Euclid's Elements stand out, among Hellenistic mathematical works, in their pedagogic intent. Yet their very end – book xiii – already suggests the ludic, and at the very end is a theorem, attached as a kind of appendix, that would have been worthy of Archimedes. The theorem is often considered to have been discovered early (though its form may be due to Euclid himself, or even to some later reader of him). However this may be, it may serve as an example of an important compositional phenomenon: the mosaic proof. Here then is the proof that there are exactly five regular solids (adapted from Heath's translation):

This, my third study on Greek mathematics, serves to complete a project. My first study, The Shaping of Deduction in Greek Mathematics (1999) analyzed Greek mathematical writing in its most general form, applicable from the fifth century bc down to the sixth century ad and, in truth, going beyond into Arabic and Latin mathematics, as far as the scientific revolution itself. This form – in a nutshell, the combination of the lettered diagram with a formulaic language – is the constant of Greek mathematics, especially (though not only) in geometry. Against this constant, the historical variations could then be played. The historical variety is formed primarily of the contrast of the Hellenistic period (when Greek mathematics reached its most remarkable achievements) and Late Antiquity (when Greek mathematics came to be re-shaped into the form in which it influenced all of later science). My second study, The Transformation of Mathematics in the Early Mediterranean (2004), was largely concerned with the nature of this re-shaping of Greek mathematics in Late Antiquity and the Middle Ages.

This study, finally, is concerned with the nature of Greek mathematics in the Hellenistic period itself. Throughout, my main concern is with the form of writing: taken in a more general, abstract sense, in the first study, and in a more culturally sensitive sense, in the following two.

The three studies were not planned together, but the differences between them have to do not so much with changed opinions as with changed subject matter.

When I teach Plato, I begin, as many of us do, with the dialogues surrounding the trial and death of Socrates. And I find that for the most part my students enthusiastically embrace what they take to be Socrates' urging - to continually question our beliefs and our claims to know. Indeed, they understand this to be what their college education involves, and to a point I agree with them. But surely that is not all education involves. As Eva Brann says, “the claim, so often made in prose and poetry that the quarry is nothing and the quest everything, turns the pursuit of truth into a mere exercise, which is, for all its strenuousness, rather idle. Why look when one does not mean to find?”

Whatever else we can say about Socrates, it is surely the case that for him the pursuit of truth was no “mere exercise.” Yet my students' view of Socrates as emphasizing questioning above all has its echoes in contemporary scholarship. Let me take as one example Dana Villa's conception of “Socratic citizenship,” which emphasizes the critical, dissident force of Socratic interaction. Here Socratic elenchus is not so much a method of philosophical investigation as a method of inducing perplexity; no one ever figures out what justice or piety really is. Socrates' activities, Villa argues, are best characterized by the notion of “dissolvent rationality,” where what is dissolved through the use of reason is the lazy or dogmatic assumption that the beliefs we hold are true. The point of the perplexity that results is a slowing down, a “greater hesitancy in action” that may help us avoid injustice.

In his ethical works, Aristotle presents an extraordinarily nuanced and insightful account of the different aspects of the good human being and the good human being's life. Yet many of Aristotle's claims about ethical virtue seem so puzzling that modern philosophers have often discarded them, or reworked them to fit more familiar modern theories that do not make ethical virtue central. My aim in this book is to re-examine Aristotle's puzzling claims, especially the much-maligned doctrine of the mean, showing how together they form a thought provoking ethic of virtue, one that deserves to be developed and refined. Aristotle's Nicomachean Ethics, the most important of his ethical treatises, appeals to anyone interested in ethical character and thought, including undergraduates, graduates, specialists in ancient Greek philosophy, and modern virtue ethicists. I should be pleased if this book does the same.

