Soul and mathematics

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UC LLUOÍN west 4 141 I. SOUL AND MATHEMATICALS , see the survey in F. M. Cornford, Plato and Parmenid es (1939) V-IX; 131-134. On Plato's system as presented in Aristotle see eg. W. Aristotle's Metaphysics, 2 vv. (1924), v. 1, p. XLV-LXX I. D. LS When Festa edited Iamblichus,asDe pe eee Isc) in Ross, scientia * (subsequently referred toFirst Prologue ol tue — oticed its similarity to the apparatus pointed out the In its brevity remarkable is E. Lask, Gesammelte Schriften, 3 vv. (1923-1924), v. 111 36-38. On Aristotle’s Platonism see W. Jaeger, Aristoteles? (1955). On Iamblichus, De communi mathematica scientia we find a few lines or words in Zeller 111/25 758-760; T. Whittaker, id ** and in the are going to discuss one ie sca ry on Eucland pote parallels. We = The Neo-Platonists? (1928) 225-228; G. Mau and W. Kroll, art. lamblich aspect of thìs similarity. us in RE: Ueberweg- Praechter!? 612-617; K. Praechter, “Richtungen und Schulen im Neuplatonismus” (see above) 128. Examples of works written with sympathy for excessive realism: H. Bett, Johannes Scotus Erigena (1925), esp, 109-115 ; G. R. G. Mure, Aristotle (1932): G. R. G. Mure, An Introduction to Hegel (1940); N. Hartmann, Aristoteles und Hegel in: Kleinere Schriften , v. II (1957) 214252, esp. 229-244. conceptual realists, ontologists, pas pr a oe ichus and Proclus_are e* l'arithmétique pythagoricienne”, in: Mémoires scientifiques, v. IL (1912) 179-201. e © uming pea i iate2., ass temiXIVne, asp. 52,inte6rmed LImir F; 54, 10-13 F). s nn : g (Ise ch. 2. A particularly penetrating discussion concerni ng the difference between the mysticism of Plato and that of Plotinus can be found in P. Friedlaender, Plato, v.1 (1958) 82-84. According to Friedlaender, in Plato ‚si bein g one with the Supreme object of contemplation, whereas in Plotinus such an identification does take place. But from the Epinomi s, whoever its author, can eYAc a += gta ‘‘trialists’. The intermediate character :ofgf ee = à stressed by Iamblichus ae pe ee a ; ch. ' P Fi ch , p.51, 11 F;Prol$4,. I, 2-19 14-25 E: ch. XIV eer Ge5-5e6,4 . Loe pF; cf. Proclus Jn Excl. , 1-6 F; ch. p. 46, 1-3F XV, p. 55, Fr; 19,12Fr; 35,7 Fr). To the nn 12: 11, 26-122, esponds gl in ri fae character of mathematicals corrdge q >E A character of mathematical knowle (/sc ch. en (991 A; 992 A). Such a learner will after his death have overcome the plurality of sensations and as re yolpxc nereiinpöra póvos wal éx mod Eva yeyovóza, cósaluova + Écecüxt (992 B). How far are we here from the formula pévos rpès uévov or its alternative els mpd Ev (or Evx) ? Admittedly, the Epinomis does not speak of such a ‘unificati on’ in this life, whereas Plotinus does, But clearly the contemplation of the One results in the soul itself becoming what she contemplates, viz. One. Thus, ect! h. XXXIII, p. 95, 5-22 F) ***. In obvious conn A hs pts also the en of being Iamblichustheoacce logy, mathematics, and ph) theoretical philosophy into the orbit of Platonthe object of his contemplation. Cf. also W. Jaeger, “The Greek Ideas of Immortality”, Harvard Theological Review 52 (1959) 135-147, esp. 144f. (in its ascent through knowledge the soul gradually becomes what it knows}, 1 ism, iple”’ diia isting. Below then ligibilia, also fullia yorsubse ot ru A. intel find the sensibilblichus and Proca re we ‚as ius de a rule, Jam I. On the general plan of lamblichus’ work on Pythagor ism see P. Tannery, Pour l'Histoire de la Science Hellène (1887) 372-374 and “Sur contrary to what Friedlaender asserts, there is within lism asserting the subsistence ( s in universainls.lamThu escaland dthew me chus bli sa d above mathemati s we have È ism space for the identification of the contemplator with a Deere h gt Lodo concerns itself witver, a un t at here w should not overlook, howe Appendix easily be seen how close such an identification is to Plato (or a Platonist). It is necessary, says the Epinomis, to espy the One that links all the uabiuata (shy éuonoyiav oboxv uixv &xév-wv). And this link (Beouèc els =2vzwv scil. of arithmeticals, geometricals, harmonicals, astronomi cals — see below, p. 89) will be revealed to the learner who ele Ev Brérov pavOavy Te “ApenCaYtPeiCd nn a= The realist-nominalist controversy I with regard to ee AD BIBLIOGRAPHICAL NOTE the climactic experience does not imply the soul's becomin a deseussiem of Filta SeaPtwa li ... cd. N.N. Festa (1891). | sentiaia liber scient librum ig e Efor Sa umFantaFR=r em Instandprim er(1873).al gia merec.Egri i= pau e ein. Friedl for will Fwi AR G. Friedlein ren = ichi i tica te character of mathematic...als©.greeea oe o. o On the intermedia atonismus””, Genethliakon und Schulen im Neupl (32, More on it later.

