5
Back to the Presocratics

I

‘Back to Methuselah’ was a progressive programme, compared with ‘Back to Thales’ or ‘Back to Anaximander’: what Shaw offered us was an improved expectation of life—something that was in the air, at any rate when he wrote it. I have nothing to offer you, I am afraid, that is in the air today; for what I want to return to is the simple straightforward rationality of the Presocratics. Wherein does this much discussed ‘rationality’ of the Presocratics lie? The simplicity and boldness of their questions is part of it, but my thesis is that the decisive point is the critical attitude which, as I shall try to show, was first developed in the Ionian School.

The questions which the Presocratics tried to answer were primarily cosmological questions, but there were also questions of the theory of knowledge. It is my belief that philosophy must return to cosmology and to a simple theory of knowledge. There is at least one philosophical problem in which all thinking men are interested: the problem of understanding the world in which we live; and thus ourselves (who are part of that world) and the way we acquire our knowledge of it. All science is cosmology, I believe, and for me the interest of philosophy, no less than of science, lies solely in its bold attempt to add to our knowledge of the world, and to the theory of our knowledge of the world. I am interested in Wittgenstein, for example, not because of his linguistic philosophy, but because his Tractatus was a cosmological treatise (although a crude one), and because his theory of knowledge was closely linked with his cosmology.

For me, both philosophy and science lose all their attraction when they give up that pursuit—when they become specialisms and cease to see, and to wonder at, the riddles of our world. Specialization may be a great temptation for the scientist. For the philosopher it is the mortal sin.

II

In this paper I speak as an amateur, as a lover of the beautiful story of the Presocratics. I am not a specialist or an expert: I am completely out of my depth when an expert begins to argue which words or phrases Heraclitus might, and which he could not possibly, have used. Yet when some expert replaces a beautiful story, based on the oldest texts we possess, by one which—to me at any rate—no longer makes any sense, then I feel that even an amateur may stand up and defend an old tradition. Thus I will at least look into the expert’s arguments, and examine their consistency. This seems a harmless occupation to indulge in; and if an expert or anybody else should take the trouble to refute my criticism I shall be pleased and honoured.1

I shall be concerned with the cosmological theories of the Presocratics, but only to the extent to which they bear upon the development of the problem of change, as I call it, and only to the extent to which they are needed for understanding the approach of the Presocratic philosophers to the problem of knowledge—their practical as well as their theoretical approach. For it is of considerable interest to see how their practice as well as their theory of knowledge is connected with the cosmological and theological questions which they posed to themselves. Theirs was not a theory of knowledge that began with the question, ‘How do I know that this is an orange?’ or, ‘How do I know that the object I am now perceiving is an orange?’ Their theory of knowledge started from problems such as, ‘How do we know that the world is made of water?’ or, ‘How do we know that the world is full of gods?’ or, ‘How can we know anything about the gods?’

There is a widespread belief, somewhat remotely due, I think, to the influence of Francis Bacon, that one should study the problems of the theory of knowledge in connection with our knowledge of an orange rather than our knowledge of the cosmos. I dissent from this belief, and it is one of the main purposes of my paper to convey to you some of my reasons for dissenting. At any rate it is good to remember from time to time that our Western science—and there seems to be no other—did not start with collecting observations of oranges, but with bold theories about the world.

III

Traditional empiricist epistemology and the traditional historiography of science are both deeply influenced by the Baconian myth that all science starts from observation and then slowly and cautiously proceeds to theories. That the facts are very different can be learned from studying the early Presocratics. Here we find bold and fascinating ideas, some of which are strange and even staggering anticipations of modern results, while many others are wide of the mark, from our modern point of view; but most of them, and the best of them, have nothing to do with observation. Take for example some of the theories about the shape and position of the earth. Thales said, we are told, ‘that the earth is supported by water on which it rides like a ship, and when we say that there is an earthquake, then the earth is being shaken by the movement of the water’. No doubt Thales had observed earthquakes as well as the rolling of a ship before he arrived at his theory. But the point of his theory was to explain the support or suspension of the earth, and also earthquakes, by the conjecture that the earth floats on water; and for this conjecture (which so strangely anticipates the modern theory of continental drift) he could have had no basis in his observations.

We must not forget that the function of the Baconian myth is to explain why scientific statements are true, by pointing out that observation is the ‘true source’ of our scientific knowledge. Once we realize that all scientific statements are hypotheses, or guesses, or conjectures, and that the vast majority of these conjectures (including Bacon’s own) have turned out to be false, the Baconian myth becomes irrelevant. For it is pointless to argue that the conjectures of science—those which have proved to be false as well as those which are still accepted—all start from observation.

However this may be, Thales’ beautiful theory of the support or suspension of the earth and of earthquakes, though in no sense based upon observation, is at least inspired by an empirical or observational analogy. But even this is no longer true of the theory proposed by Thales’ great pupil, Anaximander. Anaximander’s theory of the suspension of the earth is still highly intuitive, but it no longer uses observational analogies. In fact it may be described as counter-observational. According to Anaximander’s theory, ‘The earth … is held up by nothing, but remains stationary owing to the fact that it is equally distant from all other things. Its shape is … like that of a drum. … We walk on one of its flat surfaces, while the other is on the opposite side.’ The drum, of course, is an observational analogy. But the idea of the earth’s free suspension in space, and the explanation of its stability, have no analogy whatever in the whole field of observable facts.

In my opinion this idea of Anaximander’s is one of the boldest, most revolutionary, and most portentous ideas in the whole history of human thought. It made possible the theories of Aristarchus and of Copernicus. But the step taken by Anaximander was even more difficult and audacious than the one taken by Aristarchus and Copernicus. To envisage the earth as freely poised in mid-space, and to say ‘that it remains motionless because of its equidistance or equilibrium’ (as Aristotle paraphrases Anaximander), is to anticipate to some extent even Newton’s idea of immaterial and invisible gravitational forces.2

IV

How did Anaximander arrive at this remarkable theory? Certainly not by observation but by reasoning. His theory is an attempt to solve one of the problems to which his teacher and kinsman Thales, the founder of the Milesian or Ionian School, had offered a solution before him. I therefore conjecture that Anaximander arrived at his theory by criticizing Thales’ theory. This conjecture can be supported, I believe, by a consideration of the structure of Anaximander’s theory.

Anaximander is likely to have argued against Thales’ theory (according to which the earth was floating on water) on the following lines. Thales’ theory is a specimen of a type of theory which if consistently developed would lead to an infinite regress. If we explain the stable position of the earth by the assumption that it is supported by water— that it is floating on the ocean (Okeanos)—should we not have to explain the stable position of the ocean by an analogous hypothesis? But this would mean looking for a support for the ocean, and then for a support for this support. This method of explanation is unsatisfactory: first, because we solve our problem by creating an exactly analogous one; and also for the less formal and more intuitive reason that in any such system of supports or props failure to secure any one of the lower props must lead to the collapse of the whole edifice.

From this we see intuitively that the stability of the world cannot be secured by a system of supports or props. Instead Anaximander appeals to the internal or structural symmetry of the world, which ensures that there is no preferred direction in which a collapse can take place. He applies the principle that where there are no differences there can be no change. In this way he explains the stability of the earth by the equality of its distances from all other things.

This, it seems, was Anaximander’s argument. It is important to realize that it abolishes, even though not quite consciously perhaps, and not quite consistently, the idea of an absolute direction—the absolute sense of ‘upwards’ and ‘downwards’. This is not only contrary to all experience but notoriously difficult to grasp. Anaximenes ignored it, it seems, and even Anaximander himself did not grasp it completely. For the idea of an equal distance from all other things should have led him to the theory that the earth has the shape of a globe. Instead he believed that it had the shape of a drum, with an upper and a lower flat surface. Yet it looks as if the remark, ‘We walk on one of its flat surfaces, while the other is on the opposite side’, contained a hint that there was no absolute upper surface, but that on the contrary the surface on which we happened to walk was the one we might call the upper.

What prevented Anaximander from arriving at the theory that the earth was a globe rather than a drum? There can be little doubt: it was observational experience which taught him that the surface of the earth was, by and large, flat. Thus it was a speculative and critical argument, the abstract critical discussion of Thales’ theory, which almost led him to the true theory of the shape of the earth; and it was observational experience which led him astray.

V

There is an obvious objection to Anaximander’s theory of symmetry, according to which the earth is equally distant from all other things. The asymmetry of the universe can be easily seen from the existence of sun and moon, and especially from the fact that sun and moon are sometimes not far distant from each other, so that they are on the same side of the earth, while there is nothing on the other side to balance them. It appears that Anaximander met this objection by another bold theory—his theory of the hidden nature of the sun, the moon, and the other heavenly bodies.

He envisages the rims of two huge chariot wheels rotating round the earth, one 27 times the size of the earth, the other 18 times its size. Each of these rims or circular pipes is filled with fire, and each has a breathing-hole through which the fire is visible. These holes we call the sun and the moon respectively. The rest of the wheel is invisible, presumably because it is dark (or misty) and far away. The fixed stars (and presumably the planets) are also holes on wheels which are nearer to the earth than the wheels of the sun and the moon. The wheels of the fixed stars rotate on a common axis (which we now call the axis of the earth) and together they form a sphere round the earth, so the postulate of equal distance from the earth is (roughly) satisfied. This makes Anaximander also a founder of the theory of the spheres. (For its relation to the wheels or circles see Aristotle, De Caelo, 289b10 to 290b10.)

