# Munich - Konstanz Workshop on Philosophy of Mathematics/Logics (November, 19-20)

## Idea & Motivation

Philosophy of mathematics and philosophy of logic are two of the central topics of a formally oriented philosophy. The field is diverse, and the researchers often distributed around the world. Fortunately, both Konstanz and Munich have strong research groups working in these areas. The workshop intends to bring together researchers from both universities and to foster the relations and interaction between both research groups

## Speakers

- Nicola Bonatti (MCMP)
- Hazel Brickhill (Konstanz)
- Luca Castaldo (Bristol)

- Walter Dean (Warwick/MCMP)
- Matteo De Benedetto (Bochum/MCMP)
- Martin Fischer (MCMP)
- Leon Horsten (Konstanz)
- Deborah Kant (Konstanz)
- Hannes Leitgeb (MCMP)
- Beau Mount (Konstanz)
- Sam Roberts (Konstanz)
- Lorenzo Rossi (Turin/MCMP)
- Giorgio Venturi (Konstanz)

## Programme

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

Friday |
Ludwigstr. 28 RG, Room 025 | |

11:00 - 12:15 | Hannes Leitgeb (MCMP): A Finitistic Semantics for Mathematics | |

12:15 - 13:15 | Sam Roberts (Konstanz): Might it be indeterminate what pluralities there are? | |

13:15 - 14:30 | Lunch Break | |

14:30 - 15:30 | Matteo De Benedetto (Bochum/MCMP): Stretching Lakatos' concept-stretching | Prof.-Huber-Pl. 2, Room W401 |

15:30 - 16:30 | Deborah Kant (Konstanz): Naturalness judgements in set-theoretic practice | |

16:30 - 17:00 | Coffee Break | |

17:00 - 18:00 | Nicola Bonatti (MCMP): Extremal Axioms and the Reflective Equilibrium of Intended Models | |

18:00 - 19:00 | Giorgio Venturi (Konstanz): On ideal objects in set theory | |

19:30 | Dinner | |

Saturday |
Prof.-Huber-Pl. 2, Room W401 | |

10:00 - 11:00 | Walter Dean (Warwick): Informal rigour and the origins of forcing | |

11:00 - 12:00 | Beau Mount (Konstanz): Type-Free Ramseyan Truth Theories: A Preliminary Report | |

12:00 - 12:15 | Coffee Break | |

12:15 - 13:15 | Lorenzo Rossi (Turin/MCMP): Truth Meets Vagueness: Unifying the Semantic and the Soritical Paradoxes | |

13:15 - 14:30 | Lunch Break | |

14:30 - 15.30 | Luca Castaldo (Bristol): Saving logic from paradox | |

15:30 - 16:30 | Hazel Brickhill (Konstanz): The Middle Way: Buddhist metaphysics |

## Abstracts

### Nicola Bonatti (MCMP): Extremal Axioms and the Reflective Equilibrium of Intended Models

The literature agrees that the (quasi-)categoricity of Peano arithmetic, Hilbert geometry and Zermelo-Fraenkel set theory fails to demonstrate that there is a unique structure corresponding to our informal mathematical practice. In this talk, I will argue that the indented model of such theories is not determined by the categoricity theorem alone, but by the assumption of extremal axioms – such as the axioms of Induction, Completeness and either Constructibility or Large Cardinals. The leading idea is that extremal axioms imply a reflective equilibrium between the informal beliefs concerning the subject matter of a theory and the formal resources adopted to formalize it. I will support the claim arguing that extremal axioms are both intrinsically and extrinsically justified. On the one hand, extremal axioms are intrinsically justified because they express the properties of finite chain, extendibility and (non) constructibility which are identified through the analysis of the basic notions of, respectively, natural number, point/line and set. On the other hand, extremal axioms are extrinsically justified because they imply the categoricity of the models (if the theory is formulated in Second-order logic), thus providing a faithful formalization of the informal beliefs. I will conclude that the reflective equilibrium between the intrinsic and extrinsic justification of extremal axioms grounds the coherence of our judgments about the intended model of such theories – thus debunking the theoretical threat of non-standard

models.top

### Hazel Brickhill (Konstanz): The Middle Way : Buddhist metaphysics

Although this is less well-known in the west than other aspects of Buddhism, philosophic argument is of high importance in many Buddhist schools.

Central philosophical texts, with their denial of extremes such as existence and non-existence, may seem to endorse non-classical logic. But such an interpretation misses the point. Rather, these texts try to communicate a metaphysics that is, by their own assertion, inexpressible. I will take on this impossible task and try to give a flavour of Buddhist metaphysics, from the perspective of western philosophy.top

### Luca Castaldo (Bristol): Saving logic from paradox

In the field of theories of truth, non-classical logics are often used to avoid Liar-like paradoxes. Yet, giving up on classical reasoning has its own costs, and the goal of this talk consists in saving logic from paradox (with apologies to Hartry Field). Building on Reinhardt (1985,1986), we suggest an instrumentalist interpretation of classical logic, providing a precise (and strong) sense in which classical reasoning can be seen as a *useful but dispensable instrument*. As a case study, we consider theories of Kripke-Feferman truth, showing that theories formulated in classical logic and theories formulated in non-classical logic can reason about truth-theoretic facts in the same way. This is joint work with Martin Fischer.top

