Understanding Reflection (20-22 June 2022)
(update 21 June)
Idea & Motivation
The aim of this event is to gather philosophical logicians, epistemologists, and philosophers of mathematics to make progress on the analysis of what makes our knowledge of mathematical and logical principles possible, with a particular focus on the analysis of reflection as an epistemic process.
One type of question that will be discussed concerns the grounds on which our confidence in different kinds of reflection principles, statements that assert an implication between the provability of a class of sentences and their truth, can be justified. Can reflection principles be seen as formal expressions of an informal “process of reflection” involved in arithmetical reasoning? Do reflection principles express a kind of trust in a formal theory whose soundness is unconditionally accepted? Do our pre-theoretical intuitions suffice for adding such principles to formalised theories?
Such questions have only been addressed explicitly in the philosophy of mathematics in the last decade, while the vast literature in epistemology on epistemic entitlement and non-evidential warrants has not yet been systematically brought to bear on questions about the justification of reflection principles in the philosophy of mathematics and logic. This conference aims at approaching these kinds of questions from a perspectives that integrates philosophical logic, the philosophy of mathematics, epistemology, and the philosophy of language.
- Sharon Berry (University of Indiana Bloomington)
- Cezary Cieslinski (University of Warsaw)
- Walter Dean (University of Warwick)
- Martin Fischer (MCMP)
- Leon Horsten (University of Konstanz)
- Mateusz Łełyk (University of Warsaw)
- Carlo Nicolai (King's College London)
- Nikolaj J.L. Linding Pedersen (Underwood International College) & Matteo Zicchetti
- Lorenzo Rossi (University of Turin)
- Daniel Waxman (National University of Singapore)
- Nicola Bonatti (MCMP)
- Fabian Pregel (University of Oxford)
|09:30-10:45||Leon Horsten (University of Konstanz): What is Reflection?|
|11:15-12:30||Nikolaj Jang Lee Linding Pedersen (Yonsei University):
A tale of two ´proofs´: Moore, mathematics, and minimal realism
|14:00-15:15||Cezary Cieśliński (University of Warsaw): Accepting reflection - how and why?|
|15:15-16:30||Walter Dean (Warwick): The liar and the sorites: towards a uniform arithmetical treatment|
|17:00-18:15||Fabian Pregel (University of Oxford): Neo-Logicism and Infinitely Iterated Extensions|
|09:30-10:45||Daniel Waxman (National University of Singapore) and Lorenzo Rossi (University of Turin):
Accepting and Proving: the case of Reflection Principles
|11:15-12:30||Nicola Bonatti (MCMP): The Reflective Equilibrium of Intended Models|
|14:00-15:15||Sharon Berry (University of Indiana Bloomington):
Justifying Set-Theoretic Reflection principles
|15:15-16:30||Carlo Nicolai (Kings College London): Reflection Principles and Non-classical Truth|
|10:00-11:15||Mateusz Łełyk (University of Warsaw): Implicit commitment from the axiomatic point of view|
|11:45-13:00||Martin Fischer (MCMP): Reflecting on trustworthy theories|
In this talk I will contrast, philosophically explicate and assess three arguments for adding set theoretic reflection principles to the standard ZFC axioms for set theory.top
The categoricity theorem shows that the realist is justified in believing that the sentences of a foundational theory T -- such as Peano arithmetic, Cantor-Dedekind analysis and Zermelo-Fraenkel set theory -- have a determinate truth-value if she is independently justified in believing that T has a particular model. In the first part of my talk, I will point out an overlooked connection between categoricity and antecedent beliefs, namely that of extremal axioms -- such as the axioms of Induction, Continuity and either Constructibility or Large Cardinals. Extremal axioms define a condition of either maximality or minimality on the class of models satisfying T. Moroever, given some additional constraints, the assumption of extremal axioms implies the categoricity of T. In the second part, I will argue that the assumption of extremal axioms is justified by the reflective equilibrium between the antecedent beliefs and the formal resources adopted to formalize T. I will claim that the proposed framework meets the epistemic desiderata imposed by the reflective equilibrium method. In the last part, I will consider two case studies -- namely, that of arithmetic and real analysis -- showing that the assumption of extremal axioms is adopted to revise, respectively, the formal resources and the antecedent beliefs.top
The presentation will take as basic the believability theory framework proposed in , see also . Although a brief description of formal results is planned, the main focus will be on the philosophical aspects of the proposal.