VI

There can be no doubt whatever that Anaximander’s theories are critical and speculative rather than empirical: and considered as approaches to truth his critical and abstract speculations served him better than observational experience or analogy.

But, a follower of Bacon may reply, this is precisely why Anaximander was not a scientist. This is precisely why we speak of early Greek philosophy rather than of early Greek science. Philosophy is speculative: everybody knows this. And as everybody knows, science begins only when the speculative method is replaced by the observational method, and when deduction is replaced by induction.

This reply, of course, amounts to the thesis that, by definition, theories are (or are not) scientific according to their origin in observations, or in so-called ‘inductive procedures’. Yet I believe that few, if any, physical theories would fall under this definition. And I do not see why the question of origin should be important in this connection. What is important about a theory is its explanatory power, and whether it stands up to criticism and to tests. The question of its origin, of how it is arrived at—whether by an ‘inductive procedure’, as some say, or by an act of intuition—may be extremely interesting, especially for the biographer of the man who invented the theory, but it has little to do with its scientific status or character.

VII

As to the Presocratics, I assert that there is the most perfect possible continuity of thought between their theories and the later developments in physics. Whether they are called philosophers, or pre-scientists, or scientists, matters very little, I think. But I do assert that Anaximander’s theory cleared the way for the theories of Aristarchus, Copernicus, Kepler, and Galileo. It is not that he merely ‘influenced’ these later thinkers; ‘influence’ is a very superficial category. I would rather put it like this: Anaximander’s achievement is valuable in itself, like a work of art. Besides, his achievement made other achievements possible, among them those of the great scientists mentioned.

But are not Anaximander’s theories false, and therefore nonscientific? They are false, I admit; but so are many theories, ‘based’ upon countless experiments, which modern science accepted until recently, and whose scientific character nobody would dream of denying, even though they are now believed to be false. (An example is the theory that the typical chemical properties of hydrogen belong only to one kind of atom—the lightest of all atoms.) There were historians of science who tended to regard as unscientific (or even as superstitious) any view no longer accepted at the time they were writing; but this is an untenable attitude. A false theory may be as great an achievement as a true one. And many false theories have been more helpful in our search for truth than some less interesting theories which are still accepted. For false theories can be helpful in many ways; they may for example suggest some more or less radical modifications, and they may stimulate criticism. Thus Thales’ theory that the earth floats on water reappeared in a modified form in Anaximenes, and in more recent times in the form of Wegener’s theory of continental drift. How Thales’ theory stimulated Anaximander’s criticism has been shown already.

Anaximander’s theory, similarly, suggested a modified theory—the theory of an earth globe, freely poised in the centre of the universe, and surrounded by spheres on which heavenly bodies were mounted. And by stimulating criticism it also led to the theory that the moon shines by reflecting light; to the Pythagorean theory of a central fire; and ultimately to the heliocentric world-system of Aristarchus and Copernicus.

VIII

I believe that the Milesians, like their oriental predecessors who took the world for a tent, envisaged the world as a kind of house, the home of all creatures—our home. Thus there was no need to ask what it was for. But there was a real need to inquire into its architecture. The questions of its structure, its ground-plan, and its building material, constitute the three main problems of Milesian cosmology. There is also a speculative interest in its origin, the question of cosmogony. It seems to me that the cosmological interest of the Milesians far exceeded their cosmogonical interest, especially if we consider the strong cosmogonical tradition, and the almost irresistible tendency to describe a thing by describing how it has been made, and thus to present a cosmological account in a cosmogonical form. The cosmological interest must be very strong, as compared with the cosmogonical one, if the presentation of a cosmological theory is even partially free from these cosmogonical trappings.

I believe that it was Thales who first discussed the architecture of the cosmos—its structure, ground-plan, and building material. In Anaximander we find answers to all three questions. I have briefly mentioned his answer to the question of structure. As to the question of the ground-plan of the world, he studied and expounded this too, as indicated by the tradition that he drew the first map of the world. And of course he had a theory about its building material—the ‘endless’ or ‘boundless’ or ‘unbounded’ or ‘unformed’—the ‘apeiron’.

In Anaximander’s world all kinds of changes were going on. There was a fire which needed air and breathing-holes, and these were at times blocked up (‘obstructed’), so that the fire was smothered:3 this was his theory of eclipses, and of the phases of the moon. There were winds, which were responsible for the changing weather.4 And there were the vapours, resulting from the drying up of water and air, which were the cause of the winds and of the ‘turnings’ of the sun (the solstices) and of the moon.

We have here the first hint of what was soon to come: of the general problem of change, which became the central problem of Greek cosmology, and which ultimately led, with Leucippus and Democritus, to a general theory of change that was accepted by modern science almost up to the beginning of the twentieth century. (It was given up only with the breakdown of Maxwell’s models of the ether, an historic event that was little noticed before 1905.)

This general problem of change is a philosophical problem; indeed in the hands of Parmenides and Zeno it almost turns into a logical one. How is change possible—logically possible, that is? How can a thing change, without losing its identity? If it remains the same, it does not change; yet if it loses its identity, then it is no longer that thing which has changed.

IX

The exciting story of the development of the problem of change appears to me in danger of being completely buried under the mounting heap of the minutiae of textual criticism. The story cannot, of course, be fully told in one short paper, and still less in one of its many sections. But in briefest outline, it is this.

For Anaximander, our own world, our own cosmic edifice, was only one of an infinity of worlds—an infinity without bounds in space and time. This system of worlds was eternal, and so was motion. There was thus no need to explain motion, no need to offer a general theory of change (in the sense in which we shall find a general problem and a general theory of change in Heraclitus; see below). But there was a need to explain the well-known changes occurring in our world. The most obvious changes—the change of day and night, of winds and of weather, of the seasons, from sowing to harvesting, and of the growth of plants and animals and men—all were connected with the contrast of temperatures, with the opposition between the hot and the cold, and with that between the dry and the wet. ‘Living creatures came into being from moisture evaporated by the sun’, we are told; and the hot and the cold also administer to the genesis of our own world edifice. The hot and the cold were also responsible for the vapours and winds which in their turn were conceived as the agents of almost all other changes.

Anaximenes, a pupil of Anaximander and his successor, developed these ideas in much detail. Like Anaximander he was interested in the oppositions of the hot and the cold and of the moist and the dry, and he explained the transitions between these opposites by a theory of condensation and rarefaction. Like Anaximander he believed in eternal motion and in the action of the winds; and it seems not unlikely that one of the two main points in which he deviated from Anaximander was reached by a criticism of the idea that what was completely boundless and formless (the apeiron) could yet be in motion. At any rate, he replaced the apeiron by air—something that was almost boundless and formless, and yet, according to Anaximander’s old theory of vapours, not only capable of motion, but the main agent of motion and change. A similar unification of ideas was achieved by Anaximenes’ theory that ‘the sun consists of earth, and that it gets very hot owing to the rapidity of its motion’.

Anaximenes’ replacement of the more abstract theory of the unbounded apeiron by the less abstract and more commonsense theory of air is matched by the replacement of Anaximander’s bold theory of the stability of the earth by the more commonsense idea that the earth’s ‘flatness is responsible for its stability; for it … covers like a lid the air beneath it’. Thus the earth rides on air as the lid of a pot may ride on steam, or as a ship may ride on water; Thales’ question and Thales’ answer are both re-instituted, and Anaximander’s epoch-making argument is not understood. Anaximenes is an eclectic, a systematizer, an empiricist, a man of common sense. Of the three great Milesians he is least productive of revolutionary new ideas; he is the least philosophically minded.

The three Milesians all looked on our world as our home. There was movement, there was change in this home, there were hot and cold, fire and moisture. There was a fire in the hearth, and on it a kettle with water. The house was exposed to the winds, and a bit draughty, to be sure; but it was home, and it meant security and stability of a sort. But for Heraclitus the house was on fire.

There was no stability left in the world of Heraclitus. ‘Everything is in flux, and nothing is at rest.’ Everything is in flux, even the beams, the timber, the building material of which the world is made: earth and rocks, or the bronze of a cauldron—they are all in flux. The beams are rotting, the earth is washed away and blown away, the very rocks split and wither, the bronze cauldron turns into green patina, or into verdigris: ‘All things are in motion all the time, even though … this escapes our senses’, as Aristotle expressed it. Those who do not know and do not think believe that only the fuel is burned, while the bowl in which it burns (cp. DK, A 4) remains unchanged; for we do not see the bowl burning. And yet it burns; it is eaten up by the fire it holds. We do not see our children grow up, and change, and grow old, but they do.

Thus there are no solid bodies. Things are not really things, they are processes, they are in flux. They are like fire, like a flame which, though it may have a definite shape, is a process, a stream of matter, a river. All things are flames: fire is the very building material of our world; and the apparent stability of things is merely due to the laws, the measures, which the processes in our world are subject to.

This, I believe, is Heraclitus’ story; it is his ‘message’, the ‘true word’ (the logos), to which we ought to listen: ‘Listening not to me but to the true account, it is wise to admit that all things are one’: they are ‘an everlasting fire, flaring up in measures, and dying down in measures’.

I know very well that the traditional interpretation of Heraclitus’ philosophy here restated is not generally accepted at present. But the critics have put nothing in its place—nothing, that is, of philosophical interest. I shall briefly discuss their new interpretation in the next section. Here I wish only to stress that Heraclitus’ philosophy, by appealing to thought, to the word, to argument, to reason, and by pointing out that we are living in a world of things whose changes escape our senses, though we know that they do change, created two new problems—the problem of change and the problem of knowledge. These problems were the more urgent as his own account of change was difficult to understand. But this, I believe, is due to the fact that he saw more clearly than his predecessors the difficulties that were involved in the very idea of change.

