The rewriting of the history of early mathematics is a necessary — but risky — enterprise.

Of course one often hears the doubt as to how much in the field is still left to be said. The layperson assumes that a finite set of information, pored over for centuries, should finally reveal all of its secrets. This is of course never the case, in the history of early mathematics as in any other branch of history. To start with, there is the very well known reason why the writing of history might never end: as our perception of the world changes so our account of the past must be adjusted. So history — as such — must always be re-written. But the history of early mathematics stands in special need of re-writing. One reason for that is historiographical, the other is philosophical. I shall start with the historiographical reason that, indirectly, also leads us to consider the more philosophical reason. And this philosophical reason, in turn, will lead us to see why this enterprise is particularly risky.

The historiographical reason for the urgency of rewriting the history of early mathematics concerns the nature of the evidence. Our knowledge of early mathematics is bottom-light, top-heavy: we have rich and important single monuments in the works of Archimedes, Apollonius, and Euclid in the west (or The Nine Chapters in the east), but relatively little knowledge of the wider context within which those monuments took shape. Just what would an ordinary piece of early mathematics look like? What was the process leading to the major monuments? On this, relatively little is known. We are accustomed, as historians, to look for origins and for lines of development. But with early mathematics, we have available to us mostly the (remarkable) snapshots of science in its full form.

The fundamental data available to us lack the robustness of solid evidence: we have many isolated pieces of text, but they do not, by themselves, form a solid pattern — this table, so to speak, has legs only. And as a consequence, early mathematics has traditionally been a field where speculation was difficult to disentangle from fact. Any shift in the historical perspective tends to affect not only the hypothetical route we prefer for explaining the emergence of mathematics, but also the meaning of the main monuments themselves.

Over the last thirty years, this has been especially evident. We can note a move away from extrapolation, based on the philosophical sources in Plato and Aristotle, and into an analysis of the practices of the extant mathematical texts themselves (for this transition, see Saito 1998). From the study of concepts, research has moved to the study of practice -- which has had some major consequences. Euclid, we now believe, may not have been as important for his contemporaries as he has been to most modern readers; his very text and identity might be in doubt (Knorr 1996; Russo 1998; Djebbar, Rommevaux and Vitrac 2001). Apollonius and Archimedes now emerge in a new shape: Apollonius as a highly original author pursuing a thought-out geometrical goal (Fried and Unguru 2001), Archimedes as the master-craftsman in the specific Greek field of geometrical problems (many studies by Knorr and in particular Knorr 1986). At the same time, in the east, Karine Chemla’s studies have revealed the role of proof and argumentation, long ignored, in The Nine Chapters and its commentator Liu Hui (see esp. Chemla forthcoming); while our understanding of mathematics in the Ancient Near East has been dramatically transformed following the studies of Damerow, Hoyrup, Ritter and Robson (see especially Nissen, Damerow and Englund 1993; Hoyrup 2002; and Robson 1999) that had finally placed ancient mathematics within the context of Ancient Near Eastern civilization as a whole. At the same time, we have come to recognize the gaps in our traditional accounts: that so much is made of the transition from Classical to Modern western mathematics, for instance – while so little is still understood of the transition from Classical to Muslim mathematics (the last, indeed, may underlie the first: see Rashed 1994). In particular, Late Antiquity as a distinctive period in the history of early mathematics has come to be recognized almost in the last few years only (with studies such as Cuomo 2000; and Netz forthcoming). It is not an exaggeration to suggest that today’s scholars of early mathematics have to study, today, a field completely different from that of their colleagues of thirty years back.

This, then, is the historiographical reason why the history of early mathematics still merits rewriting, namely, that our knowledge is fragmentary. The fundamental point seems to be as follows. Bringing in the context of cultural practice as a whole, and considering it through the perspective of the longue durée, is always advisable in historical research. But with the fragmentary nature of our evidence for early mathematics, such a contextual approach is not merely preferable: it is, in a way, the only one promising any progress at all. However, once we base our interpretations on matters of cultural context, rather than technical mathematical detail, we have also lost the sense of finality that mathematics, itself, seems to offer. And thus, over the last thirty years, the history of early mathematics has undergone a true revolution.

