A New History of Greek Mathematics, by Reviel Netz [2022]
Earliest maths functions and innovations, social histories of maths, and, for crossing 5th to 4th centuries BCE ‘threshold’, the motivation was 'having fun' …
Reviel Netz wrote:
CHAPTER 1
… Before Greece
Throughout this book, I will argue that Greek mathematicians had achieved something quite unprecedented. But of course, people everywhere know some mathematics, and the Greeks specifically must have owed something to past cultures. They did not start from scratch! …
… Paulus Gerdes, as a young mathematics teacher in Mozambique, noticed that fishermen prepare their haul for sale by drying their fish near a fire built in the sand by the seashore. To make sure all the fish become dry at the same time, they follow a certain procedure. First, plant a stick in the ground, then attach a rope, and with a second stick attached at the other side of the rope, draw a circle in the sand. At this point, place all the caught fish along this drawn line, and finally, build the fire at the center. Gerdes’s idea was revolutionary – and straightforward: Instead of starting with some abstract definitions, would it not make more sense to teach the children of those fishermen the concepts of “circle,” “center,” and “circumference” based on this procedure?
Multiply this kind of example hundreds of times, and you have the discipline of ethnomathematics. Anthropologists, even apart from any application to the education of mathematics, came to be interested in the mathematical ideas available to preliterate societies; cognitive psychologists soon came to appreciate the significance of this research for the study of the universal human mind.
Thanks to the work of the ethnomathematicians, several observations emerged. First, numbers are pretty much universal. To be clear: it has been observed that the Pirahã tribe in the Amazon has no words for numerals. (There is some scholarly debate over this: Do the Pirahã words hói and hoí mean “one” and “two,” respectively, or do they mean – as the best experts now seem to believe – merely something like “small” and “larger”?) It is extremely interesting to cognitive psychologists if, indeed, even a single language could fail to develop numerical terms – and so, perhaps, number is not directly hardwired into the human brain. However, from the point of view of the anthropologist or of the historian, the example of the Pirahã is striking primarily for its freakish rarity. Everywhere you go around the globe, languages possess varied systems of counting. A few might be more impoverished (in particular, the Amazon has a number of less numerical societies, of which the Pirahã are an extreme and relatively well-studied case). But more often, simple societies have highly sophisticated numerical systems, with addition, multiplication, and iteration encoded into language itself. (Only one among these is the base-ten numerical systems now used by nearly all humans; it is nearly universal, perhaps, because it is, if anything, mathematically simpler than many of its alternatives.)
Second, geometrical terms are not as universally verbalized, but once again, one of the most persistent features of almost all cultures is some kind of attention to patterns – molded, painted, tattooed, drawn in the sand. Those patterns often display symmetries and occasionally involve more precisely drawn geometrical shapes. Does this amount, in and of itself, to geometry? Is any of this mathematics? Authors in the tradition of ethnomathematics often elide this question, and one sometimes has the impression that they try to impute to indigenous cultures geometrical knowledge concerning figures, where in fact, all that those cultures have is the habit of producing those figures. Some ethnomathematicians probably are overenthusiastic in this sense, but mostly this is a misleading framing. Once again, let us take an example from Paulus Gerdes. He describes the following pattern in Mozambique weaving baskets:
A nice geometrical pattern! But more than this, Gerdes observes, we may share this pattern in class and then proceed to discuss, with our students, how we may find here a relation between the various areas. In fact, with a little manipulation, we can derive, from this pattern, Pythagoras’s theorem itself! … It is likely, I believe, that Pythagoras’s theorem was indeed discovered around such drawings – by Babylonian teachers, working in a very different milieu. Now, Gerdes does not mean that African basket-weavers are aware of Pythagoras’s theorem; but it is nonetheless likely that the near-universal presence of patterns, of one form or another, is a significant precondition for the rise of geometrical reflection.
However, let us not get carried away. This is not yet reflective of explicit knowledge of geometrical properties, nor is the presence of a numerical vocabulary tantamount to the explicit knowledge of arithmetic. The discipline of ethnomathematics is useful for its scope – as well as for its limits. All humans, everywhere, talk about quantity and operate with shapes. But they almost never reflect on them explicitly, let alone develop a specialized craft of talking about numbers and figures. The discipline of mathematics and the profession of the mathematician are extremely rare.
