Proof-theory has traditionally been developed based on linguistic (symbolic) representations of logical proofs. Recently, however, logical reasoning based on diagrammatic or graphical representations has been investigated by logicians. Euler diagrams were introduced in the eighteenth century. But it is quite recent (more precisely, in the 1990s) that logicians started to study them from a formal logical viewpoint. We propose a novel approach to the formalization of Euler diagrammatic reasoning, in which diagrams are defined not in terms of regions as in (...) the standard approach, but in terms of topological relations between diagrammatic objects. We formalize the unification rule, which plays a central role in Euler diagrammatic reasoning, in a style of natural deduction. We prove the soundness and completeness theorems with respect to a formal set-theoretical semantics. We also investigate structure of diagrammatic proofs and prove a normal form theorem. (shrink)

The paper addresses the issue of human diagrammatic reasoning in the context of Euclidean geometry. It develops several philosophical categories which are useful for a description and an analysis of our experience while reasoning with diagrams. In particular, it draws the attention to the role of seeing-as; it analyzes its implications for proofs in Euclidean geometry and ventures the hypothesis that geometrical judgments are analytic and a priori, after all.

This book analyses the role of diagrammatic reasoning in Plato’s philosophy: the readers will realize that Plato, describing the stages of human cognitive development using a diagram, poses a logic problem to stimulate the general reasoning abilities of his readers. Following the examination of mental models in this book, the readers will reflect on what inferences can be useful to approach this kind of logic problem. Plato calls for a collaboration between writer and readers. In this book the readers will (...) examine the connection between diagrams and discovery, realizing the important epistemic role of visualization. They will recognize the crucial role that diagrams play in problem solving. The logic problem elaborated by Plato is addressed considering the epistemic function of mental models. These models introduce to an advanced stage of cognitive development, in which reasoning uses in its investigations a higher-level of mathematical complexity, represented by structuralism. (shrink)

In Part III of his 1879 logic Frege proves a theorem in the theory of sequences on the basis of four definitions. He claims in Grundlagen that this proof, despite being strictly deductive, constitutes a real extension of our knowledge, that it is ampliative rather than merely explicative. Frege furthermore connects this idea of ampliative deductive proof to what he thinks of as a fruitful definition, one that draws new lines. My aim is to show that we can make good (...) sense of these claims if we read Frege’s notation diagrammatically, in particular, if we take that notation to have been designed to enable one to exhibit the (inferentially articulated) contents of concepts in a way that allows one to reason deductively on the basis of those contents. (shrink)

This paper is concerned with scientific reasoning in the engineering sciences. Engineering sciences aim at explaining, predicting and describing physical phenomena occurring in technological devices. The focus of this paper is on mathematical description. These mathematical descriptions are important to computer-aided engineering or design programs (CAE and CAD). The first part of this paper explains why a traditional view, according to which scientific laws explain and predict phenomena and processes, is problematic. In the second part, the reasons of these methodological (...) difficulties are analyzed. Ludwig Prandtl’s method of integrating a theoretical and empirical approach is used as an example of good scientific practice. Based on this analysis, a distinction is made between different types of laws that play a role in constructing mathematical descriptions of phenomena. A central assumption in understanding research methodology is that, instead of scientific laws, knowledge of capacities and mechanisms are primary in the engineering sciences. Another important aspect in methodology of the engineering sciences is that in explaining a phenomenon or process spatial regions are distinguished in which distinct physical behaviour occur. The mechanisms in distinct spatial regions are represented in a so-called diagrammatic model. The construction of a mathematical description of the phenomenon or process is based on this diagrammatic model. (shrink)

Just before the Scientific Revolution, there was a "Mathematical Revolution", heavily based on geometrical and machine diagrams. The "faculty of imagination" (now called scientific visualization) was developed to allow 3D understanding of planetary motion, human anatomy and the workings of machines. 1543 saw the publication of the heavily geometrical work of Copernicus and Vesalius, as well as the first Italian translation of Euclid.