 Cezary Cieśliński The Epistemic Lightness of Truth. Deflationism and its Logic, Cambridge University Press 2017
 Corrigendum to The Epistemic Lightness of Truth. Deflationism and its Logic, http://cieslinski.filozofia.uw.edu.pl/Corrigendum.pdftop
The unification of the paradoxes of truth and vagueness has been a topic of recurrent philosophical interest. I will present a sequence of observations which illustrate how the liar and sorites paradoxes are formally related, culminating in the claim that they give rise to similar sorts of mathematical incompleteness results. A central tool will be the use of the arithmetized completeness theorem to provide interpretations of putatively paradoxical notions within the language of first-order arithmetic.topMartin Fischer (MCMP): Reflecting on trustworthy theories
In this talk I try to justify reflective extensions of trustworthy theories. Rather than taking the implicit commitment thesis as a general claim, I opt for a relativization to a background conception. Relative to such a background conception it seems possible to justify certain reflection principles by relying on the justification of the trustworthy theories themselves. I illustrate the use of reflective extension by considering three case studies: PRA, PA and PKF. top
In the mathematical sciences, we find proof theorists analysing axioms that they call reflection principles; we find set theorists investigating axioms that they call reflection principles; and we find probability theorists (and formal epistemologists) investigating principles that they call reflection principles.
In my talk, I want to initiate a discussion of the following questions:
- Do these different kinds of reflection principles have anything interesting in common? If so, what?
- Are we epistemically warranted in believing these kinds of principles, and, if so, wherein does this warrant consist?
- Are these kinds of reflection principles related to concepts of reflection that have played a role in the history of philosophy, and, if so, how?
These are big questions. I do not pretend to solve any of them in my talk, but merely hope to take some initial steps in exploring them, and will by doing this try to convince the audience that these questions may be worth pursuing.top
The main purpose of the talk is to argue for the following formulation of the implicit commitment thesis (ICT): ''Anyone who is justified in believing a sufficiently powerful, consistent mathematical formal system S is also implicitly committed to various additional statements which are expressible in the language of S but which are formally independent of its axioms." ICT has recently been put into question by Walter Dean, who observed that it clashes with the intuitive epistemic stability of some prominent foundational systems (e.g. PA). We state two axioms, called Invariance and Axiomatic Reflection, and argue that they should be satisfied by any notion of implicit commitment. We show that each axiom in isolation does not force the implicit commitment of a system S to be properly stronger than S (at least for natural choices of S) but taken together they prove that the Uniform Reflection Principle for S is S's implicit commitment. Lastly we compare our approach with two other proposals from the literature: Nicolai and Piazza's Semantic Core, and Cezary Cieśliński's believability theory, presenting some relatively new relevant formal results. This is (to a large extent) joint work with Carlo Nicolai.top
Theories of truth in classical logic have been traditionally employed to warrant the use of proof-theoretic reflection. More recently, it’s proof-theoretic reflection that has been employed to justify principles of truth. These strategies have been criticized for being somewhat incoherent: most theories of truth do not sit well with their closure under proof-theoretic reflection. Nonclassical theories apparently fare better against this objection; however, in axiomatizations in the usual Kleene logics the logical form of proof-theoretic reflection needs to be altered to achieve nontrivial strength. In the talk I will reconstruct the formal and philosophical context of the study of proof-theoretic reflection over nonclassical truth, and discuss some open problems and new results.top
