Why do people create extra representations to help them make sense of situations, diagrams, illustrations, instructions and problems? The obvious explanation— external representations save internal memory and com- putation—is only part of the story. I discuss seven ways external representations enhance cognitive power: they change the cost structure of the inferential landscape; they provide a structure that can serve as a shareable object of thought; they create persistent referents; they facilitate re- representation; they are often a more natural representation of (...) structure than mental representations; they facilitate the computation of more explicit encoding of information; they enable the construction of arbitrarily complex structure; and they lower the cost of controlling thought—they help coordinate thought. Jointly, these functions allow people to think more powerfully with external representations than without. They allow us to think the previously unthinkable. (shrink)

Viewed this way, the texts yield striking examples of language and notation that are irreducibly ambiguous and productive because they are ambiguous.

What is the nature of number systems and arithmetic that we use in science for quantification, analysis, and modeling? I argue that number concepts and arithmetic are neither hardwired in the brain, nor do they exist out there in the universe. Innate subitizing and early cognitive preconditions for number— which we share with many other species—cannot provide the foundations for the precision, richness, and range of number concepts and simple arithmetic, let alone that of more complex mathematical concepts. Numbers and (...) arithmetic, and mathematics in general, have unique features—precision, objectivity, rigor, generalizability, stability, symbolizability, and applicability to the real world—that must be accounted for. They are sophisticated concepts that developed culturally only in recent human history. I suggest that numbers and arithmetic are realized through precise combinations of non-mathematical everyday cognitive mechanisms that make human imagination and abstraction possible. One such mechanism, conceptual metaphor, is a neurally instantiated inference-preserving cross-domain mapping that allows the conceptualization of abstract entities in terms of grounded bodily experience. I analyze how the inferential organization of the properties and “laws” of arithmetic emerge metaphorically from everyday meaningful actions. Numbers and arithmetic are thus—outside of natural selection—the product of the biologically constrained interaction of individuals with the appropriate cultural and historical phenotypic variation supported by language, writing systems, and education. (shrink)

Philosophy of mathematics is moving in a new direction: away from a foundationalism in terms of formal logic and traditional ontology, and towards a broader range of approaches that are united by a focus on mathematical practice. The scientific research network PhiMSAMP (Philosophy of Mathematics: Sociological Aspects and Mathematical Practice) consisted of researchers from a variety of backgrounds and fields, brought together by their common interest in the shift of philosophy of mathematics towards mathematical practice. Hosted by the Rheinische Friedrich- (...) Wilhelms-Universitat Bonn and funded by the Deutsche Forschungsgemeinschaft (DFG) from 2006-2010, the network organized and contributed to a number of workshops and conferences on the topic of mathematical practice. The refereed contributions in this volume represent the research results of the network and consists of contributions of the network members as well as selected paper versions of presentations at the network's mid-term conference, "Is mathematics special?" (PhiMSAMP-3) held in Vienna 2008. (shrink)

Die Grundlagen der Arithmetik. Eine Ionisch mathematische UoterciicboDn über den Begriff der Zahl Dr. 0. Frege, ao Profeuor an der Univer»ität Jena. -. ...

Until recently, cognitive science focused on such mental functions as problem solving, grammar, and pattern-the functions in which the human mind most closely resembles a computer. But humans are more than computers: we invent new meanings, imagine wildly, and even have ideas that have never existed before. Today the cutting edge of cognitive science addresses precisely these mysterious, creative aspects of the mind.The Way We Think is a landmark analysis of the imaginative nature of the mind. Conceptual blending is already (...) widely known in research laboratories throughout the world; this book, written to be accessible to both lay readers and interested scientists, is its definitive statement. Gilles Fauconnier and Mark Turner show that conceptual blending is the root of the cognitively modern human mind, and that conceptual blends themselves are continually combined and reblended to create the rich mental fabric in which we live.The Way We Think shows how this blending operates; how it is affected by (and gives rise to) language, identity, culture, and invention; and how we imagine what could be and what might have been. The result is a bold and exciting new view of how the mind works. (shrink)

"There are books—few and far between—which carefully, delightfully, and genuinely turn your head inside out. This is one of them. It ranges over some central issues in Western philosophy and begins the long overdue job of giving us a radically new account of meaning, rationality, and objectivity."—Yaakov Garb, _San Francisco Chronicle_.

Cognitive research on metaphoric concepts of time has focused on differences between moving Ego and moving time models, but even more basic is the contrast between Ego‐ and temporal‐reference‐point models. Dynamic models appear to be quasi‐universal cross‐culturally, as does the generalization that in Ego‐reference‐point models, FUTURE IS IN FRONT OF EGO and PAST IS IN BACK OF EGO. The Aymara language instead has a major static model of time wherein FUTURE IS BEHIND EGO and PAST IS IN FRONT OF EGO; (...) linguistic and gestural data give strong confirmation of this unusual culture‐specific cognitive pattern. Gestural data provide crucial information unavailable to purely linguistic analysis, suggesting that when investigating conceptual systems both forms of expression should be analyzed complementarily. Important issues in embodied cognition are raised: how fully shared are bodily grounded motivations for universal cognitive patterns, what makes a rare pattern emerge, and what are the cultural entailments of such patterns? (shrink)

This paper discusses the connection between the actual history of mathematics and Lakatos's philosophy of mathematics, in three parts. The first points to studies by Lakatos and others which support his conception of mathematics and its history. In the second I suggest that the apparent poverty of Lakatosian examples may be due to the way in which the history of mathematics is usually written. The third part argues that Lakatos is right to hold philosophy accountable to history, even if Lakatos's (...) own view of mathematics fails that test. (shrink)

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.