INTRODUCTION (Isc ch. XV, p. 55, 8. 23 F; ch. ' XXVIII, p. 88, 19 F; ch. XXX , p. 91, 13. 24 F; ch. XXXI, p. 92, 19 F; 93, 2F). With their realism goes what we could call intuitionism, Mathematicals do not become objects of our knowledge by being abstracted by us from the sensibles in which they are embodied (Ise ch, V, p. 19, 19-20, 20 F; ch. VIII, p. 34,9 F; ch. XXVIII, p. 89,5 F; cf. Proc lus In Eucl. Prol. I, p. 11, 26-1 4, 23 Fr). Rather they are known directly *. Regardless of whet her knowledge of them begins with sensation, this knowledge most certainly does not stem from sens ation (to use Kant's language ). We even may ask whether we could “know” sensibilia at all without our knowledge of math ematicals. But whether or not the mathematicals are prior with regard to us, they are prior by nature (Isc ch. XXXIV, p. are called the object p. 44,7 F). 97, 9 F). Significantly, math ematicals matter of recollection ** (Zse ch. XI, The same holds true for the rel ation between the intelligibles and the mathematicals. The latte r are “derived” from the forme r, not the other way around. And t he intelligibles, too, are objects of direct “intuition” . It is one of the great tasks of mathematics to train the eye of our soul in the Perception of the intelligibles. This task mathematics can fulfil l because its objects too can be “seen” if one trains oneself, whil e the untrained person has an eye that sees only the sensible. lus 13 describe do Iamblichus and Proc ‘ka the three r| a of which the mathematicals are the e ae ji + "a = words most characteristic of the pe e - ch. I I, p. indivisible and divisible (/sc ch. I, p. 10,9 8) ee 14, sù Proclus In Eucl., Prol. I, p- fee isible and the hematicals are a kind of mixture of the indiv Isc. ch. III dives limit and the unlimited, one and many ath il 1 ’ 12,usa 26-13, 9 F; ch. XII, p. 46, 1-6 E). P- Ì of predicates atta is ae taras ches itself“oe= de “unlimit ed” (icen III Pa22-24 F), and “intelligible” and "sensible t are the terms sc cn. di. de "limited | gn ’ (Isc ch. XXXIII, p. 95, 5-6 de f all for a neo-Py thaIt is impossible for any one (and least o la rate r Platonist) to read the description o iate, without a pis other realms between which they media N the = ane of Plato’s Timaeus. Here > A e torments pe ds, the (world) soul is described as being ın aha er a En clus (cf. The Elements of Theology, prop. 6 ome same nd other “realms” *. How, then, could a oneal his commentary a.l.) describe mathematica!S o a, ? d by Plato to describe the world soul? "But the iplicated by problem is even somewhat Sere PORT irely A, ja » high ony Plato the a that in his psychog constitution of wor È ling manner) the a * The mathematical realism of Proclus and — by implicatio n — of lamblichus is presented in N, Hartmann, Des Proklus Diadochus philos ophische A nfangsgruende der Mathematik nach den ersten zwei Buech ern des Euklidkommentars dargestellt (1909); A. Schmekel, Die positive Philosophie in ihrer geschichtlichen Entwicklun g, 2 vv. (1938, 1914), esp. v. I 100-106, see below p, 40; A. Denkweise ? (1945) 57-61: M. Steck, Proklu passim. On the First Prolo gue see Speiser, Die mathematische s Dia dochus ... Kommentar (1945) also P, Tannery, La Géome trie grecq 1-152, ue (1887) 21, Neo-Kantians (like the early Hartm ann) are in sympathy with anti-abstractionism, but not with intuitionism and realism; they are incli ned to interpret intuitioni sm as apriorism (see below P. 77), It is only in Husserl that anti-abstractionism and intuitionism meet again; wheth er this combination implies excessive realism js a matter of controversy. Husse rl himself answered the quest ion in the negative. There is a sense in which abstractio nism and intuitionism are not opposed: see A. Hufn Die intuitive Erkenntnis agel, nach dem hl. Thomas von Aguin (1932) 49 n, 4. ** Archytas fr. 3 Diels reads: Sel Yàp À paBévra nap' Aw Y) adrów ÉEcu pévrx ... Emorápova ofa... Elcup cÜüropov xal etv Sì wh Larodvra &ropov xal oravi fardtov, pr Emoráyevoy Sè Intetv dBuv ov, Larobvra dè atov. The last words are usually translated: “for him who does not know [how] to seek it is impossible to find", It is characteristic that lambl ichus interprets them as meaning: “for him who does not know it is impossible to seek; therefore there must have been a time when we knew — obviously before our birth ” (Isc ch. XI, p. 45, 7 F), In other words, according to Jamblichus, Archytas taugh t the doctrine of anamnesis, | in F. M. ssage can be5 found d eg. € : : y of the Timaeus passa { intermediate a gi “Cosmolosy (1937) 60-66. The soul is a sagra an diversity. Corner’, PES mediate ; identity,iate andbetween in divisible ' an d indiintermediate i ” SUDSIAI bstance),h interim (being, ; ? * three essence Feel n cases, srmanently changing visiDias (NDA ¡ » identical with itself, no thing Is te aout i trulyyiis, no © hing is truly iden i ervthing truly is, every i thing srnally unchanging every every other thing. hing. This from i any other. In the rea Im of the eterna" ‘ervthing truly differs from frati 2 uns vi A or completely chang $ is truly identij cal wi th itself, ievery amplet ely changingi e.n disorder mastered, completely red, though created cosmos of 0 urs is neither e and changelessness — i.e. dis i world sad Sais,: puede. This is due to the presence of the i ger gion ot subdue d com ing). intermediate lm ts ner a of Proclus, as cos ie n . ’ “pretation is € Cornford's interpre is als out. It is also that © , ho { Hermeias, who mediate essence, do : ity and precision with great brevity and i inte ) Picduà Phaedrum Pha (Hermiac Alexandrini in she and atonis a reur , . , cn Opes diversity art ity, and intermediate; ¢ regi made diate identity, en 0600) p.had123,this7-11). ER, from 5}ave interprTheetation eS rapita that both à A Ù Hermeias may ints says:

Certainly this was not overlook by Iamblichus and Proclus. What did they think, whe n they described the math ematicals in the way in which Plato described the soul? How did they explain that entity? Plato's soul so much rese mbles a mathematical Or, to make the problem simp ler: since it is the soul whic h is described by Plato in the Timaeus as being an inte rmediate between the indivisible and the divisible in the realm of body, and since Iamblichus and Proc lus use the same terms to desc ribe the mathematicals, what is the relation between their mathematicals (intermediate) and Plato's soul (intermediate) ? After all, even a philosop her who does not start from a description of the mathematical in terms used by Plato to desc ribe the soul may, simply on read ing the Timacus, ask himself: of what is Plato actually speaki ng ? of the soul? or ofmathematic als? Indeed, we find this pro blem discussed in full in Isc and referred to in Proclus’ comm entaries on Euclid and on the Timaeus *. It is remarkable that Iamblichus in a work devoted to Philosop should discuss this problem hy of mathematics — not to an interpretation of the Timaeu s or to the study of the soul . It is remarkable that precisely the same problem is treated by Proclus in his commentary on the Tima eus, This proves clearly that it is more than a special prob lem. We can safely say: whoe ver within the orbit of Platonis m accepts either the intermed iacy of mathematicals or the intermed iacy of the soul, will have to discuss the relation between the two intermediates **, Let us discuss Tamblichus first. After having described the math ematicals as intermediate (Isc ch. I-II, p. 10, 10-24 F; 11, 3-15 Fi 11,25-12, 2 F), Tamblichus says (ch. III, p. 12, 22-13 ,9 F) that the principles of mathematicals are the limi ted and the unlimited in the form appropriate to mathematical s, these Principles being, in some form, omnipresent in all reality. Incidentally, it is somewhat misleading to use these term s (see Plato, Philebus 24 A: R. G. Bury, The Philebus of Plat o * Procli Diadochi In Platonis Timacum commentaria ed. 15 SOUL AND MATHEMATICALS SOUL AND MATHEM ATICALS E. Diehl, 3 vv, (1903-1906). 