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Theoretical Ethics

Aristotle and Mathematical Ethics for Happiness?

Raymond M. Herbenick
University of Dayton

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ABSTRACT: Philosophers since antiquity have argued the merits of mathematics as a normative aid in ethical decision-making and of the mathematization of ethics a theoretical discipline. Recently, Anagnostopoulos, Annas, Broadie and Hutchinson have probed such issues said to be of interest to Aristotle. Despite their studies, the sense in which Aristotle either opposed or proposed a mathematical ethics in subject-matter and method remains unclear. This paper attempts to clarify the matter. It shows Aristotle’s matrix of exactness and inexactness for ethical subject-matter and ethical method in the Nicomachean Ethics. Then it probes a resultant puzzle from the matrix, namely, the HL model of the happy life without consideration of mathematical justice (Bk. III) and the HJL model of the happy life with such consideration (Bk. V). Finally, it examines Aristotle’s twofold rationale for differentiating these two models in his overall moral feedback loop system: differences in the intellectual virtue of good deliberation; the priority of friendship over justice for the happy life. This suggests Aristotle saw no objection either to using mathematics as an aid to ethical decision-making for a happy life, or to mathematizing at least some parts of an ethical theory of eudaimonism.

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I. The problem of math ethics in modernity and antiquity

Mathematizing ethics to become scientific ethics has long been a dream of some philosophers, dating to both the Academy and perhaps the Lyceum. In modern philosophy Jeremy Bentham, (1) G.E. Moore, (2) and Nicholas Rescher (3) have tried to mathematize ethics. Such mathematizations square with Quine's view that mathematizing inexact things by way of exact methods marks a successful reduction of an art to a science. (4)

But many philosophers take the dream of a mathematical ethics to be a nightmare. Some have argued in principle against formal moral thermometry; others have posed practical difficulties. Mill's critique of Bentham's moral thermometry calculus is well-known and widely accepted. Recently, MacIntyre has argued for a return to Aristotelian virtue ethics from the modern metaethical impulse to mathematize ethics, not fully cognizant of Aristotle's own flirtations with mathematical ethics. (5) Jonsen and Toulmin have complained about "overintellectualism" in normative ethical theory, targeting Sidgwick's arguments for a scientific ethics. (6) This dispute about mathematical ethics has its roots in antiquity with Aristotle. While some contemporary proponents gloat over the mathematization of some of Aristotle's practical wisdom in his Economics and Politics in such fields as microeconomics, macroeconomics, and political econometrics, the received view is that Aristotle himself opposed mathematical ethics in his Nicomachean Ethics. But did he? (7)

Scholars have recently clarified Aristotle's concerns about mathematical ethics often expressed as the problem of formal exactness (akribeia) and of material inexactness (tupos). Georgios Anagnostopoulos has meticulously examined the issue of exactness and has concluded that Aristotle is inconsistent in his views, sometimes affirming and sometimes denying the inevitability of inexactness in ethical subject matter. (8) However, by assuming the parity of all virtues for Aristotle and by overlooking an ordinal relationship between friendship and justice for Aristotle, he misses the role of mathematical ethics in Aristotle's ethics.

The views of three other scholars (Sarah Broadie, (9) Julia Annas, (10) and D. S. Hutchinson (11)) better glimpse the mathematical ethics issue for Aristotle. All three suspect a connection between the virtues of friendship and justice for Aristotle's notion of the happy life. But they do not notice the proper ordinality between the virtues of friendship and justice (i.e., the nonparity of these virtues for Aristotle with friendship as prior to justice). Nor do they fully probe Aristotle's use of mathematizations in some of the special justice virtues.

To advance the discussion, I will first sketch Aristotle's matrix on exactness in subject-matter and in method. Then I will examine in order: a puzzle it presents for his ethical decision-making theory; his attempted resolution of the puzzle by a moral feedback loop system; and finally his rationale for partial mathematization of ethics.