Ethnomathematics is, of course, part of ethnography, and ethnographers tend to focus on what people do – how people interact, form kinship structures, cook, talk, sing. Anthropologists are trained to observe action, and so ethnomathematicians, quite properly, observe actions that are rich in mathematical meaning: counting, calculating, patterning. Those actions are real and form the background for the history we are about to explore in this book. Still, we should try to draw a line between an action that can be explained, by us, through our own mathematical understanding and the actors’ own mathematical knowledge. Fishermen in Mozambique draw lines in the sand to dry their fish, and it is right and proper that we describe those lines in terms of circle, center, and circumference. It is also important to draw the conclusion that those fishermen have what it takes to create geometry. And finally, it is reasonable to say that the fishermen act in a geometrically intelligent way, without possessing any knowledge of a theoretical field such as geometry.
Many of you would probably agree that drawing a circle in the sand does not display, yet, knowledge of the theoretical field of geometry. I would say that the same is true about drawing a route, from point A to point B, along a straight line. This is a geometrically intelligent practice – but not a display of geometrical theory. I would also say the same about the building of a straight canal of irrigation leading to your fields. If you construct such a canal, then it is still the case that you may, or may not, have some theoretical understanding of “lines.” I would also say the same about a straight road, faced by straight walls that form rectangular houses. And I would continue to say the same even if the houses become very imposing and perhaps assume the more complicated forms of various temples and pyramids. A pyramid, in and of itself, implies no more science than a line drawn for drying fish on the sand.
All of this is relevant to the question of the rise of mathematics as a theoretical discipline. We can find extremely sophisticated architecture and town planning around the globe – from the imperial cities of China to those of the Aztecs – and it is often assumed, especially by nonspecialists, that such imposing structures must involve theoretical mathematical knowledge. They certainly could, but the buildings, themselves, are not dispositive. And in fact, when we do find mathematics emerging, the context seems to be somewhat different.
Empire and the Invention of Mathematics
We can locate several historical moments where mathematics was independently invented. Taking them together, we may form certain conclusions about the natural context of such an invention – which brings us back to politics. The Inca empire, ruling over a vast region of the Andes in South America, left behind many monuments – but no writing. From the very beginning, Western observers noted a puzzling and rather humble artifact. Known as the “quipu,” this is a system of knotted threads (often made of cotton) that can usually be spread out as pendants – one main thread, with many others hanging on the main one; occasionally, this can become a many-layered object. Each of the threads has a pattern of knots attached to it, and throughout the twentieth century, as more of these artifacts were surfaced and analyzed, the system came to be understood as essentially numerical (and base ten). Roughly speaking, the knots on a cord form clusters. To simplify things a little, it works like this: if you have a cluster of three knots, a space, and then a cluster of two, this can stand for “32.” Such individual numbers on the hanging cords are summed up as the number recorded on the main cord. This, then, seems like an accounting device. The research leading up to this basic decipherment, based purely on a mathematical analysis of the extant quipus (of which there are now several hundred), can be found in the work of Ascher, Code of the Quipu (1981). I mention this because Ascher is also one of the most brilliant scholars in ethnomathematics and the author of the basic monograph in the field, Ethnomathematics: A Multicultural View of Mathematical Ideas (1991). For her, quipus are an example of ethnomathematics: an indigenous culture’s preliterate display of mathematical sophistication. We should, in fact, note an ambiguity: Is that display, strictly speaking, preliterate? Or was the quipu, instead, simply the Inca form of literacy? As more evidence came to light in the last generation, based on more careful excavations, we came to understand better the original function of the quipu. As was often suspected in the past, it seems to represent a tax system based on geographical allocation through subdivisions. We find that several quipus replicated each other (a guarantee of accounting consistency), and some quipus may be identified as summing up the results in other quipus (apparently, this represents lower and higher layers of the geographical subdivision). Most spectacularly – a veritable Rosetta Stone – a very late set of quipus from after the Spanish conquest was seen to match a Spanish written list of tributes from across many villages. It now seems likely that the colors of the threads were also meaningful, perhaps encoding geographical regions – thus, quipus were an even more informative system than we had ever assumed. The upshot of this research is that the Inca empire produced a specialized class of quipu masters whose job was to maintain information on the tribute required from across the empire. Now, as a matter of fact, we cannot really say how much “mathematics” those quipu masters knew, precisely because the Inca produced no writing. Whatever education was involved in the perpetuation of the quipu-master technique was purely oral and is now lost. But some education of this kind certainly existed, and so we can say this: in the Andes, prior to Pizarro, there must have been some mathematics actively produced, with people explicitly discussing rules of calculation and accounting.