Experimental studies indicate that nonhuman animals and infants represent numerosities above three or four approximately and that their mental number line is logarithmic rather than linear. In contrast, human children from most cultures gradually acquire the capacity to denote exact cardinal values. To explain this difference, I take an extended mind perspective, arguing that the distinctly human ability to use external representations as a complement for internal cognitive operations enables us to represent natural numbers. Reviewing neuroscientific, developmental, and anthropological evidence, (...) I argue that the use of external media that represent natural numbers (like number words, body parts, tokens or numerals) influences the functional architecture of the brain, which suggests a two-way traffic between the brain and cultural public representations. (shrink)

"--David Ruelle, author of "Chance and Chaos" "This is an important book, one that should cause an epoch-making change in the way we think about mathematics.

We usually consider literary thinking to be peripheral and dispensable, an activity for specialists: poets, prophets, lunatics, and babysitters. Certainly we do not think it is the basis of the mind. We think of stories and parables from Aesop's Fables or The Thousand and One Nights, for example, as exotic tales set in strange lands, with spectacular images, talking animals, and fantastic plots--wonderful entertainments, often insightful, but well removed from logic and science, and entirely foreign to the world of everyday (...) thought. But Mark Turner argues that this common wisdom is wrong. The literary mind--the mind of stories and parables--is not peripheral but basic to thought. Story is the central principle of our experience and knowledge. Parable--the projection of story to give meaning to new encounters--is the indispensable tool of everyday reason. Literary thought makes everyday thought possible. This book makes the revolutionary claim that the basic issue for cognitive science is the nature of literary thinking. In The Literary Mind, Turner ranges from the tools of modern linguistics, to the recent work of neuroscientists such as Antonio Damasio and Gerald Edelman, to literary masterpieces by Homer, Dante, Shakespeare, and Proust, as he explains how story and projection--and their powerful combination in parable--are fundamental to everyday thought. In simple and traditional English, he reveals how we use parable to understand space and time, to grasp what it means to be located in space and time, and to conceive of ourselves, other selves, other lives, and other viewpoints. He explains the role of parable in reasoning, in categorizing, and in solving problems. He develops a powerful model of conceptual construction and, in a far-reaching final chapter, extends it to a new conception of the origin of language that contradicts proposals by such thinkers as Noam Chomsky and Steven Pinker. Turner argues that story, projection, and parable precede grammar, that language follows from these mental capacities as a consequence. Language, he concludes, is the child of the literary mind. Offering major revisions to our understanding of thought, conceptual activity, and the origin and nature of language, The Literary Mind presents a unified theory of central problems in cognitive science, linguistics, neuroscience, psychology, and philosophy. It gives new and unexpected answers to classic questions about knowledge, creativity, understanding, reason, and invention. (shrink)

In a recent article, Azzouni has argued in favor of a version of formalism according to which ordinary mathematical proofs indicate mechanically checkable derivations. This is taken to account for the quasi-universal agreement among mathematicians on the validity of their proofs. Here, the author subjects these claims to a critical examination, recalls the technical details about formalization and mechanical checking of proofs, and illustrates the main argument with aanalysis of examples. In the author's view, much of mathematical reasoning presents genuine (...) meaning-dependent mathematical characteristics that cannot be captured by formal calculi. ‘…there is a conflict between mathematical practice and the formalist doctrine.’ [Kreisel, 1969, p. 39]. (shrink)

Our visual experience seems to suggest that no continuous curve can cover every point of the unit square, yet in the late 19th century Giuseppe Peano proved that such a curve exists. Examples like this, particularly in analysis received much attention in the 19th century. They helped to instigate what Hans Hahn called a ‘crisis of intuition’, wherein visual reasoning in mathematics came to be thought to be epistemically problematic. Hahn described this ‘crisis’ as follows : " Mathematicians had for (...) a long time made use of supposedly geometric evidence as a means of proof in much too naive and much too uncritical a way, till the unclarities and mistakes that arose as a result forced a turnabout. Geometrical intuition was now declared to be inadmissible as a means of proof … "Avoiding geometrical evidence, Hahn continued, mathematicians aware of this crisis pursued what he called ‘logicization’, ‘when the discipline requires nothing but purely logical fundamental concepts and propositions for its development’. On this view, an epistemically ideal mathematics would minimize, or avoid altogether, appeals …. (shrink)

Previous research has shown that na_ve participants display a high level of agreement when asked to choose or drawschematic representations, or image schemas, of concrete and abstract verbs [Proceedings of the 23rd Annual Meeting of the Cognitive Science Society, 2001, Erlbaum, Mawhah, NJ, p. 873]. For example, participants tended to ascribe a horizontal image schema to push, and a vertical image schema to respect. This consistency in offline data is preliminary evidence that language invokes spatial forms of representation. It also (...) provided norms that were used in the present research to investigate the activation of spatial image schemas during online language comprehension. We predicted that if comprehending a verb activates a spatial representation that is extended along a particular horizontal or vertical axis, it will affect other forms of spatial processing along that axis. Participants listened to short sentences while engaged in a visual discrimination task (Experiment 1) and a picture memory task (Experiment 2). In both cases, reaction times showed an interaction between the horizontal/vertical nature of the verb's image schema, and the horizontal/vertical position of the visual stimuli. We argue that such spatial effects of verb comprehension provide evidence for the perceptual–motor character of linguistic representations. (shrink)