This is significant as, in fact, thirty years ago the history of early mathematics was based primarily on this promise of mathematical – rather than historical – certitude. This is the irony: thirty years ago, the study of early mathematics appeared to be nearing finality.

Through most of the twentieth century, scholarship in early mathematics was dominated by the figure of Otto Neugebauer (1899-1990). Neugebauer’s studies in the ancient exact sciences will always remain a monument for intellect and scholarship. As his disciples – we all are, to some extent – we shall never equal his mastery of both philological and mathematical skills. But the very fact that we try is thanks to Neugebauer’s model that has turned the study of early mathematics into a rigorous academic discipline. All of this we shall always owe to Neugebauer. But it is almost hard to remember now that, a few decades ago, Neugebauer’s works appeared to have been the last possible word on their subject. Here was on offer a supremely confident philological and mathematical analysis of the contents of ancient theories – and so, what more could have been said? Once you have mastered the mathematical contents of mathematics, there was nothing else to be added – nothing, that is, save the incidental. Aside from Neugebauer’s program – so Neugebauer has effectively suggested – there remained merely the inconsequential gossip of chattering about the dead. Here was Neugebauer, introducing his monumental project of the history of mathematical astronomy, in 1975: “Many things are omitted here. The reader who wants to hear about Archimedes taking a bath or about the silver nose of Tycho Brahe can find innumerable books which dwell on these important biographical matters” (Neugebauer 1975, 1). That is, beyond the mathematical technical facts, lie such details as Brahe’s artificial nose. And yet it was precisely by going beyond the technical mathematical details themselves, to their cultural setting, that the most important research of the last thirty years was made. Brahe’s nose was found, after all, to be not trivial at all.

And so we begin to skirt the border of the historiographical and the philosophical. It is an accidental fact about early mathematics that the evidence is sparse; it thus invites interpretation through a consideration of the cultural context; which in turn implies a view of early mathematics or of the early exact sciences as a whole, where the context matters – where an account setting out the technical mathematical detail would be insufficient. And if so, the historiographical issue transforms into the following, epistemological question: is the significance of a mathematical text reducible to its culturally invariant, technical content? Or is the full significance of the text revealed only by paying attention to the context of cultural practice within which the text was produced?

This, of course, was the very question of the debate surrounding Unguru’s seminal paper, “On the Need to Rewrite the History of Greek Mathematics” (Unguru 1975). This paper was at the foundation of all culturally sensitive historiographical approaches to the history of early mathematics since. Yet it was fiercely debated and, if the debate did indeed quiet down – with the sense overall that Unguru was simply right – this removes nothing of the urgency of the question.

Which, finally, brings us to the essential, philosophical reason why the history of early mathematics must, indeed, always be rewritten. After all, there is an important reason why we should study the history of early mathematics. Early mathematics has a central role in western intellectual history. It had a major influence on philosophers from Plato and Aristotle down to, say, Spinoza, Leibniz, or Kant, so that it shaped the very modern notions of truth and rationality. And because of its seminal role for modern science, it to a large extent shaped the direction of the modern exact sciences as well. At least until the very last decades, modern science was organized as a system with mathematical physics at its center – indirectly, then, based on a certain perception of Archimedes – mathematics itself understood on the perceived model of Euclid’s axiomatic system. In short, our understanding of the nature of early mathematics – and in particular of the major works of Euclid and Archimedes – should also define our understanding of the nature of science and of rationality themselves. After all, the ideal of Euclid was created by philosophers, the ideal of Archimedes was emulated by scientists, precisely because these were perceived as being impersonal and ideally objective. It was for this good philosophical reason that Neugebauer could dismiss as frivolous any interest in “Braha’s nose”: because Braha, after all, has dealt with the objective certainties of mathematical astronomy.