For all change is the change of something: change presupposes something that changes. And it presupposes that, while changing, this something must remain the same. We may say that a green leaf changes when it turns brown; but we do not say that the green leaf changes when we substitute for it a brown leaf. It is essential to the idea of change that the thing which changes retains its identity while changing. And yet it must become something else: it was green, and it becomes brown; it was moist, and it becomes dry; it was hot, and it becomes cold.

Thus every change is the transition of a thing into something with, in a way, opposite qualities (as Anaximander and Anaximenes had seen). And yet, while changing, the changing thing must remain identical with itself.

This is the problem of change. It led Heraclitus to a theory which (partly anticipating Parmenides) distinguishes between reality and appearance. ‘The real nature of things loves to hide itself. An unapparent harmony is stronger than the apparent one.’ Things are in appearance (and for us) opposites, but in truth (and for God) they are the same.

Life and death, being awake and being asleep, youth and old age, all these are the same … for the one turned round is the other and the other turned round is the first. … The path that leads up and the path that leads down are the same path. … Good and bad are identical. … For God all things are beautiful and good and just, but men assume some things to be unjust, and others to be just. … It is not in the nature or character of man to possess true knowledge, though it is in the divine nature.

Thus in truth (and for God) the opposites are identical; it is only to man that they appear as non-identical. And all things are one—they are all part of the process of the world, the everlasting fire.

This theory of change appeals to the ‘true word’, to the logos, to reason; nothing is more real for Heraclitus than change. Yet his doctrine of the oneness of the world, of the identity of opposites, and of appearance and reality threatens his doctrine of the reality of change.

For change is the transition from one opposite to the other. Thus if in truth the opposites are identical, though they appear different, then change itself might be only apparent. If in truth, and for God, all things are one, there might, in truth, be no change.

This consequence was drawn by Parmenides, the pupil (pace Burnet and others) of the monotheist Xenophanes who said of the one God: ‘He always remains in the same place, never moving. It is not fitting that He should go to different places at different times … He is in no way similar to mortal men, neither in body nor in thought.’

Xenophanes’ pupil Parmenides taught that the real world was one, and that it always remained in the same place, never moving. It was not fitting that it should go to different places at different times. It was in no way similar to what it appeared to be to mortal men. The world was one, an undivided whole, without parts, homogeneous and motionless: motion was impossible in such a world. In truth there was no change. The world of change was an illusion.

Parmenides based this theory of an unchanging reality on something like a logical proof; a proof which can be presented as proceeding from the single premiss, ‘What is not is not’. From this we can derive that the nothing—that which is not—does not exist; a result which Parmenides interprets to mean that the void does not exist. Thus the world is full: it consists of one undivided block, since any division into parts could only be due to separation of the parts by the void. (This is ‘the well-rounded truth’ which the goddess revealed to Parmenides.) In this full world there is no room for motion.

Only the delusive belief in the reality of opposites—the belief that not only what is exists but also what is not—leads to the illusion of a world of change.

Parmenides’ theory may be described as the first hypothetico-deductive theory of the world. The atomists—Leucippus and Democritus—took it as such; and they asserted that it was refuted by experience, since motion does exist. Accepting the formal validity of Parmenides’ argument, they inferred from the falsity of his conclusion the falsity of his premises. But this meant that the nothing—the void, or empty space—existed. Consequently there was now no need to assume that ‘what is’—the full, that which fills some space—had no parts; for its parts could now be separated by the void. Thus there are many parts, each of which is ‘full’: there are full particles in the world, separated by empty space, and able to move in empty space, each of them being ‘full’, undivided, indivisible, and unchanging. Thus what exists is atoms and the void. In this way the atomists arrived at a theory of change—a theory that dominated scientific thought until 1900. It is the theory that all change, and especially all qualitative change, has to be explained by the spatial movement of unchanging bits of matter—by atoms moving in the void.

The next great step in our cosmology and the theory of change was made when Maxwell, developing certain ideas of Faraday’s, replaced this theory by a theory of changing intensities of fields.

X

I have sketched the story, as I see it, of the Presocratic theory of change. I am of course well aware of the fact that my story (which is based on Plato, Aristotle, and the doxographic tradition) clashes at many points with the views of some experts, English as well as German, and especially with the views expressed by G. S. Kirk and J. E. Raven in their book, The Presocratic Philosophers, 1957. I cannot of course examine their arguments in detail here, and especially not their minute exegeses of various passages some of which are relevant to the differences between their interpretation and mine. (See, for example, Kirk and Raven’s discussion of the question whether there is a reference to Heraclitus in Parmenides; cf. their note 1 on pp. 193 f., and note 1 on p. 272.) But I wish to say that I have examined their arguments and that I have found them unconvincing and often quite unacceptable.

I will mention here only some points regarding Heraclitus (although there are other points of equal importance, such as their comments on Parmenides).

The traditional view, according to which Heraclitus’ central doctrine was that all things are in flux, was attacked forty years ago by Burnet. His main argument (discussed by me at length in note 2 to ch. 2 of my Open Society) was that the theory of change was not new, and that only a new message could explain the urgency with which Heraclitus speaks. This argument is repeated by Kirk and Raven when they write (pp. 186 f.): ‘But all Presocratic thinkers were struck by the predominance of change in the world of our experience.’ About this attitude I said in my Open Society: ‘Those who suggest … that the doctrine of universal flux was not new … are, I feel, unconscious witnesses to Heraclitus’ originality, for they fail now, after 2,400 years, to grasp his main point.’ In brief, they do not see the difference between the Milesian message, ‘There is a fire in the house’, and Heraclitus’ somewhat more urgent message, ‘The house is on fire’. An implicit reply to this criticism can be found on p. 197 of the book by Kirk and Raven, where they write: ‘Can Heraclitus really have thought that a rock or a bronze cauldron, for example, was invariably undergoing invisible changes of material? Perhaps so; but nothing in the extant fragments suggests that he did.’ But is this so? Heraclitus’ extant fragments about the fire (Kirk and Raven, fragm. 220–2) are interpreted by Kirk and Raven themselves as follows (p. 200): ‘Fire is the archetypal form of matter.’ Now I am not at all sure what ‘archetypal’ means here (especially in view of the fact that we read a few lines later, ‘Cosmogony … is not to be found in Heraclitus’). But whatever ‘archetypal’ may mean, it is clear that once it is admitted that Heraclitus says in the extant fragments that all matter is somehow (whether archetypally or otherwise) fire, he also says that all matter, like fire, is a process; which is precisely the theory denied to Heraclitus by Kirk and Raven.

Immediately after saying that ‘nothing in the extant fragments suggests’ that Heraclitus believed in continuous invisible changes, Kirk and Raven make the following methodological remark: ‘It cannot be too strongly emphasized that [in texts] before Parmenides and his apparent proof that the senses were completely fallacious … gross departures from common sense must only be accepted when the evidence for them is extremely strong.’ This is intended to mean that the doctrine that bodies (of any substance) constantly undergo invisible changes represents a gross departure from common sense, a departure which one ought not to expect in Heraclitus.

But to quote Heraclitus: ‘He who does not expect the unexpected will not detect it: for him it will remain undetectable, and unapproachable’ (DK, B 18). In fact Kirk and Raven’s last argument is invalid on many grounds. Long before Parmenides we find ideas far removed from common sense in Anaximander, Pythagoras, Xenophanes, and especially in Heraclitus. Indeed the suggestion that we should test the historicity of the ideas ascribed to Heraclitus—as we might indeed test the historicity of those ascribed to Anaximenes—by standards of ‘common sense’ is a little surprising (whatever ‘common sense’ may mean here). For this suggestion runs counter not only to Heraclitus’ notorious obscurity and oracular style, confirmed by Kirk and Raven, but also to his burning interest in antinomy and paradox. And it runs counter, last but not least, to the (in my view quite absurd) doctrine which Kirk and Raven finally attribute to Heraclitus (the italics are mine): ‘… that natural changes of all kinds [and thus presumably also earthquakes and great fires] are regular and balanced, and that the cause of this balance is fire, the common constituent of things that was also termed their Logos.’ But why, I ask, should fire be ‘the cause’ of any balance—either ‘this balance’ or any other? And where does Heraclitus say such things? Indeed, had this been Heraclitus’ philosophy, then I could see no reason to take any interest in it; at any rate, it would be much further removed from common sense (as I see it) than the inspired philosophy which tradition ascribes to Heraclitus and which, in the name of common sense, is rejected by Kirk and Raven.

But the decisive point is, of course, that this inspired philosophy is true, for all we know.5 With his uncanny intuition Heraclitus saw that things are processes, that our bodies are flames, that ‘a rock or a bronze cauldron … was invariably undergoing invisible changes’. Kirk and Raven say (p. 197, note 1; the argument reads like an answer to Melissus): ‘Every time the finger rubs, it rubs off an invisible portion of iron; yet when it does not rub, what reason is there to think that the iron is still changing?’ The reason is that the wind rubs, and that there is always wind; or that iron turns invisibly into rust—by oxidation, and this means by slow burning; or that old iron looks different from new iron, just as an old man looks different from a child (cp. DK, B 88). This was Heraclitus’ teaching, as the extant fragments show.

