# Munich-Warsaw Workshop on the Foundations of Mathematics (30 June - 1 July)

## Idea & Motivation

This international workshop gathers philosophers and logicians working on the Foundations of Mathematics to present their novel work. Topics include implicit commitment, non-standard models of Peano arithmetic, categoricity and Truth. Our aim is to make some progress concerning philosophical, logical, and epistemological issues in the Philosophy of Mathematics. Additionally, this workshop wants to foster the collaboration between the Ludwig-Maximilians-Universität and the University of Warsaw.

## Speakers

- Cezary Cieśliński (Warsaw)
- Ali Enayat (Gothenburg)
- Marta Esteves (MCMP)
- Martin Fischer (MCMP)
- Maciej Głowacki (Warsaw)
- Leon Horsten (Konstanz)
- Mateusz Łelyk (Warsaw)
- Bartosz Wcisło (Gdansk)
- Spencer Woolfson (MCMP)
- Benjamin Zayton (MCMP)
- Matteo Zicchetti (Warsaw)

## Program

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

Friday | |

09:15-09:30 | Welcome |

09:30-10:45 | Mateus Łelyk: Recent news about reflection and implicit commitment |

10:45-10:55 | Coffee Break |

10:55-12:10 | Cesary Cieśliński and Luca Castaldo: Satisfaction classes via approximations |

12:10-14:00 | Lunch |

14:00-14:45 | Benjamin Zayton: Qualifying the Received View on Urelements |

14:45-16:00 | Matteo Zicchetti: Soundness Arguments for Consistency and Their Epistemic Value |

16:10-16:15 | Coffee Break |

16:15-17:30 | Ali Enayat: Leibnizian and anti-Leibnizian themes in the foundations of mathematics |

19:30 | Dinner |

Saturday | |

10:00-11:15 | Leon Horsten: Reinhardt and Truth Reflection |

11:15-12:00 | Marta Esteves: Towards a general Theory of Structures: The Case of Groups and The Generalization |

12:00-14:00 | Lunch |

14:00-14:45 | Spencer Woolfson: Relative Interpretation of Inconsistency Statements |

14:45-16:00 | Maciej Głowacki and Luca Castaldo: Implicit commitments of instrumental acceptance. A case study |

16:00-16:10 | Coffee Break |

16:10-17:25 | Bartosz Wcisło: Truth predicates with the full collection scheme |

17:25-17:35 | Coffee Break |

17:35-18:50 | Martin Fischer: Significant Reasoning in the Setting of Type-Free Truth |

## Abstracts

**Cesary Cieśliński and Luca Castaldo: Satisfaction classes via approximations**

Abstract: The objective is to present a construction of a satisfaction class for the language of first-order arithmetic. In the construction, the notion of a proof approximation will be crucially used. The technique of approximations will be developed as a part of proof theory.

### Ali Enayat: Leibnizian and anti-Leibnizian themes in the foundations of mathematics

Abstract: I will overview results in the metamathematics of set theory and arithmetic that in some way or another are related to Leibniz's dictum on the identity of indiscernibles. The overview will include a discussion of Leibnizian models of arithmetic and set theory, and theories of arithmetic and set theory with built-in indiscernibles.

**Marta Esteves: Towards a general Theory of Structures: The Case of Groups and The Generalization**

Abstract: Structuralism is a theory in the philosophy of mathematics which argues that “mathematics is the general study of structures” (Reck and Schiemer, 2023).

The perspective of non-eliminative structuralism, in particular, argues that mathematical structures should be understood as sui generis structures (Leitgeb 2020, Shapiro, 1997). We argue that it is important to provide an explicit formal characterization of structures as they are understood by non-eliminative structuralism. As an example, we show how such a characterisation can be provided for the particular case of group theory, in particular we present a second-order theory of groups as structures sui generis, in a similar way to what was presented for the case of unlabeled graphs in Leitgeb (2020). We then go on to present the first attempt towards a generalization of this approach,

which aims at providing a characterization of general construction principles (such as products, quotients, free objects, etc.) within a general second-order theory of structures. Finally, we will discuss the consequences of these formal results for the theory of structuralism in the philosophy of mathematics, and we will argue, in particular, that the fact that we are able to provide an explicit logical characterization of sui generis structures provides a strong argument in favour of non-eliminative structuralism.

### Martin Fischer: Significant Reasoning in the Setting of Type-Free Truth

Abstract: In the talk I consider different forms of reasoning for a type-free truth predicate. A reading of truth as an inferential tool suggests that significant reasoning can be adequately represented in a sequent system for partial logic, such as used for the system PKF. The question whether also in classical systems such as KF some forms of significant reasoning is adequately captured is critically discussed. Moreover, an analysis via unravelling trees is suggested that might help to interpret significant KF reasoning in a reflective extension of PKF. The talk is based on joint work with Luca Castaldo and Johannes Stern.