27, Qn these and related probl ems cf, L. Robin, La Théor ie platonicienne des Idee; et des Nombres d'aprés Arist ote (1908) 592-595; cf. 203-2 11 and 265 f, ackforth, Plato's Examination of Pleasure { o bee 2: La Index Aristoleheus SV. Te should think, “limit” and “unlimited would be more nr es (Plato Philebus 23C). The limited means obviously: = © idered from its circumference, not from its area or vol ve ce. We shall see later on (p. 38) bi = n of some importan o ing ti I ne we ot also causes of motion [= change]? go back to lamblichus. i 4 of yore pes gir ra ssmathematic s? And wi hy, afte 9F) *. i ea i ilosophy of eis min 2 rs unlimited be consideredWe principles com Es of ‘motion (change)? Whatever the intrinsic reason, according to Iamblichus some made these two principles a ciples of motion (change) — those, namely, who wa ES Ts Gwats existence of these two principles &v 77% buy nat nc Yu the matheo oa Ù das some connection between passage is Li matical and the soul is established. The but at least lamblichus” objection is clear. It is better; “ ve to posit the soul in a different sphere of being and = m et È the mathematical principles and the ne MI ita y line being are unmoved or unchanging (ibid., lines 12-16; es a a interprets the mie E E 7F). . his adversaries. These are (1) that the limited and the ea are principles of the soul; (2) that they [therefore] sa pera sa of motion (change), soul being obviously considered as a pi ati pe of motion (change); (3) that, therefore, the ere bé or contain principles of motion (change). en Fa ie blichus, this implies an identification of the soul with h Ea thematical to the extent that both would belong so pei sphere of being. He criticizes this identification; he ie to keep the two spheres of hina separated and to € x als, mathematic g e) from i i furt her, a word of warnin g must be pa We cannot expect a consistent terminology. W hat ane a calls divisible and indivisible (partible and impartible) an additions. my additi indicate my * Square brackets within a translation or a paraphrase indicate

may call unlimited and limited (or limit); a third, the same different from mathematicals; four or more spheres of being. To answer this question We must resume our analysis of Isc. Ch. IV ended with the assertion that mathematical principles differ from the corporeals by being immaterial; from the HS 16 and the other; a fourth, one and multitude; a fifth, ungenera ted and in the process of generation (or generated) ; a sixth, intelligible (or intellectual) and sensible, and so forth. It is obvious that we must understand the idea, whereupon we can easily see that all these pairs express one and the same dualism, though somewhat different points of view. Once we see this, we perceive clearly that the whole problem discussed by Iamblichus, whether the mathematicals are motive (see below), is connect ed with problems of the interpretation of the Timaeus. One more word of warning. We distinguish plainly between a principle of motion (change) and what is moved (changing). This distinction is not always made in Greek, An orthodox Aristotelian would be careful to distinguish; but not so lamblichus. ‘‘Mathematicals are unmoved (changeless)” often means for him that they are not principles of motion. Thus, whenever we use the adjective “motive” we use it as equivalent to: xuwmrév, xumrixév, Kıvodv, xivoduevoy, i.e, changer , changeable, changing, leaving it to the context to decide which is meant. We can now resume our discussion. Iamblichus says: we had % telligibles by their composite character; and from the principles of the soul by being unmoved. “The principles of 1 i (or to use Iamblichus’ more circumstantial description, the principles which one investigates with regard to life”) is only another expression for soul; and thus the chapter reiterates the doctrine of four different kinds of principles, mathematicals differing from soul. | This seems to wind up the topic concerning the relation between mathematicals and the soul. Ch. V gives a survey of theorems common to all branches of mathematicals and makes it clear that “common” does not mean “abstracted” and in this sense later than the specific theorems but on the contrary designates what is prior to all specific cases. Ch. VI gives a series of excerpts from the Republic and the Epinomis. Ch. VII (identical with In Nicomachi arithmeticam introductionem p.7, 3-9, 23 Pistelli and derived from Nicomachus) contains a discussion of the better assume the soul to have a separate kind of existence. This continuous and the discontinuous and introduces us toa quadrimeans that in addition to the three spheres of being which we partite mathematics (arithmetics, geometry, music, astronomy ; have met so far (and which we meet in Ise time and have to assume a fourth one. Indeed this is stressed III and IV (p. 13, 13-15 F; again) we in chapters p. 18, 13-20 F). These chapters leave us with the impression that instead of a triparti tion we should assume at least a quadripartition of being, Whatever the origin of the problem, the solution certainl y is no longer within the framework of the Timaeus. In the Timaeus there is no place for a fourth sphere of being. Whether the intermediate is interpreted as soul or as a mathematical or as both, there can be no more than one such intermediate. This can be said with confidence. Therefore the question is legitimate: how are we to reconci le the presuppositions of the problem with its solution? These presuppositions are: soul as intermediate; three spheres of being; problem as to the identity of the soul and mathem aticals. They are still well within the problems of the Timaeus . But the solution is: mathematicals not motive; soul in a sphere of being see below p.89). Ch. VIII contains an exposition of Plato's quadripartite line and in connection with | this an antiabstractionist statement as to the way in which we come to know mathematicals, and a quotation from “Brotinos” on the difference between voös and 3ávora together with a commentary on it, finally a quotation from ““Archytas” on the quadripartite line with a commentary on it, None of these topics has anything to do with the relation between mathematicals and the soul. But in ch. IX the problem emerges again. . However the point of view is this time completely different. The problem is not whether the mathematicals are motive; it is with what branch of mathematicals we should identify the soul. Tamblichus says: “Let us discuss first the doctrine held by those who refer mathematics to the soul. .... * * Or, as we could also say, utilizing the summary of this chapter (p. 4, 15-19 FJ: those who reduce the mathematical sphere of being to the soul.