There is a fine line between a work on Aristotle's ethics and a work on Aristotelian Ethics, the former being more closely tied to Aristotle's texts, the latter going into new territory, inspired by Aristotle's ideas. While I have approached Aristotle's texts from a less familiar perspective, bringing out elements in his philosophy that have not received their due, and considering them in the light of ancient and modern debates, I have refrained from venturing too far from his original views. The exception is Chapter 4, where I consider possible virtues that Aristotle himself does not consider, but, even there, I keep comments on the particular virtues short and to the point.

The passage of more than two thousand years has yielded a wealth of scholarship about Aristotle's work, and the last thirty years has seen burgeoning interest in Aristotle's ethics, with new books and articles published every year.

The aim of this chapter is to re-evaluate a central but much-maligned thesis in Aristotle's Nicomachean Ethics, the doctrine of the mean. The doctrine of the mean is a prominent thesis in Aristotle's account of virtue, and it enables other aspects of Aristotle's ethics to be understood from a new perspective, as we shall see in the chapters to come. Aristotle's doctrine of the mean has had a bad press, from being dubbed the “Goldilocks theory of Ethics” to receiving Bernard Williams's intended epitaph that the doctrine is “better forgotten”. Most of the unkind comments are based on construing the doctrine of the mean as a decision procedure, and a useless one at that. I wish to examine some aspects of the doctrine that show it in a different light, but that involves solving some long-standing puzzles in the interpretation of the doctrine itself, with special reference to how the doctrine applies to particular ethical virtues.

I argue that Aristotle's doctrine of the mean has the following three aspects. First, virtue, like health, is in equilibrium and is produced and preserved by avoiding extremes and hitting the mean; it is self-sustaining. Second, virtue is in a mean “relative to us”. Third, each virtue is in a mean between two vices, one of excess and one of deficiency.

Aristotle himself says that his account is “true, but not at all clear” (EN VI 1138b26), and, at first sight, there are certainly many unclarities. First, it is unclear what Aristotle's positive account of the mean “relative to us” is.

In the previous chapter, I argued that Aristotle thinks that it is impossible to have one ethical virtue fully without having the rest, because ethical virtue involves reason (it is meta logou) and so the reason and feelings of the good person are completely integrated. One might therefore expect the good human being to be free of inner conflict and regret, and that is exactly how Aristotle describes him (EN IX 4 1166a27–29). Indeed, according to Aristotle, the good person will feel neither shame nor disgrace because he will never voluntarily perform base actions (EN IV 9 1128b20–33). One might also expect it to be impossible for the demands of one virtue to conflict with the demands of another, and indeed, consistently with that view, the author of the Magna Moralia writes that the demands of bravery cannot conflict with those of justice, even in particular circumstances (MM II 3 1199b35–1200a). It is reasonable to assume that the ethical virtues are meant to be internally consistent as well.

All of these further claims make Aristotle's view about the unity of the ethical virtues even more controversial from both an ancient and modern point of view. It is often thought that the Greek tragedians show how life contains situations where even the good human being is faced with a tragic dilemma and must do something that is not virtuous and that will shatter her fragile goodness. It is also argued that there are situations where the good person is faced with conflicting demands and where, even if he makes the right choice, he will have done something shameful and will be left with “dirty hands” and a special sort of regret.

Tyranny and the Emergence of Historical Thinking

Herodotus and Thucydides, jointly responsible for the invention of history in the West, suggest an intriguing connection between historical thinking and the overcoming of tyrannical aspiration. On this topic, the historians should be regarded as fundamentally like-minded. Both object to the conventional tale of how Athens freed herself of her tyrants, the story of Harmodius and Aristogeiton and their alleged overthrow of the Peisistratid ruling family in 514 B.C.E. In the process of contesting this cherished tradition and replacing the tyrannicides with their own favored characters, Herodotus and Thucydides carve out a role for the historian in defining political identity. Herodotus, the Father of History, steps into the shoes of Solon, famed wise man of Athens, while Thucydides, often referred to as the Father of Objective History, assumes a Periclean role, his character of choice. Presumably each seeks to maintain control over the interpretation of these figures in a way that was not the case with the iconography of Harmodius and Aristogeiton. The historical accounts preclude any mere celebration of self and require, instead, an immersion in a complex set of particular details. Ultimately, Athens was to receive the heroes she deserved: the Herodotean Solon and the Thucydidean Pericles.

