apeiron 2014; aop Joseph Karbowski Is Aristotle’s Eudemian Ethics Quasi-Mathematical? Abstract: This paper challenges D. J. Allan’s famous claim that Aristotle employs a ‘quasi-mathematical’ method in the Eudemian Ethics. Allan believes that Aristotle’s use of posits introduced by ‘hupokeisthō’, ‘estō’, etc. in the treatise is ultimately modeled upon, or at least inspired by, their use in mathematics. However, in this paper I show that there are substantive procedural and epistemological differences between mathematical posits and those in the Eudemian Ethics and that, in any case, there is no textual basis for Allan’s thesis. When we look outside of the Eudemian Ethics, we find, in fact, that the use of posits is a rather common argumentative strategy of Aristotle. What specifically motivates his use of posits in the Eudemian Ethics, I suggest, is his desire to meet the criteria for proper philosophical arguments set out at the end of EE I.6. Keywords: Eudemian Ethics, hypotheses, mathematics, method, ethics DOI 10.1515/apeiron-2014-0046 In the first two books of the Eudemian Ethics (EE) Aristotle derives substantive definitions of happiness and character virtue from general claims about the soul, virtue, etc. which are supported by familiar examples from the crafts (see § 1). Even a cursory glance at the text confirms that his argumentative strategy in those books is carefully regimented. One thing in particular that has caught the attention of scholars is Aristotle’s use of posits flagged by ‘hupokeisthō’, ‘estō’, etc. in the EE.1 D.J. Allan was the first to highlight this aspect of the meth 1 Hupokeisthō: EE II.1 1218b37; hupekeito: EE II.1 1219a10; hupokeimenōn: EE II.1 1219a29; echetō: EE II.1 1219a8; estō: EE II.1 1219a6, a24; legōmen: EE II.7 1217a30; theteon: EE II.7 1217a40. In what follows I will refer to the claims that get set down as ‘assumptions’ or ‘posits’ and the act of setting them down as ‘assuming’ or ‘positing’. I refrain from referring to the claims set down as ‘hypotheses’ both because I will be questioning their similarity to the assumptions used in mathematics (often called ‘hypotheses’) and because calling them ‘hypotheses’ may create the misleading impression that they are purely provisional or tentative.  Joseph Karbowski: University of Notre Dame – Philosophy, 100 Malloy Hall, Notre Dame, Indiana 46556, United States, E-Mail: jkarbows@nd.edu Authenticated | jkarbows@nd.edu author's copy Download Date | 6/14/15 4:07 AM 2  Joseph Karbowski odology of the EE.2 In his short but classic paper, he famously argued that the prevalence of assumption vocabulary in the first two books of the treatise betrays Aristotle’s observance of a ‘mathematical pattern of deduction’ similar to what we find in Euclid’s Elements.3 Allan’s claim about the quasi-mathematical slant to the EE has proved very influential and is often cited approvingly in discussions of the methodology of the treatise.4 Nonetheless, the purpose of this paper is to dispute it. Not only are there substantive procedural and epistemological differences between the posits in the EE and their mathematical counterparts; there are no sign-posts to indicate that Aristotle’s argumentative strategy in the EE is influenced by any sort of mathematical procedure (§ 2). Moreover, when we extend our gaze beyond the EE we quickly find that the use of posits introduced by ‘hupokeisthō’, ‘estō’, etc. is a common argumentative trope found elsewhere in Aristotle (§ 4) and other ancient writers as well (§ 5). Emphatically, this is not to deny that there may be some broad or general similarities between the EE’s arguments and mathematical proofs. But, as I will argue below, the differences in the particular ways that posits function in both contexts are a major obstacle to considering the EE quasi-mathematical in any interesting sense. 1 Overview of the Argumentative Strategy of the EE In EE I.6, the work’s most extensive methodological discussion, Aristotle indicates that he is seeking rational conviction (pistis) on ethical matters by means  2 Allan ‘Quasi-mathematical method’. 3 Importantly, in characterizing the method of the EE as ‘mathematical’ or ‘quasi-mathematical’ Allan is not attributing to Aristotle the presumption that mathematical exactness is attainable in ethics (Allan, ‘Quasi-mathematical method’, 307). His qualification is only meant to suggest that Aristotle’s use of posits in the EE is modeled upon, or at least inspired by, their use in mathematics. 4 See Irwin, Aristotle’s First Principles, 601n.5; Kenny, The Aristotelian Ethics, 9n.2; Kullmann, Wissenschaft und Methode, 235; Monan, Moral Knowledge, 119n.2; Rowe, The Eudemian and Nicomachean Ethics, 71n.3; Zingano, ‘Aristotle and the Problems of the Method in Ethics’, 3078. Needless to say, these scholars do not necessarily agree with everything Allan has to say about the EE and its methodology, but none of them explicitly dispute his claim about the ‘quasi-mathematical’ nature of Aristotle’s procedure in the treatise. In a footnote at the end of his paper, Allan reports some doubts about his thesis from the audience at the Symposium Aristotelicum (Allan, ‘Quasi-mathematical method’, 318n.6), but these doubts have never found their way into print. Authenticated | jkarbows@nd.edu author's copy Download Date | 6/14/15 4:07 AM Is Aristotle’s Eudemian Ethics Quasi-Mathematical?  3 of arguments (logoi) which use the appearances (phainomena) as witnesses and examples (1216b26–8).5 Careful examination of the text confirms that he practices what he preaches. In the second book of the EE Aristotle arrives at substantive definitions of happiness and character virtue by constructing complex arguments (logoi) whose premises find support in claims that are ‘familiar to us’.6 Consider the argument by which Aristotle establishes his definition of happiness: This conclusion is clear from what we have laid down (ek tōn hupokeimenōn), namely that happiness is the best thing, the ends are in the soul and the best of goods, and the things in the soul are either a state or an activity. So since the activity is better than the disposition, and the best activity belongs to the best state, it is clear from what has been laid down that the activity of the soul’s virtue is the best thing. And the best thing is also happiness. Happiness, then, is the activity of the good soul. (EE II.1 1219a28–35, tr. Inwood and Woolf) This argument derives a ‘clearer (by nature)’ definition of happiness from an initial ‘unclear’ definition of it in conjunction with additional theses about the soul, virtue, etc.7 It can be represented as follows: 1. Happiness is the best human good and an end achievable in action. 2. Goods/ends in the soul are best among human goods. 