Jessica Carter – The role of Figures in Contemporary Mathematics
We note that use of figures abounds in contemporary mathematics. Figures play a role for understanding as well as tools for discovery. They even appear as a basis for proofs. The talk presents in detail an example from analysis using figures. The aim is to understand the role that figures play in mathematical reasoning comparing them with other types of representations. In order to do this I draw on the semiotics of Peirce and distinguish between 2-dimensional diagrams, that is representations of relations exploiting 2-dimensionality, and symbolic expressions.
Karine Chemla – The Proof is in the Diagram
In a specific tradition of dealing with algebraic equations in China, 11th to 13th century writings on the topic combine problems, algorithms and diagrams of several types. My talk focuses on the geometrical diagrams that some of them contain. The argument holds that the captions in these diagrams establish a specific connection with the algorithms in relation to which they are given. Accordingly, I claim that these diagrams constitute the proof of the correctness of the algorithms. Reading the diagrams as assertions is thus in my view essential to capture what is at stake in them. These diagrams disappear from another tradition of dealing with algebraic equations in China, to which writings from the second half of the 13th century and the early 14th century attest. I will suggest that these diagrams are replaced by a form of algebraic proof in an algorithmic context, which is also expressed in specific ways.
Silvia De Toffoli – Thinking with Diagrams: The Case of Mathematics
Visual representations of various kinds are ubiquitous in pure and applied mathematics, in the natural and social sciences, as well as in many other human activities. By investigating these visualizations, many questions arise: What are they? How do they function? What are the conditions of their correct use? Why are they, at times, such effective aids to cognition? What are their epistemological roles? In this talk, I will focus on diagrams in mathematics, not exclusively in geometry where diagrams are common, but in different mathematical domains, such as knot theory and homological algebra. Despite the extreme variety of representations in mathematics, I will try to distill fundamental properties of the nature and use of mathematical diagrams. I will argue that diagrams are not static illustrations simply recording information, but dynamic displays for advancing thought. An effective diagram, or a sequence of diagrams, sets the relevant reasoning into material, visual form. By manipulating these concrete external representations in prescribed ways, information about abstract mathematical structures can be obtained without going through a process of formal calculation. In this way, cognitive abilities that no doubt evolved in order to manipulate concrete objects can be re-deployed in the abstract realm of mathematics.
Paula Findlen – Projecting Nature: Agostino Scilla’s Fossil Illustrations
Agostino Scilla’s Vain Speculation Undeceived by Sense (1670) has been the subject of a growing discussion as an important contribution to the development of a new understanding of the nature of fossils as a record of the earth’s history in the mid-seventeenth century. The fact that Scilla was a painter who drew his own illustrations from his collection in Messina, subsequently engraved by the Perugian artist Pietro Santi Bartoli in Rome, makes his work even more interesting for understanding the evolution of scientific illustration. This talk explores some of the unique features of Scilla’s representations of fossils as key element of the significance of his contribution to natural history. What inspired his new approach to depicting fossils? How might we see this as an example of “visual thinking” that helped to prove his point?
Valeria Giardino – The Role of Cognitive Tools in the Sciences
In the first part of my talk, I will briefly present previous work with De Toffoli on the practice of topology. We have proposed that topologists, in order to become experts, have to learn how to use what we have defined as manipulative imagination. Such a form of imagination is central to many areas of topology, for example knot theory (De Toffoli & Giardino, 2014), low-dimensional topology (De Toffoli & Giardino, 2015) and braid theory (De Toffoli & Giardino, 2016). To clarify, in order to follow the proofs, topologists have to envisage transformations of and on the diagrams. Their interaction with the representations is therefore essential: the figures are not static, but have to be used dynamically so as to trigger a form of imagination that allows them acting on them and drawing inferences accordingly. Representations are thus cognitive tools whose functioning depends in part from pre-existing cognitive abilities and in part from specific training. If manipulative imagination exists, and possibly it is used also in other areas of the sciences, what kind of imagination is it? In the second part of my talk, I will refer to the notion of imagination as “make-believe” as proposed by Walton (1990) to give an account of the role of cognitive tools in mathematics as props. To better specify my claim, I will also rely on the notion of “affordance” as proposed by Gibson (1979) and discuss how it can be extended from concrete objects to representations.
James Griesemer – Thoughts in the Role of Semi-Diagrams in the Biological Sciences
In following causal processes that are not available to the naked eye over short periods of time, scientists typically need to observe processes by marking them in ways that help visualize and trace them. Scientific reports therefore tend to be accompanied by representations that reflect these marking and tracking activities, emphasize some aspects and deemphasize others, and communicate findings in ways that facilitate acceptance and promote adoption of successful practices. In this talk I will reflect on the role of images and diagrams in selected episodes from the history of biological practice since the mid-19th century in order to suggest the centrality of semi-diagrams (C. Otis Whitman 1887) to both the development of particular scientific projects and the history of scientific specialties. A semi-diagram combines pictorial and symbolic elements in ways that can reflect both the mundane details of tracking work and serve more theoretical roles of abstraction to general principles and empirical guidance in model-building.