II. Aristotle's solution to the problem of math ethics

In his Nicomachean Ethics (1094b 11-27; 1104a 1-9; 1141a 16-19), Aristotle lauds exact methods for exact subjects and inexact methods for inexact subjects. But he warns against using inexact methods on exact subjects and exact methods on inexact subjects. (12)

Two kinds of nonreductive methodological successes are possible for Aristotle: A) using an exact method on an exact subject-matter (e.g., geometricians constructing a proof); and B) using an inexact method on an inexact subject-matter (e.g., doctors healing this patient and navigators sailing this course). And two kinds of reductive methodological failures are possible: C) using an exact method on an inexact subject-matter (demanding of a rhetorician a demonstrative proof instead of a merely persuasive speech); and D) using an inexact method on an exact subject- matter (accepting a probable reasoning from a mathematician instead of a demonstrative proof). For Aristotle, it is a mark of an educated person to nonreductively pursue both A and B. But it is foolish to reductively pursue C and D. Respect for both method and subject-matter are needed as given by the following matrix:

Table 1

However, in Aristotle's Nicomachean Ethics a puzzle about exactness and inexactness in subject-matter and method appears. Two models of decision-making seem to conflict with one another for agent happiness. These are: the happy life in Book III without the need for character justice or institutional justice (HL hereafter); and, the happy life in Book V with the need for character justice and institutional justice (HJL hereafter). The two models may be summarized as follows.

(1) The HL Model

This is Aristotle's inexact doctrine of telos and arete to account non-mathematically for errors (hamartia) and mistakes:

(apate) in habitually choosing virtuous actions, and

feelings to fashion character states along the lines of his helmsman's inexact navigational model (kubernetes) rather than a rhetorician's inexact model (hretorike) or geometrician's exact model (geometrikos).

The HL Model in Book III shows Aristotle's inexact algorithm for a happy life through choices of actions and feelings "up to us" alone or with "partners in deliberation" we enlist on major issues (1112b 8-12) to help one fashion one's moral character. (13) While friendliness to others (not friendship) is discussed at length, justice is not. Aristotle's novel notice is that of an iterative cycle in moral decision-making by way of a non-mathematical determination of an intermediate action or feeling deliberately desired to attain this good wished for to promote the good of happiness — but without considering justice. The use of mathematical means such as the arithmetical average of 6 pounds of food by an expert trainer for Milo the legendary four-time Olympic wrestling champion and for a neophyte gymnast when 10 pounds is too much to eat and 2 pounds not enough is inappropriate. (14) Here Aristotle disallows mathematizations in ethical decision-making!

(2) The HJL Model

This is Aristotle's exact doctrine of telos and arete to account mathematically for errors (hamartia) and mistakes (apate) in choosing just actions and feelings to fashion character states along the lines of his geometrician's exact line-dividing model (diaireseis) rather than the juryman's inexact model (dikastes).

The HJL Model in Book V shows Aristotle's combined inexact and exact algorithm for a happy and just life through choices of actions and feelings to help one fashion one's moral character. The algorithm includes mathematical means for several of the specific virtues of justice such as rectificatory (fair balance / arithmetic mean of the judge to restore numeric equality, 1132b 1-5), distributive (fair share/geometric mean of the mathematician to divide proportionate equality, 1131b 5-24), and exchange (fair price/proportional reciprocity in the market place based on commensurate needs to hold a city together, 1132b 21 - 1133b 28). (15) But it also includes non-mathematically conceived kinds of justice in fair law to include decency and equity. Aristotle's novel notice is of an iterative cycle of mixes of exact and inexact justice in moral decision-making by way of mathematical determinations of some virtues of justice for a happy and just life. Choice of an intermediate action and feeling deliberately desired to attain this good wished for promotes the good of happiness with justice. Here Aristotle allows for mathematizations in ethical decision-making!

