1. See Post. Anal. I.xii, xxvii, and discussion above (chap. 4).

2. W. F. R. Hardie, Aristotle's Ethical Theory (Oxford: Clarendon Press, 1968), p. 31, for instance, claims that "what methods and procedures are proper in a science depends on its purpose . . ." but stops short of concluding that the method of ethics is nondemonstrative.

3. Post. Anal. 71b16, 92b19, 93a10, 98a20; Anim. 402a. The term , however, is often used to mean simply the way something is done, which may have nothing to do with the method an inquiry or investigation uses—e.g., Top. 156a15, Pr. Anal. 32b8.

4. Aristotle uses this term most often to signify the manner of obtaining the elements or premises of a syllogism, e.g., Pr. Anal. 53a2, 43a21. At other times, however, he uses it to signify something that is closer to the way or method of proving something, e.g., Pr. Anal. 46b24, Post. Anal. 82b30. But in many instances what Aristotle considers to be different methods are really different syllogistic forms.

5. Aristotle, using the same term, speaks again of the way to the first principles at Post. Anal. 84b23.

6. For the use of see N.E. 1099b14 (where some inquiry that studies matters pertaining to the gods is distinguished from the inquiries of ethics and politics), 1102a12, 1146b14. For the use of , see E.E. 1217a3, Top. 152b12, and for that of , N.E. 1096a11, 1112b21, Anim. 403b25.

7. At E.E. 1216b35-39 Aristotle writes, "And in every investigation [or method, ] arguments [accounts] stated in philosophical form are different from those that are not philosophical . . . since that is the philosophical way in every investigation [ ]." Although it is not altogether certain that the term means "inquiry" in its first occurrence, it most probably does. The term is, however, used at 1214a14 to clearly mean "field of inquiry."

8. See Polit. 1260b35, 1279b12, 1289a25, 1293b29, 1295a1, 1317a18, b33, 1324a2, a21.

9. M. Sintonen, The Pragmatics of Scientific Explanation (Helsinki: Acta Philosophica Fennica, vol. 37, 1984), p. 29. I quote Sintonen's statement below.

10. W. F. R. Hardie, op. cit. , p. 31, claims that "what methods and procedures are proper in a science depends on its purpose, on what are the questions to which answers are sought . . ." (my italics).

11. What Aristotle says in this passage (1098b) is not without its problems. He urges that we do not demand the cause in the same way in all cases ( ). Aristotle's words have been rendered in a variety of ways: "In all matters alike . . . an explanation of the reason why" (Rackham), "The cause in all things alike" (Apostle), "The cause in all matters alike" (Ross), "The same demand

for an explanation in all cases" (Irwin). Rackham and Irwin, in rendering as explanation, express agreement with the view defended recently by several scholars that Aristotelian causes are to be understood in terms of explanation-see J. Barnes, Aristotle's Posterior Analytics (Oxford: Clarendon Press, 1975) and R. Sorabji, Necessity, Cause, and Blame (London: Duckworth, 1980). Apostle and Ross, however, take Aristotle to be speaking about causes. These translators also go on to render differently what Aristotle says about what we might do if we are not to seek the "causes." Aristotle says that in some cases, and in particular in the case of the principles, it is sufficient to show adequately that something is so ( ). Most translators render Aristotle's words in such a way that they distinguish, as Aristotle seems to want to do, between proving or explaining (by giving the causes of) the principles on the one hand and in some way showing or establishing (Rackham, Ross) or indicating (Apostle) them on the other. Irwin, however, renders Aristotle's words by saying, "It is enough to prove that something is true without explaining why it is true." There is, at least prima facie, something puzzling with the idea that it is possible to prove something without explaining it.

12. See Eustratius's discussion, op. cit. , pp. 73-74, where he contrasts sharply what the builder does by using the T-square, the carpenter's orthogonal triangle, and the leaden weight for obtaining a straight line or a right angle to what the geometrician does.

13. "And after having given an art a single name in what has preceded, thereby making us think that it was a single art, does not the discussion now assume that the same art is two and ask whether the art as practiced by the philosopher or the non-philosopher was the more exact?" (57C).

14. Sometimes Plato's words at Philebus 57B are translated as follows: "Well, had not the discussion already found in what preceded that the various arts had various purposes [ ] and various degrees of exactness" (translated by H. N. Fowler, Loeb Classical Library). But actually what Plato says in this passage is that "different arts, dealing with different things [ ]" possess different degrees of exactness. R. Hackforth's rendering of this passage in which he attributes variation in exactness to the differences of the objects arts or disciplines deal with is the correct one— Plato's Examination of Pleasure (Indianapolis: The Library of Liberal Arts, 1945).

15. Its goals are, as Plato and Aristotle often insist, purely theoretical; it aims at knowledge for the sake of knowledge.