P. 40, 24-25si monics (cf. also Isc ,ch.theIX,ari metic, geometry, har thmetic,= the geometric where three “analogies”,merate p- 41,5-15F, descri = the harmonic, are enu to d,aritand hmeticals, geometricals, an soul It would not be reasonable to posit the soul as being just one branch [kind] of mathematicals ... Therefore the soul should not be defined either as [1] idea of the all-extended [threethe “debt” of the dimensional] or as [2] self-motive number or as [3] mathematical, subsisting harmony [attunement] nor as anything else of this kind specifically, but rather all this should be intertwined together, because the soul is, it is true, idea [form] of the numerable {determinable by number] but it also subsists according to numbers comprising harmony; and all the symmetries occurring in mathematics should be listed as belonging in common to the soul; as a result, then, the soul coexists simultaneously with the geometrical, arithmetical, and harmonical proportion, so that the soul is identical with [all] formulas of analogies [Aéyo: xar' dvañoyiav] ...” (p. 40, 9-41, 1 F). “And, to sum up the whole doctrine, we think that the soul exists in relations common to al! mathematicals .... The concept [definition] of the soul contains the complete fullness of mathematics” (p. 41, 24-42, 6 F). 19 SOUL AND MATHEMATICALS SOUL AND MATHEMATICALS = Jamblichus rejects, then, the identification of the soul with any single branch of mathematicals. Therefore we should not describe the soul as idea (form) of the all-extended (threedimensional). It is clear, and will become even more so, that whoever described it in this way, identified it with geometricals. The word “extended” sufficiently indicates it. We should not describe the soul as a self-moved number. It is clear that he who described it in this way, identified it with arithmeticals. And we should not describe it as subsisting mathematical attunement (harmony). Whoever does so, would identify it with harmonicals atSE iN explae which neewordsd.somSope hon word in our pasIt sag A ias erable (Zp{6pt0<). isa difficult nation: num sis, p. lol Y Hayduel), (In libros Aristotelis de anima paraphra to number (üpdpös: Beasobons d it who copied the passage, changethat lichus intended to Er ipidpod). But it is obvious woulamb ld be a summing “po ha > a description of the soul which According to him, w at three descriptions mentioned by him. is only their one-sidedness; is wrong with these descripteionsby lamblichus is a description mad and the only suggestion Ther must efore, his own descriptionrdan ed. sufficiently many-sid acco ce riptions. The words in"cor contain all three partialngdescattu nt (harmony) respond with numbers comprisi selfneme -moved [self-changing] number with the two descriptions ). Therefore ‘idea (form) and mathematical attunement (harmony to “form ofthe all-extended , nd of the numerable” must correspo espond to “all-extended”. . and “numerable” must corr But dpi6proc does at mean could it mean “extended”? , which is simply “body” = anything else but épOunr6c 8, p. 21, 19-21 Wachsmut h) able (cf. e.g. Moderatus in Stob. I, Pr,quan cal or in other words — geometrical tity or the geometriA x y can designate the geometricall In Latin, too, “numerabilia”(see 151, p. Inst. e.g. Cassiodorus, “stuff” *. (such as the arithmetical, geometrical, and harmonical proextended as subject to number portion). And no similar descriptions are admissible which 21 f.; 152, 1 Mynors). would identify the soul with a special part of mathematicals passage in one of the excerpts from Iamblich instead of making it a compound of all of them, because the soul is an idea (form) of the numerable (see below p. 19), i.e. has a geometrical nature; its existence is number-like, i.e., it has an arithmetical nature; and these numbers contain ratios, i.e., the soul has also a harmonic nature — in short, the soul exists according to relations common to all branches of mathematics; he who says “soul” expresses mathematics in its fullness. And the presupposition is that there are three such branches: arithme- | how How is this possible? ’Apifuros means “numerable”; tary on our However, we also have an excellent commen us’ On the Soul, preserved in Ioannes Stobaeus **. . essence “After this I am going to review those who posit the of the soul as mathematical essence. m h longer pass;age i passage, thehe muc * But even regardless of the detaili s of this the soul unites ichus Jambl to ding accor it obvious that x. eal, 1s F makeofs mathe Em u . maticals. «lf the three aspects el i AT na Ter ige fluess “Ueber , Merlan P. cf. ** On this pb 1936, 909-912, esp. 912; [A. J.] Festugiére, Inlogische M'ochenschrift d'Hermis Trismegiste, v. 111 (1953) 179-182.

Jamblichus in Isc in effect: “Only by combining the three do Now, one kind of mathematicals is [formed by] figure (which we get an adequate description of the soul”. In short, while the identity of the soul and mathematicals was denied in Isc ch.s III and part of IV, this identity is virtually proved in ch. IX.d It is wrong to identify the soul with a mathematical; it shoul is the limit of extension) and by this extension itself. The Platonist Severus defined the soul in these very terms, while Speusippus [defined it] as idea [form] of the all-extended ... [fr. 40 Lang]. Number is still another kind [of mathematicals]. The whole problem is well known to Proclus, too * His solution is rather similar to that of Iamblichus. Of the two passages in which he deals with it, one resembles the passage m lamblichus [defined the soul] as a self-moved [self-changing] number. Moderatus [defined it] as comprising ratios ... Let us further consider attunement (harmony), ... viz. mathe- On the Soul. In his commentary on Euclid Prochis defends (p. 12, 9-18, 4 Fr) the realistic point of view in various ways. He objects particularly to the theory of abstraction. Where does the soul receive its knowledge of mathematicals? Not matical. Moderatus defined the soul by it ...” (lambl. in Stob. I 49, 32, p. 363, 26-364, 20 Wachsmuth). from itself, nor from the voös alone. And it was proved before that it cannot receive it from the sensibles (it 1s remarkable how here the three spheres of being are presupposed). The The similarity of the two Iamblichus passages is palpable. In both the basic question is: with what branch of mathematicals should we identify the soul? With the help of the second passage we can find who identified the soul with arithmeticals alone, only possibility left is: the soul receives it jointly from the vo5z and from itself. After all, the soul is “iconically all that the vole is “paradigmatically”. Therefore Plato is right when he constructs the soul of al! mathematical branches [kinds] with geometricals alone, with harmonicals alone. Let us consider those, says Iamblichus, who think that the essence (substance) of the soul is mathematical. There are three and divides it numerically and binds it by proportions and branches (kinds) of mathematics: arithmetic, geometry, and harmonic, and accordingly we find definitions of the soul in harmonical ratios and places the erstwhile principles of figures init ... and makes the circles in it move in an intellectual motion. Thus, al! mathematicals exist primarily in the soul ... and the terms of arithmetic, geometry, or harmonic. Examples of the first are Xenocrates and Moderatus. The (proportions). Examples of the second are Speusippus and Severus. The former describes the soul as form of the all-extended threedimensional; the latter, as limit of the extended (dimensional). An example of the third is Moderatus again *. In On the Soul these three main mathematical interpretations of the soul are simply reported by Iamblichus. Not so, however, in Isc, We saw that here Iamblichus considered them to be one-sided and wanted to replace them by one expressing the identity of the soul with all the branches of mathematics. “Neither Speusippus, nor Xenocrates, nor Moderatus”, says * It is obvious that in a definition like ‘the soul is number comprising harmony" (or “the soul subsists according to a number which comprises harmony”, etc.) we either can stress the number element or the attunement element. Hence, lamblichus can quote Moderatus twice. | soul is the fullness of all mathematicals ... former speaks of the soul as a self-moving [self-changing] number; as of a number containing ratios . . be identified with the mathematical. Indeed, some Pythagoreans find “number” without any qualification to be a fitting description of the soul. Xenocrates, however, the latter, 21 SOUL AND MATHEMATICALS SOUL AND MATHEMATICALS VAnCarIe,A PaOeNs The soul has its essence ** in these branches of mathematicals ... (p. 16, 16-17, 6 Fr). | u Having made sure that the soul should be identified with all branches of mathematics — i.e. arithmetics, harmonics, geometry, astronomy (on this order and on the emergence of a fourth branch of mathematics see below p. 89), Proclus now adds some words of caution. We quote them, because some of the most characteristic terms reappear in them establishing a closer connection between Proclus and Iamblichus. . “Neither should we take the number as [applied] to it to be a multitude of monads, nor should we interpret [the phrase] ‘idea of the extended’ as [meaning a] body ...” (p. 17, 7-9 Fr). * See in this connection A. Speiser, Die mathematische Denkweise® (1945) 56 f. »» } suggest the verb “to essence” (the soul essences}.

