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)

Two different concepts of iconicity compete in Peirce's diagrammatical logic. One is articulated in his general reflections on the role of diagrams in thought, in what could be termed his diagrammatology — the other is articulated in his construction of Existential Graphs as an iconic system for representing logic. One is operational and defines iconicity in terms of which information may be derived from a given diagram or diagram system — the other has stronger demands on iconicity, adding to the (...) operational criterion a demand for as high a degree of similarity as possible and may be termed optimal iconicity. Peirce himself does not clearly distinguish these two iconicity notions, a fact that has caused some confusion. By isolating them, we get a clearer and more refined conceptual apparatus for analyzing iconic signs, from pictures to logic. (shrink)

Starting from the observation that small children can count more objects than numbers—a phenomenon that I am calling the “lifeworld dependency of cognition”—and an analysis of finger calculation, the paper shows how learning can be explained as the development of cognitive systems. Parts of those systems are not only an individual’s different forms of knowledge and cognitive abilities, but also other people, things, and signs. The paper argues that cognitive systems are first of all semiotic systems since they are dependent (...) on signs and representations as mediators. The two main questions discussed here are how the external world constrains and promotes the development of cognitive abilities, and how we can move from cognitive abilities that are necessarily connected with concrete situations to abstract knowledge. (shrink)

This paper presents an analysis of Seneca's 58th letter to Lucilius and Porphyry's Isagoge, which were the origin of the tree diagrams that became popular in philosophy and logic from the early Middle Ages onwards. These diagrams visualise the extent to which a concept can be understood as a category, genus, species or individual and what the method of dihairesis (division) means. The paper explores the dissimilarities between Seneca's and Porphyry's tree structures, scrutinising them through the perspective of modern graph (...) theory. Although traditional trees can be interpreted in a modern way, their terms and definitions defy modern standardisation. These distinctions are not only useful for differentiating between Seneca's and Porphyry's trees, but also for comprehending the evolution of this philosophical tradition. Thus the paper highlights the continued significance of these tree diagrams in analysing concepts and categories in philosophy and structuring objects and information in modern fields such as artificial intelligence, computer science, bioinformatics, graph theory, and many more. (shrink)

This thesis develops and defends a Peircean conception of the task of metaphysics and critically compares it with recent anti-metaphysical forms of pragmatism. Peirce characterises metaphysics in terms of its place within his hierarchical classification of the sciences. According to the classification, metaphysics depends on logic for principles and provides principles to the natural and social sciences. This arrangement of the sciences is defended by appeal to Peirce's account of philosophy as 'cenoscopy'. The dependence of the natural and social sciences (...) on cenoscopy is then argued for on the basis of Peirce's rejection of psychologism and in terms of the necessity of abductive inference. Peirce's position is then compared with recent forms of pragmatism. While it is less naturalistic, Peirce's position is defended on pragmatist grounds. An account of Peirce on truth is then developed. Peirce's account of truth in terms of an ideal limit of inquiry is defended as consistent with recent, more deflationary, approaches. The truth of 'abstract propositions' is a matter of local indefeasibility. These abstract propositions are related to the 'absolute truth', understood as a single non-abstract proposition. The truth of this proposition is then understood in terms of an identity theory. Two conceptions of Peircean metaphysics are presented. Both are 'abductive'. Their task is to explain the possibility of success in inquiry. However, only one proposal accepts the notion of an absolute truth. The 'absolutist' proposal is defended as an interpretation of Peirce and as a contemporary option for pragmatist philosophers. The thesis concludes by comparing recent anti-metaphysical arguments due to Huw Price with the Peircean position. Room for the absolutist proposal is defended by means of an account of recent exchanges between Price and Robert Brandom on dispositional modality. (shrink)

Intersemiotic translation (IT) was defined by Roman Jakobson (The Translation Studies Reader, Routledge, London, p. 114, 2000) as “transmutation of signs”—“an interpretation of verbal signs by means of signs of nonverbal sign systems.” Despite its theoretical relevance, and in spite of the frequency in which it is practiced, the phenomenon remains virtually unexplored in terms of conceptual modeling, especially from a semiotic perspective. Our approach is based on two premises: (i) IT is fundamentally a semiotic operation process (semiosis) and (ii) (...) IT is a deeply iconic-dependent process. We exemplify our approach by means of literature to dance IT and we explore some implications for the development of a general model of IT. (shrink)

Os signos e as classes dos signos estão entre os tópicos mais importantes do sistema filosófico de Charles S. Peirce. As 10, 28, e 66 classes de signos são classificações desenvolvidas especialmente a partir de 1903 e representam um grande refinamento da divisão fundamental de signos – ícone, índice, símbolo. Nossa abordagem aqui define uma estratégia de visualização das classificações dos signos, com especial atenção para as 10 e 66 classes de signos. O livro está dividido em duas partes: (i) (...) bases teóricas, (ii) diagramas para as classes de signos. Na Parte I (bases teóricas), são apresentadas introdutoriamente as noções fundamentais das categorias fenomenológicas, classes de signos, as tricotomias e diagramas, assim como implicações gerais de nossa abordagem, metodologia e aplicações. Na Parte II (diagramas para as classes de signos), são apresentados e discutidos os modelos das classes, propostos por Peirce e seus comentadores, incluindo uma introdução detalhadamente documental de nossos modelos, 10cubes e 3N3, diagramas dinâmicos para as classes de signos. (shrink)