A permissivist framework is developed to include images in the reconstruction of the evidential base and of the theoretical content. The paper uses Newton’s optical theory as a case study to discuss mathematical idealizations and depictions of experiments, together with textual correlates of diagrams. Instead of assuming some specific type of theoretical content, focus is on novel traits that are delineable when studying the carriers of a theory. The framework is developed to trace elliptic and ambiguous message design, and utilizes (...) variegated acceptance as an asset. Newton’s resources allowed for various framing modes and reconstructions, entailing various judgements concerning the theoretical content, the evidence base, and Newton’s use of mathematics. Elliptic presentation of the theory’s proof-structure and ambiguities influenced uptake, contributed to the process of opinion-polarization, and the acceptance/rejection of the theory. The study suggests that the analysed carriers of theoretical content have an argumentative function, and one of their uses is to adjust the burden of proof. (shrink)

Between the last decades of the 18th century and the middle of the 19th century, something of paramount importance happened in the history of anthropology. This was the advent of a physical anthropology that was about the classification of ‘human races’ through comparative measurement. A central tool of the new trade was diagrams. Being inherently about relations in and between objects, diagrams became the means of defining human groups and their relations to each other – the last point being disputed (...) between the monogenists and the polygenists. James Cowles Prichard, a proponent of the comparative historical approach, was able to do without images in his pioneering Researches Into the Physical History of Man of 1813, but the third edition, which appeared in five volumes between 1836 and 1847, was richly illustrated with ‘ethnic types’ and skulls, including diagrams. What was happening is a process I engage with in detail for Samuel George Morton, who collected and distributed human skulls as lithographs in Crania americana (1839) and Crania aegyptiaca (1844). Along with the paper skulls travelled detailed instructions of how to look at them through a set of lines and to set their individual parts in relation to each other as well as to those of other types. Drawing on Johann Friedrich Blumenbach and Peter Camper, the Crania thus played a pivotal role in establishing what I call a diagrammatics of race – a diagrammatics that became overtly political with Types of Mankind (1854), which was written in Morton's honour by Josiah Nott and George Gliddon. (shrink)

Our brains make up a series of signs and are engaged in making or manifesting or reacting to a series of signs: through this semiotic activity they are at the same time engaged in “being minds.” An important effect of this semiotic activity of brains is a continuous process of “externalization of the mind” that exhibits a new cognitive perspective on the mechanisms underlying the semiotic emergence of abductive processes of meaning formation. I consider this process of externalization interplay critical (...) in analyzing the relation between meaningful semiotic internal resources and devices and their dynamical interactions with the externalized semiotic materiality suitably stocked in the environment. The last part of the paper will describe some aspects of this interplay between external diagrammatization and internal recapitulation showing the specificity of their co-evolution. (shrink)

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, experimental and historical examples. (shrink)

A diagrammatic logical calculus for the syllogistic reasoning is introduced and discussed. We prove that a syllogism is valid if and only if it is provable in the calculus.

Rocco Gangle addresses the methodological questions raised by a commitment to immanence in terms of how diagrams may be used both as tools and as objects of philosophical investigation. Gangle integrates insights from Spinoza, Pierce and Deleuze in conjunction with the formal operations of category theory.

This paper proposes a diagrammatic reconstruction of Carnap's formal method of quasianalysis in the Aufbau and then explains on that basis why Quine's criticism of Carnap's constitution of physical space fails.

Ernesto Neto's installation at the Panthéon in Paris, Leviathan Toth (2006), brings us into a semiotics of intensities that does not belong to the ‘aesthetic regime’ as described by Jacques Rancière but rather to a Diagrammatic Agency of Contemporary Art. In this case study, the latter is constructed after Deleuze and Guattari – from a politics of the Body without Organs critically and clinically identified to a Body without Image.

