## Abstract

Algebra has emergent properties that are neither found in the cultural context in which mathematicians work, nor in the evolved cognitive abilities for mathematical thought that enable it. In this paper, I argue that an externalization of mathematical operations in a consistent symbolic notation system is a prerequisite for these emergent properties. In particular, externalism allows mathematicians to perform operations that would be impossible in the mind alone. By comparing

the development of algebra in three distinct historical cultural settings - China, the medieval Islamic world and early modern Europe - I demonstrate that such an active externalism requires specific cultural conditions, including a metaphysical view of the world compatible with science, a notation system that enables the symbolic notation of operations, and the ontological viewpoint that mathematics is a human endeavour. I discuss how extending mathematical operations from the brain into the world gives algebra a degree of autonomy that is impossible to achieve were it performed in the mind alone.

the development of algebra in three distinct historical cultural settings - China, the medieval Islamic world and early modern Europe - I demonstrate that such an active externalism requires specific cultural conditions, including a metaphysical view of the world compatible with science, a notation system that enables the symbolic notation of operations, and the ontological viewpoint that mathematics is a human endeavour. I discuss how extending mathematical operations from the brain into the world gives algebra a degree of autonomy that is impossible to achieve were it performed in the mind alone.

Original language | English |
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Title of host publication | Worldviews, science and us. Philosophy and complexity. |

Editors | C. Gershenson, D. Aerts, B. Edmonds |

Publisher | World Scientific, New Jersey |

Pages | 338-351 |

Number of pages | 14 |

ISBN (Print) | 978-981-270-548-8 |

Publication status | Published - 1 Mar 2007 |

### Publication series

Name | Worldviews, science and us. Philosophy and complexity. |
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### Bibliographical note

C. Gershenson, D. Aerts, B. Edmonds## Keywords

- complexity
- cultural evolution