And if so, the question is restated with renewed urgency: are science and rationality culturally sensitive? To argue for the existence of a science which is free of the subjective, we must at least defend it for early mathematics; or, alternatively, if we wish to argue that science must always be understood historically, while still trying to preserve a notion of rationality, then our most important task involves early mathematics: we must show, on cultural terms, how Euclid and Archimedes have succeeded in creating an objectively rational form of science. Here is the philosophical reason to engage in the enterprise of rewriting the history of early mathematics – and also, of course, the reason why this enterprise is so risky. Unguru, ultimately, was attacked by mathematicians because in his call to historicize mathematics, he seemed to be challenging a mathematical and a-historical notion of rationality. The task before us – Unguru’s historiographical descendents – is to show how a historicized mathematics need refine, but not destroy, our sense of rationality and truth.

Of course, not all historians and philosophers of science can actively engage in the debates concerning early mathematics. The field calls for special training, in several ancient languages and in a type of mathematics which is both technical and, in many ways, alien to contemporary mathematical practice. And yet, I shall argue, the field of early mathematics should not at all be considered marginal to the field of science studies as a whole: on the contrary, it is crucial to its most important research question. This, then, is the essential reason why, now, we should follow, with particular urgency, the rewriting of the history of early mathematics.

We collect here eight articles based on papers, mostly read for a conference that, while not intended as such, ended up celebrating Unguru’s achievement in the history of mathematics. The articles all go back to this task of rewriting the history of early mathematics. And the diverse many directions they offer for this rewriting are suggestive, taken together, of the position of the field, a quarter century following Unguru’s seminal paper.

Four of the papers face directly the fundamental historiographical challenge of early mathematics: the scarcity of the sources. How should we go about that? During the first century of the intensive study of early mathematics – from the days of Heiberg and Tannery down to the early works of Wilbur Knorr – the preferred way forward was that of rational deductive reconstruction. Once again, this historiographical approach is based on the special properties of mathematical argument. Since mathematical results need to be proved, it follows that, from a few testimonies for results one can reconstruct the entire shape of the growth of mathematical theories. We may notice immediately how this method of rational deductive reconstruction assumes – typically to any such mathematically-inspired historiography – a historical invariance of the nature of mathematical practice: for the reconstruction to go through, we must assume that the early authors involved were trying to create mathematical theories of the same shape as that of the mature science. This assumption has practically no foundation for the earliest period of Greek mathematics until as late as the end of the fourth century BC; indeed, I will myself argue for the radical thesis, that our notion of what a complete mathematical theory looks like assumes a mathematical practice that took shape only in Late Antiquity itself (Netz forthcoming). Of course, some of the research in the tradition of rational deductive reconstruction is still very valuable, and one may mention in particular in this context Knorr’s first book, The Evolution of the Euclidean Elements (1975). Something did go on, after all, at the historical moments for which we have little evidence, and mathematical validity is certainly one of the keys we have for reconstructing those gaps.

And yet, the realization of the limitation of rational deductive reconstruction is now widespread among the historians of early mathematics. There are two main reactions, each reaction represented here by two articles. One, more positive, may be seen as a move away from rational deductive reconstruction to what may be called historically sensitive reconstruction. In other words, the attempt is to suggest a reconstruction for the gap in our information, based not only on mathematical validity but also on a richer sense of the nature of mathematical practice at the time. Mueller’s and Taisbak’s articles, opening this volume, both reflect this positive reaction. The second reaction, more negative in character, stresses the negative fact of our limited knowledge, either to suggest that the gap may be real – there simply was nothing there – or to suggest that it may be irrecoverable – there may have been something there, but we shall never know what it was. Saito’s and Yavetz’ articles, following in this volume, both reflect this negative reaction.

