Mathematics considers as tf they existed separately objects that do not exist separately (ch. 3).
107717. (C) As the universal propositions in mathematics are about spatial magnitudes and numbers but not about them as such, so there may be proofs about sensible magnitudes “but not about them gwa sensible. As there are reasonings about things merely gwa movable without there being an entity ‘the movable’ either apart from or in sensible things, so there can be reasonings about movable things gua bodies, gua planes, &c.
gi. Thus we can say without qualification that mathematical objects exist, and are such as mathematicians suppose. Each science deals with objects in respect of some particular attribute and not of those incidental to it. Similarly mathematics deals neither with sensible things as such nor with separate non-sensible things, but with the attributes that belong to sensible things gwa involving lines and planes.
1078*9. The simpler the object, the more exact the knowledge ; arithmetic is more accurate than geometry, geometry than kinetics, the kinetics of simple movement than the kinetics of complex movement.
14. Harmonics and optics, again, study their objects not gua voice or sight but gva numbers and lines; so too mechanics. There is no mistake involved in supposing the objects separated from their con- comitants, any more than in the geometer’s supposition that a line is a foot long when it is not.
21. The best procedure is that of arithmetic and geometry --to suppose separate what is not really so. Their objects exist potentially though not actually.
gi. Since the beautiful may be found in unchangeable things though the good is confined to action, they err who hold that mathematics says nothing of the beautiful or the good. Even if it does not mention the beautiful it proves attributes that are the chief forms of beauty— order, symmetry, definiteness.
be, Since these are the causes of many results, mathematics in a sense treats the beautiful as a cause.
107720. % evar Svaiperd. One might have thought that divisi- bility was an essential characteristic of all the objects of mathematics. But it must be remembered that points (# 12) and units (1078 24) are among these objects.
_ 33. Aristotle said in 1, 16 of mathematical objects that otx dds
éorw. Now he Says ott éotw atA@s GAybes cimety. They donot exist in the unqualified or strict sense, but we can say in an unqualified or general way that they exist (that daA@s goes with ciety is indicated by |. 31). daAds, ‘ without qualification’, can mean ‘strictly’ or ‘ vaguely ’ according to the context.
36. The reading of EA ci iyrewov 1d Aevkdv, 4} 8 orw tyrevov, add’ éxeivov 7 éoriv Exdorov, byrewov tyvewod is unintelligible. Alex- ander read ei 76 tyvevov Aevkdv, 4 8 eorw tyrewod, add’ exeivov ob eorlv éxdory, «i tyvewov vyewod, which with Bz.’s emendations, the © reading of # for 7 and the addition of 7 after the second <i, gives a good sense. In * 36 the reading of the manuscripts «i byewov To AevKov could be kept without the sense being much affected, but it seems best to follow Alexander as far as possible throughout the passage.
1078? 1. et (7) byvewvdv Gyvewoi, ‘if of the object gua healthy, then of the healthy ’.
12. The ordinary punctuation, with a full stop after kwioews, is mis- leading. It suggests that kal uddiora dvev Kwyoews introduces a higher degree of precision than dvev peyebovs, as if the geometry of motionless bodies were more precise than arithmetic. Rather kal pddiora avev Kwyoews introduces an independent principle of distinction. ‘ And, while the science is most precise if it deals with unmovables, it is next best that it should study the primary kind of movement, and especially uniform movement of the primary kind.’ It is not clear whether rjv mpotyv is meant to distinguish locomotion from the other kinds of change (A. 1072) 8) or circular motion from other kinds of locomotion (1072> 9). Very likely it is meant to point to both distinctions. Then, just as d&vev peyeGovs rd. places arithmetic above geometry, and pddwora dvev kwycews places geometry above all sciences of motion, cay 0€ Kivyow, padioTa THY mparnv places astronomy above sublunary kinetics, which studies non-circular q@opaé, and still more above, say, biology, which studies avdénows kat pOiors.
15. ypoppat refers to optics, which is subordinate to geometry, GpiOyot to harmonics, which is subordinate to arithmetic (An. Post. 75” x5).
20. Alexander’s reading is sufficiently confirmed by N. 108% 22,
28. todrwy, humanity and indivisibility (cf. 1. 26), Alexander may have read rovrov, indivisibility (739. 14).
16 Suvarév. Alexander takes this as subject of érdpyew and supposes that it is a geometrical attribute which Aristotle states to be capable of belonging to man even if he were not indivisible. Alexander is no doubt thinking of the sense of dvvac6a. which has led to the use of the
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word ‘power’ in its arithmetical meaning. But (1) what is said dvvacGa in this sense is the line on which a square or a cube is erected, so that 8vvardy in this sense is not asuitable epithet for a man, and (2) the subject of ddpxew is undoubtedly &... aird. Bz. takes 7d dwvardy to mean ‘so far as possibility is concerned’, but this with évdéyerau is otiose, and the usage is apparently not found elsewhere. 0 dvvaroy is probably a gloss on iAuas (1, 31), perhaps introduced here by a copyist who took rovrwy to be the antecedent of g@ and thought that trdpyew needed a subject. The words are omitted by I.
30-31. Sitrdv yap... dAukas. Aristotle means that mathematical objects exist éAucds, and this, since it is opposed to évredexela, we may interpret as = duvdue. The meaning must be that mathematical objects exist neither (1) as actually and substantially present all along in sensible things (refuted 10768 38-11), nor (z) as substances actually existing apart from sensible things (refuted 1076 11 —1078%21), but (3) as potentially present in sensible things and receiving actual existence by the geometer’s act of ywpucpds. Cf. Il. 21 ff. and @, 10514 21-33. The mathematical parts into which a body can be divided are its tdy vonry, as its material elements are its vAn aicOyryn (Z. 1035% 12, 10364 9-12, > 32—1037*5).
