, Volume 22, Issue 4, pp 433456
First online:
Indexed Natural Numbers in Mind: A Formal Model of the Basic Mature Number Competence
 Wojciech KrysztofiakAffiliated withInstitute of Philosophy, University of Szczecin Email author
Abstract
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 storytasks 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 neoKantian philosophy of arithmetic.
Keywords
Cognitive arithmetic Number line Indexed natural numbers Numberaxes Title
 Indexed Natural Numbers in Mind: A Formal Model of the Basic Mature Number Competence
 Open Access
 Available under Open Access This content is freely available online to anyone, anywhere at any time.
 Journal

Axiomathes
Volume 22, Issue 4 , pp 433456
 Cover Date
 201212
 DOI
 10.1007/s1051601191499
 Print ISSN
 11221151
 Online ISSN
 15728390
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Cognitive arithmetic
 Number line
 Indexed natural numbers
 Numberaxes
 Authors

 Wojciech Krysztofiak ^{(1)}
 Author Affiliations

 1. Institute of Philosophy, University of Szczecin, ul. Krakowska 7179, 71004, Szczecin, Poland