3. Therefore, happiness must be a good (the best good) in the soul. 4. Goods in the soul are either states or activities. 5. Activities are better than states, and the best activity is correlated with the best state. 6. Therefore, happiness must be the best activity in the soul, the one correlated with the best state.  5 Aristotle follows this approach because he thinks it will help him achieve maximum agreement for his conclusions (EE I.6 1216b28–32) and discover clearer (by nature) definitions of happiness and character virtue that serve as ethical first principles (EE I.6 1216b32–5). I offer a more detailed discussion of the argumentative strategy of the EE and its central methodological chapter in forthcoming Karbowski, ‘Phainomena as Witnesses and Examples: The Methodology of Eudemian Ethics 1.6’, forthcoming in Oxford Studies in Ancient Philosophy 49. 6 Scholars often construe the phainomena to which Aristotle appeals in the EE as endoxa, but that is only partly correct. While he does cite some generally accepted beliefs in the EE (EE I.7 1217a21–2; 2.1, 1219a40) Aristotle also relies upon perceptual observation (EE II.2 1220b3–5; 2.8, 1224a20–3) and ordinary life experience (EE II.1 1218b37–1219a5; II.1 1220a22–8) in establishing the premises of his arguments. 7 Aristotle believes that the initial definition of happiness as the best good is ‘unclear’ in the sense that it does not reveal the fundamental essence of happiness, i.e. it is ‘unclear by nature’; he does not mean that it is semantically unclear (ambiguous, vague, etc.), see EE II.1 1220a13– 22. Authenticated | jkarbows@nd.edu author's copy Download Date | 6/14/15 4:07 AM 4  Joseph Karbowski 7. The best state of the soul is its excellence. 8. Therefore, happiness is the activity of the good, i.e. excellent, soul. The first premise is supported by reflective clarification of a generally accepted definition of happiness (EE I.7–8). The second is established by appeal to the popular works and universal agreement (EE II.1 1218b32–6). The fourth premise is an assumption which Aristotle thinks is obvious or unobjectionable (EE II.1 1218b36–7), and the fifth premise is established by an argument that appeals to the connection between a thing’s function (ergon) and end (telos) (EE II.1 1219a6–11). Finally, the seventh premise of the argument is established by induction (epagōgē) from particular craft examples (EE II.1 1218b37–1219a5). Aristotle reaches his preferred definition of character virtue in the EE by means of an argument that has a similar structure: Virtue is set down (hupokeitai) to be the sort of state that enables people to perform the best actions and which best orients them towards what is best; and the best and most excellent is what accords with correct reasoning. And this is the mean relative to us between excess and deficiency. It is necessary, then, that virtue of character in each case in a mean point and has to do with certain means in pleasures and pains and in pleasant and painful things. (EE II.5 1222a6–12, tr. Inwood and Woolf, slightly modified) Like its counterpart in the case of happiness, this argument derives a clearer (by nature) definition of character virtue from an initially unclear definition of it and additional theses. It can be depicted as follows: 1. Virtue is a state that enables one to perform the best actions and best orients one to what is best. 2. What is best and most excellent in a given domain is what accords with correct reasoning about that domain. 3. Correct reasoning about any domain seeks (and hits) the mean relative to us between excess and deficiency in the relevant domain. 4. Virtue of character is concerned with pleasures and pains. 5. Therefore, virtue of character is a mean state and is concerned with the mean (relative to us) in pleasures and pains. Aristotle establishes the first premise of this argument again by induction from craft examples (EE II.1 1220a29–34). The second premise is an undefended assumption of the inquiry, though it receives partial support from the defense of the third premise, which appeals to both induction and rational argument (EE II.3 1220b30–6). Finally, the fourth premise is established by appeal to the division of soul and its associated virtues (EE II.4 1221b27–1222a5). Authenticated | jkarbows@nd.edu author's copy Download Date | 6/14/15 4:07 AM Is Aristotle’s Eudemian Ethics Quasi-Mathematical?  5 Much more can be said about each of these arguments, but for our current purposes it suffices to note that they proceed from theses that Aristotle explicitly flags as posits or assumptions (‘what we have laid down’). For, this is the feature of the text that led D. J. Allan to suggest that Aristotle is deploying a ‘quasi-mathematical’ method in this part of the treatise, one that has a close affinity to what we find in Euclid’s Elements. Those familiar with the proofs in the Elements will know quite well that they have a complex structure.8 However, the feature of Euclid’s procedure of which Allan is specifically thinking is ‘the grouping of the requisite assumptions at the beginning’, i.e. his exposition of his definitions, postulates, and common notions at the start of the treatise.9 Now, Aristotle could not have been inspired by Euclid himself; the latter’s floruit (300BC) postdates the former’s death (322BC).10 However, Euclid did not operate in vacuum; we know that he drew upon the work of others in his Elements. For instance, Proclus gives Euclid credit for ‘systematizing many of the theorems of Eudoxus, perfecting many of Theatetus’, and putting in irrefutable demonstrable form propositions that had been rather loosely established by his predecessors’ (Comm. 68.7–10). But, as the following remarks from Plato and Aristotle confirm, Euclid was most certainly not the first geometer to clearly articulate his principles at the outset of discussion: You’re aware, I imagine, that when people are doing things like geometry and arithmetic, there are some things they take for granted in their respective disciplines. Odd and even, figures and the three types of angle. That sort of thing. Taking these as known, they make them into assumptions. They see no need to justify them either to themselves or to anyone else. They regard them as plain to anyone. Starting from these, they then go through the rest of their argument, and finally reach, by agreed steps, that which they set out to investigate. (Rep. 7, 510c2–d3, tr. Ferrari/Griffith) Proper too are the items which are assumed to exist and concerning which the science studies what holds of them in themselves – e.g. units in arithmetic, and points and lines in geometry. They assume that there are such items, and that they are such-and-such. As for the attributes of these items in themselves, they assume what each means – e.g. arithmetic assumes what odd or even or quadrangle or cube means and geometry what irrational or inflexion or verging means – and they prove that they are, through the  8 Typically, Euclidean proofs consist in: 1) a general statement of what is to be proved or constructed (protasis); 2) a description of a set of ‘given’ points, lines, or figures (ekthesis); 3) a redescription of what must be proved or constructed in terms of the ‘givens’ (diorismos); 4) supplementary constructions or operations (kataskeuē); 5) the proof itself (apodeixis); and 6) the conclusion (sumperasma), see Proclus, Comm., 203.1–205.8. 