Idea & Motivation
The workshop The Philosophy of Unfolding will bring together researchers from proof theory, foundations of mathematics, and the philosophy of mathematics to investigate the concept of Unfolding. The meeting is motivated by Solomon Feferman’s influential idea that mathematical and logical strength can, in certain contexts, be understood through the systematic unfolding of what is already implicit in previously accepted concepts.
The central objective of the event is to create a sustained dialogue between two communities that often engage with closely related questions but do so with different methods. On the formal side, we welcome experts working on -- among other topics -- explicit mathematics, operational set theory, generalized inductive definitions, and ordinal analysis. On the philosophical side, we invite scholars that engage with -- but are not limited to -- predicativity, epistemology of mathematics, and implicit commitment.
Mathematically, unfolding offers a principled way of describing how stronger systems may arise from weaker ones through conceptually motivated expansion. In philosophy, it bears on questions about justification, conceptual content, implicit commitment. The aim of this workshop is to foster a dialogue between mathematicians and philosophers and to make Feferman’s Unfolding framework more approachable for philosophers interested in question regarding the philosophy of mathematics.
Confirmed Speakers
- Riccardo Bruni (University of Florence)
- Andrea Cantini (University of Florence)
- Laura Crosilla (University of Florence)
- Kentaro Fujimoto (University of Bristol)
- David Hofmann (MCMP)
- Gerhard Jäger (Universty of Bern)
- Mateusz Łełyk (University of Warsaw)
- Simon Schmitt (University of Salzburg)
- Peter Schuster (University of Verona)
- Matteo Zicchetti (University of Warsaw)
Program
Thursday
10:00 - 10:15 Start
10:15 – 11:15 Gerhard Jäger: Non-Conventional Unfoldings
11:30 – 12:30 Matteo Zicchetti: Epistemic Stability and Acceptance of Foundational Theories
12:30 – 14:00 Break
14:00 – 15:00 Simon Schmitt: Unfolding Unfolding – A Predicativist Approach to Potentialism
15:15 – 16:15 Kentaro Fujimoto: On the Predicate Substitution Rule
16:30 – 17:30 Laura Crosilla: Some thoughts on predicativity
Friday
10:00 – 11:00 Mateusz Łełyk: Schemes in the Foundations of Mathematics
11:15 – 12:15 Riccardo Bruni: The risks of reflection: on the implicit commitment in mathematics
12:15 – 14:00 Break
14:00 – 15:00 David Hofmann: Conceptual Unfolding: A Predicativist Approach to Potentialism
15:15 – 16:15 Peter Schuster: On the smallest not smaller than any other
16:30 – 17:30 Andrea Cantini: Reflecting and unfolding
Abstracts
Riccardo Bruni: The risks of reflection: on the implicit commitment in mathematics
(joint work with A. Cantini and V. Halbach)
According to the Implicit Commitment Thesis (ICT for short, henceforth), by fully
accepting a system S one is committed to further statements which may not be already provable
in S. We discuss examples of S that may carry an implicit commitment to inconsistency.
These theories are not like other known examples that contain an obvious mistake. They are
arithmetically sound, and some possess ω-models. They are also reasonable and have attractive
features, as we will argue. Our examples force a decision about the thesis: if ICT is correct,
we must not accept certain arithmetically sound reasonable systems, because their acceptance
implicitly commits us to accepting all sentences (via inconsistency); if we decide that these
systems are acceptable, we have to reject ICT.
Andrea Cantini: Reflecting and unfolding
By Gödel’s First Incompleteness Theorem, any recursive and consistent formal system
capable of formalizing Peano Arithmetic is incomplete. So we need new axioms. Where could
these come from: From new intuitions about the mathematical universe; or by expanding on
what is already implicitly present in current axiomatizations, e.g. reflection principles. Gödel’s
theorem convincingly demonstrates the in-principle-inexhaustibility of pure mathematics in the
sense of the never ending need for new axioms, and it invites us to ponder the question: just
what axioms for mathematics ought to be accepted and why.