35. The épya of beauty are the facts in the nature of the universe due to rats and 76 dpurpevoy ( 2-4), i.e. to the striving of nature to attain to order and determinateness. By the Adyo of beauty Aristotle means rats, cupperpia, TO dpirpevov.
The next sentence presents them in another light, as main species (eidn) of beauty. His stricter view is that they are elements in the definition of beauty (cf. Poet, 1450 36 70 Kadov ev peyeOa kai rage éori, ‘involves size and order’), The péyefos which is mentioned in the Poetics and in Pol. 1326* 33 as an element in beauty answers to 16 dpurpévov here; the third element cupperpia is mentioned in Top. 116% 21 as being thought to constitute the beauty of musical tunes,
bi, rdgts, the spatial arrangement of the parts; oupperpta, the proportional size of the parts; 13 dpiopévoy, the limitation in size of the whole. Mathematics does not speak of beauty but it proves that certain objects have these attributes, which are the very soul of beauty.
5. év dAdo, Neither A. 7 (1072* 34), 8, ro, nor N. 4, nor the De Caelo really fulfils the promise here made, and it seems best to treat it as one of Aristotle’s unfulfilled promises.
I]. ‘Tur Forms (chs. 4, 5). Flistory and criticism of the theory of Forms (ch. 4).
1078" 7. We must first examine the doctrine, in its original orm, apart from any theory of numbers.
12, The founders of the doctrine were convinced by Heraclitus arguments that sensible things are always in flux, and inferred that there must be other things to serve as objects of knowledge.
17. Socrates was the first to seek general definitions—viz. of the virtues. Democritus had defined, in a way, heat and cold; the Pythagoreans had reduced the definitions of a few things to numbers.
2g. It was natural that Socrates should seek definitions ; for he was trying to reason, and the ‘what’ is the starting-point of reason- ing; there was at that time no dialectical power such as enables people to study contraries without knowing the ‘what’. Two things we may ascribe to Socrates are inductive arguments and general definition, both concerned with the starting-point of knowledge.
30. Socrates did not treat the universals as existing separately ; his successors did, and called them Ideas; by the same argument they involved themselves in Ideas of all universals.
34. Objections: (i) The theory merely doubles the number of things to be explained; for there is an Idea answering to every set of things with a common name; there is a ‘one over many’ both for substances and for non-substances, both for temporal and for eternal entities.
1079" 4. (ii) Of the proofs of the theory, some prove nothing, others would prove the existence of Ideas of things of which the Platonists think there are none, (a) The arguments from the existence of the sciences would prove that there are Forms of all things of which there
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are sciences. (8) The argument of ‘one over many’ would prove that there are Forms of negations. (y) The argument from the possibility of thinking when the object has perished would prove that there are forms of perishable objects. —
11. (5) Of the most accurate arguments some lead to Ideas of relative terms, others posit the ‘third man’.
14. (iii) In general the arguments about the Forms destroy what the supporters of Forms think more important than the Forms; number becomes prior to the dyad, the relative prior to number and thus to the absolute. In various ways the opinions about the Forms conflict with the first principles of the theory.
1g. (iv) According to the view on which the theory is based there will be Forms of many things besides substances (for there can be a single conception, or a science, of other things); but according to the logical requirements of the theory and their actual opinions, if the Forms are shared in there are Forms only of substances.
26. For (a) each is shared in not as an accident of something else but as something not predicated of a subject (i.e. not as anything that shares in doubleness shares in eternality because doubleness is eternal), so that the Forms must be substance. But (8) the same names must indicate substance in the sensible world as in the ideal (else what is meant by calling the Idea ‘one over many’? Ifthe Ideas and the things that share in them ave che same form, there is something com- mon, for instance, to the Idea of two and the particular two, as there is to the perishable twos and to the particular mathematical twos; if they have not the same form, they have only their name in common, as Callias and a statue may both be called ‘a man’.)
b 3. If it be suggested that the common definition applies to a Form, and only the name of that which it is the Form of has to be added, the suggestion is unmeaning. (a) To what element in the definition is this tobe added? Every element in it isan Idea, genus and differentia alike. (8) ‘Formness’ will itself be a Form present in all Forms, as ‘plane’ is present in all its species. Thus there is an infinite regress.)
1078” 7-8, St. Te dvta éoti Kal THs dvtTa (7a pabyuarixd) has been the general subject of chs. 2, 3; m@s mpdtepa kai ms of mpdtepa the subject in particular of 1077 17-20, 24—-) 11.
11. Who were ot mp@to tas iS€as pyoavtes etvar? A comparison of ll. 12-32 with A. 987%29-58 shows clearly that Aristotle means Plato. The evidence of Aristotle is against the ascription of the ideal theory to Socrates or to the Pythagoreans. It may, of course, be contended that Aristotle had no knowledge of Socrates’ views except what he got from the Platonic dialogues, and that he com-
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pletely misunderstood the dialogues in supposing that the doctrines ascribed to Socrates in them are ascribed to him for dramatic pur- poses. But that the ‘mind of the school’ misunderstood the dialogues so completely is unlikely and demands more proof than has yet been offered.