Research on concepts has concentrated on the way people apply concepts online, when presented with a stimulus. Just as important, however, is the use of concepts offline, when planning what to do or thinking about what is the case. There is strong evidence that inferences driven by conceptual thought draw heavily on special-purpose resources: sensory, motoric, affective, and evaluative. At the same time, concepts afford general-purpose recombination and support domain-general reasoning processes—phenomena that have long been the focus of philosophers. There (...) is a growing consensus that a theory of concepts must encompass both kinds of process. This book shows how concepts are able to act as an interface between general-purpose reasoning and special-purpose systems. Concept-driven thinking can take advantage of the complementary costs and benefits of each. The book lays out an empirically-based account of the different ways in which thinking with concepts takes us to new conclusions and underpins planning, decision-making, and action. It also spells out three useful implications of the account. First, it allows us to reconstruct the commonplace idea that thinking draws on the meaning of a concept. Second, it offers an insight into how human cognition avoids the frame problem and the complementary, less discussed, ‘if-then problem’ for nested processing dispositions. Third, it shows that metacognition can apply to concepts and concept-driven thinking in various ways. The framework developed in the book elucidates what it is that makes concept-driven thinking an especially powerful cognitive resource. (shrink)

Since Tolman’s paper in 1948, psychologists and neuroscientists have argued that cartographic representations play an important role in cognition. These empirical findings align with some theoretical works developed by philosophers who promote a pluralist view of representational vehicles, stating that cognitive processes involve representations with different formats. However, the inferential relations between maps and representations with different formats have not been sufficiently explored. Thus, this paper is focused on the inferential relations between cartographic and linguistic representations. To that effect, we (...) appeal to heterogeneous inference with ordinary maps and sentences. In doing so, we aim to build a model to bridge the gap between cartographic and linguistic thought. (shrink)

This paper presents examples of infinite diagrams whose use is more or less essential for understanding and accepting various proofs in higher mathematics. The significance of these is discussed with respect to the thesis that every proof can be formalized, and a “pre” form of this thesis that every proof can be presented in everyday statements-only form.

Abstract In this essay, I will argue that images can play a substantial role in argumentation: exploiting information from the context, they can contribute directly and substantially to the communication of the propositions that play the roles of premises and conclusion. Furthermore, they can achieve this directly, i.e. without the need of verbalization. I will ground this claim by presenting and analyzing some arguments where images are essential to the argumentation process. Content Type Journal Article Pages 1-14 DOI 10.1007/s10503-011-9259-y Authors (...) Axel Arturo Barceló Aspeitia, Instituto de Investigaciones Filosóficas, Universidad Nacional Autónoma de México, Cto. Mario de la Cueva s/n, Cd. Universitaria, Coyoacán, 04500 DF, Mexico Journal Argumentation Online ISSN 1572-8374 Print ISSN 0920-427X. (shrink)

This paper examines the contemporary philosophical and cognitive relevance of Charles Peirce's diagrammatic logic of existential graphs (EGs), the ‘moving pictures of thought’. The first part brings to the fore some hitherto unknown details about the reception of EGs in the early 1900s that took place amidst the emergence of modern conceptions of symbolic logic. In the second part, philosophical aspects of EGs and their contributions to contemporary logical theory are pointed out, including the relationship between iconic logic and images, (...) the problem of the meaning of logical constants, the cognitive economy of iconic logic, the failure of the Frege–Russell thesis, and the failure of the Language of Thought hypothesis. (shrink)

In the last few decades there has been a revival of interest in diagrams in mathematics. But the revival, at least at its origin, has been motivated by adherence to the view that the method of mathematics is the axiomatic method, and specifically by the attempt to fit diagrams into the axiomatic method, translating particular diagrams into statements and inference rules of a formal system. This approach does not deal with diagrams qua diagrams, and is incapable of accounting for the (...) role diagrams play as means of discovery and understanding. Alternatively, this paper purports to show that the view that the method of mathematics is the analytic method is capable of dealing with diagrams qua diagrams, and of accounting for such role. (shrink)

This paper examines Charles Peirce's graphical notation for first-order logic with identity. The notation forms a part of his system of "existential graphs," which Peirce considered to be his best work in logic. In this paper a Tarskian semantics is provided for the graphical system.