Eunsoo Lee – Toward a Critical Edition with Archetypal Diagrams
In general, classical philology is a discipline that studies transcribed texts. Philologists compare various readings of transcribed texts in order to reconstruct the archetypal text. Besides the text, diagrams are drawn in geometrical treatises. Shared diagrammatic errors found across manuscripts reveal that diagrams were also transcribed. Considering the plethora of studies on the scribal practices of copyists, it is surprising that diagrams are rarely studied in a similar manner as transcribed objects. In this talk, I will explain the methods I used to reconstruct archetypal diagrams. My talk will present preliminary research reconstructing archetypal diagrams from the Greek manuscript diagrams for 30 propositions in Euclid’s Elements. My reconstruction generally proceeds according to the following steps: 1) Investigate textual reconstruction to cross-reference with diagrams; 2) Decompose (take apart) diagrammatic attributes to find unstated shared errors between diagrams; 3) Follow the logic of textual philology to find the most plausible choice among competing candidates for the archetype. By sharing this methodology with participants in the workshop, we can work together to develop best practices in curating archetypal diagrams for inclusion in a critical edition.
Melissa Lo – Descartes without Pictures
Why are Descartes’s images so little known today? In this paper, I will briefly sketch the visual intensity of Cartesianism throughout the 17th and early 18th centuries, and the subsequent decline of Descartes’s graphism in the early decades of the nineteenth century. As Descartes was revived by French philosophers – especially Victor Cousin – his physics, and, consequently, its pictures were all but suppressed. This, I propose, was not merely a response to the professionalization of science. It was also owed to two more factors: the redefinition of what philosophy was (and what it was not); and the proliferation of images during the nineteenth century. It is this image-less Descartes, whose mind held his culture at arm’s distance, who prized mathesis above imagination, that our digital age has inherited. The metamorphoses of Descartes’s figures permits reflection on the fortunes of images from the past, their abiding cultural contingency, and the meaning of histories that recover them.
Kenneth Manders – On Geometric Thought
Geometric thought paradigmatically combines inexact spatial grasp with exact concerns; typically supported by diagramming respectively symbol manipulation.
After a quick survey of contexts of geometric thought up to present, we counter a recent century’s tendency to eliminate the roles of spatial grasp in geometrical thought by strategising on the challenge of articulating what these roles come to in broader geometric contexts of the 19th and 20th centuries.
Jennifer Pegg – Audacious Psyche: Visual Representation and the Evolutionary Ideas of John Pringle Nichol
A simple circle, a sinuous curve, and the mythical figure of Psyche ascending a ladder of snakes: these images all captured crucial elements of the evolutionary ideas of John Pringle Nichol, a mid-nineteenth-century Scottish astronomer and political economist. The principles that he intended them to illustrate included the unity of knowledge and of natural law, the organic essence of the universe, and the fundamentally transformist nature of physical and moral change, but they created just as many questions as they answered. Were the transformations of nature rigidly progressive and teleological, or contingent and meandering? To what extent was the universe guided by Providence versus natural law? Could human societies and even the human soul evolve without limits? I examine Nichol’s theory of cosmic evolution, and show how his choice of imagery reflected essential ambiguities—and the instability of evolutionary concepts and language—in pre-Darwinian evolutionary theory.
Greg Priest – Diagrams as Tools: The Case of Darwin’s “Tree of Life”
In 1837, Charles Darwin wrote in a notebook that “organized beings represent a tree, irregularly branched” and sketched a simple drawing of a tree to visualize the idea. From this seed was to grow the only illustration included in the Origin—the “tree of life” diagram. Although Darwin used the tree of life to illustrate and explain his theory, and to persuade his contemporaries to accept it, a rhetorical analysis does not exhaust the role tree diagrams played in the development of his theory. Darwin also deployed tree diagrams as tools to enable him to think through a number of the most difficult questions raised by his theory. These diagrams not only served as invaluable aids in his explorations of complex evolutionary dynamics; they also embodied unarticulated ontological and epistemological presuppositions that informed Darwin’s scientific practice. By examining these twin aspects of Darwin’s diagrammatic practices, we can better understand central aspects of his thought, and gain insight into how diagrams can contribute to, and constrain, the development of scientific theories.
M. Norton Wise – On the Stories Told by Indicator Diagrams and Carnot Diagrams
I am interested in the narrative quality of diagrams that aim to capture a development in time, particularly those that reveal the internal processes of a system and diagnose its “health” while also explaining its operation. Such was the “indicator diagram” for the operation of a steam engine, as drawn by an “indicator” attached to it. Literally, the diagram traced the motion of the piston moving up and down in the cylinder while recording its pressure. It also, however, produced a visual narrative of the working engine, including the effects of improper valve timing, readable and treatable by an experienced mechanic. This highly practical visual narrative took on a quite different form when it was idealized and abstracted by mathematical engineers and physicists to become the famous “Carnot diagram” and used to explain how the engine did its job. It told a story initially about how it produced work from the “fall” of heat and then later about how it converted heat into work. I will discuss the relation of these various narratives as told by engines.