And another remarkable observation: numeracy was so central, in this particular civilization, that it completely supplanted literacy. To explain: the tool that the state needed was some kind of numerical record. This was efficiently achieved by the quipu, and this did not give rise to literacy as a spin-off.
In other places, of course, states did rely much more on writing. Once again, it is useful to start from as far away from Greece as possible: let us get a sense of the entire range of possibilities. We may begin with China, where finally, we see a very clear tradition of theoretical mathematics. Here it is useful to focus on a relatively late work, The Nine Chapters on the Mathematical Art, a work that may have reached something like its current form under the Han dynasty (perhaps in the second century ce?). The Chinese court always required a large retinue of scholars, the bulk of whom were masters of religious rituals, but many specialized in fields such as astrology or other forms of scientific knowledge. It seems that at the latest by the Middle Ages, but perhaps even in the very earliest times, some were trained, and examined, based on their knowledge of the Nine Chapters – which is appropriately, then, seen to concentrate around accounting-like needs. The measurement of fields and of heaps of rice and grain, taxation, and distribution by proportion – all brought under a set of general, well-understood algorithms, which then become a subject of study in their own right. The needs of the state, generalized – and turned into a mathematical art. Once again, our evidence in this case is late, and it is hard to tell how mathematics first emerged in China. But more recent archeological excavations do provide us with more context and push the evidence further back. One dramatic find is that of “The Book on Numbers and Computation,” a set of inscribed bamboo strips that a civil servant took to his tomb, sometime early in the second century BCE. Much earlier, then, in Chinese imperial history – but still well after the formation of the first Chinese states – yet we see here the same kind of material as that found in the Nine Chapters. Problems that relate to concrete bureaucratic needs – solved with considerable general sophistication.
Beginning in Babylon
This brings us to the best-documented and most significant emergence of mathematics – and also, much closer to Greece itself. To the extent that the emergence of Greek mathematics was in debt to previous civilizations, it was to Babylonian mathematics.
This begins very early, along the shores of the Tigris and the Euphrates, and especially near their southern marshes. This is one of the origins of urban civilization, and from the beginning, we find a system of accounting – not unlike that of the Quipu, perhaps – based, this time, on clay. (In the steep Andes, transportation is at a premium, and one looks for light tools; in the flat, river-based civilization of Mesopotamia, heavy but durable inscriptions are favored.) Archeologists have noted small, variously shaped pieces of clay found in many sites from the late Neolithic. Schmandt-Besserat was the first to offer a general account of those tools, and although she is not without her critics, very few doubt her basic interpretation (Schmandt-Besserat’s critics mostly point out that the small pieces of clay could have been used for a variety of purposes beyond those she emphasizes; this is a reasonable critique). Most likely, different shapes stood for different commodities – so, for instance, could it have been a particular shape, say, for one head of cattle? Economic obligations – in the form of contracts or even taxation – could have been certified by an archive of such small tokens. This is all still ethnomathematics, a direct reliance on basic calculation and simple tools. And then, Schmandt-Besserat noted, something dramatic happened: it was realized that one could make impressions on clay, whose shape resembled the actual tokens. Late in the fourth millennium BCE, people in Mesopotamia began to use such tracings as economic records. A new idea, then: visual traces to mark numerical quantities. Pretty soon, instead of being tied to particular commodities, symbols emerged to represent number as such, and at this point, it took a mere step (or, if you will, a leap of genius) to begin to record other linguistic elements as well – at first, names of the objects counted and, very soon, language itself with its full vocabulary. By the end of the fourth millennium BCE, one of the major Mesopotamian languages – Sumerian – became fully written, the first ever. Literacy emerged, piggybacking on numeracy.
