Thirteen scholars reply to Reviel Netz's ‘The Place of Archimedes in World History’

The rules of the game

The rules of the game we played to produce this volume were clear to all of us.

Reviel Netz would write an essay on a thesis that, to his eyes, is crucial. The title of the essay, ‘The Place of Archimedes in World History,’ straightforwardly states the issue that Reviel wanted to address. Thirteen scholars would reply to Reviel's essay, some specializing in historical epistemology, others in economic history, and still others in the history of mathematics or that of the physical sciences; some specializing in ‘classics’ – as the usual terminology calls this field of knowledge –, others in the history of China, South Asia or the Arabic world, and still others in European studies. In the end, Reviel would have the opportunity to close the volume with a short response.

The contract between Reviel and me was no less clear.

I embarked on this project around Reviel's views on Archimedes with a warning to him that there would be strong disagreements. We all admire Reviel's ground-breaking work in the study of the mathematical practices recovered from Greek geometrical texts from antiquity and his ability to offer fresh insights into old topics. 1 However, there are a number of assertions Reviel takes as obvious, as well as a number of theses he holds, in which I do not follow him – in particular, those that he formulates in this essay about Archimedes and world history. We disagree not merely about details, but about major issues. As I understand it, this is not simply a matter of forming different opinions on the basis of the same evidence. Reviel also brings into play a number of historiographic practices that I find debatable. This was the background of the project that led to the composition of this issue of Interdisciplinary Science Reviews (ISR). From the outset, it was clear to him and to me that this volume would enable us to make this disagreement explicit and allow for a broader debate.

Reviel accepts and even enjoys debate. This was the sine qua non for this set of articles to see the light of day. We must be grateful to him that this debate, however adversarial, is possible. Deep mutual respect, especially from a moral viewpoint, is what allowed this exchange to be both friendly and candid, as we agreed from the start. In the thirteen articles that follow Reviel's essay, each of us, instead of composing an article of a classic type, draws on his or her personal historiographic experience to put assertions and methods in debate. Exercises of this type are, in our view, much needed in the history of science, and we think this volume illustrates what might be gained from them.

The main issues in the debate

Formulated without nuance, Reviel's argument can be summarized simply as this: Without Archimedes, modern science would have been different. Perhaps even, neither the scientific revolution nor the industrial revolution would have taken place. As crude as this summary of his long argument is, it will enable me to map the debate that follows his essay.

To begin with, these few words suffice to make clear that Reviel's essay presents counterfactual reasoning. Such reasoning, and how it might be conducted in history, constitutes the entry points that both Ian Morris and Walter Scheidel have chosen in their response to Reviel. From a theoretical viewpoint, Morris focuses on the logic of reasoning of this kind, making explicit rules that it should follow and criteria that it should satisfy. This analysis offers tools that Morris puts into play to examine and discuss Reviel's reasoning. Scheidel approaches counterfactuals from another angle, focusing on the interplay between counterfactual reasoning and comparative history. This comparative perspective leads Scheidel to raise doubts on some causal elements Reviel associates with Archimedes with respect to the industrial revolution, and further, to highlight elements central to this revolution and unrelated to the Syracusan mathematician.

Reviel's reasoning also involves a probabilistic element. For sure, if the probability of an Archimedes being born was high, the reasoning would not hold. Lorraine Daston undertakes to discuss the argument that the essay unfolds from that angle. This leads her to a key observation: had Reviel asserted that ‘Without Archimedes, there would have been no computation of the area of a segment of a parabola until much later,’ few people would have simply paid attention. What makes the thesis arresting is the second term that the conditional statement puts into play: ‘the Scientific Revolution’ and, more broadly, ‘modern science’ – two phenomena commonly considered as crucial in the discipline of the history of science and beyond. The importance given to these two notions in the current historiographies as well as in the widely shared representations of the history of science contrasts with the lack of agreement on what exactly they refer to, or even on whether they mean anything at all. In effect, many contributions to this issue of ISR, starting with Daston's, discuss the interpretation that can be given to these expressions, highlighting facets arguably what, for Reviel, is constitutive of modern science that would be independent of Archimedes and the second efflorescence of Greek mathematics.