1. Two routes from IC. . . All this leads us to the philosophical problem IC of implicit
commitment:
• What are we implicitly committed to in accepting a theory S and what can we justifiably
accept?
Of course this naturally leads to the question of characterizing the theory resulting from an
analysis of IC, and traditionally, this has pursued along proof-theoretic patterns:
1. by use of iteration of accepted principles, this requires wellorderings. . . ;
2. following an alternative just via operations.
As is well-known, the opening question has its roots in work by Kreisel in the Fifties and a
main problem is how to characterize IC of a given theory. Our aim is to consider two possible
answers to the IC-problem: the first one was developed over the years by Solomon Feferman,
starting already in the Sixties and the early Seventies with the work on Predicative Analysis,
and culminating in 1991 with the investigation of self-referential truth; while the first route –
reflecting – directly leads into the land of truth theories, the second one – unfolding, is more
mathematical in spirit and hinges upon a point of view, which drives us to the very notion of
operation. It arose from work by Feferman 1996 in cooperation with Thomas Strahm, and with
additional contributions by Eberhard and Buchholz.
Our talk will survey the two alternatives, focussing mainly upon the second one.
Laura Crosilla: Some thoughts on predicativity
In this talk I will review what I consider some of the most significant ideas in relation to the development of predicative philosophies of mathematics.
Kentaro Fujimoto: On the Predicate Substitution Rule
The predicate substitution rule (Subst) plays a key proof-theoretic role in Feferman’s system U(NFA) of unfolding of non-finitist arithmetic. In fact, Feferman’s preceding two other alternative “more persipicuout” theories for predicativity by Feferman also employ similar rules. I will argue that this rule, however, has a conceptual problem. Concluding my talk, I’ll discuss a few possible remedies (with no mathematical results).
David Hofmann: Conceptual Unfolding: A Predicativist Approach to Potentialism
Contemporary debates in the philosophy of mathematics predominantly conceive of
potentialism in ontological terms. Prominent potentialists, such as Øystein Linnebo and James
Studd, argue, building on Charles Parsons’s work, that the universe of sets is inherently potential.
While Parsons’s writings can be interpreted as arguing for an ontological potentialism, in this talk
we propose that Parsons can also be understood as advocating for a second kind of potentialism:
ideological potentialism. On this view, the base domain of mathematical objects is an actual,
definite totality. The indefinite extensibility resides not in the objects, but in the ideology - - the
open-ended hierarchy of predicates, concepts, and classes we can dynamically formulate over the fixed ontology. Further, we situate Solomon Feferman’s conceptual structuralism as a version of ideological potentialism. To formally explicate this, the talk introduces a modalized version of
Feferman’s Unfolding program to capture the potential character of the ideology. Ultimately, we
argue that our work provides a formally precise and philosophically robust alternative version of
potentialism.
Gerhard Jäger: Non-Conventional Unfoldings
The notion of unfolding has been introduced by Feferman in the article “Gödel’s program for new axioms: Why, where, how and what”. Later Feferman and Strahm came up with the unfoldings of non-finitist arithmetic NFA and finitist arithmetic FA and set up in these two articles a very detailed technical “unfolding machinery” which is now regarded as the standard approach to unfolding.
In this talk I will look at unfoldings from the perspective of explicit mathematics and focus on inductive definitions and the unfoldings of set theories of various strengths.
Mateusz Łełyk: Schemes in the Foundations of Mathematics
The purpose of the talk is to argue that schemes, to wit first-order sentences with a placeholder predicate or, equivalently, a free second order variable, are good tools for modelling some discussions in the foundations of mathematics, in particular the ones revolving around categoricity or determinacy issues. We discuss the concept of the open-ended acceptance of a scheme and investigate various commitments it may trigger (such as to the full second-order logic). We introduce parts of the formal framework that can be used to develop metamathematics of schemes, in particular the notions of scheme interpretability and various shades of internal categoricity.
The talk is a report on a work in progress, initiated in “Categoricity-like properties in the first-order realm” (joint with Ali Enayat) and continued in “Definiteness properties of first-order schemes” (joint with Piotr Gruza).
