Philosophy and mathematics

The relationship between philosophy and mathematics appears to be less close in the Hellenistic age than in previous centuries. The mathematicians proceeded with their work without any explicit adhesion to philosophical doctrines or any reply to the theoretical and epistemological problems raised by the Epicureans and the sceptics when dealing with mathematics. In the fifth century AD the Neo-platonist Proclus saw Euclid as a Platonist, because his Elements concluded with the construction of regular polyhedrons, the basis of the cosmic structure outlined by Plato in the Timaeus (Eucl. 68.20–3; 70.22–71.5). But there is no proof that Euclid meant to direct the whole of his writing towards the creation of a cosmology.

Another level on which Proclus tried to show Euclidean geometry's dependence on philosophy was the formal structure of the Elements. This was based on the distinction between a small number of principles, assumed at the outset without demonstration and listed in the first book as definitions, common notions and postulates, and a body of propositions reached deductively starting from the principles. Proclus highlighted a correspondence between this structure and the theory of science formulated by Aristotle, particularly in the Posterior Analytics, and also between Euclid's principles and those mentioned by Aristotle (Eucl. 76.6–77.2). It is in fact possible to make out some parallelism: for example, Aristotle includes among the principles used by mathematicians Euclid's third common notion, which is that if equals are subtracted from equals, the remainders will be equal (APo. 76a41).

[1] ATTICUS: Since we have already walked enough and you have to make a start on a new topic, why don't we move and sit down for the rest of the conversation on the island in the Fibrenus – I think that that is the name of the other river?

MARCUS: Certainly. I use that spot regularly with great pleasure, whether I am thinking something over or reading or writing something.

[2] ATTICUS: For my part, since I have just now come here, there are no bounds to my pleasure, and I have only contempt for grand villas and marble pavements and paneled ceilings. Those water channels that some people call “Nile” or “Euripus” can only arouse laughter when you have seen this spot. And just as you, in speaking of law and justice a little while ago, made nature the standard for everything, so too in seeking aids for mental relaxation and pleasure nature is best. I used to wonder – I thought that there was nothing here but rocks and mountains, basing my opinion on your own speeches and poetry – I wondered, as I said, why you took such pleasure in this place. Now I wonder why when you are away from Rome you ever go anywhere else.

[3] MARCUS: When I have enough free time, particularly at this season, I seek out the beauty and the healthfulness of this place – though that is not very often.

Foundations and first principles

Stoic ethics starts from foundations and first principles which are more explicit than those of most ancient ethical systems. Chrysippus announced in his Propositions in Physics that ‘there is no other or more fitting way to tackle the theory of good and bad things, the virtues, and happiness than on the basis of nature as a whole and the administration of the cosmos’ (Plu. De Stoicis In Platonis Rempublicam commentarii 1035c). This explicit statement about starting points puts the emphasis on nature in the cosmic sense, i.e. the nature of the entire providentially governed cosmos; but elsewhere Chrysippus turns to a more inclusive sense of nature: when he says ‘Where should I begin from and what should I take as the starting point for the appropriate and as the raw material for virtue, if I skip over nature and what accords with nature?’ (Plu. De communibus notitiis adversus Stoicos. 1069e), it is clear that the nature in question is not just cosmic. Crucial ethical concepts also find their roots in the nature of humans and in theories about what accords with human nature.

The central importance of human nature clearly goes back to the founder of the school. In the list of titles given by Diogenes Laertius (VII.4) we find a treatise On the life according to nature and one On impulse, also titled On human nature. But we have no record of an On goals or On virtue; indeed, the main evidence given for Zeno's views on the telos is the work On human nature (D.L. VII.87). Whatever the rôle of human as opposed to cosmic nature in Zeno's thinking, it is striking that the major ethical treatises (aside from the Republic) suggest a strong interest in the former.

The epistemological turn

It is generally agreed that the Hellenistic period is the great age of ancient epistemology. For a variety of reasons, many of which have nothing to do with the history of philosophy, the period is standardly deemed to start in 323 bc on the death of Alexander the Great. By a curious coincidence, two philosophers of signal and symbolic importance had connections with Alexander. The first is Aristotle, who had been tutor to the young Alexander and who died a year after his royal pupil, leaving a vast body of scientific and philosophical work which, after a period of mixed fortune, would for centuries be considered - in particular by the sceptics - as a model of dogmatic thought. The second is Pyrrho, some twenty years younger than Aristotle, who accompanied Alexander on his eastern campaign: he returned from Asia in his prime, and the words and deeds which filled the rest of his long life caused him, rightly or wrongly, to be regarded for centuries as the eponymous hero of scepticism.

