies ...’ (193 b 23 #158), he gives his reasons for [raising] the question. Thirdly, he answers the question where he says, ‘Now the mathematician...’ (193 b 31 #159). He says, therefore, first that after the uses of the name ‘nature’ have been determined, it is necessary to consider how mathematics differs from natural philosophy.
lib. 2 l. 3 n. 2Deinde cum dicit: etenim plana etc., ponit rationes ad quaestionem. Quarum prima talis est. Quaecumque scientiae considerant eadem subiecta, vel sunt eaedem, vel una est pars alterius; sed mathematicus philosophus considerat de punctis, lineis et superficiebus et corporibus, et similiter naturalis (quod probat ex hoc quod corpora naturalia habent plana, idest superficies, et firma, idest soliditates, et longitudines et puncta; oportet autem quod naturalis consideret de omnibus quae insunt corporibus naturalibus); ergo videtur quod scientia naturalis et mathematica vel sint eaedem, vel una sit pars alterius. Secundam rationem ponit ibi: amplius, astrologia et cetera. Et circa hanc rationem primo movet quaestionem, utrum astrologia sit omnino altera a naturali philosophia, vel sit pars eius. Manifestum est enim quod est pars mathematicae: unde si est etiam pars naturalis philosophiae, sequitur quod mathematica et physica conveniunt ad minus in hac parte. Quod autem astrologia sit pars physicae, probat dupliciter. Primo quidem per rationem talem. Ad quemcumque pertinet cognoscere substantias et naturas aliquarum rerum, ad eum etiam pertinet considerare accidentia illarum; sed ad naturalem pertinet considerare naturam et substantiam solis et lunae, cum sint quaedam corpora naturalia; ergo ad eum pertinet etiam considerare per se accidentia ipsorum. Hoc etiam probat ex consuetudine philosophorum: nam philosophi naturales inveniuntur determinasse de figura solis et lunae et terrae et totius mundi, circa quod insudat etiam astrologorum intentio. Sic igitur astrologia et scientia naturalis conveniunt non solum in eisdem subiectis, sed etiam in consideratione eorundem accidentium, et in demonstratione earundem conclusionum. Unde videtur quod astrologia sit pars physicae; et per consequens physica non totaliter differat a mathematica.	158. Next where he says, ‘Obviously physical bodies ...’ (193 b 23), he gives his reasons for [raising] the question. The first of these is as follows. Whenever sciences consider the same subjects, they are either the same science, or one is a part of the other. But the mathematical philosopher considers points and lines and surfaces and bodies, and so does the natural philosopher. (For he proves from the fact that natural bodies have planes, i.e., surfaces, and volumes, i.e., solidity, and lengths and points. Moreover the natural philosopher must consider all things that are in natural bodies.) Therefore it seems that natural science and mathematics are either the same or that one is a part of the other. He gives the second reason where he says, ‘Further, is astronomy...’ (193 b 25). In connection with this reason he raises the question whether astronomy is altogether other than natural philosophy or a part of it. For it is clear that astronomy is a part of mathematics. Whence, if it is also a part of natural philosophy, it follows that mathematics and physics agree at least in this part. That astronomy is a part of physics he proves in two ways. First by the following argument. To whomever it belongs to know the substances and natures of certain things, also belongs the consideration of their accidents. But it belongs to the natural philosopher to consider the nature and substance of the sun and the moon, since they are certain natural bodies. Therefore it belongs to the natural philosopher to consider their per se accidents. He proves this also from the custom of the philosophers. For natural philosophers are found to have treated the shape of the sun and of the moon and of the earth and of the whole world. And these are topics which claim the attention of the astronomers. Therefore astronomy and natural science agree not only in [having] the same subjects but also in the consideration of the same accidents, and in demonstrating the same conclusions. Whence it seems that astronomy is a part of physics, and as a result physics does not differ totally from mathematics.
lib. 2 l. 3 n. 3Deinde cum dicit: de his quidem igitur etc., solvit praemissam quaestionem. Et circa hoc duo facit: primo ponit solutionem; secundo confirmat eam, ibi: fiet autem utique et cetera. Circa primum tria facit: primo solvit quaestionem; secundo concludit quoddam corollarium ex praedictis, ibi: unde et abstrahit etc.; tertio excludit errorem, ibi: latet autem hoc et cetera.	159. Next where he says, ‘Now the mathematician ...’ (193 b 31), he answers the question raised above. Concerning this he makes two points. First he gives his solution, and secondly he confirms it, where he says, ‘This becomes plain...’ (194 a 1 #163). Concerning the first part he makes three points. First he answers the question. Secondly, where he says, ‘That is why he separates ...’ (193 b 33 #161), he concludes to a sort of corollary from the above. Thirdly, where he says, ‘The holders of ...’ (193 b 35 #162), he excludes an error.