live and intelligent paradi gms of the phenomenal [1] number s, [2] figures, [3] Proporti ons, and [4] motions ...” (p. 17, 9-11 ; 15-21 Fr). “Thus, the mathematical relations constituting the souls in their fulness are essential and self-moved ...” (p. 17, 22-24 Fr). Further and in a differen t context, “the motion peculiar to mathematics is neither local nor motion as cha nge ...; it is lifelike [vital] (p, 18, 2224 Fr). In other words, Proclus insi sts that though the soul has said to be a number [i.e. , an arithmetical], it is not containing a multiplicity otarày [i.e., a geometrical, means both geometrical a number +&v Stxorazov idéa corresponding to 0774 4), it is not a body [by whi ch Proclus body having geometri cal ed . the soul with all branches of by both Iamblichus and Proclus. Both do it with in the frame of a philosophy of mathemati cs; both move well within the orbit of the Timaeus. * On the difference betw een these two kinds of extension as insisted upon Aristotle see below p, by 98. ** On the problem of the motive character of the soul, cf,, for the time being, K. Mras, ‘*Macrobius Kommenta r zu Cicero's Somnium”, SE der Berl. Ak., Philos.-hi KI., 1933, 232-286, esp. 274 st, f, , says: ake it out to be of the monad, as < being ing i indivisible,ue os unlimited dyad, as being divisible; some others, taking the soul to be a geometrical entity, construct it of point and extension, Of the former opinion are men like Aristander and ge and very many other interpreters; of the gt Severus” (In Tim. 35 A, 187 A, v. I I 153, ; 17-25 Diehl). s q. ai ua distinguished between the discontinuou element o element of arithmetic) and the continuous (the With et to the soul both coincide (ovvtpéyer), the pela A try), he sums up by saying: rece mat 4 esomallo, as Eratosthenes did, nor should we define it a of geometrical extension, as Severus did (186 nn u : And Proclus is obviously none too sure that to mak e the soul motive (via mathemati cals as motive) is strictly Platonic **, In a different context he notices that sometimes Plat o seems to mak Thus, the identification of Tim. ae 25 Diehl) as the soul being partly somatic, partly to be mathematical, because the soul is intermediat physicals and superphysicals, some say sn catari bier! . ion in connection with harm onicals see below p. 29). Though the soul has been describe d in terms of motion (i.e., as an astr onomical, having been desi gnated as containing a plurality of circ les) this motion, being mat hematical, is neither spatial motion nor change. hematics — the pleromat ic character of the soul — is stressed ivi ;+ cf. c In ivi and undivided n the divided soul ne predecessors who make out the essence ofe the inn ratios [i.e. sometimes he lets the soul seal o nati nature of the soul and thematical ~| and are self-moved (on mot e the soul itself motive, + We should not interpret the ern DE metric body. Continuing, Proclus = to the problem o , as harmonical] these ratios are not only relations; the y subsist ive its motive character fro m the realm of intelligibil ia (p. 32, 7-10 Fr). Indeed, it is well known that of the five gen era in the Sophist (being, sameness , otherness, motion, rest ) only the first three appear in the Tim aeus. he passage in Isc. In his commentary on the Timaeus, PIRA A Pea on p. 22, i.e. that the soul is neither a sensible nor a en —_ extension and n described in terms of The second Procius passage (cf. above p. 21) is sig ores 25-28 Diehl). The objections to Eratosthenes and : e È are treated together, obviously because of the reason pe Sensuous body having extension in the ord inary sense of the word] *. Though soul has bee nti dUEa yVe, been of units. Though it has been said that the soul is iS +üv Suxr 23 SOUL AND MATHEMATICALS “In the soul are present the Wea cation and the differentiation ... Therefore the substance o t e soul is not merely arithmetical or the soul = viet ey ical, or the soul wou 1 [merely] geometrical, ous; nor is it Mr differentiated. But the soul is the one and the other at the same ea by being arithmetical, the soul en substantially [i.e., in the form of being, not in the form knowing] harmonics ... by being geometrical the soul posse astronomy, because the circles in the soul are both oi ne pe: unmoved | .. [Therefore the soul is] a substantial bond o [a branches of] mathematics {the soul not only knows ngee È the soul is mathematics]” (In Tim. 36 B, 213 D-E, v. 239, 6 Diehl). The connection with the On the Soul passage is evident ; Proclus

enumerates the opinions ident ifying the soul with single b of the mathematicals. We also know his ist “n ieh when referring to the soul, must not mean a numb ù er aes ol monads, extension must not mean “geometrical” exte end-result of this whole discussion regarding the Gil io tT 4 nature of the soul we find a little later: the soul is Gr es a. geometric, harmonic, and astronomic and na th ated e — e of mathematics — the soul conta ranches of mathematics — the stress being om WTLiaur và1 allde on “all” To d Er pale in the two passages we find all the phrases ur sé ee amblichus: all branches of math ematics: number; h: monic ratio; figure; fullness of all branches of mathematic. the idea (form) of things extended; point (corresp onding to li ni ; gi (cf. In Tim. 36 B, 213 E, v. II 239, 6-16 D me si m =nieER o g I se still a the disc ussion the relation nn + le So P. e tipi > i nn precisely: the sphe re of being “soul””] consi st of nn By what e mixture en a pica It may be, therefore, that the division between ch.s IX and X is not in complete agreement with the summary — either as the result of somebody’s slip of the pen, marking off ch. X at the wrong place or some slight inconsistency between outline and its execution (or content and subsequent summary), likely to stated ie 24 F) in following word s: In what way does = aa the nen kind of distinction could [of mathematicals] complete de reality additional within of [the principle of the soul? objects of] them be to occur in any writer. But in the present context the question of the composition of the two chapters is entirely secondary. What is important is to see that the whole inspiration of ch.s IX and X is completely different from that of ch.s III and IV. In the latter, the soul and mathematicals were said to belong to different spheres of being; in the former, the soul becomes virtually indistinguishable