I suggest that Kirk and Raven’s methodological principle ‘that gross departures from common sense must only be accepted when the evidence for them is extremely strong’ might well be replaced by the clearer and more important principle that gross departures from the historical tradition must only be accepted when the evidence for them is extremely strong. This, in fact, is a universal principle of historiography. Without it history would be impossible. Yet it is constantly violated by Kirk and Raven: when, for example, they try to make Plato’s and Aristotle’s evidence suspect, with arguments which are partly circular and partly (like the one from common sense) in contradiction to their own story. And when they say that ‘little serious attempt seems to have been made by Plato and Aristotle to penetrate his [i.e. Heraclitus’] real meaning’ then I can only say that the philosophy outlined by Plato and Aristotle seems to me a philosophy that has real meaning and real depth. It is a philosophy worthy of a great philosopher. Who, if not Heraclitus, was the great thinker who first realized that men are flames and that things are processes? Are we really to believe that this great philosophy was a ‘post-Heraclitean exaggeration’ (p. 197), and that it may have been suggested to Plato, ‘in particular, perhaps, by Cratylus’? Who, I ask, was this unknown philosopher—perhaps the greatest and the boldest thinker among the Presocratics? Who was he, if not Heraclitus?

XI

The early history of Greek philosophy, especially the history from Thales to Plato, is a splendid story. It is almost too good to be true. In every generation we find at least one new philosophy, one new cosmology of staggering originality and depth. How was this possible? Of course one cannot explain originality and genius. But one can try to throw some light on them. What was the secret of the ancients? I suggest that it was a tradition—the tradition of critical discussion.

I will try to put the problem more sharply. In all or almost all civilizations we find something like religious and cosmological teaching, and in many societies we find schools. Now schools, especially primitive schools, all have, it appears, a characteristic structure and function. Far from being places of critical discussion they make it their task to impart a definite doctrine, and to preserve it, pure and unchanged. It is the task of a school to hand on the tradition, the doctrine of its founder, its first master, to the next generation, and to this end the most important thing is to keep the doctrine inviolate. A school of this kind never admits a new idea. New ideas are heresies, and lead to schisms; should a member of the school try to change the doctrine, then he is expelled as a heretic. But the heretic claims, as a rule, that his is the true doctrine of the founder. Thus not even the inventor admits that he has introduced an invention; he believes, rather, that he is returning to the true orthodoxy which has somehow been perverted.

In this way all changes of doctrine—if any—are surreptitious changes. They are all presented as re-statements of the true sayings of the master, of his own words, his own meaning, his own intentions.

It is clear that in a school of this kind we cannot expect to find a history of ideas, or even the material for such a history. For new ideas are not admitted to be new. Everything is ascribed to the master. All we might reconstruct is a history of schisms, and perhaps a history of the defence of certain doctrines against the heretics.

There cannot, of course, be any rational discussion in a school of this kind. There may be arguments against dissenters and heretics, or against some competing schools. But in the main it is with assertion and dogma and condemnation rather than argument that the doctrine is defended.

The great example of a school of this kind among the Greek philosophical schools is the Italian School founded by Pythagoras. Compared with the Ionian school, or with that of Elea, it had the character of a religious order, with a characteristic way of life and a secret doctrine. The story that a member, Hippasus of Metapontum, was drowned at sea because he revealed the secret of the irrationality of certain square roots, is characteristic of the atmosphere surrounding the Pythagorean school, whether or not there is any truth in this story.

But among Greek philosophic schools the early Pythagoreans were an exception. Leaving them aside, we could say that the character of Greek Philosophy, and of the philosophical schools, is strikingly different from the dogmatic type of school here described. I have shown this by an example: the story of the problem of change which I have told is the story of a critical debate, of a rational discussion. New ideas are propounded as such, and arise as the result of open criticism. There are few, if any, surreptitious changes. Instead of anonymity we find a history of ideas and of their originators.

Here is a unique phenomenon, and it is closely connected with the astonishing freedom and creativeness of Greek philosophy. How can we explain this phenomenon? What we have to explain is the rise of a tradition. It is a tradition that allows or encourages critical discussions between various schools and, more surprisingly still, within one and the same school. For nowhere outside the Pythagorean school do we find a school devoted to the preservation of a doctrine. Instead we find changes, new ideas, modifications, and outright criticism of the master.

(In Parmenides we even find, at an early date, a most remarkable phenomenon—that of a philosopher who propounds two doctrines, one which he says is true, and one which he himself describes as false. Yet he makes the false doctrine not simply an object of condemnation or of criticism; rather he presents it as the best possible account of the delusive opinion of mortal men, and of the world of mere appearance—the best account which a mortal man can give.)

How and where was this critical tradition founded? This is a problem deserving serious thought. This much is certain: Xenophanes who brought the Ionian tradition to Elea was fully conscious of the fact that his own teaching was purely conjectural, and that others might come who would know better. I shall come back to this point again in my next and last section.

If we look for the first signs of this new critical attitude, this new freedom of thought, we are led back to Anaximander’s criticism of Thales. Here is a most striking fact: Anaximander criticizes his master and kinsman, one of the Seven Sages, the founder of the Ionian school. He was, according to tradition, only about fourteen years younger than Thales, and he must have developed his criticism and his new ideas while his master was alive. (They seem to have died within a few years of each other.) But there is no trace in the sources of a story of dissent, of any quarrel, or of any schism.

This suggests, I think, that it was Thales who founded the new tradition of freedom—based upon a new relation between master and pupil—and who thus created a new type of school, utterly different from the Pythagorean school. He seems to have been able to tolerate criticism. And what is more, he seems to have created the tradition that one ought to tolerate criticism.

Yet I like to think that he did even more than this. I can hardly imagine a relationship between master and pupil in which the master merely tolerates criticism without actively encouraging it. It does not seem to me possible that a pupil who is being trained in the dogmatic attitude would ever dare to criticize the dogma (least of all that of a famous sage) and to voice his criticism. And it seems to me an easier and simpler explanation to assume that the master encouraged a critical attitude—possibly not from the outset, but only after he was struck by the pertinence of some questions asked, by the pupils perhaps, without any critical intention.

However this may be, the conjecture that Thales actively encouraged criticism in his pupils would explain the fact that the critical attitude towards the master’s doctrine became part of the Ionian school tradition. I like to think that Thales was the first teacher who said to his pupils: ‘This is how I see things—how I believe that things are. Try to improve upon my teaching.’ (Those who believe that it is ‘unhistorical’ to attribute this undogmatic attitude to Thales may again be reminded of the fact that only two generations later we find a similar attitude consciously and clearly formulated in the fragments of Xenophanes.) At any rate, there is the historical fact that the Ionian school was the first in which pupils criticized their masters, in one generation after the other. There can be little doubt that the Greek tradition of philosophical criticism had its main source in Ionia.

It was a momentous innovation. It meant a break with the dogmatic tradition which permits only one school doctrine, and the introduction in its place of a tradition that admits a plurality of doctrines which all try to approach the truth by means of critical discussion.

It thus leads, almost by necessity, to the realization that our attempts to see and to find the truth are not final, but open to improvement; that our knowledge, our doctrine, is conjectural; that it consists of guesses, of hypotheses, rather than of final and certain truths; and that criticism and critical discussion are our only means of getting nearer to the truth. It thus leads to the tradition of bold conjectures and of free criticism, the tradition which created the rational or scientific attitude, and with it our Western civilization, the only civilization which is based upon science (though of course not upon science alone).

In this rationalist tradition bold changes of doctrine are not forbidden. On the contrary, innovation is encouraged, and is regarded as success, as improvement, if it is based on the result of a critical discussion of its predecessors. The very boldness of an innovation is admired; for it can be controlled by the severity of its critical examination. This is why changes of doctrine, far from being made surreptitiously, are traditionally handed down together with the older doctrines and the names of the innovators. And the material for a history of ideas becomes part of the school tradition.

To my knowledge the critical or rationalist tradition was invented only once. It was lost after two or three centuries, perhaps owing to the rise of the Aristotelian doctrine of epistēmē, of certain and demonstrable knowledge (a development of the Eleatic and Heraclitean distinction between certain truth and mere guesswork). It was rediscovered and consciously revived in the Renaissance, especially by Galileo Galilei.

XII

I now come to my last and most central contention. It is this. The rationalist tradition, the tradition of critical discussion, represents the only practicable way of expanding our knowledge—conjectural or hypothetical knowledge, of course. There is no other way. More especially, there is no way that starts from observation or experiment. In the development of science observations and experiments play only the role of critical arguments. And they play this role alongside other, nonobservational arguments. It is an important role; but the significance of observations and experiments depends entirely upon the question whether or not they may be used to criticize theories.

According to the theory of knowledge here outlined there are in the main only two ways in which theories may be superior to others: they may explain more; and they may be better tested—that is, they may be more fully and more critically discussed, in the light of all we know, of all the objections we can think of, and especially also in the light of observational or experimental tests which were designed with the aim of criticizing the theory.

There is only one element of rationality in our attempts to know the world: it is the critical examination of our theories. These theories themselves are guesswork. We do not know, we only guess. If you ask me, ‘How do you know?’ my reply would be, ‘I don’t; I only propose a guess. If you are interested in my problem, I shall be most happy if you criticize my guess, and if you offer counter-proposals, I in turn will try to criticize them.’

This, I believe, is the true theory of knowledge (which I wish to submit for your criticism): the true description of a practice which arose in Ionia and which is incorporated in modern science (though there are many scientists who still believe in the Baconian myth of induction): the theory that knowledge proceeds by way of conjectures and refutations.

