The now-classic Metaphors We Live By changed our understanding of metaphor and its role in language and the mind. Metaphor, the authors explain, is a fundamental mechanism of mind, one that allows us to use what we know about our physical and social experience to provide understanding of countless other subjects. Because such metaphors structure our most basic understandings of our experience, they are "metaphors we live by"--metaphors that can shape our perceptions and actions without our ever noticing them. In (...) this updated edition of Lakoff and Johnson's influential book, the authors supply an afterword surveying how their theory of metaphor has developed within the cognitive sciences to become central to the contemporary understanding of how we think and how we express our thoughts in language. (shrink)

In treating cognition as problem solving, Andy Clark suggests, we may often abstract too far from the very body and world in which our brains evolved to guide...

Hutchins examines a set of phenomena that have fallen between the established disciplines of psychology and anthropology, bringing to light a new set of relationships between culture and cognition.

Only human beings have a rich conceptual repertoire with concepts like tort, entropy, Abelian group, mannerism, icon and deconstruction. How have humans constructed these concepts? And once they have been constructed by adults, how do children acquire them? While primarily focusing on the second question, in The Origin of Concepts , Susan Carey shows that the answers to both overlap substantially. Carey begins by characterizing the innate starting point for conceptual development, namely systems of core cognition. Representations of core cognition (...) are the output of dedicated input analyzers, as with perceptual representations, but these core representations differ from perceptual representations in having more abstract contents and richer functional roles. Carey argues that the key to understanding cognitive development lies in recognizing conceptual discontinuities in which new representational systems emerge that have more expressive power than core cognition and are also incommensurate with core cognition and other earlier representational systems. Finally, Carey fleshes out Quinian bootstrapping, a learning mechanism that has been repeatedly sketched in the literature on the history and philosophy of science. She demonstrates that Quinian bootstrapping is a major mechanism in the construction of new representational resources over the course of childrens cognitive development. Carey shows how developmental cognitive science resolves aspects of long-standing philosophical debates about the existence, nature, content, and format of innate knowledge. She also shows that understanding the processes of conceptual development in children illuminates the historical process by which concepts are constructed, and transforms the way we think about philosophical problems about the nature of concepts and the relations between language and thought. (shrink)

This book argues against the view that mathematical knowledge is a priori,contending that mathematics is an empirical science and develops historically,just as ...

How do novel scientific concepts arise? In Creating Scientific Concepts, Nancy Nersessian seeks to answer this central but virtually unasked question in the problem of conceptual change. She argues that the popular image of novel concepts and profound insight bursting forth in a blinding flash of inspiration is mistaken. Instead, novel concepts are shown to arise out of the interplay of three factors: an attempt to solve specific problems; the use of conceptual, analytical, and material resources provided by the cognitive-social-cultural (...) context of the problem; and dynamic processes of reasoning that extend ordinary cognition. Focusing on the third factor, Nersessian draws on cognitive science research and historical accounts of scientific practices to show how scientific and ordinary cognition lie on a continuum, and how problem-solving practices in one illuminate practices in the other. (shrink)

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the classical picture of (...) mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations. (shrink)

The fundamental texts of the great classical period in modern logic, some of them never before available in English translation, are here gathered together for ...

Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set (...) theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science. (shrink)

FIRST DAY INTERLOCUTORS: SALVIATI, SA- GREDO AND SIMPLICIO ALV. The constant activity which you Venetians display in your famous arsenal suggests to the ...

This article discusses the role of diagrams in mathematical reasoning in the light of a case study in analysis. In the example presented certain combinatorial expressions were first found by using diagrams. In the published proofs the pictures were replaced by reasoning about permutation groups. This article argues that, even though the diagrams are not present in the published papers, they still play a role in the formulation of the proofs. It is shown that they play a role in concept (...) formation as well as representations of proofs. In addition we note that 'visualization' is used in two different ways. In the first sense 'visualization' denotes our inner mental pictures, which enable us to see that a certain fact holds, whereas in the other sense 'visualization' denotes a diagram or representation of something. (shrink)

Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

To analyze the task of mental arithmetic with external representations in different number systems we model algorithms for addition and multiplication with Arabic and Roman numerals. This demonstrates that Roman numerals are not only informationally equivalent to Arabic ones but also computationally similar—a claim that is widely disputed. An analysis of our models' elementary processing steps reveals intricate tradeoffs between problem representation, algorithm, and interactive resources. Our simulations allow for a more nuanced view of the received wisdom on Roman numerals. (...) While symbolic computation with Roman numerals requires fewer internal resources than with Arabic ones, the large number of needed symbols inflates the number of external processing steps. (shrink)

It is explained that, in the sense of the sociologist Erving Goffman, mathematics has a front and a back. Four pervasive myths about mathematics are stated. Acceptance of these myths is related to whether one is located in the front or the back.

The canonical history of mathematics suggests that the late 19th-century “arithmetization” of calculus marked a shift away from spatial-dynamic intuitions, grounding concepts in static, rigorous definitions. Instead, we argue that mathematicians, both historically and currently, rely on dynamic conceptualizations of mathematical concepts like continuity, limits, and functions. In this article, we present two studies of the role of dynamic conceptual systems in expert proof. The first is an analysis of co-speech gesture produced by mathematics graduate students while proving a theorem, (...) which reveals a reliance on dynamic conceptual resources. The second is a cognitive-historical case study of an incident in 19th-century mathematics that suggests a functional role for such dynamism in the reasoning of the renowned mathematician Augustin Cauchy. Taken together, these two studies indicate that essential concepts in calculus that have been defined entirely in abstract, static terms are nevertheless conceptualized dynamically, in both contemporary and historical practice. (shrink)

The well known Wiener-Kuratowski explicit definition of the ordered pair, which sets ⟨x, y⟩ = {{x}, {x, y}}, works well in many set theories but fails for those with classes which cannot be members of singletons. With the aid of the Axiom of Foundation, we propose a recursive definition of ordered pair which addresses this shortcoming and also naturally generalizes to ordered tuples of greater lenght. There are many advantages to the new definition, for it allows for uniform definitions working (...) equally well in a wide range of models for set theories. In ZFC and closely related theories, the of an ordered pair of two infinite sets under the new definition turns out to be equal to the maximum of the ranks of the sets. (shrink)

In this paper I investigate two notions of concepts that have played a dominant role in 20th century philosophy of mathematics. According to the first, concepts are definite and fixed; in contrast, according to the second notion concepts are open and subject to modifications. The motivations behind these two incompatible notions and how they can be used to account for conceptual change are presented and discussed. On the basis of historical developments in mathematics I argue that both notions of concepts (...) capture important aspects of mathematical and scientific reasoning, and, consequently, that a pluralistic approach that allows for representing both of these aspects is most useful for an adequate account of investigative practices. (shrink)