1. For example, Met. 1003b33; Met. 1054a13–14.

2. For example, Met. 1003b23–31; Met. 1054a15–16; Met. 1061a15–18.

3. The Scholastics later took being, unity, good, something, true as convertible predicates. On being and unity see Aquinas ST la Q.11 Art.1; on being and good see Aquinas ST Q.6 Arts. 1,3. Since being is transcendental then any predicates convertible with being will also be transcendental. Aristotle affirms the transcendental status of good at NE 1096a23–7, EE 1217b26–32 (though the consequence drawn in EE concerning the possibility of a single science of being is different from that drawn in Met. Γ2). On the relation of unity and being in Plato, see Parm. 142e, 144e. See also Ross, ' helpful note in Aristotle's Metaphysics I (1924) 256Google Scholar.

4. See for example Leibniz' letter to Arnauld, 30 April 1687, The Leibniz–Arnauld Correspondence (translated Mason, H. T.) (1967) 121Google Scholar; Hume, Treatise on Human Nature 1.2.2; on Zeno, Philoponus, in Phys. 42.9–43.6, 80.23–81.7Google Scholar, Simplicius, in Phys. 99.12–16Google Scholar. Compare also Aquinas' remarks on the use of this relation by Plato and the Pythagoreans, ST 1a Q. 11 Art. 1 ad 1, Comm. in Met. prs. 900–1, and by Avicenna, , Comm. in Met. prs. 556–60, 1981–1982Google Scholar. Aquinas', Commentary on Aristotle's Metaphysics is translated by Rowan, J. P., Aquinas' Commentary on the Metaphysics of Aristotle, 2 vols. (1961)Google Scholar.

5. See the passages mentioned in Part 1 below.

6. See for example Annas, , ‘Individuals in Aristotle's categories: two queries’, Phronesis 19 (1974) 146–52CrossRefGoogle Scholar; Kirwan, , Aristotle's Metaphysics Books ΓΔΕ (1971) 134Google Scholar; Annas, , Aristotle's Metaphysics, Books M and N (1976) 38Google Scholar.

7. See for example White, , ‘Aristotle on sameness and oneness’, Philosophical Review 80 (1971) 177–97CrossRefGoogle Scholar; Fred Miller disagrees with White, 's conclusions in his ‘Did Aristotle have the concept of identity?’, Philosophical Review 82 (1973) 483–90Google Scholar. White in his paper notes that Aristotle did not keep separate two uses of ‘X and Y are one’, first to mean that X is somehow identical with Y, second to mean that X and Y make up one thing. As a result Aristotle does not separate a one and a two place usage of ‘One’. For more on White's point see the text in Part 1 below.

8. Perhaps as Annas, , Aristotle's Metaphysics M and N 38Google Scholar, ‘Aristotle is the victim here of the Greek language’ so that it is unreasonable to expect him to draw the relevant distinctions. See also Schofield, ‘Plato on unity and sameness’, CQ n.s. 24 (1974) 33–45CrossRefGoogle Scholar, with an account offered of why Aristotle should have conflated ἕν and ταὐτό, especially in face of Plato's argument for their distinction at Parm. 139d, e.

9. On these issues see the collection of papers Rorty, , Schneewind, and Skinner, (eds.), Philosophy in history (1984)CrossRefGoogle Scholar, particularly the paper by Rorty, , ‘The historiography of philosophy: four genres’, 49–75Google Scholar. See also David Charles' discussion of the distinction between philosophical scholarship, traditional classical scholarship and original speculation inspired by the reading of a text in his Aristotle's Philosophy of action (1984) ix–xiGoogle Scholar. For consideration of these issues in the special case of exegesis in the face of limited textual evidence see my ‘How can we find out what ancient philosophers said?’, Phronesis 33 (1988) 121–32CrossRefGoogle Scholar.

10. At Topics 1.15 Aristotle offers, as a test for distinctions of sense ( 106a9) within a term, distinctions in its contraries.

