The Philosophy of Metamathematical Results (21-22 January 2022)
Idea & Motivation
Since Gödel published his work on incompleteness, subsequent contributions to its implications have spawned a huge area of research that is still active today. This conference will give both historical insights and a reappraisal of Gödel’s work as well as an overview of newly developing research. The conference offers peers a chance to be up to date with current research and network with other researchers in the field.
This event is part of the worldwide UNESCO World Logic Day celebration.
|Friday, 21 January
|14:30 - 15:30||Hannes Leitgeb: A Finitistic Semantics for Mathematics|
|15:30 - 16:30||Lucas Rosenblatt & Camila Gallovich: Theories of paradoxicality|
|16:30 - 17:00||Break|
|17:00 - 18:00||Graham Priest: Perspectives on the Universe|
|18:00 - 19:00||Juliette Kennedy: Book Presentation: Gödel’s Incompleteness Theorems (Spring 2022, Cambridge University Press)|
|19:00 - 20:00||Roundtable|
|Saturday, 22 January
|10:30 - 11:30||Lavinia Picollo: On Arithmetical Pluralism (with Dan Waxman)|
|11:30 - 12:30||Balthasar Grabmayr: Can we turn metamathematical results into representation-independent insights?|
|12:30 - 14:00||Lunch Break|
|14:00 - 14:30||Roundtable|
|14:30 - 15:30||Volker Halbach: Soundness and Completeness.
|15:30 - 16:30||Øystein Linnebo: Weyl and Two Kinds of Potential Domains (joint work with Laura Crosilla)|
|16:30 - 17:00||Break|
|17:00 - 18:00||Albert Visser: Interpreter Logics|
|18:00 - 19:00||Eva-Maria Engelen: The Position of Mathematics in Kurt Gödel’s Philosophical Notebooks. An Appetizer to Reading ›Maxims IV‹|
|19:00 - 20:00||Roundtable|
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 epistemic access to finitistic models of mathematics is mediated by standard infinitary ones.
A lot has been written on solutions to the semantic paradoxes, but very little on the topic of general theories of paradoxicality. The reason for this, we believe, is that it is not easy to disentangle a solution to the paradoxes from a specific conception of what those paradoxes consist in. In this talk we go some way towards remedying this situation. We first address the question of what one should expect from an account of paradoxicality. We then critically evaluate a number of accounts that have been offered in the literature. Our main goal is to argue in favor of a Kripke-inspired conception of paradoxicality. According to this conception, a statement is paradoxical if it cannot obtain a classical truth-value at any fixed-point. In order to assess this proposal rigorously we provide a non-metalinguistic characterization of paradoxicality and we analyze whether the resulting account satisfies a number of reasonable desiderata.
Joel Hamkins has advanced a well known view to the effect that there is no unique universe of sets. There is simply a plurality of such universes. We have, then, a pluriverse. A natural objection to this view is that there is still a single universe: the totality, V, in which all the members of the pluriverse find themselves. In this paper I consider a reply to the objection, to the effect that there is no such thing as V in itself. Rather, each member of the pluriverse simply gives a different perspective on what V is. This view is then generalised in the light of mathematical pluralism. What emerges is a vastly expanded, and logic-neutral view of the pluriverse.
Juliette Kennedy (Helsinki): Book Presentation: Gödel’s Incompleteness Theorems (Spring 2022, Cambridge University Press)
This talk will discuss my recent book on the incompleteness theorems for the Cabridge Element series. The book takes a deep dive into Gödel's 1931 paper, giving the first presentation of the Incompleteness Theorems. It considers the main ingredients of Gödel's proof: arithmetization, strong representability, and the Fixed Point Theorem in a layered fashion, returning to their various aspects: semantic, syntactic, computational, philosophical and mathematical, as the topic arises. It considers set-theoretical incompleteness, and finally considers some of the philosophical consequences considered in the literature.
For this talk in particular, special attention will be devoted to i) the semantic content of the Incompleteness Theorems; and ii) the question whether the incompleteness of e.g. Peano Arithmetic gives immediately the undecidability of the Entscheidungsproblem, as Kripke has recently argued.
