David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Axiomathes 22 (4):433-456 (2012)
The paper undertakes three interdisciplinary tasks. The first one consists in constructing a formal model of the basic arithmetic competence, that is, the competence sufficient for solving simple arithmetic story-tasks which do not require any mathematical mastery knowledge about laws, definitions and theorems. The second task is to present a generalized arithmetic theory, called the arithmetic of indexed numbers (INA). All models of the development of counting abilities presuppose the common assumption that our simple, folk arithmetic encoded linguistically in the mind is based on the linear number representation. This classical conception is rejected and a competitive hypothesis is formulated according to which the basic mature representational system of cognitive arithmetic is a structure composed of many numerical axes which possess a common constituent, namely, the numeral zero. Arithmetic of indexed numbers is just a formal tool for modelling the basic mature arithmetic competence. The third task is to develop a standpoint called temporal pluralism, which is motivated by neo-Kantian philosophy of arithmetic
|Keywords||Cognitive arithmetic Number line Indexed natural numbers Number-axes|
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References found in this work BETA
Mark H. Ashcraft (1992). Cognitive Arithmetic: A Review of Data and Theory. [REVIEW] Cognition 44 (1-2):75-106.
Susan Carey (2001). Cognitive Foundations of Arithmetic: Evolution and Ontogenisis. Mind and Language 16 (1):37–55.
Helen De Cruz (2008). An Extended Mind Perspective on Natural Number Representation. Philosophical Psychology 21 (4):475 – 490.
Lieven Decock (2008). The Conceptual Basis of Numerical Abilities: One-to-One Correspondence Versus the Successor Relation. Philosophical Psychology 21 (4):459 – 473.
Stanislas Dehaene (2001). Précis of the Number Sense. Mind and Language 16 (1):16–36.
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