11. I intend only to show that ‘One’ is treated differently in each of the questions, and to show this by the difference between the types of answers which are appropriate. I do not intend the answers offered to be either Aristotelian answers or correct answers.

12. This point is worth making over and above the first point, for it shows that not all uses of ‘One’ as a unity predicate are one place usages. Thus it does not follow from a particular use of ‘One’ as a two place predicate that the term is there used as an identity predicate.

13. .

14. Met. 1003b22–1004a10.

15. Met. 1003b33–6: .

16. Met. 1004a18: .

17. Met. 1018a4–5.

18. Met. 1018a7–9.

19. Met. 1004b27.

20. Mel. 1040b8–9: .

21. Met. 1040b14–15: .

22. Phys. 227a10–34.

23. Met. 1052a15–1052b1, summarized at 1052a33–5: .

24. On focal meaning see Owen, G. E. L., ‘Logic and metaphysics in some earlier works of Aristotle’ in Owen's collected papers: Nussbaum, M. (ed.), Logic, Science and Dialectic (1986) 180–99Google Scholar.

25. Met. 1003b5–10, 1060b31–1061a10.

26. Clearly the example offered here requires for its relevance that the items in non-substance categories are taken in a certain way, i.e. as non-repeatable unit properties, for example the particular white in Socrates, or the particular five foot in Callias. I will say more about this interpretation in Part III below.

27. Met. 1003b26–7: .

28. Met. 1054a29–32.

29. I give the text as printed by Ross, rather than that given by Jaeger, in the Oxford Classical Texts edition, which reads .

30. As Plato, saw, Parm. 128e–129dGoogle Scholar, applied to like and unlike, many and one.

31. Met. 1052a15–1052b1, cited earlier.

32. See Met. 1045a8–12.

33. Met. 1003b29–30: .

34. I take this to be similar to Geach's claim that count nouns (for example ‘man’) are derelativisations of equivalence relations (identity relations, for example ‘– is the same man as –’). See Geach, , Reference and generality (1968) para. 109Google Scholar, and ‘Ontological relativity and relative identity’ in Munitz, M. K. (ed.), Logic and ontology (1973) 287–302, especially 290–2Google Scholar.

35. Met. 1042b10–11: .

36. For these as types of καθ’ αὑτὰ ἓν λεγόμενα see Met. 1015b36–1016a1, Met. 1052a19–25. In these passages ἕν is treated as a unity predicate.

37. Wright, , Wittgenstein on the foundations of mathematics (1980) viiiGoogle Scholar.

38. See ST 1a Q. 11 Art.1 ad 1, Q.11 Art.2; also Comm. in Met. prs. 501, 557, 559, 560, 875, 901, 1981, 1997, 2090. Aquinas makes the distinction also in his early Commentary on the de Trinitate of Boethius (the first four questions of which are translated as Faith, reason and theology by Maurer, A. (1987))Google Scholar. See Q.4 Art.1, especially Objection 3.

39. ST 1a Q.11 Art.1 ad 1; Comm. in Met. prs. 501, 556–60, 900–1, 1981Google Scholar.

40. On Aquinas' Commentaries on Aristotle see Owens, Joseph, ‘Aquinas as an Aristotelian commentator’ in Caton, J. R. (ed.), St Thomas Aquinas on the existence of God: the collected papers of Joseph Owens (1980) 1–19Google Scholar.

41. There is a notion of division connected with, or implied by, the ἕν convertible with being, since that concept of ἕν is a concept of indivision, as at ST 1a Q.11 Art.1, unum enim nihil aliud significat quam ens indivisum. Compare also Comm. in Met. pr.501.

42. Comm. in Met. pr.2090 is a little complicated, since Aquinas says there that the notion of plurality opposed to the ἕν convertible with being (viz. plurality taken absolutely) is in a sense the genus of number. He makes a similar claim at pr.2091. On the other hand, ἕν taken as a measure is limited to the category of quantity. At this point the distinction between the ἕν convertible with being and the ἕν which is the principle of number may seem to become cloudy.

