In 1303 Wang Zhen published the agricultural treatise Nongshu 農書, in which he included in tabular form instructions for, amongst others, the best period for sowing different kinds of crop. These tables were accompanied by further textual explanations with the obvious goal of guiding the reader to agricultural improvements (Bray, this issue). In 1996, the Bank of England produced fan charts in its inflation report showing the range of uncertainty in its inflation forecasts. These charts were accompanied by an explanatory text intended to guide the reader in understanding the diagram (Boumans, this issue). When the famous ancient mathematical text Zuobi Suanjing 周髀算經 (The Gnomen of the Zhou) was re-edited for the Complete Library of the Four Treasures, the restructuring of textual elements added another layer of interpretation to a fundamental text explaining how to perform mathematical calculations (Chemla, this issue). Imperially commissioned in the early 1770s, this edition appeared around the same time Benjamin Franklin was jotting down dots into a tabular system to keep track of his moral improvements, a system he explained in his Autobiography first published, in French, in 1793 (Maas, this issue).

These four cases, together with two other articles by Hsiang-Ke Chao and Mary Morgan, and a field review by Chiara Ambrosio, constitute this special issue, “Thinking and Acting with Diagrams.” Diagrammatic reasoning has been an important topic in recent studies in the history and philosophy of science and technology. This special issue contributes to the discussion by offering the studies that treat the questions of how to think and act with diagrams in the history and philosophy of science and technology from widely different time periods and geographical areas. Diagrammatic reasoning is (at least) two-dimensional; reasoning with diagrams is a form of visual reasoning that takes place in space. This common feature of diagrams is prominently present in the six original research papers as well as the field survey article in this special issue, covering historical and contemporary examples and comparative studies of East and West.

Space is of central importance in Hsiang-Ke Chao’s comparative essay on economic geography. Chao examines different diagram-based modeling strategies of nineteenth- and twentieth-century economists and sociologists to discover regularities in the geographical distribution of towns and villages in Germany, the United States and China. By going through a wide range of examples from different scientific disciplines, especially economics, Mary Morgan provides an in-depth discussion of the cognitive and inferential work done by and with diagrams. The importance of diagrams as a form of spatial reasoning springs out as an overarching theme in Chiara Ambrosio’s historico-philosophical reflection on diagrammatic reasoning in science and technology. In this short introduction to this special issue, we would like to highlight the importance of space in diagrammatic reasoning, and how this may challenge longstanding distinctions between image and text, inductive and deductive reasoning, and the meaning of what diagrams are. Before we do so, we say a few words about the origin and gestation of this special issue.

The original impetus for this issue came from the increased and still increasing interest of historians, philosophers, and sociologists of science and technology in the role of visual artifacts in the formation and transmission of scientific and technological knowledge. To examine one of these artifacts more in depth, and also to see if new insights could be gained from a combination of scholarship on Chinese and Western science and technology, Hsiang-Ke Chao and Hsien-chun Wang organized a conference on diagrammatic reasoning at National Tsing Hua University in Taiwan from which the present collection of essays, after the normal review process, was retained. There were two important reasons to focus on diagrams and diagrammatic reasoning. The first was inscribed in recent scholarship on model-based reasoning, as can be found in the work of Michael Weisberg, Marcel Boumans, Mary Morgan, and many others (see, for example, Morgan 2012; Boumans 2004; Weisberg 2007; Matthewson and Weisberg 2009). The second, in the already-mentioned wish for a treatment of diagrammatic reasoning that would extend beyond the confines of the history of Western science and technology.

Building on A. C. Crombie’s monumental work on styles of reasoning in science, this recent literature on modeling has made scholars increasingly aware of the importance of diagrammatic systems, the investigation of their logical status in scientific discovery and justification, and their practical role as tools for action. As a result, the focus has shifted from an interest in diagrams as representational tools to their active role in actual scientific reasoning and research design, and their constructive role in the grounding of ontological and epistemic claims. Being intrinsically visual, diagrams not only serve as a certain type of representational device but also can be used to carry out model-based reasoning and action. It is therefore unsurprising that research into diagrams has come to play a privileged role in science and technology studies.