Mueller and Taisbak both deal with a crucial moment for which we have almost no evidence. The first, discussed by Mueller, is the formulation of what we know as Euclid’s Parallel Postulate. The second, discussed by Taisbak, is the creation of a geometry of “application of areas.” The Parallel Postulate is central both to the overall logical structure of Greek plane geometry, and to the nature of Greek approaches to axiomatic structure as such. The geometry of “application of areas” is where Greek mathematics comes closest to encoding quantitative relations (comparable to those of elementary algebra) into its geometrical configurations; it is also constitutive to the main advanced area of Greek mathematics, namely the theory of conic sections. The two, taken together, come to define much of the mature form of Greek mathematics. However, both the Parallel Postulate and the geometry of “application of areas” were formed in the earliest – and least known – period of Greek mathematics, from the late fifth to the late fourth centuries BC. Thus one seems to be reduced to speculation.

To reconstruct this gap, Mueller addresses directly the question of the development of mathematical practice. The basic facts seem to be as follows. Apparently something such as Euclid’s Elements I.32 – showing that the sum of interior angles in a triangle is equal to two right angles – was proved already in a context earlier and different from that of Euclid’s Elements themselves. In the mature Euclidean form of this theorem, however, it fundamentally relies on the Parallel Postulate. The argument from silence does suggest, in this particular case, that the Parallel Postulate is relatively late (perhaps even original to Euclid himself). We are thus forced to look for some kind of early argument for the theorem on the sum of interior angles in a triangle, based on something other than the Parallel Postulate. Mueller skillfully weaves together the scattered pieces of evidence from Aristotle, Proclus, and even Philoponus, to suggest an outline argument. He is searching for (what appears like the more) historically appropriate reconstruction, one which is less “axiomatic” in character and more concerned with the immediately intuitive claims of “cut and paste” techniques (where, interestingly, the object cut and pasted is not – as the phrase “cut and paste” is usually used – two-dimensional areas, but rather angles). However, a difficulty remains: the early mentions of the theorem on the sum of interior angles always refer to “two right angles” and this formulation implies, Mueller argues, that any proof would have to display a level of sophistication beyond the intuitions of cut-and-paste techniques. Indeed, a proof for the sum of angles being equal to two right angles should possess something such as the more axiomatic approach of the mature Elements. Mueller cautiously concludes that either the evidence for early knowledge of the theorem should be dismissed; or that, in point of fact, the earliest Greek mathematics was surprisingly similar to what we see in Euclid himself.

Taisbak’s study is based on a method that, in many publications, the author has made his own. Taisbak becomes a kind of historical-mathematical participant observer. By patiently developing arguments the way Greek mathematicians (of the extant sources) would develop them, Taisbak gradually builds up his case that certain assumptions about the ancient practice are likely, just because they are indeed perceived as natural, when seen from the perspective of the working mathematician. It is typical to this approach that the argument rarely relies on mathematical validity as such. Instead, what matters is the “naturalness” or “attractiveness” of a mathematical statement as it is approached by the route Taisbak describes. In other words, Taisbak’s approach tends to extend the notion of mathematical rationality to other, epistemic values, beyond those of logical validity itself.

Taisbak’s subtle mathematical analysis leads us to see that, once one is immersed in what he calls “gnomonic situations” (as early Greek mathematicians no doubt were), and if one is interested in the problem of constructing certain rectangles with given ratios to other rectangles (and everything suggests that ratio and proportion were always key interests of Greek mathematicians) then it is natural to conceive such problems in terms of the application of areas. Dealing with the problem of finding a rectangle with a given ratio to another, Taisbak shows how natural it is to reach a “shift of focus” from the rectangle standing in a ratio, to another rectangle, equal to another, and “applied” in the technical Greek sense with (possibly) an excess or deficiency. Thus Taisbak achieves two results: accounting for the origin of interest in the application of areas; and locating this interest within the wider range of Greek mathematical interests and practices.

Mueller and Taisbak thus both offer historically sensitive reconstructions for gaps in our knowledge of Greek mathematics. Both Yavetz and Saito raise doubts concerning the possibility of this enterprise.