But how can Aristotle square his apparent two virtue models (a happy life without the need for justice; a happy life with the need for justice)? The intellectual virtue of good deliberation is crucial for distinguishing two models and noting their unity. Both HL and HJL precede Aristotle's discussions of the intellectual virtues in themselves and in relation to the moral virtues. In particular HL and HJL appear before his analysis of the intellectual virtue of good deliberation. "Good deliberation is correctness that reflects what is beneficial, about the right thing, in the right way and at the right time." (1142b 28) (16) Good deliberation thus is HL-compatible. But, unconditional good deliberation is superior. "The unconditionally good deliberator is the one whose aim expresses rational calculation in pursuit of the best good for a human being that is achievable in action." (1141b 12-14) (17) Hence, it is HJL-compatible.

Good deliberations, whether unconditional as in HJL or not as in HL, suppose a common moral decision-making feedback loop system found in the Nicomachean Ethics (1111b 5 - 1113b 24) and shown in TABLE 2 below. (18) It is the first such moral decision-making feedback loop system in Greek antiquity and similar to C.S. Peirce's work on iterative decision-making. Aristotle writes that "decision seems to be most proper to virtue, and to distinguish characters from one another better than actions do." (1111b 5-6) (19) A deliberator would inquire and analyze in the way described, "as though analyzing a diagram." (1112b 20) (20)


For Aristotle, actions and feelings are "up to us" as is moral character development through iterations of excellent actions and feelings to form habits of acting well and feeling well that settle into highly moral characters. Aristotle writes that "... by having the sort of character we have we lay down the sort of end we do." (1114b 23) (21) This is character filtering of goods wished for as ends. And he notes that "... each type of activity produces the corresponding character. This is clear from those who train for any contest or action, since they continually practice the appropriate activities." (1114a 7-9) (22) This is character shaping by habituation. Character filtering and character shaping are thus both "up to us" for Aristotle in an iterative loop system amenable to a happy life with or without the need for deliberations of justice.

Why then does Aristotle differentiate the two models? Two reasons come to mind: 1) the bearing of intellectual virtues on moral virtue through good deliberation and unconditionally good deliberation; and 2) the bearing of the moral virtue of friendship on the virtue of justice.

For Aristotle, good deliberation does not require rational calculation with mathematical means as in the work of a doctor healing wounds, of a navigator sailing the Aegean, or of an athletic trainer mentoring an Olympic-bound wrestler. But unconditionally good deliberation can require it as in the work of an allocator of awards, of a magistrate trying to make things whole by restoring equality, or of an appraiser establishing a fair market price. Hence, the HL model prior to Aristotle's consideration of the difference between good deliberation and unconditionally good deliberation mirrors more the moral decision-making of a good deliberator since rational calculation of variables for action is not at issue.

The HJL model mirrors more the moral decision-making of an unconditionally good deliberator than the mere good deliberator since rational calculation of variables for action is at issue in fair distributions, rectifications, and exchanges. Hence, the applied use of mathematics in special situations is ethically required and not just tolerable. The metaethics of moral decision-making with special focus on deliberation invites partial mathematization of ethics as Aristotle saw. Indeed, Aristotle offered no 'in principle' objection to mathematization of ethics, his reaction to Speussipus' accession to the leadership role in the Academy notwithstanding. Aristotle saw mathematical ethics as a concession to exactness in handling cases of injustice for the sake of a happy and just life, an exception to the role and rule of inexactness in ethics when good deliberation is inadequate and when unconditionally good deliberation is required.

But Aristotle also mathematizes ethics in part for another very good reason as is evidenced by his two models HL and HJL. Aristotle sees a fundamental connection between friendship and justice, a connection still not widely understood nor appreciated. Anagnostopoulos, for example, indexes neither the terms 'friendship' nor 'justice in his study of the plausibility of ethical theory for Aristotle's practical wisdom. (23) The mathematical means for justice arise only when one of two conditions arise: either the virtue of friendship is lacking, or, the virtue of friendship is excessive. In such cases, injustices can clearly arise and be in need of adjudication. The use of mathematical means does not necessarily arise when the virtue of friendliness is lacking or is excessive. But it might arise. However, friendship changes based on pleasure to self, on usefulness to self, or on good character for the other can entail injustices in need of arbitration.