The idea was to square this scholarship with recent scholarship in Chinese history of science and technology that has concentrated on an analysis of tu (圖), the ancient Chinese term for visual representations such as diagrams, graphs, maps, and technical images, as witnessed in the excellent volume Graphics and Texts in the Production of Chinese Technology Knowledge: The Warp and the Weft (Bray, Dorofeeva-Lichtmann, and Métailie 2007) that has cast new light on how technical knowledge was conceived and displayed in China. This work demonstrates the crucial role that diagrams play in knowledge manufacturing and transmission. Also, Karine Chemla’s work in the history of Chinese mathematics (for example Chemla 2010) has contributed greatly to our current understanding of the Chinese usage of diagrams in mathematical proofs. By integrating these two different research strands, the aim was to contribute to a better understanding of the distinctive role of diagrams in knowledge formation, as representational and reasoning tools, and as tools for action.

One might, therefore, expect that we need to be clear about what we mean by “diagrams.” How otherwise can we point out their importance in this variety of roles? One can also expect, however, that it would have been a matter of sheer coincidence if the Chinese concept of a diagram would have tallied one-to-one with its prevailing notion in Western science. Mary Morgan starts her contribution with a definitional search for how she will understand diagrams for the purposes of her paper. She reviews several recent contributions, including Oxford English Dictionary definitions, some of which include tables, lists, and maps as also fitting the notion of a diagram, but settles on one specific kind of diagram—graph diagrams, which then come in different sorts, historical and analytical, or, using Judy Klein’s (1995) important distinction, graphs in historical and logical space. She then highlights their two-dimensionality and zooms in on the reasoning exercises that can be done with graphs on and in logical space, with Alfred Marshall’s supply and demand diagram as the famous, and widely used stellar example. She thus makes explicit that her primary focus will be on diagrams consisting of graphs, not on tables or other kinds of diagrams, even though some of them play an important role in her argument as well. Diagrams, in Morgan’s analysis, make relations visible in a real or imagined world, and manipulating the graphs in the logical space of the diagram enables what she calls “visual deductions,” a concept that sits uneasily alongside the neat philosophical distinction between inductive and deductive reasoning.

To introduce her argument, Morgan explains that she will use such traditional philosophical distinctions “loosely,” emphasizing that when looking at scientific practice, neat philosophical distinctions do not always fit so well with the messy reality of practicing scientists. A great many analytical philosophers may be inclined to tune out, because they see it as their job to introduce neat definitions to a messy world. But not necessarily, as Ambrosio shows in her survey article on Charles Sanders Peirce, who also underlined the importance of diagrammatic reasoning for his own work. She writes, “Peirce was a champion of diagrammatic reasoning in practice, as well as in theory. ‘I do not think I ever reflect in words,’ he famously reminisced in 1909, ‘I employ visual diagrams firstly because this way of thinking is my natural language of self-communion, and secondly because I am convinced that it is the best system for the purpose’” (Ambrosio, this volume). Ambrosio continues with a discussion of Peirce’s famous “existential diagrams” which are, if anything, not only visual but spatial. Peirce presents reasoning not as an internal dialogue, but as a dialogue that also involves our eyes and hands, and that is performed with pen and paper or similar devices. Reasoning is physical, graphical, material, and spatial, and visual diagrams fit that purpose best. The spatial, two-dimensional characteristic of a diagram is important, and it is in fact by making full use of its two-dimensional characteristic that the full force of diagrammatic—visual—reasoning can come to the fore.