43. See Met. 1052b18–19.

44. The interesting example of the Papal Crown is taken from Wiggins', DavidSameness and substance (1980) 73Google Scholar. However, Wiggins makes a different use of the example, saying that there is ‘no definite answer’ to the question of how many crowns the Pope has on his head.

45. Clearly at Met. 1052b15–19: . Also Met. 1016b17–19: .

46. Aristotle does not, of course, use the term ‘N-unit’ nor does he explicitly draw the distinctions I have drawn between N-units, B-units and I-units: this is part of the source of the problem I am interested in. What I mean is that the notion Aristotle is talking about is identifiable as what I have called N-unity. I will not make this disclaimer again.

47. See Post. An. 87a36, 88a33; Phys. 227a30; de Caelo 300a18; Met. 1069a12, 1084b26. Since individuals do have a position, and (at least in some categories) can be in contact, these monads cannot be individuals from one of the categories.

48. On this distinction see Phys. 219b5–9.

49. Annas, , Aristotle's Metaphysics M and N especially 26–41Google Scholar is clear and concise on Aristotle's ideas concerning numbers.

50. I do not intend that F should be limited to sortal terms. The foot is an N-unit, but ‘foot’ is not a sortal term.

51. Compare also Phys. 223b14–15, 224a3–15; Met. 1016b20–4, 1053a24–7, 1087b33–1088a14. In citing these passages, and in explicating what Aristotle says by means of the completion of number terms by general, terms, I do not mean to suggest that Aristotle wholly prefigures Frege's insight that numbers are properties of concepts. Indeed I do not mean to offer any general view on Aristotle's account of numbers and counting. Rather, the completion of number terms by general terms in explication of indivisibility and N-unity (and later as regards B-unity too) is useful for putting matters clearly; it also answers to ideas that are found in Aristotle.

52. See references at n.38 above.

53. I say ‘actual triangles’ because the kind of example I have in mind is this. The actual triangle ABC is composed of (and so divisible into) the actual triangles ADE, DBF, DEF, EFC. But the totality is countable (as five triangles) because there is an N-unit, e.g. the actual triangle ADE is indivisible into any further actual triangles. Clearly, however, ABC (or any triangle) can be divided into any number of potential triangles. Thus the order ‘count the potential triangles in ABC’ could not be followed, precisely because there is no N-unit in that case.

54. As an aside we can see why Aristotle claims, as at Met. 1052b18–20, that the primary application of this concept is in the category of quantity. If we consider the categories as dividing up different questions that might be asked about things, as Ackrill mentions in his Aristotle's Categories and de Interpretations (1963) 78–9Google Scholar, and on which compare Topics 1.9, then the question ‘how many?’ will receive an answer in the category of quantity. Counting that team as 11 (men) does not say what it is (substance) or what it is like (quality) or…, but how many it is (quantity). Note that in Cat. 6 number is repeatedly given as an example of quantity, 4b22, 4b25, 5a23, 5a30, 6a20.

55. I shall say more about this in Part III below.

56. I shall say something more about this convertibility presently, though it should now be becoming fairly clear what convertibility involves.

57. de Anima 415b13. Also GC 318b25; NE 1166a4. It is on the same grounds that Aristotle denies that a dead man is a man, de Int. 21a21–2.

58. For these examples see Met. 1042b15–1043a11.

59. Post. An. 92b13–14: .

60. Hence Quine's well-known slogan ‘no entity without identity’.

61. As Geach conjectures, ‘Ontological relativity and relative identity’ (n.34) 287–8. But that I-unity might also be convertible with being should give us no more reason to deny the distinction between I-unity and B-unity than does the convertibility of being and good give us reason to deny the distinction between I-unity and good. They will be different in definition (λόγος) as Aristotle says of ἕν and ὄν at Met. 1003b24–5.