While Aristotle seems to have two different models of ethics (one inexact for a happy life and one exact for a happy and just life), such appearances are deceiving. Aristotle's general sketch of inexact ways to happiness is primary. But it need not exclude exact ways through special justice at times. As the Peripatetic observed poignantly in his Nicomachean Ethics:

Moreover, friendship would seem to hold cities together, and legislators would seem to be more concerned about it than about justice. For concord would seem to be similar to friendship and they aim at concord above all, while they try above all to expel civil conflict, which is enmity. Further, if people are friends, they have no need of justice, but if they are just they need friendship in addition; and the justice that is most just seems to belong to friendship. (1155a 27-28) (24)

Hence, although the inexactness of friendship is studied after both the exactness and inexactness of justice, the virtue of friendship is ethically prior to the virtue of justice. This suggests that HL is ethically prior to HJL as well. It also implies for Aristotle that mathematization of ethics is a special case and one not to be dismissed lightly as some critics believe since mathematical exchange justice rather than nonmathematical friendship holds cities together (1132b 36). (25)

For Aristotle, inexactness is the general rule in normative ethics and exactness the exception, not vice-versa. His model HL is perhaps designed for life in a state of nature or on a Greek frontier without the need for exactness and thus without the need for justice due to the dominance of the needed virtue of friendship in a complete life and of the helpful virtue of friendliness or both. His model HJL is perhaps designed for life in a state of governance with the need for some exactness and thus with the need for justice due to the lack of the necessary virtuous friendships for a complete life and of the helpful virtue of friendliness. Hence, mathematical ethics need not arise when friendships flourish (and injustices do not occur) but necessarily arises when friendships flounder (and injustices occur). If Aristotle's analysis is correct, metaethics needs to account for at least the partial mathematization of ethics. (26)

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Bibliography (abbreviations for texts cited)

AGE: Georgios Anagnostopoulos, Aristotle on the Goals and Exactness of Ethics, (Berkeley: University of California Press, 1994).

AMT: Raymond M. Herbenick, "Augustine's Moral Thermometer of Human Goodness," The University of Dayton Review 22:3 (Summer 1994), 251-294.

AOC: Albert R. Jonsen and Stephen Toulmin, The Abuse of Casuistry: A History of Moral Reasoning (Berkeley: University of California Press, 1988).

CCA: D. S. Hutchinson, "Ethics" ed. by Jonathan Barnes, The Cambridge Companion to Aristotle (Cambridge: Cambridge University Press, 1995), 195-232.

ETH: G. E. Moore, Ethics (New York: Oxford University Press, 1965).

EWA: Sarah Broadie, Ethics with Aristotle (New York and Oxford: Oxford University Press, 1991).

IVT: Nicholas Rescher, Introduction to Value Theory (Englewood Cliffs: Prentice-Hall, 1969).

MOH: Julia Annas, The Morality of Happiness (New York and Oxford: Oxford University Press, 1993).

NE: Aristotle, Nicomachean Ethics, trans. by Terence Irwin (Indianapolis: Hackett Publishing Company, 1985).

PHP: Samuel Enoch Stumpf, Philosophy: History & Problems, 3rd edition (New York: McGraw-Hill, 1983).

TAT: W. V. Quine, Theories and Things (Cambridge: Harvard University Press, 1981).

TRV: Alasdair MacIntyre, Three Rival Versions of Moral Enquiry: Encyclopaedia, Genealogy, and Tradition (Notre Dame: University of Notre Dame Press, 1990).