Skipping many centuries of Mesopotamian history, we may look at the same shores of the Tigris and the Euphrates almost a millennium later. They are now dominated by different people, speaking a different language (Akkadian, a Semitic language that is somewhat similar to Hebrew or Arabic), still using the same script, the same inscriptions on clay. The technical knowledge of the Sumerians was not lost, in this and in other matters. The rivers themselves required constant attention – digging the canals and irrigating the fields. A lot of engineering, planning, and control was necessary, and throughout, Mesopotamia saw the rise of strong central authorities, powerful temple centers, and kings and their retinue. In the late third millennium, we see clear evidence for a specialized bureaucracy. Scribes were trained in writing, keeping accounts, and advising the rulers. What is most important: they did not just use the basic techniques of writing and calculation; they took pride in becoming genuine masters in all of those. Thus, besides simply writing down bureaucratic records in Akkadian, they also transcribed (a much harder task) the old literary legacy in Sumerian. And they did not just calculate, say, how many workers were required to dig a canal or how much tax should be levied on a field – they also invented particular fictional problems of a more abstract character, where one calculated volumes, plane areas, and work rates. In the Chinese Nine Chapters (or in the somewhat earlier “The Book on Numbers and Computation”), we see the end result of, perhaps, a similar trajectory: bureaucratic training becoming its own raison d’être, giving rise to the problem-set version of a mathematics, which, although quite elementary, is already sophisticated. In Mesopotamia, our evidence is much more plentiful (early Chinese writing used a variety of delicate surfaces, such as the bamboo strip; from Mesopotamia, we have the clay tablet, history’s most robust writing material). And so we get a closer sense of the entire transition: tokens, then writing, a bureaucracy, and this, finally – sublimated into mathematics. We have massive evidence, from the end of the third millennium to the beginning of the second millennium BCE. The evidence stops quite abruptly a little after 1800 BCE, for reasons we cannot quite fathom (for indeed, we no longer have substantial evidence!). It appears that the same old cities came under different sets of rulers and that the scribal traditions were disrupted. Little is known, then, for over a millennium – but clearly, there was some continuity. Beginning in the eighth century BCE, we find, once again, Mesopotamian palaces – preserving masses of clay tablets and a lot of the ancient culture. There is little mathematics to be found, though, in this later material (but plenty of astronomy; we shall return to this). The object we study, then, is fantastically distant in time: the mathematics produced early in the second millennium BCE, or roughly four thousand years ago. …
… And this is how mathematics first emerges in the historical record: the simple, clever games accompanying the education of bureaucrats [MGH: planning town squares etc., sadly no more space for diagrams in this email].
We have come a long way from the fishermen drawing a circular line on the sand. Here, surely, is mathematics. And it is precisely here that mathematics is to be found, in this particular variation on bureaucratic education. The Egyptian builders of pyramids may have been no more than the makers of glorified lines in the sand. Babylonian schoolmasters, however, created theoretical knowledge. And the difference is clear. As one builds a pyramid, one engages, throughout, in a concrete endeavor. There is no occasion to abstract away from the actual slabs of limestone to purely geometrical prisms: the slabs are what you handle throughout. But in the schoolroom, for instance, in the calculation example cited earlier, one no longer deals with actual measured fields. One deals with rules of calculation for the measurement of objects such as a square field. The schoolroom is at a remove from the field itself, and so its subject matter is not the concrete objects under calculation but, rather, the terms for calculation itself. Paulus Gerdes’s project, setting out on his campaign for ethnomathematics, was to use the concrete knowledge of the fishermen of Mozambique as a starting point for the teaching of theoretical knowledge in the classroom. And in this, he traced the very same movement through which mathematics first emerged, four thousand years before.
The Greeks: Standing Apart?
This is nearly universal to humans: an ability to calculate with integers, the manipulation of shapes and patterns. Complex states give rise to bureaucracies, and this, occasionally, may give rise to the training of a scribal elite, which, finally, provides the context for the explicit statements of mathematical facts. And so, at last, you do not just calculate or draw patterns without reflection. Instead, you produce rules for calculation and for the measurement of areas.
This is a valid broad outline of the rise of mathematics in many parts of the world, but it has to be qualified. Even the human universals have a great deal of variety in them – perhaps the Pirahã don’t even have numbers! – and the same may be true for the rise of state bureaucracies. In fact, even the three cases just mentioned – the Andes, China, and Mesopotamia – show considerable variety. We have no evidence for a more reflective geometry in the Andes. In China, reflective mathematics seems to postdate empire; in Mesopotamia, mathematics emerged almost simultaneously with the market economy itself. It might be argued that all of this is a matter of the different sources of our evidence. In the Andes, one relied less on bulky artifacts, and so we have merely the threads of the quipu to tell our stories; presumably, much more oral lore circulated and is now lost. The Chinese bamboo strip is only slightly more robust than the quipu; we must have lost a lot from the initial stages of Chinese mathematics. Mesopotamian clay, finally, is extremely durable, providing us, in such a way, a much more detailed panorama of early Mesopotamian civilization. All of this is true but perhaps misses the point. The various societies used different media because state bureaucracy was not always the same. Mesopotamia really was – at least at times and in certain places – a heavily regimented society, recording the tiniest details of property and labor. It used an abundant, robust form of writing because this is what it required. Writing is not some kind of ornament; it may be built into the very fabric of society, defining its overall practices and achievements.