An important and well-established trend in classical Greek political theory reads texts in their historical and political context as fully as the sources allow, believing that what we know about these texts' roles in the vigorous controversies of their own time helpfully informs efforts to use this material to reflect on our own time. In this chapter we bring this interpretative approach to bear on Aristotle's Constitution of Athens (hereafter AthPol), a text that tells the story of Athenian constitutional history over several centuries and presents a detailed account of the laws and institutions of the Athenian democracy of Aristotle's time. This text, recovered only in 1891, is often assumed to be of interest to historians and constitutional scholars but not to political theorists because, beyond occasional philosophical observations and/or expressions of political bias, it is said to neglect matters of normative concern. We disagree. In our view, complex normative concerns shape the AthPol, and history and theory inform each other in that text, as they do in Aristotle's ethical and political writings, more generally. For these reasons we support an integrated and continuous reading of Aristotle's practical works, including the Politics, Nicomachean Ethics, Rhetoric, Poetics, and AthPol. Although we do not address the issue of authorship directly, our argument contributes reasons for favoring Aristotle's authorship.

It is often thought that what is distinctive about ethical virtues is that they are meant to remedy certain defects, either in human nature or in the world, or both. These views are supposedly linked with the idea that if human beings were perfect, or if they lived in a perfect world, there would be no need for ethical virtues. In this chapter, I argue that Aristotle's ethical virtues are not essentially remedial. This is especially clear in the case of the nameless virtues I discussed in the previous chapter, but remedial interpretations are also at odds with aspects of Aristotle's doctrine of the mean, discussed in Chapter 1, and his famous function argument. The view that Aristotelian ethical virtues are not essentially remedial is therefore an important thesis in Aristotle's ethic of virtue. It contributes to Aristotle's account of particular virtues, to be explained in Chapter 4, and it also supports Aristotle's humanism, to be further elaborated in Chapter 6.

Modern proponents of remedial views often trace their position back to Aristotle. For example, Philippa Foot in “Virtues and Vices” says that the “virtues should be seen as correctives” and she suggests that Aristotle would agree. Christine Korsgaard in an influential paper, “Aristotle and Kant on the Source of Value”, argues that, according to Aristotle, the ethical virtues only have conditional value. In Korsgaard's words:

While the works of Plato and Aristotle draw universal attention from students of political philosophy, Thucydides' reception has been more limited and localized. Most frequently, he is sought for his contributions to international relations theory, often accessed through a small number of set pieces such as the Melian dialogue. Whatever the reasons for this relative lack of attention, it is unfortunate, for Thucydides is an important conversation partner not only with more familiar voices within the so-called Western tradition of political philosophy but also with modern political theorists who discuss the functions and disorders of political institutions and political cultures. In this connection, Thucydides may have more to contribute to democratic political theory than is often supposed. Appreciating his contributions means taking him at his word when he writes early in Book 1 of the History that he has composed “a possession forever, and not a competitive entry to be heard for the moment” (1.22). Yet the precise meaning of this very ambitious claim is unclear. Interpreting it is inseparable from coming to grips with the kind of work this is.

Assessments of the genre of Thucydides' work proliferate. He has been read as a historian who narrates and explains the most striking events of his time, as a social theorist who discovers the deepest causes of political disorder, and as a memorializer of the civic leadership of Pericles. To the extent that we are convinced by one or more of these judgments, however, Thucydides’ voice is heard as conclusive and monologic and the political thought that emerges from the pages of his work stands apart from the immediate context of political life.