(1) PHP, 346. British social reformer Bentham proposed a moral thermometer to formally measure the difference between pains and pleasures on 7 quantifiable scales netted out in a "lot" measurement of the effects of an action on relevant parties. These factors were intensity, duration, certainty, and propinquity of the pleasure itself, and also fecundity, purity and extent of the pleasure in its circumstantial context. See Bentham's An Introduction to the Principles of Morals and Legislation (1789). Bentham's assumption was that individuated pleasures and pains were quantifiable and thus both measurable in cardinal and not just ordinal ways. (This is a revised paper based on one prepared for and presented at the Aristotle and Contemporary Science International Conference at the University of Thessaloniki, Greece in September 1997. My thanks to colleagues Jane Zembaty and Paul Benson as well as to Conference participants for their critiques.)

(2) ETH, 8. Moore developed in his Ethics an agathistic utilitarianism with a scale for theoretical calculation of the total net utility of pleasure and pain effects of an action on all possible parties. Admitting his moral thermometer had little practical value, Moore did show what was required for a total utility view inclusive of graded impacts on gods and even minimally sentient creatures.

(3) IVT, 54. Nicholas Rescher set forth some of the requirements for formal and material thermometry to measure values both cardinally and ordinally, including moral values. His Introduction to Value Theory helped revive interest in the mathematical ethics program once more.

(4) TAT, 148-150 passim. This occurs when a mathematization develops within a subject-matter rather than merely being applied to it from without and is continuous with the growth of precision until blossoming with algorithms and proof procedures according to Quine.

(5) TRV, 177, 186.

(6) AOC, 7, 95. Jonsen and Toulmin have even bemoaned Augustine's "dream of an ethical algorithm" without fully understanding the North African Church Father's ordinal (rather than cardinal) measurement of morals program in response to Stoic Paradox 3 cited by Cicero. See my article on mathematical ethics in Augustine's philosophical writings in AMT, 254-257.

(7) At the direction of Plato's testamentary will, Plato's nephew Speussipus was named head of the Athenian Academy rather than the Macedonian light of the school, Aristotle. Fortunately, in my view, Aristotle read the politics of the situation correctly and left his Athenian home of twenty years for travel and empirical research rather than work under someone with a strong mathematical agenda in mind to the detriment of other areas of inquiry. Did Aristotle ever try to mathematize ethics in whole or in part upon his return to Athens where as a non-citizen he leased the Lyceum and devoted himself to studies of methodological, theoretical, practical, and productive wisdom?

(8) AGE, 156-162 passim.

(9) EWA, 17-19 passim. Sarah Broadie in particular in her book, Ethics With Aristotle, takes Aristotle to task for his views on imprecision in ethics. On the one hand, she claims he is "not promising precision or certainty in ethics" and is promising only "as much clearness as the subject-matter admits of." Hence, for Aristotle this means that "in ethics we have to be content with generalisations true only for the most part." On the other hand, she claims that he is making a statement about method in ethical philosophy: universalizing practical judgments always results in exceptions. And in this regard she thinks "his remarks are questionable." In fact, she goes on to claim that: "It would seem, then, if we distinguish between the subject-matter of philosophical ethics and the philosophical analysis of that subject-matter, that the conclusions of the analysis can (though Aristotle seems to deny it), possess the universality and exactness of mathematical truths." (EWA, 18) Ultimately, Broadie thinks Aristotle relies on a "misplaced analogy" — that pure mathematics is undermined because the physical objects to which it may be applied are not spherically exact or unequal by any exact amount. In effect, she believes that Aristotle is right on the inexactness of the subject-matter of practical judgments in ethics but either mistaken in not applying exact method to the inexact subject-matter or wrong in applying inexact method to the inexact subject-matter. As a result, she misses an important connection between justice and friendship and thus Aristotle's own partial mathematization of ethics in the directions she herself seems to argue for.

