## Abstract

This chapter examines the innate basis of our concepts of the positive integers. In practice, real valued variables are never exactly equal; nor is it easy to specify an algorithm for establishing exact equality between two random Gaussian variables. Furthermore, because number concepts must support arithmetic inference, a necessary part of the psychological foundations is the integer concept ONE. ONE is required because it is the multiplicative identity element for which no other value, approximate or exact, can be substituted. Moreover, ONE is required by the successor function, which generates all the other positive integers. It is argued that an essential constraint on any proposal for discrete (integer-valued rather than real-valued) mental symbols is computational compatibility with the real- (or rational-) valued mental magnitudes that represent continuous quantity. These constraints rule out most current proposals that postulate systems of discrete numerons or other symbols representing only very small numbers. Alternative proposals are considered.

Original language | English (US) |
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Title of host publication | Foundations and the Future |

Publisher | Oxford University Press |

Volume | 3 |

ISBN (Electronic) | 9780199868117 |

ISBN (Print) | 9780195332834 |

DOIs | |

State | Published - Jan 1 2008 |

## All Science Journal Classification (ASJC) codes

- Arts and Humanities(all)

## Keywords

- Exact equality
- Natural numbers
- Number words
- One
- Positive integers