Yavetz’ article deals with one of the most fruitful areas of rational deductive reconstruction in the early exact sciences: Eudoxus’ mathematical astronomy. Based on reports in Aristotle and in Simplicius, it was always assumed that Eudoxus in the fourth century had offered a model for the sun, moon, and five planets based on the stars as points on a system of rotating, concentric spheres (this interpretation has its radical critics, for which see Bowen and Goldstein 1983). Simplicius, in particular, offers a series of details, which seem, at first glance, to determine how the system of spheres is to be arranged. Add to this certain constraints arising from the behavior of the stars themselves, and presumably one could reconstruct Eudoxus’ model. So argued Schiaparelli, in great detail (in Schiaparelli 1875) and so the matter stood for over a century. Starting with Heglmeier (1988) and in particular in a series of publications by Mendell (1998, 2000) and Yavetz (1998), it was shown how further mathematical reconstructions may be offered, all compatible with the data in Simplicius while being mutually incompatible. In other words, we face here a classical problem of under-determinacy of the theory based on the evidence, undermining the very project of reconstructing the actual theory of Eudoxus. But the very richness of the proposals offered allows Yavetz, now, to reconsider the problem from a new angle. Considered in greater detail, it may be argued that each of the proposed reconstructions, after all, suffers from its own conceptual drawbacks. Yavetz can now therefore ask the question: do the data in Simplicius merely underdetermine the theory (so that many incompatible theories can be offered on their basis) or are they in reality incoherent so that any reconstruction offered is bound to be mathematical or conceptually deficient? Yavetz argues that the latter is indeed the case, throwing doubt on Simplicius’ basic credibility as a source.

An even more radical kind of claim is made by Saito. His article returns to the issue of ratio and proportion, already central to Taisbak’s paper (indeed, it is difficult to study Greek science without returning time and again to this theme). While Taisbak offers a reconstruction of problems using the tool of ratio and proportion, Saito discusses the (alleged) theories of ratio and proportion as such. That is: we have, in book V of Euclid’s Elements, a detailed theory where proportion is the key term, defined (in some sense) and then developed in general terms. The theory also fundamentally relies on a remarkable definition of “being in the same ratio” – V.5 – which is standardly ascribed to Eudoxus and so is generally assumed not to have been used by the earliest Greek mathematicians. Thus past scholars have offered several reconstructions of what theories of ratio and proportion Greek authors could have produced in the context of the earliest Greek mathematics of the late fifth and early fourth centuries.

The situation, then, is comparable to that studied by Mueller. The Parallel Postulate determines a certain architecture of plane geometry, that of Elements I; the Eudoxean definition of “being in the same ratio” determines a certain architecture of proportion theory, that of Elements V. It is reasonable to assume that the Parallel Postulate, as well as the Eudoxean definition of “being in the same ratio,” both postdate their respective fields of plane geometry and proportion. Thus there would have had to be some plane geometry without the Parallel Postulate, and some Proportion theorems without the Eudoxean definition. What would they look like? Mueller’s conclusion is that early plane geometry could probably have looked surprisingly similar to the mature form of Euclid’s Elements themselves. Saito’s conclusion is the opposite, and is negative in character: we have no reason to believe, he argues, in the existence of any early proportion theory. The form of proportion theory we find in Euclid’s Elements V may well be the first ever offered by any Greek mathematician; and the earlier uses of proportion theorems were just that – namely, proportion used as a tool but not as a subject of investigation on its own. In other words, the early mathematical practice could have been radically different from that of the mature Elements. This radical thesis throws in doubt a great deal of the research into early Greek mathematics, but it is more than a mere epistemological criticism, pointing out the limits of what we can know for certain. Saito, in effect, offers a positive historical reconstruction – one where we are asked to envisage a mathematical practice where terms such as “ratio” and “proportion” are freely used without the presence of a theory underpinning them. This radical thesis should have important repercussions for the future research of early Greek mathematics.