It is tempting - and conventional - to assert that, on Aristotle's death, philosophy saw itself driven from a happy paradise of epistemological innocence, and that the poison of doubt, spat out by the serpent of Pyrrhonism, would oblige any future philosopher who failed to succumb to it to earn his neo-dogmatic bread by the sweat of his brow. And this picture makes a pleasing diptych with the picture which is painted, with equal facility, of the state of ethics: before the geopolitical earthquake provoked by Alexander, the moral existence of the Greeks had been firmly framed by the ethical and political structures of the city-state; after the earthquake, the new Hellenistic schools could offer the shaken citizenry nothing more than recipes for individual salvation.

Introduction

In an age which has produced much agonizing over how to reconcile the life of the mind with a materialist physics, we are likely to feel an immediate affinity with an earlier theorist who combined the project of explaining human psychology with a commitment to the claim that ‘the totality of things is bodies and void’ (Epistula ad Herodotum 39). In common with the vast majority of modern psychologists and philosophers of mind, Epicurus was committed to atomistic materialism – and indeed, unlike that of most modern psychologists and philosophers, his commitment actually extended to arguing for the truth of that position. It was not a thesis which he accepted merely on authority. Like that of Aristotle in the generation before him, Epicurus’ psychology needs to be seen as part of an attempt to provide a complete natural philosophy. At least part of what he has to say about the psuchē is directly intended to show that his atomic theory is capable of explaining such complex natural phenomena as perception, thought and action. All of what he has to say is intended to be consistent with that physical theory.

This, however, places fewer constraints on what will count as a successful theory of the relation between the mental and the physical than is sometimes supposed. We are perhaps apt to be over-impressed by Epicurus’ espousal of atomism and to assume that, just by accepting the thesis that all material objects are divisible into atomic parts, he took on the task of showing that all the properties of material objects can be reduced to the states and movements of those parts.

The Peripatetics

Late antiquity learned two logics: categorical syllogistic and hypothetical syllogistic. Categorical syllogistic studies categorical arguments. An argument is categorical if all its components (its premisses and its conclusion) are categorical propositions. A proposition is categorical if it ‘says one thing of one thing’ – or better, if it is simple in the sense of not containing two or more propositions as components. Hypothetical syllogistic studies hypothetical arguments. An argument is hypothetical if at least one of its components is a hypothetical proposition. A proposition is hypothetical if it contains at least two propositions as components.

It is a plausible guess that this terminology was developed in the Hellenistic Peripatos. The phrase ‘hypothetical argument’ is attested for Chrysippus (D.L. VII. 196); Galen says that ‘the ancients’ spoke of hypothetical propositions (Institutio logica III. 3), Alexander that they spoke of ‘mixed’ syllogisms (In Aristotelis Analytica priora commentaria 262. 31–2); and ‘the ancients’ in such contexts are usually the Peripatetics. Philoponus says that Theophrastus used the phrase ‘wholly hypothetical syllogism’ (In Aristotelis Analytica priora commentaria 302.9).

Categorical syllogistic was thought of as essentially Peripatetic, hypothetical syllogistic as essentially Stoic; and although it was known that the Stoics and Peripatetics had disputed with one another in logic no less than in ethics and physics, it was often supposed that the two syllogistics were partners, each adequate in its own area. This irenic view is misleading. The Peripatetics thought that their categorical syllogistic embraced the whole of logic: any argument which submitted to formal treatment at all submitted to categorical syllogistic. And the Stoics held the same for their hypothetical syllogisms.

The translation of On the Commonwealth is based on C. [K.] Ziegler (ed.), M. Tullius Cicero: De re publica (7th ed., Leipzig, 1969), and (for the continuous portions of the palimpsest and the Dream of Scipio). J. Zetzel (ed.), Cicero: De re publica: Selections (Cambridge, 1995). The translation of On the Laws is based on K. Ziegler (ed.), M. Tullius Cicero: De legibus (3rd ed., rev. by W. Goerler: Heidelberg, 1979). Most departures from these editions are indicated in the notes. It should be noted that a new critical edition of both texts is being prepared by J. G. F. Powell for Oxford Classical Texts.