9 See Allan, ‘Quasi-mathematical method’, 318n.6. 10 Actually, some scholars believe that Euclid’s advancements upon his predecessors’ work in geometry were partly due to his acceptance of Aristotle’s advice about the proper organization of a science, see McKirahan, Principles and Proofs, 133–43. Authenticated | jkarbows@nd.edu author's copy Download Date | 6/14/15 4:07 AM 6  Joseph Karbowski common items and from what has been demonstrated (An. Post. 1.10, 76b3–11, tr. Barnes) Even though Aristotle could not have been appropriating Euclid’s employment of posits in the EE, these remarks raise the possibility he still might have been co-opting a similar strategy employed by mathematicians in his time. Is there reason to think that that was actually the case? Was Aristotle’s use of posits or assumptions in the EE consciously inspired by their employment in contemporary mathematics? 2 Against Allan’s Interpretation There can be little doubt that the use of posits was an integral feature of the mathematics of Aristotle’s day. However, there are strong reasons to doubt that Aristotle’s use of posits in the EE is inspired by their employment in mathematics. First, there are substantive procedural differences between the use of posits in mathematics and in the EE. As the previous quotes attest, mathematicians posit their claims at the beginning of the discussion, and they do so without any argument or justification (‘taking these as known …’). The purpose of these assumptions is to ‘make clear the meanings of the terms to be used before argumentation begins, that is, to make clear the nature of the objects to be studied’.11 One may of course set out to defend the truth of the relevant mathematical assumptions, but that defense is not part of mathematics itself (cf. Rep. 7, 510b–511d).12 By contrast, Aristotle posits claims all throughout his inquiries into happiness and character virtue in the EE (not only at the beginning), and he often defends them as soon as he assumes them. Here is a list of the claims explicitly cited as ‘posits’ in the inquiry into happiness: 1. That some goods are achievable by human action, while others are not. (EE I.7 1217a30–2) 2. That happiness is the best of the goods achievable by human action. (EE I.7 1217a39–40) 3. That all good things are in the soul or external to it, the former being more choiceworthy than the latter. (EE II.1 1218b32–6)  11 Mueller, ‘Euclid’s Elements’, 294. 12 Cf. Burnyeat, ‘Why Mathematics is Good for the Soul’, 24-33, 37–8; Mueller, ‘Euclid’s Elements’, 293–4. Authenticated | jkarbows@nd.edu author's copy Download Date | 6/14/15 4:07 AM Is Aristotle’s Eudemian Ethics Quasi-Mathematical?  7 4. That the things in the soul are states/capacities and activities/processes. (EE II.1 1218b36–7) 5. That virtue is the best state of each thing that has a use or function. (EE II.1 1218b37–1219a1) 6. That the better state has a better function. (EE II.1 1219a6) 7. That as states hold in relation to one another, so too do their respective functions (EE II.1 1219a6–8) 8. That what is best and ultimate is the end for the sake of which the others are. (EE II.1 1219a10–1) 9. That the function of the soul is to make something live, and [the function of] living is a using and being awake. (EE II.1 1219a23–5) Some of these posits are (parts of) premises of the central argument from which Aristotle deduces his clearer definition of happiness in EE II.1 (2, 4), while others play a subsidiary role and support those premises (1, 3, 5–9). Moreover, some of them are genuine assumptions or stipulations for which no argument is given (1, 4, 6, 7, 9), but others are supported by arguments of various sorts (2, 3, 5, 8). Let us take a brief look at Aristotle’s argument for claim 5 above: …let it be assumed (hupokeisthō) further concerning virtue that it is the best disposition or state or capacity of each of the things that have some use or function. This is clear from induction, since we consider things to be this way in all cases. For example, a cloak has a virtue, since it has a function and use, and its best state is its virtue. The same applies to a boat and a house, and so on, and hence to the soul, since it has a function. (EE II.1 1218b37–1219a5, tr. Inwood and Woolf) In this passage Aristotle establishes that virtue in general is the best state or disposition of a thing that has a use or function by means of induction (epagōgē) from familiar craft examples. But notice that he does this immediately after he posits that very thesis (‘let it be assumed …’). Clearly, the defense of the ethical posits in the EE is not beyond the scope of ethical inquiry. This constitutes one important difference between the posits of the EE and their mathematical counterparts. Another difference between mathematical posits and those in the EE concerns their respective epistemic statuses. The claims assumed at the outset of a mathematical discussion – at least one that proceeds synthetically – are the fundamental definitions of mathematical objects and their properties (and rules licensing operations to be performed upon them). In Aristotelian lingo, they are the first principles from which the synthetic proofs or constructions proceed. By contrast, the claims posited (and defended) in the EE are not ethical first principles; they are true claims familiar to us (as opposed to familiar by nature) from Authenticated | jkarbows@nd.edu author's copy Download Date | 6/14/15 4:07 AM 8  Joseph Karbowski which Aristotle derives his ethical principles (EE I.6 1216b32–5). Since Aristotle is generally proceeding towards first principles in the EE (cf. NE I.4 1095a30–b8), if any mathematical procedure is likely to have inspired his argumentative strategy in that treatise, it would have been the method of analysis, which is used for the discovery of the elements of mathematical proofs.13 Interestingly, Aristotle compares practical deliberation to geometrical analysis at NE III.3 1112b15–27 (quoted below), but there is no such comparison of his methodology to analysis in the EE.14 So, the text offers no evidence