Two of the greatest men who clearly saw that there was no such thing as an inductive procedure, and who clearly understood what I regard as the true theory of knowledge, were Galileo and Einstein. Yet the ancients also knew it. Incredible as it sounds, we find a clear recognition and formulation of this theory of rational knowledge almost immediately after the practice of critical discussion had begun. Perhaps our oldest extant fragments in this field are those of Xenophanes. I will present here five of them in an order that suggests that it was the boldness of his attack and the gravity of his problems which made him conscious of the fact that all our knowledge was guesswork, yet that we may nevertheless, by searching for that knowledge ‘which is the better’, find it in the course of time. Here are the five fragments (DK, B 16 and 15; 18; 35; and 34) from Xenophanes’ writings.

The Ethiops say that their gods are pug-nosed and black
While the Thracians say that theirs have blue eyes and red hair.

Yet if cattle or horses or lions had hands and could draw
And could sculpture like men, then the horses would draw their gods
Like horses, and cattle like cattle, and each would then shape
Bodies of gods in the likeness, each kind, of its own.

The gods did not reveal, from the beginning,
All things to us; but in the course of time,
Through seeking we may learn, and know things better …

These things, we conjecture, are somehow like the truth.

But as for certain truth, no man has known it,
Nor will he know it; neither of the gods,
Nor yet of all the things of which I speak.
And even if perchance he were to utter
The perfect truth, he would himself not know it: For all is but a woven web of guesses.

To show that Xenophanes was not alone I may also repeat here two of Heraclitus’ sayings (DK, B 78 and 18) which I have quoted before in a different context. Both express the conjectural character of human knowledge, and the second refers to its daring, to the need to anticipate boldly what we do not know.

It is not in the nature or character of man to possess true knowledge, though it is in the divine nature … He who does not expect the unexpected will not detect it: for him it will remain undetectable, and unapproachable.

My last quotation is a very famous one from Democritus (DK, B 117):

But in fact, nothing do we know from having seen it; for the truth is hidden in the deep.

This is how the critical attitude of the Presocratics foreshadowed, and prepared for, the ethical rationalism of Socrates: his belief that the search for truth through critical discussion was a way of life—the best he knew.

Appendix: Historical Conjectures and Heraclitus on Change

In an article entitled ‘Popper on Science and the Presocratics’ (Mind, NS. 69, July 1960, pp. 318 to 339), Mr G. S. Kirk has responded to a challenge, and to a criticism, which formed part of my presidential address to the Aristotelian Society ‘Back to the Presocratics’. Mr Kirk’s article is not, however, mainly devoted to the task of replying to my criticism. It is, largely, devoted to another task: it tries to explain how and why I am the victim of a fundamentally mistaken ‘attitude to scientific methodology’ which has made me come forward with mistaken assertions about the Presocratics and with mistaken principles of historiography.

A counter-attack of this kind might, to be sure, have its intrinsic merits and interest. And the fact that Mr Kirk has adopted this procedure shows at any rate that he and I agree at least on two points: that the fundamental issue between us is a philosophical one; and that the philosophical attitude we adopt can have a decisive influence on our interpretation of the historical evidence—such as, for example, the evidence concerning the Presocratics.

Now Mr Kirk does not accept my general philosophical attitude any more than I do his. Thus he rightly feels that he should give reasons for rejecting mine.

I do not think that he has offered any reason for rejecting my views; simply because Mr Kirk’s views on what he believes to be my views, and the devastating conclusions he draws from these views, are unrelated to my actual views, as I shall show.

There is another difficulty. The method of counter-attack which he has adopted has its own peculiar drawback: it does not seem to lend itself easily to furthering the discussion of the definite points of criticism made in my address. Kirk does not, for instance, state very clearly which of my points he accepts (for he does accept some) and which he rejects; instead, acceptance and rejection are submerged in a general rejection of what he believes to be my ‘attitude to scientific methodology’, and of some of the consequences of this imaginary attitude.

I

My first task will be to give some evidence for my allegation that Kirk’s treatment of my ‘attitude to scientific methodology’ is largely based upon misunderstandings and misreadings of what I have written, and upon popular inductivist misconceptions about natural science, which were fully discussed and dismissed in my book, The Logic of Scientific Discovery (L.Sc.D.).

Kirk rightly presents me as an opponent of the widely accepted dogma of inductivisim—of the view that science starts from observation and proceeds, by induction, to generalizations, and ultimately to theories. But he is mistaken in believing that, since I am an opponent of induction, I must be an adherent of intuition, and that my approach must be due to an attempt to defend an intuitionist philosophy, which he calls ‘traditional philosophy’, against modern empiricism. For although I do not believe in induction, I do not believe in intuition either. Inductivists are inclined to think that intuition is the only alternative to induction. But they are simply mistaken: there are other possible approaches besides these two. And my own view may be fairly described as critical empiricism.

But Kirk ascribes to me an almost Cartesian intuitionism when he presents the situation as follows (p. 319): ‘Philosophy of the traditional type had assumed that philosophical truths were metaphysical in content and could be apprehended by intuition. The positivists of the Vienna Circle denied this. In disagreeing with them Popper was asserting his belief in something not far distant from the classical conception of the role of philosophy.’ Whatever one may say about this, there certainly exists a ‘traditional philosophy’—that of Descartes or Spinoza, for example—which treats ‘intuition’ as a source of knowledge; but I have always opposed this philosophy.1 From this passage on, Kirk writes ‘intuition’, in the sense in which he uses it here, several times in quotes (pp. 320, 321, 322, 327) and several times out of quotes (pp. 318, 319, 320, 324, 327, 332, 337), yet always apparently under the impression—and certainly creating the impression—that he is citing me when ascribing to me intuitionist views; views which I have never held in my life. In fact, the only time the word ‘intuition’ occurs in my address,2 it is used in a context which is anti-inductivist and anti-intuitionist at the same time. For I write there (p. 7; this volume p. 189) about the problem of the scientific character of a theory (italics not in the original): ‘What is important about a theory is its explanatory power, and whether it stands up to criticism and to tests. The question of its origin, of how it is arrived at—whether by an “inductive procedure”, as some say, or by an act of intuition— … has little to do with its scientific [status or] character.’3

Now Kirk quotes this passage and discusses it. But the undeniable fact that this passage indicates that I am a believer in neither induction nor intuition does not prevent him from constantly ascribing to me intuitionist views. He does so, for example, in the passage on p. 319 quoted above; or on p. 324, when he discusses the question whether to accept my alleged ‘premise that science starts from intuitions’ (while I say it starts from problems; see below); or on pp. 326 f. when he writes: ‘Are we therefore to infer with Popper that Thales’s theory must have been based on a non-empirical intuition?’

Now my own view is very different from all this. As to the starting point of science, I do not say that science starts from intuitions but that it starts from problems; that we arrive at a new theory, in the main, by trying to solve problems; that these problems arise in our attempts to understand the world as we know it—the world of our ‘experience’ (where ‘experience’ consists largely of expectations or theories and partly also of observational knowledge—although I happen to believe that there does not exist anything like pure observational knowledge, untainted by expectations or theories). A few of these problems—and some of the most interesting ones—arise from the conscious criticism of theories uncritically accepted hitherto, or from the conscious criticism of the theory of a predecessor. One of the main things I set out to do in my paper on the Presocratics was to suggest that Anaximander’s theory may well have originated in an attempt to criticize Thales; and that this may well have been the origin of the rationalist tradition, which I identify with the tradition of critical discussion.

I do not think that a view of this kind has much similarity to traditional intuitionistic philosophy. And I was surprised to find that Kirk suggests that my mistaken approach might be explained as that of a speculative philosopher not sufficiently intimately acquainted with scientific practice; he suggests for example on p. 320: ‘It seems possible that his [Popper’s] view of science was not the result of an initial objective observation of how scientists proceed, but was itself, in an early application of Popper’s developed theory, an “intuition” closely related to current philosophical difficulties and subsequently compared with actual scientific procedure.’4 (I should have thought that even a reader who knows very little about science might have noticed that some at least of my problems originated within the physical sciences themselves, and that my own acquaintance with scientific practice and research was not wholly second-hand.)

The kind of critical discussion I have in mind is, of course, a discussion in which experience plays a major role: observation and experiment are constantly appealed to as tests of our theories. Yet Kirk surprisingly goes so far (p. 332; italics mine) as to speak of ‘Popper’s thesis that all scientific theories are entirely based on intuitions’.

Like most philosophers I am quite used to seeing my views distorted and caricatured. But this is hardly a caricature (which must always rely on a recognizable similarity to the original). I may remark that none of my empiricist and positivist friends, opponents, and critics, has ever criticized me for holding or for reviving an intuitionist epistemology, and that, on the contrary, they usually say that my epistemology does not significantly deviate from theirs.

It will be seen from the foregoing that Kirk offers several conjectures, not only about the content of my philosophy, but also about its origin. But he does not seem to be aware of the conjectural character of these constructions. On the contrary, he believes that he has some textual evidence for them. For he says of me that my ‘own attitude of scientific methodology … was formed as he [Popper] writes in the 1958 preface to The Logic of Scientific Discovery, in reaction against the attempts of the Vienna Circle to base all philosophical [sic] and scientific truth upon verification by experience’ (Kirk, p. 319). I need not comment here on this mistaken description of the Wittgensteinian philosophy of the Vienna Circle. But since it is a historian of philosophy who writes here about what I have written, I feel I must nip in the bud a historical myth about what I have written. For in the preface to which Kirk refers, I do not say a word about how I formed my views or my attitude; nor do I say a word about the Vienna Circle. Indeed, I could not have written anything resembling Kirk’s account, because the facts are otherwise. (Part of the story, first published in 1957, Mr Kirk might have found in a Cambridge lecture of mine, now in this volume under the title ‘Science: Conjectures and Refutations’, in which I tell how I developed my ‘attitude … in reaction against the attempts of’ Marx, Freud, and Adler, none of whom was either a positivist or a member of the Vienna Circle.) It seems unlikely that it was the Heraclitean obscurity of my style which caused this quite inexplicable misreading by Mr Kirk, for in comparing it with ‘Back to the Presocratics’ he describes (p. 318) the same 1958 preface to which his above quoted passage refers as ‘a more lucid statement’.