16. Grant in his commentary on 1098a30 (9.2) seems to me to have something like this in mind when he points to the different aspects or conditions of demonstration and the possibility that ethics may differ from mathematics because it does not satisfy all of these conditions or at least not to the degree or in the way mathematics does.

17. I do not mean to claim here that Aristotle does not identify any methods other than demonstration for investigating into certain domains or for discovering the basic principles. He clearly does. In Post. Anal. II.xix he argues that some type of induction, and not demonstration, is the way by which the basic principles are to be grasped. At Top. 101a37 he argues that the starting points of the sciences are arrived at by using dialectic. And at N.E. 1098b3 he claims that principles "are

studied by means of induction, perception, habituation and other means." But these methods, as G. E. L. Owen has pointed out, are not methods of explaining, proving, or producing scientific understanding—see G. E. L. Owen, "Aristotle: Method, Physics, Cosmology," in C. C. Gillespie (ed.), Dictionary of Scientific Biography (New York: Scribner, 1970).

18. See Post. Anal. 79a15, 88b13; Met. 1025a, 1064a.

19. See his commentary on this passage in his translation of the N.E. (Dordrecht: D. Reidel, 1975).

20. This translation is by P. Shorey in the Loeb Classical Library edition. Shorey, commenting on Plato's use of the word "ludicrous" to characterize the language of doing that is employed by some in relation to geometry, says that "the very etymology of 'geometry' implies the absurd practical conception of the science." Why the conception is, however, absurd is not obvious. Plato himself did not remove geometry from any and all practical applications (see below).

21. The goal of eliminating from mathematics the kind of language or proofs that Plato finds ludicrous has always been a part of the ideal of mathematical knowledge, even in the earliest stages of the history of the discipline of mathematics. This is brought out quite clearly in the comments of T. L. Heath on Euclid's Elements . Heath quotes the following from Schopenhauer: "I am surprised that, instead of the eleventh axiom [the Parallel-Postulate], the eight is not rather attacked: 'Figures which coincide are equal to one another.' For coincidence is either mere tautology, or something entirely empirical, which belongs, not to pure intuition, but to external sensuous experience. It presuposes in fact the mobility of figures; but that which is movable in space is matter and nothing else. Thus this appeal to coincidence means leaving pure space, the sole element of geometry, in order to pass over to the material and empirical" ( Die Welt als Wille und Vorstellung , vol. 2, 2d edition, p. 130). Euclid himself expresses reservations about the use of superposition of figures in geometry, but, as Heath observes, superposition is fundamental in geometry and an inseparable element of some of the common notions or axioms, especially 1.4—"Things which coincide with one another are equal to one another"—: "But seeing how much of the Elements depends on 1.4, directly or indirectly, the method can hardly be regarded as being, in Euclid, of only subordinate importance; on the contrary, it is fundamental. Nor, as a matter of fact, do we find in the ancient geometers any expression of doubt as to the legitimacy of the method," Euclid's Elements (New York: Dover Publications, 1956), vol. 1, p. 225. The problems associated with interpreting Plato's views on the nature of mathematical and scientific knowledge in general have been recently discussed by G. Vlastos, "The Role of Observation in Plato's Conception of Astronomy," and A. Mourelatos, "Plato's Real Astronomy: Republic VII 527D-531D,'' in J. P. Anton (ed.), Plato on Science and the Sciences (New York: Eidos Books, 1980); see also my discussion of these papers in "Plato on the Sciences," Inquiry 26 (1983), pp. 237-246.

22. According to Heath, op. cit. , p. 226, Aristotle accepts the method of superposition in geometry. But it would not be true to accuse Aristotle of failing to understand the nature of mathematical knowledge and of mathematical propositions—that is, of confusing them with empirical propositions on the basis of the

fact that he accepts the kinds of proofs that presumably Plato rejects. For he draws a sharp distinction between the proof one uses and the nature of what one proves—"the geometrician does not infer anything from the existence of the particular line which he himself has mentioned, but only from the facts which his diagrams illustrate" ( Post. Anal. 77a). See also Met. 1078a20, 1089a25, and Heath's discussion of Aristotle's views in his A History of Greek Mathematics (New York: Dover Publications, 1981), vol. 1, pp. 336-337.

23. Aristotle argues at Post. Anal. 72a30 and N.E. 1139b32 that our knowledge of the basic principles of a science must be superior to our knowledge of the propositions derived from them (its theorems).

24. See Met. 1027a: "And that the builder produces health is an accident, because it is the nature not of the builder but of the doctor to do this—but the builder happened to be a doctor. Again, a confectioner, aiming at giving pleasure, may make something wholesome, but not in virtue of the confectioner's art; and therefore we say 'it was an accident', and while there is a sense in which he makes it, in the unqualified sense he does not. For to some things correspond faculties productive of them."

