Quid sit proprium uniuscuiusque artis.
The Diverse Paths of the Arts
Though all arts share philosophy's goal, each follows its own proper way of considering things — logic through the mind's understanding, mathematics through abstracted reason.
Now although all the arts tend toward a single goal in philosophy, they don't all travel by the same path; instead, each one has its own proper ways of considering things, and it's by these that they differ from one another. The logical way of considering things looks at the mind's understanding of things — whether through the intellect, so that these things neither exist on their own nor do likenesses of them exist, or through reason, so that these things themselves don't exist but their likenesses still do. So then, logic considers the species and genera of things.
Mathematical Abstraction and Reason's Power
Mathematics considers things in a confused way through reason, abstracting lines and quantities from bodily reality, which allows continuous quantities to decrease infinitely and discrete quantities to increase infinitely.
Mathematics, on the other hand, is properly concerned with observing confused activities of things in a confused way through reason. For example, in the actual world of things, you won't find a line without surface and solidity. That's because no body is simply and solely long in such a way that it lacks width or depth — in every body these three dimensions are present together. And yet reason can attend to a line purely in itself, stripped of surface and thickness — which is the mathematical object — not because it actually exists in reality that way or could exist that way, but because reason often considers the activities of things not as they are, but as they can exist: not in themselves, but as far as reason itself is concerned — that is, as reason would be able to exist. It's by this way of considering things that it's been said that a continuous quantity can decrease infinitely, and a discrete quantity can increase infinitely.
Reason's Keen Vitality in Division
Reason's remarkable vitality is shown in its power to divide every length and breadth endlessly, generating interval from interval without any gap of its own.
Such, you see, is reason's keen vitality that it divides every length into lengths, every breadth into breadths, and so on — and, lacking no interval itself, it generates interval from interval.
Physics and the Purity of Natural Activities
Physics examines the mixed activities of bodily things without mixing them, considering the pure activities of fire, earth, air, and water in themselves apart from their concrete embodiment.
Physics, by contrast, is properly concerned with examining the mixed activities of things without mixing them — that is, keeping them distinct in the act of attention. The activities of bodily things in the world are not pure, but are composed from pure activities — and physics considers these purely and in themselves, even though they are not found purely in isolation. It judges, namely, the pure activity of fire, or of earth, or of air, or of water — and from the nature of each considered in itself, apart from the concretion and productive power of the whole.
The Unique Domain of Physics
Physics alone deals with things themselves, whereas logic deals with understandings according to predicamental structure and mathematics according to integral composition, the latter always involving imagination and thus lacking true simplicity.
This too must not be passed over: physics alone deals properly with things themselves, while all the other disciplines deal with our understandings of things. Logic deals with the understandings themselves according to their predicamental structure; mathematics, on the other hand, according to their integral composition. And so logic sometimes uses pure intelligence, whereas mathematics is never without imagination — which is why it has nothing truly simple.
The Order of Learning and the Role of Reason
Because logic and mathematics precede physics in the order of learning and serve it as instruments, they place their inquiry in unshaken reason rather than deceptive experience, so that reason may later guide the descent into empirical study.
Because logic and mathematics come before physics in the order of learning, and in a way serve it in the role of instruments — since everyone must first be informed by them before devoting effort to the study of physics — it was necessary that they place their inquiry not in the activities of things, where experience is deceptive, but in reason alone, where unshaken truth remains. Then, with reason itself going ahead as their guide, they would descend into the experience of things. Now that we've shown how the division of theoretical philosophy that Boethius sets out fits with the earlier discussion, let's briefly run through both divisions side by side, so that we can compare each term of the one with the corresponding term of the other.
Read the original Latin
Cum vero omnes artes ad unum philosophiae tendant terminum, non una tamen via omnes currunt, sed singulae suas proprias quasdam considerationes habent, quibus ab invicem differunt. logica consideratio est in rebus, attendens intellectus rerum, sive per intelligentiam, ut neque sint haec neque horum similitudines, sive per rationem, ut non sint haec sed horum tamen similitudines. considerat ergo logica species et genera rerum. mathematicae autem proprium est actus confusos inconfuse per rationem attendere. verbi gratia, in actu rerum, non invenitur linea sine superficie et soliditate. nullum enim corpus sic solummodo longum est, ut latitudine vel altitudine careat, sed in omni corpore haec tria simul sunt. ratio tamen attendit sine superficie et crassitudine lineam pure per se, quod est mathematicum, non quia in re ita vel sit vel esse possit, sed quia ratio saepe actus rerum considerat, non ut sunt, sed sicut esse possunt, non in se, sed quantum ad ipsam rationem, id est, ut ratio pateretur esse. secundum quam considerationem dictum est continuam quantitatem in infinita decrescere, et discretam crescere in infinitum.
talis est enim vivacitas rationis, ut omne longum in longa dividat, latum in lata, et cetera, utque ipsi rationi nihil carens intervallo intervallum generet. physicae autem est proprium actus rerum permixtos impermixte attendere. actus enim corporum mundi non sunt puri, sed compositi ab actibus purorum, quos physica, cum per se non inveniantur, pure tamen et per se considerat. purum scilicet actum ignis, sive terrae, sive aeris, sive aquae, et ex natura uniuscuiusque per se considerata, de concretione et efficientia totius iudicat. hoc etiam praetereundum non est, quod sola physica proprie de rebus agit, ceterae omnes de intellectibus rerum. logica tractat de ipsis intellectibus secundum praedicamentalem constitutionem; mathematica vero, secundum integralem compositionem, et ideo logica quandoque utitur pura intelligentia, mathematica autem nunquam sine imaginatione est, ideoque nihil vere simplex habet. quia enim logica et mathematica priores sunt ordine discendi quam physica, et ad eam quodammodo instrumenti vice funguntur quibus unumquemque primum informari oportet antequam physicae speculationi operam det, necesse fuit ut non in actibus rerum, ubi fallax experimentum est, sed in sola ratione, ubi inconcussa veritas manet, suam considerationem ponerent, deinde ipsa ratione praevia ad experientiam rerum descenderent. postquam igitur demonstravimus quomodo divisio theoricae, quam ponit Boethius, superiori conveniat, breviter nunc utrasque repetimus, ut singula utriusque verba divisionis invicem conferamus.
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