from Mia prg it is taken for granted, the soul consists of E es : i me pases The problem still left is to find O dn branches contribu tes to the soul so as denti one ni oe and how, on the other hand , in spite of the lira ma vie oh ye a mathematical entity. It is a mathematical entity of its own oe: the diversity of the kind, to be sure, by being an arithmetical, a geometrical, and a harmonical at the same time, but a mathematical entity it still is. All these identifications of the soul and the mathematicals may ultimately be a correct or a mistaken interpretation of the left is whether there are en. Se e whether the soul contains in itself cit pente, ical real “e ity. E = pda turn from the summary to what on » the is marked a on as chapter X, corr esponding to the Summar y » the agreement between bp* we may have some doubts as to y On 5 *(1929) en (xeparata) in lamblichus see H, Oppermann in G Pythagorae d the summary and the actual content. The last question of the summary (whether there are any mathematicals outside the soul) is indeed being discussed in ch. X; but all the preceding questions seem to be discussed in the second part of ch. IX (p. 41, 5-42, 6 F). In this section lamblichus relates the different branches of mathematics to different aspects of the soul. Its “determinate” and “defining”” character the soul receives from numbers (its “unitary” A6yos from the number One); for its capacity to discharge itself into the realm of the extended the soul is indebted to geometricals; its capacity to establish harmony, order in motion, and common measure in what is incommensurable and to elevate accord to concord (symphony to eumetry) comes to the soul from harmonicals (and this is the reason why the soul can perceive harmony, being itself harmony and its essence consisting of numbers and other similar mathematical measurables). between mathemati =n on The content of this chapter is 25 pee epee L. Deubner, “Bemerkungen zum Text d ss > er Vita st, K es lamblichos'', SB der Berl. Ak., ' Philos.-hi 824-82 ‚hist, KL, 1935, 612-690; -827, esp. 689 n.; 690: 690 n. 2. Timaeus; in any case the net result is a division of being into three spheres, the middle sphere being described in such a way as to obliterate virtually any difference between soul and mathematicals. It is this tripartition of being which occurs in lamblichus most frequently; but we saw that ch.s III and IV lead to a quadripartition of being with the soul being distinguished

mà e e a e th o int fit t no do er Ti = , = a s .s o Ch c sensibles. ste intelligibles, mathematicals, and the soul cer ss the difference between the SOUL AND MATHEMATICALS from mathematicals (whereas ch. VIII leads to still another quadripartition, viz. that corresponding to Plato's quadripartite line, i.e. into intelligibles, mathematicals, sensibles, and images). The unity of Iamblichus' book is most precarious as we can already see and as we shall see time and again. Let us now resume our discussion of ch. X. In spite of the mathematical constitution of the soul already established the question can still be asked: is the soul the product of the combination of the three branches of mathematicals? between that the sou lish any rlydifferstrencesse the wr illsl y"aabo id particallulabranches of mathfacemt aticals rather o bswii tified ee + " alone.wiThthey are therefore compatible with thie Le g. In both respects né a “n= F 7 a* pei g bein «Ba accept a au n unity of the soul or is the unity of the soul prior to the diversity Iamblichus rejects both alternatives. The first would deprive ' mattreteratis ison thofattheborelthatiton ebe hi=e ith fiLtiorn as a2 serious interp t E u that bo nl rpose to assuiemee ient for ouridepunti den o It is ifsufthfice sou l is to be t fieonde wibthranofcit,h r » bret debe identififie jus h t i th wi d no st l jéi d with all — thraskeee doronfouPp.r = a ether be idewontirdfie hee sage In other s, the quthese tipronoblem one Partie so the presupposition of in the form ol a q saprete the soul of its status which is to be the principles of mathematicals, make the branches of mathematics a scattered plurality, and the soul an almost accidental product of their concurrence, and have other odd consequences. The second would make the soul the cause of mathematicals and introduce a difference between the two according to the principle that the cause is superior to and Una. different from its effects. What is left is the third alternative: neither is prior to the other. The soul coincides (concurs) with the mathematicals (ouvrp£yer mods «dtd — one is almost tempted to translate: the soul and mathematicals form one single team of runners — only we must not imagine these runners to exist independently from the team) and coexists with them (ovvvoéornxev) with the paradoxical result of an uncomposed and undivided mixture. A complete interpenetration of the mathematicals and the soul takes place so that the soul gives complete unity to the different branches of mathematicals and in turn abandons itself to all and several of them. There are no mathematicals outside the soul. But the unity of the soul does not prevent its differentiation *. The last question of the summary (whether mathematicals have any principle in addition to the soul) has been answered in the negative. lamblichus' discussion would be provided by the question: “Is the organism prior to its parts or are the parts prior to the organism?" with the subsequent answer that neither is the case; that the whole organism is indivisibly in its parts and in turn exists only in virtue of the plurality of them, The organism is not the result of its parts; nor is it the cause of them, thematicals rather os reereswientthpuarpnuose it is not all-important t0 ascertain to ui Di nnhoi ! s m a l i p d an us ch li mb Ia d ) ni unconditional identifica In other words, is the diversity of the branches prior to the of the three branches? î o tw the ng, bei of ion tit par tri de i r e r a p iu mber of passages in Proclus. \ Or are the three branches, on the contrary, products of one soul? * A good parallel illuminating the point of 27 “ è= LI » solution (offered in ch.beIII r È ah e s d an ul so n ee tw ing iat ent fer being, dif ch. s essa si be answered by saying thate itsouislonanlyd ei gen À ice, of th virtual identification ori © a p get on ti mp su as al gin the th wi t ten consis eed gr er. ca whereas ch.s III and IV i are ind ter agreeme e th at th y sa t no es do s io ec Pr e e e er rm p = sists In the. fo en between soul and mathematicals con pne e e n ge ar ch to o i a o e cn h n ionsistence? Is it possible to assume that 7 aes je i ing not. i inc noticed the en“a ch. II an ee t is o 1 t tha is on ti es qu e a c u d e h ibe with 4 =ELpaganoroger to entrust a scr ors rather than