that Aristotle used geometrical analysis as a model for his procedure in the EE. This brings me to my third and final point. Quite generally, there is nothing in the text to suggest that Aristotle’s use of posits, or his ‘proof’ strategy more generally, in the EE is consciously or deliberately modeled upon any sort of mathematical procedure. He does briefly discuss mathematical hypotheses at EE II.6 1222b23–39, but only to contrast them with the ‘changing’ origins that he primarily cares about in that context (EE II.6 1222b41–1223a9). There is no indication in that passage or elsewhere that the use of hypotheses in mathematics exerted any influence upon Aristotle’s argumentative strategy in the EE. Arguments from silence can of course be tricky, but in this case the lack of sign-posts indicating a mathematical influence is telling. For, Aristotle is typically vocal when his procedure has a mathematical origin or analogue. It is worth dwelling upon this last point. Certain passages in the logical and scientific works confirm that Aristotle does not hesitate to admit when some doctrine or procedure in which he is interested has a counterpart in mathematics (An. Pr. I.30 46a17–22; Part. An. I.1 639b6–10). But we can also find instances of this phenomenon in passages closer to home. One case in point is Aristotle’s description of deliberation in NE III.3: … they take the end for granted and examine how and by what means it will come about; and if it appears as coming about by more than one means, they look to see through which of them it will happen most easily and best, whereas if it is brought to completion by one means only, they look to see how it will come about through this… until they arrive at the first cause, which comes last in the process of discovery. For the person who deliberates seems to investigate and analyse in the way we have said, as if  13 The classic description of the process of analysis is contained in Pappus’ Mathematical Collection. Briefly, he characterizes analysis as a procedure which begins from what one is seeking (to zētoumenon), taking it as established, and incrementally works back to something that is already known from which it can eventually be derived (Pappus, Collection 7.1–2). Admittedly, this description is something of an oversimplification; it proves extremely difficult to give a rigorous description of the logical structure of analysis, see Hintikka and Remes, The Method of Analysis; Menn, ‘Plato and the Method of Analysis’. 14 I also argue below that an ethical argument directly modeled upon mathematical analysis would have a different structure from what we find in the EE, see § 5. Authenticated | jkarbows@nd.edu author's copy Download Date | 6/14/15 4:07 AM Is Aristotle’s Eudemian Ethics Quasi-Mathematical?  9 with a diagram (and while not all investigation appears to be deliberation, as e.g. mathematical investigations are not, all deliberation is investigation); and what is last in the analysis seems to be first in the process of things’ coming about. And if people encounter an impossibility, they desist… (NE III.3 1112b15–27, tr. Broadie and Rowe) In this passage Aristotle compares deliberation to problematic analysis in geometry. In problematic analysis one posits the object (the zētoumenon) as though it were already constructed and then incrementally reasons back to a figure or object that can immediately be produced.15 This ‘last thing’ reached by problematic analysis is the very first thing generated in the construction proof. If, however, one reasons back to an impossible scenario, one gives up hope of constructing the relevant figure and generates a reductio proof. Similarly, in practical deliberation, Aristotle maintains, we start with the end (telos) to be achieved and incrementally reason back to something that it is in our power to do. If we identify an action that is possible, then it will be the first deed in a chain of events that ultimately culminates in the end. By contrast, if we reason to an action that is impossible, we desist from pursuing the goal (at least in the relevant context). The previous passage occurs in a non-common book of the NE. However, another important example of Aristotle’s forthrightness is contained in one of the books common to both ethical treatises16: The just, then, represents a kind of proportion. For the proportionate is not just a property of numbers that consist in abstract units, but of number in general; proportion is equality of ratios, and it involves at least four terms. (That discrete proportion does so is clear; but so too does continuous proportion, because it treats one term as two, mentioning it twice: e.g. as line A is to line B, so line B is to line C …) The just, too, involves at least four terms, and the ratio is the same: the people and the things are divided similarly. In that case, as A is to B, so C is to D, and so, alternando, as A is to C, so B is to D. Thus one whole (A + C), too, stands to the other whole (B + D) in the same way – which is the pairing the distribution creates, and if things are put together in this way, the pairing is done justly. Thus the combination of A with C and that of B with D is what is just in distribution, and the just in this sense is intermediate, while the unjust is what contravenes the proportionate; for the proportionate is intermediate, and the just is pro 15 ‘Problematic’ analysis attempts to discover the elements of construction proofs; ‘theoretic’ analysis discovers the elements of proofs that establish the truth of mathematical theorems, see Pappus, Collection 7.1–2. 16 Anthony Kenny cogently argued on stylistic grounds that the three books common to the EE and NE were originally composed for the former, see Kenny, The Aristotelian Ethics. His influential view has recently come under fire by Oliver Primavesi, see Primavesi, ‘Ein Blick in den Stollen von Skepsis’. I shall not take a stance on this controversial debate here. It suffices for my purposes to assume that the common books were intended by Aristotle to be part of the EE, which is corroborated by Harlfinger’s important work on the manuscripts of the EE, see Harlfinger, ‘Die Überlieferungsgeschichte der Eudemischen Ethik’. Authenticated | jkarbows@nd.edu author's copy Download Date | 6/14/15 4:07 AM 10  Joseph Karbowski portionate. (Mathematicians call this sort of proportion ‘geometrical’; for it is in geometrical proportion that one whole also stands to the other whole as each term stands to the other in a given pair.) (NE V.3 =EE IV.3 1131a29–b15, tr. Broadie and Rowe.) In this passage Aristotle offers his account of distributive justice. Briefly, his idea is that a just distribution of goods is proportional to the respective merits of the