Moore is widely known for his claim that he could rigorously prove the existence of an external world by uttering “Here is one hand” (while making a certain gesture with the right hand) and “Here is another” (while making a certain gesture with the left hand). Crispin Wright has offered an influential diagnosis of why one might find Moore’s ‘proof’ epistemically defective: it cannot be used to acquire an evidential warrant for thinking that there is an external world because it is epistemically circular. In light of this failure Wright has proposed a non-evidential notion of warrant—epistemic entitlement—applicable to acceptance of the existence of an external world. We transpose Moore’s ‘proof’ to the case of arithmetic. Crucial steps of the transposition include the articulation of relevant metaphysical and epistemological commitments—in particular: what is the fundamental nature of reality? And which faculties or capacities facilitate our thinking about it? We introduce minimal realism to answer the first, metaphysical question. According to minimal realism about a given domain D, D states of affairs are objective in the sense of being mind-independent. (Hence the realism.) At the same time, beyond mind-independence, minimal realism is neutral with respect to the metaphysical character or constitution of states of affairs. (Hence the minimalism.) To answer the second, epistemological question we adopt a quasi-perceptual account of intuition. Against the background of minimal realism and quasi-perceptualism about intuition we argue that Moore-style ‘proofs’ in the case of arithmetic are bound to fail. Intuition-driven attempts to acquire an evidential warrant for thinking that arithmetic has a subject matter are epistemically circular. In light of this failure we explore Wright-style entitlement as a candidate epistemology of basic mathematical ontology. This is joint work with Matteo Zicchetti.top
The Neo-Logicist is looking to establish that the truths of arithmetic are analytic. The challenge from incompleteness is that, given Gödel’s first incompleteness theorem, reducing the axioms of Peano Arithmetic to logic and definitions will not yield an axiom system in which one can derive all truths of Peano Arithmetic. To address the challenge, a natural idea is to generate larger and larger systems via reflection sequences of axiom systems. I outline the standard methodology for studying such reflection sequences as developed by Turing and Feferman. I then focus specifically on whether the Neo-Logicist can employ reflection sequences to generate stronger theories from Frege Arithmetic as base theory in order to address the challenge from incompleteness.top
Some authors take reflection principles for free. According to this view, if one accepts / is entitled to accept a theory S, one is also entitled to accept all instances of some kind of reflection principle for S. (The reflection-for-free view comes in many variants: some talk about flat acceptance, others of entitlement to accept, epistemic warrant, and more). Of course, reflection principles are not provable in their base theories (if the latter are consistent). Therefore, if one accepts S, according to this view, one is entitled to accept a hierarchy of stronger and stronger theories. In this paper, we will try to isolate and study the epistemic principles behind the reflection-for-free view. More specifically, we will formalize the notion of entitlement to accept at work in the reflection-for-free epistemology in a suitable modal environment, and observe its interaction with modal provability principles. Our assessment will not be entirely positive: once suitably formalized, we maintain, the reflection-for-free epistemology reveals a problematic picture of acceptance for mathematical theories. Therefore, we will close by tentatively suggest some alternatives, and offer some intermediate conclusions.top
Call for Abstracts
We invite the submission of abstracts up to 1000 words prepared for blind review. Please submit your abstract by email to the organisers: Marianna.AntonuttiMarfori@lrz.uni-muenchen.de, M.Fischer@lrz.uni-muenchen.de, email@example.com, and include in the body of the email your institutional affiliation, if you have one, and the title of your abstract.
The deadline for submission is April 1st, 2022. We aim to send decisions to all applicants by April 15th, 2022.
Submissions from underrepresented groups are particularly welcome. The conference will be organised and run under the MCMP's code of conduct.
If public health regulations will allow it the conference will be held in person, though we are preparing to hold a hybrid event should some participants prefer to participate remotely.
Please register via our online tool before June 20th. Note that participation is only possible after confirmed registration.
- Hannes Leitgeb (MCMP)
- Walter Dean (University of Warwick)
- Julien Murzi (University of Salzburg)
- Lavinia Picollo (NUS, UCL)
- Johannes Stern (University of Bristol)
IBZ (Internationales Begegnungszentrum der Wissenschaft München)
For information about practical matters and registration, please contact Marianna Antonutti Marfori at Marianna.AntonuttiMarfori@lrz.uni-muenchen.de.
This workshop is funded by a Deutsche Forschungsgemeinschaft (DFG) International Scientific Event grant n. AOBJ 666230.
Gefördert durch die Deutsche Forschungsgemeinschaft (DFG) – AOBJ 666230.