copying passages indicated by him, on loose sheets changes of a certain ni extent Iamblichus assumes the responsibility for his sources; but he does not have to make them appear entirely consistent. An example well known already reveals immediately the kind of Iamblichus' editorial activity. One and the same passage (as we know today, taken from Aristotle’s Prolrep ticus) appears in his Protrepticus and in Isc. In the former it is part of ch. VI, p. 38, 3-41, 2 Pistelli. In the latter it is part of ch. XXVI, p. 8), F. Unhesitatingly, by cuts and replacing some words of Aristotle by his own, Iamblichus adapts the original text for his purpose, but he does not mind using the same passage as a whole twice in two different books. Once more we have the inpression that Iamblichus intends to produce someth ing which 7-83, 2 SAENS Èblico reportso Iseafterch.NicV,omap. chu18, s27-In19,n1 )gezZ: o B vi IR hs TÉ * LeA L>dPiCeIS * On the subdivisions of mathematics see P, T' 2, 1004a8 (v. I 259). It is well known that in addition toa quadripartition we find also an entirely different division of mathematics in Proclus, reported by him as that of Geminus (cf. J. G. van Pesch, De Procli fontibus [1900] 87-1 13, esp. 97). Mathematics is divided into two parts, which today would be called pure and applied. Pure mathematics contains arithmeties and geometry, leaving astronomy and acoustics to applied mathematics (In Eucl., Prol. I, p. 38, 1-12 Fr; on the designation of acoustics as canonics see p. 40, 22 Fr). above o) pian ms mentary on Euclid (sec Li ni) and in his com should in one and the same me |è oil of ana nobody ate os ics contains the four branches enumer the other hand, maticals areofunmmatovehemd.atiOncs with the exclusion er thetri artition CAPA ne j e he should, cling to the interpretation ee i acceptd. s Acca ordtripingartily,te aunmove t i a d Ù his mathematicals are ; an, ola . ease ecau mathematics, iv problem of the motive charac ter of nthe soul is entirely supin se tà Ist ge ho quotes ail as self-moving (self-changing) numberst 10a ix le pe oe based sugge “N, e , E RE al as + sided But though he professes to : on the principle of xowf supreme gpl "i sa návta tH yévo of mathematicals [p. 40, 12-13 FJ, 190 three 49, 32, p. 363, 26-365,4 Wachsmuth, presenting also tire: een evn i.e., arithmeticals, ee da word ne the Spiate definition contains part =A‘em en de ATOHIVNTOS , arin : in this . ? ..” “synthetic initi definition we miss j definition of Xenocrates (number) and omits harmonics (acoustics), Tannery, La Géométrie grecque (1887) 38-52; W. D. Ross, Aristotle's Metaphysics (1924) ad gr clus is ready to a Er on which Pro ripartiti uad + ci er G, ee Lis on the Timaeus (e.g. [n Tim. 213 n sed We see this(p with particular clarity pa o.fnthe SEPPaAT cratınestwo 7 F) Neno 40, 16-1 Besides, it appears that Iamblichus is not quite insensitive to the contradiction between ch.s III and IV on one hand, ch.s IX and X on the other. In fact, he avoids in ch.s IX and X anything which would make either the soul or the mathematicals appear to be motive. We shall see this better when we investigate once more the relation between the identifi cations of the soul with mathematicals as present in Proclus and Iamblichus. The most outstanding difference between the two is as indicated. Iamblichus’ mathematics in ch.s IX and X is tripartite, with astronomy wanting. Proclus’ mathe maticals are quadripartite, with astronomy included *. Now, it was Nicomachus 19-31 4F (cf. als . is neither an original nor a florilegium. containing only arithmetics, geometry, and e that mathematicss,isdissam ee ee 4 i lenE Lg e cre uou being eithermetdisicscre)etororinconreltin cus a àseititheter per ons Em ati se (arith b da E po — ng | either r quantitymy).beiThis uous (ast nice)= incontin — i E; e rono motion ae no particular consistency can be expected to result. Nor should it be expected: Jamblichus’ book is comparable to a selection from sources rather than to an exposition of the views of one author . It is not quite a florilegium; but it is not meant to be an original either. To he quadripartition of mathejustifi and patching them up into a whole by introductions, summaries, a word every now and then, etc. It is obvious that 29 SOUL AND MATHEMATI; CALS SOU L AND MATHEMATICALS seotens da motion). Furthermore, describing the aspect 0 dos (assoni Ww nicals he = = PA casa hich the soul is indebted to harmo | 41,12-13 F), but : . om pics 1a “harmonic” — he almost could have said se:= r of harmonic motion (p. e 41, ma des the source of motion but rather of Un not E in the motions of the universe. la tr nia there be one, in ch. IX that the et a with tripartite hint because in this chapter the so ul is identified with

blichus did not bother es decided in different ways *. IamAsa result, hisi BEATSical =s cile contradictory opinio1 ns. SOUL AND MATHEMATICALS mathematics, excluding astronomicals. To this extent lamblic hus may have tried to avoid too flagrant a contradiction between ch.s III and IV on one hand, ch.s IX and X on the other. does it at the cost of suppressing the questi on of the motive character of the soul, though he does not succee d completely. At the end of ch. X we read that the soul will be differentiated in accordance with its different Suvéuerc, Ywal, and Evesyzızı (p- 43, 8-10 F). This immediately reminds us of the argument of those who tried to make mathematicals motive , according to Isc ch. III. Some, says Iamblichus here, wil] perhap s grant motion to the principles of mathematicals (i.e., the limited and the unlimited) — viz. those who posit these principles in the soul and the wat and Suvauers of the soul (p. 13, 9-12 F). On reading the passage in ch. X quoted above, one feels inclined to ask: is the mention of Cwat and Suvéuers as peculia r to the soul an indication that the mathematicals of which it consist s are indeed moved? No clear answer can be found in ch. X — we are left feeling that the relations between motion and mathematicals, motion and the soul, mathematicals and the soul as presented in ch.s III and IV, and again in ch.s IX and X are in several respects inconsistent. Let us now sum up the results of the foregoing discussion. ; his ma mes exclude motionr porn ie include, titsome,eti a— e sometimes p a sometimes tripara substantijal number oi i key He Both ¿do problems. 