individuals. More can, of course, be said about this doctrine.17 But it suffices for the current project to note three observations about this passage. First, Aristotle deliberately defends the extension of the conception of proportion (analogia) to the ethical realm: he insists that it is not restricted to abstract numbers but applies to number in general (including concrete magnitudes, like political goods). Second, in illustrating his theory Aristotle employs principles from proportion theory that have explicit counterparts in Euclid.18 At 1131b6–7 he uses the law of alternation or alternando, which states that if four magnitudes are proportional (A is to B as C is to D), then they are also proportional alternately (enallax) (A is to C as B is to D).19 Moreover, Aristotle’s subsequent claim that the wholes themselves (A + C, B + D) will be proportional (1131b7–8) is governed by the proposition that if magnitudes are proportional separately, then they are also proportional in combination (suntethenta).20 Third, and finally, Aristotle makes an explicit reference to mathematicians at 1131b12–3 and borrows their term ‘geometrical’ for the sort of proportion in question. The two passages previously examined are of crucial importance for the current project. For, they are contexts in which Aristotle ostensibly appropriates mathematical concepts and strategies for ethical purposes. Though the two contexts are different, they both contain clues or signals that indicate his indebtedness to mathematics. In NE III.3 there is an explicit comparison to problematic analysis (… ‘as if with a diagram’) and a reference to ‘mathematical investigation’. In NE V.3/EE IV.3 Aristotle goes out of his way to justify the extension of proportion (analogia) to the ethical realm, explicitly refers to ‘mathematicians’, and uses principles from proportion theory to illustrate his view of distributive justice. These observations should lead us to expect that the text would contain some clear indication if Aristotle’s use of posits in the EE were deliberately influenced by or modeled upon their employment in mathematics. The fact that  17 For further discussion of Aristotle’s view of distributive justice, see Keyt, ‘Aristotle’s Theory of Distributive Justice’. 18 Again, this should not be taken to imply that Aristotle learned these from Euclid. The anonymous author of the scholium to book V of the Elements maintains that the relevant theory of proportion was discovered by Eudoxus. 19 See Euclid, Elements, Proposition V.16. 20 See Elements, Proposition V.18. Cf. the ‘suntethēi’ at NE V.3 =EE IV.3 1131b8. Authenticated | jkarbows@nd.edu author's copy Download Date | 6/14/15 4:07 AM Is Aristotle’s Eudemian Ethics Quasi-Mathematical?  11 that treatise contains no such indications gives us strong reason to doubt that Aristotle is consciously employing a ‘mathematical pattern of deduction’ in the EE, especially when we combine this observation with the previous considerations mentioned above. 3 A Modified Version of Allan’s Interpretation If the foregoing is correct, there is little reason to suppose that Aristotle’s use of posits in the EE is inspired by their employment in mathematics. Allan’s interpretation overstates the influence of mathematics on the methodology of the EE. However, in fairness to Allan, it must be pointed out that there is a way to modify his proposal so that it circumvents the previous criticisms. Instead of claiming that Aristotle’s use of posits in the EE is deliberately modeled upon their employment in mathematics, Allan could simply maintain that the treatise is quasi-mathematical in the sense that its arguments in fact bear a striking similarity to mathematical proofs. This similarity is easily missed, he might add, if we expect the arguments in the EE to replicate mathematical proofs exactly. For, the former are not meant to be mathematical. Nonetheless, Allan could insist that their quasi-mathematical nature is more readily apparent when we compare them to the arguments in the NE.21 Before I set out to critically evaluate the revised version of Allan’s proposal I would like to make one concession and one clarification. The concession is that Aristotle does seem to be far more hostile to the mathematization of ethics in the NE than the EE. The NE contains well-known reflections about ethical precision which distance ethics from mathematics and attempt to disabuse the audience of the expectation that mathematical exactness is possible in ethics (NE I.3 1094b11–27; cf. NE II.2 1103b34–1104a9). By contrast, the EE contains no such reflections. Aristotle never remarks upon the fluctuation of ethical phenomena or claims that ethical generalizations must hold only ‘for the most part’ and ‘in outline’ in that treatise.22 As Marco Zingano rightly notes, these are nontrivial differences which color Aristotle’s attitudes to ethical inquiry and reasoning in the two treatises,23 and they must be borne in mind in our discussion. This naturally leads me to my point of clarification.  21 I think an anonymous referee for pressing me to take this modified interpretation more seriously. 22 Cf. Zingano, ‘Aristotle and the Problems of the Method in Ethics’, 308–9. 23 See Zingano, ‘Aristotle and the Problems of the Method in Ethics’, 306–9, 312–6. According to Zingano, the EE operates in a ‘realm of opinion’ which employs perfectly valid dialectical Authenticated | jkarbows@nd.edu author's copy Download Date | 6/14/15 4:07 AM 12  Joseph Karbowski In view of the important differences between the NE and EE stated above, I must emphasize that it is not my intention to wholeheartedly deny that there may be ways in which the EE’s arguments are similar to mathematical proofs, or more similar to them than those of the NE. For instance, the use of regimented and valid deductive arguments to establish (deiknunai) substantive theses (cf. EE I.6 1216b31–2) is an important similarity between the EE’s ‘proofs’ and those of mathematics.24 Indeed, the very fact that both enterprises make some use of posits is another (superficial) similarity that I do not wish to deny. What, ultimately, I aim to show is that this particular similarity, i.e. the fact that posits play a role in the EE and mathematics, proves not to be as significant or substantive as Allan claims. 