And so let us approach the Greek evidence in an open-minded way. Did the Greeks have bureaucratic state mathematics? Were they like Mesopotamians? At first, they seem so. Indeed, once again, we find the very same medium already familiar from Mesopotamia, that of the clay tablet.
Since the beginning of the twentieth century, excavations have discovered extremely ancient temples and palaces in Greece and, in particular, in Crete. Dating from the fifteenth to twelfth centuries BCE, several of those sites also yielded written tablets. For a long time, it was not even clear which civilization – or language group – occupied, in such ancient times, the lands now known as “Greek.” It was only after the decipherment of Linear B in the 1950s – a triumph of linguistic deduction made by Alice Kober and Michael Ventris – that the language of the tablets was identified as Greek. In some other ways, we can say, the decipherment produced an anticlimax. The contents of the tablets, themselves, were very mundane pieces of accounting. All this linguistic brilliance put in by Kober and Ventris – and then: “two tripod cauldron of Cretan workmanship … One tripod cauldron with a single handle . . .” An anticlimax, perhaps, but also a meaningful result. Early Greek civilization, in the Bronze Age, blended together with that of the ancient Near East as a whole. Not just the tool of the clay tablet – we find an entire cultural practice shared and perhaps transmitted: centers of political authority, where detailed numerical accounts are written down and stored. The implied sociology – the rise of some kind of state power (kings or king-priests?), with its bureaucrats, is clear enough. Did this go together with a more explicit training in numeracy? Was there, then, Bronze Age Greek mathematics? If so, it left no traces in writing. This is not where our story begins, and – the more general point – this is not, really, where the Greek story begins, in general. What the archeological evidence suggests, instead, is dramatic rupture. At about 1200, early Greek civilization comes nearly to a halt. Palaces fall down; plunderers set in. The writing stops, seems to be forgotten; cities and their civilization shrink and disappear. This rupture is not isolated and is instead seen across the Near East, where many states seem to fail at about the same time: the rare event of a civilizational near-extinction. Well, elsewhere in the Near East, the habits of the state were perhaps more powerful, and the rupture was not as total. But in the Greek-speaking world, the year 1200 could well have been a kind of year zero. The culture of Linear B would have been as puzzling to the Greeks of the year 800 as it would be, many centuries later, to Kober and Ventris.
But then, at about the year 800, something new began to stir. And here, finally, we come to our proper topic. To clarify the contrast: earlier, in the second millennium BCE, we clearly found state formation in the Greek-speaking world, along the familiar lines of kings or king-priests setting up strong centers of power. This early state formation is knocked out by the crisis of the twelfth century, and when the state begins to reemerge, it seems to take a different shape from that of the previous temples and palaces and that of the other early empires noted in this chapter.
What was remarkable about the reemerging Greek state? That it was weak and small. The usual tale of state formation is that as societies become more complex, they become more unequal. A few individuals emerge as the powerful rulers, and if lucky, they manage to consolidate power over ever-larger groups. Eventually, state becomes empire. This was definitely the case in the pre-Columbian Andes and in China, and although Mesopotamian states often controlled no more than a mere region (sometimes, not much more than a single city), control was often centralized, a king, priests, their retinue – and a mass of subjects. Greek cities just did not go along such a route. Emerging from the rubble of the post-1200 collapse, villages gradually grew in size, but they never reached a very large extent. Greece, quite simply, was not a great river civilization. It was always marked by steep mountains, sharply cutting into the sea, isolating small islands and tiny valleys. At its height, centuries later, Athens would have hundreds of thousands of residents; but for now, in the eighth or seventh century, even larger settlements would have no more than a few thousand residents. The people whose voice mattered for any kind of political or cultural life – the more or less self-sufficient male adults – would number much fewer, surely no more than a few hundred. To start with, those settlements were not very rich, and thus even the rulers could not be very rich themselves. (If you rule over a couple hundred small-scale farmers, how much wealth can you amass?) And so: scale creates habits, which can then become self-sustaining. Instead of relying on smaller armies of noble horsemen and charioteers, one relied on the larger standing army of the entire polis (polis is the Greek word for “city,” which we will use from now on; the Greek term evokes a distinctive cultural model). It is fitting that those soldiers are armed with the cheaper and more widespread material of iron – not the expensive, specialized bronze of predecessor states. Kings in the ancient Near East counted their wealth in heavy ingots of precious metals. Citizens of the Greek polis would begin to use a more manageable currency, the coin. Invented in Asia Minor, or present-day Turkey, in the seventh century, the coin would become a hallmark of the Greek world: small pieces of metal, widely owned across the social strata, easy to transport and manipulate. Also, in Mesopotamia and in Egypt, literacy was a scribal specialty, sometimes monumental, always based on arcane, difficult systems of writing. The Greeks borrowed the cheapest and most portable writing surface available from Egypt – the papyrus roll – and they borrowed the simplest and easiest-to-learn writing system from Phoenicia: the alphabet. Writing, for the Greeks, was never a matter of some rarified scribal elite.6 Not that it mattered all that much in the eighth or seventh century. Culture, quite simply, did not belong to any inward-looking court, with its established retinue of bureaucrats. There was nothing like that in early Greece. Culture belonged to the open spaces of the polis – perhaps in a festival, the youth singing together as they walk in procession; perhaps in a public square, a professional bard reciting an epic; sometimes, the richer folk, relaxing together and singing in a symposium with their guests from other poleis (because, in such a world of small poleis, there are always many other poleis, not far away, their people coming and going through your own).