Another example of misreading The Logic of Scientific Discovery is equally inexplicable—at least for anybody who has read the book as far as p. 61 (not to mention pp. 274 or 276) where I refer to the problem of truth, and to Alfred Tarski’s theory of truth. Kirk says that ‘Popper abandons the concept of absolute scientific truth’ (p. 320). He does not seem to see that, when I say that we cannot know, even of a well-corroborated scientific theory, whether or not it is true, I am actually assuming a ‘concept of absolute scientific truth’; just as somebody who says ‘I did not succeed in reaching the winning post’ operates with an ‘absolute concept of a winning post’—that is, one whose existence is assumed independently of its being reached.

It is surprising to find these obvious misunderstandings, and these occasional misquotations, in a paper by an outstanding scholar and historian of philosophy. They make a philosophical defence of my real views about science unnecessary.

II

Thus I may now turn to something more to the point—to the history of the Presocratics. In this section I shall confine myself to straightening out two of Kirk’s mistakes concerning my historical method, and my views on historical method. In section III I will deal with our real disagreements.

(1) Kirk discusses on p. 325 a remark of mine which I made in order to disclaim any competence regarding such matters as text emendation. The passage he quotes reads: ‘I am completely out of my depth when an expert begins to argue what words or phrases Heraclitus might have used, and what words or phrases he could not possibly have used.’

Commenting on this disclaimer of competence, Kirk exclaims: ‘As though “what words or phrases Heraclitus might have used”, for example, is irrelevant to the assessment of what he thought!’

But I never said or suggested that these matters are ‘irrelevant’. I merely confessed that I had not studied the linguistic usages of Heraclitus (and others) sufficiently deeply to feel myself equipped to discuss the work done in this field by such scholars as, say, Burnet or Diels or Reinhardt, and, more recently, Vlastos or Kirk himself.

Yet Kirk goes on to say that

It is these ‘words and phrases’, and the other verbatim fragments of the Presocratics themselves, and not the reports of Plato, Aristotle, and the doxographers, as Popper appears to think, that are ‘the oldest texts we possess’ … It should in fact be obvious even to an ‘amateur’ that the reconstruction of Presocratic thought must be based both upon the later tradition and upon the surviving fragments.

I cannot imagine how my disclaimer of competence in the field of linguistic criticism can have induced Kirk to suggest that such things are not ‘obvious’ even to the particular amateur in question. Moreover, he might have noticed that fairly frequently I quote, translate, and discuss, the fragments themselves (much more than the reports of Plato and Aristotle, though we now seem to agree that these are quite relevant also), both in ‘Back to the Presocratics’ and in my Open Society, where I discussed, for example, a considerable number of the surviving fragments of Heraclitus. Kirk refers to this book on p. 324. Why then does he, on p. 325, interpret my disclaimer in the sense that I disclaim interest in the surviving fragments, or in the problem of their historical status?

(2) As an example of the way, unsatisfactory in my opinion, in which Kirk answers the criticisms I made in ‘Back to the Presocratics’, I now quote the end of his reply (p. 339). He says:

More startling still, he [Popper] applies the criterion of possible truth as the test of the historicity of a theory. On page 16 he [Popper] finds that ‘the suggestion that we should test the historicity of Heraclitus’ ideas … by standards of “common sense” is a little surprising.’ Shall we [Kirk] not find his [Popper’s] own ‘test’ much more surprising— ‘But the decisive point is, of course, that this inspired philosophy [i.e. that man is a flame, etc.] is true, for all we know’ (p. 17 [in this volume, p. 199])?

The simple answer to this is that I neither said nor implied that the truth, or the possible truth, of a theory is a ‘test’ of its historicity. (This may be seen from pages 16 and 17 of my address—in this volume, pp. 198 f.—and the second paragraph of section vii; incidentally, did Kirk forget his thesis that I have abandoned the idea of truth?) And when Kirk here puts ‘test’ in quotes—thereby indicating that I have used the term ‘test’ in this context, or in this sense—then he clearly misquotes me. For all I have said or implied is that the truth of that theory of change which has been traditionally, and I think correctly, attributed to Heraclitus, shows that this attribution at least makes sense of Heraclitus’ philosophy—while I at any rate could not make sense of the philosophy attributed to Heraclitus by Kirk. Incidentally, I do think that it is an important and even an obvious principle of the historiography and interpretation of ideas that we should always try to attribute to a thinker an interesting and a true theory rather than an uninteresting or a false one, provided of course the transmitted historical evidence allows us to do so. This is neither a criterion nor a ‘test’, to be sure; but he who does not try to apply this principle of historiography is unlikely to understand a great thinker such as Heraclitus.

III

The most important disagreement between Kirk and me as far as the Presocratics are concerned was over the interpretation of Heraclitus. And here I claim that Kirk, perhaps unconsciously, has almost ceded my two main points which I am going to discuss below under (1) and (2).

My general approach to Heraclitus may be put in the words of Karl Reinhardt: ‘The history of philosophy is the history of its problems. If you want to explain Heraclitus, tell us first what his problem was.’5

My answer to this challenge was that Heraclitus’ problem is the problem of change—the general problem, How is change possible? How can a thing change without losing its identity—in which case it would be no longer that thing which has changed? (See ‘Back to the Presocratics’, sections viii and ix.)

I believe that Heraclitus’ great message was linked with his discovery of this exciting problem; and I believe that his discovery led to Parmenides’ solution that change, indeed, is logically impossible for any thing—for any being; and later to the closely related theory of Leucippus and Democritus that things do not, indeed, change intrinsically, although they change their positions in the void.

The solution of this problem which, following Plato, Aristotle, and the fragments, I attribute to Heraclitus, is as follows: there are no (unchanging) things; what appears to us as a thing is a process. In reality a material thing is like a flame; for a flame seems to be a material thing, but it is not: it is a process; it is in flux; matter passes through it; it is like a river.

Thus all the apparently more or less stable things are really in flux; and some of them—those which indeed appear stable—are in invisible flux. (Thus Heraclitus’ philosophy prepares the way for the Parmenidean distinction between appearance and reality.)

In order to appear as a stable thing, the process (which is the reality behind the thing) has to be regular, law-like, ‘measured’: the lamp which holds a stable flame has to supply to it a definite measure of oil. It seems not unlikely that the idea of a measured or law-like process was developed by Heraclitus from suggestions of the Milesians, especially of Anaximander, about the significance of the cosmic periodic changes (such as day and night, perhaps also the tides, the waxing and waning of the moon, and especially the seasons of the year). These regularities might well have contributed to the idea that the apparent stability of things, and even of the cosmos, can be explained as a ‘measuredprocess— a process ruled by law.

(1) The first of the two main points on which I criticized Kirk’s views on Heraclitus is this. Kirk suggested that Heraclitus did not believe, and that it was against common sense to believe, ‘that a rock or a bronze cauldron … was invariably undergoing invisible changes’. Kirk’s lengthy discussion (pp. 334 ff.) of my criticism ultimately arrives at a point about which he says:

At this point the argument becomes somewhat rarefied. I agree, though, that it remains theoretically possible that certain invisible changes of our experience, for example the gradual rusting of iron, cited by Popper, struck Heraclitus so forcefully that they persuaded him to assert that all things which were not in visible change were in invisible change. I do not think however, that the extant fragments suggest that this was the case (p. 336).

I do not think that the argument need in any sense become rarefied; and there are many extant fragments which suggest the theory which I attribute to Heraclitus. Yet before referring to these I must repeat a question which I raised in my address: if, as Kirk and Raven agree, fire is, as it were, the structural model or the prototype (or the ‘archetypal form’ as they have it) of matter, what else can this mean but that material things are like flames, and therefore processes?

I do not, of course, assert that Heraclitus used an abstract term like ‘process’. But I conjecture that he did apply his theory not only to matter in the abstract, or to ‘the world order as a whole’ (as Kirk says on p. 335), but also to concrete, single things; and these things, then, must be compared to concrete, single flames.

As to the extant fragments in support of this view and of my interpretation in general, there are first the fragments about the sun. It seems to me pretty clear that Heraclitus regarded the sun as a thing, or perhaps even as a new thing every day; see DK, B 6 which says6 ‘The sun is new every day’, though this may perhaps only mean that it is, like a lamp, re-kindled every day: ‘Were there no sun, it would be night in spite of the other stars’ says B 99. (See also B 26 and my remark above concerning lamps and measures and compare B 94.) Or take B 125: ‘If not stirred, the barley-brew decomposes.’ Thus movement, process, is essential to the continued existence of the thing which otherwise ceases to exist. Or take B 51: ‘What struggles with itself becomes committed to itself: there is a link or harmony due to recoil [or: to the turning back of the strings] and tension, as in the bow and the lyre.’ It is the tension, the active force, the inherent strife (a process), which makes bow and lyre what they are, and only as long as the tension is kept up, only as long as the strife of their parts goes on, do they continue to be what they are.