62. Met. 1003b29–30.

63. On the irreducibly general nature of the quantifier notation concept see Miller, Barry, ‘In defence of the predicate “Exists”’, Mind 84 (1975) 338–54CrossRefGoogle Scholar on the distinction between precisely one individual and one precise individual.

64. On Aquinas' use of esse see Anscombe, and Geach, , Three philosophers (1961) 88–100 (by Geach)Google Scholar, especially his comment at 91: ‘It is the present-actuality sense of ‘est’ that is involved in Aquinas’ discussions of ens and esse. It corresponds to the uses of the verb ‘to exist' in which we say that a thing comes to exist, continues to exist, ceases to exist.…’ Owen in his discussion of Aristotle at ‘Aristotle on the snares of ontology’ in Bambrough, (ed.), New essays on Plato and Aristotle (1965) 69–95Google Scholar expresses the tensed/tenseless distinction as between being* and being** respectively.

65. Kirwan (n.6) 135.

66. As introduced by Cartwright, Helen. See her ‘Heraclitus and the bath water’, Philosophical Review 74 (1970) 466–85CrossRefGoogle Scholar; ‘Quantities’, Philosophical Review 79 (1970) 25–40CrossRefGoogle Scholar; ‘Amounts and measures of amount’, Nous 9 (1975)Google Scholar. I do not suggest that Cartwright introduces the notion of a quantity in a trivialising way, so as to render it analytic that everything is one something: she has her eye on certain important problems about material identities.

67. We can give another example, not using a mass term, by picking an English term whose singular and plural forms are the same. ‘Sheep’ and ‘one sheep’ do not seem the same: compare ‘I own sheep’ and ‘I own one sheep’.

68. See the passages cited at n.4 above.

69. Translated by H. T. Mason, reference at n.4 above.

70. Reference at n.4 above.

71. Leibniz, , Monadology pr. 1Google Scholar.

72. Zeno _B1. On the interpretation of this argument, and its lack of reliance on the claim that every magnitude is divisible (as distinct from the claim that once a magnitude is allowed to be at all divisible then it is infinitely divisible) see my ‘The indivisibility of the atom’, Archiv fü Geschichte der Philosophie (forthcoming).

73. I undertake such a diagnosis in an unpublished paper ‘Unity’.

74. Met. 1046b26–7.

75. White (n.7 above), especially 191–5, explains Aristotle's conflation of ‘X is identical with Y’ and ‘X and Y go together to make up a single thing’ in terms of Aristotle's concentration on problems concerning identity over time. For the question ‘what makes Socrates yesterday identical with Socrates today?’ might also be put as ‘what makes Socrates yesterday and Socrates today go to make up a single thing (viz. Socrates)?’ Now White does not draw precisely the distinctions I have drawn, but, insofar as he emphasizes temporal considerations and the unity of motion, we might take his approach as leading to the view canvassed above, that the unity convertible with being is to be understood in terms of identity.

76. Ackrill, , ‘Aristotle on ‘Good” and the Categories’ in Barnes, , Schofield, and Sorabji, (eds.), Articles on Aristotle II (1977) 17–24Google Scholar, makes a similar point when considering the transcendental status of good, concerning examples such as white man and negro (a substance and a quality) and doctor (a substance and an action). See also Kosman, , ‘Animals and other beings in Aristotle’, in Gotthelf, and Lennox, (eds), Philosophical issues in Aristotle's biology (1987) 360–91CrossRefGoogle Scholar, especially 367–71.

77. For this interpretation see Ackrill (n.54) 74–75, and Anscombe (n.64) 7–19.

78. Cat. 2a19–33.

79. For this interpretation see Owen, , ‘Inherence’, Phronesis 10 (1965) 97–105CrossRefGoogle Scholar. See also Frede, M., ‘Individuals in Aristotle’ in his Essays in ancient philosophy (1987) 49–71Google Scholar.