The thread running through all of this is being spread out. It is the opposite of the concentration of Mesopotamia. The ancient Near East had a few expensive bronze chariots; hard-to-decipher hieroglyphs and cuneiform scripts; and rare, heavy ingots of gold; its culture was reproduced among a tiny group of trained scribes. Greece had plentiful iron spear tips and relatively accessible alphabetic writing on papyrus, and eventually, it would have plenty of tiny silver coins – and it had the culture of the open polis, which could be shared with almost all members of the community. The political and demographic forces that made for shared culture became entrenched in tools that had a way, in turn, of maintaining the features of this culture, even as states did become, eventually, somewhat more powerful. Silver coins would tend to preserve a market economy, just as iron spear tips would tend to preserve military and political order based on the citizen body, fighting side by side. Public performance – recorded in widely accessible script – would become the most stubbornly entrenched of all those cultural habits. We still read Homer.
Without any specialized bureaucracy, archaic Greeks surely did not develop any specialized scribal schools. It is, in fact, very hard to reconstruct anything about Greek mathematical education in the archaic era. (I shall return to this topic in Chapter 4, which discusses Greek mathematical education in the Hellenistic era and later, where some evidence is available.) All we can say is that the Greeks knew little, and what they knew must have come from elsewhere. But surely, if we want to understand the rise of Greek mathematics, we ought to make some guesses concerning the contents of such knowledge!
To get there will require a double detour, into the historiography of early Greek science and philosophy and into the historiography of the ancient Near East. Why “historiography”? Because in those early mists of history, so little is known, and so much is based on speculation, that it is impossible to discuss the past in separation from the way in which modern historians have interpreted it. And so, Thales and Pythagoras, and then, Babylon. …
[MGH: several sections skipped here]
The Threshold of Mathematics
There is no turning back from the empire to the tribe. In simple societies, there is usually a language for counting and practices of calculation, as well as pattern-making. People do things, but they do not produce theoretical reflections on such activities. Empires give rise to bureaucrats, and those (often) give rise to teachers, and teachers, finally, do produce theoretical reflections. And at this point, they begin to do something new. They engage in practices of writing, they impart rules, and they share riddles. And these, finally, will tend to reproduce outside the closed environment of the classroom. Babylon was gone. But the ancient Near East, even post-Babylon, would not have the same mathematical practice as that of an isolated tribe in the Amazon. The riddles were already out in the wild, reproducing.
What is described here is oral knowledge that, by definition, can no longer be uncovered. How many people knew just what? All we can do is point at the overall parameters. Certainly, as Greek democracy grew, more Greeks had access to education. From what we have seen in the foregoing discussion, it is likely that such an education included a modicum of mathematics. We may then follow Høyrup’s lead and suggest that mathematical knowledge spread through a few orally shared examples/riddles, and we might also suggest, then, that this knowledge, as it spread through the Greek world and became entrenched in elementary mathematical education, included the basic rules of area measurement, up to and including Pythagoras’s theorem.
Peering back into their earliest history, Greeks felt that they had to attribute the beginnings of their mathematical knowledge to particular named Greek authors: hence, the invention of Thales and Pythagoras as mathematical authors. But this was a category error, on the behalf of Greek historians as well as their modern followers. The earliest mathematical knowledge shared by the Greeks was neither authored nor Greek. It was, instead, the reflection in oral teaching of past Mesopotamian scribal schools. (Which, to be fair, many Greeks sensed, as well; Herodotus, for instance, was quite explicit that Greek knowledge of mathematics came from the East!)