Admittedly, Heraclitus likes generalizations and abstractions; and so he proceeds at once to a generalization which may well be intended as one on a cosmic scale, as in B 8: ‘The opposites agree, and from discord results the best harmony.’ (See also B 10.) But this does not mean that he loses sight of the single things, the bow, the lyre, the lamp, the flame, the river (B 12, 49a). ‘Upon those who step into the same rivers, different and again different waters flow … We step into the same rivers, and we do not step [into the same rivers]. We are, and we are not.’

Yet before becoming symbols of the cosmic processes, the rivers are concrete rivers, and beyond that, symbols of other concrete things, including ourselves. And although ‘we are, and we are not’ (which, incidentally, Kirk and Raven prefer not to attribute to Heraclitus) is, in a sense, a sweeping and perhaps cosmic generalization and abstraction, it is no doubt also meant as a very concrete appeal to every man: it is a Heraclitean memento mori, like so many other fragments which remind us that life becomes death, and death becomes life. (Compare for example B 88, 20, 21, 26, 27, 62, 77.)

If B 49a moves towards something like a generalization, B 90 moves from the general and cosmic idea of a consuming (and dying) fire to the particular: ‘Everything is an exchange for fire and fire for everything just as wares for gold and gold for wares.’

Thus when Kirk now asks (p. 336): ‘Can we then say that the conclusion that all things separately are in permanent flux is necessarily entailed by any course of reasoning followed by Heraclitus?’, then the answer is an emphatic ‘yes’, as far as we can speak at all of anything as being ‘necessarily entailed’ by a ‘course of reasoning’ in a field where everything must remain to some extent conjecture and interpretation.

Thus take for example B 126, ‘What is cold becomes hot and what is hot becomes cold; what is moist becomes dry and what is dry becomes moist.’ This may well have a cosmic significance: it may refer to the seasons, and to cosmic change. But how can it be doubted (especially if we attribute to Heraclitus ‘common sense’, whatever this may mean7) that it applies to concrete, individual things and their changes—and incidentally, to ourselves and our souls? (Cp. B 36, 77, 117, 118.)

But things are not only in flux—they are invisibly in flux. So we read in B 88: ‘It is always one and the same: what is alive and what is dead; what is awake and what is asleep; what is young and what is old. For the one turns into the other and the other turns back into the one.’ Thus our children age—as we know, invisibly; yet the parents also turn—somehow—into their children. (See also B 20, 21, 26, 62 and 90.) Or take B 103: ‘In a circle, the beginning and the end are the same.’ (The identity of opposites; opposites invisibly merging into each other; see also B 54, 65, 67, 126.)

That Heraclitus notices that these processes may indeed be invisible, and that he therefore felt that sight, and observation, were deceptive, may be seen from B 46: ‘… sight is deceptive.’ B 54: ‘Invisible harmony is stronger than visible.’ (See also B 8 and 51.) B 123: ‘Nature loves to hide.’ (See also B 56 and 113.)

I have not the slightest doubt that any one or all of these fragments may be explained away. But they do seem to me to support what it is reasonable to assume in any case, and what is, in addition, supported by Plato and Aristotle. (And though the evidence of the latter has become suspect, especially in view of the great work of Harold Cherniss, nobody thinks—and least of all Harold Cherniss—that Aristotle’s evidence has been completely discredited, including that which is supported by Plato or by the ‘fragments’.)

(2) The last point of my reply, and my second and main point about Heraclitus, concerns the general summary of his philosophy which can be found in Kirk and Raven on p. 214 under the heading ‘Conclusion’.

I quoted part of this conclusion in my address, and said that I found the doctrine attributed by Kirk and Raven to Heraclitus ‘absurd’; and in order to make quite clear what I regarded as ‘absurd’, I used italics. I repeat here my quotation from Kirk and Raven, with the italics as previously used by me. What I found ‘absurd’ is the allegedly Heraclitean doctrine ‘that natural changes of all kinds [and thus presumably also earthquakes and great fires] are regular and balanced and that the cause of this balance is fire, the common constituent of things that was also termed their Logos’. (See p. 198, above.)

I did not object to anybody’s attributing to Heraclitus the doctrine that change is ruled by law, or perhaps the more doubtful doctrine that the rule, or regularity, was their ‘Logos’; or the doctrine that ‘the common constituent of things was fire’. What I felt to be absurd were the doctrines (a) that all changes (or ‘changes of all kinds’) are ‘balanced’ in the sense in which many important changes and processes such as the fire in a lamp, or the cosmic seasons, may well be called ‘balanced’; (b) that fire is ‘the cause of this balance’; and (c) that the common constituent of things—that is, fire—‘was also termed their Logos’.

Moreover, I could find no traces of these doctrines in Heraclitus’ fragments, nor in any of the ancient sources, such as Plato or Aristotle.

Where then is the source of this summary or ‘conclusion’—that is to say, the source of the three points (a), (b), and (c) which express Kirk’s general view of Heraclitus’ philosophy, and which colour so much of his interpretation of the fragments?

Reading the chapter on Heraclitus in Kirk and Raven again, I could find only one hint: the doctrines to which I object are first formulated on p. 200, with reference to the fragment which they number 223. (See also p. 434.) Now Kirk and Raven’s 223 is the same fragment as DK, B 64: ‘It is the thunderbolt which steers all things.’

Why should this fragment make Kirk ascribe to Heraclitus the doctrines (a), (b), and (c)? Is it not quite satisfactorily explained if we remember that the thunderbolt is the instrument of Zeus? For according to Heraclitus, DK, B 32 = KR, 231, ‘One thing—the only one that is wise—wants, and does not want, to be called by the name of Zeus.’ (This seems quite sufficient to explain DK, B 64. There is no necessity to connect it with DK, B 41 = KR, 230, though this could only further strengthen my interpretation.)

But Kirk and Raven interpret on pages 200 and 434, the fragment ‘It is the thunderbolt which steers all things’ more elaborately: first by identifying the thunderbolt with fire; secondly by attributing to fire a ‘directive capacity’; thirdly by suggesting that fire ‘reflects divinity’; and fourthly by suggesting its identification with the Logos.

What is the source of this somewhat over-elaborate interpretation of a short and simple fragment? I could find no trace of it in any of the ancient sources—the fragments themselves, or Plato, or Aristotle. The only trace I could find was an interpretation of Hippolytus, whom Kirk and Raven describe on p. 2 of their book as ‘a theologian in Rome in the third century A.D.’ (almost six centuries junior to Plato) who ‘attacked Christian heresies by claiming them to be revivals of pagan philosophy’. It seems that Hippolytus, himself a schismatic Bishop, not only claimed that the Noetian heresy ‘was a revival of Heraclitus’ theory’, but that he also contributed by his attacks to the extermination of the heresy.

Hippolytus is also the source of B 64, the beautiful fragment about the thunderbolt. He quotes it, apparently, because he wants to interpret it as closely related to the Noetian heresy. In this attempt he identifies the thunderbolt first with fire; next with eternal or divine fire, endowed with a providential ‘directive capacity’ (as Kirk and Raven have it), and thirdly with prudence or reason (Kirk and Raven have ‘Logos’); and ultimately he interprets the Heraclitean fire as ‘the cause of the cosmic housekeeping’, or of the ‘directorship’ or the ‘economic government’ that keeps the balance of the world. (Kirk and Raven have it that fire is ‘the cause of this balance’.)

(The third of these identifications of Hippolytus might indeed have had a basis in the text: Karl Reinhardt, in an article in Hermes, 77, 1942, conjectures that there was a lost fragment, alluded to in Hippolytus, which read ‘pur phronimon’ or ‘pur Phronoun’. I am unable to evaluate the force of Reinhardt’s arguments though to me they do not appear very compelling. But the alleged lost fragment itself would fit perfectly well into my interpretation: since I interpret Heraclitus to mean that we— our souls—are flames, ‘thinking fire’ or ‘fire as a thought process’ would of course fit very well. But only a Christian—or heretical Christian—interpretation could render it ‘fire is providence’; and as to the ‘cause’ of Hippolytus, Reinhardt says explicitly that this is not Heraclitean. Fire as the cause of world-balance would come in—if at all—only through a Conflagration on the Day of Judgment, as the balance of justice; yet Kirk does not accept that this Conflagration is part of the teaching of Heraclitus.)

Thus the doctrine whose attribution to Heraclitus I found so unacceptable appears to be Kirk’s interpretation of an interpretation through which Hippolytus may have tried to establish the semi-Christian character of Heraclitus’ teaching—perhaps, as Karl Reinhardt suggests, in an attempt to fasten upon the Noetians heretic doctrines of pagan origin, such as the doctrine that fire is endowed with providential or divine powers.

Though Hippolytus may perhaps be a good source when he cites Heraclitus, he clearly cannot be taken very seriously when he interprets Heraclitus.

Considering its doubtful source, it is far from surprising that I could not make any sense of the quoted final summary or ‘conclusion’ of Kirk and Raven. I still feel that the doctrine there ascribed by Kirk and Raven to Heraclitus is absurd—especially the words which were italicized by me; and I am sure that I am not alone in this feeling. Yet Kirk now writes (on p. 338), referring to the passage in my address where I discuss his ‘conclusion’ and say that it is ‘absurd’: ‘Popper is indeed isolated when he asserts that such an interpretation of Heraclitus is “absurd”.’ But when we look more closely at Kirk’s present interpretation, we find that he has almost conceded my point: he now omits almost all the words which I put in italics because they seemed absurd to me (and in addition the words ‘changes of all kinds’); and he omits, more especially, the statement that the cause of the balance is fire (and ‘that was also termed their Logos’).