It is interesting that when we shift from Peirce’s existential logic to rhetoric, we find that a fifteenth-century discourse on rhetoric—that is reasoning in the public domain—explained rhetoric in diagrammatic form. Its author, Lodovico Castelvetri, explained the separate steps of a rhetorical argument by means of a table in which the grid represents the separate steps waiting to be filled with content (see Fig. 1 in Maas’s contribution). Historians of accounting Christiano Busco and Paolo Quattrone (2018) cite Mary Carruthers’s Clark Lectures in which she traced the historical linkage of rhetoric to spatial reasoning. The rhetorical notion of topoi, spaces to be filled with content (topics), betrays its etymological roots to space: rhetoric was a spatial form of reasoning known as ratiocinatio, where, still following Carruthers’s idea, the Latin ratio meant “account,” “calculation,” or “computation,” a link still present in the French expression for accounting books, livres de raison.

While Busco and Quattrone (2018) emphasize the “lost connection between rhetoric, accounting and rationality,” we would like to emphasize the lost connection in modern conceptions of reasoning with space and diagrammatic thinking. Ambrosio perceptively notes that diagrams never sat well with analytical philosophy because of their unsettling status between the empirical and the analytical, between induction and deduction, thus echoing the same reservations against neat conceptual distinctions we find in Morgan’s contribution. We might also note that before more geometrico became identified with deduction, a geometric proof was performed on paper and with instruments such as a ruler and compass. Over time, the spatial dimension of reasoning got lost in theory, but as the contributions to this volume show, not in practice. The links between diagrammatic thinking and practical accounting procedures are clearly present in Maas’s three different case studies in which individuals engage in an act of self-communion by manually filling in tabular grids to examine and evaluate their moral worth. With a notion of reasoning that entails both eyes and hands, and the material practice of moving things around to find out, space re-enters and we are once again far removed from clear-cut distinctions between the empirical and the analytical and between induction and deduction indeed. Much of this is in the spirit, if not the letter, of A. C. Crombie’s magnum opus.

This is not the place to write the history of what then becomes set in stone as “rigorous deductive reasoning.” The tensions are tangible, however, in Victorian Britain, between John Stuart Mill’s discussion of ratiocination in his Logic (Mill 1843) that shows no trace of spatiality, and George Boole’s algebraization of logic vis-à-vis John Venn’s diagrammatic logic and Alfred Marshall’s exemplary defense of the method of diagrams as an engine of discovery—as the most efficient and productive way of reasoning in economics. Cambridge’s strong attachment to mathematics’ geometric tradition may have played a role here, but even Marshall, in his defense of the method of diagrams, drew a contrast between the night train of mathematical proof and the day train of visual diagrammatic demonstration which over time became scorned by mathematicians and mathematical economists as lacking the rigor of deductive proof. We would like to suggest that much contemporary work on model-based reasoning aims to retrieve this lost connection between space and reasoning.

These links between space and diagrammatic reasoning are manifest in Morgan’s and Chao’s contributions that both concentrate on two-dimensional diagrams that, in Morgan’s words, “induce visibility” and/or aim to “visually deduce” consequences from manipulations in and of diagrams. We can see both processes at work in Chao’s contribution to this issue. Chao’s historical analysis of early studies into the location of towns and villages in relation to market structure compares the different modeling strategies—the concept enforces itself—of economic geographers and sociologists in the United States, Germany, and China. These scholars either induce visibility from fine-grained empirical research into geometric forms that best “fit” the data or visually deduce conclusions from prior assumptions to explain Chinese market structure. Chao compares Ching-Kun Yang’s pioneering study of systematic field observations of Chinese periodic markets in the 1930s, which was heavily influenced by the empirical fieldwork of the Chicago school of sociology, especially the work of Robert E. Park and Ernest Burgess, with that of the American anthropologist William Skinner, who successfully applied the central place theory of the German Location School to explain the development of China’s rural market structure. While Chao’s study shows that diagrams provide an example where the efficiency of scientific inquiry may rest on the form of the reasoning devices, Morgan points out that we must not take for granted that the particular metrics of diagrams determine entirely the logic of reasoning of a diagram. A scientist must understand both vocabulary and grammar of the diagram to use it efficiently.