On the Commonwealth differs in the format of the dialogue from On the Laws in that the latter is pure dialogue, with no narrator, and changes of speaker are marked (by convention) with the name of the speaker followed by a colon, as in dramatic texts; in On the Commonwealth, by contrast, there is a narrator, and in the Latin text speakers are often introduced by phrases such as “Then Scipio said.” To avoid extremely stilted translation, these phrases have been replaced here by the same dramatic convention as is used in On the Laws.

With respect to the order and presentation of the fragments of On the Commonwealth, there have been many departures from Ziegler's text.

Atomism was the creation of two thinkers of the fifth century B.C., Leucippus and Democritus. The former, attested by Aristotle, our primary source, as the founder of the theory, was a shadowy figure even in antiquity, being eclipsed by his more celebrated successor Democritus to such an extent that the theory came to be generally regarded as the work of the latter. Epicurus, who developed and popularised atomism in the late fourth and early third centuries B.C. (following in the tradition of various figures such as Nausiphanes and Anaxarchus, now little more than names), went so far as to deny that Leucippus ever existed. Only a little more is known about Democritus (see p. xix). The precise relation between Leucippus and Democritus is unclear. Plato never mentions either by name. Aristotle and his followers treat Leucippus as the founder of the theory, but also assign its basic principles to both Leucippus and Democritus; later sources tend to treat the theory as the work of Democritus alone. While it is clear that the theory originated with Leucippus, it is possible that the two collaborated to some extent and almost certain that Democritus developed the theory in a number of areas, for example, extending it to include a materialistic psychology, a sophisticated epistemology, and an account of the development of human society that laid particular stress on the human capacity to learn from chance experience.

Unlike other books in this series, the present volume is not a “companion” to a single philosopher but to the set of thinkers who collectively formed the beginnings of the philosophical tradition of ancient Greece. Most of them wrote little, and the survival of what they wrote or thought is fragmentary, often mediated not by their own words but only by the testimony of Aristotle, Theophrastus, and other much later authors. These remains are exceptionally precious not only because of their intrinsic quality but also for what they reveal concerning the earliest history of western philosophy and science. The fascination of the material, notwithstanding or even because of its density and lacunar transmission, grips everyone who encounters it. Two of our century's most influential philosophers, Heidegger and Popper, have “gone back” to the earliest Greek philosophers in buttressing their own radically different methodologies and preoccupations. Many of these thinkers are so challenging that the small quantity of their surviving work is no impediment to treating each of them at book length. Even so, there are reasons beyond our fragmentary sources and conventional practice for presenting these and other early Greek philosophers in a collective volume.

DOXOGRAPHI GRAECI

Because the works of the early Greek philosophers have been lost, our knowledge of their content is entirely dependent either on sparse verbatim quotations (though less sparse than for instance those relating to the early Stoics) or on various forms of reportage in all sorts of ancient authors. It has thus become customary to begin books of this kind with a critical review of our sources of information.

What is at stake is the reliability of these sources. The ideal of an objective history of philosophy is a nineteenth-century invention. In antiquity history of philosophy was part of systematic philosophy, serving a variety of purposes. The ideas of earlier philosophers were used and interpreted in many ways, and, more often than not, served merely as springboards. This holds not only for the attitude of major thinkers such as Plato and Aristotle but also for the far humbler works consisting of collections of doctrines, with or without some biographical detail, that circulated on a fairly extensive scale. Such works were used, it would seem, in the context of a primary education in philosophy and also as quarries to be exploited whenever someone writing about a philosophical issue felt he should set off his own view against those of others, to improve upon an already existing view or to replace it with another.

THE APPROACH TO HERACLITUS

Heraclitus of Ephesus must have been active around 500 B.C. Nothing is known of the external events of his life; the later biographical reports are fiction. Of Heraclitus' book, around one hundred fragments survive. It seems to have consisted of a series of aphoristic statements without formal linkage. The style is unique. Heraclitus' carefully stylized and artfully varied prose ranges from plain statements in ordinary language to oracular utterances with poetical special effects in vocabulary, rhythm, and word arrangement. Many statements play with paradoxes or hover teasingly on the brink of self-contradiction. Many seem intended as pungently memorable aphorisms. (Translations in this chapter try to capture some of the ambiguities, where this is reasonably possible.)