The first four articles, then, are all concerned with the historiographical problem of reconstructing the unknown stretches of the mathematical past. They thus all deal, naturally, with the most interesting period for this question, the earliest period of Greek mathematics. The next four articles are all related to the more philosophical question of the nature of mathematical practice (already important, of course, for many of the historical reconstructions discussed in the first four articles). If a mathematical text is not merely the sum of its mathematical results, in principle expressible outside of time and place, then what is it? Over the last thirty years, many interpretations of mathematical texts were offered where the text is seen to represent an underlying mathematical practice, specific to its time and place. The articles by Fried, Imhausen, Bernard, and Chemla all make important contributions to this recent tradition.

Fried’s article may be compared with that of Taisbak in that both seek a rich understanding of mathematical practice by following the details through which a mathematical argument is developed. This approach has a long tradition which, however, tended to be framed within the model of rational deductive reconstruction. This calls for a brief detour. As is well known, Greek mathematical proofs typically take a synthetic form where the result is constructed or shown from scratch. Sometimes, however, we are also provided an analytical version of some proofs, where the desired result is assumed proved and the conditions for its derivation are then found. A long historiographical tradition assumed, in effect, that analysis was a more or less mechanical procedure of discovery, standardly used by Greek mathematicians to discover their results. Thus the search for the “context of discovery” of a Greek mathematical proof transformed into the strictly mathematical task of discovering an analysis yielding the conditions of the proof used by the Greek author. I have criticized this understanding of Greek analysis in Netz (2000), arguing that we must consider analysis, as published by Greek mathematicians, as a strategy of presentation and not of discovery. In this, I follow upon the more recent tradition of research into Greek mathematical discovery, represented by authors such as Taisbak and Fried. For them, discovery is followed through its more plausible – and more fickle – historical route. Discovery is not the result of proof; on the contrary, proof is the result of discovery. The free process of discovery must be understood through cognitive and epistemic processes that are freer than that of proof itself.

Fried makes crucial progress in this recent tradition, in formulating a principle underlying Apollonius’ thought in the Conics : this is the principle of analogy. This is of course a principle made much use of in the philosophy of science in general (and, to some extent, also in the philosophy of mathematics). Fried’s article is thus of very wide philosophical significance, making the specific claim for a general philosophical approach. But, even more important, Fried shows the precise structure underlying a certain use of analogy in the Conics : this is based on the viewing of diagrams as analogically related to other diagrams. Further, the analogy in question is also related to precise (and separately defined) proportions connecting the diagrams. The analogy thus works simultaneously at three levels: as abstract analogy as such; as a visual relation between diagrams; and as a mathematical proportion between lines. Fried’s account brings together the philosophical notion of analogy, together with the two main tools of Greek mathematics: visualization and proportionality. In this way, we are offered a reading of Greek mathematical practice in its context, where the rationality of the text can be understood only following its historicization.

A different type of historical contextualization is offered by Imhausen in the article following. Imhausen offers a reading in historical context of several Egyptian mathematical texts. This reading in historical context is the hallmark of the most important recent tradition of Ancient Near Eastern science – a field that can be truly said to have been reborn even over the last decade. Here, indeed, rational deductive reconstruction was particularly attractive: the evidence is particularly fragmentary and inaccessible and even so as one reads the available evidence, one is attracted to a purely mathematical interpretation. It was in the reading of these documents, particularly cuneiform clay tablets, that Neugebauer has made his name, and one can only admire the unique genius that has allowed such works as Neugebauer and Sachs (1945). And yet, the drive to interpret mathematically leads, naturally, to a reading of the texts as if their function was to contribute to theoretical mathematics as understood in the modern context. Such an assumption is of course at least doubtful. As mentioned above, a series of studies by Damerow, Hoyrup, Ritter, Robson and others has now shown the historical context for the production of Ancient Near Eastern mathematics. The documents are now understood in the context of early state-formation, not so much used directly for the practice of social control, but rather as tools in the formation of identity of a bureaucratic class.