80. For a defence of the traditional interpretation against Owen I have found particularly useful and convincing Heinaman, 's paper ‘Non-substantial individuals in the Categories’, Phronesis 26 (1981) 295–307CrossRefGoogle Scholar. Other papers relevant to this controversy are Allen, , ‘Individual properties in Aristotle's Categories’, Phronesis 14 (1967) 31–9CrossRefGoogle Scholar; Duerlinger, , ‘Predication and inherence in Aristotle's Categories’, Phronesis 15 (1970) 179–203CrossRefGoogle Scholar; Jones, , ‘Individuals in Aristotle's Categories’, Phronesis 17 (1972) 107–23CrossRefGoogle Scholar; Annas, ‘Individuals in Aristotle's Categories: two queries’, Phronesis 19 (1974) 146–52CrossRefGoogle Scholar; Jones, , ‘An introduction to the first five chapters of Aristotle's Categories’, Phronesis 20 (1975) 146–72CrossRefGoogle Scholar. Hartman, , Substance, body and soul (1977)Google Scholar also makes relevant points, endorsing the traditional view at, for example, 14 n.5.

81. See, for example, Jones, , Phronesis 17 (1972) 107CrossRefGoogle Scholar. Annas, , Aristotle's Metaphysics M and N, 204Google Scholar makes the distinct, and acceptable, point that only at Met. 1089b24–8 does Aristotle explicitly consider the question of non-substantial individuals, outside Categories. But what she says there is compatible with both the traditional interpretation and Owen's interpretation.

82. Heinaman (n.80) 297–99. See also Engmann, , ‘Aristotelian universals’, CP 17 (1978) 17–23Google Scholar.

83. The most interesting are Phys. 228a3–12 (a particular person's health, a particular man's walking); Phys. 242a66–242b41 (a motion is numerically one if it proceeds from something that is numerically one to something that is numerically one, for example, from a particular white to a particular black, 242b39 ); de Long. Vit. 465a19–26 (the termination of what must be a particular ignorance or a particular knowledge); Met. 1040b25 (what is one cannot be in many things at the same time, so one brown will have to be a non-repeatable unit property, this brown); Met. 1044b21–9 (the coming-to-be of what must be a particular white); Met. 1070a21–4 (the existence of the shape of a (particular) bronze sphere). Heinaman also cites other passages, but requires some discussion to reveal in them Aristotle's endorsement of non-repeatable unit properties. For these the reader is referred to Heinaman's paper.

84. de Sensu 440b18–23, 442a12; also Cat. 12a17–19; Phys. 188b3–5; Met. 1070b19–20, which gives white as the form, black as the privation and the surface as the matter.

85. At de Sensu 439b20–440b25 Aristotle considers and rejects a juxtaposition account, where white and black are juxtaposed, each separate white and black being invisible, but producing a visible intermediate colour, and a superposition account, whereby one is seen through the other, various ratios of superposition producing intermediate colours. Aristotle prefers an account based on complete mixture, 440b14–17.

86. de Sensu 442a13–18.

87. de Sensu 442b29–30, though this is unclear.

88. de Sensu 445b25–8. Every object of sense has contrariety, as white and black in the case of colour, sweet and bitter in the case of taste. On sounds see also de Sensu 448a2–14.

89. See Cat. 6 on the explication of quantities in terms of parts, whether related to one another (as the parts of a line) or not (as the parts of a number).

90. The case of such relatives as great, small, half, double will be a more difficult one. For, as noted by Annas, , Aristotle's Metaphysics M and N (n.6) 198Google Scholar, it appears to be of such relatives as these that Aristotle says at Met. 1088a24–7 that they are least of all φύσις τις ἢ οὐσία, are posterior to quality and quantity, and are affections of quantity, .

91. The list often categories at Cat. 1b25–2a3 is repeated only at Topics 103b20–2. I have nothing to say here on whether Aristotle held there to be ten categories throughout his writings.

92. Se for example NE 1096a23–7, EE 1217b26–32.

93. Argued by Aquinas at ST 1a Q.6 Art.1.

94. I would like to thank Robert Wardy and the Editors for their helpful comments on earlier forms of this paper.