### Abstract

Hilbert and Husserl presented axiomatic arithmetic theories in different ways and proposed two different notions of "completeness" for arithmetic, at the turning of the 20th Century (1900- 1901). The former led to the completion axiom, the latter completion of rewriting. We look into the latter in comparison with the former. The key notion to understand the latter is the notion of definite multiplicity or manifold (Mannigfaltigkeit). We show that his notion of multiplicity is understood by means of term rewrite theory in a very coherent manner, and that his notion of "definite" multiplicity is understood as the relational web (or tissue) structure, the core part of which is a "convergent" term rewrite proof structure. We examine how Husserl introduced his term rewrite theory in 1901 in the context of a controversy with Hilbert on the notion of completeness, and in the context of solving the justification problem of the use of imaginaries in mathematics, which was an important issue in the foundations of mathematics in the period.

Original language | English |
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Title of host publication | 24th International Conference on Rewriting Techniques and Applications, RTA 2013 |

Editors | Femke van Raamsdonk |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 4-19 |

Number of pages | 16 |

ISBN (Electronic) | 9783939897538 |

ISBN (Print) | 9783939897538, 9783939897538 |

DOIs | |

Publication status | Published - 2013 Jan 1 |

Event | 24th International Conference on Rewriting Techniques and Applications, RTA 2013 - Eindhoven, Netherlands Duration: 2013 Jun 24 → 2013 Jun 26 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 21 |

ISSN (Print) | 1868-8969 |

### Other

Other | 24th International Conference on Rewriting Techniques and Applications, RTA 2013 |
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Country | Netherlands |

City | Eindhoven |

Period | 13/6/24 → 13/6/26 |

### Keywords

- Hilbert
- History of term rewrite theory
- Husserl
- Knuth- bendix completion
- Proof theory

### ASJC Scopus subject areas

- Software

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## Cite this

*24th International Conference on Rewriting Techniques and Applications, RTA 2013*(pp. 4-19). (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 21). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.RTA.2013.4