Classifications of animals and plants have long been represented by hierarchical lists of taxa, but occasional authors have drawn diagrammatic versions of their classifications in an attempt to better depict the "natural relationships" of their organisms. Ornithologists in 19th-century Britain produced and pioneered many types of classificatory diagrams, and these fall into three groups: (a) the quinarian systems of Vigors and Swainson (1820s and 1830s); (b) the "maps" of Strickland and Wallace (1840s and 1850s); and (c) the evolutionary diagrams of (...) the post-Darwin authors (1860 on). The quinarians distinguished between affinity and analogy and used both in their classifications, whereas Strickland rejected the quinarians' belief in numerical regularity and their use of analogy. Wallace's "maps" are easily given an evolutionary interpretation, and his approach was taken up and modified by later evolutionary anatomists. Sharpe returned to Strickland's methods and merely appended a superficial evolutionary interpretation. Contrary to common belief systematics has a rich conceptual history, and many of the conceptual developments in 19th-century systematics were made by ornithologists. (shrink)

In this paper I offer a theory about the nature of diagrammatic representation in geometry. On my view, diagrammatic representaiton differs from pictorial representation in that neither the resemblance between the diagram and its object nor the experience of such a resemblance plays an essential role. Instead, the diagrammatic representation is arises from the role the components of the diagram play in a diagramatic practice that allows us to draws inferences based on them about the ojbects they represent.

In this paper I offer a theory about the nature of diagrammatic representation in geometry. On my view, diagrammatic representaiton differs from pictorial representation in that neither the resemblance between the diagram and its object nor the experience of such a resemblance plays an essential role. Instead, the diagrammatic representation is arises from the role the components of the diagram play in a diagramatic practice that allows us to draws inferences based on them about the ojbects they represent.

In the graphical representation of ontologies, it is customary to use graph theory as the representational background. We claim here that the standard graph-based approach has a number of limitations. We focus here on a problem in the graph-based representation of ontologies in complex domains such as biomedical, engineering and manufacturing: lack of mereotopological representation. Based on such limitation, we proposed a diagrammatic way to represent an entity’s structure and various forms of mereotopological relationships between the entities.

Theorems in automated theorem proving are usually proved by formal logical proofs. However, there is a subset of problems which humans can prove by the use of geometric operations on diagrams, so called diagrammatic proofs. Insight is often more clearly perceived in these proofs than in the corresponding algebraic proofs; they capture an intuitive notion of truthfulness that humans find easy to see and understand. We are investigating and automating such diagrammatic reasoning about mathematical theorems. Concrete, rather than general diagrams (...) are used to prove particular concrete instances of the universally quantified theorem. The diagrammatic proof is captured by the use of geometric operations on the diagram. These operations are the inference steps of the proof. An abstracted schematic proof of the universally quantified theorem is induced from these proof instances. The constructive -rule provides the mathematical basis for this step from schematic proofs to theoremhood. In this way we avoid the difficulty of treating a general case in a diagram. One method of confirming that the abstraction of the schematic proof from the proof instances is sound is proving the correctness of schematic proofs in the meta-theory of diagrams. These ideas have been implemented in the system, called Diamond, which is presented here. (shrink)

According to the received view, Charles S. Peirce's theory of diagrammatic reasoning is derived from Kant's philosophy of mathematics. For Kant, only mathematics is constructive/synthetic, logic being instead discursive/analytic, while for Peirce, the entire domain of necessary reasoning, comprising mathematics and deductive logic, is diagrammatic, i.e. constructive in the Kantian sense. This shift was stimulated, as Peirce himself acknowledged, by the doctrines contained in Friedrich Albert Lange's Logische Studien (1877). The present paper reconstructs Peirce's reading of Lange's book, and illustrates (...) what, according to Peirce, was right and what was problematic in Lange's account of reasoning. It further seeks to explain how Peirce's theory of deductive reasoning was a combination of Kant's philosophy of mathematics and Lange's philosophy of logic. (shrink)