And yet, the very need to attach cultural achievements to the names of their authors is telling – and specifically Greek. We do not pause to consider this as such because this habit – the reliance on named authors – has become so natural to us. But it is, in fact, a specific historical invention, happening only in some civilizations. We do not have an author for the epic of Gilgamesh, for the Chinese Book of Odes, for the Mesoamerican Popol Vuh. Indeed, Greek bards, too, recited epic stories of the siege of Troy, never attributing them to any named author. As in so many other civilizations, these were just the stories being told. A Babylonian teacher, imparting the rule whereby one finds the sides of the rectangle from the combination of area and diagonal, did not invoke the “author of the rule.” It was just a rule: teachers knew how to apply it. Similarly, a Greek, singing the song of Achilles’s wrath, did not invoke the “author of the poem.” It was just an epic poem: bards knew how to sing it.
And then, a few Greeks started to make a name for themselves. Instead of just reciting a well-known set of epic poems, widely shared among professional bards, some Greeks started to sing more personal songs, produced from a personal angle. Others repeated those poems, still attributing them to their original authors. In this new tradition, what made a particular poem effective was precisely that one could imagine, in concrete detail, a particular person – say, Sappho or Hesiod – in a particular place, say, Lesbos or Boeotia, authoring the song. So much so that soon enough, people began to imagine a concrete person, the author of the epic poems themselves! Perhaps at some point in the sixth century, “Homer” was invented. The Greeks started to believe that there was one particular person who was responsible for just those two epics, the Iliad and the Odyssey. And now we can see: Thales and Pythagoras became, as it were, the “Homer” of mathematics. They were retrospectively reinvested with a new identity; they were reidentified as the authors of what was, in fact, an unauthored oral tradition.
For indeed, there was no turning back now. All Greek culture, from now on, would have a named author. This was a new, exciting departure, in many ways specific to the Greeks: through literature, one could make a lasting reputation for oneself.
This invention of the author is also the direct background for a new development in the late fifth century: a proliferation of writing. The real – that is, historical – Thales and Pythagoras were sages, performing orally. Sages do not write; they proclaim. Throughout the fifth century, a handful of sage-like figures began to add a book to their résumés. The proclamations, attached to a name, would spread far and wide. Such are the wise, surprising proclamations of Parmenides, Anaxagoras, Philolaus: a single book, distilling a life of wisdom. Right near the end of the fifth century, this practice of book writing becomes an avalanche. Suddenly, many authors try to make their name in prose, and several produce not one book but many. “Author” becomes, in such cases, the key identity. And so, more and more figures of the author emerge. The genres multiply. These were exciting times, and many turned to the writing of history. Political speech mattered a lot: many wrote speeches, and some produced manuals of rhetoric; teachers of speech-making, they wrote about the practice of speech – as many others now did, writing about their craft. Physicians wrote about health and disease; artists wrote about the proportions of their statues; architects, about the proportions of their buildings. And so, a few wrote on mathematics, as well. Our history proper begins here. ….
[MGH: and here we will skip to some interesting paragraphs in Chapter 2]
CHAPTER 2
… The Hypothesis of Generational Events
The social history of science is annoying. If I explain the rise of Greek mathematics in terms of “the motivation to project one’s identity as an author,” “the rise of literacy,” or “the proliferation of genres,” there’s a strong sense that I’m missing the simplest and most important point. Why did people do mathematics in the late fifth through the early fourth century? Because, to them, it was fun. This is the intuition of many readers today – most important, of many working mathematicians.
I hope it’s clear that I agree. If, as social historians of science, we lose sight of this point, we are not merely disrespectful but simply wrong. In fact, we miss an important social observation. Absent special incentives, you would expect people to do pleasurable things; this determines, then, what mathematics gets done. They didn’t need to worry about tenure, and so Greek mathematics is tilted to such mathematical questions that provide – to the mathematically inclined – direct intellectual satisfaction. More to the point, they were not schoolteachers. They were authors, seeking fame – and so, reaching for the more remarkable results. This, after all, is what made people of the following centuries go back to Greek mathematics and try to extend it. The intellectual attractiveness of Greek mathematics – forced upon it, as it were, through the absence of institutional support – is an important part of its significance as a historical presence.