Before outlining the directions explored by these discussions, it might be useful to step back and wonder why notions such as these, whose meaning is unclear, elicit so much emotion. In fact, addressing this question sheds light on the fact that the overall structure of the debate is not innocuous. As Daston emphasizes, whether we like it or not, notions such as ‘modern science’ are actually loaded with political issues. Historians would therefore be well advised to reflect on the historical and political contexts in which the historiography of science has given pride of place to ‘modern science’ and to the event of its ‘rise,’ if they want to regain some freedom of thought. Daston points out in particular the post-World War II period as well as that of the Cold War as moments in which historians, especially those active in Europe and North America, have mainly focused on this phenomenon and this event, taking them implicitly or explicitly as specific to ‘the West.’ Looking at these historiographic notions from South Asia, Agathe Keller, for her part, stresses colonization and the making of empires as processes in which historiographies of science of this kind have played a part. It is, therefore, not surprising that these historiographies have been perceived as tools of hegemony, and that, Keller continues, counter-historiographies of various kinds have been offered, as tools of resistance or emancipation. Interestingly, whether these counter-historiographies of science claim a part of ‘modern science’ for collectives other than ‘the West,’ or contribute to the definition of ‘alternative modernities,’ they do not change the terms in which the debate is generally put: ‘modernity’ remains a central term. Political issues willy-nilly loom large in historians’ work, as is demonstrated by the uses that have been made of these historiographies outside the discipline, and this as much in Europe and North America as anywhere else in the world. A reflection on the history of these historiographies might help us better understand how political issues have been involved in the various types of historiography of science that coexist.

This issue of ISR does not offer insights into other possible frameworks for new historiographies of science that might replace the old ones. However, several contributions take steps in this direction by examining how Reviel accounts for the events in the history of science that occurred outside the line he draws between Archimedes and ‘modern science.’

The structure of Reviel's argument – especially the thesis that ‘Greek mathematics’ is a ‘far outlier’, if compared to mathematics elsewhere (Netz 2022, 303) – leads him to lump everyone else together and also to put forward universal assertions. Keller points out one sweeping statement of this kind, with Reviel's assertion that ‘people whose task is to articulate statements concerning numbers, figures and the stars’ ‘appear together with the state’ (Netz 2022, 303). Indeed, the earliest Sanskrit documents, she argues – namely, the śulbasūtras – seem to call for a revision of this assertion. Moreover, she adds, these sources differ drastically from other Sanskrit mathematical documents, raising the problem of treating ‘the others’ as homogeneous blocks. In a similar vein, Geoffrey Lloyd focuses on the comparison between Greek and Chinese mathematical documents to suggest that the contrast between them that Reviel asserts might be more in degree than in kind. Distrustful of any theory of ‘Greek exceptionalism,’ Lloyd seeks to refine the description of differences offered by Reviel and calls for a contextualization that would account for them.

The structure of Reviel's argument also leads him to concentrate on a single historical line of development. To this facet of the argument, Dhruv Raina objects by noteworthily focusing on ‘contemporary mathematics and the exact sciences’ – and not on ‘modern science.’ His main point consists in arguing that we need to consider them as the result of multiple genealogies. In Raina's view, the historical line followed by Reviel might not even be the most central genealogy for the history of science. More importantly, the emphasis on a process of development as unilinear, Raina argues, may hide phenomena such as hybridization between different cultures, which played a crucial role in the advancement of science. What is more, Raina stresses that concentrating on concepts put forward by Archimedes and essential for authors like Kepler and Newton might conceal important changes that these concepts underwent and that await contextualization as well as interpretation.

Jamil Ragep's article, which offers reflections on a global history of mathematical astronomy, can be considered as dealing with some of these genealogies in which Archimedes played no role. For Ragep, the ideal of achieving as high an accuracy as possible in the prediction of astronomical phenomena as well as the practices of observing and designing mathematical models to reach this goal took shape in Babylon and Uruk in the last centuries before the Common Era. The appropriation of this goal and these practices drastically changed the astral sciences written in Greek, which, only after these contacts, demonstrate the ambition of quantifying and testing. What is more, in line with a remark by Keller, Ragep argues that calling the computations carried out by Hipparchus and Ptolemy ‘trigonometric’ prevents us from understanding the huge difference that the introduction into astronomy of truly trigonometric computations such as those found in Sanskrit and later Arabic sources represented. It is precisely this conceptual blurring that may incite us to believe in a single genealogy of mathematical astronomy – in which Greek authors would be important – when in fact hybridity played a key part in its history.