lib. 2 l. 3 n. 4Dicit ergo primo quod mathematicus et naturalis determinant de eisdem, scilicet punctis, lineis et superficiebus et huiusmodi, sed non eodem modo. Non enim mathematicus determinat de eis inquantum unumquodque eorum est terminus corporis naturalis; neque considerat ea quae accidunt eis inquantum sunt termini corporis naturalis; per quem modum de eis considerat scientia naturalis. Non est autem inconveniens quod idem cadat sub consideratione diversarum scientiarum secundum diversas considerationes.	160. He says, therefore, first that the mathematician and the natural philosopher treat the same things, i.e., points, and lines, and surfaces, and things of this sort, but not in the same way. For the mathematician does not treat these things insofar as each of them is a boundary of a natural body, nor does he consider those things which belong to them insofar as they are the boundaries of a natural body. But this is the way in which natural science treats them. And, it is not inconsistent that the same thing should fall under the consideration of different sciences according to different points of view.
lib. 2 l. 3 n. 5Deinde cum dicit: unde et abstrahit etc., concludit quoddam corollarium ex praedictis. Quia enim mathematicus considerat lineas et puncta et superficies et huiusmodi et accidentia eorum non inquantum sunt termini corporis naturalis, ideo dicitur abstrahere a materia sensibili et naturali. Et causa quare potest abstrahere, est ista: quia secundum intellectum sunt abstracta a motu. Ad cuius causae evidentiam considerandum est quod multa sunt coniuncta secundum rem, quorum unum non est de intellectu alterius: sicut album et musicum coniunguntur in aliquo subiecto, et tamen unum non est de intellectu alterius, et ideo potest unum separatim intelligi sine alio. Et hoc est unum intellectum esse abstractum ab alio. Manifestum est autem quod posteriora non sunt de intellectu priorum, sed e converso: unde priora possunt intelligi sine posterioribus, et non e converso. Sicut patet quod animal est prius homine, et homo est prius hoc homine (nam homo se habet ex additione ad animal, et hic homo ex additione ad hominem); et propter hoc homo non est de intellectu animalis, nec Socrates de intellectu hominis: unde animal potest intelligi absque homine, et homo absque Socrate et aliis individuis. Et hoc est abstrahere universale a particulari. Similiter autem inter accidentia omnia quae adveniunt substantiae, primo advenit ei quantitas, et deinde qualitates sensibiles et actiones et passiones et motus consequentes sensibiles qualitates. Sic igitur quantitas non claudit in sui intellectu qualitates sensibiles vel passiones vel motus: claudit tamen in sui intellectu substantiam. Potest igitur intelligi quantitas sine materia subiecta motui et qualitatibus sensibilibus, non tamen absque substantia. Et ideo huiusmodi quantitates et quae eis accidunt, sunt secundum intellectum abstracta a motu et a materia sensibili, non autem a materia intelligibili, ut dicitur in VII Metaphys. Quia igitur sic sunt abstracta a motu secundum intellectum, quod non claudunt in suo intellectu materiam sensibilem subiectam motui; ideo mathematicus potest ea abstrahere a materia sensibili. Et nihil differt quantum ad veritatem considerationis, utrum sic vel sic considerentur. Quamvis enim non sint abstracta secundum esse, non tamen mathematici abstrahentes ea secundum intellectum, mentiuntur: quia non asserunt ea esse extra materiam sensibilem (hoc enim esset mendacium), sed considerant de eis absque consideratione materiae sensibilis, quod absque mendacio fieri potest: sicut aliquis potest considerare albedinem absque musica, et vere, licet conveniant in eodem subiecto: non tamen esset vera consideratio, si assereret album non esse musicum.	161. Next where he says, ‘That is why he separates ...’(193 b 33), he concludes to a sort of corollary from what he has just said. Because the mathematician does not consider lines, and points, and surfaces, and things of this sort, and their accidents, insofar as they are the boundaries of a natural body, he is said to abstract from sensible and natural matter. And the reason why he is able to abstract is this: according to the intellect these things are abstracted from motion. As evidence for this reason we must note that many things are joined in the thing, but the understanding of one of them is not derived from the understanding of another. Thus white and musical are joined in the same subject, nevertheless the understanding of one of these is not derived from an understanding of the other. And so one can be separately understood without the other. And this one is understood as abstracted from the other. It is clear, however, that the posterior is not derived from the understanding of the prior, but conversely. Hence the prior can be understood without the posterior, but not conversely. Thus it is clear that animal is prior to man, and man is prior to this man (for man is had by addition to animal, and this man by addition to man). And because of this our understanding of man is not derived from our understanding of animal, nor our understanding of Socrates from our understanding of man. Hence animal can be understood without man, and man without Socrates and other individuals. And this is to abstract the universal from the particular. In like manner, among all the accidents which come to substance, quantity comes first, and then the sensible qualities, and actions and passions, and the motions consequent upon sensible qualities. Therefore quantity does not embrace in its intelligibility the sensible qualities or the passions or the motions. Yet it does include substance in its intelligibility. Therefore quantity can be understood without matter, which is subject to motion, and without sensible qualities, but not without substance. And thus quantities and those things which belong to them are understood as abstracted from motion and sensible matter, but not from intelligible matter, as is said in Metaphysics, VII:10. Since, therefore, the objects of mathematics are abstracted from motion according to the intellect, and since they do not include in their intelligibility sensible matter, which is a subject of motion, the mathematician can abstract them from sensible matter. And it makes no difference as far as the truth is concerned whether they are considered one way or the other. For although the objects of mathematics are not separated according to existence, the mathematicians, in abstracting them according to their understanding, do not lie, because they do not assert that these things exist apart from sensible matter (for this would be a lie). But they consider them without any consideration of sensible matter, which can be done without lying. Thus one can truly consider the white without the musical, even though they exist together in the same subject. But it would not be a true consideration if one were to assert that the white is not musical.