2 = tangle of many now = guide through the t follows; but even ge ‘e now have . ‘ come obvious in wha different sources from which la i ; - it is to discover the into the difference (an clear insight = once we haveaatrip artite and a quadripartite mathematics. it implies) between is of the chapter with a synopsmot And we can concludebasthis ive or the to e ic contradiction as rew, Rito ’ sno Unern passages involving th non-motive character of mathematicals. Isc ch. VII, p. 30, 25-31, 2F Ise ch. III, p. 13, 12-15F . . received as a metry It is better to assume that Geo te the spheric [astronothe mathematical principles helpma ch is the instrument of whi and the mathematical sphere my] uous quanof being [obota] are nonmotive. n Isc ch. IV, p. 18, 14-18 F The mathematical principles knowledge of contin tity in motion. Isc XII, p. 47,6-16F Most men believe that the branches of mathematics are nonmotive and that the objects of their knowledge are lamblichus and Proclus describe the mathematical s (which they take to subsist) as intermediate. The realms betwee n which they mediate are often described in terms of the divisible and the indivisible, Both are aware of the “intermediac y” of the soul, though Proclus stresses it more than Iambli chus. Both deal are nonmotive. with the problem whether methematicals and vestigating the number of motion ... and the incorporea) their nonmotive nature. ions of the soul Isc ch. XV, p.55, 14-15 F circular mot the soul are identical — Jamblichus arguing sometimes pro, someti mes contra; Proclus assuming identity. In connection with this question both assert the identity of the soul with all branch es of mathematics — three in Iamblichus, four in Proclus . These assertions are closely linked with the motive or nonmo tive character of mathematicals. Proclus asserts the former, Iambli chus sometimes the former, sometimes the latter. The solution is closely with the problem of a tripartite mathematics withou connected t astronomy, or a quadripartite one, including astronomy. This much is immediately clear: Isc is based on differen t sources, in which some of the problems treate d above were Isc ch. XIII, p. 50, 18-19 F Mathematicals differ from the realm of becoming by Mathematics prepares for theology also by being a connonmotive; this, however, 15 a wrong opinion. For there are branches of mathematics inwith which the heavenly revolutions coexist .... In such inions astronomy and templation of things stable vestigat harmonics are contained. nonmotive. p. 63, 23-64, 13F Iscch. XIX, The mathematician deals ics is concerned > ch. XXIV, p.75, 18-19 Mathemat © Cf. Zeller 111/23 (1923) 759.

ATICALS with a kind of nature stab le and void of motion. Isc ch. XXVI, p.81,11-12F Things belonging to the nonmotive mathematical kind s are limited and ordered. relations between mathematicalco the soul with regard to the with theologicals, noeric mat hematicals, the self-moved sphe re of being, and the eternal ratios (wherein they define the self-moved number), with the heavenly bodies and moti ons, :ne:aa 2 vez "nia the two pragaKohnkdi interpretedby e *. The snteresti other. They ade be pa contract eachantithesis . It is ion, nonmotive, corresponding gs s, others the character of mathematical which are exempt of mot ion. completely mn we ee the fact that the two chapters correbut nonmotive, (in acoustics and tronomy) are motive. asthe two columns, 1. The contradiction betwee n ch. III and ch. IX of Ise is only apparent, according to Loenen *. This he proves by saying that the same (apparent) contradi been present in Posidonius, soul with mathematical ction must have, at least impl icitly, For he, on one hand, identifi ed the s but on the other must have known that the soul according to Plat o was motive. How then resolve the apparent cont radiction ? By assuming when speaking of the mot Sense of the word, does Loenen that Iamblichus ive soul means soul in the ordinary whereas mathematicals he still can when he speaks of non-motive identify them with the soul , viz. with the soul as idea (ideal soul ). This distinction, says Loenen, should be of great interest to students of Plato. Loenen’s explanation is a classic example of an ad hoc hypothesis. There is not the slig htest hint of the doctrine of a double soul in /sc. As to what Pos idonius must have kno * J. H. Loenen, Mnem osyne, S. IV, vol. X/1 (1957) 80-82, wn about the ne and ch. IX from the work in which it was refute i Kohnke might object that I attribute too n a be je iyRI of pi a it a Ne an Kohnke suspectshishimeditoria ig = ABIOTLeSNHsodCEtAH gIeo.DIy, explained by his hypothesis. After all, grego » ted ch. III of Isc from the work in whic it origin 7 Pubityo ele Appendix Posidonius. spond to each other (Kohnke is right on this point) 2 Di The source of the first right-side passage is Nic omachus, Iutr, arithm. 1 3, 1; P. 6f. Hoche (cf. Festa’s adnota tion a.l.); there can hardly be much doubt that the left-side pass age is from some other source. They contradict each othe r and the sam e holds true for the rest of . ' proceeded to refute them. pa Pg even guesse In mathematics some thin are know no work whose author first presented the Shigeo ire Isc ch. XXVII, p. 86,14-15F to we in correspond to each other like thesis and rese: to be assumed that both appeared in one and= pan etc, Isc ch. XXVIII, p. 89, 2-8 F The mathematical ratios are 33 SOUL AND MATHEMATICALS i no way on which to arrive at ce N i inty. would illustrate the kind and scope of incidentally, if Kohnke guessed correctly and both c x > + are from Posidonius, this would po en kur: en Y 2er eu se in the concept of & aring in ch. JII, p. 13, mach De (vis vitalis), which we connect gora ta à. È (see K. Reinhardt, art. Poseidonios, RE XXII/ Il A ae 3. The importance of the term ouvepeyety to ci of distinct natures or the distinction in spite of gra” e n Chris Chalcedonense (e.g. in P. Schaff, The Creeds of = . Ci On the ene of Iamblichus’ ‘authorship’ see below, p. 7, later reprints] 62). * F, W. Kohnke, Gnomon 27 (1955) 157-164.