The contributions that I have mentioned above deal mainly with ancient history. The remaining ones examine Reviel's argument from the viewpoint of his claims on science in Europe in the sixteenth and seventeenth centuries. To begin with, Courtney Roby addresses the question of what reading Archimedes meant precisely at the time. She does not merely analyze the actual documents in which readers can get access to his works, but she also highlights the difficulties encountered by readers at the time. She brings these difficulties to light by examining the editions published and the commentaries added to Archimedes’ writings in these editions. Niccolò Guicciardini deals with the same issue of the interpretation of Archimedes’ works in the seventeenth century from a different perspective. His main point consists in highlighting that, at the time, different scholars read wholly different contributions in Archimedes, and that he was the point of departure for completely different traditions. For instance, for Leibniz, Archimedes had used infinitesimal methods, and Leibniz thus perceived he was pursuing this predecessor's work. For Newton, however, Guicciardini argues, it was a different Archimedes who mattered: the one who had used curves drawn by mechanical apparatus appropriate to solving a problem simply and elegantly. By contrast, pace Reviel, Apollonius’ ellipses were not central to Newton's and others’ work in celestial mechanics: scholars were aware there were perturbations, and they were looking for other curves. For this concrete work, in fact, Newton gave the main role to computation, that is, to a practice that had nothing to do with ancient Greece, but that was central to the work of engineers at the time. Rivka Feldhay pursues the latter reflection in her contribution, in which she emphasizes that too narrow a focus on the Syracusan leads Reviel to underestimate the role played by practical knowledge and men of practice in the historical process that she refuses to conceptualize as a ‘Revolution.’ Feldhay elaborates concepts to think differently about what, for her, was a ‘transformation’ rather than a ‘revolution.’ Notably, in this development she introduces the notion of ‘knowledge sites’ – a tool with which to figure out ways of analyzing what for too many historical discourses remains simply ‘Europe.’

Guicciardini's insistence on the significant role played by computation for practitioners of science in the sixteenth and seventeenth centuries echoes arguments made in several responses to Reviel's article, and notably in the last three contributions. They all emphasize important events that took place with respect to magnitude, number, quantity and computation – events that were all unrelated to ancient Greek works but that proved essential for the major changes evidenced in scientific activity at the time. My own article starts from a remark: in the current historiography of science – and Reviel is no exception – the place-value decimal system, that is, the numeration system commonly used today for calculation, is considered to have played only a minor role in the history of science. It is true that this numeration system has nothing to do with Greek works of antiquity. However, considering it as merely practical and inconsequential, as is often the case, is a short-sighted appreciation: I argue that this numeration system has had a long-term and important theoretical impact in the history of mathematics, whose echoes we still find in Newton's mathematical work on series, and beyond.

Jens Høyrup further highlights the important role played by the mathematics of counting and measuring, in which the excerpts of Greek texts available in medieval Europe did not have any major impact. In this respect, without denying the impact of Archimedes’ works in the seventeenth century, Høyrup also invites us not to underplay the significant role of algebra from al-Khwārizmī and al-Karajī – and not only al-Khayyām – through the abbacus schools and cossic algebra to Clavius and Descartes. In this line of development, whose importance for seventeenth-century mathematics cannot be underestimated, Archimedes’ works played at best a tangential role.

These new branches of knowledge also elicited dramatic transformations in the way Greek texts from antiquity – to begin with, Archimedes’ works – were read and used. This is precisely the point that Pier Daniele Napolitani's contribution to the debate highlights. In line with Raina's remarks on conceptual change, Napolitani points out several ways in which, beyond family resemblances, the mathematical objects considered in the sixteenth and seventeenth centuries were drastically different from those to which Archimedes devoted his efforts. More importantly, perhaps, Napolitani describes the lost struggle of a practitioner of mathematics like Maurolico in his attempt to ground Archimedes’ mathematical works on a notion of general quantity while remaining within the framework of ancient Greek geometrical texts. Subsequent mathematicians would break the framework, interpreting number, quantity and magnitude in ways wholly different from Euclid and Archimedes, but in ways that would demonstrate hybridization with concepts coming from other traditions of reflection.

The preceding lines should be read as an invitation to set off into the debate that this issue of ISR hosts. They only offer a reading of the principal themes of a complex debate, without exhausting the much richer set of arguments that each article deploys. To let readers choose their way between the contributions and meander within the debate, Reviel and I have decided to arrange the articles alphabetically by authors’ name. May the debate last beyond these initial thirteen contributions.