### Walter Dean (Warwick): Informal rigour and the origins of forcing

This paper presents a reconstruction of the discovery of arithmetical forcing in the framework of Kreisel's method of informal rigour. The basics of Feferman's presentation of this method will first be reviewed and located with respect to the better known (but essentially contemporaneous) work of Cohen. I will then present a reconstruction of the methodology of informal rigour on the basis of Kreisel's analogy to "philosophical proof". On this basis, I will illustrate how the first result proved by arithmetical forcing can be seen as answering a philosophical question posed by Kreisel about the characterization of predicative definability. I will then relate a parallel story about how Kreisel's attempt to axiomatize lawless sequences in intuitionistic analysis led to an informally rigorous characterization of an \omega-generic set. Time permitting, I will present some additional problems -- both philosophical and technical -- in regard to the next steps in set theoretic independence proofs.top

### Matteo De Benedetto (Bochum): Stretching Lakatos' concept-stretching

Lakatos' heuristics of proofs and refutations (Lakatos 1976) provided the first explicit model of conceptual change in mathematics, centered around the notion of 'concept-stretching'. Despite its historical significance, one can find, in the philosophical and historical literature, several examples of mathematical conceptual change that defy Lakatos' model. I will show how Lakatos' model of conceptual change can be re-fined by virtue of a self-application of Lakatos' heuristic of proofs and refutations, shielding his model of conceptual change from known counterexamples.top

### Deborah Kant (Konstanz): Naturalness judgements in set-theoretic practice

Despite the difficulty of specifying the concept of naturalness, it seems important in set theory. For example, it is suggested as a way of addressing the set-theoretic independence problem (e.g. in Bagaria 2005). In my PhD thesis, I pursue a practical approach on naturalness and analyse the results of an interview study with set theorists. An account of naturalness judgements in set-theoretic practice is developed that captures how they depend on understanding and how they relate to acceptance. Finally, natural axioms are considered in terms of this account.top

### Hannes Leitgeb (MCMP): A Finitistic Semantics for Mathematics

Is mathematics (arithmetic, analysis, set theory,...) committed to the existence of infinitely many objects? The aim of my talk will be to argue the answer is: No.

For that purpose, I will introduce a new semantics ("role semantics“) according to which mathematical statements can be understood as having only finite ontological commitments.

I will assess the semantics, show that its properties are similar to those of standard Tarskian semantics, and hence conclude that there is not much disadvantage in Interpreting mathematics by the new semantics. At the same time, the finitistic role semantics does not come with any epistemological benefit: if anything, our access to finitistic models of mathematics is mediated by standard infinitary ones.top

### Beau Mount (Konstanz): Type-Free Ramseyan Truth Theories: A Preliminary Report

Most formal work on theories of truth is in the Tarskian tradition, in which a truth predicate applies to sentences. There is an alternative tradition, deriving from Ramsey and taken up by Prior and Strawson (and, more recently, Williamson and Rumfitt), on which the truth of a sentence is defined in terms of propositional quantification and a cross-type 'expresses' predicate. Thus far, little work has been done on type-free versions of these theories. I discuss how to adapt Kripke–Feferman-like theories to this framework and show that the approach faces serious obstacles.top

### Sam Roberts (Konstanz): Might it be indeterminate what pluralities there are?

Some people are borderline bald: it is indeterminate whether they are bald. So it can be indeterminate whether certain objects have certain properties. Similarly, some people borderline surround the house: it is indeterminate whether they (collectively) surround the house. So it can be indeterminate whether certain pluralities of objects have certain properties. But might it be indeterminate what objects there are? Similarly, keeping what objects there are fixed, might it be indeterminate what pluralities there are? It is this second question that I will be concerned with in my talk. I will provide what I take to be the strongest argument that, keeping what objects there are fixed, it is a determinate matter what pluralities there are. I will then use this to argue for the determinacy of a broad class of mathematical statements.top

### Lorenzo Rossi (Turin): Truth Meets Vagueness: Unifying the Semantic and the Soritical Paradoxes

(joint work with Riccardo Bruni)

Semantic and soritical paradoxes display remarkable family resemblances. For one thing, several non-classical logics have been (independently) applied to both kinds of paradoxes. For another, revenge paradoxes and higher-order vagueness—among the most important problems targeting solutions to semantic and soritical paradoxes, respectively—exhibit a rather similar dynamics. Some authors have taken these facts to suggest that truth and vagueness require a unified logical framework, or perhaps that the truth predicate is itself vague. However, a common core of semantic and soritical paradoxes has not been identified yet, and no explanation of their relationships has been provided. Here we aim at filling this lacuna, in the framework of many-valued logics. We provide a unified diagnosis of semantic and soritical paradoxes, identifying their source in a general form of indiscernibility. We then develop our diagnosis into a theory of paradoxicality, which formalizes both semantic and soritical paradoxes as arguments involving specific instances of our generalized indiscernibility principle, and correctly predicts which logics can non-trivially solve them.top

### Giorgio Venturi (Konstanz): On ideal objects in set theory

Abstract: In this talk I will review some recent results on the construction of non-classical models of ZFC, obtained by extending the Boolean-valued constructions to algebra that are non boolean. After presenting the main results and their application to the problem of independence in set theory, we will discuss the ontological import of these constructions. We will argue that non-classical objects can be seen as ideal in the sense of Hilbert.

## Registration

Please note that participation is only possible after confirmed registration. The organizers will verify compliance with the 3G rules. To register, please write an email to M.Fischer@lrz.uni-muenchen.de by November 17 the latest.

## Organizers

- Martin Fischer (MCMP)
- Leon Horsten (Konstanz)

## Venues

Ludwigstr. 28 RG**Room 025**

Prof.-Huber-Platz 2**W401**

If you need help finding the venue or a particular room, you might consider using the LMU's roomfinder.