25. The point here is not affected by the fact that Aristotle is mistaken about the function of the brain.

26. See in this connection the important discussion by R. Kraut in his "The Peculiar Function of Human Beings," Canadian Journal of Philosophy 9 (1979), pp. 467-478. Aristotle has at times been criticized for assuming that things have unique functions or for equating the function of something with any activity that might be unique to it; see, for example, T. Nagel, "Aristotle on Eudaimonia ," in A. Rorty (ed.), Essays on Aristotle's Ethics (Berkeley and Los Angeles: University of California Press), pp. 7-14.

27. For a discussion of Putnam's example see below. That an inexact explanation, instruction, or concept may he what is required in certain contexts or for certain purposes is a view put forth by Wittgenstein in the Philosophical Investigations . Our language, Wittgenstein often argues, is just the way it should be if it is to serve the purposes it serves: "If I tell someone 'stand roughly here'—may not this explanation work perfectly? And cannot every other one fail too? But isn't it an inexact explanation?—Yes; why shouldn't we call it 'inexact'? Only let us understand what 'inexact' means. For it does not mean 'unusable' " ( Philosophical Investigations , I, 88).

28. M. Sintonen, op. cit. , p. 29; see also pp. 92-100. For "es-questions" and "se-answers" read "Explanation-seeking questions" and "Scientific-explanatory answers,'' respectively.

29. H. Putnam, Mind, Language and Reality (Cambridge: Cambridge University Press, 1975), p. 295.

30. Op. cit. , p. 296.

30. Op. cit. , p. 296.

31. Ibid.

32. H. Putnam, Meaning and the Moral Sciences (London: Routledge & K. Paul, 1979), p. 47.

33. M. Scriven, "Explanations, Predictions, and Laws," in H. Feigle and G. Maxwell (eds.), Minnesota Studies in the Philosophy of Science , vol. 3 (Minneapolis: University of Minnesota Press, 1962), p. 196.

34. B.C. van Fraassen, "Salmon on Explanation," The Journal of Philosophy , vol. 82, no. 11 (1985), p. 640.

35. The view that mathematics can, and in fact does, use less rigorous types of proofs, e.g., inductive evidence, has been recently defended by M. Steiner, Mathematical Knowledge (Ithaca, N.Y.: Cornell University Press, 1975), especially ch. 3.

36. See on this point my "Aristotle on Function and the Attributive Nature of the Good," in D. Depew (ed.), The Greeks and the Good Life (Indianapolis: Hackett, 1980).

37. The goals of ethics, then, are not merely psychological, as Hardie suggests, op. cit. , p. 31. Or, at least, they are not psychological in the sense we often understand this term, i.e., as implying something that is merely subjective.

37. The goals of ethics, then, are not merely psychological, as Hardie suggests, op. cit. , p. 31. Or, at least, they are not psychological in the sense we often understand this term, i.e., as implying something that is merely subjective.

38. Wittgenstein remarks at Philosophical Investigations , I, 88:" 'Inexact' is really a reproach, and 'exact' is praise. And that is to say that what is inexact attains its goal less perfectly than what is more exact. Thus the point here is what we call the 'goal'. Am I inexact when I do not give our distance from the sun to the nearest foot or tell a joiner the width of a table to the nearest thousandth of an inch?" In some cases, of course, the exact is what attains its goals—e.g., as Aristotle claims, an account in ethics is exact if it attains the level of detail or specificity that is needed in order to attain the goals of the discipline. It is also true, however, that sometimes we designate certain accounts, descriptions, or explanations as inexact because they fail to meet certain standards of exactness—e.g., Aristotle considers explanations in ethics to fail to meet the exactness of the explanations we give in mathematics. But, Wittgenstein retorts, "No single ideal of exactness has been laid down; we do not know what we should be supposed to imagine under this head—unless you yourself lay down what is to be so called. But you will find it difficult to hit upon such a convention; at least any that satisfies you" ( ibid. ).

39. For Aristotle's use of geometric proportion in his account of distributive justice, see the excellent discussion of D. Keyt, "Distributive Justice in Aristotle's Ethics and Politics ," Topoi 4 (1985), pp. 23-45.

40. This is, of course, Plato's line of argument in the Gorgias , where he argues in support of the view that rhetoric, medicine, cooking, and so forth cannot be left to mere sophistic treatment or guesswork. Too much depends on such arts, and if we go wrong when using them the effects can be at times fatal.

41. J. de Romilly, Magic and Rhetoric in Ancient Greece (Cambridge: Harvard University Press, 1975). See also the comments of T. Irwin in his translation of the Gorgias (Oxford: Clarendon Press, 1979).

42. H. H. Joachim, Aristotle: The Nicomachean Ethics (Oxford: Clarendon Press, 1955), p. 16.