4 The Use of Posits in the Aristotelian Corpus Allan apparently presumes that the use of posits introduced by ‘hupokeisthō’, ‘estō’, etc. in the EE requires special attention because he thinks it is an argumentative strategy unique to that treatise. At any rate, he deems it the ‘most singular feature’ of the EE’s methodology. However, upon closer inspection his presumption proves false. The use of assumptions is by no means confined to the EE; it is quite prominent throughout the Aristotelian corpus. In the logical works, Aristotle makes use of posits in proofs by reductio ad absurdum and certain possibility proofs (Pr. An. I.13 32a19; I.15 34a26; II.11  proofs of ethical matters reliant upon posits. While he admits that their appeal to endoxa distinguishes the Eudemian arguments from mathematical/scientific proofs, he nonetheless maintains that they are at least structurally similar to the latter, cf. Zingano, ‘Aristotle and the Problems of the Method in Ethics’, 307–8. By contrast, the NE forgoes dialectical argumentation and operates in a particular-oriented ‘realm of truth’. In Zingano’s view, there is even less of a connection between ethical argument and mathematics in the NE, because of the nature of the generalizations involved and the intention to merely indicate (as opposed to prove or demonstrate) the truth of the matter. I should say that there is much with which I agree in Zingano’s penetrating study. The only feature of his analysis with which I wish to take issue in this paper is his presumption (shared with Allan) that the use of posits makes the EE’s arguments substantively similar to mathematical proofs. This is a minor modification, which neither compromises the main infrastructure of Zingano’s interpretation nor, as I emphasize below, precludes other comparisons between the EE’s arguments and mathematical proofs. 24 Zingano is surely correct to emphasize that the qualification ‘in some way’ (pōs) at 1216b32 is not meant to weaken the deductive power of the relevant arguments. Instead, its purpose is to indicate that the perfectly valid arguments in the EE are not causal demonstrations, cf. Zingano, ‘Aristotle and the Problems of the Method in Ethics’, 305. Authenticated | jkarbows@nd.edu author's copy Download Date | 6/14/15 4:07 AM Is Aristotle’s Eudemian Ethics Quasi-Mathematical?  13 61a37, 61b11, 61b19).25 But more apposite for our current purpose is the use of posits in his scientific works. For they cite assumptions that Aristotle himself accepts. In his scientific works Aristotle sometimes posits theses with the intention of establishing them later. For instance, in his proof of the primacy of locomotion in Phys. II.5 he assumes (hupokeisthō) that continuous motion is possible and promises to show that it is possible later (260b24).26 Other times he argues for his posits immediately before or after setting them down. In Phys. I.2, for example, Aristotle famously assumes that natural beings are changing and then immediately proceeds to support his thesis by appeal to induction(ek tēs epagōgēs) (Phys. I.2 185a12–3).27 There are also times when Aristotle assumes theses without any argument, expecting their truth to be obvious or apparent. For instance, in his treatment of time he assumes without any defense that what is bounded by the ‘now’ is time (Phys. IV.11 219a29–30).28 The previous list noticeably omits passages from the ethical treatises, especially the NE. However, Aristotle uses posits in that treatise as well.29 For example, in the function argument he says that ‘we must posit (theteon) the active life; for this seems to be called a practical life in the more proper sense’ (NE I.7 1098a6). This is a case in which he defends or justifies the posited claim immediately after stating it. Moreover, in his account of habituation Aristotle provisionally sets it down (hupokeisthō) that ‘one should act in accordance with the correct prescription’ and offers to discuss it in more detail later (NE II.2 1103b32). Finally, in the very next chapter he sets down (hupokeitai), this time  25 See Rosen and Malink, ‘A Method of Modal Proof’, 193, nn.21 and 22. 26 For additional instances of this use of posits, see Phys.II.5 196b29–31; De Cael. II.3 286a21– 2, 30–1; Part An. IV.10 689a9–10, a12–13. 27 We already saw that induction is one strategy Aristotle uses to support the posits in the EE (see II.1 1218b37-1219a5, quoted above). But he also uses other strategies to defend his posits including an appeal to general consensus (De Cael. III.4 302a15–9), linguistic usage (NE V.1, 1129a2–5), and previous treatments of the matter (Meteor. III.2 372b9–1; De Insomniis II 460b1– 3). 28 See also Phys. IV.10 218a18–9; VIII.10 266a26–8; De Cael. II.8 289b5–6; Meteor. II.6 363a30– 2; De Insomniis 1 459a11–2. 29 One might find the use of posits in the NE surprising, especially given Aristotle’s remarks about ethical precision in NE I.3. However, it must not be overlooked that the cautionary remarks in that chapter pertain solely to our expectations about the results of ethical inquiry. Aristotle’s point is that we must not expect his conclusions (cf. the ‘sumperainesthai’ at NE I.3 1094b22) to exhibit the same sort of precision or exactness as those of mathematical inquiry. He does not say here, or anywhere else, that we cannot use similar methods or procedures to reach results of varying precision in mathematics and ethics. Besides, as we will see in the next section, the (mere) use of posits or hypotheses is not a distinctively mathematical strategy. Authenticated | jkarbows@nd.edu author's copy Download Date | 6/14/15 4:07 AM 14  Joseph Karbowski without any argument, that ‘excellence is a disposition to act in the best ways in relation to pleasure and pains’ (NE II.3 1104b27). Although it must be conceded that the use of posits is more prominent in the EE than the NE, the previous survey confirms that they also have a role in the latter treatise – and throughout the corpus. This observation, in turn, suggests that the use of posits introduced by hupokeisthō’, ‘estō’, etc. is a standard weapon in Aristotle’s methodological arsenal, not a strategy peculiar to the EE which requires a special explanation. Though this lesson must be borne in mind, by itself it does not undermine the modified version of Allan’s thesis. For, he can still insist that the EE is quasi-mathematical in virtue of its more prominent use of posits than the NE. In order to block this response, or at least significantly diminish its force, we must dissociate the use of posits from mathematical practice. Can that be done? 5 The Use of Posits by Fourth Century Authors It may seem like an uphill battle to try to sever the connection between posits/ positing and mathematical practice. For, it is a staple of mathematical proof and inquiry. However, it is important to distinguish the use of posits in general from the specific uses to which posits are put in different contexts. Mathematics is one such context, but it is by no means the only context in which posits were employed in the Ancient Greek world. A broad survey of the use of the term ‘hupothesis’ and related vocabulary by fourth century authors reveals that positing was