Arithmetical pluralism is the view that every consistent arithmetical theory is true (of some objects) and, therefore, as legitimate as any other, at least from a theoretical standpoint. Pluralist views have recently attracted much interest but have also been the subject of significant criticism, most saliently from Putnam (1979) and Koellner (2009). These critics argue that, due to the possibility of arithmetizing the syntax of arithmetical languages, one cannot coherently be a pluralist about arithmetical truth while holding that claims about consistency are matters of fact. In response, Warren (2015) argued that Putnam's and Koellner's argument relies on a misunderstanding, and that it is in fact coherent to maintain a pluralist conception of arithmetical truth while supposing that consistency is a matter of fact. In this paper we argue that it is not. We put forward a modified version of Putnam's and Koellner's argument that isn't subject to Warren's criticisms.
Balthasar Grabmayr (Haifa): Can we turn metamathematical results into representation-independent insights?
There is a well-known gap between metamathematical theorems and their philosophical interpretations. Take Tarski's Theorem. According to its prevalent interpretation, the collection of all arithmetical truths is not arithmetically definable. However, the underlying metamathematical theorem merely establishes the arithmetical undefinability of a set of specific Gödel codes of certain artefactual entities, such as infix strings, which are true in the standard model. That is, as opposed to its philosophical reading, the metamathematical theorem is formulated (and proved) relative to a specific choice of the Gödel numbering and the notation system. Similar observations apply to Gödel and Church's theorems, which are commonly taken to impose severe limitations on what can be proved and computed using the resources of certain formalisms. The philosophical force of these limitative results heavily relies on the belief that these theorems do not depend on contingencies regarding the underlying representation choices. The main aim of this talk is to put this belief under scrutiny by exploring the extent to which we can abstract away from specific representations in the formulations and proofs of several metamathematical results.
Logical consequence is defined as truth preservation under all interpretations of the non-logical vocabulary. The properties of the resulting consequence relation depend on the distinction between logical and non-logical vocabulary. In particular, I will look at the normal first-order quantifiers whose interpretation can be varied by interpretations if interpretations are assumed to specify a domain of interpretation.
The definition of logical consequence will then proceed along the lines of my "The Substitutional Analysis of Logical Consequence" (Noûs 54, 2020, 431–450). I will discuss different problems with proving soundness and completeness for first-order logic with some modifications forced by the choice of logical constants.
According to Weyl, “inexhaustability is essential to the infinite''. However, Weyl distinguishes two kinds of inexhaustible, or merely potential, domains: those that are ''extensionally determinate'' and those that are not. This article clarifies Weyl's distinction and explains its logical and philosophical significance. In particular, the distinction sheds lights on the contemporary debate about potentialism, which in turn affords a deeper understanding of Weyl.
We discuss an idea of Harvey Friedman on how to define consistency statements for finitely axiomatized theories in an arithmetisation-free way. Outside its intended range, Friedman’s idea still yields consistency-like statements. We call the sentences that the idea delivers: interpreters.
Friedman’s idea gives rise to analogues of provability logics, which we call: interpreter logics. In our talk, we give a motivated definition of interpreter logic and outline some results. In the sequential case, we have, surprisingly, still Löb’s Logic and Solovay’s Theorem, even if the interpreters are different from ordinary consistency statements.
Eva-Maria Engelen (Konstanz): The Position of Mathematics in Kurt Gödel’s Philosophical Notebooks. An Appetizer to Reading ›Maxims IV‹
One focus in Kurt Gödel’s philosophical notebook ›Maxims IV‹ is on mathematics and logic (foundations). Almost one half of the remarks in this notebook are dedicated to these subjects. This is in difference to the other philosophical notebooks. One reason for this focusing is Gödel’s project to establish a modern scientia generalis and to give mathematics and logic a place within it; other reasons are his thinking about philosophy of mathematics, the Grundlagenstreit, and possible interpretations of the ›Principia Mathematica‹. I will give an introduction to this in my talk, which is to be taken as an appetizer to reading ›Maxims IV‹.
Online (via Zoom)
Cordelia Berz (LMU/MCMP)
Armin Heydari (LMU)
David Hofmann (LMU/MCMP)
Juan Miguel López (LMU/MCMP)
Pablo Rivas-Robledo (LMU/MCMP & Genova)
Maximilian van Remmen (LMU/MCMP)