In recent years, semiotics has become an innovative theoretical framework in mathematics education. The purpose of this article is to show that semiotics can be used to explain learning as a process of experimenting with and communicating about one's own representations of mathematical problems. As a paradigmatic example, we apply a Peircean semiotic framework to answer the question of how students learned the concept of "distribution" in a statistics course by "diagrammatic reasoning" and by developing "hypostatic abstractions," that is by (...) forming new mathematical objects which can be used as means for communication and further reasoning. Peirce's semiotic terminology is used as an alternative for notions such as modeling, symbolizing, and reification. We will show that it is a precise instrument of analysis with regard to the complexity of learning and of communication in mathematics classroom. (shrink)

Peirce’s diagrammatic system of Existential Graphs (EGα)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$EG_{\alpha })$$\end{document} is a logical proof system corresponding to the Propositional Calculus (PL). Most known proofs of soundness and completeness for EGα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$EG_{\alpha }$$\end{document} depend upon a translation of Peirce’s diagrammatic syntax into that of a suitable Frege-style system. In this paper, drawing upon standard results but using the native diagrammatic notational framework of the graphs, we present (...) a purely syntactic proof of soundness, and hence consistency, for EGα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$EG_{\alpha }$$\end{document}, along with two separate completeness proofs that are constructive in the sense that we provide an algorithm in each case to construct an EGα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$EG_{\alpha }$$\end{document} formal proof starting from the empty Sheet of Assertion, given any expression that is in fact a tautology according to the standard semantics of the system. (shrink)

Journal Name: Semiotica Issue: Ahead of print.

If we are to think about diagrams closely, we must register their cognitive significance as they direct work and establish networks of relationships among multiple symbolic fields. Diagrams do not engage a simple horizon of understanding but are part of an integrative process through which structures literally appear in the world. Rather than being hermeneutic in a strict sense, diagrams are heuristic because they are accompanied by an expectation that they participate in a process that turns words and experience into (...) structure. Because diagrams may shift in status, they may be thought of as relays that both create meaning and enable symbolic translation from one mode of representation to another. While a diagram may visually present or reinforce an idea one moment, the next it may provide a means for seeing something never seen before. After commenting on how recent architectural theory has approached diagram, I consider the relationship between diagram and metaphor before turning my attention to the ways that diagrams function within a process that includes design as well as building. Finally, I ask how the use of diagram within architecture points not only to a phenomenology of invention and practice but also to the networks which constitute architecture. (shrink)

Grammatical asymmetries in possessive constructions are overtly coded in about 18% of the world’s languages according to the World Atlas of Language Structures What primarily motivates these grammatical asymmetries is controversial and has been at the crux of the “iconicity vs. frequency” debate This paper contributes to this debate by focusing on the grammatical asymmetries of Paamese possessive constructions, and looking for the primary motivating factor in their multidimensional experiential context. After a careful account of four experiential dimensions of distance, (...) the degrees of experiential distance are shown to systematically correspond to the degrees of formal distance of the possessive constructions used to refer to these experiences. Relevant anthropological and linguistic data concerning Paamese is used to explore whether this correspondence between language and experience is primarily motivated by iconicity or economy. I argue that diagrammatic iconicity is the primary motivating factor for the grammatical asymmetries in Paamese possessive constructions, and that economy can account for some, but not all cases. I also show that economy and iconicity can collaborate in motivating some cases, and thus do not necessarily need to be opposed. (shrink)

Schemes of diagrammatic representation have been so familiarly introduced into logical treatises during the last century or so, that many readers, even of those who have made no professional study of logic, may be supposed to be acquainted with the general nature and object of such devices. Of these schemes one only, viz. that commonly called "Eulerian circles," has met with any general acceptance. A variety of others indeed have been proposed by ingenious and celebrated logicians, several of which would (...) claim notice in a historical treatment of the subject; but they mostly do not seem to me to differ in any essential respect from that of Euler. They rest upon the same leading principle, and are subject all alike to the same restrictions and defects. We must therefore cast about for some new scheme of diagrammatic representation which shall be competent to indicate imperfect knowledge on our part; for this will at once enable us to appeal to it step by step in the process of working out our conclusions. I have never seen any hint at such a scheme, though the want seems so evident that one would suppose that something of the kind must have been proposed before. (shrink)