As already indicated, the grammar of diagrams differs over time and place. While Morgan confines her discussion to what is generally considered the common reference point for a diagram—a Cartesian space with line-graphs—Francesca Bray’s contribution to this volume also recognizes the importance of space in diagrams, but her focus shifts to the work done by diagrammatic tables and one of their constructive elements; the chaîne opératoire, the operational chain that ties sequences of action in time and space. The importance of space reflects the main point of her survey of the history of tu: the characteristic of tu is that they offered a way of “spatial encoding” that translates the factual and conceptual information into knowledge in spatial terms (Bray 2007). Moreover, what distinguishes tu from other kinds of Chinese visual artifacts is that tu were usually referred to as instructive images that conveyed knowledge of skills and technology. Therefore, the powers of tu are based on serving as templates for action, instead of being merely stylistic representations.

The contributions of Boumans and Chemla use spatial decoding to “reverse engineer” the conceptual and historical tenets of their respective subjects. Boumans discusses the conceptual assumptions behind the probability distribution that drives the inflation baseline prediction of the Bank of England. This baseline prediction is subsequently modified by the expert judgments of a panel of economists, which creates the spread around the baseline. He thus shows how recent efforts of the Bank of England to domesticate their ignorance about the future course of interest-rates lead to a spatial representation of ignorance and chance that results from a combination of a probabilistic benchmark projection that implicitly combines principles of Gestalt psychology and expert judgment. Pushing the boundaries of what can be considered “diagrammatic” to include textual building blocks, Karine Chemla used these diagrammatic features to trace and date the provenance of different parts of The Gnomen of the Zhou. The diagrammatic features of a Classic Chinese mathematics text thus become an unexpected historiographic research tool.

The connection between spatial representation and reasoning will also depend on the available technologies and means of communication, as witnessed most clearly in the contributions of Bray and Boumans. Bray connects her own diagrammatic efforts to understand tea-cultivation in China with two early agricultural manuals, both of which substantially use one particular diagrammatic device indicating the operational sequence of actions in land cultivation, the chaîne opératoire. She shows how after the invention of block printing, these operational sequences lead to diagrammatic representations that can be reasoned with and acted upon. Bray shows how the tu comprising the sequence of actions in the cultivation of various crops is combined with a text explaining how to read and interpret the diagram. We see the importance of print technology also in Boumans’s contribution. Reasoning on chance and ignorance expresses itself spatially in fan-charts that use color codes to indicate probabilities. Both contributions thus show how the representational function of diagrams also depends on the technologies that are available and on the textual explanation of what is represented, very much like the way the famous tableau économique of the physiocrats was embedded in its textual explanation, and maps, graphs and diagrams nowadays combine visual image and explanatory key. This may seem a trivial point, but the close relations between text and image put in question the boundaries of what counts as part of a diagram and what counts as part of a text, boundaries that are explicitly questioned in Chemla’s contribution that does not push textual features into a diagram, but makes the reverse movement of searching for diagrammatic features in texts.

From different angles, the contributions to this special issue thus highlight the importance of space in diagrammatic reasoning, and this not only for the diagrams that consist of a Cartesian space with graphs but also for diagrams consisting of combinations of geometric forms or tables or even only of configurations of text. They thus question received distinctions between inductive and deductive reasoning, the empirical and the analytical. Diagrams contain scripts for action but do so in combination with explanations of how they should be used. Thus, sometimes but certainly not always, diagrams are tools that represent the world—be this fictional or empirical—as well as tools of world-making. It is his “how-to-do” function, prominent in Bray’s contribution, that helps explain their historical and enduring importance in science, technology, and society.

Acknowledgments

The early drafts of the papers in this special issue were presented at the conference “Reasoning and Representation with Diagrams: History and Philosophy of Science and Technology in East and West,” held on 24–25 November 2016 at National Tsing Hua University, Taiwan, organized by Hsiang-Ke Chao and Hsien-Chun Wang. We thank all the participants in the conference for stimulating discussions. Financial support for the conference from Taiwan’s Ministry of Science and Technology and National Tsing Hua University is gratefully acknowledged. We also thank the authors for their contributions to this special issue and to EASTS, particularly editor-in-chief Wen-Hua Kuo and collaborating editor Pingyi Chu for their kind support.

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