The vision of machines autonomously carrying out substantive conjecture generation, theorem discovery, proof discovery, and proof verification in mathematics and the natural sciences has a long history that reaches back before the development of automatic systems designed for such processes. While there has been considerable progress in proof verification in the formal sciences, for instance the Mizar project’ and the four-color theorem, now machine verified, there has been scant such work carried out in the realm of the natural sciences—until recently. (...) The delay in the case of the natural sciences can be attributed to both a lack of formal analysis of the so-called “theories” in such sciences, and the lack of sufficient progress in automated theorem proving. While the lack of analysis is probably due to an inclination toward informality and empiricism on the part of nearly all of the relevant scientists, the lack of progress is to be expected, given the computational hardness of automated theorem proving; after all, theoremhood in even first-order logic is Turing-undecidable. We give in the present short paper a compressed report on our building upon these formal theories using logic-based AI in order to achieve, in relativity, both machine proof discovery and proof verification, for theorems previously established by humans. Our report is intended to serve as a springboard to machine-produced results in the future that have not been obtained by humans. (shrink)

What is perception doing in mathematical reasoning? To address this question, I discuss the role of perception in geometric reasoning. Perception of the shape properties of concrete diagrams provides, I argue, a surrogate consciousness of the shape properties of the abstract geometric objects depicted in the diagrams. Some of what perception is not doing in mathematical reasoning is also discussed. I take issue with both Parsons and Maddy. Parsons claims that we perceive a certain type of abstract object. Maddy claims (...) (at least at one time claimed) that perception provides the basis for intuition of mathematical sets. 1 Mathematical reasoning with diagrams 2 Do we perceive abstract objects? 3 Do we perceive mathematical sets? (shrink)

Hyperproof is one of the first systems to permit and encourage reasoning across heterogeneous media. Its advocates argue that it has merits over and above the obvious pragmatic and cognitive ones. This paper suggests analysing Hyperproof-like systems in terms of languages interpreted over a common conceptual scheme and translation relations between logical expressions in such languages. This analysis shows that, despite initial appearances, Hyperproof has no real theoretical merits apart from its admittedly important pragmatic advantages.

The general aim of this project is to fundamentally re-think the design of teaching materials in view of what is now known about cognitive deficits and about what Howard Gardner has termed ‘multiple intelligences’. The applicant has implemented this strategy in two distinct areas, the first involving the writing of an English language programme for Chinese speakers, the second involving the construction of specialized equipment for teaching elementary logic to blind students. The next phase (for which funding is sought) is (...) to test the effectiveness of the logic device, because in theory – the one to be tested – materials the design of which is informed by the above rationale will provide a richer learning experience for non-impaired users. (shrink)

Glenberg's account falls short in several respects. Besides requiring clearer explication of basic concepts, his account fails to recognize the autonomous nature of perception. His account of what is remembered, and its description, is too static. His strictures against connectionist modeling might be overcome by combining the notions of psychological space and principled learning in an embodied and situated network.

Que tipo de objetos são os diagramas, os quais podem ser manipulados lógica e matematicamente e também servem para obter conhecimento? Neste artigo, proporei uma maneira de responder a esta pergunta. Sustentarei a hipótese, inspirada por Kant e Peirce, de que os diagramas são uma classe de objetos híbridos com um aspecto intelectual e outro sensível. Os lógicos e matemáticos estão interessados e estudam uma certa estrutura que se exemplifica em um diagrama, embora não de maneira perfeita. Devido a seu (...) caráter sensível e a sua acessibilidade, tais diagramas permitem a manipulação e a experimentação, mas unicamente em virtude do tipo de relações por eles implicadas tornam possíveis juízos absolutamente universais. (shrink)