Simon Schmitt: Unfolding Unfolding – A Predicativist Approach to Potentialism
(Based on joint work with Martin Fischer (MCMP) and David Hofmann (MCMP))
Potentialism has recently gained a lot of attention in the philosophy of mathematics, with most accounts offering a potentialist characterization of the set-theoretic universe. However, these approaches still leave open certain questions: the exact interpretation of their modal operators is frequently criticized, and they typically only provide a potentialist re-reading of a fixed theory (such as ZFC) without transcending its original deductive strength.
In this talk, we develop an alternative approach to potentialism, based on predicativism. Instead of generating new mathematical objects, the potentialism we propose is driven by a systematic process of language expansion, which we model using Solomon Feferman’s Unfolding Program. Starting with the weak schematic base theory of non-finitist arithmetic (NFA), we expand the base language with predicates that can be generated by the application of logical operations to a given stock of initial predicates. This expansion process transcends the base theory and yields several natural stopping points of increasing proof-theoretic strength, namely ϵ0 (PA), φ(2, 0) (RA<ω or RT<ω), and Γ0 (the Feferman-Schütte ordinal).
Crucially, we utilize Unfolding not just as a technical device, but to provide a precise explication of the potentialist process of concept explication relative to an arithmetical base theory. Finally, we will outline how this dynamic, language-driven potentialism suggests a novel reading of “first-orderism” in the philosophy of arithmetic.
Peter Schuster: On the smallest not smaller than any other
The supremum sup(S) of a bounded subset S of a partial order P is, if it exists, the least upper bound of S in P: that is, the minimum, or least element, of the set S∗ of upper bounds of S in P. In particular, sup(S) itself is an element of S∗, i.e. an upper bound of S in P; whence a widely purported circularity. We reconsider this phenomenon especially in the case of real numbers as lower Dedekind cuts in the rational numbers, in which already Weyl constructed instead a certain sup(S) from below, as the union of S.
Emerged from suggestions by Laura Crosilla.
Matteo Zicchetti: Epistemic Stability and Acceptance of Foundational Theories
(joint work with Maciej Głowacki and Mateusz Łełyk)
The Implicit Commitment Thesis (ICT) holds that accepting a foundational theory S rationally commits agents to accept further statements independent of S. The Epistemic StabilityThesis (EST), by contrast, holds that some foundational theories carry no such commitments: Agents can rationally accept such a theory without any epistemic obligation to accept further statements independent of S. Both ICT and EST seem to be quite intuitive: consider for instance the case of ICT and the consistency statement for a theory S that we accept. Informally, ICT captures the intuition that there is something epistemically wrong in accepting S whilst refraining from accepting the consistency of S. EST seems to be motivated by the natural thought that some theories “completely capture” some mathematical domain, or mathematical concept, so that acceptance of this concept must be exhausted by the theory. Despite this intuitiveness, ICT and EST seem to be in clear tension as they disagree about what rational acceptance of S amounts to.
The aim of this talk is to argue that this tension is only apparent and that ICT and EST are in fact compatible, once the notion of acceptance involved in ICT and EST has been made precise enough. We will introduce the distinction between global and local acceptance of theories. Recently, it has been argued that global acceptance of a theory S always results in non-trivial implicit commitments to statements independent of S, violating EST. What is missing is an explanation as to why and to what degree local acceptance can result in an epistemically stable position. The talk proposes such an explanation, and an argument that local acceptance of a theory S does result in an epistemically stable position: discussing well-known foundational equivalence theses (such as Finitism and Predicativism), we will argue that local acceptance of theories capturing these foundational positions does imply epistemic stability. With this, we hope to provide a more explicit and fine-grained analysis of epistemic stability. Importantly, we will argue that epistemic stability of local acceptance is compatible with ICT in the following sense: local acceptance will generate non-trivial implicit commitments, which—as we will show—will stay within the bounds of S. If there is enough time, we will discuss some further philosophical issues and consequences of our analysis.
Organizers
Registration
For registration, please send an email including your affiliation to Hofmann.David@campus.lmu.de
Acknowledgement
The workshop is sponsored by the Fritz Thyssen Stiftung.