Mathematics: The Loss of Certainty.
About: This article is published in American Mathematical Monthly.The article was published on 1982-11-01. It has received 453 citations till now. The article focuses on the topics: Certainty.
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TL;DR: Conceptual integration—“blending”—is a general cognitive operation on a par with analogy, recursion, mental modeling, conceptual categorization, and framing that yields products that frequently become entrenched in conceptual structure and grammar.
Abstract: This is an expanded version of Gilles Fauconnier and Mark Turner. 1998. "Conceptual Integration Networks." Cognitive Science 22:2 (April-June 1998), 133-187. Conceptual integration - "blending" - is a general cognitive operation on a par with analogy, recursion, mental modeling, conceptual categorization, and framing. It serves a variety of cognitive purposes. It is dynamic, supple, and active in the moment of thinking. It yields products that frequently become entrenched in conceptual structure and grammar, and it often performs new work on its previously entrenched products as inputs. Blending is easy to detect in spectacular cases but it is for the most part a routine, workaday process that escapes detection except on technical analysis. It is not reserved for special purposes, and is not costly.In blending, structure from input mental spaces is projected to a separate, blended mental space. The projection is selective. Through completion and elaboration, the blend develops structure not provided by the inputs. Inferences, arguments, and ideas developed in the blend can have effect in cognition, leading us to modify the initial inputs and to change our view of the corresponding situations.Blending operates according to a set of uniform structural and dynamic principles. It additionally observes a set of optimality principles.
1,151 citations
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21 Jan 2008TL;DR: In this paper, the authors present a discourse on thinking about (mathematical) thinking and its relation to discourse in the context of mathematical discourse, and discuss how we mathematize and what we do about it.
Abstract: Introduction Part I. Discourse on Thinking: 1. Puzzling about (mathematical) thinking 2. Objectification 3. Commognition: thinking as communicating 4. Thinking in language Part II. Mathematics as Discourse: 5. Mathematics as a form of communication 6. Objects of mathematical discourse: what mathematizing is all about 7. Routines: how we mathematize 8. Explorations, deeds, and rituals: what we mathematize for 9. Looking back and ahead: solving old quandaries and facing new ones.
885 citations
University of Cambridge1, University of Sheffield2, University of Oxford3, University of California, Berkeley4, University of Leeds5, Technion – Israel Institute of Technology6, Imperial College London7, Queen's University Belfast8, Swansea University9, University of Exeter10, University of Sussex11, University of Aberdeen12, Bangor University13, University of Warwick14, Australian National University15, Environmental Change Institute16, University of Reading17, University of Edinburgh18, Microsoft19, Rockefeller University20, University of Zurich21, Swedish University of Agricultural Sciences22, Helmholtz Centre for Environmental Research - UFZ23
TL;DR: The 100th anniversary of the British Ecological Society in 2013 is an opportune moment to reflect on the current status of ecology as a science and look forward to high-light priorities for future work.
Abstract: Summary 1. Fundamental ecological research is both intrinsically interesting and provides the basic knowledge required to answer applied questions of importance to the management of the natural world. The 100th anniversary of the British Ecological Society in 2013 is an opportune moment to reflect on the current status of ecology as a science and look forward to high-light priorities for future work.
652 citations
TL;DR: The Mundurucu, an Amazonian indigene group with a reduced numerical lexicon and little or no formal education, mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic numbers and logarithsmic mapping when numbers were presented nonsymbolically under conditions that discouraged counting.
Abstract: The mapping of numbers onto space is fundamental to measurement and to mathematics. Is this mapping a cultural invention or a universal intuition shared by all humans regardless of culture and education? We probed number-space mappings in the Mundurucu, an Amazonian indigene group with a reduced numerical lexicon and little or no formal education. At all ages, the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic numbers and logarithmic mapping when numbers were presented nonsymbolically under conditions that discouraged counting. This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic. The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education.
532 citations