In his Pensées, Pascal introduced the very influential distinction between the subtle intelligence and the geometrical intelligence. In the first part of the present paper Pascal’s distinction is considered by looking at his famous wager argument where Pascal acts as a skeptical philosopher and at the same time as an applied mathematician. This argument employs the esprit de finesse in a way that is of fundamental significance for the epistemology of mathematics. This claim will be backed up in the second (...) part of the paper that explores Charles Sanders Peirce’s conception of diagrammatic reasoning. Peirce’s semiotically inspired epistemology of mathematics brings to the fore the significance of the “old”position of Pascal – one has to face the fundamental problems of application. (shrink)

This review article of Frederik Stjernfelt’s Sheets, Diagrams, and Realism (2022) argues that Peirce’s theory of iconicity with its subdivision into the image-diagram-metaphor triad must not be reduced to diagrammatic iconicity. The foundation of the triadic subdivision of the icon is not in Peirce’s diagrammatic logic but in Peirce’s cenopythagorean categories. A focus is on misinterpretations of Peirce’s concept of thirdness in the firstness of the icon. The paper argues that not only metaphors, but also comparisons, analogies, analogic arguments, and (...) examples belong to the third class of icons. (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)

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)

Spider diagrams are based on Euler and Venn/Peirce diagrams, forming a system which is as expressive as monadic first order logic with equality. Rather than being primarily intended for logicians, spider diagrams were developed at the end of the 1990s in the context of visual modelling and software specification. We examine the original goals of the designers, the ways in which the notation has evolved and its connection with the philosophical origins of the logical diagrams of Euler, Venn and Peirce (...) on which spider diagrams are based. Using Peirce’s concepts and classification of signs, we analyse the ways in which different sign types are exploited in the notation. Our hope is that this analysis may be of interest beyond those readers particularly interested in spider diagrams, and act as a case study in deconstructing a simple visual logic. Along the way, we discuss the need for a deeper semiotic engagement in visual modelling. (shrink)

Uma seção da Gramática Especulativa de C.S.Peirce – Dez classes de signos – recebeu, a partir de 1903, um importante tratamento diagramático. Neste artigo, são apresentados e discutidos dois diagramas desenvolvidos por Peirce para as dez classes, incluindo esboços desses modelos.

I articulate Charles S. Peirce’s philosophy of mathematical education as related to his conception of mathematics, the nature of its method of inquiry, and especially, the reasoning abilities required for mathematical inquiry. The main thesis is that Peirce’s philosophy of mathematical education primarily aims at fostering the development of the students’ semeiotic abilities of imagination, concentration, and generalization required for conducting mathematical inquiry by way of experimentation upon diagrams. This involves an emphasis on the relation between theory and practice and (...) between mathematics and other fields including the arts and sciences. For achieving its goals, the article is divided in three sections. First, I expound Peirce’s philosophical account of mathematical reasoning. Second, I illustrate this account by way of a geometrical example, placing special emphasis on its relation to mathematical education. Finally, I put forth some Peircean philosophical principles for mathematical education. (shrink)

The thesis of the paper is that semiotic processes are intrinsic to computation and computational systems. An explanation of computation that does not take this semiotic dimension into account is incomplete. Semiosis is essential to computation and therefore requires a rigorous definition. To prove this thesis, the author analyzes two concepts of computation: the Turing machine and the mechanistic conception of physical computation. The paper is organized in two parts. The first part (Sects. 2 and 3) develops a re-interpretation of (...) the Turing machine starting from Peirce’s semiotics. The author shows how this reinterpretation allows overcoming the dualism between a purist and a realist version of the Turing machine. The second part of the paper (Sect. 4) shows how a mechanistic explanation of physical computation such as the one developed by Piccinini is incomplete without considering semiotic relations. The paper intends to be a contribution to the philosophical debate on computation. (shrink)

I argue against the tendency in the philosophy of science literature to link abduction to the inference to the best explanation (IBE), and in particular, to claim that Peireean abduction is a conceptual predecessor to IBE. This is not to discount either abduction or IBE. Rather the purpose of this paper is to clarify the relation between Peireean abduction and IBE in accounting for ampliative inference in science. This paper aims at a proper classification—not justification—of types of scientific reasoning. In (...) particular, I claim that Peireean abduction is an in-depth account of the process of generating explanatory hypotheses, while IBE, at least in Peter Lipton's thorough treatment, is a more encompassing account of the processes both of generating and of evaluating scientific hypotheses. There is then a two-fold problem with the claim that abduction is IBE. On the one hand, it conflates abduction and induction, which are two distinct forms of logical inference, with two distinct aims, as shown by Charles S. Peirce; on the other hand it lacks a clear sense of the full scope of IBE as an account of scientific inference. (shrink)