And so, we need a social history. Of course, people do the things they like doing – among the options they have. Social history tells us what the options are, and by the late fifth century, those have dramatically changed. Born around the year 430 BCE, you would be surrounded by books. They come with their authors marked prominently, names from far abroad, suddenly gaining the fame once allowed only to successful public performers (are you not tempted to imitate them?). They touch on all subjects. Here is a transcribed speech on the human body (the author says that all men are made of fire and water! It must be against that other book I read last month, which said all men are made of four humors!) Here is the history of the recent war, written by an eyewitness. Here is a treatise on political theory, arguing against Athens’ democracy. And here is a papyrus roll dedicated to the measurement of the circle (would you believe this is possible?) – but wait, of course this is wrong; I could do better than that author!
And so, many more try. I imagine that, in the final few decades of the fifth century, no more than a handful of mathematical works got written. Through the first few decades of the fourth, there were probably dozens of mathematical authors who, taken together, produced many dozens of works between them (possibly, even a few hundred): a very substantial mathematical library. This is the subject of this chapter.
This chapter is concentrated on a few decades. The first few decades of a new development are always significant, setting the stage for what follows. But once again, I think there is a more specific story to be told. Because ancient science (and, with few exceptions, ancient culture as a whole) did not develop stable, impersonal institutions, it was hard for any cultural trend to persist. Many ancient cultural events can be seen as generational. More precisely, what we often find is perhaps a generation and a half – a passing fashion or, perhaps, a response to some powerful presence that, after a while, loses its power. We recall: Thales of Miletus is said to have argued, very early on (around the beginning of the sixth century?), something that posterity remembered as all is water. In the ensuing generation, Anaximander of Miletus would argue (as transcribed by posterity) that all is the unlimited; Anaximenes of Miletus, that all is air. This is the gist of the information we have based on Aristotle, and it is probably false in many ways, but one thing is clear: here was a powerful voice – that of Thales – and a generation of local, immediate responses. And then – silence. For a long while, we hear of no more philosophers in Miletus, and elsewhere, wise people seem to be engaged with rather different concerns. A powerful voice – a set of echoes – and then, more or less, silence.1 This pattern seems to be repeated very often in ancient cultural life. A prominent master is followed by a prominent disciple but with no obvious continuation into a third generation. Parmenides of Elea, and then Zeno of Elea. Leucippus, and then Democritus. In the early third century in Alexandria, there was a flourishing of scientific pursuits of all kinds (which will, of course, greatly interest us in the following chapter). Perhaps chief among this is medicine, where the foundations of modern anatomy and physiology are laid by Erasistratus and Herophilus. They seem to dissect, even vivisect, not only animals but also humans. A deeply disturbing scientific breakthrough! Modern historians of medicine are often puzzled by the fact that this flourishing is so brief. If so much new knowledge is gained, why stop cutting? For in fact, immediately thereafter and then for several centuries, apparently no one else in the Greek world pursued the practice of dissection and vivisection. But perhaps this is not so surprising. This is simply how ancient science worked. A handful of leading practitioners, competing against each other and influenced by each other, working simultaneously in the same place, contributing to the same field. For a while, this was the thing to do. But then, those leading practitioners got old, died. And so, quite naturally, things changed. You see, Herophilus and Erasistratus did not set up a research center with funds dedicated to anatomical research. And so: new people would make their names in different, new ways.
There are, of course, exceptions to this rule. Tragedy and comedy persisted, especially in the city of Athens, original and vibrant for at least two centuries. Later on, the same city of Athens had four continuous philosophical traditions – Stoic, Aristotelian, Academic, and Epicurean – once again stable for about two centuries. In both cases, this continuity was based on a social institution. The Athenian state supported dramatic competitions beginning at about the years 500 (for tragedy) and 450 (for comedy). Eventually, starting at around the year 300, there were four schools of philosophy, actual institutions with their own properties and rules (each school had a fixed place; each school had a leader – scholarch – serving for life, a new one elected when the old one died). This kind of institutional stability is the exception in ancient culture and was never approximated in mathematics.
And so, there is nothing too surprising about the hypothesis proposed here. At around the year 400, a whole generation of authors started to produce mathematical works. They were all impressed by a few original contributions; they were probably aware of each other and aware of the significance of the moment as a new development. It must have been exciting. And then, the first significant authors – and even their immediate followers – got old; something of the original excitement was gone. By midcentury, new work continued to be produced, but the pace could have slowed down, and the sense of a meaningful moment was lost. For a while, at least, fewer new authors would be attracted to this field; other pursuits became more fashionable instead. This, then, is the first generation of Greek mathematics.
The Source:
Reviel Netz, A New History of Greek Mathematics, Cambridge University Press 2022
Evolutions of social order from the earliest humans to the present day and future machine age.