a relatively common strategy with nuanced applications in rhetoric, medical theory, among other disciplines.30 This observation has important implications for our evaluation of the modified version of Allan’s thesis. A posit (hupothesis) is, literally, something ‘set down’, typically at the outset of some process or activity, which guides its subsequent development.31 The process or activity in question can be practical (action) or intellectual (reasoning). Let me start with some practical uses of the notion. Readers of Aristotle’s ethical treatises will already be familiar with this sort of use of posit vocabulary, because it occurs in his characterization of deliberation: deliberators start by ‘setting out’ (themenoi) the end and then consider what actions available to  30 Helpful discussions of this terminology can be found in Schiefsky, Hippocrates: On Ancient Medicine, 111–5, 120–26; Wolfsdorf, Trials of Reason, 158–60. Robinson’s discussion is restricted to Plato, but it is still helpful, see Robinson, Plato’s Earlier Dialectic, 93–100. 31 Cf. Schiefsky, Hippocrates: On Ancient Medicine, 112; Wolfsdorf, Trials of Reason, 158. Authenticated | jkarbows@nd.edu author's copy Download Date | 6/14/15 4:07 AM Is Aristotle’s Eudemian Ethics Quasi-Mathematical?  15 them promote it (NE III.3 1112b15-6). In fact, Aristotle explicitly calls the end of deliberation a ‘hupothesis’ in the EE (EE II.10 1227a8–9; II.11 1227b28–30). A related, albeit more grandiose, practical use of the term is found in Isocrates, when he speaks of the hupothesis of a life, i.e. a broad structuring principle which gives a life order and meaning (Epistles, 6.9–10; Orations, 1.48, 8.18, 6.90). The foundational principles of a polis are also referred to as ‘hupotheseis’ by Demosthenes (Oynthiac 2, 10.8; Epitaph, 27.4; cf. Aristotle, Pol. VI.2 1317a40– b3). Of the intellectual uses of posits, their role as starting points or principles of mathematical proofs is perhaps their most familiar (cf. An. Post. I.10, 76b3– 11; Rep. VII, 510c2–d3, quote above). But positing also plays a role in mathematical analysis.32 For, one begins analysis by positing the theorem to be proved or the object to be constructed (Pappus, Collection, VII.1–2). Posits also make an appearance in medical theory. For instance, the author of On Ancient Medicine argues at the beginning of the treatise that medicine has no need for some ‘newfangled hypothesis’ (On Ancient Medicine, I.3). Though the attitude here is critical, it suggests that others do base their medical theories on posited foundational principles, like the hot and cold or wet and dry.33 In the Platonic dialogues we also find the term ‘hupothesis’ used in a less elevated sense to denote an interlocutor’s proposal or claim (Euthyphro 11c4–5; Gorgias 454c1–4; Hippias Major 302e9–303a1). This survey is inevitably brief and incomplete, but it already reveals that posits take a variety of forms in a variety of contexts. This observation does not imply that the term ‘posit’ is ambiguous or ‘homonymous’ in Aristotelian lingo, any more than the fact that archai, ‘origins’ or ‘starting points’, take different forms in different contexts, e.g. there are practical archai and theoretical archai, makes the term ambiguous (cf. Met. V.1). For, there is a common account applicable to posits of all the aforementioned varieties, viz. something set down as a starting point of some further process of action or deliberation. Nonetheless, this variation in the forms and uses of posits confirms that the notion of a posit and the practice of using posits did not likely originate in  32 The famous ‘geometer’s technique from the Meno is modeled upon the method of analysis and its use of posits, see Menn, ‘Plato and the Method of Analysis’. This is an avowed case where philosophy appropriates a mathematical strategy, but it must be stressed that what is appropriated is a particular method or rule-governed strategy for using posits, not the use of posits in general, cf. Robinsion, Plato’s Earlier Dialectic, 99–100. 33 It is worth pointing out that what the author of On Ancient Medicine rejects is, not the use of foundational principles per se, but the need for esoteric or abstract foundations with little basis in experience, cf. Schiefsky, Hippocrates: On Ancient Medicine, 113–5. Authenticated | jkarbows@nd.edu author's copy Download Date | 6/14/15 4:07 AM 16  Joseph Karbowski any specific domain or discipline. Instead, as Richard Robinson claims, ‘it is a natural and inevitable notion that arises wherever men use any prolonged reflection’.34 Now, this is not to deny that posits play an important and distinctive role in mathematics. They certainly do. But it is also important to acknowledge that they play no less important and distinctive roles in other disciplines, e.g. rhetoric, political theory, and medicine. What this suggests is that if Allan or anyone else is to claim that the use of posits in the EE bears some interesting and significant similarities to their use in mathematics (as opposed to one of these other disciplines that also deploy posits), he or she must show that their particular application in the EE is similar to their application in mathematics. Is that the case? I believe that the answer to the previous question is negative. For, we already saw that there are substantive epistemological and procedural differences between the EE’s posits and those that serve as principles of synthetic proofs in mathematics (§ 2). These differences stem from the different directions in which these two endeavors are proceeding. The EE, recall, is moving towards the first principles. Consequently, its posits are preliminary (unclear by nature) definitions of ethical topics and supplementary claims, and Aristotle derives from them substantive definitions (clearer by nature) of the relevant concepts. By contrast, Euclid and the other mathematicians that postulate their definitions and additional claims/construction rules at the outset of discussion are proceeding from first principles. The EE’s direction, i.e. going towards the principles, suggests that the use of posits in analysis may be a more promising analogue. However, this turns out not to be the case upon closer inspection. For, in analysis what gets posited is either the theorem to be proven or the figure to be constructed. One then works back to something familiar that one can use to begin the theorematic or construction proof. The exact ethical analogue of this procedure would begin by positing some substantive definition (‘character virtue is a mean’) and then reason from that to the preliminary claims needed to establish it (‘virtue orients us to what is best’, etc.). But that is not what Aristotle does in the EE. He does not start from substantive definitions; he starts from, and posits, preliminary definitions of ethical topics which are unclear by nature and deduces from them (with the help of auxiliary claims) definitions that are clearer by nature (§ 1).  