ARISTOTLE'S ETHIC OF VIRTUE

In the preface to the first edition of his classic work on Aristotle's ethical theory, W. F. R. Hardie wrote, “In the study of Aristotle familiarity can be an obstacle to understanding; we are prone to think we know what he means before we do”. In the preceding chapters, I have examined some familiar but puzzling Aristotelian theses, especially his muchmaligned doctrine of the mean, from a less familiar perspective, arguing that Aristotle's ethics is primarily an ethic of virtue.

I have argued that the doctrine of the mean is of more interest and plays a larger role in Aristotle's ethics than has previously been thought. The doctrine of the mean combines three different aspects: (1) a doctrine of equilibrium, according to which the good human being has the emotions that are called for in a given situation; (2) a sophisticated view of the particulars of the situation; and (3) a triadic taxonomy of virtue and vices. The doctrine of the mean has substantive consequences. Aristotle's list of ethical virtues, including the important nameless ones, while quite controversial, is grounded in the doctrine of equilibrium. Moreover, the doctrine of equilibrium supports Aristotle's view that the ethical virtues are not essentially remedial, and is consistent with his claim that one cannot have one ethical virtue fully without having all the rest. The second aspect of Aristotle's doctrine of the mean enters into Aristotle's way of addressing moral dilemmas. Perhaps most surprisingly, the third aspect shows how Aristotle can reply to the immoralist, and it also contributes to Aristotle's views about democracy in the polis.

The argument of this chapter is that Aristotle's practical syllogism, despite the qualms of modern commentators, is at the heart of Aristotle's ethic of virtue. This chapter connects the different strands of Aristotle's thought on ethical virtue, motivation, and practical wisdom, and it completes the argument of Chapter 5 that ethical virtue and practical wisdom go hand in hand.

I begin by raising a long-lived puzzle about how to distinguish Aristotelian practical wisdom (phronēsis) from a technical skill (technē), and arguing that the solution comes, not from Aristotle's metaphysics, but from his account of the virtues, especially the nameless virtue of truthfulness. I then raise the further problem whether Aristotle is begging the question when he says that the difference between practical wisdom and a technical skill is that the former requires ethical virtue whereas the latter does not. In Chapter 5, I discussed Aristotle's controversial claim that it is impossible to have all the ethical virtues fully without having practical wisdom, and that it is impossible to have practical wisdom without having all of the Aristotelian ethical virtues fully (EN VI 13 1144b30–1145a1). I now turn to Aristotle's practical syllogism, which, on my interpretation, makes the connection between ethical virtue and practical wisdom even clearer.

In the sphere of formal logic, Aristotle is famous for discovering the syllogism (a piece of reasoning consisting in two premisses and a conclusion), and for categorizing all the valid forms of syllogisms with premisses containing subject and predicate terms.

As part of the longer argument towards the conclusion that ethical virtue is central in Aristotle's Nicomachean Ethics, in the previous chapter and in Chapter 5, I argued that ethical virtue requires practical wisdom and conversely in two different ways. In Chapter 5, I explained how ethical virtue involves reason (practical wisdom) as opposed to being merely in accordance with reason, and in Chapter 8, I explained how the practical syllogism that the person with practical wisdom gets right, involves the virtues of character. However, the aspects of practical wisdom I have discussed so far have been fairly minimal, so one might think that the person with practical wisdom needs to know a lot more than what I have so far discussed. Therefore, in this chapter, I raise some puzzles about what, or how much, the Aristotelian good person has to know, and what the good student needs to know in order to study ethics. The answers have far-reaching consequences for the way in which we should read Aristotle's own Nicomachean Ethics, as well as for the appropriate way to address the immoralist.

It is uncontroversial that Aristotle thinks that knowledge is insufficient for ethical virtue (e.g., EN II 1105b1–5, 12–18 cf. X 9 1179b4–10). Reading a book is no substitute for engaging in virtuous action. However, it is unclear what knowledge, if any, is necessary for being a good person, according to Aristotle. On the one hand, there are philosophers, like David Wiggins, who think that the good person can grasp the situation at hand and do the right thing without any knowledge of general principles or of other disciplines.