In this article, the role of abductive reasoning within Peirce's diagrammatic reasoning is discussed. Both abduction and diagrammatic reasoning bring in elements of discovery but it is not clear if abduction should be a part of a fully developed diagrammatic system or not for Peirce. This relates to Peirce's way of interpreting abduction in his later writings. Iconicity and perceptual elements as a basis for discoveries are analyzed, both in deductive and abductive reasoning. At the end, the role of modern (...) ideas of distributed cognition applied to the Peircean scheme is shortly delineated. (shrink)

The Emergence of Word-Meaning in Early China makes an innovative contribution to studies of language by historicizing the Chinese notion that words have "meaning" (content independent of instances of use). Rather than presuming that the concept of word-meaning had always existed, Jane Geaney explains how and why it arose in China. To account for why a normative term (yi, "duty, morality, appropriateness") came to be used for "meanings" found in dictionaries, Geaney examines interrelated patterns of word usage threading through and (...) across a wide range of genres. These patterns show that by the first millennium, as textual production exploded—and as radically different writing forms (in Buddhist sutras) were encountered—yi already functioned as an externally accessible "model" for semantic interpretation of texts and sayings. The book has far-reaching implications. Because the idea of word-meaning is fundamental to theorizing, the book illuminates not only semantic ideas and the normativity of language in Early China, but also aspects of early Chinese philosophy and intellectual history. As the internet supplants one form of media (print), thereby reducing knowledge to vast digital databases, so too, this book explains, two thousand years ago a culture that prized oral and visual balance became an "empire of the text.". (shrink)

Different situations — like school and workplace — demand different forms of knowledge. Even more important, in particular for lifelong learning, are forms of knowledge we need for managing movements between those situations. To develop a better understanding of how to ‘navigate’ knowledge boundaries, this paper analyzes, firstly, interviews with scientists interpreting familiar and unfamiliar graphs. Our goal is to identify those forms of knowledge that should receive special attention in education. Secondly, the article elaborates — based on Peirce’s semiotics (...) — an epistemologically reflected semiotic model to describe the role and conditions of knowledge necessary for crossing knowledge boundaries. (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)

This article offers an analysis of the cognitive role of diagrammatic movements in the theater. Based on the recognition of a theatrical work’s inherent ability to provide new insights concerning reality, the article concentrates on the way by which actors’ movements on stage create spatial diagrams that can provide new insights into the spectators’ world. The suggested model of theater’s epistemology results from a combination of Charles S. Peirce’s doctrine of diagrammatic reasoning and David Lewis’s theoretical account of the truth (...) value of counterfactual conditionals. I argue that in several theatrical works – in particular those whose central image is dominated by movements – the relation of what Lewis names “comparative overall similarity” between the fictional and the actual world is based on diagrammatic homology. The cognitive process involved in deciphering them is, hence, based on diagrammatic reasoning. The main emphasis of the analysis is on the previously unnoticed but important cognitive role of observation in the theater: the idea that observation takes an active role in the reasoning process that enables the spectators to form new knowledge about their actual world. Samuel Beckett’s plays Quad and Come and Go serve here as case studies. (shrink)

This thesis is concerned with establishing a bridge between matters of aesthetics and epistemology, by investigating the mechanisms through which artworks allow agents to derive knowledge through the former’s manipulation. It is proposed that, in order to understand the epistemic potential of artworks, we need to approach them as diagrams, in the sense developed by Charles Peirce. The background upon which the arguments are developed are mainly those of American Pragmatism – with a special emphasis on primary literature from Charles (...) Sanders Peirce about semiotics and inquiry, and John Dewey about aesthetics. It is introduced that the manipulation upon artworks can be an example of an inquiry process, due to artworks’ diagrammatic symbol-type-token structure, which embody the logical relations of abduction, deduction and induction. Through Peirce’s definition of inquiry, and Dewey’s definition of an aesthetic experience, it is possible to defend that artworks deliberate create states of doubt and chance as to stimulate and guide inquiry as a way to achieve an aesthetic experience, corroborating to the claim that inquiry processes and aesthetic experience share the same ontological basis: it is in the nature of aesthetic experience to confront agents and contexts with state of chance and doubt, to be further guided through inference into a state of regularity and belief. In order to develop and test the argument, I present the development of Arthur Danto’s influential theory of aesthetics based on Andy Warhol’s Brillo Boxes, by focusing on a review of Danto’s main writings about it. The central point that Danto developed in this theory is that what gives something the status of an artwork are not the immediately perceivable sensorial features of an artifact, but something else. I claim it to be a “diagrammatic structure” that can be potentially manipulated through an inquiry process. I conclude by establishing a possible development of my argument in relation to modelling processes: as diagrams, artworks can be understood as models, opening room for the emergence of new hypothesis and conjectures about the relationship between aesthetics and epistemology. (shrink)