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Switch to: Citations
References in:
Semiotic Scaffolding in Mathematics
Biosemiotics 8 (2):325-340 (2015)
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Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be less than the (...) |
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In this article we propose a theoretical framework of distributed representations and a methodology of representational analysis for the study of distributed cognitive tasks—tasks that require the processing of information distributed across the internal mind and the external environment. The basic principle of distributed representations Is that the representational system of a distributed cognitive task is a set of internal and external representations, which together represent the abstract structure of the task. The basic strategy of representational analysis is to decompose (...) |
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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 (...) |
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The production of mathematical formalism on state of the art computers is quite different than by pen and paper. In this paper, I examine the question of how different media influence mathematical writing. The examination is based on an investigation of professional mathematicians' use of various media for their writing. A model for describing mathematical writing through turn-takings is proposed. The model is applied to the ways mathematicians use computers for writing, and especially it is used to understand how interaction (...) |
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We present data and argument to show that in Tetris - a real-time interactive video game - certain cognitive and perceptual problems are more quickly, easily, and reliably solved by performing actions in the world rather than by performing computational actions in the head alone. We have found that some translations and rotations are best understood as using the world to improve cognition. These actions are not used to implement a plan, or to implement a reaction; they are used to (...) |
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Reexamines the Western philosophical tradition, looking at the basic concepts of the mind, time, causation, morality, and the self. |
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Of course, words aren’t magic. Neither are sextants, compasses, maps, slide rules and all the other paraphenelia which have accreted around the basic biological brains of homo sapiens. In the case of these other tools and props, however, it is transparently clear that they function so as to either carry out or to facilitate computational operations important to various human projects. The slide rule transforms complex mathematical problems (ones that would baffle or tax the unaided subject) into simple tasks of (...) |
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