The meaning and purpose of Heraclitus7 book has always been found to be problematic, even by those who read it in its entirety. The Peripatetic Theophrastus (D.L. IX.6) diagnosed Heraclitus as “melancholic” (manic-depressive), on the grounds that he left some things half-finished, and contradicted himself; later Greeks named him “the obscure.” Certainly Heraclitus did not always aim at expository order and clarity as usually understood. What remains shows that he often was deliberately unclear. Like a riddle or an oracle, he practised a deliberate half-concealment of his meanings, goading the reader to participate in a game of hide-and-seek.

INTRODUCTION: THE POETICS OF EARLY GREEK PHILOSOPHY?

For some readers, the very title of this chapter will seem a paradox or a provocation. After all, while the term “Presocratics” is modern, the concept has ancient roots; and from the very beginning it has been used to distinguish philosophers who, for the most part, wrote in prose, from poets who composed in verse. Such a distinction, which establishes the largely nonphilosophical character of the early Greek poets and the largely nonpoetical character of the early Greek philosophers, may seem self-evident to us, but in fact it has not always been so. Heraclitus names Hesiod and Xenophanes, in the same breath with Pythagoras and Hecataeus (DK 22 B40); Hippias wrote a treatise paralleling the opinions of poets and philosophers (DK 86 B6); Plato does not distinguish sharply between poets and philosophers among his predecessors, and he has his Protagoras claim that ancient poets were really sophists but disguised their opinions for fear of exciting hostility [Prot. 316d-e). As far as we know, Aristotle was the first author to distinguish terminologically between what he called mythologoi and theologoi on the one hand and physikoi or physiologoi on the other. On his view, the former group were really storytellers, poets narrating myths about heroes and gods, and any views about the nature of the world that might be extracted from their works were incidental, obscure, and philosophically uninteresting; the latter group, beginning with Thales, were engaged in basically the same kind of investigation of the physical world as Aristotle himself was and, even though their theories were, unsurprisingly, deficient in comparison with his own, nonetheless they were philosophically serious, that is, they were worth studying, pillaging, and refuting.

Parmenides and Melissus were bracketed in antiquity as the two great exponents of the Eleatic world-view which denies change and plurality. In modern times their treatment has been curiously unequal. Too much has been written on Parmenides - albeit the greater thinker of the two - too little on Melissus. Too much has been said about Parmenides' use of the verb “be,”while too little has been said about his detailed arguments for the individual characteristics of what-is. However, neither these nor other anomalies should disguise the immense wealth of scholarship that has furthered the reconstruction of their Eleaticism.

PARMENIDES

Around 150 lines of Parmenides' hexameter poem, written in the early- to mid-fifth century, have been recovered, most belonging to its first part. His densely metaphorical diction is replete with Homeric echoes, and presents the further difficulty of having to use the very language of change and plurality that it aims ultimately to outlaw. These are among the many aspects to which it will be impossible to do justice in the present chapter.

In the modern world Pythagoras is the most famous of the early Greek philosophers. The same was true in the fourth century B.C., when Plato wrote his Republic, some 150 years after Pythagoras left Samos in about 530, to emigrate to Croton in southern Italy, where Pythagoreanism would flourish. Plato has Socrates say that Pythagoras was “especially loved as a leader of education in the private sphere,” and that his followers

However, beginning with Plato's successors in the Academy, the reputation of Pythagoras became seriously exaggerated, and by the fourth century A.D. in the Neoplatonic tradition, he had become the greatest of all philosophers, from whom both Plato and Aristotle borrowed their central ideas.

“The idea of nature as implying a universal nexus of cause and effect comes to be made explicit in the course of the development of Presocratic philosophy”: G.E.R. Lloyd. “The conception of cause is borrowed from the language of medicine, as is clear from the word prophasis which Thucydides uses”:W. Jaeger. “The word aition is, from the Hippocratic writings on, a standard word for 'cause', and its relative aitia... meant a complaint or an accusation, but already by the time of Herodotus's book it can mean simply 'cause' or 'explanation'”: B. Williams.

These three distinguised scholars, distant though they are from one another in their intellectual orientations, seem to agree on the opinion that a precise and well-defined conception of causality is present in fifth-century philosophy, history, and medicine. This judgement is widely shared, but it needs to be corrected, or at least clarified and formulated, from two different but complementary perspectives.