A Practice-Based Approach to Diagrams

Diagrammatic Reasoning, 2015

Many types of everyday and specialized reasoning depend on diagrams: we use maps to find our way, we draw graphs and sketches to communicate concepts and prove geometrical theorems, and we manipulate diagrams to explore new creative solutions to problems. The active involvement and manipulation of representational artifacts for purposes of thinking and communicating is discussed in relation to C.S. Peirce’s notion of diagrammatical reasoning. We propose to extend Peirce’s original ideas and sketch a conceptual framework that delineates different kinds of diagram manipulation: Sometimes diagrams are manipulated in order to profile known information in an optimal fashion. At other times diagrams are explored in order to gain new insights, solve problems or discover hidden meaning potentials. The latter cases often entail manipulations that either generate additional information or extract information by means of abstraction. Ideas are substantiated by reference to ethnographic, experi...

Synthese, 2017

The paper asks whether diagrams in mathematics are particularly fruitful compared to other types of representations. In order to respond to this question a number of examples of propositions and their proofs are considered. In addition I use part of Peirce's semiotics to characterise different types of signs used in mathematical reasoning, distinguishing between symbolic expressions and 2-dimensional diagrams. As a starting point I examine a proposal by Danielle Macbeth (2014). Macbeth explains how it can be that objects "pop up", e.g., as a consequence of the constructions made in the diagrams of Euclid, that is, why they are fruitful. It turns out, however, that diagrams are not exclusively fruitful in this sense. By analysing the proofs given in the paper I introduce the notion of a 'faithful representation'. A faithful representation represents as either an image (resembling what it stands for) or as a metaphor (sharing some underlying structure). Secondly it represents certain relevant relations (that is, as an iconic diagram in Peirce's terminology). Thirdly manipulations on the representations respect manipulations on the objects they represent, so that new relations may be found. The examples given in the paper illustrate how such representations can be fruitful. These examples include proofs based on both symbolic expressions as well * I especially thank Danielle Macbeth for comments on earlier versions of this paper. Furthermore I wish to thank the anonymous referees provided by Synthese for their helpful comments.

Pragmatics & Cognition, 2015

Many types of everyday and specialized reasoning depend on diagrams: we use maps to find our way, we draw graphs and sketches to communicate concepts and prove geometrical theorems, and we manipulate diagrams to explore new creative solutions to problems. The active involvement and manipulation of representational artifacts for purposes of thinking and communicating is discussed in relation to C.S. Peirce’s notion ofdiagrammatical reasoning. We propose to extend Peirce’s original ideas and sketch a conceptual framework that delineates different kinds of diagram manipulation: Sometimes diagrams are manipulated in order to profile known information in an optimal fashion. At other times diagrams are explored in order to gain new insights, solve problems or discover hidden meaning potentials. The latter cases often entail manipulations that either generate additional information or extract information by means of abstraction. Ideas are substantiated by reference to ethnographic, experim...

Signata

The present paper is part of a larger research on the role of diagrams in mathematical "ampliative" reasoning. Leaning on Charles Sanders Peirce's work we will here focus on the semiotic aspects of the question and refer to other papers for more details on the historical and epistemological issues.

Vision Fulfilled: The Victory of the Pictorial Turn, 2019

Both ‘formal’ and ‘practical’ accounts are met with in current diagram studies. However, the supporters of these accounts, while agreeing in their opposition to the suspicious view, have also to some extent disapproved each other: the ‘formal’ view is criticised for its lack of naturalness while the ‘practical’ view is suspected of not accounting for mathematical rigor. In this article, I consider these objections. I argue that the first objection rests on the needless assumption that diagrammatic reasoning is necessarily reasoning with a (single) diagram. The second objection is answered by adopting epistemological strategies, including the much disputed ‘derivation-indicator’ principle.

Semiotica, 2000

In the first part of this paper, I delineate Peirce's general concept of diagrammatic reasoning from other usages of the term that focus either on diagrammatic systems as developed in logic and AI or on reasoning with mental models. The main function of Peirce's form of diagrammatic reasoning is to facilitate individual or social thinking processes in situations that are too complex to be coped with exclusively by internal cognitive means. I provide a diagrammatic definition of diagrammatic reasoning that emphasizes the construction of, and experimentation with, external representations based on the rules and conventions of a chosen representation system. The second part starts with a summary of empirical research regarding cognitive effects of working with diagrams and a critique of approaches that use 'mental models' to explain those effects. The main focus of this section is, however, to elaborate the idea that diagrammatic reasoning should be conceptualized as a case of 'distributed cognition.' Using the mathematics lesson described by Plato in his Meno, I analyze those cognitive conditions of diagrammatic reasoning that are relevant in this case.

Psychological Science, 1993

We report an experimental study on the effects of diagrams on deductive reasoning with double disjunctions, for example: Raphael is in Tacoma or Julia is in Atlanta, or both. Julia is in Atlanta or Paul is in Philadelphia, or both. What follows? We confirmed that subjects find it difficult to deduce a valid conclusion, such as Julia is in Atlanta, or both Raphael is in Tacoma and Paul is in Philadelphia. In a preliminary study, the format of the premises was either verbal or diagrammatic, and the diagrams used icons to distinguish between inclusive and exclusive disjunctions. The diagrams had no effect on performance. In the main experiment, the diagrams made the alternative possibilities more explicit. The subjects responded faster (about 35 s) and drew many more valid conclusions (nearly 30%) from the diagrams than from the verbal premises. These results corroborate the theory of mental models and have implications for the role of diagrams in reasoning.

The role of diagrams in mathematics is something of a mystery. On the one hand, most mathematicians deny that diagrams have any formal sta-tus, but on the other hand, diagrams are ubiquitous in mathematics texts. We hypothesize that diagrams are often used to carry meta-information about a proof; for example the proof strategy or constructions, that are to be used in carrying out the proof. To test this hypothesis, we have designed and built an automated reasoning system which can use information expressed in a diagram to guide its search for a proof. Using this system, called &/grover, we have proved some quite diicult theorems for automated reasoning systems, theorems which our system cannot prove without access to the diagram. In this paper we describe the proof of the Schrr oder-Bernstein Theorem which is diicult for contemporary automated reasoning systems. This proof relies on the use of a diagram to convey crucial information to the system in a natural way.

The use of diagrams as external aids to facilitate cognitive abilities is not new. This paper looks into cognitive studies for insight into when, why and how diagrams are effective in problem solving. A case study examines the use of diagrammatic representations as thinking tools and tools for communicating information. The purpose is to examine whether diagramming could be used as a design method, as part of the designer's creative process.

Studies in Applied Philosophy, Epistemology and Rational Ethics, 2013

In this paper I analyze the cognitive function of symbols, figures and diagrams. The analysis shows that although all three representational forms serve to externalize mental content, they do so in radically different ways, and consequently they have qualitatively different functions in mathematical cognition. Symbols represent by convention and allow mental computations to be replaced by epistemic actions. Figures and diagrams both serve as material anchors for conceptual structures. However, figures do so by having a direct likeness to the objects they represent, whereas diagrams have a metaphorical likeness. Thus, I claim that diagrams can be seen as material anchors for conceptual mappings. This classification of diagrams is of theoretical importance as it sheds light on the functional role played by conceptual mappings in the production of new mathematical knowledge.