# Workshop: Reflection in Philosophy and Mathematics (29 June 2019)

## Idea & Motivation

Since Gödel’s incompleteness theorems one of the central topics in philosophy of mathematics is the question concerning the status of independent statements. In the general case there is no clear answer to the question about the truth of some independent statements in mathematics. However, there are independent statements, which can be recognised to be true. Those sentences are soundness statements of the underlying systems. Through reflection on these soundness assumptions we can extend our formal systems in well-motivated fashion.

Therefore, from a meta-mathematical perspective reflection is an intuitive method to extend sound formal systems. Reflection is related to questions of great philosophical significance, such as the question about the epistemic status of the reflection principles adopted to extend the sound systems. In particular, it is debated whether such principles are justified, and if yes, on what grounds. Another related question concerns the possibility of a priori knowledge of statements in the new extended system entirely via reflection. The process of reflection and the use of reflection principles provide a deep connection between mathematics and philosophy. The workshop is devoted to the discussion of “reflection” from mathematical, logico-philosophical and epistemological perspectives. In particular, we will be interested in the following questions: What is the epistemic status of reflection principles? What is the connection between proof-theoretic reflection principles, known from the application over arithmetical theories, and set-theoretic reflection principles? What role do non-mathematical concepts, such as truth or necessity, play in the process of reflection?

## Speakers

- Marianna Antonutti Marfori (MCMP/LMU Munich)
- Walter Dean (Warwick)
- Martin Fischer (Scuola Normale Superiore di Pisa/MCMP)
- Volker Halbach (Oxford)
- Leon Horsten (Bristol)
- Philip Welch (Bristol)
- Matteo Zicchetti (Bristol)

## Program

## Abstracts

### Marianna Antonutti Marfori (MCMP/LMU Munich): The Epistemic Status of Reflection Principles

TThe aim of this talk is to present a way of justifying the addition of proof theoretic reflection principles to formal arithmetical theories on the basis of epistemic considerations, by employing certain uncontroversial properties of the notion of informal or absolute provability for arithmetic. I will show that the recognition of the axioms and rules of inference of PA as correctly formalising (at least some of) our informal arithmetical reasoning is sufficient to formally imply the local and uniform reflection schemes. This gives a precise sense in which reflecting on our informal methods of proof entails more arithmetical consequences than those that follow from PA alone.

### Walter Dean (Warwick): Kreisel, reflection, and the method of informal rigour

TThis talk will explore the relationship between Kreisel's so-called method of informal rigour and his program for extending mathematical theories with reflection principles so as to analyze foundational standpoints such as finitism and predicativism. I'll begin by examining the context of some of Kreisel's early technical work relative to his better known "Informal Rigour" paper (1967). I'll then describe his contemporaneous work with Levy on reflection principles and non-finite axiomatizability before turning to consider some more recent proposals for using reflection principles to justify mathematical theories of increasing strength.

### Martin Fischer (Scuola Normale Superiore di Pisa/MCMP): Connecting potentialism and predicativism

In this talk I want to consider a version of potentialism given the natural numbers. The possible worlds are considered to be projections of subsets of the natural numbers and the accessibility will be epistemically constrained. This will then be used as an interpretation of Feferman's schematic reflective closure of arithmetic.

### Volker Halbach (Oxford): Proving Reflection

While it is possible to axiomatise truth using (proof-theoretic) global reflection principles, I show that there are methodological reasons to start from compositional axioms instead. Global reflection principles should be derived and and they are far from self-evident. I discuss various reflection principles, including global reflection for logic and finitely axiomatized arithmetical theories. Not all type-free theories of truth are equally well suited for proving reflection principles. Some fail already at reflection for logic, that is, the soundness theorem for logic. I will list some desiderata for a theory useful for proving reflection principles and thereby narrow the choice of type-free truth theories.

### Leon Horsten (Bristol): On Reflection

This article gives an epistemological analysis of the reflection process by means of which you can come to know the consistency of a mathematical theory that you already believe. It is argued that this process can result in warranted belief in new mathematical principles without doing extra justificatory work.

### Philip Welch (Bristol): Reflecting on Global Reflection

The Global Reflection Principle, GRP, asserts a connection between the universe of sets of mathematical discourse V with the collection C of its classes, and a rank initial segment some V_kappa say, and its classes, so V_{kappa+1}. GRP entails many desirable large cardinal properties that other set-theoretic reflection principles do not. We give an introduction to this principle in general terms and discuss extensions involving outer and inner model reflection.

[1] "Reflecting on Absolute Infinity", Leon Horsten, J.Phil., 113, 2016.

[2] "Obtaining Woodin's Cardinals" in "Logic at Harvard: conference celebrating the birthday of Hugh Woodin", AMS Contemp. Math. Series. vol 690, 2017.

### Matteo Zicchetti (Bristol): Coherent truth theories

Very recently it has been claimed that coherent theories of truth can play a foundational role in mathematics. In my talk I investigate the notion of coherence of a truth theory from a more formal perspective. After spelling out the formal properties that truth theories should satisfy to be coherent, I show that theories of positive truth (and falsity) are coherent.

## Registration

Reduced fee: 30 EUR

Regular fee (for graduate students): 50 EUR

Members of the MCMP and LMU: participation free of charge

The conference dinner is not included.

In order to register for the conference, please send an email to matteo.zicchetti@gmail.com with the subject line: Registration: Reflection in Philosophy and Mathematics.

## Organizers

- Martin Fischer (Scuola Normale Superiore di Pisa/MCMP)
- Matteo Zicchetti (Bristol)

## Venue

Main University Building

Geschwister-Scholl-Platz 1

80539 München**Room E210**

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