Some of the most intriguing and important phenomena in modern many-body physics are explainable in terms of self-consistent quantum mechanical field theory. This is the powerful theory developed by Umezawa and co-workers and modified by Benson and Hatch in applications to ferromagnetism. It is usually lengthy and involved mathematically. Thus, it is very helpful and meaningful to see its overall step-by-step progress in simple, diagrammatic flow starting from basic principles, with a ferromagnetic model as an example. As one immediately notes, (...) there are two paths leading to very powerful physical conclusions and implications, and something most interesting is that each path implies the other. Many useful examples of applications of these methods are noted and some future possible applications are cited. (shrink)

This article aims to examine the relationship between image and narrative by means of Peirce’s first trichotomy of qualisign-sinsign-legisign or, for the purposes of the current argument, image-diagram-metaphor. It is argued that narrative, as an extended metaphor, can be examined in three modes: in the image; schematically, in the imagination; and allegorically or in a thought experiment, through hypothetic interpretation. The article outlines two kinds of diagrammatic reasoning emphasized by Peirce: corollarial deduction in which the image is ‘literally seen’ and (...) the reasoning steps are manifest in its conclusion; and theorematic deduction where the conclusion in a diagram is subject to a hypothesis which transforms the image into something new. Demonstrating the breadth of diagrammatic reasoning with reference to the 2018 film, The Shape of Water, the article seeks to explore how allegory and diagram are mutually cooperative, based on three ontological modes: the expressive, the cognitive, and the symbolic. Its primary focus, then, is not so much on the story events of the narrative, as the way that they are visualized and characterized as the fairy story unfolds. It is suggested that the interpreting activity involved in allegory and diagram ties interpretation to metacognition, ultimately (re)recognizing the image in The Shape of Water in an attempt to ascertain the meaning of love. (shrink)

Many types of everyday and specialized reasoning depend on diagrams: we use maps to fnd 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. While the linear and symbolic character of verbal language has long served as the predominant model of human thought, it is remarkable how — through a range of contexts — thinking and communication critically depend on manipulations of external, ofen non-linear, (...) and manipulable iconic-diagrammatic vehicles. In this issue of Pragmatics and Cognition we explore such diagrammatic reasoning, that is, of individual and intersubjectively distributed cognitive processes relying on external representations (i.e. diagrams in a broad sense) as tools for thinking and communicating. (shrink)

Charles S. Peirce’s semiotics uniquely divides signs into: i) symbols, which pick out their objects by arbitrary convention or habit, ii) indices, which pick out their objects by unmediated ‘pointing’, and iii) icons, which pick out their objects by resembling them (as Peirce put it: an icon’s parts are related in the same way that the objects represented by those parts are themselves related). Thus representing structure is one of the icon’s greatest strengths. It is argued that the implications of (...) scaffolding education iconically are profound: for providing learners with a navigable road-map of a subject matter, for enabling them to see further connections of their own in what is taught, and for supporting meaningful active learning. Potential objections that iconic teaching is excessively entertaining and overly susceptible to misleading rhetorical manipulation are addressed. (shrink)

The rise in computing and multimedia technology has spawned an increasing interest in the role of diagrams and sketches, not only for the purpose of conveying information but also for creative thinking and problem-solving. This book attempts to characterise the nature of "a science of diagrams" in a wide-ranging, multidisciplinary study that contains accounts of the most recent research results in computer science and psychology. Key topics include: cognitive aspects, formal aspects, and applications. It is a well-written and indispensable survey (...) for researchers and students in the fields of cognitive science, artificial intelligence, human-computer interaction, and graphics and visualisation. (shrink)