lib. 2 l. 3 n. 6Deinde cum dicit: latet autem hoc facientes etc., excludit ex praedictis errorem Platonis. Quia enim latebat eum quomodo intellectus vere posset abstrahere ea quae non sunt abstracta secundum esse, posuit omnia quae sunt abstracta secundum intellectum, esse abstracta secundum rem. Unde non solum posuit mathematica abstracta, propter hoc quod mathematicus abstrahit a materia sensibili; sed etiam posuit ipsas res naturales abstractas, propter hoc quod naturalis scientia est de universalibus et non de singularibus. Unde posuit hominem esse separatum, et equum et lapidem et alia huiusmodi; quae quidem separata dicebat esse ideas: cum tamen naturalia sint minus abstracta quam mathematica. Mathematica enim sunt omnino abstracta a materia sensibili secundum intellectum, quia materia sensibilis non includitur in intellectu mathematicorum, neque in universali neque in particulari: sed in intellectu specierum naturalium includitur quidem materia sensibilis, sed non materia individualis; in intellectu enim hominis includitur caro et os, sed non haec caro et hoc os.	162. Next where he says, “The holders of the theory...’ (193 b 35), he excludes from what he has said an error of Plato. Since Plato was puzzled as to how the intellect could truly separate those things which were not separated in their existence, he held that all things which are separated in the understanding are separated in the thing. Hence he not only held that mathematical entities are separated, because of the fact that the mathematician abstracts from sensible matter, but he even held that natural things themselves are separated, because of the fact that natural science is of universals and not of singulars. Hence he held that man is separated, and horse, and stone, and other such things. And he said these separated things are ideas, although natural things are less abstract than mathematical entities. For mathematical entities are altogether separated from sensible matter in the understanding, because sensible matter is not included in the understanding of the mathematicals, neither in the universal nor in the particular. But sensible matter is included in the understanding of natural things, whereas individual matter is not. For in the understanding of man flesh and bone is included, but not this flesh and this bone.
lib. 2 l. 3 n. 7Deinde cum dicit: fiet autem utique manifestum etc., manifestat positam solutionem dupliciter: primo quidem per differentiam definitionum quas assignat mathematicus et naturalis; secundo per scientias medias, ibi: demonstrant autem et quae magis et cetera. Dicit ergo primo quod hoc quod dictum est de diverso modo considerationis mathematici et physici, fiet manifestum si quis tentaverit dicere definitiones naturalium et mathematicorum, et accidentium eorum: quia mathematica, ut par et impar, et rectum et curvum, et numerus et linea et figura, definiuntur sine motu et materia; non autem caro et os et homo: sed horum definitio est sicut definitio simi, in cuius definitione ponitur subiectum sensibile, scilicet nasus; non autem sicut definitio curvi, in cuius definitione non ponitur aliquod subiectum sensibile. Et sic ex ipsis definitionibus naturalium et mathematicorum apparet quod supra dictum est de differentia mathematici et naturalis.	163. Next where he says, ‘This becomes plain ...’ (194 a 1), he clarifies the solution he has given in two ways, first by means of the difference in the definitions which the mathematician and the natural philosopher assign, and secondly by means of the intermediate sciences, where he says, ‘Similar evidence ...’ (194 a 7#164). He says, therefore, first that what has been said of the different modes of consideration of the mathematician and the natural philosopher will become evident if one attempts to give definitions of the mathematicals, and of natural things and of their accidents. For the mathematicals, such as equal and unequal, straight and curved, and number, and line, and figure, are defined without motion and matter, but this is not so with flesh and bone and man. Rather the definition of these latter is like the definition of the snub in which definition a sensible subject is placed, i.e., nose. But this is not the case with the definition of the curved in which definition a sensible subject is not placed. And thus from the very definitions of natural things and of the mathematicals, what was said above [#160ff] about the difference between the mathematician and the natural philosopher is apparent.