### Maciej Głowacki und Luca Castaldo: Implicit commitments of instrumental acceptance. A case study

Abstract: When accepting an axiomatic theory Th, it seems that we are implicitly committed to accepting some statements which are independent from Th, for example, reflection principles for it. The discussion on the existence and scope of implicit commitments has been focused primarily on the notion of foundational acceptance. However, the literature on implicit commitments triggered by weaker notions of acceptance is underdeveloped. In this talk, we investigate implicit commitments of instrumental acceptance and compare them with those involved in foundational acceptance. As a case study, we concentrate on Reinhardt's instrumentalism vis-a-vis Kripke-Feferman truth. We develop a formal theory of acceptance motivated by Reinhardt's Programme and analyze its formal properties.

**Leon Horsten: Reinhardt and Truth Reflection**

Abstract: It seems to me hard to argue convincingly that a mathematician is implicitly mathematically committed to accepting the uniform reflection principle for the strongest mathematical theory that she accepts. On the other hand, I believe that a mathematician can come to believe in the consistency of the strongest mathematical theory that she accepts without deriving this consistency in a stronger mathematical theory. In my talk, I will discuss the question whether a similar account holds for local reflection.

**Mateusz Łełyk: Recent news about reflection and implicit commitment**

Abstract: The talk is devoted to the exposition of recent logico-philosophical insights regarding the implicit commitment of foundational theories and reflection principles. The talk is divided into three parts. In the first, we present the outcomes of the joint work with Maciej Głowacki on proof-theoretic analysis of Cieśliński's believability theory (Bel), its iterations and the mixed system with untyped truth and believability. In the second part, we outline an approach to study implicit commitments of general sequential theories, based on a recent paper with Carlo Nicolai "Implicit Commitment in a General Setting". Finally, we present a structural result for the uniform reflection principle, which we call the reflection inversion theorem. We prove that every theory extending PA in the language with finitely many new predicate symbols which proves the full induction scheme for its own language can be axiomatized by a uniform reflection principle over some theory. The result applies inter alia to some truth theories such as TB, UTB and FS.

**Bartosz Wcisło: Truth predicates with the full collection scheme**

Abstract: In the past several years, a lot of results were obtained concerning the conservativity of extensions of the compositional truth theory without induction, CT^-. The natural extensions of this theory of truth can be mostly obtained in two ways: either as correctness or reflection principles, stating that truth is preserved under certain forms of argument or as forms of the induction scheme which say that the sets of numbers defined using truth predicates behave similarly to all other sets of numbers. One principle of the latter kind is the collection scheme which says, roughly, that a function defined in terms of the truth predicate whose domain is a bounded set of numbers has a bounded set of values. Answering a question of Kaye, we will show that compositional truth with the full collection scheme is a conservative extension of Peano Arithmetic PA.

### Spencer Woolfson: Relative Interpretations of Inconsistency Statements

Logicians use provability predicates to talk about theories inside themselves; however, internal provability predicates often do not behave as expected. Perhaps the most shocking result is that classsical theories can interpret their own inconsistency statement. However, this result does not hold for intuitionistic theories. I hope to explore this result to understand better what causes this difference.

### Benjamin Zayton: Qualifying the Received View on Urelements

In the philosophy of set theory, the received view on urelements is that urelements are necessary to account for the applicability of set theory outside of mathematics, but dispensable for theoretical purposes. In this essay, both components of this view will be qualified. As groundwork, there first is a brief recounting of the role of urelements in 20th-century mathematics, showing that the received view goes back to Zermelo.

Thereafter, three arguments for the theoretical dispensability of urelements are discussed.

The first, Reinhardt's structuralist argument, argues that all structural questions about (impure) sets can be reduced to questions about pure ordinals. However, this says little about the structural features of models of set theory, and the representational role of set theory requires consideration of non-structural features of sets.

The second argument uses bi-interpretability results between pure set theory and set theory with urelements to argue that one can omit urelements. However, the most general results with less assumptions about the number of urelements obtained by Hamkins and Yao require the philosophically problematic addition of parameters to the set-theoretic language.

The quasi-empirical third argument is based on the hitherto expressive adequacy of set theory without urelements, and can be resisted by noticing that set theory with urelements achieves other Maddian foundational goals, such as a more faithful representation of structures, as shown by Barton and Friedman. The first qualification is thus that urelements are dispensable for some foundational goals, but not for others.

Finally, the last section of the paper shows that on popular accounts of mathematical representation, namely the mapping and the DEKI account, the absence of urelements is a mere idealisation. Thus, the second qualification is that the purported necessity of urelements for accounts of application hinges on one's view on idealisations in scientific models.

## topRegistration

Please register sending an email to Office.Leitgeb@lrz.uni-muenchen.de.

## Organizer

- Martin Fischer (MCMP)
- Matteo Zicchetti (Warsaw)

## Venue

Geschwister-Scholl-Platz 1, main building,

80539 München**Room M 101**

**Acknowlegement**

The workshop is generously funded by the Münchner Universitätsgesellschaft (MUG).** **