Aristotle’s philosophy of geometry is widely interpreted as a reaction against a Platonic realist conception of mathematics. Here I argue to the contrary that Aristotle is concerned primarily with the methodological question of how universal inferences are warranted by particular geometrical constructions. His answer hinges on the concept of abstraction, an operation of “taking away” certain features of material particulars that makes perspicuous universal relations among magnitudes. On my reading, abstraction is a diagrammatic procedure for Aristotle, and it is through (...) imagined variation of diagrams that one can universalize spatial inferences. In order to shift the discussion of Aristotle’s work on geometry from ontological to methodological questions, I argue in the first section that geometrical properties for Aristotle are necessary accidents of the particulars, canonically diagrams, in which they inhere. I then consider a line of research that shows how diagrams in ancient geometry are not illustrations of the textual proof but are in part constitutive of geometrical inference, an insight I claim can shed light on Aristotle’s philosophical project. Next I substantiate this assertion by giving an alternative interpretation of abstraction according to which it is a diagrammatic procedure that allows a part of a particular diagram to stand in for a universal. In the last section, I consider Aristotle’s account of mathematical cognition, arguing that there is evidence for the view that imagination is necessary both for the construction of figures, and for their presentation as abstractions subject to universal inference. (shrink)

Diagrams figure prominently in human reasoning, especially in science. Cognitive science research has provided important insights into the inferences afforded by diagrams and revealed differences in the reasoning made possible by physically instantiated diagrams and merely imagined ones. In scientific practice, diagrams figures prominently both in the way scientists reason about data and in how they conceptualize explanatory mechanisms. To identify patterns in data, scientists often graph it. While some graph formats, such as line graphs, are used widely, scientists often (...) develop specialized formats designed to reveal specific types of patterns and not infrequently employ multiple formats to present the same data, a practice illustrated with graph formats developed in circadian biology. Cognitive scientists have revealed the spatial reasoning and iterative search processes scientists deploy in understanding graphs. In developing explanations, scientists commonly diagram mechanisms they take to be responsible for a phenomenon, a practice again illustrated with diagrams of circadian mechanisms. Cognitive science research has revealed how reasoners mentally animate such diagrams to understand how a mechanism generates a phenomenon. (shrink)

In this paper, we interpret a 19th century diagram, which is meant to visualise G.W.F. Hegel’s entire method of the `Science of Logic' on the basis of bitwise operations. For the interpretation of the diagram we use a binary numeral system, and discuss whether the anti-Hegelian argument associated with it is valid or not. The reinterpretation is intended to make more precise rules of construction, a stricter binary code and a review of strengths and weaknesses of the critique.

In this study I aim to develop a cognitive evaluation of how semantically-driven and rule-based diagrammatic reasoning were psychologically plausible for partic-ular cases of scientific practice: an actual trackless path reconstruction in bubble chamber experiments, as reported by Galison (1997). I will propose “cognitive imagery projection and manipulation” (CIPM) as the most plausible psychologi-cal (perceptual/attentional/cognitive) mechanism matching the specific explanatory requirements for the case study, outlining the most significant current theories about this mental phenomenon (Shimojima; 2011).

Parallelism has been drawn between modes of representation and problem-sloving processes: Diagrams are more useful for brainstorming while symbolic representation is more welcomed in a formal proof. The paper gets to the root of this clear-cut dualistic picture and argues that the strength of diagrammatic reasoning in the brainstorming process does not have to be abandoned at the stage of proof, but instead should be appreciated and could be preserved in mathematical proofs.

We show that the physical principle, “the adjoint associates to each state a ‘test’ for that state”, fully characterises the Hermitian adjoint for pure quantum theory, therefore providing the adjoint with operational meaning beyond its standard mathematical definition. Moreover, we demonstrate that for general process theories, which all admit a diagrammatic representation, this physical principle induces a diagrammatic reflection operation.