lib. 2 l. 3 n. 8Deinde cum dicit: demonstrant autem etc., probat idem per scientias quae sunt mediae inter mathematicam et naturalem. Dicuntur autem scientiae mediae, quae accipiunt principia abstracta a scientiis pure mathematicis, et applicant ad materiam sensibilem; sicut perspectiva applicat ad lineam visualem ea quae demonstrantur a geometria circa lineam abstractam; et harmonica, idest musica, applicat ad sonos ea quae arithmeticus considerat circa proportiones numerorum; et astrologia considerationem geometriae et arithmeticae applicat ad caelum et ad partes eius. Huiusmodi autem scientiae, licet sint mediae inter scientiam naturalem et mathematicam, tamen dicuntur hic a philosopho esse magis naturales quam mathematicae, quia unumquodque denominatur et speciem habet a termino: unde, quia harum scientiarum consideratio terminatur ad materiam naturalem, licet per principia mathematica procedant, magis sunt naturales quam mathematicae. Dicit ergo de huiusmodi scientiis, quod contrario modo se habent cum scientiis quae sunt pure mathematicae, sicut geometria vel arithmetica. Nam geometria considerat quidem de linea quae habet esse in materia sensibili, quae est linea naturalis: non tamen considerat de ea inquantum est in materia sensibili, secundum quod est naturalis, sed abstracte, ut dictum est. Sed perspectiva e converso accipit lineam abstractam secundum quod est in consideratione mathematici, et applicat eam ad materiam sensibilem; et sic determinat de ea non inquantum est mathematica, sed inquantum est physica. Ex ipsa ergo differentia scientiarum mediarum ad scientias pure mathematicas, apparet quod supra dictum est. Nam si huiusmodi scientiae mediae abstracta applicant ad materiam sensibilem, manifestum est quod mathematicae e converso ea quae sunt in materia sensibili abstrahunt.	164. Next where he says, ‘Similar evidence...’ (194 a 7), he proves the same thing by means of those sciences which are intermediates between mathematics and natural philosophy. Those sciences are called intermediate sciences which take principles abstracted by the purely mathematical sciences and apply them to sensible matter. For example, perspective applies to the visual line those things which are demonstrated by geometry about the abstracted line; and harmony, that is music, applies to sound those things which arithmetic considers about the proportions of numbers; and astronomy applies the consideration of geometry and arithmetic to the heavens and its parts. However, although sciences of this sort are intermediates between natural science and mathematics, they are here said by the Philosopher to be more natural than mathematical, because each thing is named and takes its species from its terminus. Hence, since the consideration of these sciences is terminated in natural matter, then even though they proceed by mathematical principles, they are more natural than mathematical sciences. He says, therefore, that sciences of this sort are established in a way contrary to the sciences which are purely mathematical, such as geometry or arithmetic. For geometry considers the line which has existence in sensible matter, which is the natural line. But it does not consider it insofar as it is in sensible matter, insofar as it is natural, but abstractly, as was said [#160ff]. But perspective conversely takes the abstract line which is in the consideration of mathematics, and applies it to sensible matter, and thus treats it not insofar as it is a mathematical, but insofar as it is a physical thing. Therefore from this difference between intermediate sciences and the purely mathematical sciences, what was said above is clear. For if intermediate sciences of this sort apply the abstract to sensible matter, it is clear that mathematics conversely separates those things which are in sensible matter.
lib. 2 l. 3 n. 9Et per hoc etiam patet responsio ad id quod supra obiiciebatur de astrologia. Unde astrologia est magis naturalis quam mathematica. Unde non est mirum si communicet in conclusionibus cum scientia naturali. Quia tamen non est pure naturalis, per aliud medium eandem conclusionem demonstrat. Sicut quod terra sit sphaerica demonstratur a naturali per medium naturale, ut puta quia partes eius undique et aequaliter concurrunt ad medium: ab astrologo autem ex figura eclipsis lunaris, vel ex hoc quod non eadem sidera ex omni parte terrae aspiciuntur.	165. And from this it is clear what his answer is to the objection raised above [#158] concerning astronomy. For astronomy is a natural science more than a mathematical science. Hence it is no wonder that astronomy agrees in its conclusions with natural science. However, since it is not a purely natural science, it demonstrates the same conclusion through another method. Thus, the fact that the earth is spherical is demonstrated by natural science by a natural method, e.g., because its parts everywhere and equally come together at the middle. But this is demonstrated by astronomy from the figure of the lunar eclipse, or from the fact that the same stars are not seen from every part of the earth.