34 See Plato’s Earlier Dialectic, 99. Authenticated | jkarbows@nd.edu author's copy Download Date | 6/14/15 4:07 AM Is Aristotle’s Eudemian Ethics Quasi-Mathematical?  17 6 Why Use Posits in The EE? Before concluding I wish to undertake one last order of business. Although Aristotle’s use of posits is not confined to the EE, we may still legitimately wonder why he chose to deploy them in that treatise. What purpose does the use of posits serve in the EE? What does it add to Aristotle’s arguments or his argumentative strategy? Marco Zingano has suggested that the use of posits fits nicely with the ‘dialectical’ nature of the EE’s methodology.35 This proposal has merit because posits have a role in Aristotelian dialectic (Top. III.6 119b35 ff., VII.1 152b17 ff.). But here I would like to pursue a different suggestion, which is in principle compatible with Zingano’s proposal.36 At EE I.6 1216b40–1217a17 Aristotle sets out various criteria which proper philosophical arguments must meet. Proper philosophical arguments, he maintains, must be appropriate (oikeion) to the subject matter (1216b40– 1217a10); keep the causal principles distinct from the derivative facts that they establish (1217a10–1); attend appropriately to the appearances (1217a12–4); be valid (1217a14–6); and have true premises (1217a16–7). Aristotle takes these criteria very seriously, because they effectively prevent fallacious, sophistical reasoning from derailing the investigation.37 My suggestion, then, is that the use of posits is first and foremost motivated by Aristotle’s desire to ensure that his arguments meet these various criteria. By setting down and flagging the main premises of his arguments as posits, Aristotle can clearly keep them separate from the substantive conclusions that he is inferring from them (cf. EE II.11219a28–9, II.4 1221b37–9). This not only allows him to clearly distinguish the ethical principles from what they can be used to establish; it also makes it easier for him to ensure his arguments are valid and rely upon true claims appropriate to the subject matter. Admittedly, confusing the first principles with the derivative theorems may seem like a simple-minded mistake that only a beginner would make, but it was a real problem in Aristotle’s time.38 In the Phaedo Socrates applauds the method of hypothesis for its ability to help keep us from jumbling our principle and its consequences ‘as the debaters do’ (101e1–3), which suggests that this was an ongoing problem in philosophy at the time; and there is some evidence that this  35 Zingano, ‘Aristotle and the Problems of the Method in Ethics’, 307–8. 36 This explanation is also compatible with a more scientific interpretation of the EE’s methodology, see Karbowski, ‘Phainomena as Witnesses and Examples: The Methodology of Eudemian Ethics 1.6’, forthcoming in Oxford Studies in Ancient Philosophy 49. 37 See Dirlmeier, Eudemische Ethik, 186–7. 38 See Mueller, ‘Euclid’s Elements’, 293. According to Mueller, Euclid’s ‘axiomatic’ method was developed as an antidote to just these mix-ups in mathematics. Authenticated | jkarbows@nd.edu author's copy Download Date | 6/14/15 4:07 AM 18  Joseph Karbowski problem occurred in mathematics as well (cf. An. Pr. II.16 64b28–65a9). If the current proposal is correct, Aristotle’s use of posits in the EE is primarily intended to prevent him from making a similar conflation of principles and theorems in the ethical realm. This interpretation is difficult to confirm, but it at the very least has the virtue that it entails that Aristotle held himself to the very philosophical standards described at the end of EE I.6.39 7 Conclusion Everything is similar to everything else in some way. But not all of the similarities are interesting or important. While I do not want to deny that, at a very general level, the arguments in the EE bear some similarities to those in mathematics, the specific way that posits function in the EE is not similar enough to how they function in mathematics to consider that treatise or its methodology ‘quasi-mathematical’ in any interesting sense. In concluding, I should stress that my point is not primarily terminological. I am not out to banish the application of the term ‘quasi-mathematical’ to the EE. What is ultimately driving the current project is the worry that too much preoccupation with Aristotle’s employment of posits in the EE or its putative connection to mathematics may cause us to overlook other important features of its arguments and broader methodology which are interesting in their own right. If we agree with Allan that pretty much everything that Aristotle says about ethical methodology in EE I.6 is ‘unsensational’ and consequently ignore that chapter, we will be neglecting very sophisticated and thoughtful reflections about ethical method that deserve as much attention as their counterparts in the NE. Naturally, the best way to refute Allan’s presumption about EE I.6 is to offer a broad-ranging interpretation which clearly displays the merits of the relevant method. Regrettably, that has not been done in the current paper. But I at least hope that, as Socrates said of recollection, it will make us ‘energetic and keen on the search’ for a more detailed treatment of EE I.6.40  39 I should point out that my proposal is consistent, and in fact jives quite well with, the hypothesis that the EE is directed at a more rigorous philosophical audience than the NE, cf. Simpson, The Eudemian Ethics of Aristotle. For, in that case, it would be especially important for Aristotle to use whatever means necessary to ensure that his arguments meet all of the criteria for philosophical argumentation set out in EE I.6. 40 A more detailed treatment of the EE I.6 method can be found in Karbowski, ‘Phainomena as Witnesses and Examples: The Methodology of Eudemian Ethics 1.6’, forthcoming in Oxford Studies in Ancient Philosophy 49. Authenticated | jkarbows@nd.edu author's copy Download Date | 6/14/15 4:07 AM Is Aristotle’s Eudemian Ethics Quasi-Mathematical?  19 Bibliography Allan, D. J. “Quasi-mathematical method in the Eudemian Ethics”. In Aristote et les Problèmes de Méthode, edited by S. Mansion, 303–18. Louvain: Publications Universitaires, 1961. Barnes, J. Aristotle: Posterior Analytics. Oxford: Clarendon Press, 1993. Broadie, S. and C. J. Rowe. Aristotle, Nicomachean Ethics. Oxford: Oxford University Press, 2002. Burnyeat, M. “Plato on Why Mathematics is Good for the Soul”. 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