# Foundations of Mathematical Structuralism (12-14 October 2016)

## Idea and Motivation

In the course of the last century, different general frameworks for the foundations of mathematics have been investigated. The orthodox approach to foundations interprets mathematics in the universe of sets. More recently, however, there have been other developments that call into question the whole method of set theory as a foundational discipline. Category-theoretic methods that focus on structural relationships and structure-preserving mappings between mathematical objects, rather than on the objects themselves, have been in play since the early 1960s. But in the last few years they have found clarification and expression through the development of homotopy type theory. This represents a fascinating development in the philosophy of mathematics, where category-theoretic structural methods are combined with type theory to produce a foundation that accounts for the structural aspects of mathematical practice. We are now at a point where the notion of mathematical structure can be elucidated more clearly and its role in the foundations of mathematics can be explored more fruitfully.

The main objective of the conference is to reevaluate the different perspectives on mathematical structuralism in the foundations of mathematics and in mathematical practice. To do this, the conference will explore the following research questions: Does mathematical structuralism offer a philosophically viable foundation for modern mathematics? What role do key notions such as structural abstraction, invariance, dependence, or structural identity play in the different theories of structuralism? To what degree does mathematical structuralism as a philosophical position describe actual mathematical practice? Does category theory or homotopy type theory provide a fully structural account for mathematics?

## Program

### Room Arrangement

Date | Address, Room |
---|---|

12.10. - 13.10.2016 | Leopoldstraße 13, Room 1205 (2nd floor) |

14.10.2016 | Leopoldstraße 13, Room 1503 (5th floor) |

### Wednesday (12 October 2016)

Time | Event |
---|---|

09:00 - 09:15 | Welcome and Conference Start |

09:15 - 10:45 | Steve Awodey: "Univalence and Structuralist Foundations of Mathematics" |

10:45 - 11:00 | Coffee Break |

11:00 - 12:30 | Øystein Linnebo: "Structure Abstraction" (Slides, 222 Kb) |

12:30 - 12:45 | Coffee Break |

12:45 - 13:45 | José Ferreirós: "On Feferman’s Conceptual Structuralism" |

13:45 - 15:00 | Lunch Break |

15:00 - 16:30 | James Ladyman: "Does the Univalence Axiom in Homotopy Type The- ory Encode Mathematical Structuralism?" |

16:30 - 16:45 | Coffee Break |

16:45 - 17:45 | Daniel Waxman: "Structuralism and the Epistemology of Coherence" (Handout, 48 Kb) |

### Thursday (13 October 2016)

Time | Event |
---|---|

09:00 - 10:30 | Hannes Leitgeb: "A Theory of Unlabeled Graphs as Ante Rem Structures" (Slides, 998 Kb) |

10:30 - 10:45 | Coffee Break |

10:45 - 12:15 | Erich Reck: "Towards a New Taxonomy for Mathematical Structuralism" |

12:15 - 12:30 | Coffee Break |

12:30 - 13:30 | Francesca Boccuni & Jack Woods: "Structuralist (Neo-?)Logicism" |

13:30 - 15:00 | Lunch Break |

15:00 - 16:30 | Mary Leng: "An ‘i’ for an i, a Truth for a Truth?" |

16:30 - 16:45 | Coffee Break |

16:45 - 17:45 | Josef MenÅík: "Mathematical Structuralism: Internal and External" |

### Friday (14 October 2016)

Time | Event |
---|---|

10:00 - 11:30 | Jessica Carter: "Structuralism and Mathematical Practice" |

11:30 - 11:45 | Coffee Break |

11:45 - 13:15 | Gerhard Heinzmann: "The Structuralist Roots of Mathematical Understanding: Early French Structuralism Reconsidered" (Slides, 1.62 Mb) |

13:15 - 14:45 | Lunch Break |

14:45 - 15:45 | Francesca Biagioli: "Structuralism and Mathematical Practice in Felix Klein’s Work on Non-Euclidean Geometry" (Slides, 1023 Kb) |

15:45 - 16:00 | Coffee Break |

16:00 - 17:30 | Geoffrey Hellman: "Modal-Structural Mathematics for a Multiverse" (Handout, 59 Kb) |

## Acknowledgement

The conference is funded by the Deutsche Forschungsgemeinschaft (DFG) and organized by the Munich Center for Mathematical Philosophy (LMU Munich).

## Photo Credits

Header background: Michael Fajardo, "Waves". Some right reserved (desaturated from original). Source: www.piqs.de.