Imhausen’s article, indeed, shows how a historically attentive reading is necessary for the basic reading of the contents of Ancient Near Eastern texts. She pays attention to Egyptian texts, read and interpreted already several times in the past. It turns out that the very meaning of the texts was misunderstood, not due to any mathematical incompetence of past scholars, but because the texts, immersed as they are in a bureaucratic state culture, take for granted highly specific meanings between the terms that can be recovered only by placing the texts within the wider context of administrative texts. In short, Imhausen shows that the very fundamental task of recovering the meaning of Ancient Near Eastern mathematical texts stands to gain by the application of contemporary, historicizing approaches to the history of mathematics.

Fried’s contextual reading is highly “epistemic”: he brings into his interpretation of Apollonius’ Conics a contextual understanding of the epistemic processes underlying Greek mathematics (such as analogy, visualization, and proportion). Imhausen’s contextual reading, by contrast, is highly “social”: the contextual dimensions are not epistemic but are based on extra-scientific social practice. In the article following Imhausen’s, Bernard offers a contextual reading of Greek mathematics in Late Antiquity which combines the epistemic with the social. After all, the very epistemic practices of scientists must represent the kind of social practice they are engaged in. The dual use of visualization and proportion, for instance, of the Greek mathematical diagrammatic proof, is fundamentally based on a social practice of elite literacy (notice, for instance, how the diagram uses letters of the alphabet as labels). Even more “socially,” we can always see how scientists develop their argumentative strategies according to their given social practice of debate and persuasion: the classical study in this respect is of course that of Shapin (1994).

In the late ancient Greek-speaking world, rhetoric was the central social practice of debate and persuasion. Bernard investigates the consequences of this social and epistemic practice for contemporary mathematics. Going back to the question of the context of discovery of Greek mathematical proofs, Bernard points out that “discovery,” in this cultural context, had a specific meaning within the practice of rhetorical argument: indeed, as Bernard points out, “discovery” was simply the core of rhetorical training. And if “discovery” is the goal of rhetorical training, “problem” (another key mathematical term) is the main theme around which rhetorical training revolves. Bernard sketches these examples, arguing persuasively that such observations should necessitate a rethinking of our interpretation of Late Ancient mathematics. This promises to be an important research program for the years to come.

The final article, by Chemla, brings us back to the fundamental philosophical reason to engage in the re-writing of early mathematics – and to why this enterprise is so risky. The claim is as bold as could be: that our notion of “generality” (crucial to our understanding of mathematics as a rational practice) is in itself culturally dependent. We take it for granted – following Greek models – that generality should be correlated with abstraction. Chemla’s main claim is that, in the Chinese context of Liu Hui’s commentary to The Nine Chapters, we see generality without abstraction. Thus the claim is simultaneously historical and philosophical.

It is a common claim that Chinese mathematics (as well as other non-Greek mathematical traditions) is characterized by an algorithmic, rather than a demonstrative approach: the aim, according to this interpretation, is not so much to show the truth of a claim but to illustrate the use of an algorithm. In a series of publications on the subject, Chemla has shown that many Chinese arguments are in fact dedicated to showing the validity of an algorithm, in this way counting as “proof.” In this new article she extends the analysis further. A central characteristic of a Greek mathematical proof is its general import: Pythagoras’ theorem is proved simultaneously for all right-angled triangles. So-called “algorithmic” traditions, however, typically concentrate on particular cases only, never ascending to make their statements in general terms. Often, the particular cases are also presented as specific numerical values, in sharp difference from the Greek practice where numerical values are usually avoided. Chemla shows that the statement of an argument with the aid of specific numerical terms may yet contain enough suggestions to allow an intended inference to a more general conclusion: the use of a concrete example may coincide with a general import of the argument. Thus generality is disassociated from abstraction, with major consequences for our understanding of the history and philosophy of mathematics.

The eight articles collected here all go about rewriting the history of early mathematics in many different ways. Whether in addressing the gaps in our evidence, or in contextualizing further the evidence we do possess, we see examples of a discipline re-shaping its own practice just as it is re-evaluating the practice of past science. To this project of re-writing, this volume now adds a new layer.