(10) MOH, 316. Annas notes Aristotle's refusal to reduce justice to one basic type — either of the character of the individual agent type in classical philosophy or of the institutional type in modern philosophy. But she also fails to grasp the important ordinal connection between justice and friendship for Aristotle and thus Aristotle's own partial mathematization of ethics. Also see MOH, 291. Annas writes that "justice and friendship can be exercised among family and friends, but its proper field is that of the society in which the agent lives." In fact, Annas holds that the entry point for the good of others beyond good for oneself is "most prominently justice" rather than friendship or commitment to family and colleagues (philia). See MOH, 223. Annas thinks justice is directly connected to the goods of others and thus deserves special treatment apart from friendship. These views are contrary to Aristotle's claim that friendships with family and friends are necessary for happiness and that justice need not arise in regard to family and friends. See NE 1155a 27-28.

(11) CCA, 222-224 passim. D.S. Hutchinson in his article, "Ethics," notes two kinds of institutional social justice mathematically treated by Aristotle (distributive and exchange) but fails to note that of rectificatory justice treated by the Peripatetic. He also skews Aristotle's notion of friendship towards same status collegial commitments rather than to different status collegial and family friendships. But he does find an intimate connection between justice and friendship in Aristotle. "Justice is very close to Aristotle's mind throughout his treatment of friendship." (CCA, 229) Aristotle's use of a 'mathematical approach' to a proportionate value quarrel on the worth of wisdom exchanged for a fee in a failed teacher-pupil relationship is characterized by Hutchinson as unsuccessful.

(12) NE, 3-4, 35-36, 157. See AGE, 162 for an incomplete account of this matrix.

(13) Ibid., 62.

(14) Ibid., 43.

(15) Ibid., 124-131.

(16) Ibid., 163.

(17) Ibid., 158.

(18) Ibid., 59-65.

(19) Ibid., 59.

(20) Ibid., 63.

(21) Ibid., 70.

(22) Ibid., 67.

(23) AGE, 463. Also see 363-366 for his discussion of the deniability of ethical theory as argued by the likes of Sartre, McDowell, Baier, Williams, Nussbaum, Toulmin and Jonsen.

(24) Ibid., 208.

(25) Ibid., 128.

(26) What bearing does Aristotle's partial mathematization of ethics have for contemporary thought? I think it impacts recent work on the subject-matter of moral psychology and the method of moral psychology in fundamental ways. Aristotle's HL and HJL models are highly relevant as criticisms of recent moral psychological maturation theories of Jean Piaget, Lawrence Kohlberg and Carol Gilligan. For Piaget, empirical studies have found that children understand justice in developing their moral judgment in three stages: first as obedience; then as equality; and finally as equity. This is in line with Aristotle's own observations about exact and inexact justice except for the lack of iteration in Piaget's model and the blurring of exact and inexact senses of justice in his experiments. Piaget's conception of justice fails to distinguish adequately mathematical ethics considerations from nonmathematical ones in terms of subject-matter. For Kohlberg and Gilligan, moral maturation systems tend to empirically describe, in cross-cultural and longitudinal studies, the rationales for ethical decision-making in reflectively resolving simulated moral dilemmas. This is testing the practice of simulating moral dilemmas of spectators but is not testing actual moral dilemmatic reasoning of participants. It misses the mark methodologically, or, as Broadie likens it, it is "playing at ethics" or even a "perversion." It is, as Aristotle sees in the Nicomachean Ethics, a deception, since the underlying longitudinal assumption is that someone thinks they can become good by talking about the good without doing good and without being impacted by doing what they have chosen in a moral feedback loop system. (1105b 13-17) Furthermore, such maturation theories overlook the iterative dimension of moral decision-making with feedback loops and filters in the development of moral character — including the possible use of mathematical ethics in the manner of Aristotle, who seems to have steered a middle course between complete reductive mathematization of ethics and an apriori resistance to even a partial mathematization of ethics. "Not too much and not too little!"

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