For Kirk now writes on p. 338, suggesting that this is the ‘interpretation of Heraclitus’ which I described as absurd: ‘Heraclitus accepted change in all its manifest presence and inevitability, but claimed that the unity of the world-order was not thereby prejudiced: it was preserved through the logos which operates in all natural changes and ensures their ultimate equilibrium.’

I think that even this interpretation might perhaps be formulated more happily; but it is no longer absurd. On the contrary, it seems to agree, for example, with the interpretation which I myself gave in my Open Society, where I suggested that the ‘logos’ may be the law of change. Moreover, though I strongly object to describing fire (with Kirk and Raven, or with Hippolytus) as the cause of balance, I do not object to an interpretation which lays some stress on balance or on balanced change. Indeed, if the apparently stable material things are in reality processes like flames, then they must burn slowly, in a measured way. They will, like the flame of a lamp, or like that of the sun, ‘not overstep their measure’; they will not get out of control, as a conflagration might. We may remember here that it is movement, a process, that keeps the barley brew from decomposing, separating, disintegrating; and that it is not every kind of movement that has this effect but, for example, a circular, and thus a measured movement. It is therefore the measure which may be called the cause of the balance of fire, of flames, and of things— of those processes and changes which appear as stable and as things at rest, and which are responsible for the preservation of things. The measure, the rule, the lawful change, the logos (but not the fire) is the cause of balance—including especially the balance of a fire when it is under control, such as a balanced flame or the sun or the moon (or the soul).

It is clear that according to this view most of the balanced change must necessarily be invisible: this kind of balanced or lawful change must be inferred by reasoning, by the reconstruction of the tale, the story, of how things do happen. (Perhaps this is why it is called the logos.)

This may well have been the way that led Heraclitus to his new epistemology, with its implicit distinction between reality and appearance, and its distrust of sense experience. This distrust, together with the doubts of Xenophanes, may later have helped Parmenides to arrive at his contrast between the ‘well-rounded truth’ (the invariant logos) on the one hand, and the delusive opinion, the erring thought of the mortals, on the other. Thus came about the first clear contrast between an intellectualism or rationalism, which Parmenides upheld; and an empiricism or sensualism, which he not only attacked, but which he was the first to formulate. For he taught (B 6 : 5) that the muddled horde of erring mortals, always in two minds about things, with erring thoughts (B 6 : 6) in their hearts, mistake sensations for knowledge; and that they take being and not-being for the same and yet for not the same. And against them he contended (B 7):

Never shall it prevail that things that are not are existing.
Keep back your thought from this way of enquiry; don’t let experience,
Much-tried habit, constrain you; and do not let wander your blinded
Eye, or your deafenèd ear, or even your tongue, along this way!
But by reason alone decide on the often contested
Argument which I have here expounded to you as a disproof.

This is Parmenides’ intellectualism or rationalism. He contrasts it with the sensualism of those erring mortals who hold the conventional and erroneous opinion that there is light and night, sound and silence, hot and cold; that their eyes mingle with light and night, and that their limbs mingle with hot and cold and so become themselves hot and cold; that this mixture determines the physical state or ‘nature’ of their sense organs or their limbs; and that this mixture or nature turns into thought. This doctrine according to which there is nothing in the erring intellect (the ‘erring thought’ or ‘erring knowledge’ of B 6 : 6) which was not previously in the erring sense-organs is stated by Parmenides as follows (B 16):8

This anti-sensualist theory of knowledge soon afterwards turned, practically unchanged, into a pro-sensualist theory which extolled the sense-organs (disparaged by Parmenides) as more or less authoritative sources of knowledge.

The whole of this story is somewhat idealized and, of course, conjectural. I merely try to show how epistemological and logical problems and theories might have arisen in the course of a critical debate of cosmological problems and theories.

It almost seems more than a conjecture that something like this did happen.

Notes

   The Presidential Address, delivered before the meeting of the Aristotelian Society on 13th October 1958; first published in the Proceedings of the Aristotelian Society, N.S. 59, 1958–9. The footnotes (and the Appendix) have been added in the present reprint of the address.

1 I am glad to be able to report that Mr G. S. Kirk has indeed replied to my address; see below, notes 4 and 5, and the Appendix to this paper.

2 Aristotle himself understood Anaximander in this way; for he caricatures Anaximander’s ‘ingenious but untrue’ theory by comparing the situation of its earth to that of a man who, being equally hungry and thirsty yet equidistant from food and drink, is unable to move. (De Caelo, 295b32. The idea has become known by the name ‘Buridan’s ass’.) Clearly Aristotle conceives this man as being held in equilibrium by immaterial and invisible attractive forces similar to Newtonian forces; and it is interesting that this ‘animistic’ or ‘occult’ character of his forces was deeply (though mistakenly) felt by Newton himself, and by his opponents, such as Berkeley, to be a blot on his theory.

3 I do not suggest that the smothering is due to blocking breathing-in holes: according to the phlogiston theory, for example, fire is smothered by obstructing breathing-out holes. But I do not wish to ascribe to Anaximander either a phlogiston theory of combustion, or an anticipation of Lavoisier.

4 In my address, as it was originally published, I continued here ‘and indeed for all other changes within the cosmic edifice’, relying on Zeller who wrote (appealing to the testimony of Aristotle’s Meteor. 353b6): ‘Anaximander, it seems, explained the motion of the heavenly bodies by the currents of the air which are responsible for the turning of the stellar spheres.’ (Phil. d. Griechen, 5th edn., vol. 1, 1892, p. 223; see also p. 220, n. 2; Heath, Aristarchus, 1913, p. 33; and Lee’s edition of the Meteorologica, 1952, p. 125.) But I should perhaps not have interpreted Zeller’s ‘currents of air’ as ‘winds’, especially as Zeller should have said ‘vapours’ (they are evaporations resulting from a process of drying up). I have twice inserted ‘vapours and’ before ‘winds’, and ‘almost’ before ‘all’ in the second paragraph of section ix; and I have replaced, in the third paragraph of section ix, ‘winds’ by ‘vapours’. I have made these changes in the hope of meeting Mr G. S. Kirk’s criticism on p. 332 of his article (discussed in the appendix to the present chapter).

5 This should establish that it makes sense, at any rate. I hope it is clear from the text that I appeal to truth here in order (a) to make clear that my interpretation at least makes sense, and (b) to refute the arguments of Kirk and Raven (discussed later in this paragraph) that the theory is absurd. An answer to G. S. Kirk which was too long to be appended here (although it refers to the present passage and to the present paragraph) will be found in the Appendix at the end of this paper.

   This Appendix, a reply to Mr Kirk’s article in Mind, was published in part under the title ‘Kirk on Heraclitus, and on Fire as the Cause of Balance’, in Mind, N.S. 72, July 1963, pp. 386–92. I wish to thank the editor of Mind for his permission to publish here the whole article as originally submitted to him. (In the second and some later editions of this book I have made some additions to this Appendix.)

1 Kirk quotes on p. 322 my L.Sc.D., p. 32, but a reading of what precedes my reference there to Bergson will show that my admission that every discovery contains (among other elements) ‘an irrational element’ or a ‘creative intuition’ is neither irrationalist nor intuitionist in the sense of any ‘traditional philosophy’. See also my Introduction to the present volume, ‘On the Sources of Knowledge and of Ignorance’, especially pp. 36 ff.

2 There are also casual occurrences such as ‘uncanny intuition’, ‘less formal and more intuitive reasons’ and ‘From this we see intuitively’, on pp. 17 and 5 (this volume, pp. 199 and 187). In all cases the word is used in a non-technical and almost deprecatory sense.

3 The two words in square brackets have now been added by me in order to make my meaning still more obvious.

4 It is Kirk who puts the word ‘intuition’ in quotes, thereby suggesting that it is I who use ‘intuition’ in this sense.

5 K. Reinhardt, Parmenides, 2nd edn., 1959, p. 220. I cannot mention this book without expressing my boundless admiration for it, even though I feel that I must reluctantly disagree with its fundamental doctrine: that Parmenides not only originated his problem independently of Heraclitus, but preceded Heraclitus, to whom he handed on his problem. I believe, however, that Reinhardt has given overwhelming reasons for the view that one of these two philosophers depends upon the other. I may perhaps say that my attempt to ‘locate’, as it were, Heraclitus’ problem, may be regarded as an attempt to answer Reinhardt’s challenge quoted in the text. (See also section vi of ch. 2, above.)

6 Cp. Diels-Kranz. For B 51 see G. Vlastos, AJP 76, 1955, pp. 348 ff.

7 Kirk, it seems, has misunderstood my criticism of his appeal to ‘common sense’. I criticized the view that there was in these matters a straightforward standard of common sense to which the historiographer could appeal, and I suggested (but only suggested) that my interpretation of Heraclitus may attribute to him perhaps as much, or more, common sense, than Kirk’s interpretation. (Besides, I also suggested that Heraclitus was the last man on earth whose sayings were to be measured by somebody else’s standard of common sense.) And is not the invisible change in Ovid’s ‘gutta cavat’ common sense? (Alan Musgrave has called my attention to an elaborate argument for invisible change in Lucretius, De rer. nat., i, 265–321, which may have been Ovid’s source.)

8 The significance of this passage, and my translation of it (which should also be compared with Empedocles B 108), are more fully discussed in Addendum 8 at the end of this book; see especially sections 6–10.